Taper functions

Computing the interatomic forces is the most time-consuming part in an MD simulation. The use of a cutoff radius is a standard trick of the trade that reduces computational cost by neglecting interactions between atoms separated by a distance larger than the specified cutoff. As described earlier, this truncation results in a discontinuity of both the potential and the force at the cutoff distance, but the drawback thus entailed can be avoided by implementation of either the shifted-force potential or a taper function. 

In many force fields, truncation schemes are often used to reduce the number of non-bonded electrostatic and Lennard-Jones interactions that need to be calculated. Such schemes, are readily incorporated into the.inter-action Hamiltonian either by omitting all interactions that have a distance greater than some cutoff or by multiplying the appropriate interactions by a tapering function that reduces the interactions to zero beyond a certain distance. It is to be noted that in some hybrid force fields (see, for example, [35]) the electrostatic interaction terms are not included and the QM/MM interaction is due solely to the Lennard-Jones terms (and link-atoms if they are present). This could be a reasonable approximation in non-polar systems (such as the transition metal complexes for which some of these force fields were developed) but it will not be sufficiently accurate in the general case. 

Equation (15) can be numerically integrated to generate the taper function l iy). Once a choker bar is constructed following the design equation l iy), this bar can be inserted into the die to correct the flow nonuniformities caused by manifold enlargement. 

An extrusion die is usually designed for a particular polymeric liquid. Liu et al. [1] found that the flow uniformity is quite sensitive to the rheological properties of the delivered polymeric liquid. If the rheological properties vary, the flow uniformity will deteriorate rapidly. Because the die is very expensive, it is desirable to use a die for solutions with different rheological properties. If a die is designed based on a polymeric liquid with the power-law index m, then the manifold taper function h is... 

The taper function l iy) can be obtained by numerically integrating Eq. (17). A choker bar with the taper function l iy) can be built to correct flow nonuniformities because of viscosity variations. 

Therefore, if we numerically integrate (21) to generate the taper function /3(y) and then construct a choker bar using this /3(y) as the taper function, the flow distribution can be uniform again even if the two ends of the die are blocked. 

We shall illustrate the variations of the choker bar taper function l y) by some sample calculations. The following geometric parameters of a linearly tapered coat-hanger die are selected for demonstration ... 

Liu et al. [1] found that an enlarged manifold can deteriorate the lateral flow uniformity, and higher flow rates would appear at the ends of the die. The inertial effect was neglected here and Re = 0. yo was selected to be 0.05 and /3o = 0.5cm. The effect of the power-law index n on the taper function l y) is displayed in Fig. 4. Note that here y=0 refers to the die end and... 

Liu et al. [ 1 ] found that the flow uniformity is very sensitive to the variation of the power-law index in a linearly tapered coat-hanger die. If a die was originally designed based on n=0.5 and suddenly n was changed to 0.6, a W-type flow distribution would appear. On the other hand, if n dropped to 0.4, an M-type flow pattern would appear instead. The design equation (17) can be applied to offset the ffow uniformities caused by the variation of w. We fixed =0.5 and made proper selections of ho and h. Then n was varied to n = 0.6 and 0.4. l iy) was computed based on Eq. (17) and then a choker bar with the taper function l y) could be constructed to correct ffow nonuniformities. The effect of n on l y) is displayed in Fig. 7. For the case /I=0.6 >0.5, l y) has to decrease from the die end to the die center to eliminate the ffow nonuniformities if n=0.4 <0.5, then l iy) will increase from the die end to the die center instead. If is longer, the variation of hiy) will be smaller. 

In this chapter, we have described a novel design approach to correct flow nonuniformities caused by four types of production variations in a linearly tapered coat-hanger die. The theoretical approach is based on the onedimensional lubrication approximation and can be used to predict the taper function of an adjustable choker bar. Once a choker bar is constructed based on the prediction of the mathematical model, the flow nonuniformities can be properly eliminated by inserting this bar into the die. A choker bar can be tapered in different ways as indicated in Fig. 3. The shape displayed in Fig. 3a may be easier to machine and was selected for illustration. The four production variations we have considered include (i) enlarging the manifold, (ii) including the fluid inertial terms, (iii) varying the viscosity of the polymeric liquids, and (iv) narrowing the liquid film width to meet production requirements. All these four production variations can be properly handled, but if the fluid inertia becomes dominant, or the Reynolds number is not small, the present method may not be applicable. 

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

These expressions are valid provided that the cross-section for heat flow remains constant. When it is not constant, as with a radial or tapered fin, for example, the temperature distribution is in the form of a Bessel function i26). 

The phase diagram of the tapered copolymer melt computed within the two shell approximation [42] is shown in Fig. 14. The distribution function of A monomer along the tapered copolymer chains is... 

Likelihood and severity of withdrawal are a function of dose and duration of exposure. Gradual tapering of dosage is necessary to minimize withdrawal and rebound anxiety. 

In an adiabatically tapered axially symmetric MNF, the propagation constant of the fundamental mode, p(z), is a slow function of the coordinate z along the axis of the MNF and (13.4) is modified as follows ... 

Fig. 13.8 Transmission loss as a function of the MNF diameter calculated with (13.10) for different characteristic lengths L of the MF taper (curves 1, 2, and 3) and a<, 0.5 pm. Reprinted...

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