Influence of layer height and nozzle diameter on the resistivity of additively manufactured electrically conductive structures made by material extrusion

Additive Manufacturing (AM) offers the potential for the production of components with complex shapes, hierarchical structures and integrated functions (Gibson et al., 2021; Rosen, 2007). Approaches aimed at improving the resulting product properties by leveraging these potentials are collectively referred to as Design for Additive Manufacturing (DfAM) (Gibson et al., 2021). The thermal material extrusion process of polymers (MEX-TRB/P), in particular, offers the potential for the production of multi-material components with a high degree of design freedom and material complexity, through the selective application of molten polymer. In this process, a thermoplastic feedstock is liquefied by heat in a nozzle and is extruded as a strand. The cross-section of the strand is determined by the diameter of the nozzle and the distance between the nozzle and the substrate, as well as the flow rate. The polymer melt is deposited strand by strand and layer by layer through the nozzle. The incorporation of electrically conductive polymer composites allows for the integration of a wide range of functionalities into the parts, thereby expanding the feasible design space through Design for Functionality. The incorporation of conductive additives, including but not limited to carbon black (CB), carbon nanotubes (CNT), graphite and graphene (G), into the polymer facilitates the attainment of electrical conductivity. A number of various electrical functional structures have been additively manufactured and analysed in an academic context to date. These include sensors (Goutier et al., 2023; Huang et al., 2018; Wang et al., 2019; Watschke et al., 2021; Heitkamp et al., 2024; Mousavi et al., 2020; Elgeneidy et al., 2018; Christ et al., 2019; Hohimer et al., 2020; Kwok et al., 2017; Jonkers et al., 2023; Stano et al., 2020; Daniel et al., 2018; Ragolia et al., 2021; Wolterink et al., 2020), actuators (Watschke et al., 2019; Hilbig et al., 2022; Kim and Lee, 2020; Nowka et al., 2023), conductive structures (Macdonald et al., 2014; Gardner et al., 2016; Yang et al., 2019; Flowers et al., 2017; Hampel et al., 2017; Gonçalves et al., 2018; Nassar and Dahiya, 2021) and energy storage (Flowers et al., 2017; Reyes et al., 2018; Sabato et al., 2023; Tan et al., 2022). In addition to the geometric parameters, the utilisation of additive manufacturing presents, in contrast to conventional processes, the opportunity to influence the final component properties through the strategic selection of process parameters (Watschke et al., 2019; Ansari and Kamil, 2021).

The layer height and nozzle diameter are of particular importance in the production of AM components. These two parameters not only significantly determine the volume flow of the molten polymer and thus the economic factor of production time (Chacón et al., 2017; Nancharaiah, 2011; Gurrala and Regalla, 2011; Nidagundi and Prakash, 2015; Srivastava et al., 2018; Fischer et al., 2022), but also have a significant influence on the mechanical (Rankouhi et al., 2016; Rayegani and Onwubolu, 2014; Mahendran et al., 2019; Bakhtiari et al., 2023; Tura et al., 2022; Fischer et al., 2022) and electrical (Gao and Meisel, 2020; Glogowsky et al., 2023; Hohimer et al., 2020; Paz et al., 2020; Stankevich et al., 2023; Nassar and Dahiya, 2021; Barši Palmić et al., 2020; Zhang et al., 2017; Yang et al., 2017; Nowka et al., 2024b; Abdalla et al., 2020; Dembek et al., 2022) properties of the manufactured components, as well as the part resolution (Diegel et al., 2020; Gibson et al., 2021; Gebhardt, 2016; Godec et al., 2022; Lachmayer et al., 2022) and surface quality (Anitha et al., 2001; Thrimurthulu et al., 2004; Horvath et al., 2007; Vasudevarao and Natarajan, 2000; Lin and Hu, 2007). In particular, the resolution within a layer is mainly determined by the nozzle diameter (Gibson et al., 2021), while the resolution in the build direction (stair-stepping effect) depends on the layer height (Diegel et al., 2020; Gibson et al., 2021; Gebhardt, 2016; Godec et al., 2022; Lachmayer et al., 2022). To achieve an optimal balance between the surface quality, performance and production time of the resulting components, it is essential to pay close attention to the selection of these parameters. However, the layer height and nozzle diameter are not independent parameters; rather, they must be chosen in a balanced relationship to each other. Layer heights are generally used in the range of 20%–80% (Czyżewski et al., 2022; Zemcik and Sedlak, 2019; Dembek et al., 2022) of the nozzle diameter, with a guideline value of 50% (Zemcik and Sedlak, 2019).

