Mathematical modeling of cement paste microstructure by mosaic pattern. Part II. Application
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Mathematical modeling of cement paste microstructure by mosaic pattern. Part II. Application Paul D. Tennis Department of Civil Engineering, Northwestern University, Evanston, Illinois 60208
Yunping Xi Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, Colorado 80309
Hamlin M. Jennings Department of Civil Engineering and Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208 (Received 7 March 1996; accepted 13 April 1997)
A model based on mosaic pattern analysis is shown to have the potential to describe the complex shapes and spatial distribution of phases in the microstructures of multiphase materials. Several characteristics of both micrographs of portland cement pastes and images generated using the few parameters of the model are determined and, for the most part, agreement is good. The advantage is that spatial features of the microstructures can be captured by a few parameters.
I. INTRODUCTION
Standard image analysis of micrographs provides certain characteristics, such as the area fractions, characteristic size, and some shape parameters of the various phases. However, it is difficult to use this information to model the spatial distribution of phases in portland cement and, in turn, to model properties such as hydraulic conductivity, electrical impedance, or diffusivity. For example, Coverdale et al.1,2 have shown that the impedance response and Neubauer et al.3 have shown that shrinkage is a function of spatial distribution of various phases. Digital modeling of the development of microstructure of cement phases is limited by the large amounts of computer time and memory necessary to simulate the process of hydration of even small, two-dimensional systems. Furthermore, simulation models cannot characterize real material; they only simulate a model microstructure. Mosaic pattern analysis is a new technique that is able to describe complex microstructures using relatively few parameters.4 For example, a two-phase mosaic pattern, such as that shown in Fig. 1, can be described with two parameters: l, a measure related to the average size of the individual polygons in the system, and P, the probability that a given polygon is a certain phase. For a multiphase image, n independent parameters are required, where n is the number of phases. If a cement paste is simplified to consist only of anhydrous phases, a gel consisting of C–S–H and other reaction products, calcium hydroxide, and pores (as suggested by Powers and Brownyard5 ), then the microstructure may be described with only four independent parameters. J. Mater. Res., Vol. 12, No. 7, Jul 1997
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If the structure is isotropic, three-dimensional microstructures can be modeled based on information obtained from two-dimensional images, providing a new technique for correlating two- and three-dimensional information. Thus, the mosaic pattern technique will simplify th
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