Multi-Scale Modelling of the Diffusivity of Mortar and Concrete, Proceedings of Chloride Penetration into Concrete, RILEM, 1997 Reference: D.P. Bentz, R.J. Detwiler, E.J. Garboczi, P. Halamickova, and L.M. Schwartz, Proceedings of Chloride Penetration into Concrete,edited by L.O. Nilsson and J.P. Ollivier, RILEM (1997).

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Multi-Scale Modelling of the Diffusivity of Mortar and Concrete

D.P. Bentz(1), R.J. Detwiler(2), E.J. Garboczi(1), P. Halamickova(3), and L.M. Schwartz(4)
(1) National Institute of Standards and Technology, Gaithersburg, MD 20899 USA
(2) Construction Technology Laboratories, Skokie, IL 60077 USA
(3) Yolles Partnership Inc., Toronto, Ontario M5A 1S1 CANADA
(4) Schlumberger Research, Ridgefield, CT USA


Modelling ion diffusion in concrete is complicated by the heterogeneity which exists at a variety of length scales. The cement paste fraction of the concrete is heterogeneous at the scale of micrometers, consisting of unhydrated cement, porosity, and hydration products. The concrete itself must often be considered as a three-phase composite, consisting of aggregates, bulk cement paste, and interfacial transition zone (ITZ) cement paste. In this research, separate microstructure models for cement paste and mortar or concrete are linked to estimate the diffusivity of a mortar or concrete. Once an ITZ thickness is selected, a hard core/soft shell representation of the concrete is utilized to compute the volume fractions of ITZ and bulk pastes in the mortar. A cement hydration model containing a single flat plate geometry aggregate is then employed to determine the porosities of the ITZ and bulk pastes as a function of water-to-cement ratio and degree of hydration. Based on these porosities, relative diffusivities can be estimated using previously developed empirical equations. Finally, these relative diffusivities can be used in the original hard core/soft shell representation of the mortar, where random walk algorithms are employed to estimate its diffusivity, for comparison with experimental data.
Keywords: Cement-based materials, Diffusivity, Interfacial transition zone, Microstructure, Multi-scale modelling, Random walk, Simulation.

1 Introduction

In many instances, the service life of concrete in a given environment is controlled by the resistance it presents to the ingress of deleterious materials such as chloride or sulfate ions [1]. For cases where the concrete is located in a saturated environment, the ingress of ions will be dominated by diffusive transport. Prediction of the diffusivity of concrete is complicated by its heterogeneous microstructure evolving over time. In a conventional concrete, there exists an interfacial transition zone (ITZ) surrounding each aggregate [2,3], characterized by a higher porosity and larger pores [4,5,6]. Thus, the concrete must be considered as at least a three-phase composite consisting of aggregates, bulk cement paste, and ITZ cement paste, with each phase having its own distinct diffusion characteristics [7]. At the scale of micrometers, the cement paste itself is heterogeneous consisting of capillary pores, unhydrated cement, and crystalline and gel hydration products, each with its own diffusion properties [8].

The basic premise of this paper is that if the heterogeneous nature of the concrete can be accounted for in a suite of microstructure models, the diffusion properties of a specific concrete or mortar can be predicted. In the present study, a linkage is developed between a digital-image-based microstructure model for cement hydration at the scale of micrometers [9,10] and a continuum-based hard core/soft shell spherical inclusion microstructure model for concrete and mortar at the scale of millimeters [4,7]. It should be pointed out that we are only attempting to estimate the steady-state diffusion coefficient of chloride ions in a concrete and are not accounting for the chloride binding and reaction with the aluminate phases present in the cement paste. However, the modelling techniques outlined here can be extended to consider the effects of binding and reaction [11].

2 Computer Modelling Techniques

The computer modelling techniques used to estimate the diffusivity of cement paste and concrete have been described in detail elsewhere [7,8]. Here, we shall concentrate on the linkage of the individual models to develop an integrated multi-scale model for the diffusivity of concrete as illustrated in Fig. 1. To successfully represent the microstructure of a concrete and its component paste, the following input parameters are necessary: 1) mixture proportions (aggregate content, water-to-cement (w/c) ratio by mass, and air content), 2) the particle size distributions (PSDs) of the cement and aggregates, and 3) the degree of hydration () of the cement paste in the concrete at the time of interest.


