Size scaling

Binder K 1981 Finite size scaling analysis of Ising-model block distribution-functions Z. Phys. B. Oondens. Matter. 43 119-40... 

Binder K and Landau D P 1984 Finite size scaling at Ist-order phase transitions Phys. Rev. B 30 1477-85... 

In homopolymers all tire constituents (monomers) are identical, and hence tire interactions between tire monomers and between tire monomers and tire solvent have the same functional fonn. To describe tire shapes of a homopolymer (in the limit of large molecular weight) it is sufficient to model tire chain as a sequence of connected beads. Such a model can be used to describe tire shapes tliat a chain can adopt in various solvent conditions. A measure of shape is tire dimension of tire chain as a function of the degree of polymerization, N. If N is large tlien tire precise chemical details do not affect tire way tire size scales witli N [10]. In such a description a homopolymer is characterized in tenns of a single parameter tliat essentially characterizes tire effective interaction between tire beads, which is obtained by integrating over tire solvent coordinates. 

Fracture mechanics is now quite weU estabHshed for metals, and a number of ASTM standards have been defined (4—6). For other materials, standardization efforts are underway (7,8). The techniques and procedures are being adapted from the metals Hterature. The concepts are appHcable to any material, provided the stmcture of the material can be treated as a continuum relative to the size-scale of the primary crack. There are many textbooks on the subject covering the appHcation of fracture mechanics to metals, polymers, and composites (9—15) (see Composite materials). 

Classical astronomy is largely concerned with the classification of stars without regard to the details of their constituent plasmas (63). Only more recently have sateUite-bome observations begun to yield detailed data from the high temperature regions of other stellar plasmas. Cosmic plasmas of diverse size scales have been discussed (64). 

Despite recent developments in algorithms and computer hardware, to bridge the gap between the time and size scales accessible by computer simulations and those required by experimental observations we still need to develop noble approaches. 

In numerous applications of polymeric materials multilayers of films are used. This practice is found in microelectronic, aeronautical, and biomedical applications to name a few. Developing good adhesion between these layers requires interdiffusion of the molecules at the interfaces between the layers over size scales comparable to the molecular diameter (tens of nm). In addition, these interfaces are buried within the specimen. Aside from this practical aspect, interdififlision over short distances holds the key for critically evaluating current theories of polymer difllision. Theories of polymer interdiffusion predict specific shapes for the concentration profile of segments across the interface as a function of time. Interdiffiision studies on bilayered specimen comprised of a layer of polystyrene (PS) on a layer of perdeuterated (PS) d-PS, can be used as a model system that will capture the fundamental physics of the problem. Initially, the bilayer will have a sharp interface, which upon annealing will broaden with time. 

On the other hand, when the size scale of the heterogeneities is sufficiently small (generally p,m) and uniform, another type of analysis may be used. In this case it is assumed that the interline will be able to adjust its position to... 

The aim of this chapter is to describe the micro-mechanical processes that occur close to an interface during adhesive or cohesive failure of polymers. Emphasis will be placed on both the nature of the processes that occur and the micromechanical models that have been proposed to describe these processes. The main concern will be processes that occur at size scales ranging from nanometres (molecular dimensions) to a few micrometres. Failure is most commonly controlled by mechanical process that occur within this size range as it is these small scale processes that apply stress on the chain and cause the chain scission or pull-out that is often the basic process of fracture. The situation for elastomeric adhesives on substrates such as skin, glassy polymers or steel is different and will not be considered here but is described in a chapter on tack . Multiphase materials, such as rubber-toughened or semi-crystalline polymers, will not be considered much here as they show a whole range of different micro-mechanical processes initiated by the modulus mismatch between the phases. 

Processes that occur at a size scale larger than the individual chain have been studied using microscopy, mainly transmission electron microscopy (TEM), but optical microscopy has been useful to examine craze shapes. The knowledge of the crazing process obtained by TEM has been ably summarised by Kramer and will not be repeated here [2,3]. At an interface between two polymers a craze often forms within one of the materials, typically the one with lower crazing stress. 

Micro-mechanical processes that control the adhesion and fracture of elastomeric polymers occur at two different size scales. On the size scale of the chain the failure is by breakage of Van der Waals attraction, chain pull-out or by chain scission. The viscoelastic deformation in which most of the energy is dissipated occurs at a larger size scale but is controlled by the processes that occur on the scale of a chain. The situation is, in principle, very similar to that of glassy polymers except that crack growth rate and temperature dependence of the micromechanical processes are very important. 

In this section we study a system with purely repulsive interactions which demonstrates the importance of entropy effects on the stability of phases when the effect of the corrugation potential due to the structured surface is completely neglected. The phase diagrams are determined by finite size scaling methods, in particular the methods of Sec. IV A. 

In this section we review several studies of phase transitions in adsorbed layers. Phase transitions in adsorbed (2D) fluids and in adsorbed layers of molecules are studied with a combination of path integral Monte Carlo, Gibbs ensemble Monte Carlo (GEMC), and finite size scaling techniques. Phase diagrams of fluids with internal quantum states are analyzed. Adsorbed layers of H2 molecules at a full monolayer coverage in the /3 X /3 structure have a higher transition temperature to the disordered phase compared to the system with the heavier D2 molecules this effect is... 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

P. Nielaba, J. L. Lebowitz, H. Spohn, J. L. Valles. J Stat Phys 55 745, 1989. V. Privman, ed. Finite Size Scaling and Numerical Simulation. Singapore World Scientific, 1990. 

In the case of a single patch, the size dependence of the system follows directly from the finite size scaling theory [133]. In particular, the critical point temperature scales with the system size as predicted by the equation... 

E. V. Albano. The critical behavior of dimer-dimer surface reaction models. Monte Carlo and finite-size scaling investigation. J Stat Phys 69 643-666,1992. 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

The inset illustrates the extrapolation of the simulation data to the thermodynamic limit according to mixed field finite size scaling for Na = 40 and Nb = 120. From Muller and Binder. ... 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

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