### 2.1 Fluidic behaviors of clay suspensions—steady state shear measurement

Fluidic behaviors of suspensions are classified by a relationship between shear stress τ and shear rate \(\dot{\gamma }\). The shear stress is stated precisely as the tangential force applied per unit area, and the shear rate is as the change of shear strain per unit time. The ratio of shear stress τ to shear rate \(\dot{\gamma }\) is defined as viscosity *η*. In other words, η is a measure of the resistance of suspensions to shearing flow or fluidic motion.

$$\eta = \frac{\tau }{{\dot{\gamma }}}.$$

(1)

Figure 1 schematically shows five types of fluidic behaviors: Newtonian, pseudoplastic, Bingham plastic, Bingham, and Dilatant. In the Newtonian fluid, shear viscous stress is linearly proportional to shear rate and viscosity (i.e. a ratio of shear stress over shear strain) does not depend on the shear rate. Other four types of fluids are called non-Newtonian fluids which do not exhibit a constant shear stress–shear strain ratio. In many cases, aqueous clay suspensions exhibit Bingham plastic behavior [20].

Bingham model postulates that a finite stress should be applied to initiate flow. As the shear rate increases, Bingham fluid gets close to Newtonian fluid. Hence, the resistance of Bingham fluid to the shear flow can be set as two regimes; (i) a non-Newtonian regime in which the shear stress is constant regardless of the shear rate, (ii) a Newtonian regime in which the shear stress is proportional to the shear rate. The Bingham model is quantitatively expressed as follow,

$$\uptau = \uptau_{\text{B}} + \upeta_{pl} \times \dot{\gamma }.$$

(2)

where η_{
pl
} is the plastic viscosity that is determined by the slope of the curve, and τ_{B} is the Bingham yield stress estimated from the intercept of the flow curve at high shear rate regime. The other models describing the rheological behavior of clay suspensions are Casson equation [21].

$$\uptau^{ 1/ 2} = {\text{ k}}_{0} + {\text{ k}}_{ 1} \; \times \;\dot{\gamma }^{ 1/ 2} .$$

(3)

and the Herschel–Bulkley equation [22].

$$\uptau = \uptau_{\text{y}} + K \times \dot{\gamma }^{\text{n}}$$

(4)

where yield stress τ_{y}, flow consistency *K*, and flow behavior indices n. Both models have been used to describe the consistency curves of the clay based fluids [23]. In both cases, the given suspension has an initial yield stress at low shear rates. As the shear rate increases, the viscosity of the fluid decreases and gets saturated to a certain value. This is called a shear thinning behavior.

### 2.2 Characterization of viscoelastic behavior by oscillatory shear measurement

The non-Newtonian behavior of clay suspensions is due to rearrangement of clay particles in fluids. An effective way to examine this interparticle interaction in clay suspensions is to characterize viscoelastic properties of fluids by oscillatory shear measurement [24]. This method provides quantitative information on mechanical properties of soft materials such as colloidal suspension, gel, emulsion polymer and foam. The oscillatory measurement produces sinusoidal deformation in soft materials and measures applied stress and strain of soft materials. In pure elastic materials, stress is exactly in phase with a sinusoidal change in strain over time and a proportionality constant (shear modulus) does not depend on the magnitude of stress or strain rate. If materials show an ideal viscous behavior, stress for the deformation is proportional to a rate of deformation which is called a strain rate (\(\dot{\gamma }\) = *d*γ/*d*t). As the strain rate increases, the viscosity decreases. In viscous materials, stress and strain are out-of-phase and a phase difference (δ) is π/2. A stress–strain relation of viscoelastic materials is in-between those of elastic and viscous materials. When materials are deformed, a phase difference between stress and strain is not either 0 or π/2. Measured stress of viscoelastic materials has in-phase and out-of-phase components in comparison with strain.

In the oscillatory measurement, a small magnitude of shear strain is first induced in an oscillatory mode in soft materials or fluids which are confined between two circular plates. Then, stress and strain of materials are examined simultaneously. By measuring the time lag of frequency Δ*t*, the phase angle shift δ is attained:

$$\updelta = \Delta t\omega$$

(5)

where ω is the frequency in radians per second (ω = 2π*ν*, ν is the frequency in Hz). A complex shear modulus *G** of materials under oscillatory shear is written by;

$$G^* ( \omega ) = \tau ( t ) / \gamma ( t)$$

(6)

where τ(t) is shear stress. It is noted that *G**(ω) is a function of the oscillation frequency ω. For viscous and viscoelastic systems, shear stress is in advance of strain by a phase difference of δ. Then, stress and strain have following forms;

$$\gamma (t)= \gamma_{\text{oi}} { \exp }\left( {i\omega t} \right)=\gamma_{\text{o}} { \sin }\omega t$$

(7)

$$\tau \, (t)\, \tau_{\text{oi}} { \exp }\left[ {i\left( {\omega t + \, \delta } \right)} \right]=\tau_{\text{o}} { \sin }\left( {\omega t + \delta } \right)$$

(8)

where γ_{o} and τ_{o} is the amplitude of imposed strain and measured stress, respectively. In a complete elastic system, the stress is exactly in phase with the strain (δ = 0), while in a complete viscous liquid, it is exactly out of phase with the strain (δ = 90°). For a viscoelastic system, the phase angle shift lies on a certain point between elastic and viscous systems. From the equations above, the following relations can be derived;

$$G^{\prime}= \left| {G^*} \right|{ \cos }\delta$$

(9)

$$G^{\prime\prime}= \left| {G^*} \right|{ \sin }\delta$$

(10)

$$G^* = G^\prime + iG^{\prime\prime}$$

(11)

*G*′ and *G*″ are called storage modulus and loss modulus, which are measures of stored energy and dissipated energy during the cyclic deformation. Figure 2 shows a relative change in *G*′ and *G*″ as a function of strain in a typical viscoelastic fluid system. In a strain regime where *G*′ is larger than *G*″, colloidal suspensions behave like gel and particles have strong interactions to form a network structure. If *G*′ is smaller than *G*″, the interparticle interactions are weakened and colloidal suspensions exhibit a liquid-like behavior. Small amplitude oscillation experiment also provides a transition point from gel-like to liquid-like behavior of viscoelastic fluid system (*G*′ = *G*″).