Energy storage in steady states under cyclic local energy input.

We study periodic steady states of a lattice system under external cyclic energy supply using simulation. We consider different protocols for cyclic energy supply and examine the energy storage. Under the same energy flux, we found that the stored energy depends on the details of the supply, period, and amplitude of the supply. Further, we introduce an adiabatic wall as an internal constraint into the lattice and examine the stored energy with respect to different positions of the internal constrain. We found that the stored energy for constrained systems is larger than its unconstrained counterpart. We also observe that the system stores more energy through large and rare energy delivery, comparing to small and frequent delivery.

Evaporation of freely suspended single droplets: experimental, theoretical and computational simulations.

Evaporation is ubiquitous in nature. This process influences the climate, the formation of clouds, transpiration in plants, the survival of arctic organisms, the efficiency of car engines, the structure of dried materials and many other phenomena. Recent experiments discovered two novel mechanisms accompanying evaporation: temperature discontinuity at the liquid-vapour interface during evaporation and equilibration of pressures in the whole system during evaporation. None of these effects has been predicted previously by existing theories despite the fact that after 130 years of investigation the theory of evaporation was believed to be mature. These two effects call for reanalysis of existing experimental data and such is the goal of this review. In this article we analyse the experimental and the computational simulation data on the droplet evaporation of several different systems: water into its own vapour, water into the air, diethylene glycol into nitrogen and argon into its own vapour. We show that the temperature discontinuity at the liquid-vapour interface discovered by Fang and Ward (1999 Phys. Rev. E 59 417-28) is a rule rather than an exception. We show in computer simulations for a single-component system (argon) that this discontinuity is due to the constraint of momentum/pressure equilibrium during evaporation. For high vapour pressure the temperature is continuous across the liquid-vapour interface, while for small vapour pressures the temperature is discontinuous. The temperature jump at the interface is inversely proportional to the vapour density close to the interface. We have also found that all analysed data are described by the following equation: da/dt = P(1)/(a + P(2)), where a is the radius of the evaporating droplet, t is time and P(1) and P(2) are two parameters. P(1) = -λΔT/(q(eff)ρ(L)), where λ is the thermal conductivity coefficient in the vapour at the interface, ΔT is the temperature difference between the liquid droplet and the vapour far from the interface, q(eff) is the enthalpy of evaporation per unit mass and ρ(L) is the liquid density. The P(2) parameter is the kinetic correction proportional to the evaporation coefficient. P(2) = 0 only in the absence of temperature discontinuity at the interface. We discuss various models and problems in the determination of the evaporation coefficient and discuss evaporation scenarios in the case of single- and multi-component systems.

Three-dimensional space partition based on the first Laplacian eigenvalues in cells.

We determine a partition of three-dimensional space into cells by minimization of the sum of the first Laplacian eigenvalues over the cells. This partitioning scheme emerges as a stationary state of a reaction-diffusion process taking place in a system of n different species which mutually annihilate, and simultaneously are duplicated in an autocatalytic reaction, so that the number of particles is kept constant and equal for each species. The system is considered in the limit of strong reactivity, so that the species separate each other into cells with well-defined, sharp boundaries. For a given n and fixed sizes of a periodic simulation box, this partition minimizes the aforementioned sum of eigenvalues. Further minimization is done by changing n and the side ratio of the periodic box. The global minimum is obtained for the structure with A15 symmetry, similar to the Weaire-Phelan foam. Depending on n and the side ratio, there are also many local minima, in particular: hcp (hexagonal close packed), fcc (face centered cubic), the Kelvin structure, and Frank-Kasper sigma phase.

Influence of poly(ethylene glycol) molecular mass on separation and ordering in solutions of CiEj nonionic surfactants: depletion interactions and steric effects.

We study ternary mixtures of nonionic surfactants C(i)E(j) (i = 12; j = 5, 6, 8) and poly(ethylene glycol) (PEG) in water. For sufficiently large molecular mass of PEG (M >M(sep) approximately 600), we observe a lowering of phase separation temperature with an increase in polymer concentration. The value of M(sep) is consistent with the analysis based on depletion interactions between micelles induced by polymer chains. We also demonstrate that there is another critical molecular mass of PEG (M = M* approximately 2000) necessary to induce ordering in the surfactant-rich phase. This critical molecular mass follows from two requirements: (a) PEG has to reduce the separation temperature below a temperature of hexagonal-isotropic phase transition in a binary surfactant-water mixture and (b) the PEG radius of gyration has to be larger than the size of the water channels in the hexagonal phase.

Tiling a plane in a dynamical process and its applications to arrays of quantum dots, drums, and heat transfer.

We present a reaction-diffusion system consisting of N components. The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large N, the stationary state becomes a periodic array of hexagonal cells. We present a functional of the densities of the components, which decreases monotonically during the evolution and attains its minimal value in the stationary state. This value is equal to the sum of the first Laplacian eigenvalues for all cells. Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem.

Condensation of a vapor bubble in submicrometer container.

Condensation of a spherically symmetric submicrometer size vapor bubble is studied using diffuse interface hydrodynamic model supplemented by the van der Waals equation of state with parameters characteristic for argon. The bubble, surrounded by liquid, is held in a container of constant volume with temperature of the wall kept fixed. The condensation is triggered by a sudden rise of the wall temperature. We find that in the same container and subjected to a similar increase of the wall temperature the condensation process is totally different from the opposite process of droplet evaporation. In particular, the rapid change of the wall temperature excites the wave, which hits the interface and compresses the bubble, leading to a considerable increase of the temperature inside. The condensation of the submicrometer size bubble takes tens of nanoseconds, whereas evaporation of the same size droplet lasts roughly 50 times longer. In contrast to evaporation the condensation process is hardly quasistationary.

Minimization of the Renyi entropy production in the space-partitioning process.

The spontaneous division of space in Fleming-Viot processes is studied in terms of non-extensive thermodynamics. We analyze a system of n different types of Brownian particles confined in a box. Particles of different types annihilate each other when they come into close contact. Each process of annihilation is accompanied by a simultaneous nucleation of a particle of the same type, so that the number of particles of each component remains constant. The system eventually reaches a stationary state, in which the available space is divided into n separate subregions, each occupied by particles of one type. Within each subregion, the particle density distribution minimizes the Renyi entropy production. We show that the sum of these entropy productions in the stationary state is also minimized, i.e., the resulting boundaries between different components adopt a configuration which minimizes the total entropy production. The evolution of the system leads to decreasing of the total entropy production monotonically in time, irrespective of the initial conditions. In some circumstances, the stationary state is not unique-the entropy production may have several local minima for different configurations. In the case of a rectangular box, the existence and stability of different stationary states are studied as a function of the aspect ratio of the rectangle.

Global symmetry breaking in the nonconserved order parameter system during phase ordering.

We study global symmetry breaking in the 2D system of scalar nonconserved order parameter following a quench to zero temperature. We show that the instant of time when the symmetry is broken and the final morphology is chosen corresponds to the saturation of the order parameter inside the domains. There are three possible final morphologies: the positive and negative order parameter final morphology, and the state of the coexisting positive and negative order parameter subsystems with a flat interface between them. We find also that each type of the final morphology constitutes about 1/3 of all cases, what agrees with the results obtained recently by Spirin et al. [Phys. Rev. E 65, 016119 (2001)]. Our results are pertinent for the two dimensional systems, but we suspect that there is also a way to apply similar arguments for the three dimensional ones.

Evaporation of a thin liquid film.

Evaporation of a thin (submicrometer size) liquid film confined between two solid substrates is studied using diffuse interface hydrodynamic model supplemented by the van der Waals equation of state. The time and space evolution of the basic thermodynamic quantities such as temperature, density, entropy, chemical potential, and entropy production is presented. The values of numerical parameters chosen correspond to those of argon. The time and space scales studied range from picoseconds to microseconds and from nanometers to micrometers correspondingly.

Hidden minima of the gibbs free energy revealed in a phase separation in polymer/surfactant/water mixture.

We observed a very unusual kinetic pathway in a separating C(12)E(6)/PEG/H(2)O ternary mixture. We let the mixture separate above the spinodal temperature (cloud point temperature) for some time and next cool it into a metastable region of a phase diagram, characterized by two minima of the Gibbs potential, one corresponding to the homogeneous mixture and one to the fully separated PEG-rich and C(12)E(6)-rich phases. Despite the fact that in the metastable region the thermodynamic equilibrium corresponds to the separated phases (global minimum of the Gibbs free energy), we observe perfect mixing of the initially separated phase. The homogeneous state, obtained in this way, does not separate, if left undisturbed. However, many cooling-heating cycles or full separation with visible meniscus above the cloud point temperature induce the phase separation in the metastable region. The metastable region can exist tens of degrees below the cloud point temperature. This effect is not observed in the binary mixture of C(12)E(6)/H(2)O.

Evaporation of a sub-micrometer droplet.

