# Electrokinetic Properties of 3D-Printed Conductive Lattice Structures
## 3 Continuous Medium Approximation
\(\eta(\vec{d}_{m})\) is a correction function that vanishes like \(\alpha|\vec{d}_{m}|\) at the limit \(d_{m}=|\vec{r}_{m}-\vec{r}_{0}|\to 0\) so as to balance the Coulomb singularity. Comparing Equation (5) for \(m=0\) and Equation (6) for \(\vec{r}_{m}\rightarrow\vec{r}_{0}\) allows one to set:  
\[\alpha=\frac{4\pi G_{0}R}{\rho}\quad. \tag{7}\]  
Combining Equations (4)-(7), the two-point resistance can be recast as:  
\[R_{0m}^{\text{eq}}=\frac{2V(\vec{r}_{0})-2V(\vec{r}_{m})}{I}=2G_{0}R\left(1- \frac{\eta(\vec{d}_{m})}{\alpha d_{m}}\right)\quad. \tag{8}\]  
The open circle symbols in Figure 3 represent the equivalent resistances between pairs of nodes of the simple cubic and body-centered cubic resistor lattices up to a distance of nine-times the cubic lattice parameter \(a\). Although the pairs of nodes are oriented along various crystallographic directions, the equivalent-resistance data chiefly fall around a single monotonous curve of the separation distance \(d_{m}\). For these cubic lattices, one may then ignore the existence of tiny anisotropic variations and consider that the correction function \(\eta(\vec{d}_{m})\) is a function of the distance \(d_{m}\). Setting from now on:  
\[\eta(\vec{d}_{m})=h(\alpha d_{m}) \tag{9}\]  
yields:  
\[R_{0m}^{\text{eq}}=R_{0\infty}^{\text{eq}}\left(1-\frac{h(\alpha d_{m})}{ \alpha d_{m}}\right) \tag{10}\]  
Figure 2: Half the conventional unit cell (thin black lines) of a face-centered cubic resistor lattice. The resistive rods represented in gray interconnect nodes located within the same horizontal plane. The blue rods are located inside the unit cell. The red rods, located in the vertical faces, are shared by two adjacent cubic cells. At the bond length distance \(d_{1}\), the equivalent resistance is \(R_{01}^{\rm eq}=2R/Z\)[48], with \(Z\) the lattice coordination as above. Accordingly, Equation (10) tells one that:  
\[h(\alpha d_{1})=\alpha d_{1}\left(1-\frac{2R}{ZR_{0\infty}^{\rm eq}}\right)= \alpha d_{1}\left(1-\frac{1}{ZG_{0}}\right)\quad. \tag{11}\]  
The values of \(\alpha d_{1}\) and \(h(\alpha d_{1})\) for cubic lattices are listed in Table 1.  
Having a continuous-medium approximation in mind, one considers \(d_{m}\) as a continuous parameter. At a large distance, the discrete character of the resistor lattice should play no role, with the consequence \(\lim_{x\to\infty}h(x)=1\). At the other end, it has already been established through the definition of \(\alpha\) (Equation (7)) that \(h(x)\sim x\) for \(x\to 0\). At the bond length distance, Table 1 indicates that \(h(\alpha d_{1})>1\). Summing up the information gained thus far, the function \(h(x)\) increases for small \(x\), reaches a maximum above unity, and eventually approaches one asymptotically. The shape of the function \(h(x)\) can be sketched; see the dashed-line curve plotted in \[V_{m}=r_{m0}(0+)I \tag{5}\]0 For reasons that will be made  
\[V_{m}=r_{m0}(0+)I \tag{5}\]1 Plot of the equivalent resistance \(R_{0m}^{\rm eq}\) between Sites 0 and \(m\) in a perfect resistor lattice versus separation distance \(d_{m}\). (a) and (b) panels correspond to the simple cubic (sc) lattice and the body-centered cubic (bcc) lattice, respectively. The open symbols are exact results from the literature. The red full-line curve is from Equation (10). The right-hand scale along the vertical axis of the (a) panel is for the blue dashed-line curve representing the simple expression (Equation (12)) of the correction function \(\eta(d_{m})\). clear below, one will further impose \(\int_{0}^{\infty}[1-h(x)]\,\mathrm{d}x=0\). A simple function that reproduces all these characteristics is:  
\[V_{m}=r_{m0}(0+)I \tag{5}\]2  
The full-line curves in Figure 3 illustrate how well this function works for the simple cubic and body-centered cubic resistor lattices.