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Derivation by hydrodynamics of the structural characteristics of adsorbed polymers at liquid-solid interfaces
Derivation by Hydrodynamics of the Structural Characteristics of Adsorbed Polymers at Liquid-Solid Interfaces E M I L E P E F F E R K O R N , P H I L I P P E D E J A R D I N , AND R A P H A I ~ L V A R O Q U I Centre de Recherches sur les MacromolOcules, C.N.R.S., 6, rue Boussingauh, 67083 Strasbourg Cedex, France
Received March 21, 1977; accepted June 16, 1977 The adsorption of a polyacid, the alternated copolymer of maleic acid and ethylvinyl ether, from aqueous NaCI electrolyte solutions onto cellulose ester surfaces was studied as a function of molecular weight and solvent power by increasing the electrolyte concentration. The thickness of the adsorbed layer was derived by measuring the volume flow rate of solvent through narrow-pore cellulose ester filters coated with the adsorbed molecules. The apparent hydrodynamic thickness was interpreted on the basis of the characteristic layer width of an exponential distribution of beads in polymeric loops. We discuss the linear relation between the layer thickness and the intrinsic viscosity of the polymer solution in connection with previous published theories for interacting chains in contact with an adsorbing surface. We show that the solvent power most profoundly affects the layer thickness and we establish a well-defined correlation between the Flory polymer-solvent interaction parameter and the average layer thickness. INTRODUCTION The parameters influencing the adsorption isotherms of polymers adsorbed from liquids onto a solid substrate have been extensively studied by rather simple means on a variety of s o l v e n t - p o l y m e r and solid substrates. On the other hand, only a few techniques are available for determining the conformation of the polymer chains in the interface. Ellipsometry, which is based on the reflection of polarized light from a polymeric film covering the adsorbent surface, has been used mostly for metallic adsorbents or mirror surfaces that reflect light (1). The h y d r o d y n a m i c technique, founded on the measurement of the reduction of volume flow in narrow capillaries coated with the polymer, is particularly attractive owing to its simplicity (2, 3). Moreover, the study of the permeability of porous media in relation to polyelectrolyte adsorption p h e n o m e n a has recently b e c o m e popular in connection with, for example, oil r e c o v e r y (4) and stabilization of c01loidal soils (5); also in living systems,
flows are occurring through pores that may be open or closed by a mechanism intimately dependent on structural changes of adsorbed polymers (6). The present work relates and discusses, in connection with a theoretical study (7), some experimental results of volume flow reduction caused by a polyacid adsorbed from water solutions onto cellulose ester solid surfaces. In a preceding paper, the relationship between the h y d r o d y n a m i c thickness of adsorbed polymers and the structural features of those polymers was established theoretically for the case where the m o n o m e r concentration in the layer decreases exponentially (7). We show here that this model offers a straightforward explanation of some fundamental experimental facts. EXPERIMENTAL METHODS M a t e r i a l s . The anionic polyelectrolyte was obtained by copolymerizing maleic anhydride with ethylvinyl ether. Radical polymerization yields a 1-1 alternated co-
353 0021-9797/78/0632-0353502.00/0 Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
354
PEFFERKORN, DEJARDIN AND VAROQUI
polymer of average molecular weight 3 x 105. The polymer was hydrolyzed by shaking the anhydride with water for several days to obtain a polymer of the following structure: -[-- C H - - C H - - C H 2 - - C H - - ] , ,
[
I
I
COOH COOH
[1]
OC2H5
The polymer was then fractionated with ½ tetrahydrofurane and ½ hexanol solvent mixture and cyclohexane as the precipitant. Each of the resulting fractions was characterized with light scattering and viscosity techniques in 0.5 M NaC1 aqueous solutions at pH 2.9. Light scattering was measured after dialysis against pure electrolyte solutions, in agreement with the theory for multicomponent systems (8). NaC1 of analytical grade was used throughout as well as deionized water of 18 Mf~cm -1 purified by a 0.10 /zm Millipore filter device (Millipore Super-Quality). Table I gives the intrinsic viscosity ['0] for five molecular weights and for the different electrolyte concentrations m (expressed in molarities) of the aqueous solutions. Porous adsorbing media. The porous media used as adsorbent were commercially available Millipore filters (Millipore cellulose acetate EG) with the following characteristics given by the manufacturer: pore radius, 10-5 cm; film thickness, 1.251.50 × 10-2 cm; specific surface, 30 m2/g; porosity, 71%. This material has, accordTABLE I [7] (ml g-i) Ma. x lOS:
m
50
80
160
265
335
0.00 0.01 0.03 0.07 0.10 0.30 0.50
8.15 7.58 7.20 7.00 6.77 6.35 6.30
12.00 10.84 10.38 10.09 9.82 8.73 7.22
25.00 19.50 16.75 15.17 14.45 12.30 11.i0
50.50 31.09 25.82 22.36 20.91 16.67 14.49
62.20 36.67 29.33 25.09 24.09 19.20 16.20
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
ing to the manufacturer, a narrow poresize distribution with a sharp cutoff at the maximal pore radius and as shown previously (2) is more h o m o g e n e o u s than the Pyrex glass sintered filter disk used by Rowland et al. The largest viscosity figure in Table I is 62.20; from the F l o r y - F o x formula [Ref. (9), Chap. 14] one derives an average molecular dimension (root-mean-square distance of segments from the center of the molecule) of 183 A, which is still small compared to the pore diameter of 2000 A. Weak polyacids are probably adsorbed onto cellulose esters through H-bond formation of carboxyl groups with esters or residual hydroxyl groups, though binding by hydrophobic interaction cannot a priori be excluded. However, the exact mechanism of attachment is only of borderline interest in the context of this work. Volume flow rate measurements. The cellulose ester filters were first soaked in pure water under vacuum to remove adsorbed air bubbles. Then the water-soaked filter was placed in a closed vessel containing 100 ml aqueous polymer solution at a concentration of 1.4 x 10 a g/ml. The vessel was fitted with a glass and counter electrode and the pH was monitored by means of an automatic titrator (Tacussel U-5); slight agitation was provided to accelerate the adsorption rate by faster diffusion of the polymer into the filter. After a predetermined time the filter was removed and washed with pure solvent, and the volume flow was measured according to a previously reported experimental procedure (3). All measurements were done in triplicate. The volume flow Jv ° and Jr, per unit time before and after equilibration with the polymer solution, respectively, are relative to the flow rate of pure solvent through the adsorbent. This procedure is legitimate, as no polymer was desorbed in the presence of pure solvent during a period much longer than the time of flow measurement. It is generally recognized that
POLYMER STRUCTURE AT LIQUID-SOLID INTERFACES
x.
