Chapter 1 General theory of chromatography

1

Chapter I

General theory of chromatography 1 . I . CHARACTERIZATION OF THE CHROMATOGRAPHIC METHOD Chromatography, discovered in 1903 by a Russian scientist Tswett [ l ] , is a variation of a dynamic sorption process in a two-phase system in which a mixture of substances migrating with a gas or solvent flow through a porous medium is separated into single components according to their sorption activity. Depending on the type of mobile phase, chromatography is divided into gas and liquid chromatography and, depending on the sorbents used as the stationary phase, it is divided into partition chromatography (a liquid on the surface of an inert hard support), adsorption chromatography (a very porous sorbent), ion-exchange chromatography (on ion-exchange resins) and gelpermeation chromatography (on macroporous inert sorbents). Gas chromography is used to separate volatile substances, and liquid chromatography is employed for the analysis and fractionation of thermally labile and non-volatile substances. The separation of substances in liquid chromatography is based on their interaction with the sorbent (one-phase chromatography being an exception). In the mid 1970s coarse-grained sorbents based on silica, aluminium oxide, porous polymers and ionexchange resins were replaced by sorbents with microparticles of irregular shape and microspherical silica or polymer sorbents thoroughly fractionated according to particle size, with dp from 5 to IOpm, including sorbents with the surface modified by organic groups. These sorbents exhibit low resistance to mass transfer and, adequately packed into columns, determined the progress in modern high-performance liquid chromatography (HPLC). At present, HPLC uses the following types of interactions between the macromolecules of substance and the sorbent, and the corresponding types of liquid chromatography can be distinguished: adsorption interaction with a non-polar hard sorbent (liquid-solid chromatography, LSC); hydrophobic interactions with bonded non-polar groups of the sorbent (reverse-phase chromatography, RPC); interaction with the sorbent with bonded non-polar groups (normal bonded phase chromatography, N-BPC); ion exchange (ion-exchange chromatography, IEC); exclusion interaction with a macroporous sorbent (exclusion chromatography, EC). The importance of all these types of chromatography in practical analytical work is constantly changing. Recently, the use of RPC has aroused continuous interest. Thus, according to R.E. Majors (1980) the use of various types of liquid chromatography in chemical and biological investigations was distributed as follows: LSC 5%

RPC 65%

N-BPC 10%

JEX

10%

EC 10%

Total 100%

The predominant interest in RPC is due to the availability and high quality of sorbents for this type of chromatography (mainly octyl and octadecyl silica gels),

2

GENERAL THEORY OF CHROMATOGRAPHY

TABLE 1.1 TECHNIQUES O F REVERSE-PHASE CHROMATOGRAPHY ~~

~

Primary name

Other name

Typical mobile phase

Regular

'Normal'

(A) Water + water-miscible organic solvent, e.g. acetonitrile, methanol, dioxan

Ionization control

Solvophobic chromatography

(B) As in (A) + buffer (C) As in (A) + acid or base; For weak acids, as in (A) plus acid (e.g. phosphoric, perchloric); For weak bases, as in (A) plus base (e.g. carbonate, dilute NH,)

Ion suppression

Ion pair

Soap chromatography Paired ion chromaWPhY Ion pair partition

(D) For cations as in (B) + alkyl sulphonate or sulphate (e.g. C, sulphate) For anions, as in (B) + tetraalkylammonium salt (e.g. tetrabutyl ammonium chloride)

Secondary chemical equilibria

E.g. Argentation chromatography (Ag+)

(E) As in (B) plus metal reagents, silver ion (for olefins), ligands

NARP

Non-aqueous reverse phase

(F) Automatic or methanol + tetrahydrofuran or methylene chloride

high reversibility of sorption, good mass exchange and versatility of application. The latter is mainly due to the modification of the mobile phase. Table 1.1 (by Majors) gives various types of RPC related to the modification of the mobile phase and ensuring the universal use of this method. All these types of RPC have been widely used in the analysis of lowmolecular-weight hydrophilic substances, peptides and proteins. The modification of the mobile phase in which the dissociation of ionogenic groups is suppressed and the charges are shielded by increasing the ionic strength of the eluent, leads to increasing retention. The dissociation of ionogenic groups also affects the conformation of macromolecules. This, in turn, can lead either to the increase or decrease in retention on alkyl-modified silica gels. This provides great possibilities of obtaining highly selective RPC systems for proteins and large peptides. Unfortunately, no use has yet been found for RPC in the analysis of synthetic polymers. We should like to think that this type of chromatography, in particular its NARP variation, will be useful in the analysis of polymers, mainly in analysis of the compositional inhomogeneity of copolymers and the functionality of oligomers. At present the chromatographic analysis of synthetic polymers is virtually entirely based on exclusion chromatography with respect to the determination of MWD, and on adsorption chromatography with respect to the study of the polydispersity of the chemical structure of macromolecules. Only these types ofliquid chromatography of polymers are considered in this book.

CHARACTERIZATION OF THE METHOD

3

The chromatographic separation of mixtures is generally carried out by using columns packed with granulated sorbent, plates coated with a thin layer of the sorbing substance, films made of porous materials and specific grades of paper. A chromatographic system consists of stationary and mobile phases. Owing to a tendency towards thermodynamic equilibrium, interphase exchange of the molecules of the substances to be separated takes place. During this exchange directed diffusion flows pass from one phase into the other simultaneously with random walks of molecules in each phase. In the stationary phase these walks are only of a thermal nature, whereas in the mobile phase they are also affected by hydro(aer0)dynamic conditions of the flow of a solution or a gas mixture. During their movement along the chromatographic system the molecules of the substances being separated reside for some time in the stationary phase (where their average rate in the direction of the movement is zero) and for some time in the mobile phase (where they move at the same rate as the phase). The passage of the molecules from the mobile phase into the stationary phase is called ‘sorption’ and the reverse process is called ‘desorption’. When adsorption interaction occurs, molecules are adsorbed on the surface of the sorbent (in adsorption chromatography). When there is no adsorption interaction, sorption consists of the residence of the molecule in the stationary phase (for a porous sorbent in its pore volume, as, for example, in gel-permeation chromatography). Molecules of substances undergoing chromatography spend different periods in the phases of the chromatographc system. This time depends on the chemical nature of the molecules and the ratio of their size to that of the pores of the sorbent. The former determines their interaction with the sorbent under the chromatographic conditions employed and the latter determines the penetration of the molecules into the pore volume. The molecules that are more readily sorbed reside in the stationary phase longer than those that are less easily sorbed. As a result, they move along the chromatographic system more slowly, and thus the separation of mixtures into individual components is effected. Simultaneously, the spreading of the zone of each component takes place. This is due to the thermal motion of the molecules, the stochastic (random) and non-equilibrium character of the chromatographic process and the existence of longitudinal and transverse flow-rate profiles in the mobile phase. If the spreading rate of the zones of the components is lower than the difference between the rates of their movement along the chromatographic system (this can be easily accomplished experimentally), these components can be chromatographically separated from the mixture. This condition provides the basis for the chromatographic separation of substances and is always satisfied in a chromatographic experiment. In the early stages of the development of chromatographic methods, Martin and Synge [2] proposed a model for a simple description of the chromatographic process and the characterization of its efficiency. In this model the chromatographic column was regarded as a system consisting of a large number of elementary layers of finite thickness or ‘theoretical plates’. The thickness of the layer H , called the height equivalent to a theoretical plate, HETP, is determined by the spreading of the chromatographic zone and serves as its measure: this is the dispersion ui per unit column length L

H = ui/L

GENERAL THEORY OF CHROMATOGRAPHY

4

The number of theoretical plates N in the column shows its efficiency

N = L/H In practice the H and N characteristics were found to be very convenient and are generally used, although the plate model itself has many drawbacks and does not meet present-day requirements. Usually in column chromatography the value of N is determined as follows:

N = V2/u$ (1.3) where V is the retention volume and u$ is its dispersion. The plate model may be justified by the probabilistic nature of processes leading to zone spreading. Spreading can be regarded as a result of some quasi-diffusion process occurring on the background of zone movement along the column with the solvent flow. Then a model of one-dimensional random walk with a step of length H can be used for the quantitative description of this phenomenon. In the earlier steps, substance distribution in the zone corresponds to Poisson’s law, and when the number of these steps is relatively large, one can regard it as the Gaussian distribution and use eqns. 1 .l-1.3. In this case each step of a random walk corresponds to one of the N layers into which the chromatographic system of length L is tentatively divided. This correspondence is equivalent to the assumption that in each of these layers zone spreading does not occur: it occurs abruptly at each step on passing from one layer into the next. The rnodern’theory of chromatography is based on physical laws related to the stochasticity of the chromatographic process and the peculiarities of its hydro(aero)dynamics, kinetics and statics [3-101. The stochasticity of this process is due to the probabilistic distribution of the molecules of the substance to be analysed between the phases of the system [8-lo]. After an appropriate normalization the values of concentration of the substance in each phase serve as distribution functions. The ratios of these concentrations are used t o determine the probability of the passage of the molecules from one phase to the other. One can conceive chromatographic experiments in which only one molecule of the substance takes part. Then, owing to the stochasticity of the chromatographic process, in different experiments the molecule will emerge from the system after different times rather than after a single fixed time interval. The probability of each elution time is determined by a distribution law identical to that of the distribution of the molecules according t o their times of elution from the system in one experiment in which many identical molecules undergo chromatography. The stochastic character of the chromatographic process permits a clear description of the kinetics in which the exchange of the molecules between the phases is expressed by the probabilities of their sorption and desorption. Let the probability of the sorption and desorption of one molecule of the ith kind per unit time be XI and A;, respectively. Then the kinetics of the sorption-desorption process by using the probabilities Xi and Xi [ 1I ] are described by the equation

METHODS OF THE DESCRIPTION OF THE PROCESS

5

In the simplest case, when the probabilities Xi and hi d o not depend on concentrations, the kinetics of the process are called linear. When they depend on concentrations, the kinetics become non-linear. 1.2. METHODS OF THE DESCRIPTION OF THE CHROMATOGRAPHIC PROCESS FOR SOME SPECIFIC CHROMATOGRAPHIC SYSTEMS In order to describe the chromatographic process completely, the balance equation taking into account the hydro(aer0)dynamic features of the process should be combined with kinetic eqn. 1.4:

ac'

p--'

at

=

(YXiCi

-px;c;

u'

where is the flow rate of the solvent in the channels of the mobile phase of the column and Di the 5oefficient of molecular diffusion in the solution of molecules of the ith kind. The term (U V q ) takes into account substance transfer in the mobile phase and its convective mixing, the term Di*Aci represents diffusion spreading and the term @/a)/ (acl/at) expresses mass exchange between the phases. The system of eqns. 1.5 can be simplified if it is assumed that the flow rate U is constant both across the column section and along the column and that all deviations from U = const. are considered as random and are described by a quasi-diffusion equation with the coefficient of longitudinal quasi-diffusion bLi:

ac; = - hici-h;c; at

P

The thin-layer chromatographic process can be described by a system of eqns. 1.7:

As in the column variation, we need not introduce into this system the flow-rate profile in transverse directions, i.e. throughout the thickness of the layer and along its width, and may take into account the spreading due to it by using some coefficients of quasi-diffusion D * . Moreover, in contrast to the column variation, the spreading in two directions (along the length of the plate and along its width) should be considered and it should be remembered that the rate changes in the longitudinal direction. In other words, from the standpoint of chromatographic spreading the process of TLC should be considered as two-dimensional.

6

GENERAL THEORY OF CHROMATOGRAPHY

1.3. KINETICS OF THE CHROMATOGRAPHIC PROCESS Simplification in the description of the chromatographic process considered in section 1.2 is related to various models of its hydro(aer0)dynamics. In many variations of chromatography the descriptions of the kinetics of the process may also be simplified. For this purpose the exchange of the molecules of the substance between the phases of the chromatographic system may be considered as a heterogeneous process if the transformations occurring at the interface are regarded as heterogeneous. Heterogeneous processes consist of several stages. The first stage is the transfer of the particles taking part in the process to the site of heterogeneous transformation. In chromatography this is the transfer of the molecules of the substance being investigated to the interface as a result of molecular diffusion and a combination of hydro(aer0)dynamic factors. In the second stage the heterogeneous reaction proper takes place. In chromatography this is the sorption-desorption of the molecules being eluted. In the third stage the reacted particles leave the reaction site. In chromatography this means that the sorbed or desorbed molecules move away from the interface. The overall rate of the heterogeneous process is governed by the rate of the slowest stage. When the movement of the reagents to or from the interface is the slowest stage, it is said that the reaction is characterized by diffusion kinetics. If the stage of chemical or physical transformation is the slowest stage, it determines the reaction rate. When the rates of the transfer of the reagents and those of their transformations are comparable, these reactions are termed ‘heterogeneous reactions of a mixed type’. Most chromatographic processes in which the heterogeneous transformation consists of the passage of the eluted molecules from the mobile phase into the stationary phase and vice versa are characterized by diffusion kinetics. In adsorption chromatography this passage is accompanied by energetic interaction with the sorbent surface. The diffusion kinetics in chromatographic processes, in turn, consist of two stages called external and internal. In the stage of external diffusion the molecules move in the mobile phase and are adsorbed after reaching the interface. In the stage of internal diffusion they are located inside the sorbent grains. If one of these diffusion stages proceeds much more slowly than the other, the process is limited by the former stage which determines the sorption rate. Most ion-exchange chromatographic processes are characterized by intradiffusion kinetics. In the chromatography of polymers the stage limiting the process depends on the choice of the polymer-sorbent-solvent system, on the porosity of the sorbent, the accessibility of its inner zones to the macromolecules being investigated, the flow rate of the solvent, the solution concentration and the temperature. The character of the kinetics can often be determined from the value of the distribution coefficient of the substance between the phases of the chromatographic system

K d = c’/c

(1.8)

where c and c’ are average (across the section) substance concentrations in the mobile and the stationary phases of the chromatographic column. Low values of Kd (Kd Q 1) show that the pore volume of the sorbent is either almost inaccessible to the macromolecules or very small. In particular, it may be inaccessible

