Microscopic Starting Point

even impossible to calculate the transition density exactly because it is a collective phenomenon and cannot be treated by considering the motion of a single particle while keeping the other ones fixed. An approximate theory that very successfully describes the glass transition is the mode coupling theory (MCT) [10].

Another phenomenon connected to the glass transition is aging; if the density of the particles is increased rapidly (“quenched”) to the glassy phase (or if, for soft particles the temperature is rapidly decreased below the transition temperature), the system falls out of equilibrium. It is not in the global energy minimum which it can reach only very slowly due to its glassy nature. The outcome of measurements will depend on the amount of time elapsed since the quench; the system changes slowly with time, it “ages”.

The described phenomenology can be found e.g. in Ref. [11].

While the glass transition of hard spheres (or of super-cooled liquids) has been studied for many years, it is still a great challenge to describe systems close to the glass transition under external driving. We will consider the simplest external driving, namely a steady homogeneous shear flow. This is currently also the subject of experimental studies [12–

14]. We will consider space translationally invariant systems, where the gradient of the flow velocity of the particles is constant, i.e., we do not consider phenomena like shear banding [15], where this is not the case. Also, the increase of the viscosity at higher shear rates, the so called shear thickening [16] attributed to hydrodynamic interactions, and the jamming transition at somewhat higher densities and shear rates [11, 17] will not be issues for our small-shear studies. For small shear rates, one usually observes shear thinning, i.e., the viscosity decreases with shear rate. Aging effects are absent in the time translationally invariant steady state under shear. See e.g. the reviews in Refs. [18,19].

The sheared system under consideration hence reaches a steady state, but it is out of equilibrium. This causes many new and interesting phenomena. The system under shear is ergodic even when the un-sheared system is glassy, i.e., due to the shear, the particles can explore all (phase-) space. The dynamics of the system is governed by shear at arbitrary small shear rates. Due to this, many observables are non analytic in shear rate which leads to non-trivial limiting values. The most famous example is the yield stress, which will be introduced in Sec.1.6.4. As we will see in Chap.4, the violation of the fluctuation dissipation theorem also shows such a non-trivial limit in glassy states.

1.3 Microscopic Starting Point

We consider a system of N spherical particles of diameter d, dispersed in a solvent, see Fig. 1.1. The system has volume V. The particles have bare diffusion constants D₀ =k_BT µ₀, with mobility µ₀ and k_BT is the thermal energy. The interparticle force acting on particlei(i= 1. . . N) at positionr_i is given byF_i=−∂/∂r_iU({r_j}), whereU is the total potential energy. In an experimental system, also hydrodynamic interactions between the particles are present [20]; if particle i moves, it causes the surrounding solvent to move as well. The moving solvent affects the motion of the other particles,

1 Introduction

y

z x

Figure 1.1: The considered system: A dense colloidal suspension under shear. The spacial directions are referred to as flow (x), gradient (y) and neutral (z) direction.

which in turn affects the solvent velocity and on this way particleiagain. We will neglect hydrodynamic interactions to keep the theoretical description as simple as possible. Also, one assumes that they are less important for the slow glassy systems under consideration [18]. We will compare our results mostly to computer simulations, where hydrodynamic interactions are absent.

The external driving, viz. the shear, acts on the particles via the solvent flow velocity v(r) = ˙γyx, i.e., the flow points inˆ x-direction and varies in the y-direction. ˙γ is the shear rate. Due to the solvent velocity, the friction force of the solvent on particle iis not proportional to its velocity, but proportional to the difference of this velocity to the solvent velocity at r_i. This leads to the equation of motion for the particles, namely the Langevin equation that misses in the over-damped system an inertia term,

∂r_i

∂t −v(ri) = (Fi+f_i)µ0. (1.1) Different particles are coupled by the forces F_i. f_i is the random force, representing the thermal activations by the solvent molecules. It satisfies (α and β denote directions)

Df_i^α(t)f_j^β(t^′)E

= 2k_BT

µ₀ δ_αβδ_ijδ(t−t^′). (1.2) Eq. (1.2) expresses that the random forces in different directions, on different particles, and at different times are uncorrelated. The pre-factor must be chosen such that the ran-dom force obeys the fluctuation dissipation theorem. One can show that from Eq. (1.2) the mean squared displacement of a free particle equals 2D₀t. For the theoretical de-scription of the system it is more handy to use an equivalent formulation to Eq. (1.1), namely the Smoluchowski equation. It is an equation for the particle distribution

func-1.3 Microscopic Starting Point which will be used throughout the thesis unless stated otherwise. In contrast to Eq. (1.1), Eq. (1.3) has the advantage that it contains no random element. Nevertheless, it cannot be solved exactly for the many particle system [22]. Without shear, the system reaches the equilibrium distribution Ψ_e, i.e., Ω_eΨ_e= 0. Under shear, the system is assumed to reach the stationary distribution Ψ_s with ΩΨ_s = 0. Ensemble averages in equilibrium and in the stationary state are denoted

h. . .i =

respectively. In the stationary state, the distribution function is constant but the system is still not in thermal equilibrium due to the non-vanishing probability currentj^s_i [21],

j^s_i = [−∂_i+F_i+κ·r_i]Ψ_s. (1.5) In thermal equilibrium at ˙γ = 0 the probability current vanishes, i.e., the system obeys detailed balance [21]. In the steady state only the divergence of the current vanishes, not the current itself,

X

i

∂_i·j^s_i =−ΩΨ_s= 0. (1.6)

1.3.1 Low Density Limit – Taylor Dispersion

For the case of a single particle under shear, the Smoluchowski equation for the proba-bility densityP(r,r₀, t) can be solved and shall be shortly sketched here because we will compare some of our results to the low density case. It reads (with restored units) [20]

∂

1 Introduction

From this one can determine the mean squared displacements (MSDs) of the particle in the different directions, The mean squared displacements in directions perpendicular to the shear flow are un-affected by it. In x-direction the MSD is not diffusive but grows with power t³ for long times. The physical reason is that a fluctuation of the particle in gradient direction leads to an increase of the velocity in x-direction, because the flow velocity is a function of y. The MSD in x-direction also contains a ballistic term due to the average velocity of the particle of ˙γy₀. The non-diffusive motion in flow direction is called Taylor dis-persion [23–26]. There is an additional correlation between the x- and y-direction as seen in (1.9c), which is not present without shear. In Chap.3 we will derive the Taylor dispersion for systems near the glass transition.