On a relation between roughening and coarsening

Abstract

We argue that a strict relation exists between two in principle unrelated quantities: The size of the growing domains in a coarsening system, and the kinetic roughening of an interface. This relation is confirmed by extensive simulations of the Ising model with different forms of quenched disorder, such as random bonds, random fields and stochastic dilution.PACS numbers: 05.40.-a Slow relaxation is a feature found in a variety of physical systems including randomly stirred fluids, ballistic aggregation, magnetic flux lines in superconductors [1], directed polymers and manifolds in random media [2][3][4], phase-ordering [6][7][8][9][11][12][13][14][15][16][17][18][19][20] and others. In these cases the evolution is characterized by scale-free power-law behaviors and the equilibrium state is approached on a timescale that diverges in the thermodynamic limit. In this Letter we focus on two paradigms of slow relaxation, i) the evolution a disordered ferromagnet after a temperature quench and ii) the roughening of an interface in a medium. The aim of the paper is to show a strict correspondence between i) and ii).A unified description of the two problems above can be arrived at by considering a ferromagnetic system, namely a collection of interacting Ising spins S i on the sites i of a lattice. For such a system the usual para-ferromagnetic transition occurs at some critical temperature T c . Quenched disorder, in the form of random fields or bonds, stochastic dilution or other sources of randomness, can be present provided that the low-temperature ordered phase is preserved. The two kinds of slow evolution i) and ii) mentioned above can be observed in such a model at low temperature T < T c by preparing the sample with the initial condition (at time t = 0) in the following two ways:i) The value of each spin S i = ±1 is randomly chosen and is uncorrelated from the others, corresponding to the equilibrium state of the ferromagnet at infinite temperature. This protocol amounts to the instantaneous quench of the system from the initial temperature T i = ∞ to the final temperature T < T c . As it is well known, a coarsening stage is observed [5][6][7][8][9][11][12][13][14][15][16][17][18][19][20] with growing magnetic domains of size L(t, ǫ, ℓ), where ℓ is the system size and we indicate generically by ǫ the strength of the disorder. For example, denoting by L the domains size in an infinite system (L(t, ǫ) ≡ L(t, ǫ, ℓ = ∞)), in a pure (non disordered) magnet (ǫ = 0) one usually has L(t, 0) ∝ t 1/z (with z = 2 with a non-conserved order parameter). Dynamical scaling [5] implies that L is the only dominant length at large times.ii) The system is divided into two halves, as for instance by the diagonal in a square system, and spins are set to the value S i = +1 in one half and S i = −1 in the other. Appropriate boundary conditions are provided in such a way that the spanning interface seeded by the initial condition remains at all times (e.g. if the system is divided by the diagonal anti-periodic boundary conditions are us...

Fractal character of the phase ordering kinetics of a diluted ferromagnet

“…It should be mentioned that, in principle, if the quench is made to a finite temperature T f , roughening of interfaces takes place and, on scales of the order of the roughening length R, this should result in K = 0. However, since in d = 2 the roughness of the interface is of order R(t) a(T f )R(t) 1/2[4,69], where a(T f ) is a constant which vanishes for T f → 0, this eect would produce K = 0 on scales R(t) R(t) so small that the eect cannot be observed in our data (see figure6).…”

“…The case d = d c , instead, is much less trivial, because the fractal character of In this paper we continue the study of the coarsening kinetics of diluted systems focusing on the characterisation of the geometrical of the dynamic configurations of the quenched BDIM on the square lattice. Beyond addressing some of the properties of the domains and their boundaries, an issue that has been longly discussed in system with quenched disorder [69][70][71][72][73][74][75][76][77], we also investigate the geometrical properties on scales much larger than those of the correlated regions, following a more recent research line aimed at the comprehension and characterisation of the emerging percolative features.…”