Enhanced Colloidal Stabilization via Adsorption of Diblock Copolymer

which is attributed in turn to strong segmental adsorption of the short anchor block and .... Distinguishing between dynamic yielding and wall sli...
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Langmuir 1998, 14, 7104-7111

Enhanced Colloidal Stabilization via Adsorption of Diblock Copolymer from a Nonselective Θ Solvent Ramesh Hariharan† and William B. Russel* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544 Received December 16, 1997. In Final Form: September 28, 1998 Experimental measurements, interpreted with mean field theories, provide quantitative understanding of the stability of colloidal particles coated with a highly asymmetric diblock copolymer. Adsorption of poly((dimethylamino)ethyl methacrylate)/poly(n-butyl methacrylate) onto colloidal silica particles from 2-propanol, which is nonselective and a Θ solvent for poly(n-butyl methacrylate), yields a very high surface coverage and a highly stretched brush. With increasing temperature, the adsorbed chains swell more than those in the bulk. Upon cooling, the dispersions exhibit thermodynamic stability well below the Θ temperature of the nonadsorbed block. A simple mean-field theory rationalizes the stability as a consequence of the high surface coverage, which is attributed in turn to strong segmental adsorption of the short anchor block and negligible excluded volume interactions among the stabilizing chains.

1. Introduction The goal of this paper is to probe selected features of diblock copolymer layers adsorbed on colloidal particles from a nonselective solvent. In contrast to a selective solvent in which the adsorbing block interacts unfavorably with the solvent and prefers to coat the surface,1,2 a nonselective solvent3,4 dissolves both blocks, leaving only the chemical affinity of the anchor block for the surface to drive adsorption. Fleer et al. review extensively theoretical and experimental studies exploring the effect of block sizes on surface coverage and layer thickness.5 In this paper, our goal is to understand quantitatively the adsorption isotherm and the influence of changing solvency on the adsorbed layer. For both selective6 and nonselective solvents7 high affinity isotherms are fairly typical, as for homopolymer adsorption.8 Munch and Gast1 noted that the mean distance between attachment points for poly(styrene)/poly(ethylene oxide) adsorbed from cyclopentane is about half that for end-functionalized poly(styrene) adsorbed from toluene.9 Toluene is a good solvent for the PS block whereas cyclopentane is slightly better than a Θ solvent, indicating the significant effect of solvency on surface coverage. To highlight the role of composition, Wu et al.10 adsorbed poly((dimethylamino)ethyl methacrylate)/poly(n-butyl methacrylate) (PDMAEM-PBMA) on silica from 2-propanol and found substantially higher adsorbed amounts * To whom all correspondence should be addressed. † Current address: Polymer Materials Laboratory, General Electric CR&D, Schenectady, NY 12301. (1) Munch, M. R.; Gast, A. P. Macromolecules 1990, 23, 2313. (2) Luckham, P. F.; Ansarifar, M. A. Chem. Eng. Sci. 1987, 42, 799. (3) Guzonas, D.; Boils, D.; Hair, M. L. Macromolecules 1991, 24, 3383. (4) Parsonage, E.; Tirrell, M.; Watanabe, H.; Nuzzo, R. G. Macromolecules 1991, 24, 1987. (5) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers of Interfaces; Chapman and Hall: London, 1993. (6) Killman, E.; Maier, H.; Baker, J. A. Colloid Surf. 1988, 31, 51. (7) de Silva, G. P. H. L.; Luckham, P. F.; Tadros, Th. F. Colloid Surf. 1990, 50, 251. (8) Cosgrove, T.; Heath, T.; van Lent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20, 1692. (9) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Nature 1988, 332, 712. (10) Wu, D. T.; Yokoyama, A.; Setterquist, R. L. Polym. J. 1991, 23, 709.

