J.Sep.Sci.2009,32,757–770757
Kanji Miyabe
Graduate School of Science and Engineering for Research, University of Toyama,Gofuku, Toyama,Japan Review
Moment analysis of chromatographic behavior in reversed-phase liquid chromatography
Chromatographic behavior,such as sample retention,band broadening,elution peak profile,column efficiency,separation performance,and so on,depends on both the retention equilibrium and mass transfer kinetics in the column.A great number of research works have been carried out so far on the retention equilibrium in chromatography.However,chromatographic behavior has not so abundantly been investigated from kinetic points of view because of some essential difficulties of kinetic study.Additionally,in contrast with the extensive applications of various HPLC instruments and separation media developed in recent decades,it seems that arrangements of theoretical bases and experimental strategies relating to the kinetic study are not well recognized by most chromatographers.In this review, some items of information about the moment analysis method and concrete exam-ples about mass transfer kinetics derived by the method are provided.The progress of some strategies for the kinetic study on chromatography is also introduced. Keywords:General rate model/Mass transfer kinetics/Moment analysis/RPLC/Surface diffu-sion
Received:October27,2008;revised:January6,2009;accepted:January6,2009
DOI10.1002/jssc.200800607
1Introduction
Chromatography is one of essential and powerful tools for fine separations in various fields of science[1,2].The rapid progress of chromatographic technologies depends on the pursuit of fundamental studies on the separation mechanism and on the development of some apparatuses,e.g.high pressure pumps and high effi-ciency columns.Chromatographic behavior has so far been mainly investigated from the viewpoint of the retention equilibrium because chromatography primar-ily rests on the phase equilibrium thermodynamics. Compared with the ample research work on the reten-tion behavior,the number of kinetic studies on chroma-tography has not been sufficient.Mass transfer phenom-ena taking place in HPLC columns should be quantita-tively studied in more detail although various related results about the kinetic behavior of chromatography have been obtained by means of different types of experi-mental approaches,i.e.(i)spectroscopic methods,such as the fluorescence relaxation[3–14]and the nuclear mag-netic resonance[15–21],and(ii)column operations, such as the frontal analysis[22–32],the shallow bed method[33–37],and the pulse on a plateau method[38–40].
In the community of chromatography,it has been the most conventional approach for the kinetic study on chromatography to analyze the flow rate dependence of height equivalent to a theoretical plate(HETP)(H).Some ordinary rate equations,such as the van Deemter equa-tion and the Knox equation,have extensively been used to account for the flow rate dependence of HETP[1,2, 41–47].It has also been well recognized that the band broadening(hence,the value of H)depends on the contri-butions of several mass transfer processes in the column. For example,as illustrated in Fig.1,the following four kinetic processes are considered in the general rate(GR) model of chromatography[2,48,49],i.e.(i)axial mixing in the bulk mobile phase percolating between separation media(axial dispersion),(ii)the external mass transfer of sample molecules between bulk mobile phase and the external surface of stationary phase(fluid-to-stationary phase mass transfer),(iii)the diffusive migration through pores inside packing materials(intra-stationary
Correspondence:Professor Kanji Miyabe,Graduate School of Science and Engineering for Research,University of Toyama, 3190,Gofuku,Toyama930-8555,Japan
E-mail:miyabe@eng.u-toyama.ac.jp
Fax:+81-76-445-6835
Abbreviations:GMS,generalized Maxwell-Stefan;GR,general rate;HETP,height equivalent to a theoretical plate;MA,mo-ment analysis;ML,modeling-Laplace;MN,modeling-numerical; POR,lumped pore diffusion;PP,peak parking;PR,pulse re-sponse
758K.Miyabe J.Sep.Sci.2009,32,757–770
phase diffusion),and(iv)the adsorption/desorption proc-esses at actual adsorption sites on the stationary phase surface(adsorption/desorption kinetics).It is usually assumed that the contribution of the fourth process to band broadening is negligibly small in RPLC because the real rate of the adsorption/desorption kinetics is pre-dicted to be fast enough in the case of physisorption[49]. The flow rate dependence of HETP must provide some items of significant information about the mass transfer kinetics in the column and in the stationary phase.How-ever,it is relatively hard to extract the important infor-mation about the mass transfer kinetics from the curved profile of the correlation between H and the mobile phase flow velocity.There would be the following three barriers,which prevent the progress of kinetic studies on chromatography.
At first,the kinetic study requires in principle far more experimental data than the study on the retention equilibrium.Although the latter needs the information about only first absolute moment(l1)of a peak experi-mentally measured,the former needs the information about both l1and the second central moment(l29).Addi-tionally,the whole profile of the curved correlation rep-resenting the flow rate dependence of HETP is essential to accurately determine some related kinetic param-eters.We should experimentally measure elution peaks as many times as possible for the accurate kinetic study. This situation is quite different from that of the study on the retention equilibrium.We could go so far as to say that an equilibrium parameter,such as retention equili-brium constant(K a)and retention factor(k),can be obtained from only one datum of elution peak. Second,it is more difficult to accurately measure the value of l29than that of l1.This means that errors made on the measurement of H are larger than those for K a and k.It is necessary to integrate an elution peak profile in a range of time in order to accurately determine l1and l29. However,the integration time range required for the determination of l29is wider than that of l1.The diffi-culty in the measurement of accurate values of l29leads to the error concerning the second moment analysis.The accurate measurement of the correlation between H and the mobile phase flow velocity is originally more diffi-cult than the determination of K a and k.
Third,although the conventional rate equations are quite popular[1,2,41–47],they are not sufficient for deriving the important information about the mass transfer kinetics in chromatography from the curved profile of the flow rate dependence of HETP because they are empirical and contain several fitting parameters,of which the physical definition and meanings are not nec-essarily clear.Unfortunately,they do not provide suffi-ciently quantitative information about some kinetic parameters relating to the mass transfer processes in the column.
