Paired interactions

This choice was introduced independently by Topic [83] and Zwanzig [84]. We assume that the anisotropic pair interaction can be written as... 

Thurnauer M C and Norris J R 1980 An electron spin echo phase shift observed in photosynthetic algae. Possible evidence for dynamic radical pair interactions Chem. Phys. Lett. 76 557-61... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

Additionally, to optimize task 4, we applied a conventional, atom pair interaction based multiple-time-step scheme to the force computation within Ihe innermost distance class. Here, for atom pairs closer than 5 A, the Coulomb sum is calculated every step, and for all other atom pairs the Coulomb sum is extrapolated every second step from previously explicitly calculated forces. 

Except for the high molecular weight range, nearly all substances can be separated by reversed-phase (RP) HPLC. The many different separation mechanisms in RP HPLC, based on hydi ophobic, hydi ophilic and ion-pairing interactions, and size exclusion effects together with the availability of a lai ge number of high quality stationary phases, explain its great populai ity. At present approximately 90% of all HPLC separations are carried out by reversed-phase mode of HPLC, and an estimated 800 different stationai y phases for RP HPLC are manufactured worldwide. 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

FIGURE 12.39 The proposed secondary structure for E. coli 16S rRNA, based on comparative sequence analysis in which the folding pattern is assumed to be conserved across different species. The molecule can be subdivided into four domains—I, II, III, and IV—on the basis of contiguous stretches of the chain that are closed by long-range base-pairing interactions. I, the 5 -domain, includes nucleotides 27 through 556. II, the central domain, runs from nucleotide 564 to 912. Two domains comprise the 3 -end of the molecule. Ill, the major one, comprises nucleotides 923 to 1391. IV, the 3 -terminal domain, covers residues 1392 to 1541. 

One of the major ingredient for the understanding of alloy phase stability is the configurational energy. Models have been proposed to represent the configurational energies in terms of effective multisite interactions, in particular effective pair interactions (EPls). 

Figure 1 (a) The nearest neighbor pair interactions and (b) antiphase boundary energies as functions of energy for Pdj,Vi j, alloys x=0.25, x = 0.5 and x = 0.75 ( from top to bottom). Vertical lines mark the Fermi energy for the three different concentrations. 

Figure 2 (a) The pair interaction V as a function of distance PdsoRhso alloy, (b) Spinodal curve for Pdj.Rhi j alloy system. The points indicate calculated points while the solid line is the cubic spline fit through the points. 

EXPERIMENTAL STUDY OF THE SHORT RANGE ORDER IN THE PT-V SYSTEM EFFECTIVE PAIR INTERACTIONS AS A FUNCTION OF THE CONCENTRATION... 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

Those Warren-Cowley parameters have been determined in situ above the order-disorder transition temperature by diffuse neutron scattering. From these experimentally determined static correlations, the first nine effective pair interactions have been deduced using inverse Monte Carlo simulations. 

Figure 3. Compared effective pair interaction for the two sets Pt V and PtgU in ineV.

It turns out that for some systems the GPM yields the pair interactions, particularly those between first neighbors, which do not correspond to experimental phase diagrams. It is the purpose of the present work to show some of these cases and make a comparison with results obtained by other methods, particularly by the CWIS. 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

In order to check the consistency and mutual relations of ECIs calculated by various methods, as well as to compare them with experimental data, we have performed calculations for several alloy systems, as diverse as Cu-Nl, Al-Li, Al-Ni, Ni-Pt and Pt-Rh. Here we present the results for Al-Ni, Pt-Rh and Ni-Pt alloys in some detail, because the pair interactions between the first neighbors are dominant in these alloys which makes the interpretation relatively simple. On the other hand, the pair interactions between more distant neighbors and also triplet interactions are important for Al-Li and Cu-Ni. The equilibrium atomic radii, bulk moduli and electronegativities of A1 and Ni are rather different, while Pt and Rh are quite similar in this respect. The Ni and Pt atoms differ mainly by their size. 

Fig. 1. The mixing energies agree rather well with the results of Pasturel et al. and those of Abrikosov et al. . The pair interaction i> (c) is In a very good agreement with the locally relaxed quantity given by Carlsson, and also it is rather similar to the effective pair interaction found by Pasturel et al., although it is smaller approximately by a factor 0.7. The pair interaction u (c) differs from the corresponding quantity reported by Abrikosov et al. , which can be probably attributed to different computational schemes, as we assume neutral spheres, while Abrikosov et al. suppose equal sphere radii. The pair interaction (c) closely follows u (c) for Ni-rich alloys and v (c) for Al-rich alloys, at intermediate concentrations all three quantities...

Figure 1. The effective pair interactions as functions of alloy composition for the alloy system Al-Ni. The results of the CWIS based on the FP-LAPW calculations for 5 ordered structures (full line) and on the TB-LMTO-CPA for 5 disordered alloys (dash line) are compared with the the results of the GPM (dotted line).

We have found that for some alloys (e.g. Pt-Rh and Ni-Pt), the GPM yields pair interactions which are incorrect, because their values are either too large and would lead to overestimated transition temperatures (Ni-Pt), or they have even opposite sign than that expected from the experimental phase diagram and predicted by other theoretical methods (Pt-Rh). Various explanations of these discrepancies are conceivable ... 

Figure 1 Disorder-Llo-Ll2 phase diagram [9]. The broken line indicates the (100) Spin-odal ordering locus and dotted lines e metastable phase boundaries. The temperature axis is normalized with respect to the neaiest neighbor pair interaction energy. The ordering transition temperatures of L q and LI2 phases are 1.89 and 1.92, respectively.

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