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The conductance filtration: sink-relative persistence and exact edge sensitivity for heat transport in particle-filled composites

Daisuke Yasufuku

arXiv:2609.30810v1cond-mat.mtrl-scicond-mat.dis-nncs.CE

TL;DR

Thermal-interface composites can remain far below particle conductivity, while the effective result alone does not reveal which design variable should change. The paper models particle-pair conductances and uses them for sink-relative persistence and exact sensitivity analysis. It thereby provides mechanism diagnostics and exact predictions for finite changes to a single edge on one graph representation.

  • Problem

    Composite effective thermal conductivity remains about an order of magnitude below particle conductivity, and raising packing fraction also raises viscosity exponentially.

    Source: §1 · p. 2

  • Method

    The paper models particle-pair conductances and uses conductance as the filtration axis with persistence relative to the sink boundary.

    Source: §1 · p. 3

  • Results

    The dissipation share equals the logarithmic sensitivity of effective conductance, and a finite change to a single edge is predicted exactly in closed form.

    Source: §1 · p. 3, §8 · p. 34

  • Takeaways & Limitations

    The framework provides allocation of dissipation among branch types and idealisation ratios for interface, matrix, and particle mechanisms on one graph representation.

    Source: §1 · p. 3, §8 · p. 34

    The paper does not cross-check absolute effective thermal conductivity against finite-element analysis or measurement, and does not model a finite-thickness interphase.

    Source: §7.1 · p. 32, §7.2 · p. 33

Abstract

from arXiv · show

We formulate the thermal conductance $g_e$ of a particle pair as three branch types -- contact, near-contact and bond -- within a single construction, and we introduce the filtration that takes this $g_e$ as its axis (the conductance filtration) together with persistence taken relative to the sink boundary. We then show that the dissipation share is exactly the logarithmic sensitivity of the effective conductance, and that a finite change of a single edge is predicted exactly in closed form. The allocation of the dissipation among branch types, and the idealisation ratio of each mechanism (interface, matrix and particle), are thereby obtained as diagnostic quantities on one and the same graph representation.

1 Introduction

Thermal-interface composites face a gap between particle-level conductivity and effective performance, while a measured effective conductivity alone does not identify which design variable to change. The paper connects network transport modelling with persistent homology by using particle-pair conductance as the filtration axis and returns diagnostics on one graph representation.

1 Introduction

  • Composites remain about an order of magnitude below particle conductivity, while increasing packing raises both conductivity and viscosity exponentially.Reported greases with nominal filler conductivities of 8.5 and 14.2 W/(m·K) both remain at 3–6 mm²·K/W in the same measurement system.Source: §1 · p. 2
  • The internal structure-to-transport process is computable stage by stage, but an effective-conductivity result alone does not identify which of four design variables to change.Those variables are packing fraction, particle size distribution, filler species, and surface treatment.Source: §1 · p. 2
  • Prior network and effective-property approaches return one or a few indicators per structure, not a mechanism breakdown for arbitrary packed structures.The cited mechanistic model uses an ideal lattice unit cell rather than an arbitrary packed structure.Source: §1 · p. 3
  • The paper uses particle-pair thermal conductance as the filtration axis and makes the filtration relative to the sink boundary.The conductance function and sink-boundary relativisation are the paper’s stated choices within a standard superlevel-set construction.Source: §1 · p. 3
  • A branch-conductance model, sink-relative persistence, and exact sensitivity and finite-edge-change analyses close a design loop on one graph representation.The paper also provides algorithms and a software package for the framework.Source: §1 · p. 3, Figure 1 · p. 4

2 The branch-conductance model

The branch-conductance model assigns particle pairs conductances through contact, near-contact, and bond paths, with a graph representation used to evaluate transport. Its conditions and element-wise numerical checks distinguish the modeled paths and constrain how the network is constructed.