On the other hand,there are other strategies for ana-lyzing chromatographic behavior.The moment analysis (MA)method is one of them,which is based on the GR model of chromatography[2,38–40,48–59].It is effec-tive for deducing some items of important information about the retention equilibrium and mass transfer kinetics in the column from the elution peak profiles experimentally measured.Similar to the conventional rate equations,it is also assumed in the MA theory that HETP consists of the contributions of several mass trans-fer processes in the column.In practice,the flow rate dependence of HETP is similarly analyzed in order to derive the kinetic information.On the other hand,there are some differences between the MA theory and the ordinary rate ones.The moment equations are derived on the basis of the GR model of chromatography.The physical meanings of all the parameters included in the moment equations are distinct because they are clearly defined.Quantitative information about the mass trans-fer kinetics in the column and in the stationary phase can be obtained with physically sound justification.This
Figure1.Schematic illustration of the mass transfer proc-esses in the column.Reproduced from[39],with permission from Wiley-VCH Verlag GmbH&Co.KGaA.
J.Sep.Sci.2009,32,757–770Liquid Chromatography759 is the most significant difference between the method of
moments and the ordinary rate analysis.
However,it is probably true that the MA method is not so familiar with most chromatographers.It is not recog-nized that the MA method is an effective approach for the kinetic study of chromatographic behavior.It would be difficult to find the detailed information about the fundamentals of the MA method and the practical proce-dure of moment data analysis in the conventional litera-ture and textbooks of chromatography.The main goal of this review is(i)to provide a brief explanation about a framework of the MA method in Section2,(ii)to demon-strate in Section3what kind of new information can be obtained by the MA of chromatographic peaks experi-mentally measured,and(iii)to introduce recent progress of some strategies for the kinetic study on chromatogra-phy in Section4.
There are semantic discussions about the retention mechanism of chromatography,for example,“partition”or“adsorption”in RPLC.However,in this review,it is regarded as adsorption phenomena in a wide sense that sample molecules migrate between the bulk mobile phase and the stationary phase surface and that the sam-ple molecules are consequently concentrated on the sur-face.It is not intended to discuss the retention mecha-nism.This review is concerned with some quantitative studies on the kinetic aspects of chromatography.
2MA theory
The MA method is one of effective strategies for quantita-tively analyzing chromatographic https://www.doczj.com/doc/cd14971079.html,rmation about the retention(adsorption)equilibrium and mass transfer kinetics is derived from the first absolute moment(l1)and second central moment(l29)of elution peaks,respectively.The MA approach has also been intro-duced in some previous literature[2,38–40,48,49].For the sake of readers,basic information about the frame-work of the MA method is briefly explained in the follow-ing.
2.1Moment equations
The moment equations are mathematically derived from partially differential equations of the GR model of chro-matography,which represent the material balance and the mass transfer rates in the column and in the station-ary phase[2,38–40,48,49,54–56].The moment equa-tions are represented as follows when full-porous spheri-cal particles are used as the stationary phase.At first,the position of an elution peak is correlated with the reten-tion equilibrium constant(K a)(Eq.1),l1?
Z1
CeetTtdt
Z1
CeetTdt
?
L
u0
e ete1àe eTe ite1àe iTK a
f g
? e1T
where Ce(t)is the concentration profile of the sample compound at the exit of the column as a function of t,t the time,L the length of the column,u0the superficial velocity of the mobile phase solvent,e e the void fraction of the column(external porosity),and e i the porosity of the stationary phase(internal porosity).Equation(1) appears to be somewhat complicated.However,when the elution peak is symmetric,it indicates the same cor-relation as the following equation,which is well known in the community of chromatography(Eq.2).
t R?t0e1tkTe2Twhere t R is the retention time of the sample compound,t0 the elution time of an inert tracer,and k the retention factor.When the elution peak has a symmetrical profile, l1is equal to t R.
On the other hand,l92is represented as follows(Eqs.3–8).
l92?
Z1
CeetTetàl1T2dt
Z1
CeetTdt
?
2L
ed axtd ftd dtd adsTe3Td0?e ete1àe eTe ite1àe iTK a
f ge4T
d ax?
e e D L
u0
d2
e5T
d f?e1à
e eT
R p
3k f
e ite1àe iTK a
f g2e6Td d?e1àe eTe
R2p
15D e
Te ite1àe iTK a
f g2e7Td ads?e1àe eTe1àe iT
K2a
a
e8Twhere D L is the axial dispersion coefficient,k f the external mass transfer coefficient,D e the intraparticle diffusivity, k a the adsorption rate coefficient,R p the radius of station-ary phase particle and d the contribution of each mass transfer process to l92.The subscripts ax,f,d,and ads denote the axial dispersion,external mass transfer,intra-particle diffusion,and adsorption/desorption kinetics, respectively.Similar to l1,l92is equal to a well known chromatographic parameter,i.e.variance(r2)of elution peak.Equations(3)–(8)indicate that the band broaden-ing originates from the contributions of the several mass
760K.Miyabe J.Sep.Sci.2009,32,757–770 transfer processes in the column.However,it seems that
the contribution of the adsorption/desorption kinetics to
the total mass transfer resistance is usually assumed to
be negligibly small in the case of RPLC because the reac-
tion rate of physical adsorption is fast enough[49].
2.2MA procedure
2.2.1First MA
The following Eqs.(9)and(10)are derived from Eqs.(1)
and(2).
l1àt0
e ?
L
e1àe iTK ae9T
t0?L
u0
f e ete1àe eTe i ge10T
where t0is the hold-up time,which relates to the total porosity(e t)of the column and is usually measured by injecting an inert tracer,i.e.uracil and thiourea in RPLC [60–62].A plot of the left hand side of Eq.(9)versus L/u0 should be a straight line passing through the origin of the coordinates.The value of K a is derived from the slope of the straight line.In the MA method,the pulse response (PR)experiments are carried out while changing the mobile phase flow velocity because the flow rate depend-ence of HETP is analyzed in order to extract some kinetic parameters from l29in the second MA as explained in Sec-tion2.2.2.When the linear correlation between(l1–t0)/ (1–e e)and L/u0passing through the origin is observed,it seems that the temperature conditions are homogene-ously controlled in the column.The residence time of the sample band is different in the column when the mobile phase flow rate is changed.
2.2.2Second MA
Similar to the conventional kinetic studies based on the ordinary rate equations,the flow rate dependence of HETP is analyzed in the MA procedure.The value of H is calculated from Eqs.(1)–(8)as follows.The value of HETP is expressed as H total because it consists of the contribu-tions of several mass transfer processes in the column (Eq.11).
