Scale, length

For standard problems in the physics and chemistry of condensed matter, such as simple fluids containing rare gas atoms or diatomic molecules, etc., computer simulation considers a small region of matter in full atomistic detail. For example, for a simple fluid it often is sufficient to simulate a small box containing of the order of 10 atoms, which interact with each other with chemically realistic forces. These methods work because simple fluids are homogeneous on a scale of 10 A already the oscillations in the pair distribution function then are damped out under most circumstances. Also reliable models for the effective forces are usually available from quantum chemistry methods. 

1 Length scales characterizing the structure of a long polymer coil (polyethylene is used as an example). (From Binder. ) 

Collective phenomena may even lead to much larger lengths e.g., in a polymer brush (i.e., a layer of polymers anchoring with a special end group at an otherwise repulsive wall) the height h of the brush is predicted 

In view of this discussion, it is clear that even the detailed model of eqs (1.1)-(1,3) should not be taken as a faithful description of polyethylene (PE), but rather as a prototypical schematic model of linear polymers. In the context of simulations of lipid monolayers,it has been suggested that it is necessary to shift the center of gravity of the united atom off the position of the carbon atom at the chain backbone (anisotropic united atom model). Very recent work (see also Chapter 8) suggests that it is more satisfactory to include the hydrogen atoms explicitly, if one wishes to describe PE properly. For the reasons quoted above, such work is restricted to relatively 

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Discuss the paradox in the wettability of a fractal surface (Eq. X-33). A true fractal surface is infinite in extent and a liquid of a finite contact angle will trap air at some length scale. How will this influence the contact angle measured for a fractal surface ... 

Another realistic approach is to constnict pseiidopotentials using density fiinctional tlieory. The implementation of the Kolm-Sham equations to condensed matter phases without the pseiidopotential approximation is not easy owing to the dramatic span in length scales of the wavefimction and the energy range of the eigenvalues. The pseiidopotential eliminates this problem by removing tlie core electrons from the problem and results in a much sunpler problem [27]. 

Lum K, Chandler D and Weeks J D 1999 Hydrophobicity at small and large length scales J. Phys. Chem. B 103 4570... 

In order to describe the second-order nonlinear response from the interface of two centrosynnnetric media, the material system may be divided into tlnee regions the interface and the two bulk media. The interface is defined to be the transitional zone where the material properties—such as the electronic structure or molecular orientation of adsorbates—or the electromagnetic fields differ appreciably from the two bulk media. For most systems, this region occurs over a length scale of only a few Angstroms. With respect to the optical radiation, we can thus treat the nonlinearity of the interface as localized to a sheet of polarization. Fonnally, we can describe this sheet by a nonlinear dipole moment per unit area, -P ", which is related to a second-order bulk polarization by hy P - lx, y,r) = y. Flere z is the surface nonnal direction, and the... 

Both MD and MC teclmiques evolve a finite-sized molecular configuration forward in time, in a step-by-step fashion. (In this context, MC simulation time has to be interpreted liberally, but there is a broad coimection between real time and simulation time (see [1, chapter 2]).) Connnon features of MD and MC simulation teclmiques are that there are limits on the typical timescales and length scales that can be investigated. The consequences of finite size must be considered both in specifying the molecular mteractions, and in analysing the results. 

Coarse-grained models have a longstanding history in polymer science. Long-chain molecules share many common mesoscopic characteristics which are independent of the atomistic stmcture of the chemical repeat units [4, 5 and 6]. The self-similar stmcture [7, 8, 9 and 10] on large length scales is only characterized by a single length scale, the chain extension R. 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

In the vicinity of the critical point (i.e. t < i) the interfacial width is much larger than the microscopic length scale / and the Landau-Ginzburg expansion is applicable. 

On short length scales the coarse-grained description breaks down, because the fluctuations which build up the (smooth) intrinsic profile and the fluctuations of the local interface position are strongly coupled and camiot be distinguished. The effective interface Flamiltonian can describe the properties only on length scales large compared with the width w of the intrinsic profile. The absolute value of the cut-off is difficult... 

These chain models are well suited to investigate the dependence of tire phase behaviour on the molecular architecture and to explore the local properties (e.g., enriclnnent of amphiphiles at interfaces, molecular confonnations at interfaces). In order to investigate the effect of fluctuations on large length scales or the shapes of vesicles, more coarse-grained descriptions have to be explored. 

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