Focusing Review

CHROMATOGRAPHY, Vol.31 No.1 (2010)

Focusing Review

Development of New Strategy for Analysis of Chromatographic Behavior
on the Basis of Moment Theory
Kanji Miyabe
Graduate School of Science and Engineering for Research, University of Toyama,
3190, Gofuku, Toyama 930−8555, Japan

Abstract
The moment theory was used for the analysis of chromatographic behavior. Information about the characteristics and mechanism of chromatographic separation in RPLC mode was quantitatively derived by analyzing experimental data of pulse response (PR) measurements from kinetic and thermodynamic points of view. Especially, some fundamentals of surface diffusion were studied in detail because it has a predominant contribution to the mass transfer in the stationary phase, hence a significant influence on the column efficiency at high flow velocity (fastchromatography). New moment equations were developed for various packing materials having different structural characteristics of shapes
and porosities, such as monoliths and shell-type particles, for high speed separations. The peak parking (PP) method was also used for the kinetic study on the molecular migration in the columns and for the development of some experimental techniques relating to the kinetic study
on chromatography. The PR and PP methods complement each other. The combination of the two methods and the moment theory leads to acomprehensive strategy for studying the mass transfer kinetics in chromatography.
Key Words: Moment analysis, Mass transfer kinetics, Reversed-phase liquid chromatography, Surface diffusion, Pulse response method, Peakparking method

ior has not so abundantly been investigated from kinetic points of

1. Introduction
High performance liquid chromatography (HPLC) has alreadyview because of some essential difficulties concerning the kinetic

been essential as a powerful tool for fine separations. However,study. Additionally, in contrast with the extensive applications ofstill now, some apparatuses, e.g., high pressure pumps and highvarious HPLC instruments and separation media developed in re-performance columns, have rapidly been developed in order to es-

cent decades, it seems that arrangements of theoretical bases andtablish fast chromatography techniques with high efficiency. On

experimental strategies relating to the kinetic study are not well

the other hand, the further progress of chromatographic technolo-

recognized by most chromatographers. Now, it is required to de-gies needs the pursuit of fundamental studies on the separation

velop a new strategy for the analysis of chromatographic behavior

mechanism. Chromatographic behavior, such as the sample reten-

beyond the ordinary “plate theory” and “rate theory” of chromatog-

tion, band broadening, elution peak profile, column efficiency,

raphy [1−3].

separation performance, and so on, depends on both the retention

The author has developed an analytical system based on theequilibrium and mass transfer kinetics in the column. A great num-“moment theory” [3−5], which contributes to the kinetic study on

ber of research works have been carried out so far on the retention

chromatographic behavior, and applied the new approach to the ex-

equilibrium in chromatography. However, chromatographic behav-

perimental study on separation mechanism. Some fundamentals of

Correspondence: Graduate School of Science and Engineering for Research, University of Toyama, 3190, Gofuku, Toyama 930−8555, Japan
Phone: +81−76−445−6835
Fax: +81−76−445−6835
E-mail: E−mail:miyabe@eng.u-toyama.ac.jp―１―

CHROMATOGRAPHY, Vol.31 No.1 (2010)

chromatographic separation have also been clarified from the theoretical analyses of the experimental data from the various viewpoints of the retention equilibrium, mass transfer kinetics, and thermodynamic properties. In this review, some items of information
about the moment analysis (MA) method and the concrete examples about the mass transfer kinetics derived by the MA method areprovided. The progress of some strategies for the kinetic study on
chromatography is also introduced.
In the section 2, it is explained why the kinetic study on chromatographic behavior is essentially difficult in comparison with thestudy on the retention equilibrium. A brief explanation about aframework of the MA method is provided in the section 3. In thesection 4, it is described what kind of new information about chromatographic behavior can be obtained by the MA from elution
peak profiles experimentally measured. The section 5 is devoted toindicate the results of some characteristics and mechanism of surface diffusion, which has not appropriately been recognized forseveral decades in the community of chromatography. The recent
progress of some strategies for the kinetic study on chromatography is also introduced in the section 6.
There are semantic discussions about the retention mechanism ofchromatography, 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 themobile and stationary phases and that the sample molecules areconsequently concentrated on the stationary phase surface. It is not

Figure 1.

intended to discuss the retention mechanism. The author would liketo focus the main subject of this review on the quantitative analysisof the kinetic aspects in chromatography.
2. Background of the kinetic study on chromatography

Schematic illustration of the mass transfer processes in
the column. [Reproduced with permission from [7], K.
Miyabe, G. Guiochon: Measurement of the parametersof the mass transfer kinetics in high performance liquid
chromatography. J. Sep. Sci. 2003. 26. 155−173.
Copyright Wiley-VCH Verlag GmbH & Co. KGaA].

Fundamental studies on chromatographic behavior have so far
been mainly pursued from the viewpoint of the retention equilib-

taken into account in the general rate (GR) model of chromatogra-

rium because chromatography primarily depends on the phasephy [3−10], i.e., (1) the axial mixing in the bulk mobile phase per-equilibrium thermodynamics [3]. In comparison with a great num-colating between packing materials (axial dispersion), (2) the diffu-

ber of research works on the chromatographic retention, the kinetic

sive migration of sample molecules between bulk mobile phase andstudy on chromatography has not sufficiently been carried out.

the external surface of stationary phase (external mass transfer), (3)Mass transfer taking place in HPLC columns and stationary phases

the diffusive migration of sample molecules through pores insideshould quantitatively be studied in more detail for well understand-separation media (intra-stationary phase diffusion), and (4) theing some important characteristics of chromatographic separations.

adsorption-desorption process at actual reaction sites on the station-

It is the most conventional approach for the kinetic study on

ary phase surface (adsorption-desorption kinetics). The flow rate

chromatography to analyze the flow rate dependence of HETP (H ).

