Equivalent Average Interface Number (EAIN)

• EAIN is a dimensionless parameter that combines the average interface number and refractive-index mismatch to quantify haze in transparent composites.
• It is calculated using microstructural data and validated via a scaling exponent of 0.87, providing accurate predictions in various composite systems.
• EAIN serves as a practical metric for optimizing optical designs by encapsulating complex interactions between interface density and index contrast.

Equivalent Average Interface Number (EAIN) is a dimensionless parameter introduced in "Analytical Model for Light Scattering in Transparent Composites" to combine the Average Interface Number (AIN) and the refractive-index mismatch into a unified quantity for fast haze prediction in transparent composites (Chen et al., 7 Jul 2025). In the formulation of that work, AIN, denoted $M$, is the expected number of matrix–reinforcement interface crossings encountered by a ray along a path of length $L$, while EAIN is defined as

$\mathrm{EAIN} = M^{0.87} |\Delta n|,$

with $\Delta n = n_m - n_r$. The parameter was introduced to collapse microstructural complexity and index contrast into a single haze-governing metric for systems including fiber-reinforced composites, particle-reinforced composites, and transparent wood.

1. Definition and conceptual basis

EAIN is rooted in a geometrical-optics picture of transparent composites consisting of a matrix of refractive index $n_m$ and reinforcements of refractive index $n_r$, with a small refractive-index mismatch $\Delta n = n_m - n_r$ (Chen et al., 7 Jul 2025). Rays propagating through such media undergo refraction and, for some incidence angles, total internal reflection at matrix–reinforcement interfaces. The cumulative angular deflection arises from many interface encounters rather than from a single scattering event.

The underlying microstructural quantity is the Average Interface Number, $M$, defined as the expected number of matrix–reinforcement interface crossings encountered by a ray along a path of length $L$. Physically, $M$ measures how many times a ray meets an interface where deflection can occur. EAIN combines this interface-crossing count with the magnitude of the refractive-index mismatch, using the absolute value $L$0, to provide a compact descriptor of haze.

The exponent in the EAIN definition was obtained by fitting a large transparent wood dataset. The optimized value was reported as $L$1, with robustness quantified as $L$2. The resulting scaling was presented as a compact representation of the empirical observation that haze grows nonlinearly with $L$3 and roughly linearly with $L$4 at fixed $L$5.

2. Relation to AIN and microstructural descriptors

AIN can be computed directly from microstructure. For a binary cross-sectional image, one traces a set of straight rays perpendicular to the image direction across thickness $L$6, counts the number of pixel transitions from matrix to reinforcement or reinforcement to matrix for each ray, and averages over many rays. The mean count is $L$7 (Chen et al., 7 Jul 2025).

A stereological form relates $L$8 to the interface area per unit volume $L$9:

$\mathrm{EAIN} = M^{0.87} |\Delta n|,$0

For statistically isotropic interfaces, this gives a direct link between interface density and the expected number of optical encounters. For anisotropic media, the paper states that the directional specific surface should be used.

For common idealized composite classes, the paper gives explicit expressions. For monodisperse spheres in particle-reinforced composites,

$\mathrm{EAIN} = M^{0.87} |\Delta n|,$1

where $\mathrm{EAIN} = M^{0.87} |\Delta n|,$2 is the particle volume fraction and $\mathrm{EAIN} = M^{0.87} |\Delta n|,$3 is the particle diameter. For infinite cylinders aligned parallel to one axis, with rays crossing orthogonally to the fiber axes,

$\mathrm{EAIN} = M^{0.87} |\Delta n|,$4

For transparent wood, the recommended approach is image-based counting in cross-sectional slices perpendicular to the ray direction, because analytical $\mathrm{EAIN} = M^{0.87} |\Delta n|,$5 expressions depend on detailed cell geometry and are more cumbersome.

Within this framework, EAIN inherits the dependence of $\mathrm{EAIN} = M^{0.87} |\Delta n|,$6 on $\mathrm{EAIN} = M^{0.87} |\Delta n|,$7, $\mathrm{EAIN} = M^{0.87} |\Delta n|,$8, $\mathrm{EAIN} = M^{0.87} |\Delta n|,$9, and interface morphology, but weights the resulting interface-crossing count by $\Delta n = n_m - n_r$0. A plausible implication is that EAIN functions as a single-axis summary of two distinct optical controls: how often rays encounter interfaces, and how strongly each encounter perturbs propagation.

3. Connection to angular scattering and haze

The analytical model links interface-level deflection statistics to emergent angular broadening. If $\Delta n = n_m - n_r$1 denotes the angle change at a single interface event, the cumulative variance after $\Delta n = n_m - n_r$2 encounters is

$\Delta n = n_m - n_r$3

For cylindrical reinforcements, the simplified single-interface variance used for practical prediction is

$\Delta n = n_m - n_r$4

with $\Delta n = n_m - n_r$5 (Chen et al., 7 Jul 2025). The first term accounts for refraction, and the second captures the contribution from total-internal-reflection-like events when applicable.

The emergent angle in air, $\Delta n = n_m - n_r$6, is then related to the accumulated internal variance through

$\Delta n = n_m - n_r$7

with $\Delta n = n_m - n_r$8 and

$\Delta n = n_m - n_r$9

For small angles, this reduces to $n_m$0, but the logarithmic form is retained to preserve higher-order behavior.

The paper models the emergent angular intensity as a Gaussian about the forward direction,

$n_m$1

with $n_m$2. Haze is defined according to ASTM D1003 as

$n_m$3

using a standard collection aperture half-angle of $n_m$4, described in the paper as the ratio of rays that have an angle greater than $n_m$5. Under the Gaussian model,

$n_m$6

where $n_m$7 is the standard normal CDF.

