Active and passive particle transport in dense colloidal suspensions

in Dense Colloidal Suspensions

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Dissertation

zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften

an der Universit¨at Konstanz Fachbereich Physik

vorgelegt von

Igor Gazuz

Tag der m¨undlichen Pr¨ufung: 25.08.2008 1. Referent: Prof. Dr. Matthias Fuchs

2. Referent: Prof. Dr. Rudolf Klein

Lehrstuhl f¨ ur Theoretische Physik Prof. Dr. Matthias Fuchs

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/6629/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-66299

First, I would like to thank Prof. Matthias Fuchs for giving me the opportunity to work on the very interesting subject of the mode-coupling theory of the glass transition and especially to contribute to the exciting new developments in this field. I also acknowledge the opportunity to participate in different teaching and seminar activities, which helped me to learn unconventional approaches to many problems and to broaden my physical knowledge.

I am indebted to Prof. Rudolf Klein who agreed to be the second referee of the thesis.

The fruitful collaboration with Dr. Antonio Puertas is greatfully acknowledged. It was great to benefit from his experience in computer simulations and from numerous discussions.

Thanks to Dr. Thomas Voigtmann, David Hajnal and Dr. Matthias Sperl for pro- viding their computer programs.

I am very grateful to the DFG for the financial support in the framework of the SFB (Collaborative Research Center) 513. I would like to thank the speaker of the SFB, Prof. Paul Leiderer and Dr. Atur Erbe for collaboration and discussions.

I also would like to thank all my colleagues for a nice and friendly working atmo- sphere. Matthias Kr¨uger showed much interest for my work and I acknowledge the discussions with him. Special thanks go to Oliver Henrich, who was the first phd student in the group and shared much of his knowledge with the others. He also initiated many social events (like our famous pub crawls), which made the group life better.

Last but not least, I thank my parents for their continuous support.

i

Acknowledgments i

Introduction vii

1 Linear Response Theory 1

1.1 The Smoluchowski equation . . . 1

1.2 Linear perturbation of the equilibrium distribution . . . 2

1.3 Response to the external force on the tracer . . . 3

1.4 Mode-coupling approximation of the force-force correlator . . . 6

1.5 The mode coupling theory . . . 8

1.5.1 The equations . . . 8

1.5.2 The main features . . . 10

1.6 Low density limit . . . 14

1.6.1 The structure factors . . . 15

1.6.2 The friction coefficient . . . 17

1.7 Pulling with fixed velocity . . . 18

2 Nonlinear Theory 23 2.1 Exact solution of the Smoluchowski equation . . . 23

2.2 The tracer friction coefficient . . . 24

2.3 The tracer density correlator . . . 25

2.4 Low density limit . . . 27 iii

3 Long Time Limit of the Tracer Density Correlator 33

3.1 The Fourier space picture . . . 33

3.2 The real space picture . . . 37

3.3 Comparison with simulations and experiments . . . 44

3.3.1 Simulations . . . 44

3.3.2 Experiments . . . 45

4 Schematic Models 51 4.1 Construction of the models . . . 51

4.2 The phase diagrams . . . 52

4.3 Bifurcation analysis . . . 56

4.4 Effect of the external force on the correlators . . . 58

4.4.1 Above the glass transition . . . 58

4.4.2 Below the glass transition . . . 58

4.5 The β-correlators . . . 63

4.6 Fex-Sj¨ogren model in the liquid for Fex < F_ex^c . . . 64

4.6.1 The β-relaxation . . . 64

4.6.2 The α-relaxation . . . 67

4.7 The critical correlators . . . 68

4.8 The friction coefficient . . . 72

4.8.1 F_ex-F1 model. . . 72

4.8.2 F_ex-Sj¨ogren model . . . 75

4.9 Comparison with simulations and experiments . . . 79

4.9.1 The friction coefficient . . . 79

4.9.2 The correlators . . . 80

5 Summary and Outlook 83

6 Zusammenfassung 87 A Derivation of the equation for the long time limit 89

B F_ex-dependent calculations 91

B.1 The adjoint of the full Smoluchowski operator . . . 91 B.2 Averages for the mode-coupling memory kernel . . . 92

C Numerical details 95

C.1 q-dependent calculation . . . 95 C.2 Fourier back transform . . . 99 C.3 Schematic models . . . 100

Bibliography 101

Tracking the diffusive motion of a colloidal tracer particle embedded in a soft ma- terial can be used to test its linear rheological [1] properties. Corresponding exper- imental techniques were developed during the last years [2–6] and are now widely known asmicrorheology. This techniques utilize the fluctuation-dissipation theorem [7], which connects the linear response to external fields with the time-dependent equilibrium correlation functions.

Compared to the conventional macrorheology, where a shear field is applied to a macroscopic sample, microrheology has several advantages: only a small amount of material is needed and for heterogeneous materials like e. g. biological cells, their local properties can be probed.

Linear rheology, however, does not give the full information about the mechanical properties of materials. If the structural deformation becomes strong, typical non- linear effects come into play likeshear thinning, i. e. the decay of the viscosity upon increasing the shear rate for liquid-like materials or yielding, i. e. the existence of a finite stress needed to produce a shear flow for solid-like materials.

In order to observe nonlinear phenomena in a microrheological experiment, the tracer has to be actively pulled by means of an external force. Such experiments were performed using magnetic forces [8–14] as well as optical tweezers [15–17].

As for the theoretical developments, the research concentrated up to now on con- tinuum approaches [18, 19], where the probed material is described by means of generalized viscoelastic hydrodynamic equations and on microscopic approaches in the low-density limit [20–22].

In this thesis, an attempt is made to develop microscopic theory of microrheology for colloidal suspensions at finite densities. Suspensions of colloidal particles be- came paradigmatic in the physics of soft matter as simple model systems, since many soft materials are complex fluids, containing simple atomic or molecular liq- uids as solvents for different additional mesoscopically structured constituents like polymers, membranes etc.

