Fluctuating film models

A variety of thermodynamic models qualitatively capture the central equilibrium and dynamic features of microemulsion behavior (41 5). These range from phenomenological models, which treat the oil-water interface as fluctuating membranes to lattice models which describes discrete surfactants interacting with oil and water. Membrane models range from simple cubic lattice descriptions to fluctuating film models that include curvature-dependent bending elasticity to prescribe lower limits on domain size. 

While thin polymer films may be very smooth and homogeneous, the chain conformation may be largely distorted due to the influence of the interfaces. Since the size of the polymer molecules is comparable to the film thickness those effects may play a significant role with ultra-thin polymer films. Several recent theoretical treatments are available [136-144,127,128] based on Monte Carlo [137-141,127, 128], molecular dynamics [142], variable density [143], cooperative motion [144], and bond fluctuation [136] model calculations. The distortion of the chain conformation near the interface, the segment orientation distribution, end distribution etc. are calculated as a function of film thickness and distance from the surface. In the limit of two-dimensional systems chains segregate and specific power laws are predicted [136, 137]. In 2D-blends of polymers a particular microdomain morphology may be expected [139]. Experiments on polymers in this area are presently, however, not available on a molecular level. Indications of order on an... 

On compression, a gaseous phase may condense to a liquid-expanded, L phase via a first-order transition. This transition is difficult to study experimentally because of the small film pressures involved and the need to avoid any impurities [76,193]. There is ample evidence that the transition is clearly first-order there are discontinuities in v-a plots, a latent heat of vaporization associated with the transition and two coexisting phases can be seen. Also, fluctuations in the surface potential [194] in the two phase region indicate two-phase coexistence. The general situation is reminiscent of three-dimensional vapor-liquid condensation and can be treated by the two-dimensional van der Waals equation (Eq. Ill-104) [195] or statistical mechanical models [191]. 

A passive film is stable in the region between the passivation and breakdown potentials if any part of the film is broken, it is rapidly repaired. Therefore it is necessary to derive a model that depicts the processes by which such local destruction and restoration are continuously repeated. This process can be regarded as a kind of nonequilibrium fluctuation concerning passivity. Using energetics, Sato7 analyzed such fluctuation processes as follows. 

When the film thickness is of the order of roughness heights, the effects of roughness become significant which have to be taken into account in a profound model of mixed lubrication. The difficulty is that the stochastic nature of surface roughness results in randomly fluctuating solutions that the numerical techniques in the 1970s are unable to handle. As... 

The model has been applied successfully to predicting the performances of bearings, gears, seals, and engines [10-12]. A fundamental limitation of the statistic models is their inability to provide detailed information about local pressure distribution, film thickness fluctuation, and asperity deformation, which are crucial for understanding the mechanisms of lubrication, friction, and surface failure. As an alternative, researchers paid a great interest to the deterministic ML model. 

An alternative description of membrane stability has been based on hydrodynamic models, originally developed for liquid films in various environments [54-56]. Rupture of the film was rationalized by the instability of symmetrical squeezing modes (SQM) related to the thickness fluctuations. In the simplest form it can be described by a condition [54] d Vdis/dh < where is the interaction contribution related to the dis-... 

A very different model of tubules with tilt variations was developed by Selinger et al.132,186 Instead of thermal fluctuations, these authors consider the possibility of systematic modulations in the molecular tilt direction. The concept of systematic modulations in tubules is motivated by modulated structures in chiral liquid crystals. Bulk chiral liquid crystals form cholesteric phases, with a helical twist in the molecular director, and thin films of chiral smectic-C liquid crystals form striped phases, with periodic arrays of defect lines.176 To determine whether tubules can form analogous structures, these authors generalize the free-energy of Eq. (5) to consider the expression... 