dependence of HETP must provide some items of significant infor-

Some ordinary rate equations, such as the van Deemter equation

mation about the mass transfer kinetics in the column and in theand the Knox equation, have extensively been used to account for

stationary phase. However, it is hard to extract the important infor-the dependence of HETP on the mobile phase flow velocity [1−3].

mation about the mass transfer kinetics from the curved profile rep-

It has also been recognized that band broadening rests on the con-resenting the dependence of H on the mobile phase flow velocity.

tributions of several mass transfer processes in the column. For ex-

As indicated in Table 1, it seems that fundamental studies on theample, as illustrated in Figure 1, the four kinetic processes are

mass transfer kinetics are more difficult than those of the retention―２―

CHROMATOGRAPHY, Vol.31 No.1 (2010)

Kanji Miyabe

Table 1. Comparison of the study on the retention equilibrium with that on the masstransfer kinetics in chromatography

TheoryRetention equilibrium

Mass transfer kinetics

Thermodynamics

Rate theory of chromatography
Moment analysis

Retention time (tR)First absolute moment (µ1)

Peak variance (σ2)
Second central moment (µ2’)

Experimental data
Type

Accuracy

High

Relatively low

Precision

High

Relatively low

Number

A few

Many

equilibrium. There would be the following three barriers, which(RSD) less than 1%. However, it seems that the RSD for the meas-

prevent the progress of kinetic studies on chromatography.

urement of the HETP and the number of theoretical plates isaround several percent.

At first, in principle, it is required for the kinetic study to measure far more experimental data than the study on the chroma-

Third, the study on the retention equilibrium rests on the solid

tographic retention. The latter needs the information about only thebasis, i.e., thermodynamics. On the other hand, the conventional

chromatographic retention, i.e., the retention time (tR) or first abso-

rate equations are not sufficient for the detailed analysis of thelute moment (µ1) of an elution peak. On the other hand, the former

mass transfer kinetics in chromatography from the curved profile

needs the information about the band broadening, i.e., the varianceof the flow rate dependence of HETP. Although they are quite(σ ) or second central moment (µ2’) of the peak, other than tR or µ1.

popular [1−3], they are empirical and contain several fitting pa-

Additionally, a number of chromatographic data must be measured

rameters, of which the physical definition and meanings are notin a wide range of the mobile phase flow velocity in order to suffi-

necessarily clear. Consequently, they do not provide sufficiently

ciently represent the flow rate dependence of HETP. It is required

quantitative information about some kinetic parameters relating toto experimentally measure elution peaks as many as possible be-

the mass transfer processes in chromatography.

2

cause the whole profile of the curved correlation between HETP
3. Moment Analysis Theory

and the mobile phase flow velocity is essential to accurately determine some related kinetic parameters. In contrast with the kinetic

There are other theories and models for analyzing chroma-

study, we could go so far as to say that an equilibrium parameter,

tographic behavior. The MA method is one of them. It is based onsuch as the retention equilibrium constant (Ka or K ) and the reten-

the GR model of chromatography [3−8]. The information about thetion factor (k), can be obtained from only one datum of elution

retention equilibrium and the mass transfer kinetics in the column

peak. The number of experimental data for the kinetic study on

is derived from µ1 and µ2’ of elution peaks experimentally meas-

chromatography is much more than that for the retention equilib-

ured, respectively. Similar to the ordinary manner of kinetic studies

rium.

of chromatography using the conventional rate equations, the flow

Second, accurate measurement of σ2 or µ2’ is more difficult thanrate dependence of HETP is analyzed for deriving the kinetic infor-that of tR or µ1. This means that the error for the experimental meas-

mation. It is also assumed in the MA theory that the HETP consists

urement of H is larger than that for Ka or K and k. Although theof the contributions of several mass transfer processes in the col-values of µ1 and µ2’ are obtained by integrating an elution peak pro-umn. On the other hand, the physical meanings of all the parame-

file in a range of time, the integration time range required for theters included in the moment equations are clearly defined because

determination of µ2’ is wider than that of µ1. The difficulty in thethey are mathematically derived on the basis of the GR model ofmeasurement of accurate value of µ2’ leads to the error of the sec-

chromatography. The information about the mass transfer kinetics

ond moment analysis. Ultimately, it is more difficult to accurately

in the column and the stationary phase can quantitatively be ob-measure the flow rate dependence of HETP than the determination

tained with physically sound justification. This is the most impor-of Ka or K and k. On the other hand, it is not so difficult to measuretant difference between the MA method of chromatography and thethe equilibrium parameters with the relative standard deviation

ordinary kinetic study with the conventional rate equations.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

The MA method has been introduced in some previous literature(5)[3−8]. However, it is not probably recognized that the MA method
is an effective approach for the kinetic study of chromatography. It
is not so familiar with most chromatographers because it is difficult
to find the detailed information about the fundamentals of the MA(6)method and the practical MA procedure in the conventional literature and textbooks of chromatography. For the sake of readers,some basic information about the framework of the MA method is(7)

briefly explained in the following.
3.1. Moment Equations(8)

The moment equations are mathematically derived from the basic equations of the GR model of chromatography, which are thewhere DL is the axial dispersion coefficient, kf the external mass

partially differential equations representing the material balancetransfer coefficient, De the intraparticle diffusivity, kads the adsorp-

and the mass transfer rates in the column and in the stationary

tion rate coefficient, Rp the radius of stationary phase particle and δ

phase [3−8, 11, 12]. When full-porous spherical particles are usedthe contribution of each mass transfer process to µ2’. The subscripts

as the stationary phase, the moment equations are represented asax, f, d, and ads denote the axial dispersion, external mass transfer,follows. At first, µ1 represents the position of an elution peak and isintraparticle diffusion, and adsorption-desorption kinetics, respec-

correlated with Ka.

