Microscopic model for concrete diffusivity prediction

M. Bogdan1 & F. BenboudjemaLMT-Cachan, ENS-Cachan, CNRS, UniverSud Paris PRES - France

J.-B. ColliatLaboratoire de Mecanique de Lille

Universite Sciences et Technologies Lille 1, CNRS, Ecole Centrale de Lille, Arts et Metiers ParisTech

L. Stefan1AREVA NC, BU Valorisation, Decontamination GC - France

ABSTRACT: The durability of concrete structures depends largely on the diffusivity and the permeability ofconcrete. These properties depend also, amongst others, on the concrete mix, saturation degree and temperature.In this contribution, a new multi-scale framework is presented in order to predict diffusivity properties. Startingat the micro scale, modeling and description of cement hydration is first developed. It is based upon thresholdingof random field realizations, with different correlation lengths. The use of different correlation lengths allowsa description according to a cement grains particle size distribution. Analytical equations, from the literature,are used to describe the creation and evolution of hydrates (C-S-H, CH, Afm), anhydrous grains and porositythrough time. Then, by varying the thresholds, it is possible to generate the hydrates growth at the surfaceof cement grains (for example the C-S-H), or in the pore space (for the amorphous crystals), keeping vitalinformation such as volume fractions, specific surfaces, and even topological information about the connectivityof any given phase, all along the process. We will show how the same modeling approach can and will be usedfor the description higher scales like mortar and concrete. Next we intend to present a numerical homogenizationframework that will be suited for any transport property prediction, where percolation phenomena are of primeimportance. The generated cement paste microstructures (100 x 100 x 100 µm, with an elementary size of0.5 µm) are next projected onto 3D finite element meshes. With energy based equivalence in-between scales,the diffusivity of cement paste is predicted in terms of lower and upper bounds. Calculations are made undercertain assumptions: the diffusivity of porosity (supposed to be saturated) is assumed to be equal to the one inpure water, the diffusivities of the other constituents, except for the C-S-H, are assumed to be equal to zero.At this stage, it is shown that the model is able to predict the order of magnitude and the tendencies observedin experiments. For high water to cement ratios the capillary pores are connected, and the water content ofthe porosity mainly controls the diffusivity. For low water to cement ratio, the capillary pores are no moreconnected, the diffusivity is mainly controlled by the C-S-H diffusivity properties.

1 INTRODUCTION

Cement based material modelling has always been agreat challenge, especially towards durability assess-ments. For the nuclear or construction industry, a re-liable predictive approach is still an undergoing chal-lenge. A new model based upon level set methods andrandom field thresholding is introduced here, in orderto yield realistic morphologies of heterogeneous ma-terials. This framework will have one direct applica-tion here to the hydration process for cement pastes,and will be suited for various other applications.

Following the work of Adler & Taylor (2007), the

main idea is to apply level set methods to corre-lated (i.e. spatially structured) Gaussian random fields(hereafter RF). The advantages of such methods arenumerous. First, contrary to most of the explicit mod-els, we will not be dealing with spherical or ellip-soidal inclusions, but with random shapes. Secondly,the generation of micro-structures is lighter than aver-age. Once the correlated random field is yielded, a sin-gle threshold will control the entire morphology (in-stead of four parameters for each inclusion for spherepacking methods).

2 MORPHOLOGICAL MODELLING

In this section we introduce the mathematical toolused to yield micro-structures.

2.1 Correlated random field & excursion sets

A random field is define as a random function f(x, ω)over a parameter space M (which shall always betaken here to be a bounded region of Rd) which takesvalues in R. f is taken as a specific isotropic and sta-tionary Gaussian random field with mean zero, vari-ance v and a Gaussian covariance function defined as:

C(x,y) = C(‖x− y‖) = E{f(x)f(y)}

= ve−‖x−y‖2/L2c (1)

where Lc is the correlation length. One have to keep inmind that the results presented in this section can alsobe applied with non-zero mean Gaussian distributionand to the larger family of Gaussian related randomfields with different covariance functions (only thesecond spectral moment has to be finite).

The excursion set is the morphology of a subsetof a bounded region defined by thresholding a re-alisation of a random field, creating a set of ran-dom shapes. Let f(x, ωi) = fi be a realisation i off(x, ω) : M ⊂ R

d → R define as above and D ⊂ R agiven hitting set. The underlying excursion set ADi

isdefined by the subset of M where fi takes its valuesin D. From a statistical point of view, the excursionset AD is defined by equation 2.

