Density and refractive index matched suspension model

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Classification based on particle concentration

Suspensions are usually classified by the particle volume fraction (ϕ), as the rheological property of suspension changes following the increase of particle concentration. For spherical particle suspension, the particle volume fraction can be expressed in the follow equation:
Where m is the particle number density, is the particle radius. With respect to the viscosity, suspensions are often classified into 3 sections:
a) When , the suspension is considered to be a dilute suspension. Such suspension can be treated as the suspending fluid without significant difference in viscosity.
b) When 5%≤ ≤25%, the suspension is considered as semi-dilute. Here the viscosity shows a higher order dependence on ϕ, but the behavior is still approximately Newtonian.
c) When , the suspension becomes dense or concentrated. Here, some specific phenomena are clearly shown. One example is the rapid growth of viscosity compared to that of other concentrations. Usually non-Newtonian effects can be found in such dense suspensions.

Classification based on dominant forces

Excluding the inertial force (which depends on shear rate ) when the suspension is moving, 3 main forces occur in a suspension:
a) Hydrodynamic force, which is the viscous force due to the relative motion of particles to the surrounding fluid,
b) Brownian force, which is the omnipresent thermal randomizing force,
c) Colloidal forces, such including excluded volume repulsion, electrostatic interaction and van der Waals force (Brader 2010). Colloidal forces are potential forces which depend on the particle size and the distance between particles (Qin et Zaman 2003).
For particle diameter ( ) smaller than 1nm, Brownian force and colloidal forces predominate. While for particles larger than ~10μm, hydrodynamic force plays the most significant part. For particles in the intermediate range (10-3μm < < 10μm), they are affected by a combination of hydrodynamic, Brownian motion, and inter-particle forces (Qin and Zaman 2003).
In addition, as shown in Fig 2.1, following the variation of the particle volume fraction and the flow shear rate, the dominant force is different. Under low shear rates and low concentrations, the Brownian effect has the biggest influence (zone A). When the shear rate increases, hydrodynamic force takes place (zone B). While the particle volume fraction increases, the colloidal forces are the most important (zone C). At very high shear rates, the inertial forces dominate (zone D). In highly concentrated suspension, inter-particles forces dominate (zone E, F, G, not involved in the current study).
Fig 2.1 Conceptual classification of the rheophysical regimes of a suspension as a function of shear rate and solid fraction on a logarithmic scale (Coussot and Ancey 1999).
To describe the dominant force inside a suspension, some non-dimensional numbers are used, such as the Reynold number and the Peclet number.
Where is the density of the suspending liquid, is the particle radius, is the flow shear rate, is the dynamic viscosity of the suspending liquid.
The Peclet number is the ratio between the hydrodynamic force and the Brownian force, defined as:
Where =1.38×10−23 J K−1 is the Boltzmann constant, is the absolute temperature. Moreover, the ratio of the Peclet number to the Reynolds number is useful for the analysis later, named the Schmidt number, it is defined as:

Other classifications

Classifications can also be based on particle shape (spherical particle suspension and non-spherical particle suspension), number of particle types (mono-dispersed suspension, bi-dispersed suspension, poly-dispersed suspension), or particle deformability (deformable particle suspension or solid particle suspension), attractive force between particles (aggregating and non-aggregating particle suspension). Suspension can be classified by suspended particle size as well. For suspension with particle sizes in the range from a few nanometers to a few microns, it is referred as colloidal suspension. Besides, for particles with <1μm, the Brownian force is noticeable (Zhou, Scales, and Boger 2001). So suspension can been classified as Brownian or non- Brownian, depending on their particle sizes (Qin and Zaman 2003).
In this work, we will focus on non-Brownian suspension, with particle diameter in the range of 100 to 102 μm suspended in a Newtonian fluid.

Suspension viscosity

The viscosity of particle suspension is influenced by many factors, such as particle volume fraction, shear rate, particle shape, particle size distribution, particle deformability etc. In general, the dynamic viscosity of the suspension ( ) is proportional to the dynamic viscosity of the suspending liquid ( ). Then, most rheological models are expressed in terms of the relative viscosity ( ), defined as:

