Elastic Modulus at High Frequency of Polymerically Stabilized

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Elastic Modulus at High Frequency of Polymerically Stabilized Suspensions P. A. Nommensen,* M. H. G. Duits, D. van den Ende, and J. Mellema Rheology Group, Department of Applied Physics, University of Twente, PO Box 217, 7500 AE, Enschede, The Netherlands Received June 2, 1999. In Final Form: October 27, 1999

The elastic moduli of polymerically stabilized suspensions consisting of colloidal silica particles coated with endgrafted PDMS (Mn ) 80 000) in heptane, were measured as a function of concentration. And the elastic modulus at high frequency G′∞ was quantitatively described by model calculations with microscopic parameters. In the modeling of G′∞ we essentially followed the method of Elliott and Russel [Elliott, S. L.; Russel, W. B. J. Rheology 1998, 42, 361]. To suit our case of strongly curved polymer layers, we made adaptations in the description of both the pair potential and hydrodynamic interactions. Anticipating that the predicted G′∞ is sensitive to the modeling of the polymer brush, we explored two different models to predict the pair potential and the polymer layer thickness: a numerical self-consistent field lattice model [Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214] and an analytical method based on the approach of Li and Witten [Li, H.; Witten, T. A. Macromolecules 1994, 27, 449]. The polymer models were used separately in the model calculations for G′∞ with the magnitude of their parameters obtained from (elaborate) characterizations of the particles. Both models result in a quantitative description of the experimental G′∞ and thickness of the polymer layer.

1. Introduction The prediction of the macroscopic behavior of polymerically stabilized dispersions from microscopic properties is desirable in view of numerous scientific and technological applications. This is a demanding undertaking even for well-defined and conceptually simple systems. The high-frequency elastic modulus G′∞ is a good example. A complete microscopic model for G′∞ has to take into account particle configuration, pair interactions, Brownian motion, and hydrodynamics. Such modeling has been developed so far only for hard sphere dispersions1 and charged spheres.2 For polymerically stabilized suspensions no model exists to our knowledge that has given a quantitative description for G′∞. Frith et al.3 gave a scaling analysis but Elliott and Russel4 were the first to tackle this problem using microscopic properties of the polymer layer. They achieved qualitative agreement between their theoretical predictions and the experimental results of D’Haene but the magnitude of the calculated G′∞ turned out to be too large. They ascribed the observed difference to an imperfect description of the pair potential. The pair potential Φ is an essential ingredient in their model. The potential Φ appears explicitly in expressions describing G′∞ but also indirectly via the particle configuration. This underlines the need for an adequate description for Φ when the elasticity of a hairy particle dispersion is to be calculated. In this paper we will attempt to model Φ (and from this G′∞) for a system of monodisperse particles coated with endgrafted polymers. For such a system, polymer theory can be used to calculate Φ. The pair potential Φ can be calculated by considering the change in the free energy (or Helmholtz potential) of the polymer layer Fpolymer under compression. This occurs (1) Lionberger, R. A.; Russel, W. B. J. Rheology 1994, 38, 1885. (2) Wagner, N. J. J. Colloid Interface Sci. 1993, 161, 169. (3) Frith, J. F.; Strivens, T. A.; Mewis J. J. Colloid Interface Sci. 1990, 139, 55. (4) Elliott, S. L.; Russel, W. B. J. Rheology 1998, 42, 361.

when the separation r12 between two particle centers becomes less than twice the outer radius of the particles (this is the sum of the radius of the core Rc and the equilibrium polymer layer thickness L). For particles stabilized by endgrafted polymer chains, Fpolymer can be calculated using existing theories for polymer brushes and making use of characteristic parameters of polymer and solvent. The magnitude of Fpolymer depends strongly on the distribution of segments n(r b) within the polymer layer. b) has spherical symmetry making For r12 > 2[Rc + L], n(r a one-dimensional description sufficient. But for r12 < 2[Rc + L], this symmetry is broken since the deformation is not uniform over the polymer layer. The segment distribution n(r b) now has a cylindrical symmetry. However, a calculation of Φ based on Fpolymer in two dimensions is very complex. So far no 2D-analytical expression for Fpolymer has been found. The magnitude of Fpolymer can be obtained numerically using the self-consistent field lattice theory of Scheutjens and Fleer.5-8 However the computational burden to calculate in 2D is high.8 Results of a 1D description can be used to get a quasi-2D model by considering local compression and ignoring any lateral relaxations in n(r b). In this paper we will follow this quasi2D approach since the quasi-2D approach is much more practical to use, especially for fitting experimental data. Even in this simplified scheme, one still has to calculate the radial segment density profile of a spherical polymer brush. Only a few investigators have reported (attempts to obtain) analytical expressions. Ball and co-workers9 published an approach to find it for a polymer brush on a curved surface. Unfortunately they were not able to (5) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (6) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (7) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214. (8) Wijmans, C. M.; Leermakers, F. A. M.; Fleer, G. J. Langmuir 1994, 10, 4514. (9) Ball, R. C.; Marko, J. F.; Milner, S. T.; Witten, T. A. Macromolecules 1991, 24, 693.