Figure 1: Linkages between microstructure models for mortar and cement paste for predicting diffusivity of mortar or concrete. In the mortar model (left), inclusions are red, ITZ regions are yellow, and bulk paste is blue. In the cement paste images, cement particles are yellow, water-filled porosity is black, and the flat rectangular aggregate is red. In the hydrated image (upper right), calcium hydroxide is purple and calcium silicate hydrate gel is grey. The parameters and arrows (center) indicate the flow of information between the models.

To begin, the cement PSD is used to estimate the thickness of the interfacial zone regions, tITZ, present in the concrete. Simulation studies have shown that tITZ can be estimated as the median diameter of the cement PSD on a mass basis [3]. Thus, a typical value would be on the order of 20 µm. This ITZ thickness is input into the hard core/soft shell model of the mortar or concrete [4], along with the aggregate content and the aggregate particle size distribution (based on a sieve analysis, for instance). In this model, the aggregates and air voids are modelled as inpenetrable spheres, each surrounded by a concentric ITZ region of constant thickness. The ITZ regions are free to overlap one another or portions of other inclusions. In the present work, the measured air contents of the mortar are incorporated into the model by increasing the aggregate content to represent the sum of the air and sand volume fractions. Thus, we are assuming that the air voids follow a size distribution similar to that of the sand. However, if the air void size distribution has been quantified, the total inclusion size distribution can be easily modified to represent the contributions of both the sand and air.

For mortars, the computational volume typically employed in the microstructure model is 1 cm3. This results in about 20,000 sand grains and air voids being present in a typical simulation. After the inclusions have been placed in the 3-D volume, a continuum burning algorithm is employed to determine the percolation characteristics of the ITZ regions and more importantly for the present study, systematic point sampling is employed to estimate the volume fractions of ITZ and bulk cement paste [4]. These volume fractions (VITZ and Vbulk) are then input into a cement paste hydration microstructure model containing a single aggregate particle to simulate the microstructure development in the ITZ and bulk paste regions (right side of Fig. 1).

To use the cement paste microstructure model [9,10], the user specifies the cement PSD, the w/c ratio by mass, and the desired degree of hydration (). The PSD is a discretized version of the experimentally measured curve, binned into two pixel (2 µm) increments. Particles typically range between 1 and 45 µm in diameter, so that we are truncating the PSD slightly at both ends. For a specific fixed size volume, the ratio of VITZ to Vbulk can be varied by changing the thickness of the single flat plate aggregate particle. A cellular automata-based simulation is then employed to hydrate a digital-image-based representation of the cement particles in water. The model consists of discrete cycles each comprised of dissolution, diffusion, and reaction/precipitation steps [10]. By monitoring the unhydrated cement (C3S) remaining after each cycle of hydration, the simulation can be terminated when the desired overall degree of hydration is achieved. Then, point sampling is employed to determine the variation in capillary porosity as a function of distance from the aggregate surface. Additionally, the average porosity in the ITZ and bulk paste regions can be computed. The following previously developed empirical equation [8] can be used to convert capillary porosity () into a relative diffusivity (D/Do- the ratio of the diffusivity of ions in the cement paste to their diffusivity in bulk water):

Thus, the ratio of the ITZ (DITZ / Do) to the bulk paste (Dbulk / Do) relative diffusivity can be determined.

