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The aim of this work is to use a recently developed statistical model of dispersions with non-hydrodynamic interactions (Dagréou et al., 2002) to describe the linear viscoelastic properties of suspensions of rigid hairy particles in a polymeric matrix. We first present numerical simulations of our model applied to this system; we demonstrate that taking physical interactions into account allows one to predict the complex relaxation behaviour of filled polymers. We then compare the statistical model to experimental results on suspensions of grafted silica particles in a polystyrene matrix and show that they are in reasonable agreement up to volume fractions close to percolation.
L'objectif de ce travail est d'utiliser un modèle statistique (Dagréou et al., 2002) pour décrire les propriétés viscoélastiques linéaires de polymères chargés. Nous présentons d'abord des simulations numériques du modèle appliqué à ce système; nous montrons que la prise en compte des interactions non hydrodynamiques permet de prévoir un comportement complexe en élasticité. Nous comparons ensuite le modèle avec des résultats expérimentaux obtenus pour des suspensions de particules de silice greffée dans une matrice de polystyrène; il existe un assez bon accord entre théorie et expérience, jusqu'à des fractions volumiques proches de la percolation.

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Filling can cause polymer melt to undergo a so-called fluid-to-solid transition usually assigned to filler networking effect and heterogeneously retarded polymer dynamics while theories developed nowadays could not account for the several important aspects involving in the reinforcement and dissipation of the composites including the influence of molecular weight of the matrix. Herein linear dynamic rheological behaviors of carbon black filled polystyrene (PS) was investigated, disclosing significantly influences of weight-averaged molecular weight (Mw) of PS on the reinforcement and dissipation in and beyond the hydrodynamics regime. Attempts to create rheological master curves in hydrodynamics regime, taking account for both dynamics retardation in bulk polymer phase and strain amplification effect induced by filler, disclose time-concentration superpositions. The viscoelastic contributions from the dynamically retarded polymer phase and the viscoelastic “filler phase” are thus evaluated.

A series of poly(butyl methacrylate)s (PBMAs) with various molar masses (33 000–270 000 g mol−1), which were densely grafted on fumed silica nanoparticles (PBMA–SiO2), were synthesized by surface-initiated atom transfer radical polymerization. The dynamic viscoelastic behavior of PBMA–SiO2 was systematically investigated in the solid and molten states with oscillatory strains, and compared to that of a conventional nanocomposite (PBMA/SiO2). The storage moduli of PBMA–SiO2 and PBMA/SiO2 are equivalent in the solid state, whereas the storage modulus of PBMA–SiO2 is lower than that of PBMA/SiO2 in the molten state, especially at high silica loading. This is because the formation of a network structure composed of the silica nanoparticles in PBMA–SiO2 is strongly suppressed by the polymer brushes on the particles. In contrast, even at low silica loading, the PBMA–SiO2 system exhibits a gel-like behavior resulting from a steric repulsion between the composite particles, because all of the tethered polymers behave as bound polymers. Copyright © 2011 Society of Chemical Industry

Polystyrene (PS) chains with molecular weights comprised between 15,000 and 60,000 g/mol and narrow polydispersities were successfully grown from the surface of silica nanoparticles by nitroxide-mediated polymerization (NMP). Small angle X-ray scattering was used to characterize the structure of the interface layer formed around the silica particles, and at a larger scale, dynamic light scattering was used to determine the hydrodynamic diameter of the functionalized silica suspension. In a second part, blends of PS-grafted silica particles and pure polystyrene were prepared to evaluate the influence of the length of the grafted PS segments on the viscoelastic behavior of the so-produced nanocomposites in the linear viscoelasticity domain.Combination of all these techniques shows that the morphology of the nanocomposite materials is controlled by grafting. The steric hindrance generated by the grafted polymer chains enables partial destruction of the agglomerates that compose the original silica particles when the latter are dispersed either in a solvent or in a polymeric matrix.

The aim of this work is to use a recently developed statistical model of dispersions with nonhydrodynamic interactions to describe the linear viscoelastic properties of emulsions of Newtonian liquids. None of the existing models can describe the rheological behavior of such systems, particularly the elastic properties, in the linear regime. We first present the results of numerical simulations of our model applied to emulsions. We show that taking nonhydrodynamic interactions into account allows to predict that emulsions of two purely viscous liquids have a complex viscoelastic behavior. We then compare the model to experimental results on oil/water emulsions, stabilized with ionic and nonionic surfactants. We find out that our statistical mechanical approach gives a much better description of the viscoelastic behavior of these samples than purely hydrodynamic models do. However, the elasticity observed is underestimated by our model. We indicate further theoretical developments which could improve the description of the viscoelastic properties of emulsions.

