Geometry dependence in linear interface growth.

The effect of geometry in the statistics of nonlinear universality classes for interface growth has been widely investigated in recent years, and it is well known to yield a split of them into subclasses. In this work, we investigate this for the linear classes of Edwards-Wilkinson and of Mullins-Herring in one and two dimensions. From comparison of analytical results with extensive numerical simulations of several discrete models belonging to these classes, as well as numerical integrations of the growth equations on substrates of fixed size (flat geometry) or expanding linearly in time (radial geometry), we verify that the height distributions (HDs) and the spatial and the temporal covariances are universal but geometry-dependent. In fact, the HDs are always Gaussian, and, when defined in terms of the so-called "KPZ ansatz" [h≃v_{∞}t+(Γt)^{ß}χ], their probability density functions P(χ) have mean null, so that all their cumulants are null, except by their variances, which assume different values in the flat and radial cases. The shape of the (rescaled) covariance curves is analyzed in detail and compared with some existing analytical results for them. Overall, these results demonstrate that the splitting of such university classes is quite general, being not restricted to the nonlinear ones.

Circular Kardar-Parisi-Zhang interfaces evolving out of the plane.

Circular KPZ interfaces spreading radially in the plane have Gaussian unitary ensemble (GUE) Tracy-Widom (TW) height distribution (HD) and Airy_{2} spatial covariance, but what are their statistics if they evolve on the surface of a different background space, such as a bowl, a mountain, or any surface of revolution? To give an answer to this, we report here extensive numerical analyses of several one-dimensional KPZ models on substrates whose size enlarges as 〈L(t)〉=L_{0}+ωt^{γ}, while their mean height 〈h〉 increases as usual [〈h〉∼t]. We show that the competition between the L enlargement and the correlation length (ξ≃ct^{1/z}) plays a key role in the asymptotic statistics of the interfaces. While systems with γ>1/z have HDs given by GUE and the interface width increasing as w∼t^{ß}, for γ<1/z the HDs are Gaussian, in a correlated regime where w∼t^{αγ}. For the special case γ=1/z, a continuous class of distributions exists, which interpolate between Gaussian (for small ω/c) and GUE (for ω/c≫1). Interestingly, the HD seems to agree with the Gaussian symplectic ensemble (GSE) TW distribution for ω/c≈10. Despite the GUE HDs for γ>1/z, the spatial covariances present a strong dependence on the parameters ω and γ, agreeing with Airy_{2} only for ω≫1, for a given γ, or when γ=1, for a fixed ω. These results considerably generalize our knowledge on 1D KPZ systems, unveiling the importance of the background space on their statistics.

Kardar-Parisi-Zhang growth on one-dimensional decreasing substrates.

Recent experimental works on one-dimensional (1D) circular Kardar-Parisi-Zhang (KPZ) systems whose radii decrease in time have reported controversial conclusions about the statistics of their interfaces. Motivated by this, here we investigate several one-dimensional KPZ models on substrates whose size changes in time as L(t)=L_{0}+ωt, focusing on the case ω<0. From extensive numerical simulations, we show that for L_{0}≫1 there exists a transient regime in which the statistics is consistent with that of flat KPZ systems (the ω=0 case), for both ω<0 and ω>0. Actually, for a given model, L_{0} and |ω|, we observe that a difference between ingrowing (ω<0) and outgrowing (ω>0) systems arises only at long times (t∼t_{c}=L_{0}/|ω|), when the expanding surfaces cross over to the statistics of curved KPZ systems, whereas the shrinking ones become completely correlated. A generalization of the Family-Vicsek scaling for the roughness of ingrowing interfaces is presented. Our results demonstrate that a transient flat statistics is a general feature of systems starting with large initial sizes, regardless of their curvature. This is consistent with their recent observation in ingrowing turbulent liquid crystal interfaces, but it is in contrast with the apparent observation of curved statistics in colloidal deposition at the edge of evaporating drops. A possible explanation for this last result, as a consequence of the very small number of monolayers analyzed in this experiment, is given. This is illustrated in a competitive growth model presenting a few-monolayer transient and an asymptotic behavior consistent, respectively, with the curved and flat statistics.

