Average Interface Number (AIN) in Composites

• Average Interface Number (AIN) is defined as the average count of matrix–reinforcement interfaces a light ray crosses, condensing thickness, volume fraction, and feature size effects.
• AIN serves as the key parameter controlling cumulative angular scattering in transparent composites, directly affecting haze and optical blur.
• Evaluated via image or voxel-based ray counting, AIN simplifies complex microstructures into a single, actionable metric for materials design.

Searching arXiv for the cited paper and closely related work on transparent composites, transparent wood, and light scattering. Average Interface Number (AIN) is a microstructural descriptor introduced for transparent composites as the average number of matrix–reinforcement interfaces that a light ray crosses along the propagation direction, i.e., along the sample thickness. In the analytical framework of "Analytical Model for Light Scattering in Transparent Composites" (Chen et al., 7 Jul 2025), AIN, denoted by $M$, is presented as the single dominant microstructural parameter governing angular light scattering. Its central role is to compress the effects of thickness, volume fraction, feature size, and distribution into a single count of optical interaction events experienced by a ray, while the strength of each event is controlled separately by refractive-index contrast.

1. Definition and physical meaning

AIN is defined physically as the average number of matrix–reinforcement interfaces that a light ray crosses along the propagation direction. Every transition from matrix to fiber or particle, and every return crossing from reinforcement to matrix, contributes one interface crossing. The quantity is averaged over many rays traversing the sample thickness (Chen et al., 7 Jul 2025).

The physical intuition given for AIN is based on cumulative angular deflection. Each interface can alter the ray direction through refraction or reflection, and repeated crossings cause the total angular deviation to accumulate like a random walk. Within this picture, AIN represents the number of scattering “steps” along the ray path. The resulting angular spread of transmitted light therefore scales with how many such interface-crossing events occur.

A key point in the formulation is that AIN is treated as the primary structural variable, whereas parameters such as thickness, fiber diameter, particle size, volume fraction, and distribution act indirectly through their effect on $M$. In this sense, AIN is not merely another morphological descriptor; it is the quantity used to collapse multiple geometric variables into a single optical-event count.

The paper also makes clear what AIN is not. It is not presented as a closed-form function of volume fraction and feature size, and no explicit geometric formula for $M$ is given. Instead, AIN is evaluated operationally from microstructure by counting ray-interface intersections and averaging over many rays. This avoids the misconception that the framework depends on a specific idealized geometry with a universal analytic expression for interface density.

2. Analytical role in the scattering model

The analytical model expresses the emergent angular spread in terms of AIN and the variance of the angular deflection associated with a single encounter. The main relation is

$$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$

where $\theta_e$ is the emergent ray angle at the composite–air interface, $n$ is the refractive index of the composite body in the exit-refraction treatment, $M$ is AIN, and $\operatorname{Var}(\Delta\theta_i)$ is the variance of the angular change caused by a single fiber or particle encounter (Chen et al., 7 Jul 2025).

The derivation proceeds by representing the internal angular state after $M$ interface events as a sum of independent identically distributed increments:

$$\theta_{\text{end}}=\sum_{j=1}^{M}\Delta\theta_{i,j} \tag{3}$$

Under the independence assumption, the variance grows linearly with AIN:

$M$0

The internal angle is then mapped to the emergent angle in air through the composite–air interface. For normal incidence with $M$1, the variance transformation is written as

$M$2

which leads to

$M$3

In this formulation, AIN enters only through the product $M$4. The formal significance is that $M$5 determines how many deflection events occur, while $M$6 determines how strong each event is. The paper therefore treats accumulated scattering as structurally controlled by AIN and optically weighted by refractive-index-dependent single-event statistics.

The same section also specifies the practical computation of AIN. For fiber-reinforced composites with unidirectional cylindrical fibers and rays perpendicular to the fiber axes, interface crossings are counted along the ray path in the two-dimensional cross-section. For particle-reinforced composites the same counting principle is extended to three dimensions. In practice, AIN is evaluated from images or voxel models by purely geometric ray counting and averaging, rather than by a closed-form analytical expression.

3. Single-interface deflection and refractive-index mismatch

To use the analytical scattering equation, the model requires $M$7, the variance of the angular change caused by one complete fiber encounter, including entry and exit. For cylindrical fibers under geometrical optics and Snell’s law, the paper gives an exact treatment that distinguishes the cases $M$8 and $M$9, where $M$0 and $M$1 are the matrix and reinforcement refractive indices and $M$2 (Chen et al., 7 Jul 2025).