METHOD OF STATISTICAL MOMENTS

7

because the pores are small or the macromolecules are thermodynamically incompatible with the sorbent matrix. At Kd Q 1 the time of existence of the macromolecules in the stationary phase during each sorption-desorption cycle is short compared to the time of their existence in the channels of the mobile phase of the system. Hence, it might be expected that at low values of Kd the kinetics of the process are limited by the stage of external diffusion. With increasing K d the significance of this stage decreases and that of the internal diffusion stage increases. At Kd greater than unity the latter stage becomes the limiting stage. These kinetic features of the chromatographic process are adequately described by eqn. 1.4. The values of the probabilities of sorption and desorption in this equation are inversely proportional to the average times of the existence of the molecules in each phase of the chromatographic system during one sorption-desorption cycle. Hence, they determine the rate of interphase mass exchange in each stage. In the case of equilibrium the left-hand side of eqn. 1.4 becomes zero and the ratio of the probability of sorption to that of desorption becomes proportional to the distribution coefficient under equilibrium conditions [ 111 :

In the absence of equilibrium we have

xf

a

(1.10)

Hence, the following conclusions can be drawn. First, for each section of the chromatographic system regardless of its type (column, thin-layer or film chromatography), a moment exists when the ratio of the flows of the molecules from one phase of the system into the other becomes equivalent to the equilibrium ratio. Until this moment the flow of the molecules from the mobile phase into the stationary phase in a given section predominates and the probability of sorption is higher than that of desorption. After this moment the flow from the stationary phase into the mobile phase predominates and the probability of sorption is lower than that of desorption. Secondly, maxima of concentrations ci and cf in the mobile and stationary phases of the system do not coincide, and for each fixed time moment the equilibrium exchange of the molecules between the phases occurs in a section of the system situated between these maxima. Since the concentration maximum ci moves in front of the cf maximum, it can be said that the concentration of the substance in the stationary phase which is in equilibrium with the concentration in the mobile phase lags behind it in time. 1.4 METHOD OF STATISTICAL MOMENTS IN CHROMATOGRAPHY

All the features of the chromatographic process may be studied by solving the systems of differential equations 1.5-1.7. However, this approach is very laborious and cannot be used in all cases. An alternative method may be used that permits the investigation of the

GENERAL THEORY OF CHROMATOGRAPHY

8

process by finding for systems 1.5-1.7 the statistical moments characterizing substance distribution along the chromatographic system and at its outlet. It is possible to find the moments C(k(t) of substance distribution in the coordinates at a given moment t i n each phase of the system ++m p k ( t ) = XkCk(X, t)dx, &(f) = XkC;(X, t)dx (1.11) -m

-00

and the moments &(x) of substance distributions in time for fixed values of coordinates:

0

CO

0

= c,

cb = c’,

ck#o

= c/po,

ci+O = c’/&

(1.12)

The moments determined in this manner are termed initial moments. It is often convenient to use central distribution momentsMk(t) and &(x):

1

+-=

Mk(t)

=

I

+m

[X-Pl(t)lkCk(X,t)dX,

Mi(t) =

-00

1

--m

[X-P;(t)lkCi(X,t)dX

(1.13)

m

Mk(x) =

[t-Pi(x>lkCk(X,t)dt,

0

ML(X) =

[t-L(x)Ik~r(~,t)dt, 0

k = 2,3, . . .

In many cases when the type of distribution is known, distributions themselves can be reconstructed by using the values of their statistical moments. This is the situation in chromatography. The analysis of systems of eqns. 1.5-1.7 by methods of mathematical statistics carried out with a computer [12] (Fig. 1.l) shows that all unimodal chromatographic distributions may be considered with a high reliability as Pierson’s distributions. A similar analysis of experimental chromatograms of single components leads to the same result (Fig. 1.2). This means that during chromatography the substance is spread out in the ‘Pierson manner’ and its distributions in the chromatographic system (both in

Fig. 1.1. Approximation of the solution of a system of eqns. 1.6 by Pierson’s type VI distribution.

METHOD OF STATISTICAL MOMENTS

Count

9

VR

Fig. 1.2. Approximation of the chromatogram of a narrowdisperse polymer standard (polystyrene in toluene (Mw = 98,200, Mw/Mn < 1.1) by Pierson’s type VI distribution.

coordinates and in time) are distributions belonging to the Pierson family. The Gaussian distribution to which all chromatographic distributions tend in an asymptotic limit, is one of these distributions. The Pierson family also includes y- and /3-distributions, logarithmic distributions and many others, i.e. this family contains most distributions of random values occurring in practice [ 131 . A distinguishing feature of Pierson distributions is the fact that they are four-parameter distributions. These parameters, completely determining the type of distribution and all its peculiarities, are the first four statistical moments. This distinguishes the method of statistical moments from other analytical methods used for the investigation of the chromatographic process and makes it one of the main theoretical methods in chromatography.

1.4.1.Investigation of the dynamics of spreading of the chromatographic zone by the method of statistical moments Study of the spreading of the chromatographic zone makes it possible to understand the main features of the chromatographic process, to determine its mechanism and to elucidate the problems of the interpretation of experimental data. For example, in the chromatography of polymers an understanding of the character of spreading of the eluted zone and its asymmetry is indispensable for the quantitative interpretation of chromatograms in order to determine average molecular weights and molecular-weight distributions of the samples. On the basis of this understanding it is possible to develop highly effective chromatographic systems and to find their optimal experimental characteristics which is very important for further development of chromatographic technology. Zone spreading in chromatography is related to two factors: longitudinal diffusion and mass exchange between the phases of the system. Longitudinal diffusion is usually considered to be a combination of molecular diffusion, convective mixing of substance and spreading related to the rate profile. The overall dynamics of substance behaviour in the chromatograph may be described by a system of differential equations (eqns. 1.5-1.7). Let us consider the system of eqns. 1.6 for a single component:

GENERAL THEORY OF CHROMATOGRAPHY

10

under the foflowing initial conditions: atx

>0

atx

c(x, t)lt=o = 6(x)c,

c

-+ - l o o

* 0,

ac

ax

+

0,

c’

+

0

ac ’

--to

(1.14)

ax

where co = q/a is the concentration impulse introduced into the chromatographic system, 6 (x) is the Dirac delta function and q is the amount of the substance. Applying the Laplace and Fourier transformations to eqns. 1.6, we obtain the following values for time statistical moments of the chromatographic peak at the outlet of the column of length x :

M2(x) = 2(T) h+A’

Bl

Lil(x

+%) +&&

+

%)

(1.15) (1.16) (1.17)

The distribution of the substance in the column at a given moment t is characterized by moments of the x-coordinate. For the mobile phase we obtain (1.18) (1.19) (1.20)

For the stationary phase we have

(1.21) (1.22)

METHOD OF STATISTICAL MOMENTS

Mi(r) =

(1

+

A/A’)3

A’

11

4 u 2 [(A/A’) - 11 3 (1 A/A’)3 A’

+

1

(1.23)

The expressions obtained for statistical moments of concentration distributions r) and c’(x, t ) permit the distribution of the substance in the column to be represented schematically (Fig. 1.3).

c(x,

Fig. 1.3. Distribution of the eluate in each phase of the column.

Analysis of these expressions leads to the following conclusions concerning the dynamics of the movement of the chromatographic zone, the time of elution and the width and shape of the chromatographic peak [ 14-20] : (1)The velocity ZI of the movement of the zone along the chromatographic system is constant and is determined by the ratio of the probabilities of sorption and desorption: (1.24) (2) If the length of the chromatographic system is x, the average time t of the elution of the zone is slightly longer than the value x / v as a result of the additional spreading of the trailing edge of the zone as compared to the leading front, (3) The ratio of probabilities A/A’ increases with the non-equilibrium of the process and this leads to a slower movement of the whole zone (according to eqns. 1.15, 1.18, 1.21). (4) The difference in mathematical expectations of the coordinates of the zone in the mobile and the stationary phases is not equal to zero: (1.25)

where Vo is the volume of the mobile phase, Vo = a V,, V, is the volume of sorbent

GENERAL THEORY OF CHROMATOGRAPHY

12

pores and V, is the volume of the column minus the volume of the matrix of the packing sorbent. This confirms the conclusion drawn from kinetic considerations in section 1.3 that during chromatography the zone of the substance is separated into two regions one of which (that of the stationary phase) lags behind the other (that of the mobile phase) by the value Ax:

(1.26) AX = U7/[ 1 + Kd( Vp/Vo)] Equation 1.25 permits the estimation of the value of zone separation which is the function of parameter T = l/Xr, i.e. of the degree of non-equilibrium of the chromatographic process. (5) Naturally, this separation of the zone leads to its additional spreading determined by the value of the term AM2 : (1.27) in eqns. 1.19 and 1.22. For the same reason the chromatographic peak also undergoes additional broadening. This is shown by the term AM,, :

(1.28) in eqn. 1.16. The presence of the term

2

Kd (vp/vO)2&7 U2

(1.29)

shows that the peak undergoes additional broadening owing to the spreading of the trailing edge of the zone when it is eluted from the column. (6) The fact that the third moments M3(x),M 3 ( f ) and M3((t) determined by eqns. 1.17, 1.20 and 1.23 are not equal to zero shows that the substance distribution in each phase of the column is asymmetrical and the peak is also asymmetrical. The reason for this is the non-equilibrium of the process: all three moments are proportional to parameter 7. Moreover, some contribution to the asymmetry of the peak is provided by the additional spreading of the trailing edge of the zone when it is eluted from the column (point 5). It is expressed by the term

(1.30) in eqn. 1.17. (7) It follows from eqn. 1.17 that the chromatographic peak has a slope towards the leading edge

M3(x)

>0

When the following inequality holds for the ratio of probabilities X and A’:

(1.31)

METHOD OF STATISTICAL MOMENTS

x/x’ >

13

(1.32)

1

the concentration maximum in the mobile phase is displaced in the direction of the movement, i.e. towards increasing x : M 3 ( f )< 0, whereas in the stationary phase it is displaced in the opposite direction: M3(t) > 0. When the following inequality holds:

A/h‘

<

1

(1.33)

the opposite situation is observed. Inequality 1.32 is obeyed for sorbents with a relatively large volume of accessible pores:

v,

= KdVp

s

(1.34)

V,

and inequality 1.33 is valid for sorbents with a small volume of accessible pores: V,, = V, Q Vo . (8) The degree of asymmetry of the peak can be determined by using the following equation:

Kd

A

f

M,”3(x)/M2’z(x)

= sk1I3

(1.35)

where in accordance with eqns. 1.16 and 1.17 A and A T “ ~ ,i.e. the asymmetry of chromatographic distribution decreases with increasing length of the chromatographic system and increases with the non-equilibrium of the process. It should be noted that the sk parameter (peak skewing) is the generally accepted measure of asymmetry. It can be represented in a more explicit form if the chromatographic peak is approximated by the Gaussian convolution with the exponent [21]

A

f(f>=

m

exp

: -

[--:] dt’

(1.36)

where af is the dispersion of the Gaussian function and T i s the time parameter characterizing the deviation of the peak from the Gaussian shape. In this case the expressions for moments become (1.37) and the following expression is obtained for peak skewing: (1.38) It is clear that sk can change from 0 to 2: limsk = 0,

?/lot-* 0

-

lim sk = 2

?/up

GENERAL THEORY OF CHROMATOGRAPHY

14

1.5. ANALYSIS

OF CHROMATOGRAPHIC SPREADING

1S.1. General expression for chromatographic spreading

The elucidation of the main relationships of dynamic behaviour of the chromatographic zone makes it possible to give it a fairly precise quantitative description of its spreading. It consists of the contributions of longitudinal diffusion in the channels of the mobile phase

2rDmt

C J ~=

(1.39)

mass exchange between the phases of the system 2

-

om. ex. -

k'

(1.40)

and the rate (velocity) profile in a column (or a plate) depending on packing and the differences in the cross-sectional areas and lengths of the channels of the mobile phase [ 141

(1.41) In real chromatographic systems the so-called extra-column spreading u& occurs in detectors and various pipelines of the chromatograph (in thin-layer chromatography it is determined by the size of chromatographic spots at the start). The statistical type of the chromatographic process as a whole, and of its individual components leading to the spreading of the chromatographic zone, makes it possible to express the dispersion of this spreading as the sum &

=

2

QD

2 2 + 0m.ex. + 0ve1 + &t.

In terms of the dimensionless characteristics, h = H/d, and v = Ud,/D,, becomes

h~ =

hD

hm. ex.

hvd.

hext.

(1.42) eqn. 1.42

(1.43)

where hD = 2 hm.ex.

~ 1 ~

' _ 1 D, --_ _k_ - -v (1 + k')' 30 D,

(1.44) (1.45)

Dm 1 (1.46) ---f@J Dr dp The h and v characteristics are convenient for practical use. They are dimensionless, h is independent of the sorbent grain diameter and v is related to diffusion mobility of the molecules of the substance. If the values of v are less than unity (v < l), this means that the molecules of the solvent move mainly because of diffusion. The values of v greater than unity ( v > 1) mean that the part played by diffusion in the movement of solvent molecules is less important than that of the flow.

h"d.

=

ANALYSIS OF SPREADING

15

Equation 1.45 can become more precise if it is taken into account that stagnant regions with thickness df surrounding the sorbent particles are present in the channels of the mobile phase. Mass exchange taking place between them and the mobile regions of the mobile phase leads to additional zone spreading. When this is taken into account, eqn. 1.45 becomes (1.47)

-

If appropriate expressions from ref. 1 0 are used for the determination of the diffusion coefficient in the radial direction D, and if it is assumed that f ( d , ) d i , then instead of eqn. 1.46 the expression for hwl. is obtained in the form of coupling [lo] : hvd,

= (I/Ce

(1.48)

+ 1/(CrnV)>-'

where C, C, , C, and C, are structural dynamic constants of the chromatographc layer, C, and C, representing mass exchange (C, also represents convection) and C, representing eddy diffusion [ l o ] . Knox and Parcher [22] have shown experimentally that hvel. can be adequately approximated by the equation (1.49)

hvd. = A u " ~

where A is a constant representing the quality of packing. Packing is considered to be good if A < 2. Hence, each contribution to spreading can be represented by one of the dependences 1.44,1.47 and 1.49 shown in Fig. 1.4. In combination these contributions lead to the extreme dependence of h on u exhibiting one minimum. It is possible to determine the quality of packing of the system h

9-

87.