tively. Similar to µ1, µ2’ is equal to the variance (σ2) of an elution
peak, which is one of well known chromatographic parameters.
Equations (3)−(8) indicate that the band broadening originatesfrom the contributions of the several mass transfer processes in the(1)column. However, it seems that the contribution of the adsorptiondesorption kinetics to the total mass transfer resistance is usuallyassumed to be negligibly small in the case of RPLC because the reaction rate of physical adsorption is fast enough [5].

where Ce(t) is the concentration profile of the sample compound at
the exit of the column as a function of time (t), L the length of thecolumn, u0 the superficial velocity of the mobile phase solvent, εe

3.2. Moment analysis procedure

the void fraction of the column (external porosity), and εi the po-

3.2.1. First moment analysis
The following equations are derived from Eqs. (1) and (2).

rosity of the stationary phase (internal porosity). Although Eq. (1)appears to be slightly complicated, it indicates the same correlation(9)

as the following equation when the elution peak is symmetrical.
Equation (2) is well known in the community of chromatography.
(2)

(10)

where t0 is the elution time of an inert tracer. When the elution peakhas a symmetrical profile, µ1 is equal to tR. On the other hand, µ2’ isA plot of the left hand side of Eq. (9) versus L/u0 should be a

represented as follows.

straight line passing through the origin of the coordinates. The
value of Ka is derived from the slope of the straight line. In the MA
method, the pulse response experiments are carried out whilechanging the mobile phase flow velocity because the flow rate de-(3)

pendence of HETP is analyzed in order to extract some kinetic parameters from µ2’ in the second moment analysis as explained in
the next section. When the linear correlation between (µ1−t0)/(1−εe)and L/u0 passing through the origin is observed, it seems that the(4)temperature conditions are homogeneously controlled in the column. The residence time of the sample band is different in the column when the mobile phase flow velocity is changed.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

Kanji Miyabe

3.2.2. Second moment analysis(15)

Similar to the conventional kinetic studies based on the ordinaryrate equations, the flow rate dependence of HETP is analyzed in

Surface diffusion coefficient (Ds) is calculated by subtracting thethe MA procedure. The value of H is calculated from Eqs. (1)−(8)contribution of pore diffusion to De. Pore diffusivity (Dp) is esti-

as follows. The value of HETP is expressed as Htotal because it con-mated from molecular diffusivity (Dm) and some related parameterssists of the contributions of the several mass transfer processes in[3−5]. Detailed information about these parameters can be foundthe column.

out in other literature [3−8, 13]. The second term in the right handside of Eq. (15) represents the contribution of surface diffusion to
intraparticle diffusion.

(11)

4. Analysis of chromatographic behavior on the basis of the

As shown in Eq. (11), the contribution of the axial dispersion (Hax)moment theory

can be separated from those of both the external mass transfer (Hf)and the diffusive migration of sample molecules inside the station-

Fundamental studies on the characteristics and mechanism ofary phase (Hd) by taking advantage of the difference in their flow

chromatography have been conducted in RPLC systems because it

rate dependence. The following correlation is derived from Eq.

is the most popular mode of HPLC. The pulse response (PR) and(11).

peak parking (PP) experiments were carried out under different
RPLC conditions concerning the type, concentration, and composition of the stationary phase, mobile phase, and sample compound.

(12)

The temperature conditions were also changed. Experimental data
The axial dispersion coefficient (DL) of the conventional HPLC

thus measured were analyzed by the method of moments to study

columns packed with spherical particles is usually accounted for bychromatographic behavior from the viewpoints of the retention

assuming that axial dispersion consists of two main mechanisms,

equilibrium, mass transfer kinetics, and related thermodynamici.e., molecular diffusion and eddy diffusion [3].

properties.
(13)

4.1. Retention equilibrium

where γ1 and γ2 are the two geometrical coefficients, dp the particle

The retention behavior in RPLC systems was studied on the ba-

diameter, and u the interstitial velocity of the mobile phase solvent

sis of the solvophobic theory [14]. Chromatographic equilibrium(= u0/εe). Equation (12) is modified as follows by substituting Eq.

parameters, i.e., the retention equilibrium constant and the enthalpy(13) into Eq. (12).

change due to retention, were quantitatively analyzed [15]. We also
experimentally demonstrated the influence of solvent on the retention behavior in RPLC by comparing the equilibrium parameters(14)

between gas-solid and liquid-solid phase systems [16]. On the basis
Equation (14) indicates that the correlation between (Htotal−Hf) andof these results, the retention equilibrium in RPLC was studied

u0 is represented by the same formula as the van Deemter equationfrom thermodynamic points of view. In addition, a convenient pro-with three coefficients. However, Eq. (14) is essentially different

cedure was proposed for the practical measurement of adsorption

from the conventional rate equation in terms of the following two

isotherm, which is based on the Elution by Characteristic Point

points. At first, the contribution of the external mass transfer to[17].

Htotal is also taken into account in Eq. (14). Second, the definition
4.2. Mass transfer kinetics

and physical meanings of the parameter C in Eq. (14) are clearly
explained. As shown in Eqs. (7) and (14), quantitative information

Elution peak profiles simultaneously indicate us the information

about the mass transfer kinetics in the stationary phase can strictly

about the retention equilibrium and mass transfer kinetics. How-be obtained as De from the coefficient C . These are obvious differ-ever, we have not completely derived the kinetic information fromences between the ordinary rate equations and the moment equa-chromatograms because the ordinary rate equations of chromatog-

tions.

raphy are not suitable for this purpose. The author applied the MA

As illustrated in Figure 1, the diffusive molecular migration inmethod to the derivation of the important information about the

intraparticle pore space is usually assumed to consist of the parallel

molecular migration dynamics in the column, which could not be

contributions of pore diffusion and surface diffusion [4, 5]. The

extracted by the ordinary rate equations [16, 18−28].
For example, Figure 2 [27] illustrates the flow rate dependence

value of De is accounted for as follows.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

Figure 2.

