Defensive publication · heat–charge–information

Joule-Priced Co-Optimization of a Shared Microfluidic Substrate for Physical AI

One embedded electrolyte network, optimized jointly as coolant, power medium, and iontronic computer. Pricing any current alone is dominated whenever the couplings below are nonzero.

David Jean Charlot, PhD · Michael Hiner

The Charlot Lab & The Hiner Lab, Institute for Physical AI @ JBI

Correspondence: contact@physicalai-bmi.org · physicalai-bmi.org · Interactive companion: physicalai-bmi.org/research/hiner-lab#topic-microfluidic

CC0 1.0 Universal, dedicated to the public domain. Timestamp anchor: sha256://<fill>

A single embedded electrolyte network is optimized jointly as coolant, localized power/regulation medium, and iontronic computational substrate. The three currents (heat, charge, information) share one flow and one dissipation budget. Under nonzero coupling (K1–K3, §4) the joint program below strictly dominates any pricing that optimizes cooling, power delivery, or computation in isolation. The disclosed object is the program itself: the objective, its ledger, the three couplings, and a determinism constraint that makes the readout bit-exact and safe. All constituent transfer functions are public physics; every constituent transfer function cited in §2 resolves to published physics, and this review did not locate a proprietary formulation of any of them in the sources searched (the references below, plus the primary literature they cite).

1. Decision variables

Design (static) collects the channel diameters $d_i$ and lengths $\ell_i$, the manifold topology $\mathcal{T}$, electrode areas $A_{e,i}$, porosities $\varphi_i$, node geometries $(\mathbf{x}_k,a_k)$, and the per-node quantizer LSB $\delta_k$:

$$\mathbf{g} = \{\, d_i,\ \ell_i,\ \mathcal{T},\ A_{e,i},\ \varphi_i,\ (\mathbf{x}_k,a_k),\ \delta_k \,\}.$$

Operating (control, time-varying): the flows $Q_i(t)$, inlet concentration $\sigma_{in}(t)$ and temperature $T_{in}$, drive current $j(t)$, and node gates $\mathbf{v}_k(t)$, with state fields $\mathbf{s}$:

$$\mathbf{u}(t) = \{\, Q_i(t),\ \sigma_{in}(t),\ T_{in},\ j(t),\ \mathbf{v}_k(t) \,\}, \qquad \mathbf{s}(\mathbf{x},t)=(\mathbf{u}_f,\,p,\,T,\,\{c_s\},\,\phi).$$

2. Governing constraints (physics on domain $\Omega$)

Momentum (Darcy–Brinkman; open channel and porous electrode):

$$\rho(\mathbf{u}_f\!\cdot\!\nabla)\mathbf{u}_f = -\nabla p + \mu\nabla^2\mathbf{u}_f - \tfrac{\mu}{\kappa}\mathbf{u}_f, \qquad \nabla\!\cdot\!\mathbf{u}_f=0.$$

Energy:

$$\rho c_p\big(\partial_t T + \mathbf{u}_f\!\cdot\!\nabla T\big) = \nabla\!\cdot(k\nabla T) + q_{gen} + q_{rxn}.$$

Species (Poisson–Nernst–Planck with the Soret term), per species $s$:

$$\partial_t c_s + \nabla\!\cdot\mathbf{J}_s = R_s, \qquad \mathbf{J}_s = \underbrace{c_s\mathbf{u}_f}_{\text{adv}} - \underbrace{D_s\nabla c_s}_{\text{diff}} - \underbrace{z_s\tfrac{D_sF}{RT}c_s\nabla\phi}_{\text{migr}} - \underbrace{D_s c_s S_{T,s}\nabla T}_{\text{Soret}}.$$

Poisson: $\;-\nabla\!\cdot(\varepsilon\nabla\phi)=F\sum_s z_s c_s.\;$ Electrode kinetics (Butler–Volmer, temperature-dependent exchange current):

$$j = j_0(T)\big[e^{\alpha_a F\eta/RT}-e^{-\alpha_c F\eta/RT}\big], \qquad j_0(T)=j_0^{ref}\,e^{-\frac{E_a}{R}\left(\frac1T-\frac1{T_{ref}}\right)}.$$

Iontronic readout at node $k$: $\;y_k=\mathcal{G}_k(\{c_s\},T,\mathbf{v}_k)\;$ (local conductance/current functional, analog). Quantized readout: $\hat y_k=\mathsf{Q}(y_k)$ with $\mathsf{Q}$ a Q32.32 fixed-point quantizer of LSB $\delta$. The committed observable is the word $\hat y_k$, not the analog $y_k$; sub-LSB drift is absorbed by $\mathsf{Q}$. The trajectory digest is $H=\mathrm{SHA256}\big(\,\|_k\|_t\,\hat y_k(t)\,\big)$ over the canonical serialization of all node words.

