Smearing factor

Let us suppose that the atomic positional parameters r, and the thermal smearing factors 71 have been obtained by neutron diffraction. We can then calculate the X-ray scattering of the procrystal density with spherical-atom X-ray scattering factors, using... 

Two atomistic approaches have been presented briefly above molecular dynamics and the transition-state approach. They are still not ideal tools for the prediction of diffusion constants because (i) in order to obtain a reliable chain packing with a MD simulation one still needs the experimental density of the polymer and (ii) though TSA does not require classical dynamics it involves a number of simplifying assumptions, i.e. duration of jump mechanism, elastic polymer matrix, size of smearing factor, that impair to a certain degree the ab initio character of the method. However MD and TSA are valuable achievements, they are complementary in several... 

Another useful tool for determining fault seal characteristics is the shale smear map. Although the fault seal probability calculation, derived from an empirical database which contains faults which were affected by shale smear, already incorporates a shale smear factor (see also Lindsay et al., 1993), the purpose of producing shale smear maps is to define the shale smear envelope for individual shale beds across fault surfaces. They can also be used as an independent check on the fault seal probability calculations. Only one shale smear map was produced in the study, for the Revfallet Fault Complex, where the greatest thickness of syn- to post-rift shale occurs. 

The isotropic elastic motion of the polymer matrix is specified by the spectrum of the . In the homogeiKous apjHoximation we n ect the differences between the amplitudes of elastic motkm of different polymer atoms and describe the elastic motion of all polymer atoms with an identical smearing factor . One would expect to assume a value similar to the... 

Debye-Waller factors [53] obtained from tiK scattering of X-rays or neutrons with dense polymers, ie, about 0.2-0.4 A at ambient conditions. Of course, the local elasticity of the polymer matrix may deviate somehow from the mean elasticity introduced by the isotroj homogeneous smearing factor and a refinement mi t be advisable to account for the spectrum of . 

The smearing factor . To start the TSA simulation one needs to specify the smearing factor . We can surmise that, at ambient conditions, tte value rf A > should be some fractions of an A and should not be very sensith to the chemical details of the polymer matrix. On the other hand, pven tte abo approximations involved in describing the elastic motion erf polyn r atoms, a universal method for evaluating can hardly be expected tte re icher should rather rely on intuition and experience in order to specify a suitable value of for the problem at hand. Two possibilities are reviewed in the following. 

Another possible way to evaluate the smearing factor is to match the short-time region of the vs t curves obtained from TSA with those from MD calculations. As we shall see below, the broad domain of the anomalous diffusion [48, 50, 56], where oc t"(n < 1), pertains to the dynamics of small solutes in dense polymers. The value of n is sensitive to the smearing factor employed, thus making possible to evaluate from maiq>ing the results of MD and TSA in the mutually accessible time-domain (Le., 10 -10 s). 

Tables 2 and 3 show that in all cases computed diffusion coefficients D and Henry s constant S agree with experimental data [71, 73] to within an order of magnitude. The computed Henry s constants S do not scatter much among the individual micro-structures, whereas there is a considerable scattering of the diffusion coefficients D for Ar and larger molecules. The computed Henry s constants S are not sensitive to the smearing factor employed The values...

The smearing factor As discussed above, some correlations could exist between the spectrum of the residence times Xi and the time-intonral At on which the smearing fru tor is to be evaluated from MD trajectortes of the polymer matrix (in tiK absoice cS solute molecules). It has been fouml that in some cases the value of d uc from the maximum of the distribution Xg indeed gave [SO] diffusion co ts that favourably com red with experimental data. Nevertheless, in sc ne situations this empirical ruk faited to reproduce the solute s diffusion co Sdents even to within an order-of-magnitude. 

It is important that MD and TSA are used in conjunction. As an example we mention the MD analysis of short-time rotational motion of small penetrants in dense polymers that provides the basis for the pre-averaging of these degrees of freedom in the TSA. Similarly, the duration cf the hopping events, as estimated by MD (- 2 ps), is an excellent guiddine for selecting tte smearing factor of TSA. 

The thermal fluctuations of the polymer matrix are taken into account through the smearing factor A , which is related to the MSD of the matrix segments from their equilibrium positions. 

The TST method has the advantage of extending the timescale of the observation when compared to classical dynamics however, it involves a number of assumptions. First, the polymer matrix response to the guest molecule should be elastic. This is due to the rather simplistic form of calculating the smearing factor, which limits the application of the method to the behavior of small molecules, the presence of which does not affect the polymer environment. Second, the shape of the penetrant is supposed to be isotropic. As the penetrant size increases and its shape becomes anisotropic, conventional TST fails to capture the corresponding transport behavior. 

In this work, the grid size used was set to 0.3 A, in agreement with typical values found in the literature for TST calculations. - - The smearing factor, A, was calculated for each penetrant through a self-consistent scheme involving information about the MSD of all the polymer atoms from their respective equilibrium positions. The MSD was calculated from a 50 ps MD simulation at constant number of particles, volume, and temperature (NVT). The self-consistent schane converged when the relative difference between two successive values was within 2.5%. The total duration of the kMC procedure was 10 s and the MSD was averaged over the trajectories of 1000 penetrant walkers. 

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