Contribution of the three mass-transfer processes in the
C18-silica monolithic column (contributions of axial
dispersion, Hax, external mass transfer, Hf, and intraskeleton diffusive transfer, Hd) to the correlation between HETP and u0 for the experimental set 1. The
value of the external mass-transfer coefficient, kf wascalculated using the penetration theory. Reproduced
from [27] with permission.

Figure 3.

Comparison of the contributions of axial dispersion,
external mass transfer, and intraparticle diffusion to thesecond central moment. Reproduced from [16] withpermission.

Figure 4.

Comparison of the contributions of pore diffusion andsurface diffusion to intraparticle diffusion. Reproduced
from [16] with permission.

of the contributions of the three mass transfer processes, i.e., axial
dispersion, external mass transfer, and intra-stationary phase diffusion, to Htotal of a C18−silica monolithic column. Band broadening in
the column was quantitatively studied by analyzing the mass transfer rate in the three kinetic steps.
In Figure 3 [16], the contributions of the three mass transfer
processes (δax, δf, and δd) to µ2’ were compared with each other under different experimental conditions. The values in parentheses indicate the adsorption equilibrium constant (K ). The total length ofeach bar indicates the sum of δax, δf, and δd. Figure 3 indicates thatthe contributions of the three mass transfer steps to band broadening can quantitatively be evaluated. Similar results have been reported in other literature [6−8, 27].
As indicated in Eq. (15), it has usually been assumed that themass transfer in porous adsorbents rests on the contributions of thetwo parallel mechanisms, i.e., pore and surface diffusion. Figure 4[16] illustrates the comparison of the contributions between the two
diffusive molecular migration mechanisms to the overall masstransfer in the stationary phase. The number in parentheses again
indicates the value of K. The total length of each bar represents the

Dp in all the RPLC and GC systems. This means that surface diffu-

value of De. The hatched part corresponds to the contribution ofsion plays a predominant role for the mass flux inside the station-pore diffusion. It is obvious in Figure 4 that De is much larger thanary phase and most sample molecules migrate by surface diffusion.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

Kanji Miyabe

The significant contribution of surface diffusion to the intra-

HPLC conditions of the stationary and mobile phases, sample com-stationary phase diffusion has also been reported in other literaturepounds, and others (e.g., temperature). When chromatographic be-[6−8, 27, 28]. Figure 4 indicates that the MA method is also effec-

havior is studied from kinetic points of view, we can choose sur-

tive for the kinetic study on the molecular migration in porous ma-

face diffusion as an informative rate process [35−40]. Surface dif-terials. To the best of our knowledge, surface diffusion in chroma-

fusion data experimentally measured by the PR and PP methods

tography has not so frequently been studied in detail in the field of

under various HPLC conditions were analyzed on the basis of thechromatography. Some results concerning surface diffusion will beabsolute rate theory [41].
Surface diffusion is always affected by the retention behavior of

explained in the following.

the sample molecules because they migrate in the vicinity of thestationary phase surface under adsorbed state. It is usually assumed

4.3. Thermodynamic properties
The characteristics and mechanism of the retention equilibrium

to be an activated mass transfer process taking place in a potential

and surface diffusion in RPLC systems were studied from the ther-field of adsorption [4, 5, 42, 43]. It seems that the manner of sur-modynamic points of view by analyzing related experimental data,

face diffusion directly reflects the change in the chromatographic

i.e., the enthalpy change due to retention, the frequency factor andbehavior. Additionally, as explained above, surface diffusion has

activation energy of surface diffusion, and so on [6]. The retention

an important contribution to the mass transfer in the stationary

behavior and surface diffusion in RPLC were studied from thephase. The results of a series of studies on surface diffusion let us

viewpoints of two extrathermodynamic relationships, i.e., enthalpy

to conclude that the characteristics and mechanism of surface diffu--entropy compensation (EEC) and linear free energy relationshipsion should be studied in detail.

(LFER) [29−34]. At first, the values of K and Ds were analyzed indetail in order to demonstrate that a real EEC of the retention equi-5.1. Drawbacks of the conventional model for surface diffusion

librium and surface diffusion originate from substantial physico-There have been some research subjects relating to the values ofchemical effects. Then, we developed a new thermodynamic modelDs and thermodynamic properties of surface diffusion, which havebased on the EEC to explain the LFER between the retention

not been solved. Table 2 [28] lists the experimental data of the re-equilibria, between surface diffusion phenomena, and between thetention equilibrium constant (K ), the isosteric heat of adsorption

retention equilibrium and surface diffusion under different RPLC

(Qst), Ds, and the activation energy of surface diffusion (Es) meas-

conditions. The model is effective for explaining the variation of K

ured in the RPLC system consisting of a C18-silica gel and 70 vol%and Ds due to change in the RPLC conditions and the temperature

methanol and those in the GC system using the same type of pack-dependence of the LFER.

ing materials and helium.

5. Study on the fundamentals of surface diffusion

perimental values of Es and Qst. As illustrated in Figure 5 [9], theOne of subjects is the unreasonable correlation between the exratio Es/(−Qst) should be smaller than unity because surface diffu-

Chromatographic separation is influenced by the variation in the

Table 2. Comparison of thermodynamic properties in liquid-solid and gas-solid phase systems
Phase system
Stationary phase
Mobile phase

Liquid-solid
C18-silica gel
Acetonitrile/water (70/30, v/v)
ln K a−Qst
ln DasEs(kJ mol−1)(kJ mol−1)

ln K aGas-solid
C18-silica gel
Helium−Qst
ln Das(kJ mol−1)

Es(kJ mol−1)