Inner slip, between the solid wall and an adsorbed film, will also influence the surface-liquid boundary conditions and have important effects on stress propagation from the liquid to the solid substrate. Linked to this concept, especially on a biomolecular level, is the concept of stochastic coupling. At the molecular level, small fluctuations about the ensemble average could affect the interfacial dynamics and lead to large shifts in the detectable boundary condition. One of our main interests in this area is to study the relaxation time of interfacial bonds using slip models. Stochastic boundary conditions could also prove to be all but necessary in modeling the behavior and interactions of biomolecules at surfaces, especially with the proliferation of microfluidic chemical devices and the importance of studying small scales. 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

The purpose of this article is to present a model and to calculate on its basis the metastable equilibrium thicknesses of the film as a function of the applied pressure. In section II, the interaction energy of the film was calculated, assuming planar interfaces free of thermal fluctuations. The double layer interaction was calculated by accounting for the charge recombination at the surface with increasing electrolyte concentration. An approximate... 

Fig.20. Order parameter profiles m(z)=([pA(z)-pB(z)])/([pA(z)+pB(z)]), where pA(z), pB(z) are densities of A-monomers or B-monomers at distance z from the left wall, for LxLx20 films confining a symmetric polymer mixture, polymers being described by the bond fluctuation model with N=32, ab=- aa=- bb=8 and interaction range 6. Four inverse temperatures are shown as indicated. In each case two choices of the linear dimension L parallel to the film are included. While for e/kBT>0.02 differences between L=48 and L=80 are small and only due to statistical errors (which typically are estimated to be of the size of the symbols), data for e/kBT=0.018 clearly suffer from finite size effects. Broken straight lines indicate the values of the bulk order parameters mb in each case [280]. Arrows show the gyration radius and its smallest component in the eigencoordinate system of the gyration tensor [215]. Average volume fraction of occupied sites was chosen as 0.5. From Rouault et al. [56].

Fig. 22. Phase diagrams of the confined polymer mixtures for thin films of various thicknesses D, using the bond fluctuation model for symmetric polymer mixtures for NA=NB= N=32. The symbols refer to different film thicknesses D=8,10,12,14,16,20,24,28,36 and 48 (from the bottom to the top). From Rouault et al. [55]...

The rationale of using hybrid simulation here is that a classic diffusion-adsorption type of model, Eq. (2), can efficiently handle large distances between steps by a finite difference coarse discretization in space. As often happens in hybrid simulations, an explicit, forward discretization in time was employed. On the other hand, KMC can properly handle thermal fluctuations at the steps, i.e., provide suitable boundary conditions to the continuum model. Initial simulations were done in (1 + 1) dimensions [a pseudo-2D KMC and a ID version of Eq. (2)] and subsequently extended to (2 + 1) dimensions [a pseudo-3D KMC and a 2D version of Eq. (2)] (Schulze, 2004 Schulze et al., 2003). Again, the term pseudo is used as above to imply the SOS approximation. Speedup up to a factor of 5 was reported in comparison with KMC (Schulze, 2004), which while important, is not as dramatic, at least for the conditions studied. As pointed out by Schulze, one would expect improved speedup, as the separation between steps increases while the KMC region remains relatively fixed in size. At the same time, implementation is definitely complex because it involves swapping a microscopic KMC cell with continuum model cells as the steps move on the surface of a growing film. 

Contemporary understanding of liquid film rupture is based on the Linear Stability Theory and the concept of existence of fluctuational waves on liquid surfaces [81]. According to this model the film is ruptured by unstable waves, i.e. waves the amplitudes of which increase with time. The rupture occurs at the moment when the amplitude Ah or the root mean... 

Wasan et al. (27-28) explained the process of stratification on the basis of a micelle-latticing structure model. In Figure 8 a schematic of the latticing model for film thinning is provided. By fluctuations in the structure of the micellar lamellae (i.e. the individual rows of micelles in Figure 8), the film can change its thickness by stepwise transitions, each of which are equal to the micellar-lamellae thickness. According to this model the number of transitions will depend upon the micelle concentration. 

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