.x----l/
X--~x
o--//
o o
o
=
u
u
355
x - -
09
0.8 o-
°
o
0 0.7
o
u
~
//
u
O.b
, 100
I
I 200
ffl
300
I 0
, 200
100
odsor'ptl o n
i,~ i: (h)
des or'pt io n
FIG. 1. Volume flow ratio against time: Jr~J,, ° against adsorption time for left abscissa scale; Jr~Jr ° for variable immersion times into pure solvent after adsorption equilibria was reached for right abscissa scale; Mw: 115,000 (x); 265,000 (O); 385,000 (D).
adsorption against time which was reached within a few hours irrespective of molecular weight. The right part of Fig. 1 corresponds to the following experiments: after adsorption was complete, the filter was washed and immersed into pure solvent and the flow rate Jv of pure solvent was recorded against the immersion time. Since there was no apparent change in Jr~Jr ° even after immersion for about 1 week, the rate
the kinetics of adsorption and desorption differs with smooth and porous adsorbents (10). However, regardless of the nature of the adsorbent, desorption usually was found to proceed very slowly when the polymer layer was in contact with the solvent used for adsorption. As illustration, we have plotted in Fig. 1 the variation of j j j o as a function of adsorption time. The left part of the Fig. 1 shows a plateau value of ~J_~v J~
X
X
,5
o
X
X
X
0.8
0
~
0.7
0.6
0
0
¢'~
n
D
~,
o
l
I
I
I
I
2
4
6
8
10
),
/',Px I0 ~ (bor)
FIG. 2. Volume flow ratio against applied pressure; Mw: 115,000 (×); 265,000 (O); 385,000 (D). Journal of Colloid and Interface Science, Vol.63, No. 2, February 1978
356
PEFFERKORN, DEJARDIN AND VAROQUI where the bars indicate average values. For small LH/R, Eq. [3] reduces to:
of desorption in the presence Of pure solvent must be exceedingly slow. Changes in pressure A p ranging from 0 to 10-2 bars did not affect the ratio j j j o . As reported in Fig. 2, J~ was proportional to A p and most o f the results reported here were obtained at A P = 10-2 bars (11). The h y d r o d y n a m i c thickness LH of the adsorbed polymer is expressed according to Poiseuille's law in terms of the volume flow ratio and the pore radius R:
LH = R[1 - (j,,/j,o)o.25].
L.
Jv
[2]
(g --LH)4 R--7
,
_
4
1 -
_
R3
.
~,
[3]
A x Io8 ( 9 / c ~ )
P /
/
10 // /
/
/
/
/
//
/
/
/
/
/
/
0.9
O.a
(37
0.6
0.1
0.2
[4]
Thus replacing the true value of R4/R 3 by a fairly close value amounts to multiplying Ln by a constant factor, which is of no importance when the relative variation of LH are discussed. As the pore-size distribution of our filters is rather narrow, we did not judge it necessary to develop corrections on account of the heterogeneity of pore size. The amount of polymer adsorbed per unit pore surfaces was determined by potentiometric a c i d - b a s e titration of C O O H groups using 10 -2 N N a O H . To do so, the filters were taken out of the polymer solution after equilibration time and washed with solvent to remove the nonadsorbed polymer, and the amount of adsorbed polymer per unit surface was then
The nonuniformity o f pore size need not trouble us here, for we shall only be concerned in our discussion with the relative variation of L , . Instead of Eq. [2] let us write:
Jv ° -
= -
0,3
0.4
0.5
J
o.s
FIG. 3. Adsorption amount (g cm-2) as a function of the molarity of added NaC1 electrolyte for Mw 160,000 on the left ordinate scale; Mark-Houwink exponent fl of Eq. [5] as a function of m on the right ordinate scale. Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1 9 7 8
357
P O L Y M E R S T R U C T U R E AT L I Q U I D - S O L I D I N T E R F A C E S T A B L E II /f/w x 103 A x l0 s (g/cm 2)
50 6.8
80 8.8
160 10.0
265 8.8
335 9.2
derived by titrating the residue and dividing this amount by the known total filter area. RESULTS
We report on the amount of polymer adsorbed and on hydrodynamic measurements relative to the polymer in NaC1 aqueous solutions at its low natural pH and for an equilibrium concentration of 1.4 x 10-3 g/ml.
1. Adsorption as a function of solvent power and molecular weight. Figure 3 shows the adsorption amount (g/cm 2) as a function of the molarity m of added NaC1 electrolyte for Mw 160,000. The adsorption increased steadily with m. The salt modifies the thermodynamic affinity of the solvent for the polymer in solution and, therefore, the solvent power with respect to adsorption. To illustrate further this point, we have reported on the right ordinate of Fig. 3, the /3 values of the M a r k - H o u w i n k relationship [5] as a function of m. [7] = KN~
[51
N is the degree of polymerization and /3 was determined from the values of [v/] and Mw of Table I at the given salt concentration shown on the abscissa scale of Fig. 3. /3 may be used as a criterion of the solvent power; it decreases as m increases and for m = 0.54, the solvent is a 0 solvent at 25°C. In Table II the amount adsorbed (g/cm 2) is reported as a function of molecular weight of the polymer. Apart from the lowest molecular weight, the amount adsorbed did not appreciably depart from a constant value. From the indicated values one finds that the partial molecular surface per C O O H group is between 15~3 and 22.5 /~z. Further measurements not reported here showed that
this corresponds to a maximum surface coverage in the plateau region of the adsorption isotherm.
