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D.P. BENTZ*, E.J. GARBOCZI*, H.M. JENNINGS,
AND D.A. QUENARD

* Building and Fire Research Laboratory, Building 226, Room B-350,
National Institute of Standards and Technology, Gaithersburg, MD 20899 USA

Department of Materials Science and Engineering, Northwestern University,
Evanston, IL 60208 USA

Centre Scientifique et Technique du Batiment, Saint-Martin d'Heres,
FRANCE

** ABSTRACT**

Computer modelling of the properties and performance of cement-based materials is complicated by the large range of relevant size scales. Processes occurring in the nanometer-sized pores ultimately affect the performance of these materials at the structural level of meters and larger. One approach to alleviating this complication is the development of a suite of models, consisting of individual digital-image-based structural models for the calcium silicate hydrate gel at the nanometer level, the hydrated cement paste at the micrometer level, and a mortar or concrete at the millimeter level. Computations performed at one level provide input properties to be used in simulations of performance at the next higher level. This methodology is demonstrated for the property of ionic diffusivity in saturated concrete. The more complicated problem of drying shrinkage is also addressed.

** INTRODUCTION**

Predicting the performance of concrete structures is made difficult by the complexity of the material, which has a complex microstructure [1] and exhibits composite behavior at a series of length scales. At the scale of millimeters, one can view the concrete as a composite of aggregates and air voids embedded in a continuous cement paste matrix. Even here, one may need to account for the difference in cement paste microstructure and properties in the interfacial zones surrounding each inclusion [2,3,4,5,6]. At the scale of micrometers, the cement paste is a composite of unhydrated cement particles, hydration products (crystalline and amorphous), and capillary porosity. Finally, the major hydration product of calcium silicate cements, calcium silicate hydrate (C-S-H) gel, is itself a composite of nanometer-sized "particles" and pores.

A better understanding of the physical processes occurring at each of these scales and their interactions is necessary to increase the predictability of the performance of concrete. For example, it is the capillary forces which develop in the capillary and gel pores that are mainly responsible for the drying shrinkage of concrete. Thus, to reliably predict the drying shrinkage of a field concrete, one must understand the material at the scale of micrometers and even nanometers [7]. While this large scale range cannot be easily incorporated into a single model for concrete microstructure, it is possible to develop an integrated model in which information gained at one scale level is used in computing characteristics at the next higher level [2,8,9], as is demonstrated in this paper.

** COMPUTER MODELLING TECHNIQUES**

At each scale of interest, a method for generating appropriate representative three-dimensional microstructures must be chosen. Computational techniques are then employed to compute the physical properties (diffusivity, elastic moduli, etc.) of these composite microstructures. It should be noted that application of these property computation techniques is not limited to model microstructures, as they can be equally applied to 3-D microstructural representations of real materials obtained by tomographic [10] or microscopic [11] techniques.

At the scale of nanometers, in the study described here, the C-S-H gel is modelled as a two-level structure of partially overlapping spherical particles [7]. At the macro level, the larger 40 nm spherical agglomerates shown in the top right image of Fig. 1 are each composed of smaller 5 nm diameter micro level particles, as shown in the top left image of Fig. 1. The particle sizes and the total porosity at each level have been chosen to be consistent with experimental data from small angle neutron scattering [12] and sorption measurements [13]. The models are generated in continuum space (in a three-dimensional cube with periodic boundaries) and subsequently digitized into a 3-D digital image for the evaluation of properties.

**Figure 1:** Two-dimensional slices from three-dimensional model microstructures
of the C-S-H gel at the scale of nanometers (top image: left is micro
or 25 nm by 25 nm and right is macro model or 250 nm by 250 nm), the hydrated
cement paste (middle image: 250 µm by 540 µm),
and a concrete (bottom image: 3 cm by 6 cm)

as described in the text.

At the scale of micrometers, a cellular-automata based computer model is used to model the microstructural development of cement paste during hydration [14]. Three-dimensional versions of this computer model have been developed both for tricalcium silicate (the major component of portland cement) [15] and for portland cement, consisting of tricalcium silicate, dicalcium silicate, tricalcium aluminate, tetracalcium aluminoferrite, and gypsum [14]. The cement particles are modelled as digitized spheres following a user input size distribution, with the allowable particle diameter range being from about 1 to about 40 µm. The hydration process is modelled in discrete cycles consisting of steps for random dissolution, diffusion, and precipitation/reaction. Volume stoichiometry is explicitly maintained by creating the appropriate number of hydration product volume units (pixels) for each cement pixel that dissolves and reacts. Three-dimensional digital-image-based microstructures generated using the models can be subsequently analyzed to determine phase percolation [16] or diffusion coefficients [17] as a function of water-to-cement (w/c) ratio and degree of hydration (). The middle portion of Fig. 1 shows a typical two-dimensional microstructure (w/c=0.5, 87% degree of hydration) generated using the tricalcium silicate hydration model. The unhydrated cement particles are white, capillary porosity is black, calcium hydroxide is light grey, and the C-S-H gel is dark grey.

