Next: Mathematical Analysis of
Previous: Formula for interfacial
The dilute limit of a concrete composite occurs when the aggregates are present at a very small volume fraction, so that the effect of each of the aggregates can be treated independently, without any contribution from each other. In this limit, an overall property, P, normalized by the bulk property in the absence of aggregates, Pbulk, is given by:
where c is the volume fraction of aggregates present, and < m > is a dimensionless number determined by the aggregate shape, size distribution, the ratio of P/agg / Pbulk , and the geometrical and physical properties of the ITZ. The angular brackets indicate an average over the size distribution of the aggregates. As an example, for the special case of a spherical aggregate of radius r and zero diffusivity, surrounded by an ITZ of thickness tITZ and diffusivity DITZ, embedded in a matrix of diffusivity Dbulk, the exact result for m can be found [9,10],
where the parameter α = [ (r + tITZ)/r ]3 contains all the dependence on the particle and ITZ geometry. In real concrete, the ITZ has a gradient of properties, which is treated in this paper. In the following, cases where P is either the bulk modulus, the shear modulus, the thermal expansion coefficient, or the electrical conductivity/ ionic diffusivity will be discussed.
If the problem can be worked out where the spherical aggregate is surrounded by N shells of general thickness and properties, 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 derivations make 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. The simplest case, electrical conductivity/ionic diffusivity, is discussed first.
Figure 4 shows the geometry of the problem for all the properties to be considered, where the inner 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. The multi-layered inclusion is embedded in matrix material labelled N + 1.
Figure 4: A schematic sketch of the geometry of the problem. Each shell is a spherical shell, of radius R n for the nth shell.
We will use conductivity language to do the derivation, with the understanding that electric fields are equivalent to concentration and thermal gradients, and electrical conductivities to ionic diffusivities and thermal conductivities. 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. It doesn't matter in which direction the field is applied, as the sphere and the spherical shells are isotropic. Let the conductivity of the nth shell be σn. With a uniform electric field of strength E0 in the z direction, the potential far away from the inclusion must be:
where the usual spherical coordinates (r, θ, φ) are used. Laplace's partial differential equation for steady-state conduction must be satisfied, ∇ 2 V = 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:
Two more boundary conditions are that in the aggregate, B1 = 0, since the potential cannot diverge at r = 0, and in the (N + 1)th phase, which is the matrix, AN + 1 = E0, in order to give the correct uniform field far from the aggregate. This results in only 2N unknown coefficients, which can be determined uniquely from the 2N boundary condition equations.
At r = Rn, the boundary between the nth and the (n + 1)th phase, the two boundary conditions can be expressed as:
By rearranging this equation, one can come up with the form
where Pn is a 2 x 2 matrix, given by:
By iterating eq. (12), one can then derive the equation connecting A 1 and B1 to AN + 1 and BN + 1:
Equation (14) is really two equations in four unknowns, AN + 1, BN + 1, A1, and B1, with the known coefficients H11, H12, H21, and H22. However, since AN + 1 = E0, and B1 = 0, there are really only two unknowns, so that eq. (14) may be easily solved, once the elements of the matrix H have been computed, to give:
Again iterating on eq. (12), one can then solve for any of the other coefficients desired.
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:
where I stands for the entire inclusion, aggregate plus N − 1 layers, cI is the volume fraction of the entire inclusion, and σ is the effective conductivity of the entire composite system, treating it as a uniform medium. Combining these two equations so as to eliminate the average over the N + 1 phase , and using the fact that the volume fraction of aggregate, c, is in the dilute limit given by c =cI (R1/RN )3, we obtain, with the help of eqs. (12)-(15):
so that the final result for the slope m, according to the form of eq. (6) is: