Anisotropic Vortices in High-Temperature Superconductors and the Onset of Vortex-like Excit

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Laboratory experiments for intense vortical structures in turbulence velocity fields

a r X i v :p h y s i c s /0703057v 1 [p h y s i c s .f l u -d y n ] 6 M a r 2007Physics of FluidsLaboratory experiments for intense vortical structures in turbulence velocity ﬁeldsHideaki Mouri,a Akihiro Hori,b and Yoshihide Kawashima bMeteorological Research Institute,Nagamine,Tsukuba 305-0052,Japan(Dated:February 2,2008)Vortical structures of turbulence,i.e.,vortex tubes and sheets,are studied using one-dimensional velocity data obtained in laboratory experiments for duct ﬂows and boundary layers at microscale Reynolds numbers from 332to 1934.We study the mean velocity proﬁle of intense vortical struc-tures.The contribution from vortex tubes is dominant.The radius scales with the Kolmogorov length.The circulation velocity scales with the rms velocity ﬂuctuation.We also study the spatial distribution of intense vortical structures.The distribution is self-similar over small scales and is random over large scales.Since these features are independent of the microscale Reynolds number and of the conﬁguration for turbulence production,they appear to be universal.I.INTRODUCTIONTurbulence contains various classes of structures that are embedded in the background random ﬂuctuation.They are important to intermittency as well as mixing and diﬀusion.Of particular interest are small-scale struc-tures,which could have universal features that are inde-pendent of the Reynolds number and of the large-scale ﬂow.We explore such universality using velocity data obtained in laboratory experiments.We focus on vortical structures,i.e.,vortex tubes and sheets.The former is often regarded as the elementary structure of turbulence.1,2,3At low microscale Reynoldsnumbers,Re λ<∼200,direct numerical simulations de-rived basic parameters of vortex tubes.3,4,5,6,7,8The radii are of the order of the Kolmogorov length η.The total lengths are of the order of the correlation length L .The circulation velocities are of the order of the rms velocity ﬂuctuation u 2 1/2or the Kolmogorov velocity u K .Here · denotes an average.The lifetimes are of the order of the turnover time for energy-containing eddies L/ u 2 1/2.For these vortical structures,however,universality has not been established because the behavior at high Reynolds numbers has not been known.At Re λ>∼200,a direct numerical simulation is not easy for now.The promising approach is velocimetry in laboratory experiments.A probe suspended in the ﬂow is used to obtain a one-dimensional cut of the velocity ﬁeld.The velocity variation is intense at the positions of in-tense structures.Especially at the positions of intense vortical structures,the variation of the velocity compo-nent that is perpendicular to the one-dimensional cut is intense.9,10Thus,the velocity variation oﬀers some infor-mation about intense structures,although it is diﬃcult to specify their geometry.The above approach was taken in several studies.11,12,13,14,15For example,using grid turbulence 14at Re λ=105–329and boundary layers 15at Re λ=295–TABLE I:Experimental conditions and turbulence parameters:duct-exit or incoming-ﬂow velocity U∗,coordinates x andz of the measurement position,mean streamwise velocity U,sampling frequency f s,kinematic viscosityν,mean energydissipation rate ε =15ν (∂x v)2 /2,rms velocityﬂuctuations u2 1/2and v2 1/2,Kolmogorov velocity u K=(ν ε )1/4,rms spanwise-velocity increment over the sampling interval δv2s 1/2= [v(x+U/2f s)−v(x−U/2f s)]2 1/2,correlation lengths L u=R∞0 u(x+r)u(x) / u2 dr and L v=R∞0 v(x+r)v(x) / v2 dr,Taylor microscaleλ=[2 v2 / (∂x v)2 ]1/2,Kolmogorov lengthη=(ν3/ ε )1/4,and microscale Reynolds number Reλ=λ v2 1/2/ν.The velocity derivative was obtained as∂x v=[8v(x+r)−8v(x−r)−v(x+2r)+v(x−2r)]/12r with r=U/f s.Ductﬂow Boundary layerUnits1234567891011FIG.1:Sketch of a vortex tube penetrating the(x,y)plane at a point(x0,y0).The inclination is(θ0,ϕ0).The circulation velocity is uΘ.We consider the spanwise velocity v along the x axis in the mean streamdirection.FIG.2:Mean proﬁles in the streamwise(u)and spanwise (v)velocities for the Burgers vortices with random positions (x0,y0)and inclinations(θ0,ϕ0).The u proﬁle is separately shown for∂x u>0(u+)and∂x u≤0(u−)at x=0.The position x and velocities are normalized by the radius and maximum circulation velocity of the Burgers vortices.The dotted line is the v proﬁle of the Burgers vortex for x0= y0=θ0=0,the peak value of which is scaled to that of the mean v proﬁle.uΘand strainﬁeld(u R,u Z)areuΘ∝ν4ν ,(1a) (u R,u Z)= −a0RRuΘ(R),(2a) v(x−x0)=(x−x0)cosθ0Ru R(R),(4a)v(x−x0)=−(x−x0)sin2θ0sinϕ0cosϕ0+y0(1−sin2θ0sin2ϕ0)FIG.3:Probability density distribution of the absolute spanwise-velocity increment|v(x+U/2f s)−v(x−U/2f s)| at Reλ=719,1304,and1934.The distribution is verti-cally shifted by a factor103.The increment is normalized by δv2s 1/2= [v(x+U/2f s)−v(x−U/2f s)]2 1/2.The arrows indicate the ranges for intense vortical structures,which share 0.1and1%of the total.The dotted line denotes the Gaussian distribution.bulence was almost isotropic becausethe measured ra-tio u2 / v2 is not far from unity(Table I).The vortex tubes induce small-scale variations in the spanwise veloc-ity.If we consider intense velocity variations above a high threshold,their scale and amplitude are close to the ra-dius and circulation velocity of intense vortex tubes with |y0|<∼R0andθ0≃0.To demonstrate this,mean proﬁles are calculated for the circulationﬂows uΘof the Burgers vortices with random positions(x0,y0)and inclinations (θ0,ϕ0).Their radii R0and maximum circulation veloc-ities V0=uΘ(R0)are set to be the same.We consider the Burgers vortices with|∂x v|at x=0being above a threshold,|∂x v|/3at x=0for x0=y0=θ0=0.When ∂x v is negative,the sign of the v signal is inverted before the averaging.The result is shown in Fig.2.Despite the relatively low threshold,the scale and peak amplitude of the mean v proﬁle are still close to those of the v proﬁle for x0=y0=θ0=0(dotted line).The extended tails are due to the Burgers vortices with|y0|≫R0orθ0≫0.IV.MEAN VELOCITY PROFILEMean proﬁles of intense vortical structures in the streamwise(u)and spanwise(v)velocities are extracted,FIG.4:Mean proﬁles of intense vortical structures for the 0.1%threshold in the streamwise(u)and spanwise(v)ve-locities.(a)Reλ=719.(b)Reλ=1098.The u proﬁle is separately shown for∂x u>0(u+)and∂x u≤0(u−) at x=0.The position x is normalized by the Kolmogorov length.The velocities are normalized by the peak value of the v proﬁle.We also show the v proﬁle of the Burgers vortex for x0=y0=θ0=0by a dotted line.by averaging signals centered at the position where the absolute spanwise-velocity increment|v(x+r/2)−v(x−r/2)|is above a threshold.10,13,14,15,19The scale r is the sampling interval U/f s.The threshold is such that0.1% or1%of the increments are used for the averaging(here-after,the0.1%or1%threshold).These increments com-prise the tail of the probability density distribution of all the increments as in Fig.3.20Example of the results are shown in Fig.4.20The v proﬁle in Fig.4is close to the v proﬁle in Fig. 2.Hence,the contribution from vortex tubes is dominant.The contribution from vortex sheets is not dominant.If it were dominant,the v proﬁle should ex-hibit some kind of step.12Direct numerical simulations at Reλ<∼200revealed that intense vorticity tends to be organized into tubes rather than sheets.4,5,6,7,21,22This tendency appears to exist up to Reλ≃2000.Vortex sheets might contribute to the extended tails in Fig. 4. They are more pronounced than those in Fig. 2.Here it should be noted that our discussion is somewhat sim-pliﬁed because there is no strict division between vortex tubes and sheets in real turbulence.Byﬁtting the v proﬁle in Fig.4around its peaks by the v proﬁle of the Burgers vortex for x0=y0=θ0=0 (dotted line),we estimate the radius R0and maximumTABLE II:Parameters for intense vortical structures:radius R 0,maximum circulation velocity V 0,Reynolds number Re 0=R 0V 0/νand small-scale clustering exponent µ0.We also list the threshold level τ0.Duct ﬂowBoundary layer Units1234567891011δx pδx p /2−δx p /2v t (x +r )dr.(5)For allthe data,the R 0and V 0values are summarizedin Table II.They characterize the scale and intensity of vortical structures,even if they are not the Burgers vortices.The radius R 0is several times the Kolmogorov length η.The maximum circulation velocity V 0is several tenths of the rms velocity ﬂuctuation v 2 1/2and several times the Kolmogorov velocity u K .Similar results were obtained from direct numerical simulations 3,4,6,7,8and laboratory experiments 11,12,14,15at the lower Reynolds numbers,Re λ<∼1300.The u proﬁle in Fig.4is separated for ∂x u >0(u +)and ∂x u ≤0(u −)at x =0.Since the contamina-tion with the w component 17induces a symmetric posi-tive excursion,14,23,24we decomposed the u ±proﬁles into symmetric and antisymmetric components and show only the antisymmetric components.15The u ±proﬁles in Fig.4have larger amplitudes than those in Fig. 2.Hence,the u ±proﬁles in Fig.4are dominated by the circu-lation ﬂows u Θof vortex tubes that passed the probe with some incidence angles to the mean ﬂow direction,11tan −1[v/(U +u )].@The radial inﬂow u R of the strain ﬁeld is not discernible,except that the u −proﬁle has a larger amplitude than the u +proﬁle.14,15Unlike the Burgers vortex,a real vortex tube is not always oriented to the stretching direction.4,5,6,7,8,25V.SPATIAL DISTRIBUTIONThe spatial distribution of intense vortical structures is studied using the distribution of interval δx 0between successive intense velocity increments.13,14,15,22The in-tense velocity increment is deﬁned in the same manner as for the mean velocity proﬁles in Sec.IV.Since theyare dominated by vortex tubes,we expect that the distri-bution of intense vortical structures studied here is also essentially the distribution of intense vortex tubes.Ex-amples of the probability density distribution P (δx 0)are shown in Figs.5and 6.20The probability density distribution has an exponen-tial tail 14,15that appears linear on the semi-log plot of Fig.5.This exponential law is characteristic of intervals for a Poisson process of random and independent events.FIG.5:Probability density distribution of interval between intense vortical structures for the 1%threshold at Re λ=719,1304,and 1934.The distribution is normalized by the ampli-tude of the exponential tail (dotted line),and it is vertically shifted by a factor 10.The interval is normalized by the streamwise correlation length L u .The arrow indicates the spanwise correlation length L v .FIG.6:Probability density distribution of interval between intense vortical structures for the1%threshold at Reλ=719, 1304,and1934.The distribution is normalized by the peak value,and it is vertically shifted by a factor10.The dotted line indicates the power-law slope from30ηto300η.The interval is normalized by the Kolmogorov lengthη.The arrow indicates the spanwise correlation length L v.The large-scale distribution of intense vortical structures is random and independent.Below the spanwise correlation length L v,the proba-bility density is enhanced over that for the exponential distribution.15Thus,intense vortical structures cluster together below the energy-containing scale.In fact,di-rect numerical simulations revealed that intense vortex tubes lie on borders of energy-containing eddies.6Over small intervals,the probability density distribu-tion is apower law13,22that appears linear on the log-log plot of Fig.6:P(δx0)∝δx−µ0.(6)Thus,the small-scale clustering of intense vortical struc-tures is self-similar and has no characteristic scale.22Ta-ble II lists the clustering exponentµ0estimated over in-tervals fromδx0=30ηto300η.Its value is close to unity.The exponential law over large intervals and the power law over small intervals were also found in laboratory ex-periments for regions of low pressure.26,27,28,29They are associated with vortex tubes,although their radii tend to be larger than those of intense vortical structures studied here.29VI.SCALING LA WDependence of parameters for intense vortical struc-tures on the microscale Reynolds number Reλand on the conﬁguration for turbulence production,i.e.,ductﬂow or boundary layer,is studied in Fig.7.Each quantity was normalized by its value in the ductﬂow at Reλ=1934 individually for the0.1%and1%thresholds.That is,we avoid the prefactors that depend on the threshold.When the threshold is high,the radius R0is small,the maxi-mum circulation velocity V0is large,and the clustering exponentµ0is small as in Table II.We focus on scaling laws of these quantities.The radius R0scales with the Kolmogorov lengthηas R0∝η[Fig.7(a)].Thus,intense vortical structures remain to be of smallest scales of turbulence.The maximum circulation velocity V0scales with the rms velocityﬂuctuation v2 1/2as V0∝ v2 1/2[Fig. 7(b)].Although the rms velocityﬂuctuation is a charac-FIG.7:Dependence of parameters for intense vortical struc-tures on Reλ.(a)R0/η.(b)V0/ v2 1/2.(c)V0/u K.(d)Re0.(e)Re0/Re1/2λ.(f)µ0.The open andﬁlled circles respec-tively denote the ductﬂows for the0.1%and1%thresholds. The upward and downward triangles respectively denote the boundary layers for the0.1%and1%thresholds.Each quan-tity is normalized by its value in the ductﬂow at Reλ=1934 individually for the0.1%and1%thresholds.teristic of the large-scaleﬂow,vortical structures could be formed via shear instability on borders of energy-containing eddies,6,27,28where a small-scale velocity vari-ation could be comparable to the rms velocityﬂuctua-tion.The maximum circulation velocity does not scale with the Kolmogorov velocity u K,a characteristic of the small-scaleﬂow,as V0∝u K[Fig.7(c)].Direct numerical simulations for intense vortex tubes6,7at Reλ<∼200and laboratory experiments for intense vortical structures11,15at Reλ<∼1300derived the scalings R0∝ηand V0∝ v2 1/2.We have found that these scalings exist up to Reλ≃2000,regardless of the conﬁguration for turbulence production.The scalings of the radius R0and circulation veloc-ity V0lead to a scaling of the Reynolds number Re0= R0V0/νfor the intense vortical structures:6,7Re0∝Re1/2λif R0∝ηand V0∝ v2 1/2,(7a) Re0=constant if R0∝ηand V0∝u K.(7b) Our result favors the former scaling[Fig.7(e)]rather than the latter[Fig.7(d)].With an increase of Reλ, intense vortical structures progressively have higher Re0 and are more unstable.6,7Their lifetimes are shorter.It is known30that theﬂatness factor (∂x v)4 / (∂x v)2 2 scales with Re0.3λ.Since (∂x v)4 is dominated by in-tense vortical structures,it scales with v2 2/η4.Since (∂x v)2 2is dominated by the background randomﬂuc-tuation,it scales with u4K/η4.If the number density of intense vortical structures remains the same,we have (∂x v)4 / (∂x v)2 2∝ v2 2/u4K∝Re2λ.The diﬀerence from the real scaling implies that vortical structures with V0≃ v2 1/2are less numerous at a higher Reynolds num-ber Reλ,albeit energetically more important.The small-scale clustering exponentµ0is constant[Fig. 7(f)].A similar result withµ0≃1was obtained from laboratory experiments of the K´a rm´a nﬂow between two rotating disks22at Reλ≃400–1600.The small-scale clustering of intense vortical structures at high Reynolds numbers Reλis independent of the conﬁguration for tur-bulence production.Lastly,recall that only intense vortical structures are considered here.For all vortical structures with vari-ous intensities,the scalings V0∝ v2 1/2and Re0=R0V0/ν∝Re1/2λare not necessarily expected.For allvortex tubes,in fact,direct numerical simulations3,8at Reλ<∼200derived the scaling V0∝u K.The devel-opment of an experimental method to study all vortical structures is desirable.VII.CONCLUSIONThe spanwise velocity was measured in ductﬂows at Reλ=719–1934and in boundary layers at Reλ=332–1304(Table I).We used these velocity data to study fea-tures of vortical structures,i.e.,vortex tubes and sheets. We studied the mean velocity proﬁles of intense vor-tical structures(Fig.4).The contribution from vortex tubes is dominant.Essentially,our results are those for vortex tubes.The radius R0is several times the Kol-mogorov lengthη.The maximum circulation velocity V0 is several tenths of the rms velocityﬂuctuation v2 1/2 and several times the Kolmogorov velocity u K(Table II).There are the scalings R0∝η,V0∝ v2 1/2,and Re0=R0V0/ν∝Re1/2λ(Fig.7).We also studied the distribution of interval between in-tense vortical structures.Over large intervals,the distri-bution obeys an exponential law(Fig.5),which reﬂects a random and independent distribution of intense vortical structures.Over small intervals,the distribution obeys a power law(Fig.6),which reﬂects self-similar clustering of intense vortical structures.The clustering exponent is constant,µ0≃1(Table II and Fig.7).Direct numerical simulations3,4,6,7,8,9,10and laboratory experiments11,12,13,14,15,22derived some of those features. We have found that they are independent of the Reynolds number and of the conﬁguration for turbulence produc-tion,up to Reλ≃2000that exceeds the Reynolds num-bers of the prior studies.The Reynolds numbers Reλin our study are still lower than those of some turbulence,e.g.,atmospheric turbu-lence at Reλ>∼104.Such turbulence is expected to contain intense vortical structures,because turbulence is more intermittent at a higher Reynolds number Reλand small-scale intermittency is attributable to intense vortical structures.They are expected to have the same features as found in our study.These features appear to have reached asymptotes at Reλ≃2000(Fig.7),regard-less of the conﬁguration for turbulence production,and hence appear to be universal at high Reynolds numbers Reλ.AcknowledgmentsThe authors are grateful to T.Gotoh,S.Kida, F. Moisy,M.Takaoka,and Y.Tsuji for interesting discus-sions.1U.Frisch,Turbulence,The Legacy of A.N.Kolmogorov (Cambridge Univ.Press,Cambridge,1995),Chap.8.2K.R.Sreenivasan and R.A.Antonia,“The phenomenol-ogy of small-scale turbulence,”Annu.Rev.Fluid Mech. 29,435(1997).3T.Makihara,S.Kida,and H.Miura,“Automatic tracking of low-pressure vortex,”J.Phys.Soc.Jpn.71,1622(2002). These authors pushed forward the notion that vortex tubes are the elementary structures of turbulence.4A.Vincent and M.Meneguzzi,“The spatial structure and8statistical properties of homogeneous turbulence,”J.Fluid Mech.225,1(1991).5A.Vincent and M.Meneguzzi,“The dynamics of vorticity tubes in homogeneous turbulence,”J.Fluid Mech.258, 245(1994).6J.Jim´e nez,A.A.Wray,P.G.Saﬀman,and R.S.Rogallo,“The structure of intense vorticity in isotropic turbulence,”J.Fluid Mech.255,65(1993).7J.Jim´e nez and A.A.Wray,“On the characteristics of vor-texﬁlaments in isotropic turbulence,”J.Fluid Mech.373, 255(1998).8M.Tanahashi,S.-J.Kang,T.Miyamoto,S.Shiokawa,and T.Miyauchi,“Scaling law ofﬁne scale eddies in turbulent channelﬂows up to Reτ=800,”Int.J.Heat Fluid Flow 25,331(2004).9A.Pumir,“Small-scale properties of scalar and velocity diﬀerences in three-dimensional turbulence,”Phys.Fluids 6,3974(1994).10H.Mouri,M.Takaoka,and H.Kubotani,“Wavelet iden-tiﬁcation of vortex tubes in a turbulence velocityﬁeld,”Phys.Lett.A261,82(1999).11F.Belin,J.Maurer,P.Tabeling,and H.Willaime,“Obser-vation of intenseﬁlaments in fully developed turbulence,”J.Phys.(Paris)II6,573(1996).They studied turbulence velocityﬁelds at Reλ=151–5040.We do not consider their results at Reλ>∼700,where (∂x u)3 / (∂x u)2 3/2and (∂x u)4 / (∂x u)2 2of their data are known to be inconsis-tent with those from other studies.212A.Noullez,G.Wallace,W.Lempert,es,and U.Frisch,“Transverse velocity increments in turbulentﬂow using the RELIEF technique,”J.Fluid Mech.339,287 (1997).13R.Camussi and G.Guj,“Experimental analysis of inter-mittent coherent structures in the nearﬁeld of a high Re turbulent jetﬂow,”Phys.Fluids11,423(1999).14H.Mouri,A.Hori,and Y.Kawashima,“Vortex tubes in velocityﬁelds of laboratory isotropic turbulence:depen-dence on the Reynolds number,”Phys.Rev.E67,016305 (2003).15H.Mouri,A.Hori,and Y.Kawashima,“Vortex tubes in turbulence velocityﬁelds at Reynolds numbers Reλ≃300–1300,”Phys.Rev.E70,066305(2004).16K.R.Sreenivasan and B.Dhruva,“Is there scaling in high-Reynolds-number turbulence?,”Prog.Theor.Phys.Suppl.130,103(1998).17The two wires individually respond to all the u,v,and w components.Since the measured u component corresponds to the sum of the responses of the two wires,it is contam-inated with the w component.Since the measured v com-ponent corresponds to the diﬀerence of the responses,it is free from the w component.18T.S.Lundgren,“Strained spiral vortex model for turbu-lentﬁne structure,”Phys.Fluids25,2193(1982).19For convenience,when consecutive increments are all above the threshold,each increment is taken to determine the center of a vortex.This is somewhat unreasonable but does not cause serious problems,judging from Fig.2where mean velocity proﬁles were obtained practically in the same manner.20While the experimental curves in Figs.3and4are mere loci of discrete data points,we applied smoothing to the tails of the experimental curves in Figs.5and6.21F.Moisy and J.Jim´e nez,“Geometry and clustering of in-tense structures in isotropic turbulence,”J.Fluid Mech.513,111(2004).22F.Moisy and J.Jim´e nez,“Clustering of intense structures in isotropic turbulence:numerical and experimental ev-idence,”in IUTAM Symposium on Elementary Vortices and Coherent Structures:Signiﬁcance in Turbulence Dy-namics,edited by S.Kida(Springer,Dordrecht,2006),p.3.23K.Sassa and H.Makita,“Reynolds number dependence of elementary vortices in turbulence,”in Engineering Tur-bulence Modelling and Experiments6,edited by W.Rodi and M.Mulas(Elsevier,Oxford,2005),p.431.24The positive excursion might be partially induced byﬂuc-tuation of the instantaneous velocity U+u at which a struc-ture passes the probe.Under Taylor’s frozen-eddy hypoth-esis,the velocity increment over the sampling interval U/f s is more intense for a faster-moving structure,which is more likely to be incorporated in our conditional averaging.14 Other mechanisms might be also at work.25M.Kholmyansky,A.Tsinober,and S.Yorish,“Velocity derivatives in the atmospheric surface layer at Reλ=104,”Phys.Fluids13,311(2001).26P.Abry,S.Fauve,P.Flandrin,and roche,“Analy-sis of pressureﬂuctuations in swirling turbulentﬂows,”J.Phys.(Paris)II4,725(1994).27O.Cadot,S.Douady,and Y.Couder,“Characterization of the low-pressureﬁlaments in a three-dimensional turbulent shearﬂow,”Phys.Fluids7,630(1995).28E.Villermaux,B.Sixou,and Y.Gagne,“Intense vortical structures in grid-generated turbulence,”Phys.Fluids7, 2008(1995). Porta,G.A.Voth,F.Moisy,and E.Bodenschatz,“Using cavitation to measure statistics of low-pressure events in large-Reynolds-number turbulence,”Phys.Fluids 12,1485(2000).30B.R.Pearson and R.A.Antonia,“Reynolds-number de-pendence of turbulent velocity and pressure increments,”J.Fluid Mech.444,343(2001).。

金属塑性成形6)plastic anisotropy

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6.3 Normal anisotropy and planar anisotropy
Specimen taken from different directions have different 1) The cause of anisotropy -crystallographic orientation r-value. This kind of anisotropy is called planar anisotropy. Planar anisotropy is 2) Types of by characterizedanisotropy for sheet metals
6 Plastic anisotropy*
6.2 r-value
1)A useful parameter to show plastic anisotropy of a sheet metal is the The cause of anisotropy -crystallographic orientation so-called r-value, which is the ratio of plastic strains, measured from a 2) Types of anisotropy for sheet tensile specimen of sheet metal. metals

高等传热学全英文

Transport of Heat by Conduction
7
Effect of Geometry on Temperature Profile

1D conduction in a cone (the surfaces are thermally insulated) with the cross‐sectional area Ax = ax2. Calculate the temperature profile along x direction.

3D Fourier’s law
q ' k T
Transport of Heat by Conduction
5
Thermal Conductivity K

K is the physical properties of materials

Varies with temperature Increases with temperature for gases but decreases with increasing temperature for solids Independent of direction for isotropic materials but varies with direction for anisotropic materials. Good electrical conductor often also a good thermal conductor

Conduction: Heat transfer due to atom vibration, collision or free electron diffusion (no bulk movement).

Pseudovorticella paracratera n.sp.,a new marine peritrich ciliate

Hydrobiologia515:49–57,2004.©2004Kluwer Academic Publishers.Printed in the Netherlands.49Pseudovorticella paracratera n.sp.,a new marine peritrich ciliate (Ciliophora:Peritrichida)from north ChinaDaode Ji1,Weibo Song1∗&Alan Warren21Laboratory of Protozoology,ARL,Ocean University of China,Qingdao266003,People’s Republic of China Fax:+865322032283.E-mail:wsong@(∗Author for correspondence)2Department of Zoology,The Natural History Museum,Cromwell Road,London SW75BD,U.K.Received5May2003;in revised form25August2003;accepted27August2003Key words:marine ciliate,morphology,Peritrichida,Pseudovorticella paracratera n.sp.AbstractA new marine peritrich ciliate,Pseudovorticella paracratera n.sp.,was isolated from marine waters off the coast of Qingdao,China.The morphology,infraciliature,and silverline system were studied from living and silver-impregnated specimens.It is characterized by a wide conical or inverted bell-shaped body which measures40–88µm in length×64–144µm in width in vivo,two ventrally located contractile vacuoles,and a wide peristomial lip with closely-spaced,irregular tubercles.The pellicle is tubercular with conspicuous,widely-spaced striations. There are15–18transverse silverlines between the peristomial lip and the aboral ciliary wreath and9–11between the aboral ciliary wreath and the scopula.The stalk is8–10µm wide and about5–12times the zooid length.The spasmoneme has numerous conspicuous thecoplasmic granules.IntroductionPeritrich ciliates are a ubiquitous and diverse group, which often play an important ecological role in both marine and freshwater environments(e.g.,Müller, 1776;Kent,1881;Noland&Finley,1931;Kahl,1935; Küsters,1974;Jankowski,1976,1985;Song,1986, 1991,1997;Foissner et al.,1992).However,it has long been recognised that species identiﬁcation among peritrichs is often difﬁcult,mainly because many have highly variable morphologies so that different spe-cies may appear to be very similar to one another when observed in vivo.Furthermore,many species descriptions are based only on live observations(Kent, 1881;Kahl,1935;Noland and Finley,1931;Küsters, 1974;Warren,1986,1987,1988).Major improve-ments in the taxonomy of peritrichs were initiated by Foissner&Schiffmann(1974,1975)who recommen-ded that extensive photographic documentation of live specimens,along with detailed investigations of the silverline system and infraciliature,should be carried out when taxa are described or redescribed.Pseudovorticella was established by Foissner and Schiffmann(1974)for peritrichs which are morpholo-gically similar to Vorticella,i.e.,solitary zooids borne upon a non-branching contractile stalk,but which have a reticulate silverline system with lines running ver-tically as well as horizontally.Pseudovorticella now comprises more that20species,most of which have been transferred from Vorticella(Foissner&Schiff-mann,1974,1979;Foissner,1979;Warren,1987; Song,1989,1997;Foissner et al.,1992;Sachiko& Warren,1996;Song et al.,1996;Leitner&Foissner, 1997;Song&Warren,2000).In September2001,a Pseudovorticella species was collected from the surface of the marine green alga Ulva sp.(Chlorophyta,Ulvales)off the coast of Qingdao,China.Careful observations,both in vivo and following silver impregnations,revealed that this organism is unlike any other previously described spe-cies of Pseudovorticella.We therefore consider it as a new species,P.paracratera n.sp.,and provide a detailed description.50Figures1–7.Pseudovorticella paracratera n.sp.from life.1–representative extended and contracted zooid.2,4–extended zooids at high magniﬁcation,showing,inter alia,the position of contractile vacuoles(arrows),food vacuoles(double arrowheads),and the appearance of pellicle and peristomial lip.3–telotroch.5–detail of stalk,with conspicuous thecoplasmic granules in spasmoneme and many rod-shaped bacteria on stalk surface.6,7–shape variety of macronucleus,showing variations in shape.Scale bars:1–100µm;3–50µm.Material and methodsPseudovorticella paracratera was collected on10 September2001off the coast of Qingdao,China. Individuals were isolated in the laboratory from the surface of Ulva sp.(Chlorophyta,Ulvales)and were observed in vivo using a high power oil immersion objective and differential interference contrast mi-croscopy.The infraciliature and myoneme system were revealed with protargol impregnation accord-ing to Wilbert(1975),and the silverline system by the Chatton–Lowff silver nitrate method according to Song&Wilbert(1995).51 Figures8–11.Pseudovorticella paracratera n.sp.after protargol(Figs.8,9)and silver nitrate(Figs.10,11)impregnation.8–general infraciliature and myoneme system.9–buccal infraciliature in apical view,showing polykinety,haplokinety,germinal kinety,epistomial membrane(arrow),and peniculi1-3;arrowhead marks displaced kinety of P3.10,11–silverline system,arrows show aboral ciliary wreath. Abbreviations:ACW–aboral ciliary wreath,G–germinal kinety,H–haplokinety,P1-3–peniculi1-3,Po–polykinety.Scale bar:8–40µm.Photomicrographs were obtained by using a digital camera.The drawings of impregnated specimens were made with the help of a camera lucida at×1250mag-niﬁcation.Drawings of live specimens are based on in vivo observation and on photomicrographs.Terminology is mainly according to Corliss(1979, 1994)and Warren(1987).ResultsPseudovorticella paracratera n.sp.(Figs.1–22, Table1)DiagnosisZooid wide conical or inverted bell-shaped,size in vivo40–88µm in length×64–144µm in diameter/width.Length:width ratio1:1.2–2.5.Wide peristomial lip with numerous irregular tubercles. Macronucleus J-shaped.Two small contractile vacu-52Figures12–18.Morphologically similar Pseudovorticella n.sp.and Vorticella species to Pseudovorticella paracratera.12–Vorticella cratera (after kent,1881).13–Pseudovorticella patellina(after Song,2000).14–P.monilata(a)Zooid;(b)telotroch(after Warren,1988).15–P.margaritata(after Warren,1988).16–P.stilleri(after Warren,1988).17–Vorticella fornicata(after Song,1991).18–V.campanula(a) Zooid;(b)telotroch(after Warren,1986).Scale bars:12–100µm;13–15,17,18–30µm;16–50µm.oles near ventral wall of vestibulum.Reticulate sil-verline system with15–18transverse rows between peristome and aboral ciliary wreath,and9–11rows between aboral ciliary wreath and scopula.Stalk×5–12zooid length,but maximally600µm long,8–10µm across.Individuals typically grouped together forming pseudocolonies.Marine.Type specimensOne holotype slide with silver nitrate impregnated specimens and one paratype slide with protargol-impregnated specimens are deposited in the col-lections of the Natural History Museum,London, U.K.with registration numbers2002:5:22:3and 2002:5:22:4respectively.Two paratype slides(regis-tration numbers:01090301,02062801)are deposited at the Laboratory of Protozoology,College of Fish-eries,Ocean University of China,Qingdao,P.R. China.EtymologyThe name‘paracratera’is a composite of the Greek/Latin preﬁx para-(=beside,like,similar to) and the species-group name cratera,and thus recalls the similarity in general appearance between the new species and Vorticella cratera Kent,1881.53Figures19–29.Photomicrographs of Pseudovorticella paracratera n.sp.from life(Figs.19,20,24–29)and after silver nitrate(Fig.21)and protargol(Figs.22,23)impregnation.19–a fully extended zooid at low magniﬁcation.20,24,26–apical(Fig.20)and lateral(Figs.24, 26)views of zooids at×400magniﬁcation,showing the irregular tubercles on the peristomial lip(arrows).21–silverline system in posterior body portion,arrows mark aboral ciliary wreath.22–posterior view of zooid showing aboral ciliary wreath(double arrowheads)and scopula (arrow).23–apical view of zooid showing buccal infraciliature including epistomial membrane(arrow)and peniculi(P1-3).25–zooid at ×400magniﬁcation,showing the lower contractile vacuole(arrow).27–detail of stalk,arrowheads showing thecoplasmic granules.28–living zooid,arrows showing the pellicular striations.29–a telotroch(arrow)near a contracted zooid.Scale bars:19–100µm;27–10µm.54DescriptionZooid in vivo40–88µm long×64–144µm wide, broadly inverted campaniform to wide conical,widest at peristomial area with thin and relatively rigid peristomial lip,usually80–100µm in diameter, (Figs.1,2,4,19)with extremity to64–144µm. Ratio of zooid length:width1:1.2–2.5,1:2.0on aver-age.Peristomial lip with numerous irregular,closely-spaced pellicular tubercles(Figs.2,4,20arrows,24, 26).Peristomial disc broad,ﬂattened and slightly ob-liquely elevated when cell is fully extended.Pellicle appears to be coarsely striated at low magniﬁcation, conspicuously tubercular when viewed at higher mag-niﬁcations(Figs.2,4,28).Cytoplasm colourless or greyish,usually contain-ing several food vacuoles8–10µm across packed with bacteria(Figs.2,4,19,25).Two contractile vacuoles near ventral wall of vestibulum,one above mid-body near end of funnel-shaped vestibulum,the other in posterior half of body near level of aboral cil-iary wreath(Figs.2,25,arrows),contract at different rates:anterior one about every3min,posterior one often>5min.In some zooids only one contractile vacuole was observed which was located in mid-body region(Fig.4)and contracted at shorter time intervals (<1min).Macronucleus usually J-shaped,occasionally C-shaped(Figs.6,7),posterior end usually curved but sometimes relatively straight.Micronucleus not observed.Stalk×5–12zooid length,but maximally600µm long,8–10µm across,surface smooth with nu-merous bacteria attached,contracts spirally due to Spasmoneme,which3–4µm across,with numer-ous,conspicuous,about0.8µm across,dark grey thecoplasmic granules(Figs.5,27).Individuals often closely grouped together thus forming pseudocolonies of up to50zooids.Telotroch(swarmer)roughly cylindrical,about60×30µm in size(Figs.3,29).Oral apparatus similar to that of many other perit-richs.Haplokinety and polykinety describe about1.5 turns around persistomial disc before entering vesti-bulum,where they make a further turn(Figs.8,23). Polykinety forms three peniculi in lower half of ves-tibulum,each consisting of three kineties.Peniculus 1(P1)and P2are much longer than P3.P1and P3 terminate at same level whereas P2terminates above these two(Fig.9).Inner kinety of P3displaced slightly anteriorly to the other two(Fig.9,arrowhead).Ger-minal kinety lies parallel to haplokinety within upper third of vestibulum.Epistomial membrane short,loc-ated near opening of vestibulum(Fig.9,23,arrows). Aboral ciliary wreath formed by row of double kin-eties which encircles cell in posterior region(Fig.8, 22double arrowheads).Myoneme system of P.cratera includes thick spas-moneme within stalk,and myonemes around scopula which extend anteriorly beyond mid-body(Fig.8).Silverline system reticulate;transverse lines in an-terior region of cell more widely spaced than those in posterior region(Figs.10,11,21).Aboral ciliary wreath represented by two parallel lines(Figs.10,11, 21,arrows).Number of transverse silverlines from peristome to aboral ciliary wreath,15–18;from wreath to scopula,9–11.Pellicular pores sparse. Discussion and comparisonThe present species is assigned to Pseudovorticella Foissner and Schiffmann,1974because it is solitary and has a spirally contractile stalk,and a reticulate silverline system.In terms of its tubercular peristo-mial lip and body shape,Pseudovorticella paracratera is similar to Vorticella cratera Kent1881(Fig.12), which was originally described by Ehrenberg(1838) under the name Vorticella patellina and later renamed by Kent(1881).The new species differs from V.crat-era in its habitat(marine vs.freshwater),cell size (40–88vs ca.120–130µm),pellicle(tubercular vs smooth),and number of contractile vacuoles(2vs1) (Kent1881).Pseudovorticella paracratera is also similar to P.patellina(Müller,1776)Song&Warren,2000 (Fig.13),especially in terms of body shape and size, number of contractile vacuoles,and the position and shape of the macronucleus.Nevertheless,P.paracrat-era can be distinguished from the latter by the number of transverse striations between the peristome and the aboral ciliary wreath(15–18vs19–22),the appear-ance of the pellicle(tubercular vs smooth),and the position of the contractile vacuoles(ventral vs dorsal).Pseudovorticella paracratera bears some resemb-lance to P.monilata(Tatem,1870)Foissner and Schiffmann,1974(Fig.14),particularly with respect to the tubercular pellicle,the J-shaped macronucleus, the possession of two contractile vacuoles,and the spasmoneme with thecoplasmic granules.However, the two taxa can be clearly separated by the body length:width ratio(usually1:2vs1:<1),the positions of the contractile vacuoles(one in mid-body region55 Table1.Morphometrical characterizations of Pseudovorticella paracratera.Measurements inµm.Max–maximum,Min–minimum,Mean–arithmetic mean,SD–standard deviation,SE–standard error of the mean,CVr–coefﬁcient of variation,n–sample numberCharacter Min Max Mean SD SE CVr nBody length in vivo408855.315.6 4.7028.211Body width in vivo6414410021.4 6.4621.311Number of silverlines from oral area to aboral ciliary wreath151816.3 1.030.42 6.346Number of silverlines from aboral ciliary wreath to scopula9119.80.840.378.545Table2.Morphological comparison between Pseudovorticella paracratera n.sp.and morphologically similar Pseudovorticella and Vorticella species.Measurements inµm.?–data not available.Species Body lengthin vivoExtremes(mean)Body widthin vivoExtremes(mean)Number oftransversesilverlinesfrom scopulato aboralciliarywreathNumber oftransversesilverlinesfrom aboralciliarywreath toanterior endSurface ofperistomiallipNumber ofcontractilecacuolesHabitat Data sourcePseudovorticella paracratera40–88(55.3)64–144(100)9–1115–18Tubercular2Marine Present workP.patellina55–110(81.0)50–100(70.6)13–1619–22Slightlycoarse2Marine Song&Warren,2000P.monilata45–8035–609–18(14)15–23(19)Convex2Freshwater Foissner&Schiffmann,1974P.margaritata60–14040–70??Convex2Freshwater Warren,1987 P.stilleri8580??Smooth?Freshwater Warren,1987 Vorticella cratera120–130150??Frill-like1Freshwater Warren,1986V.fornicata24–32(29)24–37(32)??Smooth1Marine Song&Warren,2000V.campanula50–16035–10027–33(29)69–77(72)Smooth1Freshwaterand marineFoissner et al.,1992;Warren,1986and one in posterior1/3of body vs both in anterior half of body),and the total number of transverse striations (24–29vs31–41).Other species of Pseudovorticella with which P.paracratera should be compared include P.mar-garitata(Fromentel,1874)Jankowski,1976(Fig.15) and P.stilleri Warren,1987(Fig.16).Pseudovorticella paracratera differs from P.margaritata in terms of its macronucleus shape(usually J-shaped vs C-shaped) and the position of the contractile vacuoles(one in mid-body region and one in posterior1/3of body vs both in anterior part of body).Likewise,it differs from P.stilleri in its ratio of body length:width(usually1:2 vs1:<1),appearance of the peristomial lip(tubercular vs smooth)and macronucleus shape(usually J-shapedvs C-shaped).Considering its general appearance in vivo,P.paracratera should also be compared with someVorticella-species which,however,lack the reticulate pellicle,respectively,silverline system.Vorticella for-nicata Dons,1915(Fig.17),for example,has a similarbody shape to P.paracratera(width>length)butis distinctly smaller(24–32×24–37µm vs40–88×64-144µm)(Song1991).Vorticella campanula Ehrenberg,1831(Fig.18)resembles P.paracrat-era with respect to its body size(68×78µm),ﬂattened peristomial disc,J-shaped macronucleus,long stalk(up to500µm long),spasmoneme withnumerous thecoplasmic granules,and its tendency56to form large pseudocolonies.Vorticella campanula, however,is constricted below peristomial lip(vs not constricted),has only one contractile vacuole(vs2 CVs),lacks pellicular tubercles(vs present)and has a typical Vorticella-style silverline system(i.e.,trans-verse silverlines only vs reticulate silverline system) with a total of96–110(vs24–29)transverse stri-ations(Warren,1986).A summary of comparisons between P.paracratera and morphologically similar Pseudovorticella spp.and Vorticella spp.is given in Table2.AcknowledgementsWe would like to thank Xiaozhong Hu for his helpful comments.This work was supported by the“Ch-eung Kong Programme”,the NSFC(Project No. 40206021)and the Royal Society(Joint Project No. Q822).ReferencesCorliss,J.O.,1979.The Ciliated Protozoa:Characterization,Clas-siﬁcation and Guide to the Literature,2nd ed.Pergamon Press, New York,455pp.Corliss,J.O.,1994.An interim utilitarian(‘user-friendly’)hier-archical classiﬁcation and characterization of the protists.Acta Protozoologica33:1–51.Dons, C.,1915.Neue marine Ciliaten und Suctorien.TromsøMuseums Aarschefter38:75–100.Ehrenberg,C.G.,1831(1832).Über die Entwicklung und Lebens-dauer der Infusionthiere;nebst ferneren Beiträgen zu einer Vergleichung ihrer Organischen systeme.Abhandlungen der Akademie der Wissenchaften der DDR,year1831:1–154. 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High-temperature potentiometric amperometric NOx sensors

Keywords: NOx sensor; Potentiometric sensor; Amperometric sensor; Mixed potential; Stabilized zirconia; Oxide electrode; Spinel-type oxide; CdCr2O4
1. Introduction In order to solve effectively the NOx (NO and NO2)related pollution problems such as acid rain and photochemical smog, we need inexpensive compact sensors to monitor NOx in the combustion exhausts and in environments. Various types of NOx sensors have been developed so far, e.g. resistive type sensors using oxide semiconductors [1–5] or metal phthalocyanines [6], potentiometric and amperometric sensors using solid electrolytes [7–16] and capacitor-type sensors using oxides [17]. However, all of these sensors can not be applied to in situ monitoring of NOx (especially NO) in high temperature combustion exhausts from automobiles because of their relatively low operation temperature and their lower sensitivities to NO compared to NO2.