Finally, the values for DITZ and Dbulk are input into a myopic random walker algorithm, implemented in the original hard core/soft shell representation of the concrete, to estimate the diffusivity of the original mortar or concrete microstructure relative to its bulk cement paste value [7], as indicated in Fig. 1. Typically, 10,000 walkers are placed at random locations in the bulk and ITZ cement paste (not in the aggregates). Each random walker maintains a record of its cumulative travel time, its starting location, and its current location. Each walker's steps are of a constant length in a random direction in 3-D space. This random walk simulation would be equivalent to the simulation of Brownian motion of an individual ion, but at a larger scale. The amount of time which elapses during a step depends on the starting and ending locations of the step (e.g., a step entirely within the ITZ paste takes less time than a step entirely within the bulk paste due to the higher relative diffusivity of the former). The walkers are not allowed to step into the aggregates or air voids, but their clock still advances when such an attempted step is made. Additionally, the probability of a walker being allowed to step from an ITZ region into a bulk paste region is biased by the ratio of its relative diffusivities in the two media [12]. By dividing the distance travelled squared by the time elapsed after a large number of steps (e.g., 200,000), an estimate of the diffusivity of the walker in the composite media can be determined. By averaging this value over the 10,000 walkers and multiplying by the total cement paste (ITZ+bulk) volume fraction, an estimate of the relative diffusivity of the mortar or concrete (DM / Dbulk) can be obtained. This value can be converted into an absolute diffusion coefficient for comparison against experimental data by multiplying it by the relative diffusivity of the bulk cement paste and the diffusion coefficient of the ions of interest in bulk water
[DM = (DM / Dbulk) (Dbulk / Do) Do].

3 Results

Few data sets exist in the literature, where all of the necessary information for executing the multi-scale modelling has been provided. An exception is the data set for a series of mortars of varying sand content provided in a recent Master's thesis [13,14]. Here, information is provided on the cement and aggregate PSDs, of the mortars, and the diffusion coefficients measured for chloride ions using an accelerated electrochemical technique. Experiments were performed for two w/c ratios (0.4 and 0.5 by mass) and four different sand volume fractions (0, 35, 45, and 55%). The air contents, while not measured for each specific sample, were assessed for both the neat cement pastes (1%) and mortars (8%).

Equation 1 can be directly applied for estimating the relative diffusivity of the neat cement paste samples. To do this, it is necessary to convert the degree of hydration values measured experimentally [13] to capillary porosity values, using the following equation:

where the value 3.2 is the specific gravity of cement and the value 1.31 represents the relative increase in solids volume due to the hydration reaction. To convert from relative to absolute diffusivities, a value of 2.0 x 10-9 m2/s is used for chloride ions in water at 25oC [15]. For the mortars, Table I summarizes the simulations employed for each of the three different sand contents. The ITZ paste fraction value indicates the fraction of the overall cement paste volume which is contained within the ITZ regions and is seen to be considerable for a sand volume fraction of 55%.

Sand volume fraction Number of inclusions Inclusion volume fraction ITZ paste fraction
0.35 17750 0.403 0.217
0.45 21770 0.494 0.307
0.55 25500 0.578 0.413

Table 1: Characteristics of simulated mortar micrstructure

The ITZ paste fractions were used to establish single aggregate simulations for the cement paste microstructure model. The flat plate aggregate thickness was adjusted to achieve the correct ratio of ITZ to bulk cement paste while maintaining a constant system size of 2203 pixels (µm) with tITZ = 20 µm, which compares favorably with the median cement particle diameter of 16.1 µm measured in [13]. After randomly placing the cement particles into the 3-D volume to achieve the correct overall w/c ratio, the w/c ratios on a mass basis of the ITZ and bulk paste regions were determined before executing any hydration. Table II summarizes the computed results. As expected, the w/c ratio of the ITZ is higher than the overall value, but one should also note that the w/c ratio of the bulk paste is significantly reduced to compensate for this higher ITZ value. The actual initial distributions of porosity as a function of distance from the aggregate surface are shown in Fig. 2, which illustrates that 20 µm is indeed a reasonable choice for tITZ.


Figure 2: Initial porosity distributions vs. distance from aggregate for the two w/c ratios by mass and three sand contents examined in this study.

Inclusion volume fraction Overall w/c ITZ w/c Bulk paste w/c
0.403 0.4 0.592 0.361
0.494 0.4 0.575 0.343
0.578 0.4 0.555 0.320
0.403 0.5 0.770 0.448
0.494 0.5 0.745 0.431
0.578 0.5 0.712 0.400

Table 2: Initial w/c ratios by mass for single aggregate simulations

Each initial microstructure was then hydrated to the points at which diffusivity measurements were performed in [13]. Once again, the capillary porosity as a function of distance from the aggregate surface was computed. In a few cases, high enough degrees of hydration could not be achieved using the hydration model, as the hydration terminates when all cement particle surfaces become covered by hydration product (upper right of Fig. 1). In these cases, the final average ITZ and bulk paste capillary porosities were extrapolated from their initial values and the ultimate values achievable using the model. The linear extrapolation employed is justified by Fig. 3 which shows a linear decrease in ITZ and bulk paste capillary porosity with degree of hydration, although the two lines do exhibit different slopes. Evidence elsewhere suggests that when the capillary porosity in the bulk paste levels off (hydration ceasing), porosity in the ITZ region may continue to decrease as ions slowly diffuse and precipitate within the more porous ITZ [5,6].


