Driven particle in a two-dimensional periodic substrate: Nonmonotonic dependence of drift velocity on temperature

Motion of a driven particle in a two-dimensional (2D) periodic potential of square symmetry is studied by means of Brownian dynamics simulations. The average drift velocity and long time diffusion coefficients are obtained as a function of driving force and temperature. For driving forces above the critical depinning force, a reduction of drift velocity is observed as temperature is increased. The drift velocity reaches a minimum for temperatures at which $k_{B}T$ is of the order of the barrier height of the substrate potential and then increases and saturates to the value of drift velocity for the substrate free case. Depending on the driving force, the drop in drift velocity can be as large as $36\%$ of its value at low temperatures. While this phenomenon is observed in 2D for different types of substrate potentials studied and for various drive directions, studies using the exact result show no such dip in drift velocity in one dimension (1D). As in the case of 1D, a peak is observed in the longitudinal diffusion coefficient as the driving force is varied at a fixed temperature. But unlike in 1D, the location of the peak is temperature dependent. Approximate analytical expressions for the average drift velocity and the longitudinal diffusion coefficient are formulated using the exact results in 1D by finding a temperature dependent effective 1D potential to model the motion in the presence of a 2D substrate. This approximate analysis is successful in qualitatively predicting the observations.

Figure 1

Substrate potential with square symmetry. (a) The plot gives the geometry of the substrate for the particle moving on a 2D periodic potential. The fixed particles that generate the substrate potential are located at the corners of the squares. The squares are drawn to indicate the symmetry of the potential. (b) The surface plot of the potential for the case when $U\left(\mathbf{r}\right)$ corresponds to the screened Coulomb potential [Eq. (6)].

Figure 2

Average drift velocity $v_d$ (in units of $\lambda \eta$) versus $k_{B}T$ (in units of barrier height, $U_b$) for various driving forces. The plots (a), (b), and (c) are for the cases when the driving force is along the edge of the square unit cell ($y$ direction) with magnitude $F_d=F_c,5F_c$, and $10F_c$, respectively. Blue squares are simulation data points, red solid lines are theoretical predictions using the potential of mean force (PMF) approach, and black dash-dotted lines are theoretical fit obtained by including the drive dependent parameter $\alpha$ in determining the effective potential. Error bars shown are standard deviation of the mean. Plot (d) gives the variation of magnitude of the dip, $\gamma$, as a function of the driving force. Red dotted line is a guide to the eye.

Figure 3

Average drift velocity, $v_d$, versus $k_{B}T$ for various driving angles. The plots (a), (b), and (c) are for the cases when the driving force $F_d=10F_c$ at an angle $\theta =6^{\circ },{15}^{\circ }$, and ${25}^{\circ }$, respectively. Error bars shown are standard deviation of the mean. Plot (d) shows the variation of $\gamma$ as a function of driving angle, $\theta$. Blue circles are simulation data points and the dotted lines are shown as a guide to the eye.

Figure 4

Parallel ($v_{\parallel }$) and perpendicular ($v_{\perp }$) components of drift velocity $v_d$ versus $k_{B}T$ for various driving angles. The plots (a),(b), (c),(d), and (e),(f) are for the cases when the driving force $F_d=10F_c$ at an angle $\theta =6^{\circ },{15}^{\circ }$, and ${25}^{\circ }$, respectively. Blue circles are simulation data points and the dotted lines are given as a guide to the eye.

Figure 5

$y$ variance (${\langle \mathrm{\Delta }y\rangle }^2$) and $x$ variance (${\langle \mathrm{\Delta }x\rangle }^2$) (in units of ${\lambda }^2$) as a function of time $t$ (in units of $1/\eta$) for various driving forces. (a) $k_{B}T=0.1U_b$: there is periodic switch to subdiffusive behavior in the direction of the driving force. The period of this switching is drive dependent (see text). The diffusion in the direction perpendicular to the drive is nearly zero. (b) $k_{B}T=1.0U_b$: the oscillatory nature of $y$ variance is absent at higher temperatures. The $x$ variance is substantially larger than at low temperature but decreases with drive. The diffusion constants $D_{\parallel }$ and $D_{\perp }$ are calculated using the slope of these curves in the long time limit. The dashed lines in both plots give the variance for free diffusion.

Figure 6

Diffusion coefficients as a function of driving force for fixed temperatures. (a) $D_{\parallel }$ (in units of $D_0$ at respective temperatures) versus driving force $F_d$ (in units of $F_c$) for $k_{B}T=0.1U_b$ and $1U_b$. Accelerated diffusion is observed in $D_{\parallel }$, wherein there is a peak in the diffusion constant at fixed temperatures. Unlike accelerated diffusion seen in 1D, the position of the diffusion peak is temperature dependent. The continuous curves are fits using the effective 1D theory, both using PMF approach and with drive dependent factor $\alpha$ (see Sec. 4). (b) $D_{\perp }$ versus $F_d$ for temperatures $k_{B}T=0.1U_b,0.5U_b$, and $1U_b$. Accelerated diffusion is not observed for $D_{\perp }$. Diffusion in transverse direction is small for $k_{B}T=0.1U_b$. At all temperatures $D_{\perp }$ decreases monotonically with driving force. The error bars give the standard deviation of the mean.

