Anal. Chem. 2001, 73, 3059-3064
A Mathematical Model for Hydrodynamic and Size Exclusion Chromatography of Polymers on Porous Particles Yves-Claude Guillaume,*,† Jean-Franc¸ ois Robert,‡ and Christiane Guinchard†
Laboratoire de Chimie Analytique and Laboratoire de Chimie Organique (Equipe de Chimie Therapeutique), Faculte´ de Me´ decine et de Pharmacie, Place Saint-Jacques, 25030 Besanc¸ on Cedex, France
When packed columns filled with porous particles are used for the separation of macromolecules, either size exclusion chromatography (SEC), hydrodynamic chromatography (HDC), or a combination of both determine the macromolecule retention mechanism. This paper develops a simple mathematical model to describe a molecular weight calibration graph, which includes both HDC and SEC. There is a transition between the HDC calibration region at higher molecular weights to an SEC region at lower molecular weights. The degree to which SEC and HDC are mixed depends on the particle diameter, the relative size of the pores, and the macromolecule size. In addition, using fractal considerations, the fractal character of the apparent selectivity between two adjacent peaks on the chromatogram is shown. This model constitutes an attractive tool to enhance the expansion of these two chromatographic techniques for the separation of biological or synthetical macromolecules. Fractal geometry has provided a mathematical formalism for describing complex and dynamic structures.1 It has been successfully applied to a variety of areas such as astronomy,2 economics,3,4 and biology.5,6 Due to its success in such a variety of areas, it would seem natural to develop fractal applications in chromatography. The fractal concept was used to study the effect of surface irregularity on the accessibility to silylating reactions.7 Grafting of surfaces is a key process in the preparation of chromatographic materials,8-11 given the ability to fine-tune the type of surface absorbate interactions by a suitable choice of the †
Laboratoire de Chimie Analytique. Laboratoire de Chimie Organique. (1) Mandelbrot, B. B. The fractal geometry of nature; Freeman, Academic: New York, 1991. (2) Martinez, V. J. Science (Washington, D. C.) 1999, 5413 (284), 445. (3) Mandelbrot, B. B. Fractals and scaling in finance: discontinuity, concentration, risk; Springer-Verlag: New York, 1997. (4) Mandelbrot, B. B. Multifractals and 1/f noise: wild-self-affinity in physics; Springer Verlag: New York, 1999. (5) Dewey, T. G. J. Chem. Phys. 1993, 98, 2250. (6) Fedorov, B. A.; Fedorov, B. B.; Schmidt, P. W. J. Chem. Phys. 1993, 99, 4076. (7) Farin, D.; Avnir, D. J. Chromatogr. 1987, 406, 317. (8) Riedo, F.; Gzencz, M.; Liardon, O.; Kovatz, E. Sz. Helv. Chim. Acta 1978, 61, 1912. (9) Korosi, G.; Kovatz, S. Z. E. Colloids Surf. 1981, 2, 315. (10) Le Ha, N.; Ungvarai, J.; Kovatz, S. Z. E. Anal. Chem. 1982, 54, 2410. (11) Gobet, J.; Kovatz, S. Z. E. Adsorpt. Sci. Technol. 1981, 1, 111. ‡
10.1021/ac010003+ CCC: $20.00 Published on Web 05/24/2001
© 2001 American Chemical Society
derivatizing agent.12 The structure of the derivatizing layer has been an issue of much debate regarding the questions of whether silanols are evenly distributed on the surface of silica13 or whether they are heterogeneously clustered.14 Lochmuller et al.15,16 used the intermolecular complexation process between ground-state and exited-state pyrene (py) to investigate this problem. The effect of surface irregularity on parameters such as surface concentration was studied by Farin and Avnir.17 Another application in chromatography was proposed to study the solute retention on immobilized human serum albumin (HSA). This study proposes a mathematical model to provide a more realistic understanding of the molecular processes that take place in the sucrose dependence of dansylamino acid binding on the HSA site II cavity.18 For the separation of macromolecules, and colloidal particles, field flow fractionation19 and capillary zone electrophoresis20 (CZE) have shown potential in the development stage and are now being introduced into analytical laboratories. Size exclusion chromatography (SEC) has been applied to ultrahigh-molecular-weight polymers by the development of particles with very large pore sizes. Hydrodynamic chromatography (HDC) on nonporous particles has developed into an inexpensive, high-speed fractionation technique for the submicrometer range, applicable to synthetic polymers, colloidal particles.21 For flexible biological or synthetical polymers, another nonequilibrium chromatography (NEC) technique, the slalom chromatography (SC) has been used.22-30 In SC, the separation process is based on the use of (12) Unger, K. K. Porous Silica; Journal of Chromatography Library Vol. 166; Elsevier: Amsterdam, 1979. (13) Unger, K. K.; Roumeliotis, P. J. Chromatogr. 