- Email: [email protected]
Effect of modulator sorption in gradient elution chromatography: gradient deformation
Chemical Engineering Science, Vol. 47, No. 1, pp. 233-239, 1992. Printed in Great Britain.
ooo!-2509pz s5.ca + 0.00 1991 PergamanPressplc
0
EFFECT OF MODULATOR SORPTION IN GRADIENT ELUTION CHROMATOGRAPHY: GRADIENT DEFORMATION AJOY VELAYUDHAN and MICHAEL R. LADISCH+ Laboratory of Renewable Resources Engineering, Department of Agricultural Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received for publication 16 May 1991) Ahstict-In gradient elution chromatography, the mobile phase composition at the column inlet is a function of time. The consequences of accounting for the adsorption of the mobile phase components themselves are investigated for the experimentally common situation of a linear inlet gradient. Surface excess adsorption data from the literature for water-acetonitrile mobile phases in reversed-phase chromatography using different octadecyl stationary phases are shown to he in good internal agreement. The resulting individual isotherms as calculated by Tani and Suzuki (1989) are fitted to Langmuir and BET forms; it is shown that the Langmuirian form fits the data well except at extremely high acetonitrile concentrations. Using these isotherm parameters, it is found that a linear inlet gradient could suffer significant distortion while moving down the column. In the extreme case, a shock front of the mobile phase modulator could result. Analytical expressions are derived describing the conditions under which such a
shock could occur within the column and its subsequent Path, assuming Langmuirian sorption. INTRODUCTION
Elution chromatography is carried out in either the isocratic mode, where the mobile phase composition
remains unchanged throughout the separation, or the gradient mode, where the mobile phase composition at the inlet varies with time (Snyder and Kirkland, 1979). For the separation of relatively small molecules, gradient elution is generally used to shorten the separation time by decreasing the retention of the more strongly retained components of the feed mixture; these components are also generally recovered in a more concentrated form than would be obtained from the corresponding isocratic elution run. The concomitant disadvantage is that resolution is lost with respect to isocratic elution; the method can therefore only be used when the initial separation factors of the feed components are sufficiently high (Snyder, 1980). On the other hand, macromolecules such as proteins, as a result of their multifunctional interactions with the stationary phase, tend to adsorb very strongly and, in the absence. of a competitive agent, frequently exhibit irreversible adsorption, at least on the time scale of chromatographic interest (Horbett and Brash, 1987; Andrade, 1985). Consequently, isocratic elution is usually not a viable option and gradient elution, along with frontal and displacement chromatography and stepwise elution, become the techniques of choice. Gradient elution has been the focus of considerable work, notably by Snyder et al. (summarized in Snyder, 1980; Snyder, 1986), and Jandera and Churacek (summarized in Jandera and Churacek, 1985). For most gradient runs these studies, which assume that none of the mobile phase components adsorb, result in an
accurate estimation of the retention time and a reasonable assessment of the resulting band widths in the effluent history. However, mobile phase components do adsorb to the stationary phase, as is well known (Snyder, 1980; Jandera and Churacek, 1985) and we attempt a quantitative analysis of the effect of such adsorption on the shape of the gradient as it passes through the column. We consider, for simplicity, a binary mobile phase consisting of a carrier and an additive which modulates adsorbate retention. We call this additive the mobile phase modulator (MPM), a general term intended to subsume the various commonly used mobile phase additives: it thus includes salts in ion exchange (and hydrophobic interaction) chromatography and organic modifiers in reversed phase chromatography. We present a quantitative discussion of the case where the modulator exhibits Langmuirian sorption behavior (justification of this particular form will be given presently), and show that, in many cases of experimental interest, the front of the modulator is appreciably distorted as a result of its sorption. When the distortion is sufficiently large, a part of the gradient front can become a shock (or shock layer); an analytical expression describing the conditions under which a shock is formed will be given later. To the extent that shock layers are characteristic of stepwise elution separations, it becomes possible for gradient elution runs to exhibit features that are usually associated with stepwise desorption schedules, e.g. ghost peaks (Yamamoto et al., 1988). PROBLEM
that
‘Author to whom
the
retention correspondence
stationary
should be addressed.
233
FORMULATION
The rationale for considering modulator modulator by
competing
frequently for
alters
binding
phase. Since the adsorbates,
sorption is adsorbate on the particularly
sites
234
ASOY VELAYUDHAN
and MICHAEL
those that are macromolecules, are typically much more retentive than the modulator, it can only compete effectively by being present at relatively much higher concentrations. At such high concentrations, it is likely to be in the nonlinear region of its singlecomponent sorption isotherm, and will consequently suffer self-interference (Helfferich and Klein, 1970). Thus, different concentrations of the modulator will move at different speeds, and the shape of the gradient at the inlet will suffer distortion as it travels down the column. This argument is probably not applicable to hydrophobic interaction chromatography, where the salt does not achieve its modulation of adsorbate retention by binding to the stationary phase. We will consider the case where the feed components are in the linear regions of their sorption isotherms. This is not only often true experimentally, but can frequently be applied even when the initial feed concentrations are high. The presence of the modulator suppresses the retention of the feed components, generating a linearizing effect: the region of a feed component’s competitive sarption isotherm that is linear can he much larger than the corresponding region of its pure isotherm (Yamamoto et al., 1988; Frey, 1990). Then the modulator bhavior can be decoupled from that of the feed. In e first approximation, the modulator can be treate$ ,, as possessing se much less no dispersion, since it is likely to dis than the feed components (whose bands I?! reading is of primary interest). Ideal chromatography being assumed, the deformation of the modulator gr,adient can then be analyzed by the method of characteristics.