AD , {x ∈M | f(x, ω) ∈ D} (2)

An example of this principle in dimension d = 1with D = [u;+∞) (u being a given threshold), isshown on figure 1. Au shall denote the excursion setfor such hitting set of threshold u.

γ

x

u

Au M

Figure 1: Schematic representation of a one-dimensional excur-sion set Au

In the following, application is made with d =3. Realizations in such space naturally define three-dimensional excursion sets. The two excursions rep-resented in figure 2 are made from the same realisa-tion with two different threshold values. It is clear

that, by changing this value, a large range of variedmorphologies can be generated. This example showsthat “low” values of u produce excursions mainlymade of handles with high volume fraction, giving a“sponge” like topology (figure 2(a)), whereas “high”values of u produce excursions made of several con-nected components with a lower volume fraction (fig-ure 2(b)).

(a) “Low” threshold (b) “High” threshold

Figure 2: Effect of threshold value on tri-dimensional excursiontopology

2.2 Morphological measures

In order to provide a global description of the suchmorphologies, the Lipschitz-Killing curvatures (here-after LKCs) are chosen. In a d-dimensional spaced+1 LKCs can be defined, where each can be thoughtof measures of the “j-dimensional size” of Au. In ourthree-dimensional case, the four LKCs, denoted byLj , j = 0..3, provide both geometrical - L1,L2,L3

- and topological - L0 - descriptions of the excursionset Au. They are defined by:

• L3(Au) is the three dimensional volume of Au.

• L2(Au) is half the surface area of Au.

• L1(Au) is twice the calliper diameter of Au.

• L0(Au) is the Euler characteristic (hereafter EC)of Au

Contrary to the other LKCs, L0(Au) is a topologi-cal measure. In three-dimension, it can be calculatedby adding the number of connected components andholes and subtracting the number of handles of Au.For example, a ball or a cube are topologically identi-cal - L0 = 1 - but differ from a hollow ball - L0 = 2- or a ring torus - L0 = 0.

Following (Adler & Taylor 2007), a probabilisticlink has been made between excursion set propertiesand random field thresholding parameters giving anexplicit formulae for the expectation of the LKCs -E{Li (Au (f,M))}.

E{L i (AD(f,M))} =

N−i∑

j=0

[

i+ jj

](

λ2

2π

)j/2

×Li+j(M)Mj(D) (3)

where λ2 is the second spectral moment of f , Mj the

Gaussian Minkowski functionals and [•] represent thecombinatorial flag coefficients. It is not the purpose ofthis paper to give details on these formulae, however,full proof and details can be found in (Adler 2008),and a full explicit development of this formula can befound in Roubin (2013) PhD’s thesis.

Figure 3 represent the excursion set’s volume andEuler characteristic for different thresholds values inthe three dimensional case.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.5 -1 -0.5 0 0.5 1 1.5

Vo

lum

e

Threshold u

(a) Volume fraction

-80

-60

-40

-20

0

20

40

60

-1.5 -1 -0.5 0 0.5 1 1.5

Eu

ler

char

acte

rist

ic

Threshold u

(b) Euler Characteristic

Figure 3: LKCs of excursion sets of Gaussian random field interm of threshold values.

The constant decreasing shape of the volume frac-tion curve in term of u clearly reflects the effect ofthe threshold level on the “size” of Au. In this case,it is defined by the tail distribution function of the

underlying Gaussian random field. Even if more pe-culiar, the Euler characteristic curve shape reflectsalso easily the effect of the threshold on excursionsets topology. For values of u lower than the lowestvalue of f , the Euler characteristic is the one of thefull cube (L0 = 1). By increasing u, several holes ap-pear, counting in positive for the Euler characteristic(L0 > 1). Then, the expansion of the holes starts toform handles which lead to a sponge like topology(L0 < 0). By increasing u even more, handles disap-pear forming a “meatball” like topology of connectedcomponents (L0 > 0). Finally, the Euler character-istic decreases to L0 = 0 when no more connectedcomponents remain.

