The β-correlators

In this Section we consider the behaviour of the correlators around the β-relaxation plateau. Here, the ansatz

φ^s(t) = f^s+G^s(t), (4.31)

φ(t) = f+G(t) (4.32)

can be made with the assumptions that f^s, f fulfill the long-time limit equation (4.11) and the β-correlatorsG^s(t), G(t) are small

|G^s(t)|, |G(t)| ≪1. (4.33) Using the same steps (partial integration etc.) as in the derivation of the equation for the long time limit in Appendix A, we rewrite (4.5) as

∂tφ^s(t) =−ω φ^s(t) +m(t)− d dt

Z _t

0 dt^′m(t−t^′)φ^s(t^′) (4.34) and insert (4.31), (4.32) into (4.34). This gives

∂tG^s(t) = −ω(f^s+G^s(t)) +vs(f^s∗+G^s∗(t)) (f+G(t))

− d dt

Z _t

0 dt^′vs(f^s∗+G^s∗(t−t^′)) (f +G(t−t^′)) (f^s+G^s(t^′)). (4.35) Using now the eq. (4.11), retaining only the terms of the order not higher then one in G^s, G and neglecting the time derivative in (4.35), we are left with

G^s(t)

Here we want to consider the Fex-Sj¨ogren model in the liquid (ε < 0) for the case that the external force is smaller than its critical value for ε= 0. This means that the β-relaxation plateau is non-zero.

4.6.1 The β-relaxation

We choosef^sto be the long time limit of the tracer correlator forε = 0 in eq. (4.36).

So, f^s 6= 0 and (4.36) can be considered as a linear inhomogeneous system of equations for G^s₁(t) and G^s₂(t) with externally given G(t).

We express ω/v_s in terms of the well known functions of F_ex, namely f₁^s and f₂^s, using the long-time limit equation (4.12): ω/vs=f^s∗f(1−f^s)/f^s. This leads to

−G^s(t)f^s∗f+G^s(t)^∗f^sf(1−f^s) +G(t)|f^s|²(1−f^s) = 0 (4.37) In terms of G^s₁, G^s₂ (for the sake of simplicity, we suppress the time dependence of the β-correlators for a while), the real and imaginary parts of eq. (4.37) read

−(G^s₁f₁^s+G^s₂f₂^s)f + (G^s₁f₁^s+G^s₂f₂^s)f(1−f₁^s)

+ (G^s₁f₂^s−G^s₂f₁^s)f f₂^s+G|f^s|²(1−f₁^s) = 0 (4.38) (−G^s₂f₁^s+G^s₁f₂^s)f−(G^s₁f₁^s+G^s₂f₂^s)f f₂^s

+ (G^s₁f₂^s−G^s₂f₁^s)f(1−f₁^s) − G|f^s|²f₂^s = 0 (4.39) This equations can be solved in a trivial way with the result:

G^s₁(t) = β1G(t), (4.40)

G^s₂(t) = β2G(t), (4.41)

with

β1 = |f^s|²(2f₁^sf₂^s2+ (2f₁^s+f₂^s2−f₁^s2) (1−f₁^s))

f(4f₂^s2f₁^s(1−f₁^s)−(2f₁^s+f₂^s2−f₁^s2) (f₂^s2−f₁^s2)) (4.42) β2 = β1f(f₂^s2−f₁^s2) +|f^s|²(1−f₁^s)

2f f₁^sf₂^s (4.43)

As a check, we set Fex = 0 and obtain f₂^s = 0 and thus

G^s₁ =G(1−f₁^s)/f. (4.44)

0.2 0.4 0.6 0.8 1.0 F

ex

3.2 3.4 3.6 3.8 4.0

Β

₁

HF

_ex

L

0.2 0.4 0.6 0.8 1.0 F

ex

0.5 1.0 1.5 2.0

Β

2

HF

ex

L

Figure 4.9: The beta-scaling functions of the Fex-Sj¨ogren model (v2 = 2.0, vs = 4.0, F_ex^c = 0.61).

10 15 20

Figure 4.10: Beta correlators, unscaled (upper panel) and scaled (lower panel) (v₂ = 2.0, vs= 4.0, ε=−10⁻¹¹).

