^{1}and Kenneth S. Schweizer

^{1,a)}

### Abstract

Liquid state theory is employed to study phase transitions and structure of dense mixtures of hard nanoparticles and flexible chains (polymer nanocomposites). Calculations are performed for the first time over the entire compositional range from the polymer melt to the hard sphere fluid. The focus is on polymers that adsorb on nanoparticles. Many body correlation effects are fully accounted for in the determination of the spinodal phase separation instabilities. The nanoparticle volume fraction at demixing is determined as a function of interfacial cohesion strength (or inverse temperature) for several interaction ranges and nanoparticle sizes. Both upper and lower critical temperature demixing transitions are predicted, separated by a miscibility window. The phase diagrams are highly asymmetric, with the entropic depletion-like lower critical temperature occurring at a nanoparticle volume fraction of , and a bridging-induced upper critical temperature at filler loading. The phase boundaries are sensitive to both the spatial range of interfacial cohesion and nanoparticle size. Nonmonotonic variations of the bridging (polymer-particle complex formation) demixing boundary on attraction range are predicted. Moreover, phase separation due to many body bridging effects occurs for systems that are fully stable at a second order virial level. Real and Fourier space pair correlations are examined over the entire volume fraction regime with an emphasis on identifying strong correlation effects. Special attention is paid to the structure near phase separation and the minimum in the potential of mean force as the demixing boundaries are approached. The possibility that nonequilibrium kinetic gelation or nanoparticle cluster formation preempts equilibrium phase separation is discussed.

This work was supported by the Division of Materials Sciences and Engineering, U.S. Department of Energy under contract with UT-Battelle, LLC. We thank Ben Anderson and Chip Zukoski for many informative and stimulating discussions.

I. INTRODUCTION

II. MODEL, THEORY, AND VIRIAL COEFFICIENTS

III. SPINODAL PHASE DIAGRAM

A. Numerical procedure and general results

B. Specific trends of phase behavior

C. Comparison with virial analysis

D. Filler PMF

IV. STRUCTURAL CHANGES APPROACHING PHASE SEPARATION

A. Real space correlations

B. Collective structure factors

C. Osmotic compressibilities

D. Local filler cage and polymer microphase order parameters

V. POTENTIAL OF MEAN FORCE AT PHASE SEPARATION AND KINETIC AGGREGATION

A. PMF depth along spinodal boundaries

B. Nonequilibrium aggregation and gelation

VI. SUMMARY AND DISCUSSION

### Key Topics

- Polymers
- 103.0
- Nanoparticles
- 68.0
- Phase separation
- 56.0
- Solubility
- 41.0
- Critical point phenomena
- 26.0

## Figures

Filler second virial coefficient normalized by its hard sphere value as a function of monomer-particle attraction strength in units of the thermal energy for and three values of spatial range.

Filler second virial coefficient normalized by its hard sphere value as a function of monomer-particle attraction strength in units of the thermal energy for and three values of spatial range.

(a) Spinodal phase diagram for and three values of . Numerical data points were determined by approaching the spinodal instability via incrementing (closed symbols) and (open symbols). Though some symbols appear close to the or 1 axes, all systems are, of course, miscible in these limits. Lines are a guide to the eye. (b) Alternate representation of the same spinodal phase boundary in terms of reduced temperature. The homogeneous miscible and phase separated depletion and bridging regimes are indicated.

(a) Spinodal phase diagram for and three values of . Numerical data points were determined by approaching the spinodal instability via incrementing (closed symbols) and (open symbols). Though some symbols appear close to the or 1 axes, all systems are, of course, miscible in these limits. Lines are a guide to the eye. (b) Alternate representation of the same spinodal phase boundary in terms of reduced temperature. The homogeneous miscible and phase separated depletion and bridging regimes are indicated.

Spinodal phase diagram for and three values of spatial range. The phase boundaries were determined in the same manner as in Fig. 2. Solid lines are a guide to the eye. Dashed lines show the spinodal predicted by the virial calculation (Ref. 25) of Eq. (8); endpoints of the dashed lines are spatial range labels.

Spinodal phase diagram for and three values of spatial range. The phase boundaries were determined in the same manner as in Fig. 2. Solid lines are a guide to the eye. Dashed lines show the spinodal predicted by the virial calculation (Ref. 25) of Eq. (8); endpoints of the dashed lines are spatial range labels.