From the comparison between theoretical valuesand measures on one realisation, we can point outthat the variability of the numerical generation is verylow. Therefore, although equation 3 gives only ex-pectations of the LKCs, for this range of excursionsets we can assume that V{Li(Au)} ≪ 1. Examplesare shown bellow (figure 4 and 5 ) with a compari-son between expected characteristics and experimen-tal ones. One has to keep in mind that the first test wasmade just over one realisation of the RF. As we cansee on figure 4, even though it is not a Gaussian RFany more, but a χ2

1 one (as mentioned earlier, Adler’stheory is transpoable to any Gaussian related RF fam-ily), the expected volume fraction for any threshold isa perfect fit with the experimental one, and the EC isa good fit. If we take a look at figure 5, we see thesame thing, but the EC experimental points have beensmoothed, and fit better with the expectations. Thiscurve was made averaging values over 5 realisationsof the same RF.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tresholds u

Vo

lum

e fr

acti

on

[−

]

0 2 4 6 8 10−150

−100

−50

0

50

100

150

Tresholds u

Eu

ler

Car

acte

rist

ic [

−]

Figure 4: Expectations (-) vs. Numerical experiments (+): Lc =10, with χ2

1 random field

2.2.1 Limitations

As explained previously, the previous morphologyyielding approach is based upon Gaussian correlatedrandom fields. However if one is to use exclusivelyGaussian random fields, some limitations will appear.

In terms of “matrix-inclusions” like morphologies,one will be limited to volume fractions around 15 %.This means that if we impose a positive EC (i.e. mor-phology with disconnected components), we will be

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Tresholds u

Vo

lum

e fr

acti

on

[−

]

0 2 4 6 8 10−150

−100

−50

0

50

100

150

Tresholds u E

ule

r C

arac

teri

stic

[−

]

Figure 5: Expectations (-) vs. Numerical experiments (+): Lc =10, with χ2

1random fields, averaged over 5 realisations

restrained on the achievable volume fraction. In orderto represent the initial state of a cement paste, we willneed much more to even come close to what can beobserved experimentally: for a classic concrete with aw/c around 0.4, one will need above 70% of volumefraction to be accurate.

(a) Gaussian ran-dom field realisa-tion: γ

(b) χ2

1random field

realisation: δ = γ2

Figure 6: Comparison between Gaussian (a) and χ2

1(b) realisa-

tions, with the same threshold.

However, as far far as those random field used areGaussian related, one can find a way to preserve theprobabilistic link (equation 3) that exists between theexcursion sets and the field parameters. As demon-strated figure 6 and figure 5, χ2 are pretty useful toincrease the volume fraction, but some other usagemight be consider in the future. They allow to dou-ble to achievable volume fraction up to 31%, keepinga disconnected morphology (EC>0).

In summary, by changing the nature of the randomfield we were able to increase the volume fraction,but another step is needed to help even more with thisissue, and at the same time allow to yield more real-istic morphologies for cement based materials, at anyscale. It is the use of several random field that will beunified as one, described in the next section.

2.3 Union and PSD curve

The general idea was to us two (or more) ran-dom fields: one with a “big” correlation length,which by thresholding will provide only few big dis-connected components to the morphology (volume

fraction improvements) ; and a second one with a“small” correlation length, which will bring small dis-connected components (connectivity improvements).When used together, such combinations not only canlift some of the previous limitations, but it also makesthe yielded morphologies much more realistic. In-deed, cement grains are anything but mono-sized,they follow a given Particle Size Distribution (PSD)curve.

In order to represent the latter, we need to take thetwo (or more) realizations of the two random fields,and make the union. This process actually only re-quire to take the maximum (or respectively minimum,according to the considered excursion set) of the tworandom fields for each point in space. The result is theunion of the above mentioned fields.

The challenge here was to be able to keep the vi-tal link between the different random field parametersand the yielded excursion sets. We have shown pre-viously that for Gaussian or Gaussian related randomfield, equation3 provides us geometrical and topolog-ical informations about the excursion. But how doesthese relations behave when we are dealing with theunion.