(1.5-14.0 mg/m2) than typical (0.8-3.5 mg/m2).4,5 The positively charged block adsorbs strongly to the negatively charged silica, while the PBMA is solvated near its Θ temperature (20.9-25 °C) in 2-propanol at room temperature. The very high adsorption is a consequence of the strong attraction of the adsorbed block to the surface and weak excluded volume interactions in the solvated layer. Here we examine the adsorption isotherms, layer thickness, and dispersion stability for one of the same polymers. Application of the mean-field theory of Alexander and de Gennes at slightly better to worse than theta conditions11 rationalizes the behavior. The following section describes the experiments, while section 3 reviews the theory for isolated and interpenetrating polymer brushes. In sections 4 and 5 we present our results and draw our conclusions. 2. Experimental Section 2.1. Materials. A stable dispersion of monodisperse silica was prepared according to the method of Stober et al.,12 which involves the hydrolysis and subsequent polymerization of tetraethyl orthosilicate (TEOS) in an alkaline ethanol solution. The number averaged diameter from transmission electron microscopy (TEM) (Figure 1) 2a ) 245 ( 6 nm is less than the intensity average of 265 ( 6 nm from dynamic light scattering. A Gaussian distribution of sizes with the mean and standard deviation from TEM yields an intensity average of 258 ( 6 nm. In addition, the silica structure is porous with a density F ) 1700 kg/m3, so imbibition of 2-propanol might cause slight swelling and account for the remaining difference. The diblock copolymer was synthesized and characterized by Wu et al.10 The composition, DMAEM18BMA769, corresponds to Mn ) 112 kg/mol with Mw/Mn ) 1.15, but gel permeation chromatography (GPC) before and after adsorption detected unreacted PBMA. Elemental analysis of a dry polymer sample (Robertson Microlit Labs, Madison, NJ) detected 0.200 wt % N2. The diblock copolymer has 0.225 wt % N2 whereas PBMA has none, implying that our sample contains 89 wt % diblock copolymer and 11 wt % homopolymer. The unreacted PBMA should not play an important role since it adsorbs negligibly on the surface10 and is dilute in the bulk. In addition to the diblock copolymer, we also examined a sample of poly(butyl methacrylate) from Aldrich (Milwaukee, WI) having Mn ) 73.5 kg/mol and Mw/ (11) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (12) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62.

10.1021/la971383q CCC: $15.00 © 1998 American Chemical Society Published on Web 11/11/1998

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Figure 1. TEM of silica particles from the Stober synthesis at a magnification of 4.3 × 104. Mn ) 4.3. Reagent grade 2-propanol from Aldrich was filtered using a 0.22 µm Teflon (Millex FG) filter from Millipore Inc. (Bedford, MA). 2.2. Procedures. Wu et al.10 provided the polymer in tetrahydrofuran. After evaporation of the solvent in a vacuum oven, the dry polymer powder was dissolved in filtered 2-propanol to obtain a 10 000 ppm solution. The dispersion prepared by Stober’s process in an ethanol-water-ammonia mixture was exchanged with 2-propanol by repeated cycles of centrifugation, supernatant removal, 2-propanol addition, and redispersion, resulting in a dilute dispersion with volume fraction φ ) 0.005. To obtain the adsorption isotherm, the ratio of the polymer added to that required for an assumed surface coverage of 5 mg/m2 was varied between 0.2 and 50.0. The polymer solution and the dispersion were mixed and placed in a wrist shaker for 2-3 days to obtain a fully equilibrated, adsorbed layer. We measured the surface coverage via thermal gravimetric analysis (TGA).13,14 To prepare a sample, the (bare or coated) particles were settled in an ultracentrifuge, washed with pure solvent, and dried in a vacuum oven. Upon heating, the polymer breaks down fully, well before reaching the highest temperature (800 °C). Therefore, the difference between the weight fraction of the residue at 800 °C for the bare (wb) and coated (wc) particles yields the surface coverage as

Γ)

