Ind. Eng. Chem. Res. 1994,33, 1391-1396
1391
RESEARCH NOTES Partition Coefficients for Distribution of Rigid Non-Axisymmetric Solutes between Bulk Solution and Porous Phases: Toward Shape-Selective Separations with Controlled-Pore Materials Johannes M. Nitsche' Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Kirk W. Limbach Rohm and Haas, Znc., 727 Norristown Rd., Spring House, Pennsylvania 19477
Perturbation arguments are used to derive an approximate formula for partition coefficients of non-axisymmetric Y-shaped molecules in circular cylindrical pores. Quantitative accuracy thereof is verified by comparison with Monte Carlo calculations for star-shaped solutes. Results are used to calculate macroscopically observable partition coefficients in polydispersed porous materials, and implications for solute shape-selective separations are discussed. 1. Introduction
Equilibrium partitioning of macromolecules and colloidal particles between a dilute solution and an adjacent porous phase, as quantified by a distribution coefficient K, plays a central role in size-exclusion chromatography (SEC) (Pecsokand Saunders, 1966;Barth and Boyes, 1990) and hindered pore diffusion processes (Anderson and Quinn, 1974; Deen, 1987). In SEC, the distribution coefficient determines the retention volume Vrebntion of the solute according to an oft seen expression involving the volumes Vwlwnt and Vwmwofthe mobile and stationary phases, u~z.,Vrebntion = Vsolvent + KVpr, (Laurent and Killander, 1964;van Kreveld and van den Hoed, 1973). Thus, differences in K between solutes are responsible for selective permeation and thereby separation. In pore diffusion, K relates the observed concentrations in the bulk and porous phases and contributes a factor to the overall effective diffusion coefficient. The superficial solute concentration (amount per total volume) within an inert, wetted porous phase is lower than the bulk concentration for two reasons. First, only a fraction of the total volume (the porosity t) is occupied by solution; second, even within the interstitial fluid only a fraction of the pore volume is accessibleto the solute center. The latter phenomenon is due to the existence of an exclusion layer representing the minimum distance between the solute center and the wall at contact, determined by the solute size. For rigid nonspherical species,the shape (and volume) of the excluded region depends on the solute orientation. Determination of K therefore hinges upon a geometrical factor 9representing the orientation-average ratio of the accessible to total pore volumes. Although the existence of attractive and repulsive forces can be accounted for by introducinga Boltzmann weighting factor into the average, attention here will be restricted to the key phenomenon of purely geometrical exclusion in inert pores. The geometrical issues are easily addressed for spherical solutes in circular cylindrical pores. As noted in passing
by Pappenheimer et al. (1951),for a solute radius a and pore radius R , 3 = (1 - a/R)2
(1)
For rigid nonspheres a landmark paper is that of Giddings et al. (1968).These authors presented, inter alia,formulas for various solute shapes in a number of pore structures. Beyond these specific results, the broad conclusion apparent in their work is the statement that 9for all solutepore combinations conforms to the asymptotic behavior 3
-
1 - h as h 4 rm/(V/S)wre-* 0
(2)
in the limit of vanishing ratios h of solute to pore radii, provided h is defined precisely as the mean projected solute radius r, divided by the volume-to-surface ratio ( V/S)pore of the pore. This conclusion follows from the observation that, for small solutes, the pore wall appears flat locally, and the orientation-average excluded volume therefore reduces to the product of the mean radius and the wall surface area. The full dependence of 3on h depends upon details of solute and pore geometry, as elucidated recently for a number of geometrical combinationsby Baltus (1989) and Limbach et al. (1989). Excepting analysesfor flexible polymer models (Casassa, 1967;Casassa and Tagami, 1969;Davidson et al., 1987), which are not directly applicable here, all theoretical studies but one (Baltus, 1989) have been restricted to axisymmetric solutes. Baltus developed a numerical integration procedure to calculate partition coefficients 3 for planar multibead complexes ranging from dumbbells to eight-membered rings in circular cylindrical pores. Numerical results for each of the six molecular shapes considered were fitted empirically with a quadratic polynomial to furnish a convenient analytical representation. With the important case of antibodies in mind (McCullough and Spier, 1990),the specific purpose of this Research Note is to present an explicit asymptotic formula for a,corroboratedby Monte Carlo calculations,applicable
American Chemical Society o a a a - 5 a a ~ ~ 9 4 ~ 2 ~ 3 3 - i 3 ~ ~ ~ o04 1994 .5o/o
1392 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
Pz - wall contact
0 = Arctan(a/bcos$)
/
(b)
(d)
I
(C)
Figure 1. Geometrical considerations for calculation of the mean projected radius of a Y-shaped solute: (a) specification in solute-fixed coordinates; (b) distance of closest approach h; (c) subdivision of the (e,$) plane; (d) appreciable thickness of the arms.