For any given geometrical or topological prop-erties, one have to satisfy equation 4, where thesubscript i stands for the chosen property, and the

1 , 2 ,. . . refers to the random fields, sorted by de-scending correlation length. In order to simplify theequations, we will deal with specific values, so Li willstand for Li/T

3

E

{

L∪i

}

= E

{

L1i

}

+E

{

L2i

}

+

· · · − E

{

L∩i

}

(4)

Since we have all the information for each inde-pendent field, all terms with i subscript are known.Hence the only approximation one have to do is onthe ∩ term. If we are interested in L3 (the volumefraction), equation 4 suggests that the volume frac-tion of the union is equal to the volume fraction of

the first excursion set 1 , plus the volume fraction of

the second excursion set 2 , from which we must de-duce the volume fraction of the intersection (part of

2 inside 1 ). Under assumptions of sufficient ergod-icity for the random field, and keeping a Lc/T ratiothat ensure a morphological REV (this study is notdeveloped in this paper), we will approximate the vol-ume fraction of the intersection by the product of the

two volume fractions 1 and 2 respectively. In otherterms, and for any interest quantity, we will approxi-mate the intersection terms by equation 5.

E

{

L∩i

}

≈ E

{

L13

}

×E

{

L2i

}

(5)

So if we are combining two random fields, equation6 will be used to describe the properties such a union.

E

{

L∪i

}

≈ E

{

L1i

}

+E

{

L2i

}

×

(

1−E

{

L13

})

(6)

The equation 6 generalization to N fields is prettystraight forward. If we keep the same assumptions(the subscript i stands for the chosen property, and

the 1 , 2 ,. . . refers to the random fields, sorted by de-scending correlation length), we will have:

E

{

L∪i

}

≈ E

{

L1i

}

+

N−1∑

j=1

E

L

j + 1

i

×

(

1−

j⋃

k=1

E

{

Lk3

}

)

(7)

where the term⋃j

k=1E

{

Lk3

}

can be re-written

according to the Inclusion-Exclusion Principle, alsoknown as the Poincare formulae:

N⋃

k=1

E

{

Lk3

}

=

N∑

j=1

(−1)k−1

×∑

1≤k1≤k2≤···≤kj≤N

Lk13 ∩ · · · ∩ L

kj3 (8)

As shown on figure 7, the main benefit is the vol-ume fraction that increases (for a given threshold) ac-cording to the number of unions. The counterpart isthat there is no big effect on the EC, except that themorphologies yielded are more connected. No mat-ter how connected a small group of inclusions can be,one big inclusion will connect them all.

As a first applied example, we will focus here ana given cement, and show how we determine the ran-dom field needed, and how we yield the correspond-ing morphologies. We chose to model a pretty stan-dard cement from LAFARGE TM, designated: “CEM-I52.5 CE CP2 N”. Its PSD is shown figure 8.

0 5 10 150

0.2

0.4

0.6

0.8

1

Threshold u

Vo

lum

e F

ract

ion

[−

]

Any χ2

1 RF

Union of any 2 χ2

1 RF

Union of any 3 χ2

1 RF

Union of any 4 χ2

1 RF

Figure 7: Unions’ volume fraction according to their threshold

The first approach used is illustrated figure 8. Givena PSD curve, we chose correlations lengths that wouldfit. For any field we noticed that that the maximalvolume fraction achievable with disconnected com-ponents was always in the same vicinity in terms ofthresholds, and the average size was always in thevicinity of the correlation lengths. We first set up thesmaller and the bigger ones, with different consider-ations. The bigger one was chose to limit as far aswe can the size effects of the inclusions. For a mod-elled cube of size T = 100µm, the biggest correlationlength will be Lmax

c = 20. Then for the smaller size,we chose to limit ourselves to a size Lmin

c = 2 to beable to numerically be able to represent them. Finally,we set up a third correlation length, that would fit inthe middle of the PSD curve between Lmax

c ad Lminc ,

here in the example Lmiddlec = 7

Then, as we wanted to have only one threshold tomanipulate the entire morphology, we did the union ofthe three random field. This way, by varying a uniquethreshold, we are able to create a morphology that fol-lows the PSD curve.

90

100

CEM I

60

70

80

%P

assi

ng

20

30

40

50

Cum

ula

tive

%

0

10

20

0 0 0 1 1 0 10 0 100 0 1000 0

C

2 7 200,0 0,1 1,0 10,0 100,0 1000,0

Equivalent particle diameter (µm)

7 20

Figure 8: PSD curve for a CEM I - 52.5 CE CP2 N from LA-FARGE TM

One can already notice that the overall frameworkwill be suited for cement pastes (cement within a ma-trix of water, and growth) as well as for higher scales:mortar (sand inclusions within a matrix of cementpaste) or even concrete (aggregates inclusions withina matrix of mortar).

