Microstructure Statistical Symmetry, and Quantification of Anisotropic Thermal Conduction in Additive Manufactured Short Carbon Fiber/Polyetherimide Composites

Abstract

This work presents a microstructure-informed pathway for assigning a material symmetry class to distinguish between tensor components and scalar effective thermal conductivity (ETC) values derived from directional measurements. The framework combines directional thermal measurements with three-dimensional statistical quantification of microstructural features (fibers and voids) to assess whether symmetry assumptions required for tensorial interpretation are justified. Three distinct microstructures of short carbon fiber-reinforced polyetherimide composite were analyzed, with the microstructure statistics altered by the melt extrusion additive manufacturing process parameters. The directional temperature-rise history in the material samples was measured using the Transient Plane Source sensor. The statistics obtained from 3D images of microstructural features were used to assess the material’s anisotropy class to justify the applicability of the transverse isotropic regression method for ETC. One microstructure exhibited characteristics consistent with a statistical transverse isotropy idealization, enabling inference of the ETC tensor; the others did not, and their directional ETC values are treated as test-specific parameters obtained from isotropic model fits. The results also demonstrate that microstructure parameters may strongly influence directional thermal transport. More broadly, this work highlights the need for microstructure-informed justification when interpreting directional measurements as tensor components rather than configuration-dependent scalars, underscoring a critical unresolved gap in the experimental characterization of general anisotropic ETC tensors.

1. Introduction

Thermal conductivity (TC) of polymers and fiber-reinforced polymer composites (FRPC) remains a largely underdeveloped subject, both from fundamental and technical perspectives, despite significant progress in theoretical analysis and experimental characterization over the past several decades [1,2]. The mechanisms of TC in polymers are acknowledged as predominantly phonon transport-controlled, yet broad theoretical understanding and the empirical evidence of the effects of molecular structure, crystallinity, temperature dependence [3]—especially near thermal transitions [4]—are arguably limited for homopolymers (e.g., [5,6,7,8]), and even more so for multi-component systems (copolymers, blends, e.g., [9,10]). Ongoing since the 1950s, efforts [11,12] in TC measurement resulted in several mainstream experimental techniques, supported by their mathematical models and instrumentation. These experimental methods can be broadly categorized [13] into steady-state and transient techniques and rely on isotropic material regression models except for the transverse isotropy option in the Transient Plane Source (TPS) method [14,15] (though the principal directions of TC must be assumed beforehand). While multiple studies demonstrated that adding conductive fibers or fillers to polymers yields anisotropic TC [14,15,16,17], measuring general anisotropic TC is a longstanding and unresolved challenge. This includes difficulty in determining when constrained anisotropy (transverse isotropy) can be legitimately assumed for composites with complex, biased phase morphologies, such that an appropriate tensorial description of TC can be applied.

Macroscopic anisotropy in TC of FRPC originates from the heterogeneity at the micro- and mesoscales. Key microstructure parameters in short FRPC include fiber volume fraction (VF), fiber length and diameter distribution, fiber orientation distribution (FOD), void VF, and void morphology. Alignment of fibers enhances conductivity along the alignment direction, and increased fiber VF typically results in a higher effective TC. Voids disrupt thermal pathways and reduce conductivity, especially if elongated or directionally aligned (as suggested by prior theoretical models [18,19,20,21,22]). Such microstructures naturally form during melt extrusion (ME) additive manufacturing (AM), as shown in Figure 1a–c, which encompasses both filament-fed (Fused Filament Fabrication, FFF) and pellet-fed (Fused Pellet Modeling/Manufacturing, FPM) 3D printing processes [23]. In these systems, the flow kinematics within the heated nozzle promote shear-induced fiber alignment during extrusion of individual printed beads. Upon deposition, fibers in the outer shell of a bead exhibit greater alignment with the print direction, while the bead core often shows more random orientation [14,24,25,26]. Porosity inevitably develops [27] in the form of (i) interbead voids at the interface between adjacent extruded paths and (ii) intrabead voids within a single path/bead. Voids in 3D-printed FRPC primarily form due to thermal gradients, viscous flow characteristics, and pressure differentials during deposition [28,29,30]. The intrabead voids are typically smaller than the interbead voids [31], and are difficult to control by changing printing parameters [32,33]. Interbead voids may be reduced using mechanical tampers or rollers [34]; however, many commercial FPM systems do not include such features. Intrabead voids, in principle, can be reduced through improved mixing (e.g., by increasing screw length or modifying shear conditions) or by raising the extrusion temperature, though at the risk of causing more fiber attrition or polymer degradation. These trade-offs make void control a non-trivial problem, highlighting the need for further research into process/structure relationships in fiber-filled ME AM systems. There is evidence in the literature that voids can align with a reinforcing [35] or processing direction [36] in response to similar flow or stress fields that orient fibers, albeit mostly in contexts other than short-FRPC. To the best of our knowledge, no prior study has quantitatively reported void orientation tensors in short-FRPC. These features, fiber orientation gradients and spatially distributed voids, define the micro- and meso-structure of the 3D-printed material and govern the direction-dependent heat conduction observed macroscopically. Given the increasing industrial use of 3D-printed short-fiber composites for high-temperature tooling [37] and structural parts, understanding the directional dependence of effective TC (ETC) is important for process design, part performance, and thermal management applications. Talabi et al. [38] demonstrated that microstructural parameters (fiber orientation, fiber length, and porosity) can be influenced by process conditions (nozzle diameter, screw speed, and temperature) through a systematic study of carbon fiber-filled ABS processed via large-format AM. Additionally, layer thickness and line width have been reported to influence thermal conduction in 3D-printed reinforced polymers [39].

In a homogeneous anisotropic medium, the constitutive relationship between the heat flux vector $\mathrm{q}\left(x,t\right)={\left\{q_1,q_2,q_3\right\}}^T$ and the temperature gradient $\nabla T$ obeys Fourier’s law, which is expressed in the Cartesian coordinate system $x=\left(x_1,x_2,x_3\right)$ by Equation (1) [40]:

$$\mathrm{q}=-\mathrm{K}\nabla T\leftrightarrow \left\{\begin{array}{c}{q}_1\\ q_2\\ q_3\end{array}\right\}=-\left[\begin{array}{ccc}{k}_{11}& k_{12}& k_{13}\\ k_{21}& k_{22}& k_{23}\\ k_{31}& k_{32}& k_{33}\end{array}\right]\left\{\begin{array}{c}\partial T/\partial x_1\\ \partial T/\partial x_2\\ \partial T/\partial x_3\end{array}\right\}$$