导热复合材料英文资料

Novel high performance Al2O3/poly(ether ether ketone)nanocomposites for electronics applicationsR.K.Goyal a,*,A.N.Tiwari b,U.P.Mulik a,Y.S.Negi c,*a Centre for Materials for Electronics Technology(C-MET),Department of Information Technology,Govt.of India,Panchwati,OﬀPashan Road,Pune411008,Indiab Department of Metallurgical Engineering and Materials Science,Indian Institute of Technology,Bombay,Powai,Mumbai400076,Indiac Polymer Science and Technology Laboratory,Department of Paper Technology,Indian Institute of Technology,Roorkee,Saharanpur Campus,Saharanpur,U.P.247001,IndiaAbstractThis paper deals with the preparation and characterization of nanocomposites of poly(ether ether ketone)(PEEK)containing nano-aluminum oxide(n-Al2O3)ﬁller up to30wt%(12vol%)loading.Nanocomposites showed improved thermal stability,crystallization, and coeﬃcient of thermal expansion(CTE).Thermogravimetric analysis showed enhanced thermal stability and char yield on increasing the n-Al2O3loading in PEEK matrix.The peak crystallization temperature is increased up to13°C for the nanocomposites as compared to pure PEEK.The CTE is decreased to a value very close to the CTE of copper at12vol%Al2O3loading.The CTE values obtained were compared with the theoretical equations in the literature.The X-ray diﬀraction showed that PEEK crystalline structure is unchanged with addition of n-Al2O3.The distribution of n-Al2O3in the PEEK matrix was studied by transmission electron microscopy and scanning electron microscopy.The results show that the prepared n-Al2O3/PEEK nanocomposites may have potential applications in electronics.Keywords:A.Polymer-matrix composites;PEEK;B.Thermal properties;D.X-ray diﬀraction;D.Transmission electron microscopy1.IntroductionHigh performance polymer composites such as poly(-ether ether ketone)(PEEK),polyethersulphone(PES), polyphenylenesulphide(PPS)and polyimides reinforced with ceramicﬁllers result in unique combination of ther-mal,mechanical and electrical properties,which make them useful for various applications.By introducing suit-able reinforcingﬁllers in polymers,composite properties can be tailored to meet speciﬁc design requirements such as low density,high strength,high stiﬀness,high damping, chemical resistance,thermal shock resistance,high thermal conductivity,low coeﬃcient of thermal expansion(CTE) and good electrical properties such as dielectric constant.It is well documented that PEEK exhibits excellent ther-mal,mechanical,electrical properties,good moisture and chemicals resistance[1].Recently,its properties have been further improved by incorporating micron size particles such as aluminum nitride(AlN)[2,3],aluminum oxide (Al2O3)[4],CaCO3[5],and hydroxiapatite(HA)[6]ﬁllers. In the last one decade,polymer based nanocomposites con-taining nanoﬁllers have been intensively investigated due to ﬁller’s much higher surface area to volume ratio,which results in much higher interface between the nanoﬁllers and the polymer matrix as compared to conventionally used micron sizeﬁllers and polymer matrix.Hence,a very low loading(<5vol%)of nanoﬁllers is required to improve the thermal,mechanical,optical,electrical and magnetic properties in contrast to high loading(>20vol%)of micronsizeﬁllers.In particular the typical micron sizeﬁllers needed for reducing the CTE of polymers are as high as 50vol%[7].As a result of highﬁller loading,the main advantages such as ease of processing and light weight of polymers get lost.Therefore,the use of nanoﬁller in poly-mer composites has attracted the attention of materials sci-entists,technologists,and industrialists for diﬀerent applications.Nevertheless,the eﬀect of nanoﬁller on prop-erties of composites depends strongly on its shape,size, aggregates size,surface characteristics and degree of dis-persion.In order to improve properties of polymer nano-composites,a homogeneous dispersion of the nanoﬁllers in the polymer matrix is essential[8–14].There are a several hundred publications on the eﬀect of ceramicﬁllers on diﬀerent polymer properties,but there is rare literature on the eﬀect of n-Al2O3ﬁller on PEEK. However,recently Kuo et al.have studied the eﬀect of n-Al2O3and n-SiO2(up to5vol%)on PEEK’s mechanical and thermal properties[15].Moreover,Wang et al.have studied the wear properties of PEEK by incorporating SiC[16],SiO2[17],Si3N4[18],and ZrO2[19]nanoﬁllers up to20wt%.Nevertheless,higher loading ofﬁllers is required to decrease the CTE of the polymer to avoid the thermal stresses and to increase the thermal conductivity of polymer to dissipate the heat generated during turning on and turning oﬀthe electronic devices.In view of the above,in present paper a systematic inves-tigation of the eﬀect of electrically insulating and thermally conducting n-Al2O3ﬁller on the PEEK nanocomposites prepared by mixing PEEK and n-Al2O3ﬁllers(up to 30wt%)in alcohol medium using mechanical stirring fol-lowed by hot compression molding was studied.The den-sity,thermal stability,melting and crystallization behavior,CTE,and crystal structure of the nanocompos-ites were characterized by using density,thermogravimetric analysis(TGA),diﬀerential scanning calorimetry(DSC), thermomechanical analyzer,ands X-ray diﬀraction tech-nique,respectively.The dispersion of the n-Al2O3ﬁllers in PEEK matrix was observed by scanning electron micros-copy(SEM)and transmission electron microscopy.2.Experimental2.1.MaterialsThe commercial PEEK,grade5300PF donated by Gharda Chemicals Ltd.Panoli,Gujarat,India under the trade name GATONE TM PEEK was used as matrix.It has a reported inherent viscosity of0.87dl/g measured at a con-centration of0.5g/dl in H2SO4.Theﬁller used in the prep-aration of nanocomposites was n-Al2O3of density4.00g/ cm3.It was used as supplied by Aldrich Chemical Company. Figs.1a and b are typical SEM micrographs of PEEK pow-der and n-Al2O3powder,respectively.As received ethanol of Merck grade was used for homogenizing the n-Al2O3 and PEEK mixture.The particle size of the PEEK deter-mined by GALAI CIS-1laser particle size analyzer was ranges from4to49l m.The mean size of the PEEK particle was25l m.The reported average particle size and surface area of n-Al2O3is39nm and43m2/g,respectively.2.2.Nanocomposites preparationNanocomposites of PEEK reinforced with n-Al2O3up to 30wt%loading were prepared using the method described in our previous paper[2].Dried powder of n-Al2O3and PEEK were well premixed through magnetic stirring at high stirring speed using an ethanol as medium and the resultant slurry was dried in an oven at120°C to remove the excess alcohol.The pure PEEK(controlled)and nanocomposite samples were prepared by using a laboratory hot press under a pressure of15MPa at a temperature of350°C. 3.Characterization3.1.DensityThe density of the nanocomposites prepared by taking appropriate amount of PEEK and n-Al2O3was increased due to higher density of n-Al2O3(4.00g/cm3)ascompared Fig.1.SEM micrographs of:(a)PEEK powder,magniﬁcation=2·103;(b)n-Al2O3powder,magniﬁcation=10·103.1803to pure PEEK(1.30g/cm3).Theoretical density(q th,c)of the nanocomposites was calculated by the rule of mixture with no voids and no loss ofﬁllers during processingq th;c¼q m V mþq f V fð1Þwhere q m,q f,V m,and V f is the density of matrix,density of ﬁller,volume fraction of matrix,and volume fraction ofﬁl-ler,respectively.Experimental density(q ex,c)of the PEEK nanocompos-ites was determined by Archimedes’s method using:q ex;c¼½W air=ðW airÀW alcoholÞ Áq alcoholð2Þwhere W air and W alcohol is the weight of the sample in air and in alcohol medium,respectively.The q alcohol is the den-sity of the alcohol medium used.3.2.Thermogravimetric analysis(TGA)The thermal stability of the PEEK nanocomposites was determined on a TGA using Mettler-Toledo TGA/SDTA 851e.The samples were heated from room temperature to 1000°C at the heating rate of10°C/min in air or nitrogen atmosphere.The maximum decomposition temperature (T m),was taken as the temperature corresponding to the maximum of the peak obtained by theﬁrst order derivative curve.The%char yield was determined at temperature of 1000°C in nitrogen atmosphere.3.3.Diﬀerential scanning calorimetry(DSC)The melting and non-isothermal crystallization behavior of PEEK nanocomposites was performed on DuPont Instruments910DSC.The samples placed in aluminum pan wereﬁrst heated from30°C to400°C at a heating rate of5°C/min and soaked isothermally at400°C for5min to allow complete melting of the polymer.The samples were then cooled to30°C at a cooling rate of5°C/min.Each sample was subjected to single heating and cooling cycles under a dry nitrogen purge.3.4.X-ray diﬀraction measurementsXRD pattern of as molded PEEK nanocomposites was recorded on Philips X’Pert PANalytical PW3040/60.Ni-ﬁltered Cu K a radiation(k=1.54A˚)generated at40kV and30mA was used for the angle(2h)ranged from10°to50°.The scan step size and time per step was0.02°and5s,respectively.3.5.Morphological examinationMorphological analysis of the PEEK powder,n-Al2O3 powder and nanocomposites pellets was conducted with a SEM(Quanta200HV,FEI).For SEM study of nanocom-posites,a small piece of the sample was cut from the pellets and mounted in a block of acrylic based polymer resin (DPI-RR cold cure).The obtained sample surfaces were manually ground and polished with successiveﬁner grades of emery papers followed by cloth(mounted on wheel)pol-ishing to remove scratches developed during emery paper polishing.Thus,obtained samples were called as polished samples in the present study.The same polished samples were also etched for5min in a2%w/v solution of potas-sium permanganate in a mixture of4vol.of orthophos-phoric acid and1vol.of water and were called as etched samples.After polishing and etching,samples were rinsed well in water and dried for examining the polished and etched samples,respectively.The morphology of PEEK and Al2O3powder was determined by suspending powder in an ethanol followed by dispersing on metal stub.Finally the samples were coated with a thin layer of gold using gold sputter coater[Polaron SC7610]to make the sample elec-trically conducting.Dispersability of the n-Al2O3ﬁller in the PEEK matrix was also observed using TEM(Philips CM30)operated at an accelerating voltage of200kV. The ultra-thin section slice($100nm thick)of the nano-composites was cut with ultramicrotome(Leica Ultracut UCT)at room temperature.The slices were mounted on 200-mesh copper grids and dried before the TEM observation.3.6.Thermo mechanical analyzer(TMA)The out-of-plane(through thickness direction)CTE of the nanocomposites were determined using Perkin–Elmer DMA7e in thermo mechanical analyzer mode.The detailed procedure of the CTE measurement was described elsewhere[20].The annealed sample was heated under pressure of50mN from30to250°C at a heating rate of 5°C/min in argon atmosphere.The sample was then cooled to30°C and reheated at5°C/min to250°C.The results were reported for the second run and an average value of CTE was determined over a speciﬁc temperature range of30–140°C,i.e.below glass transition temperature (T g)of PEEK.4.Results and discussionPEEK nanocomposites reinforced with varying weight fraction of n-Al2O3were prepared by hot compression molding technique.Resulting compositions were character-ized and discussed in details in this section.Table1showed the properties of the PEEK matrix and Al2O3ﬁller.These Table1Properties of PEEK and Al2O3Material PEEK[1]Al2O3[7] Density(g/cc) 1.30a 4.00b CTE(·10À6/°C)58a 6.6 Young’s modulus(GPa) 3.6385 Shear modulus(GPa) 1.3155Bulk modulus(GPa) 6.2247 Poisson ratio0.400.24a Experimental results.b Suppliers datasheet.1804properties were used to estimate the theoretical density and CTE of the composites.Table2showed the weight%and volume%of the n-Al2O3ﬁller added into the PEEK matrix.From the given weight fraction ofﬁller,volume fraction of theﬁller can be determined by using:V f¼W f=½W fþð1ÀW fÞÁq f=q m ð3Þwhere W f is the weight fraction of theﬁller.4.1.DensityFig.2shows the density of the n-Al2O3ﬁlled PEEK as a function of n-Al2O3content.It can be seen that the nano-composites density increased with n-Al2O3loading in a lin-ear fashion due to the higher density of n-Al2O3(4.00g/ cm3)than that of pure PEEK(1.30g/cm3).The experimen-tal density of the nanocomposites is in good agreement with the theoretical density except at12vol%nano-Al2O3.This might be an indication of the porosity free samples due to good processing conditions.The experimen-tal density of the NC-30nanocomposite is about1.3%les-ser than theoretical density.This may be due to the presence of voids,which is resulted from the n-Al2O3 agglomerates.During hot pressing the inﬁltration of melt PEEK resin,due to very high viscosity,is diﬃcult through the agglomerates,hence results in voids in theﬁnal samples.4.2.Thermogravimetric analysis(TGA)Figs.3and4show the percentage of original weight remaining as a function of temperature in nitrogen and air atmosphere,respectively.The temperature of10wt% loss was taken as the degradation temperature(T10)and tabulated in Table3.It can be seen from Table3that pure PEEK has T10in nitrogen atmosphereðT10;N2Þat570°C and in air atmosphere(T10,air)at556°C,which is attrib-uted to the decomposition of the PEEK matrix.Pure n-Al2O3powder does not show(not shown inﬁgure)any abrupt change in weight and only a slight($3–4%) decrease at500°C appears due to the loss of physisorbed water[21].It is observed that as the n-Al2O3loading increases in PEEK the degradation temperature(thermal stability)of nanocomposites is improved signiﬁcantly.The increase in thermal stability by14°C and28°C was observed for the NC-10nanocomposites in nitrogen and air atmosphere, respectively.However,on further increasing the n-Al2O3 loading to30wt%decreased the T10value to below theTable2Composition of n-Al2O3/PEEK nanocompositesSample code Al2O3in PEEK by:wt%vol% NC-000 NC-1 1.250.41 NC-2 2.50.82 NC-5 5.0 1.67 NC-77.5 2.54 NC-1010 3.46 NC-20207.46 NC-303012.14Fig.3.TG curves of the nanocomposites at the heating rate of10°C/min under nitrogen atmosphere:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e) NC-7,(f)NC-10,(g)NC-20,and(h)NC-30.Fig.4.TGA curves of the nanocomposites at the heating rate of10°C/ min under air atmosphere:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e) NC-7,(f)NC-10,(g)NC-20,and(h)NC-30.1805580°C.As the n -Al 2O 3content increased the agglomera-tion tendency of n -Al 2O 3ﬁllers increased,and thermal sta-bility decreased but it is still higher than pure PEEK.Therefore,the incorporation of n -Al 2O 3in PEEK matrix improved thermal stability of the nanocomposites in both atmospheres.The increase in thermal stability could be attributed to the interaction between the n -Al 2O 3and PEEK matrix,which hindered the segmental movement of the PEEK [22].Figs.5and 6show the derivative thermogravimetric analysis (DTG)curves of nanocomposites in nitrogen and air atmosphere,respectively.It can be seen from Fig.5that there is 6–10°C increase in maximum decompo-sition temperature (T m1)in nitrogen atmosphere.The increase in T m1did not vary much with increase in volume fraction of n -Al 2O 3.Fig.6shows two decomposition stages of PEEK nanocomposites under air atmosphere in contrast to single decomposition stage under nitrogen atmosphere.The lower maximum decomposition temperature (T m1)is probably occurred from the degradation of the PEEK mol-ecules due to thermal energy,while the higher maximum decomposition temperature (T m2)is expected to be the oxi-dation of the degraded PEEK backbone.As shown in Table 3,there is no signiﬁcant change in T m1.However,T m2is signiﬁcantly increased from 644°C for pure PEEK (NC-0)to 695°C for NC-10.This implies that thermo-oxi-dative stability of nanocomposites is improved by about 50°C.Moreover,the ﬁnal decomposition temperature (T f )in air atmosphere is increased by about 42°C from 694°C for NC-0to 736°C for NC-10.The n -Al 2O 3ﬁller,uniformly dispersed within the PEEK matrix,probably interfere with degradation mechanism hence improved the decomposition temperature.Table 3shows that the char yield of pure PEEK is about 48%,in agreement with a reported value [23].This char yield was increased to 62%for NC-30due to the increase in wt%of n -Al 2O 3,which is thermally very stable at higher temperature.Similar trend of char yield was obtained for micron size Al 2O 3incorporated PEEK composites [4].4.3.Diﬀerential scanning calorimetry (DSC)DSC measurements were carried out to determine the thermal properties such as melting temperature (T m ),heat of crystallization (H c ),degree of crystallinity,onset crystal-lization temperature (T on ),and peak crystallization temper-ature (T c )of PEEK nanocomposites.The DSC heating and cooling curves are shown in Figs.7and 8,respectively.Table 3Degradation temperature and char yield of the n -Al 2O 3/PEEK nanocomposites Sample code T d in air atmosphere (°C)T d in N 2,atmosphere (°C)Char yield,%T 10,air a T m,1T m,2T f T 10;N 2b T m,1NC-055659064466457058448NC-158059067074457859249NC-256458264267057259049NC-556758865869257059250NC-757858867873557359053NC-1058458869573658459453NC-2056758869571557559056NC-3058059068871058059062a T 10,air is the degradation temperature at 10wt%loss in air atmosphere.bT 10;N2is the degradation temperature at 10wt%loss in nitrogenatmosphere.Fig.5.DTG curves of the nanocomposites at the heating rate of 10°C/min under nitrogen atmosphere:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e)NC-7,(f)NC-10,(g)NC-20,and (h)NC-30.Fig.6.DTG curves of the nanocomposites at the heating rate of 10°C/min under air atmosphere:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e)NC-7,(f)NC-10,(g)NC-20,and (h)NC-30.1806From the recorded heating and cooling curves,thermal properties were calculated and tabulated in Table 4.The crystallinity percentage of PEEK (v c )was calculated with a value of the heat of crystallization for the 100%crystal-line PEEK as 130cal/g [2].The crystallinity of PEEK con-stituent in composite was determined by:v c ð%Crystallinity Þ¼D H c Â100=ðD H 0c w Þð4Þwhere D H 0c is the heat of crystallization (130J/g)for 100%crystalline PEEK,and w is the mass fraction of PEEK in the composites.It is seen from the curves (a–h)of Fig.7and Table 4that T m is increased by 1–6°C as the n -Al 2O 3content increased in PEEK.However,above 10wt%the signiﬁcant increase in T m was not observed.The similar increasing trend in T m was reported recently for AlN (5l m)/PEEK [2]and Al 2O 3(8l m)/PEEK [4]composites.However,a recent study has shown that the addition of nano Al 2O 3and nano SiO 2decreases slightly the T m of PEEK [15].Lorenzo MLD et al.reported that T m of the PET is decreased with the addition of untreated CaCO 3but increased with the addition of treated CaCO 3due to good adhesion between the ﬁller and matrix [24].Pingping et al.have not found sig-niﬁcant change in the T m of CaCO 3/PET composites [25].However,the decrease in T m about 5–6°C of CaCO 3/PEEK composites was observed,irrespective of ﬁller’s sur-face treatment [5].It is well known that the melting point of the polymer crystals is a function of lamellar thickness anddegree of crystal perfection [26].Therefore,the increase in T m ,in present study,may be due to the increased crystal size,and crystal perfection.Priya et al.reported that change in crystal structure/morphology of composite due to the addition of ﬁller aﬀect the T m of the polymer [27].This factor may be ruled out for the present study because the XRD results have shown that there is not any change in PEEK crystal structure.From Fig.8,it was observed that the T on ,T c ,and half time of crystallization (t 1/2)of PEEK was aﬀected by the presence of the n -Al 2O 3,which indicate that nucleation is inhomogeneous.The addition of n -Al 2O 3in PEEK shifts the T c towards higher temperature by 2–12°C depending on the n -Al 2O 3content in PEEK for a given cooling rate in comparison to pure PEEK.This implies that the addition of n -Al 2O 3into PEEK enhanced the rate of PEEK crystalli-zation.A similar enhancement of crystallization was reported for AlN/PEEK [2],CaCO 3/PP [13],SiO 2/PP [26,28,29],clay/PVDF [27],SiO 2/PET [30,31],clay/PET [32],and nanocomposites.However,our results are in con-trast to the recent study of CaCO 3/PEEK [5]and Al 2O 3/PEEK [15]nanocomposites,where decrease in T c was found with the increase of ﬁllers in PEEK matrix.This diﬀerence may be attributed to the shape,size,loading,dispersion level,adhesion,and surface morphology of the ﬁller.Never-Fig.7.DSC heating curves of the nanocomposites:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e)NC-7,(f)NC-10,(g)NC-20,and (h)NC-30.Fig.8.DSC cooling curves of the nanocomposites:(a)NC-0,(b)NC-1,(c)NC-2,(d)NC-5,(e)NC-7,(f)NC-10,(g)NC-20,and (h)NC-30.1807theless the impurities present on theﬁller’s surface may also aﬀect the crystallization behavior of the polymer.The half time(t1/2)of crystallization temperature of PEEK nanocomposites was determined by using the equa-tion[t1/2=(T onÀT c)/rate of cooling].Table4shows that t1/2value of nanocomposite decreases with the increase in n-Al2O3content in PEEK.The t1/2for the pure PEEK is 2.8min,which is decreased to about1.4min for the NC-30nanocomposites.The t1/2for the nanocomposites varies 2.6–1.4min depending on the nanoﬁller loading.The decrease in t1/2implies that the nucleation eﬀect is increased for PEEK with increase in n-Al2O3.For the same rate of cooling,there is enough time for the molecular chains of PEEK to pack into a closer arrangement. Although the enthalpy of crystallization(D H c)for nano-composites decreased slightly with the increase of n-Al2O3 as compared to the pure PEEK.Moreover,there is not any trend in D H c with n-Al2O3content.The supercooling temperature(D T)of the nanocomposites decreases with increasing n-Al2O3in PEEK,indicating that the crystalliza-tion becomes easier in the nanocomposites due to the nucleating eﬀect of the n-Al2O3.4.4.Crystal structureFig.9shows intensity versus angular position(2h)in the range10–50°of major crystallographic reﬂection for the PEEK nanocomposites.The pure PEEK and PEEK consti-tute of nanocomposites crystallizes primarily in the form-I [33]with orthorhombic crystal structure which shows dif-fraction peaks(2h)at about18.7°,20.8°,22.9°and28.9°, corresponding to diﬀraction planes of(110),(111), (200),and(211).In the studied angular range for nano-composites,there are only two weak diﬀraction peaks of n-Al2O3appearing at about39.41°,and45.815°,corre-sponding to Miller indices(222),and(400).Apparently, apart from those of pure constitutes,no new diﬀracting peaks were observed in the diﬀraction pattern of the nano-composites.Moreover,all nanocomposite samples showed the same XRD patterns with varying peak intensity in pro-portion of the constituent’s volume faction.The absence of new diﬀraction peaks showed that the presence of n-Al2O3 did not change the crystal structure of PEEK.However,in other polymer nanocomposite system a new diﬀracting peak was observed which implies new morphology of the polymer[27,34].4.5.Morphological examinationFig.1a shows micrographs of pure PEEK powder at 2000·magniﬁcation.PEEK powders have irregular parti-cles of rod like shape of length ranging from10to 50l m.In order to determine the morphology of the n-Al2O3ﬁller,it was dispersed in ethanol for15min under ultra sonic bath and observed under the SEM.Theﬁllers are seen as agglomerates in Fig.1b with sub-micron size of primary particles,which are diﬃcult to be resolved by the SEM.This is due to the fact that n-Al2O3particles have a strong tendency to form agglomerate due to Wander Wall’s forces between particle-particle.However the same can be observed well separated in composites under SEM due to interaction between n-Al2O3and PEEK,which results in well dispersion in PEEK matrix.Figs.10a and b show SEM micrographs for NC-1and NC-10after pol-ishing.Due to the nano size,ﬁllers are not distinctly visible. In order to get distinct boundary between the n-Al2O3ﬁller and the PEEK matrix,NC-1and NC-10nanocomposites were etched in2%w/v solution of potassium permanganate in a mixture of4vol.of orthophosphoric acid and1vol.of water.During etching amorphous PEEK or loosely bounded PEEK surrounding the n-Al2O3ﬁllers were etchedTable4The melting and crystallization data of n-Al2O3/PEEK nanocompositesSample T m(°C)T c(°C)T on(°C)D H c(J/g)a v c t1/2(min)D T(°C) NC-033427028431.8324.46 2.864NC-133526827730.1823.22 2.067NC-233627628827.0420.8 2.460NC-533527228529.1922.45 2.663NC-733827528530.3523.35 2.063NC-1034027328329.5622.74 2.064NC-2033727928928.7322.10 2.056NC-303362832903224.62 1.453a Normalized heat of crystallization of PEEK constituent innanocomposites.Fig.9.X-ray diﬀraction pattern of the nanocomposites.For clarity,scansof NC-1–NC-30have been displaced upward.1808out,which results in appearance of n -Al 2O 3ﬁllers in PEEK matrix.It could be seen from Figs.10c–f that n -Al 2O 3ﬁll-ers were uniformly distributed throughout the PEEK matrix.However,some n -Al 2O 3agglomerates were also seen in the PEEK matrix.Nevertheless,with increase of n -Al 2O 3content,the inter particle distance decreases which results in formation of Al 2O 3aggregates.As shown in Figs.10c–f,SEM could not provide good contrast between n -Al 2O 3and PEEK matrix.For this reason,NC-1and NC-10nanocomposites were also examined with TEM.Figs.11a and b show TEM images of pure n -Al 2O 3powder.The n -Al 2O 3particles are almost spherical in shape and its size varies between 20and 90nm.Figs.11c and d show TEM images of NC-1and NC-10nanocomposites,respec-tively.The most of the n -Al 2O 3particles remained individ-ual in NC-1nanocomposite.However,as the n -Al 2O 3content increased to 10wt%(NC-10)in PEEK,due to the particle-particle interaction some aggregates of about 100nm size was also observed with individual n -Al 2O 3par-ticles.This shows that shear forces applied during mechan-ical stirring were not capable of breaking and uniformly distributing the n -Al 2O 3in PEEKmatrix.Fig.10.SEM micrographs of:(a)polished NC-1,magniﬁcation =6·104;(b)polished NC-10,magniﬁcation =6·104;(c)etched NC-1,magniﬁca-tion =8·104;(d)etched NC-1,magniﬁcation =1.6·105;(e)etched NC-10,magniﬁcation =8·104;(f)etched NC-10,magniﬁcation =1.6·105.18094.6.Coeﬃcient of thermal expansion (CTE)The T g of the PEEK determined by inﬂection in the curve between dimension change and temperature was found about 153°C.The average out-of-plane CTE below T g for the nanocomposites is shown in Fig.12as a function of volume %of n -Al 2O 3ﬁller.The CTE of the NC-0was 58·10À6/°C and decreased with increasing n -Al 2O 3ﬁller in PEEK matrix.The CTE of the NC-30(12vol%)nanocomposite was about 23·10À6/°C.The reduction in CTE may be attributed to three reasons.First,decrease in volume fraction of the PEEK in the composite results in decreased free volume of PEEK,hence reduced room for PEEK expansion.Second,well dispersion of n -Al 2O 3ﬁller results in good interfacial area between n -Al 2O 3and PEEK.It is well known that in par-ticulate polymer composites,particles are surrounded by two regions;ﬁrst by tightly bounded polymer or con-strained polymer chain,and second,by loosely bounded polymer chains or unconstrained polymer chain.As the average inter-particle distance decreases with the incorpo-ration of more ﬁller particles,the loosely bound polymergradually gets transformed to the tightly bound polymer.Hence,the volume fraction of loosely bound polymer decreases [3,35].Hence formation of increasedcon-Fig.11.TEM micrographs of:(a)n -Al 2O 3powder as received,magniﬁcation =6.6·104;(b)n -Al 2O 3powder as received,magniﬁcation =1.15·105;(c)NC-1,magniﬁcation =3.8·104;(d)NC-10,magniﬁcation =3.8·104.1810。

