1040
Langmuir 1996, 12, 1040-1046
Thermal Expansion and Contraction of Adsorbed Diblock Copolymers near Θ Conditions Richard M. Webber,† Catharina C. van der Linden,‡ and John L. Anderson*,§ Lubrizol Corporation, 29400 Lakeland Boulevard, Wickliffe, Ohio 44092, Fysische en Kolloidchemie, Landbouwuniversiteit, Wageningen 6703 HB, Netherlands, and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received July 18, 1995. In Final Form: November 13, 1995X The hydrodynamic thickness of adsorbed layers of poly(2-vinylpyridine)/polystyrene (PVP/PS) diblock copolymers was measured as a function of temperature near the Θ point of PS in cyclohexane. Adsorption occurred from a selective solvent, toluene, with the PVP block anchoring the PS chains to the pores of mica membranes. The ratio of block sizes, NPVP/NPS, was in the range 0.03-0.16. The hydrodynamic thickness (LH) was determined by measuring the flow rate of cyclohexane at the specified temperature as a function of applied pressure difference across the membrane. The variation of LH with temperature was linear and reversible. The expansion coefficient of the layers was defined as R ) LH(T)/LH(Θ). The experimental values of dR/dT were of order 10-2 (°C)-1, which is almost an order of magnitude greater than the temperature derivative of the expansion factor of PS chains in cyclohexane solution above Θ. Predictions of LH based on Scheutjens-Fleer theory for terminally attached chains and a Brinkman model for viscous flow through polymer chains were made using literature-derived values for the fundamental system properties. The theory overpredicts LH(Θ) and underpredicts dR/dT; one possible explanation for these discrepancies is a temperature dependent, reversible adsorption of PS segments to areas of the surface not covered by the PVP block.
Introduction The extension of a polymer layer at a solid/fluid interface is a prime determinant of the effectiveness of the polymer in shielding the surface and blocking molecular transport along the surface. The extension depends not only on the size of the polymer but also on the density of chains in the adsorbed layer and the solvent quality. Evidence of the substantial changes in layer thickness that can be achieved with solvent composition are found in the literature, for example, ionic strength effects on the thickness of adsorbed polyelectrolytes1,2 and solvent/nonsolvent effects on the thickness of terminally attached chains.3-6 The effectiveness of temperature in changing layer thickness has also been studied. Israelachvili et al.7 measured the force profile between two mica surfaces with adsorbed polystyrene homopolymer at temperatures below and above the Θ point in cyclohexane. They observed a relatively long range attraction between surfaces both below and above Θ, where the shape of the profile depended on temperature. The authors suggest temperature dependent adsorption/desorption as a mechanism for some aspects of the change in profile shape with temperature. Dejardin8 measured the hydrodynamic thickness (LH) of polystyrene homopolymer in trans-decalin adsorbed to porous glass disks as a function of temperature below Θ. Contraction of the layer was linear over the range between Θ and the critical miscibility temperature with a decrease * To whom correspondence should be addressed. † Lubrizol Corporation. ‡ Landbouwuniversiteit. § Carnegie Mellon University. X Abstract published in Advance ACS Abstracts, February 1, 1996. (1) Pefferkorn, E.; Schmitt, A.; Varoqui, R. J. Membr. Sci. 1978, 4, 17. (2) Kim, J. T.; Anderson, J. L. Ind. Eng. Chem. Res. 1991, 29, 1008. (3) Webber, R. M.; Anderson, J. L.; Jhon, M. S. Macromolecules 1990, 23, 1026. (4) Auroy, P.; Auvray, L. Macromolecules 1992, 25, 4134. (5) Auroy, P.; Auvray, L. Langmuire 1994, 10, 225. (6) Marra, J.; Hair, M. L. Colloids Surf. 1988, 34, 215. (7) Israelachvili, J. N.; Tirrell, M.; Klein, J.; Almog, Y. Macromolecules 1984, 17, 204. (8) Dejardin, Ph. J. Phys. (Paris) 1983, 44, 537.
