# Modelling of Anisotropic Electrical Conduction in Layered Structures 3D-Printed with Fused Deposition Modelling

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## Abstract

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## 1. Introduction

## 2. Theoretical Model

- Prescribed Voltage: a fixed voltage at a boundary because of the presence of a terminal or ground.$${U}_{n}(0,t)={U}_{\mathrm{prescribed}}$$
- Meandering End: a connection to another traxel (normally a neighbouring traxel), causing an equal voltage and an equal but opposite current.$${U}_{n}(L,t)={U}_{n-1}(L,t)$$$${I}_{n}(L,t)=-{I}_{n-1}(L,t)$$
- Open End: a floating electrical voltage and no current flowing, since there is no end connection with other traxels or sources. This could represent stacked layers or traxels without meandering ends.$${I}_{n}(0,t)\propto \frac{\partial {U}_{n}(0,t)}{\partial x}=0$$

#### 2.1. Model Derivation

#### 2.1.1. Bulk Properties

#### 2.1.2. Contact Properties

#### 2.1.3. Combined Properties

#### 2.1.4. Solving the Equations

#### 2.2. Boundary Conditions

#### 2.2.1. Explicit Boundary Conditions

#### 2.2.2. Implicit Boundary Conditions

#### 2.2.3. Applying Boundary Conditions

#### 2.3. Dimensionless Parameters and Limit Cases

#### 2.3.1. Anisotropy Ratio

#### 2.3.2. Number of Traxels

#### 2.3.3. Aspect Ratio

#### 2.3.4. Open-Ended Resistance Approximation

#### 2.3.5. Meandering Resistance Approximation

## 3. Methods

#### 3.1. Model Implementation

#### 3.2. FEM Simulations

#### 3.3. Model Analysis

## 4. Results

#### 4.1. Model Verification

- A comparison of voltage and x-current density for both an open-ended and meandering sample with three traxels. This example is used to clearly show the basics and compare the results. The $\sigma =2\times {10}^{-2}\Omega {\mathrm{m}}^{2}$ used yields an anisotropy ratio of $0.1007$, which is well suited for anisotropic conduction.
- A comparison of the total resistance for a sample with the parameters from Table 2 with, in one case, a varying aspect ratio and, in the other case, a varying anisotropy ratio. This can be used to study the model in the range for which the FEM simulations are experimentally validated, and to give an indication of the shortcomings of the model. Furthermore, a comparison is made to the approximated resistance expressions for both the open-ended and meandering case as an analytical verification of the limit cases.
- A comparison of the total resistance for the combined aspect ratio and anisotropy ratio for different numbers of traxels, in order to verify the model over the entire range of interest. A low, medium, and large number of traxels are used of, respectively, 5, 19, and 51 traxels.

#### 4.1.1. Three Traxel Verification

#### 4.1.2. Anisotropy Ratio and Aspect Ratio Verification

#### 4.1.3. Total Resistance Verification

#### 4.2. Model Findings

#### 4.2.1. General Model Findings

#### 4.2.2. Anisotropy Ratio

#### 4.2.3. Aspect Ratio

#### 4.2.4. Number of Traxels

#### 4.2.5. Total Resistance and Effect of Meandering and Open-Ends

#### 4.2.6. Multiple Inputs

#### 4.2.7. Frequency-Dependent Behavior

#### 4.3. Sensor Application

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

CB | Carbon Black |

CNT | Carbon Nanotubes |

CPC | Conductive Polymer Composite |

CFRP | Carbon Fiber Reinforced Polymer |

FDM | Fused Deposition Modeling |

FEM | Finite Element Method |

RMS | Root-mean-square |

SEM | Scanning Electronc Microscopy |

Traxel | Track Element |

VCSEM | Voltage Contrast Scanning Electron Miscroscopy |

## Appendix A. Equivalent Contact Impedance

**Figure A1.**Equivalent circuit representation for the electrical contact properties c combined with the vertical bulk properties b of a slice of two neighbouring traxels of $\Delta x$ wide.