6. 5-

m

4-

32.

11

I 0

5

10

15

20

v

Fig. 1.4. Reduced HETP, h , vs. reduced flow rate V . Contributions to h : (1) longitudinal diffusion; (2) mass exchange between the phases; (3) flow rate profde in a column.

GENERAL THEORY OF CHROMATOGRAPHY

16

from the position of this minimum: the lower the minimum and the more to the right it is located, the better the packing. Good packing corresponds to the values of h < 3 and u > 2. Combining eqns. 1.44, 1.47 and 1.49 one obtains the equation traditionally called Knox’s equation [22,23] : 2Y h = --+Au”3 U

+cv

(1 S O )

This equation approximates particularly well the experimental dependence of h on u in the range of the values of u close to the position of the minimum. For high-quality columns this range is 1 < u Q 10. Recently, eqn. 1S O has been used more often than other equations describing zone spreading in papers dealing with the optimization of the chromatographic process. Its universal form for all solutes makes it possible in a chromatographic experiment to determine diffusion coefficients of molecules of various substances. 1S.2. Types of the chromatographic process

According to hydrodynamics and mass exchange, four main types of the chromatographic process can be distinguished when chromatography is carried out on packed columns, open-tubular capillary columns [24-261, thin layers and thin porous films. Two of these variations belong to column chromatography and the other two are thinlayer types. In column chromatography the elution rate U is constant and is determined according to Darcy’s law by the pressure drop across the column AP, its length L , sorbent grain diameter d , , permeability ko and solution viscosity r) (1.51)

-

-

For packed columns the permeability ko is profoundly affected by the packin; quality (for good packing ko and for open capillary columns (ko 0.030 t o 0.015) it is determined by their inner diameter d , and solvent viscosity r) [26]. It is clear that in column chromatography the pressure drop cannot be very low. It should be high enough to overcome the viscous friction experienced by the solvent moving along the column. Equations 1S O and 1.5 1 allow an exact evaluation of the lower limit of AP which is called by Guiochon [27] the critical value (1.52) Equation 1.51 easily shows that a limiting efficiency exists for each value of the product APd,’ (1.53) In other words, in column chromatography each chromatograph can be characterized

ANALYSIS OF SPREADING

17

by the value of the limiting efficiency of the analysis carried out with it (for each fixed grain size and for an infinitely long column). In thin-layer chromatography (TLC) the solution moves-alongthe plate (film) due to the effect of capillary forces at a variable rate, U(t) which gradually decreases as a result of viscous friction

U(t) = a

(1.54)

m

where (1.55) Here X is the surface tension coefficient and 6 the contact angle. In this case, in order to describe zone spreading quantitatively, it is convenient to use the elution rate averaged over time t.+t

U(t’)dt’

%

4a2 dP -

r0

(1.56)

L

where t is the time during which the solvent passes the distance L along the plate (film)

1

to+ t

L =

(1.57)

U(t‘)dt’

to

In other respects, from the standpoint of hydrodynamics, mass exchange and permeability, the chromatography on saturated thin layers is similar to that on packed columns and the film variation of TLC is similar to the chromatography employing open capillary columns. It should be noted that in TLC, just as in column chromatography, a limiting attainable value of the efficiency of analysis, Nlim,exists for each fixed grain size and the selected technical parameters. This value can be estimated by using eqns. 1 S O and 1.56. (1.58) It should be remembered that chromatography on open-tubular columns and porous films is distinguished by high permeability of layers and low zone spreading due to interphase mass exchange (because the layers representing the stationary phase are not very thick). Moreover, open capillary columns are capillaries with a diameter d, of 1 to 5 p m and having a porous inner wall with thickness df of 0.1 pm, and there is no spreading due to the coupling effect (eqns. 1.46, 1.48 and 1.49). In this case Knox’s equation (1 S O ) for the reduced HETP is transformed into Golay’s equation [28]

-

2 1+6k’+Ilk’Z h =--f Y 96(1 +k’)*

2

k‘

(1.59)

Hence, in this case the reduced HETP for a specific substance is described by eqn. 1.SO in which the term A Y ” ~has been omitted

GENERAL THEORY OF CHROMATOGRAPHY

18

= 2Y -+a

h

(1.60)

V

There is another variation of chromatography on packed columns in which sorbents with surface-porous particles, called pellicular particles, are used instead of common sorbents. In this case mass exchange in spreading decreases approximately 20 times. For good columns with porous particles the coefficient in the third term of eqn. 1 S O C = 0.05, and for columns with pellicular particles C = 0.003 [29,30]. In both cases the coefficient in the first term of eqn. 1 S O characterizing the tortuosity of the channels of the mobile phase, y, ranges from 1.7 to 2.0. All four groups of the dependences of h on v are shown in Fig. 1.5. h

m

m II

I

1

0

5

10

15

20

v

Fig. 1.5. Reduced HETP, h , vs. reduced flow rate v for various types of chromatographic systems: (I) open-tubular capillary columns; 01) thin-layer plates; (111) columns packed with pellicular particles; (IV) columns packed with porous particles (conventional sorbents).

1S . 3 . Dependence of chromatographic spreading on the ratio of the column to the sorbent particle diameters

Inhomogeneous packing of columns leads to additional spreading of the chromatographic zones. Spreading increases with column width. This additional spreading is caused, in particular, by a change in the flow rate across the column cross-section. This change is particularly great near the walls where the quality of packing is usually.inferior to that in the central region. This effect is undesirable and can be avoided if the sample is introduced only into the narrow central zone of the column rather than over the entire cross-sectional area. It is then possible to choose the column of such a length L that only a small part of the injected sample (e.g. less than 5%) will reach the walls during the experiment. This is expressed by the condition

d,

> 40,

(1.61)

where d, is the diameter of the column and a," the sample dispersion in the radial (transverse) direction. Spreading in the transverse direction may be characterized by the height equivalent to a radial theoretical plate

ANALYSIS OF SPREADING

19

(1.62)

It is known in practice that the dependence ofH, on the flow rate can be described to within 50% by the following equation [ 2 2 ] : Hr = 2yDm + O.15dp U

(1.63)

Neglecting the contribution of molecular diffusion (i.e. the first term in eqn. 1.63) to

H , one can write according to eqn. 1.61

(1.64)

to give

L

d C

5 0.4-d C

dP

(1.65)

Hence, eqn. 1.65 yields the relationship between the ratio of the column length L to its diameter d, and that ofd, to the grain diameter p = d,/dp at which undesirable additional spreading due to the wall effect can be avoided. Columns for which this relationship holds are called ‘columns of infinite diameter’. Thus, a column with the diameter d, = 0.21 cm packed with grains of dp = 5 lo4 cm is a column of infinite diameter if its length does not exceed 35 cm. The contribution of additional spreading of the chromatographic zone close to the wall calculated in refs. 31-33 should lead to the dependence of Giddings’ structural coefficientsXi and wi and the corresponding constants C, and C, on p [ 10,331.

1 S.4. Effect of flow turbulence on the spreading of the chromatographic zone All the foregoing considerations were made on the assumption that the solvent flow passing through the column is laminar. However, it is known from hydrodynamics that at high flow rates turbulence is possible. Its contribution to spreading is similar to that of convective diffusion: both convection and turbulence level off the flow rate profile in the column cross-section and thus induce a decrease in spreading in the longitudinal direction. Giddings has suggested a special experiment for comparative estimation of the effect of convection and turbulence in which zone spreading had to be studied simultaneously as a function of the flow rate and as that of Reynolds’ number Re = Ud,/p ( p is the kinematic viscosity of the mobile liquid or gas). This experiment was carried out by H. Kaizuma and co-workers on a non-porous sorbent by both liquid and gas chromatography at different rates [34]. In this case spreading in a laminar flow should be characterized only by the first two terms in eqn. 1S O . Turbulence would change the expression for h as follows 13.51 : (1.66)

GENERAL THEORY OF CHROMATOGRAPHY

20

where W is the weight fraction of the contribution of turbulence t o the value of h which increases with Reynolds’ number Re, as is shown in Fig. 1.6 [36] . Equation 1.SO shows that when u increases (at u > 10) the value of h l , should first increase and then decrease slowly. Thus, h as a function o f Y should have a relatively flat maximum*. The same behaviour should be expected of the value of hturb.Therefore, two dependences h = h (u) and h = h (Re) were obtained simultaneously (Figs. 1.7 and 1.8).

W

1

Fig. 1.6. Turbulence coefficient W from eqn. 1.66 vs. Reynolds’ number.

L

m .-

2

107

.-

5-

J

3-

-0

1:

0 Q u

2-

:

7: 0.5 -

,$ 0.3

~

10

100

1,000

Reduced velocity,

10,000

100,000

1,000,O’

1,

Fig. 1.7. Reduced HETP, h , vs. reduced velocity (flowrate), v.

On the one hand, these dependences confirmed the validity of the foregoing theoretical considerations and, on the other hand, they showed that in liquid chromatography the maximum of h is attained at such flow rates v at which the effect of turbulence on spreading is small (Re z 8 and, as can be seen in Fig. 1.6, the value of W is about 0.05). In gas chromatography the maximum value of h is attained at lower values of v. However, in this case Reynolds’ number is very high, Re z lo3, and therefore at these flow rates turbulence is very pronounced ( W r0.8). Hence, the results [34] show that in liquid The dependence of h on w at the values of w

< 10 will be considered in detail in section 1.6.

OPTIMIZATION OF THE PROCESS

21

u-

f

cn 10 : ._ e,

r

53-

2D e,

1-

3

D 0.5

lz a3

-

02 -

0.11

I

0.01

I

0.1

I

1

I

10

Reynolds n u m b e r , Re

I

100

I

1,000

I

10,000

Fig. 1.8. Reduced HETP, h , vs. Reynolds' number, Re.

Fig. 1.9. Tortuoaity coefficient (obstruction factor) vs. reduced flow rate.

chromatography (including GPC) the effect of turbulence can be neglected, whereas in gas chromatography this cannot be done. To complete the consideration of factors affecting the spreading of the chromatographic zone, the results obtained by Hawkes [37] should be mentioned. He investigated the dependence of the tortuosity factor 7 on the flow rate v. Figure 1.9 shows that the value of 7 depends on reduced rate v only at the values of v usually attained in gas chromatography. Hence, in liquid chromatography it is correct to consider the value of -y as independent of v .

1.6. OPTIMIZATION OF THE CHROMATOGRAPHIC PROCESS 1.6.1. introduction The main practical purpose of chromatography is the separation (at some grade of purity) of a mixture of substances for analysis or preparative fractionation. In the case of

22

GENERAL THEORY OF CHROMATOGRAPHY

analysis two problems should be solved: the identification of chromatographic peaks and their quantitative interpretation. The difficulty lies in the fact that the separation of components during chromatography is always accompanied by spreading. Hence, these conflicting factors (separation and spreading) should be taken into account in designing a Chromatographic system and choosing the conditions of operation. The specific problems to be solved should also be taken into account. Sometimes maximum sensitivity of analysis is required, and in other cases maximum speed is an indispensable condition. Often the dimensions of the chromatographic column should be very small or the analysis should be carried out at a minimum pressure drop. This purposeful choice of the characteristics of chromatographic analysis is usually called its optimization [lo, 38-48]. There are three main types of optimization: optimization according to the sensitivity of analysis (I); according to its speed (11); according to its efficiency (111). The purpose of the first type of optimization is to carry out the analysis of a given efficiency at the highest sensitivity. The column should have the minimum length permitting the desired efficiency. This optimization makes it possible to use instruments of a smaller size, sharply decreases the consumption of the sorbent and the solvent and, hence, the cost of analysis. Moreover, the amount of the sample becomes smaller, which is of particular importance for the analysis of substances of biological origin usually available in micro-amounts. The aim of optimization of the second type is to obtain a chromatographic system of the desired efficiency in which analysis takes the minimum time and the pressure drop through the column is limited. This optimization is of particuiar importance for routine industrial chromatographic analyses. The decrease in the pressure drop and column size obtained without decrease of efficiency reduces the price of chromatographs and makes them available to a wide range of customers. For this optimization it is desirable to carry out the most effective analysis. It is necessary, for example, for the fractionation of mixtures that are difficult to separate and the estimation of their precise quantitative composition. 1.6.2. Criteria for the quality of chromatographic systems

In all three types of optimization the search for optimum conditions requires the establishment of some criterion for the quality of the chromatographic analysis. It should correspond to the criterion for the quality of the chromatographic system, including such characteristics as its efficiency, selectivity, permeability and some others. 1.6.2.1. Efficiency of separation

Usually the efficiency of chromatography (or its performance) is characterized by the number of theoretical plates N : (eqn. 1.3). Evidently, the efficiency determined in this manner is inversely proportional to substance spreading per unit time

(1.67)

OPTIMIZATION OF THE PROCESS

23

Sometimes the efficiency is characterized by the so-called effective number of theoretical plates Neff =

(vR - vO)*

(1.68)

/&

where Vo is the interstitial volume of the column (also called the interparticle or void volume). The value of Neff corresponds more closely to the efficiency of the chromatographic system than N because it depends not only on substance spreading but also on its real retention volume. For GPC this is the pore volume of the sorbent V,, accessible to the molecules of a given substance

vacc v~ - vo

(1.69)

= K d ' Vp

where Kd is the interphase distribution coefficient and Vp the pore volume of the sorbent. If the capacity factor k' is introduced (1.70) eqn. 1.68 becomes (1.71) It is also of interest to express Nef, by using the retention coefficientRf which is the ratio of the velocity (flow rate) v of the substance along the chromatographic system to that of the velocity in the mobile phase U 1

V

R - - = -

u

1 +KdVP/VO

-

1 1 +k'

(1.72)

Substitution of eqn. 1.72 into eqn. 1.7 1 gives

N,,, = N ( 1 -Rf)'

(1.73)

Sometimes the number of effective theoretical plates per unit time, Neff/t, also called delivery, is used to evaluate the efficiency of separation (t is the time of analysis). Applying the equation (1.74) and eqns. 1.7 1 and 1.73 where H is the height of the equivalent theoretical plate (HETP), one obtains (1.75) Equation 1.75 shows that the value of Neff/tconsidered as the function of k' or R f reaches a maximum at k' = 2 (Rf = f ) if H is independent of k'. This is valid when the