Inserting this into (4.41) leads in the limit Fex→0 to G^s₂ =Gf₁^s2(1−f₁^s)−(1−f₁^s)f₁^s2

2f²f₁^sf₂^s , (4.45)

and so to G^s₂ → 0 and to real G^s. Using f₁^s = 1−1/(vsf) we obtain from (4.44) the well-known result of the Sj¨ogren model [33],[78]:

G^s=G 1

vsf². (4.46)

β1, β2 are plotted on Figure 4.9 as functions of Fex for fixed values of v2 and vs. Both functions increase monotonically in the (meaningful) region of the force values F_ex < F_ex^c ). At F_ex= 0, the function β₂(F_ex) starts linearly from zero, whereas the function β1(Fex) exhibits a plateau.

To check our results numerically, we plot the β-correlators, first unscaled and then scaled according to the expressions (4.40), (4.41) on Figure 4.10. We see, thatG^s₁(t), G^s₂(t) indeed collapse on the master curve given byG(t), if one is not too far away from the plateau.

4.6.2 The α -relaxation

We consider now the α-decay region, where the correlator finally relaxes from the β-plateau to zero.

The α-relaxation behaviour of the schematic model without the external force was sketched in Section 1.5.2. We saw that the α-scaling law

φ(t) = ˜φ t t^′_ε

!

, (4.47)

valid asymptotically for ε→ 0, had the consequence that so that on the plot with logarithmic time scale the curves were just shifted to larger times with decreasing

|ε|(see Figure 1.4).

As we saw in Section 4.4, the presence of the external force influences both the time scale of the α-decay and the height of the β-plateau. So, a simple scaling law like (4.47) cannot be valid any more.

We propose the generalized ansatz

φ^s(t) =f^sφ˜^s t τ(Fex, ε)

!

, (4.48)

suggesting that there is still a universal decay function but accounting for the change in the plateau value. ˜φ^s(t) is suggested to be real.

parts) for different forces Fex < F_ex^c with the amplitude rescaled by the factor f₁^s(Fex = 0)

f_i^s(Fex) (4.49)

(i= 1, 2 corresponds to the real and the imaginary part, respectively) so that that all the correlators decay from the same (F_ex= 0 real part) plateau.

We can see that our generalizedalpha-scaling law (4.48) applies only approximately:

the curves are shifted to lower times with increasing F_ex, but their shape varies slightly. The agreement between the real and the imaginary part becomes better with increasing Fex so that for Fex = 1.4 (slightly below the critical force) almost no difference can be seen.

The precision of (4.48) can be considered as acceptable if one realizes that the change of the decay time scale by several orders of magnitude has a much stronger effect than the small change in the shape of the curves.

To determine the alpha-time scale τ, we match (4.48) to the β-decay law:

φ^s_i(t) =f_i^s+βiG(t) =f_i^s(1− βi

where i= 1, 2 corresponds to the real and imaginary part, respectively.

4.7 The critical correlators

In this Section we want to consider the β-correlators at the critical force value F_ex =F_ex^c . So, we set f^s = 0 in (4.36) and obtain

G^s(t)ω+G^s∗(t)vsf = 0. (4.52) This is a linear homogeneous system of equations in terms of G^s₁(t), G^s₂(t):

(1−f vs)G^s₁(t) +FexG^s₂(t) = 0 (4.53)

−FexG^s₁(t) + (1 +f vs)G^s₂(t) = 0 (4.54) Thus, except from the trivial solution G^s₁(t) = G^s₂(t) = 0, the solution of (4.52) exists only when the determinant of the matrix



 1−f vs Fex

−Fex 1 +f vs



 (4.55)

20 22 24 26

log(t)

0 0.1 0.2 0.3 0.4

φ

s 1

(t), φ

s 2

(t), rescaled

φ₁ , F

ex = 0.0 φ₁ , F_ex = 0.5 φ₁ , F_ex = 1.0 φ₁ , F_ex = 1.2 φ₁ , F_ex = 1.4 φ₂ , F_ex = 0.5 φ₂ , F_ex = 1.0 φ₂ , F_ex = 1.2 φ₂ , F_ex = 1.4

Figure 4.11: α-decay of the correlators, rescaled in the amplitude according to their β-plateau values. The parameters are v_s = 6.0 ,v₂ = 2.0, ε=−10⁻¹¹ (this corresponds to F_ex^c = 1.445).

This is exactly the bifurcation condition. If it is fulfilled, the solution of (4.52) is not unique and represents a linear relationship between G^s₁(t) and G^s₂(t). This is just the consequence of the central manifold theorem mentioned in Section 1.5.2.

