Fluctuations of interface

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

In order to study the fluctuations of interfaces and surfaces, one first needs an expression for the free energy of an interface with a shape that is not necessarily flat. Phenomenologically, one can write that the free energy is the product of the interfacial tension, y, and the area, which in the Monge representation for the surface, z = h Xy y), reads ... 

When a phase separation into a Janus cylinder structure occurs, e.g., where the upper half of the cylinder contains the B-rich phase and the lower half the A-rich phase, we have a planar AB interface (Fig. 32a) and the quantity that we wish to record is the vector normally oriented to this interface for any monomer of the backbone. Studying the orientational correlations of this vector will yield the desired information on possible fluctuations of interface orientation (Fig. 32b). Since the AB interface at nonzero temperature is not a sharp dividing surface, but rather has a finite width, a numerical characterization of the local orientation of this interface normal is difficult. Therefore, an essentially equivalent but numerically unambiguous characterization of this Janus cylinder-type ordering has been... 

Fluctuations of interfaces are directly relevant to a number of interfacial phenomena. One example, ion transfer across a liquid-liquid interface, will be discussed in Section 6.1. Another example is the behavior of monolayers of surfactants on water surfaces. Surface fluctuations are also fundamental to several processes in water-membrane systems, such as unassisted ion transport across lipid bilayers and the hydration forces acting between two membranes. Here, however, the problem is more complicated because not only capillary waves but also bending motions of the whole bilayer have to be taken into account. Furthermore, the concept of the surface tension is less clear in this case. This topic is discussed in Molecular Dynamics Studies of Lipid Bilayers. 

A system of interest may be macroscopically homogeneous or inliomogeneous. The inliomogeneity may arise on account of interfaces between coexisting phases in a system or due to the system s finite size and proximity to its external surface. Near the surfaces and interfaces, the system s translational synnnetry is broken this has important consequences. The spatial structure of an inliomogeneous system is its average equilibrium property and has to be incorporated in the overall theoretical stnicture, in order to study spatio-temporal correlations due to themial fluctuations around an inliomogeneous spatial profile. This is also illustrated in section A3.3.2. 

In this brief review of dynamics in condensed phases, we have considered dense systems in various situations. First, we considered systems in equilibrium and gave an overview of how the space-time correlations, arising from the themial fluctuations of slowly varying physical variables like density, can be computed and experimentally probed. We also considered capillary waves in an inliomogeneous system with a planar interface for two cases an equilibrium system and a NESS system under a small temperature gradient. 

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

The fluctuations of the local interfacial position increase the effective area. This increase in area is associated with an increase of free energy Wwhich is proportional to the interfacial tension y. The free energy of a specific interface configuration u(r,) can be described by the capillary wave Hamiltonian ... 

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... 

In the mean field considerations above, we have assumed a perfectly flat interface such that the first tenn in the Hamiltonian (B3.6.21) is ineffective. In fact, however, fluctuations of the local interface position are important, and its consequences have been studied extensively [, 58]. 

As shown in Fig. 21, in this case, the entire system is composed of an open vessel with a flat bottom, containing a thin layer of liquid. Steady heat conduction from the flat bottom to the upper hquid/air interface is maintained by heating the bottom constantly. Then as the temperature of the heat plate is increased, after the critical temperature is passed, the liquid suddenly starts to move to form steady convection cells. Therefore in this case, the critical temperature is assumed to be a bifurcation point. The important point is the existence of the standard state defined by the nonzero heat flux without any fluctuations. Below the critical temperature, even though some disturbances cause the liquid to fluctuate, the fluctuations receive only small energy from the heat flux, so that they cannot develop, and continuously decay to zero. Above the critical temperature, on the other hand, the energy received by the fluctuations increases steeply, so that they grow with time this is the origin of the convection cell. From this example, it can be said that the pattern formation requires both a certain nonzero flux and complementary fluctuations of physical quantities. 

Then the diffusion equation for the fluctuation of the metal ion concentration is given by Eq. (68), and the mass balance at the film/solution interface is expressed by Eq. (69). These fluctuation equations are also solved with the same boundary condition as shown in Eq. (70). 

Capillary waves occur spontaneously at liquid surfaces or liquid liquid interfaces due to thermal fluctuations of the bulk phases. These waves have been known as surface tension waves, ripples, or ripplons for the last century, and Lamb described their properties in his book Hydrodynamics in 1932 [10]. Before that, William Thomson (Lord Kelvin) mentioned these waves in some of his many writings. 

It follows from the above that the mechanism for electrical potential oscillation across the octanol membrane in the presence of SDS would most likely be as follows dodecyl sulfate ions diffuse into the octanol phase (State I). Ethanol in phase w2 must be available for the transfer energy of DS ions from phase w2 to phase o to decrease and thus, facilitates the transfer of DS ions across this interface. DS ions reach interface o/wl (State II) and are adsorbed on it. When surfactant concentration at the interface reaches a critical value, a surfactant layer is formed at the interface (State III), whereupon, potential at interface o/wl suddenly shifts to more negative values, corresponding to the lower potential of oscillation. With change in interfacial tension of the interface, the transfer and adsorption of surfactant ions is facilitated, with consequent fluctuation in interface o/ wl and convection of phases o and wl (State IV). Surfactant concentration at this interface consequently decreased. Potential at interface o/wl thus takes on more positive values, corresponding to the upper potential of oscillation. Potential oscillation is induced by the repetitive formation and destruction of the DS ion layer adsorbed on interface o/wl (States III and IV). This mechanism should also be applicable to oscillation with CTAB. Potential oscillation across the octanol membrane with CTAB is induced by the repetitive formation and destruction of the cetyltrimethylammonium ion layer adsorbed on interface o/wl. Potential oscillation is induced at interface o/wl and thus drugs were previously added to phase wl so as to cause changes in oscillation mode in the present study. 

Fig. 3a indicates that the bubble-rise velocity measured based on the displacement of the top surface of the bubble ( C/bt) quickly increases and approaches the terminal bubble rise velocity in 0.02 s. The small fluctuation of Ubt is caused by numerical instability. The bubble-rise velocity measured based on the displacement of the bottom surface of the bubble (Ubb) fluctuates significantly with time initially and converges to Ubt after 0.25 s. The overshooting of Ubb can reach 45-50 cm/s in Fig. 3a. The fluctuation of Ubb reflects the unsteady oscillation of the bubble due to the wake flow and shedding at the base of the bubble. Although the relative deviation between the simulation results of the 40 X 40 x 80 mesh and 100 x 100 x 200 mesh is notable, the deviation is insignificant between the results of the 80 x 80 x 160 mesh and those of the 100 X 100 x 200 mesh. The agreement with experiments at all resolutions is generally reasonable, although the simulated terminal bubble rise velocities ( 20 cm/s) are slightly lower than the experimental results (21 25 cm/s). A lower bubble-rise velocity obtained from the simulation is expected due to the no-slip condition imposed at the gas-liquid interface, and the finite thickness for the gas-liquid interface employed in the computational scheme.

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