Homogeneous mean-field model

The computational requirements of a molecular dynamics model can be greatly reduced if we make use of averaging to produce a mean-field model such as those based on the concept of a molecule in a tube. Here, instead of starting from a detailed picture of the interactions between individual molecules, we focus attention on a single molecule, a test chain, and represent the effect of all the surrounding molecules by an average field of constraints. Such models can be used to calculate the response to homogeneous deformations such as step shear and steady-simple shear. 

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where / , (/<. t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4> (k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/<. t) of the form14... 

The application of this technique to symmetric binary blends is discussed in Fig. 21, where we plot the results of simulations of the bond fluctuation model. The interface tension has been obtained for different chain lengths and the results are compared to mean field theory [104,107,108]. This comparison utilizes the identification of the incompatibility obtained from the simulation of the spatially homogeneous system [65] in Sect. 3.2. 

The assumption of homogeneity can be abandoned if the continuous me m-field treatment is replaced by a discrete treatment where the positions of fluid molecules are restricted to nodes on a lattice. The discussion in Section 5.4.2 and 5.6.5 showed that the mean-field lattice density functional theory developed in Section 4.3 w as crucial in unraveling the c.om-plex phase behavior of fluids confined by chemically decorated substrate surfaces. A similar deep understanding of the phase behavior would not have been possible on the basis of simulation results alone. Nevertheless, the relation between these MC data and the lattice density functional results remained only qualitative on accoimt of the continuous models employed in the computer simulations. Thus, we aim at a more quantitative comparison between MC simulations and mean-field lattice density fimctioiial theory in the closing. section of this diaptcr. 

To parameterize the polymersome model, the identification of the virial coefficients, vaij and wapv is driven by the requirement that the amphiphiles described by (10) and (12) should create a stable bilayer with given material properties. Assuming that the hydrophobic interior should be in a melt state, the coefficients vaa and Waaa are determined such that (12) enforces the A-blocks to create a melt in equilibrium with its vapor which, in a solvent free model, represents the surrounding water. It can be shown, from (12), that the equation of state of such a homogeneous melt, within mean-field approximation, is [138]... 

Within the mean-field theory there are no spontaneous elastic deformations since any deformation increases the free energy. However, when a nematic LC is subject to interactions with the confining walls the homogeneous order can be perturbed. On the microscopic level, the molecules of the walls and of the liquid crystal attract each other via a short-range van der Waals interaction. In the macroscopic description this is modeled with a contact quadruple-quadruple interaction, known as the Rapini—Papoular model [32,33], which to the lowest order reads... 

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