Anal. Chem. 1988, 6 0 , 895-900
895
Resolution of Electrostatic and Steric Factors in Aqueous Size Exclusion Chromatography Paul L. Dubin,*’ Catherine M. Speck,’ and Jerome I. Kaplad
Department of Chemistry and Department of Physics, Indiana-Purdue University, Indianapolis, Indiana 46223
Size exclusion chromatography of sodlum poly( styrenesulfonate) (NaPSS) on CPG porous glass has been studied In order to elucidate the Influences of lntennacrolon expanslon and polybn-statlonary phase lnteractlon on elutlon behavior. Measurement of the dhnlnutlon of the chromatographlc partltlon coefflclent, KISC, for the polylon, relative to that of a nonlonk polymer (pullulan) of Identical molecular volume makes lt posslble to Isolate the contrlbutlon of polylon-substrate repulsive forces to KIIEC. Thls effect may be approxlmately attributed to a volume wlthln the pore from which the polylon Is repelled. The linear dlmenslons of thls “repulsion volume” vary Inversely wlth the square root of the lonlc strength; Le. Its thlckness, XE, Is proportional to the Debye length. However, the electrostatlc potentlal at the locus of thls “barrler” Is rather lnsensltlve to the lonlc strength. The molecular weight dependence of X , suggests two reglmes, a low molecular welght range In which X , varies wlth the net valency of the solute and a hlgh molecular weight range of constant X , In which the polylon behaves as a sphere of constant surface charge dendty. The superposltlon of measured values for X , on a almple geometrlc model for the sterlc exckgkn effect leads to Calculated values for K’, of NaPSS In very good agreement wlth experimental results.
The recognition of electrostatic interactions between ionic solutes and size exclusion chromatography (SEC) stationary phases dates back at least as far as 1953, when Wheaton and Bauman separated sodium chloride from glucose, using cation exchange resins ( I ) . The particularly rapid elution of low molecular weight salts from ionophoric packings has been referred to as “ion exclusion”chromatography(2)and has been studied in considerable detail by Neddermeyer and Rogers (3,4),Stenlund (2,5),and others (6). It is not surprising that synthetic polyelectrolytes or charged natural macromolecules also exhibit Coulombic interactions with glass or silica packings. However, it may not be evident that virtually all aqueous SEC packings bear some ionizable surface sites which may dramatically influence the elution of charged macromolecule, even when the concentration of such surface groups is relatively low. Sephadex, for example, contains-depending on gel type-between 4 and 50 pequiv/g dry gel of carboxylic acid (3, 7). PW gel, a cross-linked hydrophilic polyether, also contains COOH groups (8). A wide variety of packings based on porous silica with chemically bonded hydrophilic phases has been commercialized (9); numerous studies with low molecular weight salts,synthetic polyelectrolytes, and proteins reveal the ionic character of these substrates (IO), presumably resulting from the fact that the silanization of silica is far from stoichiometric (11). The rapid elution of anionic polyelectrolytes on porous glass or silica packings has been noted by several workers, all of whom observed the supression of this electrostatic effect at ‘Department of Chemistry. *Department of Physics.
