Effects of Boundary Conditions on Magnetic Friction

TL;DR

This study investigates how boundary conditions and sample thickness influence magnetic friction in ferromagnetic Ising models, revealing controllable frictional forces via boundary magnetic fields and critical temperature singularities.

Contribution

It demonstrates the impact of boundary conditions on magnetic friction and introduces a method to control frictional forces using boundary magnetic fields.

Findings

Friction can be controlled by boundary magnetic fields.

Critical temperature exhibits singularity dependent on velocity.

Boundary conditions significantly affect magnetic friction behavior.

Abstract

We consider magnetic friction between two square lattices of the ferromagnetic Ising model of finite thickness. We analyze the dependence on the boundary conditions and the sample thickness. Monte Carlo results indicate that the setup enables us to control the frictional force by magnetic fields on the boundaries. In addition, we confirm that the temperature derivative of the frictional force as well as that of the boundary energy has singularity at a velocity-dependent critical temperature.

Figures

Equations

$$H=H_{\mathrm{lower}}+H_{\mathrm{upper}}+H_{\mathrm{slip}}(t),$$

$$H_{\mathrm{lower}}:=$$

$$-J\sum _{x=1}^{L_x}\sum _{z=1}^{L_z/2-1}{\sigma }_{x,z}{\sigma }_{x,z+1},$$

$$H_{\mathrm{upper}}:=$$

$$-J\sum _{x=1}^{L_x}\sum _{z=L_z/2+1}^{L_z-1}{\sigma }_{x,z}{\sigma }_{x,z+1},$$

$$H_{\mathrm{slip}}(t):=$$

$$F=$$

Full text

I Introduction

Many studies have revealed the nature of friction and its applications Persson (2000); Yoshino et al. (2007); Saito and Matsukawa (2007); Goryo et al. (2007); Inui (2007); Zaloj et al. (1999); Sasaki et al. (2007); Filippov et al. (2008); Dudko et al. (2002); Braiman et al. (2003); Tshiprut et al. (2005, 2009); Guerra et al. (2008) and yet the friction has been difficult to understand from a microscopic point of view. In non-equilibrium statistical physics, it is also an unanswered question how macroscopic sliding motion of objects makes their microscopic quasi-particle excitations on the sliding surface and how the excited energy dissipates. Several experimental facts suggest that physical degrees of freedom, such as phonon Highland and Krim (2006); Coffey and Krim (2005); Torres et al. (2006); Dag and Ciraci (2004); Liebsch et al. (1999); Tomassone and Sokoloff (1999); Sokoloff and Tomassone (1998), orbital motion of electrons Sokoloff (2018); Conache et al. (2010); Park et al. (2007); Sokoloff and Tomassone (1998) and magnetic moment of spins Wolter et al. (2012); Ouazi et al. (2014), play roles of dissipation channels. Especially for the magnetic moment, Monte Carlo simulations of classical spin systems by the use of the Monte Carlo simulations and the analysis based on the Landau-Lifshitz-Gilbert equation Kadau et al. (2008); Magiera et al. (2009); Hucht (2009); Magiera et al. (2011); Hilhorst (2011); Iglói et al. (2011); Heinrich (2012); Hucht and Angst (2012); Li and Pleimling (2012); Angst et al. (2012); Magiera (2013); Li and Pleimling (2016) have revealed several facts regarding the friction due to magnetism from the viewpoints of statistical mechanics.

We here explore behavior of the magnetic friction by considering an Ising ferromagnetic system from two new points of view, namely the effects of finite thickness and boundary conditions (see Fig. 1). Many facts with the magnetic friction have been revealed, but almost all of them are related to the model of infinite size (Fig. 1(1(a))) Hucht and Angst (2012); Iglói et al. (2011); Hilhorst (2011); Hucht (2009); Li and Pleimling (2016); Angst et al. (2012), where almost exclusively non-equilibrium phase transitions are discussed. In order to understand the non-equilibrium nature of classical spin systems, however, finite-size extension is one of the most important directions. We have made the system size finite in the direction perpendicular to the slip line and apply two types of boundary conditions on the top and bottom of the system (Fig. 1(1(b))).

Our two-dimensional Ising model has two parameters, namely the temperature $T$ and the sliding velocity $v$ Kadau et al. (2008), in addition to the sample thickness and the boundary conditions. Since we are interested in the size dependence in the direction perpendicular to the slip line ($z$ direction), we take the thermodynamic limit only in the direction parallel to the slip line ($x$ direction). This process enables us to examine how extensive physical quantities depend on the thickness $L_{z}$ in the $z$ direction. We compare them under two extreme fixed boundary conditions, namely anti-parallel and parallel conditions (see Fig. 2). Because of the finite thickness, we can discuss boundary-condition dependence of physical quantities, especially the frictional force.

II Numerical Simulations

Our two-dimensional Ising model is described by the following Hamiltonian:

[TABLE]

where

[TABLE]

with the ferromagnetic interaction $J>0$ and the Ising spin variables $\sigma_{x,z}=\pm 1$ ($1\leq x\leq L_{x},1\leq z\leq L_{z}$). A slip line runs at the center between the $(L_{z}/2)$th and the $(L_{z}/2+1)$th layers, along which the interactions are replaced at a constant velocity.

In the case of $v=0$, our model exhibits the corresponding equilibrium state and various configurations of domain walls under a temperature $T$. If we add an external force on the system and make it slide, the domain walls around the slip line temporally break and thus the total energy rises. Subsequently, the system tries to return to equilibrium and thus the energy decreases. As the result of these two competing effects, the system exhibits a non-equilibrium steady state which depends both on the temperature $T$ and the sliding velocity $v$.

