Density single-chain

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

The last quantity that we discuss is the mean repulsive force / exerted on the wall. For a single chain this is defined taking the derivative of the logarithm of the chain partition function with respect to the position of the wall (in the —z direction). In the case of a semi-infinite system exposed to a dilute solution of polymer chains at polymer density one can equate the pressure on the wall to the pressure in the bulk which is simply given by the ideal gas law The conclusion then is that [74]... 

The Alexander approach can also be applied to discover useful information in melts, such as the block copolymer microphases of Fig. 1D. In this situation the density of chains tethered to the interface is not arbitrary but is dictated by the equilibrium condition of the self-assembly process. In a melt, the chains must fill space at constant density within a single microphase and, in the case of block copolymers, minimize contacts between unlike monomers. A sharp interface results in this limit. The interaction energy per chain can then be related to the energy of this interface and written rather simply as Fin, = ykT(N/Lg), where ykT is the interfacial energy per unit area, q is the number density of chain segments and the term in parentheses is the reciprocal of the number of chains per unit area [49, 50]. The total energy per chain is then ... 

The density functional theory has the stmcture of a self-consistent field theory where the density profile is obtained from a simulation of a single chain in the... 

If there were no intramolecular interactions (such as bonding or excluded volume), then V(R) = 0, and the next guess for the density profile can be obtained directly from Eq. (75). The presence of V(R) necessitates either a multidimensional integration or (more conveniently) a single-chain simulation. 

One way to obtain predictions for the density profile is to use a Picard iteration procedure. In this method one starts with an initial guess for the density profile. The field A,(r) is then calculated using Eq. (76), and a new estimate for the density profile is obtained using Eq. (77). The latter requires the simulation of a single chain with intramolecular interaction E(R) in an effective field x(R) = density profile is then calculated from... 

A more efficient way of solving the DFT equations is via a Newton-Raphson (NR) procedure as outlined here for a fluid between two surfaces. In this case one starts with an initial guess for the density profile. The self-consistent fields are then calculated and the next guess for density profile is obtained through a single-chain simulation. The difference from the Picard iteration method is that an NR procedure is used to estimate the new guess from the density profile from the old one and the one monitored in the single-chain simulation. This requires the computation of a Jacobian matrix in the course of the simulation, as described below. 

Equilibrium properties are surprisingly accurately predicted by molecular-level SCF calculations. MC simulations help us to understand why the SCF theory works so well for these densely packed layers. In effect, the high density screens the correlations for chain packing and chain conformation effects to such a large extent that the properties of a single chain in an external field are rather accurate. Cooperative fluctuations, such as undulations, are not included in the SCF approach. Also, undulations cannot easily develop in an MD box. To see undulations, one needs to perform molecularly realistic simulations on very large membrane systems, which are extremely expensive in terms of computation time. 

Most physical quantities of interest can be expressed in terms of the anci 0f fhe G(m) = G -mK To give an example we note the density-density correlation of a single chain system, which takes the form... 

In later chapters we will see a few more examples of explicit calculations. Specifically we will present the calculation of the single-chain density correlation function in Appendix A 15.2 and the calculation of the osmotic pressure in Appendix A 17,1. 

If we look at a single chain in a polymer solution, the average number density of its segments, rjCoii, he., their number contained in unit volume of the average space occupied by the chain, is given by... 

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