Field-theoretic simulation

To summarize, the example of homopolymer/copolymer mixtmes demonstrates nicely how field-theoretic simulations can be used to study non-trivial fluctuation effects in polymer blends within the Gaussian chain model The main advantage of these simulations is that they can be combined in a natural way with standard self-consistent field calculations. As mentioned earlier, the self-consistent field theory is one of the most powerful methods for the theoretical description of polymer blends, and it is often accurate on a quantitative level hi many regions of the parameter space, fluctuations are irrelevant for large chain lengths (large Jf) and simulations are not necessary. Field-theoretic simulations are well suited to complement self-consistent field theories in those parameter regions where fluctuation effects become important. 

Field theoretical simulations [74,75,80] avoid any saddle point approximation and provide a formally exact solution of the standard model of the self-consistent field theory. To this end one has to deal with a complex free energy functional as a fimction of the composition and density. This significantly increases the computational complexity. Moreover, for certain parameter regions, it is very difficult to obtain reliable results due to the sign problem that a complex weight imparts onto thermodynamical averages [80]. We have illustrated that for a dense binary blend the results of the field theoretical simulations and the EP theory agree quantitatively, i.e., density and composi-... 

Lee J, Popov YO, Fredrickson GH (2008) Complex coacervation a field theoretic simulation study of polyelectrolyte complexation. J Chem Phys 128(22) 224908. doi 10.1063/1.2936834... 

Popov YO, Lee J, Fredrickson G (2007) Field-theoretic simulations of polyelectrolyte complexation. J Polym Sci Pol Phys 45 3223-3230... 

The thermodynamic integration scheme can be appUed to different models including coarse-grained, partide-based models of amphiphihc systems and membranes [133, 134] (e.g., soft DPD-models [135-137], Lennard-Jones models [138,139], or solvent-free models [140-142] of membranes) as well as field-theoretic representations [28]. It can be implemented in Monte Carlo or molecular dynamic simulations, as well as SCMF simulations [40-42, 86], field-theoretic simulations [28], and external potential dynamics [27, 63, 64] or dynamic density functional theory [143, 144]. 

Similar attempts to include fluctuations beyond one-loop in SC FT have also been exercised in the context of neutral polymers using field theoretical simulations [55,90] or by bridging SCFT with Monte Carlo techniques [91]. However, these techniques have not been applied for the case of polyelectrolytes with counterions and added salt ions due to very high computational cost. 

A catalyst may play an active role in a different sense. There are interesting temporal oscillations in the rate of the Pt-catalyzed oxidation of CO. Ertl and coworkers have related the effect to back-and-forth transitions between Pt surface structures [220] (note Fig. XVI-8). See also Ref. 221 and citations therein. More recently Ertl and co-workers have produced spiral as well as plane waves of surface reconstruction in this system [222] as well as reconstruction waves on the Pt tip of a field emission microscope as the reaction of H2 with O2 to form water occurred [223]. Theoretical simulations of these types of effects have been reviewed [224]. 

Figure 4.4. Mossbauer spectra of Hox-2.10 hatched marks) prepared from the spectra of D. vulgaris hydrogenase poised at -310 mV. The data were recorded at 4.2 K in a magnetic field of 0.05 T A, or 8 T B applied parallel to the y-rays. Theoretical simulations for the individual iron sites of the [2Fe]H cluster dotted-and-dashed lines, component A dashed lines, component B) and of the [4Fe-4S]e cluster (dotted-and-dashed lines, component C dashed lines, component D) are shown above the experimental data. Solid lines, superpositions of these four simulated spectral components.

This scheme has yet to be fully tested in a high-field ESR experiment but initial experimental tests and theoretical simulations by the Pisa group appear very promising. We have also tested this scheme in our own laboratory with very encouraging results and expect this type of resonator to become a common feature in high field systems. 

This question has a considerable relevance not only for the comparison with experiment but also from a purely theoretical perspective. If we want to acquire insight into the mechanism of the ionization reaction while it unfolds, we need to devise meaningful ways of inspecting the wavefunction also before the interaction with the external laser field is over. The unique ability of theoretical simulation to constantly monitor the wave packet is arguably helpful to devise efficient control protocols of the course of the fragmentation process itself. 

FIGURE 10.5 Contributions to in the theoretical simulation of spin relaxation in the radical pairs from DCA-POZ and DCA-PSZ evaluated under the assumption of 0 (data points). The full simulations are represented by the curves denoted / -POZ/DQtot.A 0.45 and fc-PSZ/DQtot, respectively. The contribution from the esdi mechanism (k-esdi )=6E-7) corresponds to an effective translational diffusion constant of D = 6 x 10 cm s. The curves indicated as / -POZ/DQ.A represent the contributions of the ahfi mechanism, the indicating the factor by which the theoretical anisotropy parameter A is reduced. The curve indicated hy k-PSZ/DQ.g denotes the contribution due to the g tensor anisotropy in the PSZ radical. The constant values c POZ and c PSZ represent the field-independent contributions to k. For details of the calculation cf. Ref. 23. 

A control mechanism has been proposed on the basis of the joint analysis of the experimental and theoretical information. The control scheme leading selectively to the formation of CpMn(CO)3+ is represented in Figure 6(b). The experiment realized with an optimal laser field is simulated by one pump pulse (at 3.49 eV) followed by a probe pulse (at 4.716eV) designed with the adequate properties of phase, frequency, and duration. Within these specific conditions, the quasi-bound state c A is populated selectively and the CO dissociation... 

A number of theoretical transfer functions have been reported for specific experiments. However, analytical expressions were derived only for the simplest Hartmann-Hahn experiments. For heteronuclear Hartmann-Hahn transfer based on two CW spin-lock fields on resonance, Maudsley et al. (1977) derived magnetization-transfer functions for two coupled spins 1/2 for matched and mismatched rf fields [see Eq. (30)]. In IS, I2S, and I S systems, all coherence transfer functions were derived for on-resonance irradiation including mismatched rf fields. More general magnetization-transfer functions for off-resonance irradiation and Hartmann-Hahn mismatch were derived for Ij S systems with N < 6 (Muller and Ernst, 1979 Chingas et al., 1981 Levitt et al., 1986). Analytical expressions of heteronuclear Hartmann-Hahn transfer functions under the average Hamiltonian, created by the WALTZ-16, DIPSI-2, and MLEV-16 sequences (see Section XI), have been presented by Ernst et al. (1991) for on-resonant irradiation with matched rf fields. Numerical simulations of heteronuclear polarization-transfer functions for the WALTZ-16 and WALTZ-17 sequence have also been reported for various frequency offsets (Ernst et al., 1991). 

Theoretical simulations are widely, and increasingly, used in numerous fields of science. The rapid development of computing power facilitates simulations of problems that continue to grow in scale and complexity. Combining this with the fact that heterogeneous catalysts play an enormous economic role in industry and environmental control, it is not surprising that computer simulations are used to great extent in the study of catalytic processes on solid surfaces. 

Hopefully, we have hinted that there are interesting nonequilibrium systems that can be explored using homogeneous, field-driven simulations. Because these approaches are often extensions of equilibrium methods, it is natural to first present the equilibrium foundation, as is done in the next section. From there, we will be able to develop the theoretical basis for NEMD simulations and practical guidelines for implementing them. Once the tools are in place, we will discuss applications and the kinds of question that can be tackled by NEMD methods. 

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