H total?l29
l21
L?
2e e D e
u0
t
2u0
d20
d ft
2u0
d20
d d
?H axtH ftH de11TAs shown in Eq.(11),the contribution of the axial dis-persion(H ax)can be separated from those of both the external mass transfer(H f)and the diffusive migration of sample molecules inside the stationary phase(H d)by tak-ing advantage of the difference in their flow rate depend-ence.The following correlation(Eq.12)is derived from Eq.(11).H totalàH f?H axtH d?
2e e D L
u0
t
2u0
d20
d de12T
The axial dispersion coefficient(D L)of the conventional HPLC columns packed with spherical particles is usually accounted for by assuming that axial dispersion consists of two main mechanisms,i.e.molecular diffusion and eddy diffusion(Eq.13)[2].
D L?c1D mtc2d p u0e13Twhere c1and c2are the two geometrical coefficients,d p the particle diameter,and u the interstitial velocity of the mobile phase solvent(=u0/e e).Equation(12)is modi-fied as follows by substituting Eq.(13)into Eq.(12)(Eq.
14).
H totalàH f?
2e e c1D m
u0
t2c2d pt
2u0
d2
d d
?
B
u0
tAtCu0e14TEquation(14)indicates that the correlation between (H total–H f)and u0is represented by the same formula as the van Deemter equation with three coefficients,i.e.A, B,and C'.However,Eq.(14)is essentially different from the conventional rate equation in terms of the following two points.At first,the contribution of the external mass transfer to H total is also taken into account in Eq.(14).Sec-ond,the definition and physical meanings of the param-eter C in Eq.(14)are clearly explained.As shown in Eqs.
(7)and(14),quantitative information about the mass transfer kinetics in the stationary phase can strictly be obtained as D e from the coefficient C.These are obvious differences between the ordinary rate equations and the moment equations.
As illustrated in Fig.1,the diffusive molecular migra-tion in pores is usually assumed to consist of the parallel contributions of pore diffusion and surface diffusion[48,
49].The value of D e is accounted for as follows(Eq.15).
D e?D pte1àe iTK a D se15TSurface diffusion coefficient(D s)is calculated by sub-tracting the contribution of pore diffusion to D e.Pore dif-fusivity(D p)is estimated from molecular diffusivity(D m) and some related parameters[2,48,49].Detailed infor-mation about these parameters can be found out in other literature[38–40,48,49,63].The second term in the right hand side of Eq.(15)represents the contribution of surface diffusion to intraparticle diffusion.Another expression for the contribution was also proposed[64,
65](Eq.16).
D e?D pt
2ae p c p
p
KD se16T
where r p is the average pore diameter,e p the internal porosity,and c p the particle obstructive factor.The
J.Sep.Sci.2009,32,757–770Liquid Chromatography761 parameter a is the length factor,which represents the
average total thickness of the hydrophobic layer and the
neat silica.Its value was reported as2.41nm for a C18-
silica gel.Gritti and Guiochon[64]used Eq.(16)to derive
D s from D e after subtracting the contribution of pore dif-
fusion to the intraparticle mass transfer.Although Eq.
(16)resembles Eq.(15),the second terms in the right
hand side are different between the two equations.Gritti
and Guiochon explain that the second term in Eq.(16)
expresses the surface diffusion flux inside the particle as
the number of sample molecules that cross a unit length
(along the surface of the pores)per unit time because sur-
face diffusion is restricted onto the surface of the bonded
silica[64].They reported that the values of D s are meas-
ured in the range of the order of10–6to10–5cm2N s–1
and independent of the modification concentration of
alkyl ligands on the stationary phase surface[64].
3Results of MA of chromatographic
behavior
There are strong demands for the development of fast
separation techniques with high efficiency.It is impor-
tant to acquire accurate and quantitative information
about the chromatographic behavior taking place in the
column and in the stationary phase in order to design appropriate separation systems and packing materials. The MA method is one of suitable strategies for the pur-pose.In the following,it is indicated how the informa-tion about the chromatographic behavior is derived by the MA procedure.In this review,chromatographic behavior of a C18-silica monolith is taken as an example because monolithic separation media have extensively been studied as one of powerful tools for the fast separa-tion with high efficiency[66–74].
3.1MA of flow rate dependence of HETP
Kele and Guiochon[68]studied the correlation between HETP and the mobile phase flow velocity under six sets of different experimental conditions(#1–#6)of the mobile phase,sample compound,and column temperature.The flow rate dependence of HETP exhibits ordinary profiles, i.e.convex downward,in all the cases.As an example,the flow rate dependence of H total of butylbenzene in the RPLC system(Set#1)consisting of C18-silica monolithic column(#19)and a methanol/water mixture(80/20,v/v) was analyzed by the MA method,i.e.Eq.(14)[75].Figure2 illustrates the correlation between(H total–H f)and u0. There are three different series of plots in Fig.2(solid circle,triangle,and square).The three curved profiles were calculated from the flow rate dependence of H total [68].The difference between the three curved profiles depends on the selection of the equations for estimating the k f value.Because no suitable correlation has been pro-posed for the external mass transfer of sample molecules between the mobile phase and the surface of cylindrical stationary phase like silica monoliths,the k f value was estimated by using the equation of the penetration theory[76,77],the Wilson–Geankoplis equation[78], and the equation proposed by Kataoka and co-workers [79].The experimental data points in Fig.2were fitted to Eq.(14)including the three coefficients,i.e.A,B,and C,of which values were determined by the least square regres-sion method.From their values,the contribution of each mass transfer process to the column efficiency was quan-titatively analyzed.
3.2Mass transfer in C18-silica monolithic column It is indicated in Eq.(11)that H total is the sum of the contri-butions of the three mass transfer processes in the col-umn.However,no information about each individual contribution can directly be obtained because we can experimentally measure only the value of H total.As indi-cated in Eq.(14),the coefficients A and B correspond to the contribution of eddy diffusion and axial molecular dispersion,respectively.The coefficient C is correlated with the mass transfer resistance in the monolithic sta-tionary phase.The contributions of the three kinetic processes to H total are compared in Fig.3.In the low flow rate range(u0a ca.0.25cm N s–1),the contribution of H ax to H total is always larger than those of H f and H d.Both the
Figure 2.Dependence of the value of(H total–H f)on the superficial velocity(u0)(Set#1).Penetration theory(solid circles and solid line):A=4.1610–4l4610–5,B=2610–5 l 6.3610–7,C=1.95610–3l 1.7610–4,R2=0.99147.