We investigate the nanostructure and the linear rheological properties of polybutylacrylate (PBA) filled with Stöber silica particles grafted with PBA chains. The silica volume fractions range from 1.8 to 4.7%. The nanostructure of these suspensions is investigated by small-angle neutron scattering (SANS), and we determine their spectromechanical behavior in the linear region. SANS measurements performed on low volume fraction composites show that the grafted silica particles are spherical, slightly polydisperse, and do not form aggregates during the synthesis process. These composites thus constitute model filled polymers. The rheological results show that introducing grafted silica particles in a polymer matrix results in the appearance of a secondary process at low frequency: for the lowest volume fractions, we observe a secondary relaxation that we attribute to the diffusion of the particles in the polymeric matrix. By increasing the silica volume fraction up to a critical value, we obtain gellike behavior at low frequency as well as the appearance of a structure factor on the scattering intensity curves obtained by SANS. Further increasing the silica particle concentration leads to composites exhibiting solidlike low-frequency behavior and to an enhanced structure peak on the SANS diagrams. This quantitative correlation between the progressive appearance of a solidlike rheological behavior, on one hand, and a structure factor, on the other hand, supports the idea that the viscoelastic behavior of filled polymers is governed by the spatial organization of the fillers in the matrix.

The frequency and time dependent moduli of up to 60% glass bead filled polydimethyl siloxane (PDMS) were studied for four different molecular weight materials with a rotational, compressional, and lubricated squeezing flow rheometer. Measurements of the storage and loss modulus taken with a conventional rotational instrument show good qualitative and quantitative agreement with those taken with a newly developed oscillatory compressional rheometer for PDMS of the lowest molecular weight studied, and show trends similar to those reported previously for similar materials. The agreement between the two instruments is better for both unfilled and 35% filled material of a higher molecular weight. Good agreement was also observed between measurements of the relaxation modulus in lubricated squeezing flow and measurements of the storage modulus with the new compressional device for a high molecular weight PDMS at all filler levels. The modulus derived from oscillatory measurements showed good quantitative agreement with that measured in lubricated squeezing flow for identical materials. The effect of polymer molecular weight on the relative behavior of the loss and storage modulus with increasing filler content was also studied. It was shown that as the polymer molecular weight increases, the ratio of the loss to storage modulus becomes nearly independent of the filler Volume fraction over the frequency range studied. The effect of an increase in filler amount on the elasticity of the material was shown to depend both on the polymer molecular weight and the frequency, and is explained in terms of the influence of the filler on the Deborah number of the system. (C) 2001 The Society of Rheology.

The linear viscoelastic behavior in the melt of PMMA/PS blends and various blends of PMMA containing rubbery latex particles has been characterized by dynamic shear rheometry. For the rubber-toughened PMMA samples, the influence of rubber content, structure of the latex particles, and aggregation of the particles in the PMMA matrix have been investigated. Morphologies of the dispersed type or of the aggregated type were produced by performing the blending in the melt or in solution. The data for G' and G'' have been analyzed in the frame of a rheological emulsion model which is found to account for the behavior of the PMMA/PS blends and of rubber-toughened PMMA at low rubber content. At high rubber concentrations the model does not predict the secondary plateau in G' which arises at low frequencies for these systems. Therefore, this plateau cannot be attributed to the deformability of dispersed inclusions as in PMMA/PS blends, but is shown to depend on the extent of aggregation of the dispersed particles, and to be most important in a well dispersed morphology where the particles form a network-type structure.

We present a new approach to describe the rheological properties of dispersions with non-hydrodynamic interactions (steric, electrostatic and Van der Waals interactions) in the linear viscoelastic domain. Our model is based on the calculation of additional stresses resulting from interaction potentials between spheres and Brownian motion. We start from the statistical mechanical approaches which have been developed by Batchelor and Green and later Lionberger and Russel, to model the viscoelastic properties of emulsions and suspensions. We have extended their calculations to the more general case of viscoelastic deformable inclusions in a viscoelastic matrix. Our contribution lies in the computation of the hydrodynamic functions involved in the term describing interaction stresses. This computation is based on Palierne's results on the deformation field around a viscoelastic inclusion embedded in a viscoelastic matrix. We have also rewritten the conservation equation in the case of interest, over the whole frequency domain. We finally express the complex shear modulus of the dispersion as the sum of two terms : Palierne's complex shear modulus gives the purely hydrodynamic contribution; the interaction contribution depends on both the hydrodynamic properties and the interaction potential.