2.1 The branch-conductance model

  • The model maps particles to isothermal graph vertices and classifies pair edges as near-contact, contact, or bond according to surface contact and solid continuity.Source and sink boundary slabs are assigned temperatures 1 and 0, and their total heat flow defines effective conductance.Source: §2.1 · pp. 4–5

2.2 The near-contact branch

  • For near-contact pairs, heat crosses the matrix film; the thin-film annular integral incorporates interface resistance as added effective-gap length and adds a saturation length for particle diffusion.The annular-integral expression is stated under the thin-film condition h ≪ er; the saturation term is fixed to match the point-contact saturation value at h = 0 and aK = 0.Source: §2.2 · p. 5, §2.2.1 · p. 7, §2.2.2 · p. 7

2.3 The contact branch

  • For contact pairs, the solid spot path is parallel to the surrounding matrix annulus, and the spot resistance combines two constriction resistances with the contact-face interface resistance.The constriction terms arise as streamlines inside each particle converge toward the contact face.Source: §2.3 · p. 7, §2.3.1 · p. 8

2.3.2 The matrix annulus gann

  • The contact-branch matrix annulus follows the near-contact annular integral with adjusted lower limit, and its closed-form conductance does not depend on contact radius.Source: §2.3.2 · p. 8

2.4 The bond branch

  • A bond represents a pair within the same continuous solid and uses axial bulk conduction without contact constriction resistance.A separated pair in the same solid is treated as near-contact rather than bond.Source: §2.4 · pp. 8–9

2.5 The through-particle resistance

  • Through-particle resistance is added in series with the whole edge, not inside the contact spot; it is omitted as a separate term for bond edges.The cross-section is the particle’s great circle to allow three-dimensional diffusion.Source: §2.5 · p. 9

2.6 Conditions of validity

  • The model requires continuity where contact and near-contact branches meet and an interaction cut-off to prevent counting the same matrix region for multiple pairs.The cut-off removes edges below ηcut = 1, with an absolute 0.5d cutoff as a safeguard; removing edges does not compensate for their lost conduction paths, setting a lower applicability bound.Source: §2.6 · p. 9

2.7 The contribution of each element

  • In 89.4% of conditions the matrix annulus carries more than 50% of an edge; relaxing ηcut from 1 to 0.1 raises conductivity to 3.1–10.6 times effective medium theory.Adding through-particle resistance typically changes the result by just under 20%, and relaxing the cutoff raises the same-lattice edge count from 0–18 to 54–176; the latter is a validity premise, not an accuracy-tuning range.Source: §2.7 · p. 10, Table 2 · p. 11

3 The conductance filtration and sink-relative persistence

The section replaces a purely geometric filtration axis with particle-pair conductance and counts components by whether they have reached the sink. It compares the resulting edge order with distance and identifies the source–sink connection threshold as a descriptor.

3.1 Choosing the filtration function

  • Conductance is chosen over geometry as the filtration function because edge orderings can differ when particle size or material properties vary.For equal 20µm spheres, contact and near-contact conductances join continuously at h = 0; across diameters of 2–100µm, conductances at h = 0.1µm span 2.23 decades.Source: §3.1 · p. 11, Figure 4 · p. 12

3.2 Introducing the conductance filtration

  • The conductance filtration uses particle-pair conductance as its edge function and tracks connected components as the conductance threshold decreases.The descriptor depends on material as well as structure: changing filler species or surface treatment changes the barcode for the same arrangement.Source: §3.2 · pp. 12–13

3.3 When the two axes are order isomorphic

  • Distance and conductance orderings coincide for equal-sized particles in a uniform material, but size distributions can disrupt that agreement.Kendall’s τ is 0.461 for a bimodal 5:1 distribution and −0.216 for a broad continuous distribution.Source: §3.3 · p. 14, Figure 6 · p. 14
  • Changing filler species or surface treatment leaves D* unchanged but shifts the critical conductance level across 1.737 decades.Across the three conditions, D* is 19.235295µm while θ* is 3.339, 4.894, and 3.157, with corresponding g* values of 4.6×10^-4, 1.3×10^-5, and 7.0×10^-4 W/K.Source: §3.3 · p. 15, Figure 7 · p. 15