In our case the codimension of the bifurcation, i. e. the dimension of the critical space is one, as was shown in Section 4.3. Thus at the bifurcation, the correlators G^s₁(t),G^s₂(t) are proportional to the critical eigenvector of the stability matrix and thus to each other.

Considering also the next-to-leading (quadratic) terms in (4.35) and still assuming f^s = 0, we obtain side of (4.56) vanishes due to (4.52) and we obtain the following equation forG^s(t):

0 = d dt

Z t

0 dt^′G^s(t^′)G^s∗(t−t^′). (4.57) This equation can be solved by means of the power-law ansatz

G^s(t) =t^x+i t^y. (4.58)

We get under the integral in (4.57) the expression

t^′x(t−t^′)^x+t^′y(t−t^′)^y+i [t^′y(t−t^′)^x−t^′x(t−t^′)^y]. (4.59)

where Γ(x) is the gamma function, we see that the imaginary part of the right-hand side in (4.57) vanishes. The choice x=y=−1/2 lets also the real part of the right-hand side in (4.57) vanish, since then the denominator in (4.60) diverges.

We found thus the power lawsolution

G^s(t) =t^−1/2+i t^−1/2 (4.61) of the equation (4.57), which gives the critical β-correlator of the Fex-F1 model.

Figure 4.12 shows the critical correlators for different values of the parameter vs.

1e-05 1 1e+05

t

0 0.2 0.4 0.6 0.8 1

φ

s 1

(t), φ

s 2

(t)

φ

^s₁

, v

s

= 1.5 φ

^s₁

, v

_s

= 2.0 φ

^s₁

, v

_s

= 2.5 φ

^s₂

, v

_s

= 1.5 φ

^s₂

, v

_s

= 2.0 φ

^s₂

, v

s

= 2.5

1 1e+05 1e+10

t

1e-06 0.0001 0.01 1

φ

s 1

(t), φ

s 2

(t)

φ

^s₁

, v

_s

= 1.5 φ

^s₁

, v

_s

= 2.0 φ

^s₁

, v

_s

= 2.5 φ

^s₂

, v

_s

= 1.5 φ

^s₂

, v

_s

= 2.0 φ

^s₂

, v

_s

= 2.5 t

^-1/2

Figure 4.12: The critical correlators of the Fex-F1 model for different values of vs. The double logarithmic plot demonstrates the validity of the asymptotic power law.

imaginary part indeed holds asymptotically for large times.

Actually, the solution (4.61) can still be multiplied by an arbitrary prefactor. This expresses the scale invarianceof the eq. (4.57). The correct prefactor can be found by matching to the initial decay.

If we consider now the Fex-Sj¨ogren model, we get from eqs. (4.56), (4.52) G^s∗(t)G(t) = d

dt

Z _t

0 dt^′G^s(t^′)G^s∗(t−t^′). (4.62) This equation has no simple power-law solution, since with the power-law ansatz, the imaginary part of its right-hand side would vanish, whereas the left-hand side would still have a non-vanishing imaginary part.

4.8 The friction coefficient

Within the framework of the schematic models, where no wave vector dependence of the correlators is present, we define the friction coefficient increment following our considerations in Section 2.2 as the time integral over the product of the real part of the tracer correlator and the bath correlator:

∆ζ =

Z ∞

0 dt φ^s₁(t)φ(t) (4.63)

4.8.1 F

_ex

-F1 model

We start by considering first the Fex-F1 model, since exact analytical results are available here. Eq. (4.10) reads in the Laplace space:

−i(zφ^cs(z) + 1) =−(1−i Fex)φ^cs(z)−vs(zφ^cs(z) + 1)φ^cs∗(−z). (4.64) We use the following definition of the Laplace transform

f(z) =ˆ LT[f(t)](z)≡i

For the calculation of the friction coefficient, only the imaginary part of φ^cs(z = 0) we obtain the following system of equations

φc^s₁+Fexφc^s₂ = −vsφc^s₁ (4.71) 1 +vsφc^s₂ = φ^cs₂−Fexφc^s₁, (4.72) which yields

∆ζ =φ^cs₂ = 1 +vs

F_ex² + 1−v²_s (4.73)

So, we have an exact analytical result² and see that the friction coefficient exhibits thinning behaviour with increasing external force.