Benzene

0.73

6.7

−12.8

19.4

0.43

5.8

−12.0

14.6

6.08

35.3

−12.4

13.5

Toluene

1.28

8.7

−13.0

20.5

0.79

5.8

−12.3

16.2

7.42

40.7

−12.7

17.5

Ethylbenzene

1.75

9.7

−13.2

22.0

1.12

6.1

−12.6

16.6

8.30

43.8

−13.2

23.2

p-Xylene

1.87

10.3

−13.1

23.4

1.17

6.5

−12.6

13.9

8.48

44.6

−13.0

21.1

n-Pentane

2.35

11.8

−b

−b

1.78

6.2

−b

−b

4.66

31.7

−12.0

10.2

n-Hexane

2.90

12.6

−b

−b

2.19

8.2

−b

−b

5.75

33.4

−12.1

14.1

14.8

b

b

−

9.6

b

−

−b

6.94

38.2

−12.2

14.4

b

b

b

b

17.1

n-HeptaneaLiquid-solid
C18-silica gel
Methanol/water (70/30, v/v)
−Qst
ln DasEs
ln K a(kJ mol−1)(kJ mol−1)

3.43

−

2.62

n-Octane

3.98

17.2

−

−

3.05

10.9

−

−

8.16

43.4

−12.5

Cyclohexane

2.45

10.4

−13.6

22.2

1.88

6.7

−b

−b

5.92

33.0

−12.5

15.6

Chlorobenzene

1.20

8.1

−13.0

22.6

0.77

5.9

−12.4

13.8

7.74

40.5

−13.4

25.7

At 298 K. b Not determined.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

Figure 5.

Schematic illustration of the thermodynamic properties
relating to the retention and surface diffusion of sample
molecules on the stationary phase surface. Reproduced
from [9] with permission.
Figure 6.

sion is a mass transfer process of adsorbate molecules under adsorbed state. The molecules do not need to be completely desorbed
from the surface to the bulk liquid phase. Some characteristics andComparison of the dependence of surface diffusion coefficient on the mobile phase composition with that ofmolecular diffusivity. Reprodeced from [44] with permission.

mechanism of surface diffusion have ordinarily been investigatedon the basis of an Arrhenius type equation [5].

sion by analyzing (1) the dependence of Ds on the retention
strength [38], (2) the correlation between Ds and Dm [40], (3) some(16)

thermodynamic and extrathermodynamic properties of surface diffusion [30, 31, 33] on the basis of the absolute rate theory [41], andwhere α is the ratio Es/(−Qst), R the gas constant, and T the abso-(4) the difference of surface diffusion in the liquid-solid (RPLC)

lute temperature. However, Table 2 indicates that Es is larger than -and the gas-solid (GC) systems [16, 19, 20]. These are new ap-

Qst in the RPLC systems [6]. This means that surface diffusion does

proaches for the study on surface diffusion.
Figure 6 [44] illustrates the influence of the composition of

not take place.
Another subject is related with the value of Ds under weak reten-methanol (φ) in the mobile phase ranging from 60 to 80 vol% on Ds

tion conditions. It has been observed that the frequency factor ofand Dm. It is indicated that Ds increases with increasing φ irrespec-

−3

−2

2

−1

surface diffusion (Ds0) ranges between ca. 10 and 10 cm s intive of the sample compounds and temperature. In addition, mostthe RPLC systems [6]. Equation (16) indicates that, if the value ofdata points fluctuate around the solid straight lines. Figure 6 also

α is constant, Ds approaches Ds0 when Qst tends toward zero. Thisshows the correlation of Dm in the mobile phase solvents of the dif-means that Ds is several orders of magnitude larger than molecularferent compositions. The Dm values were estimated by the Wilke-

diffusivity (Dm) when the sample retention is weak because Dm is

Chang equation [13]. All the plots of Dm are located on the linear

usually of the order of ca. 10−6−10−5 cm2 s−1 in liquid phase systems

dashed lines, of which the slope is almost unity. Similar to Ds, Dm[1, 3−5, 13]. This conclusion is unreasonable because surface diffu-

increases with increasing φ. This result is reasonable because thesion must be slower than molecular diffusion. Neither theory nor

viscosity of an aqueous solution of methanol decreases almost line-

model has been developed for comprehensively explaining the un-arly with increasing φ in the range between 60 and 80 vol% [3].

reasonable experimental results, i.e., Ds larger than Dm and the cor-

The two solid lines are the extrapolation of the corresponding

relations between Es and Qst.

dashed lines, suggesting that the variation in Ds due to the change

In this study, we analyzed some experimental data from various

in φ results primarily from the corresponding variation in Dm. The

points of view in order to clarify some important characteristics

results in Figure 6 demonstrate that the influence of the mobile

and mechanism of surface diffusion [35−40]. At first, the influencephase composition on surface diffusion and molecular diffusion areof temperature and the amount of adsorbate adsorbed on Ds has or-similar and that there is an intimate correlation between Ds and Dm.

dinarily been studied [18]. Additionally, we studied surface diffu-

The same conclusion is also obtained in other RPLC systems [35−―８―CHROMATOGRAPHY, Vol.31 No.1 (2010)

Figure 7.

Kanji Miyabe

Ratio of the surface diffusion coefficient to the molecular diffusivity versus the adsorption equilibrium constant. Inset: enlargement of the plots in the range of thesmall K values. Reproduced from [38] with permission.
Figure 8.

