Structure, Interactions, and Dynamics in Tethered Chain Systems

These micelles provide a valuable model system to study chains tethered to curved ... chain length through variations in the diblock copolymer degree ...
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Langmuir 1996, 12, 4060-4067

The Langmuir Lectures Structure, Interactions, and Dynamics in Tethered Chain Systems Alice P. Gast Department of Chemical Engineering, Stanford University, Stanford, California 94305 Received December 14, 1995. In Final Form: June 7, 1996X We use the self-assembling nature of diblock copolymers in selective solvents to produce monodisperse spherical micelles. These micelles provide a valuable model system to study chains tethered to curved interfaces. They provide a means to alter the topology of the anchoring interface as well as the tethered chain length through variations in the diblock copolymer degree of polymerization. A polymer chain layer that is thin compared to the core radius has highly extended chains resembling a polymer brush structure. Polymers in micelles having small spherical cores have a structure similar to that found in star polymers. We study polymeric micelles and their interactions via small angle light, X-ray, and neutron scattering. We model the tethered layer structures via self-consistent field theory and combine this with liquid state theory to predict the structure in concentrated suspensions of micelles. We compare the liquid structure determined from small angle neutron scattering to that predicted from our statistical mechanical models. The disorder to order transition is profoundly influenced by small changes in the micellar structure; micelles having larger cores form face centered cubic arrays while more starlike structures order into body centered cubic crystals. The crystal structure has a great influence on the suspension rheology. We briefly review the structural transitions occurring during simple shear of these arrays.

1. Introduction The need to prevent the aggregation of colloidal particles under a variety of solvent conditions has fueled the study of polymeric stabilizers over the past 20 years.1-3 This process, termed steric stabilization, requires the attachment of a strongly bound layer of polymer to the colloidal surface to prevent aggregation due to van der Waals attractions. The desire to produce a thick polymer layer has led to the investigation of polymers attached to surfaces, either physically or chemically, by one end only. It is these chains anchored by one end that have come to be known as “tethered chains”. Block copolymers in a selective solvent have a tendency to self-assemble at surfaces and into micelles.4-7 This interfacial activity makes block copolymers an attractive system for the study of tethered chains. Studying their adsorption onto solid surfaces yields information about layer structure8-12 and interactions13-16 while the micelles X

Abstract published in Advance ACS Abstracts, August 1, 1996.

(1) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983. (2) R. Buscall, T. C.; Stageman, J. Polymer Colloids; Elsevier: Amsterdam, 1985. (3) Clarke, J.; Vincent, B. J. Colloid Interface Sci. 1981, 82, 208. (4) Tuzar, Z.; Kratochvil, P. Adv. Colloid Interface Sci. 1976, 6, 201. (5) Riess, P. B.; Hurtrez, G. Encyclopedia of Polymer Science and Engineering; Wiley: New York, 1985; Vol. 2, Chapter Block Copolymers, pp 324-434. (6) Gast, A. P. Scientific Methods for the Study of Polymer Colloids and Their Applications; Kluwer: Dordrecht 1990, Chapter Block Copolymers at Interfaces, pp 311-328. (7) Halperin, A.; Tirrell, M.; Lodge, T. P. Adv. Polym. Sci. 1992, 100, 31. (8) Cosgrove, T.; Heath, T. G.; Phipps, J. S.; Richardson, R. M. Macromolecules 1991, 24, 94. (9) Field, J. B.; Toprakcioglu, C.; Ball, R.; Stanley, H.; Dai, L.; Barford, W.; Penfold, J.; Smith, G.; Hamilton, W. Macromolecules 1992, 25, 434. (10) Cosgrove, T.; Ryan, K. Langmuir 1990, 6, 136. (11) Auroy, P.; Auvray, L.; Leger, L. Macromolecules 1991, 24, 2523. (12) Auroy, P.; Mir, Y.; Auvray, L. Phys. Rev. Lett. 1992, 69, 93. (13) Hadziioannou, G.; Patel, S.; Granick, S.; Tirrell, M. V. (14) Tirrell, M.; Patel, S.; Hadziioannou, G. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 4725. (15) Taunton, H. J.; Toprakcioglu, C.; Fetters, L. J.; Klein, J. Nature 1988, 332, 712.