2.4 Morphologies’ evolutions

We have seen in the previous section that thresholdinga realisation of a random field yields a two phase mor-phology. We will now consider three phase morpholo-gies. Example is shown on figure 9. The same wayapplying a threshold will separates the random fieldvalues in two distinct subsets (figure 9(a)) and cre-ate a two phase morphology (figure 9(b)), two thresh-olds will separate it in three (figure 9(c)) and createa three phase material (figure 9(d)). We will now beable to represent the matrix, inclusions, and a concen-tric phase around the latter.

(a) One randomfield realisation,thresholded at u

(b) Initial morphol-ogy: cement grainsand water

(c) Same realisation,with v ≥ u, andw ≤ u

(d) Cement hydra-tion: cement grains,water, and hydrates

Figure 9: From the random field creation to cement paste mor-phology yielding - grey: unreacted cement; blue: water; orange:hydration products

For each one of those three phase, equation 3 and 7can be adapted, so we will keep the probabilistic linkbetween the random field, the thresholds, and the ge-ometrical and topological properties of each phase ofthe actual morphologies. Instead of using the excur-sion set D = Du = [u;+∞), we had to adapt equa-tion 3 to D =Du = [u;v] and D =Du = [0;u] (sincewe use χ2 random fields, the lower limit will be 0,for Gaussian random field we would have had the lasthitting set D = Du = [−∞;u]). Equation 7 stays un-changed for any hitting sets.

We can notice that an extension to n-phase mor-phologies will be straight forward, each additionalthreshold will be adding a concentric phase aroundthe previous excursion.

3 HYDRATION MODELLING

In this section we will introduce the model used todescribe the hydration reactions.

3.1 Jennings and Tennis’ model

To be able to describe properly the hydrated cementpastes morphologies, we need here a model that de-scribes the volume fraction changes through time. Itneeds to account for the unreacted cement as well asthe different hydration products. We chose here to useJennings & Tennis (1994) model. The overall descrip-tion of the hydration is given by equation 9 for 1g offresh cement paste .

c = 1

e/c+1

VC = c(1− ξtotal)1

ρcVCH = c(0.189ξC3SpC3S + 0.058ξC2SpC2S)VAFm

= c(0.849ξC3ApC3A + 0.472ξC4AFpC4AF )VC−S−HS

= c(0.278ξC3SpC3S + 0.369ξC2SpC2S

Vcap = (1− c)−∑4

i=1(ξipi∆i)

VC−S−HP= 0.219VC−S−HS

Vtot pores = Vcap + VC−S−HP

(9)

where pi and ξi are respectively the initial volumefraction and hydration degree for each phase, Vi thevolume fraction of phase i during the hydration pro-cess, c is the mass fraction of anhydrous cement atthe initial state, and finally, C, CH, AFm, C-S-HS ,cap, C-S-HP and tot pores represent respectively theanhydrous cement, the Portlandite, the mono-sulfo-aluminates, the solid C-S-H, capillary pores, C-S-H’spores and lastly the total pores volume. In addition,the term ∆i accounts for the Le Chatelier contrac-tion. The ξi are estimated thanks to an Avrami likelaw (equation 10), and the overall hydration degreeξtotal will be a weighted average of the ξi.

3.2 Avrami like law revised

In the (Jennings & Tennis 1994) and (Tennis & Jen-nings 2000) works, each individual anhydrous phasehas its own hydration kinetics. Their hydration degreeare set to follow Avrami like laws, as in equation 10,with table 1 coefficients. Those laws are known to bewell suited to represent nucleation growth phenom-ena. However, for very early ages (less than a day),they present an non-finite derivative at the origin. Inthe spirit of on-going development aiming at takinginto account mineral additions, where the early agekinetics of hydration is of prime importance, we de-cided to resolve this issue upfront by modifying equa-tion 10.We decided to add a term to equation 10 andtransform it into equation 11.

ξi = 1− exp (−ai(t− bi)ci) (10)

0 5 10 15 20 250

10

20

30

40

50

60

70

80

90

100

time [days]

Deg

ree

of

rect

ion, %

by m

ass

ξC

3A

(t)

ξC

3S

(t)

ξC

4AF

(t)

ξC

2S

(t)

Figure 10: Time evolution of the degrees of reaction for eachanhydrous phase.