In the Monte Carlo simulation, we realize the sliding motion of the system by means of discrete sliding over a lattice constant every $1/v$ unit time and realize relaxation to the equilibrium by a sequence of single-spin flips (see Fig. 3). We implement our Monte Carlo simulations with single-spin-flip dynamics according to Ref. Kadau et al. (2008). The actual Monte Carlo schedule is as follows: (i) We try to relax the system to equilibrium; (ii) We slide the upper part of the system by a lattice constant against the lower; (iii). We perform single-spin flips $N/v$ times. Repeating these steps $v$ times corresponds to one Monte Carlo sweep, which therefore consists of $v$ times of slides and $N$ times of single-spin flips.

After the non-equilibrium steady state is achieved, the average energy increase due to the discrete slides per a unit time is equal to the average power done by the external force, which is the reaction to the frictional force in the steady state. Therefore, the frictional force is given by the expectation value of the energy change due to the slides over a unit time:

[TABLE]

where $\Delta H_{\rm slip}$ denotes the change due to the discrete slides over a unit time and $\left\langle\bullet\right\rangle_{\rm st}$ the steady-state expectation value. We extrapolate the thermodynamic limit in the $x$ direction so that the size dependence of $F/L_{x}$ may disappear.

III Results

In the following Monte Carlo results, we took the average over 480 samples of continuous measurements over 3200 sweeps after the initial relaxation of 3200 sweeps. We adopted as the initial configuration a domain wall on the slip line in the case of the anti-parallel boundary conditions and the completely magnetized configuration in the case of parallel boundary conditions. We chose these initial states because they are the most natural ground states that are consistent with their boundary conditions.

For each value of $L_{z}$, we estimated $F/L_{x}$ in the cases of $L_{x}=30\times L_{z},40\times L_{z},50\times L_{z}$, from which we judged that all the data well approximate the limit $L_{x}\to\infty$ within the error-bar range (see Fig. 4 (4(a))). We find a clear difference between the estimates for the two types of boundary condition (see Fig. 5), until it virtually vanishes for $L_{z}=64$.

The following reasons may explain why it is achieved that the frictional force under the anti-parallel condition becomes larger than that under the parallel condition with $T\simeq T_{\rm c}$ and sufficiently small $L_{z}$: When the system is approximately two-dimensional, namely in the limit $L_{z}\to\infty$, the ratio between the correlation length and the size, $\xi_{z}(L_{z})/L_{z}$, approaches 1 around $T\simeq T_{\rm c}$. This effect is not diminished even when the size $L_{z}$ is sufficiently small as long as the condition $T\simeq T_{\rm c}$ holds, and hence the system has a strong correlation in the $z$ direction. Therefore the anti-parallel boundary condition stabilizes the domain wall along the slip line in the initial state through the long-range correlation, competing with the temperature fluctuation. The spatial fluctuation of the domain wall increases the number of crossing points between the domain wall and the slip line (Fig. 6), which raises the frictional force.

Our results also suggest transitions between two different steady states due to switching of boundary conditions: the frictional force can be enhanced or reduced by switching the boundary condition between the parallel and anti-parallel ones.

Incidentally, the temperature derivative of the frictional force indicates a peak growing for larger size $L_{z}$ (see Fig. 7). We observed that the peak location converges to $T\simeq 2.4$ in the limit $L_{z}\to\infty$, which is consistent with the non-equilibrium critical temperature $T_{\rm c,eq}(v)|_{v=10}\simeq 2.4$ reported in Ref. Hucht (2009). We also measured the energy density of the whole system $E/(L_{x}L_{z})$ as a bulk quantity (see Fig. 8 and Fig. 4 (4(b))). Its temperature derivative $(\partial E/\partial T)/(L_{x}L_{z})$ exhibits a peak at the temperature $T\simeq 2.26$ (see Fig. 9), which is consistent with the exact solution of the two-dimensional Ising model.

IV Summary and Discussions

We considered a two-dimensional cylindrical Ising model with a one-dimensional circular slip line, and made its size in the perpendicular direction to the slip line finite, in order to introduce non-trivial conditions on the two open boundaries. We then discussed the boundary-condition dependence of physical quantities.

We found that the boundary condition imposed on the top and bottom edges of the system has a non-negligible effect on the frictional force when the thickness $L_{z}$ is small and the temperature $T$ is close to the non-equilibrium critical temperature $T_{\rm c,eq}(v)$. The frictional force under the anti-parallel condition is greater than the case of the parallel condition.

A large value of the ratio $\xi_{z}/L_{z}$ of the correlation length $\xi_{z}$ along the $z$ direction and the size $L_{z}$ in the vicinity of the equilibrium critical temperature $T_{\rm c,eq}^{(2)}=2.26$ makes the difference of the frictional forces greater around the critical temperature. This effect diminishes around $L_{z}\simeq 64$ and thus the system is almost two-dimensional for $L_{z}=64$.

We expect that our model is applicable to technical control of friction in practical situations by aligning selectively the spins on the boundaries. The magnetic contribution of friction estimated from actual magnetic metal may be comparable to other contributions, such as phonon and orbital motion of electrons Kadau et al. (2008). According to Monte Carlo simulations, this method of control is effective when two boundaries are close enough.

As future works, we plan to discover other examples in which switching the boundary conditions for finite-thickness systems makes a remarkable difference on the frictional force, suggest an experimental method of effectively controlling the frictional force by switching the boundary conditions at the temperature with the maximum difference of frictional forces between the two boundary conditions, and make the model calculation more precise to realize such an experiment.