GENERAL THEORY OF CHROMATOGRAPHY

24

spreading of the chromatographic zone is determined largely by mass transfer of the substance in the mobile phase. Figure 1.10 shows that the optimum value of delivery, Neff/t,lies in the range of values of k' E [ 1.5; 41 or RfE [&; $1. If the capacity and retention coefficients are beyond these ranges, Nef&decreases sharply. Hence, these are optimum ranges. Usually the efficiency of a GPC system is expressed by the value o f N determined by eqn. 1.3 for a low-molecular-weightsubstance with Kd = 1. The values of k' and Rfvary from k' 2 1 and RfY 4 for rigid sorbents (silica gels and macroporous glasses) to k' 2 3 and R f Y f for soft swelling gels. It is assumed that when some standard method of column packing is employed the Vp/Vo ratio is independent of the column length and diameter and the size of sorbent particles. t

v: 1.5' 12 I

OAO

'4

6

8

10

k'

0.20

0.14 0.i4

011

0.09

Rf

1

Fig. 1.10. Productivity of the chromatographic system and the value of Rf.

as a function of capacity coefficient k'

1.6.2.2. Selectivity of separation The selectivity of the chromatographic system is generally determined by using the value of 6 (1.76) This value ranges from unity when retention volumes of the components to be separated are equal (V, = V, and Ak' = 0) to infinity when the first component is eluted at the interstitial volume Vl = Vo,i.e. k' = O,Kd = 0. Apart from the 8-value, selectivity can also be characterized by the parameter S. It is defined as the relative difference between retention volumes of two components being separated (1.77) It can be expressed by capacity coefficients

s = - Ak'

1+k;

and is related to the value of 8

(1.78)

OPTIMIZATION OF THE PROCESS

25

(1.79) The value of S ranges from zero when k; = k; and Ak' = 0 and there is no selectivity to k; /( 1 k ; ) when k', = 0 and k' = k; and the system exhibits the highest selectivity

+

When k; increases, S tends to unity, i.e. 0 Q S Q 1.

1A.2.3. Resolution of separation The combination of the values of N and S determines the third (and more general) chromatographic characteristic, the resolution coefficient K R (1.80) According to eqn. 1.80, in order to attain a given value of K R it is possible to use a system of high selectivity (S)and low efficiency ( N ) or, conversely, a system of low selectivity and high efficiency. In both cases the quality of separation of two neighbouring components of a mixture will be equal if it is evaluated in terms of the resolution coefficient. However, these two methods are not equivalent with respect to the requirements which the analysis should meet. The first method is usually chosen for the separation of simple mixtures with a small number of components. The second method is preferred for the separation of complex multicomponent mixtures, since under usual chromatographic conditions (without using temperature and solvent gradients) it is impossible to attain equally high selectivity for all pairs of components. However, when the efficiency of the system increases, the separation of all the components of the mixture is improved.

1.6.2.4. Productivity (delivery)of the column When the quality of chromatographic columns is compared it is convenient to use the characteristics of separation per unit time (1.81) Equation 1.81 shows that the value of K k combines the three important characteristics of the chromatographic system: selectivity (S), efficiency ( N ) and the time of analysis (t).By using eqns. 1.51 and 1.74 the value of K k can be written as a function of all the above-mentioned characteristics of the chromatographic system and its main operating parameters AP,L,d , and 77 (1.82)

26

GENERAL THEORY OF CHROMATOGRAPHY

Equation 1.82 permits the formulation of the following thesis. The resolution COefficient per unit time KA can be considered as a generalized criterion for the quality of chromatographic systems. Of any two systems being compared, the superior system is that which ensures higher resolution ( K k ) per given time. If the resolutions of two systems are equal, the superior system is that ensuring the attainment of this resolution during a shorter time (at a lower pressure or in a shorter column, depending on the aim of optimization). According to eqn. 1.82, the required value of K k can be obtained by varying the characteristics and parameters of the system. The choice of these values for each specific task should be made separately. For example, as already mentioned, in the analysis of a two-component mixture the chromatographic system should have high selectivity S. Hence, while retaining the quality of the column (KA value) the greatest possible increase in S should be attained at the price of its efficiency N and the length of the column L . In the analysis of a multicomponent mixture the efficiency of the system should be increased as far as possible without paying too much attention to the possible decrease in selectivity, provided the quality criterion K ; is kept at an adequate level.

1.6.2.5. Peak capacity as a separation characteristic For the quantitative evaluation of the separation of multicomponent systems it is also convenient to use another criterion, the peak capacity Q [lo]. This is the number of peaks corresponding to the components of the mixture resolved at a purity K R 2 1 during a given time (i.e. on a given chromatogram). If it is assumed that the number of theoretical plates for all components of a given mixture N is the same (i.e. independent of k ’ ) , the peak capacity can be determined as follows:

4 = 1 + 0.6N”210g(l +kL)

(1.83)

where k; is the capacity coefficient of the last component. Hence, it might be concluded that to improve the separation of multicomponent systems the efficiency of the system N and the capacity coefficient of the last component should be increased. In high-speed analysis it is important that the last component should not be retained too long and that the first component should emerge as early as possible, i.e. with the interstitial volume Vo. In conventional GPC the maximum value of the peak capacity is only 2.5-3. In adsorption chromatography q5 is greater than 3 because, depending on the quality of the solvent and the operating temperature, capacity coefficients k‘ (and distribution coefficients K d ) can be greater than unity. Applying eqn. 1.74 one can rewrite eqn. 1.83 for @ (1 34)

where to is the elution time of the unretained component to = L/U. Equation 1.84 shows the extreme character of the peak capacity q5 as a function of t o : the value of @ passes through a maximum when the retention time t is approximately 7.4 times greater than t o ,i.e.

OPTIMIZATION OF THE PROCESS

a4- - 0 at0

21

t atln-=2

(1.85)

to

This condition corresponds to the capacity coefficient of the most highly retained component of the mixture k’ = e2 - 1 = 6.39 and the peak capacity

=, ,$

d

fl

= 1+-

(1.86)

2

Like the value of Kk ,the peak capacity attained per unit time 1 - 4 $J

= - = [l

t

ko APd; + 0.6N1’21~1(1 + k’)] 1 + k’ L 2 q

(1.87)

can serve as a criterion for the quality of chromatographic columns. 1.6.3. Choice of characteristics of the chromatographic system

In planning an experiment one should always decide beforehand which are the required characteristics of the chromatographic system and how they should be related to each other for the analysis to be carried out at an appropriate level. The answers to these questions will be different in each specific case. 1.6.3.1. Choice of selectivity and efficiency depending on resolution For the chromatographic separation of the components of a mixture with the isolation of substances in a pure form (preparative chromatography) the resolution coefficient K R should not be less than unity. If this condition is fulfdled, no more than 2% of the areas of neighbouring peaks will overlap at the outlet of the column. In this case eqn. 1.80 leads to the following inequality: 1

< $s”’2

(1.88)

permitting the variation of the selectivity and efficiency of the system. Evidently, it is useless to increase very greatly the distance between the peaks of the components on the chromatogram. Moreover, it is convenient to stop the process as soon as the separation attains the required resolution, such as K R = 1. The question arises which are the ranges of the value o f N in this case. Substitution of the maximum value of S = S,, = k; /( 1 k;) into eqn. 1.88 gives

+

(1.89)

The highest value of the capacity coefficient in GPC is equal to the V J V . ratio. It varies between 1 and 3 depending on the nature of the sorbent. The corresponding limiting lowest value of N varies over the following range: 28

< Nmin < 64

These values correspond to the case when one of the two components to be separated

28

GENERAL THEORY OF CHROMATOGRAPHY

is eluted with the interstitial volume Vo and the other is eluted with the maximum volume V,. The lower limit corresponds to soft swelling sorbents of the Sephadex type ( V,/Vo 3), and the upper limit refers to rigid silicate sorbents of the porous glass type
-

A VR VR Substitution of this value into eqn. 1.80 at K R = 1 gives

smin ‘

=

- = 0.002

Nmaw = 4. lo6 Consequently, the separation of two components at a resolution KR = 1 by using a system with the minimum selectivity equal to the relative reproducibility of retention volumes for recent chromatographs requires extremely high efficiency which in some cases has already been attained [49].

I . 6.3.2.Relationship of selectivity and efficiency to reproducibility In the section 1.6.3.1. the case of the limiting highest purity of separationKR 2 1 was considered. The opposite limit, KR Q 1, is also of some interest. It is important for checking the individuality of chromatographic peaks*, their identification and the quantitative interpretation of the composition of the irresolvable mixture when the system is calibrated for single components. In this case it is important to determine the lowest limit of the resolution coefficient K R and the corresponding limiting values of selectivity and efficiency. This limit has already been established for selectivity: it is equal to the reproducibility of chromatographic data given by the abscissae. It will be shown that the limiting value of the resolution coefficient K R may be related to the reproducibility given by the ordinates of the chromatograms. For this purpose it will be assumed that the main reason for a change in the ordinates of the chromatogram is its shift along the abscissa, i.e. elution volumes. Then the meansquare relative error ( x ’ ) ’ ’ ~ in the determination of the ordinates F(V) of Gaussian chromatographic peaks should be found:

(1.90)

We call a peak ‘individual’ if it corresponds to one substance rather than to a mixture of substances.

OPTIMIZATION OF THE PROCESS

29

(1.91) If A F is the change in the ordinates of the peak related to the change in the corresponding abscissae, i.e. (1.92) then the substitution of eqns. 1.91 into eqn. 1.90 gives (1.93) The value of (x’)~’’ may be considered as the mean-square relative deviation of the ordinates F ( V R ) and F(VR + AVR) of two neighbouring peaks, the distance between the maxima of’which is A VR. The multiplication and division of the right-hand side of eqn. 1.93 by V i gives [SO]

x2

=

(x) 0’ AVR V i

= S2N = 16Ki

(1.94)

The last equality in eqn. 1.94 is written in accordance with eqn. 1.80. In other words, the resolution coefficient is the square root of the mean square of the relative difference between the ordinates of the peaks of two neighbouring separable components divided by 16. Its minimum observed value exceeding the experimental error is KRmin= (XAin/ 16)’” where (Xiin)’” is the reproducibility of chromatographic data given by the ordinate. The evaluation of this reproducibility with an up-to-date chromatograph gives for KRmln the value of the order of magnitude of 0.06 (with the reproducibility AF/F at the peak maximum of the order of magnitude of 0.01). In accordance with eqns. 1.80 and 1.94 the efficiency of separation Nlim should correspond to minimum resolution KRminand minimum selectivity &in Nlim

= (4KRmin/Smin)2

(1.95)

The substitution of numerical values into eqn. 1.95 gives Nlim 1 14,500 plates

This is the limiting value of N for most recent chromatographs, making it possible to distinguish the chromatographic peaks at the limit of their reproducibility. For problems related to the identification of chromatographic peaks and their subsequent quantitative interpretation (if an appropriate calibration is available) further increase in N is useless because it will not yield any new information’. It should be noted that under these conditions the separation of components can be visually imperceptible. However, if computer interpretation is used, the separation of peaks can also be reliably made on a column with Nlim = 14,500 plates in the cases where the resolution is close to the reproducibility of the instrument.

GENERAL THEORY OF CHROMATOGRAPHY

30

Hence, the following relationships can be formulated: (1) Two chromatographic peaks are assigned to different substances if their maxima are at a distance ofAVR/VR = Sminapart. This distance should not be less than the reproducibility of the instrument, and the efficiency of the systemN should not be lower than (KRmin/4Smin)2

N 2

(KRmin/4Smin)2

(1.96)

In other words, the higher the selectivity of the chromatographic system S, the lower is the efficiency N required for the correct identification and interpretation of chromatograms. (2) A chromatograph for which the reproducibility given by the abscissa is Sminand that given by the ordinate is xLin, can be used most effectively if a system of columns with a number of theoretical plates Nlim= (KRfi,,/4Smin)2 is available. This system makes it possible to distinguish components with the minimum distance between their peaks A V for a given chromatograph. Further increase in N will only make the peaks of the components more narrow, i.e. the reliability of identification will not increase. At N
1.6.4.Choice of effickncy criteria in GPC In GPC, only the precision of the molecular-weight distributions (MWD) and their average characteristics (AMW) can serve as criteria for the efficiency of chromatographic analysis. This precision depends not only on the quality of the system but also on calibration data and the method used for the interpretation of chromatograms. Generally the errors in calibration data on molecular weights (and, hence, on the corresponding retention volumes) are much higher than the instrumental errors previously considered. Therefore, the latter are not of decisive importance for the chromatographic determination of molecular-weight characteristics of the polymer. They are closely related only to the correction of chromatograms for instrumental spreading. For this reason this correction cannot be of any desired precision since the number of peaks resolved (owing to correction) within one chromatogram is limited by the existence of the limiting value KRlim. However, if the characteristics of the system are adequately chosen, the precision of the determination of the MWD and AMW parameters can become comparable to the errors of absolute methods, the data of which are used for the calibration of GPC columns. Therefore, in GPC it is often important to choose an efficiency N permitting an approximate determination of the MWD and AMW of polymers at a given selectivity of the system. This efficiency should ensure the determinations of AMW with an error not exceeding that in molecular-weight calibration dependence, even if instrumental spreading is not taken into account. Since this dependence is plotted by using polymer standards characterized according to AMW to within 5%, it is natural to restrict the interpretation of GPC data to this precision.