0743-7463/96/2412-1040$12.00/0
of 3% in LH per °C drop. Zhu and Napper9 measured the effective diameter of polystyrene latex spheres with poly(N-isopropylacrylamide) (NIPAM) chains grafted to the surface as a function of temperature near the Θ point for NIPAM in water. In this system the solvent quality decreases as temperature increases. They measured a gradual decrease in effective sphere diameter as T was increased toward Θ, consistent with a modest contraction of LH for NIPAM as the solvent quality was reduced; however, a sharp decrease in sphere diameter down to near the “core” dimension of the latex was observed over a range of 15 °C above Θ. Zhu and Napper interpeted this apparent collapse of LH as a coil to globule transition of the NIPAM layer. Auroy and Auvray4 reported data for changes in layer thickness above and below the Θ temperature for poly(dimethylsiloxane) chains terminally grafted at high density on silica spheres dispersed in styrene. Their data indicate a linear dependence between layer thickness and temperature. This paper is the fourth in a series devoted to the study of poly(2-vinylpyridine)/polystyrene (PVP/PS) adsorbed from a selective solvent (toluene) to the capillary pores of muscovite mica membranes. In this system the PVP preferentially adsorbs to the mica10 and the PS chains are solvated. The first three papers3,11,12 deal with the relationship between the thickness of the adsorbed layers and the size of the PVP and PS blocks. Here we are concerned with changes in the thickness of the adsorbed polymer layer effected by changes in temperature about the Θ point for the solvated (PS) chains. This is a good experimental system from a theoretical point of view in that the Θ state for isolated PS chains in solution is well studied and hence the material parameters of any theory should be accessible from bulk solution properties. After the experiments are first described and the results are presented for hydrodynamic thickness versus tem(9) Zhu, P. W.; Napper, D. J. Colloid Interface Sci. 1994, 164, 489. (10) Parsonage, E.; Tirrell, M.; Watanabe, H.; Nuzzo, R. G.; Macromolecules 1991, 24, 1987. (11) McKenzie, P. F.; Webber, R. M.; Anderson, J. L. Langmuir 1994, 10, 1539. (12) Webber, R. M.; Anderson, J. L. Langmuir 1994, 10, 3156.
© 1996 American Chemical Society
Expansion and Contraction of Adsorbed Copolymers
Langmuir, Vol. 12, No. 4, 1996 1041
Table 1. Summary of Dataa polymer
LHT
73/444 28/150 25/167 25/551 25/778
176 80 109 232 360
LHH
LHΘ
dLH/dT
79
2.13
163 222
3.14 2.12
21 12 68
a All values of L in Å, T in °C. The superscript on L designates H H the liquid: T ) toluene, H ) n-heptane, no superscript ) cyclohexane. The data for toluene and n-heptane were taken at room temperature (25 °C). Θ ) 34.5 °C for cyclohexane.
perature, a comparison is made with theoretical predictions. The polymer segment profile was calculated from the Scheutjens-Fleer theory, applied to terminally attached chains,13 using a literature-derived value for the variation of the excluded-volume parameter with temperature. These profiles were then substituted into a Brinkman model for flow through polymer layers14 to obtain the predicted values of LH versus temperature. The comparison between prediction and experiment is not particularly good; possible reasons for the disagreement are suggested. Experimental Section The PVP/PS diblock copolymers were adsorbed from toluene to the pores of track-etched mica membranes. More details of this system are found in other references.11,12,15 The membranes were thin (≈7 µm thick) disks of muscovite mica with capillary type pores formed by first irradiating the disk with a collimated beam of fission fragments and then etching the tracks into pores. The pores were uniform, straight capillaries with a cross section having the geometry of a 60° rhombus. The pore radius was determined from the liquid flowrate (Q) of toluene versus the applied pressure difference (∆P) using a modified Poiseuille equation:
nπR4 ∆P Q ) (0.68) 8ηl
(1)
where η is the viscosity of the liquid (toluene, cyclohexane, or n-heptane) adjusted for temperature, n is the number of pores, and l is the thickness of the membrane. The factor 0.68 accounts for the noncircular cross section of the pore; R is the radius of a circle having an area equal to the rhomboidal pore. The pore radius before adsorption of polymer is designated by R0; it was always more than 10 times larger than the thickness of the polymer layer. The hydrodynamic thickness of the polymer layer, LH, was calculated from R0 and the pore radius determined after adsorption (R) using the following relation:
LH )
R0 - R 1.21
(2)
where the factor 1.21 again accounts for the rhomboidal geometry of the pore. The uncertainty in LH is estimated to be (20 Å; the reproducibility was better than (10 Å. The PVP/PS polymers were synthesized by anionic polymerization.15 The polymers are listed in Table 1. The number designation 73/444 means 73 monomers of 2-vinylpyridine and 444 monomers of styrene in the diblock copolymer. The polydispersity index of each PS block was about 1.1 or less as determined by size exclusion chromatography.15 The relative size of the PS to PVP blocks was determined by elemental analysis (Desert Analytics, Inc.). Adsorption was achieved by contacting a membrane with a solution of the copolymer at 30-40 µg/cm3 in toluene at room temperature for at least 92 h, which was found to be sufficient time to form a stable layer. The adsorbing (13) Cosgrove, T.; Heath, T.; Van Lent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20, 1692. (14) Anderson, J. L.; McKenzie, P. F.; Webber, R. M. Langmuir 1991, 7, 162. (15) Webber, R. M. Ph.D. Thesis, Carnegie Mellon University, 1991.