to rigid Y-shaped bodies of arbitrary aspect ratio, in order that prototypical non-axisymmetric species may be shape characterized by permeation. It should be noted explicitly that our focus is solute geometry, so that attention is restricted to the limit of dilute solutions in inert pores, for which the phenomena of solute-solute exclusion (see, e.g., Fanti and Glandt, 1989) and energetic interactions with the walls (see, e.g., Lin and Deen, 1990) are absent. The contents of the text are as follows: An explicit algebraic formula for the mean projected radius of Y-shaped bodies is derived in section 2, and this result is built into a perturbation analysisof second-order wall curvature effects in section 3 for the special case of a circular cylindrical pore. Section 4 shows that the present asymptotic formula agrees well quantitatively with Monte Carlo calculations, as well as with the previous work by Baltus (1989) for a three-bead solute. Section 5 assembles available results for a number of solute shapes and demonstrates by direct calculation the possibility of considerably enhanced solute shape selectivity with materials having controlled-pore microstructures.
y’, z’) centered at (xo, yo,
2. Mean Projected Radius
P3:
In solute-fixed coordinates (x’, y’, z‘) a thin Y-shaped entity may be characterized by specifying the positions of its three termini, which we shall take to be the points PI, (x’, Y’, 2’) = (O,O,a); P2, (x’, Y’, 2’) = (O,O,-a); P3, (x’, Y’, z’) = (b, 0,O)(see Figure la). Arbitrary solute configurations are quantified by a triple of Eulerian angles (0 E (0, r ) ,4 E (0,BT), $ E (0,274) giving the orientation of the solute-fixed axes relative to their space-fixed ( x , y, z ) counterparts. Following the convention of Goldstein (1950),the polar and azimuthal angles 6 and 4 determine the orientation of the solute-fixed 2’ axis as in the usual definition of spherical polar coordinates, and $ specifies rotation about this axis. It follows that the space-fixed coordinates of a point with solute-fixed coordinates (x’,
In order to observe the steric constraint imposed by a solid wall and prevent penetration by any one of the three ends, the projected radius h at any orientation must be identified with the maximum of the three values of ZO. It is evident from (3) and (4) that 4 does not affect any elevations above the plane z = 0; moreover, considerations of symmetry indicate that integrations over 0 and $ may respectively be restricted to the intervals (0, d 2 ) and (0, 7).Thus, the orientations average of h reduces to
ZO)
are given by
+ (cos 8 cos 4 cos $ - sin 4 sin $)z’ (cos 8 cos 4 sin $ + sin 4 cos $)y’ + (sin 8 cos $12’ y = yo + (cos 8 sin 4 cos J, + cos 4 sin $)x’ + (-cos 0 sin 4 sin + cos 4 cos $)y’ + (sin 0 sin 4)z’ z = zo - (sin 0 cos $)x’ + (sin 0 sin $)y’ + (cos e)z’ (3) x = xo
The mean projected radius rm represents the average height h of the locator point (solute “center”) (xo, yo, ZO) above the plane z = 0 for conditions of solute-wall contact (Figure lb). In view of the last member of (31, wall contact for each of the three termini occurs at the followinglocatorpoint elevations:
P,:
(z0ll= -a
COS
e = f,ce)
(zOl3= b sin B cos $ E f3(e,$)
(4)
It will be observed that solute-wall contact occurs at PZ
Ind. Eng. Chem. Res., Vol. 33, No.5,1994 1393 be solved explicitly, we settle for an answer in the form of a perturbation expansion
PI
3 = 1+ 3,(a/R) + 3,(a/R)' + O((a/R)') (8) for small values of A,, E alR, which turns out to be quite accurate quantitatively. The projection of the "Y" centered at ( X O , yo, 20) is a skewed triangle with vertices at
locaior point (ZO, YO)
'T
(x,, YJ = ( x 0 + a sin &yo)
P,:
-
(xZ,y2)= (no
P$
Q
sin 4yo)
+
+
(x3,y3)= (no b cos 0 cos $,yo b sin $)
P$
(9)
as shown in Figure 2a. The set of points accessible to the locator point ( X O , yo) is bounded by the piecewise smooth dashed curve in Figure 2b comprising arcs of three distinct circles representing contact of one of the vertices Pi with the circular boundary (given by the constraints Xi2 + yi2 = R2). If polar coordinates (FO, (YO) are used to specify the position of the locator point instead of Cartesian coordinates ( X O , yo), then the equation of each circle i can be expressed in the form of the following power series in A,, = a/R: Figure 2. Geometrical considerations for the analysis of wall curvatureeffecta in a circularcylindrical pore: (a)a typical projection of the "Y";(b) set of positions accessible to the locator point.