Very small layer heights have the potential to cause production issues, as they can result in pressure build-up within the nozzle (Sukindar et al., 2016; Coogan and Kazmer, 2017). In contrast, an excessively high layer height can result in inadequate adhesion between the layers, due to the insufficient pressure to effectively press the polymer melt onto the surrounding surfaces (Coogan and Kazmer, 2017). Within this range, it is essential to adjust the parameters with careful consideration.

Prior studies have investigated the effects of the layer height and/or strand width on the resistivity within a layer (xy-plane) (Gao and Meisel, 2020; Glogowsky et al., 2023; Hohimer et al., 2020; Paz et al., 2020; Stankevich et al., 2023; Nassar and Dahiya, 2021; Zhang et al., 2017; Nowka et al., 2024b), as well as in the z-direction (Gao and Meisel, 2020; Hohimer et al., 2020; Paz et al., 2020; Zhang et al., 2017). The impact of the layer height on the resistivity along the z-direction is uniformly clarified. Higher layer heights reduce the resistivity along the z-direction (Gao and Meisel, 2020; Hohimer et al., 2020; Paz et al., 2020; Yang et al., 2019; Nassar and Dahiya, 2021). Although Zhang et al. did not observe a significant effect of layer height, a significant influence of strand width was found. Since a correlation between strand width and layer height linked by the nozzle diameter, this observation further supports the others (Zhang et al., 2017). The studies attribute the positive effect of higher layer heights on the reduction of contact interfaces between layers, thus reducing the portion of contact resistance to the overall resistance (Gao and Meisel, 2020; Hohimer et al., 2020; Wolterink et al., 2018). Further beneficial effects are the potential increase in contact interface area (Zhang et al., 2017) and the reduction in porosity (Hohimer et al., 2020). The literature on the resistivity measured in the xy-plane yields contradictory results. While some studies report a reduction in resistivity with increasing layer height (Glogowsky et al., 2023; Hohimer et al., 2020; Paz et al., 2020; Yang et al., 2019; Stankevich et al., 2023; Stano et al., 2020; Barši Palmić et al., 2020; Dembek et al., 2022) other studies report the opposite (Abdalla et al., 2020; Gao and Meisel, 2020; Zhang et al., 2017; Dembek et al., 2022).

Stano et al. suggest that an increase in layer height results in a greater contact area between the strands and a reduction in air gaps, which subsequently reduces the resistivity (Stano et al., 2020). Gao et al. attribute the higher conductivity for lower layer heights to a smaller proportion of voids (Gao and Meisel, 2020). Abdalla et al. hypothesize that the geometric constraints of low layer heights lead to better ordering of the filler, which supports the formation of the conductive network (Abdalla et al., 2020). Hohimer et al. propose that bigger strand cross-sections are inherently more conductive (Hohimer et al., 2020). A possible explanation for this is the lower volume fraction of the less conductive skin layer (Wolterink et al., 2018; Nowka et al., 2024b). Additionally, depending on the study the strand width has a different influence on the resistivity within the xy-plane. Stano et al. and Zhang et al. found no significant effect of the strand width on the resistivity (Stano et al., 2020; Zhang et al., 2017), whereas Nowka et al., Stankevich et al., Palmic et al. and Paz et al. reported decreasing resistivities (Nowka et al., 2023; Stankevich et al., 2023; Barši Palmić et al., 2020; Paz et al., 2020). The authors attribute this to the deterioration of surface quality with decreasing nozzle diameter (Paz et al., 2020; Barši Palmić et al., 2020). Yang et al. observed similar effects for smaller layer heights, which are typically used in conjunction with narrower strand widths (Yang et al., 2019). Larger nozzle diameters, according to Nowka et al., result in a more uniform melt flow, leading to fewer defects (Nowka et al., 2024b). Dembek et al. found no clear trend across the gradations for the nozzle diameters investigated (Dembek et al., 2022).

