The water budget of a hydrating paste
Mix one gram of cement with
and powers_alpha_max supplies it to the rate laws. This page is about what that 0.42 is made of, which part of it a Gibbs energy minimization can predict, and why the other part is not a thermodynamic quantity.
1. Three destinations for the mixing water
At a degree of hydration
| destination | symbol | what it is |
|---|---|---|
| chemically bound | water written into hydrate formulas — C-S-H, portlandite, AFm, AFt. It is no longer water | |
| the volume the reaction lost | hydrates occupy less than the water and clinker they came from, so voids open: chemical shrinkage | |
| still liquid | the remainder | pore solution, held more or less tightly depending on the pore it sits in |
Both coefficients are computed, not assumed. From certified equilibria on a CEM I at imposed
each constant in
2. What the minimization predicts: the stoichiometric threshold
A Gibbs minimization under a conservation budget answers one question: what is the lowest-energy composition reachable with the atoms available. Water appears in it as hydrogen and oxygen, and nothing more. So the threshold it predicts is the point at which the hydrogen runs out — that is,
Measured on the w/c example's species list, at equilibrium:
| clinker left at equilibrium | |
|---|---|
| 0.15 | 54.6 % |
| 0.28 | 2.0 % |
| 0.30 and above | 0 % |
That is a real prediction and it is a stoichiometric one. Below
Below that threshold the calculation also leaves its own domain
At solvent_fraction and SOLVENT_FRACTION_FLOOR exist to name, and no activity model repairs it.
3. What Powers' 0.42 actually contains
Powers splits his coefficient himself: about 0.23 g/g of non-evaporable water — what survives D-drying, his operational definition of chemically bound — plus about 0.19 g/g of gel water, which is water present in the paste and unusable by the reaction.
So the gap between the two thresholds,
is exactly the gel water, and it is the whole of the question. A paste at
Written as a budget, the arrest condition is
where
4. Why the gel water is not a thermodynamic effect
The natural hypothesis is that confined water is thermodynamically unavailable: water held in a fine pore has a reduced activity, so hydrates that consume it become less stable, and the arrest should fall out of the minimization. The magnitude is wrong by two orders of magnitude, and the arithmetic is short enough to do here:
using Printf
R, T = 8.31446261815324, 298.15
a_w = 0.80 # the internal humidity a sealed paste arrests at
ΔG_water = R * T * log(a_w) # J per mole of water
n_water_per_alite = 3.3 # mol H₂O per mol C3S → C-S-H + CH
@printf("RT ln a_w = %8.1f J/mol of water\n", ΔG_water)
@printf("× %.1f mol water per mol alite = %8.2f kJ per mol of alite\n",
n_water_per_alite, n_water_per_alite * ΔG_water / 1000)
@printf("hydration ΔG, order of magnitude = %8.0f kJ per mol of alite\n", -100.0)
@printf("\nactivity that would null it = %.2e\n", exp(-100_000 / (n_water_per_alite * R * T)))RT ln a_w = -553.2 J/mol of water
× 3.3 mol water per mol alite = -1.83 kJ per mol of alite
hydration ΔG, order of magnitude = -100 kJ per mol of alite
activity that would null it = 4.91e-06An activity of that size corresponds, through Kelvin, to a meniscus radius smaller than a water molecule. And the measurement agrees with the arithmetic: imposing a water activity from saturation down to 0.80 through CapillaryWater leaves the equilibrium assemblage of a CEM I paste unchanged to six digits, with a certificate on every answer — that is the first negative control on the self-desiccation page.
So the capillary lowering of the water activity is real, representable, and energetically negligible. What stops a real paste is elsewhere.
It is a question of scale, which a 0D calculation does not have
The reaction needs water at the surface of a clinker grain, and dissolved ions carried away from it. As hydration proceeds:
the coarse capillary pores empty first, because they hold water most loosely;
what remains sits in gel pores — nanometers wide, inside the C-S-H — as films one or two molecules thick;
the permeability of the paste falls by orders of magnitude, and the liquid path between the remaining water and an unreacted grain, tens of microns away, is broken.
The reactants stop meeting. That is a statement about geometry and transport across four orders of magnitude of length, and a 0D model has no length at all: it assumes one well-mixed solution in contact with every solid, so a water molecule in a 1 nm gel pore is the same object as one in a 10 μm capillary pore. No activity model, however good, can distinguish them — the distinction is not thermodynamic.
There is a numerical hint of that boundary inside the theory itself. The Debye screening length of a cement pore solution at
5. So what can be predicted, and how
Three routes, and they differ in what has to be assumed.
Impose powers_alpha_max. This is also what GEM-Selektor and Reaktoro offer, since they are 0D equilibrium codes too.