Fa(wb - wc) 3wc

(1)

with Fa/3 the mass per unit area for the bare particles. The surface coverage then determines the polymer concentration in the supernatant via mass balance.15 The diblock copolymer chains in the bulk are isolated coils, while those at the surface stretch to a layer thickness L depending on Γ and the interactions within the brush. We measure the hydrodynamic size of the chains in the bulk and at the surface using dynamic light scattering.16 For a dilute monodisperse suspension, the intensity autocorrelation function [F(τ)] decays exponentially. A second moment fit to the autocorrelation function

F(τ) ) exp[-2q2D0τ(1 - bq2D0τ)]

Figure 2. Schematic representation of an isolated polymer brush and assumed segment density profile. and b to account for polydispersity through the O(τ2) term. Only correlation functions with less than 0.1% difference between the measured and calculated baselines were accepted. The diffusion coefficient is related to the hydrodynamic radius rh by the StokesEinstein relationship

D0 ) kT /6πµrh

(3)

with µ the viscosity of the solvent and kT the thermal energy. The difference between the hydrodynamic radii of coated and bare particles is defined as the hydrodynamic layer thickness.

3. Mean-Field Theory for Polymer Brushes The effect of varying solvent quality on the layer thickness and the stability of a dispersion covered by a dense polymer brush is elucidated at least qualitatively by mean-field theory.11 Consider a surface fully covered at density σ ) NaΓ/M by polymer chains with N segments of length l (Figure 2). The excluded volume per segment v is related to the Flory χ parameter as v ) w1/2(1 - 2χ) where w1/2 ) C∞m0/(NaF) is the physical volume per segment, Na is Avogadro’s number, F is the bulk density of the polymer, and C∞ and m0 are the characteristic ratio and molecular weight per bond, respectively. 3.1. Isolated Brushes. The free energy per chain in the brush

(

)

3 L2 A Nl2 Nvn Nwn2 ) + + 2 -2 + 2 kT 2 Nl 2 6 L

(4)

(2)

is employed with q ) [4πn0 sin(θ/2)/λ], n0 the refractive index of the solvent, λ the wavelength of the laser, θ the scattering angle, (13) Herd, J. M.; Hopkins, A. J.; Howard G. J. J. Polym. Sci., Part C 1971, 34, 211. (14) Ayub, A. L.; Ling, S. S. J. Appl. Polym. Sci. 1990, 41, 419. (15) Hariharan, R. Ph.D. dissertation, Princeton University, 1996. (16) Pusey, P. N. In Photon Correlation and Light Beating Spectroscopy; Cummins, H. Z., Pike, E. R., Eds.; Plenum Press: New York, 1974.

incorporates the entropic or elastic contribution for ideal chains, plus contributions from the excluded and physical volumes, respectively. Assuming a constant segment density within the polymer brush17,18 such that n ) Nσ/L permits the bulk osmotic pressure to be equated with that in the layer as

∂ A )0 ∂L kT

( )

(5)

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Hariharan and Russel

h - Li ) Li, and the equilibrium is dictated by the value of Li that minimizes the free energy at a given separation. The total free energy per chain now can be assembled from eq 4 by summing the contributions from the entropy and excluded and physical volume interactions from each region to obtain

[

][

Ai 3 Li2 Nl2 N(h - Li)vn1 ) + 2 -2 + + 2 kT 2 Nl 2Li Li

][

2

N(h - Li)wn1 6Li

+

]

N(2Li - h)vn2 N(2Li - h)wn22 + 2Li 6Li

[

]

[

2

7φp 3 3z H 3H ) [Ri2 + Ri-2 - 2] + 1+ 12 2 2Ri 3Ri 7R 6Ri i

]

where the dimensionless layer thickness Ri ) Li/N1/2l depends on the dimensionless separation H ) h/N1/2l, in addition to φp and z. A minimum in the free energy with respect to the layer thickness

( )

∂ Ai )0 ∂Ri kT

(9)

is determined by

[

2

]

2

7φp Hφp zH z ) 0 (10) Ri + Ri5 - Ri2 - 1 + 2 9 3 2

Figure 3. Schematic representation of interpenetrating polymer brushes with piecewise continuous segment densities and homogeneous stretching.