1978, 149, 211. (14) Lochmuller, H.; Wilder, D. R. J. Chromatogr. Sci. 1979, 17, 574. (15) Lochmuller, C. H.; Colborn, A. S.; Hunnicut, M. L.; Harrius, J. M. J. Am. Chem. Soc. 1984, 106, 4077. (16) Lochmuller, C. H.; Colborn, A. S.; Hunnicut, M. L.; Harris, J. M. Anal. Chem. 1983, 55, 1344. (17) Farin, D.; Avnir, D. J. Phys. Chem. 1987, 91, 5517. (18) Peyrin, E.; Guillaume, Y. C. Anal. Chem. 1999, 71, 1496. (19) Kuhr, W. G. Anal. Chem. 1990, 62, 403. (20) Caldwell, W. D. Anal. Chem. 1988, 60, 959. (21) McHugh, A. J. CRC. Crit. Rev. Anal. Chem. 1984, 15, 63. (22) Dimarzio, L. A.; Guttman, C. M. Macromolecules. 1970, 2, 131. (23) Small, H. J. J. Colloid Interface Sci. 1974, 48, 147. (24) Hoangand, D. A.; Prudhomme, R. K. Macromolecules 1989, 22, 775. (25) Stegeman, G.; Kjraak, J. C.; Poppe, H. J. Chromatogr. 1991, 550, 721. (26) Stegeman, G.; Kjraak, J. C.; Poppe, H.; Tijssen, R. J. Chromatogr., A 1993, 657, 283. (27) Venema, E.; Kraak, J. C.; Poppe, H.; Tijssen, R. J. Chromatogr., A 1996, 740, 159.
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001 3059
the laminar flow, which occurs in the interstiticial spaces created between the particles packed in the column. Separation in SC has been reported for flexible macromolecules such as doublestranded DNA molecules.28,30 In HDC, macromolecules are separated on the basis of their size due to the hydrodynamic effect. The center of mass of large molecules is excluded from the lowvelocity flow regions near the particle wall. As a result, large molecules eluate from the column with a higher mean velocity than smaller molecules that can come closer to the wall. In SEC, the separation mode equally depends on the size of the macromolecules. The smallest macromolecules tend to be included in the particle pores packed in the column and the largest macromolecules are obviously excluded. SEC has many features in common with HDC, such as the elution order and the limited elution volume range. This would indicate that SEC and HDC can therefore be modeled by a mathematical treatment. This paper proposes to both describe this simple model and, using fractal considerations, show the fractal character of the apparent selectivity between two adjacent peaks on the chromatogram. THEORY In this model, at local equilibrium in a small volume element of the column containing n macromolecules, (1) the small macromolecules will be uniformly distributed in the pores of the material packing the column and in the regions near and far from the wall, (2) the medium-sized macromolecules will be restricted in the pores and in the regions near the wall, and (3) only the large macromolecules (i.e., those far from the wall) will be excluded. For a macromolecule with an effective radius r, when r is increased by dr, n increased by dn in accordance with a SEC mechanism and decreased by dn in accordance with a HDC mechanism. The differential equation describing this phenomenon can be written as
dn/dr ) (λ1 - λ2)n
(1)
where λ1 and λ2 are two positive constants which characterize the magnitude of SEC and HDC in the global separation mechanism. For a hypothetical macromolecule for which r f 0, the total molar fraction ξ of macromolecules excluded from the particle pores f 0. In addition, for a value of r above a critical value rc (r . rc)ξ f 1. The second following differential can be written
dξ/dr ) λ3n
is no interaction between the SEC and HDC mechanism (only the HDC mechanism exists in this case). Combining eqs 1 and 3, the following is obtained:
(
)
ξ dn n ) λ1 1 dr ξc
(4)
dn dn dξ ) dr dξ dr
(5)
and
Introducing eq 2 into eq 5 yields
dn dn ) λ3n dr dξ
(6)
Combining eqs 4 and 6 yields
(
)
dn λ1 ξ ) 1dξ λ3 ξc
(7)
Integrating eq 7 gives
∫ dn ) λ ∫ (1 - ξξ ) dξ λ1
n
0
ξ
0
3
(8)
c
The solution of this integration yields
n)
(
)
λ1 ξ2 ξλ3 2ξc
(9)
Combining eqs 2 and 9, the following differential equation is obtained:
(
)
dξ ξ2 ) λ1 ξ dr 2ξc
(10)
Rearrangement of eq 10 gives
λ1 dr )
dξ ξ(1 - ξ/2ξc)
(11)
(2) The integration of eq 11 gives
Finally, as the curve ξ ) f(r) is a sigmoidal curve (for r .> rc, i.e., ξ .> ξc, ξ f 1), for the inflection point (rc, ξc) for which d2ξ/dr2 ) 0, it can be assumed that
λ2 ) λ1ξ/ξc
(3)
In this equation, if ξ ) ξc, λ2 ) λ1 and if λ2 ) 0, ξ ) 0, then there (28) Hirabayashi, J.; Kasai, K. In Molecular Interactions in Bioseparations; Ngo, T. T., Ed.; Plenium Press: New York, 1993; Chapter 5, p 69. (29) Boyes, E.; Walker, D. G.; McGeer, P. L. Anal. Biochem. 1988, 170, 127. (30) Peyrin, E.; Guillaume, Y. C.; Grosset, C.; Ravel, A.; Villet, A.; Alary, J.; Favier, A. J. Chromatogr., A, in press.