high ionic strength ( p ) (12-14). It may not, however, be assumed that “ideal SEC” (unperturbed by polymer-substrate interactions) can always be achieved in this manner. Even at ionic strengths as large as 0.4 M, the chromatographic partition coefficients (K)of globular proteins on SW columns (a hydrophilic bonded-phase silica) increase with p for acidic proteins and decrease with p for basic proteins (15). Furthermore, the value of K for globular proteins may display a minimum with p , indicating the development of other solute-substrate interactions, e.g. hydrophobic, at high salt concentrations (10). Attractive forces between strong polycations and highly charged packings cannot be supressed by the addition of salt; interactions between such polyions and anionic packings are evident even in the presence of moderate ionic strength mobile phase (16). Consequently, the problem of the electrostatic contribution to K is not subject to a consistently reliable remedy. The ionic strength dependence of the elution volume of polyelectrolytes results from two effects: intrachain expansion and polyion-substrate interactions. When the polyelectrolyte and the stationary phase are of like charge, e.g. anionic polyelectrolytes on glass or silica packings, both effects result in diminished elution volumes at low ionic strength. Resolution of these two effects is possible since “universal calibration” plots of [VIMvs V , or K (17) take into account intrapolyion expansion. Naturally, such resolution is not necessary for “rigid” solutes, such as proteins or latex particles. However, such solutes commonly display adsorptive effects at high ionic strength, complicating the interpretation of chromatographic behavior (10, 18). We previously reported universal calibration curves for sodium poly(styrenesu1fonate)(NaPSS), pullulan (Pul), and dextran on CPG porous glass, in mobile phases of varying ionic strength, p, and pH (19). At ionic strengths of 0.5 M or more, the universal calibration curves for NaPSS and the neutral polysaccharides are congruent. The progressive divergences of these curves with increasing pH or with decLeesingionic strength were taken as evidence of polyion-substrate repulsion, which in turn is a function of both the glass surface charge density and the degree of ionic screening. The repulsion of polyion from the charged pore was envisaged as an “electrostatic barrier” to polyion permeation, the thickness of which depends on the degree of ionization of the glass and on the ionic strength. At ionic strength 20.5 M (for pH ;=8) the mean thickness of this hypothetical barrier, XE,becomes negligible so that universal calibration curves of ionic polymers converge with the nonionic reference plots. The value of XEmay be experimentally determined, inasmuch as XEA = V,, where A is the pore surface area of the packing and V , is the volume within the column which is penetrable to nonionic solute but impenetrable to polyion (19). A is virtually identical to the total packing surface area, calculated from BET adsorption measurements. Under most conditions, the ratio X , / a p , where ap is the mean pore diameter, is sufficiently small so that curvature may be neglected and XEmay be equated to V J A . We formerly determined VI by assuming that, at constant [7]M,Kpu1 and Kpss differ only because the effective column pore volume for NaPSS,
0003-2700/88/0360-0895$01.50/00 1988 American Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 9, MAY 1, 1988
V,,’, differs from the geometric column pore volume V, (which determines the elution of nonionic polymers). In other words, the conventional expression for K
Table I. Pullulan Standards
where V, is the measured elution volume, V, the column pore volume, and V, the column interstitial volume, is replaced, in the case of the polyion, by a
It was postulated (19) that universal calibration curves for
NaPSS and pullulan would converge at all ionic strengths if [VIMwere plotted vs K’.Thus, forced fit of the curves yielded values for V, and hence XE.While XEdepends strongly on pH and p, it was expected that the electrostatic potential at
