Computer simulations transformations

Metallurgists originally, and now materials scientists (as well as solid-state chemists) have used erystallographic methods, certainly, for the determination of the structures of intermetallic compounds, but also for such subsidiary parepistemes as the study of the orientation relationships involved in phase transformations, and the study of preferred orientations, alias texture (statistically preferential alignment of the crystal axes of the individual grains in a polycrystalline assembly) however, those who pursue such concerns are not members of the aristocracy The study of texture both by X-ray diffraction and by computer simulation has become a huge sub-subsidiary field, very recently marked by the publication of a major book (Kocks el al. 1998). 

Equation (1.15) indicates that our ultimate focus in calculating A A is on determining the ratio Qi/Qo - or equivalently Zi/Zo - rather than on individual partition functions. On the basis of computer simulations, this can be done in several ways. One approach consists in transforming (1.16) as follows ... 

To illustrate how stratification works in the context of free energy calculations, let us consider the transformation of state 0 into state 1 described by the parameter A. We further assume that these two states are separated by a high-energy barrier that corresponds to a value of A between Ao and Ai. Transitions between 0 and 1 are then rare and the free energy estimated from unstratified computer simulations would converge very slowly to its limiting value, irrespective of the initial conditions. If, however, the full range of A is partitioned into a number of smaller intervals, and... 

Computational efficiency remains a central question. For a given amount of computer time, how good are nonequilibrium estimates of free energy differences compared to estimates from equilibrium methods Overall, evidence is mounting that nonequilibrium methods are less efficient than equilibrium methods [13, 20, 38], However, new approaches have been suggested that use long time steps [25]. For relatively slow transformations, it has been shown [20] that for a given amount of computer simulation time, one obtains more-accurate results for few slow transformations than for many fast transformations. At the other extreme, i.e., in the limit of... 

For a cycled feed (a chain of interfering pulses) the interpretation requires some mathematical transformations, but results such as Fig. 7 can also be used as qualitative indicators of certain processes (35). The complete computer simulation is quite tedious and has so far been done for relatively few reactions thus the partial exploitation of the data in the spirit of preceding work is often all that is attempted in practice. 

Another, promising avenue to understand silk protein conformation and assembly is the use of model peptides. Although not recent (Fraser and MacRae, 1973 Lotz et al., 1974), studies of silk-based peptide from chemical synthesis, DNA recombinant technology, and computer simulation (Anderson et al., 1994 Asakura et al., 2003 Fahnestock et al., 2000 Fossey et al., 1991 Heslot, 1998 Kaplan, 1998 Wilson et al., 2000) have shown that selected repeats of silk proteins can be transformable hydrogels, elastomers, or regular thermoplastics and that with a proper design they can function as diverse molecular machines (Altman et al., 2003 Heslot, 1998 Kaplan, 1998 Urry, 1998). 

Fig. 7. The excitation profiles (n= — 2 to 2) by a periodic pulse of / Jt sin(jr//7) /Jt, where the solid lines are computer simulated results, solid circles are calculated with the effective RF fields, and open circles are obtained from the Fourier transformation of the RF shape. In the computer simulation, the pulse is composed of 4001 steps and has a pulsewidth r = 5 ms. Reprinted from Ref. 27 with permission from Elsevier.

ESE envelope modulation. In the context of the present paper the nuclear modulation effect in ESE is of particular interest110, mi. Rowan et al.1 1) have shown that the amplitude of the two- and three-pulse echoes1081 does not always decay smoothly as a function of the pulse time interval r. Instead, an oscillation in the envelope of the echo associated with the hf frequencies of nuclei near the unpaired electron is observed. In systems with a large number of interacting nuclei the analysis of this modulated envelope by computer simulation has proved to be difficult in the time domain. However, it has been shown by Mims1121 that the Fourier transform of the modulation data of a three-pulse echo into the frequency domain yields a spectrum similar to that of an ENDOR spectrum. Merks and de Beer1131 have demonstrated that the display in the frequency domain has many advantages over the parameter estimation procedure in the time domain. 

The other technique which has poved valuable in this area is computer simulation. When the kinetic data become very complicated, as with oscillating reactions involving two elementary steps, it is still possible to obtain rate constants from the data by doing computer simulation. That is actually not as outlandish as it might appear. It is really in the same category as the Fourier transform approach. I think this is an area that will make a considerable impact upon inorganic kinetic studies in the future. 

Let us instead turn our attention to the consequences of sampling the function at evenly spaced intervals of x. Consider the A function and its transform, a sine function squared, shown in Fig. 3. Suppose that we wish to compute that transform numerically. First, let us replicate the A by convolving it with a low-frequency III function. Now multiply it by a high-frequency III function to simulate sampling. We see a periodically replicated and sampled A. The value of each sample is represented as the scaled area under a Dirac <5 function. 

In the last few years, computer graphics with colour display are being more commonly used not only to visualize complex structures better, but also to examine unusual structural features, defects and transformations as well as reactions. In Fig. 1.45, we show the presence of a Nal" cluster within the sodalite cage of zeolite Y as depicted by computer graphics the cluster fits well within the cavity bounded by the van der Waals surface (net) of the framework atoms. The immense power of computer graphics has been exploited widely in recent years. Structural transitions in solids and sorbate dynamics in zeolites are typical areas where computer simulation and graphics have been used (Ramdas et al., 1984 Rao et al., 1992). 

When the concentration in the donor compartment remains constant during the experiment, the concentration in the receptor compartment increases regularly at first, then attains a plateau. For an ideal system with the thickness of a biological membrane a maximum increase in concentration of 130 times can be predicted by computer simulation. This system gives an experimental physicochemical example of a transformation of scalar chemical energy into a vectorial catalysis effect. ... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

The complexity of the physical properties of liquid water is largely determined by the presence of a three-dimensional hydrogen bond (HB) network [1]. The HB s undergo continuous transformations that occur on ultrafast timescales. The molecular vibrations are especially sensitive to the presence of the HB network. For example, the spectrum of the OH-stretch vibrational mode is substantially broadened and shifted towards lower frequencies if the OH-group is involved in the HB. Therefore, the microscopic structure and the dynamics of water are expected to manifest themselves in the IR vibrational spectrum, and, therefore, can be studied by methods of ultrafast infrared spectroscopy. It has been shown in a number of ultrafast spectroscopic experiments and computer simulations that dephasing dynamics of the OH-stretch vibrations of water molecules in the liquid phase occurs on sub-picosecond timescales [2-14],... 

A shape of such clusters or domains containing alternatively A or B particles transforms rapidly to the trapezoidal profile, in a complete agreement with both computer simulations and theory presented in [27]. Note also that in the asymmetric case (broken lines and r.h.s. scale in Fig. 6.15(b) immobile particles A form much more dense and compact clusters them mobile B s [26],... 

Unfortunately, the simple importance sampling as described above cannot be used to sample multidimensional integrals over configuration space as in Eq. (1.1). The reason is simply that we do not know how to construct the transformation in Eq. (1.7) that will enable us to generate points in configuration space with a probability density as given by the Boltzmann factor. In fact, in order to do so, we must be able to compute analytically the partition function of the system. If we could do that, there would hardly be any need for computer simulations ... 

The availability of powerful three-dimensional flow computer simulation packages and personal computers capable of handling them is gradually transforming profile die design from an empirical trial-and-error process to one where design optimization benefits from computational results. Sebastian and Rakos (83) were the first to utilize realistic computational fluid-mechanical results in the design of profile dies. 

The atomic density of the hex phase is about 20% higher than that of the 1 x 1 phase. As could be demonstrated by scanning tunneling microscopy (STM) (49), during the CO-induced hex — 1 x 1 transformation, these additional atoms are squeezed out from the surface layer, on top of which they are aggregating to new small I x 1 patches, a result which could also be successfully modeled by computer simulations (50). 

Correlator hardware and linearly-spaced sample time limitations are discussed. Data transformation from an intensity-weighted to a mass-weighted size distribution is demonstrated. The artificial width of broad distributions is discussed. The use of multiangle measurements is considered, and the effect of baseline error is shown using computer simulated data. 

In a given work computer simulations devoted to study of nanostructure of abovementioned cryogenic amorphous phases of ice, mechanisms of their transformations, and properties to accumulate methane and hydrogen was realized within the theoretical concepts thermo field dynamics [5] and quantum-field chemistry [6-9]. We developed two models of nanostructures corresponding to HDA-ice and LDA-ice, respectively. Some computations of energetic barriers locking molecules CH4 and H2 inside of amorphous ice were fulfilled. 

The extensive studies on the structure [72, 89] and Raman and Brillouin spectra [68-70, 73], as well as computer simulation results [77-82] have revealed that in the 8-50 GPa pressure range and at room temperatures, the silica glass is subject to a broad transformation accompanied by a change in the short-range order structure and an increase in the coordination number from 4 to 6. It should be noted that during coordination transformation at intermediate pressures, many silicon atoms have a fivefold coordination. The main part of the transformation takes place in a narrower pressure range of 10-40 GPa. 

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