3060
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
λ1r ) ln
ξ +β (1 - ξ/2ξc)
(12)
where β is a constant. Usually, in SEC or HDC, the relative retention volume τ of a macromolecule can be expressed by the equation26
τ ) VR/VH
(13)
where VR is its retention volume and VH the retention volume of a macromolecule totally excluded from the particle wall (corre-
sponding to the void fraction). VH was determined using the macromolecule with the highest molecular weight (MH). τ is a chromatographic parameter similar to the relative migration factor classically used in HDC26 and depends on both the macromolecule molecular weight and the geometrical characteristics of the material packing the column. Using a phenomenological approach, if τˆ is the relative retention volume for a hypothetical macromolecule for which r .> rc (large macromolecule totally excluded from the particle wall in practice) and if τˇ is the relative retention volume for a macromolecule for which r ,< rc (small macromolecule in practice), then the τ value of a macromolecule of radius r is a linear combination between τˆ and τˇ :
τ ) ξτˆ + (1 - ξ)τˇ
(14)
From eq 12, the ξ value is given by
ξ)
exp(λ1r - β) 1 1+ exp(λ1r - β) 2ξc
(15)
Combining eqs 14 and 15 the following equation is obtained:
ξ)
2ξc(τˆ - τˇ ) exp(λ1r - β) + (2ξc + exp(λ1r - β))τˇ 2ξc + exp(λ1r - β)
(16)
For a linear random coil polymer in a good solvent, the effective polymer radius, r, can be calculated from the radius of gyration rg. For the example used in this work, the relationship between rg and the weight-average molecular weight noted, M, is known. For r, the following is obtained:31
r ) 1/2 xπrg ) aMb
fragment size (level 0, size M0) to the highest fragment size (level n - 1). Mn-1 corresponding obviously to MH. At level L, the fragment size has an ML size. The parameter
(17)
where a and b are constants. Therefore, combining eqs 16 and 17, the following is obtained:
τ)
Figure 1. Correlation between predicted (eq 18) and experimental τ values. The slope is 0.98 with an r2 ) 0.997 as determined by linear regression.
2ξc(τˆ - τˇ ) exp(λ1aMb - β) + (2ξc+ exp(λ1aMb - β))τˇ 2ξc + exp(λ1aMb - β) (18)
pL ) ML/ML-1
where pL > 1 and 1 e L e n - 1 were introduced to characterize the fragment size in the series. Assuming a self-similar fractal nature of the fragment series implies that pL must be independent of L: pL ) p ) constant. Thus, at level L, the fragment size can be expressed as
ML ) M0pL Equation 18 shows that τ is a function of the macromolecule molecular weight in such a way that a unique calibration curve for SEC and HDC can be obtained. In addition, to characterize in practice the separation quality between two peaks on a chromatogram, the apparent selectivity was defined as
R ) τ2/τ1
(20)
(21)
and n-1
∑M
(22)
M0pn - M0 p-1
(23)
MT )
k
k)0
(19) and
where τ2 and τ1 represent the relative retention volume of the two macromolecules on the chromatogram. Fractal geometry (FG) was used in this paper to study the effect of the fragment size, M, on the apparent selectivity, R. A series of n fragments was introduced. This series was composed of n terms from the smallest (31) Van Kreveld, M. E.; Van den Hoed, N. J. Chromatogr. 1973, 83, 111.