ADSORPTION
lSOTHERM
OF THE MODULATOR
For specificity, we restrict attention to reversedphase chromatography with acetonitrile-water as the mobile phase. This system is widely used in many practically important contexts (e.g. analytical chromatography of proteins and polypeptides, preparative chromatography of amino acids and oligonucleotides). In what follows, we will consider octadecyl silica (C- 18) sorbents although analogous calculations can be made for other stationary phases. We first need to specify the modulator’s (acetonitrile) isotherm. However, the modulator is part of a binary mixture and it is well known that measurement of adsorption from a binary liquid solution can only unambiguously yield the so-called excess isotherm (e.g. Adamson, 1967). This represents the amount taken up by the adsorbent as the difference between the equilibrium bulk compositions before and after contacting with the adsorbent. Consequently, only a net sorption is known, and an additional model of the sorption process is required in order to extract the individual isotherm of the modulator. The simplest model is to assume that the water does not adsorb. Then the surface excess data can be converted into the individual isotherm for acetonitrile, as was done by Slaats et al. (1981), for Lichrosorb RP-18 (10 p particles). However, the resulting individual
R.L~ursc~
isotherm went through a maximum, which they suggested is physically unlikely. McCormick and Karger (1980) also reported an adsorption maximum. We offer another argument to indicate the physical improbability of such an isotherm. Consider a batch system in which the equilibrium mobile phase concentration of acetonitrile is such that its stationary phase concentration is at its maximum value. If a small amount of acetonitrile is now added to the mobile phase, the system will shift to a new equilibrium at which the stationary phase concentration will be lower than it was previously, since it was initially at its maximum value. Consequently, some acetonitrile will be expelled from the stationary phase into the mobile phase. But this increase in mobile phase concentration will trigger a further shift in equilibrium, which further decreases the stationary phase concentration. A similar instability exists for all points on the decreasing branch of the isotherm, contradicting the assumption-implicit in an isotherm-that the stationary and mobile phase concentrations are in equilibrium along this branch. In this context it may be noted that individual isotherms that exhibit maxima and minima have been reported in the literature for the adsorption of surfactants (Trogus et al., 1979). However, micelles are formed in these instances and it is this additional phenomenon that gives rise to adsorption extrema. We assume that the kinds of modulators used in chromatography do not form micelles or undergo other reactions in solution that could affect the shape of the individual isotherm. Therefore, it seems necessary to assume that the water also adsorbs. Another argument in favor of this assumption is that some surface excess data on reversed-phase materials [e.g. the work of LeHa et al. (1982). on C-14, and of SIaats et al. (1981), on C-S] exhibit azeotropy and negative net adsorption. It is clearly impossible for this to occur unless the water also adsorbs. While the C-18 data does not exhibit such azeotropy, it is unlikely that increasing the hydrophobic character of the stationary phase somewhat will completely suppress the sorption of water. Tani and Suzuki (1989). who also measured ACN-H,O sorption on C-18, therefore resorted to the layer model to calculate individual isotherms (Everett, 1986). This gives truly multicomponent isotherms, and is patterned after the model of Kipling and Tester (1952) who required that the total stationary phase concentration always remain at its saturation level. In what follows, we use the individual isotherm data of Tani and Suzuki (1989). It may be noted that the surface excess data of Slaats et al. (1981) agrees very well with that of Tani and Suzuki (see Fig. l), although they were obtained on different C-18 stationary phases. Similar agreement was found by LeHa et al. for two different C-14 stationary phases, even though one was branched and the other linear. Such indifference to the exact nature of the surface might perhaps be attributed to the lowenergy nature of these materials (LeHa et al., 1982).
Effect of modulator sorption in gradient elution chromatography For the purposes of using such individual isotherms in the mass balances, it is convenient to introduce functional forms. The data in Fig. 2 were fitted to the well-known Langmuir and BET forms by nonlinear regression. It must be emphasized that this is a purely formal process; since the ACN individual isotherm represents its multicomponent adsorption in the presence of water, there is no mechanistic basis to either the Langmuir or the BET form. T’he resulting fits are shown in Fig. 2, and the isotherm parameters in Table 1. It can be seen that the Langmuir form fits the data much better at lower and intermediate concentrations; as is to be expected from the Type II nature of the data, the BET form is somewhat better at very high levels of ACN. For the purpose of analysis, the Langmuir isotherm will be used in what follows; however, a numerical simulation of both the BET and Langmuir forms will be offered later for comparison.
BEHAVIOR
235
OF THE MODULATOR
DEFORMATION
AND SHOCK
BAND-
FORMATION
We consider a modulator whose single component isotherm is Langmuirian. An analogous formula will hold for univalent salts in idea1 ion-exchange chromatography (when activity coefficients are unity, Donnan exclusion effects are neglected, etc.). We will consider the experimentally common case of a linearly increasing gradient at the inlet. In the absence of dispersive effects, the mass balance for the modulator may be written as
(l+##~)!!k+v~=o
(1)
where cM and qM are the mobile and stationary phase modulator concentrations, 11the mobile phase velocity and 4 the volumetric phase ratio. These last two terms are given by U 0=
4
00
0
IIt if
1
&bbp (1 - e,)(l - 4 &b+ t1 - Ebb,
_
(3
where u is the superficial velocity, and t+,and E,,are the bulk and intraparticulate porosities. The single-component Langmuirian isotherm gives
l
2
(2)
&b+ t1 -
O
I
1
l
0
which when substituted l
0.
om
I
l
I
0.75
0.50
0.25
o.Ml
ahf9.f
qM= 1 +
ACNmolehreton, M.P.
(4)
b,c,
into eq. (1) yields
&b
ah
%+” 1 at
_-1 ’ + (1 +
l.M)
b,c,)z
z
= ”
(3
Using the dimensionless variables 5 = x/L, z = vt/L, k=wb,.ep (where L is the column length), eq. (5)
Fig. 1. Surface excess adsorption data of water-acetonitrile
on C-18 reversed-phase adsorbents. “Slaats” refers to Slaats
et al. (1981). and “Tani” to Tani and Suzuki (1989).
Il +
+aM
(1 +
The initial condition
4x.
t) = CM,Cl
x=
i
.-a-
att=O
(7a)
!I! ‘I
la
/
Langmuir isotherm
I C
(
x0
at z = 0.
VW
Table 1. Langmuir and BET parameters for the individual isotherm data of acetonitrile in Fig. 2. All stationary phase concentrations are in mmol/g, all mobile phase concentrations in M. To convert the mobile phase concentrations of Fig. 2 into molar units, multiply by 18.78.
20
1
0.25
0.50
G holumo fmcdal~
0.75
1.0
Fig. 2. Individual isotherm for acetonitrile from the layer model, after Taai and Suzuki (1989). Surface area of the stationary phase = 167.49 m’/g. Best-fit Langmuir and BET isotherms arc also shown. CES 47:1-p
is
0.
30
3
aX+aX=o.
x)* 1 ar a<
BET isotherm
4=-
ac
1 + bc a = I.48 mmol/(g-M) b = 0.26 M-’ AKbc
’ = (1 -
bc)Il + (K - l)bc]
A - 3.461 mmol/g ; = fOt; (c$ensionless) = .
236
AJOYVELAYUDHANZUI~MICHA~LR.LADISCH
The boundary gradient is
condition
Atx=O,eM
for a linearly increasing
= = )12 + =
CM.0,
o<
cM.O
Ytt - timj), t*.jG t
+
t < t,.i
cC l+ (1
4%
.+ xc + /Ip*)Z
1 (1’5)
where p z (9 - rinj). These characteristics cross when
where y is the gradient steepness at the inlet. This form of boundary condition is chosen to represent the beginning of the gradient only after the ftid has been completely introduced into the column over a time tinj. In dimensionless form this gives
(171 The point of intersection has the coordinates
*~(1+xo+BPl~2~1+xo+BP2~2 9%fm z_
'II+(I
By the method of characteristics,
(19) Letting q1 -+ qZ + q gives a point on the envelope (e.g. Kaplan, 1958), where (T
7 = tl +
(1 +
=
x0 +
(1 + x0 + BP13 Wad BP)
4% + (1 +
7=?J
{* = (1 +
&I2 1*
xd3
(22)
Wad
This results in
7=4+r [
l+(l+# +abf
1
from which x can be calculated at any 5 and z for z 3 rinj_ However, these characteristics will cross eventually, since the inlet concentrations increase with time and higher concentrations have smaller slopes in the T< plane, as can be seen from eqs (9) and (13). Thus, a shock must be introduced to avoid the physically unrealistic situation of crossing characteristics. (They correspond to a multivalued concentration for some values of 5, t, which is physically unacceptable.) The velocity and strength of the shock can be calculated as follows. An analogous calculation has been carried out by Rhee et al. (1986), for the case where the initial concentration of the modulator and the injection time are zero. If we consider two adjacent characteristics emanati ing from the r-axis (representing boundary data) for r > rinj, they will be of the form
c[ l +
4aM
(1 + x0 + j?p,)Z
5. = r in,
(13)
and
41+
x0 +
2&Q
[.