For vs >1, i. e. above the glass transition, expr. (4.73) can be rewritten as

∆ζ = 1 +vs

F_ex² −F_ex^{c 2}, (4.74)

with

F_ex^c =^qv_s²−1 (4.75)

(according to eq. (4.20) with f = 1).

Note that eq. (4.74) applies only if Fex > F_ex^c , otherwise the tracer is localized and

∆ζ =∞holds. Due to the identity

F_ex² −F_ex^{c 2} = (F_ex−F_ex^c )(F_ex+F_ex^c ), (4.76)

∆ζ diverges at F_ex^c according to the asymptotic power law

∆ζ ∼ 1

Fex−F_ex^c . (4.77)

On Figure 4.13 we plot the numerical and the analytical values of ∆ζ for different vs and observe quite good agreement.

2This nice finding, unfortunately, cannot be transferred toz6= 0, except forFex = 0 [33] or for the (physically uninteresting) model (see the the footnote on p.56) without complex conjugation in the memory function.

0 0.5 1 1.5 2 2.5

t

0 2 4 6 8 10 12

∆ ζ

vs = 0.3 v_s = 0.6 vs = 0.9 vs = 1.0 v_s = 1.2 vs = 1.5

Figure 4.13: Friction coefficient increment from the F_ex-F1 model. The circles show the numerical values, and the continuous lines show the analytical values calculated from eq. (4.73).

4.8.2 F

ex

-Sj¨ ogren model

For the Fex-Sj¨ogren model, the numerical results for the friction coefficient incre-ment are shown on Figure 4.14. We observe the thinning with increasing force and see that for large forces, all the curves collapse on the same master curve. This master curve corresponds to the large Fex limit of the schematic model, where the ω-term, which contains Fex dominates and the memory term can be neglected so that one obtains a 1/F_ex² decay law for ∆ζ.

For ε ≥ 0, the curves diverge at the critical value of force, which increases with ε. Forε →0⁻, two different decay regimes can be distinguished: the strong decay from the initial (linear response) plateau forF_ex < F_ex^c (ε= 0) and the further decay for Fex> F_ex^c (ε= 0), which follows the ε = 0 master curve.

In the initial decay region, on the logarithmic plot the curves just shift upwards with increasing ε so that rescaling their amplitude again leads to the collapse on the same master curve (see Figure 4.15). This result can be explained with the α-decay law derived in Section 4.6.2, since the time integral over the correlators is dominated by the α-decay region.

Due to (4.63), (4.48) and the α-scaling law for the bath correlator we have

∆ζ ≈

From (4.51) we see that theα-decay time factorizes into anε- and anF_ex-dependent term. Thus, by means of rescaling the time, theε-dependence can be factorized out of the integral³

To study the divergence of the curves for ε > 0 at the critical force, we present a double logarithmic plot for ∆ζ as function of Fex−F_ex^c on Figure 4.16. We can see that the power law

∆ζ ∼ 1

(Fex−F_ex^c )^δ (4.79)

holds over several decades in ∆ζ and more than one decade in Fex−F_ex^c . For the model parameters v2 = 2.0, vs = 4.0, the power law fit to the ε = 0 curve yields the value 2.13 for the power law exponent δ, whereas the exponents for higherεare smaller and lie around 1.1. With the other model parameters the ε = 0-exponent seemed to increase. An exact study, however, is complicated here by the numerical instabilities, which occur (especially at the critical force value) already when the parameter vs increases from 4.0 to 6.0.

3Whereas theFex-dependencecannot, since the bath correlator does not depend on Fex.

0 0.2 0.4 0.6 0.8 1 1.2

F

_ex

1e+05 1e+10 1e+15

∆ ζ

ε = -1.0e-3 ε = -1.0e-4 ε = -1.0e-5 ε = -1.0e-6 ε = -1.0e-7 ε = -1.0e-8 ε = 0

ε = 1.0e-4 ε = 1.0e-3

Figure 4.14: Friction coefficient increment from theF_ex-Sj¨ogren model (v₂ = 2.0, v_s = 4.0).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

F

_ex

1e+08 1e+09 1e+10 1e+11 1e+12 1e+13

∆ζ , scaled

ε = -1.0e-9, scaled ε = -1.0e-8, scaled ε = -1.0e-7

Figure 4.15: Scaled friction coefficient increment from the F_ex-Sj¨ogren model (v₂ = 2.0, vs= 4.0) for Fex< F_ex^c . The curves collapse on a master curve.

