^{1,2}, M. A. J. Michels

^{1}and A. V. Lyulin

^{1,a)}

### Abstract

We have performed molecular-dynamics simulations of atactic polystyrene thin films to study the effect of shear rate, pressure, and temperature on the stress-strain behaviour, the relevant energetic contributions and non-affine displacements of polymer chains during constant-shear deformation. Under this deformation sliding motion is observed at high shear rates between the top substrate and top polymer layer, which disappears when the shear rate decreases. At low shear rates stick-slip motion of the whole film with respect to the bottom substrate takes place. We found that at low shear rates the yield stress logarithmically depends on the shear rate; this behaviour can be explained in terms of the Eyring model. It was also observed that an increase in the normal pressure leads to an increase in the yield stress in agreement with experiments. The contributions to the total shear stress and energy are mainly given by the excluded-volume interactions. It corresponds to a local translational dynamics under constant shear in which particles are forced to leave their original cages much earlier as compared to the case of the isotropic, non-sheared film. Moreover, it was observed that under constant-shear deformation the polymer glass is deformed non-affinely. As a result, the middle part of the film is much more deformed than the layers close to the supporting substrates, meaning that the well-known effect of shear-banding occurs.

This study is a part of the research program of the Dutch Polymer Institute, Project No. 654. The work was also sponsored by the Stichting Nationale Computerfaciliteiten (National Computing Facilities Foundation, NCF) for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scientific Research, NWO).

I. INTRODUCTION

II. SIMULATION DETAILS

A. Simulation model

B. Polymer-substrate interactions

C. Preparation of capped films

D. Deformation protocols

III. SIMULATION RESULTS

A. Stress-strain dependencies: Shear modulus and yield peak

B. Energy and stress partitioning

C. Local translational mobility during shear deformation

IV. CONCLUSIONS

### Key Topics

- Shear deformation
- 52.0
- Polymers
- 51.0
- Yield stress
- 26.0
- Stress strain relations
- 20.0
- Plasticity
- 18.0

## Figures

Schematic illustration of a typical stress-strain response of a glassy polymer upon shear deformation. Here the shear modulus *G* is defined as a slope in a linear elastic regime.

Schematic illustration of a typical stress-strain response of a glassy polymer upon shear deformation. Here the shear modulus *G* is defined as a slope in a linear elastic regime.

The stress-strain curves for the 16-chains aPS capped film at normal pressure 32 MPa (circles) and 52 MPa (squares) at temperature 300 K (a) and at pressure 52 MPa and at two temperatures 300 K (filled squares) and 370 K (open squares) (b). In both figures the measured shear stress is taken at the highest simulated shear rate s^{−1}.

The stress-strain curves for the 16-chains aPS capped film at normal pressure 32 MPa (circles) and 52 MPa (squares) at temperature 300 K (a) and at pressure 52 MPa and at two temperatures 300 K (filled squares) and 370 K (open squares) (b). In both figures the measured shear stress is taken at the highest simulated shear rate s^{−1}.

The stress-strain curves for the 16 aPS capped film at normal pressure 32 MPa (circles) and 52 MPa (squares) at temperature 300 K (a) and pressure 52 MPa at two temperatures 300 K (filled squares) and 370 K (open squares) (b). In both figures the shear stress is taken at the lowest simulated shear rate s^{−1}.

The stress-strain curves for the 16 aPS capped film at normal pressure 32 MPa (circles) and 52 MPa (squares) at temperature 300 K (a) and pressure 52 MPa at two temperatures 300 K (filled squares) and 370 K (open squares) (b). In both figures the shear stress is taken at the lowest simulated shear rate s^{−1}.

Shear modulus as a function of the shear rate at two normal pressures and fixed temperature 300 K (a) and two temperatures at fixed normal pressure 52 MPa (b). The dashed lines in both figures denote roughly the transition from the sliding regime to the stick-slip regime.

Shear modulus as a function of the shear rate at two normal pressures and fixed temperature 300 K (a) and two temperatures at fixed normal pressure 52 MPa (b). The dashed lines in both figures denote roughly the transition from the sliding regime to the stick-slip regime.

Yield stress as a function of the shear rate at two temperatures (a) and two normal pressures (b). Insets: the yield stress as a function of temperature (a) and normal pressure (b). The vertical arrows in the insets denote the increase of the shear rate from s^{−1} to s^{−1}. The dashed lines in both figures denote roughly the transition from the sliding regime to the stick-slip regime.

Yield stress as a function of the shear rate at two temperatures (a) and two normal pressures (b). Insets: the yield stress as a function of temperature (a) and normal pressure (b). The vertical arrows in the insets denote the increase of the shear rate from s^{−1} to s^{−1}. The dashed lines in both figures denote roughly the transition from the sliding regime to the stick-slip regime.

Applied work *W* and internal energy *U* at high s^{−1} (a) and low s^{−1} (b) shear rates, temperature 300 K, and pressure 52 MPa.

Applied work *W* and internal energy *U* at high s^{−1} (a) and low s^{−1} (b) shear rates, temperature 300 K, and pressure 52 MPa.

Various contributions to the energy (a) and to the shear stress (b) during the constant shear deformation at low shear rate, s^{−1}, temperature 300 K, and pressure 32 MPa.

Various contributions to the energy (a) and to the shear stress (b) during the constant shear deformation at low shear rate, s^{−1}, temperature 300 K, and pressure 32 MPa.

(a) Change of the average *X* coordinate for the top layer upon shear deformation with different shear rates. (b) Change of the average *X* coordinate for different layers in a capped film upon shear deformation at low, s^{−1}, shear rate. In both figures the calculations were made at 300 K and 52 MPa.

(a) Change of the average *X* coordinate for the top layer upon shear deformation with different shear rates. (b) Change of the average *X* coordinate for different layers in a capped film upon shear deformation at low, s^{−1}, shear rate. In both figures the calculations were made at 300 K and 52 MPa.

(a) Ratio of the averaged velocity of the polymer segments in different layers of simulated film to the actual velocity of the top substrate at normal pressure 52 MPa, temperature 300 K, and for high and low shear rates. The dashed line denotes the velocity profile in the case of the ideal affine shear deformation. (b) *X*-component of the mean squared displacements of all particles in the top layer during shear deformation at high and low shear rates and also for non-sheared film (diamonds). The slopes 2 and 0.5 denote the ballistic and sub-diffusive motion of polymer segments.

(a) Ratio of the averaged velocity of the polymer segments in different layers of simulated film to the actual velocity of the top substrate at normal pressure 52 MPa, temperature 300 K, and for high and low shear rates. The dashed line denotes the velocity profile in the case of the ideal affine shear deformation. (b) *X*-component of the mean squared displacements of all particles in the top layer during shear deformation at high and low shear rates and also for non-sheared film (diamonds). The slopes 2 and 0.5 denote the ballistic and sub-diffusive motion of polymer segments.

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