above the curve a cos 0 = b sin 0 cos $ and at P3 below this curve in the (e, $) plane (Figure IC).Thus,
Pj-wall contact: ( F ~ R )=? 1 + AiXa + BiX:
+ O(X:)
with A, = -2 cos a. sin 8, B, = (cos' a0- sin' a0)sin20
A, = 2 cos a0sin 0, B, = (cos2a. - sin2ao)sin20 A, = -2p(cos a. cos 0 cos $ + sin a. sin $), B, = ~ '(cos' t a. - sin' cue) (cos' e cos' $ - sin2$1 4 cos a. sin a. cos 8 cos +sin $3 (11)
+
A straightforwardbut lengthyevaluationof integralsyields the surprisingly simple final result
rm = (1/4)[a
+ (a' + b')'/'I
(7)
As a check, it may be observed that rmlb=o= a12 and ?'mla=O = b14, both statements consistent with the known fact (Giddings et al., 1968) that the mean projected radius of an infinitesimally thin rod is one-fourth of ita total length. Equation 7 quantifies the steric "bulk" of a "Y". If its arms are capsules ending in spherical beads of nonnegligibleradius s,say (Figure Id), then r, is obtained simply by adding s to the right-hand side of (7). 3. Perturbation Analysis of Wall Curvature Effects Here we seek an expression for @ for the special case of a "Ywin a straight circular cylindrical pore of radius R, aligned, say, with the z axis. The task at hand then involves calculating the area accessible to the center (no,yo) as a function of the solute orientation, and averaging the result over all orientations. Although the accessible region depends on 4, ita area is independent of 4 and so may be evaluated with 4 = 0 (Figure 2) without loss of generality. Moreover, symmetry considerations again allow us to reduce the angular intervals,sothat orientational averaging is now carried out over the domain 0 5 0 I?r/2,0 I $ I 6 2 . Because the analysis leads to equations that cannot
where @ bla. The three circles intersect each other at special angles (Y0,ij = Uij + Vijb + ... determined by the condition that rdR for Pi-wall contact equals r d R for P,wall contact. The lowest-order approximations are given by
u,, = -TI2 U13 = Arcsin U23 = T
t
sin 0 - p COS 6 COS $ [(p sin +I' + (sin e - p cos e cos $)'I'/'
-
Arcsin
{[(B sin $)z
sin 0 + p COS e COS + (sin e + B cos os:
$)21'/2
]
1 (12)
By definition of 3 in terms of the orientation average accessiblearea (evaluatedby polar-coordinate integration),
1394 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
Since the ao-interval from a0,12 to ~ O J Z + 27r generally comprises three subsets in which a different one of the three vertex-wall contacts furnishes the minimum locatorpoint radial distance ro (cf. eq lo),
Expansion of the right-hand side in powers of A, then yields the perturbation approximation (8) with
0.4
’
0.2
’
0.0
0.2
0.6
0.4
0.8
‘ 1.o
ClR
Figure 3. Comparison of perturbation analysis and exact values of Q, obtained via the Monte Carlo method for equilateral Y-shaped solutes.