Figure 1. Hydrodynamic thickness versus PS block size for three different solvents: toluene (b), cyclohexane at Θ temperature (0), and n-heptane (4). The data for toluene and heptane were taken at room temperature (25 °C). NPVP ) 25 for all the polymers except NPS ) 150 for which NPVP ) 28. The lines through the data are best fits to the expression LH ∼ (NPS)ν. solution was then flushed from the membrane cell, and polymerfree toluene was introduced. After several hours the pressureflow experiments were begun. There was no evidence of desorption of the polymer from the membrane even after exposure to polymer-free toluene for 2 weeks. Because homopolymers of polystyrene showed negligible adsorption from toluene under conditions similar to those of the copolymer adsorption, we conclude that the PVP block adsorbed to the pores and the PS block was solvated, which is consistent with another study10 of PVP/PS diblocks adsorbing from toluene to mica. The solvents were toluene, cyclohexane (Θ solvent conditions at 34.5 °C), and n-heptane (nonsolvent for both PS and PVP). Solvent quality was changed by altering either the solvent itself or the temperature. When the solvent was changed, at least 2 h was allowed for the layer to relax to its new steady state conformation. Initially PVP/PS layers were contacted with the new solvent for several days, but it was found that LH did not change appreciably after the first 2 h of flushing the membrane with the new solvent. LH was reversible to changes in the solvent; the same values of LHT were obtained before and after measurement of LH in heptane.15 Temperature was controlled by submerging the experimental apparatus in a heated water bath. LH was measured upon both increasing and decreasing the bath temperature. In both cases, LH was measured as a function of time in order to determine the minimum equilibration time required to achieve a steady state condition for the polymer layer as indicated by time-invariant LH measurements. Upon increasing temperature, a steady state was achieved in less than 2 h, while upon decreasing temperature, greater than 10 h was required;15 the kinetics was not investigated further. For all the results presented here, LH as a function of temperature was measured in a sequence of increasing temperatures. At least two cycles of measuring LH while increasing temperature were made over several days; the collinearity of the data for these cycles confirms that the layers were at a steady state conformation when LH was determined and that the layer thickness was reversible to the changes in solvent quality caused by changes in temperature.
Results and Discussion Figure 1 shows the dependence of hydrodynamic thickness on the solvent quality and the number of PS monomers at constant NPVP. The thickness of the polymer layer decreases as the solvent is changed from toluene to cyclohexane to heptane because the quality of these solvents for the PS block changes from good to Θ to nonsolvent, respectively. Heptane, a nonsolvent for both PS and PVP, probably collapses the PS block to near its bulk density as evidenced by small measured values of LHH; surface coverages calculated using LHH values and bulk densities for PVP and PS are similar to those measured by a different technique on a similar system.10-12 As discussed in the Experimental Section, the effects of solvent quality illustrated in Figure 1 were reversible.
1042 Langmuir, Vol. 12, No. 4, 1996
Webber et al. Table 2. Comparison between the Expansion of PS Chains Terminally Attached to a Surface (LH ) Hydrodynamic Thickness) and PS Chains in Solution (RH ) Hydrodynamic Radius) as the Solvent Quality Is Increaseda NPVP/ LHT/ LHT/ LHΘ/ RHT/ LH(Θ+10)/ RH(Θ+10)/ NPS LHΘ RHT RHΘ RHΘ n1 LHΘ RHΘ 73/444 2.23 3.27 1.62 1.10 8.5 25/551 1.42 3.82 3.00 1.12 3.3 25/778 1.62 4.89 3.45 1.14 3.8
Figure 2. Hydrodynamic thickness versus temperature for the adsorbed polymer 73/444 exposed to toluene (b) and cyclohexane (O).
Figure 3. Hydrodynamic thickness versus temperature for the adsorbed polymer 25/551 (0) and 25/778 (4) exposed to cyclohexane.
