Functional Renormalization Description of the Roughening Transition

We reconsider the problem of the static thermal roughening of an elastic manifold at the critical dimension $d=2$ in a periodic potential, using a perturbative Functional Renormalization Group approach. Our aim is to describe the effective potential seen by the manifold below the roughening temperature on large length scales. We obtain analytically a flow equation for the potential and surface tension of the manifold, valid at all temperatures. On a length scale $L$, the renormalized potential is made up of a succession of quasi parabolic wells, matching onto one another in a singular region of width $\sim L^{-6/5}$ for large $L$. We also obtain numerically the step energy as a function of temperature, and relate our results to the existing experimental data on $^4$He. Finally, we sketch the scenario expected for an arbitrary dimension $d<2$ and examine the case of a non local elasticity which is realized physically for the contact line.Comment: 21 pages, 2 .ps figures. Submitted to E.P.J.

Comment on ``Roughening Transition of Interfaces in Disordered Media''

Emig and Nattermann (Phys. Rev. Lett. 81, 1469 (1998)) have recently investigated the competition between lattice pinning and impurity pinning using a Renormalisation Group (RG) approach. For elastic objects of internal dimensions $2 < D < 4$, they find, at zero temperature, an interesting second order phase transition between a flat phase for small disorder and a rough phase for large disorder. These results contrast with those obtained using the replica variational approach for the same problem, where a first order transition between flat and rough phases was predicted. In this comment, we show that these results can be reconciled by analysing the RG flow for an arbitrary dimension $N$ for the displacement field.Comment: Submitted to Phys. Rev. Let

Domain wall roughening in dipolar films in the presence of disorder

We derive a low-energy Hamiltonian for the elastic energy of a N\'eel domain wall in a thin film with in-plane magnetization, where we consider the contribution of the long-range dipolar interaction beyond the quadratic approximation. We show that such a Hamiltonian is analogous to the Hamiltonian of a one-dimensional polaron in an external random potential. We use a replica variational method to compute the roughening exponent of the domain wall for the case of two-dimensional dipolar interactions.Comment: REVTEX, 35 pages, 2 figures. The text suffered minor changes and references 1,2 and 12 were added to conform with the referee's repor

Wandering of a contact line at thermal equilibrium

We reconsider the problem of the solid-liquid-vapour contact-line on a disordered substrate, in the collective pinning regime. We go beyond scaling arguments and perform an analytic computation, through the replica variational method, of the fluctuations of the line. We show how gravity effects must be included for a proper quantitative comparison with available experimental data of the wetting of liquid helium on a caesium substrate. The theoretical result is in good agreement with experimental findings for this case.Comment: 24 laTex pages with 5 EPS figures included. submitted to Phys. Rev

Dissipation in Dynamics of a Moving Contact Line

The dynamics of the deformations of a moving contact line is studied assuming two different dissipation mechanisms. It is shown that the characteristic relaxation time for a deformation of wavelength $2\pi/|k|$ of a contact line moving with velocity $v$ is given as $\tau^{-1}(k)=c(v) |k|$. The velocity dependence of $c(v)$ is shown to drastically depend on the dissipation mechanism: we find $c(v)=c(v=0)-2 v$ for the case when the dynamics is governed by microscopic jumps of single molecules at the tip (Blake mechanism), and $c(v)\simeq c(v=0)-4 v$ when viscous hydrodynamic losses inside the moving liquid wedge dominate (de Gennes mechanism). We thus suggest that the debated dominant dissipation mechanism can be experimentally determined using relaxation measurements similar to the Ondarcuhu-Veyssie experiment [T. Ondarcuhu and M. Veyssie, Nature {\bf 352}, 418 (1991)].Comment: REVTEX 8 pages, 9 PS figure

Width distribution of contact lines on a disordered substrate

We have studied the roughness of a contact line of a liquid meniscus on a disordered substrate by measuring its width distribution. The comparison between the measured width distribution and the width distribution calculated in previous works, extended here to the case of open boundary conditions, confirms that the Joanny-de Gennes model is not sufficient to describe the dynamics of contact lines at the depinning threshold. This conclusion is in agreement with recent measurements which determine the roughness exponent by extrapolation to large system sizes.Comment: 4 pages, 3 figure

Roughening Transition in a Moving Contact Line

The dynamics of the deformations of a moving contact line on a disordered substrate is formulated, taking into account both local and hydrodynamic dissipation mechanisms. It is shown that both the coating transition in contact lines receding at relatively high velocities, and the pinning transition for slowly moving contact lines, can be understood in a unified framework as roughening transitions in the contact line. We propose a phase diagram for the system in which the phase boundaries corresponding to the coating transition and the pinning transition meet at a junction point, and suggest that for sufficiently strong disorder a receding contact line will leave a Landau--Levich film immediately after depinning. This effect may be relevant to a recent experimental observation in a liquid Helium contact line on a Cesium substrate [C. Guthmann, R. Gombrowicz, V. Repain, and E. Rolley, Phys. Rev. Lett. {\bf 80}, 2865 (1998)].Comment: 16 pages, 6 encapsulated figure

Exact Renormalization Group Equations. An Introductory Review

We critically review the use of the exact renormalization group equations (ERGE) in the framework of the scalar theory. We lay emphasis on the existence of different versions of the ERGE and on an approximation method to solve it: the derivative expansion. The leading order of this expansion appears as an excellent textbook example to underline the nonperturbative features of the Wilson renormalization group theory. We limit ourselves to the consideration of the scalar field (this is why it is an introductory review) but the reader will find (at the end of the review) a set of references to existing studies on more complex systems.Comment: Final version to appear in Phys. Rep.; Many references added, section 4.2 added, minor corrections. 65 pages, 6 fig

Quasi-long-range order in the random anisotropy Heisenberg model: functional renormalization group in 4-\epsilon dimensions

The large distance behaviors of the random field and random anisotropy O(N) models are studied with the functional renormalization group in 4-\epsilon dimensions. The random anisotropy Heisenberg (N=3) model is found to have a phase with the infinite correlation radius at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law < m(x) m(y) >\sim |x-y|^{-0.62\epsilon}. The magnetic susceptibility diverges at low fields as \chi \sim H^{-1+0.15\epsilon}. In the random field O(N) model the correlation radius is found to be finite at the arbitrarily weak disorder for any N>3. The random field case is studied with a new simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization group equations.Comment: 12 pages, RevTeX; a minor change in the list of reference