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arxiv: 1906.11022 · v1 · pith:554FE5XNnew · submitted 2019-06-26 · ❄️ cond-mat.stat-mech · cond-mat.other

Surface thermodynamics, surface stress, equations at surfaces and triple lines for deformable bodies

Juan Olives (CINaM) This is my paper

Pith reviewed 2026-05-25 15:25 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.other
keywords surface thermodynamicssurface stresstriple linesYoung's equationLaplace equationdeformable bodiescapillarityGibbs thermodynamics
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The pith

Surface strain as a thermodynamic variable yields new equilibrium equations at surfaces and a modified Young's law at triple lines.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper refines Gibbs' approach to surface thermodynamics for deformable bodies by showing that the surface state is fully determined by temperature, chemical potentials, and the surface strain tensor. This identification supports a new definition of surface stress and produces mechanical equilibrium conditions at the surface itself, whose normal part generalizes the classical Laplace pressure relation. At a body-fluid-fluid triple line the derivation supplies two distinct balance equations: one for the forces when the line is fixed to the body, and a second governing the line's motion relative to the body. The motion equation replaces Young's classical capillary relation with a strain-dependent form. The results matter for any system in which surfaces can sustain strain, such as viscoelastic solids or viscous fluids, because they replace ad-hoc surface-tension treatments with thermodynamically consistent stress and equilibrium conditions.

Core claim

Following and refining Gibbs, the local thermodynamic variables of the surface are only temperature, chemical potentials, and the surface strain tensor. A new definition of surface stress is introduced and the corresponding thermodynamic equations are written. The mechanical equilibrium equation at the surface, involving this stress, is obtained and is analogous to the Cauchy equation in the volume; its normal component generalizes the Laplace equation. At a triple contact line two equations appear: force equilibrium for a line fixed on the body, and equilibrium relative to motion of the line with respect to the body. The latter strongly modifies Young's classical capillary equation.

What carries the argument

The surface strain tensor treated as a true thermodynamic variable that, together with temperature and chemical potentials, fixes the surface state and determines the surface stress.

If this is right

  • The normal component of the surface equilibrium equation supplies a generalized Laplace relation for the pressure jump across a curved surface.
  • Force balance at a fixed triple line is expressed directly in terms of the three surface stresses meeting at the line.
  • A separate equation governs the equilibrium motion of the triple line relative to the underlying body and replaces the classical Young's angle condition.
  • The same thermodynamic framework applies uniformly to viscoelastic solids and viscous fluids because both are described by the strain tensor as the relevant state variable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Contact-angle data collected on soft or strained substrates would require reinterpretation once the strain dependence in the triple-line equation is included.
  • The motion equation could be tested in dynamic-wetting experiments where the triple line advances across a pre-strained solid surface.
  • The generalized Laplace relation suggests that curvature-induced pressure jumps should be re-measured on surfaces whose strain is independently controlled.

Load-bearing premise

The local thermodynamic variables of the state of the surface are only the temperature, the chemical potentials and the surface strain tensor.

What would settle it

Direct measurement of contact angles or force balance at a moving triple line on a deformable solid under controlled surface strain that fails to satisfy the derived motion equation would falsify the modification of Young's law.

Figures

Figures reproduced from arXiv: 1906.11022 by Juan Olives (CINaM).