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themselves can be viewed as model spherical colloidal particles carrying a high density of tethered chains. Our interest has been in understanding the structure of tethered chains and the implications that structure has on the interactions between and dynamics of tethered chain layers. With this goal we have studied the adsorption of block copolymers onto solid surfaces,17-20 the structure of micelles of block copolymers suspended in a selective solvent,21-24 and the interactions and rheology of concentrated suspensions of block copolymer micelles.25-28 In this lecture I review some of our findings about the structure, interactions, and dynamics in model systems of tethered chains: polymeric micelles. While there are many interesting applications of tethered chains for colloidal stabilization, adhesion, surface modification, and lubrication, there are also several compelling fundamental questions we seek to understand. The first issue is the structure of polymer chains anchored by one end to a highly curved surface and crowded by its neighbors. This issue can be addressed through the synthesis and study of star-branched polymers or micelles having very small core blocks.6,22,23,29-31 The second question to address is the nature of the interaction (16) Taunton, H. J.; Toprakcioglu, C.; Klein, J. Macromolecules 1988, 21, 3333. (17) Munch, M. R.; Gast, A. P. Macromolecules 1988, 21, 1360. (18) Munch, M. R.; Gast, A. P. Macromolecules 1988, 21, 1366. (19) Gast, A. P.; Munch, M. R. Polym. Commun. 1988, 30, 324. (20) Leermakers, F.; Gast, A. Macromolecules 1991, 24, 718. (21) Cogan, K. A.; Gast, A. P. Macromolecules 1990, 23, 745. (22) Cogan, K.; Capel, M.; Gast, A. Macromolecules 1991, 24, 6512. (23) Vagberg, L.; Cogan, K.; Gast, A. Macromolecules 1991, 24, 1670. (24) Cogan, K.; Leermakers, F.; Gast, A. Langmuir 1992, 8, 429. (25) McConnell, G. A.; Gast, A. P.; Huang, J. S.; Smith, S. D. Phys. Rev. Lett. 1993, 71, 2102. (26) McConnell, G. A.; Lin, E. K.; Gast, A. P.; Huang, J. S.; Lin, M. Y.; Smith, S. D. Faraday Discuss. R. Soc. Chem. 1995, 98, 121. (27) McConnell, G. A.; Lin, M. Y.; Gast, A. P. Macromolecules 1995, 28, 6754. (28) Lin, E. K.; Gast, A. P. Macromolecules 1996, 29, 390. (29) Adam, M.; Fetters, L. J.; Grassley, W. W.; Witten, T. A. Macromolecules 1991, 24, 2434.

© 1996 American Chemical Society

The Langmuir Lectures

between layers of tethered chains in curved geometries. To elucidate these properties, we investigate the liquidlike structure produced by solutions of micelles. We combine small angle neutron scattering with statistical mechanical models of micellar structure and interactions.26,28 The ability to contrast match the core or corona of the micelle for small angle neutron scattering is a key feature of these studies. Finally, like most molecular and colloidal particles having predominantly repulsive interactions, polymeric micellar colloids will undergo a disorder to order transition when the concentration exceeds a critical value. We study this ordering via small angle X-ray scattering25 and find the formation of both bodycentered and face-centered cubic structures depending on the micellar architecture. We follow the structural transitions occurring when an ordered array of micelles is subjected to simple shear with small angle neutron scattering.27 The behavior of polymeric micelles is quite similar to that found in charged colloidal suspensions, and thus we rely on useful analogies with those systems. The combination of scattering experiments with statistical mechanics gives us a colloid-level view of the interplay between composition, structure, interactions, and dynamics in these tethered chain systems.

Langmuir, Vol. 12, No. 17, 1996 4061

Figure 1. Star polymer model for chains tethered to highly curved surfaces. Chains are represented as a string of close packed blobs of size ξ(r) extending radially from the core of the star.

2. Tethered Chain Structure in Block Copolymer Micelles Polymers tethered to surfaces by one end only will be stretched from their random coil configuration by interactions with their neighbors. The details of the structure they form depend on the nature of the solvent and the geometry of the tethering surface. A polymer chain layer tethered to a large spherical micelle such that it is thin compared to the radius of curvature has highly extended chains resembling a polymer brush.32 Polymers tethered to small spherical cores have a structure similar to that found in star polymers. Over a decade ago, Daoud and Cotton provided the scaling analysis of chains tethered to a small spherical core.33 Consideration of the tethered arms as confined to a string of close-packed blobs of size ξ(r) provides a starlike polymer density profile that decays from the center as F(r) ≈ (r/b)1/ν-3 where ν is the Flory exponent reflecting the quality of the solvent as illustrated in Figure 1. In a good solvent, ν ) 0.6, the density falls off as 1/r4/3 while in a theta or ideal solution where ν ) 0.5, the profile decays as 1/r. This profile is supported by molecular dynamics simulation,34 Monte Carlo simulations,35 and lattice calculations.24,36 We investigated the starlike structure of polymeric micelles via small angle X-ray and neutron scattering.22 The system we studied comprised polystyrene/poly(ethylene oxide) (PS/PEO) diblock copolymers in cyclopentane or deuterated cyclohexane. We chose temperatures for these solvents such that the polystyrene is at its theta point. We were able to take advantage of the sensitivity of these micelles to water, caused by the hydrophilicity of the PEO core block,21 and use trace amounts of water to alter the number of polymer chains (30) Bauer, B. J.; Fetters, L. J.; Graessley, W.; Hadjichristidis, N.; Quack, G. Macromolecules 1989, 22, 2337. (31) Richter, D.; Jucknischke, O.; Willner, L.; Fetters, L. J.; Lin, M.; Huang, J. S.; Roovers, J.; Toporovski, C.; Zhou, L. L. J. Phys. IV-C8 1993, 3, 3. (32) Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. (33) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982, 43, 531. (34) Grest, G.; Kremer, K.; Witten, T. Macromolecules 1987, 20, 1376. (35) Toral, R.; Chakrabarti, A. Phys. Rev. E 1993, 47, 4240. (36) van Lent, B.; Scheutjens, J. M. H. M. Macromolecules 1989, 22, 1931.

