Calculation of ternary diagrams by mass balance calculations in MS Excel

The phase assemblages that form from the hydration of Portland cement over time, or from changes in composition, can be predicted by thermodynamic modelling or simply by mass balance calculations with some knowledge of the relative stability of the hydrate phases. Thermodynamic modelling can deal with the complexity of cementitious systems and is well suited to predicting the effect of changes in single or several components at once. To successfully apply thermodynamic modelling, some knowledge of geochemistry (e.g. which phases can reasonably precipitate or dissolve under the conditions of equilibrium) as well as practical knowledge of the modelling software being used is needed.

For a general overview of the effect of different chemical compositions, ternary phase diagrams provide an effective means of graphical representation of the thermodynamic predictions. Ternary diagrams follow the same laws as thermodynamic modelling, including of course the phase rule; P+F=C+2, where P is the number of phases, F is the number of degrees of freedom, and C is the number of components. By fixing temperature and pressure it can be condensed to P+F=C. An introduction to how these ternary diagrams are constructed and illustrations of their usefulness in predicting the phase assemblage of hydrated cements can be found in the chapter ternary diagrams by Herfort and Lothenbach in the recently published book  “A Practical Guide to Microstructural Analysis of Cementitious Materials”.


An Excel file for performing the phase assemblage calculation and automatically plotting the composition onto the two sub-ternary diagrams is available here. Typical compositions for Portland cement, limestone, fly ash, silica fume and slag are used, but the user is free to enter any composition of interest within the 8 component system, and replace the fly ash with other alumino-silicate pozzolans. Gypsum can also be used to test the effect of increased sulfates. There are two different versions of the excel sheet available, one corresponding to ambient conditions. To include the formation of thaumasite, which is especially important at lower temperatures, a second set of ternary plots was constructed.


The compositions of the hydrate phases used to construct the diagrams have been taken from the literature. The boundary compositions for C-A-S-H (C1.75SH4, C1.75A0.05SH4, C1.3A0.1SH3, C0.67A0.05SH2, C0.67SH2) have been chosen based on the range of data determined for Portland cements (Ca/Si = 1.7-2.0; Al/Si = 0.05-0.1), blended cements (Ca/Si = 1-1.5, Al/Si = 0.03-0.2) and alkali activated slags (Ca/Si = 0.7-1; Al/Si = 0.1-0.3)  and the water content of C-S-H in hydrating Portland cements. The invariant C-A-S-H compositions can be modified as new data becomes available, and of course, these diagrams provide a framework to design experiments for this purpose.

Dr. Barbara Lothenbach

Dr. Barbara Lothenbach
Senior Researcher / Projektleiterin / Adjunct Prof. NTNU

Phone: +41 58 765 4788