0.001 0.01

F

_ex

- F

^c_ex

10000 1e+06 1e+08

∆ζ

ε = 0 ε = 1.0e-5 ε = 1.0e-4 ε = 1.0e-3

Figure 4.16: The power law in the friction coefficient increment as function ofF_ex−F_ex^c .

10¹ 10²

Figure 4.17: Fit to simulations

4.9 Comparison with simulations and experiments

4.9.1 The friction coefficient

The Fex-Sj¨ogren model fits to the simulational [65] and experimental [8] curves for the dimensionless tracer friction coefficient increment are presented on Figures 4.17 and 4.18. The bath volume fractions ϕ and the values of the parameters ε and vs

of the schematic model are specified. The value of the parameter v2 is chosen to be v₂ = 2.0 (see Section 1.5.2). The insets show the separation parameterεas function of ϕ. Two ϕ-independent scaling factors for the force and the friction were used.

From Figure 4.17 we see that the schematic model describes the divergence of the linear response friction value with ε→0 as well as the strong decay from the linear response plateau over more than one decade in the force and in ∆ζ/ζ0. The second (much less density dependent), high force plateau is not contained the schematic model.

The experimental curves do not show the linear response plateau, because the corresponding velocities were to small to be resolved. However, the range (10 till 10⁵) of the variation of the friction is much larger than in the simulations. This

0.01 0.1 1 10

Figure 4.18: Fits to experiments

is probably related to the larger ratio of the tracer and the bath particle radii.

The schematic model is still able to capture the strong divergence of the friction coefficient near the threshold force. This requires, however, the coupling parameter vs also to be varied over much longer range than in the simulation fits.

It would be very interesting to test the predictions of the schematic model for the divergence of the friction coefficient at the critical force. However, the statistics of the present experimental and sumulational data is not yet good enough for such a test to be reliable.

4.9.2 The correlators

The simulation results [65] for the tracer density correlator are shown on Figure 4.19. The wave vector (corresponding to the average particle separation) was taken in the force direction. The schematic model parameters are the same as in the fit from the last section, and the force parameter of is given in scaled units with the scaling factor used for the fit.

We see that the β-plateau goes down with increasing force continuously and for F > 30, roughly above the estimative critical force value (see Section 3.3), the

0.1 1 10 100

Figure 4.19: Tracer correlators from the simulation (upper plots) and from the schematic model (lower plots).

curves start to oscillate.

Whereas the overall features of the simulation curves are well captured by the schematic model, the details (like e. g. the oscillation frequency for largeFex) depend on the wave vector magnitude (and direction) and should be compared to the full microscopic version of MCT equations.

Summary and Outlook

The main objective of this work was to extend the standard mode-coupling theory for the motion of a tracer particle in a dense colloidal suspension near the glass transition to the case, where the tracer experiences the external force F_ex^c , which cannot be assumed to be small compared to the internal units of the system. This means, an attempt is made to go beyond the linear response regime. We follow the ideas of the integration through transientsapproach [56], recently developed for sheared systems.

In Chapter 1, the usual linear response theory is reviewed. The difference between the two pulling regimes is studied (constant force and the constant velocity pulling), which in the low density limit was discussed in Ref. [20]. As a new result, we predict that for a hard sphere system near the decoupling transition of the tracer from the surrounding glassy matrix, the difference in the observed tracer friction coefficient can become very large (Section 1.7).

Chapter 2 starts with the nonlinear theory. In Section 2.1, an exact relation is derived for the response of an observable to the external force on the tracer, gen-eralizing the Green-Kubo relations well known from the linear response theory. In Section 2.2, we generalize the mode-coupling approximation for the tracer friction coefficient and in Section 2.3, we consider the time-dependent tracer density mode correlator φ^s_q, which is the central quantity in our mode-coupling approach.

The presence of the external force leads to a drastic difference compared to the linear response case: the tracer density correlator becomes complex. This is the consequence of the fact that instead of the unperturbed Smoluchowski operator Ω0, the full operator Ω containing the external force enters φ^s_q. Ω turns out to be non-hermitian with respect to the equilibrium average, as the consequence of the fact that we consider an open system. Interestingly, in the mode-coupling theory under shear the density correlators are prevented from becoming complex due to the reflection symmetry k→ −k, special for the geometry of a simple shear flow.