Effect of particle volume fraction

The particle volume fraction is one of the most important factors for suspension viscosity. Here the study is begun with a simple case: suspensions of mono-dispersed hard spheres. Hard spheres are defined as rigid spherical particles, with no inter-particle forces other than infinite repulsion in contact (Genovese 2012). The viscosity of hard-sphere suspensions is affected by hydrodynamic forces, Brownian motion, and the excluded volume of the particles.
In the dilute regime, the relative viscosity of hard-sphere suspensions was first addressed theoretically by Einstein (1956). He defined the following linear dependency:
Where B is the ‘Einstein coefficient’ or ‘intrinsic viscosity’, which takes the value B=2.5 for hard spheres.
For semi-dilute suspension, Batchelor and Green(1972) extended Einstein’s equation to second order:
Where B1 = 6.2 for Brownian suspensions in any flow, and B1 = 7.6 for non-Brownian suspensions in pure straining flow (Batchelor 1977; Batchelor and Green 1972).
At higher concentrations, the distance between particles is much closer, the probability of collision increases. The resulting relative viscosity shows a significant positive deviation from the prediction of the equation (2.6). In this case, , the maximum volume fraction or maximum packing fraction of particles should be considered (Genovese 2012). When particle concentration approaches , there is no longer sufficient fluid to lubricate the relative motion of particles, jamming occurs and consequently the viscosity rises to infinity (Metzner 1985). For mono-dispersed spherical particle suspension, the theoretical value of is 0.74 (in a face centered cubic arrangement). However, there is no consensus on how to define the value of in real suspensions, even with mono-dispersed hard spheres. In general, observed values can range from 0.55 to 0.68 (Qi and Tanner 2011). Some experimental observations have shown that loose random packing is about 0.60, and the random close packing is close to 0.64 (McGEARY 1961; Qin and Zaman 2003; Quemada 2002).
Taking into account , equation (2.7) evolves to another form. One of the most accepted expressions is the semi-empirical equation of Krieger and Dougherty(1959) for mono-disperse suspensions:
Fig 2.2 Relative viscosity vs. particle volume fraction predicted by Einstein’s equation for dilute hard-sphere suspensions (Equation (2.6) with B = 2.5), and Krieger–Dougherty’s equation for concentrated hard-sphere suspensions (Equation (2.9) with ϕm = 0.6).
The product B× in equation (2.7) is often around 2 for a variety of experiments (Maron and Pierce 1956; Quemada 2002; Russel and Sperry 1994). Therefore Krieger– Dougherty’s equation is usually simplified to:
As described in Fig 2.2, at low concentrations, the relative viscosity is close to the prediction of Einstein’s equation (2.6). After a certain value of concentration the viscosity increases rapidly, following the prediction of Krieger and Dougherty’s equation (2.9). Finally, when the concentration approach , it rises to infinity.
Other factors, such as shear rate, particle shape, particle size distribution, and particle deformability can also affect the relative viscosity (and ) of suspensions (Genovese, Lozano, and Rao 2007; Zhou, Scales, and Boger 2001). For example, when the particles are bi-dispersed hard-spheres, the maximum packing fraction will deviate from the theoretical value, as the smaller particles can occupy the space between larger particles.
In this situation, an effective maximum packing fraction ϕm-eff , should replace in equation (2.9), which gives:
Equation (2.10) is a generally used to describe non hard-sphere suspensions. ϕm-eff is defined as the maximum packing fraction of non-hard sphere particle suspension. As ϕm-eff changes in each particular system, the influence of other factors can then be analyzed based on this equation.