Figure 2 shows LH versus temperature for an adsorbed layer of 73/444 in contact with toluene and cyclohexane. There is no detectable effect of temperature with the toluene data, the expected result for a good solvent. In the case of cyclohexane, LH increased in a linear fashion when the temperature was increased from about 20 °C to above Θ. The linear variation of LH with temperature above and below Θ, seen in Figures 2 and 3 for all three diblock copolymers, is consistent with the observation of Auroy and Auvray4 for a different polymer/solvent system. The experimental results for LH are summarized in Table 1. Table 2 compares the effects of solvent quality on LH and the hydrodynamic radius (RH) of PS chains. RH is determined from the diffusion coefficient using the Stokes-Einstein equation. The following empirical correlations fit the data of Varma et al.16 for PS in the molecular weight range 0.238-5.47 × 106 (RH in Å):
toluene (30 °C) cyclohexane (34.5 °C)
cyclohexane (44.5 °C)
RH ) 1.77NPS0.56
(3a)
RHΘ ) 2.31NPS0.50 (3b) RH(Θ+10) ) RHΘ 1 + 0.000517NPS0.537 (3c)
As shown in Table 2, LH was greater than RH in both toluene and cyclohexane (Θ), and the ratio LH/RH increased as NPS increased at NPVP ) 25. In an earlier paper11 we noted that in toluene LH increased more strongly with NPS than RH. Using eq 3a and the best-fit straight line shown in Figure 1, LHT/RH ) 0.838NPS0.26; Figure 1 hints (16) Varma, B. K.; Fujita, Y.; Takahashi, M.; Nose, T. J. Polym. Sci., Polym. Phys. Ed. 1984, 22, 1781.
1.27 1.93 1.10
1.014 1.015 1.018
n2 17.6 11.6 5.0
a The superscript T denotes the solvent toluene; otherwise, the solvent is cyclohexane. The values of RH were determined from eqs 3a-c, which represent the data of Varma et al.16 The exponents n1 and n2 are defined by eqs 4a and 4b.
that a similar dependence for LH in cyclohexane on NPS may also exist. Table 2 shows that the increase in the relative extension (LH/RH) of the 73/444 layer in going from cyclohexane (Θ) to toluene was larger than for the polymers with the larger ratios of NPS/NPVP, thus indicating that in cyclohexane the 73/444 layer was not so extended. The adsorbed copolymer layers show significantly greater percentage changes due to changes in solvent quality, both by solvent and temperature, then has been observed for free PS coils in solution; this behavior might be due to the different dimensionality of moderately dense terminally attached polymer layers versus coils in solution.17-19 One basis for comparison is the relative change in apparent monomer concentration upon change in solvent quality. Consider two cases, the changes in going from a good solvent (toluene) to a Θ solvent (cyclohexane) and the effects of temperature in the Θ solvent:
( ) ] [ ]
LHT RHT ) LHΘ RHΘ
[
n1
LH(Θ+10) RH(Θ+10) ) LHΘ RHΘ
(4a)
n2
(4b)
If the relative change in segment density is similar for the expansion of polymer layers compared to coils in solution, then for a one-dimensional expansion of the layer we would expect n1 and n2 to equal 3. Table 2 shows the experimentally derived values of n1 and n2. For changes in solvent, values of n1 for the 551 and 778 PS chains are close to 3, suggesting that confinement of the polymer chains results in approximately the same relative change in monomer concentration as observed with coils free in solution. For the 444 PS layer, n1 . 3 indicates a greater change in relative monomer concentration compared to PS in free solution. The comparison between Θ and Θ + 10 °C in cyclohexane gives n2 . 3 for all copolymers, thus indicating that changes due to temperature near Θ conditions are much greater for the adsorbed PS chains compared to the hydrodynamic radius of PS in solution even allowing for dimensionality. Furthermore, the fractional increase in LH decreases as NPS increases, whereas the opposite trend occurs for PS chains in solution. From these data we conclude that thermally induced changes in the hydrodynamic thickness of terminally attached polymers above but near Θ are fundamentally different than for chains in solution. (17) Zhulina, E. B.; Borisov, O. V.; Pryamitsyn, V. A.; Birshtein, T. M. Macromolecules 1991, 24, 140. (18) Wijmans, C. M.; Scheutjens, J. M. H. M.; Zhulina, E. B. Macromolecules 1992, 25, 2657. (19) Carignano, M. A.; Szleifer, I. J. Chem. Phys. 1994, 100, 3210.
Expansion and Contraction of Adsorbed Copolymers
Langmuir, Vol. 12, No. 4, 1996 1043
Table 3. Temperature Dependence of the Expansion Factor for the Adsorbed Polymer Layersa polymer
σ × 10-12
[dR/dT]exp
dR*/dT
[dR/dT]SF
(LHΘ)SF/ (LHΘ)exp
73/444 25/551 25/778
2.37 2.33 1.58
0.0270 0.0193 0.0095
0.0085 0.0085 0.0103
0.0054 0.0054 0.0059
3.29 1.90 1.67
a T is expressed in °C. The chain density (σ, adsorbed molecules per cm2) was calculated from eq 6, which is a correlation of the experimental results obtained by Parsonage et al.10 R* is the expansion factor for the brush height, as predicted from the meanfield model of Zhulina et al.17 (see eq 16). The subscript SF denotes predictions from the segment profiles calculated from the Scheutjens-Fleer lattice model for terminally attached chains, using σ and eq 13, and the numerical solution of eq 17 with the Brinkman coefficient given by eq 18.