Particle-Based Anisotropic Surface Meshing

Particle-Based Anisotropic Surface MeshingZichun Zhong∗Xiaohu Guo*Wenping Wang†Bruno L´e vy‡Feng Sun†Yang Liu§Weihua Mao¶*University of Texas at Dallas†The University of Hong Kong‡INRIA Nancy-Grand Est §NVIDIA Corporation¶UT Southwestern Medical Center atDallasFigure1:Anisotropic meshing results generated by our particle-based method.AbstractThis paper introduces a particle-based approach for anisotropic sur-face meshing.Given an input polygonal mesh endowed with a Rie-mannian metric and a speciﬁed number of vertices,the method gen-erates a metric-adapted mesh.The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one,called “embedding space”.The vertices of the mesh are generated by uni-formly sampling the surface in this higher dimensional embedding space,and the sampling is further regularized by optimizing an en-ergy function with a quasi-Newton algorithm.All the computations can be re-expressed in terms of the dot product in the embedding space,and the Jacobian matrices of the mappings that connect d-ifferent spaces.This transform makes it unnecessary to explicitly represent the coordinates in the embedding space,and also provides all necessary expressions of energy and forces for efﬁcient compu-tations.Through energy optimization,it naturally leads to the de-sired anisotropic particle distributions in the original space.The tri-angles are then generated by computing the Restricted Anisotropic V oronoi Diagram and its dual Delaunay triangulation.We compare our results qualitatively and quantitatively with the state-of-the-art in anisotropic surface meshing on several examples,using the stan-dard measurement criteria.∗{zichunzhong,xguo}@,†{wenping,fsun}@cs.hku.hk,‡bruno.levy@inria.fr,§thomasyoung.liu@,¶weihua.mao@ CR Categories:I.3.5[Computer Graphics]:Computational Ge-ometry and Object ModelingKeywords:Anisotropic Meshing,Particle,and Gaussian Kernel. Links:DL PDF1IntroductionAnisotropic meshing offers a highlyﬂexible way of controlling mesh generation,by letting the user prescribe a direction and densi-tyﬁeld that steers the shape,size and alignment of mesh elements. In the simulation ofﬂuid dynamics,it is often desirable to have e-longated mesh elements with desired orientation and aspect ratio given by a Riemannian metric tensorﬁeld[Alauzet and Loseille 2010].For surface modeling,it has been proved in approxima-tion theory that the L2optimal approximation to a smooth surface with a given number of triangles is achieved when the anisotropy of triangles follows the eigenvalue and eigenvectors of the curvature tensors[Simpson1994;Heckbert and Garland1999].This can be easily seen from the example of ellipsoid surface in Fig.2where the ratio of the two principal curvatures K max/K min is close to 1near the two ends of the ellipsoid and is as high as100in the middle part.Anisotropic triangles stretched along the direction of minimal curvatures in the middle part of the ellipsoid provide best approximation,while isotropic triangles are needed at its two ends. In this paper,we propose a new method for anisotropic meshing of surfaces endowed with a Riemannian metric.We rely on a particle-based scheme,where each pair of neighboring particles is equipped with a Gaussian energy.It has been shown[Witkin and Heckbert 1994]that minimizing this pair-wise Gaussian energy leads to a u-niform isotropic distribution of particles.To compute the anisotrop-ic meshing on surfaces equipped with Riemannian metric,we uti-lize the concept of a higher dimensional“embedding space”[Nash 1954;Kuiper1955].Our method optimizes the placement of the vertices,or particles,by uniformly sampling the higher dimension-al embedding of the input surface.This embedding is designed in such a way that when projected back into the original space(usual-Figure 2:Isotropic and anisotropic meshing with 1,000output vertices of the ellipsoid surface.The stretching ratio (deﬁned in Sec.2.1)is computed as √K max /K min ,where K max and K min are the two principal curvatures.Note that the “End Part”is ren-dered with orthographic projection along its long-axial direction,to better show the isotropy.ly 2D or 3D),a uniform sampling becomes anisotropic with respect to the input metric.Direct reference to the higher dimensional em-bedding is avoided by re-expressing all computations in terms of the dot product in the high-dimensional space,and the Jacobian matri-ces of the mappings that connect different spaces.Based on this re-expression we derive principled energy and force models for ef-fective computation on the original manifold with a quasi-Newton optimization algorithm.Finally,the triangles are generated by com-puting a Restricted Anisotropic V oronoi Diagram and extracting the dual of its connected components.This paper makes the following contributions for efﬁciently gener-ating high-quality anisotropic meshes:•It introduces a new particle-based formulation for anisotropic meshing.It deﬁnes the pair-wise Gaussian energies and forces between particles,and formulates the energy optimization in a higher dimensional “embedding space”.We show further how anisotropic meshing can be translated into isotropic meshing in this higher dimensional embedding space (Sec.3.1).The energy is designed in such a way that the particles are uni-formly distributed on the surface embedded in this higher di-mensional space.When the energy is optimized,the corre-sponding particles in the original manifold will achieve the anisotropic sampling with the desired input metric.•It presents a computationally feasible and efﬁcient method for our energy optimization (Sec.3.2).The high-dimensional energy function and its gradient is mapped back into the o-riginal space,where the particles can be directly optimized.This computational approach avoids the need of computing the higher dimensional embedding space.Such energy opti-mization strategy shows very fast convergence speed,without any need for the explicit control of particle population (e.g.,inserting or deleting particles to meet the desired anisotropy).2Background and Related Works2.1Deﬁnition of AnisotropyAnisotropy denotes the way distances and angles are distorted.Ge-ometrically,distances and angles can be measured by the dot prod-uct:⟨v ,w ⟩,which is a bilinear function mapping a pair of vectors v ,w to R .The dot product is symmetric,positive,and deﬁnite (SPD).If the dot product is replaced with another SPD bilinear for-m,then an anisotropic space is deﬁned.We consider that a met-ric M (.),i.e.an SPD bilinear form,is deﬁned over the domain Ω⊂R m .In other words,at a given point x ∈Ω,the dot product between two vectors v and w is given by ⟨v ,w ⟩M (x ).In practice,the metric can be represented by a symmetric m ×m matrix M (x ),in which case the dot product becomes:⟨v ,w ⟩M (x )=v T M (x )w .(1)The metric matrix M (x )can be decomposed with Singular Value Decomposition (SVD)into:M (x )=R (x )T S (x )2R (x ),(2)where the diagonal matrix S (x )2contains its ordered eigenvalues,and the orthogonal matrix R (x )contains its eigenvectors.We note that a globally smooth ﬁeld R (x )may not exist for surfaces of arbitrary topology.For the metric design,we use the following two options:(1)In some of our experiments,we start from designing a smooth scaling ﬁeld S (x )and a rotation ﬁeld R (x )that is smooth in re-gions other than those singularities,and compose them to Q (x )=S (x )R (x )and M (x )=Q (x )T Q (x ),which is the same as Du et al.[2005].They are deﬁned on the tangent spaces of the surface.Suppose s 1and s 2are the two diagonal items in S (x )correspond-ing to the two eigenvectors in the tangent space,and s 1≤s 2.Wesimply call s 2s 1as the stretching ratio .This process will play a role later when the user speciﬁes the desired input metric (Sec.5).(2)Note that if M (x )is given by users,the decomposition to Q (x )is non-unique.An equivalent decomposition M (x )=Q O (x )T Q O (x )is given by any matrix Q O (x )=O (x )Q (x ),where O (x )is a m ×m orthogonal matrix.In other words,Q (x )is unique up to a rotation .However,it is easy to show that if a SPD metric M (x )is giv-en,its square root Q ′(x )=√M (x )is also a SPD matrix,and such decomposition is unique (Theorem 7.2.6of [Horn and John-son 1985])and smooth (Theorem 2of [Freidlin 1968]).Q ′(x )is a symmetric afﬁne mapping:Q ′(x )=R (x )T S (x )R (x ),and M (x )=Q ′(x )Q ′(x ).In Sec.5.1,we use the “Mesh Font”ex-ample to show that Q ′(x )can work well in our framework,given a user speciﬁed smooth metric ﬁeld M (x ).It is interesting to note that if the metric tensor ﬁeld is given as:M (x )=ρ(x )2m ·I ,(3)where ρ(x ):Ω→R and I is the identity matrix,then M (x )deﬁnes an isotropic metric graded with the density function ρ(x ).Given the metric ﬁeld M (x )and an open curve C ⊂Ω,the length of C is deﬁned as the integration of the length of tangent vector along the curve C with metric M (x ).Then,the anisotropic distance d M (x ,y )between two points x and y can be deﬁned as the length of the (possibly non-unique)shortest curve that connects x and y .2.2Previous WorksAnisotropic Voronoi Diagrams:By replacing the dot product with the one deﬁned by the metric, anisotropy can be introduced into the deﬁnition of the standard no-tions in computational geometry,e.g.,V oronoi Diagrams and De-launay Triangulations.The most general setting is given by Rie-mannian V oronoi diagrams[Leibon and Letscher2000]that replace the distance with the anisotropic distance d M(x,y)deﬁned above. Some theoretical results are known,in particular that Riemannian V oronoi diagrams admit a valid dual only in dimension2[Boisson-nat et al.2012].However,a practical implementation is still beyond reach[Peyre et al.2010].For this reason,two simpliﬁcations are used to compute the V oronoi cell of each generator x i:V or Labelle(x i)={y|d xi (x i,y)≤d xj(x j,y),∀j}V or Du(x i)={y|d y(x i,y)≤d y(x j,y),∀j}where:d x(y,z)=√(z−y)T M(x)(z−y).(4)Theﬁrst deﬁnition V or Labelle[Labelle and Shewchuk2003]is eas-ier to analyze theoretically.The bisectors are quadratic surfaces, known in closed form,and a provably-correct Delaunay reﬁnemen-t algorithm can be deﬁned.The so-deﬁned Anisotropic V oronoi Diagram(A VD)may be also thought of as the projection of a higher-dimensional power diagram[Boissonnat et al.2008a].The second deﬁnition V or Du[Du and Wang2005]is best suited to a practical implementation of Lloyd relaxation in the computation of Anisotropic Centroidal V oronoi Tessellations.Centroidal Voronoi Tessellation and its Anisotropic Version:A Centroidal V oronoi Tessellation(CVT)is a V oronoi Diagram such that each point x i coincides with the centroid of its V oronoi cell.A CVT can be computed by either the Lloyd relaxation[L-loyd1982]or a quasi-Newton energy optimization solver[Liu et al. 2009].It generates a regular sampling[Du et al.1999],from which a Delaunay triangulation with well-shaped isotropic elements can be extracted.In the case of surface meshing,it is possible to gener-alize this deﬁnition by using a geodesic V oronoi diagram over the surface[Peyre and Cohen2004].To make the computations simpler and cheaper,it is possible to replace the geodesic V oronoi diagram with the Restricted V oronoi Diagram(RVD)or Restricted Delau-nay Triangulation(RDT),deﬁned in[Edelsbrunner and Shah1994] and used by several meshing algorithms,see[Dey and Ray2010] and the references herein.Hence a Restricted Centroidal V oronoi Tessellation can be deﬁned[Du et al.2003].With an efﬁcient algo-rithm to compute the Restricted V oronoi Diagram,Restricted CVT can be used for isotropic surface remeshing[Yan et al.2009]. CVT was further generalized to Anisotropic CVT(ACVT)by Du et al.[2005]using the deﬁnition V or Du in Eq.(4).In each Lloyd iteration,an anisotropic Delaunay triangulation with the given Rie-mannian metric needs to be constructed,which is a time-consuming operation.Valette et al.[2008]proposed a discrete approximation of ACVT by clustering the vertices of a dense pre-triangulation of the domain.This discrete version is much faster than Du et al.’s continuous ACVT approach,at the expense of slightly degraded mesh quality.Sun et al.[2011]introduced a hexagonal Minkows-ki metric into ACVT optimization,in order to suppress obtuse pared to these ACVT approaches,our particle-based scheme avoids the construction of A VD in the intermediate itera-tions of energy optimization,thus shows much faster performance as shown in Sec.6.1.Surface Meshing in Higher Dimensional Space:Uniformly meshing surfaces embedded in higher dimensional space has also been studied in the literature[Ca˜n as and Gortler2006;Ko-vacs et al.2010;L´e vy and Bonneel2012].The work of L´e vy and Bonneel[2012]is most related to ours,since both can be considered as using the framework of energy optimization in a higher dimen-sional embedding space.They extended the computation of CVT to a6D space in order to achieve a curvature-adaptation.In partic-ular,the anisotropic meshing on a3D surface is transformed to an isotropic one on the surface embedded in6D space,which can be efﬁciently computed by CVT equipped with V oronoi Parallel Lin-ear Enumeration[L´e vy and Bonneel2012].However,it does not provide users with theﬂexibility to control the anisotropy via an in-put metric tensorﬁeld.Our approach is designed to handle the more general anisotropic meshing scenario where a user-desired metric is speciﬁed.Reﬁnement-Based Delaunay Triangulation:Anisotropic versions of point insertion in Delaunay triangula-tion has been successfully applied to many practical application-s[Borouchaki et al.1997a;Borouchaki et al.1997b;Dobrzynski and Frey2008].Boissonnat et al.[2008b;2011]introduced a De-launay reﬁnement framework,which is based on the goal to make the star around each vertex x i to be consisting of the triangles that are exactly Delaunay for the metric associated with x i.In order to“stitch”the stars of neighboring vertices,reﬁnement algorithm-s are proposed to add new vertices gradually to achieve theﬁnal anisotropic meshing.Our approach is different and consists in op-timizing all the vertices of the mesh globally.Another difference is that we compute the dual of the connected components of the RVD[Yan et al.2009]instead of the RDT.The results are com-pared in Sec6.3.Particle-Based Anisotropic Meshing:Turk[1992]introduced repulsive points to sample a mesh for the purpose of polygonal remeshing.It was later extended by Witkin and Heckbert[1994]who used particles equipped with pair-wise Gaussian energy to sample and control implicit surfaces.Meyer et al.[2005]formulated the energy kernel as a modiﬁed cotangen-t function withﬁnite support,and showed the kernel to be near-ly scale-invariant as compared to the Gaussian kernel.It was lat-er extended to handle adaptive,isotropic meshing of CAD mod-els[Bronson et al.2012]with particles moving in the parametric space of each surface patch.All these methods are only targeting isotropic sampling of surfaces.To handle anisotropic meshing,Bossen and Heckbert[1996]incor-porated the metric tensor into the distance function d(x,y),and use f(x,y)=(1−d(x,y)4)·exp(−d(x,y)4)to model the re-pulsion and attraction forces between particles.Shimada and his co-workers proposed physics-based relaxation of“bubbles”with a standard second-order system consisting of masses,dampers,and linear springs[Shimada and Gossard1995;Shimada et al.1997; Yamakawa and Shimada2000].They used a bounded cubic func-tion of the distance to model the inter-bubble forces,and further extended it to anisotropic meshing by converting spherical bubbles to ellipsoidal ones.Both Bossen et al.and Shimada et al.’s works require dynamic population control schemes,to adaptively insert or delete particles/bubbles in certain regions.Thus if the initialization does not have a good estimation of the number of particles needed toﬁll the domain,it will take a long time to converge.The method proposed in this paper is very similar to the idea of Adaptive Smoothed Particle Hydrodynamics(ASPH)[Shapiro et al.1996]which uses inter-particle Gaussian kernels with an anisotropic smoothing tensor.However,as addressed in Sec.3.3, ASPH directly formulates the energy in the original space without using the embedding space concept.To compute the forces between particles,the gradient of the varying metric tensor has to be ignored due to numerical difﬁculty.This treatment will lead to inaccurateanisotropy in the computed mesh as shown in Fig.4,when there are mild or signiﬁcant variations in the metric.Relation with the Theory of Approximation:It has been studied in the theory of approximation [D’Azevedo 1991;Shewchuk 2002]that anisotropy is related to the optimal ap-proximation of a function with a given number of piecewise-linear triangular elements.The anisotropy of the optimal mesh can be characterized,and optimization algorithms can be designed to best approximate the given function.The continuous mesh concept in-troduced by Loseille and Alauzet [2011a;2011b]provides a rela-tionship between the linear interpolation error and the mesh pre-scription,which has resulted in highly efﬁcient anisotropic mesh adaptation algorithms.The relationship between anisotropic mesh-es and approximation theory has also been studied for higher-order ﬁnite elements [Mirebeau and Cohen 2010;Mirebeau and Cohen 2012],which leads to an efﬁcient greedy bisection algorithm to generate optimal meshes.Other Related Works:This paper only focuses on anisotropic triangular meshing,which is different from other works handling anisotropic quad-dominant remeshing [Alliez et al.2003;Kovacs et al.2010;L´e vy and Liu 2010;Zhang et al.2010].The notion of anisotropy has also been applied to the blue noise sample generation [Li et al.2010].3The Particle ApproachConsidering each vertex as a particle,the potential energy be-tween the particles determines the inter-particle forces.When the forces applied on each particle become equilibrium,the particles reach the optimal balanced state with uniform distribution.To han-dle anisotropic meshing,we utilize the concept of “embedding s-pace”[Nash 1954;Kuiper 1955].In such high-dimensional em-bedding space,the metric is uniform and isotropic.When the forces applied on each particle reach equilibrium in this embedding space,the particle distribution on the original manifold will exhibit the desired anisotropic property.Basic Framework:Given n particles with their positions X ={x i |i =1...n }on the surface Ωwhich is embedded in R m space,we deﬁne the inter-particle energy between particles i and j as:Eij=e−∥x i −x j ∥24σ2.(5)Here σ,called kernel width ,is the ﬁxed standard deviation of the Gaussian kernels.In Sec.4.1we will discuss how to choose an appropriate size of σ.Clearly,E ij =E ji .The gradient of E ij w.r.t.x j can be considered as the force F ij applied on particle j by particle i :Fij=∂E ij∂x j =(x i −x j )2σ2e−∥x i −x j ∥24σ2.(6)Analogous to Newton’s third law of motion,we have F ij=−F ji.We want to note that the formulation of Eq.(6)is similar to the particle repulsion/attraction idea of Witkin and Heckbert [1994].By minimizing the total energy E =∑i ∑j =i Eijwith L-BFGS [Liu and Nocedal 1989],we can get a uniform isotropic sam-pling,where the forces applied on each particle reach equilibrium.It is shown in the supplementary Appendix that this particle-based energy formulation is fundamentally equivalent to Fattal’s kernel-based formulation [2011],for the uniform isotropic case.However,Fattal’s method does not handle anisotropic case.For non-uniform isotropic case,our analysis in Appendix shows the difference with respect to Fattal’s approach,from both theoretical viewpoints and experimentalresults.Figure 3:A simple example of an embedding function that trans-forms an original 2D anisotropic surface Ω(left)into the surface Ω(right)embedded in a higher dimensional space (3D in this exam-ple)where the metric is uniform and isotropic.In the general case a higher number of dimensions is required for Ω.3.1Anisotropic CaseThe top-left image of Fig.3shows a representation of a 2D metric ﬁeld M .The ﬁgure shows a set of points (black dots)and their associated unit circles (the bean-shaped curves,that correspond to the sets of points equidistant to each black dot).The bottom-left image of Fig.3shows the ideal mesh governed by such metric ﬁeld:the length of the triangle edges,under the anisotropic distance,are close to be equal.For this simple example of Fig.3,one can see that the top-left im-age can be considered as the surface in the top-right image “seen from above”.In other words,by embedding the ﬂat 2D domain as a curved surface in 3D,one can recast the anisotropic meshing problem as the isotropic meshing of a surface embedded in higher-dimensional space.In general,for an arbitrary metric M ,a higher-dimensional space will be needed [Nash 1954;Kuiper 1955].We now consider that the surface Ωis mapped to Ωthat is embedded in a higher-dimensional space R m .We simply call R m as the embedding space in this pa-per.Suppose the mapping function is ϕ:Ω→Ω,where Ω⊂R m ,Ω⊂R m ,and m ≤m .Let us denote the particle positions on this surface Ωby X ={x i |x i =ϕ(x i ),i =1...n }.A unifor-m sampling on Ωcan be computed by changing the inter-particleenergy function E ij of Eq.(5)as follows,hence deﬁning E ij:Eij=e−∥x i −x j ∥24σ2.(7)The gradient of E ijw.r.t.x j ,i.e.,the force F ijin the embeddingspace,can be deﬁned similarly as:Fij=∂Eij∂x j =(x i −x j )2σ2e −∥x i −x j ∥24σ2.(8)3.2Our Computational ApproachWe show in this subsection how to optimize E ijwithout referringto the coordinates of Ωin the embedding space.From the introduction of Sec.2.1,we have seen that introducing anisotropy means changing the deﬁnition of the dot product.If we consider two small displacements v and w from a given lo-cation x∈Ω,then they are transformed into v=J(x)v and w=J(x)w,where J(x)denotes the Jacobian matrix ofϕat x. The dot product between v and w is given by:⟨v,w⟩=v T J(x)T J(x)w=v T M(x)w.(9) In other words,given the embedding functionϕ,the anisotropy M corresponds to theﬁrst fundamental form ofϕ.If we now suppose that the anisotropy M(x)is known but not the embedding function ϕ,it is still possible to compute the dot product between two vectors in embedding space around a given point.3.2.1Computing the Energy FunctionWe now consider the inter-particle energy function in Eq.(7).Con-sider neighboring particles i and j.We use the Jacobian matrix evaluated at their middle point:x i+x j2.In the following we de-note J ij=J(x i+x j2),M ij=M(x i+x j2),Q ij=Q(x i+x j2),andQ′ij=Q′(x i+x j2)(see Sec.2.1),for notational simplicity.Sincethe middle point is close to both x i and x j,it is reasonable to make the following approximation:x i−x j=ϕ(x i)−ϕ(x j)≈J ij(x i−x j).(10) Thus the exponent in the term E ij can be approximated as:∥x i−x j∥2=⟨x i−x j,x i−x j⟩≈(x i−x j)T J T ij J ij(x i−x j)=(x i−x j)T M ij(x i−x j).(11) Our inter-particle energy function can be approximated by:E ij≈e−(x i −x j)T M ij(x i−x j)4σ2.(12)The total energy is simply:E=n∑i=1n∑j=1,j=iE ij(13)3.2.2Computing the Force FunctionUsing Eq.(10)and Eq.(11),the inter-particle forces of Eq.(8)be-comes:F ij≈J ij(x i −x j)2σ2e−(x i−x j)T M ij(x i−x j)4σ2.(14)Here,for a particle i,different neighbors j have different J ij,which essentially encodes the variation of the metric.The total force applied on each particle i is simply:F i=∑j=iF ji.(15)Note that the expression in Eq.(14)still depends on the Jacobian matrix J ij.In our case,neither the embedding functionϕnor its Jacobian is known.Therefore,we propose below an approximation of Eq.(14)that solely depends on the anisotropyﬁeld M(x).We denote the set of particle i’s neighbors as N(i),and denote the vectors v ij=x i−x j,j∈N(i).To better understand J ij,let us explore the relationship between the matrices J ij and Q ij.J ij is am×m matrix,where m is the dimension of the embedding space, and m is either2or3,depending on whetherΩis a2D domain or a3D surface.Consider the QR decomposition:J ij=U ij[P ij], where U ij is a m×m unitary matrix(i.e.a rotation matrix in R m), P ij is a m×m matrix,and0is a(m−m)×m block of zeros. Then:M ij=J T ij J ij=P T ij P ij,(16) since U T ij U ij=I.As mentioned in Sec.2.1,if both S ij and R ij are given by users, then we can compose them and deﬁne Q ij=S ij R ij;if a smoothmetric M ij is given by users,we can use its square root Q′ij=√M ij.In the following derivation,both Q ij and Q′ij will lead to the same approximation technique.So we simply use Q ij in the following discussion.From Eq.(16)we can see that P ij is exactly Q ij up to a rotation, i.e.,P ij=O ij Q ij where O ij is a m×m rotation matrix.We can simply represent J ij as:J ij=U ij[O ij Q ij]=W ij[Q ij],(17) where W ij is a rotation matrix in R m:W ij=U ij[O ij00I].(18)If the metricﬁeld M(x)is smooth,then it is reasonable to approx-imate the rotation matrix W ij with W i,where W i is the rotation matrix of Eq.(18)evaluated at x i.Thus for j∈N(i),the m-dimensional vectors J ij v ij in Eq.(14)can be approximated by:J ij v ij=W ij[Q ij]v ij≈W i[Q ij v ij].(19) Then the force vector on particle i in Eq.(15)becomes:F i=∑j=iJ ij v ij2σ2e−(x i−x j)T M ij(x i−x j)4σ2≈∑j=i12σ2W i[Q ij v ij]e−(x i−x j)T M ij(x i−x j)4σ2=W i∑j=i12[Q ij v ij]e−(x i−x j)T M ij(x i−x j)4σ2.(20) If we deﬁne the m-dimensional forces:F ij=Q ij(x i−x j)2σ2e−(x i−x j)T M ij(x i−x j)4σ2,(21)andF i=∑j=iF ji,(22) then the m-dimensional force F i in Eq.(20)is simply:F i≈W i[F i]=V i F i,(23) where V i=W i[I m×m],and I m×m is a m×m identity matrix. Note that F i is the gradient in the higher-dimensional space R m, while F i is in the original space R m.They are related by the ma-trix V i in Eq.(23),which builds up a bijection between them.We can see that they can guide the optimization to arrive at the same equilibrium,since F i=0⇔ F i=0.Thus for the energy op-timization purpose,we can simply replace Eq.(15)with Eq.(22) which can be computed directly on the original surfaceΩ.The idea behind our force approximation can be interpreted as fol-lows.At a given particle i,different neighboring pairs(i,j1)and(i,j2)may be equipped with different metrics M ij1and M ij2(aswell as different Jacobians J ij1and J ij2).The difference betweenJ ij encodes the variation of the metric locally around particle i. J ij includes both“metric”part(Q ij)and“embedding rotation”part(W ij)(Eq.(19)).W ij transforms the tangent plane at x ij in the original space into the tangent plane in embedding space.Our approach uses the exact variation of neighboring metric Q ij,and approximates the embedding rotation W ij with W i in Eq.(19). Thus,the variation of embedding rotation is ignored in each parti-cle’s neighborhood,but the variation of metric is accounted.In summary,we can optimize the uniform isotropic sampling onΩwith the approximated energy of Eq.(12)and force of Eq.(21)us-ing L-BFGS optimizer.They are both computed using the particle positions X onΩ,together with the metric M.If M is given by users,we use its square root Q′instead of Q in Eq.(21).Although we utilize the elegant concept of“embedding space”to help devel-op our formulation for anisotropic meshing,we do NOT need to compute such an embedding space.3.3Importance of the Embedding SpaceAnisotropic meshing is deﬁned by the Riemannian metric M,to lo-cally afﬁne-transform triangles into a“unit”space while enforcing the transformed triangles to be uniformly equilateral.Thus it is nat-ural to directly deﬁne the energy optimization problem in this“unit”space.However,the metrics on each point can be different.With-out establishing a coherent“unit”space,we cannot describe how these local afﬁne copies of“unit”spaces can be“stitched”together. Our approach coherently considers all these local“unit”spaces by embedding the surfaceΩinto high-dimensional space.Our energy in Eq.(7)is designed exactly by the deﬁnition of“anisotropy”–the afﬁne-transformed triangles inΩshould be uniformly equilateral (the particles should be uniformly distributed).This deﬁnition also leads to very efﬁcient computations of forces in Eq.(21).We want to emphasize that:without using this embedding space, the deﬁnition of energy function and the corresponding force for-mulation would be inconsistent with the deﬁnition of anisotropic mesh and thus lead to incorrect results.If we do not use this high-dimensional embedding space,the most intuitive formulation of en-ergy will be Eq.(12).We elaborate on that and give some compar-isons below.Ignoring the Gradient of Metric(ASPH Method):We need to note that the metric M ij in Eq.(12)is dependent on the positions of particles x i and x j.Therefore,the force formula-tion will involve the gradient of M ij w.r.t.x j,which is numerically very difﬁcult to compute.In the method of Adaptive Smoothed Par-ticle Hydrodynamics(ASPH)[Shapiro et al.1996],they use inter-particle Gaussian kernels and incorporated an anisotropic smooth-ing kernel to deﬁne the potential energy between particles,which is similar to Eq.(12).However,it is mentioned in their paper(Sec.2.2.4of[Shapiro et al.1996])that the gradient of metric term is ignored when computing the gradient of such inter-particle energy. Thus it leads to the following ASPH force formulation:F ij≈M ij(x i−x j)2σ2e−(x i−x j)T M ij(x i−x j)4σ2.(24)It is easy to see that Eq.(24)differs to Eq.(21)by only replacing Q ij with M ij.Thus if the metricﬁeld is not constant,these two forces will lead to different local minima.Our method in Eq.(21)only ignores the variation of embedding ro-tation in each particle’s neighborhood,while the variation of metric is accounted.As conﬁrmed in our experiments in Fig.4,this has a measurable inﬂuence on the quality of generated meshes. Ignoring the Variation of Jacobian Matrix:Another approximation is to apply the pseudo-inverse of Jacobian matrix in the expression of Eq.(14).In Eq.(14),J ij is different for different neighbors j.If we approximate J ij with J i in Eq.(14), and then apply the pseudo-inverse of J i,we arrive at the formula-tion(without the leading M ij or Q ij)as follows:F ij≈(x i−x j)2σ2e−(x i−x j)T M ij(x i−x j)4σ2.(25)We emphasize the difference with our method:this variation is ap-proximating J ij with J i in Eq.(14),while our method is approxi-mating W ij with W i in Eq.(19).As mentioned above,J ij con-tains both“metric”part Q ij and“embedding rotation”part W ij. Thus the approximation of J ij with J i will potentially“erase”the variation of metric between neighboring particles.To see their different effects on anisotropic mesh generation,we conduct the energy optimization in a2D square domain using the following three choices of forces:(1)our force in Eq.(21);(2)the ASPH force in Eq.(24);and(3)the force in Eq.(25).As shown in Fig.4,the2D square domain is equipped with the background tensorﬁeld:M(x)=diag{Stretch(x)2,1},where theﬁeld of Stretch(x)is in the range of[0.577,9].In this experiment,we use a spatially nonuniform metricﬁeld–if M(x)is spatially uniform, then all the three forces will lead to the same particle conﬁguration. The comparative measurements of the quality of the generated anisotropic mesh are shown in Fig.4,with triangle area quality G area,angle histogram,G min,G avg,θmin,θavg,and%<30◦, which are all deﬁned in Sec.4.5.The color-coded triangle area quality of our method shows that the areas of triangles computed using our force are uniform(all close to1),which means the tri-angle sizes are conforming to the desired density deﬁned by the metric tensor.From this experiment,we can see that performing energy optimization using our force in Eq.(21)generates the ideal anisotropic mesh,while optimizing the energy using the other two alternative forces in Eq.(24)and Eq.(25)cannot,which illustrates that formulating the energy optimization in the embedding space with our approximation leads to a principled formulation of inter-particle forces.4Implementation and Algorithm DetailsOur particle-based method is summarized in Alg.1below.To help reproduce our results,we further detail each component of the al-gorithm and the implementation issues.4.1Kernel WidthThe inter-particle energy as deﬁned in Eq.(5)depends on the choice of theﬁxed kernel widthσ.The slope of this energy peaks at dis-tance of√2σand it is near zero at much smaller or much greater distances.Ifσis chosen too small then particles will nearly stop spreading when their separation is about5√2σ,because there is almost no forces between particles.Ifσis chosen too large then nearby particles cannot repel each other and the resulting sampling pattern will be poor.In this work,we chooseσto be proportional to the average“radius”of each particle when they are uniformly dis-tributed onΩ:σ=cσ√|Ω|/n,where|Ω|denotes the area of the surfaceΩin the embedding space,n is the number of particles,and cσis a constant coefﬁcient.Note that our goal is to let the particles。