(1)

where $k_{ij}\left(i,j=1, 2, 3\right)$ are the components of the second-order TC tensor $\mathrm{K}$; the TC tensor is symmetric (in the absence of magnetic fields) $k_{ij}=k_{ji}\left(i,j=\mathrm{1,2},3\right)$ [41,42]; $k_{ii}>0; k_{ii}{k}_{jj}>k_{ik}^2 \left(i\ne j\right)$, to ensure the positive-definiteness of the quadratic form. There exists a new rectangular coordinate system $\xi =\left({\xi }_1,{\xi }_2,{\xi }_3\right)$ that transforms a given TC tensor to a diagonal form, as shown in Equation (2):

$${\mathrm{K}}_{\left(x_1,x_2,x_3\right)}=\left[\begin{array}{ccc}{k}_{11}& k_{12}& k_{13}\\ \dots & k_{22}& k_{23}\\ symm& \dots & k_{33}\end{array}\right]\to {\mathrm{K}}_{\left({\xi }_1,{\xi }_2,{\xi }_3\right)}=\left[\begin{array}{ccc}{k}_1& 0& 0\\ 0& k_2& 0\\ 0& 0& k_3\end{array}\right]$$

(2)

where $k_1$, $k_2$, and $k_3$ are called the principal conductivities along the principal directions of TC (or, principal thermal transport axes), ${\xi }_1$, ${\xi }_2$, and ${\xi }_3$, respectively. The principal directions $\xi$ of TC are not known a priori and generally are not coincident with arbitrarily selected reference directions $x$, as schematically shown in Figure 1d,e. Experimental characterization of anisotropic TC means measuring a fully populated ${\mathrm{K}}_{\left(x_1,x_2,x_3\right)}$ matrix in selected reference axes $\left(x_1,x_2,x_3\right)$ and determining the principal directions and values to ensure the data is usable and transferable for further analysis through tensor transformations. In general, the extent of anisotropy within macroscopic TC is controlled by the directional biases of the microscale heterogeneity. In crystallography, Neuman’s principle [43] applies to perfect crystals and states that “the symmetry elements of any physical property tensor must include the symmetry elements of the crystal’s point group”. This principle is a specific case of Curie’s broader symmetry principle (cause and effect must share common symmetries) [44]. Neumann’s principle was extended to stochastic and heterogeneous media [45,46]: any symmetry (or lack thereof) in the spatial statistics of a material will be reflected in the symmetry of its effective property tensors [47]. In contrast to perfect crystalline solids, where symmetry is defined through exact geometric repetition and point-group operations, statistical symmetry refers to invariance in the ensemble-averaged structure of a heterogeneous material. For a short-FRPC, the microstructure is inherently irregular (individual fibers vary in position, length, orientation, and local spacing), but one can still describe its large-scale structure using statistical descriptors, such as the FOD.

A short-FRPC microstructure is said to exhibit statistical symmetry if the probability distribution governing fiber orientation state is invariant under certain symmetry operations [48]. When materials exhibit structural symmetry, the symmetry axes should coincide with the principal directions of the ETC tensor, and materials can be classified as isotropic, transversely isotropic, or orthotropic (general anisotropic) [49]. If the fiber orientations are equally likely in all directions, the composite is considered statistically isotropic, i.e., TC is expected to appear independent of direction at the macroscopic scale ($k_{ii}=k; k_{ij}=0,i\ne j$). With three orthogonal preferred directions in the FOD, the short-FRPC may be approximated as statistically orthotropic ($k_1\ne k_2\ne k_3$). If fibers are preferentially aligned along one axis (the axis of symmetry) but randomly distributed in the transverse plane (perpendicular plane of rotational symmetry), the composite is transversely isotropic and only two distinct ETC values are expected—along the axis of symmetry and in the plane perpendicular to it. Assuming a particular symmetry class (e.g., transverse isotropy) for the ETC tensor requires quantitative justification. For example, Shemelya et al. [50] presented experimental evidence of direction-dependent TC for FFF-printed ABS-based composite, but their analysis implicitly assumed transverse isotropy (print direction vs. build plane) without quantitatively justifying this symmetry class. In general, the orientation of statistical symmetry axes may not coincide with the chosen reference frame and must be inferred from the microstructure itself. If such symmetry cannot be established, the material must be treated as fully anisotropic, in which case the ETC tensor is orthotropic, and its principal directions may not coincide with the selected reference directions.

Experimental characterization of TC is an inverse problem [40]. The temperature response of a specimen is measured and then used to infer the material’s diffusivity/conductivity through regression against an analytical or semi-analytical solution of the heat conduction problem, which is emulated by the experimental setup. No closed-form analytical solutions exist for the general 3D heat conduction problem in fully anisotropic solids, and, as a result, there is no existing experimental method capable of determining all components of the TC tensor in orthotropic materials. The only exception is the TPS method configured for transverse isotropy, which can extract two independent conductivity values (along and across the axis of symmetry, $\mathrm{K}=diag\left(k_1,k_2,k_2\right)$, but only when the principal directions of TC are known and aligned with the experimental setup.

One common practice found in the literature (e.g., [20,51,52,53,54]) is taking an anisotropic material and performing directional measurements of a transient or steady-state thermal response in a defined coordinate system using one of the standard techniques, with a common pitfall of applying data regression that assumes the material is isotropic—i.e., experiments fit each directional test to an isotropic model, effectively treating the material as if it had a single TC in that orientation. The literature providing these directional measurements is valuable evidence that heterogeneous composites exhibit anisotropic heat conduction, often linking this behavior to microstructural features. However, such “TC numbers” must be interpreted with care. It is important to recognize that the reported TC from such tests is not a fundamental material constant in the rigorous heat-transfer sense when the material is anisotropic. A single scalar number obtained by an isotropic fit or one-directional test on an anisotropic sample does not correspond to any one tensor component (i.e., they are not immutable material properties under Fourier’s law); instead, it yields what is essentially an ETC parameter/coefficient for that test configuration, which lumps together material’s directional responses and the boundary conditions of the experiment.

In this work, we investigate short carbon fiber (CF)-reinforced polyetherimide (PEI) composites fabricated via ME AM, using both FFF and FPM approaches, aiming to characterize their anisotropic ETC in relation to microstructure. We focus on experimentally measuring ETC along different directions using the TPS technique, while directly quantifying the FOD and void morphology using 3D imaging. While μCT-based microstructure characterization and TPS measurements of directional thermal conduction have been employed in prior studies, the contribution of this work lies in the interpretation of three-dimensional microstructure statistics. Based on this microstructural evidence, we assess whether statistical symmetry conditions are sufficient for assuming transverse isotropy. For the FFF sample, we quantitatively justify the assumption of statistical transverse isotropy based on the measured microstructure and directional temperature response, which allows us to infer the components of the TC tensor. In contrast, for two FPM-printed samples with different layer heights (1.4 mm and 2.4 mm), the microstructure does not support transverse isotropy, and the directional ETC values are interpreted as test-specific parameters derived from isotropic model fits. The observed differences in ETC between the FPM samples are attributed to variations in fiber alignment and void morphology, demonstrating that process parameters can be actively used to tailor directional thermal transport in 3D-printed short-FRPCs. While directional ETC parameters reflect the influence of microstructure, these results also point to an unresolved gap in how general anisotropic TC tensors can be experimentally characterized—a challenge that remains open for future work.