The dimensionless layer thickness R ) L/N1/2l then depends on the excluded volume parameter

z ) N3/2σv/l

(6)

and the dimensionless surface density

φp ) Nσw1/2/l

(7)

through

[

R3 - 1 +

]

φp2 -1 z R ) 9 6

(8)

Conversely, eq 8 provides an estimate of the excluded volume v from known values of physical volume, surface coverage and layer thickness of an isolated polymer brush. 3.2. Interpenetrating Brushes. Along with adsorption and layer thickness we also probe the dispersion stability. Prediction or interpretation of stability, for a given solvent quality and surface coverage, requires the free energies associated with interactions between polymer brushes at separation h < 2L (Figure 3). If the brushes are simply compressed, the chain length Li is merely half the separation, Li ) h/2. Otherwise we assume uniform stretching of the chains through two regions with piecewise continuous concentrations: n1 ) Nσ/Li for 0 e x e h - Li or Li e x e h and n2 ) 2Nσ/Li for h - Li < x < Li. Compression is a special case with no interpenetration or (17) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (18) de Gennes, P. G. Macromolecules 1980, 13, 1069.

The interaction potential Φ per unit area is the free energy difference between surfaces at separation H and those at infinity and follows from the free energies above as

[

]

Ai(H,z,φp) A(z,φp) Φ ) 2σ kT kT kT

(11)

For our experimental conditions, we will present calculations for Ri and Φ for varying values of H and compare the calculated interparticle potential with observed stability of the dispersions. 4. Results and Discussion 4.1. Isotherms. Thermal gravimetric analysis spans 30-800 °C, in which the polymer decomposes completely, as shown in Figure 4 for a concentrated polymer solution. First 2-propanol evaporates from 30 to 150 °C, then the polymer decomposition occurs between 320 and 460 °C. Representative TGA plots of the bare and coated particles (Figure 5) yield wb ) 0.854 and wc ) 0.774 which with 3/aF ) 0.0135 m2/mg results in a surface coverage Γ ) 7.6 mg/m2 (eq 1). The adsorbed amounts and supernatant concentrations for varying ratios of added polymer mass to particle surface area (Figure 6) demonstrate that the adsorbed amount quickly reaches a plateau, implying strong adsorption. The free energy penalty for confining a chain on the surface is clearly overwhelmed by the attraction of the PDAEM block to the surface. Once the surface becomes crowded, excluded and physical volume interactions extract a stiff penalty for every additional chain at the surface, explaining the plateau. The adsorbed amount at the plateau (8.0 ( 0.5 mg/m2) is slightly higher than that reported by Wu et al.,10 for the same system (≈6.8 mg/m2). This could reflect different surface charge densities on the silica or a lower concentration at which the adsorbed amount was measured. In any case, both plateau coverages are much

Enhanced Colloidal Stabilization

Figure 4. Weight loss of a concentrated polymer solution measured by thermal gravimetry.

Langmuir, Vol. 14, No. 25, 1998 7107

Figure 7. Plot of layer thickness measured by dynamic light scattering (filled circles) versus supernatant concentration compared with the prediction from the mean-field theory with γ ) 4.7 and w1/2/l3 ) 0.78.

substantially reducing the energy penalty due to excluded volume interactions. The crowding of chains on the surface can be gauged by the ratio of the area per chain in the bulk to that at the surface, Nl2σ. With Γ ) 8 mg/m2, Nl2 σ ) 7, implying that the chains are highly crowded because they strongly prefer the surface to the bulk. Beyond C ) 6000 ppm, the surface coverage apparently rises well above the plateau value. One may suspect this to reflect a rise in the bulk chemical potential, but that should occur only beyond C*, the transition from the dilute to the semidilute regime, given by

C* ≈

Figure 5. Weight loss of vacuum-dried, bare (dotted line) and coated (solid line) silica particles measured by thermal gravimetry.

