Kardar-Parisi-Zhang interfaces bounded by long-ranged potentials

We study unbinding transitions of a nonequilibrium Kardar-Parisi-Zhang interface in the presence of long-ranged substrates. Both attractive and repulsive substrates, as well as positive and negative Kardar-Parisi-Zhang nonlinearities, are considered, leading to four different physical situations. A detailed comparison with equilibrium wetting transitions as well as with nonequilibrium unbinding transitions in systems with short-ranged forces is presented, yielding a comprehensive picture of unbinding transitions and of their classification into universality classes. These nonequilibrium transitions may play a crucial role in the dynamics of the wetting or growth of systems with intrinsic anisotropies.

Figure 1

(Color online) Effective potentials as derived from Eq. (3) in the pinned $(a<0)$ and the depinned $(a>0)$ phases and at coexistence $(a=0)$. (a) Repulsive walls $(b,c>0)$ and (b) attractive walls ($b<0$, $c>0$).

Figure 2

(Color online) Phase diagrams for $\lambda <0\left(a\right)$, $\lambda >0\left(b\right)$. Paths labeled 1 correspond to nonequilibrium complete wetting transitions; 2, critical or first-order unbinding transitions (not studied); 3, first-order unbinding transitions; 4, unbinding transition in the directed percolation universality class. For $\lambda <0$ and $b<b_w$ (attractive substrates), two-phase coexistence is observed in the area delimited by the two lines.

Figure 3

(Color online) Effective potentials in terms of $n$ obtained from the numerical integration of the interaction part of Eq. (10). Left and right panels correspond to, respectively, repulsive $(b=1)$ and attractive $(b=-1)$ interactions. For both $b$ we show results for negative and positive $\lambda$ and different values of $a$ corresponding to the bound and unbound phases, as well as at coexistence. Note that for $\lambda <0$ $(>0)$ the unbound phase corresponds to a minimum of $n$ at 0 $\left(\infty \right)$. Transitions for attractive walls can be either first order or continuous (directed percolation).

Figure 4

(Color online) Features common to all simulations. (a) Roughness vs $t$ gives ${\beta }_W=0.33\left(1\right)$; (inset) $-a_c\left(L\right)$ vs ${10}^3{L}^{-1}$ falls on a straight line that yields $\nu \sim 1$ (data for $\lambda =-1$ and $p=2$). (b) Saturation time vs system size leads to $z=1.48\left(4\right)$ (data for $\lambda =1$ and $p=2$).

Figure 5

(Color online) Log-log plot of the time evolution at $a_c$ of $⟨h⟩$ (upper, red curves), $-\ln ⟨n_{\mathit{OP}}⟩$ (middle, green curves), and the width $w$ (lower, black curves), in the mean-field regime $p=0.5$, for $\lambda <0$ (main) and $\lambda >0$ (inset). Irrespective of the sign of $\lambda$, $⟨h⟩$ and the roughness may be fitted to a power law with the predicted exponents ${\theta }_h=1∕(p+2)$ and ${\beta }_W=1∕3$, respectively. $-\ln ⟨n_{\mathit{OP}}⟩$ falls on a straight line in a double logarithmic plot, confirming the stretched-exponential behavior of the order parameter.

Figure 6

(Color online) Log-log plot of the time evolution at $a_c$ of $⟨h⟩$ (dashed, black curve) and $-\log ⟨n_{\mathit{OP}}⟩$ (solid, red curve) in the fluctuation regime $(p=2)$ and for $\lambda >0$. From the slopes of the straight-line fits one finds ${\theta }_h=0.32\left(2\right)$ (upper curve) and ${\theta }_{\mathit{OP}}=0.228\left(6\right)$ (lower curve). Upper inset: finite-size scaling of $⟨h⟩$ yielding ${\beta }_h∕\nu =0.46\left(2\right)$. From the lower inset one obtains ${\beta }_{\mathit{OP}}∕\nu =0.34\left(2\right)$. These exponents agree with those of the MN2 universality class.

Figure 7

(Color online) Main: order-parameter multiplied by the expected power law $t^{0.228}$. For systems with $b=1$, long transients that depend on $p$ are observed for systems with $1<p<2$. Inset (a): the crossover times are reduced as $b$ decreases; compare the upper, red curve for $b=0.05$ with the lower, black one for $b=1$ $(p=2)$. Inset (b): order-parameter at $t={10}^6$ vs $p$ $(b=1)$. $p=2$ marks the boundary of the strong transient region as illustrated by the different behaviors observed above and below $p=2$.

Figure 8

(Color online) Results for $\lambda <0$ and $p=2$. (a) Log-log plot of the time decay of the order-parameter. The straight line is a guide to the eye and has a slope ${\theta }_{\mathit{OP}}=1.19\left(1\right)$. (b) Log-log plot of the average distance from the wall vs time, leading to ${\theta }_h=0.33\left(1\right)$. (c) The scaling of the saturation value of the order parameter yields ${\beta }_{\mathit{OP}}=1.76\left(3\right)$. (d) Log-log plot of the gap (between contact points) distribution function at $t=2^{10}$. The initial slope gives $z{\theta }_{\mathit{OP}}\approx 1.75\left(10\right)$.

Figure 9

(Color online) Time decay of the order-parameter at the critical point $a=0.38448$. From the slope in the log-log plot we find $\theta =0.161\left(2\right)$. Inset: finite-size scaling of the order parameter. From the slope in the log-log plot we estimate $\beta ∕\nu =0.26\left(2\right)$, in agreement with directed percolation values.

Figure 10

(Color online) Time growth exponents ${\theta }_h$ in the mean-field-like regime for $⟨h⟩$ at the critical point, as results from the discrete interfacial model. Blue circles (red squares) stand for $\lambda >0$ $(<0)$ data points, and the solid line is the predicted curve $1∕(p+2)$.