arXiv:1001.1260v1 [cond-mat.mtrl-sci] 8 Jan 2010 Dynamical transitions and sliding friction of the phase-field-crystal model with pinning J.A.P. Ramos 1 , $^{2}$, E. Granato 2 , $^{3}$, S.C. Ying $^{3}$, C.V. Achim 4 , $^{5}$, K.R. Elder $^{6}$, and T. Ala-Nissila 3 , 4 1 Departamento de Ciˆ encias Exatas, Universidade Estadual do Sudoeste da Bahia, 45000-000 Vit´ oria da Conquista, BA,Brasil $^{2}$Laborat´ orio Associado de Sensores e Materiais, Instituto Nacional de Pesquisas Espaciais,12245-970 S˜ ao Jos´ e dos Campos, SP, Brazil $^{3}$Department of Physics, P.O. Box 1843, Brown University, Providence, RI 02912-1843, USA $^{4}$Department of Applied Physics and COMP Center of Excellence, P.O. Box 1100, Helsinki University of Technology, FI-02015 TKK, Espoo, Finland 5 Institut f¨ ur Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universit¨ at D¨ usseldorf, Universit¨ atsstraße 1, D-40225 D¨ usseldorf, Germany and $^{6}$Department of Physics, Oakland University, Rochester, Michigan 48309-4487, USA We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers. PACS numbers: 64.60.Cn, 68.43.De,64.70.Rh, 05.40.-a ticle density while still retaining information on atomic length scales. By extending the PFC model to take into account the effect of an external pinning potential $^{13}$, a two-dimensional version of the model has been used to describe commensurate-incommensurate transitions in the presence of thermal fluctuations 14 and the driven re- sponse of pinned lattice systems without thermal fluctu- ations and inertial effects. $^{15}$. However, in order to study fully the sliding friction behavior, both inertial effects and thermal fluctuations need to be taken into account. In this work, we study sliding friction of an adsorbed layer via the nonlinear response of the PFC model to an external force in the presence of a pinning potential. To this end, we first derive the general stochastic dynamical equations for the particle and momentum density fields taking full account of the thermal fluctuations and iner- tial effects. The resulting coupled equations are studied numerically. At low temperatures, we find that the veloc- ity response of an initially commensurate layer shows hys- teresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear re- sponse are similar to the results obtained previously with molecular dynamics simulations of particle models. How- ever, the dynamical melting and freezing mechanisms are significantly different. In the PFC model, nucleation oc- curs via stripes rather than closed domains found in the particles model. Recently, a phase-field-crystal (PFC) model was introduced 10–12 that allows for both elastic and plastic deformations within a continuous description of the par- In recent years, considerable attention has been given to the study of driven adsorbed layers in relation to sliding friction phenomena between surfaces at the mi- croscopic level $^{1–6}$. The nonlinear response of the ad- sorbed layer is a central problem for understanding ex- periments on sliding friction between two surfaces with a lubricant 4 or between adsorbed layers and an oscillat- ing substrate $^{7,8}$. Various elastic and particle models have been used to study the driven dynamical transitions and the sliding friction of adsorbed monolayers $^{1–3,5}$. A fun- damental issue in modeling such systems is the origin of the hysteresis and the dynamical melting and freezing transitions associated with the different static and kinetic frictional forces when increasing the driving force from zero and decreasing from a large value, respectively. In driven lattice systems, hysteresis can occur in the under- damped regime where inertial effects are present or in the presence of topological defects such as dislocations and thermal fluctuations $^{9}$. Topological defects can be auto- matically included in a full microscopic model involving interacting atoms in the presence of a substrate potential using realistic interaction potentials. However, numeri- cal observation of the full complexities of the phenomena is severely limited by the small system sizes that can be studied, even when simple Lennard-Jones potentials are employed $^{3}$. I. INTRODUCTION