Granular Matter

, 21:103 | Cite as

Semiphenomenological model to predict hardening of solid–liquid–liquid systems by liquid bridges

  • Toru IshigamiEmail author
  • Chisato Tokishige
  • Tomonori Fukasawa
  • Kunihiro Fukui
  • Shin-ichi Kihara
Original Paper


In this study, we developed two semiphenomenological models to quantitatively predict the rheological properties of capillary suspensions. These models can be used to estimate the critical volume fraction of an additional immiscible fluid above which liquid bridges have formed between all neighboring particles, causing gelation. The models can also be used to estimate the numerical value of the yield stress. This model was derived from the mass balance between the net volume of the liquid bridge and the volume fraction of the secondary fluid assuming monodisperse particles and a cylindrical liquid bridge. The yield stress was constructed based on Rumpf’s equation. The calculation results demonstrated good agreement with the experimental data. We also used the developed model to investigate the effect of particle size on the yield stress. This model qualitatively described the experimental data and reference data, and revealed the mechanism of the particle size dependence of the yield stress. The model applies to capillary suspensions in the pendular state.


Capillary suspension Yield stress Rumpf’s equation Rheology Pendular state 



This study was supported, in part, by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 17K14852 and 19K05122, and grants from the Information Center of Particle Technology, Japan, and Foundation, Oil & Fat Industry Kaikan. We thank Dr. K. Tsurusaki and Ms. R. Takeda of the Kanagawa Institute of Industrial Science and Technology Center for their technical help and advice.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10035_2019_959_MOESM1_ESM.docx (27 kb)
Supplementary material 1 (DOCX 27 kb)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Chemical Engineering, Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan

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