XE,\kE, migh be relatively constant. The value for \kE was obtained by interpolation among tabulated numerical solutions of the Poisson-Boltzmann equation in radial geometry (20). In 0.05 M pH 8.0 phosphate buffer, X E = 38 f 6 A and \kE = 4.8 f 2 mV. Assuming an identical value for \ k in ~ 0.035 M pH 6.1 buffer, we calculated XE= 26 f 4 A; the measured value was 34 f 6 A. While the preceding results provided partial but somewhat ambiguous support for the use of the foregoing description as a zero-order model, this preliminary study required development along several lines. From an experimental perspective, the use of a mixed pore-size (70, 172,324, and 1038 A) column obscured the relationship between macroscopic and microscopic pore volumes. The five NaPSS samples used defined only a portion of the universal calibration curve. Interpolations among literature values for the surface charge of silica (21) introduced very large errors into the surface potential values. Two of the three ionic strengths employed, 0.001 15 and 0.510 M, corresponded, respectively, to the extreme situations-near-total exclusion, and complete supression of repulsion. With regard to the model and related calculations, the assertion that K = K’at constant [VIM,and the concominant assumption of constant V , and XE throughout the molecular weight range must be questioned. Finally, interpolations among the tabulated numerical solutions of ref 20 were rendered even more awkward by the use of an asymmetric salt, sodium phosphate, as the mobile phase electrolyte. The current work represents extension along the lines indicated by the preceding remarks. In addition to addressing the experimental deficiencies noted above, we have taken into account the variation in repulsion volume with polyion size. We have also considered, in some detail, relationships previously put forward for nonionic polymers vis-a-vis our measurements of K and [VIMfor pullulan. EXPERIMENTAL SECTION Materials. Pullulan standards (Showa Denko, Ltd., New York) are listed in Table I along with molecular weight data available from the supplier. NaPSS (MWs 1600,4OOO,6500,16OOO, 31 OOO, 65000, 177000, and 354000) was from Pressure Chemical Co. (Pittsburgh, PA). Blue dextran was from Pharmacia (Piscataway, NJ). NaCl and Na,HPO, were reagent grade. All water was deionized. Methods. The chromatographic system was comprised of a Milton Roy minipump, Rheodyne (Model 7010) injector with a 100-pL sample loop, and a Waters R401 differential refractometer. The 50 cm X 0.60 cm i.d. column blank with 2-pm end frits (Supelco) was dry-packed with nominally 379-A pore diameter 200-400 mesh CPG controlled pore glass (Electronucleonics, Fairfield, NJ). The total packing surface area was 4.5 X IO6 cm2, according to the weight of CPG glass in the column and the
code
MW x 10-4
[TI: cmSg-’
P-5 P-10 P-20 P-50 P-100 P-200 P-400 P-800
0.58 1.22 2.37 4.80 10.0 18.6 38.0 85.3
6.4 10.5 16.3 26.2 42.8 64.8 105 180
In uure H,O.
manufacturer’s value for the specific surface area. The column efficiency, determinedwith DzO,was lo00 plates m-l. The mobile phase was 9:l NaCl-NaHzP04adjusted to pH 5.0 and the desired ionic strength. Elution volumes were obtained by injection of polymers at 5 mg mL-’, at a flow rate of 1.0 mL min-‘, determined to within 2 ppt by weighing timed collection of solvent. Vo, the exclusion volume, was obtained from the elution volume of blue dextran, and V,, the total column volume, from the elution of DzO. Intrinsic viscosities of NaPSS were measured at 25.00 f 0.05 “C with a Schott AVS 300 automaticviscometer over a wide ranqe of ionic strengths and molecular weights. The surface charge density of CPG was determined by pH titration of acid-washed glass. About 2 g of CPG was equilibrated with 0.2 M HC1 and then washed exhaustively with deionized water and dried. A weighed sample of the dried CPG was titrated with 0.100 M NaOH to pH 5.00, in the appropriate unbuffered mobile phase solvent, using a Radiometer Model M26 pH meter equipped with a combination pH electrode. The surface charge density of the glass was obtained as follows. When acid-washed glass is combined with NaCl solution, the pH decreases, due to SiOH dissociation, until an equilibrium value, “pH$ or “the natural pH”, is reached. The number of milliequivalentsof SiOH groups dissociated at pHo is equal to the number of milliequid e n t s of HCl required to bring a glass-freeblank from its initial pH (“pHin”)to pHo. Since the pH of interest, 5.00, is less than the “natural pH”-corresponding to the addition of more acid-the number of SiO- groups formed is less than the amount obtained from the blank titration from pHh to pHo described above. This diminution in the quantity of SiO- is given by the amount of HC1 required to titrate the glass and solvent from PI-& to pH 5.00, less the amount required to titrate the blank over the same pH range. Thus mequiv of SiO- = mequiv of HC1~p~-pHd - [mequiv of HClTz$p~5,m)- meqUiV Of HC@$,,H~,~)] = mequiv of HClt$i-pH6,m) mequiv of H C l ~ p ~ p H 5 . m ) (3)