40, 44].
Figure 7 [38] illustrates Ds as a function of K . The inset showsthe enlargement of the plots in the range of K smaller than 5. Fig-

Correlation between the ratio of the surface diffusion
coefficient to the molecular diffusivity and the isostericheat of adsorption. Reproduced from [39] with permission.

ure 7 indicates three important points concerning the correlation

there is an intimate correlation between surface diffusion and mo-

between Ds and K . At first, all the data measured under various

lecular diffusion and that surface diffusion is regarded as the mass

RPLC conditions are plotted around a single curved line. Thistransfer, which is originally similar to molecular diffusion, but re-means that the mechanism of surface diffusion is similar irrespec-

stricted in the potential field of adsorption. This conclusion is quite

tive of the sample compounds and the modification conditions ofcontrast with the ordinary concept and model of surface diffusion

the stationary phase surface. Second, Ds decreases with increasing[4, 5, 42, 43]. It has commonly been recognized that surface diffu-

K , suggesting that surface diffusion is restricted by the retention

sion is completely different from molecular diffusion and that there

strength. Finally, although the extrapolation is not so exact, it

is no correlation between the two mass transfer mechanisms. On

seems that Ds approximately approaches corresponding value of Dm

the other hand, the thermodynamic study on surface diffusion indi-

as K decreases. This result also suggests that surface diffusion iscates that Es consists of the contributions of the hole-making andintimately correlated with molecular diffusion.

jumping steps, which are correlated with the evaporation energy

The ratio of Ds to Dm is plotted against −Qst in Figure 8 [39]. Al-(∆Ev) of the mobile phase solvent and −Qst in liquid-solid phasethough Figure 8 shows some scatter, the plots fluctuate around asystems, respectively [35, 36]. In addition, the comparison of thesingle straight line, indicating the presence of a linear correlation

surface diffusion data measured in both the RPLC and GC systems−1

between ln (Ds/Dm) and −Qst. The ratio Ds/Dm at Qst = 0 kJ mol isprovided the information about the influence of the mobile phase

0.54. Although Ds is not completely equal to Dm at Qst = 0 kJ mol−1,

solvents on the chromatographic behavior in RPLC [16].

it is likely that Ds is of the same order of magnitude with Dm. Fig-

On the basis of the experimental results, a surface-restricted mo-ure 8 also indicates the restriction of surface diffusion due to thelecular diffusion model [6−8, 28, 35−40, 44] was proposed as asample retention and the intimate correlation between surface dif-first approximation for the mechanism of surface diffusion and for-fusion and molecular diffusion. It seems that the small difference

mulated by applying the absolute rate theory [41].

between the intercept and unity is due to some specific structural(17)and chemical conditions of the mobile phase solvent in the vicinityof the stationary phase surface, in which surface diffusion takes
place. It is easily imaged that the conditions near the stationary

where Em is the activation energy of molecular diffusion and β thephase surface are different from those in the bulk mobile phase.

proportional coefficient. In the absolute rate theory [41], it is as-

The results of these studies ultimately lead to the conclusion thatsumed that surface diffusion consists of two mechanisms, i.e., a―９―CHROMATOGRAPHY, Vol.31 No.1 (2010)hole-making process and a jumping (bond-breaking) one. In Eq.

kJ mol−1. This means that Es is smaller than −Qst when −Qst is larger(17), Em and β(−Qst) correspond to the contribution to Es of the holethan about 40 kJ mol−1. The explanation is supported by a few ex--making and the jumping (bond-breaking) one, respectively [41].

perimental data in Figure 9. By contrast, the value of −Qst in the
RPLC systems usually ranges from ca. 10 to 20 kJ mol−1, as listed
in Table 2, which is smaller than the threshold. On the other hand,

5.2. Interpretation of various correlations between Es and Qst

there is no solvent effect on Qst in the gas-solid phase adsorption.

Figure 9 [28] illustrates the experimental data of Es against −Qst.
As listed in Table 2 [28], the values of Es are larger than those of −This is the reason why the ratio of Es/(−Qst) in gas-solid adsorption

Qst in the RPLC system consisting of a C18-silica gel column andis smaller than unity as listed in Table 2 [28, 30].

aqueous mixtures of methanol or acetonitrile (70 vol%). A similar

The surface-restricted molecular diffusion model is also effective

situation is also observed in other liquid-solid adsorption systems

for explaining the second subject concerning the Ds values under[6, 28]. No surface diffusion is expected under such conditions. On

weak retention conditions. Equation (17) suggests that Ds is equal

the contrary, Es smaller than −Qst has also been reported even in

to Ds0exp(−Em/RT ) under such conditions. This value is of the sameliquid-solid phase adsorption [6, 28]. The contradictory two corre-

order of magnitude with Dm. This is consistent with the results inlations between Es and Qst in liquid-solid phase systems can be ex-

Figures 7 and 8 that Ds approaches the corresponding value of Dm

plained on the basis of Eq. (17).

when K and Qst tend toward zero. In addition, the surface-restricted

The two dashed lines in Figure 9 indicate that Em is around 15 kJ

molecular diffusion model is also effective for the quantitative ex-

mol−1. This value is reasonable because it is not required to com-planation of some intrinsic characteristics and mechanism of sur-pletely eliminate solvent molecules when the hole is constructed

face diffusion [6−8, 28, 35−40, 44].

for diffusive molecular migration. The ratio of Em/∆Ev is about 1/3−
1/2 because ∆Ev of various solvents is usually ranging from about

6. Advance in analytical strategy of chromatographic behavior

30 to 40 kJ mol−1. On the other hand, the dashed lines indicate that

Different types of separation media having various structural

β is about 0.5. This result is also reasonable because it is not re-characteristics have been developed for attaining fast chromatogra-

quired that molecules adsorbed are completely desorbed from thephy with high efficiency, e.g., monolithic stationary phases [45],stationary phase surface to the bulk mobile phase when they mi-pellicular (shell) particles, and non-porous particles. Alternatively,

grate by surface diffusion. It is a diffusive molecular migration

ultra-high pressure LC systems have also been developed for thisprocess under adsorbed state.

purpose, which use columns packed with full-porous spherical particles of the diameter around 2 µm. We need to quantitatively ana-

The dashed lines intersect the diagonal line at around Qst = −40

lyze chromatographic behavior in the different stationary phases
under very high efficiency conditions. Some items of information
relating to the topics are picked up in this section.
6.1. New moment equations for chromatography
About 30−40 years ago, the moment equations have alreadybeen proposed for the conventional columns packed with fullporous spherical particles [3−8, 11, 12]. However, they cannot be
used for the different separation media because their structuralcharacteristics are significantly different from those of full-porous
spherical particles. The author has systematically developed aframework of new moment equations for chromatography using
the various types of packing materials having different structuralcharacteristics, i.e., shape (spherical particle, cylindrical fiber, flat
plate, and hollow tube) and porous structure (full-porous, superficially porous (pellicular or shell), and non-porous) [27, 45−48].
Figure 10 [47] shows the schematic illustration of the structuralcharacteristics of the different packing materials.
Figure 9.