Since < and o in eqs (20) and (21) are monotonically increasing functions of p, the incipience of the shock corresponds to the cusp of the envelope, when p = 0. The coordinates of this point are therefore
y=o
==
(20)
121)
with the initial condition P;ts=o,
(18)
+x0 + BPd21 4%m + 2x0+ NPI + P2)l .
(9)
(11)
m, + P2)l
+ Xo+BP2)2C4%f+v
where B = yLbJv = yb&, and thj = v/L t,,,, = timj/ to, where t, ( = L/v) is the residence time of an unretained component.
+ 2X0+
+
(1 + x0)
444 + (1 +
xo12
2bad
[
(Z3)
1-
These expressions reduce to those obtained by Rhec et al. (1986) when zinj = x0 = 0. To calculate the path of the shock, we use the expression for the shock velocity (Lax, 1957; Helfferich and Klein, 1970; Rhee et al., 1970). 1 l+@J
(24) Ccl
where [q] = qu - q,, and [c] = c, - cd, the subscripts u and d representing values upstream and downstream of the shock. The concentration downstream of the shock will always be x0; upstream, it will vary as it meets different concentrations from the boundary. Thus, the shock “slowness” (the reciprocal of the velocity) is given by
9% = l + (1 + x0)(1 + XI’ Shock
125)
The same concentration x is borne from the r-axis by the characteristic given by
1 (15)
(26)
E&ct of modulator sorption in gradient elution chromatography
Using
7’ =
7 -
~,~j
-
e,
eq. (26) becomes
dr’ I dx baM dt.=jidt+(I-X)2z-
Wad
dx
(27)
The shock slowness equation [eq. (25)] hecomes dz’
&n de = (I + X)(1 + x0)’
w3)
From eqs (27) and (28)
[; -
#aH
iE$ $ =
4aM
(1 + x)(1. + x0) - (1 + x)2
1
(29)
or
(x - x0) (I+x
2
dC+
(30)
=&.
(1+
This is a linear tit-order ordinary differential equation in 5 with respect to x, with the “initial” condition e = <* at x = x0 (since x = x0 at the inception of the shock). The solution is
< _ (1 + x0)(1 +
xl2
29a.d
7l”j
+
W+B
7
.
(32) Equations (31) and (32) give the coordinates of the shock path in parametric form, 1 being the parameter. Eliminating x between these expressions gives (1 + x0) 7 = %I] -7+dZ+
C
which is part of a parabola. = 0 recovers the solution of If the inlet gradient is kept a certain time, i.e. if the inlet cy =
CH.0.
0 Q t G ti,j Ytt
+
-
thj)r
(33)
Again, setting rini = x0 Rhee et al. (1986). at a constant value after boundary condition is
cM,O
hnj
6
! =na.s= CM.0 + r(t, - G”,),
t 6
t/
(34)
(1 + x0)(1 + %/I2 WaMB
(35)
and =f
=
751.1 +
(1 + x,) + (%I - x0) + (1 + x0)(1 + %,)z 28
(1 + ad3
(
(37)
1
*
DISCUSSION
For the purposes of comparison, we have used plate simulations (Velayudhan and Ladisch, 1991) to assess the shape of the gradient at the column outlet for three cases: when the modulator was unretained (classical Snyder and Jandera-Churacek theory), when sorption followed the Langmuir form, and when sorption followed the BET form. The inlet gradient was linear and the column parameters were as in Table 2. The effluent histories are shown in Fig. 3 for an initial slope of 2.35 M/min (which is somewhat steep but still realistic), for an initially modulator-free column. It can be seen that a shock is formed when modulator sorption is accounted for, and the unretained profile is significantly different from the retained profiles. Equation (37) is the basis for shock formation within the column. It can be seen that three kinds of terms are present: isotherm parameters, which are functions of the mobile phase adsorbent system; the phase ratio, which is a geometric property of the packing; the
t/ 6 t
the shock will eventually reach a constant strength of The coordinates of this point can he x,( = &c,.~). easily calculated by substituting x = xI into eqs (31)-(33X which gives <, =
=
244
(1 + x) + (x - x0) + (1 + x0)(1 + XY 2s
travel at the constant speed corresponding to the concentration 2,. Thus, for a sufficiently long column, any gradient that is linear at the inlet will become a discontinuous step by the time it reaches the outlet. When the dispersion and nonequilibrium contributions (such as pore diffusion and mass transfer across the particle boundary layer) of the modulator are taken into account, the shock will be smoothened into a shock layer (von Mises, 1950: Rhee et al., 1971). Nevertheless, since the modifier is usually a small molecule, the nonequilibrium terms will be relatively small; further there will be a self-sharpening effect (Helfferich and Klein, 1970) that will also act to mitigate the dispersive effects. Consequently, the ideal treatment given above may be considered as a reasonable approximation. In actual experiments with a column of finite length, the column may be too shbrt for the shock to form at all; the condition for the shock to occur can easily be stated from eq. (22) to be #$’
*
Using eq. (31) in eq. (26) gives the corresponding along the shock:
=
237
WaMB
.
(36) Beyond this ail characteristics from the origin will
Table 2. Inlet gradient steepnessrequired for the point of shock incipience to occur precisely at the column outlet. <* is set to 1 in e-q. (37) and the required gradient steepness calculated.