We formulate a theory for the nonequilibrium structure and stresses in a sheared suspension with a fluid rest state. Many body interactions are handled exactly in the thermodynamics but truncated at the pair level for the hydrodynamics. Evaluation for hard spheres in weak flows demonstrates the importance of stresses arising from the nonequilibrium structure and explains the shear rate dependence observed at volume fractions greater than 0.25–0.30.

The singular nature of the hard sphere potential combined with lubrication stresses near contact poses interesting issues with respect to the high frequency viscoelastic behavior. Dilute theories demonstrate clearly that soft potentials and/or lubrication stresses that reduce the relative mobility to zero at contact lead to a well defined plateau in G′ as ω → ∞, whereas a hard sphere potential without hydrodynamic interaction produces G′ ≈ ω+ 1/2 $/ in this limit. The former follows from a small deformation of the equilibrium structure due solely to the oscillator convection and the latter from a diffusional boundary layer near contact required to satisfy the no-flux boundary condition. Two sets of data that delineate the high frequency response for colloidal hard spheres at high volume fraction appear to differ in this regime, suggesting different physics for the interactions at small separations. Here we apply our nonequilibrium theory to extend the existing treatments to high volume fractions to predict both limits quantitatively and provide a possible interpretation for the experimental results. The two experimental systems only differ in the surface modification of the particles and the high frequency modulus is the only rheological property sensitive to this difference. The predictions of our theory with varying extent of hydrodynamic interaction illustrate the link between the behavior of the high frequency modulus and the hydrodynamic properties very near the particle surface.

The high-frequency shear modulus of a monodisperse colloidal suspension is derived from both linear response theory and a zero-time correlation function. The results are shown to reduce to the classical expression valid for molecular fluids in the limit of no hydrodynamic interaction. The expressions with an additional pairwise additive stress approximation agree with previous work on dense suspensions, thus clarifying the approximations in previous derivations. A phenomenological model based on a lattice structure is also recovered through suitable approximation for the neighbor distribution and neglect of hydrodynamic interactions. Sample calculations for a suspension of charged, Brownian spheres demonstrate that the intercolloidal parameters obtained from fitting the lattice model are not in agreement with the calculation here. Further, predictions of this theory, without adjustable parameters, are in good agreement with wave-rigidity measurements on suspensions of charged, polystyrene spheres. The role and influence of hydrodynamic interactions are discussed and the validity of the pairwise additive assumption is also tested.

The rheological behavior of narrow polybutadiene samples filled with rigid silica spheres has been investigated. Generally the rheological behavior of biphasic blends of polymers can be interpretated in terms of emulsion models when the concentration of the dispersed phase is low enough so as to avoid interparticle interactions. The main feature is the presence of a relaxation domain which appears in the low frequency zone of the complex shear modulus. The most general emulsion model proposed by Palierne [Palierne, J. F., Rheologica Acta, 1991, 30, 497.] explains that occurrence by the shape relaxation of the viscoelastic dispersed phase. But the model fails when the dispersed phase is very little deformable or non deformable (mineral fillers). Nevertheless a slow relaxation process is also seen experimentally in that case which can not be assumed due to shape relaxation. Those slow relaxation processes were measured and analyzed as a function of polybutadiene molecular weight and size and volume concentration of the silica spheres. They have been interpretated by an adsorption of polybutadiene chains on the silica surface that creates a monomolecular layer whose thickness is comparable to the bulk radius of gyration of the chains and whose relaxation time scales in the same way as branches of star polymers.

The configurational free energy of random flight polymer chains adsorbed
by one end onto a plane surface as a function of the distance from a
parallel plane surface is expressed to a good approximation in simple
analytic form. The result is used to discuss the stabilization of a
colloid suspension by adsorbed polymer. According to this theory two
types of aggregation of colloid particles may occur. If Ll < AS/2π
3NkT, where l is the link length and L the contour length of
a polymer chain, A is the Hamaker constant, N/S is the number of
adsorbed polymer chains per unit area and kT is the Boltzman constant
multiplied by temperature, the particles adhere closely, but if AS/2π
3kT < lL < AS/8π kT lg 2N a looser association is
formed. It is expected that the presence of excluded volume effects
would greatly increase the stability against the looser association.

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø2, where ø([double less-than sign] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order φ2 is found to be
\[
\mu^{*} = \mu(1+2.5\phi + 6.2\phi^2).
\]
The value of the coefficient of ø2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[cdot B: small middle dot]6, which indicates a small degree of ‘strain thickening’ in the ø2-term.

Emulsions of incompressible viscoelastic materials are considered, in which the addition of an interfacial agent causes the interfacial tension to depend on shear deformation and variation of area. The average complex shear modulus of the medium accounts for the mechanical interactions between inclusions by a self consistent treatment similar to the Lorentz sphere method in electricity. The resulting expression of the average modulus includes as special cases the Kerner formula for incompressible elastic materials and the Oldroyd expression of the complex viscosity of emulsions of Newtonian liquids in time-dependent flow.

