Korea-Australia Rheology Journal, 30(3), 189-197 (August 2018) www.springer.com/13367 DOI: 10.1007/s13367-018-0019-5 Rheological analysis of particle aggregation in a colloidal suspension of carbon black particles Noori Kim and Sangkyun Koo* Department of Chemical Engineering and Materials Science, Sangmyung University, Seoul 03016, Republic of Korea (Received May 8, 2018; final revision received July 3, 2018; accepted July 10, 2018) We examine scaling theories to estimate microstructural parameters of fractal aggregates in a colloidal sus- pension. The scaling theories are based on fractal theories and rheological properties of the colloidal sus- pension. Rheological measurement in oscillatory and steady shear modes is performed for colloidal suspensions of 56 nm carbon black particles in Newtonian ethylene glycol at the particle volume fractions φ ranging from 0.005 to 0.05. Elastic modulus G’ of the colloidal suspension at φ = 0.02-0.05 in the state of colloidal gel is used to estimate fractal dimension df of the aggregates. Steady-shear measurement gives yield stress τy as a function of φ. Shear dependence of the aggregate radius R is given by a power-law scal- −m ing, i.e., R∝S , where S is the shear rate. The power-law exponent m is estimated from df and a scaling relation between τy and φ. The estimation gives df = 2.14 and m = 0.33. The parameters df and m which can be determined by either direct measurement or theoretical calculation are used to establish a microrheo- logical model for predicting shear viscosity of aggregated suspension as a function of φ and S. Both the con- centration dependence and the shear dependence of aggregates are combined to obtain an expression for the shear viscosity. Hydrodynamic interaction effect among the aggregates is roughly considered in calculating average shear stress on the aggregate. It is found that this consideration critically contributes to behavior similarity with experimental result. It is shown that the predictions by the model reasonably agree with the experimental result. Keywords: colloidal aggregate, fractal dimension, elastic modulus, yield stress, intrinsic viscosity, effective- medium approximation 1. Introduction centration. One is low particle concentrations at which the interactions among the aggregates are negligible. Intrinsic Understanding of theoretical relation between aggregate viscosity due to aggregates is a useful measure to roughly structure and physical properties of colloidal suspensions estimate hydrodynamic size of aggregates (Smith and Bruce, has long been a central issue in colloidal science due to its 1979). Its scaling with the shear rate gives estimates for a significance in academic as well as industrial applications. combined structural parameters of fractal dimension and Since introduction of fractal concept due to self-similarity shear dependency of aggregate size (Lee and Koo, 2014). of the aggregate structure, theoretical studies on colloidal The other is the state of the colloidal gel at which the aggregation have become more popular. According to the aggregates become inter-connected to form a network as fractal concept, the mass of an aggregate scales as the the particle concentration increases. In this case, studies number of particles in the aggregate and further it scales on the scaling relations between structural parameters of with the aggregate size in a power-law form. The power- fractal aggregates and solid-like properties such as elastic law exponent is defined as the fractal dimension df which modulus and yield stress are noteworthy. Shih et al. (1990) is also related with aggregation rate and shape of aggre- have proposed a useful theory to predict the structural gate as well as dimensionality of the aggregate. parameters of the fractal aggregates using the elastic prop- The structural parameters of aggregate can be described erties. They distinguished between two regimes for col- by scaling laws including the rheological properties due to loidal gels depending on relative magnitude of inter- interactive relation between macroscopic properties and aggregate (floc) link to intra-aggregate (floc) link. When microscopic structure of a colloidal suspension. Numerous the inter-aggregate link is stronger, it is called strong-link studies have been carried out to find the relations between regime and elastic property of colloidal suspension is gov- particle aggregation and rheological properties of the col- erned by intra-aggregate link. In the weak-link regime, loidal suspension. The studies have been recently summa- intra-aggregate link is stronger and inter-aggregate link rized by Mewis and Wagner (2012). The particle aggregation determines the elastic property. The elastic property is rep- is often treated divided