Kataoka equation(solid triangles and dashed line): A=4.3610–4l4610–5,B=2610–5l 6.3610–7, C=1.67610–3l1.7610–4,R2=0.99184.Wilson–Geanko-plis equation(solid squares and dotted line):A=3.8610–4l 4610–5,B=2610–5l6.7610–7,C=1.1610–3l1.8610–4, R2=0.99209.Reproduced from[75]with permission of the American Chemical Society.
762K.Miyabe J.Sep.Sci.2009,32,757–770
contributions of H f and H d increase with increasing flow rate.However,the slope of H d is larger than that of H f by a factor of two or more.In the high flow rate range(u0A ca.
0.25cm N s–1),the contribution of H d becomes larger than that of H ax.
As illustrated in Fig.3,the relative contributions of the three kinetic processes to H total change with the mobile phase flow velocity.Figure4illustrates the comparison of the contributions to H total of the three mass transfer processes in five different experimental sets(#1–#5)of RPLC[68].In the previous study[68],the flow rate dependence of HETP was also measured in the experi-mental set#6using thiourea as the sample compound, which is not retained under conventional RPLC condi-tions.The values in parentheses indicate the retention equilibrium constant(K a).Figure4shows the values of d ax,d f,and d d calculated under three different flow rate conditions.The results at u0=1.0cm N s–1are obtained from hypothetical calculations because no datum was experimentally measured at this flow rate.The total length of the bars corresponds to the sum of d ax,d f,and d d.Although the contribution of the three kinetic proc-esses to H total(hence,l92)vary depending on u0,it is con-cluded that none of them can be neglected in the RPLC system because,roughly speaking,the values of d ax,d f, and d d are of almost the same order of magnitude.Simi-lar results have been reported in other literature[38,39, 73,80,81].
Figure4indicates that the value of d d exhibits no flow rate dependence because it corresponds to the diffusive mass transfer of sample molecules in the stationary phase.It is characteristic that d d is constant irrespective of u0.However,the relative importance of d d increases with increasing u0and reaches approximately60–70%at u0=1.0cm N s–1for all the experimental systems#1–#5. It is a demand of the time to develop HPLC for fast separa-tions with high efficiency.It is required to study the mass transfer kinetics of sample molecules in the station-ary phase because intra-stationary phase diffusion has a predominant contribution to the column efficiency under high flow rate conditions as indicated in Figs.3 and4.
3.3Mass transfer in C18-silica monolithic
stationary phase
As indicated in Eq.(15),it is frequently assumed that the mass transfer in porous adsorbents consists of two paral-lel mechanisms,i.e.pore diffusion and surface diffusion [48,49].Figure5illustrates the comparison of the contri-butions of the two kinetic processes to the overall mass transfer in the monolithic stationary phase.Each set of three bars corresponds to the experimental systems#1–#5.The total length of each bar represents the value of D e.The hatched part corresponds to D p.In Fig.5,the value of D e depends on that of k f,which is estimated by the three different equations as described above.The value of D e is larger than that of D p by a factor of about3.3–7.4in the five experimental systems.This means that most sample molecules between ca.77and88%migrate inside the monolith stationary phase by surface diffusion.The significant contribution of surface diffusion has been
Figure3.Contribution of the three mass transfer processes in the C18-silica monolithic column(contributions of axial dis-persion,H ax,external mass transfer,H f,and intraparticule dif-fusive transfer,H d)to the correlation between HETP and u0 for the experimental set1.The value of the external mass transfer coefficient,k f,was calculated using the penetration theory.Reproduced from[75]with permission of the Ameri-can Chemical
https://www.doczj.com/doc/cd14971079.html,parison of the mass transfer resistance con-tributions of the three kinetic processes in the C18-silica monolithic column(contributions of axial dispersion,d ax, external mass transfer,d f,and intraparticule diffusive trans-fer,d d)to the second central moment,l92,at different mobile phase flow velocities for the experimental sets#1–#5.The value of the external mass transfer coefficient,k f,was calcu-lated using the penetration theory.Reproduced from[75] with permission of the American Chemical Society.
J.Sep.Sci.2009,32,757–770Liquid Chromatography763
reported for the intra-stationary phase mass flux[38,39, 80,81].It is concluded that surface diffusion plays the predominant role for the mass transfer in the stationary phase.To the best of the author's knowledge,there would be few investigation based on the conventional rate equations,which quantitatively discusses the contri-butions of the external mass transfer,pore diffusion,and surface diffusion to H total and the values of related kinetic parameters,i.e.k f,D p,and D s.
Horvμth and co-workers made a great number of extremely significant contributions to the progress of kinetic study as well as of other research fields of chro-matography.Horvμth and Lin proposed a rate equation, in which the contributions of the axial dispersion,exter-nal mass transfer,intraparticle diffusion,and actual adsorption/desorption kinetics to band broadening are taken into account[44,46].Intraparticle diffusivity is also represented using the internal porosity and tortuos-ity factor.The rate equation originates from some impor-tant results due to the progress of the kinetic studies in chromatography.However,regarding intraparticle diffu-sion,there is only one difference between the model illustrated in Fig.1[39]and that proposed by Horvμth and Lin[46].No description about surface diffusion is observed in the model proposed by Horvμth and Lin[46], although pore diffusion is represented as“intraparticu-lar diffusion”.The description in the model proposed by Horvμth and Lin[46]probably indicates that many kinetic studies in chromatography have so far been car-ried out without taking the presence of surface diffusion into account.There have been no conventional kinetic theories and rate equations,which express the contribu-tion of surface diffusion to chromatographic separa-tions.Although the significance of surface diffusion was pointed out by Giddings as one of the important mass transfer processes more than40years ago[41],surface diffusion itself and its significant contribution to the mass transfer kinetics(hence,to the column efficiency) have not sufficiently been recognized in the community of chromatography.
