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E.J. Garboczi and D.P. Bentz
National Institute of Standards and Technology
Building Materials Division, 226/B350
Gaithersburg, Maryland 20899
The interfacial transition zone (ITZ) that exists in the cement paste near sand particles and rocks in concrete require concrete to be considered as a (at least) three-phase composite: (1) bulk cement paste, (2) ITZ cement paste, and (3) rock and sand, collectively called aggregates. In reality, the ITZ contains a gradient of properties, but can be approximated as a single property shell with some finite thickness. This paper discusses how this gradient of properties can be quantitatively mapped into a single property shell, for the case of ionic diffusivity (electrical or thermal conduction), using an exact analysis of the dilute composite limit, in which the volume fraction of (spherical) aggregates is small.
It is now well-established experimentally that interfacial transition zones (ITZ's) exist around aggregate (rock, sand) particles in concrete. This is mainly because the cement paste matrix is itself particulate. When the cement grains encounter the "wall" of the aggregate, a region of higher porosity near the aggregate surface will appear, due to the "packing" constraints imposed by the aggregate surface [1,2]. Because the average aggregate diameter is much larger than the average cement grain diameter, the aggregates on average will appear locally flat to the cement grains, so the ITZ thickness will depend on the median size of the cement grains, and not on the aggregate size . The median diameter of most cements in common use is around 10-30 micrometers, so this is typically the kind of width one finds associated with ITZ's.
This kind of restrained placement of cement around aggregates results in a gradient of porosity, and therefore a gradient of properties, around each aggregate. The high volume fraction of aggregates in a typical concrete (60-75%) means that the spacing between adjacent aggregates is only a few times the typical ITZ thickness. This fact implies that the cement paste in the ITZ's can have a significant volume fraction and can be percolated  and therefore can have a significant effect on properties. Of particular interest are elastic moduli, compressive strength, chloride and sulfate diffusivity, electrical and thermal conductivity, shrinkage, and creep. This paper is restricted to the property of ionic diffusivity, which mathematically is the same as thermal or electrical conductivity, in the following sense. If a piece of concrete is subjected to a gradient in termperature or ionic concentration or elecrical potential, with a known distribution of thermal conductivity, diffusivity, or electrical conductivity, then the equation that needs to be solved that determines the effective property of the concrete is Laplace's equation, which is the same for all three properties [5,6]. Diffusivity and conductivity language will be used throughout the rest of this paper.
If concrete were only a three-phase composite, consisting of aggregates, ITZ's, and bulk cement paste, all with known and uniform properties, then the question of determining the effective diffusivity of the concrete has been solved, at least for random collections of spherical aggregates . Effective medium theory has also been shown to adequately describe the effective diffusivity .
However, this is not the case. Concrete is an "interactive composite," in the following sense. Suppose a concrete is made by mixing an 0.5 w/c cement paste with enough aggregates so that the final volume fraction of aggregates is 60%. If the ITZ is 20 micrometers thick, then all cement paste within this distance from an aggregate will have a higher porosity and therefore a higher w/c ratio. But the only way this can happen is for the cement paste outside the ITZ region to have a lower porosity and therefore a lower w/c ratio. If the ITZ volume fraction is small, then this effect is small, which would be the case for a small volume fraction of aggregate and/or aggregate with a small surface area. However, as the volume and surface area of aggregate increases, the ITZ volume fraction also increases, and this effect becomes quite appreciable. So for this hypothetical concrete, it would be incorrect to treat an 0.5 w/c ratio cement paste as the matrix or bulk phase for the concrete. The actual bulk paste, where "bulk" means that paste outside the ITZ region, would have a lower w/c ratio. Therefore, the volume and surface area of the aggregates will influence the properties and amounts of the cement paste phases.
A multiscale modelling approach has been developed to approximately take this effect into account, by combining models to actually compute this redistribution of w/c ratio that takes place . First the median cement particle diameter, according to the cement PSD, is used to establish the interfacial zone thickness, tITZ . The first key step is then to place aggregate particles, following the aggregate PSD of interest, into the concrete volume. Systematic point sampling is then used to determine the volume fractions of ITZ (VITZ) and bulk (Vbulk) paste. The approximation of the interfacial zone as a fixed width region, rather than a region with a gradient of properties, is used at this point.
