Droplet shapes on structured substrates and conformal invariance

We consider the finite-size scaling of equilibrium droplet shapes for fluid adsorption (at bulk two-phase co-existence) on heterogeneous substrates and also in wedge geometries in which only a finite domain $\Lambda_{A}$ of the substrate is completely wet. For three-dimensional systems with short-ranged forces we use renormalization group ideas to establish that both the shape of the droplet height and the height-height correlations can be understood from the conformal invariance of an appropriate operator. This allows us to predict the explicit scaling form of the droplet height for a number of different domain shapes. For systems with long-ranged forces, conformal invariance is not obeyed but the droplet shape is still shown to exhibit strong scaling behaviour. We argue that droplet formation in heterogeneous wedge geometries also shows a number of different scaling regimes depending on the range of the forces. The conformal invariance of the wedge droplet shape for short-ranged forces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.

Correlation function algebra for inhomogeneous fluids

We consider variational (density functional) models of fluids confined in parallel-plate geometries (with walls situated in the planes z=0 and z=L respectively) and focus on the structure of the pair correlation function G(r_1,r_2). We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform G(z_\mu, z_\nu;q) and the zeroth moment G_0(z_\mu,z_\nu) at different positions z_1, z_2 and z_3. These relations form an algebra which severely restricts the possible form of the function G_0(z_\mu,z_\nu). For the common situations in which the equilibrium one-body (magnetization/number density) profile m_0(z) exhibits an odd or even reflection symmetry in the z=L/2 plane the algebra simplifies considerably and is used to relate the correlation function to the finite-size excess free-energy \gamma(L). We rederive non-trivial scaling expressions for the finite-size contribution to the free-energy at bulk criticality and for systems where large scale interfacial fluctuations are present. Extensions to non-planar geometries are also considered.Comment: 15 pages, RevTex, 4 eps figures. To appear in J.Phys.Condens.Matte

Physical education as Olympic education

Introduction In a recent paper (Parry, 1998, p. 64), I argued that the justification of PE activities lies in their capacity to facilitate the development of certain human excellences of a valued kind. Of course, the problem now lies in specifying those ‘human excellences of a valued kind’, and (for anyone) this task leads us into the area of philosophical anthropology. I suggested that the way forward for Physical Education lies in the philosophical anthropology (and the ethical ideals) of Olympism, which provide a specification of a variety of human values and excellences which: •have been attractive to human groups over an impressive span of time and space •have contributed massively to our historically developed conceptions of ourselves •have helped to develop a range of artistic and cultural conceptions that have defined Western culture. •have produced a range of physical activities that have been found universally satisfying and challenging. Although physical activities are widely considered to be pleasurable, their likelihood of gaining wide acceptance lies rather in their intrinsic value, which transcends the simply hedonic or relative good. Their ability to furnish us with pleasurable experiences depends upon our prior recognition in them of opportunities for the development and expression of valued human excellences. They are widely considered to be such opportunities for the expression of valued human excellences because, even when as local instantiations, their object is to challenge our common human propensities and abilities. I claimed that Olympic ideals may be seen not merely as inert ‘ideals’, but living ideas which have the power to remake our notions of sport in education, seeing sport not as mere physical activity but as the cultural and developmental activity of an aspiring, achieving, well-balanced, educated and ethical individual. This paper seeks to make good that claim by trying to develop a case for Physical Education as Olympic Education. I begin by setting out various accounts and conceptions of the Olympic Idea; then I suggest a unifying and organising account of the philosophical anthropology of Olympism; and this is followed by the practical application of that account in two examples of current ethical issues. Finally, I seek to present an account of Physical Education as Olympic Education

Local functional models of critical correlations in thin-films

Recent work on local functional theories of critical inhomogeneous fluids and Ising-like magnets has shown them to be a potentially exact, or near exact, description of universal finite-size effects associated with the excess free-energy and scaling of one-point functions in critical thin films. This approach is extended to predict the two-point correlation function G in critical thin-films with symmetric surface fields in arbitrary dimension d. In d=2 we show there is exact agreement with the predictions of conformal invariance for the complete spectrum of correlation lengths as well as the detailed position dependence of the asymptotic decay of G. In d=3 and d>=4 we present new numerical predictions for the universal finite-size correlation length and scaling functions determining the structure of G across the thin-film. Highly accurate analytical closed form expressions for these universal properties are derived in arbitrary dimension.Comment: 4 pages, 1 postscript figure. Submitted to Phys Rev Let

Effect of Gravity and Confinement on Phase Equilibria: A Density Matrix Renormalization Approach