Figure 6. Plot of surface coverage measured by thermal gravimetry (filled circles) versus supernatant concentration of diblock copolymer. The smooth curve is the fit of the mean-field theory with γ ) 4.7 and w1/2/l3 ) 0.78.

higher than those typically (1-3.5 mg/m2) reported elsewhere4,19 for adsorbing diblock copolymers of comparable molecular weights and asymmetries. In our case the PBMA adsorbs from a Θ, rather than a good solvent,

6M ≈ 105 ppm πN3/2l3Na

(12)

Since C* is roughly 17 times the concentration associated with the increase in surface coverage, we have no mechanistic explanation for the effect, but the correlation with layer thickness described below demonstrates the validity of the measurement. Dynamic light scattering provided hydrodynamic radii of the bare as well as coated particles for deducing the hydrodynamic thickness of the adsorbed layer. For the bare particles a )132.5 nm with a polydispersity b < 0.06 and typically a 5% change over scattering angles of 30° to 160°. The scattering from particles easily overwhelms that from free chains due to the greater size and refractive index mismatch, yielding no detectable difference in hydrodynamic size with or without the supernatant polymer. The coated particles exhibited slightly higher polydispersity (b ≈ 0.1) and typically less than 10% change in the hydrodynamic size from 30° to 160°. The layer thickness from measurements at θ ) 90° (Figure 7) exhibits an increase with polymer concentration that resembles the trend in surface coverage in Figure 6, a quick rise to a plateau that remains roughly constant from 1000 to 5000 ppm. The layer thickness at the plateau (≈20 nm) is about 1.4N1/2l, indicating that the chains stretch toward the solution seeking to reduce the dense packing near the surface. At concentrations greater than 5000 ppm, the layer thickness increases further, in parallel with the rise in surface coverage (Figure 6). To check the consistency, we (19) Motschmann, H.; Stamm, M.; Toprakcioglu, C. Macromolecules 1991, 24, 3681.

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Hariharan and Russel

Figure 8. Plot of layer thickness versus surface coverage along with the best fit straight line (slope ≈ 0.53).

plot layer thicknesses versus surface coverage in Figure 8 revealing a reasonable power law dependence as L ≈ Γ0.5(0.1. (The uncertainty in the slope arises solely due to possible errors in measuring the diameter of the coated particle. About 50% of the error bars in Figure 8 from systematic uncertainty in the diameter of the bare particle should not contribute.) To understand this, one need only return to eq 8 and neglect the excluded volume term due to the proximity to the Θ temperature of PBMA, leaving for φp . 1

R ∼ φp1/2

(13)

Figure 9. Hydrodynamic layer thickness (filled circles) and diameter of isolated chains in the bulk (filled squares) as a function of temperature: N1/2l ) 14.6 nm (dotted line), Θ temperature20 (thick line), and cloud points of PBMA sample (filled triangle up) and diblock copolymer (filled triangle down).

) kT ln(C/F), must equal that for adsorbed chains as

()

2Nσl2

w l6

1/2

- NAγ ) kT ln

L ∼ σ1/2 ∼ Γ1/2

(14)

Thus, the slope expected from the theory agrees well with the slope observed in Figure 8. Earlier, we presented a simple extension to the Marques-Joanny mean-field theory to cover adsorption of a diblock copolymer from a Θ solvent.15 Here we simplify that formulation to a case of a short adsorbing block. The theory incorporates physical volume and recognizes that the adsorbing block probably lies flat on the surface. We express the free energy of adsorbed chains as

[

]

( )

Aa Nl2 Nw Nσ 3 L2 ) -NAγ + + 2 -2 + 2 kT 2 Nl 6 L L

2

(15)

where γ is the segmental adsorption free energy and NA and N are the number of segments per chain of the adsorbing (PDMAEM) and solvated (PBMA) blocks, respectively. Minimization with respect to L at fixed σ predicts the layer thickness for strong stretching or high surface coverage as

L)

()

w 1 Nl 6 l x3

1/4

xσl2

(16)

and the associated free energy per chain as

()

Aa w ) -NAγ + N 6 kT l

1/2

σl2

(17)