into two cases of the particle con- resented by the elastic modulus at the plateau region and the strain at the upper linear limit of the plateau. Concen- *Corresponding author; E-mail: [email protected] tration dependence of the plateau elastic modulus and the © 2018 The Korean Society of Rheology and Springer pISSN 1226-119X eISSN 2093-7660 189 Noori Kim and Sangkyun Koo linear limit of strain provides a useful scaling relation for quantitative characterization of aggregate but also in rhe- fractal dimensions of aggregate and backbone chain of ological modeling of the aggregated suspensions. Once particles in the aggregate. these two parameters are determined by either experimen- Yield stress also represents solid-like behavior of gelled tal measurement or theoretical calculation, it allows us to colloidal suspension, which is related with interparticle predict shear viscosity of aggregated colloidal suspension attraction. There have been some investigations on the as a function of the particle volume fraction and the scaling relations between the yield stress as a function of imposed shear rate. The prediction of the shear viscosity the particle concentration and the aggregate parameters is based on an expression for concentration dependence of including a combined parameter of fractal dimension and suspension viscosity such as Quemada’s equation (Que- a power-law exponent representing shear dependence of mada, 1977). In aggregated colloidal systems, the particle aggregate size (Buscall et al., 1988; Snabre and Mills, volume fraction is replaced by the volume fraction of the − 1996). The power-law exponent m is given by R∝S m where aggregates in the suspension which can be incorporated R is the radius of gyration for an aggregate and S is the with shear dependence of aggregate size. This incorpora- shear rate. It is known that the plateau elastic modulus and tion is led to an expression for shear viscosity of aggre- the yield stress scale as the particle volume fraction in a gated colloidal suspensions as a function of the particle power-law form (van der Aerschot and Mewis, 1992; Piau volume fraction and the shear rate. et al., 1999; Potanin, 1992). In this study we measured In this study we first consider scaling theories to esti- steady-shear yield stress as well as the elastic modulus. mate the structure parameters of df and m. The scaling the- Scaling behavior of the yield stress is compared with that ories relate the structure parameters and the rheological for the plateau elastic modulus. properties of colloidal suspension. Experimental measure- Since the colloidal suspensions in industrial processes ment of the rheological properties is carried out for a col- are handled under shear action, the parameter m is signif- loidal suspension comprising of carbon black particles and icant in predicting variation of aggregate size in the pro- ethylene glycol. The carbon black is of 56 nm in average cesses. The parameter m can be determined from experi- diameter. The volume fraction of the particles ranges from mental measurement using optical technique or numerical 0.005 to 0.05 which covers two concentration zones, i.e., simulation. Sonntag and Russel (1986) obtained the below and above colloidal gelation point. In the state of parameter m using optical measurement for the suspension colloidal gel, plateau elastic modulus and linear limit of of aggregated polystyrene particles in glycerol and water strain of the colloidal suspension are measured in oscil- under shear. The parameter m was found to be 0.35 for df latory shear mode. These properties are used to obtain the = 2.1. Potanin (1993) numerically calculated m for two fractal dimensions of aggregate and backbone chain in the types of aggregates which are so-called irreversible, rigid aggregate. From steady-shear measurement, we obtain aggregate and reversible, soft aggregate, respectively. The shear yield stress by extrapolating shear stress-shear rate value for m was 0.23-0.29 for the rigid aggregate and 0.4- data. The shear yield stress is utilized to estimate the 0.5 for the soft aggregate. These values were found to be parameter m(3-df). The scaling behavior of the yield stress quite close to 1/3 and 1/2 each predicted by scaling theory. is compared with that of the plateau elastic modulus. At Recent studies on the variation of aggregate structure in low particle concentrations below gelation point, we obtain shear flow have employed numerical simulation to include intrinsic viscosity from viscosity-concentration data. Scal- calculation of hydrodynamic interaction in multiparticle ing behavior of the intrinsic viscosity with the shear rate systems. Eggersdorfer et al. (2010) calculated m for col- gives the parameter m(3-df) which is equivalent to that