Figure 7

Plots (a) and (c) give the density distribution of the particle within a substrate unit cell at $k_{B}T=1.0U_b$ for the drives $F_d=F_c$ and $F_d=50F_c$, respectively, for the screened Coulomb potential. The hopping in lateral direction is less for higher drives but the width of the channel is more for larger drive (${\delta }^{\prime }>\delta )$. (d) The variation of the $x$ coordinate with time for the same parameters as above (I is for $F_d=F_c$ and II for $F_d=50F_c$). (b) The variation of the drive dependent parameter $\alpha$ with driving force $F_d$. The main plot shows the $\alpha$ values for the modified Bessel function and screened Coulomb potential substrates [(i) and (ii), respectively] and the inset gives the $\alpha$ variation for the egg-carton potential substrate.

Figure 8

Diffusion coefficients as a function of temperature for fixed driving force. (a) $D_{\parallel }$ versus $k_{B}T$ for $F_d=F_c$. A kink in the value of $D_{\parallel }$ is observed at $k_{B}T\approx 0.2U_b$. The inset shows an enlarged view of the region where the kink is present. (b) $D_{\perp }$ versus $k_{B}T$ for $F_d=F_c$. (c) $D_{\parallel }$ versus $k_{B}T$ for $F_d=10F_c$. For $F_d=10F_c$ an enhanced diffusion peak in $D_{\parallel }$ is observed at $k_{B}T\approx U_b$. (d) $D_{\perp }$ versus $k_{B}T$ for $F_d=10F_c$. The continuous curves are fits using the effective 1D theory, both using PMF approach and with drive dependent factor $\alpha$ (see Sec. 4). The error bars shown are the standard deviation of the mean.

Figure 9

Average drift velocity $v_d$ versus $k_{B}T$ for various driving forces for the case where $U\left(\mathbf{r}\right)$ is the modified Bessel function of the second kind. Plots (a), (b), and (c) are for the cases when the driving force is along the edge of the square unit cell ($y$ direction) with magnitude $F_d=F_c,10F_c$, and $100F_c$, respectively. The blue squares are simulation data and the continuous curves are from theoretical fits discussed in Sec. IV. The error bars are standard deviation of the mean. Plot (d) shows the variation of $\gamma$ as a function of driving force (the dotted line is a visual guide to the eye).

Figure 10

$v_d$ versus $k_{B}T$ for various driving angles for the case when $U\left(\mathbf{r}\right)$ is the modified Bessel function of the second kind. Plots (a), (b), and (c) are for the cases when the driving force $F_d=4F_c$ is applied at an angle $\theta =5^{\circ },{10}^{\circ }$, and ${20}^{\circ }$, respectively, with respect to the $x$ direction. The blue squares are data points and the error bars are the standard deviation of the mean. Plot (d) shows the variation of $\gamma$ as a function of driving angle $\theta$. Blue circles are simulation data points. The dotted lines shown in all the plots are guides to the eye.

Figure 11

Average drift velocity $v_d$ versus $k_{B}T$ for various driving forces for the case where $U_s\left(\mathbf{r}\right)$ is the 2D egg-carton substrate potential. Plots (a), (b), and (c) are for the cases when the driving force is along the edge of the square unit cell ($y$ direction) with magnitude $F_d=0.1F_c,F_c$, and $5F_c$, respectively ($F_c=\frac{2\pi U_0}{d}$). The blue points are simulation data and the continuous curves are from theoretical fits discussed in Sec. 4. The error bars are standard deviation of the mean. Plot (d) shows the variation of $\gamma$ as a function of driving force (the dotted line is a visual guide to the eye).

Figure 12

$v_d$ versus $k_{B}T$ for various driving angles for the case when $U_s\left(\mathbf{r}\right)$ is the 2D egg-carton substrate potential. Plots (a), (b), and (c) are for the cases when the driving force $F_d=0.1F_c$ is applied at an angle $\theta =2^{\circ },{10}^{\circ }$, and ${40}^{\circ }$, respectively, with respect to the $x$ direction. The error bars are the standard deviation of the mean. Plot (d) shows the variation of $\gamma$ as a function of driving angle $\theta$. Here the blue points are simulation data. The dotted lines shown in all the plots are guides to the eye.

Figure 13

$\gamma$ as a function of $U_1$ for different values of drive (dotted line is a visual aid to the eye). It is observed that for low values of driving force the value of $U_1$ at which the dip appears approaches unity. Plots (b), (c), and (d) are contour plots for $U_1=0,U_1=0.5$, and $U_1=1$, respectively.

Figure 14

$v_1$ versus $k_{B}T$ for three different 1D potentials for driving forces (a) $F_d=0.1F_c$, (c) $F_d=1.0F_c$, and (d) $F_d=50F_c$. In all the plots, the solid blue line is for $V_1\left(x\right)=V_{0}cos(2\pi x/d)$, the red dash-dotted line is for $V_2\left(x\right)=V_0{e}^{cos(2\pi x/d)}$, and the black dotted line is for $V_3\left(x\right)={\sum }_{n=1}^5{C}_{n}cos(2\pi x/d)$ (where $\left\{C_n\right\}$ are random numbers). The parameter values used are $V_0=1$ and $d=1$. Plot (b) shows the graphical representation of the three potentials (shifted vertically for clarity). No dip in $v_1$ is observed for any of the potentials studied.