3.4Significance of study on surface diffusion Chromatographic behavior is influenced by the variation in various experimental conditions concerning the sta-tionary and mobile phases,the sample compounds,and others(e.g.temperature).When we study chromato-graphic behavior from kinetic points of view,we can choose surface diffusion as an informative rate process because it takes place in a potential field of adsorption [48,49,82].Surface diffusion is always affected by the retention behavior of sample molecule because it migrates in the vicinity of the stationary phase surface under adsorbed state.The manner of surface diffusion would directly reflect the change in chromatographic behavior.In addition,as explained earlier,surface diffu-sion has a predominant contribution to the intraparticu-late mass transfer.Important characteristics and migra-tion mechanism of surface diffusion should be studied in detail because it is related to the development of fast chromatography with high separation efficiency. Surface diffusion or lateral diffusion has been studied by using various methods,such as fluorescence spectro-scopy[3–14],shallow-bed adsorption[33–37],and chro-matographic approaches[38–40].Similarly,some kinetic studies on the mass transfer including surface dif-fusion have been carried out in different modes of chro-matography,i.e.(i)anion exchange chromatography of BSA[27,29],(ii)the chiral separation of S-Tr?ger's base on cellulose triacetate[28],(iii)the enantiomeric separa-tions on molecularly imprinted stationary phases[30, 83,84],and(iv)RPLC using C18-silica gel particles and C18-silica monoliths[32,85–88].The results of a series of studies on surface diffusion indicate that surface diffu-sion has an important role for the diffusive migration in the stationary phase.
4New approaches for kinetic study on chromatography
Some new theoretical and experimental strategies have been studied in order to derive accurate and quantitative information about the mass transfer kinetics in chroma-tography.Several new topics are picked up in this sec-tion.
https://www.doczj.com/doc/cd14971079.html,parison of the contributions of pore diffusion (D p)and surface diffusion(D s)to the mass transfer of the sample molecules in the C18-silica monolithic stationary phase for the experimental sets#1–#5.The three different literature correlations were used for estimating the external mass transfer coefficient,k f.Reproduced from[75]with per-mission of the American Chemical Society.
764K.Miyabe J.Sep.Sci.2009,32,757–770
4.1New moment equations for chromatography
Various types of packing materials having different struc-
tural characteristics have been developed for achieving
fast and high efficiency separations,e.g.monolithic sta-
tionary phases,pellicular(shell)particles,and non-
porous particles.It has well been recognized that the col-
umn efficiency under high flow rate conditions primar-
ily depends on the mass transfer resistance in the station-
ary phase.The packing materials described above are
designed on the concept that the band broadening due
to intra-stationary phase diffusion should be suppressed.
It is important to clarify the mass transfer kinetics in the
stationary phase when the performance and characteris-
tics of the separation media are evaluated.The MA
method can provide quantitative information about the
mass transfer kinetics in the stationary phase and in the
column,which is not necessarily obtained by the conven-
tional kinetic studies using the ordinary rate equations.
The moment equations have already been developed
for the conventional columns packed with full-porous
spherical particles about30–40years ago[2,38,39,48–
56].Until recently,we can use only the moment equa-
tions.However,they cannot be used for the various types
of separation media because their structural characteris-
tics are remarkably different from those of full-porous
spherical particles.Recently,a framework of new
moment equations was systematically developed for
chromatography using the various types of separation
media having different structural characteristics,i.e.
shape(spherical particle,cylindrical fiber,flat plate,and
hollow tube)and porous structure(full-porous,superfi-
cially porous(pellicular or shell),and non-porous)[73,
75,89–91].Figure6illustrates the structural characteris-
tics of the various packing materials[90].
Equations(17)and(18)are the moment equations of l1
and l92for the full-porous stationary phases,i.e.spherical
particles,cylindrical fibers,and flat plates.
l1às 2
L u ?1t
1àe e
e e
e ite1àe iTK a
f ge17T
l92às2 12
L u
?
2D L
u2
?1t
1àe e
e e
f e ite1àe iTK a 2
t
2e1àe eT
e e
?e1àe iT
K2
a
k a
t
$
h k f
t
$2
r D e
f e ite1àe iTK a g2 e18T
where s is the width of the rectangular sample injection pulse,u the interstitial velocity,n the diffusion distance, and h and r the numerical coefficients.At first,the first moment equation is completely identical irrespective of the shapes of the packing materials.On the other hand, the second moment equation depends on their shapes. The basic formula is the same for all the packing materi-als.However,the numerical coefficients,i.e.h and r, which are attached with the external mass transfer coef-ficient(k f)and the intra-stationary phase diffusivity(D e)
Figure6.Schematic illustration of the structural characteris-tics of various packing materials and separation media. Reproduced from[90]with permission of the American Chemical Society.
J.Sep.Sci.2009,32,757–770Liquid Chromatography765
are systematically changed,i.e.3and15for the spherical particles,2and8for the cylindrical fibers,and1and3 for the flat plates.The values of h,i.e.3,2,and1,well rep-resent the dimension of the mass transfer in the station-ary phases.The ratios of r to h are also systematically changed,i.e.5,4,and3.The numerical values reflect the geometrical difference between the packing materials. The diffusion distance(n)is also different,i.e.radius for the spherical particles and for the cylindrical fibers,and thickness for the flat plates.
Regarding spherical particles having different poros-ities,the moment equations are represented as the same formulas.However,there are some differences between them.At first,the moment equations for full-porous spherical particles are represented by Eqs.(17)and(18). Then,the moment equations are more simply repre-sented for non-porous particles because there is no intra-particle diffusion.In Eqs.(17)and(18),the contribution of intraparticle diffusion to band broadening can be negli-gible and e i=0.On the other hand,the moment equations are more complicated in the case of the“pellicular”or “shell”particles because their structure is more complicated.Kaczmarski and Guiochon also derived the equations for the first and second moments for chroma-tography using shell(pellicular)particles[92].They eval-uated the performance of shell particles for the separa-tion of various sample compounds,such as BSA,peptides, proteins,phenol,naphthalene,and anthracene[93,94]. Now,we have the systematic framework of the new moment equations,which can be used for quantitatively analyzing chromatographic behavior of the various sep-aration media in detail from the viewpoints of the reten-tion equilibrium and mass transfer kinetics.They are also useful for the preliminarily evaluation of new types of packing materials for fast HPLC.