A set of basic equations of the GR model of chromatography,

Correlation between activation energy of surface diffusion and isosteric heat of adsorption. Reproduced from[28] with permission.

which represent the mass balance and the mass transfer kinetics ina column, were analytically solved in the Laplace domain. The mo-―１０―

CHROMATOGRAPHY, Vol.31 No.1 (2010)

Kanji Miyabe

ment equations in the real time domain for µ1 and µ2’ of elution

between the packing materials. The diffusion distance (ξ) is also

peaks were derived from the analytical solution in the Laplace do-different depending on the shape of the separation media, i.e., ra-main. For example, Equations (18) and (19) are the moment equa-dius for the spherical particles and cylindrical fibers, and thickness

tions of µ1 and µ2’ for chromatography using the full-porous sta-for the flat plates.

tionary phases, i.e., spherical particles, cylindrical fibers, and flat
6.2. Peak parking method

plates [47, 48].

It was demonstrated that the peak parking (PP) (stopped flow orarrested flow) method is effective for the quantitative analysis of(18)

chromatographic behavior, as well as the pulse response (PR)method. The PR method has been used as one of powerful strategies for the kinetic study on chromatography. It is carried out underdynamic conditions by intentionally selecting a range of moderate
to high flow velocities of the mobile phase solvent. A nonequilibrium situation of the solute distribution between the mobile
and stationary phases is intentionally generated. It is a “dynamic(19)

method” because the information about intra-stationary phase diffusion is derived by analyzing the band broadening, which takes

where τ is the width of the rectangular sample injection pulse, u theplace under the non-equilibrium conditions. On the other hand, theinterstitial velocity, ξ the diffusion distance, and θ and σ the nu-

PP method is a “static method”. In contrast with the PR method,merical coefficients. At first, the first moment equation is com-the band broadening due to axial molecular diffusion in the PP ex-

pletely identical irrespective of the shapes of the packing materials.

periments takes place under equilibrium conditions of the solute

On the other hand, the second moment equation depends on theirshapes, although their basic formula is the same for all the packing
materials. Only two numerical coefficients, i. e., θ and σ, which areattached with the external mass transfer coefficient (kf) and theintra-stationary phase diffusivity (De) are systematically changed, i.
e., 3 and 15 for the spherical particles, 2 and 8 for the cylindrical fibers, and 1 and 3 for the flat plates. It seems that the θ value well
represents the dimension of the mass transfer in the stationary
phases. The ratio of σ to θ is also systematically changed, i. e., 5, 4,
and 3. The numerical values must reflect the geometrical difference

Figure 10.

Schematic illustration of the structural characteristics of various packing materials and separation media. Reproduced from [47]with permission.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

distribution between the stationary and mobile phases because
there is no convective flow of the mobile phase during the peakparking period (tp) [10, 49−51].
Figure 11 [51] illustrates the comparison of the PP method with
the PR method. In the case of the PR approach, the chromatographic process is continuously conducted after the injection of asmall sample pulse. On the other hand, in the case of the PP
method, the elution of the sample band identically injected into thecolumn is intentionally interrupted for a while in the course of thechromatographic process. The sample band naturally diffuses in the
Figure 11.

longitudinal direction of the column during tp. Subsequently, theelution chromatography is resumed and continued until the chromatographic peak is completely eluted. There are some differencesin the elution peak profiles measured by the two methods. At first,
as illustrated in Figure 11, the difference in the elution time of thetwo peaks is equal to tp. Second, the elution peak measured by the
PP method is broader than that by the PR method (dotted peak in
Figure 11). The information about the mass transfer kinetics in thestationary phase is derived by analyzing the additional band broadening due to the diffusive axial molecular migration during tp.
The characteristics of the two methods are contrary with each

Schematic illustration of the band broadening phenomena and the elution peak profiles in the pulse response and peak parking experiments. (a) Intermediate situation in the pulse response method. The mobile phase flows continuously through the column, at
a constant flow rate. (b) Intermediate situation in thepeak parking method. The mobile-phase flow is arrested for the peak parking period. (c) Elution peak
profile in the pulse response method. (d) Elution peak
profile in the peak parking method (solid line) compared to the elution peak profile in elution, shifted forthe peak parking time (dotted line). Reproduced from[51] with permission.

other because their basic concept for the analysis of molecular dynamics is completely opposite. Table 3 [51] lists the comparison of

Table 3. Comparison of the peak parking − moment analysis method with the pulse response − moment analysis method
Pulse response − moment analysis method

Peak parking − moment analysis method[Disadvantages]
1. The procedure of the data analysis is relatively complicated.
2. Estimation of kf and Dp is required for the data analysis.
3. Accurate values of µ1 and µ2’ are essential.
4. Some corrections are necessary to determine the accurate valuesof µ1 and µ2’.
(1) Asymmetric (tailing and fronting) peak profiles(2) Extra-column pipes

5. It is hard to derive the kinetic information about the mass transfer in small packing materials.
6. Acquire of many experimental data is preferable.

[Advantages]
1. The procedure of the data analyses is quite simple.
2. No estimation of kf and Dp is required for the data analysis.
3. Accurate values of µ1 and µ2’ are not necessarily essential.
4. The corrections are not necessarily required.
(1) The asymmetric peak profiles and extra-column pipes provide
little influence on the data analysis.
(2) The longer the peak parking time is, the smaller their influenceon the results becomes, because the sample band symmetricallybroadens out in the axial direction of the column during the peakparking period.