Especially, the property of dense polydisperse suspensions to undergo glass tran-

vii

nomenon is not yet fully understood. The glassy state shares the properties of solids (mechanical stiffness, ability to support transversal phonons) and the liquids (lack of long range order). Much insight into the structural relaxation of colloidal sus- pensions near the glass transition was gained by means of dynamic light scattering experiments [23–32].

A first-principles, microscopic explanation of the glass transition was provided by the mode-coupling theory (MCT) [33]. It describes the transition as kinetic arrest of density fluctuations, resulting from the nonlinear backflow feedback mechanism, which confines the particles in thecagesof their nearest neighbors so that the system becomes nonergodic. MCT provides a self-consistent, semi-quantitative description for such typical features in the vicinity of the glass transition like asymptotic scaling laws and factorization properties.

Besides describing the glass transition in simple hard sphere systems, MCT could also explain the phenomena observed in polymer melts [34–39], in mixtures of hard sphere particles of different size [40–45] and in systems with attractive interactions [46–52].

Putting a dense colloidal suspension into a strong external driving field poses a great challenge for the nonequilibrium statistical mechanics [53–55], since in this situation neither low-density no linear-response approximations are appropriate. One could think of extending the original mode-coupling theory, which allowed to predict the behaviour of the kinetic coefficients of dense colloidal suspensions in the vicinity of the glass transition in linear response (i. e. for small deviations from equilibrium) to the states lying far away from equilibrium.

Such an extension of MCT indeed became possible for the case of macrorheology [56–64], where a simple shear flow was imposed to the suspension. The theory succeeded in explaining shear-thinning and yielding in a unified microscopic frame- work.

The ideas of the mode-coupling approach to macrorheology are adopted and de- veloped here to study the effect of the external field given by the space and time independent external force on just a single colloidal tracer particle. Even though the technical details turn out to be quite different, our theory predicts the effects analogous to the yielding and the shear-thinning in macrorheology. These are the depinning of the tracer from the surrounding glassy matrix (so that the cage gets broken) if the applied force exceeds a certain threshold value (the critical force) and the decay of the tracer friction coefficient with increasing external force (force thinning).

Besides from the purely theoretical interest, the motivation for this work comes from the simulational studies [65] on overdamped Newtonian quasi-hard sphere systems and from the experiments [8] on colloidal suspensions of spherical PMMA particles,

where the nonlinear effects mentioned above were observed.

In the following, the outline of the thesis is presented.

We present the (well-known) linear responsetheory in the framework of the Smolu- chowski equation in Chapter 1. After deriving the Green-Kubo relation for the tracer friction coefficient, its mode-coupling approximation is presented, serving as a basis for the extensions to the non-linear case in the next chapter. The difference between pulling with constant force and with constant velocity is discussed. Our discussion elucidates the new effects in dense suspensions near the glass transition.

Chapter 2 introduces the general formalism of the nonlinear theory. The general- ized Green-Kubo relations valid for nonlinear response are presented. We derive the MCT equations for the time-dependent tracer density mode correlators, which appear in the mode-coupling approximations for the response functions. In contrast to the linear response case, the MCT equations contain now the external force. The consequence is, that the correlators exhibit some strikingly new properties: the are not only no longer positive definite, but even become complex. The low density limit of the theory (which already exhibits force thinning) is discussed analytically and compared to the exact low density results.

In Chapter 3, we study the long time limit of the tracer density correlator. The long-time limit allows to distinguish between localized (i. e. trapped in the cage) and delocalized states of the tracer. We observe the depinning transition mentioned above and make a quantitative comparison of the results for the critical force with simulations and experiments. The Fourier back transform of the long time limit from the q-space to the real space yields the probability density of finding the tracer at a certain space position for large times and is related to the shape of the cage. We discuss the deformation of this shape with increasing external force.

Chapter 4 is devoted to the simplified schematic models for the tracer density cor- relator. The wave vector dependence drops here and the equations have the virtue of being easily tractable, in some special cases even analytically. Schematic models help to understand the behaviour of the full MCT equations in the presence of the external force. Despite the very strong simplifications made, the depinning transi- tion and the force thinning behaviour are still present. We discuss the correlators of the schematic models and the resulting friction coefficient in detail and fit the simulational and experimental curves for the friction coefficient vs. external force.

In Chapter 5we summarize our results and give an outlook for the possible future work.

Linear Response Theory

1.1 The Smoluchowski equation

We start by writing down the Smoluchowski equation, which will provide the basis for all the considerations in this thesis:

∂tΨ = ΩΨ. (1.1)

Ω is the Smoluchowski operator

Ω = ^X

i=1, ...,N

Di∂_i·(∂_i−F_i) (1.2) Eq. (1.1) describes the time evolution of the N-particle phase space probability density Ψ(r1, . . . ,r_N, t). The particles are colloids performing Brownian motion with diffusion coefficients Diin the solvent, additionally interacting with each other by potential forces F_i = −∂_iV(r1, . . . ,r_N) (∂_i denotes ∂/∂ri). The hydrodynamic interactions will be neglected throughout this thesis.

We choose the theoretical units with kBT ≡ 1 for convenience. For comparisons with experiments or simulations, the factor kBT has to be reintroduced. For the Smoluchowski operator, this means that the forces should be divided by k_BT. The formal solution of (1.1) with the initial condition

Ψ(t = 0) = Ψ0 (1.3)

reads

Ψ(t) =e^ΩtΨ0 (1.4)

By the knowledge of Ψ(t), one can determine the mean value of an observable A(r1, . . . ,r_N) at time t:

hA(t)i=

Z

dΓ Ψ(Γ, t)A(Γ), (1.5)

1

If one uses (1.4) in (1.5), one can perform spatial integration with respect to the phase space variables in order to transfer the time dependence from Ψ to A:

hA(t)i=

Z

dΓ Ψ0A(Γ, t). (1.6)

Here A(Γ, t) =e^ΩBtA(Γ) and the backwardSmoluchowski operator Ω^B= ^X

i=1, ...,N

D_i(∂_i+F_i)·∂_i (1.7) was introduced. It is just the adjoint of Ω with respect to the unweighted scalar product hA|Bi = ^R dΓA^∗(Γ)B(Γ). Ω^B has the virtue of being self-adjoint with respect to the equilibrium-weighted scalar product hA|Bi^eq=^R dΓΨeqA^∗(Γ)B(Γ).

