# Chapter 7. Transport and elastic properties of mortar and concrete

This chapter covers the transport properties of mortar and concrete, starting with the basic hard core soft shell microstructure model, and then building up to the full multi-scale diffusivity model. It builds on the base set up in Chapter 5, that of cement paste transport properties.

As this image shows, computing the transport through the cement paste surrounding the aggregates is a difficult problem. In the image, the model aggregate have been rendered transparent, so that only the cement paste matrix is seen.

Using myopic random walkers to solve for the diffusivity of hard core soft shell models of mortar and concrete is covered in the following two sections. The first is more directly oriented towards mortar and concrete, and presents results for several models of mortar. The second gives more mathematical details, looks at the development of two effective medium theories, differential and self-consistent, and applies them to numerical results.

This paper describes how the multi-scale model of concrete diffusivity is implemented, combining models from the micrometer scale (cement paste) and the millimeter scale (concrete) to give an overall prediction for the diffusivity of concrete based on mix design and degree of hydration. It also includes the results of a statistical designed experiment, to see the variation of results expected when the parameters of the problem (kinds of concrete, kinds of curing) are varied. An empirical equation is fit to the main variables, determined from analysis of the designed experiment.

Several of the key numerical steps, as previously described for the multi-scale model, can be replaced by analytical formulas. This should make it easier for the model to be used by concrete experimentalists and mixture designers. This work is described in this section of the chapter.

A set of experiments was conducted on mortars, measuring their conductivity as a function of sand content. This enabled comparison with the theory of section (4a), and gave insight into the transport properties of the ITZ.

The differential effective medium theory described in the previous work (4a) has been improved, eliminating arbitrary parameters and increasing the accuracy of the prediction of concrete diffusivity.

Experimental data that is needed to check the multi-scale theory must include degree of hydration measurements, which is rarely seen in the literature. That is the reason that most data sets in the literature are inadequate for validating the multi-scale theory. However, some data obtained on mortars, where the degree of hydration was simultaneously measured, does exist, and is discussed in this section.

This section analyzes the above data and systematically compares it to the predictions of the multi-scale theory, with generally good agreement.

Short overall introduction to the multi-scale theory, some general application to ideas of shrinkage in concrete.

This section describes in detail how to exactly solve for the dilute limit, when a spherical aggregate is surrounded by a gradient of porosity and therefore properties, for the case of diffusion/electrical conductivity. This calculation is then applied to test one of the hypotheses of the multi-scale model, that of using a flat aggregate to compute the ratio between the bulk and the interfacial zone diffusivity.

This section describes in detail how to exactly solve for the dilute limit, when a spherical aggregate is surrounded by a gradient of porosity and therefore properties, for the case of diffusion/electrical conductivity, linear elasticity, and thermal expansion. It also briefly describes the Lu-Torquato formalism for computing total interfacial transition zone volumes.

This section demonstrates how better understanding of transport processes, on a fundamental level, is absolutely necessary to properly understand transport in cement paste, mortar, and concrete.

The first paper describes how careful impedance spectroscopy measurements can be used to quantitatively evaluate the rapid chloride test. Accurate, physically-based transport measurements are vital in order to use concrete transport properties to help predict concrete service life. It is shown that the rapid chloride test can be used to measure concrete resistivity only, using only a few seconds of current, instead of the full six hour test.

This next paper describes how, by properly applying and solving the electrochemical equations of ionic transport in concentrated electrolytes, which is what concrete pore fluid really is, the formation factors of simple porous materials (porous ceramics) can be rationally understood. The only important variables that need to be known about the porous media, in addition to the ionic content of the pore fluid, is the formation factor of the pore space and the porosity.

The following paper shows how ionic changes in the pore fluid can be properly understood by solving the correct electrochemical ionic transport equations for the pore fluid.

A proposed method for estimating the electrical conductivity of cement paste pore solution at 25 ºC is based on the concentrations of OH, K + and Na+. The approach uses an equation that is a function of the solution ionic strength, and requires a single coefficient for each ionic species. To test the method, the conductivity of solutions containing mixtures of potassium hydroxide and sodium hydroxide with molar ratios of 4:1, 2:1 and 1:1, and having ionic strengths varying from 0.15 to 2.00 mol/l were measured in the laboratory and compared to predicted values. The proposed equation predicts the conductivity of the solutions to within 8% over the concentration range investigated. By comparison, the dilute electrolyte assumption that conductivity is linearly proportional to concentration is in error by 36% at 1 mol/l and in error by 55% at 2 mol/l. The significance and utility of the proposed equation is discussed in the context of predicting ionic transport in cement-based systems.

A description of ionic transport in unsaturated porous materials due to gradients in the electro-chemical potential and the moisture content is developed by averaging the relevant microscopic transport equations over a representative volume element. The complete set of equations consists of time-dependent equations for both the concentration of ionic species within the pore solution and the moisture content within the pore space. The electrostatic interactions are assumed to occur instantaneously, and the resulting electrical potential satisfies Poisson's equation. Using the homogenization technique, moisture transport due to both the liquid and vapor phases is shown to obey Richards' equation, and a precise definition of the moisture content is found. The final transport equations contain transport coefficients that can be unambiguously related to experimental quantities. The approach has the advantage of making the distinction between microscopic and bulk quantities explicit.

This section develops a simple 2-D model of a mortar, that gives some insight into how a different elastic modulus and shrinkage characteristics in the interfacial transition zone could affect the overall elastic and shrinkage properties of a mortar. For 3-D, an analytical exact results for the dilute limit is given. Qualitative comparisons are made to experimental data, where possible.

This section describes how a series of conventional and high-performance concrete mixtures, with and without silica fume additions, were characterized with respect to their heat signature. The measured responses were compared with the NIST cement hydration model.

This section describes computer modelling and experimental studies of the influence of incorporating conducting fibers into a cement paste matrix on the impedance response of the composite.

This section describes further computer modelling and experimental studies of the influence of incorporating short conducting fibers into a weakly conducting matrix on the impedance response of the composite.

This section contains a review of the current state (as of 1999) of the computational modelling of the microstructure and effect on transport of the interfacial transition zone (ITZ) in concrete.

### Go to Chapter 8. Curing and Autogeneous shrinkage of concrete Go back to Chapter 6. Mortar and concrete microstructure

References
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