Prior studies used a variety of different polymers [thermoplastic polyurethane (TPU) (Stano et al., 2020; Hohimer et al., 2020; Wolterink et al., 2018), Polylactic Acid (PLA) (Stano et al., 2020; Gao and Meisel, 2020; Abdalla et al., 2020; Yang et al., 2019), Polycaprolacton (PCL) (Barši Palmić et al., 2020), Acrylonitrile butadiene styrene (ABS) (Paz et al., 2020; Zhang et al., 2017), Polyvinylidene fluoride (PVDF) (Stankevich et al., 2023)] and conductive additives [CB (Stano et al., 2020; Gao and Meisel, 2020; Abdalla et al., 2020; Zhang et al., 2017; Wolterink et al., 2018), CNT (Stano et al., 2020; Hohimer et al., 2020; Yang et al., 2019), graphene (Paz et al., 2020; Stankevich et al., 2023) and copper particles (Barši Palmić et al., 2020)] for sample fabrication. Due to the differing composite properties, the influence of layer height and nozzle diameter on resistivity may vary. Table 1 provides an overview of the study results, highlighting the contradictory findings regarding the influence of layer height on resistivity and the largely unexplored impact of strand width.

Layer height and track width are closely linked. However, most of the previous studies have examined their effects on resistivity as isolated parameters. Also these studies derived conclusions about the influence of these two parameters from only a few parameter sets. One exception is the study by Dembeck et al., which analysed different layer heights in combination with different nozzle sizes. However, the used specimen geometry inevitably changed the number of layers in the specimen and the corresponding layer transitions (Dembek et al., 2022), affecting the measurement results due to inter-layer contact resistance. Additionally, the chosen infill pattern is suboptimal for determining electrical properties (Dembek et al., 2022).This reduces the reliability of the findings, leading to contradictory interpretations. The studies provide general guidelines regarding individual parameter selection, but these are based on data sets that do not cover the entire range of feasible parameter combinations. In contrast, this study investigates, for the first time systematically, the combined influence of nozzle diameter and layer height with fine 50 µm step size on single-layer specimens. This approach provides new insights into how these parameters together influence electrical conductivity without the impact of the inter-layer contact resistance and surface roughness. The combined parameter space of layer height and track width examined in this study represents the vast portion of the usable range due to the fine gradations of layer heights and nozzle diameters. This allows, for the first time, an assessment/manufactured conductive structures. To limit the scope of the experiment, material variation will be excluded, and all samples will be fabricated using a single composite material.

In this study, the electrically conductive composite (Ciceri de Model Srl., Ozzero, Italy), provided by the company FILOALFA®, is used. This composite was chosen because it is one of the most electrically conductive commercially available filaments with carbon-based fillers (Nowka et al., 2024b). Alfaohm is available both as granule and as filament. In this study, the filament is used as the feedstock. According to the manufacturer, the composite is a PLA/CNT composite (Ciceri de Mondel S.r.l. a Socio Unico, 2019). Furthermore, the composite contains CB and graphite undeclared by the manufacturer (Contreras-Naranjo et al., 2021; Nowka et al., 2023; Nowka et al., 2024b; Nowka et al., 2024a). Contreras-Naranjo determined a content of approximately 3 Wt.% multiwalled CNTs with a 1:10 ratio of CNT to CB (Contreras-Naranjo et al., 2021). The manufacturer specifies the resistivity, measured according to ASTM D 257 (D09 Committee, 2021), as 15 Ωcm along the xy-plane and 20 Ωcm along the z-direction (Ciceri de Mondel S.r.l. a Socio Unico, 2019). As part of the sample preparation, a mask made of adhesive tape (Kapton, Du Pont, Wilmington, DE, the USA) is applied to the additively manufactured specimens, and colloidal silver ink EMS#12640 (Electron Microscopy Sciences, Hatfield, PA, the USA) is used as an electrical bonding agent between the specimen and the measurement setup.