## Appendix B. Extended Model Cases

- Addition of anisotropic bulk properties, for example for filler particles with large aspect ratios (e.g., carbon nanotubes or even carbon fiber):$$\Gamma =\frac{\left(\frac{{\rho}_{x}}{1+j\omega {\rho}_{x}{\u03f5}_{x}}\right)}{\left(\frac{{\rho}_{y}}{1+j\omega {\rho}_{y}{\u03f5}_{y}}\right)+\left(\frac{\sigma /W}{1+j\omega \sigma {C}_{0}}\right)}$$
- Extending the model to 3D, by stacking layers of traxels. The system equations can be obtained through a similar derivation as for the 2D-expressions:$$\begin{array}{c}\hfill \frac{{\partial}^{2}{\widehat{U}}_{m,n}(x,\omega )}{\partial {x}^{2}}+{\Gamma}_{y}\left(\omega \right)({\widehat{U}}_{m-1,n}(x,\omega )-2{\widehat{U}}_{m,n}(x,\omega )+{\widehat{U}}_{m+1,n}(x,\omega ))\\ \hfill +{\Gamma}_{z}\left(\omega \right)({\widehat{U}}_{m,n-1}(x,\omega )-2{\widehat{U}}_{m,n}(x,\omega )+{\widehat{U}}_{m,n+1}(x,\omega ))=0\end{array}$$
- Including inductive effects and extending impedance properties to next neighbours in the model in case of more dominant capacitive and inductive properties.

## Appendix C. Error Approximated Resistance

**Figure A2.**The error of the approximated open-ended resistance compared to the FEM simulations. The RMS-error for the different numbers of traxels is respectively $15.52\%$, $25.29\%$ and $31.71\%$.

**Figure A3.**The error of the approximated meandering resistance compared to the FEM simulations. The RMS-error for the different numbers of traxels is respectively $28.39\%$, $25.25\%$ and $30.59\%$.

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**Figure 1.**Scanning Electron Microscopy (SEM) image of the Carbon Black (CB) distribution in unprinted filament, showing CB particles and aggregates. The sample is prepared by the cryo-fracturing of PI-eTPU 85-700+ filament and imaged with an FEI Quanta 450.

**Figure 2.**The SEM images of a cross section of a 3D-printed sample with cross-ply infill, prepared with cryo-fracturing: (

**a**). the full cross section (scale bar 3 $\mathrm{m}$$\mathrm{m}$), (

**b**). several traxels (scale bar 300 $\mathsf{\mu}$$\mathrm{m}$), (

**c**). void between traxels (scale bar 20 $\mathsf{\mu}$$\mathrm{m}$), and (

**d**). void close-up showing carbon black particles in the void (scale bar 2 $\mathsf{\mu}$$\mathrm{m}$).

**Figure 3.**The schematical drawing of the 3D-printed traxels present in the model, with length L, width W, and height H. The cross section is drawn as a rectangular shape to clearly indicate the main dimensions; however, the actual cross section looks more like a trapezoid with rounded corners, like in Figure 2b.

**Figure 4.**Schematic interaction of the voltage and current between two neighbouring traxels slices of $\Delta x$ wide.

**Figure 5.**The possible boundary conditions for traxels: 1. Prescribed Voltage, 2. Meandering End, and 3. Open End.

**Figure 7.**Equivalent circuit representation for the horizontal electrical bulk properties of a slice of traxel $\Delta $x wide.

**Figure 8.**The equivalent impedance circuit representation for the electrical contact properties combined with the vertical bulk properties of a slice of $\Delta x$ wide of two neighbouring traxels.

**Figure 9.**A schematic image of an example with three traxels, indicating the layout for the implicit boundary condition example.

**Figure 10.**The results of the Matlab implementation of the model, showing the voltage, current density in x-direction, and the power density. The values presented in Table 2 are used as the model parameters, and traxels are modeled to be meandering at $x=0$ (except for traxel 1,

**top**) and $x=L$ (except for traxel 19,

**bottom**). This yields an anisotropy ratio of $\Gamma =0.528$, indicating a small amount of anisotropic conduction.

**Figure 11.**The FEM simulation results in COMSOL with the parameters from Table 2. The left figure presents the voltage distribution and the right one shows the current density norm.

**Figure 12.**Comparison between the model and simulations of the dc voltage and current density in x-direction in the traxels of a sample with three traxels with open ends. The parameters shown in Table 2 are used and the number of traxels, $N=3$, and inter-traxel resistance, $\sigma =2\times {10}^{-2}\Omega {\mathrm{m}}^{2}$, are changed. The total resistance from the model is $140.3$ $\mathrm{k}$$\Omega $, which gives an error of $0.53\%$ with the FEM simulation.

**Figure 13.**A comparison between the model and simulations of the dc voltage and current density in x-direction in the traxels of a sample with three traxels with meandering ends. The parameters listed in Table 2 are used and the number of traxels, $N=3$, and inter-traxel resistance, $\sigma =2\times {10}^{-2}\Omega {\mathrm{m}}^{2}$, are changed. The total resistance from the model is $137.4$ $\mathrm{k}$$\Omega $, which gives an error of $0.23\%$ with the FEM simulation.

