Simulations Size effects

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. 

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

The important issue of size effects was addressed by Karaborni and Siepmann [368]. They used the same chain model and other details employed in the Karaborni et al. simulations described earlier [362-365] and the 20-carbon chain. System sizes of 16, 64, and 256 molecules were employed with areas of 0.23, 0.25 and 0.27 nm molecule simulations with 64 molecules were also performed for areas ranging from 0.185 to 0.40 nm molecule . The temperature used was 275 K, as opposed to 300 K used in the previously discussed work by Karaborni et al. with the 20-carbon chain. At the smaller areas no significant system size dependence was found. However, the simulation at 0.27 nm molecule showed substantial differences between N = 64 and N = 256 in ordering and tilt angle. The 64-molecule system showed more order than the 256-molecule system and a slightly lower tilt angle. The pressure-area isotherm data for these simulations are not... 

Thus we have found that the screening should be more efficient than in the Debye-Hiickel theory. The Debye length l//c is shorter by the factor 1 — jl due to the hard sphere holes cut in the Coulomb integrals which reduce the repulsion associated with counterion accumulation. A comparison with Monte Carlo simulation results [20] bears out this view of the ion size effect [19]. 

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... 

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... 

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. 

Lastly, we would like to mention here results of the two kinds of large-scale computer simulations of diffusion-controlled bimolecular reactions [33, 48], In the former paper [48] reactions were simulated using random walks on a d-dimensional (1 to 4) hypercubic lattice with the imposed periodic boundary conditions. In the particular case of the A + B - 0 reaction, D = Dq and nA(0) = nB(0), the critical exponents 0.26 0.01 0.50 0.02 and 0.89 0.02 were obtained for d = 1 to 3 respectively. The theoretical value of a = 0.75 expected for d = 3 was not achieved due to cluster size effects. The result for d = 4, a = 1.02 0.02, confirms that this is a marginal dimension. However, in the case of the A + B — B reaction with DB = 0, the asymptotic longtime behaviour, equation (2.1.106), was not achieved at all - even at very long reaction times of 105 Monte Carlo steps, which were sufficient for all other kinds of bimolecular reactions simulated. It was concluded that in practice this theoretically derived asymptotics is hardly accessible. 

There have been a number of computer simulations of block copolymers by Binder and co-workers (Fried and Binder 1991a,ft), and this work was reviewed in Binder (1994). Although computer simulations are limited due to the restriction on short chain lengths that can be studied, finite size effects and equilibration problems at low temperatures, the advantages are that the models are perfectly well characterized and ideal (monodisperse, etc.) and microscopic details of the system can be computed (Binder 1994). In the simulations by Binder and co-workers, diblocks were modelled as self- and mutually-avoiding chains on a simple cubic lattice, with chain lengths N = 14 to 60 for/ = 1.A purely repulsive pairwise interaction between A and B segments on adjacent sites was assumed. A finite volume fraction of vacancies was included to speed the thermal equilibration process (Binder 1994). 

Table HI compiles MC results obtained over the years for the critical temperature and critical density of the RPM. Table in includes also results from the cluster calculations of Pitzer and Schreiber [141]. In a critical assessment of earlier work [40, 141, 179-181, 246], Fisher deduced in 1994 that T = 0.052-0.056 and p = 0.023-0.035 represent the best values [15]. Since then, however, the situation has substantially changed. Caillol et al. [53,247] performed simulations of ions on the surface of a four-dimensional hypersphere and applied finite-size corrections. Valleau [248] used his thermodynamic-scaling MC for systems with varying particle numbers to extract the infinite-size critical parameters. Orkoulas and Panagiotopoulos [52] performed grand canonical simulations in conjunction with a histogram technique. All studies indicate an insufficient treatment of finite-size effects in earlier work. While their results do not agree perfectly, they are sufficiently close to estimate T = 0.048-0.05 and p = 0.07-0.08, as already quoted in Eq. (6). Critical points of some real Coulombic systems match quite well to these figures [5]. The coexistence curve derived by Orkoulas and Panagiotopoulos [52] is displayed in Fig. 9.

First this aspect of the NIRM strategy compounds a problem inherent in all simulation studies of this problem (or, indeed, any other in many-body physics) finite-size effects. This issue deserves a section devoted to itself, and it gets one (Section VI.C). Here we note simply that such effects are harder to assess when... 

ESPS thus shares a number of the advantages that ESIT has with respect to integration methods (Sections IV.A.2 and IV.C.2). It is pleasingly transparent The evolution with temperature of the relative stability of fee and hep LJ crystals can be read off from Fig. 7 and the LJ freezing pressure can be seen in Fig. 9. Apart from finite-size effects, uncertainties are purely statistical. The fact that both phases are realized within the same simulation means that finite-size effects can be handled more systematically this seems to be a particular advantage of the ESPS approach to the liquid-solid phase boundary. 

---