This experimental study investigates the influence of strand width (w_s) in interaction with layer height (h_l) on electrical resistivity. For each strand width, a nozzle with a similiar diameter is used. The strand width increments are determined by the commercially available nozzle diameters. To ensure sufficient mechanical adhesion, an overlap of the toolpaths is required. During the generation of machine commands (G-code) by slicer software, two options are available: achieving the nominal strand width through over-extrusion or reducing the strand width by offsetting the toolpath towards the previous strand. In this study the default slicer option, an offset of the toolpaths was applied, resulting in a higher number of strands than would be achieved by simply dividing the specimen width by the nozzle diameter. Table 2 provides an overview of the number of strands for an ideal path planning without overlap, as well as a comparison with the actual resulting number of strands with overlap, for various strand widths at a constant specimen width of 24 mm.

For simplicity of description, it is assumed hereafter that the nozzle diameter corresponds to the strand width.

To enable direct comparison between different strand widths, the layer height was normalized by the strand width. This normalization yielded percentage values, allowing for effective comparison. Marion et al., Kanarska et al. and Abdalla et al. have observed that nozzle geometry influences the resulting conductivity through filler distribution and swelling behaviour at the nozzle exit (Marion et al., 2024; Kanarska et al., 2019; Abdalla et al., 2020). Therefore, all experiments are conducted using the same internal nozzle geometry. The variation in nozzle exit diameters is determined by the manufacturer’s available increments. To prevent an increase in exit diameter due to wear caused by the abrasive composite, hardened nozzles are used. These requirements are met by the Nozzle X (E3D-Online, Chalgrove, Oxfordshire, the UK), which is available in 250 µm, 300 µm, 350 µm, 400 µm, 500 µm, 600 µm and 800 µm variants. The 350 µm nozzle diameter is highly uncommon and is therefore not investigated. The lower limit of the layer height is theoretically independent of the strand width and is process-limited by the MEX manufacturing machine, primarily by the pressure buildup of the polymer melt within the nozzle. Based on preliminary experiments, the minimum layer height for this composite is limited to 100 µm due to low process reliability below this layer height. For strand widths exceeding 600 µm, the initial layer height of 100 µm results in an h_l/w_s ratio below 20%. Nevertheless, attempts are made to fabricate these parameter sets. The upper limit is set to 100% of the strand width, as exceeding this limit results in insufficient build plate adhesion or layer bonding. The layer height step size is set to 50 µm. The experimental plan, including the layer height limits varying with the strand width, is presented in Table 3.

The process parameter sets are divided into feasible and not feasible regions based on the criterion of the normalized layer height. Parameters with ratios h_l/w_s ≤ 100% are considered feasible (highlighted in grey in Table 3), while those with ratios > 100% are categorized as not feasible (white). The fabrication of AM specimens is conducted using combinations of layer height and strand width deemed feasible. Apart from layer height and strand width, all other parameters remain constant. The extrusion temperature is set to 215°C, the build platform temperature to 60°C, the printing speed to 30 mm/s and the flow rate to 100%. The specimens are manufactured in a single layer with 100% infill and without shells. The experimental plan A yields 48 process parameter sets, each comprising seven specimens.

Due to the varying nozzle diameters and, consequently, strands widths in experimental plan A, the number of parallel strands forming the specimens differs. The variation in strands width results in a differing number of adjacent strands due to the constant specimen width (see Table 2). To determine the influence of the number of adjacent strands on the resistivity, and thus as an additional factor influencing the primary experimental plan, specimens with 1, 2, 3, 5, 8, 13, 21 and 34 strands are fabricated. The gradation is selected on the basis of the Fibonacci sequence, as it is anticipated that the greatest influence will be observed with a small number of strands. The specimens in this experimental plan are manufactured with a constant nozzle diameter of 400 µm and a constant layer height of 200 µm. The remaining parameters (v_print = 30 mm/s, T_nozzle = 215°C, T_buildplate = 60°C) as well as the fabrication and testing conditions are identical to those in experimental plan A. Seven specimens are produced for each number of strands.

The filament is dried at 60°C for 48 h prior to additive processing. The specimens are additively manufactured by a Toolchanger system (E3D-Online). The system is equipped with Hemera direct drive filament extruders (E3D-Online) and hardened, coated Nozzle X nozzles (E3D-Online) with diameters selected according to the experimental plan outlined in Table 3.