The expansion factor of the adsorbed layers in cyclohexane, R, is defined by
LH(T) R≡ LHΘ
(5)
The slope of R versus temperature, obtained from the data listed in Table 1, is presented in Table 3 for three adsorbed copolymers. These values are compared with a priori theoretical predictions, as discussed below. Comparison with Theory. Before comparing our data with the theories based on lattice and analytical mean field models for terminally attached chains, we must first determine the numerical values of two input parameters for the theories: the number of polymer molecules adsorbed per unit area (σ) and the temperature dependence of the excluded volume (v) for PS in cyclohexane. Because the pores of our membranes (track-etched muscovite mica) were not readily accessible and the total surface area represented by the pore walls was small, we were not able to accurately measure the amount of polymer adsorbed. Parsonage et al.10 adsorbed PVP/PS block copolymers to the surface of muscovite mica from toluene under conditions very similar to ours. They studied a wide range of molecular weights and block ratios. For nearly all the polymers the adsorbed amount was in the range 1-2 mg/m2. Their results are very well correlated by the following empirical expression:11
0.0687λ1.058 σ ) σ0 1 + 0.00467λ1.226
σ0 )
1 π(aNPS0.595)2
(6)
where λ ) NPS6/5/NPVP2/3 and a ) 1.86 Å. σ0-1 is the area per molecule at which uniformly spaced, terminally attached PS chains would begin to overlap in a good solvent. The surface chain densities for our polymers, as estimated from eq 6, are listed in Table 3. The temperature dependence of the excluded-volume parameter, v ) (0.5 - χ) where χ is the Flory-Huggins coefficient, is estimated from literature data for the change in the radius of gyration of PS in cyclohexane near the Θ point. For small excluded volume (v , n-1/2),
Rs2 ≡
( ) rg rgΘ
2
≈ 1 + 1.276z
z)
[(
3 2π
3/2
)
n1/2v
]
(7)
where n is the equivalent number of “units” in the PS chain, and v is normalized by b3 where b is the effective length of each segment. n and b are defined so that the
classical formula for a random walk gives the actual radius of gyration at Θ conditions:
1 2 nb ) rgΘ2 6
(8)
We take b to equal the mean size of the solvent, (Vs/Navo)1/3 ) 5.65 Å where Vs is the molar volume of cyclohexane. Note that this value of b will serve as the lattice dimension when we make predictions based on the Scheutjens-Fleer theory. (Had we chosen Vs to be the molar volume of bulk PS, b would equal 5.5 Å.) From light-scattering measurements, Varma et al.16 obtained the following for PS in cyclohexane at the Θ temperature:
rgΘ ) 2.98NPS0.50 (Å)
(9)
By combining eqs 7-9, the following is obtained: 2 1.84 d(Rs ) dv ) dT N 1/2 dT
(10)
PS
Park et al.20 experimentally determined Rs below Θ for PS in cyclohexane. From Figure 6 of their paper we estimate the following:
d(Rs2) M1/2 ≈ 0.01 dT Θ
(11)
Substituting this experimental value into eq 10 gives dv/ dT ≈ 0.0006 K-1. An estimate of dv/dT above Θ is made from the experimental measurements of Orofino and Mickey,21 who found
(
z ) 0.0098M1/2 1 -
Θ T
)
(12)
Using this relation with eq 7 for the definition of z, we have dv/dT ≈ 0.0008 K-1. In using the theories described below, we assume the following temperature dependence of excluded volume for PS in cyclohexane:
v ) 1 × 10-3 (T - Θ)
(13)
for temperatures above and below Θ. Mean Field Theory for Polymer Brushes: Brush Thickness. Wijmans et al.18 compared the lattice-based self-consistent field model for terminally attached chains13 with the analytical self-consistent field model of Zhulina et al.17 The segment profiles from the two models were in good agreement for highly stretched layers (L/rg . 1). The layers in our experiments were not “highly stretched”, as indicated by the values LH/rgΘ which are of order unity; however, we feel it is worthwhile to compare the theoretical prediction for the temperature dependence of the expansion factor with our experimental results. We choose here the theory of Zhulina et al.17 because it has an analytical form at small excluded volumes, though other mean field theories are available (e.g., ref 19). Near Θ conditions the theory of Zhulina et al.17 predicts the following for the expansion factor (R* ) L/LΘ where L is the layer thickness):
2 R* ) 1 + β + O(β2) π
(14)
(20) Park, I. H.; Wang, Q.-W.; Chu, B. Macromolecules 1987, 20, 1965. (21) Orofino, T. A.; Mickey, J. W. J. Chem. Phys. 1963, 38, 2512.