Universality and dependence on initial conditions in the class of the nonlinear molecular beam epitaxy equation.

We report extensive numerical simulations of growth models belonging to the nonlinear molecular beam epitaxy (nMBE) class, on flat (fixed-size) and expanding substrates (ES). In both d=1+1 and 2+1, we find that growth regime height distributions (HDs), and spatial and temporal covariances are universal, but are dependent on the initial conditions, while the critical exponents are the same for flat and ES systems. Thus, the nMBE class does split into subclasses, as does the Kardar-Parisi-Zhang (KPZ) class. Applying the "KPZ ansatz" to nMBE models, we estimate the cumulants of the 1+1 HDs. Spatial covariance for the flat subclass is hallmarked by a minimum, which is not present in the ES one. Temporal correlations are shown to decay following well-known conjectures.

Width and extremal height distributions of fluctuating interfaces with window boundary conditions.

We present a detailed study of squared local roughness (SLRDs) and local extremal height distributions (LEHDs), calculated in windows of lateral size l, for interfaces in several universality classes, in substrate dimensions d_{s}=1 and 2. We show that their cumulants follow a Family-Vicsek-type scaling, and, at early times, when ξ≪l (ξ is the correlation length), the rescaled SLRDs are given by log-normal distributions, with their nth cumulant scaling as (ξ/l)^{(n-1)d_{s}}. This gives rise to an interesting temporal scaling for such cumulants as 〈w_{n}〉_{c}∼t^{γ_{n}}, with γ_{n}=2nß+(n-1)d_{s}/z=[2n+(n-1)d_{s}/α]ß. This scaling is analytically proved for the Edwards-Wilkinson (EW) and random deposition interfaces and numerically confirmed for other classes. In general, it is featured by small corrections, and, thus, it yields exponents γ_{n} (and, consequently, α,ß and z) in good agreement with their respective universality class. Thus, it is a useful framework for numerical and experimental investigations, where it is usually hard to estimate the dynamic z and mainly the (global) roughness α exponents. The stationary (for ξ≫l) SLRDs and LEHDs of the Kardar-Parisi-Zhang (KPZ) class are also investigated, and, for some models, strong finite-size corrections are found. However, we demonstrate that good evidence of their universality can be obtained through successive extrapolations of their cumulant ratios for long times and large l. We also show that SLRDs and LEHDs are the same for flat and curved KPZ interfaces.

Atomic Force Microscopy of spermidine-induced DNA condensates on silicon surfaces.

In the present work, we show that oxidized silicon may be successfully used to image multivalent cation-induced DNA condensates under the Atomic Force Microscope (AFM). The images thus obtained are good enough, allowing us to distinguish between different condensate forms and to perform nanometer-sized measurements. Qualitative results previously obtained using mica as a substrate are recovered here. We additionally show that the interactions between the cation spermidine (the condensing agent) and the DNA molecules are not significantly disturbed by the silicon surface, since the phase behavior of an ensemble of DNA molecules deposited on the silicon substrate as a function of the cation concentration is very similar to that found in solution.

Quantitative assessment of the interplay between DNA elasticity and cooperative binding of ligands.

Binding of ligands to DNA can be studied by measuring the change of the persistence length of the complex formed, in single-molecule assays. We propose a methodology for persistence length data analysis based on a quenched disorder statistical model and describing the binding isotherm by a Hill-type equation. We obtain an expression for the effective persistence length as a function of the total ligand concentration, which we apply to our data of the DNA-cationic ß-cyclodextrin and to the DNA-HU protein data available in the literature, determining the values of the local persistence lengths, the dissociation constant, and the degree of cooperativity for each set of data. In both cases the persistence length behaves nonmonotonically as a function of ligand concentration and based on the results obtained we discuss some physical aspects of the interplay between DNA elasticity and cooperative binding of ligands.