Vortices in Schwinger-Boson Mean-Field Theory of Two-Dimensional Quantum Antiferromagnets

a rXiv:c ond-ma t/95471v119Apr1995Vortices in Schwinger-Boson Mean-Field Theory of Two-Dimensional Quantum Antiferromagnets Tai-Kai Ng department of Physics,Hong Kong University of Science and Technology,Clear Water Bay Road,Kowloon,Hong Kong (February 6,2008)Abstract In this paper we study the properties of vortices in two dimensional quantum antiferromagnets with spin magmitude S on a square lattice within the frame-work of Schwinger-boson mean-ﬁeld theory.Based on a continuum descrip-tion,we show that vortices are stable topological excitations in the disordered state of quantum antiferromagnets.Furthermore,we argue that vortices can be divided into two kinds:the ﬁrst kind always carries zero angular momen-tum and are bosons,whereas the second kind carries angular momentum S under favourable conditions and are fermions if S is half-integer.A plausible consequence of our results relating to the RVB theories of High-T c supercon-ductors is pointed out.PACS Nos,75.10.Jm,75.30.Ds,79.60.-iTypeset using REVT E XI.INTRODUCTIONIn the past few years there has been a lot of interests in the study of quantum anti-ferromagnets in two dimension on a square lattice,stimulated by the discovery of High-T c superconductors[1,2].Among others,one approximated approach to the quantum antiferro-magnet problems is the Schwinger-Boson Mean-ﬁeld Theory(SBMFT).[2].In this approach, the quantum spins are represented as Schwinger bosons and an approximated ground state is constructed by bose condensation of spin-singlet pairs.The mean-ﬁeld theory can also be formulated in a large-N expansion as the saddle point solution to a generalized SU(N) quantum spin model[2,3].In the limit N→∞,the mean-ﬁeld theory becomes exact. SBMFT predicts that in one dimension,Heisenberg antiferromagnet is always disordered, with a non-zero spin gap in the excitation spectrum.The theory is found to give an adequate description for integer spin chains,while it fails to describe the massless state of half-integer spin chains.This is because the topological Berry phase term which plays crucial role in the latter case is not taken into account correctly in SBMFT.[4]However,in two dimension topological term does not exist and SBMFT is more reliable.In particular,the theory pre-dicts that the disordered(spin gap)phase exists only when the spin magnitude S is small enough,when S<S c∼0.19.It turns out that SBMFT oﬀers a fairly accurate description for the magnetic properties of the High-T c compounds in the low doping regime[5].However, at large doping levels,the generalization of SBMFT which includes holes as fermions[6], does not seem to describe the High-T c compounds correctly.For example,photo-emission experiments[7]indicate that the compounds have”large”Fermi-surfaces which satisfy Lut-tinger theorem,whereas the generalization of SBMFT to include holes predicts a”small”Fermi surface with area proportional to concentration of holes.In the large doping regime, it turns out that the slave-boson mean-ﬁeld theory[8]which treats the spins as fermions and holes as bosons produces a lot of properties of High-T c compounds correctly[9]and the SBMFT is only superier in describing the low-doping,antiferromagnetic state.Thus a natural question is:what is the relation between the two theories?Can one understand thetwo theories within a single framework?How does nature crossover from one description tothe other as concentration of holes increases?In SBMFT,the spin operator S i on site i is expressed in terms of Schwinger bosons S=¯Z i σZ i,where σis the Pauli matrix,Z i(¯Z i)are two component spinors¯Z i=(¯Z i↑,¯Z i↓), ietc.Notice that in order to represent a spin with magnitude S,there should be2S bosonsper site[2].The Hamiltonian can then be represented in terms of Schwinger bosons,and amean-ﬁeld theory can be formulated by introducing order parameters∆i,i+η=<Z i↑Z i+η↓−Z i↓Z i+η↑>(η=±ˆx,±ˆy)[2].Alternatively,SBMFT can also be formulated in a large-N expansion as the saddle point solution to a generalized SU(N)quantum spin model.To derive the large-N theory,we divide the square lattice into A and B sublattices and consider the following transformation for the Schwinger bosons on B-sublattice,Z B j↑(¯Z B j↑)→−Z B j↓(¯Z B j↓),Z B j↓(¯Z B j↓)→Z B j↑(¯Z B j↑),for all sites j on the B sublattice.The Schwinger bosons on the A sublattice remainsunchanged in the above transformation.The Lagrangian of the Heisenberg model can thenbe represented in the transformed boson coordinates as[3]L= iσ[¯Z A iσ(d dτ+iλj)Z B jσ−i2Sλj]+J i,η=±ˆx,±ˆy|∆i,i+η|2−J iσ,η=±ˆx,±ˆy[∆∗i,i+ηZ A iσZ B i+ησ+H.C.](1) whereλi’s are Lagrange multiplierﬁelds enforcing the constraint that there are2S bosons per site,∆i,i+η’s are Hubbard-Stratonovichﬁelds introduced in decoupling of H.σis the spin index.A SU(N)spin theory can be formulated with the above Lagrangian if the usual SU(2)spins↑and↓are extented to the SU(N)case where N-spin species are introduced. SBMFT can be considered as the saddle point solution of the path integral over L which becomes exact in the limit N→∞[3].For inﬁnite system with no defects,the saddle point solution has position independent∆i,i+η’s andλ’s and the solution is formally verysimilar to the BCS solution for superconductors,except that in present case the pairing objects are Schwinger bosons but not electrons.Another important diﬀerence is that in the present case,the pairing bosons are always located on diﬀerent sublattices,whereas no such distinction is found in the case of superconductors.The similarity between SBMFT and BCS theory leads us to ask the natural question of whether vortex-like solutions can be found in SBMFT as in the case of superconductors. More precisely,one may ask the question of whether one can construct a stable solution in SBMFT where∆i,i+ηhas a phase structure∆i,i+η∼|∆i,i+η|e iθ(i+η/2),whereθ( x)is a smooth function of x and has a singular point at x′∼0such that for distance| x|>>0,θ( x)→θ( x)+2πif one moves vector x around a close loop C which enclosed the point x′=0.As in the case of superconductors,the simplest way to address this question is to construct a continuum description for SBMFT(analogous to Ginsburg-Landau theory for superconductors)and study the possibility of having vortex solution in the continuum approximation.In the following we shall perform such a study in the disordered state in SBMFT where the Schwinger bosons are not bose-condensed and spin is a good quantum number.In realistic High-T c cuprates long range antiferromagnetic order is destroyed by introduction of charge carriers(holes).We shall not consider complications introduced by holes here and shall assume simply a disordered magnetic state with realistic spin magnitudes which can be described by the spin gap phase of SBMFT.We shall examine the properties of vortices within the continuum description.In section II we shall derive the continuum theory.Based on the continuum equations,we shallﬁrst study the properties of a single,non-magnetic impurity in the disordered state of SBMFT,where we shall point out the important diﬀerence between perturbations which are symmetric to the A and B sublattices,and perturbations which distinguish the two sublattices.In section III,we shall study static vortex excitations in our model.Based on the continuum theory,we shall argue that vortices are stable topological excitations in the disordered state of2D quantumantiferromagnets.We shall then show that there exists two kind of vortices in the continuum theory,corresponding to vortices centered on the mid-point of a plaquette,and vortices centered on a lattice point.We shall show that the properties of these two kinds of vortices are very diﬀerent,because of the diﬀerent symmetry with respect to the two sublattices.The ﬁrst kind of vortex which is symmetric to the two sublattices can carry only zero angular momentum and is a boson,whereas the second kind of vortex distinguishes between the two sublattices can carry angular momentum S under favourble conditions and is a fermion if S is half-integer.In section IV we shall concentrate ourselves at the case S=1/2which is the physical case of interests and shall examine properties of a”liquid”of fermionic vortices. Based on simple symmetry arguments,we shall argue that the eﬀective theory of a liquid of fermionic vortices has precisely the same form as the slave-boson mean-ﬁeld theory for spins in the undoped limit.The case withﬁnite concentration of holes will also be discussed.The ﬁndings in this paper will be summarized in section V,where we shall discuss a plausible scenerio of how the system crossover from a state described by SBMFT to a state described by slave-boson meanﬁeld theory upon doping.II.CONTINUUM DESCRIPTION FOR SBMFTTo derive the continuum theory we following Read and Sachdev[3]and consider the representation where our system has two sites per unit cell and introduce the’uniform’and ’staggered’components in the∆andλﬁelds,where∆i,i±η=12)+q±η(i±η2)+A±η(i±η2[λA( k)+λB( k)],(2b) Aτ( k)=1Theφ( x)ﬁeld describes uniform antiferromagnetic correlations whereas q( x)ﬁeld de-scribes spin dimerization(spin-Peierls)eﬀects[3].θ( x)and Aη( x)areﬁelds describing the corresponding uniform and staggered phaseﬂuctuations,respectively.Allﬁelds are slowly varying on the scale of a unit cell.Notice that the reason why four independent(real)ﬁelds are needed to describeﬂuctuations in one unit cell is precisely because we have divided the system into A and B sublattices.In the case of usual superconductors such a distinction is not present and only two realﬁelds(or one complex scalarﬁeld)enters into the Ginsburg-Landau equation.Notice that Vµ( x)=∂µθ( x)and Aµ( x)can be considered as U(1)gauge ﬁelds coupling to the Schwinger boson Z’s.The’uniform’gaugeﬁeld Vµ( x)couples to the bosons on the two sublattices with same gauge charge,whereas the gauge charges for the ’staggered’gaugeﬁeld Aµ( x)are opposite on the two sublattices.A single vortex solution centered at x′=0corresponds to a solution of SBMFT with boundary conditionVµ( x)dxµ=2nπ,or in terms of the gaugeﬁeld,there is a’uniform’gaugeﬂux of n/2ﬂux quantum passing through the origin,similar to the case of superconductors.Notice that in SBMFT where<∆i,i+η>=0,the gauge symmetry associated with the’uniform’gaugeﬁeld Vµ( x) (Z A→Z A e iΓ,Z B→Z B e iΓ,Vµ→Vµ−2∂µΓ)is broken,whereas the’staggered’gauge symmetry(Z A→Z A e iΓ,Z B→Z B e−iΓ,Aµ→Aµ−∂µΓ)remains intact in SBMFT.The existence of vortex solution is tied with the broken symmetry gaugeﬁeld Vµ( x)as in usual superconductors.At distance much larger than lattice spacing,ﬂuctuations associated with the’uniform’variablesφandθτare unimportant.Thus we shall neglect the∇φ(x)and∇θτ(x)terms in the following and derive a continuum theory for the rest of the variables.Notice that the Vµvariable is kept in our continuum theory since the term is essential for studying of vortices. In the continuum limit,the Lagrangian L becomesL→dτd 2x σ{¯Z A σ(x )∂∂τZ B σ(x )−µ=ˆx ,ˆy φ(x )(1−12φ(x )(D µ¯Z A σ(x ))(D ∗µ¯Z B σ(x ))+12 µ[φ(x )2+q µ(x )2]−4Sθτ(x ) ,(3)where D µ=∂µ+iA µand D ∗µ=∂µ−iA µ.The continuum Lagrangian Eq.(3)can befurther simpliﬁed by introducing new boson ﬁeldsψσ=12(Z A σ−¯Z B σ).The πσﬁeld can be integrated out safely at large distance and low energy [3],leaving eﬀective Lagrangian for ψσﬁeld,L =σ dτd 2x φ(x ) |D ∗µψσ|2+m (x )2|ψσ|2 +(φ1(x ))−1(D τ−q µD µ)ψ+σ(D ∗τ+q µD ∗µ)ψσ+18(V µ(x )V µ(x )))+m (x )2 (4)where φ( x )m ( x )2=2θτ( x )−4[1−(1/8)(V µ( x )V µ( x )]φ( x )and and φ1( x )=(1/2){θτ( x )+2[1−(1/8)(V µ( x )V µ( x )]φ( x )}.In the limit q µ=0and A µ=0,and neglecting gradient terms ∇φ,∇θτ,the above Lagrangian can be easily diagonalized resulting in an eﬀective Lagrangian L eﬀin terms of the φ( x )and V µ( x )ﬁelds.The dynamical eﬀect of the remaining terms can be obtained by looking at Gaussian ﬂuctuations of the ﬁelds around the saddle point solution A µ,q µ=0.[3]In particular,in the small m ( x )∼m →0limit,we obtain (see Appendix)L eﬀ=dτ d2x (a−4(2S+1))φ+φ2+(1φ)qµqµ+(2S+1)φF2µν+icFµτqµ+O(m2) ,(5) 2e2where e2∼m,a(<4(2S+1)),b and c are constants of order O(1).The precise values of a,b,c depend on the underlying lattice structure and cannot be obtained in a continuum theory.We have consider the realistic case N=2in deriving the above expression.Notice that similar eﬀective Lagrangian has been obtained by Read and Sachdev previously in studying eﬀect of instantons[3].The only diﬀerence here is that the VµVµterm which was not considered by Read and Sachdev is now retained.Recall that Vµis the’uniform’U(1) gaugeﬁeld coupling to the bosons and the VµVµterm in Eq.(5)just represents the Meissner eﬀect associated with nonzero order parameter∆in SBMFT.Notice that as in case of usual superconductors,we have choosen the London gauge∇. V=0in our derivation.The term can also be written in a gauge invariant way by replacing Vµby the gauge invariant object Vµ−2∂µΓin Eq.(5).This term does not contribute to the instanton eﬀect discussed by Read and Sachdev[3]but is crucial to the study of vortices.To understand more bout the continuum theory weﬁrst consider the properties of a singlenon-magnetic impurity in the disordered state of SBMFT.[10]We shall assume that a non-magnetic impurity simply replaces a spin at site i by a non-magnetic object.The simplestway to model this in our formalism is to is replace the constraint equation¯ZZ=2S onsite i by¯ZZ=2S(1−n i)=0,where n i is the number of non-magnetic impurity on site i.It is than easy to see that the eﬀect of nonzero n i in our eﬀective Lagrangian Eq.(5)is tointroduce in the Coulomb gauge an extra term−(2Sn i e i)Aτ,where e i=±1,depending onwhether the impurity is located on the A-or B-sublattices,[10]i.e.,non-magnetic impuritiesappear as eﬀective(staggered)gauge charges of magnitude2Se localized at the impuritysites.A corresponding electrostatic potential V(r)∼(2Se i)ln(r/ξ)(whereξ∼m−1)will beinduced around the impurity.To lower the electrostatic energy,2S bosons will be nucleatedout from the vacuum to screen the electricﬁeld,resulting in formation of local magneticmoment of magnitude S around a non-magnetic impurity[10].Notice that this eﬀect hasbeen observed experimentally in the High-T c compounds in the underdoped region upon substitution of Cu by Zn in the conducting planes[11].This result can also be understood in an alternative way by observing that the same physical eﬀect should have occurred if instead of replacing a spin on site i by an non-magnetic impurity,we let the Heisenberg coupling J i to go to zero for those bonds joining to site i. In this case the’impurity’site i behaves as an non-magnetic impurity as far as the rest of the system is concerned.The only diﬀerence is that a free spin of magnitude S now remains on site i.However,the eﬀect of this approach on our eﬀective Lagrangian Eq.(5)looks rather diﬀerent compared with our previous approach.Instead of introducing a’staggered’electric charge of magnitude2Se on site i,a boundary condition∆i,i+η=0(η=±ˆx,±ˆy)is now imposed on our ing Eq.(2a),the corresponding boundary condition in the continuum theory becomesqˆn(x i)=±ˆnφ(x i)where i→x i in the continuum limit.ˆn is a unit vector.±depends on whether site i is on A-or B-sublattices.The properties of the system around the non-magnetic impurity is obtained by minimizing the free energy Eq.(5)under this boundary condition.Performing the calculation,weﬁndﬁrst of all thatφ(x i)=0and correspondingly q(x i)= 0.The reason for this behaviour can be understood easily from Eq.(5)by noticing that the coeﬃcient in front of the qµqµterm becomes negative asφbecomes smaller than2b, implying that spontaneously dimerization will occur whenφbecomes small enough.Because of stability requirement,it can be shown that dimmerization| q|cannot have value largerthan q o=(√2)φ(see Appendix).Anyway bothφ(xi)and q(x i)∼ q o should be nonzeroin the continuum solution.The reason why q should be nonzero can also be understood from symmetry consideration.Notice that theφ(x)ﬁeld is symmetric under exchange of A-and B-sublattices whereas qµ(x)ﬁeld is antisymmetric.Thus a solution with qµ(x)ﬁeld identically zero would not distinguish between the two sublattices.However,the single non-magnetic impurity problem certainly distinguishes the two sublattices since the impuritycan only be placed on either one of the two sublattices.Thus we expect qµ(x)to be nonzero around a single impurity.Minimizing the free energy with respect to the qµﬁeld weﬁnd also that E(x)∼ q(x),where Eµ=iFµτis the’staggered’electricﬁeld in the system.With the boundary condition q∼(±ˆn)φaround the impurity weﬁnd that electricﬁeld radiating out (or in)from the impurity is obtained in our solution,i.e.,depending on whether the impurity is located on the A-or B-sublattices,it behaves as source or sink for the electricﬁeld,in exact agreement with our previous approach which predicts that non-magnetic impurity behaves as a single staggered gauge charge added to the system.One may also extend our approach to discuss the case when the Heisenberg coupling J′joining to the site i is small but nonzero,so that the spin at site i still couples weakly with the surrounding environment.The previous discussions still apply,except that the spin at site i appears now as a localized spinon state occupied by2S spinons in the system (notice that at two dimension,an arbitrary small attractive interaction is enough to generate a bound state[12]).The spinon state again behaves as a(staggered)gauge charge(with magnitude2S),and will bind2S other spinons with opposite gauge charge(or localized on opposite sublattice)next to itself as in the case with J′=0.However for nonzero J′the localized spinons on the two sublattices will interact antiferromagnetically because of the underlying antiferromagnetic correlation in the system,[13]forming at the end a localized spin singlet around site i.It is only in the limit J′→0that the two spinons are decoupled and free local magnetic moments appear.Our discussion can also be extended to study other forms of local defects.For example, one may study the situation when the Heisenberg coupling J i,i+ˆx between sites i and i+ˆx is set to zero,i.e.,a single bond is being removed from the system.Notice that the perturbation is symmetric with respect to interchange of A-and B-sublattices.It is straightforward to generalize our previous discussion to this case where weﬁnd that the perturbation now appears as an eﬀective electric dipole moment in L eﬀ.No local moment is expected to form around the impurity bond in this case because of the follow reason:because of the symmetry between A-and B-sublattices bosons(spinons)can be nucleated out from vaccuum only inpairs,with one localized on the A sublattice and one on the B sublattice.The interaction between the nucleated spinon pairs will again be antiferromagnetic and has a magnitude ∼Je−x/ξ,where x is the distance between the two spinons andξ∼m−1is the’size’of the spinon wavefunction.For the single bond defect,the distance x between the two spinons is of order of lattice spacing<<ξ.Thus the eﬀective interaction between the two spinons is of order J,i.e.,they will form a strong local spin singlet and no isolated magnetic moment will appear.Experimentally,the Cu ion in the conducting plane of high-T c cupartes can be substituted by Zn(non-magnetic impurity)[11]and magnitude of Heisenberg exchange J can be modiﬁed locally by introducing impurities out of conduction planes or substitution of O ions.It is observed that while substitution of Cu by Zn in the conducting plane introduces local magnetic moments in the underdoped(spin gap)phases of High-T c cuprates[11],no such eﬀect is observed in other ways of introducing defects.Our theoretical investigation on disordered state of quantum antiferromagnets based on continuum description of SBMFT is in satisfying agreement with experimental results.[10]III.STATIC VORTICES IN DISORDERED STATE OF SBMFTIn the continuum theory,a unit vortex centered as x=0corresponds to a solution ofδL eﬀ=0,withθ(r,Ω)=Ω,where r andΩare the distance and angle in the(2D) polar co-ordinate.The nonzero vorticity introduces a diverging kinetic energy through the VµVµ∼(∇θ)2term in L eﬀ,which supresses the magnitude ofφaround the vortex core r→0,as in the case of usual superconductors.Minimizing the free energy with respect√2)φ,weﬁnd thatφ=| q|=0for toφ,and keeping in mind that for smallφ,| q|∼(r<r o∼[(8(2S+1)+4b−2a)/(2S+1)]1/2,but not going to zero smoothly as r→0, as in the case of usual superconductors.This result is an artifact of our continuum theory where the(∇φ)2term is not included in L eﬀ.Nevertheless,the qualitative eﬀect whereφis being suppressed around the vortex core is clear.For large r,φ(r)→φo=(4(2S+1)−a)/2and q→0,indicating that vortices are stable topological excitations in SBMFT.The behaviour of vortices in SBMFT is very similar to vortices in usual superconductors at large r.However,the small r behaviours are very diﬀerent.The suppression ofφand qﬁelds around vortex core reﬂects the fact that around the center of vortex,the bond amplitude ∆i,i+µ’s are being suppressed.As has been discussed in the previous section,suppression of local bond amplitudes may lead to generation of localized magnetic moments,depending on the detailed bond conﬁguration.To understand the properties of a vortex,it is thus important to understand the underlying bond structure at the vortex core.Before studying the bond structure at vortex core,let us explainﬁrst how local magnetic moment binding to vortex core modiﬁes the properties of a vortex.For a unit vortex the magneticﬂux seen by a Schwinger boson is halfﬂux quantum.(recall that Vµis an’uniform’gaugeﬁeld which has same gauge charge for bosons on both sublattices)Therefore the orbital angular momentum of a Schwinger boson around a unit vortex is1the VµVµterm.Following similar analysis as in previous section weﬁnd that the sup-pressed bonds introduce in the continuum theory an eﬀective electric quadrapole structure surrounding the vortex center.Notice that as in the case of single broken bond the structure is symmetric with respect to the A-and B-sublattices.Thus we expect that localized spinon pairs may form around the center of vortex but isolated magnetic moments cannot occur. Thus this kind of vortex carries zero angular momentum and is a boson.The second kind of vortex is centered at a lattice point and as a result,it is expected that the four bonds joining to the vortex center(see Fig.1b)will be largely suppressed. This situation is very similar to the case of single non-magnetic impurity discussed in the last section.First of all,2S spinons will be found localized at the vortex center because of the suppressed bonds.The spinons behaves as a’staggered’gauge charge with magnitide 2S and2S other spinons with opposite gauge charge will be nucleated from the vacuum to screen the’staggered’electricﬁeld generated by the spinons localized at the vortex cen-ter.The spinons on opposite sublattices interact antiferromagnetically through an eﬀective Heisenberg exchange of order J′∼J|∆c|2,where∆c is the bond amplitude of the four bonds joining to the vortex center.As a result,a spin singlet will be formed.The vortex carries zero angular momentum and is again a boson.Notice that this is true even in the limit J′→0since integer number of bosons=4S are found binding to the vortex.The situation is however quite diﬀerent if we consider aﬁnite density of the second kind of vortices.We shall show that when density of vortices is large enough,it may become energetically unfavourable to nucleate spinons from vacuum to screen the staggered gauge ﬁeld and as a result,only2S spinons will be found binding to vortex center resulting in fermionic vortices when S is half-integer(odd number of bound bosons).For simplicity we shallﬁrst consider two vortices seperated by distance l,with one vortex on each sublattice. First let us consider the energy when spinons are nucleated from the vacuum to screen the staggered gauge charge on each vortex.The total energy will be sum of three terms:(i) the electrostatic energy which is of order2(2Se)2ln(l o/ξ),where l o∼ξis the’size’of the nucleated spinon wavefunction,(ii)The exchange energy between the spinons localized asvortex center and nucleated spinons,which is of order−2S(S+1)J′e−l o/ξ∼−2S(S+1)J′, and(iii)The energy needed to nucleate the spinons from the vacuum,which is of order 4Sm.The sum of the three terms is of order2S(2m−(S+1)J′).The energy for the second case when no spinons are nucleated from vacuum consists of two terms:(i)the electrostatics energy which is of order(2Se)2ln(l/ξ)and(ii)the exchange energy between spinons located at the center of the two vortices which is or order−S(S+1)J′e−l/ξ.In2D,e2∼m and the electrostatic energy is of order4S2m.ln(l/ξ).For l>>ξ,it is certainly energtically more favourable to nucleate bosons from vacuum to screen the vortex gauge charge.However, when l≤ξ,the energy cost of the second case is of order−S(S+1)J′,and is energetically more favourable if S(S+1)J′<4Sm.Notice that J′∼J|∆c|2is expected to be very small because of suppresion eﬀect around vortex core.Thus the condition can be satisﬁed easily even with a relatively small m.Forﬁnite density of vorticesδ,l∼δ1which are rather easy to acheive.We cannot,however,be absolutely sure whether’fermionic’vortices exist because of the natural limitation of a continuum theory which gives only or-der of magnitude estimates.Notice also one important distinction between the vortices we study in this paper and usual vortices in superﬂuids.In the present case,vortices cannot be distingished by sign of vorticity since vortices carrying1/2(orπ)and−1/2ﬂux quanta should be considered as identical.On a lattice where the order parameterﬁeld∆i,i+µis a link-variable,πand−πvortices can be related by a pure gauge transformation.The usual singularity encountered in the gauge transformation in continous space does not arise here because of the discretized lattice structure.It is important to clarify the diﬀerent roles played by the’uniform’and’staggered’gaugeﬁelds in deciding the properties of vortices.The gaugeﬁelds arise fromﬂuctuation in phases of the order parameter∆i,i+µ’s.The uniform gaugeﬁeld couples to bosons on the two sublattices with same gauge charge,and the corresponding gauge symmetry is broken in SBMFT.Vortices are stable because of this broken gauge symmetry,and corresponds to solutions with”uniform”magneticﬂux penetrating center of vortices.The’staggered’gaugeﬁeld couples to bosons on opposite sublattices with opposite gauge charge.It plays no role in generating a stable vortex solution,but has strong eﬀects in determining the precise properties of the vortex core.Without the’staggered’gaugeﬁeld,spinons need not be nucleated from vacuum to screen the’staggered’charges generated at,for example,the center of the second kind of vortex.In this case,for half-integer spin systems,the second kind of vortices will always be fermions independent of density,since only odd number of bosons are found binding to center of each vortex(corresponding to the spin S at center of vortex).Similarly,local magnetic moments will not be generated by non-magnetic impurities if’staggered’gaugeﬁeld does not exist.It is also interesting to point out that similar vortex excitations have been considered by Read and Chakraborty[15]in a short-ranged RVB wavefunction for S=1/2quantum antiferromagnets.They considered also the two kinds of vortices we discussed here.The statistics of the vortices were examined by direct Berry phase computations where similar。

