Polymeric systems ensemble

Schonherr H (1999) From functional group ensembles to single molecules scanning force microscopy of supramolecular and polymeric systems. Ph.D. Thesis, University of Twente, Enschede, The Netherlands... 

Escobedo and de Pablo have proposed some of the most interesting extensions of the method. They have pointed out [49] that the simulation of polymeric systems is often more troubled by the requirements of pressure equilibration than by chemical potential equilibration—that volume changes are more problematic than particle insertions if configurational-bias or expanded-ensemble methods are applied to the latter. Consequently, they turned the GDI method around and conducted constant-volume phase-coexistence simulations in the temperature-chemical potential plane, with the pressure equality satisfied by construction of an appropriate Cla-peyron equation [i.e., they take the pressure as 0 of Eq. (3.3)]. They demonstrated the method [49] for vapor-liquid coexistence of square-well octamers, and have recently shown that the extension permits coexistence for lattice models to be examined in a very simple manner [71]. 

Mes and coworkers compared TDA, DLS, HDC, and SEC and showed that all four methods can be used effectively to determine diffusion coefficients of systems with low polydispersities by measuring a series of styrene acrylonitrile (SAN) copolymers. Although these are polymeric systems, it is possible to apply the findings to supramolecular ensembles. The characterization of samples of low polydispersity was achieved best with TDA and DLS, since they both allow the rapid and absolute determination of the diffusion coefficient. However, TDA has the disadvantage that it is subject to interference due to the presence of low-molecular-mass chromophoric compounds. DLS, on the other hand, is influenced much more by the polydispersity of the sample than TDA. Furthermore, the use of DLS enables direct measurements of the Z-average diffusion coefficient of a polydisperse sample but requires a relatively large amount of the sample and is concentration dependent. Unlike TDA, DLS is especially suited for the analysis of high-molecular-mass systems, such as supramolecular systems, and is not disturbed by the presence of low-molecular-mass impurities. 

The extent of fluorescence polarization at a time t following a brief excitation pulse is directly related to the time dependent, ensemble averaged probability that an excitation resides on the chrom-ophore where it was created at t = 0 [4,6,7]. This probability, which will be designated G (t), depends strongly on the density and distribution of chromophores, and can be obtained from a theoretical analysis of the many body EET problem in the polymeric system. A fit of transient fluorescence depolarization data to a theoretical expression for G (t) provides parameter values that are directly related to chromophore distribution. G (t) and G (t) are portions of the total Green function for excitation transport. 

There has been some criticism of the study by Jang et a/., which address the importance of the equilibration procedure for polymeric systems. Elliott and Paddison have argued that the difference in density between the MD simulated system and the experimentally measured values are due to incomplete chain relaxation or an erratic force field. Since later MD simulations of Nafion using similar force field provided good agreement in density, the latter can probably be ruled out. Jang et al. used a rather complex equilibration process, were the MD boxes were heated, and expanded and compressed several times in the NPT ensemble. The resulting densities (1.60 and 1.67 g/cm ) are 5-10% lower than experimental values, which can partly be explained by the lack of semicrystalline domains in the MD box, which do exist in real membranes and contribute to a somewhat increased density. 

If a fiVT ensemble simulation can be turned into a ( quasi ) NPT ensemble-type simulation (e.g., a pseudo- FT ensemble), the inverse transformation (a pseudo-NPT ensemble) is also possible. The key relationship for a pseudo-NPT ensemble technique is Eq. (5.1) [78]. Such a reverse strategy can be practical only if molecular insertion and deletion moves can be performed efficiently for the system under study (e.g., by expanded ensemble moves for polymeric fluids). Replacing volume moves by particle insertions can be advantageous for polymeric and other materials that require simulation of a large system (due to the sluggishness of volume moves for mechanical equilibration of the system) such an advantage has been clearly demonstrated for a test system of dense, athermal chains [78]. 

A straightforward, but tedious, route to obtain information of vapor-liquid and liquid-liquid coexistence lines for polymeric fluids is to perform multiple simulations in either the canonical or the isobaric-isothermal ensemble and to measure the chemical potential of all species. The simulation volumes or external pressures (and for multicomponent systems also the compositions) are then systematically changed to find the conditions that satisfy Gibbs phase coexistence rule. Since calculations of the chemical potentials are required, these techniques are often referred to as NVT- or NPT- methods. For the special case of polymeric fluids, these methods can be used very advantageously in combination with the incremental potential algorithm. Thus, phase equilibria can be obtained under conditions and for chain lengths where chemical potentials cannot be reliably obtained with unbiased or biased insertion methods, but can still be estimated using the incremental chemical potential ansatz [47-50]. 

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