E. Bruck, As in figures 11(a) and (c) but showing the zigzag curves consisting only of the global minima of the free energy density, for hexagonal patches in domains with {L}_{{\rm{h}}}=4{L}_{{\rm{c}}}^{{\rm{h}}} to 16{L}_{{\rm{c}}}^{{\rm{h}}} as given in the legend. In section 3.3 we consider states on a square domain with periodic boundary conditions. Hexagonal patches in the nonconserved Swift–Hohenberg equation grow in the same manner [64]. For n > 1 the stationary periodic states consist of a larger number of single phase regions with a correspondingly larger number of interfaces. Note that the coefficient of | {\rm{\nabla }}\phi {| }^{2} is now negative and so favors gradients in the order parameter, while the system is regularized by the strictly positive higher order term that limits the steepness of the variations in . This option allows users to search by Publication, Volume and Page. Shortly after turning, the heterogeneous state becomes the global free energy minimum. the growth or decay rate of a harmonic perturbation as function of its wave number q = q. The second model, the PFC model, reveals additional complexity. M. Ozbolt, and We use cookies on this site to enhance your user experience. In this paper we have revisited the Maxwell construction that predicts the location of a first order phase transition in thermodynamic systems at finite temperature. The Deutsche Physikalische Gesellschaft (DPG) with a tradition extending back to 1845 is the largest physical society in the world with more than 61,000 members. In particular, we examine in detail how this construction emerges as the system size becomes larger and larger, thereby gaining additional insight into this construction. Phase diagram of the PFC model in the plane spanned by the mean concentration 0 and the undercooling r on (a) one-dimensional (1D) and (b) two-dimensional (2D) domains. The thick solid lines denote the states of minimum free energy at each value of 0. More modern classifications define first order transitions as transitions that involve latent heat or as transitions where an order parameter changes discontinuously [1, 2]. T. Gottschall, The conserved Swift–Hohenberg equation with cubic nonlinearity, also known as the PFC model [19, 37], provides the simplest phenomenological microscopic mean-field continuum description of the dynamics of the transition between a liquid state and a crystalline state. However, in 2D new perturbation modes are available and therefore their linear stability properties differ from those in 1D. Figure 9(a) shows that the localized states corresponding to global free energy minima are clustered within a band of width Δμ ≈ 0.031 08 within the snaking region, until superseded by the periodic state at large 0. In section 3.1 it is mentioned that a homogeneous state  = h loses stability when ∂ f = 0. These states exhibit the phenomenon of slanted snaking [13, 61, 62] but take the form of standard homoclinic snaking when the order parameter 0 is plotted as a function of the chemical potential, a property that is expected to carry over to other models with a conserved order parameter. However, the case of fully 2D hexagonal structures is different (figure 13), since there the zigzag segments have different lengths, although all appear once again to have the same slope. the global free energy minimum. As a result, they have identical bifurcation curves and identical behavior as L\to \infty as the 1D states. that the conditions for thermodynamic equilibrium hold. E. Bruck, “, Direct measurement of the magnetocaloric effect in MnFe(P,X)(X = As, Ge, Si) materials, M. Katter, Here we discuss the behavior of the Cahn–Hilliard equation for a finite domain size combining linear stability and bifurcation analyses, and focus on how the Maxwell construction emerges as the domain size is increased. L. F. Cohen, “, Identifying the critical point of the weakly first-order itinerant magnet DyCo, V. Basso, This fraction is determined by the symmetry of the state considered. The temperature dependence of the specific heat of SbSI at zero E field and atmospheric pressure has been measured accurately from 83 to 330 K using an ac calorimetric method. the limit of taking the system volume V = Ld to infinity, where d is the dimension of the system, whilst proportionally increasing the particle number, so that 0 remains constant as the limit is taken. A branch of nearly rotationally invariant localized target-like solutions (dark green dashed curve) emerges from this branch at very small amplitude, even though the domain does not have this symmetry (see figure 12(a)). You will only need to do this once. A. Waske, We present results for the specific heat of a Nd1/2Sr1/2MnO3 crystal. Figure 1. The resulting representation of the phase transition in the phase plane spanned by 0 and T and the respective continuous and discontinuous dependence of δ on T are shown in figure 3. Hysteresis is observed on warming and cooling, which increases in a magnetic field. localized regions of hexagons with only one extended direction, exhibit the same Δμ scaling as the strictly 1D problem. At the transition point, the curves of both phases intersect - on crossing the transition point, the other phase becomes the thermo­dynamically stable one. S. Russek, the case L = 16Lc in figures 6(c) and (d)). On further increasing L, the cluster state branch ceases to be monotonic between the two saddle-node bifurcations at \pm {\phi }_{\mathrm{sn}1}^{\mathrm{cluster}} and undergoes a hysteresis bifurcation, at 0 = 0, generating two further saddle-node bifurcations (at \pm {\phi }_{\mathrm{sn}2}^{\mathrm{cluster}}, see e.g. The horizontal red line indicates the Maxwell construction as obtained in section 4.1. In the context of buckling the corresponding concept is the Maxwell load [14]. G.-J. for the solid–liquid phase transition, the derivative with respect to pressure becomes discontinuous, implying a jump in the density. Images of such states can be found, e.g.

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