| 01 Sep 2026
A unified explicit formula for temperature-, pressure-, and composition-driven reaction-front propagation
Abstract. Metamorphic reactions, phase transitions, and fluid–rock interactions play key roles in many geodynamic processes. Such reactions can be driven by changes in temperature, pressure, or chemical composition, and hence by different thermodynamic variables, yet commonly propagate through rocks as moving reaction fronts. However, no unified theory predicts their propagation. Here, we show that reaction front propagation can be described within a single conservative moving-boundary framework. The theory couples diffusive transport with local equilibrium thermodynamics through the Rankine–Hugoniot condition and yields explicit analytical expressions for reaction-front position and velocity. Unlike classical Stefan-like solutions, which generally involve error functions and numerical solutions of a transcendental equation, our formulation directly relates front propagation to transport properties and equilibrium thermodynamics. The theory predicts the characteristic square-root-of-time, √t, scaling observed for temperature-, pressure-, and composition-driven fronts and shows that the diffusivity governing front propagation generally differs from the physical transport diffusivity. Instead, an effective front diffusivity combines diffusive transport with the thermodynamic transformation across the front. This combination is quantified by a dimensionless L number linking equilibrium thermodynamics to front dynamics. Consequently, slow front propagation does not necessarily imply slow transport but may reflect large changes in thermal energy or mass associated with the reaction. Furthermore, sharp reaction fronts can propagate under diffusion-dominated conditions because finite thermodynamic jumps are explicitly represented at the moving interface, unlike conventional advection–diffusion–reaction models, in which sharp fronts typically require advection-dominated transport. The framework provides a unified basis for predicting and interpreting reaction-front propagation in experiments and geological systems.
I enjoyed this study, with a clean 1D focus and meaningful assumptions leading to an analytical solution capturing the effective diffusivity at play in the case of diffusion and equilibrium reaction. The thermodynamical aspects of the reaction associated to the three cases considered (temperature-, pressure-, and composition-controlled) slow down the diffusion, but maintain and rescale the typical sqrt(t) relationship. The derivations are very detailed and well explained, which makes the paper accessible to a wide audience despite its mathematical approach. Overall, I found the paper well within the scope of the journal and the message of interest for a wide range of geoscientists.
My only concern is about the pitch of the manuscript, which tends to overstate both the novelty and generality of the result.
Regarding the novelty, terms such as “new explicit analytical solution”, “we derive”, and “here, we show” tend to suggest that the entire pressure-driven result, the moving-boundary construction, and the sqrt(t) scaling originate in this manuscript. This is particularly problematic because the pressure result has already been presented in the authors’ other preprint, egusphere-2026-3107. The manuscript should identify consistently what is previously established, what is reformulated here, and what is genuinely new. It should demonstrate novelty through explicit comparison rather than through general statements that no unified theory exists. I would welcome a broader literature comparison beyond the geodynamic examples presently cited, as I would expect closely related results in the moving-boundary, dissolution, filtration, corrosion or reactive-transport literature.
Regarding the scope, the assumptions should be placed more explicitly in a Péclet–Damköhler framework from the beginning. The authors should discuss the domain of validity of the diffusion-dominated and fast-reaction limit, as well as the expected departures when advection or finite kinetics becomes important. The wording needs to be more accurate. The authors seem to argue that their framework explains why sharp fronts can propagate under diffusion-dominated conditions because it explicitly represents a finite jump at the interface, but the sharpness of the front is an assumption of their model.
The term “unified” also requires qualification. The manuscript demonstrates a common representation of three separate scalar limiting cases. It does not demonstrate a coupled theory for fronts simultaneously affected by temperature, pressure and composition, nor a general theory spanning diffusion-, advection- and reaction-controlled regimes. The title, abstract and conclusions should better reflect this distinction. Natural reaction fronts are commonly affected by simultaneous variations in all three. The actual thermodynamic driving force is the combined reaction affinity, while temperature, pressure and composition are variables that contribute to it.
The manuscript is a bit repetitive in its current form. The central sequence, conservation law, jump condition, quasi-steady gradient, effective diffusivity and sqrt(t) integration, is restated separately for each application and then repeated again in the discussion and conclusions. I suggest presenting the generic derivation once, followed by a compact substitution for each case that identifies u, A, K, [A] and the relevant boundary conditions. (A summary table could potentially replace a substantial amount of repeated prose.) The individual examples should then concentrate on the genuinely case-specific physics, assumptions and limitations. This restructuring would make the common mathematical pattern more apparent, shorten the manuscript, and create space for a stronger discussion of prior literature, regime validity, simultaneous driving variables, and departures from the equilibrium diffusion-controlled limit.
Specific comments:
I look forward to reading the authors’ next steps on resolving the advective boundary-layer structure.
Thomas Poulet