3.4 Persistence relative to the sink boundary

  • Sink-relative persistence counts components not yet connected to the sink, and each bar records how long a component remains disconnected from it.The source boundary is excluded from the sink set so its connection to the sink remains a death event identifying the critical level.Source: §3.4 · pp. 15–16, Figure 8 · p. 17

3.5 Equivalent representations and stability

  • The construction matches absolute persistence on the graph with the sink contracted, and stability applies to edge-weight changes only when the complex remains fixed.Coordinate perturbations can change the interaction-cutoff edge set, so the stability theorem gives no guarantee for that case.Source: §3.5 · pp. 16, 18

3.6 The critical conductance level and the descriptor

  • The critical conductance level is the threshold at which source and sink first connect; its corresponding g* is the bottleneck conductance of the percolating path.For the example structure, θ* = 3.339 and g* = 4.58×10^-4 W/K; the network still has more than 600 components after the first path appears.Source: §3.6 · p. 18

4 The dissipation share as a logarithmic sensitivity

The section distinguishes conductance-based bottleneck structure from edges that actually limit effective conductance, which requires solving the temperature field. It defines dissipation shares as exact relative sensitivities, then extends the analysis to finite edge changes and mechanism-level idealisation ratios.

4.1 Why the temperature field must be solved

  • Edge conductance alone identifies a percolation bottleneck, not which edges limit effective thermal conductivity; that requires solving the temperature field under the boundary conditions.A weak edge may carry no heat, so its conductance magnitude alone does not determine its transport.Source: §4.1 · p. 19

4.2 Setting

  • Effective conductance is the minimum Dirichlet energy, whose minimizer is the steady temperature field and whose value equals total heat flow at unit boundary temperature difference.Edge dissipation is defined as conductance times the squared temperature difference across the edge; with normalized boundaries, effective conductance equals total dissipation.Source: §4.2 · pp. 19–20

4.3 The dissipation share equals the logarithmic sensitivity

  • The dissipation share equals exactly the relative response of effective conductance to a relative change in that edge’s conductance.This follows by combining the derivative of effective conductance with respect to an edge weight and the sum rule that total dissipation equals effective conductance.Source: §4.3 · pp. 20–21
  • Ordering edges by conductance scarcely overlaps their ordering by dissipation share, and conductance-based selection is indistinguishable from random selection.The dissipation-share ordering is an exact ordering by sensitivity.Source: §4.3 · p. 21

4.4 An exact closed form for a finite change

  • For any finite change to one edge, effective conductance has an exact closed-form update using the edge’s temperature difference and its resistance when source and sink are shorted.The expression is exact, not first-order; it approaches the derivative as the change tends to zero and the shorted-edge value as the change tends to infinity.Source: §4.4 · pp. 21–22

4.5 The idealisation ratio of each mechanism

  • The idealisation ratio bounds the conductance achievable by separately idealising interface, matrix, or particle properties, but a large ratio only indicates possible room for improvement.A ratio near unity means further improvement of that mechanism leaves effective conductance unchanged; the ordering of mechanisms can reverse with configuration.Source: §4.5 · p. 22

4.6 Relation to previous work

  • The graph-sparsification quantity geReff(e) measures sensitivity of a different object, while the dissipation share gives the exact response of effective conductance to an edge-conductance change.The two measures therefore generally rank different edges at the top.Source: §4.6 · pp. 22–23

5 Implementation, computational cost, and verification

The implementation turns particle coordinates, radii, and material assignments into a conductance-weighted graph and computes filtration, dissipation, and finite-change quantities with passes over the edges and sparse linear algebra. Timings are near-linear in particle count, while the comparison with effective medium theory and the conductance-level estimate show where structure-sensitive information adds value.