83

tem. We observe thinningbehaviour (Figure 2.2) of the friction coefficient with the external force and obtain the correct scaling with the relevant parameters (eq. 2.39) and also the correct dimensionless external force parameter (eq. 2.41). Section 2.5 is devoted to the general discussion of the conditions, which are necessary in order that the observables like the friction coefficient stay real in our theory.

InChapter 3 we calculate the long-time limit of the tracer density correlator for the hard sphere system in the glass. In Section 3.1 the existence of the critical forceF_ex^c is shown, at which a continuous bifurcation transition of the long-time limit of the tracer density correlator occurs. For Fex > F_ex^c , the long-time limit becomes zero and thus the cage surrounding tha tracer breaks, it becomes delocalized and can be pulled though the suspension. The critical force increases with the bath packing fraction (Figure 3.3).

In Section 3.2 the transformation from the Fourier to the real space is performed, giving the probability density of finding the tracer at a certain space point for t → ∞after it was localized at the origin fort= 0. This probability density remains positive over a considerable spatial range within our mode-coupling approximation.

On Figures 3.7, 3.8, 3.9 one can see how the cage, being rotationally symmetric in the absence of the external force, deforms if the external force is increased.

The comparison to simulations and experiments (Section 3.3) yields reasonable agreement in the value of the critical force and in the deformation of the cage shape upon increasing the external force.

In Chapter 4 we construct the schematic models by considering only two wave vectors parallel to the external force. Two different models are considered: the

“F_ex-Sj¨ogren model”, extending the Sj¨ogren model [78] for the tracer coupled to a bath and the “Fex-F1 model”, extending theF1 model of standard MCT, which was used to describe the tracer in a matrix of immobile particles (the Lorentz model).

In Section 4.2 the long-time limit of the tracer correlator is calculated analytically and the resulting phase diagrams are presented (Figure 4.2). The bifurcation transi-tion at the critical forceFex =F_ex^c is studied in detail in Section 4.3. The bifurcation is shown to have the codimension one.

The time-dependent equations of motion of the schematic models are solved numer-ically (see Appendix C.3 for details) to study the effect of the external force on the tracer correlator (Section 4.4).

In Section 4.5 we derive the general equation for theβ-correlators (eq. (4.35)), which is used in the following sections. In Section 4.6 we consider the F_ex-Sj¨ogren model in the liquid for the fore smaller than the critical one, where the β-plateau is still present. In 4.6.1 we calculate the β-correlators to the lowest order. In 4.6.2 we consider the α-decay region and propose a generalized α-scaling relation (eq. 4.48),

which together with the results from 4.6.1 allow us to derive an expression for the α-scaling time (eq. 4.51).

In Section 4.7, theβ-correlators exactly at the bifurcation transition are considered.

For theFex-F1 model an exact power law solution with the exponent−1/2 is found.

The force dependence of the tracer friction coefficient increment ∆ζ within the schematic models is studied in Section 4.8. Generally, the thinning behaviour is observed similar to the shear thinning in macrorheology. For the Fex-F1 model, an exact analytic expression (eq. 4.73) can be derived, showing a power law divergence with the exponent −1 at the critical force.

For the Fex-Sj¨ogren model, we use the expression (4.51) for the α-scaling time to show that for the initial strong decay from the linear response plateau, the ∆ζ vs. Fex curves for different ε in the limit ε → 0⁻ differ just by an ε-dependent prefactor so that after appropriate rescaling all the curves collapse on a master curve (Figure 4.15). For ε >0, a power law divergence at the critical force is found numerically (Figure 4.16).

In Section 4.9 a comparison to simulation and experimental results is made. The tracer correlators from the simulation are compared to that of theFex-Sj¨ogren model in 4.9.2. For q-vectors parallel to the external force, the simulations results are in qualitative agreement: the β-plateau goes down with increasing force and for large forces, the correlators start to oscillate (Figure 4.19).

In 4.9.1, we fit the results for the tracer friction coefficient (Figures 4.17, 4.18).

The strong decay from the linear response plateau for Fex < F_ex^c is captured by the schematic model. For lager forces the schematic model predict the friction coefficient increment to decay to zero, whereas in simulations and experiments the non-zero second plateau is observed.

This thesis represents (to our knowledge) the first attempt to construct a mode-coupling theory for the forced tracer motion beyond linear response. We discussed

This thesis represents (to our knowledge) the first attempt to construct a mode-coupling theory for the forced tracer motion beyond linear response. We discussed

Im Dokument Active and passive particle transport in dense colloidal suspensions (Seite 75-0)