Phase ratio I$ (dimensionless) Isotherm parameter a (dimensionless) Isotherm parameter b (M-l) Initial modulator concentration c, (M) Required scaled gradient steepness /T (dimensionless) to (tin) Required inlet gradient steepness Y (M/min)
C&e1
case2
0.67 1.85 0.26 0.00
0.40
0.67 1.85 0.26 4.70 4.43
1.44 1.07
1.44 11.83
AJOY VELAYIJDHAN and MICHAEL R. LADSCH
238
regard for eq. (37), a shock might occur in the large column, and affect the separation significantly. The present work is a theoretical analysis of the effect of modulator sorption. We have reported (Velayudhan and Lad&h, 19911 numerical results on how adsorbate peaks could be affected by gradient deformation, and particularly by modulator shock layers. Experimental work to ascertain that shocks do occur, and to examine how they could affect separations, is needed. CONCLUSIONS 01
/
0
I
2
1
Time
I
3
I
I
4
5
(dimansionless)
Fin. 3. Gradient shaces at the outlet aecordina to the unr&&d, Langmuir *and ‘BET descriptions or modulator sorption. Inlet gradient is linear, with a slope of 2.35 M/min. Gradient begun at the dimensionlesstime of 0.1 (corresponding to Q = 0.1). initial modulator concentration and the (scaled) gradient slope at the inlet, which are the experimentally adjustable parameters. A larger initial slope or a larger curvature of the isotherm causes the shock to be formed earlier, which is physically reasonable. Similarly, a larger phase ratio, which corresponds to a greater uptake of modulator, and a steeper initial slope also lead naturally to shock formation. The initial modulator concentration is probably the most important variable because of its cubic relation to the point of shock incipience. For the isotherm parameters in Table 1 and a representative value of 0.67 for the phase ratio, a sample calculation shows that even for an initially modulator-free column [which favors shock formation, from eq. (37)], the shock would occur at the column outlet for a gradient of O-100% acetonitrile in 17 min, when the dead time of the column corresponds to analytical scale (Table 2). This is a moderately steep gradient for an analytical column. However, when the initial concentration of the modulator is 25% acetonitrile (by volume), a gradient of 25-100% acetonitrile must be run in 1.4 min for the shock to form at the outlet (Table 2), which is far too steep to be realistic. Consequently, as the initial modulator level increases, the likelihood of shock formation becomes increasingly poor. However, these calculations apply to an analytical column, with a void volume of 1.44 ml [this gave a dead time of 1.44 min for the flow rate of 1 ml/min in Table 21. For a preparative column of much larger volume, the required slope for a shock to form within the column, even at a relatively high initial modulator concentration, could quite easily be within the usual operating range. Thus, the effects deriving from accounting for modulator sorption could well become extremely important for large-scale operation. This could also have implications for scale-up; if the separation is optimized at laboratory scale under conditions where no modulator shock forms, and then scaled up without
The sorption of the mobile phase modulator has been quantitatively examined, with specific reference to acetonitrile as the modulator in reversed-phase chromatography. Excess isotherm data from the literature were used to fit Langmuir and BET forms of the individual isotherm for acetonitrile. Gradient deformation was analyzed for the Langmuir isotherm under ideal chromatographic conditions. The ‘condition for modulator shock formation was calculated explicitly. Using the Langmuir data, it was shown that such shocks are likely to form in analytical columns for zero initial modulator level and relatively steep inlet gradients. For larger cohunns, shocks could form under less stringent conditions; this could affect the scale-up of gradient elution separations. Acknowledgements-The material in this work was sueported by the National Science Foundation through grants BCS8912150 and EET8613167A2. We thank Dr. Paul Westgate and Dr. Tingyue Gu for their helpful comments during preparation of this manuscript. NOTATION
64
h CY L 4M t
Cm
to
V
x
Lanamuir isotherm ,narameter; dimensionless mm01 _ m Table 1 g-M Langmuir isotherm parameter, M- ’ mobile phase modulator concentration, M (units of volume fraction in Fig. 2) column length, m stationary phase modulator concentration, mmoljg time, s duration of sample injection, s residence time of an unretained component ( = JV), s mobile phase velocity, m/s distance into the column, m
Greek letters scaled inlet ‘gradient steepness, dimensionless inlet gradient steepness, M/s Y interstitial porosity, m3 bed voids/m3 column &b volume intraparticulate porosity, m3 particle voids/m3 % particle volume position of shock incipiena, dimensionless t+ dimensionless distance into column, x/L c -c dimensionless time, a/L
B
Effect of modulator ?inj
r*
+ X
dimensionless
duration
of
sample
in gradient elution chromatography
sorption
injection,
%I& time of shock incipience, dimensionless volumetric phase ratio, dimensionless dimensionless modulator concentration,
b,c,
Adamson, A. W., 1967, Edition; pp. 414-418. Andrade, J. D.. 1985, Biomedical Polymers.
Phvsical Chemistry of Surfaces, 2nd Interscience,
New-York.
_
Surface and Interfacial Aspects
of
Plenum Press, New York. _ _ Everett, D. H., 1986, Reporting data on adsorption from solution at the solid/solution interface. Pure appl. Chem.
S8,967-984. Frey, D. D., 1990, Asymptotic relations for preparative gradient elution chromatography of biomolecules. Biotechnol. Bioengn6 35,1055-1061. Helfferioh, F. and Klein, G, 1970. Tlteory of Multicomponent Chromatography. Marcel Dekker, New York. Horbett, T. A. and Brash, J. L., 1987, Proteins at Inte@ces, ACS Symposium Series, American Chemical Society, Washington, DC. Jandera, P. and Churacek, J., 1985, Gradient Elution in
Column Liquid Chromatography.
tion of dead volume in reversed-phase liquid chromatography. Anaiyt. Chem. 52,2249-2257. Rhae, H.-K., Aris, R. and Amundsoa, N. R., 1970, On the theory of multicomponent chromatography. Phil. Trans. R. Sot. (bntdon) 267,419+55. Rhea, H.-K., Aris, R. and Amundson, N. R., 1986, First-order Partial Differential Equations: Volume I. Theory and Ap &cation of Sidle Euuations. PP. 251-257. Prentice-Hall. ‘Englewooi kXks, Nj. ’ -_ Rhee, H.-K., Bodin, B. F. and Amundson,
REFERENCES
Theory and Practice.
Elsevier, Amsterdam. Kaplan, W., 1958, Ordinary Dt@wttial Equations, pp. 334-336. Addison-Wesley, Reading, MA. Kipling, J. J. and Tester, D. A., 1952, Adsorption from binary mixtures: determination of individual adsorption isotherms. J. Chem. Sot. 4123-4133. Lax, P. D., 1957, Hyperbolic systems of conservative laws: II. Comtn. Pure Applied Math. 10, 537-566. LeHa, N., Ungvaral, J. and Kovats, E. sx.. 1982, Adsorption isotherm at the liquid-solid interface and the interpretation of chromatographic data. Analyt. Chtnn. 54, 2410-2421. McCormick, R. M. and Karger, B. L., 1980. Distribution phenomena of mobile phase components and detennina-
239
N. R., 1971, A
study of the shock layer in equilibrium exchange systems. Chem. Engng Sci. 26, 1571-1580. Slaats, E. H., Markoviski, W., Fekete, J. and Poppe. H., 1981,
Distribution equilibria of solvent components in reversedphase liquid chromatographic columns and relationship with the mobile phase volume. J. Chromat. 207,299-323. Snyder, L. R., 1980, Gradient elution in high-performance liauid chromatoaranhv. _ __in Hiah-performance Liauid Chromatography, AdGames and Peripktiees (Edited by Cs. Horvath), Vol. 1, pp. 208-3 16. Academic Press, New York. Snyder, I,,. R., 1986, High-performanoe liquid chromatography’separations of large molecules: a general model, in
High-performance Liquid Chromatography-Advances and Perspectives (Edited by (Cs. Horvath), Vol. 4, pp. 195-312. Academic Press, New-York. .__ Snyder, L. R., and Kirkland, J. J., 1979, Introduction to Modern Chromatography, 2nd edition, pp. 662-719. Wiley, New York. Tani, K. and Suzuki, Y., 1989, Comparison of the isotherms for monomeric and polymeric C-18 bonded stationary phases. J. Chromat. Sci. 27, 698-703. Trogus, F. J., Schechter, R. S. and Wade, W. H., 1979, A new internretation of adsorntion maxima and minima. J. Co&id Inter&cial Sci. 76, 293-305. Velavudhan, A. and Lad&h. M. R., 1991 Analvt. Chem. (accepted for publication). _ von Mises, R., 1950, On the thickness of a steady shock wave. J. Aeronaut. Sci. 17,551-594. Yamamoto. S., Nakanishi, K. and Matsuro, R., 1988. fonexchange Chromatography of Proteins. Marcel Dekker, New York.