Evaporation of a spherically symmetric sub-micrometer size liquid droplet is studied using a diffuse interface hydrodynamic model supplemented by the van der Waals equation of state with parameters characteristic for argon. The droplet, surrounded by saturated vapor, is held in a container with the temperature of the walls kept fixed. The evaporation is triggered by a sudden rise of the temperature of the walls. Time and space evolution of the basic thermodynamic quantities is presented. The time and space scales studied range from picoseconds to microseconds and from nanometers to micrometers, respectively. We find that the temperature and chemical potential are both continuous at the interface on the scale larger than the interfacial width. We find that at long times the radius R of the droplet changes with time t as R(2)(t) = R(2)(0) - 2tkappa(v)(T(w) - T(l))/ln(l), where kappa(v) is the heat conductivity of the vapor, n(l) and T(l) are the density and the temperature of liquid inside the droplet, respectively, l is the latent heat of transition per molecule, and T(w) is the temperature of the ambient vapor.

Ordering in surfactant mixtures induced by polymers.

We studied ternary mixtures of nonionic surfactant (C12E6, n-dodecyl hexaoxyethylene glycol monoether), polymer (PEG, polyethylene glycol), and water. A small amount of PEG induces demixing into the polymer-rich and surfactant-rich phases in the ternary PEG/C12E6/water mixture. Above a certain concentration and/or molecular weight of PEG, the surfactant-rich phase orders, even in a solution consisting of a few percent of surfactant. The phase boundary acts as a semipermeable membrane, and the equilibrium is determined by the chemical potential of water in two phases. The explicit expression for the amount of PEG needed to order C12E6 water solution is given and verified experimentally. The analysis of the coexistence conditions leads to the conjecture that only two oxygen atoms in the outward part of the hydrophilic surfactant head strongly affect the chemical potential of water. Our methodology is generic, i.e., on the same basis one can design a similar experiment for any surfactant/polymer/water system and find the right proportions of polymer that induce order in a surfactant-rich phase.

Relaxation processes in mixtures of liquid crystals and polymers near phase boundaries and during phase separation.

We present experimental studies of the relaxation of concentration fluctuations in a semidilute solution of polystyrene (PS) (30% by weight) in 4-cyano-4'-n-octyl-biphenyl (8CB) (70% by weight) using the photon correlation spectroscopy (PCS). In the homogeneous phase there are two modes of relaxation. The slow one (typical time scale is taus = 0.001 s) is due to the diffusion of polymer chains (of molecular mass 65,000) in the LC matrix (of molecular mass 290), while the fast one has the time scale of the order of tauf approximately 0.00001 s. The amplitude of the fast mode is much weaker than the one for the slow mode. Moreover it does not depend on the scattering wave vector, q. The value of the diffusion coefficient, Dc = 1/(tausq2) for the slow mode decreases with temperature according to the Arhenius law until we reach the coexistence curve. Its value close to the coexistence is Dc = 4 x 10(5) nm2/s and the activation energy in the homogeneous mixture is Ec=127 kJ/mol. If we gradually undercool the mixture below the coexistence into the metastable two-phase region without inducing the phase separation we find unexpectedly that Dc does not change with temperature even 4 degrees below the coexistence curve. The characteristic time of the fast mode does not depend on the scattering wave vector indicating that it is related to the transient gel structure. We have shown that it is possible to measure the short time relaxation of concentration fluctuations during the phase separation in the mixture. At low temperature close to the isotropic-nematic phase transition we have observed that the relaxation is well separated in time from the typical time of the domain growth. This relaxation mode is characterized by the large diffusion coefficient D = 2 x 10(8) nm2/s. The mode probably comes from the coupling between the orientational dynamics of liquid crystals and the transient gel structure of polymers.

Percolation-to-droplets transition during spinodal decomposition in polymer blends, morphology analysis.

Phase separation kinetics of the off-critical mixture of polystyrene and poly(methylphenylsiloxane) is studied by the time-resolved light scattering and optical microscopy. The results from the light scattering experiments are correlated with the images obtained by the optical microscopic observation in order to find characteristic features of the scattering intensity during the percolation-to-droplets morphology transition. At the beginning of the spinodal decomposition process only a bicontinuous network is present in the system and the light scattering intensity has only one peak. The network coarsens and at the same time small droplets appear in the system resulting in a growth of the scattering intensity at very small wave vectors. When the large network starts to break up into disjoint elongated domains a second peak in the scattering intensity appears. Finally, both peaks merge into a single peak at zero wave vector, indicating a complete transformation of elongated domains into spherical droplets of variable sizes. The comparison of the direct microscopic observations with the light scattering spectra shows that the process of breaking up of the bicontinuous network starts when the growth of the first peak, corresponding to the bicontinuous pattern, becomes very slow (essentially pinned down).

Polymer domain growth in ordered liquid crystalline matrices.