5.1 Implementation

  • Coordinates, radii, and material assignments feed neighbour search, branch classification, graph construction, and four procedures for the barcode, critical conductance, dissipation share, and finite change.The procedures sort edges by conductance, use union–find for the barcode, solve once for dissipation, and factorise once for finite-change evaluations.Source: §5.1 · p. 23

5.2 Computational cost

  • Log–log timing slopes were 1.08 for preprocessing, 1.12 for barcode construction, 1.36 for dissipation share, and 1.49 for finite-change evaluation, all close to linear in N.The measurements covered N = 200 to 8,000 particles, with three runs at each size, on a single machine.Source: §5.2 · p. 24, Figure 11 · p. 25
  • Preprocessing takes 35.9%–64.2% of preprocessing, barcode, and dissipation time; it is largest through N ≈4000, after which the sparse solve overtakes it.Barcode construction time divided by |E|log|E| is nearly constant, consistent with sorting dominating its cost.Source: §5.2 · p. 25

5.3 Comparison with effective medium theory and an estimate of the effective conductance

  • Against effective medium theory, the median relative difference is 25.2% and the median knet/kEMT ratio is 1.07, with no systematic bias.The ratio rises with φ/φmax for every lattice and is nearly flat for κf /km above 3.Source: §5.3 · p. 26, Figure 12 · p. 27
  • Among 27 packings at φ = 0.600, log10 Geff spans 1.0 decade, while θ* gives a leave-one-family-out residual band of ±0.250 decades versus 0.348 for φ alone.The θ* relation is an empirical level estimate: coefficients depend on structure family, and one jittered-lattice family is fitted poorly.Source: §5.3 · pp. 26–28, Figure 13 · p. 27

6 A numerical demonstration

The demonstration varies particle diameter, filler species, and surface treatment across eight synthetic conditions at fixed packing fraction. On a shared structure, material and treatment changes strongly affect effective conductivity, with branch-level dissipation and the critical conductance level distinguishing the mechanisms and conditions.

6.1 Computation conditions

  • The demonstration varies particle diameter, filler species, and surface treatment across eight conditions while holding packing fraction at 0.700.The structures are synthetic and bimodal; four material conditions at the same diameter and seed share an identical structure, changing only the material.Source: §6.1 · pp. 28–29, Table 7 · p. 29

6.2 Results

  • Changing only material and surface treatment on one structure varies keff by up to a factor of 23, with larger treatment effects at smaller particle diameters.The treatment ratio is computed per seed on the same structure, rather than as a quotient of median keff values.Source: §6.2 · p. 29, Figure 14 · p. 30, Table 8 · p. 31
  • The limiting branch type depends on filler: AlN heat flow is mostly through contact branches, while SiO2 near-contact matrix branches carry more than a third of dissipation.For SiO2, lowering interfacial resistance leaves keff almost unchanged.Source: §6.2 · p. 29, Figure 14 · p. 30, Table 8 · p. 31
  • For AlN, untreated, d = 25µm, 63 edges—the top 1% of 6,289—carry 49% of total dissipation.Dissipation spreads out on the side where the matrix is limiting.Source: Figure 14 · p. 30, §6.2 · p. 31, Table 8 · p. 31
  • With diameter and treatment held fixed, filler alone shifts θ* by more than 1.5 decades; the surface-treatment shift is smaller.Because θ* is obtained without solving a Laplacian, the paper presents it as a first-pass material screen.Source: Figure 14 · p. 30, §6.2 · p. 31, Table 8 · p. 31
  • For AlN at d = 5µm, interface, matrix, and particle mechanisms are comparable; surface treatment lowers the interface ratio and raises the matrix ratio.For SiO2, the interface gain remains near unity, consistent with its near-contact dissipation share.Source: §6.2 · pp. 29, 31, Figure 14 · p. 30, Table 8 · p. 31

7 Scope and limitations

The claims are bounded by the computed conditions and by model omissions, including far-field matrix paths, validation of absolute conductivity, and assumptions about particle and surface representation. The assumed particle-contact resistance is especially consequential for high-conductivity fillers, while the results are more robust for low-conductivity fillers.