7. The experimental procedure is sometimes tedious and time−consuming.
8. Accurate information about the size of the packing materials is
required.

5. The kinetic information about the mass transfer in small separation media can be obtained.
6. The kinetic information can be derived from a small number ofexperimental data.
7. Although the experimental procedure is time−consuming, it iseasy to conduct and labor-saving.
8. No information about the size of packing materials is required
for the data analysis.

[Advantage]
1. The kinetic information about the mass transfer in large packing
materials can be obtained.

[Disadvantage]
1. It takes long time to derive the kinetic information about themass transfer in large separation media.

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CHROMATOGRAPHY, Vol.31 No.1 (2010)

Kanji Miyabe

some characteristics of the PR and PP methods. It is obvious thatciency [3]. The column radial heterogeneity is an important source

the two methods are complementary to each other. A disadvantage

of asymmetry of elution peaks. We do not know how columns canof one method can be covered by an advantage of the otherhomogeneously be prepared. It is probably impossible to com-method. The combination of the two methods leads to a compre-

pletely eliminate this source of peak asymmetry due to the radial

hensive strategy for the kinetic study on chromatographic behavior

distributions of the flow velocity and the column efficiency.

irrespective of HPLC conditions. According to experimental condi-

At first, we considered the mechanism of the peak distortion bytions and research objects, we can choose one of them as a suitablea numerical approach [54−57]. The numerical calculation of elu-

approach. The PP method has been applied to the analysis of sur-tion peaks shows how the peak asymmetry is related to the charac-

face diffusion [51] and to the experimental measurements of vari-teristics of the distributions of the flow velocity and local column

ous kinetic parameters as follows.

efficiency in the radial direction of the column. Then, we developed an inverse numerical method, which allows the estimation of
the true column efficiency and of the column radial heterogeneity

6.3. Development of various related methods for the kinetic

from some parameters of peaks experimentally recorded even whenstudy on chromatography

they show asymmetrical profiles [57].

In order to establish a framework of the experimental methods
for the kinetic study on chromatography, we developed some re-7. Conclusion

lated methods, which are essential for the accurate analysis of chromatographic behavior. For example, the methods for the experi-

Still now, the development of HPLC technologies quite rapidly

mental measurement of the following kinetic parameters were de-proceeds. In the future, chromatographic behavior could not com-veloped, i.e., (1) molecular diffusivity in liquid phase (the PP

pletely be represented by the conventional models nor properly

method using capillary flow channels) [52], (2) external mass

analyzed on the basis of the ordinary HPLC theories. It must be re-transfer coefficient (the PR method using a column packed withquired to establish an appropriate analytical strategy for the ad-

non-porous particles and the combination of the PR and PP meth-vanced chromatography and to more accurately understand the fun-ods using a column packed with full-porous particles) [53], and (3)damentals of chromatographic separation. In this review, the MA

pore diffusivity (the PP method in SEC mode using benzene andmethod was introduced as one of the candidates for constructing

polystyrene of large molecular weight).

the effective strategy for the kinetic study on chromatography. The
new theories and models must contribute to the development ofsome innovatory technologies of chromatography from theoretical

6.4. Numerical method for the analysis of asymmetrical peak

points of view. The author wishes that some results of this study

profilesA numerical approach was proposed for the estimations of thecontribute to the further progress of fundamental research works

true efficiency and of the radial heterogeneity of a column from theconcerning the mechanism of HPLC separations and to the more

asymmetrical (tailing and fronting) profiles of elution peaks experi-

accurate understanding of some essential characteristics of chroma-

mentally measured [54−57].

tography.

Experimental chromatographic data are usually analyzed on the
Acknowledgment

assumption that the peak profiles under linear equilibrium isotherm
conditions are also represented by the Gaussian distribution func-

The author would like to express his sincere gratitude to Profes-tion. In addition, most theoretical models of chromatography are

sor Georges Guiochon (The university of Tennessee and Oakbased on this assumption. In practice, however, asymmetrical (tail-

Ridge National Laboratory). He kindly evaluates the researching or fronting) peaks are frequently observed even under linear

works and gives him a great number of beneficial suggestions andisotherm conditions. In such a case, the information about the true

many pieces of instructive advice as a co-researcher. The author is

performance and characteristics of chromatographic behavior can-also grateful to Professor Katsumi Goto (Professor Emeritus, Uni-

not properly be evaluated from the analysis of the asymmetricalversity of Toyama) and Professor Motoyuki Suzuki (Professor

peak profiles because these profiles are not Gaussian curves. We

Emeritus, The university of Tokyo) for their kind supervision. The

need to develop a new strategy for appropriately analyzing asym-

author gratefully acknowledges all collaborators, coworkers, andmetrical peak profiles in order to derive the accurate informationcolleagues for their kind supports and comments in the course ofabout chromatographic behavior because the packing structure of

the research works.

most columns are not homogeneous in the column radial direction.

Financial supports from the Ministry of Education, Culture,

It has been shown previously that most columns exhibit the radial

Sports, Science and Technology, the Japan Science and Technol-

distributions of the mobile phase flow velocity and the local effi-

ogy Agency (JST), University of Toyama, and also from private―１
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CHROMATOGRAPHY, Vol.31 No.1 (2010)

5171.

companies are gratefully acknowledged.

[25] Miyabe, K. Anal. Chem. 2002, 74, 2126−2132.

The author would like to make a grateful acknowledgment for
the Society for Chromatographic Sciences (SCS) for presenting the[26] Miyabe, K.; Okada, A. Analyst 2002, 127, 1420−1426.

Award of SCS in 2009 to him and for giving him an opportunity to[27] Miyabe, K.; Cavazzini, A.; Gritti, F.; Kele, M.; Guiochon,
G. Anal. Chem. 2003, 75, 6975−6986.

publish this review paper.

[28] Miyabe, K.; Guiochon, G. J. Chromatogr. A 2010, 1217,
1713−1734.

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