(-4, - Al)lao,u,2V121 sin dB drC, (16)
It is unnecessary to actually compute the integrals in (151, for the linear term must equal-2rJR in conformance with the asymptotic behavior (2). The second integral in (16) has been included for completeness, but it vanishes because the lowest-order approximations Uij to the angles a0,ij are determined precisely by the requirements that Ai = Aj. Substitution of the expressions (11)for the Bi and lengthy but straightforward evaluation of integrals then yield the unexpectedly simple final result = fl/27r.Thus, we conclude that ip has the perturbation expansion
a = 1- (1/2)[1+ (1+ p 2 ) 1 / 2 1 ~+~(p/27r)~,2+ o(x,~) (17)
As a check it is worth noting that (17) reduces to the known expansion
* = 1- A, + 0x2 + O(X,3)
(18) for a thin rod of length 2a ( b = 0). For the special case where all three arms of the “Y” have equal length (c, say),
+
+
@ = 1- [3(31’2)/4](~/R) [3(31/2)/8~](~/R)2
o ( ( ~ / R )=~M~,RR) ) (19) 4. Comparison with Numerical Calculations
Monte Carlo calculations have been performed to obtain numerical values of the single-pore partition coefficient @ for solutes of arbitrary size (not necessarily small) in circular cylindrical pores. The algorithm employed is similar to that of Davidson et al. (1987) and Limbach et al. (1989). For a given solute size and shape, a random number generator is used to assign locator-point positions within the circular cross section as well as solute orientations. Given the non-axisymmetric geometry, generation of the orientation involves two steps, uiz., assignment of an axis direction from points uniformly distributed over a sphere and subsequent random selection of an angle of twist about the axis. The single-pore partition coefficient a, representing the fraction of accessible configurations,
0.0
0.2
0.6
0.4
0.8
1.0
ClR
Figure 4. Comparison of single-pore partition coefficienta @ for thin rods, Y-shaped solutes, 5-armed starsand circular disks. Results for the first and fourth shapes are available in the literature (Giddings et al., 1968;Limbach et al., 1989. Results for the second and third were obtained by the Monte Carlo method.
is evaluated as the ratio of the number of trials for which all points of the solute lie within circular boundary to the total number of trials. Typically, 104-10s trials are performed to obtain converged numerical results. For a Y-shaped (or more generally,star-shaped) solute, it suffices to check only the positions of the terminal points of the arms in classifying a configuration as accessible or inaccessible. Figure 3 compares the perturbation formulas and exact numerical values for @ for Y-shaped solutes with arms of equal length. The tangency of the exact and approximate curves in the limit X 0 corroborates the stated formula (7) for the mean projected radius. It is seen that inclusion of the quadratic term in eq 19 affords a substantial quantitative improvement and yields accurate approximations to @ so long as 10.5or so. This statement is roughly indicative of the accuracy of eq 17 for arbitrary values of the aspect ratio j3 = b/a. Figure 4 compares single-pore partition coefficients 9 for solutes having the shapes of a rod, a “Y”, a five-armed star and a circular disk, all having the same radial dimension c. It is seen that, insofar as steric constraints
-
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1395 1.01
are concerned, addition of arms quickly fills the space effectively occupied by the solute. The circular disk can be regarded u the limiting c u e for which there are infinitely many arms. These results fill in the gap between one- and two-dimensional objects represented by the extremes of a rod and a disk. If the arms of the equilateral "Y" terminate in spherical beads of radius s (Figure Id), then the partition coefficient is given by a simple modification of (19), uiz., @ = (1 - s/R)~~*(c,R - S)
0.4 .
