Non-axial deformation

So far, the following situation has been modeled: a particle is supported on a substrate and deformed axially from the top pole. Now, let us consider the deformation of a particle suspended in liquid, i.e. deformation by shear forces.

Generally, when a solid body obeying Hooke’s law is sheared, a linear relation of the form:

σ_shear =Gγ (2.18)

can be established. It relates the applied shear stressσ_shear to the shear strain γ with the proportionality factorG being the shear modulus. This is in analogy to eq. 2.2 for uniaxial tension or compression. The other extreme would be shearing a liquid. Here, the response to the applied stress is no longer elastic deformation but viscous flow and depends on the shear rateγ˙, i.e. the derivative of γ with respect to time:

σ_shear=ηγ˙ (2.19)

The proportionality constant η is the liquid’s (dynamic) viscosity.

For systems in-between these two boundary cases, such as polymers in solution, gels or colloidal suspensions, the description naturally becomes more complex. They behave viscoelastic, meaning that G becomes time-dependent. In the simplest case, i.e. linear viscoelasticity, the shear modulus can be expressed as:

G(t) = G₀·e^−λt (2.20)

Here, G₀ is the equilibrium shear modulus andλ=η/G₀ the characteristic relaxation time. For the investigation of the dynamic properties of viscoelastic samples oscillatory shear experiments are performed where the applied strain is varied periodically at a given frequencyω like:

γ(ω, t) = γ₀·sin(ωt) (2.21)

Now, the shear stress can be described by:

σ_shear =γ₀(G⁰(ω) sin(ωt) +G⁰⁰(ω) cos(ωt)) (2.22)

where G’ is the storage modulus, accounting for the elastic contribution, and G” is the loss modulus, representing the system’s viscous response.

The so far presented formulas and more detailed descriptions on the rheology of bulk systems can be found in a range of textbooks, e.g. [77–80].

Let us now turn to the specific problem of particles in shear flow. First studies in this area date back to the 1930s when Taylor examined the deformation of liquid droplets suspended in another liquid under different flow conditions (simple shear and plane hyperbolic flow).⁸¹ He observes an elongation of the initially spherical drop to assume an elliptic shape (cf. fig. 2.5). This shape change is quantified by a geometrical parameter (later denoted as Taylor parameter,D) containing the drop’s longest and shortest diameter, L and B, respectively. It is related to the dimensionless parameter F like:

D= L−B

L+B ≈F = γη˙ _cR

σ_intf (2.23)

Here, η_c is the viscosity of the continuous phase, R the radius of the droplet and σ_intf the interfacial tension. Comparison with experimental data has shown that eq.

2.23 and its more general form derived by Cox⁸² are valid for small deformations and Newtonian liquids.⁸³ Later publications provide more refined models for the deformation of liquid drops suspended in another liquid, also accounting for other flow conditions and non-Newtonian liquids.^84–96

In contrast to liquid droplets (which are not in our focus), individual solid elastic particles under shear flow have not been studied as extensively. This discrepancy may have its origin in the incompatibility of models typically used to describe solid and liquid phases and difficulties in determining the position of the interface between these two phases.⁹⁸

Figure 2.5. An initially spherical particle suspended in liquid (left) is subjected to linear shear flow (right) causing a certain deformation (length L, width B) and spacial alignment (angleΘ). The flow is created in Couette geometry (two co-axial, counter-rotating cylinders); small arrows indicate local flow direction and velocity. Reproduced from [97] by permission of The Royal Society of Chemistry.

Interestingly, in his theoretical analysis Murata reports that the shape equations for elastic full spheres resemble very much the ones obtained by Taylor and others for fluid drops under simple shear, hyperbolic or Poiseuille flow.⁹⁹ However, quantitative descriptions of the mechanics of single solid particles in shear flow remain scarce. Early works on this problem considered the contribution of the particles to the overall rheological properties of the suspension.^100–102 Following Brunn¹⁰³ and Van-der-Reijden-Stolk¹⁰⁴ we can write the equations given below for the deformation of a homogeneous, isotropic, incompressible elastic particle in elongational shear flow:

r

Here,R and r are the initial and deformed radius of the particle, respectively,εF_ij the deformation tensor, e the strain tensor, E the elastic modulus of the particle and η

the viscosity of the continuous liquid phase. For steady-state shear conditions eq. 2.24 and 2.25 can be further simplified to yield eq. 2.26 for the deformation and 2.27 for the shape of the particle: r and the major deformation direction.

More recently, Gao and co-workers have complemented the earlier works by a compre-hensive investigation of shape and motion of initially sperical or elliptic, Neo-Hookean particles in shear flow.^105,106,98 Their analytic approach is based on a polarization tech-nique and the results obtained for single particles are used to describe the rheological properties of dilute suspensions built from them.

In contrast to full spheres, the literature on the deformation of individual hollow capsules in shear flow is much more extended.^107,108 In the simplest case, i.e. for a homogeneous, isotropic shell material following Hooke’s law and Newtonian liquids as internal and continuous phase, the deformation of a capsule in linear shear flow can be described according to:^109,97

D= 25ηcrγ˙ 4E_S = 25

4 C_a (2.28)

Here, D is the above introduced Taylor parameter, r the initial capsule radius, η_c the viscosity of the surrounding liquid and E_S the two-dimensional elastic modulus which may be approximated as the product of shell thickness and Young’s modulus. C_a is the capillary number (for a droplet suspended in liquid E_S would be replaced by the surface

tension). Eq. 2.28 resembles eq. 2.23, showing that, under the given assumptions, the deformation of a capsule in shear flow can be modeled in a similar way as a droplet.

Thus, the droplet-solution may be interpreted as a boundary scenario for a capsule with vanishing shell thickness and the modulus of the shell material being replaced by the interfacial tension.

For linear elastic shells the tilt angle denoted by Θ in fig. 2.5 was shown to assume a constant value of 45° while for linear viscoelastic capsules it depends on the applied shear rate.¹¹⁰ Finally, for a viscous shell with negligible elastic contributions no steady state is reached but the capsule’s shape changes in a periodic manner.¹¹⁰

In analogy to the above discussed axial capsule deformation, the deformation of capsules in shear flow depends as well on a range of parameters including shape,^111–113 shell stiffness,^114–117 the shell material’s constitutive law,118–120,110,121 viscosity of core and surrounding liquid,^122,123 wrinkling,124,107,125 buckling,126,118,127 pre-stress^128,129 and flow conditions.130–132,107,133