The second key step is to use a cement hydration model  to choose the thickness of a single flat aggregate particle so as to match the just determined ratio of VITZ / Vbulk for the chosen cement PSD in the cement paste model volume. Since the thickness of the interfacial region is fixed, and the overall size of the cement hydration model is chosen and held fixed, the amount of space the aggregate takes up can be then used to adjust how much of the volume is taken up by bulk cement paste. Figure 1 shows how this is accomplished.
Figure 1:Showing a flat aggregate system used in step 2) of the multi-scale model.
Figure 1:Showing a flat aggregate system used in step 2) of the multi-scale model.
Cement particles are placed into this computational volume to achieve the desired total w/c ratio. Of course, the w/c ratios in the two ) regions will be different, because of the wall effect of the aggregate surface. The hydration model is then executed to achieve a chosen degree of hydration. The porosity is measured as a function of distance from the aggregate surface and converted to relative diffusivity values using Eq. (1) 
|D = 0.001 + 0.07φ2 + 1.8H(φ − 0.18)(φ − 0.18)2||(1)|
where Do is the free diffusivity of the ion of interest in bulk water, D is the diffisivity of the ion as measured in the cement paste, φ is the capillary porosity, and H(x) is one for x > 0, and zero otherwise. The constant term represents a limiting rate of diffusion through gel pores, the term with H is a percolation term, which assumes the capillary porosity percolation point is 18% volume fraction, and the second term of eq. (1) comes from fitting between these two limiting behaviors.
In past formulations , these diffusivity values have been averaged in two subsets, those lying within a distance tITZ of the aggregate and those in the "bulk'' paste. Averaging in this way assumes that the diffusive flow in the two phases is locally parallel to the aggregate surface, and so each layer can be added up in this way. This paper explores different ways of performing this average, using an exact solution of a single spherical aggregate with a gradient of properties around it.
The third key step is to use the ratio of these two diffusivities, DITZ / Dbulk, as an input back into the original concrete model, where random walk (myopic ant) numerical techniques are employed to estimate the diffusivity Dconc / Dbulk of the overall concrete system, which consists of aggregates with a diffusivity of 0, bulk paste with a diffusivity of 1, and ITZ's with a diffusivity of DITZ / Dbulk [7,8]. This value can be converted into an absolute chloride ion diffusivity for the concrete, Dconc, by multiplying Dconc / Dbulk by Dbulk / Do, the bulk cement paste diffusivity determined from eq. (1), and then by Do, the free diffusion coefficient of chloride ions in water.
A previous paper  has focused on steps 1) and 3) above, using analytical approximations to replace the necessary supercomputer analysis. This paper focuses on step 2), and in particular the method used to determine the value of DITZ / Dbulk.
The dilute limit of a composite occurs when the second phase, usually called the inclusion phase, is present at a very small volume fraction, so that the effect of each of the inclusion phase particles can be treated independently, without any contribution from neighboring particles. In this limit, the overall diffusivity, D, normalized by the bulk diffusivity in the absence of inclusions, Dbulk, is given by
If the case can be worked out where the spherical aggregate is surrounded by N shells of general thickness and diffusivity, then any type of gradient of properties can be handled, simply by using as many shells as is necessary to mimic the gradient function. The following derivation makes use of an idea originally developed for the equivalent elastic problem, that of a transfer matrix approach . The bulk modulus  and the Stokes friction and intrinsic viscosity  have also been found in the case where the gradient of properties takes on a specific power law form.
Figure 2: An N=6 layered inclusion, where the first layer is the black aggregate
Figure 2 shows a version of the problem to be considered, where N = 6 and the inner black sphere, which represents the aggregate, is counted as number 1. Then the radius of the aggregate is R1, the radius of the first shell is R2, and so on, with the radius of the last shell being RN, or in this case, R6. We will use conductivity language to do the derivation, with the understanding that electric fields are equivalent to concentration gradients, and conductivities to diffusivities. We want to apply an electric field in the z direction, and work out the effective conductivity of the multi-layered inclusion embedded in the matrix. Let the conductivity of the nth shell be σn. With a uniform electric field in the z direction, the potential far away from the inclusion must be
|V = −Eo rcos(θ)||(4)|
where the usual spherical coordinates (r, θ, φ) are used. Laplace's equation for steady-state conduction must be satisfied, ∇2V = 0, with the boundary conditions that the potential and normal electrical current are continuous across each of the N boundaries. This gives 2N conditions that must be satisfied. The potential in the nth phase can be shown to be of the form
At r=Rn, the boundary between the nth and the (n+1)th phase, the two boundary conditions can be expressed as:
To get the slope m, one must still derive the effective conductivity, which involves averages over the entire inclusion of electric field and current. The following equations are completely general for the overall field and current averages:
The main point of this work was to use the exact solution in eq. (14), and map an ITZ with a gradient of properties into a specified thickness, uniform property ITZ. This is done by equating the exact result for m in eq. (14) to the form in eq. (3), and then solving for DITZ / Dbulk for a value of tITZ that was determined from the cement particle size distribution. This step then allows the computation of Dconc / Dbulk in step 3) of the multi- scale model .