The specimens are rectangular plates (60 × 24 × h mm) fabricated as a single layer with a height as specified in the design of experiment plan A (Table 3) on microscope slides. The specimens remain attached to the slides and are not removed in post-processing. This eliminates the potential for measurement results to be influenced by mechanical stress during removal of the specimen from the buildplate. To eliminate the influence of contact resistance and other interfacial effects between the layers, all specimens are produced as mono-layer specimen, with the specimen height directly corresponding to the layer height parameter in the experimental plan A.

The G-code is generated using SuperSlicer 2.4 (based on PrusaSlicer, Prusa Research, Prague, Czech Republic) with varying layer height and strand width (see Table 3). The infill orientation significantly affects the resistivity (Masarra et al., 2022; Stankevich et al., 2023; Glogowsky et al., 2023; Watschke et al., 2019; Hilbig et al., 2022). For infill orientations other than 0° relative to the current flow, the number of contact points between the electrical measurement contact surfaces varies. To avoid the influence of contact resistance as a non-constant disturbance variable, an infill orientation parallel to the current flow (α = 0°) is selected (see Figure 1).

The areas of the specimens intended for electrical contact with the test rig during resistance measurements are coated with a conductive bonding agent (colloidal silver). This guarantees the flow of current and the taking of measurements of potential at geometrically defined areas. A minimum of 24 h elapse between the application of the bonding agent and the resistance measurement, allowing the solvent to evaporate completely.

The strand width and layer height influence surface roughness of specimens. Since deviations from an ideal planar surface can directly affect the actual electrically conductive cross-section, surface roughness is measured. Furthermore, the surface resistance is also affected, although this influence on the measured resistance is deemed to be negligible due to the high conductivity of the composite material. A VR-5100 optical profilometer (Keyence, Neu-Isenburg, Germany) is used for contact free surface roughness measurements. The resolution in the z-direction is 0.1 µm. Measurements are conducted in macro mode with the smallest magnification (12x) in high-resolution mode, yielding a measurement accuracy in the xy-plane of ±5 µm. This results in a maximum scanning area of 24 × 18 mm.

The surfaces of the MEX specimens are scanned in the middle area between the two inner electrical contacts. The roughness R_z and R_a is calculated from the scanned profile by averaging 30 individual line profiles distributed across the entire scan width for each specimen. The scan lines run perpendicular to the deposited strands. Figure 2 illustrates a specimen with an overlaid schematic three-dimensional scan profile and the corresponding derived line depth roughness.

To avoid any distortion of the results due to the edge region, 1 mm from each end of the specimen is excluded when determining the line roughness. Subsequently, an automated inclination correction of the profile is conducted. This serves to eliminate any inclination of the surface in relation to the measuring system, thereby ensuring that the values obtained for roughness remain unaffected. For analysis, the average of the roughness depth R_z is calculated for each specimen from the 30 individual line measurements.

The determination of the resistivity is conducted according to DIN EN ISO 3915:2022-5 (DIN, 2022). Resistance measurements are performed using a Keithley 2460 Sourcemeter (Keithley Instruments, Solon, OH, the USA) in a four-wire configuration at room temperature (23 ± 1°C). The constant measurement current is set to 100 µA, ensuring that the power dissipation remains well below 100 mW, thereby minimizing any temperature influence on the measured resistance due to heating. Electrical contact between the specimen and the sourcemeter is made via spring-loaded contacts. The contact force is kept consistent across all measurements using a test rig. Figure 3 depicts a specimen with the measurement setup schematically overlaid.

Specimen thickness variations are accounted for by taking three measurements with a micrometer screw (QuantuMike® 293-140-30, Mitutoyo Corporation, Kawasaki, Japan) prior to applying the electrical bonding agent. Specimens with deviations exceeding within the tolerance limit of ±5% set by DIN EN ISO 3915:2022-5 are excluded and replaced with dimensionally accurate specimen (DIN, 2022). The resistivity ρ is calculated from the conductive cross-sectional area A, the distance between the measurement contacts L and the measured resistance R:

The inverse of the resistivity is the electrical conductivity $σ$⁠.

During specimen fabrication, the limitations of the process and machine became apparent. Specimen with a strand width of 250 µm proved challenging to produce, regardless of layer height, as the nozzles frequently became clogged. The scanning electron microscope (SEM) micrograph (Helios G4 CX, 3 keV, captured with a secondary electron detector, Field Electron and Ion Company, OR, the USA) shown in Figure 4 displays the fracture surface of a cryo-broken filament piece.