1.2 Linear perturbation of the equilibrium distribu- tion

The canonical equilibrium distribution

Ψeq ∝e^−V (1.8)

is a (trivial) solution of (1.1).

We consider now a situation, where the system described by the Smoluchowski operator Ω0 is in equilibrium until at time t= 0 an external perturbation is turned on, corresponding to a change ∆Ω in the Smoluchowski operator, so that for t >0 we have

∂tΨ = (Ω0 + ∆Ω)Ψ (1.9)

Our aim is to find the solution of (1.9) to the lowest order in ∆Ω. If we write Ψ(t) = Ψeq+δΨ(t), then eq. (1.9) yields:

∂tδΨ = Ω0δΨ + ∆Ω(Ψeq+δΨ) (1.10) Since δΨ is assumed to go to zero for ∆Ω → 0, it’s expansion in ∆Ω starts with the linear order and the term ∆ΩδΨ is O(∆Ω²) in (1.10). So, to the leading order in ∆Ω, the equation

∂tδΨ = Ω0δΨ + ∆Ω Ψeq (1.11) remains.

It is an inhomogeneous linear differential equation for δΨ. The general solution of the homogeneous part is given by

δΨhom=e^Ω0tA (1.12)

with time-independent A. A particular solution of the inhomogeneous equation can be found by the usual method of variation of constants, setting

δΨinh =e^Ω0tB, (1.13)

where B can depend on time.

We obtain

e^Ω0t(∂_tB+ Ω₀B) = Ω₀e^Ω0tB + ∆Ω Ψ_eq (1.14) for B.

Multiplying both sides of (1.14) with e^−Ω0t from the left, we get the equation

∂tB =e^Ω0t∆Ω Ψeq, (1.15)

which can be immediately solved:

B =

Z _t

0 dτ e^−Ω0τ∆Ω Ψeq. (1.16)

Altogether, the general solution of (1.11) reads:

δΨ = δΨhom+δΨinh =e^Ω0tA+e^Ω0t

Z t

0 dτ e^−Ω0τ∆Ω Ψeq. (1.17) Combined with the initial condition δΨ(t= 0) = 0, we get A= 0 and finally,

δΨ(t) =e^Ω0t

Z _t

0 dτ e^−Ω0τ∆Ω Ψeq. (1.18) This is the sought-after deviation of the distribution function Ψ(t) from the equi- librium distribution function Ψ_eq for the system described by the Smoluchowski operator Ω0 tolinear order in the external perturbation ∆Ω.

1.3 Response to the external force on the tracer

After the general consideration of the last Section let us turn now to the problem of our interest. We consider a single, distinguished tracerparticle with the diffusion coefficientDssurrounded by N identicalbathparticles with diffusion coefficientD0. The tracer is pulled by the external force F_ex (see Figure 1.1).

F

Figure 1.1: The colloidal “tracer” particle is pulled through the suspension of “bath”

particles by means of the external forceF_ex

The unperturbed Smoluchowski operator of the problem reads:

Ω0 =D0

X

i=1, ...,N

∂_i·(∂_i−F_i) +Ds∂_s·(∂_s−F_s). (1.19)

The perturbation is given by

∆Ω =−DsF_ex·∂_s. (1.20)

We note that F_ex will be always assumed to be constant in space and time in the following.

The deviation of the (nonequilibrium) mean value of an observableAat timetfrom it’s equilibrium mean value can be expressed inlinear responseapproximation using eqs. (1.5), (1.18):

δhA(t)i=hA(t)i − hAi^eq =−Ds

Z

dΓA(Γ)

Z t

0 dτ e^−Ω0τ(Fex·∂_s) Ψeq. (1.21) The term ∂_sΨ_eq can be calculated explicitely (see (1.8)) with the result

∂_sΨeq =F_sΨeq. (1.22)

Using this and changing to the backward Smoluchowski operator, one finally gets:

δhA(t)i=−DsF_ex·

Z t

0 dτhF_se^{ΩB0 (t−τ)}Ai^eq, (1.23) where h. . .i^eq means phase-space averaging with the equilibrium distribution Ψeq. In the following, we shall only use the backward Smoluchowski operator and drop the superscript “B” in the notation.

We are especially interested in the mobility of the tracer, defined as µ= hv_si

F_ex, (1.24)

where v_s is the tracer velocity. In the framework of the Smoluchowski equation, where the particle motion is overdamped, the velocity is proportional to the total force on the tracer

v_s=D_s(F_ex+F_s). (1.25)

SinceF_ex is given externally and does not depend on the phase space of the system, the problem reduces to calculating the average of the forceF_sfrom the bath particles on the tracer. Due to (1.23), it is given by

hF_si(t) =−1 3DsF_ex

Z _t

0 dτhF_s·e^Ω0(t−τ)F_si^eq. (1.26) Expression (1.26) is formally correct but it can lead to unphysical results if used in this form for approximations. Especially for approximations of mode-coupling-type, presented in the next Section, it is known [66, 67] that the correlators have to be rewritten in terms of the irreducible Smoluchowski operator before approximating them. Doing this for the force-force correlator hF_se^ΩtF_si, leads for the mobility to the result:

µ = 1

D_s⁻¹+^R₀^∞dt C(t), (1.27) C(t) = 1

3hF_se^ΩirrtF_si, (1.28) Eq. (1.27) suggests that it is more convenient to work with theinverseofµ, i. e. with the tracer friction coefficient,

ζ = F_ex

hv_si, (1.29)

For the friction coefficient in the absence of the bath particles, ζ0, the Stokes- Einstein relation ζ0 =D_s⁻¹ holds and as we see from eq. (1.27),

ζ = ζ0+ ∆ζ, (1.30)

∆ζ =

Z ∞

0 dt C(t), (1.31)

when the bath particles are present. So, the time integral over the bath-tracer force autocorrelation function C(t) is related to theincrementof the tracer friction coefficient due to the presence of the bath particles.

correlator

In this Section we perform the mode-coupling approximation for the force-force correlation function (1.28). It is the strategy within the mode coupling approach to project the dynamics of the time-dependent correlators to the dynamics of the density mode correlators.

The mentioned modes are the Fourier transforms of the tracer and the bath densities

ρ^s(r) = δ(r−r_s), (1.32)

ρ(r) =

XN i=1

δ(r−r_i). (1.33)

Our convention for the Fourier transform of a function X(r) will be X(k) =

Z

dre^ik·rX(r), (1.34)

implying

X(r) = 1 (2π)³

Z

dke^−ik·rX(k) (1.35)

for the back-transform.

So, we have

ρ^s_k = e^ik·rs (1.36)

ρk =

XN i=1

e^ik·ri. (1.37)

The simplest possible approximation of this kind is to insert the projectors P₂^s= ^X

k,p,k^′,p^′

ρ^s_kρpig(k,p,k^′,p^′)hρ^s_kρp (1.38) before and after the time evolution operator:

hF_se^ΩirrtF_si ≈ hF_sP₂^se^ΩirrtP₂^sF_si. (1.39) At least a product of one bath and one tracer density correlator is needed, since the equilibrium average with the force on the tracer hF_s . . .i appearing in (1.39) is zero both for a single tracer and for a single bath correlator, as will become clear from the following.

The normalization matrix g has to obey the equation

X

n,m

hρ^s_kρp|ρ^s_nρmig(n,m,k^′,p^′) =δ_k,k^′δ_p,p^′, (1.40) expressing the condition that a vector from the space of the tracer and bath density products should be left invariant upon application of P₂^s.

As the next approximation, the four-point correlators hρ^s_kρpe^Ωirrtρ^s_k^′ρp^′iare factor- ized into products of the two-point ones [33]:

hρ^s_kρpe^Ωirrtρ^s_k^′ρp^′i ≈δk+k^′,0δp+p^′,0hρ^s_ke^Ωtρ^s_k^′i hρpe^Ωtρp^′i. (1.41) Note also that as an additional approximation, the irreducible operator on the left hand side of (1.41) was replaced by the normal one on the right hand side.

Applied for t = 0, this factorization approximation allows us also to calculate the normalization matrix. The condition (1.40) then reads

X

n,m

δ_k,nδ_p,mNS_pg(n,m,k^′,p^′) =δ_k,k^′δ_p,p^′, (1.42) leading to

g(k,p,k^′,p^′) = 1

NS_pδk,k^′δp,p^′. (1.43) Here, the definition of the bath static structure factor was used:

S_p = 1

N hρpρ_−pi. (1.44)

Expression (1.39) contains also static correlators of the form hF_sρ^s_kρpi. These can be reduced to the tracer-bath static structure factor

S_k^s= hρ^s_kρ−ki (1.45)

by means of partial integration hF_sρ^s_kρpi=−1

Z

Z

dΓ (∂_se^−V)ρ^s_kρp = 1 Z

Z

dΓe^−V∂_s(ρ^s_kρp) =ikδ_k+p,0S_k^s, (1.46) whereZis the statistical sum. Now we see why the projection to tracer-bath density products was necessary: a single tracer or a single bath density mode would yield zero in the above average with the force F_s, since the equilibrium average of a single density mode vanishes due to the translational invariance of the system.

Introducing the notation

φ^s_k(t) = hρ^s_ke^Ωtρ^s_−ki, (1.47) φk(t) = 1

NSkhρke^Ωtρ−ki (1.48)

obtain the mode-coupling approximation of the tracer force autocorrelator hF_se^ΩirrtF_si=^X

k

1 NSk

k²S_k^s2φ^s_k(t)φ−k(t). (1.49)

1.5 The mode coupling theory

In the last Section, the mode-coupling approximations allowed us to reduce the problem of finding the tracer force autocorrelation function to the calculation of the static structure factors and the time-dependent tracer and bath density correlators.

The latter are described by the mode-coupling theory, to which we turn now.

1.5.1 The equations

The starting point is the Mori-Zwanzig memory equation for the correlator φ^s_q :

∂tφ^s_q(t) =−ωqφ^s_q(t)−

Z _t

0 dt^′Mq(t−t^′)φ^s_q(t^′). (1.50) Here

ωq = −hρ^s∗_q Ωρ^s_qi, (1.51)

M_q^s(t) = hρ^s∗_q ΩQ e^{QΩQ t}QΩρ^s_qi (1.52) and

Q= 1−P^s, (1.53)

with

P^s =^X

k

ρ^s_kihρ^s∗_k, (1.54)

the projector on the space spanned by the tracer density modes.

To calculate the frequency ωq, we have to know the result of action of Ω on the tracer density mode ρ^s_q=e^ik·rs:

Ωρ^s_q =Ds(∂²

s +F_s·∂_s)e^iq·rs =Ds(−q²+iq·F_s)ρ^s_q (1.55) Since the equilibrium average of the force from the bath particles on the tracer is zero, hF_si= 0, we obtain

ωq =Dsq². (1.56)

So far, no approximations were made, since the Mori-Zwanzig equation (1.50) is just an identity. The physical idea behind the formalism is that the dynamics of the hydrodynamic modes like the density correlators is “slow”, since these have to obey conservation laws. The slow dynamics is supposed to be “projected out”

by the projector Q so that the memory function contains only the fast dynamics.