Phase a b cC3S 0.25 0.90 0.70C2S 0.46 0 0.12C3A 0.28 0.90 0.77C4AF 0.26 0.90 0.55

Table 1: Equation 10 coefficients

Figure 10 shows the new hydration degrees fittedwith equation 11 on one side and Tennis and Jennings(2000) data on the other side. The coefficients aregiven in table 2. The derivatives at the origin are nowclearly finite, and the hydration rates are preservedover the available data.

ξi = (1− exp (−ai(t− bi)ci)) exp

(

−(α

t

)β)

(11)

3.3 The simplification for diffusion

Since the model is aimed to predict diffusivity in ce-ment pastes, only the diffusive hydrates are consid-ered: the C-S-H. All the remaining species will beconsidered as the unreacted cement, since they arecrystalline phases we consider that they do not par-ticipate in the diffusion process. Hence the hydrationdescription can be simplified to equation 12, where Dstands for the diffusive specie, and ND for the non-diffusive one.

c = 1

e/c+1

VC = c(1− ξtotal)1

ρcVHND

= VCH + VAFm

VHD= VC−S−HS

+ VC−S−HP

Vtot pores = 1− VC − VHND− VHD

(12)

Phase a b c α βC3S 0.622 0.0 0.280 0.871 10.67C2S 0.119 0.0 0.462 0.928 11.64C3A 0.712 0.0 0.230 0.788 3.40C4AF 0.502 0.0 0.281 0.861 19.22

Table 2: Equation 11 coefficients

In addition, a reflection was carried out on the finalhydration degree and final porosity in cement pastes.In (Jennings & Tennis 1994) a maximal hydration de-gree was set to follow Mills (1966) model. The im-portant point for us was that the model predicts ac-curately the porosity of fully hydrated cement pastes.Due to new experimental data, we chose to considerthe maximal hydration model introduced by Waller(1999) (equation 13)

ξ∞ = 1− exp (−A ·w/c) (13)

Usually, the constant term A is taken to be A = 3.3.Parrott et al. (1990) Justnes et al. (1992) Justnes

et al. (1992) measured experimentally the final hydra-tion degree on cement pastes for different w/c ratios,after long enough periods and figure 11 summerizethe data for the final hydration degree according tow/c. As we can notice the upgraded Jennings model,with the final hydration modification stands at theupper bound of the experimental cloud, but followsthe global tendencies well for the maximal hydrationstages.

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Po

rosi

ty[-

]

w/c [-]

Jennings model - ξ28dJennings model - ξmax

Figure 11: Porosity measured versus modeled at final hydration:squares: Richet et al. 1997: Hg porosimetry, bullets: Richet etal model, up-triangle : Tognazzi (1999) H20 porosimetry, down-triangles Bejouai (2006)

3.4 The hydration process

In summary, thanks to the parameter c in equation 12,which represents the initial volume fraction of the ce-ment grains, we are able to generate a representativeinitial morphology. It will be obtain by generatinga proper random field and thresholding it at a givenscalar u. This threshold will be directly linked to thehydration degree ξ. As ξ grow from 0 to 1, the thresh-old will move from an initial value to +∞. Then inorder to model the appearing new phase, we will needto determine a second threshold, which will yield the“concentric” phase around anhydrous cement. Thissecond threshold will be chosen so that the overall ex-pected volume fraction of the new phase matches ourhydration model predictions.

The output for the initial morphology can be seenfigure 12. We can notice first the cement grain repar-tition on figure 12(a), then by adding the complemen-tary phase (figure 12(b)) we model water. Then, fora given hydration degree, we modify accordingly toequation 3 and 12 the thresholds, and we yielded fig-ure 12(c) and 12(d).

(a) Initial cementgrains

(b) Initial morphology

with ξ = 0

(c) Cement paste mor-

phology for ξ = 0.2(d) Cement paste mor-

phology with ξ = 0.8

Figure 12: Cement paste morpholgy yielding and evolution.

4 UPSCALLING: AN ENERGY BASEDFRAMEWORK

The last step to preform properties prediction will beto have a suitable homogenization framework to en-sure that the upscaling is consistent with general ther-modynamic principles.