The phase diagram of the 2D Ising model confined between two infinite walls and subject to opposing surface fields and to a bulk "gravitational" field is calculated by means of density matrix renormalization methods. In absence of gravity two phase coexistence is restricted to temperatures below the wetting temperature. We find that gravity restores the two phase coexistence up to the bulk critical temperature, in agreement with previous mean-field predictions. We calculate the exponents governing the finite size scaling in the temperature and in the gravitational field directions. The former is the exponent which describes the shift of the critical temperature in capillary condensation. The latter agrees, for large surface fields, with a scaling assumption of Van Leeuwen and Sengers. Magnetization profiles in the two phase and in the single phase region are calculated. The profiles in the single phase region, where an interface is present, agree well with magnetization profiles calculated from a simple solid-on-solid interface hamiltonian.Comment: 4 pages, RevTeX and 4 PostScript figures included. Final version as published. To appear in Phys. Rev. Let

A symmetric polymer blend confined into a film with antisymmetric surfaces: interplay between wetting behavior and phase diagram

We study the phase behavior of a symmetric binary polymer blend which is confined into a thin film. The film surfaces interact with the monomers via short range potentials. We calculate the phase behavior within the self-consistent field theory of Gaussian chains. Over a wide range of parameters we find strong first order wetting transitions for the semi-infinite system, and the interplay between the wetting/prewetting behavior and the phase diagram in confined geometry is investigated. Antisymmetric boundaries, where one surface attracts the A component with the same strength than the opposite surface attracts the B component, are applied. The phase transition does not occur close to the bulk critical temperature but in the vicinity of the wetting transition. For very thin films or weak surface fields one finds a single critical point at $\phi_c=1/2$. For thicker films or stronger surface fields the phase diagram exhibits two critical points and two concomitant coexistence regions. Only below a triple point there is a single two phase coexistence region. When we increase the film thickness the two coexistence regions become the prewetting lines of the semi-infinite system, while the triple temperature converges towards the wetting transition temperature from above. The behavior close to the tricritical point, which separates phase diagrams with one and two critical points, is studied in the framework of a Ginzburg-Landau ansatz. Two-dimensional profiles of the interface between the laterally coexisting phases are calculated, and the interfacial and line tensions analyzed. The effect of fluctuations and corrections to the self-consistent field theory are discussed.Comment: Phys.Rev.E in prin

Interfacial fluctuations near the critical filling transition

We propose a method to describe the short-distance behavior of an interface fluctuating in the presence of the wedge-shaped substrate near the critical filling transition. Two different length scales determined by the average height of the interface at the wedge center can be identified. On one length scale the one-dimensional approximation of Parry et al. \cite{Parry} which allows to find the interfacial critical exponents is extracted from the full description. On the other scale the short-distance fluctuations are analyzed by the mean-field theory.Comment: 13 pages, 3 figure

Damage Spreading at the Corner Filling Transition in the two-dimensional Ising Model

The propagation of damage on the square Ising lattice with a corner geometry is studied by means of Monte Carlo simulations. It is found that, just at $T=T_f (h)$ (critical temperature of the filling transition) the damage initially propagates along the interface of the competing domains, according to a power law given by $D(t) \propto t^{\eta}$. The value obtained for the dynamic exponent ($\eta^{*} = 0.89(1)$) is in agreement with that corresponding to the wetting transition in the slit geometry (Abraham Model) given by $\eta^{WT} = 0.91(1)$. However, for later times the propagation crosses to a new regime such as $\eta^{**} = 0.40 \pm 0.02$, which is due to the propagation of the damage into the bulk of the magnetic domains. This result can be understood due to the constraints imposed to the propagation of damage by the corner geometry of the system that cause healing at the corners where the interface is attached.Comment: 22 pages, including figures Submited to J. Phys.: Condens. Matte

Semiclassical form factor for chaotic systems with spin 1/2

We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory.Comment: 20 pages, no figure

Casimir forces in binary liquid mixtures

If two ore more bodies are immersed in a critical fluid critical fluctuations of the order parameter generate long ranged forces between these bodies. Due to the underlying mechanism these forces are close analogues of the well known Casimir forces in electromagnetism. For the special case of a binary liquid mixture near its critical demixing transition confined to a simple parallel plate geometry it is shown that the corresponding critical Casimir forces can be of the same order of magnitude as the dispersion (van der Waals) forces between the plates. In wetting experiments or by direct measurements with an atomic force microscope the resulting modification of the usual dispersion forces in the critical regime should therefore be easily detectable. Analytical estimates for the Casimir amplitudes Delta in d=4-epsilon are compared with corresponding Monte-Carlo results in d=3 and their quantitative effect on the thickness of critical wetting layers and on force measurements is discussed.Comment: 34 pages LaTeX with revtex and epsf style, to appear in Phys. Rev.