10.1021/la9906828 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/11/2000

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solve the interesting, for us, case of a spherical polymer brush on a solid sphere in a solvent. Another method was given by Li and Witten,10 who used a variational approach to minimize the free energy. They used this technique for polymer brushes on a flat surface and for star polymers i.e., spherical brushes on a sphere with radius zero. Their results for the flat surface case are equivalent to those of Milner.11,12 In the case of the star polymers they needed extra assumptions to solve the problem. In this paper we will extend their results for star polymers by introducing a nonzero core radius to obtain an analytical expression for Fpolymer. In addition to the analytical approach to obtain Fpolymer we also perform 1-dimensional self-consistent field lattice calculations and use these to get another expression for Φ. Both models for Φ are used separately to calculate G′∞ and the equilibrium layer thickness. Subsequently we compare the calculated results with our experimental data. In this fashion we can test if the reasoning of Elliott and Russel about Φ is correct. We are also able to compare the two polymer models to experimental results. This paper is further organized as follows. In the next section we briefly summarize the model of Elliott and Russel. We modified the modeling for Φ and parts of the expressions for the hydrodynamic interaction. Both these modifications are treated in detail. In Section 3., the characterization of the experimental system is given together with a description of the experimental setup. In Section 4., the results of the calculations are compared with each other and with the experimental data. 2. Theoretical Formulation Our model calculations for G′∞ are based on the work of Elliott and Russel (ER).4 ER find an expression for G′ in the limit of small amplitude oscillatory shear at high frequencies. Their model incorporates Brownian, thermodynamic, and hydrodynamic interactions which are all strongly influenced by the polymer layer on the particles. Overlap of the polymer layers of two neighboring particles gives rise to a repulsive pair potential Φ. In addition, the polymer layer hinders the solvent flow, which changes the hydrodynamic and Brownian interactions between particles. ER evaluate the bulk stress of the suspension in a similar fashion as Lionberger and Russel1 did for hard sphere suspensions, by introducing conditionally averaged pair hydrodynamic functions. Following this approach, one can determine the changes in the microstructure when oscillatory shear with high frequency and small amplitude is applied. Due to these changes in the microstructure the free energy of the suspension increases, producing a stress. With these assumptions ER calculated the stress component in phase with the shear, resulting in an expression for the elastic modulus G′∞

gs r2 dr ∫(W - (1 - A)r12dΦ/kT dr12 ) i 12 12

R3c G′∞ 3φ2 ) 3 kT R 5π c

(1)

with φ the particle core volume fraction and r12 the center to center separation. The product g si reflects the changes (10) Li, H.; Witten, T. A. Macromolecules 1994, 27, 449. (11) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (12) Milner, S. T. Europhys. Lett. 1988, 7, 695.

Figure 1. Configuration of the polymer layers when two particles are pushed together.

in microstructure, with g(r12) the equilibrium radial pair distribution function, which is obtained using Monte Carlo simulations and

d ln g si ) W + (1 - A)r12 dr12

(2)

where W and A are hydrodynamic functions which will be discussed in Section 2.2. The quantities Φ, A, W and g depend on properties of the polymer layer. The modeling of ER for the first three of these functions was developed for particles with thin polymer layers (L , Rc). To get a description for particles with thicker layers it was necessary to modify a number of expressions. In addition our pair potential was obtained in an entirely different fashion. These changes are discussed in the following subsections. 2.1. Pair Interaction Potential. When the distance between two particle centers becomes smaller than twice the outer radius h, the polymer layers deform. The herewith occurring compression increases the Helmholtz potential of the polymer layer Fpolymer thus causing a repulsive interaction. From the change in Fpolymer the particle pair potential Φ can be obtained in a direct way. When calculating Fpolymer, one has to keep in mind that the compression of the polymer layer is not uniform over the entire polymer layer as can be seen in Figure 1. The compression is maximal along the center to center line and decreases when moving away from this line. The nonuniform compression breaks the spherical symmetry of n(r b) which increases the complexity of the calculation. However we will show that it is possible to relate Φ to Fpolymer for the uniformly compressed polymer layer. To do this we assume that no lateral relaxation of the polymer chains occurs which is justified for dense brushes under small compression. For the moment we also assume no interpenetration of the brushes (this assumption will be dropped in Sections 2.1.1. and 2.1.2.). The description then simplifies to that of a polymer layer on a sphere that is pushed against a flat surface with polymer chains only compressed in the radial direction (see Figure 2). The local segment density profile along the radial direction with local layer thickness Lloc is similar to the segment density profile under a uniform compression with layer thickness equal to Lloc. The averaged free energy of a polymer chain in a compression zone with local layer thickness Lloc can thus be related to Fpolymer under a uniform compression

