Acta Polytechnica Vol. 52 No. 6/2012

Physical and Chemical Aspects of the Nucleation
of Cement-Based Materials

Pavel Demo1,2, Alexey Sveshnikov1,2, Šárka Hošková1, David Ladman1, Petra Tichá1

1Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha 6, Czech Republic
2Czech Technical University in Prague, Faculty of Civil Engineering, Thákurova 7, 166 29 Praha 6, Czech Republic

Corresponding author: demo@fzu.cz

Abstract
A theoretical model of the nucleation of portlandite is proposed, and the critical size of a portlandite cluster and
the energy barrier of nucleation are determined. The steady state nucleation rate and the time lag of the nucleation
of portlandite are estimated for a pure solution of Ca(OH)2 in water. Possible connections with the corresponding
properties for cement paste are discussed. A new method is developed for experimentally determining the concentration
of Ca2+ ions during the initial stage of hydration of a cement paste. The time dependence of Ca2+ ions is measured for
various water-to-cement ratio values. The results are discussed from the point of view of existing models of the induction
period.

Keywords: cement paste, induction period, calcium concentration, nucleation rate, time lag.

1 Introduction
Cementitious materials are one of the most widely-
used types of materials in modern civil engineering.
Concrete has many advantages, such as very high
compressive strength, good resistance to corrosion
and fire, and good acoustic damping. Workability
occupies a special place among these properties of
concrete. Workability refers to the ability of a fresh
cement paste to fill the desired shape without reduc-
ing the quality of the concrete. High workability of
cement paste over a period of several hours signifi-
cantly reduces construction costs, and enables both
easier shaping of the material and pre-casting of con-
crete that can then be delivered to the construction
site. Having in mind the great importance of the
workability of cement paste, extensive research has
been carried out in order to understand the under-
lying mechanism and to study the possibility of ex-
tending the workability period without a significant
impact on the properties of the final material. De-
spite the efforts that have been made, there are still
many unanswered questions due to the very complex
nature of the physico-chemical transformations tak-
ing place in the hydrating cement paste.
The hydration process of cement paste can be di-

vided into a number of main stages. The first stage is
the initial wetting of the cement particles and rapid
dissolution of a small amount of the cement minerals,
especially the calcium aluminates and the alkali sul-
fates. This stage is accompanied by significant heat
production and lasts only few minutes. During this
stage the hydrating tricalcium silicate also releases a

large number of calcium ions into the solution, due
to the very high intrinsic solubility of calcium. After
the initial stage of exothermic reactions, the cement
paste “falls asleep”, and for the next few hours prac-
tically no changes are observed in the state of the
paste. The duration of this so-called dormant (or in-
duction) period directly determines the workability,
since the strength of the paste begins to grow steadily
after the dormant period finishes and the concrete
begins to set. Thus, it is important to understand
the reasons for the existence of the induction period
and possible ways to influence its duration without a
negative outcome.
Several hypotheses have been proposed in order to

explain the induction period. Taylor has grouped
them into four main classes [9]:

1. The initial nuclei of C-S-H and CH are somehow
poisoned by SiO2 and cannot grow until the so-
lution becomes supersaturated [8].

2. The membrane hypothesis assumes, that dur-
ing the first stage of paste hydration a semiper-
meable membrane forms around the particles,
thus separating the solution into an inner part
and an outer part. The membrane prevents the
ions dissolved in the inner solution from reaching
the main volume of the solution. According to
this hypothesis, the induction period ends when
the osmotic pressure inside the membrane grows
high enough to break it [1, 6].

3. The protective layer hypothesis is similar to the
membrane hypothesis, but supposes that the

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Acta Polytechnica Vol. 52 No. 6/2012

Oxide CaO SiO2 Al2O3 Fe2O3 MgO SO3 Na2O K2O MnO

Comp. [wt.%] 63.77 20.51 4.74 3.3 1.05 3.07 0.15 0.95 0.09

Table 1: Oxide composition of CEM I 42.5R, Mokrá, Czech Republic

Mineral C3S C2S C3A C4AF C2F SO3-bound CaO

Comp. [wt.%] 58.34 14.8 6.81 10.31 0 2.149

Table 2: Phase composition of CEM I 42.5R, Mokrá, Czech Republic (Bogue)

semipermeable layer is formed directly on the
surface of the grains. As in the case of the
membrane hypothesis, the induction period ends
when the protective layer is somehow removed or
made permeable.

4. The nucleation hypothesis attributes the exis-
tence of the induction period to the very mech-
anism of the formation and growth of the nu-
clei. The growing particles of the new phase
have to overcome a certain energetical barrier
by means of fluctuations, and this cannot hap-
pen instantly. This hypothesis explains the in-
duction period as an intrinsic time lag of the
nucleation process [3, 10].