1044 Langmuir, Vol. 12, No. 4, 1996
Webber et al.
β is a dimensionless excluded-volume parameter defined by
β)
vp 32
1 1 w3/4 (σa2)1/2
1/4
()
(15)
where w is the third virial coefficient, assumed to equal 1/ , a2 is the equivalent cross-sectional area of a PS chain, 6 and p is the stiffness parameter defined as Lk/a where Lk is the Kuhn equivalent length. a2 is found by equating its product with the projected monomer length for PS (≈2.5 Å) to the volume per monomer of bulk PS; the result is a2 ) 65.8 Å2. The Kuhn length for PS at Θ conditions is about 21 Å. Combining eqs 14 and 15 we have the following estimate for the temperature dependence of the expansion factor for terminally attached PS chains near Θ conditions:
T ) Θ:
1.05 dv dR* ) dT (65.8σ)1/2 dT
(16)
where σ is expressed in chains per Å2. Predictions from the Zhulina et al.17 model are compared with the experimental results for dR/dT in Table 3. With the exception of one polymer, the largest polymer with the 778 PS block which forms the most highly extended layer, the quantitative agreement is not particularly good in that the theory underpredicts dR/dT and does not predict that dR/dT decreases with increasing NPS (even at constant NPVP) as was observed. The Zhulina model should apply best to highly stretched polymer layers; however, the disagreement in the NPS dependence of dR/dT suggests that agreement between theory and experiment for the 778 PS chain may be fortuitous. Also note that R and R* may not be directly comparable because R is the expansion factor for the hydrodynamic thickness while R* is the expansion factor for the “brush height” which is not uniquely defined in the theory. Because the Brinkman parameter (κ, see eq 18) is a function of temperature, one cannot conclude that LH is proportional to a brush height that is defined in a thermodynamic sense only. Theoretical Prediction of LH from the ScheutjensFleer Lattice Model. Effective medium models for viscous flow through polymer layers are based on the premise that Brinkman’s equation22,23 can be applied to a porous structure that is nonhomogeneous on the length scale of the layer thickness. The Brinkman parameter κ is taken to be a functional of the polymer segment density F(y) at distance y from the surface, κ ) κ[F(y)]. Accepting this assumption, Brinkman’s equation leads to the following equation which must be solved in order to compute the hydrodynamic thickness of the polymer layer:14
d2G - κ2G ) 0 2 dy y ) 0: G ) 0 y f ∞:
dG f -1 dy
(17)
LH ) lim [G+y] yf∞
Assuming the validity of the Brinkman equation with a spatially dependent κ, the above relation is exact for LH in the limit of flat surfaces (LH , radius of curvature). We (22) Varoqui, R.; Dejardin, P. J. Chem. Phys. 1977, 66, 4395. (23) Cohen Stuart, M. A.; Waajen, F. H. W. H.; Cosgrove, T.; Vincent, B.; Crowley, T. L. Macromolecules 1984, 17, 1825.
Table 4. Comparison between LM and (LH)SFa LM/(LH)SF polymer
Θ - 10 °C
Θ
Θ + 10 °C
73/444 25/551 25/778
0.82 0.81 0.79
0.80 0.79 0.78
0.79 0.78 0.77
a The PS segment profiles were computed from SF mean-field theory. (LH)SF was computed from eqs 17 and 18. LM is the distance from the surface which contains 99% of the PS segments.