2. Hydrodynamic thickness of the polymer layer. Figure 4 shows the hydrodynamic film thickness of the polyacid adsorbed from 0.5 M NaC1 solutions as a function of [~7]. Under the near 0 conditions the relationship is linear. A similar finding was reported previously (2). Figure 5 illustrates the solvent's effect on LH. Measurements were made as follows: the polyacid was first adsorbed from 0.5 M aqueous polymer solutions, and after adsorption equilibrium was reached, the flow rates of aqueous solutions at different salt molarities were measured. The corresponding LH are reported in Fig. 5 as a function of the molarity m. Each curve corresponds to a given molecular weight of the polymer. In the absence of desorption, the steep variation of LH at low molarities indicates large structural changes of the adsorbed polymers, especially for the higher molecular weights for which LH undergoes a fourto fivefold change. The reversibility of the deformation was tested by recording first LH vS m with ionic strength increasing up to 0.54 m and then recording LH again for decreasing m values. The curves after several forward and backward runs were identical.
LH (h) ~0C
50
// 5
1o
15
[~] e
FIG. 4. Relationship between the hydrodynamic thickness Ln and the intrinsic viscosity ['0]0 for different molecular weights. Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
358
PEFFERKORN, DEJARDIN AND VAROQUI monomers from the capillary wall for R >> b. The set of Eqs. [6] and [7] could be solved and the result for the hydrodynamic thickness LH was shown to have the following form:
soot L~ ~, 400~ 3
0
0
~
200
LH = R × F(b/l; R/b); 12 = ~/fno
100 t÷-*--
0
~,
0.1'
×-0.5 m
0,3
FIG. 5. Hydrodynamic thickness of adsorbed polymer as a function of the molarity m of added NaC1 electrolyte; M.,: 55,000 (+); 80,000 ((3); 160,000 (x); 265,000 (0); 335,000 (A).
DISCUSSION We recently derived the relation between the parameter LH and the structural features of the adsorbed polymers (7) and here we shall briefly recall the main resuits of the theory. After adsorption of polymers, the flow velocity field of the solvent deviates from the original undisturbed Poiseuille flow because of the supplementary friction forces experienced by the monomers of the polymeric loops which protrude from the surface into the solution. Evaluation of the ratio Jr~Jr ° is then related to the problem of finding the velocity v of the liquid everywhere in the pore space; this can be done by solving the modified N a v i e r Stokes equation:
l d(r dr) -r ~ -~r - z(r)fv = -
[grad p[.
[6]
f is a friction coefficient and grad p the pressure differential. The main point is the local density z(r) in space of the monomers, r being the radial distance. A density profile that drops off exponentially from the surface for long chains and not too strong adsorption was found independently by several authors (12-14):
z(r) = no e x p [ - ( R - r)/b].
[7]
no is the bead density at r = R, R is the pore radius, and b is the average distance of Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
[8]
F is a function of the shielding ratio parameter b/l, where l is defined as the Debye shielding length. The exact form of F is given in Ref. (7) (Eqs. [19] to [21]). It turns out that for a large b/l ratio, the resulting curves of LH vs b where almost straight lines (see for example Fig. 2 of Ref. [7]). Marked deviation from a straight line occurred only at very small b and large l values. De Gennes (13) gave a limiting expression Of LH for 2.3 l < b ~< R:
LH = 2b[ln (b/l) + 0.577].
[9]
Because b/l comes in only through a logarithm, LH clearly tends to vary in proportion to the thickness b of the layer. Excluding multilayer adsorption and to the extent that the effective distribution is of the form of Eq. [7], we can discuss our experimental results bearing in mind the important fact that LH is approximately proportional to the distance b. The linear relation between LH and b accounts for the experimental finding of LH proportional to [rt] or N °5. H o e v e (15) showed that the root-mean-square distance of segments from the interface must be proportional to N °~ for 0 solvent conditions. Silberberg (16) using a step function for the distribution and a lattice model has also found a characteristic width proportional to N°'L Stromberg and coworkers (17) found that in a 0 solvent, the average root-mean-square distance o f polymer segments from the interface increases approximately in proportion with N°.L Let us for instance use Eq. [27] o f H o e v e ' s paper which can be written in the slightly different form in connection with his Eq. [29]:
POLYMER STRUCTURE AT LIQUID-SOLID INTERFACES b = 0.25N °'2~ x
[
/312In ~_L (1 + 7 . 0 9 c f l l b )
1
[10]
~ and ~ being, respectively, the volume fraction of polymer in the first surface layer (trains) and in the solution, c a flexibility parameter, and /3~-1 a characteristic length related to [~]0 by (18); /~1--2 __
2(r2}0
.
3N
'
4,(r )o3'2M -1,
[11]
q5 = 2.24 x 1023. Introducing the experimental values K o = 0.26, ~ 1 - 1 = 3.8 A and ¢ = 8.46 x 10 -4, and noting that the linear form of LH vs [~] (Fig. 4) gives in conjunction with Eq. [5] L , ( A ) = 1.95N °.5 - 29,
359
tentative interpretation of the structural changes as a function of solvent power. In the absence of desorption we conjecture that the number of trains and the number of monomers per train are not affected by changing the electrolyte concentration; the number of loops being invariable, the effect of m on LH will, therefore, only modify the conformational state of the loops. Assuming this hypothesis, it is necessary to have at least one approximate theory for b as a function of the solvent power. Adsorption as a function of solvent power was studied in light of the F l o r y - H u g g i n s theory (14, 15) for polymers. We adopt a similar point of view. We have, with Eq. [12] and with the definition of the geometrical expansion factor 3,, the following relationship: LH + 29 3' = b/bo ~
[12]
L m o + 29
,
[14]
where bo and Lmo refer again to 0 solvent values. According to Flory, the value of 3/ LH(A) = 2.3b[ln 1.18 x 103~s must result from a compromise between the repulsion between m o n o m e r s and the x (1 + 1.84cb)] 1/2 - 29. [131 stretching entropy of chains. The former Equation [13] is merely another way of effect gives a free energy contribution per expressing the result of Fig. 5 provided unit area of the form [20]. Eq. [i0] is correct. It is gratifying, however, that the hydrodynamic theory gives the A F = __kT T(r)]2vd r 2 same trend for the variation of L u with b as Eq. [13]. The proportionality of LH with y-2ng oe-2(R-")l~b°dr, [15] [v/] was in earlier investigations explained 2 ~0 by assuming that polymers are only weakly adsorbed with no large change in their A F = k Tvng,obo/4y , [16] conformational topology from the random v is the excluded volume factor. The stretchcoil solution state to the surface state. ing energy per unit surface area is given by But recent theories show conclusively that Flory's formula: adsorption from the solution state results in high-volume fractions in the surfaceA F e l = kTm[a/2(y 2 - 1) - 3 In 3,] [17] contained trains (19). This statement does not conflict with the fact that flow meas- By Eq. [15] it is assumed that the exponenurements yield large apparent layer thick- tial distribution is expanded uniformly by ness values [this point is discussed in y, and m of Eq. [17] is the number of loops per square centimeter surface. Adding Ref. (7)]. We shall now, in connection with the Eqs. [16] and [17] and minimizing the total LH variations shown in Fig. 5, propose a free energy with respect to 3' gives: we obtain:
;)
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
360
PEFFERKORN, DEJARDIN AND VAROQUI
20
15
10
0.04
0.02
[.C~ _ ~,1] N -°'s
Fie. 6. Relationship between the geometric expansion factor "y defined by Eq. [14] and the viscosity expansion coefficient an defined by Eq. [21]; Mw: 55,000 (A); 80,000 (×); 160,000 (O); 265,000 (©); 335,000 (11).