2. Experimental Methods

2.1. Materials and Sample Manufacturing

Short CF/PEI (SCF/PEI) composites were processed using two ME AM methods, FFF and FPM. Commercially available feedstock was used in both cases, sourced from different suppliers. The feedstock materials were printed as received, without modification of fiber content or formulation. Four sets of samples were prepared for TC measurements using the TPS method: neat PEI and SCF/PEI printed via FFF, and SCF/PEI printed via FPM at two different layer heights. Additional single-layer SCF/PEI prints were fabricated to prepare samples for microstructure imaging. Throughout this work, a right-handed Cartesian reference frame is used to define directions: $x_1$, $x_2$, and $x_3$, denote the print, transverse, and build directions, respectively (see Figure 1). TPS samples were printed with unidirectional bead orientation. FFF samples were printed directly to 20 × 20 × 20 mm³; FPM samples were printed as larger plates, from which 20 × 20 × 20 mm³ cubes were cut and milled to achieve flat surfaces. All sample faces were polished to remove surface roughness and ensure proper contact with the sensor during TPS testing.

FFF samples were printed using the AON M2+ printer (housed at UBC). The materials included ThermaX™ ULTEM 9085 (neat PEI) and CarbonX™ ULTEM 9085 (SCF/PEI containing 15 wt.% short carbon fiber) (both manufactured by 3DXTECH, Grand Rapids, MI, USA). Before printing, filaments were dried at 90 °C for 4 h and stored in a dry box with molecular sieve desiccant to prevent moisture uptake during printing. Print parameters are listed in

Table 1. Printing parameters used for each type of sample.

	FFF	FFF	FPM 1.4 mm	FPM 2.4 mm
Feedstock	Neat PEI	SCF/PEI	SCF/PEI	SCF/PEI
Bed Temperature (°C)	175	175	90	90
Chamber Temperature (°C)	135	135	N/A	N/A
Nozzle Temperature (°C)	380 (First layer) 365	380 (First layer) 365	400	400
Nozzle Size (mm)	0.6	0.6	12	12
Bead Width, W (mm)	0.65	0.65	15 to 18	15 to 18
Layer Height, H (mm)	0.3	0.3	1.4	2.4
Print Speed (mm/s)	40.0	40.0	14.6	14.6

. While the slicing software did not report volumetric flow rate directly, it was estimated from bead width, layer height, and print speed, yielding approximately 7.8 mm³/s.

FPM samples were printed using the CEAD AM Flexbot system equipped with a G25 extruder with an integrated melt pump (housed at the Carbon Composites Chair at the Technical University of Munich). The feedstock was Airtech Dahltram I-350CF (Airtech, Differdange, Luxembourg), a SCF/PEI composite containing 20 wt.% carbon fiber. Pellets were dried at 125 °C for 8 h before printing. Two sets of samples were produced using different single-layer heights: 1.4 mm and 2.4 mm. Corresponding volumetric flow rates were estimated at approximately 337 mm³/s and 578 mm³/s, respectively. Print settings are detailed in Table 1. Throughout the text, FPM-1.4 and FPM-2.4 refer to samples printed via the FPM process using 1.4 mm and 2.4 mm layer heights, respectively. Figure 1c provides a representative view of the FPM-printed layers, where the relative layer height, bead width, and nozzle size can be visually discerned from the known printing parameters.

Since the PEI used in the FFF and FPM feedstocks originated from different suppliers, Fourier Transform Infrared Spectroscopy (FTIR) was performed to assess chemical consistency between the samples. Spectra were collected using a PerkinElmer Frontier FT-IR spectrometer with standard parameters: scan range 4000–650 cm⁻¹, resolution 4 cm⁻¹, and 32 scans per sample. The similarity of spectral profiles across all PEI-containing samples supports the conclusion that the same base polymer (ULTEM 9085) was used throughout.

The specific heat used in TPS inversion was measured experimentally by Differential Scanning Calorimetry (DSC) using a TA Instruments Discovery DSC 2500 (TA Instruments, New Castle, DE, USA) for neat PEI, as well as for FFF and FPM SCF/PEI samples. The DSC was calibrated for specific heat using a sapphire reference following the instrument manufacturer’s procedure. To collect the data, samples were heated at 10 °C min⁻¹ to 350 °C with a 5 min isothermal hold, cooled to −10 °C, and reheated at 2 °C min⁻¹ to 50 °C. The room-temperature specific heat averaged 1.26 ± 0.09 J g⁻¹ K⁻¹ for neat PEI, 1.00 ± 0.08 J g⁻¹ K⁻¹ for FFF SCF/PEI, and 0.87 ± 0.06 J g⁻¹ K⁻¹ for FPM SCF/PEI. The lower specific heat of the composites relative to neat PEI reflects the lower heat capacity of carbon fibers, while the difference between FFF and FPM samples is attributed to their different fiber VF. Within each manufacturing route, the measured specific heat showed no systematic variation beyond experimental uncertainty.

2.2. Microstructure Imaging

Two types of microstructure imaging techniques were used in this work: X-ray micro-computed tomography (µCT) and optical microscopy. µCT enables non-destructive three-dimensional mapping of internal features, allowing identification and statistical characterization of voids and fibers. Optical microscopy complements this by providing high-resolution two-dimensional images with finer morphological details (sharper view of void shapes) on polished cross-sections.

For microscopy analysis, printed composite samples were sectioned perpendicular to the reference directions. The sections were mounted in standard casting epoxy and polished using a conventional sequence, which included SiC grinding with decreasing grit sizes, followed by rough, intermediate, and final polishing steps. The prepared surfaces were examined under a Nikon Eclipse MA200 reflected-light optical microscope. Imaging was performed at multiple magnifications. Example micrographs are shown in Figure 2. Figure 2a shows a cross-section ($x_2{x}_3$ plane) of a single extruded filament from the FFF process. Figure 2b shows the $x_1{x}_3$ plane across multiple FFF printed layers. Figure 2c,d shows selected regions of cross-sectional cuts from FPM-1.4 and FPM-2.4 samples, respectively. These represent segments of single-layer (single-bead) extrudates, each approximately 15 mm wide.