note that although eq 17 looks rather simple to solve, it is a boundary-value equation that is very sensitive to the guess of G at y ) 0, and hence “shooting” techniques will often fail. The method of solving eq 17 used here is described in the Appendix. A prediction of LH from eq 17 requires two pieces of information: the segment density F(y) and the dependence of κ on F. For the segment density of the solvated PS block we apply the Scheutjens-Fleer (SF) theory as modified for terminally attached chains.13 The lattice dimension is taken equal to 5.65 Å, the excluded-volume parameter is given by eq 13, the interaction between terminally attached PS blocks and the surface is set to 0, and the chain density (σ) is taken from Table 3. The profiles for the three adsorbed polymers at Θ are plotted in Figure 4. Note the diffuse outer edge of the profiles; the outer edge has a significant effect on the calculated value of LH because of the low permeability of polymer to solvent even at low segment densities.14 The profiles at (10 °C from Θ are qualitatively the same. For the dependence of κ on F we use the data by Mijnlieff and Jaspers24 for poly(R-methylstyrene) (PAMS) in cyclohexane (Θ ≈ 35 °C). These results were obtained from measurements of the sedimentation rate of polymer in solution at concentrations in the semidilute region where the chains overlap. The Brinkman parameter is a function of solvent quality. The polymer concentration (F, gm/cm3) is related to volume fraction (φ) by φ ) 0.90F. For the range 0.002 < F < 0.02 gm/cm3 and Θ < T < Θ + 15 °C, the experimental results of Mijnlieff and Jaspers are correlated by the following:
κ ) [1 - 0.0368(T - Θ) + 0.000943(T - Θ)2]-1/2κΘ (18a) κΘ ) (3.45 × 107)φ0.858
(18b)
Although the data upon which the above empirical correlation is based were all taken above Θ, we assume that eq 18 holds below Θ as well. Although the segment densities predicted from the mean-field lattice model are sometimes higher than 0.02, errors in LH resulting from the use of eq 18 are negligible because there is essentially no solvent flow at densities greater than 0.02. The predictions from the SF/Brinkman model are listed in Table 3. As with the analytical mean field theory of Zhulina et al.,17 the values of [dR/dT]SF are too small and the trend with NPS is opposite to that of the experiments. The final column of Table 3 also shows that the theoretical values of LHΘ are too large. The reasons for the significant discrepancy between the SF-based theory and experiment are not clear, but we will comment on some potential sources. The theory depends on the adsorbed amount of polymer as σ1/2, so it is doubtful that errors in estimating σ could account for the differences. Assuming that dR/dT varies as dv/dT, then the prefactor in eq 13 must be (24) Mijnlieff, P. F.; Jaspers, W. J. M. Trans. Faraday Soc. 1971, 67, 1837.
Expansion and Contraction of Adsorbed Copolymers
Langmuir, Vol. 12, No. 4, 1996 1045
increased by a factor of 2 or more to bring the theoretical predictions into reasonable agreement with experiment. However, data from refs 20 and 21 indicates that the prefactor is actually slightly less than 10-3, and this correction would obviously not affect the trend with NPS. Finally, one might point to the hydrodynamic calculations in the theoretical computations as a potential source of discrepancy. We think that this is not the explanation, as the Brinkman parameter is derived from experiments with a very similar polymer (R-methylstyrene) and, furthermore, the hydrodynamic effects partially cancel in determining the expansion factor R. Figure 4 shows that a significant portion of the profile, the outer edge, is diffuse with low segment concentrations. As noted above, previous investigations into the Brinkman model indicate that the outer edge of the polymer layer, the tails, contributes significantly to LH.14,22,23 We tested the sensitivity of the SF/Brinkman model to the outer edge of the segment profile to see if errors there could account for the discrepancy between theory and experiment. If the segment density profile is redefined to include only the inner 99% of segments, a new hydrodynamic layer thickness (LM) can be calculated and compared to (LHΘ)SF values. Table 4 shows the effect of the diffuse outer portion of the polymer layer on (LH)SF at three temperatures Θ 10 °C, Θ, and Θ + 10 °C. For all cases LM/(LH)SF ≈ 0.8; thus, the diffuse edge, though significant to LH, is probably not a cause of the large discrepancy in LH and dR/dT between theory and experiment. The discrepancy between the SF-based theory and experiment could be explained by a reversible temperature dependent adsorption of PS segments to bare areas of the pore walls not covered by the PVP block. PS is known to adsorb to mica near the Θ temperature in cyclohexane (albeit to a different plane of mica)7. SF calculations of the configuration of terminally attached chains show that including a non-zero interaction between polymer segments and the surface significantly affects the polymer layer configuration and average thickness.13 In addition, a significant fraction of the pore walls may not be covered with the adsorbing PVP block thus allowing PS adsorption. An estimate of the fraction (f) of pore walls that was bare can be made by assuming that the PVP block adsorbed as spheres having the same density as bulk PVP (F ) 1.17 g/cm3):
f ) 1 - (3.40 × 10-15)σNPVP2/3
(19)
where σ is expressed as chains per cm2. Using this formula and the values of σ listed in Table 3, we have f ) 0.86, 0.93, and 0.95, respectively, for the three copolymers. If the PVP blocks aggregated to form clusters, the values of f would increase closer to unity; on the other hand, if the PVP block flattened upon adsorption, then f would be smaller than predicted by eq 19. If this crude model of PVP adsorption is even approximately correct, it suggests that much of the pore wall was not covered by PVP. The segment profiles in Figure 4 were calculated assuming no interaction energy between styrene monomers and the surface. Inclusion of a temperature dependent, attractive styrene/surface interaction in the SF model would shift the segment profiles in Figure 4 toward the surface13 and decrease LH with respect values calculated here. Furthermore, if PS segments desorbed as T increased because of improved solvent quality, this would have the effect of increasing dR/dT with respect to values calculated here. A temperature dependent, reversible adsorption of PS to the mica pore walls is plausible and would serve to explain, at least qualitatively, the dis-
Figure 4. Volume fraction (φ) of PS segments versus distance (y) from the surface at T ) Θ in cyclohexane. These profiles were calculated from SF theory as applied to terminally attached chains.13 The y coordinate has been normalized by the lattice size (5.65 Å) and φ ) 0.9F where F is the mass density of segments (gm/cm3).