Figure 2. A Kratky plot for small angle X-ray scattering of PEO/PS 170/1730 micelles and PS ) 1540 chains. The dilute micellar solutions have varying trace amounts of water causing the aggregation number to change from 20 (0) to 100 (4). The position of the average blob size for each micelle and the radius of gyration for the homopolymer are indicated by the vertical line.

per micelle, known as the micellar aggregation number. Since the starlike polymer model predicts a concentration profile for ideal tethered chains to decay as 1/r, we expect the scattering from these chains to display a characteristic decay with the scattering vector, q ) 4π/λ sin(θ/2), as I(q) ∝ q-2. This behavior is expected in the intermediate scattering regime between low angles q < 1/Rg where one will measure the overall micelle dimensions (or radius of gyration, Rg) and large q > 1/b where molecular details (b ) segment size) of the chain structure are elucidated. In order to illustrate this scattering signature, we create a Kratky plot, showing the intensity multiplied by the expected scaling, in this case q2, and look for a plateau indicating that the scaling is appropriate. This is shown in Figure 2 where dilute solutions of micelles of PS/PEO diblocks of 1700 repeat units of PS and 170 units of PEO are studied. One expects to see the same signature scaling for a homopolymer undergoing a random walk under the same conditions. The difference between the Kratky plots in this case shows that the signature scaling is achieved at a scattering vector q ≈ 1/Rg for the homopolymer while the micelles show the scaling at higher q corresponding to 1/ξ or one over the average blob size for the micelle. Hence, increasing the micellar aggregation number causes the plateau behavior to move to higher q since the average

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layers by Dan and Tirrell37 to determine the pair interaction potential between curved layers. In our self-consistent field calculations we describe each tethered polymer as a freely jointed chain with N statistical segments of length b. The configurational statistics of the chain are modeled as a random walk within a mean field potential. The potential in turn depends upon the polymer segment distribution, thus requiring a selfconsistent solution in analogy to other physical problems such as Hartree-Fock.39 The fundamental quantity we calculate is the polymer segment probability distribution functions, G(z,z′|s), representing the probability of finding segment s at the position, z, given that the chain started at the position z′. This probability distribution function is governed by the forced diffusion equation Figure 3. A Kratky plot from small angle X-ray and neutron scattering of PEO/PS 65/80 micelles and PS ) 85 chains. The scattering from the micelles decays as q-1.5 and thus shows no plateau in this Kratky representation.

blob size decreases. This signature scaling is a good indication of the validity of the starlike model for these systems. This starlike behavior is not observed, however, for a polymer having 80 units of PS and 65 PEO. The Kratky plot shown in Figure 3 illustrates this lack of scaling for both small angle X-ray and neutron scattering. The comparable homopolymer achieves the q-2 scaling while the micelles show q-1.5. This scaling is intermediate between random walk behavior and the scaling one would expect from straight linear chains radiating from the core, q-1. We interpret this behavior as indicative of the severe stretching of the tethered chains occurring near the cores of these micelles. These small chains are not long enough to achieve the random walk behavior found at the larger radial distances from the core. In the 170/1700 system, the scattering is dominated by the largest blobs where the scaling is observed. In the 65/80 system, the largest blobs at the periphery of the micelles are only 6.3 nm and would thus contain only 17 statistical segments, too few to show random walk behavior. The starlike model is a reasonable picture of the structure of spherical micelles having long coronal chains tethered to very small cores. This scaling breaks down when the core size of a polymeric micelle becomes too large or when the coronal chain length is too short. We were thus motivated to study the structures of polymers tethered to more moderately curved micellar cores. 3. Influence of Curvature on Structure and Interactions in Tethered Chains In many colloidal systems polymers are tethered to curved surfaces. Usually, the desired tethered chain length is comparable to or smaller than the radius of the tethering surface rendering the star polymer structure inadequate. Little work has concentrated on probing the curvature regime in between the limits of polymer brushes on planar interfaces and star polymers. Dan and Tirrell37 performed self-consistent field (SCF) calculations elucidating the transition between the parabolic polymer brush profile and the star polymer density profile. These results are supported by recent Monte Carlo simulation results35 and lattice calculations.38 We recently presented a theoretical investigation of the structure and interactions between tethered chain layers on curved interfaces.26,28 We extended the continuum SCF treatment39,40 of curved (37) Dan, N.; Tirrell, M. Macromolecules 1992, 25, 2891. (38) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214. (39) Edwards, S. F. Proc. Phys. Soc. 1965, 85, 613. (40) Hong, K. M.; Noolandi, J. Macromolecules 1981, 14, 727.