7.1 Scope of the claims

  • Claims are limited to the computed range because thresholds for the boundaries of applicability are unavailable.The particle-pair network omits parallel heat paths through matrix far from particles and increasingly underestimates effective conductivity as packing fraction falls.Source: §7.1 · pp. 31–32
  • The absolute effective thermal conductivity was not cross-checked against finite-element analysis or measurement.Agreement with the finite-element method reported for resistor-network models generally is not validation of this model.Source: §7.1 · p. 32
  • The paper makes no claim about structure or edge rankings; θ* estimates the effective-conductance level, while its critical-path correspondence is not in the asymptotically exact regime here.For structures matched in effective conductance, θ* hardly separated the pairs.Source: §7.1 · p. 32

7.2 Model assumptions and the limits of applicability

  • The isothermal-node assumption requires through-particle resistance to be small relative to interparticle resistance; representing high-aspect-ratio fibres as overlapping spheres can undercount contact scale and underestimate resistance.The paper also omits finite-thickness interphases, for which branch exchange becomes discontinuous at h = 2tip and matrix and coating design cannot be decided independently.Source: §7.2 · pp. 32–33
  • With 30% of particles untreated, edge ordering is largely preserved and a uniformly treated surrogate only slightly degrades top-edge selection.This supports robustness to treatment non-uniformity within the tested setting.Source: §7.2 · p. 33
  • The model treats steady state only, excludes heat capacity, temperature-dependent properties, and many-body screening, and its descriptor does not encode heat-flow direction.The stated stability bound applies to edge-weight perturbations, not sink-position or coordinate perturbations that change the edge set.Source: §7.2 · p. 33

7.3 Propagation of the uncertainty

  • For high-conductivity fillers, uncertainty in assumed Rc passes almost unchanged into the result, making measurement of Rc an open problem; low-conductivity fillers are more robust within the assumed range.The assumed untreated-particle range is 2.5×10^-9 to 5×10^-8 m2·K/W.Source: §7.3 · p. 33, Table 9 · p. 34

8 Conclusion

The paper combines a branch-conductance model and sink-relative conductance filtration with exact dissipation sensitivities and closed-form single-edge changes. The filtration estimates a percolation bottleneck level, while adding omitted matrix paths, validation, and directional descriptors remains future work.

8 Conclusion

  • The paper models particle-pair conductance across branch types and introduces sink-relative persistence on the conductance axis.It also establishes exact equality between dissipation share and logarithmic sensitivity, plus a closed form for a finite single-edge change.Source: §8 · p. 34
  • Dissipation allocation among branch types is obtained as an exact sensitivity, while each mechanism’s idealisation ratio is an exact value obtained by re-solving.The filtration, in contrast, returns the bottleneck-to-percolation level rather than an exact effective-conductance prediction.Source: §8 · p. 34
  • Open problems include adding parallel matrix paths, cross-checking absolute conductivity, extracting barcode information beyond θ*, and describing heat-flow direction in anisotropic structures.For matched effective conductances, θ* hardly moves while the gain ratio from dissipation share spreads widely.Source: §8 · p. 34

A Derivations for the branch-conductance model

The derivations construct contact, near-contact, and bond conductances from the particle geometry and heat-flow paths. They also specify where approximations and branch-specific assumptions matter.

A.1 The contact radius

  • The contact face is the intersection circle of overlapping hard spheres, with its geometry derived from the overlap depth and sphere radii.The resulting relation underpins the stated continuity of the branch-conductance model.Source: §A.1 · p. 35

A.2 The annular integral of the near-contact branch

  • For near-contact pairs, interface and matrix resistances are combined in series within each radial flux tube before the tubes are combined in parallel.Reversing this order makes the gap area vanish as the gap closes and would drive conductance to zero; the outer cutoff is not unique.Source: §A.2 · pp. 35–36

A.3 The matrix annulus of the contact branch

  • The contact branch's matrix annulus uses a parabolic gap profile, with its lower integration limit approximated by the contact radius when overlap is small.A saturation shift suppresses the logarithmic divergence at zero interface resistance, and the solid constriction and matrix annulus are treated as consistent terms.Source: §A.3 · pp. 36–37

A.4 The bond branch and discretisation

  • The contact-branch constriction resistance cannot represent a bond: subdividing a solid into N spheres would make its chain resistance grow with N, unlike the bond conductance model.With the bond model, subdivision shortens each interval proportionally, leaving total chain resistance unchanged.Source: §A.4 · p. 37

B Proofs for the persistence and sensitivity results

This passage only states that the section uses the notation of Section 4.2.