7 a -
7
=
[18]
n~,obov/12m.
The relevant geometrical expansion coefficient a for a free coil can be derived according to the same lines assuming a gaussian distribution of segments in the coil; a is given by the classical expression [Ref. (9), Chap. 12]: a 5 -
a 3 = avN
against ( a , ~ - a n 3 ) N 0.5 for five molecular weights. Within experimental uncertainty, linear relationships were found as expected from inspection of Eq. [18] and ¢'Nx 10~"
[19]'
°'5,
where a is a numerical constant. From Eqs. [18] and [19] one obtains: 73
-
7 =
a-~n~,ob~N-°'5(a
5 -
a3)/12m.
[20]
We have taken a equal to the viscosity expansion coefficient a n defined by: a.~~ = [r/]/[r/]0.
[21]
In Fig. 6, the experimental variation of 7z-7 defined by Eq. [14] is plotted Journal of Colloid and Interface Science,
Vol. 63, No. 2, February 1978
'~
½
.~
Nxl0 ~
FIG. 7. Slope o-N of the straight line thrown through points in Fig. 6 against the degree of polymerization N.
POLYMER STRUCTURE AT L I Q U I D - S O L I D INTERFACES
[ 19], furthermore the slope ovN of the straight lines were proportional to the molecular weight as seen in Fig. 7; therefore since bo ~ N °'5, no,o2/m should be proportional to N °'5.
2. 3. 4.
CONCLUSION
As for free chains, hydrodynamic measurements offer a fruitful and quite simple way to explore the characteristics of thin polymer films at the liquid-solid interface. We have demonstrated the importance of the effect of solvent power on the amount of adsorption and on the configuration of chains in the interface. The solvent power was made continuously variable by increasing the electrolyte concentration, and the polymer-solvent interaction parameter v could be correlated with the expansion of the adsorbed film by choosing an exponential distribution of segments in the interface. The latter distribution, which is the underlying fact of the model, was chosen on the basis of theoretical results. It predicts realistically the large hydrodynamic thickness of adsorbed molecules and, as shown here, the sharp variation in film thickness with solvent power. It may nevertheless be desirable to devise in the future a more direct method to verify Eq. [7] in connection with the number of attachments of a polymer to the surface. ACKNOWLEDGMENTS
5. 6.
7. 8. 9. 10. 11.
12. 13. 14. 15. 16. 17.
We are indebted to Dr. P. Gramain for several discussions. This work was supported through Project 1184, ATP "Physico-chimie des Surfaces" of the Centre National de la Recherche Scientifique Franqaise.
18.
REFERENCES
20.
1.(a) Killmann, E., and Strasser, H. J., Angew. Makrom. Chem. 31, 169 (1973); (b) Gebhard, G., and Killmann, E., Angew. Makrom. Chem. 53, 17 (1976); (c) Stromberg, R. R.,
19.
361
Smith, L. E., and McCrackin, F. L., Syrup. Faraday Soc. 4, 142 (1970). Rowland, F. W. and Eirich, F. R., J. Polym. Sci. 4, 2033 (1966); 4, 2401 (1966). Gramain, P., Makromol. Chem. 176, 1875 (1975). Chauveteau, G., and Kohler, N., "Polymer Flooding: The Essential Elements for Laboratory Evaluation," paper SPE 4745 presented at SPE Symp. on Improved Oil Recovery, Tulsa, Okla., Apr. 22-24, 1974. Fleer, G. J., Koopal, L. K., and Lyklema, T., Kolloid. Z. Z. Polym. 250, 689 (1972). Singer, J., and Tasaki, S., "Biological Membranes, Physical Fact and Function" (D. Chapman, Ed.), p. 347. Academic Press, New York, 1968. Varoqui, R., and Dejardin, P., J. Chem. Phys. 66, 4395 (1977). Eisenberg, H., J. Chem. Phys. 36, 1837 (1962). Flory, J. P., "Principles of Polymer Chemistry." Cornell Univ. Press, Ithaca, New York, 1953. Lipatov, Y. S., and Sergeeva, L. M., "Adsorption of Polymers" Wiley, New York, 1972. Chauveteau, G., and Moulu, J. C. (personal communication) have found a decrease in L~/as a function of the flow rate by increasing the pressure differential. The authors were concerned with the adsorption of partially hydrolyzed polyacrylamides onto various sand beads in relation to the problem of oil recovery. At high shear flows, the chain could conceivably be distorted and loops could even unwind abruptly under very high velocity gradients as predicted by De Gennes [J. Chem. Phys. 60, 5030 (1974)]. Hoeve, C. A. J., J. Chem. Phys. 43, 3007 (1965). De Gennes, P. G., Rep. Progr. Phys. 32, 187 (1969). Rubin, R. J., J. Chem. Phys. 43, 2392 (1965). Hoeve, C. A. J., J. Chem. Phys. 44, 1505(1966). Silberberg, A., J. Chem. Phys. 48, 2835 (1968). Stromberg, R. R., Tutas, D. J., and Passaglia E., J. Chem. Phys. 69, 3955 (1965). c being of the order of 10-z or less [see for example Ref. (19)], we have neglected 2.61 C against unity in computing the fraction p of segments in the first surface layer. Hoeve, C. A. J., "On the Theory of Polymer Adsorption at One Interface," in press. Our calculation proceeds via Eq. [7] which was established for adsorption onto a plane surface. The largest polymer dimensions is of the order of 200 A for a pore radius of 1,000 A and by deriving Eq. [15] we ignore the effect of the curvature of the pores.