∂G b2 - ∇ + ω(z)G ) δ(s)δ(z - z′) ∂s 6

(1)

where ω is the mean field potential due to the interaction of the chain with all other polymer chains. The volume fraction profile of the tethered chain layer is determined from G as

φA(z) )

FA(z) C ) F0A F0AW

∫0N ds G(z,R|s) ∫R∞ dz′ × G(z,z′|N - s)

where

W)

∫R∞ dz G(z′,R|N)

(3)

is the configurational partition function representing the configurations of a tethered polymer chain, F0A is the bulk number density of the polymer, and FA(z) is the local number density at z. The self-consistency in the equations arises from the direct relationship between the volume fraction profiles and the mean field potential, ω, through

ω(z) )

F0S {-ln φS(z) + χ[φS(z) - φA(z)]} F0A

(4)

where F0A is the bulk number density of the chain segments, F0S is the bulk number density of solvent molecules, and χ is the Flory-Huggins interaction parameter.41,40 We illustrate the results of these calculations with volume fraction profiles for N ) 200 and σ ) 0.1 under theta (χ ) 0.5) conditions with different core curvatures in Figure 4. These profiles demonstrate the large influence of core curvature on the structure of the tethered chain layer. The shape of the volume fraction profiles varies smoothly from the power law decay of a star polymer when R/b ) 4 to the parabolic profile characteristic of polymer brusches at a flat interface when R ) ∞, in good agreement with the profiles determined by Dan and Tirrell37 and Wijmans and Zhulina.38 The distribution of chain ends is also of interest in SCF calculations. As the curvature of the anchoring surface increases, the chain ends are excluded from a region near the interface.42,37 The end-distribution also becomes narrower as illustrated in Figure 4. Since the tethered polymer density profiles change qualitatively with increasing curvature from a parabolic form to a power law decay, we expect the resulting (41) Whitmore, M. D.; Noolandi, J. Macromolecules 1990, 23, 3321. (42) Ball, R. C.; Marko, J. F.; Milner, S. T.; Witten, T. A. Macromolecules 1991, 24, 693.

The Langmuir Lectures

Langmuir, Vol. 12, No. 17, 1996 4063 Table 1. Parameters for Polystyrene-Polyisoprene Polymeric Micelles Studied via Small Angle Neutron Scatteringa Rgcorona

R (Å) N dPS/PI

f

134/140 170/75 180/94 295/206 320/337 393/206 357/375 402/422

80 270 235 335 245 420 90 130

P(q) solid R/b σ ) fb2/4πR2 SANS (Å) SCF (Å) 74 116 117 150 142 204 106 137

75 121 119 160 145 186 108 127

8.9 14.0 14.1 18.1 17.1 24.6 12.8 16.5

0.08 0.11 0.09 0.08 0.07 0.06 0.04 0.04

158.6 155.2 182.9

234.7

143.1 164.5 171.7 246.6 276.9 295.7 231.2 276.6

a Core radii from the form factor are compared with space-filling spheres, and the corona radius of gyration is compared with that determined from self-consistent field calculations.

