Finite Size Effect

These are film thickness D/(aN1/2), interaction parameter %/%c the chemical potential difference NAp, and bare surface energies N1/2/a fsL and N1/2/a fsR. They enable [60] an easy comparison of finite size effects observed in different systems. Sometimes two such parameters, D/(aN1/2) and %/%c, are combined into the film thickness-to-interfacial width ratio D/w=3(7/7r-l)1/2D/(aN1/2). 

The effect of GDM thickness on the capillary pressure curves has also been investigated.21 A pronounced shoulder was observed at low capillary pressures in the curves obtained for very thin samples. This feature was diminished for thicker samples and virtually non-existent in the capillary pressure curves obtained from multilayer stacks. The dependence of this feature on the sample surface area-to-volume ratio is a clear demonstration of finite-size effects.70 

One of the main characteristics of microelectrodes that makes them so interesting is the unusual mass transport properties that they exhibit. To explain this behaviour, it is simplest to start by considering a simple reduction reaction such as 

A result that will be familiar to all polarographers. This equation clearly has two limiting forms. At short times 

---

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

Challa MSS, Landau D P and Binder K 1986 Finite-size effects at temperature-driven Ist-order transitions Phys. Rev. B 34 1841 -52... 

Large deformation contacts and finite size effects... 

The finite size effects in the contact between a spherical lens of polyurethane and a soft flat sheet of crosslinked polyfdimethyl siloxane) (PDMS) has been addressed by Falsafi et al. [37]. They showed that for deformations corresponding to contact diameters larger than the sheet thickness, the compliance of the system was affected by the glass substrate supporting the soft sheet. In order to minimize the finite size effects in the adhesion measurement of small elastomeric lenses, Falsafi et al. [38] and Deruelle et al. [39] used relatively thick elastic sheets to support their samples. 

E. V. Albano. Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface Scahng approach and Monte Carlo investigation. Phys Rev B 42 R10818-R10821, 1990. 

The study of how fluids interact with porous solids is itself an important area of research [6], The introduction of wall forces and the competition between fluid-fluid and fluid-wall forces, leads to interesting surface-driven phase changes, and the departure of the physical behavior of a fluid from the normal equation of state is often profound [6-9]. Studies of gas-liquid phase equilibria in restricted geometries provide information on finite-size effects and surface forces, as well as the thermodynamic behavior of constrained fluids (i.e., shifts in phase coexistence curves). Furthermore, improved understanding of changes in phase transitions and associated critical points in confined systems allow for material science studies of pore structure variables, such as pore size, surface area/chemistry and connectivity [6, 23-25],... 

K. B. Migler 2002, (Layered droplet microstructures in sheared emulsions finite-size effects), /. Colloid Interface Sci. 255, 391. 

Mon, K. K. Binder, K., Finite size effects for the simulation of phase coexistence in the Gibbs ensemble near the critical point, J. Chem. Phys. 1992, 96, 6989-6995... 

Fisher, M. E. Barber, M. N., Scaling theory for finite-size effects in the critical region, Phys. Rev. Lett. 1972, 28, 1516-1519... 

Gelb, L. D. Gubbins, K. E., Studies of binary liquid mixtures in cylindrical pores phase separation, wetting and finite-size effects from Monte Carlo simulations, Physica A 1997, 244, 112-123... 

The D band, the disorder induced mode, normally appears between 1250 and 1450 cm. This band is activated in the first-order scattering process of sp2 carbons by the presence of in-plane substitutional hetero-atoms, vacancies, grain boundaries or other defects and by finite-size effects [134], The G band is the second-order overtone of the D band. 

Let us consider, for example, a flat symmetrical bilayer of which the area is large, so that end-effects can be ignored. Finite size effects are important, and will be discussed in the following section. The membrane is freely floating in solution, i.e. it is not supported by a frame. Combination of the first and second laws of thermodynamics gives for the difference of internal energy dl/of a bulk system with membranes with area A ... 

There are several commercial packages that realise the above strategy for molecularly realistic systems. It is useful to discuss some of the limitations. Ideally, one would like to do simulations on macroscopic systems. However, it is impossible to use a computer to deal with numbers of degrees of freedom on the order of /Vav. In lipid systems, where the computations of all the interactions in the system are expensive, a typical system can contain of the order of tens of thousands of particles. Recently, massive systems with up to a million particles have been considered [33], Even for these large simulations, this still means that the system size is limited to the order of 10 nm. Because of this small size, one refers to this volume as a box, although the system boundaries are typically not box-like. Usually the box has periodic boundary conditions. This implies that molecules that move out of the box on one side will enter the box on the opposite side. In such a way, finite size effects are minimised. In sophisticated simulations, i.e. (N, p, y, Tj-ensembles, there are rules defined which allow the box size and shape to vary in such a way that the intensive parameters (p, y) can assume a preset value. 

Let us now consider the more realistic situation in which one takes into account the energy cost due to finite size effects in creating a drop of deconfined... 

The numerical efficiency of the viscosity lower bound method has allowed calculations on considerably longer chains. The long chain Hmit results for 12-arm stars without intramolecular interactions and with EV (up to 325 beads) and in the theta region (up to 145 beads) [164] are close to the previous estimates with shorter chains (the extrapolated ratio g obtained in this study is also included in Table 4). The lower bound method has also served to characterize globule-coil transitions of 12-arm star chains from intrinsic viscosity calculations [143], though finite size effects are considerably more important than in the characterization of this transition from the radius of gyration data (see Fig. 17). This is due to the noticeable increase in the solvent permeabiHty associated with the chain expansion in better solvent conditions. However, the permeability effects are smaller in the more compact star chains than in their linear counterparts. 

In the next section, we give some details on typical models that are studied, or problems that arise (correlations between subsequently generated configurations, finite size effects, self-averaging of observables, etc.) and how they are overcome (more details can be found in Refs. 2-4). 

We now briefly consider finite size effects at first-order phase transitions. The easiest case is transitions driven by the field conjugate to the order parameter in systems at T < T. Then Eq. (40) is easily generalized by introducing Boltzmann weight factors for the two states < according to their Zeemann energies... 

On the other hand, extensive viscosity measurements on PB by Colby et al. [88] show that above about 300 entanglement lengths AT-dependence (the slope of /] vs. N) turns from to AP, indicating that the deviations from reptation are due to finite size effects (see also very recent results by Pyckhout et al. [89]). 

---