Journal of Colloidand InterfaceScience, Vol. 63, No. 2, February 1978
Received March 21, 1977; accepted June 16, 1977 The adsorption of a polyacid, the alternated copolymer of maleic acid and ethylvinyl ether, from aqueous NaCI electrolyte solutions onto cellulose ester surfaces was studied as a function of molecular weight and solvent power by increasing the electrolyte concentration. The thickness of the adsorbed layer was derived by measuring the volume flow rate of solvent through narrow-pore cellulose ester filters coated with the adsorbed molecules. The apparent hydrodynamic thickness was interpreted on the basis of the characteristic layer width of an exponential distribution of beads in polymeric loops. We discuss the linear relation between the layer thickness and the intrinsic viscosity of the polymer solution in connection with previous published theories for interacting chains in contact with an adsorbing surface. We show that the solvent power most profoundly affects the layer thickness and we establish a well-defined correlation between the Flory polymer-solvent interaction parameter and the average layer thickness. INTRODUCTION The parameters influencing the adsorption isotherms of polymers adsorbed from liquids onto a solid substrate have been extensively studied by rather simple means on a variety of s o l v e n t - p o l y m e r and solid substrates. On the other hand, only a few techniques are available for determining the conformation of the polymer chains in the interface. Ellipsometry, which is based on the reflection of polarized light from a polymeric film covering the adsorbent surface, has been used mostly for metallic adsorbents or mirror surfaces that reflect light (1). The h y d r o d y n a m i c technique, founded on the measurement of the reduction of volume flow in narrow capillaries coated with the polymer, is particularly attractive owing to its simplicity (2, 3). Moreover, the study of the permeability of porous media in relation to polyelectrolyte adsorption p h e n o m e n a has recently b e c o m e popular in connection with, for example, oil r e c o v e r y (4) and stabilization of c01loidal soils (5); also in living systems,
flows are occurring through pores that may be open or closed by a mechanism intimately dependent on structural changes of adsorbed polymers (6). The present work relates and discusses, in connection with a theoretical study (7), some experimental results of volume flow reduction caused by a polyacid adsorbed from water solutions onto cellulose ester solid surfaces. In a preceding paper, the relationship between the h y d r o d y n a m i c thickness of adsorbed polymers and the structural features of those polymers was established theoretically for the case where the m o n o m e r concentration in the layer decreases exponentially (7). We show here that this model offers a straightforward explanation of some fundamental experimental facts. EXPERIMENTAL METHODS M a t e r i a l s . The anionic polyelectrolyte was obtained by copolymerizing maleic anhydride with ethylvinyl ether. Radical polymerization yields a 1-1 alternated co-
353 0021-9797/78/0632-0353502.00/0 Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.
354
PEFFERKORN, DEJARDIN AND VAROQUI
polymer of average molecular weight 3 x 105. The polymer was hydrolyzed by shaking the anhydride with water for several days to obtain a polymer of the following structure: -[-- C H - - C H - - C H 2 - - C H - - ] , ,
[
I
I
COOH COOH
[1]
OC2H5
The polymer was then fractionated with ½ tetrahydrofurane and ½ hexanol solvent mixture and cyclohexane as the precipitant. Each of the resulting fractions was characterized with light scattering and viscosity techniques in 0.5 M NaC1 aqueous solutions at pH 2.9. Light scattering was measured after dialysis against pure electrolyte solutions, in agreement with the theory for multicomponent systems (8). NaC1 of analytical grade was used throughout as well as deionized water of 18 Mf~cm -1 purified by a 0.10 /zm Millipore filter device (Millipore Super-Quality). Table I gives the intrinsic viscosity ['0] for five molecular weights and for the different electrolyte concentrations m (expressed in molarities) of the aqueous solutions. Porous adsorbing media. The porous media used as adsorbent were commercially available Millipore filters (Millipore cellulose acetate EG) with the following characteristics given by the manufacturer: pore radius, 10-5 cm; film thickness, 1.251.50 × 10-2 cm; specific surface, 30 m2/g; porosity, 71%. This material has, accordTABLE I [7] (ml g-i) Ma. x lOS:
m
50
80
160
265
335
0.00 0.01 0.03 0.07 0.10 0.30 0.50
8.15 7.58 7.20 7.00 6.77 6.35 6.30
12.00 10.84 10.38 10.09 9.82 8.73 7.22
25.00 19.50 16.75 15.17 14.45 12.30 11.i0
50.50 31.09 25.82 22.36 20.91 16.67 14.49
62.20 36.67 29.33 25.09 24.09 19.20 16.20
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
ing to the manufacturer, a narrow poresize distribution with a sharp cutoff at the maximal pore radius and as shown previously (2) is more h o m o g e n e o u s than the Pyrex glass sintered filter disk used by Rowland et al. The largest viscosity figure in Table I is 62.20; from the F l o r y - F o x formula [Ref. (9), Chap. 14] one derives an average molecular dimension (root-mean-square distance of segments from the center of the molecule) of 183 A, which is still small compared to the pore diameter of 2000 A. Weak polyacids are probably adsorbed onto cellulose esters through H-bond formation of carboxyl groups with esters or residual hydroxyl groups, though binding by hydrophobic interaction cannot a priori be excluded. However, the exact mechanism of attachment is only of borderline interest in the context of this work. Volume flow rate measurements. The cellulose ester filters were first soaked in pure water under vacuum to remove adsorbed air bubbles. Then the water-soaked filter was placed in a closed vessel containing 100 ml aqueous polymer solution at a concentration of 1.4 x 10 a g/ml. The vessel was fitted with a glass and counter electrode and the pH was monitored by means of an automatic titrator (Tacussel U-5); slight agitation was provided to accelerate the adsorption rate by faster diffusion of the polymer into the filter. After a predetermined time the filter was removed and washed with pure solvent, and the volume flow was measured according to a previously reported experimental procedure (3). All measurements were done in triplicate. The volume flow Jv ° and Jr, per unit time before and after equilibration with the polymer solution, respectively, are relative to the flow rate of pure solvent through the adsorbent. This procedure is legitimate, as no polymer was desorbed in the presence of pure solvent during a period much longer than the time of flow measurement. It is generally recognized that
POLYMER STRUCTURE AT LIQUID-SOLID INTERFACES
x.