Thus the usual approximation for the memory function should be a fast decaying function. The mode-coupling theory introduces a further non-trivial step of coupling the dynamics of the memory function back to the density correlators. In this way a self-consistent system of equations comes out.

An additional subtlety comes in since we are using the approach of the Smolu- chowski equation, where the velocities of the particles are integrated out. This makes it necessary to introduce a second projection step, leading to the irreducible memory function. It can be understood as the remnant of the projections onto particle current modes present in the more general approach of the Fokker-Planck equation after integrating out the currents. As the result, the memory equation (1.50) acquires the form

∂_tφ^s_q(t) = −ω_qφ^s_q(t)− 1 ωq

Z _t

0 dt^′M_q^{s, irr}(t−t^′)∂_t^′φ^s_q(t^′), (1.57) where

M_q^{s, irr}(t) =hρ^s∗_q ΩQ e^ΩirrtQΩρ^s_qi (1.58) is the irreducible memory function, evolving with the irreducible Smoluchowski operator Ω^irr, which exact form is not of interest for our purposes here. The crucial point, which allows to make the approximations like the mode-coupling one for the memory kernel in a safe way is the appearance of the time derivative before the density correlator in the memory integral.

We can proceed now with the above-mentioned mode-coupling approximation for the memory function. Similar to the approximation made in (1.39), it consists in inserting the projector P₂^s (see eq. (1.38)) before and after the time evolution operator and factorizing the four-point correlators:

M_q^{s, irr}(t) ≈ hρ^s∗_q ΩQ P₂^se^ΩirrtP₂^sQΩρ^s_qi

≈ ^X

k,p

hρ^s∗_q ΩQ ρ^s_kρpihρ^s∗_kρ^∗_pQΩρ^s_ki 1 NSp

φ^s_k(t).φp^′(t) (1.59)

The operator Ω is self-adjoint, so the equilibrium averageshρ^s∗_q ΩQ ρ^s_kρpiandhρ^s∗_kρ^∗_pQΩρ^s_qi are at least complex conjugated. The calculation (see Appendix B.2, whereF_ex= 0 has to be set) yields that they are real and therefore equal and given by

hρ^s∗_kρ^∗_pQΩρ^s_qi=hρ^s∗_q ΩQ ρ^s_kρpi=δq,k+pq·(q−k)hρ^s_k−qρpi (1.60)

perscript “irr” in the further notation in this Chapter, since only the irreducible memory function will be used from now on):

M_q^s(t) = Ds

q²

X

k+p=q

1 NSp

S_p^s2 (q·p)² φ^s_k(t)φp(t). (1.61)

We see that the dynamics of the tracer couples to the dynamics of the bath. The latter itself is determined by the corresponding memory equation, which we just quote here without derivation:

τq∂tφq(t) +φq(t) +

Z _t

0 dt^′Mq(t−t^′)∂t^′φq(t^′) = 0 (1.62) Here τq =Sq/(D0q²) and the memory function is given by

Mq(t) = 1 2q⁴

X

k+p=q

n S_qS_kS_p (q·(kc_k+pc_p))², (1.63) where n=N/V is the number density of the bath particles and cq is the Ornstein- Zernike direct correlation function:

Sq = 1/(1−ncq). (1.64)

1.5.2 The main features

The standard mode coupling theory of the glass transition is described in the original papers [68, 69] as well as in the reviews [33, 70–75].

Here, we only very briefly sketch the main features of the theory which are necessary in order to understand the new developments to which this thesis is devoted.

We restrict ourself here to the hard sphere tracer-bath system (see Figure 1.2), where the control parameters are the ratioα of the radii of the tracer and the bath particles

α= a

b (1.65)

and the volume fraction of the bath particles ϕ = 4

3πb³n. (1.66)

The main highlight of the theory is the prediction of the bifurcation of the long-time limit of the bath density correlator. At the critical volume fraction ϕ = ϕc (ϕc ≃ 0.516), the long-time limit jumps from zero to a finite value. This is interpreted as

b a

Figure 1.2: The hard sphere system

theideal glass transition. In the glassy state, the system is nonergodic: the particles are arrested in the cages of their next neighbors and cannot explore the whole phase space, the density correlator thus cannot decay to zero.

With the reduced volume fraction ε defined as ε = ϕ−ϕ_c

ϕc

, (1.67)

the long-time limit of the density correlator for small positive ε obeys fq =f_q^c +hq

s σ

1−λ, (1.68)

where

σ =Cε (1.69)

is the separation parameter, and the exponent parameterλ and the constant C can be calculated from the mode-coupling functional.

The time dependence of the density correlators in the vicinity of the glass transition for ε <0 exhibits a typical two-step relaxation scenario: first the correlator relaxes from the initial value to the beta relaxation plateau and then from the beta relax- ation plateau to zero (see Figure 1.3). The first decay is known as the β-process and the second one as the α-process. Forε >0, only the β-process is present.

The asymptotic behaviour of the correlators around the β-plateau is described be the β-correlators Gq(t)

φq(t) =fq+Gq(t), (1.70)

which exhibit the following factorization property

Gq(t) =hqG(t). (1.71)

This is the consequence of the general mathematical theorem (the central manifold theorem [76, 77]) which tells that at the bifurcation, the dynamics is reduced to the

0.0001 1 10000 1e+08 1e+12 1e+16

t

0 0.2 0.4 0.6 0.8

φ (t)

ε = -0.4 ε = -1.0e-2 ε = -1.0e-3 ε = -1.0e-4 ε = -1.0e-5 ε = -1.0e-6 ε = 0 ε = 1.0e-3 ε = 1.0e-2

Figure 1.3: The typical two-step decay of the density correlators in MCT.

critical space, which is spanned by the eigenvectors of the stability matrix with the eigenvalue zero. One can show that for the MCT equations, the dimension of the critical space is one, thus all the correlators are proportional to each other.