In our paper we analyze a nucleation hypothesis of
the induction period. The hydrating cement paste is
an extremely complex system. For our analysis, we
concentrated our attention on one of the main phases
in the hydrating paste — portlandite Ca(OH)2. We
measured the concentration of Ca2+ in a hydrating
cement paste, and combined this with theoretical
modelling of the nucleation of a water solution of cal-
cium hydroxide to estimate the nucleation time lag
of portlandite formation.

2 Experimental

Materials. In our experiments we used an ordinary
Portland cement (OPC) CEM I 42.5R, Mokrá, Czech
Republic. The oxide and mineralogical composition
of the cement is given in Tables 1 and 2.

Chemicals. All chemical agents applied are of pu-
rity p.a. 0.05 M EDTA, disodium ethylenediaminete-
traacetate, 0.05 M MgSO4, Schwarzenbach buffer,
Eriochrome black T.

Method. The concentration of Ca2+ ions at an
early stage of hydration of the cement paste was
measured. The method for determining free calcium

ions [5] is essentially based on the use of a chelatomet-
ric titration method, namely determining the concen-
tration of Ca2+ ions by disodium ethylenediaminete-
traacetate (EDTA, Chelaton III), a basic volumet-
ric (titration) method in analytical chemistry. It is
well known that chelatometric determination utilizes
the fact that EDTA forms very strong compounds
(chelates) with certain metals belonging to the II, III
and IV group. Calcium is one of these metals. How-
ever, in our method for determining calcium ions in
the first two hours of setting of the cement paste,
EDTA plays a more complex role.
During the first two hours after the cement is

mixed with water, the reactions run very quickly and
the release of Ca2+ can be very rapid. The reaction
is exothermic, as shown by the calorimetric curves
(see Figure 1).
In order to measure the time dependence of the

concentration of the calcium ions in the hydrating ce-
ment pastes, it is neccessary to stop the reaction at a
chosen instant. The hydration reaction was stopped
with the use of EDTA, which is simultaneously a
strong retardant for the reaction of cement with wa-
ter. Thus this acid serves both as the retardant and
as the volumetric reagent for calcium ions. To im-
prove the accuracy of the method, we used indirect
titration, i. e. the amount of excess EDTA was de-
termined by reaction with magnesium ions, and the
Ca2+ concentration was calculated from the differ-
ence.
The concentration of Ca2+ ions as a function of

time was determined for various water-to-cement ra-
tios (w/c = 0.35; 0.40; 0.45; 0.50; 0.60), and the
results are presented in Figure 2. All the curves have
a peak at the very beginning of the hydration pro-
cess, which corresponds to the first stage of intense
exothermic reactions. Afterwards the concentration
drops. This decrease in free ion concentration can be
explained as the formation of some protective layer
or membrane. When the osmotic pressure exceeds
a certain threshold, the membrane bursts and re-
leases the Ca2+ ions held within, leading to a later
increase in concentration. At this instant, the nu-
cleation mechanism kicks in, and portlandite nuclei
begin to form. From the point of view of nucleation

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Acta Polytechnica Vol. 52 No. 6/2012

H
ea
t
(m

W
/g
)

Time [min]

Figure 1: Initial stage of the hydration process in HCP of Portland cement, Mokrá, (w/c = 0.40; 0.45) measured
using a SETARAM C 80 calorimeter.

theory, the Ca2+ ions serve as monomers for future
portlandite clusters. These monomers are absorbed
in the nucleation process, leading to a new decrease
in calcium ion concentration and the establishment
of the second peak on the curves in Figure 2.

3 Model
According to nucleation theory, the initial stage of
the process of the first order phase transition con-
sists of two steps - nucleation and growth. Before
the first particles of the new, stable phase can start
to grow steadily, they have to appear somehow in
the metastable phase, and this process is not easy
for them. Although the resulting phase is more
favorable from the energetical point of l view, the
high surface-to-volume ratio of small particles com-
bined with positive excess interface energy makes
them unstable. Thus, nucleation theory distinguishes
nucleation itself, when the embryos of the stable
phase are stochastically produced in the volume of
the metastable phase by means of fluctuations, and
growth, when the nuclei increase in size deterministi-
cally. Depending on the system under consideration,
the establishment of a steady growth rate can take
a significant period of time, and this is called the
time-lag of nucleation [11].
Portlandite clusters grow from a solution contain-

ing Ca2+ and OH− ions. We assume that hydroxyl
ions are present in abundance, because they are not
only a product of Ca(OH)2 dissociation, but also a
product of dissociation of water itself. Thus, the for-
mation of portlandite clusters is controlled by the

supersaturation of calcium ions:

S =
Cact
Ceq

, (1)

where Cact represents the actual concentration of
Ca2+ ions within the solution and Ceq is their equilib-
rium concentration. The Gibbs free energy of port-
landite cluster formation is

∆G = ∆GV + ∆GS, (2)

where ∆GV stands for the volume term, and ∆GS is
the surface term. The volume term is negative, be-
cause the solid phase as a whole is stable and energet-
ically more favorable than the solution. Conversely,
the surface term is positive and reflects the general
tendency of a system to eliminate internal interfaces
and become homogeneous.
The exact interplay between the volume and the

surface terms in (2) depends on the shape of the
growing cluster. It has been observed [2] that small
crystals of portlandite are shaped as hexagonal plates
(see Figure 3). The fact that the radius of crystal r
is much larger than its height h suggests that the
surface tension in this case is strongly anisotropic.
Denoting the surface tension on the corresponding
planes as σr and σh, we obtain

∆G = −kT
3
√

3r2h
2v

ln S + 3
√

3r2σr + 6rhσh, (3)

where k = 1.38 × 10−23 J/K is the Boltzmann con-
stant, T stands for absolute temperature, and v =

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Acta Polytechnica Vol. 52 No. 6/2012

Figure 2: Concentration of Ca2+ ions in HCP vs. time related to the total mass of HCP (t = 25◦C).

σr

σh

r

h

Figure 3: The assumed shape of a portlandite cluster and the corresponding surface energies.

1.65 × 10−28 m3 is the volume of a unit cell of port-
landite, which is hexagonal with basis cell dimensions
a = 0.3585 nm and c = 0.4895 nm.

Under given supersaturation S, expression (3) is a
function of two variables, r and h. This expression
describes the energetical landscape on which growing
clusters have to find the optimal way to the valley,
representing the stable solid phase. The optimal way
leads through a saddle point of the energetical sur-
face (3), and corresponds to the best thermodynam-
ical approximation to the shape of a cluster. If we
take kinetic aspects of cluster growth into account,
it is possible that the dynamically changing shape of
a growing crystal does not pass exactly through the
saddle point of the energetical landscape. However,
this deviation is negligible in most situations, so in
the following text we assume that the shape of a crit-
ical cluster is determined from the pair of equations

∂∆G
∂r

∣∣∣∣
rc,hc

= 0,
∂∆G
∂h

∣∣∣∣
rc,hc

= 0. (4)

Solution of the system (4) yields

rc =
4σhv√

3kT ln S
, hc =

4σrv
kT ln S

. (5)

Clusters which are smaller than the critical clus-
ter tend to dissolve, while larger clusters can grow
freely. This last conclusion is valid for an open sys-
tem, but in a closed system the situation is more
complicated. In a closed system, the concentration
of monomers is not constant, as they are consumed
by the growing clusters. Thus, supersaturation S is
constantly decreasing, which leads to an increase in
the critical sizes rc and hc. It may even be possi-
ble that in later stages of the nucleation process, the

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Acta Polytechnica Vol. 52 No. 6/2012

N
uc

le
at
io
n
ra
te
,a

rb
.
un

it
.

Supersaturation S

Figure 4: Dependence of the steady state nucleation rate of portlandite on the supersaturation of Ca2+ ions.

critical size grows faster than the front of the distri-
bution function of the cluster. Then the clusters that
have already originated and are overcritical may be-
come undercritical again and dissolve. This effect of
depletion of Ca2+ monomers due to their consump-
tion by growing portlandite is shown in Figure 2 as
a decrease in concentration after it achieves the sec-
ond maximum. However, during the whole process
the ratio of the radius of the critical platelet to its
height remains constant and independent of super-
saturation S:

rc
hc

=
σh√
3σr

. (6)

Since rc � hc, lateral surface tension σh must be
much larger than basal σr. When a macroscopic mea-
surement of the surface tension of portlandite is per-
formed, it yields the surface energy value, averaged
over the surface of the whole cluster. It can eas-
ily be shown from (6) that the contribution of the
basal surface energy to the total energy of the cluster
is four times smaller than the corresponding contri-
bution of the lateral surface energy. Consequently,
macroscopic measurements of the surface energy of
portlandite provide values for σh.