At the scale of millimeters, mortar or concrete is modelled as a
continuum of cement paste containing rigid (nonoverlapping) spherical
[18] or ellipsoidal [19] aggregate particles following
a size distribution corresponding to that of a real mortar or concrete. Each
aggregate can be surrounded by an interfacial zone of some specified
thickness, as shown in the bottom image of Fig. 1.
Here, the aggregates are shown in dark grey, bulk cement paste is black,
the interfacial zones are white, and the single rebar is light grey. One can
clearly see the modification of the aggregate distribution in the vicinity of
the labelled rebar as well
as at the top surface of the concrete and near the larger aggregate
particles. Additionally, the interfacial zone cement
paste is seen to occupy a large fraction of the total cement paste volume.
Typically, we model mortar using a sample volume 1x1x1 cm^{3},
while for
concrete, a sample volume of 3x3x3 cm^{3} is employed. Aggregate
particle diameters typically range from 75 µm to 19.05 mm (3/4'').
With this range of particle sizes, a concrete simulation can require as
many as 800,000 separate particles. For this reason, supercomputers
are often employed for generating these structures.
Once again, this continuum structure can be digitized into a 3-D
digital image for the computation of properties.

Properties are generally computed using finite difference or finite element techniques [17,20]. That is, each pixel in the 3-D microstructure is mapped into a node in either a finite element or finite difference analysis. The resultant physical equations are then solved utilizing conjugate gradient or other fast solution techniques to determine the property of interest such as an elastic modulus or diffusivity. For determining diffusivity, the Nernst-Einstein equation is used to determine the diffusivity by solving the equivalent electrical problem for the electrical conductivity of the composite material [17]. Alternatively, random walk simulations may be employed to estimate the diffusivity of a species in a composite medium, as has been utilized to estimate the diffusivity of a mortar relative to its component cement paste [3,4,5].

To utilize these solution techniques, the properties of each phase in a microstructure must be known or assumed. For example, to compute an elastic modulus of a cement paste microstructure, one must know the elastic properties of the unhydrated cement, the calcium hydroxide, and the C-S-H gel. Often, our approach is to utilize properties computed at one scale level of modelling as input into the computation procedures employed at a higher scale. For example, as is demonstrated in the results, the conductivity/diffusivity computed for the C-S-H gel nanostructure model can be used as an input property into the cement paste microstructure model, so that the conductivity/diffusivity of cement paste can be computed as a function of w/c and α. Likewise, the diffusivity computed for cement paste can be utilized in the model of concrete at the level of millimeters, to compute the diffusivity of a concrete, the value of actual interest to a designer or structural engineer.

** RESULTS AND DISCUSSION**

Transport Property Example: Diffusivity

The overall approach is demonstrated first for the case of computing
the relative diffusivity of concrete. The relative diffusivity is defined
as the ratio of the diffusivity of ions in a composite medium to their
diffusivity in bulk water. Based on the Nernst-Einstein relation, the
relative diffusivity (D/D_{o}) is equivalent to the relative conductivity
(σ/σ_{o}). The results of this computation are summarized in
Fig. 2, which is referred to in the discussion
which follows.

**Figure 2:** Relative diffusivity computed using structural models at each
of the three scale levels shown in Fig. 1.

The first step is the computation of the relative diffusivity of the C-S-H gel. Using the electrical analogy and the nanostructural model shown in the upper right of Fig. 1, a value of 1/300 is computed for the relative diffusivity of the gel. Here, we assume that diffusive transport occurs only in the cluster-level pores and not in the much smaller pores shown in the particle-level model in the upper left of Fig. 1. This is a reasonable assumption, since these smaller, nanometer size pores are on the order of the size of a water molecule and would be virtually inaccessible to many diffusing ions. A further assumption is that this relative diffusivity value is characteristic of all C-S-H gel regardless of cement composition or when during the hydration process the C-S-H is produced.