Fchain(huni) )

Fpolymer(huni) σ4πR2c

(3)

with huni ) Rc + Lloc the radius of the uniform compressed particle and σ the grafting density. The particle pair potential Φ now follows from an integration of Fchain(huni) - Fchain(hequi) over all polymers

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of the tth segment for a chain with its free end at ro, v an excluded volume parameter, and n the number density of segments given by h o ∫Max[r,r ]( r )

r

n(r) )

Figure 2. Configuration of the polymers before (a) and after (b) two particles are pushed together: chains are confined to cone-shaped volume elements that are flattened off by the symmetry plane of the particle pair.

in the deformation zone

Φ(r12) ) 2

∫0θ ∫02π[Fchain(h(θ)) max

Fchain(hequi)] σR2c sin(θ) dθdφ )

∫r

1 2

hequi 12

2

r12 [Fpolymer(h) - Fpolymer(hequi)] 2 dh (4) h

For the calculation of G′ using eq 1 the derivative of Φ with respect to r12 is needed. Differentiating eq 4 results in

Fpolymer(r12/2) - Fpolymer(hequi) Φ(r12) ∂Φ )+ ∂r12 r12 r12

(5)

In the following subsections we present two different methods to calculate Fpolymer under a uniform compression. 2.1.1. Polymer Brush on a Curved Surface: An Analytical Approach. The analytical description of the spherical polymer brush we give here is an extension of a method reported by Li and Witten.10 This LW method allows a calculation of the Helmholtz potential of a star polymer as a function of the layer thickness. We extended the LW method by incorporating a hard core with a nonzero radius Rc. LW use a mean field expression for the free energy of a star polymer with many arms in a marginal solvent. It is assumed that the arms are in the strong stretching regime (in which spatial fluctuations of chain segments are unimportant). So only the radial stretching of the polymers has to be considered. It is further assumed that the radial position rs of the tth segment is given by a function which depends only on the index of that segment and the position of the segment at the free end of the chain. The positions of these free ends (designated as ro) need not be the same for all arms, which makes the use of a number density distribution F(ro) necessary. The expression for the free energy of the polymers

Fpolymer kT

∫rh∫ c

(

)

N 1 drs(ro,t) 0 dt 2a2

2

dt F(ro) 4πr2o dro +

∫0hn2(r) 4πr2dr

ν 2

(6)

consists of an elastic term and an excluded volume interaction term where N is the number of segments per arm, a the length of one segment, rs(ro,t) the radial position

2

c

dt(r,ro) dro dr

F(ro)

(7)

The excluded volume term originates from a virial expansion where all higher order interactions are neglected which is justified for small values of n. The first integral in the first term is from rc to the layer height h (both measured from the particle center). In this fashion the insights of Semenov13 are followed which predict an exclusion zone for free ends up to a certain position rc. Fpolymer is a functional of rs(ro,t), F(ro), and n(r). The functions rs(ro,t), F(ro), and n(r) can be found by variation of Fpolymer with the constraint that the total number of segments in the polymer layer is constant. However carrying out such a minimization in a rigorous manner would be a formidable task. LW did not do the full variation but used an ansatz function for rs. They inverted eq 7 to get an expression for F and found a relation for n(r) by minimizing Fpolymer using the above-mentioned constraint. Substituting rs, F, and n back into eq 6 then gives an expression for Fpolymer as a function of h and rc, which can be further minimized. LW showed that the value for rc where Fpolymer is minimal, is also the only value that yields a continuity of n(r) at rc. The layer thickness hequi which minimizes Fpolymer can subsequently be found by setting

∂Fpolymer |h)hequi ) 0 T n(hequi) ) 0 ∂h

(8)

We have extended the algorithm given above to solid particles with a spherical polymer brush around them, since a star polymer can be seen as a limiting case of this kind of particles, i.e., the radius of the core equals zero. Since the core of the particle excludes segments, a few alterations are necessary. The first segment of each chain is now positioned on the core surface so the stretching function must be adjusted. Also the lower integration limit of the excluded volume term in eq 6 must be changed from zero to Rc. The relation for Fpolymer then becomes

F h polymer )

∫rjhh ∫01 c

(

drjs(rjo,th) dth

)

2

dthFj(rjo) 4πrj2o drjo +

∫Rhhh nj 2(rj) 4πrj2drj

1 2

c

(9)

in which we used the nondimensional parameters defined as follows

( ) ( )

Fpolymer f 7Nν2 -1/5 , rj ) r(fN3a2ν)-1/5, kT a6 f 2 -1/5 f 2 -1/5 t j)n 4 6 3 ht ) , n , Fj ) F 9 6 3 (10) N Naν Naν

F h polymer )

(

)

where we recall that f is the number of arms per particle. LW found the ansatz for rs(ro,t) considering the scaling arguments of Daoud and Cotton14 for star polymers. We use a similar strategy by demanding that the stretching function for the limiting cases of Rc ) 0 and Rc f ∞ equals (13) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733. (14) Daoud, M.; Cotton, J. P. J. Phys. 1982, 43, 531.