ooo!-2509pz s5.ca + 0.00 1991 PergamanPressplc
0
EFFECT OF MODULATOR SORPTION IN GRADIENT ELUTION CHROMATOGRAPHY: GRADIENT DEFORMATION AJOY VELAYUDHAN and MICHAEL R. LADISCH+ Laboratory of Renewable Resources Engineering, Department of Agricultural Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received for publication 16 May 1991) Ahstict-In gradient elution chromatography, the mobile phase composition at the column inlet is a function of time. The consequences of accounting for the adsorption of the mobile phase components themselves are investigated for the experimentally common situation of a linear inlet gradient. Surface excess adsorption data from the literature for water-acetonitrile mobile phases in reversed-phase chromatography using different octadecyl stationary phases are shown to he in good internal agreement. The resulting individual isotherms as calculated by Tani and Suzuki (1989) are fitted to Langmuir and BET forms; it is shown that the Langmuirian form fits the data well except at extremely high acetonitrile concentrations. Using these isotherm parameters, it is found that a linear inlet gradient could suffer significant distortion while moving down the column. In the extreme case, a shock front of the mobile phase modulator could result. Analytical expressions are derived describing the conditions under which such a
shock could occur within the column and its subsequent Path, assuming Langmuirian sorption. INTRODUCTION
Elution chromatography is carried out in either the isocratic mode, where the mobile phase composition
remains unchanged throughout the separation, or the gradient mode, where the mobile phase composition at the inlet varies with time (Snyder and Kirkland, 1979). For the separation of relatively small molecules, gradient elution is generally used to shorten the separation time by decreasing the retention of the more strongly retained components of the feed mixture; these components are also generally recovered in a more concentrated form than would be obtained from the corresponding isocratic elution run. The concomitant disadvantage is that resolution is lost with respect to isocratic elution; the method can therefore only be used when the initial separation factors of the feed components are sufficiently high (Snyder, 1980). On the other hand, macromolecules such as proteins, as a result of their multifunctional interactions with the stationary phase, tend to adsorb very strongly and, in the absence. of a competitive agent, frequently exhibit irreversible adsorption, at least on the time scale of chromatographic interest (Horbett and Brash, 1987; Andrade, 1985). Consequently, isocratic elution is usually not a viable option and gradient elution, along with frontal and displacement chromatography and stepwise elution, become the techniques of choice. Gradient elution has been the focus of considerable work, notably by Snyder et al. (summarized in Snyder, 1980; Snyder, 1986), and Jandera and Churacek (summarized in Jandera and Churacek, 1985). For most gradient runs these studies, which assume that none of the mobile phase components adsorb, result in an
accurate estimation of the retention time and a reasonable assessment of the resulting band widths in the effluent history. However, mobile phase components do adsorb to the stationary phase, as is well known (Snyder, 1980; Jandera and Churacek, 1985) and we attempt a quantitative analysis of the effect of such adsorption on the shape of the gradient as it passes through the column. We consider, for simplicity, a binary mobile phase consisting of a carrier and an additive which modulates adsorbate retention. We call this additive the mobile phase modulator (MPM), a general term intended to subsume the various commonly used mobile phase additives: it thus includes salts in ion exchange (and hydrophobic interaction) chromatography and organic modifiers in reversed phase chromatography. We present a quantitative discussion of the case where the modulator exhibits Langmuirian sorption behavior (justification of this particular form will be given presently), and show that, in many cases of experimental interest, the front of the modulator is appreciably distorted as a result of its sorption. When the distortion is sufficiently large, a part of the gradient front can become a shock (or shock layer); an analytical expression describing the conditions under which a shock is formed will be given later. To the extent that shock layers are characteristic of stepwise elution separations, it becomes possible for gradient elution runs to exhibit features that are usually associated with stepwise desorption schedules, e.g. ghost peaks (Yamamoto et al., 1988). PROBLEM
that
‘Author to whom
the
retention correspondence
stationary
should be addressed.
233
FORMULATION
The rationale for considering modulator modulator by
competing
frequently for
alters
binding
phase. Since the adsorbates,
sorption is adsorbate on the particularly
sites
234
ASOY VELAYUDHAN
and MICHAEL
those that are macromolecules, are typically much more retentive than the modulator, it can only compete effectively by being present at relatively much higher concentrations. At such high concentrations, it is likely to be in the nonlinear region of its singlecomponent sorption isotherm, and will consequently suffer self-interference (Helfferich and Klein, 1970). Thus, different concentrations of the modulator will move at different speeds, and the shape of the gradient at the inlet will suffer distortion as it travels down the column. This argument is probably not applicable to hydrophobic interaction chromatography, where the salt does not achieve its modulation of adsorbate retention by binding to the stationary phase. We will consider the case where the feed components are in the linear regions of their sorption isotherms. This is not only often true experimentally, but can frequently be applied even when the initial feed concentrations are high. The presence of the modulator suppresses the retention of the feed components, generating a linearizing effect: the region of a feed component’s competitive sarption isotherm that is linear can he much larger than the corresponding region of its pure isotherm (Yamamoto et al., 1988; Frey, 1990). Then the modulator bhavior can be decoupled from that of the feed. In e first approximation, the modulator can be treate$ ,, as possessing se much less no dispersion, since it is likely to dis than the feed components (whose bands I?! reading is of primary interest). Ideal chromatography being assumed, the deformation of the modulator gr,adient can then be analyzed by the method of characteristics.