The surface charge density, u (MCrn-,), may then be calculated as (21) u
= -AF/S
(4)
where A is mequiv of SiO-/g of SiOz,S is the surface area of the glass (m2/gof Si02),and F is Faraday’s constant, giving u in pC m-’. The pore size distribution of the CPG packing was determined by mercury porosimetry using a Micromiretics Pore-Sizer Model 9305, taking the contact angle between mercury and glass to be 140’. A value of 500 f 10 A was obtained for the mean pore diameter. This result was also in very close agreement with the equivalent hydrodynamicdiameter for pullulan extrapolated (22) to K = 0, i.e. at total exclusion. Consequently this value was used in preference to that supplied by the manufacturer. RESULTS AND DISCUSSION NaPSS Viscosity. Intrinsic viscosities were measured for selected NaPSS samples at the extremes Qf molecular weight and ionic strength in order to verify literature relationships
ANALYTICAL CHEMISTRY, VOL. 60, NO. 9, MAY 1, 1988
897
1
31
Flgure 3. Schematic representations of charged (left) and uncharged (right) polymers of identical equivalent hydrodynamic radii within cyiindrical pores of radius a p. X E represents the thickness of hypothetical electrostatic barrier to permeation.
I
5
4
6
LOG M Flgure 1. Mark-Houwink plots for NaPSS at varying ionic strengths (as shown) (intrinsic viscosity in om3 g-I). Lines are from relationship of ref 23, from top to bottom, along right axis, 0.01, 0.02, 0.05, and 0.50 M. Symbols show selected experimental resuits for (e) 0.01 M and (I 0.5 ) M ionic strength solvents.
X
4
0
0.2
0.6
0.4
0.8
K‘ Figure 4. Forced fit of modified universal calibration curves, symbols as in Figure 2.
0:
7
8
9
le
1’1
12
I3
114
ve
Flgure 2. Universal calibration plots (intrinsic viscositjl in cm3 g-I): (X) pululan in pure H20;( 0 ,0, A, 0) NaPSS in 0.5, 0.05, 0.02, and 0.01 M ionic strength eluants. Filled or bold symbols represent sodium citrate, except (e)represents D,O.
(several of which disagree). Results a t all ionic strengths agreed with the [VI-molecularweight relationships reported by Takahashi et al. (23). However, as shown in Figure 1, the Mark-Houwink plot for 0.01 M ionic strength crosses the other lines, leading to the implausible result that [q] increases with p a t low molecular weight. Huggins plots of q,,/c vs c for the low molecular weight samples at = 0.01 M had negative slopes, presumably because changes in polymer concentration correspond to changes in ionic strength. Values for [q] at p = 0.01 M were therefore obtained by extrapolation of plots (24) of [q] vs I-1/2.All other viscosity values were calculated by using the Mark-Houwink relations reported by Takahashi et al. Universal Calibration Plots and the “Mean Repulsion Volume”. Universal calibration plots for pullulan in pure water and for NaPSS in mobile phases with ionic strengths of 0.50, 0.05, 0.02 and 0.01 M, all adjusted to pH 5.00, are shown in Figure 2. Also included are results for sodium citrate. The convergence of the NaPSS p = 0.50 M plot with that for pullulan confirms the supression of polyion-ubstrate interactions at this ionic strength. As previously noted (19), the effect of ionic strength on either [q] or V , for pullulan is negligible. The divergence of the curves at lower p is attributable to electrostatic repulsion of NaPSS from the pore of the packing. To a first approximation,one might propose that this effect could be accounted for by a reduced effective total
