# Nonlinear Rheology of Unentangled Polymer Melts Reinforced with High Concentration of Rigid Nanoparticles

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## Abstract

A scaling model is presented to analyze the nonlinear rheology of unentangled polymer melts filled with high concentration of small spherical particles. Assuming the majority of chains to be reversibly adsorbed to the surface of the particles, we show that the emergence of nonlinearity in the viscoelastic response of the composite system subjected to a 2D shear flow results from stretching of the adsorbed chains and increasing desorption rate of the adsorbed segments due to the imposed deformation. The steady-state shear viscosity of the mixture in nonlinear shear thinning regime follows the power lawOpen image in new windowwhereOpen image in new windowis the applied shear rate. At large strain amplitude γ _{0,} the storage and loss moduli in strain sweep tests scale asOpen image in new windowandOpen image in new windowrespectively.

### Keywords

Rheology Polymer nanocomposites Melt Scaling relations Nonlinearity## Introduction

Stable dispersion of sub-micron solid particles in polymer melts and solutions can be achieved in the presence of favorable polymer–particle interactions. Studying the rheological behavior of polymer composites reinforced with well-dispersed nanoparticles, often referred to as polymer nanocomposites (PNCs), has been the subject of a growing number of experimental and theoretical studies in the past decade [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. A notable observation is the qualitative resemblance between the thermomechanical response of PNCs and that of polymer thin films confined between planar surfaces [11, 12, 13]. This similarity implies that the adsorption of polymer chains on the surface of nanoparticles alters the mobility of the chains far into the bulk, and therefore the mixture cannot be simply envisioned as a dispersion of hard particles interacting in a matrix.

PNCs generally exhibit strong nonlinear viscoelastic behavior in the response to dynamic inputs. Typical examples of such nonlinear characteristics are strong shear thinning at relatively low shear rates or strain-dependent viscoelastic moduli at low strain amplitudes (Payne effect). In the present Letter, we use a scaling model to elucidate the nonlinear rheological features of the nanofilled unentangled polymer melts by considering the effect of energetic affinities between polymer and particles. The model is particularly relevant to PNCs with high filler concentration where the majority of polymer chains in matrix can interact energetically with the adhesive surface of dispersed particles. Structural and flow properties of colloidal suspensions of this kind have been the subject of recent experimental studies of Anderson and Zukoski [14, 15].

## Scaling Analysis

*N*statistical segments with length

*a*and excluded volume υ, such that at equilibrium the chain’s radius of gyration is

*R*

_{g}~

*aN*

^{1/2}. The energetic affinity

*E*

_{ad}between the polymer segments and the surface of particles is short range and weak, on the order of the thermal energy

*k*

_{B}

*T*(henceforth,

*k*

_{B}

*T*is taken to be the unit of energy). As a result, polymer chains reversibly bind to the solid surface and build a layer around particles which can be described as a continuous succession of loops and tails at different sizes (Fig. 1a). At equilibrium, the reversible adsorption of flexible chains onto an adhesive solid surface leads to a fluffy layer with a self-similar structure [16]. The description of the curvature effects on the self-similar structure of the adsorbed layer was initially proposed by Aubouy and Raphaël [17]. These authors reported that when the characteristic size of the polymer,

*R*

_{g}, is much smaller than the particle size

*R*, the extent of the self-similar structure is limited by polymer dimensions alone. In other words, the colloid curvature is not relevant, and the polymer can be considered as if adsorbed on a flat surface. In the real experimental cases, the condition

*R*≫

*R*

_{g}is granted considering the size of flexible polymer chains in unentangled regime (a few nanometer) compared with the smallest size of commercially available nanoparticles (tens of nanometer). Henceforth, this condition is assumed to be valid in this paper.

Let us assume that the composite system undergoes a plane Couette flow with shear rate ofOpen image in new windowwhere the velocity field followsOpen image in new windowand *u*_{y} = *u*_{z}= 0. The polymer matrix is comprised from a population of two types of chains: chains that are adsorbed to the surface of the particles, and chains that are not adsorbed (at least on the time scale ofOpen image in new window). The average shear stress produced in the polymer matrix can be approximated asOpen image in new windowwhere *f* is the fraction of adsorbed chains, andOpen image in new windowandOpen image in new windoware the average shear stress produced by adsorbed and free chains, respectively.

The average friction acting on a free segment in the bulk readsOpen image in new windowwhere τ _{0} represents the average relaxation time of a free segment in the bulk. Due to the adhesion, the adsorbed segments undergo a larger frictionOpen image in new windowand maintain a longer residence time τ_{1} on the adhesive colloidal surface. In equilibrium, the relation between τ_{1} and τ_{0} is given byOpen image in new windowThe principal Rouse relaxation time of the adsorbed chains,Open image in new windowis also expected to be larger than that of the free chains,Open image in new windowwhereOpen image in new windowandOpen image in new windowindicate the friction on the adsorbed and free chains, respectively.

When the shear deformation rateOpen image in new windowexceedsOpen image in new windowthe adsorbed chains are expected to be tilted and stretched by the shear flow (Fig. 1b). Each stretched chain can be pictured as a string of *N*_{c} Pincus blobs of size *R*_{b}[18]. Inside each blob, the chain statistics is still Gaussian, i.e., for a blob containing *N*_{b} segments (*N*_{b} = *N/N*_{c}), Open image in new window with Rouse time of Open image in new window where Open image in new window is the friction acting on each blob. Since the polymer segment within each blob remains relaxed, the tension on each blob exerted by neighboring blobs can be determined by the condition Open image in new window This way, the friction experienced by each blob forming the dragged polymer is Open image in new window Since Open image in new window represents the velocity difference between the top and bottom of each blob, the force in the blob is Open image in new window or Open image in new window Accordingly, the shear rate-dependent size of the blob reads Open image in new window implying that the blob size shrinks as the applied shear rate on the system increases.