Figure 1
Figure 1. Figure 1: The studied system: a deformable body in contact with various fluids. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The various surfaces (Sb1, Sb2 and Sbf) defined at the body-fluid interface, and the actual and ‘ideal’ transformations of the body (Fa and Fi) between the initial and final states (see definitions in the text). the body is homogeneous up to Sb1 and that this surface is close to the interface. On Sb1, the volume mass density ρc of the substance c of the body has some value ρc,b. Then, crossing the interfac… view at source ↗
Figure 3
Figure 3. Figure 3: The image of Sbf by the ideal transformation Fi is equal to S0 bf (see proof, in the text). if A1 is filled with only the substance c, with a constant density ρc, equal to ρc,b = ρc(x1)). Note that this property does not depend on the choice of the parallelepiped A (i.e., its face on Sb1 and the orientation of −−→x1x2). Under the transformation Fa, the mass of c contained in A is equal to that contained in… view at source ↗
Figure 4
Figure 4. Figure 4: Displacement δX of the bff0 triple line in space, between the present and the varied states, and displacement δX0 of this line with respect to the body, in the reference state. The terms of the last line of (11) may be calculated in the (ideal) reference state of the body (i.e., on S0,bf). With the help of the above assertions 2 (S0 bf = Fi(Sbf), Sbf = F0,i(S0,bf) and S0 bf = F 0 0,i (S0,bf)) and 4 (F 0 0,… view at source ↗
Figure 5
Figure 5. Figure 5: Displacement δX of the bff0 triple line, expressed as wbf + δXbf on the bf side, or wbf0 + δXbf0 on the bf0 side (δXbf and δXbf0 are indicated by dotted arrows; see text). will also be discontinuous at the triple line. From the bf side point of view, and if δX0 = 0, 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

The thermodynamics and mechanics of the surface of a deformable body are studied here, following and refining the general approach of Gibbs. It is first shown that the 'local' thermodynamic variables of the state of the surface are only the temperature, the chemical potentials and the surface strain tensor (true thermodynamic variables, for a viscoelastic solid or a viscous fluid). A new definition of the surface stress is given and the corresponding surface thermodynamics equations are presented. The mechanical equilibrium equation at the surface is then obtained. It involves the surface stress and is similar to the Cauchy equation for the volume. Its normal component is a generalization of the Laplace equation. At a (body-fluid-fluid) triple contact line, two equations are obtained, which represent: (i) the equilibrium of the forces (surface stresses) for a triple line fixed on the body; (ii) the equilibrium relative to the motion of the line with respect to the body. This last equation leads to a strong modification of Young's classical capillary equation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript refines Gibbs' approach to surface thermodynamics for deformable bodies (viscoelastic solids or viscous fluids). It identifies the local thermodynamic state variables of the surface as temperature, chemical potentials, and the surface strain tensor; introduces a new definition of surface stress; derives the corresponding thermodynamic relations and the mechanical equilibrium equation at the surface (whose normal component generalizes the Laplace equation); and obtains two equilibrium conditions at a body-fluid-fluid triple line, one for fixed-line force balance and one for relative motion of the line, the latter yielding a modified form of Young's capillary equation.

Significance. If the derivations are correct, the work supplies a thermodynamically consistent, parameter-free framework for surface stress and mechanics that applies uniformly to solids and fluids. The explicit construction from the restricted variable set (T, μ_i, strain tensor) and the two equilibrium conditions at the triple line, without additional work terms, is a strength and could provide testable predictions for capillarity on soft or viscoelastic materials.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'strong modification of Young's classical capillary equation' is used without stating the explicit form of the new equation or the key difference from the classical result; adding one sentence with the modified equation would improve accessibility.
  2. The manuscript refers to 'the initial section' that establishes the variable set, but the numbering or heading of that section is not indicated in the provided text; consistent section numbering would aid cross-referencing.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no specific major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation follows from stated thermodynamic variables

full rationale

The abstract and description establish the local thermodynamic variables (T, chemical potentials, surface strain tensor) as the starting point before presenting surface stress definitions and deriving equilibrium equations at surfaces and triple lines. No equations are provided that reduce a claimed result to a fitted parameter or self-referential definition by construction. No self-citations are invoked as load-bearing uniqueness theorems, and the modification to Young's equation is presented as following directly from force balance and relative motion equilibrium conditions. The paper is self-contained against external benchmarks with no evident circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5701 in / 990 out tokens · 20855 ms · 2026-05-25T15:25:58.674380+00:00 · methodology

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Reference graph

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