The chemical potential of polymer chains in the bulk, µb

C F

(18)

The mean field theory will fail in the dilute limit before a uniform layer forms but should capture the plateau. At high concentrations the balance between the adsorption energy and the physical volume term yields

()

l6 1 Nσl2 ≈ NAγ 2 w

or via eq 7

()

1/2

(19)

Figure 6 shows that the theory captures the plateau with the physical volume of PBMA calculated from w1/2 ) c∞v˜ m0/Na ) 0.78l3 and γ ) 4.7. The corresponding layer thickness, calculated from eq 16 with the measured adsorbed amount and w1/2/l3 ) 0.78, falls well within the uncertainty of the measured values (Figure 7) through the plateau region but misses the secondary rise. Thus both the plateau in both the adsorbed amount and layer thickness are described by the theory if γ ≈ 4.7. 4.2. Effect of Temperature. Since the Θ temperature of PBMA, 20.9-25 °C,20 is close to room temperature, we have a convenient model system for studying changes in layer thickness and dispersion stability when passing through the Θ temperature. Our observations for isolated polymer coils in the bulk provide a benchmark for understanding the adsorbed layer. We measured the cloud point of a polydisperse PBMA sample (Mn ) 73.5 kg/mol and Mw/Mn ) 4.3) as well as the diblock copolymer (Mn ) 112.0 kg/mol and Mw/Mn ) 1.15) by lowering the temperature of a 1000 ppm solution in 0.1 °C increments and looking for a significant increase in the scattering intensity. The cloud point for the homopolymer, 19.1 °C, is slightly lower than the Θ temperature (20.9-25 °C) as shown along the x-axis of Figure 9, since the two coincide only at infinite molecular weight.21 The cloud point for the diblock copolymer, 11.7 °C, is significantly lower than the homopolymer, because of higher polarity imparted by (20) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; WileyInterscience: New York, 1989. (21) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.

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Langmuir, Vol. 14, No. 25, 1998 7109

the amino group of the PDMAEM block and the lower total molecular weight. The hydrodynamic diameters of diblock copolymer chains in solution, from dynamic light scattering around 300 ppm (,C*), are remarkably insensitivity to temperature from 13 to 41.2 °C (Figure 9). In stark contrast, the layer thickness for particles on the plateau of the adsorption isotherm (Γ ) 8.35 mg/m2 corresponding to C ) 4518 ppm in Figure 6) increases monotonically with increasing temperature. The error bars for layer thickness here are about half of those in Figure 7, because the uncertainty in bare particle radius ((3 nm) is systematic in nature. It must be pointed out that the surface coverage is constant in these experiments despite increasing temperature. We have measured the same value of layer thickness at room temperature before and after long heating or cooling cycles in the range of 5-35 °C. At constant temperature, for coated particles surrounded by solvent alone, the chains show no tendency to desorb as validated by constant layer thickness over a very long time (up to 3 months). Both observations suggest that the chains are trapped in steep energy wells and cannot jump out to attain global equilibrium. The size and layer thickness vary with temperature as excluded and (to a lesser extent) physical volume interactions change. Both are functions of local segment densities (eq 4), suggesting two factors as responsible for the greater sensitivity of polymer chains at the surface compared to those in the bulk: (1) The polymer segments at the surface are more densely packed than those in the bulk. The ratio of segment densities can be estimated from the known surface coverage, layer thickness, and the bulk polymer dimension as

[ ]/[ Nσ L

]

6N ) 3.6 πN3/2l3

(20)

for Γ) 8.0 mg/m2, L ) 20.0 nm, and N1/2l ) 14.6 nm. (2) The swelling of polymer chains in the bulk is isotropic, while those at the surface are confined laterally by neighboring chains and must extend away from the surface, magnifying the sensitivity. We also observe that dispersions of coated particles are stable well below the Θ temperature of PBMA. At a volume fraction φ ≈ 10-4 in 2-propanol of viscosity µ ≈ 2 × 10-3 Pa‚s, the time scale for doublet formation11

td )