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that of a star polymer and a polymer brush on a flat surface, respectively. The function 5/3 t + R5/3 rs(ro,t) ) (r5/3 0 - Rc ) c N

[

]

3/5

(11)

has such limiting behavior and eq 7 can still be inverted as is needed to find a relation for F(ro). Substituting eq 11 in eq 9 after using eq 10, and minimizing eq 9 with respect to n j gives

{

n(rj,R h c) ) -λ-

() rj jrc

9 (11rjc, 50

-4/3

5/3

- 10rj4/3R h c1/3 - R h c5/3)(rj5/3 - R h 5/3) jr4/3

n j -(rjc,R h c)

h c) ) n j -(rjc, R

for rjc e rj < h h

as a mirror this depletion zone disappears. A chain that is reflected by the wall is similar to a chain that penetrates the polymer layer of the other particle. Using the mirror to compress the polymer layer causes a decrease of Fpolymer of more than an order of magnitude. Unfortunately we cannot use these results to say anything about the occurrence of interpenetrations due to the depletion at the impenetrable wall. The Helmholtz potential can be used to obtain an expression for Φ by using eq 4, where we used the mirror to compress the polymer layer in all the model calculations presented in this paper. 2.2. Hydrodynamic Interactions. Elliott and Russel found an expression for the hydrodynamic function (1 - A)

for rj < rjc

(12) 2 5/3 5/3 h 5/3 h 5/3 λ rjc - R 9 (rjc - R c c ) 5 rj4/3 (rj1/3 - R h 1/3) 50 rj4/3 c

c

c

1+ (13)

c

with n j - the magnitude of the segment density at rjc just inside the exclusion zone that minimizes F h polymer. rjc is found in a similar fashion as in the star polymer case and λ is a Lagrange multiplier that takes care of the number of segments constraint. The magnitude of λ follows from solving

∫Rhhh nj (rj)4πrj2drj ) 1 c

(1 - A) )

(14)

When calculating Fpolymer in eq 9 the integration is truncated at h h which reflects the confinement of the polymer layer into a sphere with radius h h . To check the no-interpenetration assumption, we varied the degree of interpenetration from zero to maximal. Doing this no change in the local segment density occurs so the excluded volume interaction term in eq 6 remains constant. But the elastic term increases since a polymer chain that penetrates into the polymer layer of the other particle is extra stretched. Hence the net change of the Helmholtz potential is positive. So this approach predicts no interpenetration when two polymer layers are pushed together. 2.1.2. Polymer Brush on a Curved Surface: SelfConsistent Field Lattice Model. This model is an extension of the polymer adsorption theory of Scheutjens and Fleer5,6 and is already published by Wijmans and Zhulina.7 Here we only briefly summarize the basic concepts. A polymer-solvent system at an interface is described using a lattice where each lattice site is occupied by either a polymer segment or a solvent molecule. The equilibrium distribution is calculated by taking into account all possible conformations, each weighted by its Boltzmann probability factor. Spherical surfaces with terminally attached (“endgrafted”) polymers can be modeled by restricting the first segment of a chain to the layer adjacent to the surface15 and using a curved geometry.16 The Helmholtz potential for uniformly compressed layers can be calculated directly without the need for a virial expansion. Special attention must be paid to the wall that compresses the polymer layer. If this wall is chosen to be impenetrable, a depletion zone for the segments occurs near the wall confining the polymer layer to a smaller volume then was intended. If the wall acts (15) Cosgrove, T.; Heath, T.; Van Lent, B.; Leermakers, F.; Scheutjens, J. Macromolecules 1987, 20, 1692. (16) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Phys. Chem. 1989, 93, 7417.