(20)
rod
(cf. the consideration of capsule-shaped molecules vis-8vis infinitesimally thin rods by Giddings et al. (1968)).For
0.2 ' I
the case where the beads are so large that they just touch each other [c = (2/3ll2)s1, this formula agrees very well quantitatively with a corresponding approximation given by Baltus (1989, eq 9 and Table 1111,
+
Q = 1 - 2.408(1.509~/R) 1.379(1.509~/R)' (21)
over the range of applicability of the latter, i.e.,O I1.509slR I0.5. The character of these two formulas is different. Equation 20 represents an asymptotic analysis leading to a series in which the first few coefficients have been determined explicitly. Equation 21 represents an empirical fit to "exact" data generated by numerical integration over a prescribed interval in s/R. The close quantitative match is reassuring. 5. Shape Selectivity by Permeation
Permeation into a microporous gel discriminates between molecules on the basis of the value of the bulk partition coefficient K. At a given porosity e, the key to enhancing selectivity is to magnify differences in a. With large micropores, these differences reflect primarily the solute size. With sufficiently small pores, wall curvature effects (manifested in the shape-dependent quadratic and higher-order terms in eq 8) contribute materially to K.An ideal protocol for macromolecular or particulate separations and analysis might involve a determination of size using a gel with large micropores, followed by permeation into a finer micropore material for shape selectivity. For the latter step, appropriate choice of the pore size is critical. It must be sufficiently large to allow significant uptake, yet sufficiently small to allow the wall curvature to come into play. On the basis of the preceding calculations and existing results in the literature (Giddings et al., 1968; Limbach et al., 1989), it is possible to assemble expressions for 0 in circular cylindrical pores in the form @(rm, R) = 1 - 2rm/R+ C(r,/R)'
+ O(r,/~3)~
for a number of solute geometries, where C is a shapedependent coefficient having, inter alia, the following known values: 0, thin rod 8/3(3)'12r r 0.490, thin equilateral "Y" (23) c = { 8/r2 0.811, circular disk 1, sphere For a solid phase with a pore size distribution monodispersed a t the single radius R, K(r,) = t W m , R)gives the observed ratio of superficial solid-phase to solution concentrations at equilibrium. For a polydispersed pore size distribution characterized by a density function N(R)
2.0
f R&
= l0OA
2.2
2.4 log,,
2.6 ( L x / . J . )
2.8
3.0
4 kU= lO00.J.
Figure 5. Comparison of bulk partition Coefficients K for solutes with r, = 25 A of various shapes in a polydispersed solid having a uniform distribution of pore radii from R = 100 A to R = R,.
(defined such that N(R) dR is the number of pores with radius between R and R + dR),the experimentally observed partition coefficient corresponds to the weighted average
KO-,) = e(SR:,N(R)R' MI-' SR:,@(rmP) WW2dR (24)
Figure 5 shows partition coefficients for various solutes all having r m = 25 A as a function of the maximum pore radius R, for a porous solid with uniform distribution of radii N(R) = constant from R = 100 A to R = R,. Thus, only with R, = 100 A does the solid have a monodispersed pore size distribution. The conclusion to be drawn is that a controlled-pore solid with small pores significantly magnifies the relative differences in K. Indeed, the percentage difference in K between the rod and the sphere is only 0.2% for R, = 1000 A but 11 % for Rm, = 100 A. The latter value lies well within the range of what is separable by chromatography. Webster et al. (1985) make the pertinent observation that commonly used porous carriers for enzyme immobilization possess essentially unengineered micropore structures. Some checking with manufacturers of chromatographic packings confirms this same statement for SEC. Whereas gel particles are manufactured to strict standards of size uniformity to ensure uniform macropore flow, their internal micropore structures only satisfy exclusion limits (corresponding to specifiedvalues of R,). A significant increase in selectivity could likely be realized by strategic engineering of the micropores themselves. Particularly promising in this regard are anodized alumina membranes, which have already been the subject of careful permeability and diffusion characterization experiments (Dalvie and Baltus, 1992). These inorganic materials possess nearly monodispersed distributions of straight, roughly circular cylindrical pores arranged in a highporosity array looking very much like a thin-walled honeycomb (Dalvie and Baltus, 1992), hence our special interest in circular pores. Moreover, the membranes have a thickness in excess of the diameters of fine commercial gel beads and therefore look to have sufficient volumetric bulk to constitute a porous solid phase. The secondary purpose of this note is to underscore the importance of