3. Objective: net exergy destroyed per relevant outcome

$$\boxed{\;\min_{\mathbf{g},\,\mathbf{u}(t)}\;\; J_{\text{rel}}= \frac{\displaystyle\int_0^\tau\Big(\,P_{\text{comp}}+P_{\text{pump}}-P_{\text{fc}}-\eta_{\text{ex}}^{\text{self}}\dot X_{\text{th}}^{\text{self}}-\eta_{\text{ex}}^{\text{ext}}\dot X_{\text{th}}^{\text{ext}}\,\Big)\,dt} {\displaystyle\int_0^\tau \Lambda\big(\{y_k\},t\big)\,dt}\;}$$

Term	Definition	Role
$P_{\text{comp}}=\sum_k v_k i_k$	iontronic switching + CMOS	compute cost; floored by Landauer
$P_{\text{pump}}=\tfrac1{\eta_p}\sum_i Q_i\,\Delta p_i$	hydraulic work	debit
$P_{\text{fc}}=\int_{A_e} j\,(E_{eq}-\eta_{act}-\eta_{ohm}-\eta_{conc})\,dA$	flow-cell delivery	charge credit
$\dot X_{\text{th}}^{\text{self}}$	outlet availability recirculated to upstream electrode	self-recovery credit: closes in the network, drives K1
$\dot X_{\text{th}}^{\text{ext}}$	outlet availability exported to a named external sink	export credit: weaker; defaults to $0$
$\Lambda \le N_{node}/\tau_{ion}(T,\mathbf{g})$	utility-weighted throughput	relevant outcomes / s

with $\dot X_{\text{th}}^{\text{self}}+\dot X_{\text{th}}^{\text{ext}}=\dot m\,c_p\big[(T_{out}-T_0)-T_0\ln\tfrac{T_{out}}{T_0}\big]$. Only the self term is defensible without a counterparty; a reader cannot collapse export into self. Set $\eta_{\text{ex}}^{\text{ext}}=0$ for the standalone-die claim.

4. The three coupling terms

K1: thermal → charge, positive (the exergy-recovery mechanism):

$$\frac{\partial P_{\text{fc}}}{\partial T}>0 \quad\text{via}\quad \frac{\partial j_0}{\partial T}>0,\ \ \frac{\partial E_{eq}}{\partial T}\ (\text{Nernst}).$$

Removed heat upgrades delivered charge; dissipation becomes a credit, not pure loss.

K2: flow → cooling vs. power vs. pumping, adversarial:

$$\frac{\partial T_{\max}}{\partial Q}<0, \qquad \frac{\partial \xi}{\partial Q}<0, \qquad \frac{\partial P_{\text{pump}}}{\partial Q}>0.$$

One scalar $Q$, three opposed partials ($\xi$ = residence-limited utilization). A second adversarial scalar is the per-node quantizer LSB $\delta_k$:

One scalar (the flow rate Q) sets cooling, power delivery, and pumping cost at once. What follows?

Flow rate Q 0.35

Interactive figure. K2, drawn: raising Q lowers peak temperature (good), lowers residence-limited utilization ξ (bad), and raises pumping power (bad). Because the three partials pull against one another through a single knob, the net exergy cost J has an interior minimum; optimizing any one of the three alone lands away from it. That is exactly why the paper insists the program be solved jointly rather than per-subsystem.

Show the computation

∂T_max/∂Q < 0     cooling improves with flow
∂ξ/∂Q     < 0     residence time falls, utilization drops
∂P_pump/∂Q > 0    hydraulic work rises (∼Q²)
J(Q) = P_pump(Q) + thermal_penalty(T_max(Q)) − credit·ξ(Q)   → minimized in the interior
Illustrative parameterization only, chosen to make the three opposed signs visible:
  T_max(Q) = 0.15 + 0.85/(1 + 3.2 Q)   ξ(Q) = 1/(1 + 1.35 Q)   P_pump(Q) = 0.30 Q²
  J(Q) = P_pump + 1.15·T_max^1.6 − 0.55·ξ
Q and J are dimensionless. No device is modelled; the coefficients are not drawn from any measurement, and Q* is a property of these coefficients, not of any electrolyte network.

$$\frac{\partial\,(\text{margin}_5)}{\partial \delta_k}>0, \qquad \frac{\partial \Lambda}{\partial \delta_k}<0 \ \ (\Lambda\propto \log_2(\text{range}/\delta_k)).$$

Constraint 5 lower-bounds it: $\delta_k\ge\delta_k^{\min}(\mathcal{F})=2\sup_{\mathcal{F}}|y_k-\bar y_k|$, and $\Lambda$ wants it small, so the optimum pins $\delta_k=\delta_k^{\min}(\mathcal{F})$; quantization rides the invariance floor. Any widening of the feasible envelope $\mathcal{F}$ for K2 cooling/power freedom inflates $\sup_{\mathcal{F}}|y_k-\bar y_k|$, raises $\delta_k^{\min}$, and pays in $\Lambda$. Cooling headroom is charged, through invariance, against readout resolution (not through the energy ledger).