For the β-correlator G(t) the equation λ G²(t) +σ= d dt

Z _t

0 dt^′G(t−t^′)G(t^′), (1.72) holds, known as the β-scaling equation. For σ = 0 it has two different power law solutions

t^−a, (1.73)

−t^b, (1.74)

where the exponents a, b are related to λ by λ= Γ²(1−a)

Γ(1−2a) = Γ²(1 +b)

Γ(1 + 2b). (1.75)

The solutions of the eq. (1.72) for ε <0 asymptotically follow (1.73), (1.74). While the first solution describes theβ-relaxation, the second one corresponds to the initial phase of the α-decay

φq(t) =f_q^c−hq

t t^′_ε

!b

(1.76)

where the ε-dependence is contained in the α-scaling time t^′_ε given by

t^′_ε ∼ |ε|^−γ (1.77)

with

γ = 1 2a + 1

2b. (1.78)

For the α-decay, the following scaling law

φq(t) = ˜φq(t/t^′_ε) (1.79) holds with the ε-independent master function ˜φq(t) so that on the plot with the logarithmic time scale the curves just shift to the right with decreasing |ε| . This is shown on Figure 1.4.

Since the microscopic information is contained only in the parameters λ, and C in (1.72), one can introduce the simplifiedschematicmodels, which have the same type of the bifurcation and the same β-scaling equation, but no wave vector dependence.

The common models are the F12 model [33] for the bath

∂_tφ(t) =−φ(t)−

Z t

0 dt^′m(t−t^′)∂_t^′φ(t^′) (1.80) with

m(t) =v₁φ(t) +v₂φ²(t) (1.81) and the Sj¨ogren model[78] for the tracer

∂tφ^s(t) =φ^s(t)−

Z _t

0 dt^′m^s(t−t^′)∂t^′φ^s(t^′) (1.82) with

m^s(t) =vsφ^s(t)φ(t). (1.83) For the parameters of the F12 model the relation holds

v₁^c =v₂^c( 2

√v₂^c −1), (1.84)

for v^c₁, v₁^c lying on the bifurcation line, separating the liquid phase from the glass phase.

The parameter ε which measures the distance from the glass transition line and enters the β-scaling equation is related to v2 =v₂^c +δv2 and v1 =v₁^c+δv1 by

ε= δv1f^c+δv2f^c2

1−f^c (1.85)

10 15 20 25

log(t)

0 0.05 0.1 0.15 0.2 0.25

φ (t)

ε = -1.0e-6 ε = -1.0e-7 ε = -1.0e-8 ε = -1.0e-9 ε = -1.0e-10 ε = -1.0e-11

Figure 1.4: The alpha scaling law.

with

f^c = 1− 1

√v₂^c. (1.86)

In the following (especially in Chapter 4), the parameters will be chosen such that v2 = v₂^c, so that δv2 = 0. Thus, specifying the values of v2 and ε will completely determine the parameters of the schematic model according to the eqs. (1.85), (1.84).

The phase diagram of the Sj¨ogren model is schematically shown on Figure 1.5. For f > f^c, the bath is in the glassy state. Nevertheless, if the coupling parameter vs of the tracer to the bath becomes smaller than 1/f, the tracer decouples from the bath and becomes mobile. For the hard sphere system, the parameter vs corresponds to α. The critical value of α, at which this decoupling transition occurs, is given by α^c ≃0.15 [79].

1.6 Low density limit

We would like to consider now the limiting case of very low densities. Then, as the zeroth order approximation in bath particle density, the memory integrals can be completely neglected in the memory equations (1.57, 1.62) for the tracer and bath density correlators. One obtains the equations

∂_tφ^s_q(t) = −ωkφ^s_k(t), (1.87)

τq∂tφq(t) = −φq(t) (1.88)

v _s

f _c

f _c f

1 /

fluid

tracer mobile bath tracer localized

Figure 1.5: Phase diagram of the Sj¨ogren model.

with the obvious solutions

φ^s_q(t) = e^−Dsq2t, (1.89)

φq(t) = e^−D0/Sqq2t. (1.90) According to the expressions (1.30), (1.31), (1.28), (1.39), the mode-coupling re- sult for the increment of the tracer friction coefficient due to the presence of bath particles is given by

∆ζ = 1 3

Z ∞

0 dt ^X

k

1

NS_kk²S_k^s2φ^s_k(t)φ−k(t). (1.91)

1.6.1 The structure factors

The only missing input in (1.91) now are the static structure factorsS_k^s,Sk. For the hard sphere system, these can be easily calculated to the leading order in the volume fraction of the bath particles. We start with the tracer-bath structure factor:

S_k^s =h

XN i=1

e^{ik·(ri−rs)}i ≈Nhe^{ik·(r1−rs)}i. (1.92) The approximation made here consists in considering just the tracer and a single bath particle and multiplying the result by N. This means, we treat the system as a conglomerate of independent two-particle clusters.

r_1s

rs

r₁

Figure 1.6: The coordinates for the calculation of the tracer-bath structure factor

For two hard spheres we get he^{ik·(r1−rs)}i= 1

Z

Z

dr1drse^{−V(r1−rs)}e^{ik·(r1−rs)}. (1.93) We change the coordinates from (rs, r₁) to (rs, r_1s) (see Figure 1.6), where r_1s = r₁−r_s (the Jacobian of the transformation is equal to one) and note that since we are dealing with hard spheres,

e^−V(r1s) =

( 1, r1s > d

0, r1s < d (1.94)

applies, where

d=a+b (1.95)

is the sum of the tracer and the bath radii.