We used three porosity gradients generated in a previous paper , using cement grains packed against a flat aggregate. The flat aggregate simulates a large aggregate particle. The smallest aggregate size that we used in the concrete model had a radius of 75 micrometers, which is still much larger that the average cement particle size we used. The three cement pastes used were: a) w/c=0.3, 50% degree of hydration, tITZ = 10 micrometers, b) w/c=0.6, 50% degree of hydration, tITZ = 30 micrometers, and c) w/c=0.3, 70% degree of hydration, tITZ = 30 micrometers. The porosity gradient in each case was converted into a diffusivity gradient using eq. (1) and "put around" a spherical aggregate of radius r in micrometers. The exact value of m was computed for this gradient and the result equated to eq. (3), with the chosen value of tITZ and the given value of r. The value of DITZ / Dbulk was then solved for in a simple linear equation. Figure 3 shows the results for the value of DITZ / Dbulk plotted against the aggregate radius r for system a). The results for the other systems were similar. Figure 3 indicates how for large aggregates, the results go to the parallel result, shown as the dot-dashed line, which is independent of r. The dashed line shows the result of first doing a proper radial average of the porosity, then determining the diffusivity ratio using eq. (1). This average becomes equal to the parallel average in the limit of large r, as it should, since the curvature K=2/r of the spherical aggregates is becoming closer to that of a flat surface, K=0. It is clear that the two ways of mapping the diffusivities in the gradient to a simple ratio for a uniform thickness ITZ differ for smaller radii aggregates, but by radii greater than about 200 micrometers or so, the results only differ by about 10% or less. This was also true for systems b) and c).
Figure 3: Effective value of DITZ / Dbulk found by matching exact solution for
gradient of properties to single uniform ITZ result, vs. radius r of aggregate
The most important question arising from this work is how much difference will using the two different kinds of mapping make in the computed values for Dconc / Dbulk? Clearly, there is a large number of small particles, whose assigned ITZ diffusivities should probably be lower than the simple parallel average results suggests. But there is only a small volume fraction of these kinds of particles.
It would be very difficult to run the myopic ant numerical routine for the case where each ITZ had a different diffusivity, since it would be impossible to decide which diffusivity to use in cases of ITZ overlap. The effective medium theory [8,12] can take this fact easily into account, however. Since the effective medium theory has been shown to be a reasonable substitute for the supercomputer myopic ant computations , it becomes possible to incorporate the results of Fig. 3 and others like them into the multi-scale model. However, when we inserted the results shown in Fig. 3 into the effective medium theory, the computed values of Dconc / Dbulk only decreased by about 1%. This is due to the fact that the small aggregates only take up a fairly small portion of the total aggregate volume, and the diffusivity ratio of their ITZ's are only decreased by 10-20%, making the change in the overall diffusivity small.
It would seem more physically reasonable to map the ITZ diffusivity in the manner described in this paper, since in the actual concrete flow does go around the aggregates through the radially distributed porosity gradient. It is clear, though, that the approximation of simply averaging the diffusivity over the flat aggregate porosity gradient is perfectly reasonable and accurate enough for the purposes of the multi-scale model. If the model is extended to other concretes, especially ones with a large amount of small aggregates, it would be good to use the full analysis described in this paper in order to generate new results.
Other properties, like elastic moduli, can probably also be treated in this multi-scale approach. In the elastic case, the elastic moduli are cubic, rather than quadratic, functions of the porosity . This would tend to lead to sharper gradients in elastic moduli across the ITZ. The fact that the aggregates are significantly stiffer than the cement paste makes the elastic problem more complicated than the diffusivity problem, in which the aggregate diffusivity is zero, and so probably demands that the correct gradient mapping described in this paper be used for the elastic case. The theoretical foundation necessary for this mapping has already been carried out . The ability to carry out step 3) of the multi-scale model, where the elastic moduli of the concrete model are determined, has, however, not yet been developed.
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