EAIN was introduced specifically to accelerate haze prediction. For transparent wood, the paper fits

$n_m$8

where $n_m$9 is a tenth-order polynomial calibrated to a large simulated dataset. The coefficients are dataset-specific and are not printed, but the dependence was reported as robust across polynomial orders.

4. Computation and operational use

The computation of EAIN follows a short sequence. First, one computes AIN, $n_r$0. For unidirectional fiber-reinforced composites with rays orthogonal to the fiber axis,

$n_r$1

For spherical particle-reinforced composites,

$n_r$2

For transparent wood or general composites, one uses either $n_r$3 with an appropriate directional $n_r$4, or image-based counting of phase transitions along straight rays (Chen et al., 7 Jul 2025).

Second, one computes the refractive-index mismatch,

$n_r$5

Third, one forms

$n_r$6

This quantity is sufficient for fast haze estimation when a calibrated $n_r$7 relation is available.

If angular scattering is also required, the paper prescribes numerical evaluation of $n_r$8 by uniformly sampling $n_r$9 across its allowable range for cylindrical fibers, then computing $\Delta n = n_m - n_r$0 and $\Delta n = n_m - n_r$1. Haze may then be estimated either from the Gaussian angular model with the ASTM-like threshold at $\Delta n = n_m - n_r$2, or from the empirical polynomial relation in EAIN.

The paper gives a worked fiber-composite example with $\Delta n = n_m - n_r$3, $\Delta n = n_m - n_r$4, $\Delta n = n_m - n_r$5, $\Delta n = n_m - n_r$6, $\Delta n = n_m - n_r$7, and $\Delta n = n_m - n_r$8. From

$\Delta n = n_m - n_r$9

one obtains $M$0, and then

$M$1

The example is used to illustrate how microstructural and optical inputs are collapsed into a single dimensionless haze predictor.

5. Assumptions, regime of validity, and limitations

The analytical framework containing EAIN is explicitly restricted to the geometrical-optics regime, with feature sizes on the order of micrometers, so wavelength-scale Mie and Rayleigh effects are neglected (Chen et al., 7 Jul 2025). Interfaces are assumed to be geometrically smooth at micrometer scale, and scattering is attributed only to refractive-index mismatch between matrix and reinforcement. Defects such as voids are neglected.

Additional assumptions include random positional distribution of reinforcements in the cross-sectional plane, independence of interface encounters, parallel entry and exit surfaces, and orthogonal incident rays. In long-fiber systems, fibers may be aligned while their positions remain random.

The paper states that the model performs well when $M$2 and reports overestimation at larger scattering. In three-dimensional particle composites, curvature-induced total internal reflection events may not be fully captured by the two-dimensional formulation, although the reported errors remain acceptable. The model may fail when reinforcement sizes approach the optical wavelength, or when strong absorption, coherent effects, or substantial surface roughness are present.

Surface roughness is not included in the analytical model. The paper states that, if roughness is significant, it will increase $M$3 and should be folded into the single-interface variance term, but that extension is not covered by the current model. For anisotropic microstructures such as transparent wood, the use of directional AIN is recommended, and the paper notes that some structures, such as ray cells, can inflate $M$4 without contributing equally to scattering in the measured direction.

6. Validation, feature importance, and practical implications

The analytical framework in which EAIN is embedded was validated against ray-tracing simulations and literature experimental trends for fiber-reinforced composites, particle-reinforced composites, and transparent wood (Chen et al., 7 Jul 2025). For unidirectional GFRP, the linear fit slope between simulated $M$5 and analytical prediction was reported as $M$6, close to the ideal value of $M$7, with overestimation increasing at large scattering. For GPRP, the fit slopes were $M$8 for the $M$9-projection and $L$0 for the $L$1-projection. For transparent wood, the slope was $L$2, with slight overestimate attributed to ray cells contributing to $L$3 but weakly to radial-direction scattering.

A DNN analysis was used to assess feature importance in transparent wood haze prediction. The paper reports that AIN, together with $L$4 and $L$5, yielded the smallest mean-square error on the test set among networks using other single structural parameters. Networks using all features performed best, but the three-feature network based on AIN, $L$6, and $L$7 achieved low error with minimal inputs. This was presented as confirmation that AIN is the dominant microstructural parameter governing haze.

Reported dataset ranges further contextualize EAIN. In the GFRP dataset, $L$8 ranged between approximately $L$9 and $M$0 with $M$1 up to $M$2. In the GPRP dataset, $M$3 ranged between approximately $M$4 and $M$5 with $M$6 up to $M$7. For transparent wood, haze increased with both $M$8 and $M$9, and the EAIN scaling with exponent $L$00 effectively collapsed haze variations across the dataset.

The practical use of EAIN is as a target for optical design and optimization. The paper states that haze can be reduced by reducing EAIN: minimizing $L$01 through index matching, reducing $L$02 by lowering $L$03, increasing $L$04, or reducing $L$05, and, in fiber composites, aligning rays parallel to the fiber axes so that effective interface crossings in the measured direction are drastically reduced. It also states that reducing total-internal-reflection events through smoother interfaces and reduced curvature at incidence angles near the critical angle reduces $L$06. The associated trade-offs are explicit: index matching can affect polymer chemistry and mechanical properties; decreasing $L$07 lowers stiffness and strength; increasing $L$08 may compromise toughness and lead to larger defects. Within those constraints, EAIN serves as a clean target metric for comparing optical performance across transparent structural materials.