1 F h lub

(15)

Rη′r,∞

by balancing the (hydrodynamic) forces on two particles in a concentrated suspension undergoing shear flow. Tangential forces (i.e., not directed along the line connecting the particle centers) were not taken into account. The direct interaction between the two particles is modeled by the lubrication force F h lub between the particles (scaled by 6 πµRc Ur with µ the solvent viscosity and Ur the relative velocity of the particles). Interactions due to other particles are approximated in a mean field sense by taking them to be proportional to the relative high-frequency viscosity η′r,∞. The presence of the polymer layer changes these interactions with respect to the case of hard spheres of radius equal to that of the core. The parameter R accounts for this. The properties of the polymer layer must also be taken into account in the modeling of F h lub. In this paper we use a different modeling for Flub than ER did. We here adopt our approach17 in which we extended the work of Potanin and Russel18 by improving the geometric description of the polymer layer. In this fashion we found good agreement between experimental results and modeling for the high shear viscosity. Our expression used for the viscosity is more simple than the one used by ER. It is a summation over the solvent viscosity, the single particle contribution to the viscosity, and the viscosity due to lubrication

(

5 L ηr,∞ ) 1 + φ 1 + 2 Rc

)

3

+

µlub µ

(16)

where the latter can be obtained from F h lub

2Rc µlub )9 F h (H ) µ 2Rc + Hav lub av

(17)

with Hav the mean separation between core surfaces. It can be related to the volume fraction of the cores φ using a geometric argument

Hav ) 2Rc

[( ) φmax φ

1/3

-1

]

(18)

Anticipating that the particle configuration under small amplitude oscillatory shear is different from the configuration under steady shear; we used a different maximum (17) Nommensen, P. A.; Duits, M. H. G.; Ende, D. van den; Mellema, J. Phys. Rev. E 1999, 59, 3147. (18) Potanin, A. A.; Russel, W. B. Phys. Rev. E 1995, 52, 730.

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core volume fraction φmax for the present study. φmax was found to be 0.72 by fitting the experimental high-frequency viscosity of hard spheres as found in ref 19. The hydrodynamic function W can directly be related to A if tangential forces can be ignored

W ) -3A - r

dA dr

(19)

3. Experimental Section 3.1. Materials. All measurements were performed with a suspension of particles (coded SMD65-ps347.5) that consist of a silica core coated with a layer of end-grafted polydimethylsiloxane (PDMS) in the solvent heptane at a temperature of 25 °C. Both synthesis and characterization are elaborately reported by Nommensen et al.20 Here we summarize these results briefly. The particles were synthesized in two steps. First bare silica cores were prepared according to the method of Sto¨ber et al.21 PDMS molecules with number averaged polymer molecular weight Mn ) 8 × 104 g/mol and polydispersity Mw/Mn ) 1.4 were grafted in a separate step to the bare particles using the method of Auroy et al.22 This 2-step method allows the characterization of the silica particles before and after the PDMS is grafted to them. The radius of the silica cores (Rc) was obtained using both transmission electron microscopy and static light scattering giving respectively 80 nm (polydispersity of 8%) and 82 ( 5 nm. The hydrodynamic radius of the grafted particles (in dilute solutions) Rh was determined using dynamic light scattering. Its magnitude was 140 ( 5 nm indicating a polymer layer thickness L of 58 ( 7 nm. The hydrodynamic specific volume (in dilute solutions) qh was determined via intrinsic viscosity measurements using an automated (Schott) Ubbelohde capillary viscometer where we set the intrinsic viscosity equal to that of a hard sphere suspension. Comparing qh with the specific volume of the bare particles qc (determined from massdensity measurements) the ratio Rh/Rc can be calculated. We obtained the value of 1.8 ( 0.1 in agreement with the 1.7 ( 0.1 found with light scattering. For the modeling as described in Section 2.1., some additional input data characterizing the properties of the polymer chains (in the used solvent) are necessary. We estimated the averaged number of polymer chains per particle to be (5.3 ( 0.6) × 103 by combining results for the mass-fraction polymer/particle wpp, Mn, qc, and Rc. From this the averaged lateral distance between the grafting sites of the polymer chains can also be obtained ((4 nm) which is much smaller than the layer thickness. This indicates that the molecules are strongly stretched and hence form a brushlike structure. In Section 2.1. the polymers are modeled as freely jointed chains. PDMS can be modeled as such by setting the segment length a to 0.84 nm and the segment (molar) mass mo to 192 g/mol.23 This gives 4.2 ( 102 segments per chain. The magnitude of the excluded volume parameter v could not be unambiguously determined for our system. (19) Werff, J. C. van der; Kruif, C. G. de; Blom, C.; Mellema, J. Phys. Rev. A 1989, 39, 795. (20) Nommensen, P. A.; Duits, M. H. G.; Lopulissa, J. S.; Ende, D. van den; Mellema, J. Prog. Colloid Polym. Sci. 1998, 110, 144. (21) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (22) Auroy, P.; Auvray, L.; Leger, L. J. Colloid Interface Sci. 1992, 150, 187. (23) Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Ed.; John Wiley & Sons: New York, 1989.