MT )
Equations 20 and 23 reflect the fractal nature of the fragment series. The corresponding apparent selectivity between two Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
3061
Figure 2. Theoretical macromolecule size dependence on τ for a column packed with a particle diameter fixed at (A) 500 and (B) 1000 Å.
adjacent fragments in the fractal series was
RL ) τML-1/τML
(24)
where τML-1 and τML represent the relative retention volume of the fragments with a respective size equal to ML-1 and ML. The variation curve RL vs L is an image of the variation in the apparent selectivity versus the fragment size. As L increases, the apparent selectivity is determined between two adjacent fragments in the fractal series, i.e., between two fragments for which both their sizes and their size differences increase. RESULTS AND DISCUSSION With a weighted nonlinear regression (WNLIN),18,32,33 which was used in an earlier chromatographic study, the data reported by Vivilecchia et al.34 were fitted to eq 18. The chromatographied compounds were a series of polystyrene standards (ps), and the columns used were packed with phases with pore diameters of 500 and 1000 Å. The WNLIN regression method was used to calculate the optimum parameter values by minimizing the χ2 function with respect to each of the parameters simultaneously35 (i.e., τˆ , τˇ , ξc, and λ1 characteristics of the column). In other words, the procedure optimized the parameters in order to obtain the least-error sum of squares in predicting τ. The accuracy of the (32) Peyrin, E.; Guillaume, Y. C.; Guinchard, C. Anal. Chem. 1998, 70, 4236. (33) Guillaume, Y. C.; Peyrin, E. Anal. Chem. 1999, 71, 1326. (34) Vivilechia, R. V.; Lightbody, B. G.; Thimot, N. Z.; Quinn, M. M. J. Chromatogr. Sci. 1979, 15, 424. (35) Bevington, P. R. Data reduction and error analysis for the physical sciences; McGraw-Hill: New York, 1969.
3062
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
calculated parameters depends on the accuracy of the experimental data and the precision of the measured data. The WNLIN regression procedure requires the starting value significations for the parameters to start the iterative algorithm to locate the parameter space for the optimum values. The outcome of the WNLIN regression depends on the selected values for the parameters. The procedure seemed to be reliable as reasonable variations in the starting parameter values did not influence the results. It should be noted that one of the difficulties is the possibility of the existence of more than one local minimum for χ2 within a reasonable range for the parameters. In such a case, a coarse grid mapping of the parameter space to locate the global optimum would be advantageous.35 After the WNLIN procedure, the calculated τˆ , τˇ , ξc, and λ1 parameters were used to estimate the τ values with the measured values for the various polystyrene fragments at the three different pore radii of the column. The correlation between all the predicted and experimental τ values exhibited slopes equal to 0.98 with r2 > 0.99 (Figure 1). This good correlation between the predicted and experimental values can be considered to be adequate to verify the model. The τ values reported previously34 were plotted against the polystyrene size for the two columns packed respectively with particle pores fixed at 500 and 1000 Å, and the corresponding molecular weight calibration graphs were obtained (Figure 2). The zones of the sigmoidal-like curve corresponded to the two cases reported in the Theory section for the polymer progression into the column, i.e., r . rc and r , rc. The critical ξc (eq 18), for a pore radius equal to 500 and 1000 Å were respectively equal to 0.51 and 0.54. The data reported previously were used to determine the Rc versus
Figure 3. Selectivity RL as defined by eq 24: (A) p ) 1.05 and (B) p ) 1.07 for a column packed with particle pore size fixed at 500 Å.
Figure 4. Selectivity RL as defined by eq 24: (A) p ) 1.05 and (B) p ) 1.07 for a column packed with particle pore size fixed at 1000 Å.
L plots for two columns packed with phases with pore diameters 500 (Figure 3) and 1000 Å (Figure 4). The plots were divided into two regions, one for levels M over Mc and a second for levels below Mc: (I) For L .> Lc corresponding to M .> Mc, the
fragment size was high in such a way that ML-1 and ML f MH. Thus, RL f 1. (II) For L < Lc corresponding to M < Mc and at a sufficiently low level of L, the τ value of a fragment was proportional to its size explaining that, for relatively low p values, Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
3063
corresponding to a weak increase in ML in relation to ML-1 (p ) 1.05 for example), RL was relatively independent of L and f 1. This was clearly objectivized by using a weak p value, (p ) 1.05), which at the two extremities of the two regions I and II (ML-1 and ML , Mc and ML-1 and ML . Mc), the apparent selectivity RL was relatively independent of the level L value. This reveals the fractal character of the apparent selectivity for both the smallest and the largest fragments. In summary, a novel mathematical treatment of hydrodynamic and size exclusion chromatography associated with fractal considerations has been proposed to model both the relative retention volume and the apparent selectivity between two adjacent pss fragments on a chromatogram. It was shown that there is a
3064
Analytical Chemistry, Vol. 73, No. 13, July 1, 2001
transition between the HDC (corresponding to the highest molecular weights) and the SEC (corresponding to the lowest molecular weights) mechanisms. The degree to which SEC and HDC are mixed depends on the diameter of the particles, the relative size of their pores, and the fragment in question. In addition, the fractal character of the apparent selectivity was demonstrated by using a fractal distribution of the ps fragments.
Received for review January 5, 2001. Accepted March 27, 2001. AC010003+
