Liquids with internal surfaces at and out of equilibrium: the homogeneity index☆
Introduction
Morphology of surfaces given by the local curvatures is an important factor in the phase, transition [1]. The curvatures of interfaces between the domains, which are formed during phase transitions, appear to be the driving force for the growth of the domains especially in the late stage. Moreover, the coarsening process (the growth of the domains) is associated with the topological transformation of the interface, quantitatively described by the Euler characteristic and/or genus. The genus, g, of a closed surface is equal to the number of holes in it. The Euler characteristic is defined as χ=2(1−g). It is equal to 2 for a sphere (since g=0), 0 for a torus (g=1), and −2 for two tori joined by a handle (passage) (g=2). The Euler characteristic for a system of disjoint surfaces is equal to the sum of the Euler characteristics of individual surfaces. If we join two surfaces by a passage, the Euler characteristic of a system will change by −2, which is easy to see. Let us take two spheres for which we have χ=4. If we join them by a passage we will get a single closed surface with χ=2. Therefore, a passage changed the χ of the system by −2. If a droplet appears in a system, the Euler characteristic changes by +2. Therefore, we have two typical topological elements: a droplet and a passage. Finally, it is possible to define the Euler characteristic for the flat interface in a periodic box. Such a surface is topologically equivalent to a tori and, therefore, its Euler characteristic is zero. There is the Gauss–Bonnet theorem, which relates the Euler characteristic to the Gaussian curvature (K=1/R1R2, where Ri are principal radii of curvature) and the surface area, S,where ∫dS denotes the integral over the surface, and γ is twice the inverse of the volume of a (d−1)-dimensional sphere of the unit radius (γ=1/2π for d=3).
In this article we will study the process of phase separation in the AB binary mixture of two homopolymers. The interface between the A-rich domains and the B-rich domains has a large and negative Euler characteristic at the beginning of the separation process, as the interface is highly interconnected and the initial structure is bicontinuous. At the end of the process the Euler characteristic is +2 or 0, since either we have at the end a droplet of the minority phase in the sea of majority phase or a flat interface between two phases. Moreover, the system should exhibit scaling phenomena at late times [2], i.e. a morphological pattern of the domains at earlier times looks statistically similar to a pattern at later times apart from the global change of scale implied by the growth of L(t) – the domain size. Quantitatively it means, for example, that the correlation function of the order parameter (density, concentration, magnetization etc.)whereis the characteristic length scale in the system, scales algebraically with time t with the exponent n different for different universality classes [2]. Assuming the scaling hypothesis we can also derive all the scaling laws for different morphological measures such as: the Euler characteristic, χ(t), surface area, S(t), per unit volume, i.e.
The first law follows from the congruency of the domains [3]. We can namely treat the domains as spheres of diameter L(t) touching each other. The total number of domains, ntot, is then ntot=[L0/L(t)]3, where L0 denotes linear size of the system. Since the total area of the interface is proportional to the product ntotL(t)2, the scaling relation Eq. (4) is obtained. The scaling law Eq. (5) results from the Gauss–Bonnet theorem. Sinceand S(t)∼L(t)−1 we find the scaling Eq. (5). Finally, we shall define the homogeneity index for the three-dimensional system d=3:where V is the total volume in the system. Note here that the quantity given by Eq. (7) is related to the existing measures [4], [5], [6], [7], [8] used to characterize the features of phase interfaces. In particular, HI is proportional to the parameter introduced by Anderson et al. [4] to describe the topology of various types of triply periodic minimal surfaces.
The morphology of the phase interface in various systems has recently been the subject of many experimental as well as theoretical works [9], [10], [11], [12], [13], [14], [15], [16]. Before we study the complex surfaces, which are formed and coarsen during phase separation, let us discuss some equilibrium surfaces formed at equilibrium in the mixtures of water, oil and surfactants. The topological methods have been used for the classification of the extremely complex structures corresponding to the local minima of the Landau-Ginzburg Hamiltonian [17], [18], [19]. Such structures consist of the channels filled either with water or with oil and having the surface between them. The structures are translationally ordered. They are periodic in all three spatial directions. Some surfaces (P, G, D, and I-WP) are shown in Fig. 1. Such structures have HI between 0.21 (for very complex structures) and 0.31 (for a P-surface) [19]. The surfaces are homogeneous, i.e. the local mean curvature is zero at every point of the surface. One can imagine bicontinuous structures, which are not homogeneous. For example, we can use body center cubic (bcc), face center cubic (fcc) and simple cubic (sc) lattices on which we can place the spheres, such that they fill the available space. If two spheres touch each other, then we make a small channel between them. The structure obtained in this way has a large and negative Euler characteristic but is not homogeneous, because it has regions with large and positive Gaussian curvature at every point (we have large spheres) with regions of high and negative Gaussian curvature (channels) (see also the Gauss-Bonet theorem). Let us compute the Euler characteristic for these structures. For the sc structure the spheres are located in the vertices of the cube. The radius of each sphere is r=a/2, where a is the size of the unit cell. There is one sphere per unit cell and three channels per cell. It gives S=πa2/2 per cell, V=a3 per cell and χ=2×number of spheres−2×number of channels=−4 per cell. Therefore, HIsc=4/π3≈0.13. For the bcc lattice we have 2 spheres per unit cell of radii and we have 8 channels per unit cell. Therefore, χ=−12 per cell and we find HIbcc=0.46. Finally, for the fcc lattice we have 4 spheres per unit cell of radii and 24 channels per unit cell. Thus, we find χfcc=−40 and HI=2.58. As can be seen from this short presentation, HI can assume very different values.
However, we will show in this article that HI is a very sensitive measure of scaling in the systems undergoing phase transitions. From , , it follows that HI approaches a constant if scaling is obeyed.
The article is organized as follows: in the next section we will discuss the equations for dynamics of the AB binary homopolymer mixture and the evolution of HI as a function of time after the quench below the consolute temperature. In Section 3, we will present the results for the order–disorder phase transitions in a three-dimensional system of the scalar non-conserved order parameter using the Landau-Ginzburg free energy and fully dissipative dynamics and compare HI obtained in this case to the previous case. The concluding remarks are contained in the summary.
Section snippets
Conserved order parameter
We consider a homopolymer blend (AB) consisting of the molecules of type A and B. The average volume fraction of the A component in the system is φ0. The state of the system is described by the local volume fraction of the component A, φ(r,t), at all points, r, of the system and at time t. The phenomenological mesoscopic dynamic equation that relates a temporal change of φ(r,t) to a local current of A component, J, is governed by the following Cahn–Hilliard–Cook (CHC) equation:
Non-conserved order parameter
The second example of a system exhibiting the phase ordering kinetics [2], [20], [26] analyzed in this articlearticle is a ferromagnet quenched from a temperature above its critical temperature Tc to a temperature below Tc. After lowering the temperature, such a system is brought into thermodynamically unstable, two-phase region. The two phases are characterized by positive and negative magnetization, respectively. The system starts to evolve towards one of the two equilibrium states. Since
Conclusions
In this article, we have investigated the homogeneity index for two systems undergoing phase separation below the critical point: spinodal decomposition in a binary homopolymer mixture and the order-disorder phase transitions in a three-dimensional system of the scalar non-conserved scalar order parameter after the quench below the critical temperature. We have found, that the phase interface formed at the early stages of the spinodal decomposition in homopolymer mixture can be approximated as
Acknowledgements
This work was supported by the KBN grant 5P03B09421 and 5P03B01121. P.G. acknowledges a stipendship from the Polish Foundation for Sciences.
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This work is dedicated to Prof. M. Holovko on the occasion of his 60th anniversary.