OPTIMIZATION OF THE PROCESS

31

When the efficiency criterion in GPC is chosen, it should be taken into account that the highest precision in the interpretation of chromatograms is attained for a linear (with respect to log M) calibration dependence VR = c1 - c 2

1ogM

(1.97)

which also ensures a clear and simple interpretation. Any deviations from the linearity lead to additional errors in the interpretation caused by less accurate approximation for calibrations obtained with various polynomials of the type (1.98) if the set of polymer standards is limited. Moreover, as will be shown below, linear calibration ensures the same selectivity of separation according to logM over its entire range of action. Hence, two requirements should be met in choosing the criterion for the efficiency of the GPC system: (1) The contribution of instrumental spreading to the AMW values of polymers should not exceed 5%. ( 2 ) The calibration of the system according to logM should be linear. We will now choose the desired criterion in accordance with these requirements. By using the results of Bake and Hamielec [51] it is possible to obtain the relationship between the values of average molecular weights M k (u) and M k(0) found from chromatograms W( V ) and F( V ) corrected and uncorrected for instrumental spreading respectively. (1.99) When k is 1, Mk means the number-average molecular weight, M,, when k is 2, Mk means the weight-average molecular weight,M,, etc. Hence, at k = 1 and 2 we have (1.100) In the derivation of eqn. 1.100 the expansion of the exponent from eqn. 1.99 in a series was used, and subsequently all the terms of this series except the first two were neglected as being very small. Instrumental spreading decreases the true value of the number-average molecular weight of the polymer MI(0)=Mnand increases that of its weight-average molecular weight M2 (0)= M W .This changes the sign of the difference Mk(u)-Mk(O), and, therefore, its absolute value was used in eqn. 1.99. Now, if it is assumed that M 1 / M 1 = AMz/Mz < 0.05 (i.e. that the error inM, and& is 5%), eqn. 1.100 gives

and,hence,c2/u>l.3.

(1.101)

GENERAL THEORY OF CHROMATOGRAPHY

32

Inequality 1.lo1 can be regarded as one of the criteria for efficiency in GPC. It combines the values of the slope of calibration dependence 1.97 and substance spreading in a given chromatographic system (u) (Fig. 1.1 1). Multiplication and division of the lefthand side of eqn. 1.101 by V, gives

c2 v, = -N"2 c2 VR

0

2 7.3

(1.102)

VR

Fig. 1.11. Graphic illustration of the parameters of linear calibration dependence.

The cz/VR ratio in eqn. 1.102 can be expressed by the coefficients of calibration dependence (1.97) (1.103)

(1.104)

It is clear that the higher the value of cl/cz, the higher should be the value o f N i n order to satisfy the chosen efficiency criterion. To evaluate the cl/czratio in the calibration Kd =

A-BlogM

(1.105)

can be used. It corresponds to calibration 1.97. It can be seen that the coefficients of these calibrations are related to each other as (1.106)

Let us evaluate their values for two ranges of molecular weights M E [ lo2 ; lo'] and M E [lo2; lo6] differing in width, and assume for simplicity that Vo/Vp is unity. Figure

OPTIMIZATION OF THE PROCESS

33

1.12 shows that coefficient B in eqn. 1.105 is inversely proportional to the width of the range of log M. If MI and M2 are the end points of this range, we have

B = 1/l%(MZ/MI) (1.107) According to this, over a wide range of the logarithms of molecular weight l o g M E [ 2 ;71 one obtains CI/CZ

= 12

and over a narrower range of l o g M E [ 2 ;61 one obtains c,/cz

Kd

= 10

I

2

3

6

7

Fig. 1.12. Relationship between the efficiency of the calibration dependence (its slope) and the width of the range of separated molecular weights.

Hence, when the range of separated molecular weights becomes wider, the value of cl/cz increases. Therefore, it follows from eqn. 1.104 that the wider the range o f M , the greater should be the efficiency of the chromatographic system t o satisfy the criterion in eqns. 1.101 and 1.102. Accordingly, one can write

NMt,,

> (7,3)2

(1.108)

where M I , = (c /cz - log M)-'. According to definition 'the cz coefficient is given by c2

AVR - A VR A log M AM/M

=---

(1.109)

where AvR is the difference between the retention volumes of components whose logarithms of molecular weights differ by A logM = M / M . Equation 1.109 shows that the c z / V R ratio in criteria (1.102) is equivalent to the selectivity (1.110)

GENERAL THEORY OF CHROMATOGRAPHY

34

Ifeqn. l.llOisused,eqn. 1.102 becomes SN'12 2 7.3AM/M

(1.111)

Since it is required that in GPC the error in the determination of molecular weight should not exceed 5%, the molecular weights of neighbouring resolvable components will also differ by this value. Hence, eqn. 1.1 11 becomes SN'"

2 0.365

(1.112)

where (1.1 13) When calibration is linear over the range of molecular weight M E [lo'; l o 6 ] then, according to eqn. 1.113, for a high-quality GPC analysis of a polymer with M = lo5 it is possible to attain selectivity S = 0.01. For a wider molecular weight range M E [ lo2 ; lo'] the attainable selectivity is lower: S M 0.007. Hence, according to eqn. 1.1 12, in the latter case highquality analysis corresponds to a higher efficiency Nz than in the former case, N 1

N,/N' = (S,/S2)2= 2 For a low-molecular-weight substance with K d = 1 under the optimum conditions of chromatography for this substance the calculation of efficiencies N 1 and Nz gives N 1 = 9500 and Nz = 19,000. In this case on the basis of the data obtained by Guiochon and co-workers [48] it was assumed that the spreading of the zone of the low-molecularweight substance is approximately seven times less than that of the zone of the polymer with M = 10'. 1.6.5. Requirements for the degree of asymmetry of chromatographic peaks

All the foregoing considerations were based on the assumption of the symmetrical shape of chromatographic peaks (in particular, the Gaussian shape). However, one of the factors affecting the quality of the analysis is the peak asymmetry. In high-quality columns the asymmetry should be as low as possible. For the quantitative description it is convenient to use the asymmetry factor r/u (Fig. 1.1 3) [21], where r is the shift in the mathematical expectation of the chromatographic peak with respect to the position of its maximum. This shift leads to the distortion of the calibration curve [21, 511. As a result, additional error (AM/M), arises in the determination of molecular weights of homopolymers

AV

_ -r

(1.1 14)

For high-quality columns this error should not exceed 5%. When eqn. 1.108 is applied, this condition leads to the inequality 7

U

5 0.4

(1.115)

OPTIMIZATION OF THE PROCESS

a

35

Goussion component, U

Exponential tailing component, T T/O‘ Rotlo related to peak skew

Fig. 1.13. Example of an asymmetrical chromatographic peak; u 2 -dispersion of the Gaussian component, T - shift in mathematical expectation with respect to the Gaussian component.

1.6.6. Requirements for the visual observation of the separation

The attainment of visual observation of the separation of two components can serve as an alternative criterion for the quality of the chromatographic system. The limiting case of this separation is the existence of three extremes in the chromatogram: two maxima and a minimum between them. The resolution KR leading to this effect is a function of the ratios of amplitudes A1/A2 and dispersions u:/a; of the components [50]. The shape of this function can be seen in Fig. 1.14 where a family of curves K R = KR (A1/A2) is shown at various ratios u:/u:. The characteristic feature of this family is the existence of an envelope, the ordinates of which give a value of KR sufficient for the visual separation of the components with the ratio of amplitudes of A E A1/A2 whereas dispersions can have any value. Each curve in Fig. 1.14 represents the minimum value of K R for a given (o:/u;) 1’2 E u ratio. Another feature which is of considerable practical importance is the fact that the function KR = KR ( A , a) is not unique. It is clear that at an A 1/A2 ratio higher than the ratio u1/u2 f 1 three KR values exist at which the condition of visually observed separation is fulfilled. It begins to be fulfilled even when the distance between the components in the chromatogram is small (KR > K R , ). When this distance increases, the separation first becomes even more apparent and then begins to decrease. At a value K R = K R , only a horizontal ‘shoulder’ is observed and then it also begins to disappear. Beginning only from the third value of KR = K R , almost twice as high as the first value three extremes are again observed, and further increase in the distance between the components leads t o their better visual separation on the chromatogram. Figures 1.14 and 1.15 show that two ranges exist in which visual separation is achieved: KRI < K R < K R , and KR > K R , and two ranges in which this separation is not achieved: KR < KR, and KR,
36

GENERAL THEORY OF CHROMATOGRAPHY

Fig. 1.14. Resolution K R vs. ratio of amplitudes A = A , / A 2 for various values of u = (U;/U:)''~. The values of K R ] , K R t and KR, determining the range of visual separation of components are shown as examples at A = 7 and u = 6.

that if the impurity is eluted before the main sample (Fig. 1.16), the leading edge of the chromatogram is steeper than the trailing edge. If it is eluted after the main sample (Fig. 1.17), the leading edge should be less steep than the trailing edge (if there are not other reasons leading to the asymmetry). It is clear that in order to detect an impurity it is sufficient to attain a slightly higher resolution thanKR = K R , , and a higher resolution, KR >KR, ,is not needed. It is possible to find the efficiency required for visual separation (for a low-molecularweight substance with Kd = 1) from eqns. 1.80, 1.97,l .lo5 and 1 .lo6and a given value of K R (obtained from Fig. 1.14). (1.116)

where cu"= 1 - Vo/V,, Mlim and Mo are the limits of linear calibration and M1 and M 2 the molecular weights of the polymers that should be visually separated (h,/h is the ratio of reduced HETP for a high polymer to that for a low-molecular-weight substance at the same elution rate v). For example, for the visual separation of samples with molecular weights MI = lo4 and M2 = 10' on a column with a linear calibration in the range [ lo2; 10'1 the required efficiency is N = 1400 at h,/h = 7. If M1 = lo', and M2 = 2 * lo', the efficiency required for the same purpose is approximately 1 1 200. The main concepts considered in this section can be formulated as follows.

OPTIMIZATION OF THE PROCESS

31

Fig. 1.15. Superposition of the Gaussian functions (with amplitude ratio A = A , / A , = 3.3 and standard deviation ratio u = o , / u , = 3) as a function of their mutual position: ( 1 , 5 , 6 ) separation is not visually observed; (2,3,4,7) separation is visually observed.

Fig. 1.16. Superposition of the Gaussian functions (A = 3.3, u = 3). The admixture is placed to the left of the main component.

In GPC the selectivity of the chromatographic system is expressed by eqn. 1.113; it is proportional to the slope of calibration dependence (1.97) and inversely proportional to the width of the range of molecular weights over which this dependence holds. Average molecular weights are determined to within 5% if one of the conditions (1.101)-(1.104) and (1.1 11) is fulfilled. Moreover, the required efficiency of the system increases proportionally to the width of the range of molecular weights.

38

GENERAL THEORY OF CHROMATOGRAPHY

Fig. 1.17. Superposition of the Gaussian functions (A = 3.3, u = 3). The admixture is placed to the right of the main component.

Resolution and efficiency required for the visually observed separation of components can be determined from the plot in Fig. 1.14 and eqn. 1.1 16. 1.6.7. Choice of sorbent for attaining the required selectivity of the chromatographic system in GPC with a linear calibration dependence

The selectivity of the chromatographic system in GPC is profoundly affected by the structure of the pore volume of the sorbent used for packing the column. The narrower the pore size distribution on $(r) in a sorbent, the higher is its selectivity over the corresponding narrow range of molecular weights. This can be clearly seen in Figs. 1.1 8 and 1.19 which show unimodal distributions $(r) and the corresponding calibrations based on log M .

Fig. 1.18. Examples of unimodal pore size distributions of the sorbent.

Unimodal sorbents always yield S-shaped calibration dpendences that can be considered to be more or less linear only over a narrow range close to the maximum of distribution $(r) which is also the range of maximum selectivity of the sorbent. To make this range wider, one should use several sorbents with narrow distributions Gi(r) mutually displaced along the co-ordinate r and thus overlapping the required wide range of r. Figure 1.19 shows calibration curves obtained with these sorbents. It is clear that sorbent I exhibits the highest selectivity over the range of molecular weights [MI ; M 2 ] , sorbent I1 exhibits the highest selectivity at [M2 ; M 3 ] etc. , Over the same ranges the Calibration of each sorbent is close to linear. A reasonable combination of sorbents with narrow distributions $&) permits the preparation of chromatographic systems with a

OPTIMIZATION OF THE PROCESS

39

Fig. 1.19. Calibration dependences for columns packed with manoporous glasses with various narrow pore size distributions shown in Fig. 1.18.

linear calibration dependence (1.97) and high selectivity over any range of molecular weights (M). It was first shown by Yau et al. 1521 that over a wide range o f M a linear calibration dependence can be obtained by using only two sorbents with narrow distributions J/l (r) and $ z ( r ) or, identically, by using one bimodal sorbent with the distribution $(r) = q l J / l ( r ) + (1 -ql)J/2(r), where q1 is the weight fraction of the distribution with subscript 1 and (1 -- 4 ) is that of the distribution with subscript 2. The same result has been obtained [53] from the integral equation relating the observed distribution coefficient as a function of the size of macromolecules (K,(I?)) t o the distribution I)@): (1.117) where Kd(I?/r) is the experimental universal dependence of the distribution coefficient on the ratio of hydrodynamic dimensions of the molecule = ( M [ V ] ) ” ~to the pore radius of the sorbent r. On the basis of a unique solution of this equation for a specific type of kernel &(I?/r) the following relationship can be formulated. Over each range of sizes of macromoleculesE and, hence, over each range of molecular weights M only one pore size distribution $(r) exists that permits the derivation of the calibration dependence Kd =f(logM) which is the closest to the linear dependence. This relationship leads to the following conclusion. For any group of sorbents with unimodal pore size distributions Jli(r)for which the ranges of calibration dependences based on logM overlap, it is always possible to choose the only combination of sorbents that will simultaneously ensure the required (but not the best) approximation to the linear calibration dependence and its efficiency (i.e. its highest slope) over a given range of molecular weights. Figures 1.20 and 1.21 show examples of calibration dependences obtained with biand trimodal sorbents.

40

GENERAL THEORY OF CHROMATOGRAPHY

Fig. 1.20. Approximation of linear calibration dependences obtained on sorbents with bimodal pore size distributions.(a) Pore size distribution;(b) calibration dependence.

M

Fig. 1.21. Approximation of linear calibration dependences obtained on sorbents with trimodal pore size distributions.(a) Pore size distribution;(b) calibration dependence.