.x----l/
X--~x
o--//
o o
o
=
u
u
355
x - -
09
0.8 o-
°
o
0 0.7
o
u
~
//
u
O.b
, 100
I
I 200
ffl
300
I 0
, 200
100
odsor'ptl o n
i,~ i: (h)
des or'pt io n
FIG. 1. Volume flow ratio against time: Jr~J,, ° against adsorption time for left abscissa scale; Jr~Jr ° for variable immersion times into pure solvent after adsorption equilibria was reached for right abscissa scale; Mw: 115,000 (x); 265,000 (O); 385,000 (D).
adsorption against time which was reached within a few hours irrespective of molecular weight. The right part of Fig. 1 corresponds to the following experiments: after adsorption was complete, the filter was washed and immersed into pure solvent and the flow rate Jv of pure solvent was recorded against the immersion time. Since there was no apparent change in Jr~Jr ° even after immersion for about 1 week, the rate
the kinetics of adsorption and desorption differs with smooth and porous adsorbents (10). However, regardless of the nature of the adsorbent, desorption usually was found to proceed very slowly when the polymer layer was in contact with the solvent used for adsorption. As illustration, we have plotted in Fig. 1 the variation of j j j o as a function of adsorption time. The left part of the Fig. 1 shows a plateau value of ~J_~v J~
X
X
,5
o
X
X
X
0.8
0
~
0.7
0.6
0
0
¢'~
n
D
~,
o
l
I
I
I
I
2
4
6
8
10
),
/',Px I0 ~ (bor)
FIG. 2. Volume flow ratio against applied pressure; Mw: 115,000 (×); 265,000 (O); 385,000 (D). Journal of Colloid and Interface Science, Vol.63, No. 2, February 1978
356
PEFFERKORN, DEJARDIN AND VAROQUI where the bars indicate average values. For small LH/R, Eq. [3] reduces to:
of desorption in the presence Of pure solvent must be exceedingly slow. Changes in pressure A p ranging from 0 to 10-2 bars did not affect the ratio j j j o . As reported in Fig. 2, J~ was proportional to A p and most o f the results reported here were obtained at A P = 10-2 bars (11). The h y d r o d y n a m i c thickness LH of the adsorbed polymer is expressed according to Poiseuille's law in terms of the volume flow ratio and the pore radius R:
LH = R[1 - (j,,/j,o)o.25].
L.
Jv
[2]
(g --LH)4 R--7
,
_
4
1 -
_
R3
.
~,
[3]
A x Io8 ( 9 / c ~ )
P /
/
10 // /
/
/
/
/
//
/
/
/
/
/
/
0.9
O.a
(37
0.6
0.1
0.2
[4]
Thus replacing the true value of R4/R 3 by a fairly close value amounts to multiplying Ln by a constant factor, which is of no importance when the relative variation of LH are discussed. As the pore-size distribution of our filters is rather narrow, we did not judge it necessary to develop corrections on account of the heterogeneity of pore size. The amount of polymer adsorbed per unit pore surfaces was determined by potentiometric a c i d - b a s e titration of C O O H groups using 10 -2 N N a O H . To do so, the filters were taken out of the polymer solution after equilibration time and washed with solvent to remove the nonadsorbed polymer, and the amount of adsorbed polymer per unit surface was then
The nonuniformity o f pore size need not trouble us here, for we shall only be concerned in our discussion with the relative variation of L , . Instead of Eq. [2] let us write:
Jv ° -
= -
0,3
0.4
0.5
J
o.s
FIG. 3. Adsorption amount (g cm-2) as a function of the molarity of added NaC1 electrolyte for Mw 160,000 on the left ordinate scale; Mark-Houwink exponent fl of Eq. [5] as a function of m on the right ordinate scale. Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1 9 7 8
357
P O L Y M E R S T R U C T U R E AT L I Q U I D - S O L I D I N T E R F A C E S T A B L E II /f/w x 103 A x l0 s (g/cm 2)
50 6.8
80 8.8
160 10.0
265 8.8
335 9.2
derived by titrating the residue and dividing this amount by the known total filter area. RESULTS
We report on the amount of polymer adsorbed and on hydrodynamic measurements relative to the polymer in NaC1 aqueous solutions at its low natural pH and for an equilibrium concentration of 1.4 x 10-3 g/ml.
1. Adsorption as a function of solvent power and molecular weight. Figure 3 shows the adsorption amount (g/cm 2) as a function of the molarity m of added NaC1 electrolyte for Mw 160,000. The adsorption increased steadily with m. The salt modifies the thermodynamic affinity of the solvent for the polymer in solution and, therefore, the solvent power with respect to adsorption. To illustrate further this point, we have reported on the right ordinate of Fig. 3, the /3 values of the M a r k - H o u w i n k relationship [5] as a function of m. [7] = KN~
[51
N is the degree of polymerization and /3 was determined from the values of [v/] and Mw of Table I at the given salt concentration shown on the abscissa scale of Fig. 3. /3 may be used as a criterion of the solvent power; it decreases as m increases and for m = 0.54, the solvent is a 0 solvent at 25°C. In Table II the amount adsorbed (g/cm 2) is reported as a function of molecular weight of the polymer. Apart from the lowest molecular weight, the amount adsorbed did not appreciably depart from a constant value. From the indicated values one finds that the partial molecular surface per C O O H group is between 15~3 and 22.5 /~z. Further measurements not reported here showed that
this corresponds to a maximum surface coverage in the plateau region of the adsorption isotherm.
2. Hydrodynamic thickness of the polymer layer. Figure 4 shows the hydrodynamic film thickness of the polyacid adsorbed from 0.5 M NaC1 solutions as a function of [~7]. Under the near 0 conditions the relationship is linear. A similar finding was reported previously (2). Figure 5 illustrates the solvent's effect on LH. Measurements were made as follows: the polyacid was first adsorbed from 0.5 M aqueous polymer solutions, and after adsorption equilibrium was reached, the flow rates of aqueous solutions at different salt molarities were measured. The corresponding LH are reported in Fig. 5 as a function of the molarity m. Each curve corresponds to a given molecular weight of the polymer. In the absence of desorption, the steep variation of LH at low molarities indicates large structural changes of the adsorbed polymers, especially for the higher molecular weights for which LH undergoes a fourto fivefold change. The reversibility of the deformation was tested by recording first LH vS m with ionic strength increasing up to 0.54 m and then recording LH again for decreasing m values. The curves after several forward and backward runs were identical.
