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Compaction of bi-dispersed granular packing: analogy with chemical thermodynamics

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Abstract

The study of particle-packing models for bi-dispersed packings is important in the field of granular materials, from both theoretical and practical perspectives. Several particle-packing models have been developed for predicting the packing density (or specific volume) of a bi-dispersed packing. Most of the currently available models are phenomenological, which predict the specific volumes of a bi-dispersed packing as a function of fraction of species, and have applied to various fields, such as in concrete, pharmaceutical, soil engineering, etc. In this study, we analyze the packing densities of granular mixtures using an analogy to the thermodynamic theory for chemical solutions. The thermodynamic theory for chemical solutions provides the connections among the bulk solution density, the chemical interaction activities between species, and the concentration of each species in the solution. Parallel to the chemical potential of each species in the solution, we introduce an “excess free volume potential” for each granular species. With the interaction activities of two species in a bi-dispersed granular system, we explain the volume compaction behavior of a granular system from a new context. Subsequently, using the second law of thermodynamics, an analytical method is proposed to quantify the excess free volume potentials and to predict the density of a granular mixture. The developed analytical method is then validated by the experimental results of bi-dispersed packing mixtures of glass beads and silica sands. The performance of the analytical method and its validity are demonstrated.

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Abbreviations

\(V\) :

Total volume of a bi-dispersed packing

\(\upsilon\) :

Specific volume of a bi-dispersed packing

\(V_{1}^{0} ,V_{2}^{0}\) :

Total volumes of each component in mono-dispersed packing state

\(\upsilon _{1}^{0}, \upsilon _{2}^{0}\) :

Specific volumes of each component in mono-dispersed packing state

\(N_{1} ,N_{2}\) :

Total number of large particles and small particles in a bi-dispersed packing

\(y_{1} ,y_{2}\) :

Solid volume fraction of each component in a bi-dispersed packing

\({\text{v}}_{1}^{g}, {\text{v}}_{2}^{g}\) :

Particle solid volume of each component in a bi-dispersed packing

\(d_{1} ,d_{2}\) :

Particle size of each component in a bi-dispersed packing

\({\text{v}}_{1}^{m}, {\text{v}}_{2}^{m}\) :

Particle volume potential of the mth particle of each component in a bi-dispersed packing

\({\text{v}}_{1}^{0} ,{\text{v}}_{2}^{0}\) :

Average particle volume potential of each component in mono-dispersed packing state

\({\text{v}}_{1} ,{\text{v}}_{2}\) :

Average particle volume potential of each component in a bi-dispersed packing

X :

Compactivity (granular temperature)

S :

Granular entropy

\(V^{\prime}, V\) :

Internal volume potential, Gibbs volume potential

\(\zeta\) :

Extent of reaction

\(\Delta V\) :

Excess free volume potential of the bi-dispersed packing

\(\Delta {\text{v}}_{1} ,\Delta {\text{v}}_{2}\) :

Excess free volume potential for each component in a bi-dispersed packing

\(\alpha _{1}, \alpha _{2}\) :

Interaction activity coefficients for each component in a bi-dispersed packing

\(r_{i}\) :

Particle size ratio

\(\eta _{1}, \eta _{2} (\eta)\) :

Material parameter for each component (averaged value)

\(\bar{x}\) :

Internal state variable

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Acknowledgements

This work was supported by the National Science Foundation of the United States under a research grant (CMMI-1917238). The support is greatly acknowledged.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Massachusetts, Amherst, MA, 01003, USA

    Ching S. Chang & Yibing Deng

  2. Department of Municipal and Ecological Engineering, Shanghai Urban Construction Vocational College, Shanghai, 200438, China

    Yibing Deng

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  1. Ching S. Chang
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Chang, C.S., Deng, Y. Compaction of bi-dispersed granular packing: analogy with chemical thermodynamics. Granular Matter 24, 58 (2022). https://doi.org/10.1007/s10035-022-01219-5

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