1h.8. General relationships of optimization

The main features of the chromatographic process are closely related not only to the characteristics of the chromatographic system just considered but also to the three groups of operating parameters. The first group consists of the parameters determining the intensity of the process. They are its delivery or productivity N , t , the time of analysis t and sensitivity 6 at a given K R . The second group includes the parameters determining the economy of the process, i.e. the cost of analysis; they are the column length L , the pressure drop AZ'and the diameter of sorbent particles d p. These parameters can be varied during optimization. The third group includes technical parameters: the column diameter d,, the viscosity of the solvent 77, the operating temperature T , the capacity coefficient k', the diffusion coefficient of the substance being analysed D , coefficients in an equation relating HETP to the elution rate and depending on the method and quality of column packing and some other parameters.

OPTIMIZATION OF THE PROCESS

41

Usually, the search for optimum conditions is made for a chromatographic system with the required technical parameters and suitable S and N characteristics. These characteristics are selected according to eqn. 1.80 so as to ensure the required resolution K R and meet all other requirements with which the analysis should comply. The parameters of the second group, L , AP and d,, are varied in such a manner as to ensure optimum values of the parameters of the first group t , N / t and 6 depending on them. At the same time the L and AP parameters should have the lowest possible values and d, should not be very low. In other words, the time of analysis should be as short as possible and the productivity and sensitivity should increase with the simultaneous decrease in the L and AP values, whereas d , should not greatly decrease. This selection of operating parameters and characteristics is based on their close relationship to the spreading of the chromatographic zone. This relationship is adequately described by a known equation which in column chromatography can be rewritten by using the determination of HETP (1 .l) and Darcy’s law (1.5 1)

APdiko

___.-

NqD v

-2y + v

~ ~ 1 /+ c 3v

(1.118)

For packed columnsA = 1 + 2 and for open capillary columnsA = 0. It should be taken into account that the value o f N in eqn. 1.1 18 is determined by the aims of chromatographic separation on the basis of the required values of K R , $ or c2 / O estimated by eqns. 1.80, 1.83 or 1.104 respectively. The first step in optimization is the choice of the required values made by using these equations. The second step consists in choosing the operating parameters of the chromatographic experiment ( L , d , and k’) with the aim of attaining maximum productivity (delivery) (N/t) or sensitivity ( 6 ) of analysis at given values of the economical (L, AP and d,) and technical ( d c , T , q, etc.) parameters of the process and at a given value of N. When optimization of efficiency N is carried out, it is desirable to choose the operating parameters in such a manner as to obtain the required high value o f N . As will be seen below, the analysis of eqn. 1.1 18 and the dependences calculated from it and relating to each other various operating parameters, indicate that for each type of optimization some definite value of the reduced elution rate v corresponds to the optimum conditions (1.119) This value depends on the combination of parameters on the left-hand side of eqn. 1.I 18. This combination should be considered as a generalized parameter k (Fig. 1.22). (1.120) The values of v and k are uniquely related to each other. Hence, the value of k may serve as a criterion for the degree of optimization of the system, just as does that of v .

GENERAL THEORY OF CHROMATOGRAPHY

42

I

5

0

10

15

2b

0

5

10

15

20

25

30

40

Fig. 1.22. (a) Reduced HETP vs. reduced rate corresponding to Knox's equation (1.50) at the parameter values y = 0.9, A = 1.7 and C = 5 * 10.'. (b) Reduced flow rate vs. value of generalized optimization parameter k = APdGINxD, x = r71ko.

Comparison of eqns. 1 S O and 1 . 1 18 gives

k = hv

(1.121)

It can be seen that the parameter k is the double ratio of the dispersion o2 of chromatographic spreading of the substance during time t to the dispersion 2Dt of diffusion spreading in a free solution

k

= 202/2Dt

( 1.I 22)

In TLC the observed value of reduced HETP can be conveniently related to the average reduced elution rate 9 by the equation analogous to eqns. 1 . 1 18

(1.123) where a is determined from eqn. 1.55, R f from eqn. 1.72 and h = LRf/Nd, (L is the length of solvent migration along the plate or film). As in column chromatography, in TLC for every type of optimization one can find the values of F corresponding to optimum conditions (1.124)

and uniquely related to the generalized optimization parameter (1.125)

J5k

OPTIMIZATION OF THE PROCESS

43

1.6.9. Optimization of the sensitivity of analysis in column chromatography

The sensitivity of chromatographic analysis 6 is one of its most important characteristics. It can be defined as the ratio of the signal of the instrument to the amount of the substance q

6 = - signal 4

(1.126)

The value of the signal depends on the concentration of the substance c, at the maximum of the chromatographic zone and decreases or increases with it according to a law specific for each type of detector. The concentration c, depends on the amount of sample q and the degree of spreading during chromatography. Spreading, in turn, is determined by the parameters of the column: its length and diameter, efficiency and the degree of retention of a given substance (e.g. its capacity factor k'). If an amount of substance q is introduced into the column, its concentration at the maximum of the chromatographic zone at the outlet of the column (if a Gaussian type of spreading is assumed) is given by c,

--

q -

(I. I 27)

Multiplication and division of the right-hand side of eqn. 1.127 by the retention volume V, gives

(I. 1 28) where Vo is the interstitial volume in the column and A , is the effective cross-section of the mobile phase. Evidently, the aim of the optimization with respect to sensitivity is to decrease substance spreading during analysis. In other words, if cZin is the minimum concentration to which a given detector responds, the minimun amount of the substance in a sample, qmin, leading to this concentration will correspond to the highest efficiency of the column per unit length (NIL). At a given efficiency of analysis N and the corresponding technical parameters, sensitivity can be increased only by decreasing the column size. Aspects related to the decrease in column cross-section A , were discussed in section 1.5. Here we will consider the possibility of optimization of the column length. Equation 1.I28 shows that maximum sensitivity of analysis at a given efficiency N is achieved at a minimum column length L . Since L/N = H,this also corresponds to the minimum value of HETP, H . When columns are packed by a standard method, the N value is fixed and the technical parameters are chosen, H is the function of two variables only, AP and dp,and exhibits a minimum for each variable. Hence, the H value considered as a function of v passes through a minimum at various values of v (v = v 1 and v = v2) depending on whether the rate is altered by varying the pressure drop or by varying the size of the sorbent particles (Fig. 1.23). Each of these minima corresponds to the chromatographic process optimum

GENERAL THEORY OF CHROMATOGRAPHY

44

0

5

10

15

20

u

Fig. 1.23. Dependence of HETP,H on u at a fixed column length L : (1) v is changed as a result of the variation in pressure drop A P at d p = l0gm; (2) u is changed as a result of the variation in particle sue d , at AP = 10MPa. At the intersectionpoint of curves 1 and 2 the values of AP and d , for these curves coincide.

for the sensitivity of analysis. Hence, there are two modes (combinations of conditions) optimum for sensitivity. They are attained at the rates v1 and vz . In a system of dimensionless coordinates (h, v) the function h = h(v) passes through a minimum at the value of v = vo*

a q v =

o

(1.129)

and at Y = v l (vl < vo) the following condition is fulfilled:

qah/av)v

+h

=

o

(1.130)

Both cases correspond to the condition

aLlav

=

o

(1.131)

Equation 1.120 shows that the optimum values of v 1 and v2 correspond to a strict relationship between the parameters AP, d , and L at given values of ko ,q and D . If one of the parameters is vaned and at the same time it is desired to maintain the optimum value of v, the other two parameters should be varied correspondingly. For example, if dp is decreased, AP should be increased andlor L should be decreased. At the same time one should maintain the invariable efficiency of analysis. This means that one criterion, the value of v, is insufficient for characterization of the optimum mode. A second criterion should also be used: the value of the generalized optimization parameter k (k = kl or k = k2). Being a unique function of v, this parameter also explicitly includes the N value. Hence, in order to attain the mode optimum for analysis sensitivity by varying the particle size or the elution rate U one should obtain such a value of reduced elution rate v at which the value of H is at a minimum. At the same time the relationship between the

* In column chromatography the values of uo and v 2 coincide.

OPTIMIZATION OF THE PROCESS

45

50

40

i, 30.1 I 1

20.

10

’

I

~

(b)

10

20

-

-0

30dptprn

-----__ 2

3 AP, MPO

Fig. 1.24. Column length L vs. (a) particle size d , at various fixed values of pressure drops AP, (b) pressure drop AP at various fixed values of particle size d,. (----) - the envelopes of the families of curves L = L (dp, AP = const) and L = L (AP, d, = const).

parameters PP, dp and N should ensure the optimum value of the generalized parameter

k. In this case the column length L considered as a function of one of two variables A P or d, will also pass through a minimum. Evidently, by varying the parameters influencing the value of L one obtains (at N = const) two families of curves with minima (Fig. 1.24). These families are related to each other fairly strictly. Each of them has an envelope along which one of the optimum conditions is fulfilled ( k = k l and v = v 1 or k = k2 and v = vz). It is characteristic of these curves that each point of the envelope of one family corresponds to a minimum of one curve of the second family. The envelope of the family of curves i n Fig. 1.24a is their common tangent. Its equation is given by

L (dp) = hmin Ndp The envelope of the family of curves in Fig. 1.24b is a parabola.

(1.132)

L(AP) = const/JP,

(1.133)

const = ( ( k l ~ 1 > ~ ” / ~ l ) ( ~ o / ~ ~ ) - ’ ”

If one moves along any curve of these families towards the envelope, the system becomes ‘tuned’ in the optimum manner. If one moves away from it, the system becomes ‘untuned’. It is clear that one optimum particle size corresponds to each pressure drop (1.134) and one optimum pressure drop corresponds to each particle size. (1.135) If the conditions corresponding to the minimum in one of the curves in Fig. 1.24b (where L is a function of APat fixed d,) are chosen as operating conditions, one obtains

46

GENERAL THEORY OF CHROMATOGRAPHY

the chromatographic mode with the optimum sensitivity of analysis for a given fixed value of d,. It is characterized by some values of u = u l , N = N1, d , = d,, , A P = AP, , L = L 1 and t = t l . Moreover, the value of AP1 should be determined by eqn. 1.130. In Fig. 1.24a this mode corresponds to the tangent point of the envelope of the family shown in this figure with the curve characterized by the value AP = AP, = const. Now if the particle size decreases t o a value of d , = d P 2 determined by eqn. 1 .I 29 without varying the pressure drop, i.e. if, in Fig. 1.24b, one descends t o the point of the envelope with the coordinates ( A P , , d , , ) , the conditions of the second optimum mode ( u = u 2 ) are obtained; they are characterized by a lower L value (and, hence, by higher sensitivity). In Fig. 1.24a the transition t o this mode corresponds t o the descent along the curveL = L ( d , ; A P = A P , ) toitsminimum. Now if one descends t o the envelope in Fig. 1.24a at a fixed coordinate d , = dP2(or along the curve L = L ( A P , d , = d,,) in Fig. 1.24b t o its minimum), the preceding optimum mode (u = u , ) is attained again at lower values of L and d, and a higher value of AP. There is no common minimum of function L for both variables dp and AP. 1.6.10. Choice of optimum operating parameters and characteristics of the

chromatographic system for optimization based on the sensitivity of analysis

For selecting optimum chromatographic conditions one should proceed, first, from a generalized criterion for the quality of the chromatographic system determined by eqn. 1.82. The quality of the system and its main characteristics, such as selectivity S, permeability ko and efficiency N should be chosen depending on the aims of the analysis. Then the optimization curve h = h(u) should be plotted and the values of k l and k2 and the corresponding values of u1 and u2 satisfying optimum conditions 1.129 and 1.130 should be found. Subsequently, by using eqns. 1.134 and 1.135 it is easy t o choose optimum pairs of d, and AP values and the length of the column L . If the data obtained in ref. 48: y = 0.9, A = 1.7, C= 5 * lo-', ko = 8.46 D = 3.5 * m/sZ,77 = 0.4 CPand k' = 2, are taken as an example, then for optimum reduced rates u and the corresponding generalized optimization parameters k one obtains: u1 = 1, u2 = 2.1, kl = 3.55, k2 = 6.55. For the convenience of practical work the values of t, L , d , , AP and N are given in nomograms 1 and 2 (Figs. 1.25 and 1.26) for a system with optimized sensitivity. (They are given in nomogram 1 at k = kl and u = u1 and in nomogram 2 at k = k2 and u = vz.) All these values are obtained under the conditions when eqn. 1.50 is in accord with experimental data at the selected values of coefficients 7,A and C. Each straight line in the nomogram corresponds t o one fixed particle size. Pressure drop can be measured along these lines. Each parabolic curve corresponds t o one value of pressure drop. Particle size can be measured along these curves. The ordinate gives both the column length and the time of analysis and the abscissa gives the efficiency. For example, nomogram 2 indicates that for carrying out the analysis under optimum conditions at N = 5000 plates and LP= 0.6 MPa, one requires a column 14 cm in length packed with a sorbent withd, = 9.6 pm. The time of analysis is 9.5 min. On passing t o the

-

OPTIMIZATION OF THE PROCESS

47

Fig, 1.25. Nomogram 1 representing the relationship between the main parameters of the system: analysis time, column length, plate number, pressure drop and sorbent grain size under optimum conditions with respect to the sensitivity of the analysis. The left-hand vertical axis gives analysis time (outer side) and the product of column length in cm and sorbent grain diameter in pm (inner side). The lower horizontal axis gives the plate number. Each inclined straight line of the nomogram corresponds to a single value of sorbent grain diameter given at the outer sides of the right-hand vertical and upper horizontal axes at the points of their intersection with the inclined straight lines. The inner sides of these axes give the values of pressure drop in the column that do not change dong parabolic curves. The nomogram corresponds to the value of the generalized optimization parameter k equal to k, = 3.55.

second set of optimum conditions at the same values of N and d,, the following values are required: L = 13 cm, d, = 7 pm and t = 12.5 min. When experimental technical parameters are varied or when one passes to the other optimum mode, optimum characteristics of the system can also be found from nomogram 1 (or 2). For this purpose the data obtained must be multiplied by the corresponding correcting fact ors. For example, if under new conditions the efficiency of analysis N and the particle size d, remain unaltered, the optimum values of L , AP and analysis time r can be found from nomogram 1 according to the equations ( 1 .136)

In this case N , = N , = const and d,,, = d,, = const.

48

GENERAL THEORY OF CHROMATOGRAPHY

t, min

Fig. 1.26. Nomogram 2 representing the relationship between the main parameters of the system: analysis time, column length, plate number, pressure drop and sorbent grain size under optimum conditions with respect to analysis speed (same designations as in Fig. 1.25, nomogram 1).