The µCT scans were conducted on single-bead segments of FFF, FPM-1.4, and FPM-2.4 samples. The scanned segment dimensions were 2 mm in length for the FFF filament, 2.8($x_1$) × 2.8($x_2$) × 1.4($x_3$) mm³ for FPM-1.4, and 3($x_1$) × 2($x_2$) × 2.4($x_3$) mm³ for FPM-2.4. Scans were performed using a Zeiss Xradia 520 Versa µCT system (Zeiss, Oberkochen, Germany). Each sample was rotated through 360°, acquiring a series of 2D projection images. Detailed scan parameters are provided in

Table 2. µ-CT scanning parameters of SCF/PEI samples.

Scanning Parameters\Sample ID	FFF	FPM-1.4 mm	FPM-2.4 mm
Resolution (µm)	0.7	3	3
Number of Projections	1601	3201	3201
Exposure Time (s)	1.2	0.5	1
Voltage (kV)	60	40	40
Power (W)	5	3	3

. Tomographic reconstruction of raw 2D radiographic projections was performed using XMReconstructor (Zeiss, Oberkochen, Germany), yielding 2D grayscale image stacks that represent cross-sectional slices of the scanned volume. These were imported into VGStudio Max 2024.1 [55] for alignment, visualization, and quantitative analysis of the resulting 3D data maps of local relative density. Figure 3a,b shows two examples of 2D view (slices) from the $x_2$-directions of the FFF and FPM-1.4 samples to illustrate the contrast: the distinct variations in density among the fibers, voids, and polymer are evident. The fibers appear denser and are represented by brighter regions, the polymer matrix is seen in intermediate shades, and the voids appear as dark areas. The resolution of the density map is sufficient to distinguish fibers and voids, making these images suitable for further quantitative analysis.

2.3. Microstructure Parameters Characterization Methods

2.3.1. Fiber Length Distribution

Individual fibers were isolated from printed composite samples using a matrix burn-off process. Small pieces of material were placed in ceramic boats and subjected to controlled heating in a Thermolyne 1500 furnace (Thermo Fisher Scientific, Waltham, MA, USA): ramped to 350 °C with a 5 min hold, then increased to 500 °C and held for one hour to fully decompose the PEI matrix without damaging the carbon fibers. The remaining fibers were dispersed in silicone oil on glass slides and imaged under a Nikon Eclipse MA200 (Nikon, Tokyo, Japan) reflected-light microscope at 20× magnification. Representative micrographs are shown in Figure 4a,b. Only fibers fully contained within the micrographs were included in the analysis. ImageJ 1.54 was used to quantify fiber lengths from optical images. Measurements were conducted on both FFF and FPM samples. For FPM, only one set was analyzed, as both print conditions used the same feedstock. A total of 110 short carbon fibers were measured in the FFF sample, and 171 short carbon fibers were measured in the FPM sample.

2.3.2. Fiber Orientation Distribution

FOD was characterized using µCT data for all SCF/PEI samples, while optical micrographs were analyzed for the FFF sample only. The 3D volumes were post-processed in VGStudio Max 2024.1 using Adaptive Gauss and Non-Local Mean filters to reduce voxel-level noise and enhance segmentation accuracy. Fibers were segmented via surface determination using gray value thresholds. An example of segmented fiber (red) and void (blue) phases overlaid on the raw scan is shown in Figure 5b, derived from the grayscale µCT image in Figure 5a. The collection of unit vectors $\mathit{p}={\left[p_1,p_2,p_3\right]}^T={\left[\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta \right]}^T$, Figure 5c, aligned with the fiber axes, defines the FOD. Fiber orientation was quantified using VGStudio’s Fiber Composite Material Analysis module, which assigns orientation vectors to identified fiber voxels based on density gradient analysis and outputs the second-order orientation tensor a_ij [48], the number-average of dyadic products of fiber orientation vectors ${\left\{p_n\right\}}_{n=1}^N$, as defined by Equation (3):

$$\begin{gathered} a_{ij}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ \dots & a_{22}& a_{23}\\ symm& \dots & a_{33}\end{array}\right]={\displaystyle \frac{1}{N}}{\sum }_{n=1}^N{p}_n⨂p_n; \\ {a}_{11}+a_{22}+a_{33}=1;0\le a_{ii}\le 1;-0.5\le a_{ij}\le 0.5\left(i\ne j\right) \end{gathered}$$

(3)

From optical micrographs, FOD was quantified with an ellipse-based method implemented in MATLAB R2024a. A single 5× reflected-light image of a $x_2{x}_3$ cross-section of the FFF sample (approx. 3.2 × 2.2 mm², spanning several printed layers) was divided into 84 equal regions of interest (12 rows × 7 columns) for analysis. Within each ROI, fiber cross-sections were segmented with watershed and morphological filters, then fitted with ellipses using MATLAB regionprops function [56]. The fitted major (M) and minor (m) axes yield the apparent tilt angle $\phi$ of each fiber, Figure 5d, $\phi ={\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(m/M\right)$. Because a 2D $x_2{x}_3$ cross-section provides no information about the out-of-plane component, analysis assumes in-plane alignment (${\theta }_n=\pi /2;a_{33}=0$). Moreover, a fiber at angle $\phi$ is indistinguishable from one at $\phi +\pi$; the sign of $p_1{p}_2$, therefore, cannot be resolved, so $a_{12}$ is not evaluated. The procedure thus delivers only the in-plane diagonal terms of the second-order orientation tensor: $a_{11}={\displaystyle \frac{1}{N}}{\sum }_{n=1}^N{\mathrm{c}\mathrm{o}\mathrm{s}}^2{\phi }_n; a_{22}=1-a_{11}$.

2.3.3. Void Morphology and Orientation Distribution

Voids were analyzed as low-density features using the same inclusion framework applied to fibers. Although voids are not inclusions in a physical sense, they were characterized statistically in terms of orientation and morphology. The µCT datasets for each sample were processed in VGStudio Max 2024.1 using the Porosity/Inclusion module. Void regions were segmented by gray-value thresholding and filtered to exclude features smaller than five voxels. Orientation tensors were computed using the Fiber Composite Material Analysis module, treating voids as elongated features aligned in 3D space. Morphological characterization of voids was based on three geometric parameters: sphericity, compactness, and elongation. Sphericity is defined as the ratio of the surface area ($A_{sphere}$) of a sphere with the same volume as the void to the actual surface area of the void ($A_{void}$), Figure 6a. Compactness is defined as the ratio of the void volume ($V_{void}$) to the volume of the circumscribed sphere ($V_{sphere}$), Figure 6b. Elongation is quantified from the bounding box dimensions of each void, as the ratio of the largest dimension to the smallest. All parameters were computed by VGStudio and exported for statistical analysis.