Close the budget with a measured isotherm. Take
Integrate a rate law that reads the humidity. humidity_factor implements the empirical cut that stops hydration below about 80 % RH, and PoreHumidity computes that humidity from the current saturation, so a kinetic run arrests on its own. The arrest becomes a result of the integration rather than a criterion applied afterwards — while the criterion stays empirical. That is the honest ceiling of a 0D framework.
Beyond 0D. Making the arrest a genuine prediction needs a model with a length in it: a microstructural one, where the arrest emerges because the liquid path percolates no longer, or a transport calculation on the pore network. That is a different package.
6. The same argument for a blended binder
Nothing in sections 3 and 4 mentions alite. The coefficient
is built from the water the hydrates bind and the water the gel holds at arrest; it is a property of the pore space, not of the grain that made it. A slag particle or a fly ash sphere sitting in the same paste is in the same predicament as an unhydrated clinker core: the water that remains is in gel pores two molecules wide, and it is not going to reach a grain a thousand times larger.
So the ceiling transposes, and powers_alpha_max is applied to every constituent of a blended binder on the pages for CEM II, CEM III, CEM IV and CEM V. Two qualifications go with it, and both matter:
It is the water/binder ratio that is used. Powers measured
For a slag or an ash the water is usually not what binds. The ceiling says how far the reaction can go. What decides how far it has gone at an ordinary age is the glass's own dissolution rate, and that is much the lower of the two: the RILEM TC 238-SCM round robin (Durdziński et al., 2017) reports, at 28 days and by SEM image analysis, 38–49 % for two ground granulated slags and about 20 % for a siliceous fly ash — against a water ceiling of 0.95 at
This is the assumption an equilibrium code hides best
A Gibbs minimization reacts whatever budget it is handed, without comment. A page that hands it the whole binder has asked what the paste becomes after every grain has dissolved — a question about geological time — and will get a confident, certified, completely unphysical answer: on a CEM V at 48 % replacement, a pH of 14.4 and an element balance that cannot close because no assemblage in the database will hold the alkalis and aluminum released. The failure is in the question. The CEM V page shows both.
7. Curing is a boundary condition, and there are two of them
Everything above is a sealed specimen: it exchanges nothing with its surroundings, the volume that chemical shrinkage empties becomes gas-filled porosity, the saturation falls, and the paste desiccates itself. That is one boundary condition. The other is a specimen kept under water after setting, which draws in what the shrinkage empties and never desiccates.
Powers gives both, and the difference between them is exactly the chemical shrinkage:
the 0.06 g of water per gram of cement being the volume the reaction loses because the hydrates are denser than the reagents that made them. Sealed, that volume has to come out of the paste's own water; immersed, it comes from the bath. So the same mix reaches full hydration from a lower mixing water content when it is cured under water — which is why curing is specified, and why a strength result quoted without its curing regime is incomplete.
powers_alpha_max(0.32) # 0.762 -- sealed
powers_alpha_max(0.32; curing = :saturated) # 0.889 -- under watercuring = :saturated moves the ceiling, which is what a kinetic run needs. It does not by itself open the specimen: a coupled run is a closed system, so the water the bath supplies is not in its balance.
For an equilibrium the specimen can be opened, and that is SaturatedCuring — the mirror image of CapillaryWater. Where the sealed constraint lets the saturation fall and lowers the water activity by the Kelvin term, this one holds the specimen's total volume at the fresh paste's and draws water in to make up what chemical shrinkage empties:
one equation, linear in the composition, with the amount of water imbibed as its unknown. That amount is not a numerical device: it is the chemical shrinkage, the quantity a chemical-shrinkage test measures by watching a specimen drink.
And the shrinkage is computed, which makes 0.06 a prediction rather than a constant
| shrinkage from the molar volumes | 0.0606 cm³ |
| water the cured specimen drew in | 0.0604 g |
| Powers, as | 0.0600 g |
The two internal routes agree with each other to 0.3 %, and both land within 1 % of a number measured on pastes in 1948. They are not obliged to: the assemblage is a declared species list and not a real paste's, the molar volumes are ideal, and Powers' coefficient is an average over the cements he had. Which is what makes the agreement a check on the volume data rather than a restatement of it — and what makes an empirical coefficient intelligible rather than merely used. The w/c example runs it.
A cure is not FixedActivity("H2O@", 1.0)
The tempting way to write "kept under water" is to prescribe unit water activity, and it is wrong. A cement pore solution sits near
See also
Self-desiccation — the budget closed, with the numbers computed and the negative controls executed
The w/c ratio — the stoichiometric threshold, scanned
Activity models — why the screening length matters here