Subscript 1 refers to the parameters obtained directly from nomogram 1 in Fig. 1.25 and to the optimum characteristics of the corresponding chromatographic column. Subscript 2 refers to the parameters of the specific column used and the characteristics of the required optimum mode. Similarly, when N and AP are fixed, under new conditions the optimum values of L , d p and t can be found from nomogram 1 according to the equations dp) = d P , d G K >

L2 = L , ( h 2 l h , > & x l ,

t2 = t,h:Ih:

(1.137)

= AP2 = const.

In this case N1= N 2 = const and

1.6.11. Optimization for the sensitivity of analysis in thin-layer chromatography In TLC at the chosen technical parameters the elution rate depends only on two variables: the length of the plate (film) L and the diameter of sorbent particles dp.Hence, in this case the search for optimum conditions can be made by varying the d , parameter alone. It is necessary to select such a value of dp at which the required efficiency of analysis N will be obtained on a plate (film) of minimum length L. The corresponding dependences L = L(d,, N = const) shown in Fig. 1.27 are plotted according to the optimization curve taken from ref. 54 and shown in Fig. 1.28. Figures 1.27 and 1.28 and eqns. 1.124 and 1.125 for F and show that the optimum particle size and the corresponding plate length increase with the efficiency required for a given analysis. It could

x

OPTIMIZATION OF THE PROCESS

49

L, cm

2ot

t

,

0

, 5

,

,

,

,

, 10

,

,

,

,

, , , , 15 d,, p

Fig. 1.27. Plate length L vs. particle size d, at fixed values of analysis efficiency N (1) N = 1000; ( 2 ) N = 1500. (- - -) - the curve passing through the minima of curves of the family L = L ( d , ; N = const).

be seen that a similar situation was also observed in column chromatography (CC). However, in contrast to CC, in TLC there is only one mode optimum for the sensitivity of analysis. It corresponds to the following conditions (here and below, for simplicity, we use symbols u and k in place of Tand k): k

Fig. 1.28(a) Reduced HETP, h , vs. reduced flow rate u in TLC. (b) Generalized optimization parameter k vs. u in TLC.

GENERAL THEORY OF CHROMATOGRAPHY

50

-aL= - =aL

av

ad,

O,

aH aH _ - - =o, av

ad,

ah 2-v+h=O av

(1.138)

The value of v = v 1 satisfying eqn. 1.138 lies to the left of the minimum of the dependence h = h(v). For TLC the generalized optimization parameter differs from that in column chromatography,

k

= hv = ( 4 d / D ) R f d p / N

( 1 .139)

If its value* is determined and the parameters a, D, R f , d , and N are varied in such a manner that k = k l remains invariable, the optimum conditions for the sensitivity of analysis will be maintained. TLC is also characterized by the absence of pressure drop, and the variable elution rate. The average value of the elution rate is used in calculations and the efficiency is determined according to the ratio of the migration length along the plate (film) to its dispersion N = (RfL/v)’

(1.140)

Combined information on the characteristics and the parameters of the system operating under the optimum conditions for the sensitivity of analysis can be found from nomogram 3 shown in Fig. 1.29. Here the abscissa gives the plate length and the analysis time divided by the particle diameter, and the ordinate gives the efficiency of analysis, the particle diameter and the retention coefficient. Each parabolic curve corresponds to a single value of R f . For carrying out the analysis at a given efficiency the parameters of the plate and the time taken by the experiment are determined as follows. The required value of N is plotted on the left-hand vertical axis and the intersection of this ordinate with the parabolic curve corresponding to the given value of R f is found. The abscissa of this point gives the plate length on the lower horizontal axis and the ratio of analysis time to the particle diameter on the upper axis. The ordinate of the point of intersection of this abscissa with the upper parabolic curve in Fig. 1.29 gives the d, value on the extreme left-hand vertical axis. For example, if the efficiency of analysis should be N = 1500 at the retention coefficient R f = 0.4, the plate length L should be 8.5 cm, the particle diameter d , should be 7 pm and the time of experiment will be 17.5 min. This nomogram can be used to determine optimum parameters for any specific chromatographic system if the corresponding correction factors are known. The values of these factors are closely related to solvent viscosity, its surface tension, the wettability of the sorbent and the quality of the packing bed, i.e. to all the factors that affect the shape of the optimization curve h = h(v) and the position and height of its minimum. If subscript 1 refers to the characteristics of the curve corresponding to the nomogram in Fig. 1.29 and subscript 2 to those of the curve obtained for a specific system, the ~

u,

At the selected technical parameters the value of k corresponding to u = u , is k = k, = 3.28 and = 1.16.

OPTIMIZATION OF THE PROCESS

51

Fig. 1.29. Nomogram 3 representing the relationship between the main parameters of the TLC chromatographic system optimized for analysis sensitivity. By moving from the selected value of efficiency N along the broken line in the direction shown by arrows onecan determine (see text) the required plate length L , the particle sized, and the analysis time t for each value of the retention coefficient Rf.If the particle size dp is chosen as the starting-point, then in order to determine L , N and t one should move in the direction shown by arrows along the solid line.

following equation gives the optimum values of dp, L and analysis N :

r at a given efficiency

of

(1.141)

where dp, , L and predetermined.

t1 are

multiplied by correction factors, the values of which should be

1.6.12. Optimization for speed of analysis in column chromatography When the efficiency of analysis is fixed, its speed optimization is reduced decreasing the time of the experiment to a minimum. It is clear that as the elution rate increases, the elution time of the substance decreases and its spreading increases. This leads t o the existence of a minimum time required for obtaining a given number of theoretical plates. The equation for the analysis time t

GENERAL THEORY OF CHROMATOGRAPHY

52 t . min 100.

t, min 50

(0)

11

80-

(b)

6040-

200-

Fig. 1.30. Analysis time vs. (a) particle size d, at various fixed values of pressure drop AP, (b) pressure drop aP at various fixed values of particle size d,. (- - -) - the envelope of the family of curves t = t (AP; d, = const).

N 2 7) t = - - (1

AP ko

+ k')h2

(1.142)

in fact shows that t decreases with increasing AP (Fig. 1.30b), and when d, is varied ? passes through a minimum (Fig. 1.30a), the position of which coincides with the minimum of HETP (1.143) Hence, the desired condition of optimization is reduced to the choice of such an elution rate v that corresponds to the position of the minimum of the dependence h = h(u). Consequently, the characteristics v and k of the chromatographic mode optimum for the speed of analysis coincide with analogous characteristics of one of the two optimum modes for the sensitivity of analysis. These modes differ in that the optimum conditions for the speed of analysis are attained at the minimum of the function t = t (d,; AP = const) for the variable d, and the optimum sensitivity corresponds to the minimum of the function L = L(M; d, = const) for the variable AP*. Thus, the optimization for the speed of analysis (as in its sensitivity) consists in the choice of an optimum particle size for each pressure drop

2) 112

dip' = g k 2

(1.144)

It is clear that eqn. 1.144 for dgpt differs from the similar eqn. 1.134 only in the value of the generalized optimization parameter k = k2.

* It should be noted that both these cases occur when the minimum of the function h = h(u) for the variable u is attained.

OPTIMIZATION OF THE PROCESS

53

If the column length is the extreme function of AP at a fixed d,, the time of analysis is a monotonically decreasing function of AP. When the value of d , is chosen closer to dgPt, this dependence (of t on A P ) becomes more pronounced. At d, = dgPt the desired efficiency of analysis is attained at time t at the lowest pressure drop (among possible values). Figures 1.30a and 1.30b show that the dependences t = t (AP,N ; d , = const) and t = t (d,, N ; A P = const) are the families of mutually intersecting curves. At d, = const. they are hyperbolas and a t AP = const. they are parabolas, the apexes of which coincide with the minimum values of t. The coordinates of the apexes of the parabolas (Fig. 1.30a) correspond t o points on the envelope of the family of hyperbolas in Fig. 1.30b. This means that along this envelope the conditions of chromatography are optimum for the speed of analysis. If one moves along each hyperbola in Fig. 1.30b towards its point tangent t o the envelope, the chromatographic system is optimized for time of analysis, i.e. it is ‘tuned’ and when one moves away from it, the system is ‘untuned’. Hence, the dependence t = t(AP, N ; d, = const) is steep t o the left of the tangent point and shallow t o the right of it. Evidently, by fixing the values of k = kopt and d, = d i P t it is possible to vary in correlation with each other the values of AP and N at fixed values of k o , 77, D and R f maintaining the time of analysis at a minimum. However, this minimum will have different values at different combinations of AP and N. Let the function r = r(d,) expressed by eqn. 1.74 be at a minimum at d, Ed:Pt = d l , and at fixed values of AP = P 1 and N = N 1 . By varying the parameters AP and N and assuming them equal t o Pz and N2 it can be found that dependence 1.74 is a t a minimum at the same value of d , = dl if the PIN ratio remains unchanged, i.e. at P 1IN, = P , /N2. In this case the value of the function t = f(dp) at the point of the minimum changes by a factor of P , /P1 according t o eqns. 1.74 and 1.5 1

For example, if the value of APdiko k =k =- const NqD

is constant and the values of AP and N are varied from P1 and N1 t o P2 and N2, it follows that t will pass through a minimum at the same value of d , = d I if the equality P , /Nl = P , /Nz is fulfilled. In this case the value of t at the minimum point will change by a factor of P I /P2 or N1IN,. (1.145)

In general, when the condition of optimization is obeyed and all three parameters, P , N a n d d,, are changed, the following ratio between the times of analysis will be found: (1.146)

GENERAL THEORY OF CHROMATOGRAPHY

54

When one of the two values, P or N , is varied and the other remains fixed, the optimum value of particle size dpOPt is displaced ( 1.147)

(1.148) Hence, when the pressure drop increases by a factor in at a fixed N , the value dpOPt decreases by a factor and the time of analysis decreases by a factor m. When the efficiency increases by a factor m , the value dpOPt increases by a factor fiand the time of analysis increases by a factor mz. In general, for the technical parameters selected by us the optimum choice of operational parameters corresponds t o the value of the generalized parameter k = 6.55.

4th-

(1.149) Arbitrary variation in P and N leads to a displacement in d , = dpOPt (1.150) For obtaining optimum conditions of chromatography it is convenient t o use nomogram 2 (Fig. 1.26). The correction factors for different chromatographic columns can be found according t o eqns. 1.136 and 1.137.

1.6.13. Optimization for speed of analysis in thin-layer chromatography Owing to the specific features of thin-layer chromatography considered in sections 1.2 and 1 S . 2 the dependence of time of analysis on operational parameters differs from that for column chromatography

N3 16a4 R: D

1 = -- h

(1.151)

3 ~

Hence, when the value of N is fixed, the minimum o f f as a function of u corresponding t o the optimum mode satisfies the following condition: 3 h ’ ~ +h = 0,

h’

= ah/&

(1.152)

The value of u = u2 satisfying eqn. 1 .I52 is less than u = u,, at which the function h = h(u) passes through a minimum. At the same time u2 is greater than the value of ul which determines in TLC the mode optimum for analysis sensitivity, i.e. VI

< u2 < vo

( 1. I 53)

Hence, in contrast to column chromatography, in TLC both optimum modes are characterized by elution rates u lying t o the left of the minimum of the optimization curve h = h ( u ) . The value of uz corresponds to the generalized parameter kz. At the

OPTIMIZATION OF THE PROCESS

55

t. s

Fig. 1.31. Analysis time r vs. particle size d , in TLC at various fixed values of the efficiency of analysis N : (1) N = 1000, (2) N = 1500. (---) - the curve passing through the minima of the family of curves t = r (dp;N = const).

chosen parameters this optimum mode corresponds t o the following values of v and k : ~2

= 1.70,

k2

= 4.10

Figure 1.31 shows the dependence of time of analysis on the size of sorbent particles for two different values of efficiency N = 1000 and N = 1500. It is clear that the time t increases with N and the position of its minimum as a function of d , is displaced towards higher values of d,. Nomogram 4 in Fig. 1.32 permits the calculation of the optimum combination of operational parameters according t o eqns. 1.152.

I .6.14. Optimization of analysis efficiency The optimization based on the speed of analysis also leads to optimum productivity (or delivery) N / t when, other conditions being equal, the greatest number of theoretical plates, i.e. the greatest efficiency or performance per unit time is attained. We will now consider the change in the efficiency of the system depending on the values of parameters d,, AP and t when one of these parameters is varied and the other two are fixed. 1.6.14.1Dependence of the efficiency of the system on the size of sorbent particles when

the values of pressure drop in the column and the time of analysis remain fixed The desired dependence is plotted in Fig. 1.33. It is clear that this dependence is extreme.

GENERAL THEORY OF CHROMATOGRAPHY

56

Fig. 1.32. Nomogram 4 representing the relationship between the main parameters of the TLC chromatographic system optimized for speed of analysis (symbols as in Fig. 1.29).

(0)

7000.

0

25

50

Fig. 1.33. Efficiency of the system as a function of sorbent grain size at various values of: (a) pressure drop at 1 = 5 min, A P = (1) 0.1 MPa, (2) 1 MPa, (3) 3 MPa; (b) analysis time at A P = 1 MPa, t = (1) 1 min, (2) 5 min,(3) 10 min.

OPTIMIZATION OF THE PROCESS

51

It passes through a maximum at the same value of u at which the function h = h(v) passes through a minimum. When the pressure drop AP is increased and the values o f t are fixed, the maximum of N is displaced towards lower values of d,. When the pressure drop is fixed and the time of analysis increases, the maximum of N is displaced towards higher values of d,. Equations 1.5 1 and 1.74 readily yield ( 1.154)

It is clear that if the time is fixed and the pressure drop is increased by a factor m , the maximum value of N will increase by a factor fiand the corresponding particle size d , will decrease by a factor If the time of analysis is varied at a fixed pressure drop, then, if t is increased by a factor m , N,, will increase by a factor fi and the corresponding particle size d, will increase by a factor fi.

G.