The Relevant Scale Parameter in the High Temperature Phase of QCD

a rXiv:h ep-ph/9411293v114Nov1994CTP#2360INFNCA-TH-94-24The Relevant Scale Parameter in the High Temperature Phase of QCD Suzhou Huang (1,2)∗and Marcello Lissia (1,3)∗(1)Center for Theoretical Physics,Laboratory for Nuclear Science and Department of Physics,Massachusetts Institute of Technology,Cambridge,Massachusetts 02139(2)Department of Physics,FM-15,University of Washington,Seattle,Washington 98195†(3)Istituto Nazionale di Fisica Nucleare,via Negri 18,I-09127Cagliari,Italy †and Dipartimento di Fisica dell’Universit`a di Cagliari,I-09124Cagliari,Italy (October 1994)Abstract We introduce the running coupling constant of QCD in the high temper-ature phase,˜g 2(T ),through a renormalization scheme where the dimen-sional reduction is optimal at the one-loop level.We then calculate the rel-evant scale parameter,ΛT ,which characterizes the running of ˜g 2(T )with T ,using the background ﬁeld method in the static sector.It is found that ΛT /ΛI.INTRODUCTIONAt high temperatures QCD is expected to undergo a partial dimensional reduction[1,2], namely static correlations at distances larger than the thermal wavelength(1/T)can be reproduced by a three dimensional Lagrangian,where only the static modes of the original theory are present.This reduced Lagrangian can be computed perturbatively up to a speciﬁc order in the QCD running coupling constant.In fact,non-perturbative infrared phenomena (e.g.thermo-mass generation)prevent the complete reduction,i.e.reduction to all orders in the QCD running coupling,from taking place[2].Consequently,observables can be reproduced only up to corrections of a speciﬁc order,before non-perturbative physics begins to dominate.Even though a complete dimensional reduction is not possible,the partial dimensional reduction of QCD still provides a simpliﬁed physical picture.However,phenomenological applications of this picture depends crucially on how high is the temperature above which this picture begins to take place.Since we expect corrections to vanish with some power of the QCD coupling,and since at zero temperature the asymptotic freedom starts dominating QCD physics at typical scales of about10to20timesΛMS .Contrary to this expectation,there are strong evidences[3–5]that the dimensional re-duction picture is already valid at temperatures as low as two or three times the critical temperature T c(the deconﬁning transition in the pure Yang-Mills case or the chiral restora-tion in full QCD).Since T c is numerically not very diﬀerent fromΛMS .In fact,the deﬁnition of a scale parameter thatcharacterizes the approach to the dimensional reduction regime implies the deﬁnition of a suitable coupling constant,˜g2(T),that yields a sensible perturbative expansion at high temperature,i.e.an expansion whose coeﬃcients contain minimal contribution from non-static modes.In this paper,we use the backgroundﬁeld approach to deﬁne and compute the relevant coupling constant,and hence the scale parameterΛT.More speciﬁcally,we calculate the one-loop eﬀective action for the backgroundﬁeld in the static sector,and deﬁne the renormal-ization scheme by requiring that dimensional reduction be optimal for this gauge-invariant quantity.Furthermore,we verify by an explicit computation that this same renormaliza-tion scheme is also optimal for lattice perturbative calculations at high T,and therefore it provides a natural scale also for lattice simulations.In section II we introduce the renormalization scheme that deﬁnes the scale parame-terΛT within the backgroundﬁeld approach.In section III,we apply this deﬁnition and calculateΛT/Λthe conclusions.Several technical points pertinent to the lattice perturbative calculation at high T are discussed in the Appendix.II.DIMENSIONAL REDUCTION AND OPTIMAL RENORMALIZATIONSCHEMEThe standard SU(N)Yang-Mills gauge theory reduces at the tree level to the three dimensional Yang-Mills theory with adjoint Higgs(φa≡Q a0)L RD=−12(D iφ)a(D iφ)a,(1) where F a ij=∂i Q a j−∂j Q a i−g3f abc Q b i Q c j and(D iφ)a=∂iφa−g3f abc Q b iφc.The couplingg3is related to the four dimensional coupling through g23=g2T.Since L RD is a super-renormalizable theory in three dimensions and there is no other dimensionful scale around, all the dynamical scales must be set by the coupling constant g23=g2T.Of course,once loop corrections are included the reduced theory in Eq.(1)would acquire new vertices and the coupling constant g23would depend on the original coupling g2in a more complicated way.For example,g23would receive corrections,such as g4T and so on.However,due to the asymptotic freedom of QCD(g2∼1/ln T)we still expect that dynamical scales are set by g23≈g2T,provided the scale parameter is chosen in a suitable way.Therefore,we believe that the concept of dimensional reduction involves two equally important aspects.On one hand,there is the possibility of a simpliﬁed description by using a theory L RD with less degrees of freedom in lower dimensions.On the other hand,the evolution of the parameters of L RD as a function of temperature should be dictated by the original theory.The main concern of our present work is to determine this evolution,which in turn determines the temperature dependence of the relevant physical observables.A.Background Field Method in the Static SectorIt is well known that the eﬀective action calculated using the backgroundﬁeld method[6] is gauge invariant for the background gaugeﬁeld at T=0.This gauge invariance guarantees that the coupling constant renormalization is related to the wavefunction renormalizationof the backgroundﬁeld through Z g=Z−1/2A .Hence,the calculation of the quadratic partof the eﬀective action,i.e.the two-point function for the backgroundﬁeld,is suﬃcient to renormalize the coupling[6].Moreover,to the leading order,there is no magnetic mass generation atﬁnite T.There-fore,the one-loop eﬀective action for the magnetic sector is invariant under time-independentgauge transformations also atﬁnite T,insuring that the relation Z g=Z−1/2A still holds forthe static backgroundﬁeldA a0(τ,x)=0,A a i(τ,x)=A a i(x).(2) The same conclusion can also be reached more formally by applying,for instance,the meth-ods of Ref.[6]to the backgroundﬁeld of Eq.(2).The residual gauge invariance in the staticsector implies that,in order to compute the coupling constant renormalization atﬁnite T, we only need to compute the two-point function of the backgroundﬁeld A a i in the static sector.We can still use the zero temperature Feynman rules,as given for instance by Abbott[6]. The only diﬀerence in the calculation is that time-components of all momenta become dis-crete Matsubara frequencies(2πnT),and the corresponding integrals become discrete sums.B.Subtraction ScaleAs exhaustively discussed by Landsman[2],the decoupling of the non-zero modes at high-T is maximal only in some speciﬁc renormalization schemes,such as the BPHZ scheme.In the backgroundﬁeld method we only need toﬁx one renormalization condition:we demand that the two-point function for the backgroundﬁeld in the low external momentum(relative to T)limit coincides with the contribution solely from zero modes.Landsman[2]uses aﬁnite temperature renormalization group approach,since he dis-cusses thermal reduction in a more general context where several couplings are present. Thanks also to the backgroundﬁeld approach,we deal with a simpler situation where only one coupling needs renormalization.Therefore,we can directly implement the renormalization condition by using the freedom in the choice of the subtraction scale,µ,which becomes a function of T.Intuitively,we expect µto be of order of T.The purpose of our paper is toﬁnd out what is the proportionality constant.Then the reduced theory,Eq.(1),with the T-dependent coupling g23=g2(µ(T))T, reproduces the full two-point function up to corrections of order of p2/T2at the one-loop level.Due to the gauge invariance,the two-point function for the static backgroundﬁeld A a i must have the form(δij p2−p i p j)δabΠM(p2,T,µ).(3) Speciﬁcally,we chooseµby requiring the following renormalization condition for the non-static contribution toΠM(p2,T,µ):ΠNS M(p2=0,T,µ(T))=0.(4) The procedure is best explained by directly going through the calculation in the next section.III.CALCULATION OF THE SCALE PARAMETERA.In the ContinuumIn the continuum calculation we use dimensional regularization in the spatial dimensions, that isd4k,(5)(2π)3−2ǫand theg2(µ)− 2132α+1√11+O(p2/T2),(6)where g2(µ)is the running coupling deﬁned in theβ0ln(T2/Λ2T)=g2(µ) µ=4πT e−(γE+c),(8) which deﬁnes the scale parameterΛT=e(γE+c)MS.(9)This results has a clear physical interpretation.The non-static modes decouple in the high-T limit,but their presence is nevertheless revealed by the appearance of the new scale ΛT in the reduced theory(without any reference to the original theory,the only scale would be T).While this new scale is obviously related to the scaleΛprocess,represented by a diﬀerent set of Feynman graphs,yields a diﬀerent c in Eq.(9). However,we believe that typically|c|<∼1,and a diﬀerent choice should not modify the scale ratio in Eq.(9)in an essential way.For example,Landsman[2]calculated the temperature dependent coupling renormalization factor Z g by imposing maximal dimensional reduction on the two-and three-point functions in the conventional eﬀective action(where the rela-tionship Z g=Z−1/2A no longer holds).He did not express his result explicitly in terms ofthe scale ratio.But if we do it,weﬁnd that his result is quite close to ours,i.e.Eq.(9)with c=0.Another example that clearly shows the necessity of using an optimal dimensional re-duction scheme for deﬁning the relevant scale at high-T can be found in the Gross-Neveu model[7].In that model a similar strategy makes the sub-leading correction to the screening mass of order of˜g6(T),rather than˜g4(T),demonstrating that˜g2(T)is a sensible expansion parameter.Of course,the optimal dimensional reduction criterion is not the only way to deﬁne a temperature dependent coupling constant.For example,the quark-antiquark potential at a distance of order of1/T is used to deﬁne g2(T)in Ref.[3].While it is certainly legitimate to make such a choice,it is also true that,because the reduced theory is meant to reproduce the full theory only at distances much larger than1/T(spatial momenta small compared to T),deﬁnitions of the couplings made by matching short distance properties of the full and reduced theories do not necessarily deﬁne a scale that correctly characterizes the approach to the asymptotic high-T regime.At last,let us consider the eﬀect of quarks on our result.If N f light quarks are present in the theory,results of Eqs.(7),(8)and(9)still apply,but withβ0=(11N−2N f)/(48π2) and c=(N/2−2N f ln4)/(11N−2N f),where we have adopted the convention for the trace of the Dirac-matrix:Trγµγν=−4δµν.For the phenomenological relevant case of N=N f=3,we get the value c=−0.2525, which corresponds toΛT/Λa suitable expansion parameter for lattice perturbative calculation at high temperature,and the necessity of using expansion parameters diﬀerent from the bare lattice coupling g20(a) for perturbative calculations at zero temperature[8].In the following we verify that optimal dimensional reduction for the lattice eﬀective action computed in the backgroundﬁeld method deﬁnes indeed the same scale parameter we have found in the continuum calculation.For the sake of concreteness,we perform the calculation for the pure SU(N)Wilson action,but the same result is expected to hold for other actions as well.In general,the coupling deﬁned in the lattice backgroundﬁeld method should have the following dependence on the bare lattice coupling up to one-loopg2L(T)≡g20(a)+g40(a)β0 −ln(a2T2)+c T L .(10)We want to show that c T L is such that g2L(T)=˜g2(T).Since we have expressed˜g2(T)in terms of g2(µ)in theMS scheme[9,10]g20(a)=g2(µ)−g4(µ)β0 −ln(a2µ2)+c0L ,(11) and express also g2L(T)in Eq.(10)in terms of g2(µ)g2L(T)=g2(µ)−g4(µ)β0 −ln(µ2/T2)−c T L+c0L +O(g6(µ)).(12)By comparing Eq.(8)and Eq.(12)we see that to show g2L(T)=˜g2(T)is equivalent to show thatc T L=c0L+2γE−2ln(4π)+111−11γE+2f11+3f00+6f10−1+24π2z10+6π2−6π2/N2 .(14) The constants f ij and z ij are deﬁned asf ij≡(4π)2 ∞dx x e−8x I20(2x)I i(2x)I j(2x)−θ(x−1)expansion on the lattice.The lattice correspondent of the continuum result of Eq.(6)in the Feynman gauge(α=1)isΠL M(p2,T,a)=116NTp2−β0−ln(a2T2)+c T L +O p2/T2,a|p|,aT ,(17)with c T L given byc T L=1(4πx)3/2.(19)Since,as shown in the Appendix,f′ij=f ij−γE−3ln4,this complete the proof of Eq.(13) and,therefore,of the fact that g2L(T)=˜g2(T).In other words,if we use˜g2(T)as the expansion parameter,the lattice eﬀective action in the high-T limit takes the following formΠL M(p2,T,a)=116NTp2+O p2/T2,a|p|,aT ,(20)which is the same as its continuum counterpart,if we use the same coupling˜g2(T)(see Eq.(6)withα=1andµgiven by Eq.(7)).In both cases we have been able to absorb in the coupling constant all leading local corrections due to non-static modes,while the non-local ones are reproduced by the reduced theory.PARISON TO LATTICE RESULTIn the preceding section,we have demonstrated thatΛT is the relevant scale parameter in the high-T limit.Our argument is yet only perturbative in nature.However,as we emphasized earlier,the determination of the scale parameter is largely one-loop eﬀect.Now let us compare our result with the scale parameter determined from a non-perturbative method:lattice measurement of the spatial string tension at high T.The primary reason for choosing the spatial string tension[4]rather than the heavy quark potential at distances of order of1/T[3]is that the concept of dimensional reduction only makes sense for large distance(low momentum)quantities.Bali et al.[4]measured the spatial string tension in SU(2)gauge theory as a function of temperatureσs(T).Then theyﬁtted their result to the expected form of the string tension in the three-dimensional SU(2)Yang-Mills theory12π2ln(T/ΛT)+17Even though the simulation process knows nothing about the dimensional reduction,the ﬁtting formula Eq.(21)in fact deﬁnes the optimal three-dimensional coupling g23=g2(T)T through the string tension,similar in spirit to what we have done for the backgroundﬁeld eﬀective action.As a result,theirﬁtted value ofΛσT=(0.076±0.013)T c is theﬁrst,to our knowledge,non-perturbative determination of the scale that characterizes the high-T regime for the SU(2)gauge theory.In the scaling regime we expect that the critical temperature behaves likeT c=ΛL24π2βc5111N2βc ,(23)whereβc=2N/g20(a).From their critical couplingβc=2.74at Nτ=16,and the knownratioΛMS :T c=1.62ΛMS,(24) which is remarkably close to our resultΛT=eγE+1/22MS≈0.148Λlattice perturbative calculation at high T.Our results areΛT=0.148ΛMS for N=N f=3.We have argued that this scale is typical in the high-T regime,even if its precise value depends on the speciﬁc deﬁnition.The consequence of our result is that the high-T regime ofQCD,where the dimensional reduction picture appears to take place,sets in at temperatures as low as a few times of the critical temperature.Our calculation is in very good agreement with the non-perturbative determination ofthe scale parameter in the lattice simulations[4]in the SU(2)Yang-Mills theory,therefore reinforcing the advantage of the renormalization scheme based on the optimal dimensional reduction criterion.It would be of great interest to have other lattice measurements of the scale parameter using other observables,such as the ones related to the gluonic Debye-screening mass andthe deviations of the mesonic and baryonic screening masses from their free values.This work was supported in part by funds provided by the U.S.Department of Energy (DOE)under contract number DE-FG06-88ER40427and cooperative agreement DE-FC02-94ER40818.APPENDIX:In this appendix we discuss several points of the high temperature expansion in per-turbative lattice calculations.First we use the ghost bubble-graph to illustrate the general method,then we prove that f′ij−f ij=−γE−3ln4,andﬁnally discuss the convergence of the frequency sums to the corresponding zero temperature integrals.From the lattice action,see for instance Ref.[10],we derive the following expression for the ghost bubble-graphBµν(p)≡Nλ(1−cos kλa) ρ(1−cos(kρ−pρ)a),(A1)whereΩis the space-time volume and p=(0,p).Exponentiating the denominator and converting the spatial momentum sums into integrals(we work in the inﬁnite spatial volume limit),we obtainBµν(p)=N(2π)3∞dαdβe−(α+β)(4−cos2πnα2+β2+2αβcos pλa cos(kλ−φλ)×e−i(kµ+kν−pνa)+e i(kµ+kν−pµa)−e i(kµ−kν−pµa+pνa)−e−i(kµ−kν) ,(A2)whereφλis implicitly deﬁned by tanφλ=βsin(pλa)/[α+βcos(pλa)].Now we perform the spatial momentum integrals,yielding the modiﬁed Bessel functions.For the sake of concreteness,let us consider the componentµ=1andν=2B′12(p)=NNτ) 2 λ=1I1(α2+β2+2αβcos p3a) e−i(φ1+φ2−p2a)+e i(φ1+φ2−p1a)−e i(φ1−φ2−p1a+p2a)−e−i(φ1−φ2) .In Eq.(A3)B′is just B without the n=0term in the frequency sum.This static term is in fact the one that is directly reproduced by the reduced theory,and should be excluded from the contribution due to non-static modes.In the limit of|p|a≪1and|p|≪T(we are interested in the small lattice spacing and high-T limit),Eq.(A3)further simpliﬁesB′12(p)=−N p1p2NτNτ−1n=1e−x(1−cos2πn12∞dx x 1Nτ) e−3x I21(x)I0(x)−112∞dx x 1Nτ) 1NτNτ−1n=11π2a2T2+O(aT),(A6)and obtainB′12(p)=−Np1p2x−√√any power dependence on T trivially,and take the continuum limit of the frequency sums. Mathematically,this is guaranteed by the fact that the convergence of the limitlim Nτ→∞1Nτ= 1dx f(cos2πx)(A10)is exponential,at least when f(z)can be expanded as a power series in z,which includes the cases we are concerned with.Note that the terms with n=0should be included in these tadpole-like graphs,since they are not reproducible by the reduced theory.REFERENCES∗E-mail:shuang@ and lissia@†Present address.[1]T.Appelquist and R.D.Pisarski,Phys.Rev.D23(1981)2305;S.Nadkarni,Phys.Rev.D27(1983)917;A.N.Jourjine,Ann.Phys.155(1984)305;Alvarez-Estrada,Phys.Rev.D36(1987)2411;Ann.Phys.174(1987)442.[2]ndsman,Nucl.Phys.B322(1989)498;[3]A.Irback,cock,ler,B.Petersson and T.Reisz,Nucl.Phys.B363(1991)34;T.Reisz,Z.Phys.C53(1992)169;cock,ler and T.Reisz,Nucl.Phys.B369(1992)501;[4]G.S.Bali,et al.,Phys.Rev.Lett.71(1993)3059.[5]V.Koch,E.V.Shuryak,G.E.Brown and A.D.Jackson,Phys.Rev.D46(1992)3169;T.H.Hansson and I.Zahed,Nucl.Phys.B374(1992)227;S.Schramm and M.-C.Chu,Phys.Rev.D48(1993)2279;V.Koch,Phys.Rev.D49(1994)6063;M.Ishii and T.Hatsuda,UTHEP-282,July1994.[6]B.S.DeWitt,Phys.Rev.162(1967)1195,1239;J.Honerkamp,Nucl.Phys.B48(1972)269;G.’t Hooft,Nucl.Phys.B62(1973)444;L.F.Abbott,Nucl.Phys.B185(1981)189;D.G.Boulware,Phys.Rev.D23(1981)389.[7]S.Huang and M.Lissia,MIT preprint,CTP#2359(1994).[8]G.P.Lepage and P.B.Mackenzie,Phys.Rev.D48(1993)2250.[9]A.Hasenfratz and P.Hasenfratz,Phys.Lett.93B(1980)165;A.Hasenfratz and P.Hasenfratz,Nucl.Phys.B192(1981)210;R.Dashen and D.J.Gross,Phys.Rev.D23(1981)2340;P.Weisz,Phys.Lett.100B(1981)331;H.Kawai,R.Nakayawa and K.Seo,Nucl.Phys.B189(1981)40;[10]A.Gonzalez-Arroyo and C.P.Korthals Altes,Nucl.Phys.B205(1982)46.[11]M.Gao,Phys.Rev.D41,(1990)626.。

