Midplane approach

The Hele-Shaw equation for the determination of pressure has been derived for a two-dimensional geometry. To solve the pressure problem for a thin eavity of general planar geometry in three-dimensional space, we use a finite-element (FE) representation on the midplane of the cavity (Fig. 8.1). Each element is assigned a thickness. The Hele-Shaw equation is diseretized on eaeh element using the local coordinate system associated with that element. The unknown node pressure and the volumetric flow rate are all scalar quantities and they are not linked to the coordinate system. In addition, we use a finite dififerenee (FD) method to discretize the time- and gap-wise coordinates to solve the energy equation for the temperature field. In the following derivation of the FE/FD equations, only the cavity planar flow is considered. Derivation of the axisymmetrie form of the equations for the runner flow can be done in the same manner. This approach deals with a 2-D pressure field, eoupled to a 3-D temperature field, and therefore it is called a 2.5D simulation. 

Let us assume a triangular element with nodes 1, 2 and 3 is located in the global coordinate system Xi, X2 and X3, as schematically shown in Fig. 8.2. We define two vectors Z3 and I2 as 

In Fig. 8.2, xi, X2 and X3 represent the local coordinates, where node 1 is the origin. The local xi-axis is along the direction of vector i.e., the direction of node 1 pointing to node 2. The X2-axis lies in the plane defined by nodes 1, 2 and 3, and points toward node 3 perpendicular to the xi-axis. The 3-axis is defined by h X which is normal to the element plane. The transformation from the global coordinates to the local element coordinates is as follows  

To describe the finite element formulation of the discretized Hele-Shaw equation, we write the Hele-Shaw equation in the following form 

We now discretize the domain Q into a set of non-overlapping finite elements, for example, a collection of triangles denoted by 

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Note that under these conditions, the maximum value (C ) of c, which occurs at the midplane, approaches the bulk value exponentially... 

Table 3 Mean-force calculations optimal Ar and corresponding a(-Fhs) for fh surface approach and optimal Ah and corresponding cr(-Fideai) for the midplane approach in units of Rm and kT/Ryi at two different macroion separations. See text for further details...

Fig. 8 a Mean force as a function of the macroion separation for System II evaluated using the surface approach (filled symbols with error bars) and the midplane approach (solid curve with nearly invisible error bars) and b uncertainty of the hard-sphere and electrostatic contributions of the surface approach (filled circles) and of the ideal and electrostatic contributions of the midplane approach (open circles). Rcy = 4J M,fcyi = 12Fm, and Npass = 10 . The uncertainty estimates are based on a division of the simulations into 20 blocks... 

Figure 8 shows the mean force and the uncertainties of the force components for System II as a function of the macroion separation. First, we conclude that the two approaches predict the same mean force within the statistical uncertainties. Second, the mean force calculated using the surface approach (Fig. 8a, filled circles) displays considerable scattering at all separations, whereas the corresponding data using the midplane approach (Fig. 8a, solid curve) are smooth (uncertainty bars are smaller than the thickness of the line). Regarding the surface approach, the uncertainty in F arises entirely from the uncertainty in the determination of Fhs (cf. curves with filled circles in Fig. 8b). In the midplane approach, the uncertainty in the determination of f ideal dominates the uncertainty in F (cf. curves with open circles in Fig. 8b). A similar difference in the accuracy of the two approaches for highly coupled systems has been reported [28].

Table 4 Mean-force calculations uncertainties in Feiec and Fhs for the surface approach and in Feiec and Fideai for the midplane approach in kT/Ru units ...

A) Ample numerical evidences with theoretical understanding show that the mean-force determination using the midplane approach is more efficient than the surface approach as the accumulation of the small ions near the macroions becomes large. A direct determination of the pmf by a macroion-separation sampling provides results with similar or slightly higher efficiency than the midplane mean-force approach. 

Hence, all three approaches have their own merits. If (i) the mean force is the primary target, (ii) the interaction is needed for only a limited interval, or (iii) a separation into different force components is desired, a mean-force sampling using the midplane approach is the approach of choice. However, if the electrostatic coupling is weak and, in particular, for systems with more complex composition, the surface approach could be an alternative. The approach involving macroion-separation sampling has its strength when full interaction curves are needed, in particular if the macroion displacements can be made efficiently. 

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