Ratio of solid volume fraction to the maximum packing

The viscosity of a dilute suspension is linearly proportional to the proportion of solid particles present (ɸ), (Equation 2.1, Einstein, 1906):

ηr = ηs (1+ [k] ɸ) (Equation 2.1)

The assumptions made in Equation 2.1 are that the solid particles suspended in the liquid phase contribute to the viscosity of suspensions (Marti et al., 2005) and that the viscosity of the suspending fluid is Newtonian (Pabst, Gregorova, et al., 2006). In this equation, [k] is the intrinsic viscosity of particles in suspension (Brenner,

___________________________________________________________________

43 viscosity of the suspension (Jeffrey & Acrivos, 1976). For hard spherical particles, [k] ɸ equals to 2.5. This equation only predicts viscosities to values of ɸ/ɸmax of less than 0.6 (Figure 2.12).

Figure 2.12: The relationship between relative viscosity and the ɸ/ɸmax (adapted from Stickel & Powell, 2005) using hard polystyrene spheres and polymethyl methacrylate beads suspended in polymer solutions such as polyethylene glycol-ran-propylene glycol monobutylether.

A suspension containing hard sphere particles of uniform size (15 μm) at ɸ > 0.2, considered a concentrated suspension in which particles are close to being in contact, and the ɸ approaches ɸmax (ɸ/ɸmax ~0.6). In such suspensions, the viscosity is too great to be measured by a rheometer (Stickel & Powell, 2005).

The most recognised semi-empirical model to predict viscosity over a wide range of ɸ/ɸmax was proposed by Krieger and co-workers (Krieger & Dougherty, 1959).

ηr = (1 – (ɸ/ɸmax))-[k] ɸmax (Equation 2.2)

Equation 2.2 was developed by suspending small, rigid, spherical particles (15 nm) in a Newtonian fluid. By adding solid particles, the viscosity of the suspension

ηr

approaches infinity

Exponential phase begins

___________________________________________________________________

44 increases. The relationship between ηr and (ɸ/ɸmax) is non-linear (Krieger & Dougherty, 1959) and typical relationships are presented in Figure 2.12 where ηr approaches infinity as ɸ approaches ɸmax. At this point the rheological properties of the suspension are dominated by the interaction between the solid particles (Brown & Jaeger, 2009).

Equation 2.2 has been widely used to predict viscosity of concentrated suspensions containing solid particles of varying size and shape (Fischer, Pollard, Erni, Marti, & Padar, 2009), including suspensions of fibres and starch granules (Pabst, Berthold, & Gregorov, 2006). However, the estimation of the value [k] in the Equation 2.2 is difficult for suspensions containing non-spherical particles. In such situations, a generalised version of the Equation 2.2 is used, where [k]ɸmax = 2 (Maron & Pierce,

1956).

ηr = (1 – (ɸ/ɸmax))-2

(Equation 2.3)

To date, Equation 2.3 remains one of the most important equations used to predict the viscosity of concentrated suspensions containing particles of various sizes and shapes and it is used extensively in this thesis.

2.3.2.3. Aspect ratio of particles sizes

For a given ɸmax, the ηr of a suspension increases with the aspect ratio (R) of the suspended particles (Kitano, Kataoka, & Shirota, 1981; Pabst, Gregorova, et al., 2006), where R is the ratio of the longest to shortest axis of the particle (Brenner, 1974), spherical particles have an R of 1.

___________________________________________________________________

45 The ɸmax of suspensions containing uniform spherical particles (R = 1) varies from 34% to 74%, depending on the size distribution of the particles in suspension (Petford, 2009). However, the ɸmax of suspensions that contain particles with R > 1 increases as the R of solid particles increases (Pabst, Gregorova, et al., 2006; Wierenge & Philipse, 1996). The elongated particles rotate “end over end” with applied shear and ɸmax decreases when the R of suspended particles (radius of gyration) increases (Brenner, 1974; Mueller, Llewellin, & Mader, 2010). At very high shear rates, the alignment of elongate particles with the induced flow will reduce ηr (Boek et al., 1997; Mueller et al., 2010). Therefore, the ηr of elongated particles in suspensions such as digesta is dependent on the shear rate in the gut (Fermin & Riley, 2010; Genovese et al., 2007).

Digesta is a polydisperse suspension in which the shape of the particles is diet- dependent (Lentle & Janssen, 2008). Studies that quantified solid particles isolated from the small intestine using image analysis reported that the mean diameter of these particles are about 0.24 mm with a width of 0.06 mm, thus R values of these particles were estimated to range from 2 to 8 (Jalali, Nørgaard, Weisbjerg, & Nadeau, 2012; Krämer, Nørgaard, Lund, & Weisbjerg, 2013).

If it is assumed that the effect of the degree of polydispersity of the particles can be described by a mean value of R, then the only unknown in equation 2.3, ɸmax, can be calculated by fitting R and ɸmax into a general linear equation in the form of y = c + mx (Equation 2.4). Using reasonably homogenous fibre suspensions with R values between 6 and 27 (Kitano et al., 1981), the following relationship between R and ɸmax has been proposed (Equation 2.4):

___________________________________________________________________

46 Y = c – m X (Equation 2.4)

ɸmax = 0.54 – 0.0125 R (Equation 2.5)

However, when particles with R values ranging from 1 to 16 were suspended in a 60% sucrose solution (Pabst, Gregorova, et al., 2006), Equation 2.4 became

ɸmax= 0.51 – 0.0223 R (Equation 2.6)

Equations 2.5 and 2.6 indicate that estimation of ɸmax of a suspension is dependent on the slope (m) and the y-intercept (c) which vary with the average values of R of

solid particles in suspension. However, the determination of R of solid particles in the suspension is time consuming; hence tools such as image analysis must be used for effectively estimating R for suspensions of heterogeneous fibres such as are found in digesta. To date, little study has been carried out to relate values of R from heterogeneous distributions of plant fibres and ɸmax, and so estimate ηr for suspensions.

2.3.3. Viscosity of starch suspensions