B. Proofs for the persistence and sensitivity results

B.1 Equivalent representations

The section relates sink-relative persistence to two graph representations: the absolute barcode after contracting the sink boundary, and widest-path bottleneck values. Bars closed by the elder rule are distinct from those that reach the sink.

B.1 Equivalent representations

  • The relative H0 barcode equals the absolute H0 barcode of the quotient graph formed by contracting the sink boundary to one point.The contraction preserves vertices and edges outside the boundary, so component births, mergers, and sink connections occur at the same filtration times.Source: §B.1 · p. 37
  • For a vertex v, the filtration value when its component first reaches the sink equals the bottleneck value of its widest path to the sink.In conductance terms, this is the path maximizing its minimum edge conductance.Source: §B.1 · pp. 37–38
  • Elder-rule bars that close by merging into an older component do not have death times given by widest-path values.The widest-path correspondence applies to bars that close by reaching the sink.Source: §B.1 · p. 37, Figure 15 · p. 38

B.2 The envelope theorem and the logarithmic sensitivity

The proofs connect edge dissipation to the logarithmic sensitivity of effective conductance, give a closed form for changing one edge, and contrast that sensitivity with a spanning-tree indicator.

B.2 The envelope theorem and the logarithmic sensitivity

  • An edge's dissipation share equals the logarithmic sensitivity of effective conductance to that edge's conductance.The argument uses the minimum defining effective conductance and fixed boundary values; the envelope-theorem step removes the derivative through the minimizing potential.Source: §B.2 · p. 39

B.3 The closed form for a finite change

  • The section establishes a closed form for the effective-conductance change caused by a finite change to one edge.The expression uses the effective resistance between the edge's endpoints in the circuit with source and sink shorted; effective conductance is increasing and concave in each edge conductance.Source: §B.3 · pp. 39–40

B.4 Contrast with an existing sensitivity indicator

  • The indicator based on edge conductance times ordinary effective resistance is a logarithmic derivative of the weighted spanning-tree count, not of effective conductance.Its normalization is also described as graph-size-dependent and independent of edge weights.Source: §B.4 · p. 40

C Structure families

The structure-family tables document generation rules and generated properties for the paper’s 23-family ensemble. The 69 packings, including a legacy family, form the basis for regression, cross-validation, and the reported residual.

C Structure families

  • The ensemble comprises 23 structure families generated with three random seeds each, yielding 69 packings for regression and cross-validation with a residual of 0.250 decades.Table 10 gives generation rules, while Table 11 reports generated properties as medians over three seeds.Source: C Structure families · p. 40, Table 10 · p. 41, Table 11 · p. 42
  • Sedimentation uses s = 0 for none and s = 1 for complete, while stratification is specified by z′ = L(z/L)^(1+p).Source: C Structure families · p. 42
  • Packing fraction is targeted except for fiber_z, fiber_xy, and jittered_ref; the fiber families add 24 fibres of six spheres each to a 0.600 packing, reaching 0.696 and 1,044 particles.The fibres are added to an isotropic packing.Source: C Structure families · p. 42
  • poly_wide’s size ratio of 22.5 follows from σln = 0.5, unlike the design-value ratios 3.00 and 5.00 of the bimodal families.Simulation-box dimensions vary by family; jittered_ref alone is non-cubic at 355×93×355 µm.Source: C Structure families · p. 42
  • jittered_ref is a legacy family generated by a different method, but it remains included among the 69 packings.Its non-cubic specimen dimensions are 355×93×355 µm.Source: C Structure families · pp. 42–43
  • coat_mix uses lowk_coat as a hypothetical extreme unfavorable case, with tip = 0.05 µm, kip = 0.05 W/(m·K), and Rc = 1 × 10−6 m2·K/W.It is the only Table 1 exception to tip = 0.Source: C Structure families · p. 43
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