pore volume for the polyion, V,,’ = V, - VI, where V,, as noted above, is the geometric column pore volume, defined in turn by the difference between V, (i.e. V, for DzO)and V , (i.e. V , for an excluded polymer, such as blue dextran) and V, is the total column volume within the pores rendered impermeable to the polyion by electrostatic repulsion. These macroscopic quantities are related to the microscopic mean pore volume, up, and the microscopic mean repulsion volume, u,, in that V, = nu,, etc., where n is the number of pores within the column. V,, which is experimentally accessible, is connected to the microscopic model by recognizing, as stated in the introduction, that the ratio of V, to the total pore surface area A defies X E , the thickness of a hypothetical electrostatic barrier at the glass surface (see Figure 3). An approximate value of V, may be arrived at by assuming that-at constant [q]M-Kpd and KPSSdiffer only because V i # V , and, consequently, that “adjusted” values for Kpss, i.e. K’, defined by eq 2, would coincide with Kpd at constant [VIM. Thus, V , could be obtained by forced fit of modified universal calibration curves. The result of such a procedure is shown in Figure 4. As we shall see, the assumption that VI is constant through the molecular weight range is unrealistic. Nevertheless the values of the “mean repulsion volume”, VI, generated by this approach are of some interest. For p = 0.01, 0.02, 0.05, and 0.50, the corresponding values for Vr (in cm3)are 3.74 f 0.14, 2.64 f 0.10, 1.45 f 0.05, and, of course, zero. As noted above, the mean thickness of the hypothetical repulsion layer XEis given by 7,A-l(the assumption of planar geometry is justified since XEis small compared to the pore radius). Values for XEat p = 0.01,0.02, and 0.05, in A units, are 83 3,59 f 2, and 32 f 1. It is of considerable interest to note that the corresponding values for X E K , i.e. XEin Debye lengths, are 2.8 2.7, and 2.4, respectively ( K is the reciprocal Debye length in A-1). Approximate values for the electrostatic potential ‘kEat XE may be estimated in the following way. Since the mean pore radius exceeds K - ~by an order of magnitude or more, planar
*
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 9, MAY 1, 1988 3
Table 11. Experimental and Calculated Quantities Involved in the Determination of *E
X 6,p C
0.020 M
0.010 M
0.045 f 0.002
0.044 f 0.0020 0.7 f 0.1 0.82 f 0.01 3.5 f 0.4 90 f 10 0.18 f 0.02 4.7 f 0.6
0.044 f 0.002 0.5 f 0.1 0.88 f 0.02 4.3 f 0.6 110 f 15 0.16 f 0.02 4.2 f 0.5
cm-* 0.8 f 0.1
I Yo
q,,, mV ?!E
0.050 M
qE,mV
0.63 & 0.005 2.0 f 0.2 50 f 5 0.16 f 0.02 4.0 f 0.05
geometry may be assumed. The pore may then be viewed as equivalent to a spherical surface of large radius, a situation for which numerical solutions to the Poisson-Boltzmann equation have been tabulated by Loeb, Overbeek, and Wiersma (LOW) (20). At each ionic strength, the glass surface charge density u, obtained by pH titration, leads to a value for the gradient I of the potential at the surface. From tabulated values of I (as a function of the dimensionless surface potential yo = e\ko(kr)-l) linear plots of I vs y o were constructed. These plots were used to interpolate values for y o (and hence \ko) corresponding to the value for I obtained at each ionic strength. We are primarily interested in the potential yE a t a distance X E from the LOW particle surface. The values of XEare sufficiently large and the potential a t XEsufficiently small so that the following approximation is valid (25): Y = Yo(904-’) e x p H 9 - 9011
(5)
where q is the reduced distance from the particle center (in Debye lengths) and qo the corresponding value at the surface. Some of the results of this approach are shown in Table 11, where z is the reciprocal reduced (unitless) distance from the center of the LOW particle, defined as x-l = K(CY + XE), and CY is the LOW particle radius, chosen somewhat arbitrarily as 20 ( K - ~ ) to ensure near-planar geometry. (Values for u were obtained by glass titrations in conjunction with data from ref 21: the values shown are extrapolations from data in ref 21, while the error limits reflect differences among these results and our repeated titration.) As noted above, \ko is the surface potential, yo and yE are dimensionless variables corresponding to \ko and \kE, obtained by dividing the latter quantities by k T / e . I , the gradient of the potential at the surface, is obtained from u and p, assuming a = m. The error limits shown are obtained by propagation of errors and primarily arise from the various graphical interpolations. One notes that the values for GE are nearly identical within experimental error. Indeed, the agreement is better than expected given the crudity of the various approximations. The small values obtained for +E may be rationalized by recognizing that a large number of polyion segments must penetrate the hypothetical potential barrier at XE in order for the electrostatic repulsive energy to exceed thermal energy. Dependence of the Repulsion Volume on Molecular Weight. According to the model depicted in Figure 3, at constant [VIM,AV, = V?’ - Vepss should depend on the effective radius of the two polymers, the pore radius and XE. An expression can be obtained for AV, if we choose a welldefined shape for the pores and treat the polymers as rigid equivalent hydrodynamic spheres. Some justification for this second assumption will be presented later. For the moment, let us provisionally treat the polymers as impenetrable spheres of radius Rh within cylindrical pores of radius up. Then (26)
K = ( F )
2
9
Q
1
0
LOG Rh Figure 5. Relationship between AVe = V p ’ - VePSSand I?,(A) at ionic strengths shown.
identical Rh values, but the effective pore radius for the former is ap - XE. Then
A v , = ConSt(2a,X~- ~ R & E- X ~ ~ ) a p - ~ (7) where the constant depends on the total pore volume of the column, 7.10 cm3in the present case. If X Ewere independent of Rh a t a given ionic strength, then, according to eq 7 , AV, should decrease linearly with increasing Rh, the slope depending on XEa;2 and the column pore volume. Figure 5 shows the dependence of AV, on Rh. Rh is related to [VIM by
where [v] is expressed in cm3g-’ and Rh in A. The region of large molecular weight (Rh> 90 A) is not particularly meaningful since the pullulan and NaPSS calibration curves must converge at v, = vo,i.e. AV, 0 as Rh becomes large. At lower molecular weight one observes two linear regimes with an abrupt change in slope in the molecular weight region ea. 1 X lo4to 3 X 104 (about 35-50 8, for Rh). These results clearly show that X E depends on molecular weight. We now proceed to evaluate XE According to eq 1 and 2, the permeation volumes for nonionic and ionic polymers, are, respectively V , = V o KV, (1’)
-
(&)a
+
and
V,l= Vo + KYp‘
(2’)
At constant molecular size ([77]M), the effective pore volume for the polyion, u; = V:/n: is smaller than the geometric pore volume, up = Vp/n. Consequently, the relative probability of finding the polyion in the pore, K’, is smaller than K. As a result, V i is smaller than V,. To evaluate V,, and thus XE, we consider a polyion of size Rhf which exhibits the same elution volume as a neutral polymer with Rh > Rd. As before, we provisionally identify Rh with the equivalent hydrodynamic radius, according to eq 8. For reasons shown below, we assume cylindrical geometry for the pores. At constant elution volume (V,’ = V,) KV, = K’V,’ (9) where
2
K = (ap- Rh)’/a;
(10)
and At constant [VIMboth charged and uncharged polymers have
K’ = (ap- XE - Rh’)2/(ap - XE)’
(11)
899
ANALYTICAL CHEMISTRY, VOL. 60, NO. 9, MAY 1, 1988 80
We note that
70
r
I
i
where n is the number of cylindrical pores of mean geometric radius apand mean length i in the column. Combining eq 9-13 yields
u
or simply
XE = Rh - Rh‘
‘ 0
20
40
(15)
According to eq 15, the condition of constant K (or constant V,) requires that the geometric dimensions of the polyion be reduced relative to those of the neutral polymer by exactly XE. If XE were considered to be exclusively due to the double layer around the solute (as in ref 181,eq 15 would merely state that the geometric radius of the charged solute plus its effective ionic radius is equal to the geometric radius of a coeluting neutral solute. eq 9-15 obviate the approximation of XE > 1,the large deviations near K = 0 are expected. For K < 0.5, the more general solution of ref 27 produces values of K somewhat larger than the results from the limiting form (eq 17, curve c in Figwe 7) but still 20-50% smaller than Kh in the range 0.2 < Kh < 0.5. Thus, as noted in ref 27 and 28, the statistical model leads to values of K at least 20-25% smaller than the experimental results. For both theories, cylindrical pore geometry provides better results. Most strikingly, the geometric model with cylindrical pores reproduces not only the correct shape of the calibration curve but the absolute values, with considerable fidelity. It is worth noting that no adjustable parameters are contained in either eq 16 and 17.