In strong shear regimes, whenOpen image in new windowadsorbed chains elongate to their maximum length (~*Na*) and the size of adsorbed blobs becomes comparable with *a*. The time required for this elongation is on the order ofOpen image in new windowAt this stage, the entropic force developed within each blob of dragging chains reachesOpen image in new windowduring a time scale much smaller than the chain’s Rouse time. As a result, the residence time of the adsorbed blobs reduces toOpen image in new windowwhere λ is an activation length (similarly, we haveOpen image in new windowFor the weak and short range polymer–particle energetic interactions considered here, we have *E*_{ad} ∼ 1 and *λ* ∼ *a*. Hence, at strong shear rates ofOpen image in new window, the imposed shear rate regulates the unzipping of adsorbed segments from the adhesive particles. During such shear meditated desorption, the blob force *F*_{b} is counterbalanced by the friction force acting on the unzipping segments. Hence, the shear stress developed inside the layer of adsorbed blobs isOpen image in new windowwhere *c* is the steady-state surface density of blobs attached on the particle at a given shear rate. The mean number of the adsorbed blobs (per unit colloidal surface area) is given byOpen image in new windowwhere τ* is the average desorption time of a blob andOpen image in new window(i.e., colloidal surface is assumed to be saturated at equilibrium). Assuming λ ~ *a*, the desorption time τ* can be estimated asOpen image in new windowwhich leads toOpen image in new windowThe contribution of strongly stretched free chains in stress production can be estimated using the virial equationOpen image in new windowwhere index *n* refers to the *n* th blob along the highly stretched free chain, andOpen image in new windowindicates ensemble averaging. Note that for highly filled systems with small particles, *f* ~ 1 and the contribution of free chains in stress production is negligible.

_{xy}, can be calculated from the bulk stress usingOpen image in new windowwhere function

*h*accounts for hydrodynamic interaction between pairs of the rigid particles. For an incompressible polymer melt, filled with rigid particles,

*h*is a function of the volume fraction Φ and geometry of the particles [19]. This way, we have Open image in new window, and therefore the overall viscosity scales as Open image in new window This conclusion is compared with the recent experimental results by Anderson and Zukoski [14]. Figure 2 shows the steady-state relative viscosity of unentangled polyethylene oxide (PEO) melt with molecular weight of 400 and 1,000, reinforced with 48 vol% and 45 vol% of spherical silica nanoparticles (with average diameter of 43 nm), respectively. The line with slope −1/2 shows the slope of shear thinning predicted by our scaling analysis which is in reasonable agreement with the slope of the experimental data.

The scaling theory described here can also be used to rationalize the strain-dependent storage and loss moduli of PNCs, a phenomenon which is often referred to as Payne effect. Reduction of the viscoelastic moduli at large amplitude oscillatory flows ensues essentially for the same reason as discussed above: at large amplitudes, stretching chains along the direction of the flow enhances the desorption of the segments from the particle surface, thereby contributing in dissipation of the applied energy. Consider an oscillatory strainOpen image in new windowwith frequency ω and strain amplitude γ_{0} imposed on the composite with *f* ~ 1. At low frequenciesOpen image in new windowthe loss modulus followsOpen image in new window[20] where viscosity scales as Open image in new window as demonstrated above. Therefore, at large amplitude Open image in new window and concurrent with deformation-mediated desorption of the blobs, we have Open image in new window If δ(ω) shows the phase angle between stress and strain, the storage and loss moduli of the system can be connected to each other using Open image in new window where at low frequencies Open image in new window The time scale τ_{a}^{s} represents the relaxation time of the stretched adsorbed chains; that is the time required for an adsorbed stretched chain to diffuse a distance comparable with *Na*[21]. The speed of diffusion along the length of stretched chain is Open image in new window with Open image in new window This way Open image in new window and consequently the storage modulus at large amplitudes scales as Open image in new window These findings are comparable with experimental observations of Anderson and Zukoski [15] where the storage and loss moduli of highly filled PEO (molecular weight of 400) are measured to decrease with strain amplitude as Open image in new window and Open image in new window respectively.

## Conclusion

In summary, the presented analysis shows that owing to the reversible adsorption of the polymer chains onto the solid surface of adhesive particles (and therefore deceleration of chain relaxation), applied energy by external deformations during a fast flow can be stored in the stretched polymer chains, instead of being dissipated by their chain-scale Rouse-like relaxation. The nonlinear response of the system is mediated by unbinding the elongated polymer chains from the particles. As the relaxation process is suppressed in adsorbed chains, the nonlinearity is expected to emerge at smaller strain rates and amplitudes compared with the onset of nonlinearity in the neat polymer. The overall viscoelastic properties of the system in nonlinear regimes exhibit a universal dependence on the applied deformation (rate) in highly filled systems, where majority of the polymer chains are in contact with the surface of the particles. The resulting scaling relations are expected to be valid for the systems with spherical particles or non-spherical particles having low aspect ratio in geometry.

## Notes

### Acknowledgments

I would like to thank Professor C.F. Zukoski and Dr. B.J. Anderson for sharing their experimental data of PEO-silica nanocomposites. The financial support provided by Department of Mechanical Engineering and Office of Vice President for Research at the University of Maine is gratefully acknowledged.

**Open Access**

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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