The growth of polymer domains in the isotropic, nematic, and smectic matrices is studied by the light scattering. In the smectic and isotropic matrices the growth is diffusive, and in the nematic matrix it is influenced by the elastic forces. The scaling is obeyed. A crossover to the wetting fast-mode hydrodynamic regime is also observed at long times. In order to perform these measurements we had to eliminate the multiple scattering of light.

Periodic surfaces of simple and complex topology: comparison of scattering patterns.

We compute scattering patterns for six triply periodic minimal surfaces formed in oil/surfactant/water solutions: Three surfaces of a simple topology, Schwarz P (Im3m), Schwarz D-diamond (Pn3m), and Schoen G-gyroid (Ia3d), and three surfaces of a complex topology, SCN1 (Im3m), CD (Pn3m), and GX6 (Ia3d). We show that in the case of the complex structures, scattering intensity is shifted towards the higher hkl peaks. This might cause their misidentification and wrong estimates about the cell size of the structure.

Mechanisms for facilitated target location and the optimal number of molecules in the diffusion search process.

We investigate the number N of molecules needed to perform independent diffusion in order to achieve bonding of a single molecule to a specific site in time t(0). For a certain range of values of t(0), an increase from N to kN molecules (k>1) results in the decrease of search time from t(0) to t(0)/k. In this regime, increasing the number of molecules is an effective way of speeding up the search process. However when N> or =N0 (optimal number of N) the reduction of time from t(0) to t(0)/k can be achieved only by an exponentially large increase in the number of molecules [from N to N exp(ck) for some c>0].

Scaling of the Euler characteristic, surface area, and curvatures in the phase separating or ordering systems.

We present robust scaling laws for the Euler characteristic and curvatures applicable to any symmetric system undergoing phase separating or ordering kinetics. We apply it to the phase ordering in a system of the nonconserved scalar order parameter and find three scaling regimes. The appearance of the preferred nonzero curvature of an interface separating +/- domains marks the crossover to the late stage regime characterized by the Lifshitz-Cahn-Allen scaling.

Influence of the free-energy functional form on simulated morphology of spinodally decomposing blends

The spinodal decomposition of a binary mixture has been studied within several mesoscopic models. It has been found that the form of the equilibrium free energy has a crucial effect on the morphological development in asymmetric blends. We have shown that the principal quantity that determines the topology of the interface (and type of morphology) is the equilibrium minority phase volume fraction, while the transition from bicontinuous to droplet morphology can be treated as a percolation. The concentration dependence of the square gradient coefficient attributed for the Flory-Huggins-de Gennes free energy has no significant influences on the average domain growth, but can be distinguished experimentally from its constant-coefficient alternative by measuring the maximum wave vector of the scattering intensity as a function of the minority phase volume fraction for spinodally decomposing asymmetric blends. The concentration dependence of the Onsager coefficient has the weak, systematic effect of slowing down the morphological development. The local shape of the interface is not affected considerably by the concentration dependence of the square gradient and Onsager coefficient.

Coupling between meniscus and smectic-A films: circular and catenoid profiles, induced stress, and dislocation dynamics

In this paper we discuss the formation and shape of the meniscus between a free-standing film of a smectic-A phase and a wall (in practice the frame that supports the film). The wall may be flat or circular, and the system with or without a reservoir of particles. The formation of the meniscus is always an irreversible thermodynamic process, since it involves the creation of dislocations in the bulk (therefore it involves friction). The four basic shapes of meniscus discussed are the following: exponential, algebraic (x(3/2)), circular, and catenoid. Three principal regions of the whole meniscus must be distinguished: close to the wall with a high density of dislocations, away from the wall with medium density of dislocations, and far from the wall (i.e., close to the film) with a low density of dislocations (vicinal regime). The region with medium density of dislocations is observable using a microscope, and is determined by the competition between surface tension, energy of dislocations, and pressure difference set by the mass of the meniscus or by the reservoir. Its profile is circular as observed in recent experiments [J.-C. Geminard, R. Holyst, and P. Oswald, Phys. Rev. Lett. 78, 1924 (1997)]. By contrast, the vicinal regime with low density of dislocations is never observable with an optical microscope. In the regime with a high density of dislocations, the reasons why the dislocations tend to gather by forming giant dislocations and rows of focal conics are discussed. Finally, we discuss the stability of a smectic film with respect to the formation of a dislocation loop. We show experimentally that the critical radius of the loop is proportional to the curvature radius of the meniscus in its circular part, in agreement with the theory. In addition, we show that the mobility of edge dislocations measured in thick films is in agreement with that found in bulk samples from a creep experiment. This result confirms again our model of the meniscus.