µa3 φkT

≈ 10 s

(21)

is far exceeded by our experimental times (up to several days), implying that the observed stability is thermodynamic. Measuring the dimensions of coated particles successfully to temperatures as low as 6.5 °C opposes the traditional view22 that polymerically stabilized dispersions flocculate at the Θ temperature. We attribute this enhanced stability to the physical volume, which provides a positive contribution to the free energy that offsets the negative excluded volume. 4.3. Interaction Potentials from the Mean-Field Theory. To quantify the effects of temperature on the layer thickness and stability, we turn to the mean-field theory for isolated and interpenetrating polymer brushes. The surface coverage and the parameters characterizing the polymer listed in Table 1 determine the chain density (22) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983.

Table 1. Parameters Characterizing PBMA from Brandrup and Immergut20 molecular weight per bond m0 ) 0.071 kg/mol projected length of bond l0 ) 0.126 nm characteristic ratio C∞ ) 8.5 statistical segment length l ) C∞l0 ) 1.071 nm number of segments per chain N ) 180.9 bulk density F ) 1050[1 - 6 × 10-4(t, °C - 20)] kg/m3 Table 2. Variation of Excluded Volume with Temperature from Measured Layer Thicknesses t, °C

L, nm

φp

R

z

v/l3

6.5 16.2 24.4 33.8

19.4 21.6 25.2 26.2

7.13 7.18 7.23 7.30

1.35 1.50 1.75 1.82

-14.97 -6.71 8.74 13.26

-0.12 -0.05 0.07 0.11

as σ ) ΓNa/M, which with the physical volume parameter w1/2 yields φp (eq 7). To estimate the temperature dependence of the excluded volume, we apply the mean-field result for an isolated brush (eq 8) to determine z and v/l3 (via eq 6) from the temperature dependence of the dimensionless layer thickness R (Figure 9) and φp (Table 2). The decreasing polymer density has a small effect on φp, so the change in R is almost fully due to changes in the excluded volume parameter z or, equivalently, v/l3. The estimated values for v/l3 are negative below the Θ temperature and positive at higher temperatures. Therefore, the mean field theory successfully captures the variation of layer thickness with temperature with acceptable values of v/l3. We now fix the values of φp and z at the temperatures studied and calculate the interaction potential as a function of separation 0 < H < 2R via the theory explained earlier in section 3.2. The thickness Ri of the interacting layers is bounded by H/2 and H, i.e., between compression and complete interpenetration. The minimum free energy can lie at either of these extremes or an intermediate value dictated by eq 10. This polynomial has five solutions, but for the values of z and φp in Table 2 either one or no real solutions lie between H/2 and H. The other solutions are either complex, out of the physical range, or correspond to maxima in the free energy. If a minimum is found, the free energy is determined and compared with that at the extremes Ri ) H/2, H. The Ri corresponding to the lowest free energy determines the equilibrium conformation. For all four temperatures, the minimum free energy corresponds to Ri ) H/2 or H (Figure 10). Thus, the theory predicts either complete interpenetration (0 e H e 21/2) with Ri ) H or compression (21/2 e H e 2R) with Ri ) H/2, with the transition always occurring at H ) 21/2, independent of z or φp. This is readily verified from eq 9, by setting

Ai(H/2) Ai(H) ) MkT MkT

(22)

and obtaining a unique solution for H at all values of z and φp. This abrupt transition is undoubtedly an artifact of the mean field theory. The interaction potential Φ between flat plates follows from eq 11 for varying surface separation. For Ri from Figure 10, Φ/(σkT) is positive for all four temperatures (Figure 11), implying that the dispersions should be thermodynamically stable for T g 6.5 °C in accord with our observations. Thus, below the Θ temperature the high surface coverage (Γ ) 8.35 mg/m2) generates enough

7110 Langmuir, Vol. 14, No. 25, 1998

Figure 10. Dimensionless layer thickness as a function of dimensionless separation at 6.5 °C (solid line), 16.2 °C (dotted line), 24.4 °C (dashed line), and 33.8 °C (long dashed line).