Figure 4. Top: Volume fraction profiles for chains of length N ) 200 tethered at a surface density σ ) 0.1 under theta (χ ) 0.5) conditions. The core curvatures are R/b ) 4, 8, 16, 32, 64, ∞ from left to right. Bottom: Segment end distributions for the same conditions.

produce softer repulsions persisting over a longer range while the chains tethered to larger cores produce nearly hard-sphere interactions. In order to assess the influence of curvature on the interactions between layers of tethered chains, we studied the liquid structure produced in concentrated solutions of micelles. We create spherical micelles by suspending a series of monodisperse deuterated polystyrene-polyisoprene (dPS/PI) diblock copolymers26,43 in n-decane, a preferential solvent for polyisoprene. The molecular weights and the polydispersity indices for these polymers are listed in Table 1. We prepared solutions in mixtures of deuterated and protonated n-decane to match the scattering length density of either the deuterated polystyrene core or the polyisoprene corona. We measured the small angle neutron scattering on beamline NG7 at the National Institute of Standards and Technology (NIST).44 Measurements were taken at an incident wavelength of λ ) 7.00 Å (∆λ/λ ) 0.11 at full width, half maximum). The resulting scattered intensities are a product of intraand intermicellar interference

I(q) ) nP(q)S(q)

(5)

where P(q) is the form factor accounting for intramicellar interference, n is related to the number of scatterers, and S(q) is the static structure factor. The static structure factor indicates the short range correlations in the micellar suspension and depends explicitly on the radial distribution function, g(r) Figure 5. Pair interaction potential energies scaled on kT against center-to-center separation scaled on particle diameter for spherical particles carrying chains of length N ) 200 tethered at a surface density σ ) 0.1 under theta (χ ) 0.5) conditions. The core curvatures are R/b ) 4, 8, 16, 32, and 64 from left to right.

interactions between these structures to change as well. We used the concentration profile calculations described above to predict the pair interaction energy between two approaching curved layers. Unlike the one-dimensional isolated layer, the case of interacting spherical particles is a two-dimensional problem. To reduce the computational cost, we approximate the particle pair interaction potential, u(r) from the free energy for a one-dimensional problem. We improve upon the usual Derjaguin approximation by combining the configurational statistics from the curved geometry with a modified Derjaguin calculation of the interaction potential.28 We show the pair interaction potentials in Figure 5 for the curved layers depicted in Figure 4. The more highly curved interfaces

∫

S(q) ) 1 + F e-iq‚r[g(r) - 1] dr

(6)

We measure P(q) from dilute micellar solutions and then obtain the static structure factor from experiments at higher concentrations where interactions between micelles become important. Like classical or simple monatomic liquids, the short range correlations in polymeric micelles are indicative of their interactions. We use integral equations from liquid state theory to calculate g(r) from the interaction potential u(r) determined from the SCF calculations.45 We use the Rogers-Young closure to the Ornstein-Zernike equation,46 a hybrid approach mixing the Percus-Yevick (PY) and Hyper-Netted Chain (HNC) closures, able to accurately model the liquid structures in both soft and hard (43) Smith, S. D. In Polymer Solutions, Blends, and Interfaces; Noda, I., Rubingh, D. N., Eds.; Elsevier: Amsterdam, 1992; pp 43-64. (44) Hammouda, B.; Krueger, S.; Glinka, C. J. J. Res. Natl. Inst. Stand. Technol. 1993, 98, 31. (45) McQuarrie, D. A. Statistical Mechanics; Harper & Row: New York, 1976. (46) Rogers, F. J.; Young, D. A. Phys. Rev., A 1984, 30, 999.

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spheres. The expression

(

g(r) ) exp[-βu(r)] 1 +

)

exp[γ(r)f(r)] - 1 f(r)

(7)

where 0 e f(r) e 1, with f(0) ) 0 and f(∞) ) 1 mixes PY and HNC as a function of radial separation and reduces to the PY equation when r ) 0, f(0) ) 0 and the HNC closure as r increases and f(r) approaches unity. The mixing equation is

f(r) ) 1 - exp(-Rr)

(8)

