Run several film materials on the same track — same substrate, same slit pattern — to see, stopwatch in hand, which one drains heat fastest. This guide explains the principle, what the tool assumes, and how to read its results without being a thermal engineer.
You have several possible candidates for a conductive film — Ginestium™, CVD graphene, CVD diamond — and a simple question: which one spreads heat best, in practice, on this substrate and this ablation pattern? The tool holds one edge of the plate at a fixed temperature and watches, for each material taken separately, how fast and how strongly heat progresses toward the other edge.
Materials are compared on rigorously equal footing: same substrate, same slit pattern, same thermal conditions — only the film changes.
The thermal anisotropy simulator (the companion tool) answers "what is this pattern's effective conductivity?" at steady state. This one answers a different, more concrete question: "starting from a heat source at a given moment, how long before a sensor at a given distance feels the difference?" — useful for comparing materials against each other under a realistic propagation scenario.
It is also, very directly, a due-diligence tool: every default value is sourced (see §07), so a comparison can be defended in front of a technical partner rather than quoted from memory.
Finally, it is the only one of the three thermal tools to carry a time axis — and therefore the only one where a material's heat capacity comes into play. This point, a frequent source of questions, is covered in the note (§10).
The scenario is deliberately simple: a 50 × 50 mm plate, a left edge held at a fixed temperature (120 °C by default), and the rest of the plate warming up gradually as heat seeps to the right. A virtual "sensor" is placed at a chosen distance from the edge, and its temperature is watched rising over time.
This progression is computed directly by a closed-form formula (no step-by-step simulation needed), provided one key number is known per material: its effective diffusivity, written αeff — how fast heat propagates through it, accounting for both its conductivity and its capacity to store heat.
The film being compared may have a spectacular conductivity — but it is only a few hundred nanometres thick, against tens to thousands of micrometres for the substrate underneath. The system's effective diffusivity is a thickness-weighted average of the two: the thinner the film relative to the substrate, the more the final result resembles the substrate's own properties, regardless of the film's performance.
This is why the tool also shows each film's k·t (conductivity × thickness, along x): the real figure of merit for a film meant to spread heat laterally — a highly conductive but too-thin material can be beaten by a less conductive but thicker one.
The slit pattern (same geometry as the anisotropy simulator) is homogenised by the same calculation engine — a design that favours x stays faster here too, but the effect shows up as a difference in propagation speed rather than a k_xx/k_yy ratio. What's specific to this tool: the time-propagation formula, the heat capacity feeding it, and the ability to compare up to 4 materials side by side on diffusion maps at the same physical scale.
The left edge is assumed to be at uniform temperature across its full height: heat therefore advances the same way regardless of where on the plate you look (no top/bottom edge effects). This is what makes a closed-form calculation possible, with no step-by-step simulation.
The plate is treated as extending indefinitely to the right: no adiabatic right edge, no convective or radiative losses. Valid as long as the front hasn't reached the opposite edge — beyond that, the formula underestimates the real heating.
Film and substrate are assumed to sit at the same local temperature at every point x: the model does not track the film's warming and the substrate's separately, nor any lag between the two. This is the so-called equivalent-plate assumption, directly tied to how interfaces are handled (see §10).
Interface resistance (film/substrate) is computed and shown in the out-of-plane figures, but it does not enter the lateral propagation-front calculation — a deliberate modelling choice, identical to the reference this tool is based on. Detail and justification in §10.
In the zones hollowed out by the slits (air, vacuum, aerogel…), heat capacity is treated as negligible: only the genuinely solid fraction of film counts in the effective diffusivity calculation.
Density, heat capacity and conductivity are taken at room temperature and do not vary with temperature during the transient, even though the plate goes from 20 to 120 °C in the default scenario.
For a catalogue material, thickness and in-plane conductivity are freely editable; out-of-plane conductivity, CTE, phonon mean free path, density and heat capacity stay at their catalogue values.
Each catalogue material shows, beneath its thickness and conductivity fields, a "Lit. range" line giving a low–high spread from published measurements. Hovering over it shows the full justification (measurements cited, assumptions, units) — in the interface's active language.