BOO
ANALYTICAL CHEMISTRY, VOL. 60, NO. 9, MAY 1, 1988 8
8
1
b i
5 7
E
5
..‘.’..\ ,. \ ,
4
a
a 2
a 6
a 4
0 8
1
u
3a
I
K
Figwe 7. Comparison of experimental universal calibration plots for p u l W In pure water (symbols) with calculated plots (intrbrslc vkcosky in cm3 g-’); (a) eq 16 for cylindrical pore; (b) eq 16 for spherical pore; (c) eq 17 for cyllndricai pore; (d) eq 17 for spherical pore; (e) numerical solutions for complete expression from ref 27 for cyiindricai pore.
If the geometric model is accepted-and the goodness of the fit to experimental data is difficult to disregard-one may proceed to incorporate the electrostatic effect by replacing up in the numerator of eq 6 with ap-Xe,i.e.
Values for XE obtained in the preceding section were used, namely 53, 33, and 20 A, for p = 0.01, 0.02, and 0.05 M, respectively. Note that K’ represents the fraction of the geometric pore volume into which the polyion may permeate. Experimental values for K’ and values calculated according to eq 19 are shown for two of the three ionic strengths in Figure 8. The agreement is good for the region log [VIM2 5.5, beyond which the calculated values are too low. The lower molecular weight range which the calculated and experimental curves diverge corresponds to the molecular weight range of Figure 6 in which X , diminishes. Introduction of smaller values of XE into eq 19 would bring the calculated values of KsECcloser to the experimental results. However, the significance of both XE and the geometrical model itself are questionable in the low molecular weight range, where eq 18 also loses validity. CONCLUSIONS The chromatographic partition coefficient for sodium poly(styrenesu1fonate) on porous glass depends on both polymer molecular weight and solvent properties. The influence of polymer dimensions is accounted for by the hydrodynamic radius relative to the pore dimensions. The roles of solvent ionic strength and pH are incorporated into the variable XEwhich describes the effective reduction of pore dimensions arising from electrostatic repulsion between polyion and packing. For polymers with molecular weight in excess of about 3 x 104, \kE appears to be constant with respect to molecular weight, suggesting that the repulsive effect can be modeled as the interaction between two planar surfaces. In this molecular range, the potential due to the packing at the apparent “repulsion distance” is nearly independent of ionic strength. The foregoing treatment is intended as a first approximation since it essentially neglects the potential due to the polyion. Smith and Deen have described a treatment of a more detailed
8.2
0.4
0.6
8.8
i
K Flgure 8. Comparison of experimental universal calibration data (symbols) with calculated plots from eq 19 (Intrinsic viscosity in cm3 g-I), ionic strengths as shown.
model, which accounts for the potentials of both the macromolecule and the stationary phase (31). Comparisons of this theory with our experimental data are currently in progress. ACKNOWLEDGMENT The support of the Johnson’s Wax Fund is gratefully acknowledged. The authors thank Professor E. F. Casassa for his most helpful comments on the manuscript. Registry No. NaPSS, 9080-79-9;dextran, 9004-54-0; pullulan, 9057-02-7. LITERATURE CITED (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (28) (27) (28) (29) (30) (31)
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RECEIVED for review August 31, 1987. Accepted December 18, 1987.