where R is adjusted to achieve thermodynamic consistency in the osmotic compressibility. We combine our numerical model of the interaction potential from the SCF calculation with the Rogers-Young closure, solving for the radial distribution function via Gillan’s method.47 The liquid state theory thus allows us to take the SCF u(r), and determine the radial distribution function g(r). Taking the Fourier transform of (g(r) - 1), we have the structure factor S(q) for comparison to the scattering experiments. We apply our theoretical model to the series of dPS/PI diblock copolymer micelles described in Table 1. The core radii, R, determined by fitting the core-contrast dilute solution (0.5 wt % polymer) scattering profiles to polydisperse solid sphere form factors with a Schulz distribution are presented in Table 1. The reasonable agreement between the polydisperse solid sphere form factor and experimental data suggests that the micelles have dense spherical cores with relatively sharp interfaces. The polydispersity in the hard sphere form factor (standard deviation ) 0.1 for both systems) is partly due to the neutron wavelength dispersity (∆λ/λ ) 0.11). The aggregation number, f, calculated from the absolute intensity at zero angle through Zimm analysis48 provides the radius of a space-filling core of polystyrene chains (Table 1). Agreement between these radii suggests that the solvent swelling of the core is minimal, less than 10% by volume. The surface density σ ) fb2/4πR2 of tethered polyisoprene is based upon the core radius determined from P(q) and the aggregation number from Zimm analysis. These diblock copolymer systems have a range of core radii and chain lengths forming tethered chain structures that fall between the limiting cases of a starlike structure and the planar parabolic structure. The profiles calculated with the above parameters can be compared to the measured radii of gyration, Rg, of the micellar coronas from SANS on dilute suspensions in corona contrast decane as shown in Table 1. The radii of gyration calculated from the chain profiles compare favorably with the available experimental values; all are within the 10% uncertainty in the experimental parameters used in the SCF model. The interaction potentials are repulsive with a softsphere character; short-range potentials generally correspond to micelles with brushlike coronal layers smaller than the core diameter and softer potentials correspond with more starlike micelles. These potential energies provide the theoretical input for our characterization of the liquid structure in strongly interacting suspensions. We compare two diblock copolymers, dPS/PI 393/206 and dPS/PI 402/422 to investigate the interactions in parabolic and starlike systems, respectively. The structure factors for both micellar systems are shown in Figures 6 and 7. The oscillatory form of S(q) clearly reflects a liquid-like structure. As expected, we see that S(q) approaches unity as q increases, indicating (47) Gillan, M. J. Mol. Phys. 1979, 38, 1781. (48) Glatter, O.; Kratky, O. Small Angle X-Ray Scattering; Academic Press: New York, 1982.

Figure 6. Structure factors for dPS/PI micelles of 393/206 repeat units at core volume fractions of 0.012 (4), 0.02 (+), 0.03 (*), 0.04 (4), and 0.05 (O). The lines are the theoretical fits from the SCF interaction potentials and the Rogers Young closure to the OZ equation.

Figure 7. Structure factor for dPS/PI micelles of 402/422 repeat units at core volume fractions of 0.006 (O), 0.013 (0), and 0.019 (4). The lines are the theoretical fits from the SCF interaction potentials and the Rogers Young closure to the OZ equation.

that our division of the intensity by P(q) is reasonable. At higher concentrations the peak heights grow indicating enhanced correlations in the suspension; increasing the concentration eventually leads to S(q) for the ordered micellar structure. We compare the experimental S(q) with structure factors calculated with the SCF potentials and the RogersYoung closure. We first look at the micellar system formed from 393/206 dPS/PI diblocks where the core is relatively large compared to the corona at polymer concentrations from 3 to 12.5 wt %. The resulting interactions are of relatively short range and should be well-characterized with the SCF theory and the Derjaguin approximation. A direct comparison of theoretical and experimental structure factors for core volume fractions from 0.012 to 0.050 is shown in Figure 6. We find excellent agreement between our experimental and theoretical structure factors. The excellent agreement, even for the highly correlated liquid suspension at a core volume fraction of 0.050, suggests the appropriateness of the pair-interaction potential. The importance of the decaying character of the potential is seen when S(q) is fit with an effective hard sphere potential; hard spheres produce a much poorer fit to the data where the hard sphere radius must decrease with increasing concentration. The second system, dPS/PI 402/422, has softer interactions due to its long polyisoprene chains and relatively

The Langmuir Lectures

small core. We have fewer experimental structure factors for this system. The strong correlations in S(q) occur at smaller core volume fractions because of the longer range of the interactions. Again, as Figure 7 demonstrates, we find reasonable agreement with our liquid state theory fit with the SCF potential and experiment. A comparison of the two structure factors at the highest concentration shows a mismatch in the phase of oscillation. In this strongly correlated liquid, we can see that some error exists in the estimate of the pair-interaction potential. This may be attributed to the Derjaguin approximation for the interactions in the highly curved geometry or to errors in the mean field approximation for the more dilute regions at the edge of the large corona. Despite these limitations, they are inviting results useful for moderate, liquid-like concentrations. A more detailed model relaxing the Derjaguin approximation or using a full two-dimensional solution of the SCF theory may be necessary to provide a truly accurate estimate of the pair-interaction potential applicable at all liquid concentrations.

Langmuir, Vol. 12, No. 17, 1996 4065

Figure 8. A small angle X-ray diffraction pattern from micelles of dPS/PI 402/422 in decane at 12.5 wt %. The vertical lines indicate the anticipated peak locations for a BCC crystal with a lattice dimension of 872 Å.