2.4. TPS Measurements of ETC

TC measurements were conducted using the TPS 2500S system (Hot Disk, Gothenburg, Sweden) at 22 °C. A Kapton-insulated sensor (Hot Disk 5465) having a nickel double-spiral heater with a radius ($r$) of 3.189 mm was used for all tests. Each test configuration involved placing the sensor between the same orthogonal faces ($x_1{x}_2$, $x_1{x}_3$,$x_2{x}_3$) of two identical specimen blocks (2 × 2 × 2 cm³), oriented such that the reference direction of interest was normal to the sensor plane. This configuration ensures that heat flows along the target direction during the measurement. The specimen dimensions satisfy the recommended criteria of at least 5× the sensor radius in width and 2× in thickness. A constant clamping force of 25 N was applied to ensure complete contact between the sensor and specimen surfaces. During each test, the sensor simultaneously heated the material and recorded its own temperature rise $∆T$ over time $t$, via calibrated resistance changes. The measurement time ($t_{meas}$), input heating power ($P_0)$, and sensor size were preset, and the measured sensor temperature rise is used to compute thermal conductivity through inverse analysis.

TPS analysis of TC relies on solving an inverse problem based on the analytical solution of the transient heat transfer problem developed by Gustafsson [57], which predicts the temperature rise $∆T\left(t\right)$ of a spiral heat source embedded in an infinite medium with thermal conductivity $k$, diffusivity $\alpha$, and constant power input $P_0$. The model describes $∆T\left(t\right)=f\left(t,r,P_0,k,\alpha \right)$ as a function of time $t$, power $P_0$, sensor radius $r$, and material properties, as shown in Figure 7a,b. The TPS sensor acts as both heater and thermometer, Figure 7c,d, enabling measurement of real-time $∆T\left(t\right)$ curves. These experimental curves are used to iteratively adjust model parameters until the solution aligns with the observed response, yielding estimates of $k$ and $\alpha$. The TPS model assumes either isotropic or transversely isotropic behavior; accordingly, measurements may be regressed using either assumption to yield a scalar conductivity value $k$ or a transverse isotropic conductivity tensor $\mathbf{K}=diag\left(k_1,k_2,k_2\right)$.

Multiple measurements were performed for each direction of each sample, with cooldown periods between runs. The software exported $∆T\left(t\right)$ data at 200 equally spaced time intervals per run. Heating time ($t_{meas}$) and power input ($P_0$) were tuned to maintain $∆T$ within 0.4 °C and 4 °C, which is the recommended range for accurate readings without overheating the sensor. Figure 8a shows the grand mean and 95% confidence interval of $∆T\left(t\right)$ over 50 identical measurements in the $x_1$-direction on an FFF sample, as an example demonstrating measurement reproducibility. The $∆T\left(t\right)$ curves exhibit two regimes: Region 1 (initial) is dominated by the contact resistance and thermal mass of the sensor, while Region 2 reflects the true thermal response of the material. The first 20–25 data points (Region 1) are excluded from regression due to the mentioned artifacts.

In the isotropic case, the average temperature rise in the double-spiral heater can be expressed [57] as Equation (6):

$$∆\overline{T}\left(\tau \right)={\displaystyle \frac{P_0}{{\pi }^{3/2}rk}}D\left(\tau \right)$$

(4)

where $\tau =\sqrt{t/\theta }$ is the so-called dimensionless time, with $\theta =r^2/\alpha$ being the characteristic time, and $D\left(\tau \right)$ is a dimensionless function determined by Gustafsson’s analytical solution. The dimensionless time $\tau$ arises from the Green’s function solution to the transient heat conduction equation and enters the analytical expression for temperature rise through a known integral function $D\left(\tau \right)$. The relationship between $∆\overline{T}\left(\tau \right)$ and $D\left(\tau \right)$ is theoretically linear, and thermal properties are extracted by the TPS software (Hot Disk, Gothenburg, Sweden) via an internal iterative optimization routine that adjusts thermal diffusivity $\alpha$ to achieve this linearity. Once the best-fit line is obtained, its slope is used to compute thermal conductivity $k$, according to Equation (4). Figure 8b shows a representative fit of $∆\overline{T}\left(\tau \right)$ vs. $D\left(\tau \right)$. Each measurement is validated against two constraints to ensure consistency with the analytical solution assumptions: (i) the probing depth ${∆}_p$, Equation (5), should be less than about half the smallest sample dimension perpendicular to the sensor plane (see Figure 7e), and (ii) that the total-to-characteristic time ratio (TTCT, Equation (6)) must fall within recommended limits. The TPS software module reports both parameters. Equations (5) and (6) are as follows:

$${∆}_p=2\sqrt{\alpha ·t_{meas}};W>2{∆}_p+2r;H>2{∆}_p$$

(5)

$$0.33<TTCT=\alpha ·t_{meas}/r^2<1$$

(6)

For a transversely isotropic material, where $x_1$ is the axis of symmetry (out-of-plane or longitudinal direction) corresponding to the TC of $k_1$, and the $x_2{x}_3$ plane is the transverse plane of isotropy (i.e., $k_2=k_3$), the TPS model fits the experimental temperature rise using the corresponding Gustafsson’s solution, Equation (7) [58]:

$$∆\overline{T}\left({\tau }_1\right)={\displaystyle \frac{P_0}{{\pi }^{3/2}r{\left(k_1{k}_2\right)}^{1/2}}}D\left({\tau }_1\right)$$

(7)

where ${\tau }_1=\sqrt{t/{\theta }_1}$ with ${\theta }_1=r^2/{\alpha }_1$. As in the isotropic case, the system solves an optimization problem over the measured $∆T\left(t\right)$ data to determine thermal diffusivity ${\alpha }_1$ by achieving a linear fit of $∆\overline{T}\left({\tau }_1\right)$ vs. $D\left({\tau }_1\right)$. With known volumetric heat capacity $C$, both TC values $k_1$ and $k_2$ can be extracted from the slope of Equation (7).