Based on Fick’s equations (equation 14), we here-after define an energy dissipation due to the diffusionprocess. Analogously to thermal dissipation, the localdissipation is defined by equation 15.

q = −Di ·∇c

∂c

∂t+ divq = 0

(14)

where Di is the local diffusivity matrix, q the flux vec-tor, ∇ the gradient operator, and c the concentrationof the diffusive specie.

−φ =q ·∇c

c(15)

The effective material properties are determined bysubjecting a REV to boundary conditions satisfyingHill’s energy criterion (Hill 1994) which relates themicroscopic energy and the mesoscopic one. For me-chanical elasticity, it stipulates that for a REV, the spa-tial average of the local energy (equation 15) must beequal to the upper scale energy computed with aver-aged fields. In our case that will mean that equation 16must be verified to be sure that the scale separation islarge enough to be a REV.

〈q ·∇c〉!= 〈q〉 · 〈∇c〉 (16)

where 〈•〉 denotes volume average over a representa-tive volume element. After some transformation, thisrelation yields two kind on boundary conditions:

• c= 〈∇c〉 ·x on Γ: Kinematic Uniform BoundaryCondition (KUBC)

• q · n = 〈q〉 · n on Γ: Static Uniform Boundaryconditions (SUBC).

These KUBC and SUBC conditions will allow usto estimate respectively an upper and a lower boundof the property of interest.

The first implementation that has been made con-cerns calculations made with Dirichlet boundary con-ditions, the so-called KUBC. If we summarize Fick’sequations, time independently, we will have from amacroscopic point of view:

q = −Deff ·∇c (17)

So by applying unitary gradients , we will have thediffusivity coefficients (the diffusivity matrix Deff )by measuring the resulting macroscopic flux q. It cor-responds to the so-called KUBC case. A result isshown below, equation 18. It is a calculation madeon 100 × 100 × 100µm cube, for an ordinary Port-land cement paste (cf. PSD curve in section 2.3, withw/c = 0.5 and a hydration degree ξ = 0.8 (basedon the simplified Jennings’ model introduced section3.3). Material properties were set as summarised intable 3.

As expected, equation 18 diagonal terms are greaterby at least one order of magnitude, and the matrix issymmetrical. This result would give us a diffusivitycoefficient Deff = 4.2× 10−10m2.s−1 when we aver-age over the three main directions.

Deff =

(

438 2.4 −5.72.4 414 10−5.7 10 408

)

× 10−12m2.s−1 (18)

Phases Di [m2.s−1]

Anhydrous cement 2.0× 10−15

Hydration products 2.0× 10−12

Porosity 2.0× 10−9

Table 3: Material properties

5 RESULTS

All the calculations presented were made with thehelp of FEAP (Taylor 2008) software and COFEAP

(Kassiotis and Hautefeuille 2008) middle-ware. Es-timations were made on 100× 100× 100µm cubes,with always the same properties as listed in table 3.

In order to bridge the gap between the almost con-tinuous morphological model described and the dis-crete structures on which one can perform calculation,an operation of projection have to be handled. Thisoperation will allow us the “project” material proper-ties from the model onto a FE meshe. It was designedin an E-FEM (Sukumar et al. 1999) framework. De-veloped first with truss elements (Benkemoun et al.2010) and extended to meshes by (Roubin 2013) itallows the use of non adapted meshes. This proper-ties projection is less time consuming than re-meshingexplicitly an entire morphology for any given hydra-tion advance, and also allow the used of previouslychecked meshes. Within this framework, the bound-aries between materials do not need to be meshed ex-plicitly, and the elements that are split into two ma-terials are locally enriched to bear two material prop-erties. Hence for a three phase morphology, we willhave six type of elements: 1 - matrix, 2 - inclusions,3 - hydrates around the inclusions, and the three kindof possible interfaces 1-2, 1-3 and 2-3).

Each results reported on the following figures areaveraged values of the 3 diagonal terms (as explainedpreviously with equation 18), and then averaged over5 or 10 realisations of the random fields. Some ex-perimental data to compare with is plotted, and takenfrom (Bejaoui and Barry 2006, Mejlhede 1999, Ngala1995, Tognazzi 1999), amongst others.

All the presented results were obtained on fullyhydrated morphologies (according to equation 13),for various w/c ratio. We used the methodology de-scribed in section 2.3, with the union of three randomfields, chosen according to a PSD curve. Based on thesame correlation lengths, we produced fully hydratedmorphologies. The material properties used for eachphases were set as presented in table 3.