4. Influence of Tethered Chain Structure on the Disorder to Order Transition A more profound influence of the structure of tethered chains appears when the micelles reach a concentration where they order. The disorder to order transition occurs in monodisperse spherical systems having purely repulsive interactions. Known as the Kirkwood-Alder transition, this crystallization was discovered through computer simulations of hard spheres by Alder et al.49 Hard spheres order into face-centered cubic (fcc) arrays; several model colloidal systems provide experimental support for the simulations.50 In the 27 years since the discovery of hardsphere ordering, many systems having repulsive interaction potentials of finite range have been studied both theoretically and experimentally. Generally, repulsions acting over a long range can produce body-centered cubic (bcc) arrays while short range repulsions always favor the fcc. Particles interacting via a power law repulsion, u(r) ) (1/r)n, order into FCC arrays for powers n >6, but form BCC arrays for n e 6.51,52 Electrostatically stabilized colloidal particles interact via a screened Coulombic repulsion well-described by a Yukawa potential.53 Experimental and computational studies of Yukawa systems show ordering into fcc structures when the repulsions are of short-range and into bcc structures when the range of the repulsion greatly exceeds the particle size.50,53-55 It is reasonable to expect spherical particles interacting through tethered chains to order into cubic arrays. As we increase the concentration of solutions of dPS/PI micelles, they order into cubic lattice structures as illustrated in Figures 8 and 9. The soft-sphere star like micelles form bcc arrays as seen in Figure 8. The difference in degeneracy for the diffraction orders causes the third peak to be higher than the second and provides clear identification of this structure as body-centered cubic. The minimum in this diffraction pattern corresponds well with the first minimum in the micelle form factor indicating that the micellar form or shape has not changed appreciably from its structure in dilute solution. The micelles having larger cores and thus steeper repulsions crystallize (49) Alder, B.; Hoover, W.; Young, D. J. Chem. Phys. 1968, 49, 3688. (50) Pusey, P.; van Megen, W. Nature 1986, 320, 340. (51) Hoover, W. G.; Young, D. A.; Grover, R. J. Chem. Phys. 1972, 56, 2207. (52) Laird, B. B.; Haymet, A. Mol. Phys. 1992, 75, 71. (53) Monovoukas, Y.; Gast, A. P. J. Colloid Interface Sci. 1989, 128, 533. (54) Robbins, M. O.; Kremer, K.; Grest, G. S. J. Chem. Phys. 1988, 88, 3286. (55) Carlson, R.; Asher, S. Appl. Spectrosc. 1984, 38, 297.

Figure 9. A small angle X-ray diffraction pattern from micelles of PS/PI 442/187 in decane at 22 wt %. The vertical lines indicate the anticipated peak locations for an fcc array with a lattice dimension of 1007 Å.

into an fcc structure as shown in Figure 9. Again a minimum in the diffraction pattern where a peak is absent corresponds to the first node in the form factor for the spherical micelles in dilute solution. Thus, the systems examined above form fcc or bcc arrays depending on the range and steepness of their repulsions.25 Prediction of the disorder to order transition from the pair potential energies is challenging; the spatial correlations in a liquid near the disorder-order transition are very sensitive to the details of the interaction potential. We are currently applying density functional models to this problem.56-60 5. Micellar Dynamics: Response to Shear The molecular motions of tethered chains is an interesting area of study due to its relevance to block copolymer adsorption kinetics20 and lateral mobility. Upon assessing the collective motion of chains tethered to the outside of a micellar core via neutron spin-echo spectroscopy,61 we found that the motion could only be represented by a spectrum of relaxation times. The relaxation is welldescribed by a model of collective motion developed by deGennes and termed “breathing modes”.61,62 (56) Curtin, W. Phys. Rev., B 1989, 39, 6775. (57) Curtin, W.; Ashcroft, N. Phys. Rev., A 1985, 32, 2909. (58) Curtin, W.; Ashcroft, N. Phys. Rev. Lett. 1986, 56, 2775. (59) Marr, D.; Gast, A. P. Phys. Rev., E 1993, 47, 1212. (60) McConnell, G. A.; Gast, A. P. Submitted for publication in Phys. Rev., E. (61) Farago, B.; Monkenbusch, M.; Richter, D.; Huang, J. S.; Fetters, L.; Gast, A. Phys. Rev. Lett. 1993, 71, 1015. (62) deGennes, P. G. C. R. Acad. Sci. Paris Ser. II 1986, 302, 765.

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Figure 10. Diffraction images taken at various steady shear rates, γ˘ , for fcc micelles formed from dPS/PI 295/206 diblocks at 15 wt % suspended in core contrast decane, (a) γ˘ ) 0.00 s-1, aligned by inserting stator (b) γ˘ ) 0.66 s-1 and (c) γ˘ ) 200.0 s-1. For these experimental diffraction images qv and qe range from 0.0280 to -0.0280 Å-1.