The exact form is written as

$M$3

Because real ray tracing and imperfect geometry introduce total internal reflection even when $M$4, the paper adopts a simplified unified expression for use throughout the model:

$M$5

Several assumptions specify how this quantity is evaluated. The ray impact positions are assumed uniformly distributed over the fiber cross-section. This produces a uniform distribution of $M$6 over $M$7 when $M$8, and over $M$9 when $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$0, reflecting the domain in which refraction rather than total internal reflection occurs. The variance term

$$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$1

is evaluated numerically for fixed $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$2 and $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$3.

The physical interpretation given in the paper is explicit. For small $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$4, the deflection per fiber is small but nonzero. Larger $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$5 increases $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$6, and therefore increases $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$7 at fixed AIN. When $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$8, total internal reflection contributes strong angular deviations through the extra terms in the expression above. AIN therefore quantifies event frequency, while refractive-index mismatch quantifies event intensity.

4. Equivalent Average Interface Number and haze prediction

For transparent wood, the paper introduces the Equivalent Average Interface Number (EAIN) as a scalar combination of AIN and refractive-index mismatch intended for fast haze prediction. The stated motivation is that haze depends primarily on $$\sigma(\theta_e)\approx \sqrt{\operatorname{Var}(\theta_e)} \approx \sqrt{-\frac{1}{2}\ln\!\left(1-n^2\bigl(1-e^{-M\,\operatorname{Var}(\Delta\theta_i)}\bigr)\right)} \tag{1}$$9 and $\theta_e$0, and that these two governing variables can be reduced to one descriptor for design and comparison (Chen et al., 7 Jul 2025).

EAIN is defined as a power-law combination

$\theta_e$1

where $\theta_e$2 is determined by fitting a large simulated transparent-wood dataset comprising 10,000 cross-sections. Polynomial fits of different orders are tested, and for each order the value of $\theta_e$3 minimizing the RMS fitting error is selected. The reported optimum is stable across polynomial orders at approximately $\theta_e$4, leading to the final definition

$\theta_e$5

The haze relation is then represented as a single-valued curve

$\theta_e$6

obtained with a 10th-order polynomial fit. The data points are described as clustering close to this curve, with a green band in the figure indicating small residuals.

In this formulation, EAIN functions as a one-parameter descriptor that almost fully determines haze in transparent wood across a broad range of thickness, microstructure, and refractive-index mismatch. The paper states that materials with the same EAIN are predicted to have very similar haze even if their detailed microstructures differ. This does not replace AIN in the scattering derivation itself; rather, it provides a compact empirical reduction specifically for haze in transparent wood.

The same section clarifies the relation between angular spread and haze. Haze is defined following ASTM D1003 as the fraction of transmitted light scattered more than $\theta_e$7. For a roughly Gaussian angular distribution, larger $\theta_e$8 implies larger haze. The paper states that haze is highly correlated with the standard deviation of the emergent angle and that larger haze usually denotes larger $\theta_e$9.

5. Scaling behavior and parameter dependence

The analytical and numerical exploration of the model identifies characteristic scaling trends for angular scattering as functions of AIN and refractive-index mismatch (Chen et al., 7 Jul 2025). At fixed $n$0, the standard deviation $n$1 is nonlinear in AIN. For small $n$2, the growth is described as roughly proportional to $n$3; at larger values it saturates through the logarithmic structure of the analytic expression.

At fixed AIN, and specifically for $n$4 in the explored regime, $n$5 increases almost linearly with $n$6 for the small refractive-index mismatch range considered. This behavior is consistent with the decomposition of the model into an event-count term $n$7 and a single-event variance term governed by refractive-index contrast.

For transparent wood simulations, the reported trends are that haze increases with $n$8 at given AIN and increases with AIN at given $n$9. The EAIN collapse consolidates these dependencies into the single relation $M$0.

The design interpretations stated in the paper follow directly from these trends. Doubling thickness at fixed microstructure roughly doubles AIN and nonlinearly increases haze. Reducing refractive-index mismatch reduces $M$1, thereby reducing EAIN and haze even when AIN is high. The formulation therefore distinguishes between structural pathways for controlling the number of interface crossings and optical pathways for reducing the deflection induced by each crossing.

A further conceptual clarification arises here. Parameters such as fiber diameter, volume fraction, and thickness are not treated as independent predictors in the model once AIN is known. The paper explicitly describes them as indirect parameters. This suggests that AIN is intended as a structural coarse-graining variable rather than an additive term in a multivariate empirical regression.