LH (h) ~0C
50
// 5
1o
15
[~] e
FIG. 4. Relationship between the hydrodynamic thickness Ln and the intrinsic viscosity ['0]0 for different molecular weights. Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
358
PEFFERKORN, DEJARDIN AND VAROQUI monomers from the capillary wall for R >> b. The set of Eqs. [6] and [7] could be solved and the result for the hydrodynamic thickness LH was shown to have the following form:
soot L~ ~, 400~ 3
0
0
~
200
LH = R × F(b/l; R/b); 12 = ~/fno
100 t÷-*--
0
~,
0.1'
×-0.5 m
0,3
FIG. 5. Hydrodynamic thickness of adsorbed polymer as a function of the molarity m of added NaC1 electrolyte; M.,: 55,000 (+); 80,000 ((3); 160,000 (x); 265,000 (0); 335,000 (A).
DISCUSSION We recently derived the relation between the parameter LH and the structural features of the adsorbed polymers (7) and here we shall briefly recall the main resuits of the theory. After adsorption of polymers, the flow velocity field of the solvent deviates from the original undisturbed Poiseuille flow because of the supplementary friction forces experienced by the monomers of the polymeric loops which protrude from the surface into the solution. Evaluation of the ratio Jr~Jr ° is then related to the problem of finding the velocity v of the liquid everywhere in the pore space; this can be done by solving the modified N a v i e r Stokes equation:
l d(r dr) -r ~ -~r - z(r)fv = -
[grad p[.
[6]
f is a friction coefficient and grad p the pressure differential. The main point is the local density z(r) in space of the monomers, r being the radial distance. A density profile that drops off exponentially from the surface for long chains and not too strong adsorption was found independently by several authors (12-14):
z(r) = no e x p [ - ( R - r)/b].
[7]
no is the bead density at r = R, R is the pore radius, and b is the average distance of Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
[8]
F is a function of the shielding ratio parameter b/l, where l is defined as the Debye shielding length. The exact form of F is given in Ref. (7) (Eqs. [19] to [21]). It turns out that for a large b/l ratio, the resulting curves of LH vs b where almost straight lines (see for example Fig. 2 of Ref. [7]). Marked deviation from a straight line occurred only at very small b and large l values. De Gennes (13) gave a limiting expression Of LH for 2.3 l < b ~< R:
LH = 2b[ln (b/l) + 0.577].
[9]
Because b/l comes in only through a logarithm, LH clearly tends to vary in proportion to the thickness b of the layer. Excluding multilayer adsorption and to the extent that the effective distribution is of the form of Eq. [7], we can discuss our experimental results bearing in mind the important fact that LH is approximately proportional to the distance b. The linear relation between LH and b accounts for the experimental finding of LH proportional to [rt] or N °5. H o e v e (15) showed that the root-mean-square distance of segments from the interface must be proportional to N °~ for 0 solvent conditions. Silberberg (16) using a step function for the distribution and a lattice model has also found a characteristic width proportional to N°'L Stromberg and coworkers (17) found that in a 0 solvent, the average root-mean-square distance o f polymer segments from the interface increases approximately in proportion with N°.L Let us for instance use Eq. [27] o f H o e v e ' s paper which can be written in the slightly different form in connection with his Eq. [29]:
POLYMER STRUCTURE AT LIQUID-SOLID INTERFACES b = 0.25N °'2~ x
[
/312In ~_L (1 + 7 . 0 9 c f l l b )
1
[10]
~ and ~ being, respectively, the volume fraction of polymer in the first surface layer (trains) and in the solution, c a flexibility parameter, and /3~-1 a characteristic length related to [~]0 by (18); /~1--2 __
2(r2}0
.
3N
'
4,(r )o3'2M -1,
[11]
q5 = 2.24 x 1023. Introducing the experimental values K o = 0.26, ~ 1 - 1 = 3.8 A and ¢ = 8.46 x 10 -4, and noting that the linear form of LH vs [~] (Fig. 4) gives in conjunction with Eq. [5] L , ( A ) = 1.95N °.5 - 29,
359
tentative interpretation of the structural changes as a function of solvent power. In the absence of desorption we conjecture that the number of trains and the number of monomers per train are not affected by changing the electrolyte concentration; the number of loops being invariable, the effect of m on LH will, therefore, only modify the conformational state of the loops. Assuming this hypothesis, it is necessary to have at least one approximate theory for b as a function of the solvent power. Adsorption as a function of solvent power was studied in light of the F l o r y - H u g g i n s theory (14, 15) for polymers. We adopt a similar point of view. We have, with Eq. [12] and with the definition of the geometrical expansion factor 3,, the following relationship: LH + 29 3' = b/bo ~
[12]
L m o + 29
,
[14]
where bo and Lmo refer again to 0 solvent values. According to Flory, the value of 3/ LH(A) = 2.3b[ln 1.18 x 103~s must result from a compromise between the repulsion between m o n o m e r s and the x (1 + 1.84cb)] 1/2 - 29. [131 stretching entropy of chains. The former Equation [13] is merely another way of effect gives a free energy contribution per expressing the result of Fig. 5 provided unit area of the form [20]. Eq. [i0] is correct. It is gratifying, however, that the hydrodynamic theory gives the A F = __kT T(r)]2vd r 2 same trend for the variation of L u with b as Eq. [13]. The proportionality of LH with y-2ng oe-2(R-")l~b°dr, [15] [v/] was in earlier investigations explained 2 ~0 by assuming that polymers are only weakly adsorbed with no large change in their A F = k Tvng,obo/4y , [16] conformational topology from the random v is the excluded volume factor. The stretchcoil solution state to the surface state. ing energy per unit surface area is given by But recent theories show conclusively that Flory's formula: adsorption from the solution state results in high-volume fractions in the surfaceA F e l = kTm[a/2(y 2 - 1) - 3 In 3,] [17] contained trains (19). This statement does not conflict with the fact that flow meas- By Eq. [15] it is assumed that the exponenurements yield large apparent layer thick- tial distribution is expanded uniformly by ness values [this point is discussed in y, and m of Eq. [17] is the number of loops per square centimeter surface. Adding Ref. (7)]. We shall now, in connection with the Eqs. [16] and [17] and minimizing the total LH variations shown in Fig. 5, propose a free energy with respect to 3' gives: we obtain:
;)
Journal of Colloid and Interface Science, Vol. 63, No. 2, February 1978
360
PEFFERKORN, DEJARDIN AND VAROQUI
20
15
10
0.04
0.02
[.C~ _ ~,1] N -°'s
Fie. 6. Relationship between the geometric expansion factor "y defined by Eq. [14] and the viscosity expansion coefficient an defined by Eq. [21]; Mw: 55,000 (A); 80,000 (×); 160,000 (O); 265,000 (©); 335,000 (11).