ADSORPTION
lSOTHERM
OF THE MODULATOR
For specificity, we restrict attention to reversedphase chromatography with acetonitrile-water as the mobile phase. This system is widely used in many practically important contexts (e.g. analytical chromatography of proteins and polypeptides, preparative chromatography of amino acids and oligonucleotides). In what follows, we will consider octadecyl silica (C- 18) sorbents although analogous calculations can be made for other stationary phases. We first need to specify the modulator’s (acetonitrile) isotherm. However, the modulator is part of a binary mixture and it is well known that measurement of adsorption from a binary liquid solution can only unambiguously yield the so-called excess isotherm (e.g. Adamson, 1967). This represents the amount taken up by the adsorbent as the difference between the equilibrium bulk compositions before and after contacting with the adsorbent. Consequently, only a net sorption is known, and an additional model of the sorption process is required in order to extract the individual isotherm of the modulator. The simplest model is to assume that the water does not adsorb. Then the surface excess data can be converted into the individual isotherm for acetonitrile, as was done by Slaats et al. (1981), for Lichrosorb RP-18 (10 p particles). However, the resulting individual
R.L~ursc~
isotherm went through a maximum, which they suggested is physically unlikely. McCormick and Karger (1980) also reported an adsorption maximum. We offer another argument to indicate the physical improbability of such an isotherm. Consider a batch system in which the equilibrium mobile phase concentration of acetonitrile is such that its stationary phase concentration is at its maximum value. If a small amount of acetonitrile is now added to the mobile phase, the system will shift to a new equilibrium at which the stationary phase concentration will be lower than it was previously, since it was initially at its maximum value. Consequently, some acetonitrile will be expelled from the stationary phase into the mobile phase. But this increase in mobile phase concentration will trigger a further shift in equilibrium, which further decreases the stationary phase concentration. A similar instability exists for all points on the decreasing branch of the isotherm, contradicting the assumption-implicit in an isotherm-that the stationary and mobile phase concentrations are in equilibrium along this branch. In this context it may be noted that individual isotherms that exhibit maxima and minima have been reported in the literature for the adsorption of surfactants (Trogus et al., 1979). However, micelles are formed in these instances and it is this additional phenomenon that gives rise to adsorption extrema. We assume that the kinds of modulators used in chromatography do not form micelles or undergo other reactions in solution that could affect the shape of the individual isotherm. Therefore, it seems necessary to assume that the water also adsorbs. Another argument in favor of this assumption is that some surface excess data on reversed-phase materials [e.g. the work of LeHa et al. (1982). on C-14, and of SIaats et al. (1981), on C-S] exhibit azeotropy and negative net adsorption. It is clearly impossible for this to occur unless the water also adsorbs. While the C-18 data does not exhibit such azeotropy, it is unlikely that increasing the hydrophobic character of the stationary phase somewhat will completely suppress the sorption of water. Tani and Suzuki (1989). who also measured ACN-H,O sorption on C-18, therefore resorted to the layer model to calculate individual isotherms (Everett, 1986). This gives truly multicomponent isotherms, and is patterned after the model of Kipling and Tester (1952) who required that the total stationary phase concentration always remain at its saturation level. In what follows, we use the individual isotherm data of Tani and Suzuki (1989). It may be noted that the surface excess data of Slaats et al. (1981) agrees very well with that of Tani and Suzuki (see Fig. l), although they were obtained on different C-18 stationary phases. Similar agreement was found by LeHa et al. for two different C-14 stationary phases, even though one was branched and the other linear. Such indifference to the exact nature of the surface might perhaps be attributed to the lowenergy nature of these materials (LeHa et al., 1982).
Effect of modulator sorption in gradient elution chromatography For the purposes of using such individual isotherms in the mass balances, it is convenient to introduce functional forms. The data in Fig. 2 were fitted to the well-known Langmuir and BET forms by nonlinear regression. It must be emphasized that this is a purely formal process; since the ACN individual isotherm represents its multicomponent adsorption in the presence of water, there is no mechanistic basis to either the Langmuir or the BET form. T’he resulting fits are shown in Fig. 2, and the isotherm parameters in Table 1. It can be seen that the Langmuir form fits the data much better at lower and intermediate concentrations; as is to be expected from the Type II nature of the data, the BET form is somewhat better at very high levels of ACN. For the purpose of analysis, the Langmuir isotherm will be used in what follows; however, a numerical simulation of both the BET and Langmuir forms will be offered later for comparison.
BEHAVIOR
235
OF THE MODULATOR
DEFORMATION
AND SHOCK
BAND-
FORMATION
We consider a modulator whose single component isotherm is Langmuirian. An analogous formula will hold for univalent salts in idea1 ion-exchange chromatography (when activity coefficients are unity, Donnan exclusion effects are neglected, etc.). We will consider the experimentally common case of a linearly increasing gradient at the inlet. In the absence of dispersive effects, the mass balance for the modulator may be written as
(l+##~)!!k+v~=o
(1)
where cM and qM are the mobile and stationary phase modulator concentrations, 11the mobile phase velocity and 4 the volumetric phase ratio. These last two terms are given by U 0=
4
00
0
IIt if
1
&bbp (1 - e,)(l - 4 &b+ t1 - Ebb,
_
(3
where u is the superficial velocity, and t+,and E,,are the bulk and intraparticulate porosities. The single-component Langmuirian isotherm gives
l
2
(2)
&b+ t1 -
O
I
1
l
0
which when substituted l
0.
om
I
l
I
0.75
0.50
0.25
o.Ml
ahf9.f
qM= 1 +
ACNmolehreton, M.P.
(4)
b,c,
into eq. (1) yields
&b
ah
%+” 1 at
_-1 ’ + (1 +
l.M)
b,c,)z
z
= ”
(3
Using the dimensionless variables 5 = x/L, z = vt/L, k=wb,.ep (where L is the column length), eq. (5)
Fig. 1. Surface excess adsorption data of water-acetonitrile
on C-18 reversed-phase adsorbents. “Slaats” refers to Slaats
et al. (1981). and “Tani” to Tani and Suzuki (1989).
Il +
+aM
(1 +
The initial condition
4x.
t) = CM,Cl
x=
i
.-a-
att=O
(7a)
!I! ‘I
la
/
Langmuir isotherm
I C
(
x0
at z = 0.
VW
Table 1. Langmuir and BET parameters for the individual isotherm data of acetonitrile in Fig. 2. All stationary phase concentrations are in mmol/g, all mobile phase concentrations in M. To convert the mobile phase concentrations of Fig. 2 into molar units, multiply by 18.78.
20
1
0.25
0.50
G holumo fmcdal~
0.75
1.0
Fig. 2. Individual isotherm for acetonitrile from the layer model, after Taai and Suzuki (1989). Surface area of the stationary phase = 167.49 m’/g. Best-fit Langmuir and BET isotherms arc also shown. CES 47:1-p
is
0.
30
3
aX+aX=o.
x)* 1 ar a<
BET isotherm
4=-
ac
1 + bc a = I.48 mmol/(g-M) b = 0.26 M-’ AKbc
’ = (1 -
bc)Il + (K - l)bc]
A - 3.461 mmol/g ; = fOt; (c$ensionless) = .
236
AJOYVELAYUDHANZUI~MICHA~LR.LADISCH
The boundary gradient is
condition
Atx=O,eM
for a linearly increasing
= = )12 + =
CM.0,
o<
cM.O
Ytt - timj), t*.jG t
+
t < t,.i
cC l+ (1
4%
.+ xc + /Ip*)Z
1 (1’5)
where p z (9 - rinj). These characteristics cross when
where y is the gradient steepness at the inlet. This form of boundary condition is chosen to represent the beginning of the gradient only after the ftid has been completely introduced into the column over a time tinj. In dimensionless form this gives
(171 The point of intersection has the coordinates
*~(1+xo+BPl~2~1+xo+BP2~2 9%fm z_
'II+(I
By the method of characteristics,
(19) Letting q1 -+ qZ + q gives a point on the envelope (e.g. Kaplan, 1958), where (T
7 = tl +
(1 +
=
x0 +
(1 + x0 + BP13 Wad BP)
4% + (1 +
7=?J
{* = (1 +
&I2 1*
xd3
(22)
Wad
This results in
7=4+r [
l+(l+# +abf
1
from which x can be calculated at any 5 and z for z 3 rinj_ However, these characteristics will cross eventually, since the inlet concentrations increase with time and higher concentrations have smaller slopes in the T< plane, as can be seen from eqs (9) and (13). Thus, a shock must be introduced to avoid the physically unrealistic situation of crossing characteristics. (They correspond to a multivalued concentration for some values of 5, t, which is physically unacceptable.) The velocity and strength of the shock can be calculated as follows. An analogous calculation has been carried out by Rhee et al. (1986), for the case where the initial concentration of the modulator and the injection time are zero. If we consider two adjacent characteristics emanati ing from the r-axis (representing boundary data) for r > rinj, they will be of the form
c[ l +
4aM
(1 + x0 + j?p,)Z
5. = r in,
(13)
and
41+
x0 +
2&Q
[.