6. Validation, feature importance, and practical use

The paper validates the AIN-based analytical model against geometrical ray tracing for fiber-reinforced composites, particle-reinforced composites, and transparent wood, and reports strong agreement across all three classes (Chen et al., 7 Jul 2025). For two-dimensional glass-fiber-reinforced polymer microstructures, 1000 microstructures with varying thickness, fiber volume fraction, and fiber sizes are generated, with AIN ranging from approximately 15 to 75 and $M$2. The simulated-versus-analytical plot for $M$3 yields a nearly linear relation with slope $M$4, with good agreement up to approximately $M$5 rad and slight analytical overestimation at larger scattering.

For particle-reinforced composites, one large three-dimensional microstructure of $M$6 voxels is generated and cropped to obtain varying thicknesses. With AIN ranging from approximately 14 to 27 and $M$7, the model—although derived in two dimensions—is compared with projected emergent-angle statistics in the $M$8 and $M$9 directions. The reported slopes are $\operatorname{Var}(\Delta\theta_i)$0 and $\operatorname{Var}(\Delta\theta_i)$1, which the paper takes as evidence of strong predictive power despite three-dimensional effects such as curvature-induced total internal reflection.

For transparent wood, 10 large three-dimensional structures of $\operatorname{Var}(\Delta\theta_i)$2 voxels are generated with varied microstructural parameters including fiber size, cell wall thickness, vessels, and ray cells. From these, 10,000 cross-sections are sampled with thicknesses randomly distributed from 501 to 2991 pixels and $\operatorname{Var}(\Delta\theta_i)$3. The simulated-versus-analytical comparison for $\operatorname{Var}(\Delta\theta_i)$4 gives slope $\operatorname{Var}(\Delta\theta_i)$5. The slight overestimation at higher scattering is attributed partly to ray cells increasing AIN, and therefore the counted number of interfaces, without equally contributing to radial scattering.

The paper also compares model trends with experimental literature on transparent wood. It cites work by Montanari et al. and Yaddanapudi et al. showing that haze increases with sample thickness and with refractive-index difference $\operatorname{Var}(\Delta\theta_i)$6. These trends are presented as consistent with the analytical prediction that scattering grows with both AIN and refractive-index mismatch, and with the EAIN description of haze control.

A further layer of validation is provided by deep neural network regressors trained on 10,000 transparent-wood samples with simulated haze labels. The features considered include $\operatorname{Var}(\Delta\theta_i)$7, $\operatorname{Var}(\Delta\theta_i)$8, AIN, lumen volume ratio, average fiber cross-sectional area, standard deviation of fiber area, sample thickness in pixels, sample width, vessel number, and fiber number. Multiple DNNs are trained using always $\operatorname{Var}(\Delta\theta_i)$9 and $M$0 plus one additional feature, alongside a full-feature benchmark network. The reported result is that the DNN using $M$1 achieves much smaller test-set MSE than DNNs using any other single structural parameter as the third input, except fiber number. The full-feature DNN performs best but only modestly better, while the good performance of fiber number is described as partly an artifact because sample width can change fiber count without affecting scattering. The stated conclusion is that AIN is the most critical structural predictor of haze and that other morphological descriptors add only marginal value once AIN and refractive indices are known.

The practical implications are correspondingly direct. To reduce haze or scattering, the paper recommends reducing AIN by using thinner samples, reducing the number of interfaces per unit thickness, lowering reinforcement volume fraction, using larger reinforcement features, or increasing phase continuity. To increase controlled haze, it recommends increasing AIN through greater thickness, higher volume fraction, or smaller and more numerous reinforcements. The practical workflow is stated as follows: compute AIN from a microstructural image or model by ray counting along the intended propagation direction; combine it with known $M$2 and $M$3 to obtain $M$4; use Eq. (1) together with Eq. (9) to estimate $M$5 and thereby haze; then adjust structural parameters until the target AIN and scattering are reached.

For transparent wood in particular, EAIN is presented as a quick scalar for comparative design and optimization. Candidate wood species and polymers can be compared through their AIN and $M$6, then ranked by EAIN, with lower EAIN implying lower haze. The same model is also used for fast image rendering through scattering composites: instead of tracing rays through the full three-dimensional microstructure, one computes $M$7 from AIN and $M$8, assumes a homogeneous slab, and samples angular perturbations from the corresponding analytical distribution during backward ray tracing. The reported consequence is very fast rendering of blurred images through glass-fiber-reinforced polymer or transparent wood, with increasing AIN or increasing object–composite distance producing greater blur.

In summary, AIN is presented as the principal structural coordinate for light scattering in transparent composites: a count of average interface-crossing events that converts complex internal morphology into a compact optical descriptor. In combination with the single-interface variance term, it determines the emergent angular spread, and in transparent wood its power-law combination with refractive-index mismatch yields EAIN, a one-parameter predictor of haze.