Fractal dimensions of capillary force induced particle networks determined from rheology and confocal microscopy

1. Introduction: The rheological properties of a particle suspension can be substantially altered by adding a small amount of a secondary fluid that is immiscible with the bulk phase [1]. The substantial changes in the strength of these capillary suspensions arise due to the capillary forces induced by the added liquid leading to a percolating particle network. Capillary suspensions can be used for various applications, amongst them being the preparation of novel food products, improved polymer blends and precursors for highly porous glass or ceramic membranes. The capillary suspension networks are unique from other types of particulate networks due to the nature of the capillary attraction. Different three-phase contact angles of the secondary fluid towards the particle surface in bulk phase environment lead to very different microstructures. Contact angles smaller than 90° lead to pendular networks of particles connected with single capillary bridges or clusters comparable to the funicular state in wet granular matter, whereas a different clustered structure, the capillary state, can form for angles larger than 90°. Basic rheological features like the low-frequency shear moduli or the apparent yield stress of such ternary solid-liquid-liquid systems are influenced by particle interactions and interfacial properties, which should be precisely controlled for intelligent material design. The rheological properties of these systems are tightly connected to the microstructure of the particle network which can be quantified by structural parameters such as the fractal dimension. 2. Rheology: In pendular state systems, the apparent yield stress rapidly increases with small amounts of added secondary fluid, then reaches a plateau and shows a decrease due to spherical agglomeration when the secondary phase amount is increased further [2]. This behavior is demonstrated for samples using aluminum oxide particles in paraffin oil with sucrose solution as secondary fluid – a system that can be used as a highly porous ceramic membrane precursor. In the plateau region, where the particle network is fully formed, the yield stress shows a power law scaling with particle volume fraction and reciprocal particle size. A master curve for the yield stress as function of the ratio of secondary fluid to particle volume can be established independent of particle volume fraction and size. This scaling can be reproduced for a different system containing calcium carbonate particles. While a mean scaling exponent of 3.7 is used for the particle volume fraction dependence of the yield stress for the aluminum oxide system, a closer look reveals a weak dependence of this exponent on the particle size. When the mean particle radius x50,3 is increased from 0.33 to 3.10 µm, the scaling exponent also increases from 3.5 to 4.0. Using a scaling theory by P. G. de Gennes, the fractal dimension can be deduced from the yield stress as function of the solid volume fraction [3, 4]. Exponents obtained from the plateau yield stress indicate an increase in the networks fractal dimension Df from 1.8 to 2.0 with increasing particle size. The change in dimensionality hints at a transition from a diffusion limited aggregation for the small particles to a more reaction limited network formation for the larger particles. This transition may be explained by the corresponding reduction in the capillary force with increasing particle size. The modulus G0 in the plateau region is also determined for these samples, which is constant over a wide frequency range as the capillary-induced interaction forms very strong particle gels. The plateau modulus of the aluminum oxide sample also shows a power law scaling with particle volume fraction. Thus, the de Gennes scaling theory can also be applied, leading to calculated fractal dimensions of 1.8 for systems with a mean particle radius of 0.70 µm. This is ~0.1 lower than the value determined from the associated yield stress data, but is in the same range demonstrating the applicability of the de Gennes scaling for these systems in principle and is consistent with results from other capillary suspensions [5]. 3. Microscopy: Structural information of oil-based model capillary suspensions with silica microspheres is obtained in situ using confocal microscopy [6]. A two-dimensional example confocal image of a pendular state system is shown in Fig. 1, but the samples can also be imaged in 3D. Thus, it is possible to visualize the different types of percolating structures of capillary suspensions spatially. The confocal images of samples with contact angles smaller than 90° demonstrate a transition between pendular systems consisting only of binary bridges to funicular-like systems with bridges binding more than two particles. This transition can occur due to geometrical reasons when the contact angle is decreased even if the secondary fluid volume is kept constant. The capillary state structure is imaged for systems with contact angles larger than 90°, which comprises some singly-connected droplets that do not contribute to the overall network strength. Radial particle pair distribution functions are obtained by image analysis, which can capture some of these microstructural differences. Using a method described by Dinsmore et al. [7], the fractal dimension of the networks can be computed from a power-law fit to the pair distribution functions. By using different sized silica microspheres, the fractal dimensionality can be determined directly as a function of particle size. Additionally, the modulus G0 is measured for this microscopy model system, and thus both methods for determining the fractal dimension are compared. 4. Summary: The rheology of capillary suspensions strongly depends on their microstructure. The fractal dimension, an important quantitative structure parameter, is determined here by different methods. It is directly computed by image analysis of 3D confocal images and additionally deduced from yield stress and shear modulus measurements by applying appropriate scaling laws. The determined fractal dimension values give insight into structural changes in capillary suspension systems, e.g. when the particle size is varied. These methods can be used for a range of different capillary suspension systems, as shown here using a model system containing spherical glass beads as well as a more application-oriented system with non-spherical aluminum oxide particles. References: [1] E. Koos, Curr. Opin. Colloid Interface Sci., 19(6), 575-584 (2014). [2] J. Dittmann and N. Willenbacher, J. Am. Ceram. Soc., 97(12), 3787-3792 (2014). [3] P. G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1980. [4] J. M. Piau et al., J. Rheol., 43(2), 305-314, (1999). [5] T. Domenech and S. S. Velankar, Soft Matter, 11(8), 1500-1516 (2015). [6] F. Bossler and E. Koos, Langmuir, 32(6), 1489-1501, (2016). [7] A. D. Dinsmore et al., Appl. Opt., 40(24), 4152-4159 (2001).status: publishe