7 a -
7
=
[18]
n~,obov/12m.
The relevant geometrical expansion coefficient a for a free coil can be derived according to the same lines assuming a gaussian distribution of segments in the coil; a is given by the classical expression [Ref. (9), Chap. 12]: a 5 -
a 3 = avN
against ( a , ~ - a n 3 ) N 0.5 for five molecular weights. Within experimental uncertainty, linear relationships were found as expected from inspection of Eq. [18] and ¢'Nx 10~"
[19]'
°'5,
where a is a numerical constant. From Eqs. [18] and [19] one obtains: 73
-
7 =
a-~n~,ob~N-°'5(a
5 -
a3)/12m.
[20]
We have taken a equal to the viscosity expansion coefficient a n defined by: a.~~ = [r/]/[r/]0.
[21]
In Fig. 6, the experimental variation of 7z-7 defined by Eq. [14] is plotted Journal of Colloid and Interface Science,
Vol. 63, No. 2, February 1978
'~
½
.~
Nxl0 ~
FIG. 7. Slope o-N of the straight line thrown through points in Fig. 6 against the degree of polymerization N.
POLYMER STRUCTURE AT L I Q U I D - S O L I D INTERFACES
[ 19], furthermore the slope ovN of the straight lines were proportional to the molecular weight as seen in Fig. 7; therefore since bo ~ N °'5, no,o2/m should be proportional to N °'5.
2. 3. 4.
CONCLUSION
As for free chains, hydrodynamic measurements offer a fruitful and quite simple way to explore the characteristics of thin polymer films at the liquid-solid interface. We have demonstrated the importance of the effect of solvent power on the amount of adsorption and on the configuration of chains in the interface. The solvent power was made continuously variable by increasing the electrolyte concentration, and the polymer-solvent interaction parameter v could be correlated with the expansion of the adsorbed film by choosing an exponential distribution of segments in the interface. The latter distribution, which is the underlying fact of the model, was chosen on the basis of theoretical results. It predicts realistically the large hydrodynamic thickness of adsorbed molecules and, as shown here, the sharp variation in film thickness with solvent power. It may nevertheless be desirable to devise in the future a more direct method to verify Eq. [7] in connection with the number of attachments of a polymer to the surface. ACKNOWLEDGMENTS
5. 6.
7. 8. 9. 10. 11.
12. 13. 14. 15. 16. 17.
We are indebted to Dr. P. Gramain for several discussions. This work was supported through Project 1184, ATP "Physico-chimie des Surfaces" of the Centre National de la Recherche Scientifique Franqaise.
18.
REFERENCES
20.
1.(a) Killmann, E., and Strasser, H. J., Angew. Makrom. Chem. 31, 169 (1973); (b) Gebhard, G., and Killmann, E., Angew. Makrom. Chem. 53, 17 (1976); (c) Stromberg, R. R.,
19.
361
Smith, L. E., and McCrackin, F. L., Syrup. Faraday Soc. 4, 142 (1970). Rowland, F. W. and Eirich, F. R., J. Polym. Sci. 4, 2033 (1966); 4, 2401 (1966). Gramain, P., Makromol. Chem. 176, 1875 (1975). Chauveteau, G., and Kohler, N., "Polymer Flooding: The Essential Elements for Laboratory Evaluation," paper SPE 4745 presented at SPE Symp. on Improved Oil Recovery, Tulsa, Okla., Apr. 22-24, 1974. Fleer, G. J., Koopal, L. K., and Lyklema, T., Kolloid. Z. Z. Polym. 250, 689 (1972). Singer, J., and Tasaki, S., "Biological Membranes, Physical Fact and Function" (D. Chapman, Ed.), p. 347. Academic Press, New York, 1968. Varoqui, R., and Dejardin, P., J. Chem. Phys. 66, 4395 (1977). Eisenberg, H., J. Chem. Phys. 36, 1837 (1962). Flory, J. P., "Principles of Polymer Chemistry." Cornell Univ. Press, Ithaca, New York, 1953. Lipatov, Y. S., and Sergeeva, L. M., "Adsorption of Polymers" Wiley, New York, 1972. Chauveteau, G., and Moulu, J. C. (personal communication) have found a decrease in L~/as a function of the flow rate by increasing the pressure differential. The authors were concerned with the adsorption of partially hydrolyzed polyacrylamides onto various sand beads in relation to the problem of oil recovery. At high shear flows, the chain could conceivably be distorted and loops could even unwind abruptly under very high velocity gradients as predicted by De Gennes [J. Chem. Phys. 60, 5030 (1974)]. Hoeve, C. A. J., J. Chem. Phys. 43, 3007 (1965). De Gennes, P. G., Rep. Progr. Phys. 32, 187 (1969). Rubin, R. J., J. Chem. Phys. 43, 2392 (1965). Hoeve, C. A. J., J. Chem. Phys. 44, 1505(1966). Silberberg, A., J. Chem. Phys. 48, 2835 (1968). Stromberg, R. R., Tutas, D. J., and Passaglia E., J. Chem. Phys. 69, 3955 (1965). c being of the order of 10-z or less [see for example Ref. (19)], we have neglected 2.61 C against unity in computing the fraction p of segments in the first surface layer. Hoeve, C. A. J., "On the Theory of Polymer Adsorption at One Interface," in press. Our calculation proceeds via Eq. [7] which was established for adsorption onto a plane surface. The largest polymer dimensions is of the order of 200 A for a pore radius of 1,000 A and by deriving Eq. [15] we ignore the effect of the curvature of the pores.
Journal of Colloidand InterfaceScience, Vol. 63, No. 2, February 1978