The fracture surface reveals numerous graphite particles, some reaching sizes of approximately 100 µm. It is likely that individual large graphite particles or their aggregates can obstruct the narrow nozzle diameter of 250 µm. Consequently, only the parameter set with a layer height of 100 µm was fully manufactured due to the short service life of individual nozzles. In contrast, only five specimens from the parameter set with a layer height of 150 µm were completed, while none of the sets with layer heights of 200 µm and 250 µm were manufactured using nozzles with a diameter of 250 µm.

Additionally, it was not possible to fabricate specimens with a strand width of 800 µm and a layer height of 100 µm. This parameter set features a ratio of h_l/w_s = 12.5%, the smallest ratio in this experimental plan, which falls below the empirical threshold of 20%. The 3 Wt.% CNT content in Alfaohm significantly reduces the melt flow index (MFI) compared to pure PLA, a behaviour also reported for other CNT containing composites (Dul et al., 2018;Sezer and Eren, 2019; Yang et al., 2019). As a result, higher pressures are required than the extruder can apply without grinding the filament with the extrusion gears.

The numerical measurement results for the roughness depth R_z and the arithmetic roughness R_a are presented in Table A2 and Table A3 in the  Appendix. The evaluation is based on the roughness depth R_z, as it does not involve averaging, in contrast to the arithmetic roughness R_a. The influence of the explanatory variables strand width (w_s) and normalized layer height (h_l/w_s) on R_z, is investigated using a linear regression model generated with Minitab® Version 21 (Minitab GmbH, Munich, Germany). The explanatory variables are considered in the model as linear and quadratic terms as well as in their interactions. Non-significant terms are removed from the model through a backward elimination process using a threshold of α = 0.1 via an analysis of variance (ANOVA). 14 out of 327 data points were excluded from the evaluation as outliers. These excluded data points all exhibit an h_l/w_s ratio over 91%, which is associated with manufacturing defects and significant deviations in surface roughness. These outliers were likely, as the recommended range ends at 80% (Czyżewski et al., 2022; Zemcik and Sedlak, 2019; Dembek et al., 2022). Consequently, the model’s accuracy is significantly improved for data sets within the typical range of h_l/w_s ratios between 20% and 80% from R² = 51.85% (complete data set) to 77.05% (without outliers). An overview of the model results is provided in Table 4.

The statistical analysis shows that all model terms derived from h_l/w_s and strand width have a statistically significant effect on R_z with p = 0.000. The explained variance (R² = 77.05%) of the model is acceptable. Figure 5 illustrates the relationship between the h_l/w_s ratio on the x-axis and the roughness depth R_z on the y-axis.

From Figure 5, it becomes evident that for each strand width, the roughness depth increases with higher layer heights in the absence of corrective measures, such as adjusting the extrusion factor. The line roughness for nozzle diameters of 300 µm, 400 µm and 500 µm exhibits a noticeably smaller influence from increasing layer heights compared to nozzle diameters of 600 µm and 800 µm. For the final data point with a nozzle diameter of 600 µm at an h_l/w_s ratio of 100%, no standard deviation can be calculated due to the high number of outliers.

For the sake of completeness, it should be noted that the R_z results should be interpreted as indicative only. The nozzle-to-build platform distance settings tend to result in slight under-extrusion, which subtly but consistently affects the roughness measurements.

The layer height was varied in 50 µm increments, resulting in a different number of data points for each nozzle diameter until the criterion of h_l/w_s = 100% was reached. The measurement data sets that have already been identified as outliers in the analysis of surface roughness are also excluded from this analysis. The measurement data is presented as a plot in Figure 6, with detailed numerical values provided in Table A3 in the  Appendix. Additional, the  Appendix includes the (non-normalized) plots of the resistivity as a function of layer height in Figure A1 and extrusion width in Figure A2.

The plot reveals two distinct trends. Firstly, the curves exhibit a parabolic shape with a slight plateau near h_l/w_s ≈ 100%. Both very low and very high layer heights produce higher resistivities across all strand widths compared to the midrange values. Additionally, the resistivity decreases with increasing strand width. The minimum resistivity is observed within the range of 40–60% for the layer height-to-strand width ratio.