It is related to the Flory-Huggings parameter χ according to

(21 - χ)

v ) a3

(20)

For PDMS in heptane χ values of 0.41 to 0.45 are reported in the Polymer Handbook.23 However, these values were obtained at PDMS volume fractions ranging from 0.4 to 0.8, whereas in the grafted layers on our particles, the segment volume fraction ranges from 0 to 0.3 (an estimation based on the modeling described in Section 2.1.). Using the results of swelling experiments with PDMS networks,24 we can get an estimate for the value of χ at low concentrations. To this end we have used

χ)-

∆Gel + ln(1 - φp) + φp φ2p

(21)

with φp the polymer volume fraction of the network under equilibrium conditions and substituted herein

∆Gel )

Ge V φ1/3 RT s p

(22)

as suggested in ref 24 to obtain a relation between χ, φp and the elastic modulus of the network Ge (with T and Vp known). To find Ge we have used the data of Baney et al. for PDMS in toluene at φp ) 0.24, and inserted χ ) 0.515.23 Analyzing the swelling data in heptane with the same Ge, χ was found to be equal to 0.33 at φp ) 0.15. We realize that this is still only an estimation. Based on all these foregoing considerations the most appropriate conclusion seems to be that χ is most likely between 0.33 and 0.45. 3.2. Sample Preparation. All sample suspensions were made from a single stock. Concentrated samples were made by centrifuging a weighed amount of stock, pipetting off the calculated weight of heptane, and subsequent vigorous shaking of the suspension. From the particle weight fractions w, weight concentrations c were calculated according to 1/c ) (vp - vs) + vs/w with vp and vs the gravimetric specific volumes of respectively the particles and the solvent. The particle core volume fraction is calculated from the weight concentration using the relation φ ) (1 - wpp)qcc. 3.3. Rheological Measurements. The elastic modulus as a function of frequency G′(ω) was measured with a Bohlin VOR rheometer using a cone-plate geometry of diameter 60 mm and angle 1°. The samples were presheared at 81 s-1 with short test measurements between to monitor the time dependence of the elastic modulus G′ at 1 Hz. Measurements were started as soon as this quantity became stable. Frequency sweeps were carried out from 0.001 to 10 Hz after the critical shear amplitude, marking the end of the linear regime, had been determined at 1 Hz. The Bohlin was equipped with a homemade vapor lock to prevent a change in concentration during measurements. Especially concentrated suspensions are sensitive to solvent evaporation. Using the vapor lock no change in concentration was detected even after more than 12 h. Two types of frequency behavior of G′ were observed in our suspensions depending on the concentration as can be seen in Figure 3. A change occurred at a critical concentration which we identify as the concentration where the averaged distance between neighboring particle surfaces equals twice the layer thickness. Below this (24) Malone, S. P.; Vosburgh, C.; Cohen, C. Polymer 1993, 34, 5149.

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Figure 4. The segment volume fraction in the polymer layer ()n‚a3) as a function of the distance from the core surface, calculated with SF (solid) and analytical approach (dashed). The arrow indicates a slight kink in the dashed line.

Figure 3. The elastic moduli as a function of frequency for several core volume fractions in the range of 7.7% (O) to 15% (*). The markers in both plots correspond to the same volume fraction.

overlap concentration Maxwellian like behavior, i.e., G ≈ ω2 at low frequency and G′ frequency virtually independent at high frequency, is observed. The highest frequency we were able to access with the Bohlin was not high enough to see this high-frequency plateau. We did see a leveling off, see also ref 20. Above the overlap concentration G′ is almost independent of the frequency over the full frequency range accessible to the Bohlin. In the previous section a microscopic model for high frequency G′ plateau is given. The experimental results for samples above the overlap concentration can be compared to the modeling directly. For samples with concentration below the overlap concentration an extrapolation to high frequencies must be made. We did not make this extrapolation but used a worst case estimate for the error by setting the value of G′ at the highest accessible frequency as a lower bound for G′∞. The upper bound for G′∞ at low concentrations was set equal to G′∞ of the suspension with a concentration just above the overlap concentration, assuming that G′ increases monotonically with concentration. 4. Results and Discussion Before we compare the model calculations with experimental results we examine the differences between the two polymer theories that are relevant to the present study. To this end we plot in Figure 4 the calculated polymer segment density n under equilibrium conditions as a function of the separation from the core surface. The properties of the polymer layer in both calculations are identical and were chosen on the basis of the characterization of the particle and polymer in the solvent (see Section 3.). The calculated profiles are in fairly good correspondence although there are some typical differences. The selfconsistent field lattice model (SF) shows a tail at the end of the layer and a small depletion zone near the core surface. These effects are absent in the analytical approach. Due to the strong stretching assumption in the

Figure 5. The layer thickness as a function of the FloryHuggings parameter χ calculated with SF (b) and analytical approach (]). The difference in layer thickness becomes more pronounced when the solvent quality approaches theta conditions (χ f 0.5). The shaded area marks the window corresponding with experimental indications for L/Rc.