After substituting (5) into (3), we obtain an ex-
pression for the energy barrier of nucleation:

∆G =
16
√

3σ2hσrv
2

(kT ln S)2
. (7)

If we take the values for calcium hydroxide super-
saturation from [4], S ∈ (1.75, 3), then it is possible
to obtain the following estimations for the critical size
of a portlandite cluster at T = 298 K:

rc ∈ (5.1 nm, 10 nm), hc ∈ (1 nm, 2 nm). (8)

The corresponding energy barrier

∆G ∈ (7.2 eV, 26.5 eV). (9)

However, these estimations were performed using
the equilibrium concentration of Ca2+ ions based on
the solubility curve of calcium hydroxide in water.
Data for the calcium hydroxide solution in the ce-
ment paste is not available. We can only assume that
the nucleation of portlandite is the dominant process
during the induction period and, consequently, the
equilibrium concentration of Ca2+ does not differ sig-
nificantly from the pure solution. It is still an open
question, though, how to relate the water-to-cement
ratio of the cement paste to supersaturation of the
solution with respect to portlandite.
Despite the relatively small variations in the criti-

cal size and the energy barrier, the steady state nu-
cleation rate can be quite sensitive to supersaturation
and, thus, to the water-to-cement ratio. This is ex-
plained by the fact that the nucleation rate (i. e. the
number of new supercritical clusters appearing per
unit volume of the system per unit time) is exponen-
tially dependent on the energy barrier of nucleation:

I ∼ exp
(
−

∆Gc
kT

)
. (10)

A detailed knowledge of the kinetics of cluster
growth is necessary for calculating the exact pro-
portionality coefficient in (10). Nevertheless, rela-
tion (10) enables us to estimate the relative change
in the nucleation rate under normal conditions (see
Figure 4):

Imin
Imax

' 80. (11)

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Acta Polytechnica Vol. 52 No. 6/2012

T
im

e
la
g
of

nu
cl
ea
ti
on

,a
rb
.
un

it
.

Supersaturation S

Figure 5: Time lag of the nucleation of portlandite in dependence on supersaturation of Ca2+ ions.

Equation (10) gives the nucleation rate in the
steady state. However, as discussed above, due to
the stochastic nature of the nucleation process, the
nucleation rate does not achieve its steady state value
immediately after the creation of supersaturation in
the system. The corresponding time lag of nucleation
can be estimated as [7]

τ =
1

2.8Z2R
, (12)

where Z is the so-called Zeldovich factor, which de-
scribes the curvature of the energetical landscape (3)
at its saddle point, and R stands for the rate at which
monomers are incorporated into critical-size clusters.
The Zeldovich factor Z is equal to

Z =
√
−1

2πkT
∂2∆Gc
∂n2

, (13)

where n is the number of monomers inside the cluster.
Figure 5 shows the dependence of the time lag of
portlandite nucleation on supersaturation S.

4 Conclusion
We have proposed a new method for experimentally
determining the concentration of Ca2+ ions in hy-
drating cement paste during the first few hours after
mixing cement with water. The method is based on
a standard chelatometric titration method, but uses
EDTA both as the retardant of the hydration pro-
cess and as the volumetric reagent for calcium ions.
The evolution of calcium ion concentration has been
studied bye this method for several w/c ratios. The
resulting curves (see Figure 2) express in general a bi-
modal behavior. The existence of the first maximum

is attributed to the release of Ca2+ ions during the
initial fast process of hydration of C-S-H, followed by
the formation of some sort of protective layer around
the cement grains. During the induction period, the
concentration of calcium ions slowly grows due to
the gradual hydration process. However, at later
times the concentration goes down again because of
the consumption of Ca2+ by nucleating and growing
portlandite clusters.

Homogeneous nucleation theory is applied to port-
landite cluster formation. We derive expressions for
critical size and the energy barrier. Unfortunately,
no data is available on the equilibrium concentration
of Ca2+ ions in a cement paste. We estimate the crit-
ical size and the energy barrier of nucleation on the
basis of the solubility curve of portlandite in water.
Next, the expressions for the steady state nucleation
rate and the time lag are obtained. The time lag of
nucleation is strongly dependent on supersaturation
(see Figure 5).

In the future, we plan to use our theoretical and
experimental results to find the equilibrium concen-
tration of Ca2+ ions in a cement paste, with a cor-
responding recalculation of the critical size and the
energy barrier. If we assume that the major part of
the induction period is due to the nucleation time
lag, the comparison of the experimental data for the
dependence of the duration of the induction period
on the water-to-cement ratio, together with Figure 5,
can provide information about the supersaturation of
portlandite in a cement paste. In combination with
data about the concentration of Ca2+ ions (Figure 2),
it may be possible to calculate the solubility curve of
calcium hydroxide in a cement paste.

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Acta Polytechnica Vol. 52 No. 6/2012

Acknowledgements
This study was supported by the Ministry of Ed-
ucation, Youth and Sport of the Czech Repub-
lic, project No. CEZ MSM 6840770003 and grant
SGS10/125/OHK1/2T/11.

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