采暖通风与空气调节术语标准中英文对照

采暖通风与空气调节术语标准中英文对照AA-weighted sound pressure level A声级absolute humidity绝对湿度absolute roughness绝对粗糙度absorbate 吸收质absorbent 吸收剂absorbent吸声材料absorber吸收器absorptance for solar radiation太阳辐射热吸收系数absorption equipment吸收装置absorption of gas and vapor气体吸收absorptiong refrige rationg cycle吸收式制冷循环absorption-type refrigerating machine吸收式制冷机access door检查门acoustic absorptivity吸声系数actual density真密度actuating element执行机构actuator执行机构adaptive control system自适应控制系统additional factor for exterior door外门附加率additional factor for intermittent heating间歇附加率additional factor for wind force高度附加率additional heat loss风力附加率adiabatic humidification附加耗热量adiabatic humidiflcation绝热加湿adsorbate吸附质adsorbent吸附剂adsorber吸附装置adsorption equipment吸附装置adsorption of gas and vapor气体吸附aerodynamic noise空气动力噪声aerosol气溶胶air balance风量平衡air changes换气次数air channel风道air cleanliness空气洁净度air collector集气罐air conditioning空气调节air conditioning condition空调工况air conditioning equipment空气调节设备air conditioning machine room空气调节机房air conditioning system空气调节系统air conditioning system cooling load空气调节系统冷负荷air contaminant空气污染物air-cooled condenser风冷式冷凝器air cooler空气冷却器air curtain空气幕air cushion shock absorber空气弹簧隔振器air distribution气流组织air distributor空气分布器air-douche unit with water atomization喷雾风扇air duct风管、风道air filter空气过滤器air handling equipment空气调节设备air handling unit room空气调节机房air header集合管air humidity空气湿度air inlet风口air intake进风口air manifold集合管air opening风口air pollutant空气污染物air pollution大气污染air preheater空气预热器air return method回风方式air return mode回风方式air return through corridor走廊回风air space空气间层air supply method送风方式air supply mode送风方式air supply (suction) opening with slide plate插板式送（吸）风口air supply volume per unit area单位面积送风量air temperature空气温度air through tunnel地道风air-to-air total heat exchanger全热换热器air-to-cloth ratio气布比air velocity at work area作业地带空气流速air velocity at work place工作地点空气流速air vent放气阀air-water systen空气—水系统airborne particles大气尘air hater空气加热器airspace空气间层alarm signal报警信号ail-air system全空气系统all-water system全水系统allowed indoor fluctuation of temperature and relative humidity室内温湿度允许波动范围ambient noise环境噪声ammonia氨amplification factor of centrolled plant调节对象放大系数amplitude振幅anergy@angle of repose安息角ange of slide滑动角angle scale热湿比angle valve角阀annual [value]历年值annual coldest month历年最冷月annual hottest month历年最热月anticorrosive缓蚀剂antifreeze agent防冻剂antifreeze agent防冻剂apparatus dew point机器露点apparent density堆积密度aqua-ammonia absorptiontype-refrigerating machine氨—水吸收式制冷机aspiation psychrometer通风温湿度计Assmann aspiration psychrometer通风温湿度计atmospheric condenser淋激式冷凝器atmospheric diffusion大气扩散atmospheric dust大气尘atmospheric pollution大气污染atmospheric pressure大气压力（atmospheric stability大气稳定度atmospheric transparency大气透明度atmospheric turblence大气湍流automatic control自动控制automatic roll filter自动卷绕式过滤器automatic vent自动放气阀available pressure资用压力average daily sol-air temperature日平均综合温度axial fan轴流式通风机azeotropic mixture refrigerant共沸溶液制冷剂Bback-flow preventer防回流装置back pressure of steam trap凝结水背压力back pressure return余压回水background noise背景噪声back plate挡风板bag filler袋式除尘器baghouse袋式除尘器barometric pressure大气压力basic heat loss基本耗热量hend muffler消声弯头bimetallic thermometer双金属温度计black globe temperature黑球温度blow off pipe排污管blowdown排污管boiler锅炉boiller house锅炉房boiler plant锅炉房boiler room锅炉房booster加压泵branch支管branch duct(通风) 支管branch pipe支管building envelope围护结构building flow zones建筑气流区building heating entry热力入口bulk density堆积密度bushing补心butterfly damper蝶阀by-pass damper空气加热器〕旁通阀by-pass pipe旁通管Ccanopy hood 伞形罩capillary tube毛细管capture velocity控制风速capture velocity外部吸气罩capturing hood 卡诺循环Carnot cycle串级调节系统cascade control system铸铁散热器cast iron radiator催化燃烧catalytic oxidation 催化燃烧ceilling fan吊扇ceiling panelheating顶棚辐射采暖center frequency中心频率central air conditionint system 集中式空气调节系统central heating集中采暖central ventilation system新风系统centralized control集中控制centrifugal compressor离心式压缩机entrifugal fan离心式通风机check damper(通风〕止回阀check valve止回阀chilled water冷水chilled water system with primary-secondary pumps一、二次泵冷水系统chimney(排气〕烟囱circuit环路circulating fan风扇circulating pipe循环管circulating pump循环泵clean room洁净室cleaning hole清扫孔cleaning vacuum plant真空吸尘装置cleanout opening清扫孔clogging capacity容尘量close nipple长丝closed booth大容积密闭罩closed full flow return闭式满管回水closed loop control闭环控制closed return闭式回水closed shell and tube condenser卧式壳管式冷凝器closed shell and tube evaporator卧式壳管式蒸发器closed tank闭式水箱coefficient of accumulation of heat蓄热系数coefficient of atmospheric transpareney大气透明度coefficient of effective heat emission散热量有效系数coficient of effective heat emission传热系数coefficient of locall resistance局部阻力系数coefficient of thermal storage蓄热系数coefficient of vapor蒸汽渗透系数coefficient of vapor蒸汽渗透系数coil盘管collection efficiency除尘效率combustion of gas and vapor气体燃烧comfort air conditioning舒适性空气调节common section共同段compensator补偿器components(通风〕部件compression压缩compression-type refrigerating machine压缩式制冷机compression-type refrigerating system压缩式制冷系统compression-type refrigeration压缩式制冷compression-type refrigeration cycle压缩式制冷循环compression-type water chiller压缩式冷水机组concentratcd heating集中采暖concentration of narmful substance有害物质浓度condensate drain pan凝结水盘condensate pipe凝结水管condensate pump凝缩水泵condensate tank凝结水箱condensation冷凝condensation of vapor气体冷凝condenser冷凝器condensing pressure冷凝压力condensing temperature冷凝温度condensing unit压缩冷凝机组conditioned space空气调节房间conditioned zone空气调节区conical cowl锥形风帽constant humidity system恒湿系统constant temperature and humidity system恒温恒湿系统constant temperature system 恒温系统constant value control 定值调节constant volume air conditioning system定风量空气调节系统continuous dust dislodging连续除灰continuous dust dislodging连续除灰continuous heating连续采暖contour zone稳定气流区control device控制装置control panel控制屏control valve调节阀control velocity控制风速controlled natural ventilation有组织自然通风controlled plant调节对象controlled variable被控参数controller调节器convection heating对流采暖convector对流散热器cooling降温、冷却（、）cooling air curtain冷风幕cooling coil冷盘管cooling coil section冷却段cooling load from heat传热冷负荷cooling load from outdoor air新风冷负荷cooling load from ventilation新风冷负荷cooling load temperature冷负荷温度cooling system降温系统cooling tower冷却塔cooling unit冷风机组cooling water冷却水correcting element调节机构correcting unit执行器correction factor for orientaion朝向修正率corrosion inhibitor缓蚀剂coupling管接头cowl伞形风帽criteria for noise control cross噪声控频标准cross fan四通crross-flow fan贯流式通风机cross-ventilation穿堂风cut diameter分割粒径cyclone旋风除尘器cyclone dust separator旋风除尘器cylindrical ventilator筒形风帽Ddaily range日较差damping factot衰减倍数data scaning巡回检测days of heating period采暖期天数deafener消声器decibel(dB)分贝degree-days of heating period采暖期度日数degree of subcooling过冷度degree of superheat过热度dehumidification减湿dehumidifying cooling减湿冷却density of dust particle真密度derivative time微分时间design conditions计算参数desorption解吸detecting element检测元件detention period延迟时间deviation偏差dew-point temperature露点温度dimond-shaped damper菱形叶片调节阀differential pressure type flowmeter差压流量计diffuser air supply散流器diffuser air supply散流器送风direct air conditioning system 直流式空气调节系统direct combustion 直接燃烧direct-contact heat exchanger 汽水混合式换热器direct digital control (DDC) system 直接数字控制系统direct evaporator 直接式蒸发器direct-fired lithiumbromide absorption-type refrigerating machine 直燃式溴化锂吸收式制冷机direct refrigerating system 直接制冷系统direct return system 异程式系统direct solar radiation 太阳直接辐射discharge pressure 排气压力discharge temperature 排气温度dispersion 大气扩散district heat supply 区域供热district heating 区域供热disturbance frequency 扰动频率dominant wind direction 最多风向double-effect lithium-bromide absorption-type refigerating machine 双效溴化锂吸收式制冷机double pipe condenser 套管式冷凝器down draft 倒灌downfeed system 上分式系统downstream spray pattern 顺喷drain pipe 泄水管drain pipe 排污管droplet 液滴drv air 干空气dry-and-wet-bulb thermometer 干湿球温度表dry-bulb temperature 干球温度dry cooling condition 干工况dry dust separator 干式除尘器dry expansion evaporator 干式蒸发器dry return pipe 干式凝结水管dry steam humidifler 干蒸汽加湿器dualductairconing ition 双风管空气调节系统dual duct system 双风管空气调节系统duct 风管、风道dust 粉尘dust capacity 容尘量dust collector 除尘器dust concentration 含尘浓度dust control 除尘dust-holding capacity 容尘量dust removal 除尘dust removing system 除尘系统dust sampler 粉尘采样仪dust sampling meter 粉尘采样仪dust separation 除尘dust separator 除尘器dust source 尘源dynamic deviation动态偏差Eeconomic resistance of heat transfer经济传热阻economic velocity经济流速efective coefficient of local resistance折算局部阻力系数effective legth折算长度effective stack height烟囱有效高度effective temperature difference送风温差ejector喷射器ejetor弯头elbow电加热器electric heater电加热段electric panel heating电热辐射采暖electric precipitator电除尘器electricradian theating 电热辐射采暖electricresistance hu-midkfier电阻式加湿器electro-pneumatic convertor电—气转换器electrode humidifler电极式加湿器electrostatic precipi-tator电除尘器eliminator挡水板emergency ventilation事故通风emergency ventilation system事故通风系统emission concentration排放浓度enclosed hood密闭罩enthalpy焓enthalpy control system新风〕焓值控制系统enthalpy entropy chart焓熵图entirely ventilation全面通风entropy熵environmental noise环境噪声equal percentage flow characteristic等百分比流量特性equivalent coefficient of local resistance当量局部阻力系数equivalent length当量长度equivalent[continuous A] sound level等效〔连续A〕声级evaporating pressure蒸发压力evaporating temperature蒸发温度evaporative condenser蒸发式冷凝器evaporator蒸发器excess heat余热excess pressure余压excessive heat 余热cxergy@exhaust air rate排风量exhaust fan排风机exhaust fan room排风机室exhaust hood局部排风罩exhaust inlet吸风口exhaust opening吸风口exhaust opening orinlet风口exhaust outlet排风口exaust vertical pipe排气〕烟囱exhausted enclosure密闭罩exit排风口expansion膨胀expansion pipe膨胀管explosion proofing防爆expansion steam trap恒温式疏水器expansion tank膨胀水箱extreme maximum temperature极端最高温度extreme minimum temperature极端最低温度Ffabric collector袋式除尘器face tube皮托管face velocity罩口风速fan通风机fan-coil air-conditioning system风机盘管空气调节系统fan-coil system风机盘管空气调节系统fan-coil unit风机盘管机组fan house通风机室fan room通风机室fan section风机段feed-forward control前馈控制feedback反馈feeding branch tlo radiator散热器供热支管fibrous dust纤维性粉尘fillter cylinder for sampling滤筒采样管fillter efficiency过滤效率fillter section过滤段filltration velocity过滤速度final resistance of filter过滤器终阻力fire damper防火阀fire prevention防火fire protection防火fire-resisting damper防火阀fittings(通风〕配件fixed set-point control定值调节fixed support固定支架fixed time temperature (humidity)定时温（湿）度flame combustion热力燃烧flash gas闪发气体flash steam二次蒸汽flexible duct软管flexible joint柔性接头float type steam trap浮球式疏水器float valve浮球阀floating control无定位调节flooded evaporator满液式蒸发器floor panel heating地板辐射采暖flow capacity of control valve调节阀流通能力flow characteristic of control valve调节阀流量特性foam dust separator泡沫除尘器follow-up control system随动系统forced ventilation机械通风forward flow zone射流区foul gas不凝性气体four-pipe water system四管制水系统fractional separation efficiency分级除尘效率free jet自由射流free sillica游离二氧化硅free silicon dioxide游离二氧化硅freon氟利昂frequency interval频程frequency of wind direction风向频率fresh air handling unit新风机组resh air requirement新风量friction factor摩擦系数friction loss摩擦阻力frictional resistance摩擦阻力fume烟〔雾〕fumehood排风柜fumes烟气Ggas-fired infrared heating 煤气红外线辐射采暖gas-fired unit heater 燃气热风器gas purger 不凝性气体分离器gate valve 闸阀general air change 全面通风general exhaust ventilation (GEV) 全面排风general ventilation 全面通风generator 发生器global radiation总辐射grade efficiency分级除尘效率granular bed filter颗粒层除尘器granulometric distribution粒径分布gravel bed filter颗粒层除尘器gravity separator沉降室ground-level concentration落地浓度guide vane导流板Hhair hygrometor毛发湿度计hand pump手摇泵harmful gas andvapo有害气体harmful substance有害物质header分水器、集水器（、）heat and moisture热湿交换transfer热平衡heat conduction coefficient导热系数heat conductivity导热系数heat distributing network热网heat emitter散热器heat endurance热稳定性heat exchanger换热器heat flowmeter热流计heat flow rate热流量heat gain from lighting设备散热量heat gain from lighting照明散热量heat gain from occupant人体散热量heat insulating window保温窗heat(thermal)insuation隔热heat(thermal)lag延迟时间heat loss耗热量heat loss by infiltration冷风渗透耗热量heat-operated refrigerating system热力制冷系统heat-operated refrigetation热力制冷heat pipe热管heat pump热泵heat pump air conditioner热泵式空气调节器heat release散热量heat resistance热阻heat screen隔热屏heat shield隔热屏heat source热源heat storage蓄热heat storage capacity蓄热特性heat supply供热heat supply network热网heat transfer传热heat transmission传热heat wheel转轮式换热器heated thermometer anemometer热风速仪heating采暖、供热、加热（、、）heating appliance采暖设备heating coil热盘管heating coil section加热段heating equipment采暖设备heating load热负荷heating medium热媒heating medium parameter热媒参数heating pipeline采暖管道heating system采暖系统heavy work重作业high-frequency noise高频噪声high-pressure ho twater heating高温热水采暖high-pressure steam heating高压蒸汽采暖high temperature water heating高温热水采暖hood局部排风罩horizontal water-film syclonet卧式旋风水膜除尘器hot air heating热风采暖hot air heating system热风采暖系统hot shop热车间hot water boiler热水锅炉hot water heating热水采暖hot water system热水采暖系统hot water pipe热水管hot workshop热车间hourly cooling load逐时冷负荷hourly sol-air temperature逐时综合温度humidification加湿humidifier加湿器humididier section加湿段humidistat恒湿器humidity ratio含湿量hydraulic calculation水力计算hydraulic disordeer水力失调hydraulic dust removal水力除尘hydraulic resistance balance阻力平衡hydraulicity水硬性hydrophilic dust亲水性粉尘hydrophobic dust疏水性粉尘Iimpact dust collector冲激式除尘器impact tube皮托管impedance muffler阻抗复合消声器inclined damper斜插板阀index circuit最不利环路indec of thermal inertia (valueD)热惰性指标（D值）indirect heat exchanger表面式换热器indirect refrigerating sys间接制冷系统indoor air design conditions室内在气计算参数indoor air velocity室内空气流速indoor and outdoor design conditions室内外计算参数indoor reference for air temperature and relative humidity室内温湿度基数indoor temperature (humidity)室内温（湿）度induction air-conditioning system诱导式空气调节系统induction unit诱导器inductive ventilation诱导通风industral air conditioning工艺性空气调节industrial ventilation工业通风inertial dust separator惯性除尘器infiltration heat loss冷风渗透耗热量infrared humidifier红外线加湿器infrared radiant heater红外线辐射器inherent regulation of controlled plant调节对象自平衡initial concentration of dust初始浓度initial resistance of filter过滤器初阻力imput variable输入量insulating layer保温层integral enclosure整体密闭罩integral time积分时间interlock protection联锁保护intermittent dust removal定期除灰intermittent heating间歇采暖inversion layer逆温层inverted bucket type steam trap倒吊桶式疏水器irradiance辐射照度isoenthalpy等焓线isobume等湿线isolator隔振器isotherm等温线isothermal humidification等温加湿isothermal jet等温射流Jjet射流jet axial velocity射流轴心速度jet divergence angle射流扩散角jet in a confined space受限射流Kkatathermometer卡他温度计Llaboratory hood排风柜lag of controlled plant调节对象滞后large space enclosure大容积密闭罩latent heat潜热lateral exhaust at the edge of a bath槽边排风罩lateral hoodlength of pipe section侧吸罩length of pipe section管段长度light work轻作业limit deflection极限压缩量limit switch限位开关limiting velocity极限流速linear flow characteristic线性流量特性liquid-level gage液位计liquid receiver贮液器lithium bromide溴化锂lithium-bromide absorption-type refrigerating machine溴化锂吸收式制冷机lithium chloride resistance hygrometer氯化锂电阻湿度计load pattern负荷特性local air conditioning局部区域空气调节local air suppiy system局部送风系统local exhaustventilation (LEV)局部排风local exhaust system局部排风系统local heating局部采暖local relief局部送风local relief system局部送风系统local resistance局部阻力local solartime地方太阳时local ventilation局部通风local izedairsupply for air-heating集中送风采暖local ized air control就地控制loop环路louver百叶窗low-frequencynoise低频噪声low-pressure steam heating低压蒸汽采暖lyophilic dust亲水性粉尘lyophobic dust疏水性粉尘Mmain 总管、干管main duct通风〕总管、〔通风〕干管main pipe总管、干管make-up water pump补给水泵manual control手动控制mass concentration质量浓度maximum allowable concentration (MAC)最高容许浓度maximum coefficient of heat transfer最大传热系数maximum depth of frozen ground最大冻土深度maximum sum of hourly colling load逐时冷负荷综合最大值mean annual temperature (humidity)年平均温（湿）度mean annual temperature (humidity)日平均温（湿）度mean daily temperature (humidity)旬平均温（湿）度mean dekad temperature (humidity)月平均最高温度mean monthly maximum temperature月平均最低温度mean monthly minimum temperature月平均湿（湿）度mean monthly temperature (humidity)平均相对湿度mean relative humidity平均风速emchanical air supply system机械送风系统mechanical and hydraulic联合除尘combined dust removal机械式风速仪mechanical anemometer机械除尘mechanical cleaning off dust机械除尘mechanical dust removal机械排风系统mechanical exhaust system机械通风系统mechanical ventilation机械通风media velocity过滤速度metal radiant panel金属辐射板metal radiant panel heating金属辐射板采暖micromanometer微压计micropunch plate muffler微穿孔板消声器mid-frequency noise中频噪声middle work中作业midfeed system中分式系统minimum fresh air requirmente最小新风量minimum resistance of heat transfer最小传热阻mist雾mixing box section混合段modular air handling unit组合式空气调节机组moist air湿空气moisture excess余湿moisure gain散湿量moisture gain from appliance and equipment设备散湿量moisturegain from occupant人体散湿量motorized valve电动调节阀motorized (pneumatic)电（气）动两通阀-way valvemotorized (pneumatic)-way valve电（气）动三通阀movable support活动支架muffler消声器muffler section消声段multi-operating mode automtic conversion工况自动转换multi-operating mode control system多工况控制系统multiclone多管〔旋风〕除尘器multicyclone多管〔旋风〕除尘器multishell condenser组合式冷凝器Nnatural and mechanical combined ventilation联合通风natural attenuation quantity of noise噪声自然衰减量natural exhaust system自然排风系统natural freguency固有频率natural ventilation自然通风NC-curve[s]噪声评价NC曲线negative freedback负反馈neutral level中和界neutral pressure level中和界neutral zone中和界noise噪声noise control噪声控制noise criter ioncurve(s)噪声评价NC曲线noisc rating number噪声评价NR曲线noise reduction消声non azeotropic mixture refragerant非共沸溶液制冷剂non-commonsection非共同段non condensable gas 不凝性气体non condensable gas purger不凝性气体分离器non-isothermal jet非等温射流nonreturn valve通风〕止回阀normal coldest month止回阀normal coldest month累年最冷月normal coldest -month period累年最冷三个月normal hottest month累年最热月（３）normal hottest month period累年最热三个月normal three summer months累年最热三个月normal three winter months累年最冷三个月normals累年值nozzle outlet air suppluy喷口送风number concentration计数浓度number of degree-day of heating period采暖期度日数Ooctave倍频程/ octave倍频程octave band倍频程oil cooler油冷却器oill-fired unit heater燃油热风器one-and-two pipe combined heating system单双管混合式采暖系统one (single)-pipe circuit (cross-over) heating system单管跨越式采暖系统one(single)-pipe heating system单管采暖系统pne(single)-pipe loop circuit heating system水平单管采暖系统one(single)-pipe seriesloop heating system单管顺序式采暖系统one-third octave band倍频程on-of control双位调节open loop control开环控制open return开式回水open shell and tube condenser立式壳管式冷凝器open tank开式水箱operating pressure工作压力operating range作用半径opposed multiblade damper对开式多叶阀organized air supply有组织进风organized exhaust有组织排风organized natural ventilation有组织自然通风outdoor air design conditions室外空气计算参数outdoor ctitcal air temperature for heating采暖室外临界温度outdoor design dry-bulb temperature for summer air conlitioning夏季空气调节室外计算干球温度outdoor design hourly temperature for summer air conditioning夏季空气调节室外计算逐时温度outdoor design mean daily temperature for summer air conditioning夏季空气调节室外计算日平均温度outdoor design relative humidityu for summer ventilation夏季通风室外计算相对湿度outdoor design relative humidity for winter air conditioning冬季空气调节室外计算相对湿度outdoor design temperature ture for calculated envelope in winter冬季围护结构室外计算温度outdoor design temperature ture for heating采暖室外计算温度outdoor design temperature for summer ventilation夏季通风室外计算温度outdoor design temperature for winter air conditioning冬季空气调节室外计算温度outdoor design temperature for winter vemtilation冬季通风室外计算温度outdoor designwet-bulb temperature for summer air conditioning夏季空气调节室外计算湿球温度outdoor mean air temperature during heating period采暖期室外平均温度outdoor temperature(humidity)室外温（湿）度outlet air velocity出口风速out put variable输出量overall efficiency of separation除尘效率overall heat transmission coefficient传热系数ouvrflow pipe溢流管overheat steam过热蒸汽overlapping averages滑动平均overshoot超调量Ppackaged air conditioner整体式空气调节器。