For all three sample types, 50 repeated measurements were performed in each of the reference directions ($x_1$, $x_2$, $x_3$). For the FFF samples, the heating power was set to 20mW, 15 mW, and 15 mW with measurement times of 40 s, 20 s, and 20 s for the $x_1$-, $x_2$-, and $x_3$-directions, respectively. For the FPM-1.4 mm samples, the power settings were 25 mW, 15 mW, and 20 mW, and the measurement times were 20 s, 10 s, and 10 s for the $x_1$-, $x_2$-, and $x_3$-directions, respectively. For the FPM-2.4 mm samples, the power was set to 20 mW, 15 mW, and 15 mW with corresponding times of 20 s, 10 s, and 10 s for the $x_1$-, $x_2$-, and $x_3$-directions, respectively. Representative $∆T\left(t\right)$ transients for each sample type and measurement directions are shown in Figure 9. In all cases, output power and test duration were selected to ensure compliance with system calibration limits and probing depth constraints. Interpretation of the measured $∆T\left(t\right)$ data and extraction of TC values are presented in the following chapter.

3. Results and Discussion

3.1. Comparative Analysis of Microstructure Parameters in SCF/PEI AM Composites

The number-average, ${\overline{L}}_n={\sum }_i{n}_i{L}_i/{\sum }_i{n}_i$, and weight-average, ${\overline{L}}_w={\sum }_i{n}_i{L}_i^2/{\sum }_i{n}_i{L}_i$, fiber lengths (where $n_i$ is the number of fibers of length $L_i$) were calculated from the measured fiber length datasets, whose distributions are shown as histograms in Figure 4c. For the FFF SCF/PEI sample, ${\overline{L}}_n=67 \mathsf{\mu }\mathrm{m}$ and ${\overline{L}}_w=106 \mathsf{\mu }\mathrm{m}$; for the FPM SCF/PEI samples, ${\overline{L}}_n=128 \mathsf{\mu }\mathrm{m}$ and ${\overline{L}}_w=161 \mathsf{\mu }\mathrm{m}$. The number-average length of FPM SCF/PEI was nearly twice that of FFF SCF/PEI. The narrower fiber length distribution in the FPM material is evident from the lower ${\overline{L}}_w/{\overline{L}}_n$ ratio and the visual shape of the histogram in Figure 4c.

The FOD in the FFF SCF/PEI composite was quantified using both $x_2{x}_3$ cross-section micrographs ($a_{ij}^{FFF/micrograph}$) and volumetric CT data ($a_{ij}^{FFF/\mu CT}$):

$$a_{ij}^{FFF/micrograph}=\left(\begin{array}{ccc}0.8& N/A& N/A\\ \dots & 0.2& N/A\\ symm& \dots & N/A\end{array}\right); a_{ij}^{FFF/\mu CT}=\left(\begin{array}{ccc}0.79& -0.01& 0.01\\ \dots & 0.11& 0\\ symm& \dots & 0.1\end{array}\right).$$

While the ellipse method provides only the in-plane components $a_{11}$ and $a_{22}$, the agreement of $a_{11}$ with µCT data gives confidence in the automated VG Studio analysis. The full tensor from µCT shows $a_{22}$ and $a_{33}$ are close, and off-diagonal terms are small, suggesting that the FOD principal directions are approximately aligned with the reference directions. The µCT-measured orientation tensors for FPM-1.4 and FPM-2.4 were as follows:

$$a_{ij}^{FPM-1.4/\mu CT}=\left(\begin{array}{ccc}0.68& -0.07& -0.16\\ \dots & 0.25& 0.02\\ symm& \dots & 0.07\end{array}\right); a_{ij}^{FPM-2.4/\mu CT}=\left(\begin{array}{ccc}0.53& 0& 0.07\\ \dots & 0.35& -0.02\\ symm& \dots & 0.12\end{array}\right).$$

The FPM-1.4 sample shows greater alignment along the print direction ($x_1$) compared to FPM-2.4. This trend is consistent with expected flow behavior during deposition: for the same print speed and travel distance, a thinner layer height results in higher shear rates due to steeper velocity gradients, promoting stronger fiber alignment in the print direction. The off-diagonal terms are not negligible, given that $a_{ij}\in \left[-0.5; 0.5\right]$ for $i\ne j$, and $a_{22}\ne a_{33}$.

The reconstructed volumes of FPM samples were sliced into layers of thickness $∆x_3=0.1 \mathrm{m}\mathrm{m}$, and each slice was analyzed using VG Studio. Figure 10 shows the variation in fiber VF and the degree of alignment with the print direction ($a_{11}$) along the $x_3$-axis of the print bead. In both FPM-1.4 and FPM-2.4 samples, a gradient is evident: higher fiber alignment near the top and bottom surfaces of the bead, and lower values toward the center. This variation is consistent with the shear-driven alignment mechanism: higher shear stress at the bead boundaries during deposition causes stronger alignment, while the shear diminishes in the core region, allowing more random orientation. Micrographs shown in Figure 2c,d provide qualitative confirmation of these trends. In these images, the gray regions represent the polymer matrix, the fibers appear as white spots due to their higher reflectivity compared to the surrounding matrix; the dark areas are voids. Near the print bed (top of the image), a densely packed layer of well-aligned fibers is visible, followed by a porous region with more randomly distributed fibers. This gradation supports the trends extracted from the µCT data. Together, these observations reinforce confidence in the fiber orientation statistics (extracted alignment) and porosity distributions obtained from µCT reconstructions, justifying their use in subsequent discussion. Interestingly, the void orientation tensors extracted from the µCT data match the FOD tensor for each sample type, indicating that voids tend to align with the local fiber network. This suggests that the formation and elongation of voids are governed by the same flow-induced deformation field that aligns the fibers.

Void volume fraction values extracted from the µCT data were 25%, 32.8%, and 29.7% for the FFF, FPM-1.4, and FPM-2.4 samples, respectively. Figure 5b highlights the segmented voids identified by the VGStudio/VGDefX algorithm. The staircase graphs in Figure 11a show the variation in void VF along 0.1 mm increments in the $x_3$-direction for the FPM-1.4 mm and FPM-2.4 mm samples. The large regions of high porosity can be visually confirmed in the micrographs in Figure 2. The reconstructed µCT scans show distinct differences in the void morphology between the FPM-1.4 and FPM-2.4 samples. Figure 11b–d reports statistical distributions of their void shape metrics, namely, sphericity, elongation, and compactness. A sphericity value of 1 indicates a perfect sphere, while values less than one indicate increasing deviation from a spherical shape. High sphericity values suggest more regular, rounded voids, whereas lower values indicate more irregular, elongated, or flattened shapes. Compactness helps in understanding the void’s spatial distribution, i.e., how closely the void’s shape approximates a sphere. Compactness values close to 1 imply that a void’s volume closely matches that of its circumscribed sphere, while low values suggest irregular or extended shapes. Importantly, compactness < 1 does not imply elongation specifically; low compactness can arise from any form of irregularity (branching, flattened, or jagged features). Figure 6c,d shows two representative voids from the FPM-2.4 sample, along with their corresponding metrics, to visualize the shape descriptors. The void shape metrics in Figure 10a–b indicate that the FPM-2.4 sample has a sphericity distribution shifted toward higher values, with most voids centered between 0.6 and 0.9, while the FPM-1.4 distribution is broader and centered closer to 0.5. The elongation distribution for FPM-1.4 spans a broader range and extends beyond six, whereas FPM-2.4 shows a narrower distribution concentrated between one and two. Compactness values in FPM-2.4 span a broader range but are more evenly distributed, while FPM-1.4 has a sharp peak below 0.1, indicating a large population of low-compactness voids. These statistical trends are consistent with the more irregular and extended void structures visible in the µCT reconstructions of the FPM-1.4 sample in Figure 3d.

