The water budget of a hydrating paste

Before this page

Thermochemistry §6, where volumes and porosity are defined, and Rate laws, and every parameter in them.

Mix one gram of cement with grams of water, seal it, and wait. Some of the water ends up inside hydrate formulas, some stays in pores the hydrates leave behind, and the reaction stops well before the cement is gone. Powers' rule of thumb summarizes the arrest as

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 , the mixing water has gone to three places:

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 — the calculation is executed on the self-desiccation page — this species list gives

each constant in to four digits, which is what makes a budget linear in legitimate for this system rather than merely convenient.   cm³ per gram of cement is the chemical shrinkage that cement chemistry reports for an ordinary Portland cement, and here it is a consequence of the standard molar volumes in the database.

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   clinker survives because there is not enough water to convert it, and the number 0.30 is nothing but plus the little the solution retains.

Below that threshold the calculation also leaves its own domain

At   the free water goes to the solver's floor, the solvent mole fraction falls to 0.21 and the ionic strength reaches hundreds of mol/kg. The composition is arithmetically a solution and physically not one — every molality is per kilogram of a solvent that is no longer there. This is what 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   has enough hydrogen to consume all its clinker — the minimization says so, correctly — and does not do it.

Written as a budget, the arrest condition is

where is the degree of saturation at which hydration stops. Two of the three ingredients are thermodynamic and computed; is neither.

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 ChemistryLab, Printf          # `R_GAS` is the package's, not a literal
R, T = R_GAS, 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-06

An 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   mol/kg is 0.55 nm — the thickness of two water molecules, and the width of the pores in question. Where the screening length and the pore are the same size, the continuum-dielectric and mean-field assumptions that produce every activity coefficient on that page are both strained. The formulas keep returning numbers; they have simply left the picture they were derived in.

5. So what can be predicted, and how

Three routes, and they differ in what has to be assumed.

Impose . Give the minimization a degree of hydration and let it compute the assemblage — the construction of (Lothenbach and Winnefeld, 2006), used by the w/c example below its stoichiometric bound and by the self-desiccation page throughout. Honest, and is an input: either measured, or taken from 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 and from the minimization and from a published desorption isotherm at an assumed arrest humidity. Then is predicted up to that one empirical threshold — and inverting it is more informative than quoting it: with Powers' own  ,   corresponds to an arrest at 77.5 % relative humidity, the window sealed pastes are independently reported to stop in. The self-desiccation page does this, and shows why the proportionality   is structural and therefore no evidence at all.

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 and on Portland cement pastes, and a slag binder's C-A-S-H is not his C-S-H: it has a lower Ca/Si, it binds a different amount of water, and nobody has published the equivalent coefficients per supplementary material. Using treats every constituent as drawing on the same water, which is the conservative reading and is stated as an approximation rather than presented as a measurement.

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  . So the reacted fraction is the smaller of the two ceilings, and for the glasses it is the kinetic one. The same study is worth quoting on how well any of this is known: the precision of determination is "rather low, at best ± 4-5 %".

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 water

curing = :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

is a difference of standard molar volumes — the same data that fixed the assemblage — so the package can be asked a question it was never fitted to answer: does it reproduce the coefficient Powers measured? On a w/c = 0.40 paste, per gram of reacted cement:

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   from its dissolved salts alone, so prescribing 1 would draw water in until the solution was dilute enough to reach it — which never happens, and the constraint would imbibe without bound. A bath does not fix the activity inside the specimen. It fixes the availability: the pore space stays full. That is a volume statement, and it is why the constraint is written on the volume.

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

• powers_alpha_max, PoreHumidity, CapillaryWater

Measuring it: what thermogravimetry gives, and what it needs

Bound water is the quantity a thermogram integrates to, which makes thermogravimetry the natural second observable beside calorimetry — and the one the calibration example asks for by name, because heat constrains three combinations of six kinetic parameters and a measurement that sees the phases breaks correlations heat cannot.

ignition_loss computes the total from the formulas alone:

It counts hydrogen, not formula water, and the difference is not pedantry: portlandite is Ca(OH)₂, has no H₂O written in it, and loses one water per formula unit on ignition. A rule that searched for H₂O would report zero for the second most abundant hydrate in a paste.

The aqueous phase is excluded, which is the distinction this whole page is about: pore solution is water and is not bound water.

From the total to a curve

A total is not a thermogram, and the difference is what identifies phases: C-S-H, AFt and AFm all release below 200 °C and are told apart by the shape of the release, not by its size. That needs, per phase, a temperature window —

— and a window is not a consequence of a formula. It comes from one of two places, and DecompositionWindow makes a curve say which:

1. A publication. Then the two numbers are PROV_PUBLISHED and carry their source.

2. A measured thermogram. The windows are identifiable from one, which is the whole reason thermogram is written as a smooth function of them: window_parameters hands them to an optimizer, and identifiability says afterwards which of them the curve actually determined.

Neither is invented. A window given as a bare number is PROV_UNSTATED — the weakest claim there is — and one written to get a picture on the screen should say PROV_PLACEHOLDER and keep saying it until a measurement replaces it.

A phase can need more than one window. Gypsum loses its two waters in two steps, CaSO₄·2H₂O → CaSO₄·½H₂O → CaSO₄, and C-S-H does not leave in one piece either; a window per stage with fraction splitting the release is how that is written. The fractions are checked to sum to one, because two windows each accounting for all of a phase would release its mass twice and the only symptom would be a curve integrating to more than ignition_loss — a silent doubling rather than an error.

Overlapping peaks are where this earns its keep

Two phases releasing in the same window is the ordinary case in a paste, and a fit that reported four numbers there would be reporting two. Running identifiability on the windows is not a formality: for two peaks 5 K apart it returns a rank below four and a condition number in the tens, which is the measurement saying so.

phases_without_windows is the other half of the honesty: a phase with no window contributes to the starting mass and never leaves, so a curve computed without noticing integrates to less than ignition_loss and says nothing about it. It reports phase => product pairs rather than phases, because a phase can need two windows — a carbonated hydrate carries hydrogen and carbon, releases water and carbon dioxide, and does so at different temperatures. Counting coverage per phase would call such a phase done when half of it is, and a hemicarboaluminate is not an exotic case in a cement.

windows_without_phases is the mirror, and it is the one that catches a typo: a window on a phase that releases nothing contributes nothing and raises nothing, so its only symptom is a peak that is not there.