We arrive at

he^{ik·(r1−rs)}i= 1 Z

Z

drs

Z

r1s>ddr1se^ik·r1s (1.96) The first integral in (1.96) gives just V, the volume of the system. The statistical sum can be calculated using the same coordinate transformation as above and (1.94) to obtain

Z =

Z

dr₁dr_se^{−V(r1−rs))}=

Z

dr_s

Z

r1s>ddr_1s =V(V −V₀), (1.97) whereV0 is the volume of the sphere with radiusd. Since we consider large systems in the thermodynamic limit here, we can assume V0 ≪V to get

Z =V². (1.98)

Changing to spherical coordinates for the second integral in (1.96) and using (1.92), we obtain

S_k^s = N V

Z _∞

d dr1sdθ dφ r²_1ssinθ e^{i k r1scosθ} (1.99)

and after straightforward transformations, S_k^s = 4πN

V

Z ∞ d dr1s

r1s

k sin(k r1s). (1.100)

Taking the standard integral in (1.100) leads us finally to S_k^s= 4π nd³

x

cosx

x − sinx x²

, (1.101)

Note that the delta function corresponding to the forward scattering was omitted and the notation

x=kd (1.102)

was introduced. We see that the tracer-bath structure factor is proportional to the number density n = N/V of the bath particles in the lowest order approximation, as one would expect.

As for the bath structure factor, it can be written as

Sk = 1 +S_k^s (1.103)

where the tracer is assumed to be identical with the bath particles. This follows directly from the definition of Sk (see eq. 1.44) since all the particles in the system are equivalent. So, to the lowest order in n

Sk ≈1 (1.104)

comes out.

1.6.2 The friction coefficient

Now everything is ready to calculate the friction coefficient. Using (1.89, 1.90), performing the time integral and changing from the k-sum to the integral over the k-space:

X

k

→V /(2π)³

Z

dk, (1.105)

we obtain from (1.91)

∆ζ = 1

3 (2π)³(D0+Ds)n

Z

dkS_k^s2 Sk

. (1.106)

Changing to spherical coordinates, we get with (1.101), (1.104) and (1.102)

∆ζ = 2n

π(D0+Ds) 4 3πd³

Z _∞

0 dx

sinx

x² − cosx x

2

. (1.107)

Figure 1.7: Different pulling regimes. In the fixed force regime (left picture), the tracer can fluctuate, whereas in the fixed velocity regime (right picture) it is forced to move along a fixed path.

The integral can be calculated exactly and is equal to π/6. Writing d³ = (a+b)³ as b³(1 +α)³ withα from (1.65) and realizing that the volume fraction of the bath is given by ϕ = 4/3πb³n, we get the final low density result

∆ζ = 1

3ϕ (1 +α)³ D0+Ds

. (1.108)

1.7 Pulling with fixed velocity

Up to now we considered the case of the fixed external force applied on the tracer.

The velocity of the tracer was thus a fluctuating quantity and we could use our theory to calculate its mean value in order to obtain the friction coefficient. Exper- imentally, this would correspond to a tracer e. g. sedimenting under the influence of the gravitation or to a magnetic particle pulled by means of an external magnetic field.

One could imagine also another kind of experiments where the tracer velocity is fixed and the applied force fluctuates. This is the microrheological analog of the fixed strain or fixed stress measurements in shear rheology. The difference between the two cases is demonstrated on Figure 1.7.

To account theoretically for the fixed velocity case, we have to assume (following [20]) the tracer to be non-diffusive:

Ds = 0. (1.109)

The perturbation of the Smoluchowski operator has to be written now as

∆Ω =−v_s·∂_s. (1.110)

In contrast to the fixed force case (see eq. (1.20)) the externally given tracer velocity v_s enters here instead of the force.

The task is now to calculate the average friction force that the tracer experiences. It is easily done if one realizes that the friction forceF^{f r} results from the contribution of the solvent and of the bath particles

F^{f r} =F^{f r}_solv+F^{f r}_bath. (1.111) The solvent contribution is the same as for the fixed force

F^solv_{f r} =−ζ0v_s, (1.112) whereas the bath contribution is given by

F^{f r}_bath =F_s (1.113)

due to the definition of F_s (see Section 1.3).

To calculate hF_si, we still can use our linear response formula (1.26) after replacing DsF_ex with v_s and bearing in mind the change in the unperturbed Smoluchowski operator imposed by the condition (1.109). We obtain for the steady state value

hF_si=−1 3v_s

Z ∞

0 dthF_se^ΩtF_si. (1.114) The average force from the bath particles on the tracer is opposite to its velocity, as well as the friction force from the solvent. So the increment of the friction coefficient due to the bath is given by

∆ζv = 1 3

Z _∞

0 dthF_se^ΩtF_si, (1.115) which is merely the same expression as the one for the fixed force case. It is however notable that no need arises to introduce the irreducible operator arises here, since the expression (1.115) is already positive definite.

We can now use the same steps that led us to the expression (1.108) to get the low density result for the fixed velocity regime. The only difference is the fact that the tracer is non-diffusive now. So, we set Ds = 0 in (1.108) and obtain

∆ζv = 1

3ϕ(1 +α)³

D₀ . (1.116)

It is worth to compare our results with that of Squires and Brady [20]. They treat the low density limit exactly and obtain the same results up to the change in the prefactor from 1/3 to 1/2. It is a well-known fact that the mode-coupling theory

0.1 0.2 0.3 0.4 0.5

φ

1 10

∆ζ V / ∆ζ

α = 0.25 α = 0.3 α = 0.4 α = 0.5 α = 1 α = 2 α = 10

Figure 1.8: Ratio of friction coefficient increments for fixed velocity and fixed force vs. bath volume fraction for different values of α.

does not treat the low-density limit numerically correct [80]. However, the scaling with the relevant parameters ϕ and α is reproduced properly by MCT.

