# Electrokinetic Properties of 3D-Printed Conductive Lattice Structures
## 2 Background  
Figure 1 shows a model of a mesh having the diamond structure designed for 3D printing technology. The mesh can be viewed as a periodic duplication of the structural unit detailed in Panel (b). The mesh is composed of conducting rods interconnected at nodes. Each rod has a prismatic shape (hexagonal prism in Figure 1). A node is defined as the volume where crossing rods meet. In Figure 1, the nodes are truncated tetrahedra. The mesh can be treated as a discrete network of resistors under the assumption that each node is an equipotential body. The latter assumption is valid either when the lateral dimensions of the rods are much smaller than their length or if each node is made of a perfectly-conducting medium. Then, a single potential can be attributed to each node. Under this condition, the resistance of a rod is \(R=\sigma_{0}^{-1}/\Omega\) with \(l\) and \(\Omega\) being the length and cross-sectional area (see Figure 1b), respectively, and \(\sigma_{0}\) being the conductivity of the polymer nanocomposite used for the fabrication. The mesh can therefore be considered as a network of resistors [38]. The nodes of the mesh receive discrete labels \(i=1\cdots N\). If \(l_{i}\) is the current injected at node \(i\) from an external source, combining Kirchhoff and Ohm's laws yields:  
\[I_{i}+\sum_{j}L_{ij}V_{j}=0 \tag{1}\]  
where \(L_{ij}=1/R_{ij}\) if \(i\) and \(j\) are connected by a rod of resistance \(R_{ij}\), \(L_{ij}=0\) otherwise, and \(L_{ii}=-\sum_{j\
eq i}L_{ij}\). Obviously, the admittance matrix \(\mathbf{L}\), also called the Laplacian matrix [39], is singular. This prevents one to solve Equation (1) directly for \(V_{j}\). The difficulty can be tackled by assuming that each node is connected to an external ground electrode, whose potential is set to zero, by a constant admittance \(y\). The limit \(y\to 0+\) will be taken at the end of the calculations. With this trick in hand, Equation (1) is replaced by:  
\[\sum_{j}(y\delta_{ij}-L_{ij})V_{j}=I_{i}\quad. \tag{2}\]  
Now, the set of equations (2) for \(i=1\cdots N\) can be solved for the node potentials \(V_{i}\) in the form:  
\[V_{i}=\sum_{j}r_{ij}(y)I_{j} \tag{3}\]  
with \(r_{ij}(y)\) being the elements of the resolvent matrix \(\mathbf{r}(y)=(y-\mathbf{L})^{-1}\). The resolvent matrix is symmetric. It is perfectly defined whenever \(y\) is outside the set of eigenvalues of \(L\), in particular for any real positive value of \(y\). When the mesh is a regular infinite lattice where all nodes are equivalent and all rods have the same resistance \(R\), the resolvent matrix is related to the topological lattice Green's function [40]\(\mathbf{G}(x)=(x-\mathbf{C})^{-1}\) where \(\mathbf{C}\) is the adjacency matrix, defined as \(C_{ij}=1\) when the nodes \(i\) and \(j\) are directly connected and zero otherwise including when \(i=j\). The relation between the matrices is \(\mathbf{r}(y)=R\mathbf{G}(Z+yR)\) where \(Z\) is the lattice coordination number. With the help of [PERSON]'s theorem, a spectral representation of \(\mathbf{G}(x)\) for 2D and 3D lattices is available in the form of 2D and 3D integrals in Fourier space, to which belong the so-called Watson integrals [41]. Due to the translational symmetry of the lattice, the diagonal elements \(G_{ii}(x)\) all take on the same value. For 2D lattices, the diagonal elements of \(\mathbf{G}\) diverge like \(-ln(x-Z)\) when \(x\to Z+\), which corresponds to the limit \(y\to 0+\) of relevance here. For 3D lattices, the limit of \(G_{ii}(x)\) for \(x\to Z+\) is a finite number \(G_{0}\) listed in Table 1. There is a simple \(1/4\) factor between the \(G_{0}\) values of the fcc and diamond lattices [42]. The equivalent resistance of the mesh seen from two nodes \(m\) and \(n\), also called two-point resistance [43], is defined as the ratio \(R_{mn}^{\text{eq}}=(V_{m}-V_{n})/I\) when the current \(I\) is injected at node \(m\) and removed from node \(n\).