In addition to visual analysis, a linear regression model was used to investigate the explanatory variables of strand width (ws) and normalized layer height (hl/ws). The variables were analysed as linear and quadratic terms, including their interactions. Using ANOVA with a backward elimination procedure, nonsignificant terms were removed from the model with a threshold of α = 0.1. A summary of the model results is presented in Table 5.

The statistical analysis indicates that both the normalized layer height and strand width, along with their nonlinear interactions, have a significant impact on resistivity. All p-values equal 0.000, below the significance level of 0.05, except P(w_s) = 0.152, confirming that linear and quadratic effects, as well as their interactions, significantly influence resistivity. Figure 7 isolates the effects of the two main factors.

Figure 7(1) demonstrates that ratios of layer height to strand width below 40% and above 60% substantially contribute to increased resistivity. Conversely, as seen in Figure 7(2), strand widths below 500 µm show a variable impact on resistivity, while widths above 500 µm exhibit a nearly constant influence.

The explained variance (R² = 75.30%) of the model is acceptable, indicating that nozzle diameter and layer height account for approximately 75% of the variance in resistivity. This suggests that either an influential factor was omitted from the model or the high variability of the first layer, combined with a relatively small sample size of seven per parameter set, limits the statistical robustness of the analysis regarding the resistivities influencing factors. A potential influencing factor that has not been considered in the modelling process is the filament itself. The material deposition is controlled by the machine under the assumption of a constant filament diameter. Consequently, local diameter fluctuations may lead to over- or under-extrusion, which, in turn, can affect the resulting resistivity both through the altered electrically conductive cross-section and the contact area with adjacent tracks. Due to thickness measurements of each sample, geometric deviations are limited to ±5%. Nowka et al. demonstrated that process parameters during filament production influence not only the electrical conductivity of the filament but also the conductivity of additively manufactured parts produced from it (Nowka et al., 2023). Therefore, it is likely that filament conductivity also fluctuates due to minor variations in manufacturing parameters during filament production. The inclusion of outliers in the model decreases the explained variance to R² = 68.25%.

In Figure 8, the resistivity is plotted for varying numbers of adjacent strands.

The lowest resistivity is achieved with a single strand. Up to five strands, the resistivity increases linearly. Beyond this, no clear trend is observed, with the values stabilizing between 4.5 Ωcm and 5.5 Ωcm for strand counts greater than eight. This range aligns with the results shown in Figure 6 for a strand width of 400 µm and an h_l/w_s ratio of 50% (corresponding to a layer height of 200 µm).

The number of adjacent strands varies depending on the nozzle diameter. Previous studies by Truman et al. and Zhang et al. have shown that resistivity increases when strands are deposited side by side and fused together (Truman et al., 2020; Zhang et al., 2017). Our results indicate that for our experimental setup with strand counts of 30 or more, no discernible trend is evident, suggesting that the effect of strand count can likely be excluded. It seems reasonable for used experimental setup to conclude that the differing numbers of strands resulting from the variation in nozzle diameter do not exert a significant influence on the resistivity.

The parabolic curve in Table A1 with a plateau can be attributed to a combination of material and process-related factors. The high conductivity of the composite used in the specimens is mainly due to the CNT content. In CNT composites, which initially exhibit a well-conductive network, shear forces during melt processing improve CNT dispersion but reduce conductivity (Alig et al., 2012; Lellinger et al., 2011; Skipa et al., 2010). Higher shear rates result in greater conductivity losses (Lellinger et al., 2011; Obrzut et al., 2007; Kharchenko et al., 2004). These losses are most pronounced immediately after the onset of shear stress and decrease asymptotically thereafter (Lellinger et al., 2011; Skipa et al., 2010). Network reformation after nozzle exit is inhibited by rapid cooling, freezing the state of the melt (Patanwala et al., 2017).

Smaller layer heights require higher nozzle pressures (Sukindar et al., 2016; Coogan and Kazmer, 2017), as do smaller nozzle diameters due to restricted melt flow. Both factors increase shear rates within the polymer melt during extrusion, reducing conductivity. Conversely, larger nozzle diameters decrease the extrusion pressure due to the larger flow cross-section, thereby reducing shear within the melt and mitigating conductivity loss.