Figure 11. Diffraction images taken at various steady shear rates, γ˘ , for a bcc micellar crystal formed from dPS/PI 357/375 diblocks at 10 wt % suspended in core contrast decane: (a) γ˘ ) 0.3 s-1 after subjecting to a preshear above 50 s-1, (b) γ˘ ) 80.0 s-1, (c) γ˘ ) 322.0 s-1. For these diffraction images qv and qe range from 0.036 to -0.036 Å-1.

Consideration of the behavior of tethered chain layers subjected to flow is important for understanding the rheology of polymerically stabilized suspensions.1,63-65 A number of groups have investigated the behavior of tethered chain layers on flat surfaces through experiment66 and theory67-70 and simulation.71 The uncertainty surrounding the nature of the microstructural evolution in an ordered array of colloidal particles and the link between the structure and rheological response motivate our interest in the long range order of polymeric micelles subjected to a steady shear. Our system of monodisperse dPS/PI micelles forms a unique model system to address fundamental issues in tethered chain interactions as well as questions of flow deformation of ordered arrays. We performed small angle neutron scattering (SANS) experiments to examine the long range order in polymeric, micellar crystals subjected to linear steady shear.27 We again study the (dPS/PI) diblocks forming both bcc and fcc micellar crystals. As we increase the shear rate, the fcc crystals undergo a transition from polycrystallinity to 〈111〉 sliding layers as illustrated by the diffraction patterns in Figure 10. This transition is marked by a (63) Mewis, J.; Frith, W. J.; Strivens, T. A.; Russel, W. B. AIChE J. 1989, 35, 415. (64) Ploehn, H. J.; Goodwin, J. W. Faraday Discuss. Chem. Soc. 1990, 90, 77. (65) Genz, U.; D’Aguanno, B.; Mewis, J.; Klein, R. Langmuir 1994, 10, 2206. (66) Klein, J.; Perahia, D.; Warburg, S. Nature 1991, 352, 143. (67) Hatano, A. Polymer 1983, 25, 1198. (68) Rabin, Y.; Alexander, S. Europhys. Lett. 1990, 13, 49. (69) Barrat, J. Macromolecules 1992, 25, 832. (70) Harden, J. L.; Cates, M. E. Phys. Rev. E 1996, 53, 3782. (71) Doyle, P.; Gast, A. P.; Shaqfeh, E. S. G. To be submitted for publication.

significant hysteresis in the steady shear stress versus shear rate data. For higher shear rates, we observe 〈111〉 layers normal to the shear gradient slipping past each other. As the shear rate increases, the particles no longer hop from one registry site to the next and thus specific diffraction spots are suppressed as illustrated in Figure 10c. The bcc crystals subjected to linear shear demonstrate a more continuous deformation of the local crystalline lattice illustrated in Figure 11. The natural twinning planes of the bcc lattice are normal to the 〈110〉 direction, thus providing no easy slipping plane for the flow. Very small shear rates produce significant stress within the bcc twin plane causing the structure to take on a distorted paracrystalline structure resembling hexagonally close packed (hcp) layers as shown in Figure 11a. The structure develops into a sliding plane structure, again suppressing certain spots in the paracrystalline hcp diffraction pattern as shown in Figure 11b. At higher shear rates, Figure 11c, we observe shear melting. The structures observed in these flowing micellar suspensions are analogous to those found in charged and hard-sphere colloidal suspensions.72 What is striking about this system is how sensitive the flow behavior is to small changes in intermicellar interactions. Small differences in our block copolymer composition produce significant changes in the ordering and consequently alter the rheology. Another important difference between these systems and the charged colloids is that here the intermicellar interactions are occurring over the same length scale as the hydrodynamic interactions; thus these suspensions are always hydrodynamically concentrated. This concentration causes some of the structural transitions to occur over more experimen(72) Chen, L.; Zukoski, C.; Ackerson, B.; Hanley, G. S.; Barker, J.; Glinka, C. Phys. Rev. Lett. 1992, 69, 688.

The Langmuir Lectures

tally accessible shear rates than in the charged colloidal suspensions. Further study of these polymeric micelles as model tethered chain systems will improve our understanding of suspension rheology. Acknowledgment. This research was performed with an enthusiastic group of students and visitors including Kathleen Cogan-Farinas, Lena Vagberg, Frans Leermakers, Eric Lin, and Glen McConnell. Additional help with this work came from Stephen Nilsen and David Marr. We have benefitted from collaborations with Steven D. Smith,

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John S. Huang, Min Y. Lin, and Malcolm Capel. This work was partially supported by the NSF-MRL Program through the Stanford Center for Materials Research. Additional financial support from the Exxon Educational Foundation is gratefully acknowledged. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, and the Department of Energy for providing the facilities used in these studies. LA951538Z