Effect of shear rate

At low particle concentrations, the viscosity of hard-sphere suspensions is independent of shear rate (Equation (2.6)). At higher concentrations, the effect of shear rate is noticeable. Usually, the behaviors of the suspension relative viscosities can be separated into 4 domains depending on the shear rate: 1) at very low shear rates, they behave like a Newtonian fluid, with a constant zero-shear viscosity; 2) at intermediate shear rates they show shear-thinning effect; 3) at high shear rates the viscosity attains a limiting and constant value, and 4) after a certain limit of shear rate ( ), the suspension is estimated to be shear-thickening (Barnes 1989; Stickel and Powell 2005). The behavior beyond the shear-thickening region is not clear, but some studies indicate shear-thinning will appear again (Barnes, Hutton, and Walters 1989; Hoffman 1972). Based on the above description, Fig 2.3 shows the estimated variations of relative viscosity with respect to shear rate. At low particle concentrations, they are Newtonian. Following the increase of particle concentration, the effect of shear-thinning and shear-thickening appear and become more apparent. The behaviors after shear-thickening are represented by dashed line, as they are not clear yet.
Fig 2.3 Representation of relative viscosity versus shear rate for a fluid suspension (Stickel and Powell 2005).
Shear-thinning is a common case for suspension, which is linked to the alignment of suspended particle following the direction of the flow (Fig 2.4). While shear-thinning followed by shear-thickening behavior is not completely confirmed for all the suspensions, though it has been observed in highly concentrated suspensions (ϕ > 0.4– 0.5) (Barnes 1989; D’Haene and Mewis 1994; Picano et al. 2013; Brown and Jaeger 2014; Wyart and Cates 2014; Cheng et al. 2011). One example is shown in figure (Fig 2.5), for suspension with 1.25μm particles at a concentration varying from 47% to 57%, after the shear rate reaches a certain critical value ( ), the viscosity begins to increase again (Hoffman 1992). Shear-thickening can be divided into two categories: discontinuous shear-thickening and continuous shear thickening (Brown and Jaeger 2014). Continuous shear-thickening is that the re-augmentation of viscosity is mild (perhaps up to several tens of percent over the few decades of shear rates). Discontinuous shear-thickening means that the viscosity increases abruptly after a certain shear rate, for example, in Fig 2.5, the viscosities of the suspensions at ϕ =57% and 51% show a discontinuity after a certain shear rate respectively.
Barnes (1989) reviewed shear-thickening behavior on non-aggregating solid particle suspensions. He summarized that shear-thickening is affected by particle volume fraction, particle size, particle size distribution, particle shape, and inter-particle interactions. He inferred that decreased with increasing values of ϕ, and that increased rapidly at ϕ 0.5 (Fig 2.6). It is therefore experimentally more and more difficult to attain when ϕ decreases below 50%. But not finding does not mean it does not exist, as he wrote “so many kinds of suspensions show shear-thickening that one is soon forced to the conclusion that given the right circumstances, all suspensions of solid particles will show the phenomenon.”
As shear-thickening has been observed experimentally to become important at Re≥10-3, (Barnes 1989; Hoffman 1972), based on the Reynold number (Equation (2.2)) and the Peclet number (Equation (2.3)), Stickel and Powell (2005) proposed a dimensional criterion to characterize the behavior of suspension rheological behavior. Depending on the values of Re and Pe, they classified the rheological behavior of suspension into 4 regions: Shear-thinning, Newtonian, shear-thickening and an unknown region (Fig 2.7). They estimated that following the increase of shear rate, the behavior of suspension changes from shear-thinning to Newtonian, and finally to shear-thickening. In addition, they thought that a suspension might be expected to behave as a Newtonian fluid for greater ranges of shear rate, as particle size and fluid viscosity increased, such that Sc (Equation (2.4)) is far bigger than unity.
Recently, shear-thickening phenomena have caught more attention (Brown and Jaeger 2014; Mari et al. 2014; Wyart and Cates 2014). Picano et al. (2013) thought shear-thickening can be due to the augmentation of exclude volume of particles. Fluid inertia causes strong microstructure anisotropy that result in the formation of a shadow region with no relative flux of particles. As shown in Fig 2.8, 2a is the least distance between two particles of radius a, the region with vanishing probability to find another particle in relative motion increases at higher Reynolds numbers. This regions act as an increase of the effective volume of particles: the geometrical volume occupied by the particles plus the volume of the region, thus leading to the augmentation of the viscosity (Picano et al. 2013; Fornari et al. 2016).
Brown and Jaeger (2014) summarized that there are 3 mechanisms for shear-thickening:
1) Hydro-clustering formation of particles (N. J. Wagner and Brady 2009). Above a critical shear rate, particles stick together transiently by the lubrication forces and can grow into larger clusters (Fig 2.9). The large clusters result in a larger relative viscosity.
2) Order-disorder transition (Hoffman 1974). Following the increase of shear rate, the arrangement of particles changes from ordered layers to a disordered state, thus the viscosity increases.
3) Particle dilatancy (Brown and Jaeger 2012). When particles are sheared, they try to go around each other, but often cannot approach directly, so the packing volume of particles increases (dilates).
In addition, they found that discontinuous shear-thickening may be related to the jamming effect of suspensions.

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Other effects

Yield stress has mostly been observed at high concentrations (ϕ > 0.5) and at low shear rates (Dabak and Yucel 1987; Dzuy and Boger 1983; Heymann, Peukert, and Aksel 2002; Hoffman 1992; Jomha et al. 1991; Zhu and Kee 2002). Although the concept of yield stress and its experimental measurement has been a matter of debates (Barnes 1999; Heymann, Peukert, and Aksel 2002; Moller et al. 2009; Nguyen and Boger 1992), most works acknowledged the existence of yield stress in fluids. One focal point of the argument is how to clarify and measure the yield stress experimentally. An apparent yield stress can be observed in a suspension that means the viscosity tends toward infinity at very small shear rates, or there is a finite shear stress without deformation over long experimental time scales (Moller et al. 2009).