4.2Derivation of general HETP equation for HPLC The classical rate equations have conventionally been used so far in the community of chromatography[41–47].In addition,now we can use the new moment equa-tions introduced in the previous section for quantita-tively analyzing the kinetic behavior in chromatography. Recently,Gritti and Guiochon derived a more detailed general HETP equation for chromatography[65].Gritti and Guiochon[87]analyzed experimental data in detail using the general HETP equation and kinetic parameters estimated with literature correlations[2,48,49,63,78, 95,96].They provided some important conclusions con-cerning the mass transfer kinetics in chromatography. (i)The axial dispersion in the column at very low linear velocities must be analyzed with considering the contri-bution of intraparticle diffusion(pore diffusion and sur-face diffusion),as well as the molecular diffusion in the external void(interparticulate)space in the column.
(ii)The three contributions due to trans-channel, short-range interchannel,and long-range interchannel effects on the axial dispersion are important.The contri-bution of eddy diffusion to the band broadening is repre-sented as the ratio of two third degree polynominals.It is indicated that the eddy diffusion term depends on the mobile phase flow velocity and the chemical nature of the stationary phase surface.
(iii)The mass transfer processes are closely related.For example,surface diffusion contributes to both the axial dispersion and the trans-particle mass transfer.In addi-tion,in liquid/solid phase systems,it is predicted that sol-vent molecules are adsorbed around the stationary phase surface to form an adsorbed multilayer.The thickness of the external film affects both the eddy diffusion term and the fluid-to-particle mass transfer resistance.
(iv)Sample molecules more than ca.90%migrate by surface diffusion in the conventional C18-silica packing materials in the range of C18ligand density from2to 3l mol N m–2,demonstrating the predominant contribu-tion of surface diffusion to the mass transfer in the RPLC stationary phases.
(v)The mechanism of surface diffusion is the same irre-spective of the length and density of alkyl ligands because there is no clear trend between the activation energy of surface diffusion(E s)between ca.33and 36kJ N mol–1and the modification conditions.The con-tribution of the sample retention to E s is smaller than the corresponding enthalpy change.
(vi)The ratio of the frequency factor of surface diffu-sion to that of molecular diffusion is much(three orders of magnitude)larger than that previously reported[97]. The large discrepancy between the two results would be attributed to the difference in the second terms in the right hand side of Eqs.(15)and(16)and to the difference in some experimental conditions.
4.3GR model with generalized Maxwell–Stefan
(GMS)equations
The MA of elution peak profiles provides quantitative information about the mass transfer kinetics in the sta-tionary phase,especially surface diffusion,which is not obtained by using the ordinary rate equations of chroma-tography.Surface diffusion has been abundantly studied in gas/solid and liquid/solid adsorption systems because of its predominant contribution to the mass transfer flux in adsorbents[48,49,82,98].Some intrinsic kinetic and thermodynamic characteristics of surface diffusion have also been studied in the field of chromatography.A series of studies on surface diffusion in RPLC provides the fol-lowing information,i.e.(i)there is an intimate correla-tion between surface diffusion and molecular diffusion, (ii)surface diffusion originally corresponds to molecular diffusion,and(iii)surface diffusion is restricted by the
766K.Miyabe J.Sep.Sci.2009,32,757–770
sample retention.These experimental data of surface dif-fusion were quantitatively analyzed by applying the absolute rate theory[99].On the basis of the results,a sur-face-restricted molecular diffusion model was proposed as a first approximation of the mechanism of surface dif-fusion for quantitatively interpreting the fundamental aspects of surface diffusion[38,39,97,100–108].The model is effective for quantitatively explaining some intrinsic characteristics of surface diffusion and for dis-cussing the mass transfer mechanism by surface diffu-sion.However,the information about surface diffusion in HPLC systems usually rests on the experimental data measured in single-component chromatography under linear isotherm conditions,although the concentration dependence of D s has also been studied by the pulse on a plateau method[101,109].On the other hand,it is essen-tially important for preparative chromatography to study the mass transfer phenomena in multi-component systems under non-linear isotherm conditions[2]. Cavazzini et al.[110]studied the separation behavior of two enantiomers of1-phenyl-1-propanol in a chromato-graphic system consisting of cellulose tribenzoate coated silica and a mixture of hexane and2-propanol(97/3,v/v). It was reported that an increase in the concentration and injection volume of the sample solution of the racemic mixture is accompanied with an increase in the effective pore diffusion coefficient of S-1-phenyl-1-propanol.Oppo-sitely,the effective pore diffusion coefficient of R-1-phe-nyl-1-propanol decreases,suggesting that surface diffu-sion has an important contribution to the total diffusive flux in this case.However,such a concentration depend-ence of1-phenyl-1-propanol cannot be explained by the Fickian model of diffusion.Kaczmarski et al.[111]studied the effectiveness of the GR model of chromatography coupled with the GMS equations describing the surface diffusion flux for interpreting the chromatographic behavior of the two enantiomers in the chiral separation system.They also made a similar study on the enantio-meric separation of1-indanol on the same stationary phase[112].The GMS formulation was originally applied to the description of intraparticle diffusion and surface diffusion of multicomponent gaseous mixtures in mac-roporous and microporous adsorbents by Krishna[113–115].Kaczmarski et al.[111,112]extended the GMS approach to the kinetic study of surface diffusion in the enantiomeric separations in the liquid/solid phase adsorption systems.
Equation(15)indicates that intraparticle diffusion consists of pore diffusion,which is molecular diffusion without attractive interactions,and surface diffusion. Kaczmarski et al.assumed that the flux by pore diffusion can be estimated by a Fick equation.On the other hand, the flux of two enantiomers due to surface diffusion(J s) was calculated by using the GMS equations including surface diffusion coefficients,which are the Fickian dif-fusivities of the GMS equations for counter-sorption dif-fusivity of the two enantiomers.The values of the surface diffusion coefficients were calculated from several related parameters,i.e.the fractional surface coverage of adsorbates,the counter-exchange diffusion coefficient, and the Maxwell–Stefan diffusivities,which describe the interaction between each component and the adsorbent surface.The GR model was numerically solved by using the method of orthogonal collocation on finite elements. The values of the related diffusivities were estimated so that the calculated profiles agree with the elution peak profiles experimentally measured.Kaczmarski et al. experimentally measured elution peak profiles of the two enantiomers of1-phenyl-1-propanol while changing both the concentration and injection volume of the race-mic sample mixture[111].They also numerically simu-lated chromatographic behavior of the chiral separa-tions under the different conditions.The calculated pro-files were in good agreement with the experimental peaks.It is suggested that surface diffusion has a predom-inant contribution to the intraparticle mass transfer and that the GMS equations properly represent the concen-tration dependence of surface diffusion.