Since < and o in eqs (20) and (21) are monotonically increasing functions of p, the incipience of the shock corresponds to the cusp of the envelope, when p = 0. The coordinates of this point are therefore
y=o
==
(20)
121)
with the initial condition P;ts=o,
(18)
+x0 + BPd21 4%m + 2x0+ NPI + P2)l .
(9)
(11)
m, + P2)l
+ Xo+BP2)2C4%f+v
where B = yLbJv = yb&, and thj = v/L t,,,, = timj/ to, where t, ( = L/v) is the residence time of an unretained component.
+ 2X0+
+
(1 + x0)
444 + (1 +
xo12
2bad
[
(Z3)
1-
These expressions reduce to those obtained by Rhec et al. (1986) when zinj = x0 = 0. To calculate the path of the shock, we use the expression for the shock velocity (Lax, 1957; Helfferich and Klein, 1970; Rhee et al., 1970). 1 l+@J
(24) Ccl
where [q] = qu - q,, and [c] = c, - cd, the subscripts u and d representing values upstream and downstream of the shock. The concentration downstream of the shock will always be x0; upstream, it will vary as it meets different concentrations from the boundary. Thus, the shock “slowness” (the reciprocal of the velocity) is given by
9% = l + (1 + x0)(1 + XI’ Shock
125)
The same concentration x is borne from the r-axis by the characteristic given by
1 (15)
(26)
E&ct of modulator sorption in gradient elution chromatography
Using
7’ =
7 -
~,~j
-
e,
eq. (26) becomes
dr’ I dx baM dt.=jidt+(I-X)2z-
Wad
dx
(27)
The shock slowness equation [eq. (25)] hecomes dz’
&n de = (I + X)(1 + x0)’
w3)
From eqs (27) and (28)
[; -
#aH
iE$ $ =
4aM
(1 + x)(1. + x0) - (1 + x)2
1
(29)
or
(x - x0) (I+x
2
dC+
(30)
=&.
(1+
This is a linear tit-order ordinary differential equation in 5 with respect to x, with the “initial” condition e = <* at x = x0 (since x = x0 at the inception of the shock). The solution is
< _ (1 + x0)(1 +
xl2
29a.d
7l”j
+
W+B
7
.
(32) Equations (31) and (32) give the coordinates of the shock path in parametric form, 1 being the parameter. Eliminating x between these expressions gives (1 + x0) 7 = %I] -7+dZ+
C
which is part of a parabola. = 0 recovers the solution of If the inlet gradient is kept a certain time, i.e. if the inlet cy =
CH.0.
0 Q t G ti,j Ytt
+
-
thj)r
(33)
Again, setting rini = x0 Rhee et al. (1986). at a constant value after boundary condition is
cM,O
hnj
6
! =na.s= CM.0 + r(t, - G”,),
t 6
t/
(34)
(1 + x0)(1 + %/I2 WaMB
(35)
and =f
=
751.1 +
(1 + x,) + (%I - x0) + (1 + x0)(1 + %,)z 28
(1 + ad3
(
(37)
1
*
DISCUSSION
For the purposes of comparison, we have used plate simulations (Velayudhan and Ladisch, 1991) to assess the shape of the gradient at the column outlet for three cases: when the modulator was unretained (classical Snyder and Jandera-Churacek theory), when sorption followed the Langmuir form, and when sorption followed the BET form. The inlet gradient was linear and the column parameters were as in Table 2. The effluent histories are shown in Fig. 3 for an initial slope of 2.35 M/min (which is somewhat steep but still realistic), for an initially modulator-free column. It can be seen that a shock is formed when modulator sorption is accounted for, and the unretained profile is significantly different from the retained profiles. Equation (37) is the basis for shock formation within the column. It can be seen that three kinds of terms are present: isotherm parameters, which are functions of the mobile phase adsorbent system; the phase ratio, which is a geometric property of the packing; the
t/ 6 t
the shock will eventually reach a constant strength of The coordinates of this point can he x,( = &c,.~). easily calculated by substituting x = xI into eqs (31)-(33X which gives <, =
=
244
(1 + x) + (x - x0) + (1 + x0)(1 + XY 2s
travel at the constant speed corresponding to the concentration 2,. Thus, for a sufficiently long column, any gradient that is linear at the inlet will become a discontinuous step by the time it reaches the outlet. When the dispersion and nonequilibrium contributions (such as pore diffusion and mass transfer across the particle boundary layer) of the modulator are taken into account, the shock will be smoothened into a shock layer (von Mises, 1950: Rhee et al., 1971). Nevertheless, since the modifier is usually a small molecule, the nonequilibrium terms will be relatively small; further there will be a self-sharpening effect (Helfferich and Klein, 1970) that will also act to mitigate the dispersive effects. Consequently, the ideal treatment given above may be considered as a reasonable approximation. In actual experiments with a column of finite length, the column may be too shbrt for the shock to form at all; the condition for the shock to occur can easily be stated from eq. (22) to be #$’
*
Using eq. (31) in eq. (26) gives the corresponding along the shock:
=
237
WaMB
.
(36) Beyond this ail characteristics from the origin will
Table 2. Inlet gradient steepnessrequired for the point of shock incipience to occur precisely at the column outlet. <* is set to 1 in e-q. (37) and the required gradient steepness calculated.