1396 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
developing these materials in the form of granules for permeation, for this would greatly enhance the presently achievable permeation selectivity. 6. Summation
The preceding text extends previous studies by presenting an approximate, rigorously derived asymptotic formula for the partitioning of a prototypical nonaxisymmetric solute between a bulk solution phase and a circular cylindrical pore. On the basis of this and previous results, a direct calculation of macroscopic average partition coefficients for polydispersed porous materials shows clearly the enhancement in selectivity (to the point of discriminating between equisized macromolecules purely on the basis of shape differences) to be gained from monodispersed micropore structures in stationary phases used for permeation. Acknowledgment
Support of this work from the National Science Foundation under Grant No. CTS-9111034is gratefully acknowledged. The authors thank Professor Ruth E. Baltus for bringing previous work to their attention and for other valuable commenta rendered as a referee for the manuscript. Literature Cited Anderson, J. L.; Quinn, J. A. Restricted Transport in Small Pores: A Model for Steric Exclusion and Hindered Particle Motion. Biophys. J. 1974, 14,130. Baltus, R. E. Partition Coefficients of Rigid, Planar Multisubunit Complexes in Cylindrical Pores. Macromolecules 1989,22,1775. Barth, H. G.; Boyes, B. E. Size Exclusion Chromatography. Anal. Chem. 1990,62,38lR. Casasaa, E. F. Equilibrium Distribution of Flexible Polymer Chains Between a MacroscopicSolution Phase and Small Voids. J.Polym. Sci., Part E Polym. Lett. 1967,5,773. Casassa, E. F.; Tagami, Y. An Equilibrium Theory for Exclusion Chromatography of Branched and Linear Polymer Chains. Macromolecules 1969,2,14.
Dalvie, S. K.; Baltus, R. E. Transport Studies with Porous Alumina Membranes. J. Membr. Sci. 1992,71,247. Davidson, M. G.; Suter,U. W.; Deen, W. M. Equilibrium Partitioning of FlexibleMacromoleculesBetween Bulk Solution and Cylindrical Pores. Macromolecules 1987,20,1141. Deen, W. M. Hindered Transport of Large Molecules in LiquidFilled Pores. AIChE J. 1987,33,1409. Fanti, L. A.; Glandt, E. D. Partitioning of Spherical Solutes into Sponge-Type Materials. AIChE J. 1989,35,1883. Giddings, J. C.; Kucera, E.; Russell, C. P.; Myers, M. N. Statistical Theory for the Equilibrium Distribution of Rigid Molecules in Inert Porous Networks. Exclusion Chromatography. J. Phys. Chem. 1968,72,4397. Goldstein, H. Classical Mechanics; Addison-Wesley: Cambridge, MA, 1950; p 108. Laurent, T. C.; Killander, J. A Theory of Gel Filtration and ita Experimental Verification. J. Chromatogr. 1964,14,317. Limbach, K. W.; Nitache, J. M.; Wei, J. Partitioning of Nonspherical Molecules Between Bulk Solution and Porous Solids. AIChE J. 1989,35,42. Lin, N. P.; Deen, W. M. Effects of Long-Range Polymer-Pore Interactions on the Partitioning of Linear Polymers. Macromolecules 1990,23,2947. McCullough, K. C.; Spier, R. E. Monoclonal Antibodies in Biotechnology: Theoretical and Practical Aspects, Cambridge Studies in Biotechnology 8; Cambridge University Press: New York, 1990;pp 1-89. Pappenheimer, J. R.; Renkin, E. M.; Borrero, L. M. Filtration, Diffusion and Molecular Sieving Through Peripheral Capillary Membranes: A Contribution to the Pore Theory of Capillary Permeability. Am. J. Physiol. 1951, 167, 13. Pecsok, R. L.; Saunders, D. A Critical Evaluation of Gel Chromatography. Sep. Sci. 1966,1,613. van Kreveld, M. E.; van den Hoed, N. Mechanism of Gel Permeation Chromatography: Distribution Coefficient. J.Chromatogr. 1973, 83, 111. Webster, I. A.; Schwier, C. E.; Bates, F. S. Using the Rotational Masking Concept to Enhance Substrate Inhibited Reaction Rates: Controlled Pore Supports for Enzyme Immobilization. Enzyme Microb. Technol. 1985, 7,266. Received for review October 25, 1993 Accepted February 8,1994O ~~
e Abstract
1994.
published in Advance ACS Abstracts, March 15,
