Molecular structures finite-size scaling

Traditionally, the thermodynamics of fluids used in engineering is essentially macroscopic. Fluids are treated as homogeneous molecular structure and fluctuations are ignored. Size and surface effects disappear in the thermodynamic limit in which the volume V and the number of particles N tend to infinity while the molecular density of the substance, p = NjV, remains finite. Macroscopic thermodynamics often eliminates the size of the system by reducing the extensive thermodynamic properties by the number of particles, mass, or volume. The actual scale is restored only in the stage of engineering design. 

The conclusion that we draw from the discussion in this chapter is that the structural behavior of self-interacting semiflexible polymers cannot be adequately described by the wormlike-chain model that does not allow for the description of stmctural transitions. As we have seen, stmctural transitions also occur for semiflexible polymers and have to be taken into account. As in the case of purely flexible polymers, finite-size effects are responsible for a variety of solid, i.e., ordered, crystalline phases. This implies that the detailed stmctural behavior needs to be considered in the understanding of biomolecular processes on short length scales. It is therefore also relevant for the nanofabrication of molecular devices. 

Molecular calculations provide approaches to supramolecular structure and to the dynamics of self-assembly by extending atomic-molecular physics. Alternatively, the tools of finite element analysis can be used to approach the simulation of self-assembled film properties. The voxel4 size in finite element analysis needs be small compared to significant variation in structure-property relationships for self-assembled structures, this implies use of voxels of nanometer dimensions. However, the continuum constitutive relationships utilized for macroscopic-system calculations will be difficult to extend at this scale because nanostructure properties are expected to differ from microstructural properties. In addition, in structures with a high density of boundaries (such as thin multilayer films), poorly understood boundary conditions may contribute to inaccuracies. 

Network polymers can also be made by chemically linking linear or branched polymers. The process whereby such a preformed polymer is converted to a network structure is called cross-linking. Vulcanization is an equivalent term that is used mainly for rubbers. The rubber in a tire is cross-linked to form a network. The molecular weight of the polymer is not really infinite even if all the rubber in the tire is part of a single molecule (this is possible, at least in theory), since the size of the tire is finite. Its molecular weight is infinite, however, on the scale applied in polymer measurements, which require the sample to be soluble in a solvent. 

A major impetus for the design and construction of a finite molecular assembly is to create function not realized by the individual components.3 The size, shape, and functionality of each component, which are achieved via methods of organic syntheses, are thus amplified within a final functional structure. The components may be synthesized, e.g., to give an assembly with cavities that host ions and/or molecules as guests.3 The components may also react to form covalent bonds.1 That a molecular assembly is, de facto, larger than a component molecule means that the components may be designed to assemble to form functional assemblies that reach nanometer-scale dimensions, and beyond.4... 

Particle methods (Molecular Dynamics, Dissipative Particle Dynamics, Multi-Particle Collision Dynamics) simulate a system of interacting mass points, and therefore thermal fluctuations are always present. The particles may have size and structure or they may be just point particles. In the former case, the finite solvent size results in an additional potential of mean force between the beads. The solvent structure extends over unphysically large length scales, because the proper separation of scale between solute and solvent is not computationally realizable. In dynamic simulations of systems in thermal equilibrium [43], solvent structure requires that the system be equilibrated with the solvent in place, whereas for a structureless solvent the solute system can be equilibrated by itself, with substantial computational savings [43]. Finally, lattice models have a (rigorously) known solvent viscosity, whereas for particle methods the existing analytical expressions are only approximations (which however usually work quite well). 

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