Phase ratio I$ (dimensionless) Isotherm parameter a (dimensionless) Isotherm parameter b (M-l) Initial modulator concentration c, (M) Required scaled gradient steepness /T (dimensionless) to (tin) Required inlet gradient steepness Y (M/min)
C&e1
case2
0.67 1.85 0.26 0.00
0.40
0.67 1.85 0.26 4.70 4.43
1.44 1.07
1.44 11.83
AJOY VELAYIJDHAN and MICHAEL R. LADSCH
238
regard for eq. (37), a shock might occur in the large column, and affect the separation significantly. The present work is a theoretical analysis of the effect of modulator sorption. We have reported (Velayudhan and Lad&h, 19911 numerical results on how adsorbate peaks could be affected by gradient deformation, and particularly by modulator shock layers. Experimental work to ascertain that shocks do occur, and to examine how they could affect separations, is needed. CONCLUSIONS 01
/
0
I
2
1
Time
I
3
I
I
4
5
(dimansionless)
Fin. 3. Gradient shaces at the outlet aecordina to the unr&&d, Langmuir *and ‘BET descriptions or modulator sorption. Inlet gradient is linear, with a slope of 2.35 M/min. Gradient begun at the dimensionlesstime of 0.1 (corresponding to Q = 0.1). initial modulator concentration and the (scaled) gradient slope at the inlet, which are the experimentally adjustable parameters. A larger initial slope or a larger curvature of the isotherm causes the shock to be formed earlier, which is physically reasonable. Similarly, a larger phase ratio, which corresponds to a greater uptake of modulator, and a steeper initial slope also lead naturally to shock formation. The initial modulator concentration is probably the most important variable because of its cubic relation to the point of shock incipience. For the isotherm parameters in Table 1 and a representative value of 0.67 for the phase ratio, a sample calculation shows that even for an initially modulator-free column [which favors shock formation, from eq. (37)], the shock would occur at the column outlet for a gradient of O-100% acetonitrile in 17 min, when the dead time of the column corresponds to analytical scale (Table 2). This is a moderately steep gradient for an analytical column. However, when the initial concentration of the modulator is 25% acetonitrile (by volume), a gradient of 25-100% acetonitrile must be run in 1.4 min for the shock to form at the outlet (Table 2), which is far too steep to be realistic. Consequently, as the initial modulator level increases, the likelihood of shock formation becomes increasingly poor. However, these calculations apply to an analytical column, with a void volume of 1.44 ml [this gave a dead time of 1.44 min for the flow rate of 1 ml/min in Table 21. For a preparative column of much larger volume, the required slope for a shock to form within the column, even at a relatively high initial modulator concentration, could quite easily be within the usual operating range. Thus, the effects deriving from accounting for modulator sorption could well become extremely important for large-scale operation. This could also have implications for scale-up; if the separation is optimized at laboratory scale under conditions where no modulator shock forms, and then scaled up without
The sorption of the mobile phase modulator has been quantitatively examined, with specific reference to acetonitrile as the modulator in reversed-phase chromatography. Excess isotherm data from the literature were used to fit Langmuir and BET forms of the individual isotherm for acetonitrile. Gradient deformation was analyzed for the Langmuir isotherm under ideal chromatographic conditions. The ‘condition for modulator shock formation was calculated explicitly. Using the Langmuir data, it was shown that such shocks are likely to form in analytical columns for zero initial modulator level and relatively steep inlet gradients. For larger cohunns, shocks could form under less stringent conditions; this could affect the scale-up of gradient elution separations. Acknowledgements-The material in this work was sueported by the National Science Foundation through grants BCS8912150 and EET8613167A2. We thank Dr. Paul Westgate and Dr. Tingyue Gu for their helpful comments during preparation of this manuscript. NOTATION
64
h CY L 4M t
Cm
to
V
x
Lanamuir isotherm ,narameter; dimensionless mm01 _ m Table 1 g-M Langmuir isotherm parameter, M- ’ mobile phase modulator concentration, M (units of volume fraction in Fig. 2) column length, m stationary phase modulator concentration, mmoljg time, s duration of sample injection, s residence time of an unretained component ( = JV), s mobile phase velocity, m/s distance into the column, m
Greek letters scaled inlet ‘gradient steepness, dimensionless inlet gradient steepness, M/s Y interstitial porosity, m3 bed voids/m3 column &b volume intraparticulate porosity, m3 particle voids/m3 % particle volume position of shock incipiena, dimensionless t+ dimensionless distance into column, x/L c -c dimensionless time, a/L
B
Effect of modulator ?inj
r*
+ X
dimensionless
duration
of
sample
in gradient elution chromatography
sorption
injection,
%I& time of shock incipience, dimensionless volumetric phase ratio, dimensionless dimensionless modulator concentration,
b,c,
Adamson, A. W., 1967, Edition; pp. 414-418. Andrade, J. D.. 1985, Biomedical Polymers.
Phvsical Chemistry of Surfaces, 2nd Interscience,
New-York.
_
Surface and Interfacial Aspects
of
Plenum Press, New York. _ _ Everett, D. H., 1986, Reporting data on adsorption from solution at the solid/solution interface. Pure appl. Chem.
S8,967-984. Frey, D. D., 1990, Asymptotic relations for preparative gradient elution chromatography of biomolecules. Biotechnol. Bioengn6 35,1055-1061. Helfferioh, F. and Klein, G, 1970. Tlteory of Multicomponent Chromatography. Marcel Dekker, New York. Horbett, T. A. and Brash, J. L., 1987, Proteins at Inte@ces, ACS Symposium Series, American Chemical Society, Washington, DC. Jandera, P. and Churacek, J., 1985, Gradient Elution in
Column Liquid Chromatography.
tion of dead volume in reversed-phase liquid chromatography. Anaiyt. Chem. 52,2249-2257. Rhae, H.-K., Aris, R. and Amundsoa, N. R., 1970, On the theory of multicomponent chromatography. Phil. Trans. R. Sot. (bntdon) 267,419+55. Rhea, H.-K., Aris, R. and Amundson, N. R., 1986, First-order Partial Differential Equations: Volume I. Theory and Ap &cation of Sidle Euuations. PP. 251-257. Prentice-Hall. ‘Englewooi kXks, Nj. ’ -_ Rhee, H.-K., Bodin, B. F. and Amundson,
REFERENCES
Theory and Practice.
Elsevier, Amsterdam. Kaplan, W., 1958, Ordinary Dt@wttial Equations, pp. 334-336. Addison-Wesley, Reading, MA. Kipling, J. J. and Tester, D. A., 1952, Adsorption from binary mixtures: determination of individual adsorption isotherms. J. Chem. Sot. 4123-4133. Lax, P. D., 1957, Hyperbolic systems of conservative laws: II. Comtn. Pure Applied Math. 10, 537-566. LeHa, N., Ungvaral, J. and Kovats, E. sx.. 1982, Adsorption isotherm at the liquid-solid interface and the interpretation of chromatographic data. Analyt. Chtnn. 54, 2410-2421. McCormick, R. M. and Karger, B. L., 1980. Distribution phenomena of mobile phase components and detennina-
239
N. R., 1971, A
study of the shock layer in equilibrium exchange systems. Chem. Engng Sci. 26, 1571-1580. Slaats, E. H., Markoviski, W., Fekete, J. and Poppe. H., 1981,
Distribution equilibria of solvent components in reversedphase liquid chromatographic columns and relationship with the mobile phase volume. J. Chromat. 207,299-323. Snyder, L. R., 1980, Gradient elution in high-performance liauid chromatoaranhv. _ __in Hiah-performance Liauid Chromatography, AdGames and Peripktiees (Edited by Cs. Horvath), Vol. 1, pp. 208-3 16. Academic Press, New York. Snyder, I,,. R., 1986, High-performanoe liquid chromatography’separations of large molecules: a general model, in
High-performance Liquid Chromatography-Advances and Perspectives (Edited by (Cs. Horvath), Vol. 4, pp. 195-312. Academic Press, New-York. .__ Snyder, L. R., and Kirkland, J. J., 1979, Introduction to Modern Chromatography, 2nd edition, pp. 662-719. Wiley, New York. Tani, K. and Suzuki, Y., 1989, Comparison of the isotherms for monomeric and polymeric C-18 bonded stationary phases. J. Chromat. Sci. 27, 698-703. Trogus, F. J., Schechter, R. S. and Wade, W. H., 1979, A new internretation of adsorntion maxima and minima. J. Co&id Inter&cial Sci. 76, 293-305. Velavudhan, A. and Lad&h. M. R., 1991 Analvt. Chem. (accepted for publication). _ von Mises, R., 1950, On the thickness of a steady shock wave. J. Aeronaut. Sci. 17,551-594. Yamamoto. S., Nakanishi, K. and Matsuro, R., 1988. fonexchange Chromatography of Proteins. Marcel Dekker, New York.