**Figure 14.**A comparison of the total resistance as function of aspect ratio for open-ended and meandering traxels between the model, FEM simulation, and resistance approximation results. Errors between model and FEM results occur at an aspect ratio around 1 due to 2D-conduction effects and for the meandering case for large aspect ratios due to the increased importance of the resistance of the meandering parts.

**Figure 15.**A comparison of the total resistance as a function of anisotropy ratio for open-ended and meandering traxels between the model, FEM simulation, and resistance approximation results. The errors between the model and FEM results occur at an anisotropy ratio that is close to 1 due to 2D-conduction effects.

**Figure 16.**The error of the total resistance for the open-ended case with different numbers of traxels. The RMS-error for the different numbers of traxels is, respectively, $4.75\%$, $2.51\%$, and $1.97\%$.

**Figure 17.**The error of the total resistance for the meandering case with different numbers of traxels. The RMS-error for the different numbers of traxels is, respectively, $24.33\%$, $10.26\%$, and $4.70\%$.

**Figure 18.**A close-up of the data in Figure 16 for anisotropy ratios close to 1.

**Figure 19.**A close-up of the data in Figure 17 for anisotropy ratios close to 1.

**Figure 20.**The model results for a meandering sample with the parameters from Table 2 and an inter-traxel resistance of $\sigma =2\times {10}^{-4}\Omega {\mathrm{m}}^{2}$. This gives an anisotropy ratio of $0.918$, showing bulk conduction.

**Figure 21.**Model results for a meandering sample with the parameters from Table 2 and an inter-traxel resistance of $\sigma =2\times {10}^{-2}\Omega {\mathrm{m}}^{2}$. This gives an anisotropy ratio of $0.100$, showing mixed conduction.

**Figure 22.**The model results for a meandering sample with the parameters from Table 2 and an inter-traxel resistance of $\sigma =2\times {10}^{-1}\Omega {\mathrm{m}}^{2}$. This gives an anisotropy ratio of $0.0111$, showing traxel conduction.

**Figure 23.**The model results for a meandering thick structure with with traxels of 5 $\mathrm{m}$$\mathrm{m}$ long and an aspect ratio of $3.04$. The anisotropy ratio is taken to be $\Gamma \approx 1$ by taking $\sigma =2\times {10}^{-8}\Omega {\mathrm{m}}^{2}$. The voltage drop is mainly from top to bottom.

**Figure 24.**Model results for a meandering, long, slender structure with traxels of 45 $\mathrm{m}$$\mathrm{m}$ long and an aspect ratio of $0.338$. The anisotropy ratio is taken to be $\Gamma \approx 1$ by taking $\sigma =2\times {10}^{-8}\Omega {\mathrm{m}}^{2}$. The voltage drop is mainly from left to right.

**Figure 25.**The model converges to a solution of the Laplacian for homogeneous anisotropic materials in the case of large numbers of traxels, in this case 99 traxels.

**Figure 26.**The total resistance for the open-ended model as a function of aspect ratio and anisotropy ratio for 5, 19, and 51 traxels. The resistance is approximately the same for the different numbers of traxels.

**Figure 27.**The total resistance for the meandering model as a function of aspect ratio and anisotropy ratio for 5, 19 and 51 traxels. The resistance shows the same qualitative behavior for the different numbers of traxels with a shift in resistance.

**Figure 28.**Connecting multiple inputs and outputs is possible, like connecting all corners. In this specific example, open-ended traxels are used with an inter-traxel resistance of $\sigma =2\times {10}^{-2}\Omega {\mathrm{m}}^{2}$ combined with the other parameters from Table 2, providing an anisotropy ratio of $0.1007$.

**Figure 29.**A demonstration of the ac model, showing the voltage plots. Traxel conduction occurs at 1 $\mathrm{Hz}$ and bulk conduction occurs at 1 $\mathrm{M}$$\mathrm{Hz}$, while using the parameters ${C}_{0}=2.8\times {10}^{-5}\mathrm{F}{\mathrm{m}}^{-2}$, ${\u03f5}_{\mathrm{r}}=5$, $\rho =2.8\Omega \mathrm{m}$, and $\sigma =0.2\Omega {\mathrm{m}}^{2}$. According to this model, the frequency-dependence can significantly influence the electrical conduction through 3D-prints.

**Figure 30.**The principle of constriction-resistance strain sensor. The sensing principle makes use of the large resistance change going from bulk conduction to traxel conduction upon straining of the gaps between the traxels. The white arrows indicate the current flow through the sensor, whereas the black arrows indicate the force applied on the sensor.