The aforementioned material effects are additionally influenced by path-planning factors, which may explain the observed rise in resistivity at higher ratios h_l/w_s ratios (Figure 6). As the layer height approaches h_l/w_s = 100%, lower extrusion pressures result in weaker compression of the melt onto the build platform or adjacent strands (Coogan and Kazmer, 2017). This leads to larger voids (Fischer et al., 2022), increasing surface roughness (see Figure 5). Contrary to our findings, Palmic et al. reported reduced surface quality with lower layer heights (Barši Palmić et al., 2020). Increased voids reduce the electrically conductive cross-section and raise measured resistance, leading to an overestimated resistivity based on the assumption of a rectangular conductive cross-section for the calculation. Additionally, reduced strand contact area increases inter-strand contact resistance.

In electrically conductive multi-layer AM parts, stacked layers with different infill orientations create alternative paths through adjacent layers, bypassing poorly conductive regions. This phenomenon reduces overall resistivity (Truman et al., 2020). A schematic in Figure 9 illustrates the resistance network within a single layer.

In a single-layer specimen, if a strand section (e.g. R_2,2) has poor conductivity, the (balancing) current (I_B) must flow through adjacent strands, crossing poorly conductive boundary layers with inter-strand contact resistance (R_C). Poor strand contact and increased defects raise resistivity.

Another potential influence on the resistivity is the higher thermal input associated with increased material flow, which improves strand welding (Altan et al., 2018). This effect intensifies with larger nozzle diameters and layer heights, enhancing electrical strand bonding and reducing resistivity. Conversely, larger nozzle diameters do not disrupt the well-formed conductive network through a reorientation of particles. Simultaneously, the higher volumetric flow rate promotes the welding of strands, which collectively reduces the inter-strand resistance. However, this improvement is counterbalanced by a deterioration in conductivity due to increasingly larger voids at with greater nozzle diameters. The interplay of these effects results in a minimum resistivity.

All in all this study represents the most comprehensive investigation to date into the effects of strand width and nozzle diameter respectively layer height on the resistivity of electrically conductive structures. The findings demonstrate that both factors significantly influence the resistivity, emphasizing the importance of careful parameter selection during the machine setup and slicing. By adjusting these parameters, the resistivity can vary by more than 50%. Specifically, increasing the nozzle diameter decreases resistivity. The highest conductivity is achieved when the layer height is 40%–60% of the nozzle diameter. This aligns with the widely used empirical non application specific guideline of maintaining a 50% ratio between layer height and strand width (Zemcik and Sedlak, 2019). Larger nozzle diameters also reduce the resistivity. In the context of DfAM, this ratio provides a straightforward guideline for selecting the optimal slicer parameter values for minimizing the resistivity of electrically conductive structures.

Based on the results, the following design guidelines can be derived, each improving both the robustness of the manufacturing process and the electrical conductivity:

• Choose the largest possible nozzle diameter.

• Select a layer height between 40% and 60% of the nozzle diameter.

In this study, the strand width was determined by the nozzle diameter, requiring reconfiguration of the AM machine for each change. As mentioned earlier, strand width can also be adjusted to some extent purely through by controlling the flow rate. Further investigations are necessary to determine whether this approach yields comparable results.

In addition to resistivity, the surface roughness of all specimens was analysed. It was found that surface roughness depth increases with both higher layer heights and wider strands, potentially impacting the resistivity.

Finally, this study focused exclusively on single-layer samples. Future research should investigate the additional influences that may arise in multi-layer components to further expand these findings and needs to confirm the found trends in other composite materials, particularly those with spherical fillers such as CB.

The authors would like to thank Alexander Dijkshoorn for the inspiring exchange, Marijn Goutier for his assistance with the use of the statistical software and Christopher Gassen for his help with the photography.

Maximilian Nowka: conceptualization, methodology, formal analysis, investigation, data curation, writing – original draft, review and editing, visualization. Katja Ruge: data curation, investigation, visualization, writing – original draft. Karl Hilbig: writing – original draft, supervision, project administration, funding acquisition, resources. Lukas Schulze: writing – original draft. Thomas Vietor: supervision, project administration, funding acquisition, resources.