**2. Rationale**

In moldings with 50% weight fraction (wt.%) long fiber reinforced PA66, Bailey and Kraft observed significantly higher fiber length in the core compared to the shell region (LN(core) = 1.46 mm, LN(shell) = 0.55 mm) [13]. O'Regan and Akay also identified longer fibers in the core region (LN(core) = 0.86 mm, LN(shell) = 0.7 mm) for 60 wt.% long-fiber reinforced PA66 samples [34]. The standard technique for sample isolation used in these studies involved selecting a small amount of fibers with tweezers after matrix removal. Aside from the risk of short fibers being dropped or fibers breaking, the fiber population in these studies was very low (800–3500 fibers). However, to have

statistical confidence, large fiber populations are required, specifically when characterizing LFTs for which the fibers' aspect ratio can vary over two orders of magnitude.

Since matrix removal is usually achieved through pyrolysis, the shortest fibers tend to fall towards the bottom as the matrix melts and burns-off [14]. Therefore, to measure FLD in either the core or the shell, such region should be isolated from the complete sample before the pyrolysis step. As the shell is generally thicker than the core and more accessible [24], we propose measuring fiber length in the shell and indirectly calculating the fiber length in the core. The extraction of the shell is addressed in Section 3.3. Fiber length in the core can be determined as follows.

FLD data are often given as an average value. However, to properly describe this type of distributions both the number- and the weight-average should be reported.

Similar to the molecular weight distribution, the number-average fiber length LN is expressed as

$$\mathbf{L}\_{\rm N} = \frac{\sum \mathbf{N\_i l\_i}}{\sum \mathbf{N\_i}} \,\tag{1}$$

the weight-average fiber length LW as

$$\mathcal{L}\_W = \frac{\sum \text{N}\_{\text{i}}\text{l}\_{\text{i}}^2}{\sum \text{N}\_{\text{i}}},\tag{2}$$

and the total fiber length is described as

$$\mathbf{L}\_{\rm T} = \sum \mathbf{N}\_{\rm i} \mathbf{l}\_{\rm i} \tag{3}$$

For the arbitrary LFT sample A shown in Figure 1, the averages are calculated from the complete population of fibers inside the sample's volume. Thus, it is valid to re-formulate Equations (1) and (2) by grouping the addends into sub-volumes B (shells) and C (core). The number average of the entire sample LN(A) can then be expressed as

$$\mathbf{L}\_{\mathbf{N}(\mathbf{A})} = \frac{(\sum \mathbf{N}\_{\mathbf{i}} \mathbf{l}\_{\mathbf{i}})\_{\mathbf{B}} + (\sum \mathbf{N}\_{\mathbf{i}} \mathbf{l}\_{\mathbf{i}})\_{\mathbf{C}}}{(\sum\_{\mathbf{i}} \mathbf{N}\_{\mathbf{i}})\_{\mathbf{A}}} \tag{4}$$

**Figure 1.** Schematic of a core–shell structure.

Assuming the sample's width and length are constant, Equation (4) can be formulated in terms of the local number-average fiber length

$$\mathbf{L}\_{\mathrm{N(A)}} = \frac{\mathbf{L}\_{\mathrm{N(B)}}\mathbf{t\_{B}}\boldsymbol{\Phi}\_{\mathrm{B}} + \mathbf{L}\_{\mathrm{N(C)}}\mathbf{t\_{C}}\boldsymbol{\Phi}\_{\mathrm{C}}\boldsymbol{\Phi}\_{\mathrm{C}} + \mathbf{L}\_{\mathrm{N(B)}}\mathbf{t\_{B}}\boldsymbol{\Phi}\_{\mathrm{B}}}{\mathbf{t\_{B}}\boldsymbol{\Phi}\_{\mathrm{B}} + \mathbf{t\_{C}}\boldsymbol{\Phi}\_{\mathrm{C}} + \mathbf{t\_{B}}\boldsymbol{\Phi}\_{\mathrm{B}}} \text{ or } \frac{\sum \mathbf{L}\_{\mathrm{N(K)}}\mathbf{t\_{K}}\boldsymbol{\Phi}\_{\mathrm{K}}}{\sum \mathbf{t\_{K}}\boldsymbol{\Phi}\_{\mathrm{K}}} \text{ }, \tag{5}$$

where the index K represents individual layers along the thickness of the sample. The changes in fiber content (φK) have to be accounted for in order to satisfy mass conservation; that is, LT should remain unchanged. Since the objective is determining the length in the core (LN(C)), and both the global sample length (LN(A)) and the shell sample length (LN(B)) can be measured experimentally, Equation (5) can be solved for LN(C)

$$\mathcal{L}\_{\text{N(C)}} = \frac{\mathcal{L}\_{\text{N(A)}} (2\mathbf{t}\_{\text{B}}\boldsymbol{\phi}\_{\text{B}} + \mathbf{t}\_{\text{C}}\boldsymbol{\phi}\_{\text{C}}) - 2\mathcal{L}\_{\text{N(B)}}\mathbf{t}\_{\text{B}}\boldsymbol{\phi}\_{\text{B}}}{\mathbf{t}\_{\text{C}}\boldsymbol{\phi}\_{\text{C}}} \tag{6}$$

The weight-average fiber length in the core (LW(C)) can be calculated in the same way. This approach requires knowledge of the thickness of each layer and the through-thickness fiber content. This information can be obtained from μCT analysis.

#### **3. Materials and Methods**