The GMS model needs to use the coefficient of counter-sorption diffusion,which is a new kinetic parameter describing the influence of the presence of one compo-nent on the molecular migration of another one.Kacz-marski et al.discussed the influence of this kinetic parameter on the separation behavior of the two enan-tiomers of1-phenyl-1-propanol[111].As a result,it was indicated that the influence of the change in the value of the counter-sorption diffusion coefficient is clearly observed in the range,where the two enantiomers are co-eluted and their elution peaks are overlapped.Almost negligible influence is observed in the front shock of the first elution peak(S-1-phenyl-1-propanol)and in the rear boundary of the second elution peak(R-1-phenyl-1-propa-nol).This observation is reasonable and easily explained. In the front of the S-1-phenyl-1-propanol peak,the con-centration of R-1-phenyl-1-propanol is negligibly small. Similarly,in the tailing part of the R-1-phenyl-1-propanol peak,quite few molecules of S-1-phenyl-1-propanol exist. In such cases,it is not necessary to use the counter-sorp-tion diffusion coefficient for representing the manner of surface diffusion of each enantiomer.On the contrary, the value of the counter-sorption diffusion coefficient significantly affects the elution peak profiles of both S-1-phenyl-1-propanol and R-1-phenyl-1-propanol in the range between the two bands.In this range,the tailing part of the first peak(S-1-phenyl-1-propanol)is over-lapped with the front part of the second one(R-1-phenyl-1-propanol),suggesting that the S-1-phenyl-1-propanol molecules are diffusing out from the particle,whereas the R-1-phenyl-1-propanol molecules are oppositely dif-fusing into it.The molecular migration of S-1-phenyl-1-
J.Sep.Sci.2009,32,757–770Liquid Chromatography767
propanol and that of R-1-phenyl-1-propanol are influ-enced with each other because the direction of their migration is completely opposite.These are complicated situations of surface diffusion in the chiral separation. The application of the Maxwell–Stefan model is one of important approaches for elucidating surface diffusion phenomena because it would be effective for expanding the study on surface diffusion of a single component under linear isotherm conditions to the preparative sep-arations.Still now,we have little accurate information about surface diffusion under preparative conditions. 4.4Modeling method for determination of
intraparticle diffusivity
Although the MA method is effective for studying the mass transfer kinetics in chromatography,its data anal-ysis procedure is relatively complicated[116].Hong et al. [117]proposed a modeling method based on the lumped pore diffusion(POR)model[2,59]as an alternative for deriving the information about intraparticulate mass https://www.doczj.com/doc/cd14971079.html,pared with the GR model of chromatogra-phy,in the modeling method,the mass transfer resist-ance is more simply accounted for with an overall mass transfer coefficient(k0).The differential mass balance equations of the POR model are solved under linear iso-therm conditions using analytical and numerical meth-ods[2,59].The modeling method is divided into two approaches[117].One is the modeling-Laplace(ML) method,which is based on the analytical solution of the POR model in the Laplace domain.The other is the mod-eling-numerical(MN)method,which is based on the results of the numerical calculation performed using the method of orthogonal collocation of finite elements.The appropriate value of the overall mass transfer coefficient is estimated by the ML and MN methods so that the calcu-lated profile is in agreement with the experimental data. Then,the internal mass transfer coefficient is estimated by subtracting the contribution of the external mass transfer coefficient to the overall mass transfer coeffi-cient.Finally,the intraparticle diffusion coefficient(D e) is calculated from the internal mass transfer coefficient. In the following,some advantages and disadvantages of the modeling method are compared with those of the MA method.
(i)The number of experimental data required for the data analysis is less for the modeling method than for the MA method.The former needs only the whole profile of an elution peak.No information about some intrinsic characteristics of the peak profile is necessary such as l1 and l29.
(ii)The data analysis procedure of the modeling method is probably simpler than that of the MA method. However,mathematical treatments(i.e.iterative curve-fitting or numerical calculation)are always required for determining D e by the modeling method.A high perform-ance computer system and a suitable program are also essential for the numerical calculation of the elution peak profiles.
(iii)The modeling method needs a deconvolution cor-rection for the contribution to the peak profile due to some instrumental factors and sample injection pulse [117].Roughly speaking,the modeling method needs only one elution peak profile measured at a given flow rate to derive D e.It is just required for the deconvolution to carry out only one referential experiment without a column at the same flow rate.
(iv)The modeling method can be used irrespective of the size and shape of the packing materials.The differen-tial mass balance equations of the POR model include only one parameter,which depends on both the size and shape of the separation media.
(v)It is necessary for the modeling method to estimate the external mass transfer coefficient(i.e.k f)and D L.It is reported that10%increase in the values of D L and the external mass transfer coefficient results in3.0–11.2% decrease and1.6–6.3%increase in the D e values,respec-tively.Experimental conditions must carefully be con-trolled to satisfy a critical requirement for deriving an accurate value of D e because the error in the estimation of D L provides a significant influence on the resulting D e value[117].
Hong et al.[117]measured D e of rubrene in a RPLC sys-tem consisting of a Symmetry C18column and methanol/ water mixtures(90/10–100/0,v/v)by applying the three methods,i.e.the MA,ML,and MN methods.The D e values derived by the three different methods are in good agree-ment,demonstrating that the three approaches can pro-vide the correct information about intraparticle diffu-sion[117].The PR-modeling method and the PR-MA method have different characteristics.The modeling method is the inverse method,which searches the best value of the kinetic parameter involved in the basic equa-tions of the POR model,knowing the information about the linear isotherm.A similar inverse method has also been proposed for the determination of adsorption iso-therm.A best isotherm is determined so that the peak profile calculated by a kinetic model using suitable rate parameters agrees with the profile experimentally meas-ured.On the other hand,the MA method extracts the information about the adsorption equilibrium and mass transfer kinetics from l1and l29of the elution peak pro-files[2,38–40,48,49].