Figure 3: ITZ and bulk paste capillary porosity vs. degree of hydration for w/c=0.40 by mass and an inclusion volume fraction of 58%.

Once the capillary porosity distribution after hydration is known, equation (1) can be employed to determine DITZ / Do and Dbulk / Do. When the complete porosity distribution is available, all of the values as a function of distance from the aggregate are substituted into equation 1 and a final average value computed for the ITZ (< 20 µm) and bulk paste regions (> 20 µm). When the extrapolation technique is employed, only the average ITZ and bulk paste porosity values are used in determining each phase's relative diffusivity. Where possible, both averaging techniques were employed, with the former resulting in diffusivities which are on the average 30% higher than those computed using the extrapolated average porosities.

Because of the nature of the relationship described by equation 1, the ratio of ITZ to bulk paste relative diffusivity exhibits a maximum when plotted against degree of hydration or against capillary porosity as shown in Fig. 4. Initially, as the bulk paste capillary porosity approaches 20 %, the bulk paste relative diffusivity falls rapidly so that the ratio of ITZ to bulk diffusivity rises rapidly. Conversely, once the bulk capillary porosity falls below 20 %, it decreases much less rapidly to an asymptotic value while the higher porosity ITZ diffusivity continues to fall so that the ratio of ITZ to bulk diffusivity falls once again. As can be seen in Fig. 4, the curves for the two different w/c ratios overlap for a large range of bulk capillary porosity values, with a maximum ratio on the order of 6 being observed in these simulations. The practical implication of this observation is that there will exist an intermediate curing age at which the relative increase in diffusivity due to the presence of ITZs will be maximized.


Figure 4: Ratio of ITZ to bulk paste relative diffusivity vs. bulk capillary porosity for two w/c ratios by mass and an inclusion volume fraction of 58%.

Finally, the ratio of DITZ to Dbulk was input into the random walk algorithm to estimate the overall diffusivity of the mortar, DM. These values could then be directly compared to those measured experimentally [13]. Figures 5 and 6 compare the experimental and model results for the two w/c ratios of 0.40 and 0.50 by mass. For the pastes (0% sand), early hydration corresponds to 50-55% while late hydration is in the range of 55-60%. For the mortars, hydration is accelerated, with early hydration in the range of 50-60% and late hydration corresponding to 65-75% [13]. For the model results, those obtained using the full porosity vs. distance distribution are presented whenever available. In general, the agreement between model and experiment is quite good. The experimentally determined diffusivities are generally larger, but within a factor of 2, of the model results. Furthermore, in several cases, the model values are within 10% of the experimentally measured ones. Given the assumptions made concerning the air void size distribution, etc. and the expected accuracy of the experimental measurements [13], this agreement is quite reasonable.


Figure 5: Comparison of experimental and model mortar diffusion coefficients for w/c = 0.40 by mass.


Figure 6: Comparison of Experimental and Model Mortar Diffusion Coefficients for w/c = 0.50 by mass.

4 Conclusions

Two different scale microstructure models have been linked to provide a multi-scale modelling of the diffusivity of mortar and concrete. Quantitative results are transferred between the two levels to allow an estimation of diffusivity based on the actual experimental details (PSDs, mixture proportions, and ). In this preliminary comparison, the agreement between experimental results and model predictions is quite good. It is envisioned that the multi-scale modelling approach presented here will allow the diffusivity of a mortar or concrete at a specific to be computed, thus providing rapid estimates for service life prediction models and design codes.


- degree of hydration
- capillary porosity
D - diffusivity
H(x) - Heaviside function = (1: x >0; 0 otherwise)
t - thickness
V - volume fraction
w/c - water to cement ratio
bulk - bulk cement paste
0 - bulk solution
ITZ - interfacial transition zone cement paste
M- mortar

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