A theory is presented which relates the colloidal interactions to the microstructure of a Brownian suspension under weak shear and then to the bulk stresses via a new technique for renormalizing the thermodynamic contribution. Further derivations of the interparticle stress provide an independent test of the accuracy of requisite closures. The results are very sensitive to the coupling between equilibrium and nonequilibrium distribution functions in the three-body closures; a closure in the spirit of the Percus-Yevick equation provides the most consistent results while superposition predicts aphysical results. Comparison with the available measurements on hard-sphere systems indicates that the Brownian stresses, renormalized into a hydrodynamic function, are responsible for the divergence in the low shear limiting viscosity in dense suspensions. However, pairwise additive hydrodynamics adequately predict neither the high frequency limiting complex viscosity nor the steady shear viscosity in dense suspensions.

A system of chemical reactions has been developed which permits the controlled growth of spherical silica particles of uniform size by means of hydrolysis of alkyl silicates and subsequent condensation of silicic acid in alcoholic solutions. Ammonia is used as a morphological catalyst. Particle sizes obtained in suspension range from less than 0.05 μ to 2 μ in diameter.

This reference describes the role of various intermolecular and interparticle forces in determining the properties of simple systems such as gases, liquids and solids, with a special focus on more complex colloidal, polymeric and biological systems. The book provides a thorough foundation in theories and concepts of intermolecular forces, allowing researchers and students to recognize which forces are important in any particular system, as well as how to control these forces. This third edition is expanded into three sections and contains five new chapters over the previous edition. • starts from the basics and builds up to more complex systems • covers all aspects of intermolecular and interparticle forces both at the fundamental and applied levels • multidisciplinary approach: bringing together and unifying phenomena from different fields • This new edition has an expanded Part III and new chapters on non-equilibrium (dynamic) interactions, and tribology (friction forces).

It is shown that the osmotic pressure of a colloidal dispersion can be interpreted as the isotropic part of the macroscopic particle stress in the suspension. The particle stress is in turn expressible in terms of hydrodynamic interactions among the suspended particles. Thus, there is a completely mechanical definition of the osmotic pressure, just as there is for the pressure in a molecular fluid. For an equilibrium suspension of colloidal particles subjected to thermal Brownian forces, this mechanical definition is shown to be exactly equal to the usual ``thermodynamic'' one. The derivation given here places the equilibrium and nonequilibrium properties of macroparticle fluids on the same mechanical foundation that underlies the statistical mechanics of simple liquids. Furthermore, through this development the relationship between hydrodynamics and kinetic-theory-like descriptions of colloids is explained.

A simple model for the rheological behavior of concentrated colloidal dispersions is developed. For a suspension of Brownian hard spheres there are two contributions to the macroscopic stress: a hydrodynamic and a Brownian stress. For small departures from equilibrium, the hydrodynamic contribution is purely dissipative and gives the high-frequency dynamic viscosity. The Brownian contribution has both dissipative and elastic parts and is responsible for the viscoelastic behavior of colloidal dispersions. An evolution equation for the pair-distribution function is developed and from it a simple scaling relation is derived for the viscoelastic response. The Brownian stress is shown to be proportional to the equilibrium radial-distribution function at contact, g(2;phi), divided by the short-time self-diffusivity, D0(s)(phi), both evaluated at the volume fraction phi of interest. This scaling predicts that the Brownian stress diverges at random close packing, phi(m), with an exponent of -2, that is, eta0' approximately eta(1 - phi/phi(m))-2, where eta0' is the steady shear viscosity of the dispersion and eta is the viscosity of the suspending fluid. Both the scaling law and the predicted magnitude are in excellent accord with experiment. For viscoelastic response, the theory predicts that the proper time scale is a2/D0s, where a is the particle radius, and, when appropriately scaled, the form of the viscoelastic response is a universal function for all volume fractions, again in agreement with experiment. In the presence of interparticle forces there is an additional contribution to the stress analogous to the Brownian stress. When the length scale characterizing the interparticle forces is comparable to the particle size, the theory predicts that there is only a quantitative contribution from the interparticle forces to the stress; the qualitative behavior, particularly the singular scaling of the viscosity and the form of the viscoelastic response, remains unchanged from the Brownian case. For strongly repulsive interparticle forces characterized by a length scale b (much greater than a), however, the theory predicts that the viscosity diverges at the random close packing volume fraction, phi(bm), based on the length scale b, with a weaker exponent of -1. The viscoelastic response now occurs on the time scale b2/D0s(phi), but is of the same form as for Brownian dispersions.