Our general expressions (1.31), (1.115) agree with that of Ref. [20] in the low density limit but apply also for arbitrary densities. However the expressions of [20] can also be applied for the nonlinear response.

The virtue of our approach comes into play when considering dense systems in the vicinity of the glass transition (see Section 1.5.2). A particularly interesting situation results, when the parameters of the hard sphere system are such that the decoupling transition (see the last paragraph of Section 1.5.2) of the tracer from the bath is approached. Whereas in the fixed force regime the tracer indeed decouples, in the fixed velocity regime it is not “allowed” to do so, since it’s dynamics is fixed.

So, a strong difference in the friction coefficients for the different regimes arises.

This is shown on Figures 1.8, 1.9.

0 2 4 6 8 10 0 α

2 4 6 8

∆ζ V / ∆ζ

low density limit φ = 0.1

φ = 0.2 φ = 0.3 φ = 0.4 φ = 0.5

Figure 1.9: Ratio of friction coefficient increments for fixed velocity and fixed force vs. ratio of the tracer and bath particle radii for different values of ϕ.

Nonlinear Theory

2.1 Exact solution of the Smoluchowski equation

We start by writing down the full Smoluchowski operator Ω = Ω0 + ∆Ω in the presence of the external force F_ex on the tracer (see eqs. (1.19), (1.20)):

Ω =

XN i=1

D0∂_i·(∂_i−F_i) +Ds∂_s·(∂_s−F_s)−Ds∂_s·F_ex (2.1)

Let us consider the same situation as in Section 1.2. For times t <0, the system is equilibrated and there is no external force. At t= 0, the external force is switched on driving the system far from equilibrium. However, instead of assuming that the perturbation is small, like in the linear response case, we consider the general case now.

We have the initial condition for the Smoluchowski equation

Ψ(t= 0) = Ψeq, (2.2)

where

Ψ_eq =e^{−V({ri},rs)} (2.3)

is the equilibrium distribution of the unperturbed system.

The solution of the Smoluchowski equation can now be written down formally as:

Ψ(t) =e^ΩtΨeq (2.4)

Using the operator identity

e^Ωt= 1 +

Z _t

0 dt^′e^Ωt′Ω (2.5)

23

Ψ(t) = Ψeq−

Z t

0 dt^′e^Ωt′∆ΩΨeq (2.6) Using now the same steps which led to the eq. (1.23) in the Section 1.3, we arrive at the expression

hAi(t) =hAi^eq−DsF_ex·

Z _t

0 dt^′hF_se^Ω†t′Ai^eq (2.7) for the (now) non-linear response of an observable A at time t.

Again, the adjoint of the operator Ω with respect to the unweighted scalar product hA|Bi=^R dΓA^∗(Γ)B(Γ) (the backward Smoluchowski operator) appears in (2.7):

Ω^B =

XN i=1

D0(∂_i+F_i)·∂_i+Ds(∂_s+F_s)·∂_s+DsF_ex·∂_s (2.8) and the superscript “B” in the notation will be dropped in the following.

In contrast to the linear response expression (1.23), the fullSmoluchowski operator Ω containing the external force instead of just the unperturbed one enters (2.7).

This makes the nonlinearnature of the derived exact result explicitely evident. So, appreciating the simple form of (2.7) we should be aware that the modifications of the theory with respect to the linear response case will be quite dramatic.

The external force obviously breaks the rotational symmetry of the system. Besides this, it draws the backward Smoluchowski operator nonhermitic with respect to the equilibrium-weighted scalar product, with quite unusual consequences for the correlators of the mode-coupling theory, as we shall see in the following.

2.2 The tracer friction coefficient

In this Section we consider the friction coefficient of the tracer. The relations (1.28), (1.30), (1.31) from the last Chapter still apply, with the replacement Ω₀ → Ω.

This is so since no special properties of the unperturbed operator (like e. g. the hermiticity with respect to the equilibrium-weighted scalar product) were used in the derivations of eqs. (1.28), (1.30), (1.31). The friction coefficient of the tracer is thus given by

ζ = ζ0+ ∆ζ, (2.9)

∆ζ =

Z _∞

0 dt C(t), (2.10)

where

C(t) = 1

3hF_se^ΩirrtF_si, (2.11) amd ζ₀ is the friction coefficient of the solvent.

So far, the expressions are formally the same. However, the mode-coupling approx- imations could introduce new aspects like the advection of the wave vector, as was shown for the case of an imposed shear flow [56, 57]. This issue will be reviewed in the next Section, at the moment we just claim that in our case nothing changes and the MCT approximation for the force-force correlator remains:

hF_se^ΩirrtF_si=^X

k

1 N Sk

k²S_k^s2φ^s_k(t)φ−k(t) (2.12)

Since the static structure factors are equilibrium quantities do not contain Ω, the difference to the linear response case comes in only via the time-dependent density mode correlators

φ^s_k(t) = hρ^s_ke^Ωtρ^s_−ki, (2.13) φk(t) = 1

NSkhρke^Ωtρ−ki. (2.14) Taking into account the fact that in the thermodynamic limit, the impact of the tracer on the bath dynamics is negligible, we can approximate the bath correlator φk(t) by its F_ex= 0 value, well known from the standard mode-coupling theory.

Despite its vanishing effect on the bulk dynamics, the tracer is still able to locally perturb its neighborhood, and the effect of the external force on this perturbation is contained in φ^s_k(t). This becomes evident if we remember that the tracer-bath density mode products appeared after the factorization approximation of the four- point correlators

hρ^s_kρpe^Ωirrtρ^s_k^′ρp^′i (2.15) and these are correlators of the Fourier modes of the bath density ρ(r−r_s) with respect to the tracer position as the origin.

2.3 The tracer density correlator

The first question to clarify for the time-dependent correlator

hρ^s_ke^Ωtρ^s_k^′i (2.16) of two tracer density modes is, for which pairs of wavevectors k, k^′ is it nonzero?