Three materials, three statuses:
For diamond and graphene, default values are now those of real thin films (nanocrystalline for diamond, supported few-layer for graphene) — not those of a bulk crystal or a suspended monolayer measured in a lab, which are far more favourable cases than an actual deposited film. For Ginestium™, absent a published measurement, the central value is framed by analogy with comparable graphitic carbons; the upper end of the range remains the best internally measured case, not the default average to assume.
Note: the "literature" status does not necessarily cover every property of a material. For Ginestium™ in particular, density and heat capacity are stand-in values, not measurements — detail in the note (§10).
Summary (CSV) — summary table of the last comparison's results, ready for a spreadsheet.
Settings (JSON) — all the tool's settings (materials, pattern, substrate, thermal conditions), not the results. Re-imports identically, handy for freezing a reference configuration and circulating it.
This section is the guide's technical reference: it describes the formulas actually executed by the server engine and justifies each modelling choice. It is not required to use the tool.
Before any question of time, the slit pattern is homogenised exactly as in the thermal anisotropy simulator, with the same server module and the same formulas — hence the same numbers as if driving that simulator by hand with this material. The chain is:
The detail and justification for these six steps are in §09 of the thermal anisotropy simulator's guide. What follows is specific to this tool.
Volumetric heat capacity is an extensive quantity: two materials stacked together store the sum of what each stores. A thickness-weighted average is therefore the right combination, and it's exactly the same structure used for K_xx — numerator and denominator thus concern the same object, which is essential for their quotient in step 3 to make sense.
The (1 − ψ) factor encodes an explicit assumption: the medium filling the slits (air, vacuum, aerogel) stores nothing useful at these scales. Only the remaining solid material counts.
This is the very definition of thermal diffusivity: α = k/(ρ·c_p). It measures the competition between a material's ability to transport heat (numerator) and its ability to store heat along the way (denominator). A material that stores a lot slows the front's progression even if it conducts well — precisely the information conductivity alone does not give, and exactly why this tool exists.
K_xx is used, not the film's own k_xx: it's the whole plate, film plus substrate, that conducts heat to the right.
Since the source is uniform across the left edge's full height, the problem is left with a single spatial variable. It reduces to the one-dimensional heat equation:
This problem has an exact solution, obtained through the similarity variable η = x / (2√(α·t)):
This is one of the few transient-conduction problems to have an exact analytical solution, and the uniform-source assumption is precisely what unlocks it. The consequences are concrete: no time step, so no stability condition to respect and no accumulating numerical error; the result at 3 hours costs exactly the same calculation as the result at 3 seconds; and all four materials compute in a few microseconds. The erfc function comes from the standard library, which also removes the polynomial-approximation error a JavaScript implementation would have had to accept.
A reading note: the solution depends on x and t only through the grouping x/√(α·t). Two materials whose diffusivities differ by a factor of 4 produce the same profile, one simply twice as early as the other. This is what makes α_eff directly comparable from one material to another.
"Delay to 10%" and "delay to 50%" are the inverse of (4): the fraction reached is fixed, and the time is sought.
There is no inverse-erfc function in the standard library: z is obtained by bisection over [0 ; 8], in 80 iterations — since erfc is strictly decreasing, bisection is exact to machine precision well before the 80th step.
The delay varies as x², not as x. Doubling the sensor's distance does not double the arrival time: it quadruples it. This is the signature of any diffusive phenomenon, and it explains why an excellent heat spreader can still be ineffective beyond a few millimetres.
This quantity does not enter the transient model: it is computed separately, as an indicator. It decides whether a film carries any weight at system level, independent of the propagation scenario — since the system averages film and substrate in proportion to their thickness, a film whose k·t is far below the substrate's is diluted to invisibility, however high its conductivity.
Finally, the diffusion map samples (4) at 41 points across the plate's 50 mm, normalised between 0 and 1. Every row of the map is identical: this is not a display simplification, it's what the solution genuinely predicts for a uniform source.
Two questions recur regularly about the deposited materials: where does their heat capacity come in, and what is done about interfacial thermal resistances. Straight answers, with the assumptions actually made across the three thermal tools.
The regime decides. At steady state, heat capacity has no effect at all: once the temperature field has settled, only conductivity governs how heat is distributed. In the transient regime, by contrast, it is heat capacity that, together with conductivity, sets the thermal diffusivity α = k/(ρ·c_p) and hence the speed at which temperature evolves.