**Figure 31.**The modelled resistance versus inter-traxel resistance for a constriction-resistive strain sensor, showing a large difference between the lowest and highest resistance. For very low inter-traxel resistance values, the anisotropy ratio is almost 1 and the total resistance stays the same. For very large inter-traxel resistance values, the anisotropy ratio becomes small and, eventually, the total resistance levels out when approaching pure traxel conduction.

**Figure 32.**The modeled traxel resistance to bulk resistance ratio as a function of the number of traxels (e.g., the ratio between the maximum and minimum total resistance in Figure 31), for several aspect ratios. Hence, increasing the number of traxels and decreasing the aspect ratio will increase the traxel/bulk resistance ratio of the sensor. In theory, the ratio is limited by the number of traxels, as given in Equation (62).

**Figure 33.**The model can be used to calculate the conduction in 3D-prints from the sample geometry and material properties, over the whole range of conduction behavior from isotropic to anisotropic.

**Table 1.**A summary of the different boundary conditions, where, for the outer x-positions, we have $\xi =0\vee L$.

Condition | Voltage BC | Current BC |
---|---|---|

Applied voltage | ${\widehat{U}}_{i}(\xi ,\omega )={\widehat{U}}_{\mathrm{in}/\mathrm{out}}$ | - |

Open-ended | - | $\frac{\partial {\widehat{U}}_{n}(\xi ,\omega )}{\partial x}=0$ |

Meandering | ${\widehat{U}}_{i}(\xi ,\omega )={\widehat{U}}_{i+1}(\xi ,\omega )$ | ${\widehat{I}}_{i}(\xi ,\omega )=-{\widehat{I}}_{i+1}(\xi ,\omega )$ |

**Table 2.**The parameters used for modeling and simulations. Data from [42].

Variable/ Sample, Units | Values |
---|---|

Resistivity $\rho $, $\Omega \mathrm{m}$ | $2.8$ |

Inter-traxel resistivity $\sigma $, $\Omega {\mathrm{m}}^{2}$ | $2\times {10}^{-3}$ |

Traxel width W, $\mathrm{m}\mathrm{m}$ | $0.8$ |

Traxel length L, $\mathrm{m}\mathrm{m}$ | 15 |

Traxel height H, $\mathsf{\mu}\mathrm{m}$ | 200 |

Number of traxels N, - | 19 |

Anisotropy Number ${\Gamma}_{\mathrm{DC}}$, - | 0.528 |

Aspect Ratio $AR$, - | 1.013 |

Number of Traxels: | 5 | 19 | 51 |
---|---|---|---|

RMS-error open-ended: | $4.75\%$ | $2.51\%$ | $1.97\%$ |

RMS-error meandering: | $24.33\%$ | $10.26\%$ | $4.70\%$ |

**Table 4.**Model parameters used for the constriction-resistive strain sensor calculations, based on the work conducted by Mousavi et al. [47].

Parameter | Values |
---|---|

Resistivity $\rho $ | $6.8$$\mathrm{m}\Omega \mathrm{m}$ |

Inter-traxel resistivity $\sigma $ | 2 × 10^{−6} $\Omega $$\mathrm{m}$${}^{2}$–2 × 10^{3} $\Omega $$\mathrm{m}$${}^{2}$ |

Traxel width W | 450 $\mathsf{\mu}$$\mathrm{m}$ |

Traxel length L | 10 $\mathrm{m}$$\mathrm{m}$ |

Traxel height H | 200 $\mathsf{\mu}$$\mathrm{m}$ |

Gauge Length | 35 $\mathrm{m}$$\mathrm{m}$ |

Infill density | 95% |

Number of traxels N | $0.95*35\mathrm{m}\mathrm{m}/0.45\mathrm{m}\mathrm{m}=74\approx 75$ |

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**MDPI and ACS Style**

Dijkshoorn, A.; Schouten, M.; Stramigioli, S.; Krijnen, G.
Modelling of Anisotropic Electrical Conduction in Layered Structures 3D-Printed with Fused Deposition Modelling. *Sensors* **2021**, *21*, 3710.
https://doi.org/10.3390/s21113710

**AMA Style**

Dijkshoorn A, Schouten M, Stramigioli S, Krijnen G.
Modelling of Anisotropic Electrical Conduction in Layered Structures 3D-Printed with Fused Deposition Modelling. *Sensors*. 2021; 21(11):3710.
https://doi.org/10.3390/s21113710

**Chicago/Turabian Style**

Dijkshoorn, Alexander, Martijn Schouten, Stefano Stramigioli, and Gijs Krijnen.
2021. "Modelling of Anisotropic Electrical Conduction in Layered Structures 3D-Printed with Fused Deposition Modelling" *Sensors* 21, no. 11: 3710.
https://doi.org/10.3390/s21113710