4.5Peak parking(PP)method for determination of
surface diffusion coefficient
The PR-MA method is effective for deriving the informa-tion about the intraparticulate mass transfer including surface diffusion[2,38–40,48,49].However,this
768K.Miyabe J.Sep.Sci.2009,32,757–770
approach also has some drawbacks as explained earlier [116].It was tried to develop the PP method as an alterna-tive strategy for determining intraparticle diffusivity(D e) and surface diffusion coefficient(D s),which is also called arrested flow or stopped flow method.
Knox and McLaren[118]introduced a new elution method for the determination of diffusion coefficient and obstructive factor in GC.They injected ethylene as an unretained tracer into a column and eluted it part way along the column.Then,the carrier gas(nitrogen) stream was stopped and the sample band was allowed to spread by diffusion in the axial direction of the column for different periods of time.The carrier gas stream was then resumed to elute the sample band from the column, of which width was measured.The diffusion coefficient and the obstructive factor in the gaseous system were derived from the systematic measurements of the peak width as a function of the parking time.They determined the diffusion coefficient of ethylene in nitrogen as 1.65610–1cm2N s–1at291K and100kPa and the value of obstruction factor between0.46and0.74for different GC packing materials.The arrested flow or stopped flow method has been used for some kinetic properties in vari-ous GC and LC systems[119–127].
Figure7compares the PR-MA method and the PP-MA one.In the former,a small sample pulse is introduced into the column and continuously eluted under an iso-cratic condition at a constant velocity until it completely leaves the column.In the latter,a small sample pulse is similarly injected into the column and eluted until the sample band reaches at around a longitudinally middle position of the column.Then,the band elution is inter-rupted for a while.This is the PP period(t p),during which the sample band diffuses in the axial direction of the col-umn.At the end of t p,the band elution is resumed under the same isocratic conditions until the elution peak pro-file is completely recorded.There are some differences between the elution peak profiles measured by the two methods.First,the difference in the elution time is equal to t p.Second,the peak recorded in the PP method is broader than that recorded in the PR method(see Fig. 7(d)).Kinetic information about intraparticulate mass transfer can be derived from the analysis of the addi-tional band broadening due to the axial dispersion of the sample molecules during t p.In this case,the band broad-ening takes place under equilibrium conditions of the solute distribution between the stationary and mobile phases because there is no convective flow during t p,in contrast with the conditions under which the PR experi-ment is carried out,under non-equilibrium conditions due to the intentional choice of high flow velocities of the mobile phase.
In spite of the oppositely different conditions of exper-imental measurements,it is concluded on the basis of experimental data that the same values of D s can be obtained by the PR-MA method and the PP-MA method [116,127].The D s data also show the same tendency con-cerning kinetic and thermodynamic properties.Both the advantages and disadvantages of the two methods are compared in the following[116].The PR-MA method has several drawbacks as follows.
(i)The data analysis procedure of the PR-MA method is somewhat complicated.
(ii)It requires the information of the related kinetic parameters,i.e.k f and D p.
(iii)Accurate values of l1and l29are necessary for the correct analysis of the mass transfer kinetics.
(iv)It is required to correct the influence of some fac-tors on the moment values,e.g.the extra-column pipes and the asymmetry profile of the elution peaks.
(v)It is hard to accurately analyze the mass transfer kinetics in fine stationary phase by the PR-MA method.
In contrast,the PP-MA method has some advantageous characteristics,which correspond to the drawbacks of the PR-MA method described above.
(i)The data analysis procedure of the PP-MA method is relatively simple and easy.
(ii)No information about k f and D p is necessary for the kinetic study on chromatographic behavior.
(iii)Accurate values of l1and l29are not necessarily required.
(iv)The correction for the influence of the peak asym-metry and the extra-column pipes on l1and l29is not required.
(v)The PP-MA method permits the accurate measure-ment of D e and D s in small-sized separation media.
As described above,the two methods complement each other very well.The most suitable method can be
Figure7.Schematic illustration of the band broadening phe-nomena and the elution peak profiles in the PR and PP experiments.(a)Intermediate situation in the PR method. The mobile phase flows continuously through the column,at a constant flow rate.(b)Intermediate situation in the PP method.The mobile phase flow is arrested for the PP period.
(c)Elution peak profile in the PR method.(d)Elution peak profile in the PP method(solid line)compared to the elution peak profile in elution,shifted for the PP time(dotted line). Reproduced from[116]with permission of the American Chemical Society.
J.Sep.Sci.2009,32,757–770Liquid Chromatography769
chosen depending on the experimental conditions.The combination of the two methods provides a comprehen-sive strategy for measuring the mass transfer kinetics in the stationary phase.Additionally,as described in Sec-tion4.4,the PR-modeling method has also been proposed as an alternative for the kinetic study on chromatogra-phy[117].Now,we can use the three independent strat-egies for the study on the intra-stationary phase diffu-sion,i.e.the PR-MA method,PP-MA method,and PR-mod-eling method,which are based on the completely differ-ent theoretical backgrounds.According to experimental conditions and research objects,we can choose one of them as a suitable approach.The combination of the three methods leads to a comprehensive strategy for the kinetic study on chromatography.
5Concluding remarks
It is a demand of the time to develop fast HPLC tech-niques with high efficiency.We can list some concrete examples for the purpose,such as the ultra-high pressure HPLC systems using superfine full-porous particles,the monolithic columns,and superficially porous spherical particles.In these cases,one of the most important sub-jects is the reduction of band broadening in the column at high flow rate conditions.It is essential to study in detail the mass transfer kinetics in the columns.How-ever,we still do not have sufficient information about the mass transfer processes taking place in the columns, by contrast with the innumerable studies devoted to the retention equilibrium in HPLC.In this review,the MA method was introduced as one of effective strategies for the kinetic study on chromatography.On the other hand,as described in Section4,there are also some sub-stantial progresses in the theoretical bases and the exper-imental strategies relating to the mass transfer kinetics in chromatography.They will contribute to the develop-ment of new techniques of chromatography from kinetic points of view.
The author declared no conflict of interest.
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