1.6.14.2.Dependence of the efficiency of the system on pressure drop inthe column at fixed time of analysis and particle size The dependence of efficiency on pressure drop at fixed values of d p and t is important in practice. In this case the increase in N is accompanied by an increase in delivery, N / t . It can be seen in Fig. I .34 showing this dependence, that at a fixed time of analysis,

0

0.5

1

1.5 P,

N

Fig. 1.34. Efficiency of the system as a function of pressure drop at various values of sorbent grain size and fixed analysis time; t = 5 min, dp: (1) 5 pm;(2) 10 pm; (3) 15 pm; (4) 20 pm. The envelope of a family of curves r = t ( A P ; d , = const) is shown by a broken line.

GENERAL THEORY OF CHROMATOGRAPHY

58

efficiency increases with pressure drop. A single curve N = N ( A P ) corresponds t o each particle size d,. These curves plotted at various values of d, intersect. As in the variants of optimization previously considered, the distinguishing feature of the dependence of N on AP is the existence of the envelope of a one-parameter family of curves N = N ( A P ; d, = const) where d, is a parameter. The envelope is described by the equation (1.155) The existence of the envelope means that for each pressure drop AP it is possible to select such a particle size d , at which the maximum efficiency of the system is attained, the time of analysis being fixed. The generalized parameter k = k 2 remains invariable along the envelope. The tangent point of the dependence N = N ( A P ) with the envelope at a fixed value of d, divides the pressure range into two regions. If this point is approached from the left (pressure drop is increased), the system gradually becomes increasingly optimized and its maximum efficiency at a given pressure drop is achieved. By moving away from the tangent point to the right (continuing t o increase the pressure drop) we continue to increase the efficiency of the system. However, it is now no longer the highest efficiency. At each value of pressure drop t o the right of the point of tangent it is possible t o attain a still greater increase in efficiency up t o the optimum value. It should be noted that the increase in efficiency with increasing pressure drop (at a fixed time of analysis) occurs simultaneously with the increase in column length L .

1.6.14.3. Dependence of the efficiency of the system on the time of analysis and the column length at fixed values of pressure drop and particle size The dependence of the efficiency of the system N on the time of analysis t and the column length L at fixed values of pressure drop AP and the particle size d, is very significant. As seen in section 1.5.2, a limiting efficiency Nlim exists for each pair of values of AP and d,; it is attained asymptotically with an infinite increase in t and L . This is clearly seen in Fig. 1.35 which shows that t o attain high values of the efficiency of analysis one should not only increase the column length (maintaining its quality) but mainly use greater pressure drops. The increase in particle size also leads t o an increase in Nlim and in the length of the steep part of dependences N = N ( t ) and N = N(L), but this results in a considerable inclease in f and L . 1.6.14.4. Dependence of the efficiency of the system on pressure drop and particle size

at a fixed column length In practice the choice of the chromatographic mode a t a fixed column length is very important. In this case the efficiency of analysis is given by (1.156)

OPTIMIZATION OF THE PROCESS

59

10000 '

0

50

100 L,cm

Fig. 1.35. Efficiency of analysis N vs. column length L at various fixed values of particle size d , and pressure drop AP. Horizontal broken lines - limiting values of efficiency Nlim for each p& of values of AP and d,. Parabolic broken lines - the envelopes of the families of curves N = N (15, d,; AP = 1 MPa) and N = N ( L , d,; & = 10 MPa).

It is clear that for each chosen particle size d, the value of N is at a maximum when the reduced HETP (h) passes through a minimum. Hence, the highest efficiency on a column of a given length is attained for the optimum mode characterized by the values of k = k z and v = v2. Since this mode can be attained at different combinations of N , AP and d,, the following general trend should be borne in mind: the higher the pressure drop AP and the smaller the particle size d,, the greater is the efficiency of analysis N that can be attained on a column of a given length L . However, in this case AP and d , should be varied in accordance with each other so as to maintain the optimum values of parameters k and v. Taking in account these considerations, one can see from eqn. 1.156 that an m-fold decrease in particle size leads to an rn-fold increase in efficiency, provided the pressure drop increases m1l3 times. Hence, the following equalities hold: (1.157) at L and k = const. 1.6.14.5. Optimizationfor the efficiency of analysis in thin-layer chromatography

In TLC the optimization for the efficiency of analysis is of particular interest. The following cases may be considered here: (1) Maximum efficiency N should be attained by an appropriate choice of particle size at a fixed plate (film) length or limited time of analysis t . (2) Maximum efficiency N should be attained by varying the plate length and the time of analysis. The first case is represented by the dependences shown in Figs. 1.36 and 1.37 plotted for different values of L and t. They exhibit a pronounced extreme shape with the maxima lying on one straight line. These straight lines satisfy the equation

GENERAL THEORY OF CHROMATOGRAPHY

60

0

5

10

15

20

*PI P W

25

Fig. 1.36. EfficiencyN vs. particle size d , at various fixed values of plate length L : (1) 5 cm; (2) 10cm; (3) 20 cm. (- - -) - the curve passing through the maxima of dependences N = N (d,; L = const).

( 1.I 5 8)

where i = 1 at L = const and i = 2 a t t = const (as previously, k, = 3.28 and k2 = 4.10). The value of N = N,, for a plate (film) of length L is given by ( 1 . 159)

Equation 1.159 is obtained from eqn. 1.158 if d, is replaced by an equal expression: LRf/hiN=d,. N

Fig. 1.37. Efficiency N vs. particle size d p in TLC at fixed values of analysis time f : (1) 10min; (2) 15 min; (3) 20 rnin.

OPTIMIZATION OF THE PROCESS

61

Fig. 1.38. Efficiency of analysis N vs. plate length L for various values of particle size d,: (1) 5 gm; (2) 10 pm.(- - -) - the envelope of the family of curves N = N ( L ;d , = const)..

The second case is represented by the families of curves shown in Figs. 1.38 and 1.39. Their specific feature is the existence of envelopes. The points of the envelope shown in Fig. 1.38 correspond to the maxima of dependences shown in Fig. 1.36. This means that the envelope corresponds to a mode optimum not only for the efficiency of analysis but also for its sensitivity at L = const. The points of the envelope shown in Fig. 1.39 correspond to the maxima of dependences shown in Fig.1.37. Hence, this envelope corresponds to a mode optimum not only for the efficiency of analysis at r = const but also for its speed. A specific feature of the dependences shown in Figs. 1.38 and 1.39 is the fact that for each value of d , a limiting efficiency exists which is asymptotically attained at L and t

1500-

1000 -

I

0

I

I

5

I

I

I

10

I

I

I

.

l

15

.

.

.

.

I

.

.

.

2o t, min

.

I

I

I

25

Fig. 1.39. Efficiency N vs. analysis time r in TLC at various values of particle size d,: (1) 5 urn; (2) 7 pm; (3) 10 pm. (- -) - the envelope of the family of curves N = N ( I ;d, = const).

-

62

GENERAL THEORY OF CHROMATOGRAPHY

tending to infinity. It can be seen that at the chosen technical parameters over the ranges of L from 5 to 15cm and t from 5 to 15min the efficiency N markedly increases with increasing L and t. Beyond these limits the increase in L and t leads only to a slight increase in N . Hence, the essence of optimization for the efficiency of analysis in TLC can be formulated as follows: at the chosen technical parameters (including solvent viscosity, surface tension, the wettability of the sorbent, etc.) each value of dp corresponds to an optimum plate (film) length satisfying the equation ( 1 . 160)

where v1 corresponds to condition 1.138. The optimum value of L in Fig. 1.38 corresponds to the tangent point of the curve N = N(L ; d , = const) with the envelope of the family of these curves. A decrease in L at a given d , compared to its value required by eqn. 1.160, will lead to a drastic decrease in N , whereas an increase in L will affect N only slightly although it will still increase. Hence, in conventional TLC one should not try to attain very high efficiency. It can be achieved at a much lower cost by using multipledevelopment TLC [55]. 1.6.15. Pressure drop in the chromatographic column as a function of

operating parameters

Martin et al. [48] have drawn attention to the importance of optimization based on pressure drop in the chromatographic column. Appropriate analysis was carried out and it was shown that a directed choice of such parameters as column length L and particle size d , at a given value of N makes it possible to carry out fractionation by using a minimum pressure drop P. This choice corresponds to the value of reduced velocity at which the dependence (1 SO) h = h(v) passes through a minimum. This is due to the fact that at fixed values of t, N and technical parameters, pressure drop is a function of h 2 only and, hence, passes through a minimum simultaneously with h . Thus, in the optimization of the speed of analysis the choice of a minimum pressure drop is carried out automatically. Figure 1.40 shows that AP as a function of both L and d , is of an extremum character with a distinct minimum. Along each curve in this figure the values of L and d, change simultaneously. However, the minima of all curves obey one condition: they correspond to the minimum values of reduced height h as a function of reduced velocity v. 1.6.16 General conclusions on optimisation of the chromatographic process

The results of the optimization obtained in the preceding sections can be formulated as the following relationship. Two types of optimum conditions of chromatographic analysis exist. One of them leads to the optimum sensitivity of analysis and the other yields its optimum rate and efficiency. Each type is uniquely determined by the optimum value of the reduced elution rate v or the corresponding value of the generalized optimization parameter k = hv.

OF'TIMIZATION OF THE PROCESS

63

3

1.

0

'

-

1

.

10

20

30 L , c m

Fig. 1.40. Pressure drop in a column as a function of its length.

In the optimization based on the sensitivity of analysis, minimum dimensions of columns are attained. The optimization for the speed of analysis leads to the maximum productivity of the system N / f at a minimum pressure drop in the column AP. The optimum value of uo corresponds to a minimum of reduced HETP, h = hmh. The generalized optimization parameter k is expressed by the operating parameters of the system as follows:

1.6.17 Optimization of the chromatographic process in which extracolumn spreading is taken into account

The optimization of the chromatographic process considered in sections 1.6.1-1.6.16 is based on eqn. 1 .SO with constant coefficients 7 , A and C. This equation holds over the velocity range u E [0.1; 1001 at the size of sorbent particles dp E [ 5 ; 1001 (pm) and column length L > 1Ocm. As the column length decreases, the results of analysis will be increasingly affected by extracolumn spreading (ES) in various assemblies of the chromatograph and it should be taken into account in the optimization calculations. Here, ES decreases the efficiency of the system, i.e. the number of theoretical plates N , and increases HETP. This is due to additional spreading of the chromatographic zone caused by ES. Let us assume that in the absence of ES the number of theoretical plates N is attained during time c on a column of length L with HETP equal to H and dispersion equal to uz .

H

= LJN = ozlL

(1.161)

GENERAL THEORY OF CHROMATOGRAPHY

64

Here, ES increases HETP for a given system by the value AH up to

B

w: (1.162)

= H+AH

This is caused by the increasing dispersion of spreading

aZ =

(1.163)

o2 + P o z

Combining eqns. 1.161-1.163 gives u2 A o z H- = a2 - = -+L L L

(1.164)

From eqn. 1.164 it is possible to find the equation for calculating A H

Aa2 = -Poz A02 A f f = -L flB - R ( H + A H )

(1.165)

Hence,

Aa2 (AH>’+AH.H--== 0 N

(1.1 66)

The solution of eqn. 1.166 has the form*

Aff =

1. ( - - H + d H Z + 4AaZ/fl)

(1.167)

The value of Pa2 can be estimated experimentally and the value of fl in eqn. 1.167 should be chosen bearing in mind that all the optimization calculations in sections 1.6.1 1.6.16 should remain valid. For this purpose only one condition should be fulfilled: the desired efficiency of analysis should be expressed by the number of ‘conditional’ theoretical plates, Ncond satisfying the ratio Ncond/N = N / N , i.e.

1= LJR

= L/(H

+ AH) = (L/H)/(l + AHJH) = 1 + NAH/H

(1.168)

Hence,

N / I = 1 + AHJH

(1.1 69)

and Thus, when Ao2 is estimated experimentally and AH is estimated from eqn. 1.167, where 1is replaced with N , it is possible to leave as invariable the results of the optimization obtained in sections 1.6.1-1.6.16 under the condition that the value of N in the calculations is replaced with N c o d calculated according to eqn. 1.1 70, and that the value of H is increased by AH. Evidently, ES leads to an increase in the values of L,t and AP obtained in sections 1.6.1-1.6.16 and changes the optimum values of generalized parameters and reduced elution rates. However, the general relationships of optimiz-

* The solution of eqn. meaning.

1.166 with the negative value of A H is neglected because it has no physical

REFERENCES

65

ation described in sections 1.6.1 -1.6.16 are not affected by ES, and their quantitative recalculation is not difficult. 1.6.18. Recommendations for the choice of optimum chromatographic conditions

The following order of operations may be recommended for carrying out the optimization of each type. (1) The value of N is determined proceeding from the required values of K R , $, AM/M and S. (2) The dependences h = h(u) and u = u(M) characteristic of a given type of column packing are obtained. (3) The dependence h = h(u) is approximated by the function determined by eqn. 1S O . The coefficients of this equation are found. (4)The optimum value of kopt and the corresponding value of uoPt are calculated from the minimum of the dependence h = h(u). (5) Dependences L = L(d,) and r = t(dp) are calculated for one set of fixed values of other parameters. (6)The optimum values, kept, uoM, are found for the extreme points of these dependences. (7) The desired combination of all the main parameters of the system is determined. (8) According to these data the chromatographic system is designed depending on the chosen conditions of optimization, and the required analysis is carried out. 1h.19 ‘Boxcar’ chromatography as a kind of optimization

In conclusion it should be noted that the problems of optimization of the chromatographic process are very urgent and attract the attention of research workers, both in the field of conventional chromatography on packed columns [56-581 and in that of chromatography on open-tube (capillary) columns [59]. One of the latest advances in optimization is the ‘boxcar’ chromatography proposed by Snyder et al. [60] .It is a new approach to attaining increased rates of analysis and very large column plate numbers. It is particularly efficient for routine serial analyses of polymers, greatly reducing the time taken. The method is fairly simple,’being a new form of column-switching. First, one or several components of interest are separated from the mixture on a column of small length and moderate efficiency. They are diverted to a second, longer column of higher efficiency. Then the first column is ready for the injection of the next portion of the polymer. As a result, highefficiency separation (lo’ < N < lo’) is possible within shorter times.

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