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a rX iv:c ond-ma t/54767v1[c ond-m at.s upr-con]28Apr25Anisotropic Vortices in High-Temperature Su-perconductors and the Onset of Vortex-like Exci-tations above the Critical Temperature O.Hudak 1Department of Theoretical Physics,Faculty of Mathematics and Physics,Comenius University,Mlynska dolina F2,SK -84501BratislavaAbstractIn our recent papers we have found that a three-dimensional superconduct-ing state with anisotropic vortices localized at the vortex-lattice points is a stable state in zero and nonzero external magneticﬁeld for the layered high-temperature superconductivity materials.There exists a phase transition at the temperature T v c from the normal phase to the vortex superconducting state which is of theﬁrst order.Note that the transition is in zero magneticﬁeld. Theﬁrst order phase transition shows overheating and overcooling eﬀects.Nu-cleation of the superconducting phase in the normal phase thus may occur at temperatures higher than the transition temperature T v c.Then the onset of the vortex-like excitations above the transition temperature T v c occurs in our theory. The onset of the vortex-like excitations in Nerst signal and some other exper-imental evidence for these excitations above the transition temperature T v c in LSCO,YBCO and in other similar high-temperature materials may be explained thus by our theory.The vortex-like excitations above and below the transition temperature T v c in LSCO,YBCO and in other similar high-temperature mate-rials continuously evolve.This fact may be explained within our theory.21IntroductionSine-Gordon vortices may exist in superconductors[1]and[2].These vortices may be present in superconducting states also in zero magneticﬁelds.Their topological charge is preserved.We studied a magnetic-ﬁeld-induced supercon-ductive state in heavy fermion systems[3]and[4].The strength of the coupling between the order parameter and magneticﬁeld drives a superconductive state induced by a magneticﬁeld,[3]and[4].Thus such a state may be induced increasing the magneticﬁeld.Experimentally it was reported that in CeP b3 there exists a ferromagnetic phase and simultaneously there is an evidence for presence of superconductivity.We studied the Kondo-lattice superconductivity in this connection,[4]and[5].Superconducting states of diﬀerent symmetry s, p and d were found to exist in the Kondo-lattice systems.A phase transition from the normal phase to the superconducting phase in CeP b3type materials occurs in our phenomenological model through creation of a vortex lattice,in which vortices are superconducting regions induced by an external magnetic ﬁeld.Increasing the magneticﬁeld the density of such vortices and their shape becomes larger and the superconducting phase increases its volume in the ma-terial.In high-temperature superconductors,which are layered materials,the interlayer distance is larger than the in-layer lattice constant.The interlayer distance is comparable with the in-layer penetration length.Our results[3]-[5] lead to study of formation of a vortex in the plane,and to study of interaction of such vortices in neighbouring layers.In the presence of an in-layer vortex in-plane and out-of-plane magneticﬁeld structures exist due to currents which are present.The vortex state which we described for a plane superconductor in 1982may exist in zero magneticﬁeld[6].This fact is very important diﬀerence from usual vortices studied in superconductors,which are present due to an ex-ternal magneticﬁeld.The stability of the vortex in zero magneticﬁeld[6]is due to the topological charge of this vortex which is preserved.The magneticﬁeld generated by this stable vortex is inﬂuencing neighbouring planes and the plane in which it is localized.Thisﬁeld is coupled with the superconducting order parameter due to symmetry.As we discussed above the strength of the coupling between the magneticﬁeld and the order parameter may drive a superconduc-tive state with vortices induced by the magneticﬁeld of this vortex,[3]and[4]. Thus in layered high-temperature superconductors coupling between the mag-neticﬁeld generated by this stable vortex and the order parameter may lead to generation of other vortices(antivortices).This is a selfconsistent process.The ﬁeld of these vortices inﬂuences this stable vortex.Thus then we may study vortices and their in-plane and out-of-plane interactions due to the inductance coupling.Thus the strength of the coupling between the order parameter and magneticﬁeld drives a superconductive state induced by a magneticﬁeld as mentioned above,[3]and[4],and in layered high-temperature superconductors drives a superconducting state with vortices.Note that also in the zero external magneticﬁeld such a selfconsistent process may exists.Creation of a vortex in one plane leads to magneticﬁelds which inﬂuence the neighbouring planes,in these neighbouring planes theseﬁelds may induce creation of vortices the mag-3neticﬁelds of which inﬂuence the original vortex in the plane:its existence and properties.Creation of a vortex in the plane leads also to magneticﬁelds which inﬂuence the same plane,in this plane theseﬁelds again may induce creation of vortices the magneticﬁelds of which inﬂuence the original vortex in the same plane:again its existence and properties.Thus it follows that in layered materials one may expect that formation of such vortices may stabilize the superconducting state of diﬀerent from classi-cal symmetry in zero external magneticﬁeld.Note that in nonzero external magneticﬁelds the self-consistent process may be modiﬁed but still present.The vortex in the superconductor plane which we studied[6]is anisotropic vortex.This anisotropy(symmetry diﬀerent from symmetry of isotropic vor-tices)then leads to coupling to other quantities of the system for which an invariant contributing to free energy expansion may exist.Integrating out these other quantities from the free energy of the superconducting material we ob-tain the eﬀective free energy expansion for the superconducting order parameter with anisotropy terms present.Recently we described and studied the mentioned mechanism and formu-lated a general problem:to study behaviour of such anisotropic vortices in zero and non-zero magneticﬁelds.The vortices of the in-and inter-layer type should be compared as concerning their theoretical properties in zero and non-zeroﬁelds with existing experiments concerning study of vortices in high-temperature superconductors,[7],[8],[9]and[10].It was found experimentally that the anisotropy of the interplane-vortices in these materials is intrinsic. This is natural,the vortex is parallel to layers,layers are anisotropic.Planes are involved in such vortex conﬁguration,we are speaking about the inter-plane vortex induced by an external magneticﬁeld.As far as we know the anisotropy of such vortices is interpreted as due to anisotropy of layers.The vortices which we were studied theoretically are intrinsically anisotropic. However this anisotropy is present for vortices which are present within an isotropic plane.Their anisotropy is not generated due to anisotropy of the material.Thus it is important to compare experimental results concerning anisotropy properties of the vortices in the layered materials with those proper-ties of vortices which we found theoretically.This is the reason why we reviewe diﬀerent superconducting states with anisotropic vortices:the superconduct-ing states in which single vortex exists,the superconducting states in which lines and lattices of vortices exist,the superconducting states in which pairs of vortex-antivortex type are present and form a kind of liquid.At larger distances between vortex-antivortex in the pair the energy is linearly dependent on this distance.This is a kind of conﬁnement phenomena known from quarks forming baryons.For smaller distances there exists a minimum distance for which the vortex-antivortex pair exist in a ground state.The pair vortex-antivortex thus does not annihilate.Further we considered in our papers the superconducting states in which solitons and strings of vortices,which represent some kind of structures induced by internal mechanisms/twinning of the crystals,etc./or by surface,may exist.Intrinsic Josephson eﬀect in layered systems was described in[11]and a sum-4mary of the Josephson eﬀect in the high-temperature superconductors is given in[12].The time dependent equations for the gauge invariant phase diﬀerences of the layered order parameters for superconductive states and for a general vector potential of the electromagneticﬁeld we discussed in[1]and[2].The interlayer coupling within the mentioned model equations was studied with its inﬂuence on the properties of vortices in layers.These vortices in neighbouring layers are coupled.They are forming a string of vortices perpendicular to the layers in which the vortices exist.The string goes through the centers of these vortices.We used in our study a small parameter expansion method,and an exact method of solution of the corresponding Lagrange-Euler equations to compare diﬀerent macroscopic superconducting states.We have found their free energy and discussed their properties.The corresponding Lagrange-Euler equations are nonlinear coupled sine-Gordon equations in the approximation of the constant amplitude of the order parameter.Let as note that the nonlinearity describing the vortex conﬁgurations in nonzero/and in zero/magneticﬁeld is neglected due to diﬃculties of the mathematical origin and mainly due to expectation that the nonlinearity is negligible around the core of the vortex,which is assumed to be large.However we have found that nonlinearities in order parameter in-teractions are physically essential,[6].The Lagrange-Euler equations for the superconducting state order parameter and for the vector potential of the elec-tromagneticﬁeld in the constant amplitude approximation for the scalar(s-type)superconducting order parameter reduced in the model studied in our papers cited above to equations which are similar to the two-dimensional sine-Gordon equation.They are generalized coupled two-dimensional sine-Gordon equation.p and d symmetric superconducting states were not studied until now.The two-dimensional sine-Gordon equations occur in other physical systems, in quantum antiferromagnets,in classical two-dimensional XY model with the in-plane magneticﬁeld[13]-[14],in models describing crystal growth[15]and [16],and in models describing defects in incommensurate systems[17]and[18]. It is useful to use analogy and mathematical properties of solutions of these equations for study of anisotropic vortices in superconductors.In our paper from1982we used a new method for solving a two-dimensional sine-Gordon equation[6].We transformedg this equation into a set of two second order ordinary nonlinear non-coupled equations.Multi-(Resonant-Soliton)so-lutions and vortex-like solutions of the two-and three-dimensional sine-Gordon equations were studied in[19].The structure of a vortex in terms of the solitons was studied in[20].The vortex is formed from solitons which are intersecting. The multi-vortex solutions of the sine-Gordon equation were studied in[21]. Exact rational-exponential solutions of this equation were studied in[22].Vor-tices and the boundary value problem were studied by the inverse scattering transform method for this type of equation in[23].The singular solutions of the elliptic sine-Gordon equation were studied numerically in[24].Numerical studies of dynamical isoperimeter pattern are in[25].Vortex-like excitations in La2−x Sr x CuO4at temperatures signiﬁcantly above5the critical temperature were found in the Nerst eﬀect signal[26].In overdoped La2−x Sr x CuO4(LSCO)the upper criticalﬁeld H c2does not end at T c0but at a much higher temperature.These results imply,according to[27],to a loss in phase rigidity rather than in a vanishing of the pairing amplitude.Nerst measurements in Y Ba2Cu3O4(YBCO)and in Bi2Sr2−y La y CuO6,[28],and in LSCO inﬁelds up to33T show the existence of vortex-like excitations high above T c0.This anomaly is related to the key question of whether the Meissner transition in zeroﬁeld is caused by the collapse of long-range phase coherence or by the vanishing of the pairing amplitude.In zero magneticﬁeld there is equal number of vortices with plus and minus topological charge.In nonzeroﬁelds there is nonequal number of vortices with plus and minus topological charge.Preformed pairs as superconductorﬂuctuations may exist in Bose conden-sation of localized Cooper pairs with short coherence length,[29],[30]and[31]. Pairing correlations without phase coherence are described in[32]and[33].In our papers[1]and[2].new results were found as concerning the inﬂu-ence of neighbouring planes on the vortex state in a given plane and new ex-act solutions of the coupled Lagrange-Euler sine-Gordon equations were found, their properties and free energy were compared.We have found that the three-dimensional superconducting state with vortex lattice,which is a stable state in zero external magneticﬁeld for the layered materials.There exists a phase transition at the temperature T v c from the normal phase to the vortex super-conducting state which is of theﬁrst order.Depending on the parameters of the material the transition temperature T v c may be higher than zhat in the case of the second order phase transition from normal phase to the homogeneous superconducting phase without vortices.The transition temperature T v c would be zero for zero topological charge of vortices.Note that the transition to vortex lattice state is present also in zero magneticﬁeld.Theﬁrst order phase transition shows in experiments usually overheating and overcooling eﬀects.Nucleation of the superconducting phase in the normal phase occurs at temperatures higher than the transition temperature T v c.And vice-versa.Nucleation of the nor-mal phase in the superconducting phase occurs at temperatures lower than the transition temperature T v c.Thus the onset of the vortex-like excitations above the transition temperature T v c occurs in our theory due to nucleation of the su-perconducting phase with vortex lattice(or parts of this lattice)in the normal phase at temperatures higher than the transition temperature T v c.The onset of the vortex-like Nerst signal above the transition temperature T v c in LSCO, YBCO and in other similar high-temperature materials may be thus explained by our theory.In this paper we review our results on vortex conﬁgurations in two-dimensional sine-Gordon systems in connection with high-temperature superconductors.We will use this results in our discussion concerning anisotropic vortices and the onset of Nernst signal and of some other signals corresponding to presence of vortex-like excitations above the critical temperature.The superconducting planes are coupled by the Josephson eﬀect.The Lawrence-Doniach functional leads to the Lagrange-Euler equations for the order parameter.In[1]and[2]. We discussed the vortex states using exact solutions of these coupled Lagrange-6Euler sine-Gordon equations.Properties of the exact solutions and the free en-ergy of the corresponding order parameter conﬁgurations weres compared with the normal phase state and the homogeneous superconducting state.Stable vor-tex conﬁgurations in zero external magneticﬁeld for the layered materials exist, the same holds for a lattice of vortices case.Theﬁrst order phase transition overheating and overcooling eﬀects are present here.Nucleation of the supercon-ducting phase in the normal phase,which occurs at temperatures higher than the transition temperature T v c,exists.The onset of the vortex-like excitations above the transition temperature T v c,which occurs in our theory,is discussed. The onset of the vortex-like Nerst signal above the transition temperature T v cin LSCO,YBCO and in other similar high-temperature materials is not well understood in experiments.The vortex-like excitations above the transition temperature T v c in LSCO,YBCO and in other similar high-temperature ma-terials continuously evolve into vortex states below the transition temperatureT v c,this fact may be explained within our theory.Our theory may explain this fact and some of other experimentally observed properties of high-temperature superconductors described above.2Superconducting Planes Coupled by the Joseph-son EﬀectSuperconducting layers in layered superconductors of the high-temperature type are coupled due to intrinsic Josephson eﬀect.The Lawrence-Doniach free energy functional for the order parameter for such a system has the form[1]F(Ψn(r),A(R))=H2csΦ0A n)Ψn)|2−(1)−|Ψn)|2+18πThe order parameter in the n-th layer isΨn(r)=|Ψn(r)|exp(iΦn](r)),here z=ns,and s is an interlayer distance,R=(r,z),∇=∂|e|,A n(r)=(A nx,A ny)is an average value of the vector potential over the distance[(n−12)s along the c-axis,B=rot(A), andχn,n+1=2πζ2c=s2r8π2H2cζ2ab(3) The anisotropy ratioλabλc =ζabλ2Jsin(φn,n+1)=0(5)where the summ over m is over layers.The interlayer inductance L n,m between the layers n and m has the form L n,m= 2π0dq2(1−cos(q))+s2s(1−s4The Distance s is Comparable with the Pene-tration DepthλabWhen the distance s is comparable with the penetration depthλab as it is the case in high-temperature superconductors the mutual interlayer inductance L n,n(which is in this case when m=n the inlayer inductance in fact)is nonzero and much larger than the other inductances(interlayer,because m and n are diﬀerent)L n,mλabL n,n=sin(φn,n+1)=0(9)λ2Jwhere the lower order terms in L n,msin(φ)(12)λ2pThis form of the two-dimensional sine-Gordon equation was studied by the author in[6].5The Distance s is Comparable with the Pene-tration Depthλab-Inﬂuence of Next Layers In the previous section we described lagrange-Euler equations for the case when the distance s is comparable with the penetration depthλab.We neglected the inﬂuence of neighbouring layers.Let us consider now the eﬀect of nearest neighbour layers.The interlayer inductance for neighbouring layers in this case L n,n±1is9L n,n±1=λabλab)(13)The equation for the phase diﬀerencesφm,m+1has the form−L n,n∇2φn,n+1−L n,n+1∇2φn+1,n+2−L n−1,n∇2φn−1,n+1L n,n(15) The two-dimensional sine-Gordon equation(14)takes now the form−∇2φn,n+1−l∇2φn+1,n+2−l∇2φn−1,n+1λab)1(17) We left the exponent1in order to indicate how the next to the nearest layers will inﬂuence the Lagrange-Euler equations.In this form of the two-dimensional sine-Gordon equation three neighbouring planes are coupled explicitly.The solution of the equation(16)is again looking in the homogeneous form,in which we again introduce the variableφ=φn,n+1(18) for every n.This again may be done due to expected homogeneity in the sense mentioned above as concerning the states in planes in this limit.The equation(16)takes now the form−∇2φ+11+2l(20) which is larger thanλp due to the fact that l is positive.Note that the larger l the larger renormalised Josephson length.This homogeneous solution considered in this section has the characteristic lengthλ′p larger than in the one-layer case lengthλp due to the inter-layer coupling via interlayer inductance.The free energy of the single vortex in every layer and for the same homoge-neous state(homogeneous as concerning dependence on the the number n of the layer,the vortex state is oﬀcourse nonhomogeneous state)in all layers depends linearly on L,the system linear dimension1023λ′p−2k B T N ln(Lλ′′2psin(φ)=0(23) where the Josephson characteristic length has now the formλ′′p=λJ√2NπJ22La0)(25)where again N is the number of layers,the cutoﬀconstant a0is a constant of the order of the lattice constant.If the inductance coupling l between layers is taken into account then we have found that the free energy of the state in which the sign of the angleφchanges in neighbouring layers is higher with respect to the state in which the sign of the angleφdoes not change in neighbouring layers.The free energy of the state in which the sign of the angleφchanges in neighbouring layers is also higher with respect to the state in which the layers are not coupled at all.The small parameter l may be taken as an expansion parameter when solving the equation(16).Oneﬁnds in that case that the homogeneous in above sense11state with a vortex in one layer and with other vortices in other layers with their core center in the same point in the plane ab/a line of vortices in the direction c/is more stable.Single-vortex state in one layer and the same state in all layers give lower free energy due to the interlayer inductance coupling.The interlayer inductance coupling plays stabilisation role for these vortices.One can ask now that if many-vortex states in planes are realized,which distance between vortices is such that the vortex superconducting state becomes more preferable than the normal state,or than the homogeneous superconducting state without vortices(usual superconducting state in classical superconductors).The free energy of the single vortex in a given layer and the same(homogeneous case as concerning the signs of the phase variable)state with a vortex in neighbouring layers depends linearly on L,the system linear dimension and the(critical)value of the parameter l.At a given temperature it is given by the condition that the free energy with the vortex at the same place in every layer is the same as the free energy of the superconducting state without vortexF(vortex)=23λ′′p−2k B T N ln(Lβ.The Lawrence-Doniachfunctional for the order parameter withΦconstant givesF(Φhom)=−H2c sL2Nα24πβ(28)Here we consider the constant amplitude approximation|Ψ|2=|α|λas a contribution to the bulk critical temperatureT c.Note that the case with non-zero magneticﬁeld may be easily described and will not be considered here.12The normal state is given byΨ=0and its free energy is0.The super-conducting state with the homogeneous(in the classical sense)superconducting order parameter without vortices is given by|Ψ|2=|α|8π=−JNL2α22πJq2L va0)](30)The inter-vortex distance L v=L2λ′p k B T2λ′p k B T√2λ′p k B T λ′2p k B T−π2J2q42λ′p k B T2βζ2ab(34)As can be seen the transition at the transition temperature T cv from the normal phase to the N v-vortex superconducting state occurs,it is of theﬁrst order.Depending on the parameters of the material the transition temperature T cv may be higher for this transition than the transition temperature in the case of13the second order phase transition from the normal phase to the homogeneous superconducting phase without vortices.It is given byT cv =πJq 2a 02λ′p k B (35)Note that the transition temperature T cv would be equal zero for zero topo-logical charge q.Note further that the transition temperature T cv increases when the interlayer inductance increases.We will now discuss several solutions of the two-dimensional Sine-Gordon equations.6Single-Vortex ConﬁgurationsIn[6]we have found the single vortex conﬁguration as a solution of the two-dimensional sine-Gordon equation,and thus of the order parameter.It has the formφ=±tan −1( (2−a )sinh( a (x −x 0)sinh( 2(y −y 0)(y −y 0))(37)In isotropic material systems in which anisotropic vortices are present and which are large enough,e.i.in which λ<<L ,the anisotropy of the vortex will be detectable.7Multi-Vortex ConﬁgurationsMulti -(Resonant-Soliton)-Soliton solutions and vortex-like solutions in two and three dimension for the sine-Gordon equation were studied in [19].For the two-and three-dimensional sine-Gordon equations there exist exact multi -(resonant-soliton)-soliton solutions and vortex-like solutions,in addition to exact multi-soliton and resonant -soliton solutions.In [20]it was shown that14the quasi-vortex type solutions from[6]can be derived from the multiple soliton solutions by the proper procedure.Thus it follows from that there exist multiple vortex-like solutions.Moreover an anisotropic vortex conﬁguration far from the core of the vortex has the form of almost unchanging order parameter regions separated by soliton-like walls.8Static Lattice of Vortices,Multi-vortex Solu-tionsLet us brieﬂy overview solutions of the sine-Gordon equations which represent static lattice of vortices and multi-vortex solutions.These solutions and space periodic vortex chains and lattice and breather solutions were studied in[21]. The boundary problem for two-dimensional sine-Gordon equation was studied by the inverse scattering method in[34].Relation between vortex solutions of the two-dimensional sine-Gordon equation and solitons of the same equations were studied in[20].Single vortex solution[6]can be derived from the known multiple soliton solutions by a proper procedure.This fact shows possibility to ﬁnd multi-vortex solutions of the two-dimensional sine-Gordon equations due to existence of multisoliton solutions.Vortex-type lattice solutions found by Hirota method and by Backlund trans-formation represent alternating vortices with q=±4and form a tetragonal or square lattice.Interaction of the spin-wave and vortex lattice are similar to the soliton-wave interactions.Vortex-antivortex pair with the vorticity q=±4is given by1−x+sinh(y)exp(−x)φ=tan−1(analysis shows that the minimum distance between centers of vortices with op-posite topological charges in dipole state exists and is given by approximately 1.1in lambda units.These two vortices cannot annihilate in spite of zero total topological charge of the dipole.9Dynamic Properties of Two-Dimensional Sine-Gordon VorticesThe two-dimensional sine-Gordon equation with time dependence was studied in[25],and it was found that a particular dynamical pattern in a perturbed two -dimensional sine-Gordon systems exists.The stability of vortex-like solutions were checked.The main topological invariant of the vortex-like solutions is due to the total length of the±2πwave fronts entering as elementary kink-like patterns in the constitution of the whole conﬁguration.This corresponds to the energy conservation law.In[25]it was also shown that a particular dynamical pattern in a perturbed or not two-dimensional sine-Gordon system exists.Its stability with respect to perturbations and in time was studied.Their main topological invariant is the total length of the kink wave fronts forming the pattern and the invariant is consistent with the conservation of energy in all situations.In[19]the time dependent sine-Gordon equation without damping was stud-ied.The procedure like that used in[20]leads to a two-soliton solution,thus intersection of these solitons corresponds with a moving vortex.On the other hand the author[19]shows that three or higher number soliton solutions are known to exist only with limited parameter range.Thus time dependence with-out damping here does not lead to a vortex-lattice in general,this may exist only in a limited range of parameters.10Summary and DiscussionIn our papers[1]and[2]we studied layered superconducting materials.Vortex-like states were described and compard with normal metalic phase and supercon-ducting state without vortices.Results were found as concerning the inﬂuence of neighbouring planes on the vortex state in a given plane and new exact so-lutions of the coupled Lagrange-Euler sine-Gordon equations were found,their properties and free energy were compared.Three-dimensional superconducting state with a vortex in every plane with cores localized in the same point on a line perpendicular to planes is a stable state in zero external magneticﬁeld for the layered materials.The same holds for a lattice of vortices case.There exists a phase transition at the temperature T v c between the normal phase and the vortex superconducting state which is of theﬁrst order.Depending on the parameters of the material the transition temperature T v c may be higher than in the case of the second order phase transition from normal phase to the homogeneous(conventional)superconducting phase without vortices.The16。