The three tools split accordingly:
| Tool | Regime | Film ρ·c_p |
|---|---|---|
| Thermal Diffusion Comparator | transient | enters explicitly |
| Thermal Anisotropy Simulator | steady state | does not enter |
| Parametric Sweep | steady state | does not enter |
For the two steady-state tools, this is therefore neither an oversight nor an implicit inclusion: they solve an equation — ∇·(k∇T) = 0 — in which ρ·c_p does not appear, and neither carries these fields in its parameters at all. The material catalogue is shared by all three tools and does record ρ and c_p, but only this tool consumes them.
Here, capacity enters through the effective diffusivity — equations (1) and (2) from §09, restated together:
Two explicit assumptions are made here about the deposited layers. First, the ablated fraction of the film stores nothing: the slit medium (air, vacuum, aerogel) is treated as having no meaningful capacity at these scales, hence the (1 − ψ) factor. Second, ρ and c_p are taken at 300 K and held constant — no temperature dependence is modelled, even though the default scenario takes the plate from 20 to 120 °C.
Values come from the shared server catalogue:
| Material | ρ (kg/m³) | c_p (J/kg·K) | Data status |
|---|---|---|---|
| Ginestium™ | 2,100 | 710 | stand-in — oriented synthetic pyrolytic carbon at 300 K; not a Ginestium™ measurement |
| Few-layer CVD graphene | 2,200 | 700 | literature |
| Nanocrystalline CVD diamond | 3,400 | 536 | measured on an NCD film at 302 K |
| Copper (control) | 8,960 | 385 | literature, bulk copper |
| Substrate | user-entered (defaults 3,900 / 880) | — | |
A "custom" material has neither ρ nor c_p: in that case the tool explicitly returns "no diffusivity" and shows neither a map nor a delay, rather than guessing a value.
Thickness-weighting crushes the film's contribution. With the default settings — Ginestium™ 400 nm on 10 µm of substrate — the film accounts for only ≈ 1.6% of (ρc)_sys; push the substrate to 2 mm and its share drops below 0.01%. Effective diffusivity is therefore, in practice, "the system's lateral conductivity divided by the substrate's capacity."
A direct consequence for interpretation: the front is very insensitive to the film's c_p. Not because it's neglected — it's genuinely in the formula — but because the stack's geometry leaves it little weight. The film matters through its k·t, in the numerator, far more than through its capacity in the denominator. As a corollary, the ρ and c_p values you enter for the substrate deserve real attention: they are what drive the front's speed.
They are not ignored, but their treatment is partial, and out-of-plane only.
What exists. This tool's R_th interface slider (0 to 100 ×10⁻⁶ m²·K/W, default 10; the same parameter in the anisotropy simulator) enters in series into the out-of-plane resistance:
This is a single, global film/substrate interface. The geometric model has only two layers: there is no multilayer stack with one resistance per interface, and no lateral resistance between grains or between pattern features.
A useful order of magnitude: for graphene (k_z = 6 W/m·K over 1.675 nm), the film term is ≈ 3×10⁻¹⁰ m²·K/W, five orders of magnitude below a typical interface resistance. In R_z, it is always the interface and the substrate that dominate, never the film.
What is neglected, and this is a deliberate assumption. R_th,if enters neither the k_xx/k_yy homogenisation, nor K_xx, nor therefore α_eff — the lateral diffusion front ignores it completely. The tool states this explicitly under the relevant slider, rather than letting you believe the setting affects the maps. The parametric sweep, for its part, carries no interface resistance at all: it is a purely in-plane tool.
For lateral transport, interface resistance is not a series resistance on the heat's path: it's a coupling between two parallel paths, the film and the substrate. Neglecting it is exactly equivalent to assuming perfect coupling along z — film and substrate at the same local temperature at every abscissa. This is the equivalent-plate assumption already flagged in §03, and it is inseparable from the model's very shape: (4) describes only one temperature T(x, t), not one for the film and one for the substrate.
The regime where it breaks is identifiable: short times, or a highly conductive film on a poorly conductive substrate. There, the film would race ahead of the substrate instead of dragging it along, and the sensor would see heat arrive earlier than the model predicts — an effect the model structurally cannot represent. For the thicknesses and observation times targeted here, the approximation stays reasonable; it is nonetheless an assumption, not a consequence.