analytical approach the polymer density must be maximal at the core and decrease monotonically when moving away from the surface. The kink in the analytical approach curve marks the end of the exclusion zone for the free ends. At that position there is a sudden jump in the free end distribution. Considering the layer thickness, it is important to note that SF predicts a more extended layer than the analytical approach. This can clearly be seen in Figure 5 where the layer thickness is plotted as function of the solvent quality. For the layer thickness we have chosen a definition which in our opinion is the most appropriate one for comparison with experimental observations. As described in Section 3., estimates for the layer thickness were obtained from the hydrodynamic specific volume of the polymer coated particle at low concentrations. It can be expected that this volume is slightly smaller than the volume of the smallest sphere that encloses all polymer segments belonging to the particle. Especially when the segment density decreases very gradually in the outermost part of the layer, this will be the case. For this reason we have replaced the tail in the SF-profile by fitting the convex part of the profile using a parabolic function. Within this scheme the layer thickness is defined as the separation between the core surface and the position where the segment density becomes zero (using the original profile for the analytical approach and the modified one for SF). We have checked that this replacement of the SF-tail does not affect any of the qualitative differences as discussed above. The difference between the calculated thicknesses becomes more pronounced when the solvent quality approaches χ ) 0.5. This can partly be explained by the use of the virial expansion in the analytical approach. Including higher order terms results in an increase of the

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Figure 6. Comparison between experimental G′∞ (b) and calculations based on the analytical approach potential for χ ) 0.33 (0), 0.36 (]), and 0.45 (3). The best agreement is obtained for χ ) 0.36.

Figure 7. Comparison between experiment G′∞ and modeling based on the SF potential with the same values for χ as in Figure 6. The best agreement is obtained for χ ) 0.45.

calculated layer thickness, the more so for solvent qualities around the theta value. We use the SF and analytical approach separately as a model for Φ and will now investigate which polymer model predicts both G′∞ and the layer thickness best. All parameters needed to calculate the hydrodynamic interactions and Φ were obtained from experimental characterizations of particles, polymer, and solvent. A problem is the uncertainty in χ. In a mean field theory as used in this paper, it is usually assumed that χ is constant for a given combination of polymer and solvent. However in reality this is often not the case. In most polymer/solvent systems χ shows a concentration dependence and near a surface it can differ from its bulk value. In our calculations we use an effective χ in which all these dependences are averaged out. So it may be expected that the magnitude for χ which is most appropriate to describe our particles can differ from its magnitude value under bulk conditions. The G′∞ calculations are rather sensitive to the magnitude of χ. We show this by plotting in Figures 6 and 7 the results for values for χ ) 0.33 and 0.45, (respectively the literature value and the swelling result) together with an intermediate value for χ ) 0.36. Good quantitative agreement is found for both SF and analytical approach calculations with respectively χ ) 0.45 and χ ) 0.36. The corresponding calculated layer thicknesses are 0.65 Rc and 0.69 Rc, both in agreement with the light scattering result of (0.7 ( 0.1)‚Rc. The first conclusion that can be drawn from these results is that for the first time, a quantitative modeling of G′∞ has been achieved for a suspension of polymerically stabilized colloidal spheres. This improvement in comparison to earlier attempts4 for PMMA/PHS particles dispersed in Decalin seems to

Nommensen et al.

Figure 8. Calculated pair potentials using the parameters that give the best description of G′∞. The solid line is the SF calculation with χ ) 0.45, and the dashed line is the analytical approach calculation with χ ) 0.36. The shaded area marks the range of distances for which we found nonzero values in the radial pair distribution (using Monte Carlo simulations with the SF potential at φ ) 0.14).

confirm the importance of an adequate description of the particle pair potential. The magnitude of χ that gives quantitative description is for both SF and the analytical approach in the expected range (0.33-0.45). So despite the different χ values, it is not possible to discriminate between the polymer models on the basis of this study. In this context it should be remarked that in the analytical approach the excluded volume parameter v is overestimated to compensate for the neglect of the higher order terms. This results in a smaller χ in the analytical approach compared to χ in the SF calculations where no virial expansion is used. Adding higher order terms in the analytical approach increases the excluded volume interaction term in eq 6 which will result in a larger equilibrium layer thickness. Also the magnitude for χ that gives the best description for G′∞, will then become larger. Hence incorporating higher order terms in the analytical approach would narrow down the range for χ somewhat. This was not pursued in this study for computational reasons. It is also interesting to compare the two different model predictions at the level of the particle pair potential Φ. This is shown in Figure 8. The two potentials differ for very small and deep compression. But in the range around Φ ≈ 1 kT there is good correspondence between the two model calculations. To further investigate this point, we have also plotted in Figure 8 the range in which the particle pair interaction is probed, calculated from the concentrations that were used. In this calculation we simply made use of the already obtained g(r). The polymer layers are never really deeply compressed so the G′∞ calculations are not very sensitive to the details of Φ at small separation. For separations where Φ ≈ 1 kT, the interparticle force is no longer significant compared to the Brownian motion which interaction strength is typically of order 1 kT. Summarizing these findings, the similarity of both models in the description of G′∞ is also seen at the level of Φ(r). This corroborates the key role of Φ(r) in the scheme of ER. Finally we discuss assumptions that pertain to the analytical and the SF approach. In our quasi 2D calculation of Φ we neglected lateral relaxation of the polymer chains. We estimated the influence of lateral relaxation of the polymer chains by allowing, in the analytical approach, the polymers to tilt. The tilted angle depended on the position of the polymer relative to the line that connects the two particles. The function that relates the angle to the position of the polymer was found by minimizing the free energy of the entire polymer layer using a numerical scheme. The difference between Φ with and without lateral relaxation seems to be negligible. Even for intermediate compression, which corresponds to particle concentrations

