Crossover behavior in interface depinning

We study the crossover scaling behavior of the height-height correlation function in interface depinning in random media. We analyze experimental data from a fracture experiment and simulate an elastic line model with nonlinear couplings and disorder. Both exhibit a crossover between two different universality classes. For the experiment, we fit a functional form to the universal crossover scaling function. For the model, we vary the system size and the strength of the nonlinear term and describe the crossover between the two universality classes with a multiparameter scaling function. Our method provides a general strategy to extract scaling properties in depinning systems exhibiting crossover phenomena.

Figure 1

Crossover scaling in fracture roughness [18]. The inset shows experimental data for the height-height correlation function $C\left(r\right)=\langle {[h(x+r)-h\left(x\right)]}^2\rangle$ of a 2D fracture front, generated by pulling apart two pieces of PMMA that have been sand-blasted and sintered together [18]. The three curves differ in the size of the sand-grain beads; the relation between the bead size and the toughness fluctuations in the PMMA were not measured. The dashed lines show two different power-law critical regimes, with $C\left(r\right)\sim r^{2{\zeta }_{-}}$ and $C\left(r\right)\sim r^{2{\zeta }_{+}}$, governing the short- and long-distance scaling behavior: the crossover between these regimes is evident. Our fit gives ${\zeta }_{-}=0.63$ and ${\zeta }_{+}=0.32$, within the experimentalists' suggested range ${\zeta }_{-}=0.6\pm 0.05$ and ${\zeta }_{+}=0.35\pm 0.05$. The main figure shows a scaling plot of $r^{-2{\zeta }_{-}}C\left(r\right)$ versus $r$, with the curves shifted vertically and horizontally to best collapse. The thick black curve is a one-parameter fit of the universal scaling function to the functional form in Eq. (4).

Figure 2

Crossover of qKPZ to qEW model. Fronts generated from 128 $\times$ 256 simulations with the nonlinear KPZ term coefficients set to (a) $\lambda =0$, (b) $\lambda =0.001$, (c) $\lambda =0.1$, (d) $\lambda =5$. The random colors (shades) represent the area between each pinned front. The morphology of the interfaces changes dramatically as $\lambda$ increases.

Figure 3

Local log slope. The measured local-log slope, $dlogC/dlogr$ of the height-height correlation function for varying $\lambda$ and $k=0.01$. The lower dashed red line is ${\zeta }_{\mathrm{KPZ}}=0.63$ as expected for the KPZ universality class. The upper dashed blue line is ${\zeta }_{SR}=1.0$, the largest growth allowed for super-rough interfaces. The thin lines show the predictions of our fits (Sec. 4), and the dashed black line is the fit value of ${\zeta }_{\mathrm{KPZ}}$. The sharp cutoff in the curves at large $r$ is due to the periodic boundary condition, which forces ${\zeta }_{\mathrm{eff}}=0$ at $r=L/2$; the larger $L$ curves have later cutoffs. As expected, for small $\lambda$ the curves are described by the EW behavior ${\zeta }_{SR}=1$ at small $r$, while for large $\lambda$ they are well described by ${\zeta }_{\mathrm{KPZ}}$. For intermediate values of $\lambda \sim 0.1$, we observe a clear transition from EW behavior at small $r$ to $\mathrm{KPZ}$ behavior at intermediate $r$, before being cut off by the finite size effects.

Figure 4

Height-height correlation function. The numerics generated with an automata code (symbols) are well described by Eq. (16) (black curves) with fit parameters $\varphi =1.0\pm 0.4, {\zeta }_{\mathrm{KPZ}}=0.65\pm 0.04, {\zeta }_{\mathrm{EW}}=1.1\pm 0.15, n_{\mathrm{Cross}}=1.0\pm 0.7, B=2.5\pm 6.0$. $n_{\mathrm{EW}}=0.27\pm 0.03, n_{\mathrm{KPZ}}=1.1\pm 0.6, A_0=2.7\pm 7, A_1=770\pm 900$, and $A_{\infty }=0.3\pm 1$. (A fit constrained to the expected values of ${\zeta }_{\mathrm{EW}}$ and ${\zeta }_{\mathrm{KPZ}}$ give slightly worse, but acceptable fits, with the other parameter estimates within the quoted ranges). The legends denote $L$ and $\lambda$ for each simulated correlation function; all runs had $k=0.01$. The errors quoted are a rough measure of the systematic error [48], as described in the text, and are representative of the differences we find using different weightings and functional forms. (They are much larger than the statistical errors). The amplitude dependence is captured by the scaling form. Three of the 12 parameters (${\zeta }_{\mathrm{EW}}, {\zeta }_{\mathrm{KPZ}}$, and $\varphi$) are universal critical exponents, three ($A_0, A_1$, and $A_{\infty }$) describe the nonuniversal dependence of an overall height scale on parameters, two describe finite-size effects, and only two ($n_{\mathrm{Cross}}$ and $B$) are needed to describe the universal crossover function to the accuracy shown.

Figure 5

Rescaled spectral function $S\left(q\right|L,\lambda )$ for the height fluctuations. Rescaling by the naive RG power $q^{1+2{\zeta }_{\mathrm{EW}}}$ does collapse the data for vanishing $\lambda$ (up to a significant noise): the anomalous scaling for the super-rough interface in real space is not manifested in Fourier space. The theory curves are given by the Fourier transform of the same best fit shown in previous figures.

Figure 6

Scaling collapse of the height-height correlation function. The correlation-function data of Fig. 4 are collapsed to illustrate the crossover between EW and KPZ-dominated lengths ($r<r^{*}$ and $r>r^{*}$, respectively; see also Refs. [28, 29]). Here $r^{*}=L{\left[B{\left(L{\lambda }^{\varphi }\right)}^{2({\zeta }_{\mathrm{EW}}-{\zeta }_{\mathrm{KPZ}})}\right]}^{-1/(2-2{\zeta }_{\mathrm{KPZ}})}$ is the distance where the EW and KPZ components of the correlation function are equal in magnitude, and $C^{*}={\lambda }^{-2\varphi ({\zeta }_{\mathrm{EW}}-{\zeta }_{\mathrm{KPZ}})}{r}^{2{\zeta }_{\mathrm{KPZ}}}$ factors out the dependence expected in the KPZ regime (hence yielding flat behavior for $r\gg r^{*}$). (b) Effects of analytic corrections to scaling are also factored out; all of the curves lie on a universal curve apart from the effects of the finite system sizes (causing each curve to drop on the right). (a) The analytic correction to scaling has a significant impact on the scaling collapse: ignoring it in the analysis would produce significant errors in critical exponents and scaling functions. The thick blue curve shows the scaling function prediction for the crossover [which, if $x=r/r^{*}$, can be shown to be $C/C^{*}=B{x}^2{(x^{2n_{\mathrm{Cross}}}+x^{2n_{\mathrm{Cross}}{\zeta }_{\mathrm{KPZ}}})}^{-1/n_{\mathrm{Cross}}}]$.