The first concern was the value of Dhyd. Since thisvalue is not yet well known, we performed a sensibil-ity study. The idea was to change the ration Dhyd/Dw,and see its influence. Results are shown figure 13. Aspreviously explained, the diffusivity coefficient in wa-ter was taken to be 2 × 10−9m2.s−1. By varying theratio, we were able to compute the numerically ho-mogenized diffusivity at the macro scale for differentw/c ratio, at a fully hydrated state. For high w/c ra-tios, as the 0.8 on figure 13, we can clearly see that

there is no considerable changes. This is due to thefact that even at the end of hydration, there is stillsome water left, and the diffusive specie will take the“easiest” path, and thus preferably go through water.On the other hand, when the w/c ratio decreases, theinfluence of the ratio is more noticeable. We decidedto keep a value of Dhyd = 2× 10−12m2.s−1, since itseemed be the values underneath which the results donot fluctuate noticeably. Also it was a good compro-mise with literature values.

1e-11

1e-10

1e-09

1e-5 1e-4 1e-3 1e-2

Eff

ecti

ve

dif

fusi

vit

y[m

2.s

−1]

Dh/De

w/c = 0.5w/c = 0.8

Figure 13: Porosity measured versus modeled at final hydration

Figure 14 shows our first results. First we can no-tice that almost all of our computed values are oneorder of magnitude too high, especially for high w/c.This can be explained by the simplification we madeon Jennings’ model (equations12). We overestimatethe water in different ways: we do not into accountthe chemical shrinkage, we assign its volume fractionto water, and the gel water is included into the freewater. Also, our morphologies are always supposed tobe fully saturated, which is barely the case real situa-tions. On the positive side, we can notice the changeof slop with the decreasing w/c. It is an encourag-ing sign, since it shows that the diffusivity has tworegimes. The first is for low w/c, when the water israre and hence not connected, and thus the diffusiv-ity is mainly controlled by the hydrates phase. On theopposite side, for high w/c, it is the water that con-trols the diffusivity. It shows that the phase or porepercolation is a critical phenomenon, and needs to beexplicitly modelled.

No calculation could be made for w/c ≤ 0.4, sinceat the time we could not yield initial morphologieswith sufficient disconnected volume fractions of in-clusions.

One more series of test is actually under assess-ment, regarding the number of field used in the union.We are trying to establish if three is enough, or if willbe more realistic to use more or less fields.

1e-13

1e-12

1e-11

1e-10

1e-09

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Eff

ecti

ve

dif

fusi

vit

y[m

2.s

−1]

w/c [-]

Bejaoui (2006)Tognazzi (1998)Mejhede (1999)

Numerical simulationNgala (1995)

Figure 14: Porosity measured versus modeled at final hydration

6 CONCLUSIONS

We first presented the morphological tool used tocreate morphologies: the excursion sets of correlatedGaussian random field. After seeing the limitationsin terms of achievable volume fractions, connectivity,we went on some solutions to overcome them, mainlythanks to χ2 random fields, and unions of numerousrandom fields. We are now able to yield morpholo-gies that follow a given PSD curve, and make themevolve with concentric phases. We intend to push fur-ther investigation to be more realistic, geometricallyspeaking. An effort will also be made in the future toextend the model to n-concentric phases, and next tobe able to yield new phases in a non-concentric way(within the pore network only, for example).

Then, we developed the chemical models used todescribe the cement paste evolution, with chemicaland geometrical considerations. We described a mod-ified Jennings model, controlled by one and only oneparameter. This model already have an intended up-grade; we will add a layer to be able to model clinkersubstitution by by-products from other industries suchas blast furnace, silica fumes, . . . This developmenthas already started.

Finally, after introducing the energy based up-scalling framework, we presented our first results.Even if incomplete (only the upper bound so far,and for volume elements we can compute only ontwo phase morphologies) they are encouraging. It hasbeen showed that the model is able to predict the or-der of magnitude and the tendencies observed in ex-periments. For high water to cement ratios the cap-illary pores are connected, the diffusivity is mainlycontrolled by the porosity. For low water to cementratio, the capillary pores are no more connected, thediffusivity is mainly controlled by the C-S-H diffu-sivity. The next step, as already mentioned will be toestimate the lower bounds, and then, the extension tomortar and concrete materials.

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