Nanoparticle potential of mean force in the dilute two particle limit for and the four indicated values of attraction strength for (a) and (b) .

Nanoparticle potential of mean force in the dilute two particle limit for and the four indicated values of attraction strength for (a) and (b) .

Nanoparticle potential of mean force in the dilute two particle limit (dashed lines) and at (solid lines) for , , and three values of interfacial cohesion range.

Nanoparticle potential of mean force in the dilute two particle limit (dashed lines) and at (solid lines) for , , and three values of interfacial cohesion range.

Nanoparticle potential of mean force for various volume fractions and fixed , , and . The curve is the pure hard sphere result at a total packing fraction of 0.4, computed using the HNC closure.

Nanoparticle potential of mean force for various volume fractions and fixed , , and . The curve is the pure hard sphere result at a total packing fraction of 0.4, computed using the HNC closure.

Monomer-filler pair correlation function for various filler volume fractions for , , and . The , 0.25, 0.5, 0.75, and 0.9 lines are labeled by a square, diamond, triangle, , and circle, respectively, and have contact values , 3.5, 3.0, 2.7, and 3.2, respectively. The inset shows the nonrandom part of the pair correlation function, , weighted by the surface area factor .

Monomer-filler pair correlation function for various filler volume fractions for , , and . The , 0.25, 0.5, 0.75, and 0.9 lines are labeled by a square, diamond, triangle, , and circle, respectively, and have contact values , 3.5, 3.0, 2.7, and 3.2, respectively. The inset shows the nonrandom part of the pair correlation function, , weighted by the surface area factor .

Monomer-monomer pair correlation function for various filler volume fractions and , , and . The inset shows the nonrandom part of the pair correlation function, , weighted by the surface area factor .

Monomer-monomer pair correlation function for various filler volume fractions and , , and . The inset shows the nonrandom part of the pair correlation function, , weighted by the surface area factor .

(a) Nanoparticle collective structure factor for various volume fractions for , , and . The curve corresponds to pure hard sphere result at a total packing fraction of 0.4. (b) Small angle regime of the monomer-monomer collective structure factor for the same systems.

(a) Nanoparticle collective structure factor for various volume fractions for , , and . The curve corresponds to pure hard sphere result at a total packing fraction of 0.4. (b) Small angle regime of the monomer-monomer collective structure factor for the same systems.

(a) Inverse of the dimensionless filler osmotic compressibility as a function of volume fraction for various interfacial cohesion strengths, , and . The black solid curve corresponds to pure hard spheres at a total packing fraction of . (b) Analogous results for the inverse of the dimensionless polymer osmotic compressibility.

(a) Inverse of the dimensionless filler osmotic compressibility as a function of volume fraction for various interfacial cohesion strengths, , and . The black solid curve corresponds to pure hard spheres at a total packing fraction of . (b) Analogous results for the inverse of the dimensionless polymer osmotic compressibility.

(a) Amplitude of the nanoparticle collective cage peak as a function of volume fraction for various interfacial strengths, , and . The smooth black curve (no data points) corresponds to the pure hard sphere fluid result at a total packing fraction of . Inset shows the peak location. (b) Analogous results for the microphase peak intensity of the collective polymer structure factor. Inset shows the peak location.

(a) Amplitude of the nanoparticle collective cage peak as a function of volume fraction for various interfacial strengths, , and . The smooth black curve (no data points) corresponds to the pure hard sphere fluid result at a total packing fraction of . Inset shows the peak location. (b) Analogous results for the microphase peak intensity of the collective polymer structure factor. Inset shows the peak location.

Value of the global minimum in the nanoparticle potential of mean force at the spinodal phase separation boundary for (a) , , 10, and 15 systems and (b) , , 0.5, and 1.0 systems. Open symbols correspond to the high (bridging) side of the phase diagrams in Figs. 2 and 3, and closed symbols correspond to the low (depletion) side.

Value of the global minimum in the nanoparticle potential of mean force at the spinodal phase separation boundary for (a) , , 10, and 15 systems and (b) , , 0.5, and 1.0 systems. Open symbols correspond to the high (bridging) side of the phase diagrams in Figs. 2 and 3, and closed symbols correspond to the low (depletion) side.

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