# Electrokinetic Properties of 3D-Printed Conductive Lattice Structures
## 3 Continuous Medium Approximation  
The aim of this section is to transform the electrokinetic equations of a continuous medium so as to reproduce the main characteristics of an infinite discrete 3D lattice of resistors with constant resistance \(R\). It is implicitly assumed that all the nodes of the lattice are equivalent (mono-atomic-like crystal lattice). The first objective is to assign an apparent bulk conductivity to the resistor lattice. For a simple cubic lattice in \(d\)-dimensional-space and parameter \(a\), the apparent resistivity is \(\rho=Ra^{d-2}\)[44]. Figure 2 sketches how an apparent resistivity can be obtained for a resistive mesh with fcc lattice. Half a conventional cubic unit cell is shown. The nodes located in the basal plane are grounded to potential zero. The nodes located in the upper horizontal plane are at potential \(V\). The current flows across the non-horizontal rods. All have the same resistance \(R\) and are assembled in parallel. Each of the height red rods, being shared by two adjacent cells, must be viewed as two parallel resistors of resistances \(2R\), one per each face-sharing unit cell. Accordingly, the total resistance experienced by the current flowing through half the unit cell shown in Figure 2 is \((4/R+8/2R)^{-1}=R/8\). The same half unit cell, viewed as a continuous medium of resistivity \(\rho\), height \(a/2\), and basis area \(a^{2}\) would have a resistance \(\rho/2a\), where \(a\) is the fcc lattice parameter. Consequently, \(\rho=Ra/4\) for the fcc mesh. The apparent resistivity obtained the same way for other lattices is given in Table 1.  
If the current \(I\) is injected on a given node, referred to as \"Node 0\", Equation (3) gives the potential \(V_{m}=r_{m0}(y)I\) on any node \(m\) compared to the zero potential of the ground electrode each node is connected through the fictitious admittance \(y\). Accordingly, for any positive value of \(y\), the current \begin{table}
\begin{tabular}{c c c c c} \hline
**Lattice** & **Diamond** & **Simple Cubic** & **bcc** & **fcc \& hcp** \\ \hline \(Z\) & 4 & 6 & 8 & 12 \\ \(\rho\) & \(Ra\) & \(Ra\) & \(Ra/2\) & \(Ra/4\) \\ \(G_{0}\) & 0.448220394 & 0.252731010 & 0.17415049 & 0.11205510 \\ \(\alpha\) & 5.632 \(a^{-1}\) & 3.176 \(a^{-1}\) & 4.377 \(a^{-1}\) & 5.632 \(a^{-1}\) \\ \(\alpha d_{1}\) & 2.439 & 3.176 & 3.790 & 3.983 \\ \(h(\alpha d_{1})\) & 1.078 & 1.082 & 1.070 & 1.021 \\ \hline \end{tabular}
\end{table} 
Table 1: Coordination \(Z\), apparent resistivity \(\rho\), value \(G_{0}\) of the diagonal elements of the topological lattice Green’s function (values from [40]), scaling factor \(\alpha\) (Equation (7)), \(\alpha\) times the bond length \(d_{1}\), and the correction function at \(d_{1}\) (Equation (11) for a few 3D infinite lattices. \(R\) is the bond resistance, and \(a\) is the lattice parameter of the conventional cubic cell. cannot spread on the network to infinity, and \(V_{m}\) vanishes when the distance between Nodes \(m\) and \(0\) increases to infinity. This property remains at the limit \(y\to 0+\) with the consequence:  
\[V_{m}=r_{m0}(0+)I \tag{5}\]  
Setting \(m=0\), one obtains the potential of the contact node \(V_{0}=r_{00}(0+)I=G_{0}RI\), which is finite for a 3D lattice as mentioned above (see Table 1). In the same condition, the potential at location \(\vec{r}\) produced in a continuous medium by a point source injecting the current \(I\) at position \(\vec{r}_{0}\) is \(V(\vec{r})=\frac{\vec{p}I}{4\pi}\frac{1}{\vec{r}-\vec{r}_{0}|}\). Similar to the case of the discrete lattice, this expression for the potential vanishes at infinity. Unlike the case of the discrete lattice, the potential at the contact point is infinite. Therefore, the Coulomb-like law has to be corrected for short distances, by writing:  
\[V(\vec{r}_{m})=\frac{\rho I}{4\pi}\frac{\eta(\vec{r}_{m}-\vec{r}_{0})}{|\vec{ r}_{m}-\vec{r}_{0}|} \tag{6}\]  
with \(\vec{r}_{m}\) and \(\vec{r}_{0}\) the positions of Sites \(m\) and \(0\) on the lattice.