The second step is to use the value computed for the relative diffusivity of the C-S-H gel in a computation of the relative diffusivity of cement pastes of various w/c ratios and degrees of hydration. A diffusivity value of 1.0 is assigned to the capillary porosity, while the unhydrated cement and calcium hydroxide are assigned diffusivities of 0.0 since they contain no porosity. Again using the electrical analogy, computations performed for a variety of microstructures have resulted in the development of an equation which relates relative diffusivity to the capillary porosity of the cement paste as shown in Fig. 2. Good agreement has been observed in comparing these computed model values to ones measured experimentally both for chloride ion diffusivity [17,21] and cement paste conductivity [22].

Finally, the relative diffusivity values computed for cement paste can
be used as input into a structural model for mortar to determine the
effect of aggregates and their surrounding interfacial zones on the
diffusivity of the mortar (D_{M}). At this level, one must select the
thickness of the interfacial zone paste and the value of its
diffusivity (D_{ITZ}) compared to that
of the bulk paste (D_{P}).
Fig. 2 provides equations which approximate the relative
diffusivity of a specific ** model** mortar as a function of aggregate
content for three ratios of D_{ITZ} to D_{P} and
an interfacial zone
thickness of 20 µm [3,4,5]. The mortar
is considered to be a model mortar in that only
four different size aggregates were utilized in generating its structure.
However, the computational techniques described in [3] would be
equally applicable to a real mortar specimen, although the computational
times will increase in proportion to the number of particles in the
simulation.

Mechanical Property Example: Drying Shrinkage

The problem of drying shrinkage in concrete is much more complex from a modelling standpoint than that of ion diffusion under saturated conditions. Because much of the drying shrinkage is induced by capillary forces in the pore water [23,24], one must be able to calculate the locations of water in a microstructure at a given relative humidity, as well as the response of this microstructure to the stresses induced by the capillary tension in the water and changes in surface energy of the adsorbed water layer [25]. A computer algorithm has been developed to simulate the adsorption/desorption of water in a two-dimensional [26] or a three-dimensional [7] porous structure based on a geometrical analysis according to the Kelvin-Laplace equation, which relates a critical pore size for capillary condensation to a partial pressure (relative humidity):

.

Using this equation, one can study the effects of a change in
relative humidity, temperature, or liquid surface tension on
the critical pore radius, **r**, below which capillary condensation
will occur. For a fixed set of these parameters, a geometrical
algorithm can be utilized to determine the locations of all liquid
water in a given microstructure [7] if the drying/wetting
history of the sample is known.

Once the locations of all "liquid" water in a microstructure have been determined, capillary-tension-induced stresses (ψ = 2 γ / r) can be located at all solid-liquid interfaces. Then, finite element techniques can be used to compute the drying shrinkage which would be caused by these capillary forces. As shown in Fig. 3, once again, a hierarchical computation can be employed to relate the processes occurring in the C-S-H gel pores to the actual shrinkage measured on a concrete structure. At the concrete level, preliminary two-dimensional investigations have been conducted on the role of the interfacial zone in influencing elastic and shrinkage properties [6]. Depending on its local environment, at this level, one may also need to consider the distributions of temperature and moisture within a concrete structure.

**Figure 3:** Approach to computing drying shrinkage using
structural models at each

of the three scale levels shown in Fig. 1.

** CONCLUSIONS**

The use of a set of multi-scale microstructural models to investigate transport and mechanical properties of concrete has been demonstrated. By relating microstructure to properties at all relevant size scales, a more complete understanding of the influence of microstructure and the underlying physical processes on the performance of these composite materials can be obtained. For the case of chloride diffusivity, a complete set of results has been presented. For the more complicated phenomenon of drying shrinkage, much work is still needed, but the basic conceptual groundwork for modelling has been developed. As the structural modelling and computational capabilities described herein continue to evolve, their use as a tool in the design of a concrete with desired properties and service life should become a reality.

Symbols

α - degree of hydration

γ - liquid surface tension

φ - porosity

ψ - stress

σ - conductivity

**c,d** - constants in equation to estimate mortar relative diffusivity

**D** - diffusivity

**H(x)** - heaviside function = (1: x **>**0; 0 otherwise)

**P** - pressure

**r** - pore radius

**R** - universal gas constant

**T** - absolute temperature

**V** - volume fraction

**w/c** - water to cement ratio

Subscripts

**Agg** - aggregate

**0** - bulk solution

**ITZ** - interfacial zone cement paste

**m** - molar

**M**- mortar

**P** - bulk cement paste

** ACKNOWLEDGMENTS**

The authors would like to thank the National Science Foundation Science
and Technology Center for Advanced Cement-Based Materials for partial
financial support of much of the work described in this paper.

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