crepancy between theory and experiment for LHΘ and dR/ dT; however, it is not obvious how PS adsorption would address the discrepancy in the effects of NPS. We are not able to suggest an experiment that would directly test the hypothesis of temperature dependent PS adsorption. Summary The hydrodynamic thickness of terminally attached PS chains, anchored by strong adsorption of a PVP block, varies linearly with temperature over a range of 15 °C below and above the Θ temperature with cyclohexane. There is no “collapse” of the layer under poor solvent conditions, T < Θ, as seen by Zhu and Napper;9 however, our system has a positive third virial coefficient A3 while the system of Zhu and Napper has a negative A3. The temperature derivative of the expansion factor (R) of the PS layers in cyclohexane varies from 1 to 3% per °C (see Table 3), which is an order of magnitude greater than the expansion of PS chains of the same molecular weight in solutions of cyclohexane. Predictions of dR/dT based on Scheutjens-Fleer theory for the segment profile and the Brinkman equation for calculating LH are factors of 2-5 less than the experimental values. The theory of Zhulina et al.17 shows reasonable agreement with the largest PS chain studied here. Both theories fail to capture the experimental result that dR/dT decreases with increasing NPS. A possible explanation for the discrepancy between theory and experiment is based on a temperature dependent, reversible adsorption of PS segments to the bare area of the mica pore wall not covered by the anchoring PVP block of the copolymer. Acknowledgment. We thank Y. Solomentsev for obtaining the numerical solutions of eq 17 and providing the Appendix to this paper. Support for this research was received from National Science Foundation Grant CTS9122573. J.L.A. gratefully acknowledges support from the Dutch Research Organization NWO during his stay at Landbouwuniversiteit Wageningen. We appreciate very helpful discussions with G. Berry, M. Cohen Stuart, I. Szleifer, C. Wijmans, and K. Zhulina.
1046 Langmuir, Vol. 12, No. 4, 1996
Webber et al.
Appendix: Numerical Solution of Eq 17 The goal is to solve the following boundary value problem:
d2G(y)
- K(y) G(y) ) 0
(A1)
G(0) ) 0
(A2)
dG(y) ) -1, y f ∞ dy
(A3)
dy2
where K(y) ) κ2 is a known function which is calculated from a polymer segment density φ(y). Because the segment density is computed here from a lattice theory, K is known only at equally spaced points. We use a thirdorder spline approximation of φ(y) to obtain a continuous representation; K(y) is then calculated from this approximation. A shooting procedure is combined with a Runge-Kutta numerical method. The parameter B is a guess of the missing condition at the solid surface:
y ) 0:
dG(y) )B dy
Unfortunately the above strategy often fails due to the fact that very small changes in B cause large changes in G′∞(B). In other words, the bisection algorithm used to solve eq A6 fails when the step size becomes less than machine precision. Therefore, the procedure was modified as follows. An arbitrary point y* inside the layer is chosen, and the following problems are solved:
0 < y e y*:
dy
dG-(y) ) d1 dy
(A8)
y > y*:
d2G+(y) dy2
- K(y) G+(y) ) 0
(A9)
dG+(y) ) d1 dy
(A10)
y ) y*: G+(y) ) d,
where the parameters d and d1 are determined to satisfy boundary conditions A2 and A3:
dG+(y) G-(0) ) 0; lim
yf∞
(A5)
In principle, G and B are determined by solving eq A1 subject to eqs A2 and A4 and the following:
G′∞(B) ) -1
(A7)
(A4)
dG(y,B)
yf∞
- K(y) G-(y) ) 0
dy2
y ) y*: G-(y) ) d,
The derivative at infinity is now a function of B:
G′∞(B) ) lim
d2G-(y)
(A6)
) -1 dy
(A11)
The solution to eqs A7-A11 is less susceptible to numerical instabilities resulting from initial guesses of boundary conditions than is the direct solution of A1-A3. The solution G(y), and hence LH as calculated from eq 17, should be independent of the choice of y*. For the three polymers considered here, we found that values of y* between 5 and 30 gave the best numerical stability. LA950597+