Figure 11. Scaled interaction potential as a function of dimensionless separation at 6.5 °C (solid line), 16.2 °C (dotted line), 24.4 °C (dashed line), and 33.8 °C (long dashed line).

repulsive energy via physical volume to overcome the effect of attraction caused by negative excluded volume. Since our dispersions are thermodynamically stable for Γ ) 8.35 mg/m2 and v/l3 ) -0.12 at 6.5 °C, two questions arise naturally: (1) If the solvent quality and temperature are held fixed, at what surface coverage will the thermodynamic stability break down? (2) If the surface coverage and temperature are held fixed and a poor solvent is mixed with 2-propanol, at what effective value of v/l3 will the thermodynamic stability break down? We answer these questions within the approximations of the mean field theory in Figures 12 and 13, respectively. Figure 12 displays the scaled interaction potential for Γ ) 8.35, 3.0, 2.5, and 2.0 mg/m2, which affects both φp and z. The consequences are reflected in Φ, which develops an attactive minimum for Γ < 2.5 mg/m2. With Γ ) 8.35 mg/m2, we find that v/l3 < -0.39 will induce instability (Figure 13). The summary of dimensionless parameters for the various surface coverages and excluded volumes in Table 3 indicates that incipient flocculation corresponds to φp2 ≈ -z. In conclusion, polymeric stabilization below

Hariharan and Russel

Figure 12. Scaled interaction potential as a function of dimensionless separation with v/l3 ) -0.12 and Γ ) 8.35 (solid line), 3.0 (dotted line), 2.5 (dashed line), and 2.0 (long dashed line) mg/m2.

Figure 13. Scaled interaction potential as a function of dimensionless separation with Γ ) 8.35 and v/l3 ) -0.12 (solid line), -0.3 (dotted line), -0.39 (dashed line), and -0.43 (long dashed line). Table 3. Parameters for Calculations with Varying Surface Coverages and Excluded Volumes Γ, mg/m2

v/l3

φp

z

-z/φp2

8.35 3.0 2.5 2.0 8.35 8.35 8.35

-0.12 -0.12 -0.12 -0.12 -0.30 -0.39 -0.43

7.13 2.56 2.13 1.71 7.13 7.13 7.13

-14.97 -5.38 -4.48 -3.58 -37.6 -48.9 -53.9

0.29 0.82 0.98 1.22 0.74 0.96 1.06

Φmin/(σkT) f0 f0 -0.65 f0 -1.54

Θ conditions can be realized with a high density of adsorbed or grafted chains. 5. Conclusion Diblock copolymer adsorption onto silica from a nonselective Θ solvent, following previous work,10 generates extraordinarily high surface coverage and leads to the following conclusions: (1) The surface coverage and layer thickness at the plateau can be described via a simple mean-field theory in the spirit of Alexander and Marques and Joanny with

Enhanced Colloidal Stabilization

reasonable values for the adsorption energy and excluded volume, provided the physical volume or third virial term is included in the free energy. (2) The chain dimension at the surface is much more sensitive to temperature than that in the bulk, due to the higher segment densities and one-dimensional swelling at the surface. (3) The effect of temperature on the layer thickness is understood via a simple mean-field model11 with reasonable values of excluded volume parameters. (4) The main result of this paper is the finding that thermodynamic stability well below the Θ temperature can follow from the physical volume of the adsorbed chains,

Langmuir, Vol. 14, No. 25, 1998 7111

which overcomes the negative excluded volume at sufficiently high adsorbed amounts in qualitative accord with earlier predictions.11 Acknowledgment. The authors thank D. T. Wu, A. Yokoyama and R. L. Setterquist of E. I. Dupont Nemours & Co. for synthesis and characterization of the diblock copolymer, G. Chavauteau, C. Biver, and M. Tirrell for insightful discussions, and the National Science Foundation through NSF/CTS-9107025 and the MRSEC/ DMR9400362 for their financial support. LA971383Q