K3: thermal ↔ information, bidirectional (Soret):

$$\mathbf{J}_s^{\text{Soret}}=-D_s c_s S_{T,s}\nabla T \ \Rightarrow\ \nabla T \text{ writes } c_s \text{ writes } y_k; \qquad q_{gen}=P_{\text{comp}}\ \text{feeds } T.$$

The thermal-management field is a computational input, and computation is a distributed heat source. $\nabla T$ is a controlled design input, not a nuisance to be uniformly suppressed. Two disjoint regimes are claimed: (a) write mode, $\nabla T$ shaped at node $k$ to program $c_s$ through the Soret flux (the cooling field doubles as the write bus); (b) invariance mode, $\nabla T$ held flat within tolerance so readout depends only on $\mathbf{v}_k$. Constraint 2 bounds $T_{\max}$ in both; it does not force $\nabla T\!\to\!0$ globally. (Secondary: $\partial\tau_{ion}/\partial T<0$; warmth also accelerates $\Lambda$.)

5. Constraints

Physics: all §2 PDEs hold on $\Omega$ with boundary conditions.
Reliability: $T(\mathbf{x},t)\le T_{\max}\ \forall\,\mathbf{x},t$.
Hydraulic: $\Delta p_i\le\Delta p_{\max}$, $\ Q_i\ge0$.
Electrochemical: $\sigma\in[\sigma_{\min},\sigma_{\max}]$, $\ j\le j_{\text{lim}}(Q,c)$ (mass-transport limit).
Determinism / hardware-invariance (hard, non-tradeable): bit-exact, hash-checkable, no tolerance band. $H(\mathbf{u},\mathbf{g})=H^{\text{ref}}\Leftrightarrow \hat y_k=\hat y_k^{\text{ref}}\ \forall\,k,t,\ \forall(\mathbf{g},\mathbf{u})\in\mathcal{F}$. The reference is a fixed-point word, so invariance is verified by SHA-256 equality of the trajectory digest, not an analog $\epsilon$; $\delta$ must exceed the worst-case spread, $\sup_{\mathcal{F}}|y_k-\bar y_k|<\delta/2$. Operating point may change speed and cost, never the emitted word. This is the physical-AI-safety constraint; it does not trade against $J_{\text{rel}}$.
Thermodynamic floor: $J_{\text{rel}}\ge k_B T\ln 2$ per irreversible bit (feasibility bound).
Manufacturability: $\mathbf{g}\in\mathcal{G}_{\text{fab}}$ (open-PDK / soft-litho realizable).

6. Claim

The three currents share one network and one dissipation budget. Under $K1,K2,K3\neq 0$, any pricing that optimizes cooling, power delivery, or computation independently is dominated by the joint program above, by the following argument: an isolated objective is $J_{\text{rel}}$ with the coupling credits ($P_{\text{fc}}$, $\eta_{\text{ex}}^{\text{self}}\dot X_{\text{th}}^{\text{self}}$) and the shared-$Q$ constraint deleted, so its feasible set is a superset in which those terms are unpriced; its optimum therefore upper-bounds $J_{\text{rel}}^{*}$, with equality only when $K_1=K_2=K_3=0$. This is an argument, not a formal proof; a proof under stated regularity conditions on $\Lambda$ and $\mathcal{F}$ is not given here.. The disclosed object is the program itself (objective, ledger, K1–K3, and the determinism constraint), priced in exergy rather than energy. All constituent transfer functions ($j_0(T)$, $S_{T,s}$, $\tau_{ion}(T)$, $E_{eq}(T,\sigma)$) are public physics; every constituent transfer function cited in §2 resolves to published physics, and this review did not locate a proprietary formulation of any of them in the sources searched (the references below, plus the primary literature they cite).

Selected prior art

R. van Erp, R. Soleimanzadeh, L. Nela, G. Kampitsis, E. Matioli. Co-designing electronics with microfluidics for more sustainable cooling. Nature 585, 2020. doi:10.1038/s41586-020-2666-1.
T. M. Kamsma, W. Q. Boon, T. ter Rele, C. Spitoni, R. van Roij. Iontronic Neuromorphic Signaling with Conical Microfluidic Memristors. Phys. Rev. Lett. 130, 268401, 2023. arXiv:2301.06158.
T. M. Kamsma et al. Brain-inspired computing with fluidic iontronic nanochannels. PNAS 121, e2320242121, 2024.
R. Landauer. Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 1961.

AI-use disclosure. Preparation of this defensive publication used a large language model (Claude, Anthropic) for drafting, editing, typesetting the equations, and the interactive companion. The authors reviewed the content, verified the cited prior art, and are solely responsible for the work. Consistent with ICMJE, COPE, and IEEE guidance, the model is a tool and is not credited as an author.

Institute for Physical AI @ JBI · The Charlot Lab & The Hiner Lab
503 McKeever Rd, Arcola, TX 77583, USA
physicalai-bmi.org · contact@physicalai-bmi.org

Defensive Publication DP-2026-01
CC0 1.0 Universal, dedicated to the public domain.
No rights reserved. Published as prior art.