Proceedings of the International rilem conference Materials, Systems and Structures in Civil Engineering 2016

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International RILEM Conference on Materials, Systems and Structures in Civil Engineering 

Conference segment on Service Life of Cement-Based Materials and Structures 

22-24 August 2016, Technical University of Denmark, Lyngby, Denmark 

 

 

 

Figure 1: (a) Layer of particles representing a cement grain surface, in red, and hydration 

phase HP particles formed near it, in blue. (b) Same configuration showing also the trial 

particle for KMC insertion in yellow. 

 

 

The rate of particle insertion is coarse grained from the molecular scale following the classical 

theories of Classical Nucleation Theory (CNT) and crystal growth [24]: 

 

 

(1)

 

Details on the derivation of Eq. 1 and on the corresponding equation for the deletion rate R

del

 

are given in ref. [25] (notice that the chemical conditions considered for the simulations in 

this paper will be such that the number of dissolution events will be negligible alongside their 

effect on the rate, so the results that we will show will be reproducible even without 

implementing a deletion step at all). In Eq. 1,  C

cn

 is the concentration of possible particle 

nucleation sites in solution (e.g. the concentration of ions), V

box

 is the volume of the 

simulation box, and M is the number of trial particles assumed as uniformly distributed in 

V

box

. In the curly bracket, the first term is the characteristic time to form a critical nucleus as 

per CNT, while the second integral term is the time to grow the radius of the critical nucleus 

by single-molecule growth reactions until reaching the wanted particle size R

part

 (here 

diameter = 10 nm, thus R

part

 = 5 nm). Z is the Zeldovich factor, expressing the probability that 

a critical nucleus will indeed start growing rather than dissolving back.  G

CNT

 is the 

difference in free energy between critical nucleus and solution. For spherical particles: 

 

 

(2)

 

R

nucl

 is the radius of the critical nucleus, given by the condition d G

CNT 

/ dR  = 0, k

B

 is the 

Boltzmann constant, T is the temperature in Kelvin degrees, and   is the interfacial energy 

between cement HP and water.  U

nucl

 is the change in total interaction energy in  the system 

in case the critical nucleus appears; its relationship to the nucleus size depends on the type of 

interaction potential employed. Here we use a pairwise interaction potential U

ij

(r) that 

depends only on the distance r between the particles and has been shown to capture well the 

mechanical properties of cement HP at the 500 nm mesoscale [13, 18, 23]: 

 

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24

International RILEM Conference on Materials, Systems and Structures in Civil Engineering 

Conference segment on Service Life of Cement-Based Materials and Structures 

22-24 August 2016, Technical University of Denmark, Lyngby, Denmark 

 

 

(3)

 

(R) is the energy strength, which is assumed to scale as the volume of the particle. A 

different hypothesis on the scaling of   is discussed in ref. [25], in relation to the prediction of 

the rate of early cement hydration. 

 

Going back to Equation 1, a is the linear size of a molecule of cement HP (here we assume a 

= 0.645 nm, thus a

3

 = 0.267 nm

3

, as for C—S—H [26]). r

0

*

 is a kinetic constant (rate per unit 

surface) that contains the activity coefficient of the activated complex and the standard free 

energy barrier of the C—S—H formation reaction [24]. 

1rx

 and  U

1rx

 are the change of 

particle surface and interaction energy caused by one molecular reaction of HP product 

forming on the surface of a growing nanoparticle. Each molecular reaction changes the 

particle volume by a

3

 and consequently the radius R; both 

1rx

 and  U

1rx

 therefore depend 

on the current R. The rate of particle deletion can be obtained in a very similar manner as R

in

 

and contains neither the supersaturation term   nor the sampling-related prefactor before the 

curly brackets. Deletion will be considered in the simulation but the rate expression is omitted 

here for brevity and because we will only consider values of beta that are sufficiently high for 

deletion to be negligible. 

 

To simulate the formation of cement HP, we consider   = 86.7 mJ/m

2

 [26]. The interpretation 

of experiments based on CNT indicates that the size of the critical nucleus in cement hydrates 

is as small as one single molecule [27]. This means that the CNT term in Equation 1 is small 

compared to the crystal growth term (the integral one) and we will neglect it. For cement 

hydration, both C

cn

 and r

0

*

 are unknown, hence the two will be combined and treated as a 

single effective parameters that is used to fit the right timescale (actually, if one decides to 

identify C

cn

 with the concentration of calcium or silicon ions in solution, then literature data 

could be used to set a value for it, e.g. [28]). It is important to notice that differently from the 

existing simulations, the only scaling of time here is linear. This means that any nonlinearity 

in the rate can only be a true reflection of the formation mechanism and is not imposed ad-

hoc. 

 

The particles of cement hydrates that we insert during the simulations are monodisperse even 

though just after insertion, a small random change of the diameter by 5% is imposed to avoid 

crystallization. It is known in the cement literature that realistic details of the heterogeneities 

at the sub-micrometer scale can only be captured using polydiperse nanoparticles [16, 23], 

and there are experiments that suggest that anisotropic shapes of the particles are more 

realistic [29]. However, spherical and monodisperse particles are sufficient to discuss the 

qualitative kinetics of collective precipitation-aggregation mechanisms, as shown in the 

literature [19] and in line with the objective of this work. 

 

As a final note for this section, it is interesting to note that the coarse grained rate expression 

in Equation 1 combines some terms that are related to the chemical kinetics of the process 

(e.g. r

0

*

) with other terms that are related to the thermodynamics. Among the latters, there are 

terms that are related to the chemistry of the system (e.g.  ) and terms that are related to the 

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