3.2. Thermal Conductivity Measurements in SCF/PEI Composites

3.2.1. Interpretation of the TPS-Measured Temperature Rise for Macroscopically Anisotropic Materials

In this analysis, we interpret the FOD symmetry as representative of the microstructural symmetry class (since the $a_{ij}$ tensors computed for fibers and voids are numerically identical to three decimal places). The orientation tensor $a_{ij}$ represents the second moment of the FOD function, capturing the statistics of the average alignment state within a representative volume. Its eigenvectors define the principal directions of orientation, and its eigenvalues quantify the degree of alignment along those directions. When the eigenvalues of $a_{ij}$ differ, they indicate preferred orientation axes, i.e., anisotropy in the microstructure’s statistical structure. In this sense, the principal directions of $a_{ij}$ reflect the statistical symmetry axes of the fiber network. These symmetry axes, in turn, constrain the form of the ETC tensor, which is expected to share the same principal directions under homogenization.

In the TPS method, interpreting the measured temperature rise relies on solving a forward heat conduction problem with known symmetries. Since analytical solutions are available for isotropic and transversely isotropic materials, it is the user’s responsibility to justify whether the assumed homogenized formulation is appropriate. This justification must be based on both microstructural statistics (e.g., orientation tensor eigenvalues) and any directional dependence observed in the measurement data. No general solution exists for orthotropic TC in the TPS framework; even if the principal directions of anisotropy are known, there is no analytical model to extract the full TC tensor from the temperature rise data. Using an isotropic regression model to extract directional TC values assumes the same scalar diffusivity applies in all directions and that the temperature field is spherically symmetric. In an orthotropic material, heat spreads anisotropically, and the temperature field is inherently direction-dependent. The analytical solution used in isotropic regression does not capture this behavior. Even if the measurement directions align with the principal axes, the underlying physics is not governed by the isotropic heat equation. As a result, fitting an isotropic model to anisotropic data leads to physically invalid estimates of the tensor components. Without a valid forward model for orthotropic conduction, the inversion is not mathematically justified.

3.2.2. Transverse-Isotropic Thermal Conduction of FFF SCF/PEI Composite

Since the fiber orientation tensor $a_{ij}$ of the FFF sample is nearly diagonal in the chosen reference frame, the principal directions of FOD coincide with the reference axes, and the diagonal elements $a_{11}$, $a_{22}$,$a_{33}$ can be interpreted as the eigenvalues ${\lambda }_1$,${\lambda }_2$,${\lambda }_3$. In this case, ${\lambda }_1>{\lambda }_2\cong {\lambda }_3$, indicating a strong alignment of fibers along the print direction ($x_1$) with no preferential orientation within the transverse plane ($x_2{x}_3$). When two eigenvalues are equal, and their associated eigenvectors lie in a plane (here, $x_2{x}_3$), this implies rotational invariance of the orientation distribution within that plane, i.e., fibers are equally likely to be oriented in any direction in the transverse plane (i.e., azimuthally uniform). This is the statistical equivalent of isotropy in the $x_2{x}_3$ plane. While the FOD of the FFF sample is not fully isotropic (as would be the case if all three eigenvalues were equal to 1/3), it exhibits planar statistical isotropy. Together, these features support interpreting the SCF/PEI FFF microstructure as statistically transversely isotropic, with the axis of symmetry aligned with the dominant fiber orientation along $x_1$.

The rotational symmetry in the $x_2{x}_3$ plane is expected to produce an isotropic macroscopic thermal response within that plane. This is confirmed by the TPS measurements: when the sensor is oriented normal to the $x_2$- and $x_3$-directions (i.e., placed between the $x_1{x}_3$ and $x_1{x}_2$ faces of the samples, respectively), the resulting temperature rise curves—recorded under identical power and duration—overlap, see Figure 9a, indicating equivalent thermal behavior. The agreement between the statistically inferred microstructural symmetry and the directional temperature response further confirms that the FFF sample can be appropriately assumed transversely isotropic. This justifies the use of Gustafsson’s solution for transversely isotropic media to extract ETC components from the temperature-time data with the sensor normal to the symmetry axis. The resulting ETC tensor for the SCF/PEI FFF sample is then expressed as follows (as the average ± standard deviation across repeated measurements):

$${\mathrm{K}}^{FFF}=diag\left(k_1,k_2,k_2\right)=diag\left(0.26\pm 0.011,0.20\pm 0.015,0.20\pm 0.015\right) \mathrm{W}/\mathrm{m}\mathrm{K}$$

Figure 11 shows the extracted $k_1$ and $k_2$ values from repeated measurements, plotted against the TTCT and ${∆}_p$ parameters. The probing depth constraint appears to be satisfied across all measurements. While several TTCT values fall below the commonly cited lower bound of 0.33 (originally recommended for the isotropic Gustafsson model), this threshold does not directly apply to the transversely isotropic formulation. No irregularities were observed in the fitted curves or extracted values, so all measurements were retained.

3.2.3. General Anisotropy in Thermal Conduction of FPM SCF/PEI Composites

For both FPM-1.4 and FPM-2.4 samples, the fiber orientation tensor $a_{ij}$ is fully populated in the reference coordinate system. Standard eigen-analysis of the fiber orientation tensor $a_{ij}$ was used to determine the principal orientation axes in both FPM-1.4 and FPM-2.4 samples. The eigenvalues for FPM-1.4 are (0.73, 0.24, 0.03), and for FPM-2.4 are (0.54, 0.34, 0.12), with all three values clearly distinct. This excludes the possibility of assuming (statistical) transverse isotropy. TPS measurements conducted along the $x_2$- and $x_3$-directions, shown in Figure 9b,c, further confirm directional differences in the thermal response. The off-diagonal components of $a_{ij}$ imply that the principal directions are not aligned with the reference frame. The required frame rotations to align the reference axes with the principal axes of FOD are approximately (1°, 9°, 14°) for FPM-1.4 and (3°, 5°, 11°) for FPM-2.4, along the print, transverse, and build directions, respectively. These results indicate that both FPM samples must be treated as having a fully anisotropic (orthotropic) TC tensor. However, such a tensor cannot be extracted using the TPS method.