G′∞ of Polymerically Stabilized Suspensions

much higher than our experimental data, the difference in Φ is smaller than 30%. Another assumption is strong stretching of the polymer chains, which is necessary in the analytical approach. A direct consequence of this is the absence of interpenetration since this would raise the free energy. The degree of stretching of the chains depends strongly on the local segment density. If the segment density becomes too low, the argument given above does not hold any longer and interpenetrations can occur. For multiarm star polymers in solution25 these interpenetrations were observed using light scattering techniques, indicating that the analytical approach cannot be used for those systems. We were not able to use these light scattering techniques to measure the degree of interpenetration in our suspensions. The cores of the particles are strong scatterers causing multiple scattering at core volume fractions of interest. Also the scattering of the polymer layer is hardly detectable in the background of the core scattering. To get an indication of the degree of stretching we used the SF approach since there no assumption is made about strong stretching. The inflection point at relative separation 0.7 (see Figure 4) marks the end of the strong stretching regime. When two particles are pushed to separations where the pair potential is of order kT, the segment density at the midpoint becomes larger than the segment density at the inflection point of the uncompressed case. From this we conclude that the strong stretching assumption can safely be used and our quasi 2D calculation is justified for the particles used in this study. More generally, it can be used for particles with a dense brush as long as the compression is not very deep. 5. Conclusions For the first time, the high-frequency elastic modulus G′∞ for suspensions of polymerically stabilized particles has been quantitatively described (as a function of concentration) with a model based on the molecular properties of the polymer layer. This was achieved with a scheme proposed by Elliott and Russel, after adaptations to take into account our case of a polymer brush with a strongly curved geometry. The crucial role of the particle pair potential in the modeling of G′∞ was corroborated. Two different methods to obtain a theoretical pair potential were explored: one analytical and one numerical. In both methods a mean field approach was used, with the solvent quality parameter χ treated as an unknown within a certain range. Very comparable pair potentials and G∞ functions were obtained, but with different values for χ. Part of this difference can be attributed to the neglect of higher order terms in the excluded volume interactions in the analytical model. (25) Seghrouchni, R.; Petekidis, G.; Vlassopopulos, D.; Fytas, G.; Semenov, A. N.; Roovers, J.; Fleischer G. Europhys. Lett. 1998, 42, 271.

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Acknowledgment. This work is part of the research program of the Foundation for Chemical Research (SON) with financial support from The Netherlands Organization for Scientific Research (NWO). We thank J. van Male for performing the SF calculations, J.S. Lopulissa for synthesizing the silica/PDMS particles, S.L. Elliott, C. Marquest, and W.B. Russel for valuable discussions, and S.L. Elliott for providing us with her computer code for doing the Monte Carlo simulations. 7. List of Symbols A a Fchain Fpolymer Flub f G′∞ g Hav h k L N n nRc r12 rc ro rs(ro,t) si T t Ur wpp W R η′r,∞ λ ν F σ Φ φ φmax

hydrodynamic function ([1 - A] is mobility) (Kuhn) length of polymer segment averaged free energy of a single chain free energy of the polymer layer lubrication force number of chains attached on a particle high frequency elastic modulus equilibrium pair radial distribution function averaged separation between core surfaces core radius + local layer thickness Boltzmann constant polymer layer thickness number of Kuhn segments per polymer polymer segment density value of n at end exclusion zone radius of the core center to center separation end coordinate of the exclusion zone radial position of free end radial position of tth segment of polymer with free end at ro nonequilibrium microstructure temperature number of segment relative velocity of two particles weight fraction of polymer per particle hydrodynamic function (Brownian contribution) correction of friction factor (due to presence of polymer layer) relative high-frequency viscosity Lagrange multiplier excluded volume parameter density of free ends grafting density pair potential particle core volume fraction maximum particle core volume fraction

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