Effect of size distribution

As demonstrated in many experimental or simulation studies of bi-dispersed or poly-dispersed suspensions, at the same particle packing fraction, the bi-dispersed or poly-dispersed suspension has a lower viscosity (Chingyi Chang 1994; D’Haene and Mewis 1994; Qi and Tanner 2011; Spangenberg et al. 2014). One widely accepted explanation is the increase of effective maximum volume fraction ϕm-eff. Since small particles may occupy the space between larger particles, a higher effective packing fraction can be achieved (Metzner 1985). Then, according to equation (2.10), when ϕm-eff increases, the viscosity decreases. To understand this physically, the small particles can act as lubricants for the flow of the larger particles, thereby reducing the overall viscosity (Servais, Jones, and Roberts 2002).
In addition, ϕm-eff not only depends on the number of discrete size bands (mono-, bi-, tri-, tetra dispersed, etc.), but also depends on the size ratio of the diameter of large particles ( ) to that of the smaller ones ( ) in the next particle class ) . For a given particle size distribution, ϕm-eff increases with increasing , and reaches a maximum value at infinite diameter ratio, which can be considered as a mono-dispersed suspension.
Models of maximum volume fraction versus particle size distribution can been found in many articles (Chang and Powell 1994; Chong, Christiansen, and Baer 1971; Shapiro and Probstein 1992; Dörr, Sadiki, and Mehdizadeh 2013; Ouchiyama and Tanaka 1981; Zou et al. 2003; Farris 1968; Qi and Tanner 2011). Their results confirm the trend described above.
When the particles are non-spherical, there is an extra energy dissipation when the suspension is under flow, which result in an increase of viscosity (Genovese 2012), and its contribution to suspension viscosity depends on the orientation of non-spherical particles. The use of arbitrary shape particles will change the maximum packing fraction and the intrinsic viscosity value (Equation (2.8)). Therefore, to determine the viscosities of non-spherical particle suspensions is to determine their effective maximum packing fraction and the changed intrinsic viscosity value. Generally, particle non-sphericity induces an increase of the intrinsic viscosity value B and a decrease of the effective maximum packing fraction ϕm-eff . But their products remain ≈ 2 (Barnes, Hutton, and Walters 1989).
For example, Kitano et al. (1981) measured the viscosity of non-spherical particle suspensions. They used equation (2.10) to describe the viscosity of non-spherical particle suspensions. They found that the maximum packing fraction decreased with the increasing of the length ( ) to diameter ( ) ratio ( ). When (spheres), ϕm-eff = 0.68, when (crystals), ϕm-eff =0.44. The same tendency was found by Mueller et al. (2010), by measuring the viscosity of mono-dispersed prolate and oblate particle suspension.

Table of contents :

1 Literature review
2 Suspensions
2.1.1 Introduction
2.1.2 Classification of suspensions
2.1.3 Suspension viscosity
2.2 Flow geometries
2.2.1 Introduction
2.2.2 Drag flow geometries
2.2.3 Pressure driven flow geometries
2.3 Dense suspension flow dynamics
2.3.1 Introduction
2.3.2 Secondary flow
2.3.3 Apparent slip
2.3.4 Shear induced particle migration
2.4 Measurement techniques for suspension flow
2.4.1 Introduction
2.4.2 Micro-PIV
2.4.3 Particle locating method
2.5 Positioning of the study
3 Density and refractive index matched suspension model
3.1 Suspension preparation process
3.2 Suspension rheology
3.2.1 Measurement conditions
3.2.2 Results
3.2.3 Suspension rheology discussion
4 Experimental set-up and measurement techniques
4.1 Experimental set-up
4.1.1 Flow generation system
4.1.2 Flow measurement system
4.1.3 Environment control system
4.2 Velocity profile measurement
4.2.1 Determination of the velocity profile
4.2.2 ratio
4.2.3 Micro-PIV system parameters
4.3 Validation of experimental set-up
4.3.1 Threshold velocity of secondary flow
4.3.2 Velocity profiles of 0% particle suspension
4.4 Particle concentration measurement
4.4.1 Measurement principle
4.4.2 Particle locating program
4.4.3 Local particle concentration calculation
5 Experimental results
5.1 Suspension velocity profiles
5.1.1 Evolution of velocity profiles
5.1.2 Relative difference of shear rates
5.2 Apparent slip
5.2.1 Apparent slip characterization
5.2.2 Apparent slip discussion
5.3 Local particle concentration
5.3.1 value
5.3.2 Particle position determination
5.3.3 Local particle concentration calculation
5.4 Discussion
5.4.1 Particle migration
5.4.2 The influence on velocity profile measurement
6 Conclusion and perspective

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