3.3. Correlation Between Microstructure Parameters and Thermal Conductivity Measurement Values

We investigated three SCF/PEI composite microstructures, each with distinct statistical characteristics. Even though no valid analytical solution exists for interpreting transient temperature rise in materials with a fully anisotropic (orthotropic) TC tensor, as is the case for the FPM samples, we still aim to compare the TC performance of the different samples. To this end, the isotropic model was fit to the directional temperature rise data $∆T_{x_i}\left(t\right)$, where $x_i$ corresponds to the reference direction normal to the sensor plane, yielding apparent TC values $K_{x_i}$. While the resulting values $K_{x_i}$ cannot be interpreted as true tensor components (and, therefore, are not physical properties under anisotropic Fourier’s law) but instead reflect a direction-specific mapping of the temperature response under an invalid symmetry assumption, they serve as consistent, directionally resolved estimates that reflect the influence of processing and microstructure. Figure 12 and Figure 13 show the $K_{x_i}$ values for the FFF and FPM-2.4 samples.

Since the FFF TC tensor is available as a reference, we can directly assess the consequences of assuming isotropy during data reduction (see Figure 12). As seen in the data, the $K_{x_i}$ values deviate from the true tensor components extracted using the appropriate anisotropic model; the values inferred from the isotropic model are consistently higher. This difference reflects the overestimation introduced by fitting the temperature rise with a spherical heat conduction model in a directionally anisotropic material. Despite not being interpretable as components of a tensorial material physical property, the fitted values $K_{x_i}$ from the isotropic model correctly reflects the direction-dependent thermal response of the FFF sample: $K_{x_1}>K_{x_2}$, manifesting the effect of fiber alignment. The $K_{x_2}$ and $K_{x_3}$ values are close, and their ~8% difference reflects a departure from perfect transverse isotropy, consistent with the measured FOD ($a_{22}\approx a_{33}$), and measurement uncertainty.

Figure 14 summarizes the $K_{x_i}$ parameters across all SCF/PEI samples and also reports the thermal conductivity of neat PEI. It is evident that the addition of carbon fibers enhances thermal conductivity compared to neat PEI. In all SCF/PEI composites, the print direction $x_1$ corresponds to the axis of highest fiber alignment and consistently exhibits the highest apparent TC parameter $K_{x_1}$. As seen in Figure 14, $K_{x_1}>K_{x_2}>K_{x_3}$, consistent with the measured FOD ($a_{11}>a_{22}>a_{33}$) for all SCF/PEI samples. The observed correspondence between fiber orientation bias and directional thermal conduction aligns with qualitative predictions of idealized micromechanical homogenization models for misaligned short-fiber composites [59,60], which associate increasing effective thermal conductivity with higher degrees of fiber alignment and longer mean fiber length, and with prior experimental studies reporting enhanced heat transport along preferential fiber alignment directions in additively manufactured [14] and molded [61] composites.

The FPM samples (20 wt.% CF; ${\overline{L}}_n=128 \mathsf{\mu }\mathrm{m}$) show higher $K_{x_i}$ values than FFF (15wt.% CF; ${\overline{L}}_n=67 \mathsf{\mu }\mathrm{m}$), consistent with their higher carbon fiber content and longer average fiber length. FPM-2.4 exhibits a higher $K_{x_1}$ than FPM-1.4, despite having a lower $a_{11}$ value. This difference is attributed to void morphology. While both FPM variants have similar void VF, FPM-1.4 contains more elongated voids along the print direction. This interpretation is consistent with numerical studies of closed-pore media [18,21] showing that effective conductivity is insensitive to spatial distribution and clustering of spherical pores, whereas elongated pores lead to anisotropic response; particularly, when aligned with the heat-flow direction, more elongated pores produce a larger reduction in ETC at comparable porosity. More generally, analytical treatments of anisotropic conductivity demonstrate that inclusion shape and orientation constitute the relevant descriptors governing directional ET, rather than volume fraction alone [62].

Both FPMs have principal axes that do not coincide with the reference frame, and the $a_{22}\approx 3a_{33}$ ratio in both suggests a stronger probability of fiber orientation along one transverse axis than the other, which should translate to $K_{x_2}>K_{x_3}$ in both cases. In the FPM-2.4 sample, this difference is well pronounced. In FPM-1.4, the smaller difference between $K_{x_2}$ and $K_{x_3}$ may be due to its principal directions being more rotated relative to the reference frame, blending the responses in the $x_2$ and $x_3$ measurements. These results make clear that while the isotropic regression offers a way to track directional trends across samples, it does not yield material TC in absolute terms. However, the data remains informative, provided their interpretive limitations are recognized.

4. Conclusions

This work examined the relationship between microstructure and anisotropic ETC in SCF/PEI composites fabricated via ME AM. By combining directional TPS measurements with three-dimensional quantification of fiber orientation and void morphology, we assessed when a transverse-isotropy assumption is justified and extracted ETC tensor components only where supported by statistical microstructural evidence.

Two main contributions emerge. First, the study clarifies the interpretation of TPS-based directional measurements by proposing a microstructure-informed approach to justify the assumption of constrained anisotropy (transverse isotropy). For microstructures lacking such symmetry, scalar ETC values obtained from isotropic fits are not intrinsic material properties under Fourier’s law but test-specific descriptors; while still useful for comparative analysis (e.g., to identify dominant effects of microstructural features), their limitations must be recognized when ranking materials or when using such values as inputs for numerical simulations, for example as proxies for tensorial properties.

Second, the results provide insight into structure–property relationships in commercially available SCF/PEI feedstocks (15–20 wt.% CF). Measurable differences in directional ETC arise from processing-induced variations in microstructure. Adjusting layer height introduces bias in fiber orientation, while void shape and orientation are also affected. Notably, fiber alignment and void morphology can exert competing influences on effective thermal conduction, such that similar degrees of alignment do not necessarily produce equivalent thermal responses. These observations are consistent with prior analytical and numerical homogenization studies developed for idealized multiphase systems, while highlighting the limitations of such models when applied to complex, process-induced microstructures.

Overall, the findings point to the need for improved methods for anisotropic ETC analysis. This includes both experimental frameworks capable of correctly regressing measured transient responses to infer the full tensorial ETC, and computational frameworks that incorporate detailed microstructure statistics to capture the interaction and competition of multiple morphological features governing anisotropic thermal transport. Finally, this study primarily focuses on the correlation between process parameters, microstructure, and thermal conductivity anisotropy; the influence of temperature dependence on the thermal conductivity tensor will be explored in future research.