Kirkwood formula

The shear stress Is uniform throughout the main liquid slab for Couette flow ( ). Therefore, two Independent methods for the calculation of the shear stress are available It can be calculated either from the y component of the force exerted by the particles of the liquid slab upon each reservoir or from the volume average of the shear stress developed Inside the liquid slab from the Irving-Kirkwood formula (JA). For reasons explained In Reference (5) the simpler version of this formula can be used In both our systems although this version does not apply In general to structured systems. The Irvlng-Klrkwood expression for the xy component of the stress tensor used In our simulation Is... 

The introduction of branching in the Kirkwood formula and the KR calculations can be accomplished in a relatively easy way if Gaussian statistics corresponding to ideal chains are maintained. This description cannot, however, be very accurate in molecules with centers of high functionality because of the presence of cores with a high density of polymer units, which profoundly perturbs the internal distribution of distances. Stockmayer and Fixman [81 ] employed the Kirwood formula and Gaussian statistics to calculate h in the case of uniform stars, obtaining an analytical formula. They also performed a KR evaluation of the viscosity and proposed that g could be evaluated from the approximation... 

Ganazzoli et al. [53] performed calculations for the hydrodynamic radius (based on the Kirkwood formula), and also for the intrinsic viscosity [87] of uniform stars, using a generahzed version of the ZK method that incorporates non-Gaussian intramolecular distances. These distances were obtained according to... 

The Kirkwood Formula. A particular case of the JE, Eq. (40), is the Kirkwood formula [47, 48]. It corresponds to the case where the control parameter only takes two values Xq and Xi. The system is initially in equilibrium at the value Xq and, at an arbitrary later time t, the value of X instantaneously switches to Xj. In this case Eq. (37) reads... 

Substituting the expression of the integrand g(r, X) obtained from Eq. (81) into the Kirkwood formula yields... 

This is summed over all probe (p)-target (t) atom pairs and is a function of the distance, rj, between the atoms in the pair, which have van der Waals radii, Rj, and R,. A = 0.5C(Rj, + R,) and C is given by the Slater-Kirkwood formula [18] and is dependent on atomic polarizability and the number of effective electrons per atom. 

The stress tensor a for an electrorheological fluid can be obtained from the Kirkwood formula [see Eq. (1-42)],... 

The response of the interface to an external perturbation is analyzed using the interfacial stress tensor S. For a pairwise-additive interaction, S is given by the Kirkwood formula (Doi and Edwards, 1986 Wijmans and Dickinson, 1999a)... 

There are two basic problems associated with the MC or MD calculation of the dielectric constant. First, the relationship between e and the mean square moment obtained in the computer calculation will depend on exactly how the dipolar interactions are handled. For example, the Kirkwood formula (3.7a) only holds if A" (/ ) is that of an infinite sample, and hence does not apply in most computer situations. To find the correct relationship for a given simulation method is not a trivial problem, but for several commonly applied procedures the appropriate formulas are now known. The second, and perhaps more fundamental question, concerns whether the dielectric constant given by a particular simulation is really the true infinite-system value, or whether it is it seriously influenced by the approximate methods used in the calculation. In the absence of exact results, this question is obviously difficult to answer fully, but a detailed and, we hope, useful examination of the problem appears in Section III.D.2. 

The Stokes radius Rn of a spring-bead chain can be calculated by the Kirkwood formula (see Ref. [2] of Chapter 2 for its details) if is... 

Equation (4.199) indicates that the Kirkwood formula (4.198) cor-respon[Pg.120]

Given the upper and lower bounds, one can estimate the error of Ae preaveraging approximation. Fixman showed that the Kirkwood formula gives quite accurate estimation for Dq for flexible polymers in 0 condition, the error is of the order of a few per cent. 

This formula can also be directly derived from the Kirkwood formula for the diffusion constant eqn (4.102). 

The first results concern the comparison between the Kirkwood formula for the diffusion constant and the diffusion constant obtained from pre-averaged BD simulations. Within the error bars, the numbers are identical. However, for fluctuating hydrodynamics a diffusion constant systematically above the Kirkwood value was found, at variance with the variational bound. Since the data seem to be rather accurate, this might be an indication that something is wrong with the rigorous bound. For a discussion, see Ref. 38. These questions must be regarded as completely unresolved today. 

The polymer center-of-mass diffusion coefficient follows either via the GK relation from the velocity autocorrelation function or by the Einstein relation from the center-of-mass mean square displacement. According to the Kirkwood formula [104,105, 111]... 

Simulation results for the hydrodynamic contribution, Du = D-Do/Am, to the diffusion coefficient are plotted in Fig. 10 as a function of the hydrodynamic radius (85). In the limit Am > 1, the diffusion coefficient D is dominated by the hydro-dynamic contribution Du, since Dh Am. For shorter chains, Do/Am cannot be neglected, and therefore has to be subtracted in order to extract the scaling behavior of Dh. The hydrodynamic part of the diffusion coefficient Z>h exhibits the dependence predicted by the Kirkwood formula and the Zimm theory, i.e., Dh l/f H-The finite-size corrections to D show a dependence D = D , — const./L on the size... 

The Kirkwood formula neglects hydrodynamic flucmations and is thus identical with the preaveraging result of the Zimm approach. When only the hydrodynamic part is considered, the Zimm model yields the diffusion coefficient... 

Yamakawa and YoshizaM presented an interpolation expression of L/Rh for HW cylinders including the KP limit (not shown here). A theoretical equation of L/Rh for touched-bead HW chains is also available. It is equivalent to the Kirkwood formula for Do and for its actual use the summation over the mrmber of beads has to be taken numerically. 

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