Single chain simulations

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. 

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. 

Other moves are specific for dilute solutions (or single chain simulations) and very congested systems (as melts). Some complex rules involving different chains have been developed for the equilibrium study of melts of linear chains, such as the cooperative motion algorithm [105] where beads are moved cooper-... 

An alternative suggestion," similar to the Rubinstein-Colby model, is to determine the rate of CR self-consistently by an iterative procedure. One can run a single-chain simulation and measure the distribution of release times, then start deleting the slip-links according to this distribution. Since the CR affects the rate of chain diffusion, one should repeat this loop several times to achieve self-consistency. This algorithm should lead to exactly the same results however, it is more difficult to implement, especially for polydisperse systems. In this chapter, we shall use the algorithm described in the previous paragraph. 

The combination of DFT with simulation has been developed from various aspects. The famous combination is the Car—Parrinello QM/MM approach (Car and Parrinello, 1985) in which the QDFT is incorporated into MD simulation. Since its foundation, there is tremendous desire to perform mixed QM/MM (Burke, 2012). Regarding combination of polymeric DFT with simulation, it has been demonstrated that a single-chain simulation can be performed for collecting the intrachain correlation information, which is utilized in polymeric DFT for the investigation of polymer systems with finite chain concentration (Cao et al., 2006 Chen et al., 2008). In the following, we demonstrate two other examples on the combinations of atomic DFT and MDFT with simulation. 

Cao D, J iang T, Wu J A hybrid method for predicting the microstructure of polymers with complex architecture combination of single-chain simulation with density functional theory, J Chem Phys 124(16) 164904, 2006. 

Figure 8. Predictions for two diagonal partial structure factors of vinyl chain melts of 33 monomers. The points are from the multiple-chain Monte Carlo simulations of Yethiraj and co-workers. The curves are from single-chain simulations in which repulsive interactions between sites separated more than 2 bonds are screened (set to zero). The BB structure factors are similar to AA and were omitted for clarity.

The six intermolecular radial distribution functions for i-PP were then deduced from PRISM calculations using the single-chain simulation... 

Other Single-Chain Simulation Approaches to Polymer Melts Slip-Link and Dual Slip-Link Models... 

Since the solvation potential requires knowledge of Cay (r) and hay (r) obtained from PRISM theory and a>ay (k) is used as input to the PRISM equation, a self-consistent approach must be utilized. Initially, a guess is made for the matrix elements of the solvation potential, Way (r), and single chain simulations are performed to obtain Say (k) for each ay pair. The PRISM equation and closure are solved for Cay (r) and hay (r) and a new estimate of the solvation potential is obtained. This sequence is repeated until Way (r) converges onto a solution. 

As discussed above, our implementation of SC/PRISM theory makes use of a single chain simulation and hence is nearly exact for a given solvation potential for the intramolecular part of the problem. An alternative to SC/PRISM theory, exact at zero density, is the two-chain equation for g(r) [95, 143]. This equation was originally suggested by Laria, Wu, and Chandler (LWC) [95] and later derived by Donley, Curro, and McCoy (DCM) [143] using density functional techniques. For a single site model, they showed that g(r) can be written in the form... 

Single Chain Simulations. Monte Carlo simulations have been used to calculate thermoelastic results through the temperature coefficient of the unperturbed dimensions (224). In the case of networks of the protein elastin, such results were used to evaluate alternative theories for the molecular deformation mechanism for this bioelastomer (225). 

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