Scaling laws in the rheology of colloidal dispersions

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Scaling laws in the rheology of colloidal dispersions

Diploma Thesis by

David Hajnal

Department of Physics University of Konstanz, Germany

April 2, 2007

Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3800/

URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-38002

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Zusammenfassung

In der vorliegenden Arbeit wird, ausgehend von einem schematischen Modenkopplungsmodell, die Dynamik und das Flieÿverhalten von kolloidalen Dispersionen unter stationärer Scherung nahe am Glasübergang untersucht. Bei dem gewählten Zugang werden sowohl analytische als auch numerische Methoden eingesetzt. Die wichtigsten Kontrollparameter des Modells sind die Scherrate und der Separationsparameter, welcher den Abstand zum Glasübergangspunkt misst.

Ein negativer Separationsparameter deniert einen üssigkeitsartigen Zustand während ein pos- itiver Wert für den Separationsparameter zu einem glasartigen Zustand führt. Nach Diskussion der grundlegenden Eigenschaften des Modells und des dazugehörigen numerischen Algorithmus gliedert sich der Hauptteil der Arbeit in drei Kapitel:

Der β -Prozess

Nach Einführung der Bewegungsgleichung und Diskussion des Gültigkeitsbereiches wird eine sehr nützliche Eigenschaft, das Zweiparameterskalengesetz, diskutiert. Dies erlaubt eine präzise Denition des Flüssigkeits-, Übergangs- und Glasbereiches und die Einführung natürlicher Zeit- skalen. Es werden analytische Ausdrücke zur Beschreibung der Dynamik in den drei Bereichen für alle relevanten Zeitskalen abgeleitet und die Ergebnisse werden numerisch überprüft. Die Dynamik kann durch verschiedene verallgemeinerte Potenzreihen und einer Exponentialfunk- tion beschrieben werden. Vier dieser Potenzreihen sind notwendig um die Dynamik im Flüs- sigkeitsbereich zu beschreiben. Die Dynamik im Übergangsbereich kann durch zwei Potenzrei- hen beschrieben werden. Um die Dynamik im Glasbereich auf allen Zeitskalen zu beschrieben, werden drei Potenzreihen und eine Exponentialfunktion benötigt. Die Kurzzeitasymptoten und die scherdominierte Langzeitasymptoten hängen nicht vom Separationsparameter ab.

Flieÿkurven

In diesem Kapitel wird die stationäre Scherspannung als Funktion der Scherrate untersucht. Nach Diskussion der grundlegenden Eigenschaften der Flieÿkurven führt eine asymptotische Entwick- lung unter Zuhilfenahme des analytischen Ergebnisses für die Langzeitdynamik und des Zweipa- rameterskalengesetzes für den β-Prozess auf einen analytischen Ausdruck für die Flieÿkurven, dem Λ-Modell. Dieses Modell kann in drei Spezialfällen analytisch ausgewertet werden. Die Ergebnisse sind Potenzreihen mit nicht ganzzahligen Exponenten. Im allgemeinen Fall sind Näherungen durch numerisch motivierte Formeln notwendig. Für kleine positive Separationspa- rameter beschreibt dasΛ-Modell die Flieÿkurven für hinreichend kleine Scherraten korrekt. Für kleine negative Separationsparameter beschreibt dasΛ-Modell die Flieÿkurven in endlichen Fen- stern für die Scherrate korrekt. Schlieÿlich wird der bei negativen Separationsparametern und bei doppelt logarithmischer Auftragung der Flieÿkurven auftretender Wendepunkt untersucht. Die Approximation der Flieÿkurve durch die entsprechende Wendetangente führt zu einem eektiven Potenzgesetz.

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Analyse experimenteller Daten

Im letzten Kapitel werden experimentelle Daten zum Flieÿverhalten und zur linearen Viskoe- lastizität einer thermosensitiven Kern-Mantel-Dispersion analysiert. Es wird ein erweitertes schematisches Modell eingeführt um die Flieÿkurven und das Speicher- und Verlustmodul si- multan an die Daten anzupassen. Die allgemeinen Eigenschaften dieses erweiterten Modells und das Verfahren zur Datenanpassung werden ausfühlich erläutert. Bei niedrigen eektiven Packungsbrüchen und hinreichend groÿen Scherraten beziehungsweise Frequenzen, bei denen die Probe keine Kristallisation aufweist, können die Flieÿkurven und die dazugehörigen Module unter Verwendung des erweiterten schematischen Modells qualitativ und quantitativ korrekt an die experimentellen Daten angepasst werden.

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Abstract

In this thesis, starting from a schematic mode-coupling model, the dynamics and the ow behavior of colloidal dispersions under stationary shearing close to the glass transition is analyzed.

Both analytical and numerical methods are applied. The most important control parameters of the model are the shear rate and the separation parameter measuring the distance from the glass transition point. A negative separation parameter denes a liquid-like state and a positive value for the separation parameter leads to a glassy state. After discussing the basic properties of the model and the corresponding numerical algorithm, the main part of the thesis subdivides into three chapters:

The β -relaxation process

After motivating the equation of motion and discussing the range of validity, a useful aspect, the two-parameter scaling law, is discussed. This allows a precise denition of the liquid-, transition- and the yielding glass region and the introduction of natural time scales. Analytical expressions are derived to describe the dynamics in the three dierent regions for all relevant time scales and the results are tested numerically. The dynamics can be described by dierent generalized power series and an exponential function. Four of these power series are necessary to describe the dynamics in the liquid region. The dynamics in the transition region can be described by two power series. In the yielding glass region, three power series and an exponential function are necessary to describe the dynamics on all time scales. The short time asymptotes and the shear-dominated long time asymptotes are not dependent on the separation parameter.

Flow curves

In this chapter the steady state shear stress as a function of the shear rate is analyzed. After discussing the basic properties of the ow curves, an asymptotic expansion using the analytical expression for the long time dynamics and the two-parameter scaling law for the β-relaxation process leads to an analytical expression for the ow curves, the Λ model. This model can be evaluated analytically in three special cases. The results are power series with non-integer ex- ponents. In the general case approximations using numerically motivated formulae are required.

For small positive separation parameters theΛ model describes the ow curves correctly for suf- ciently small shear rates. For small negative separation parameters the Λ model describes the ow curves correctly in some nite shear rate windows. Finally, the inection point occurring for negative separation parameters in a double logarithmic plot of the ow curves is analyzed.

An approximation of the ow curve by the corresponding inection tangent leads to an eective power law.

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Analysis of experimental data

In the last chapter experimental data for the ow behavior and the linear viscoelasticity of a thermosensitive core-shell dispersion are analyzed. An extended version of the schematic model is introduced to t the ow curves and the storage- and loss moduli simultaneously. The general properties of the extended model and the tting procedure are discussed in detail. For low eective packing fractions and suciently large shear rates respectively frequencies, such that no crystallization occur in the sample, the ow curves and the corresponding moduli can be tted qualitatively and quantitatively correctly by the extended schematic model.

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Acknowledgments

There have been some people who supported me during my work on this thesis. First of all I want to thank Professor Matthias Fuchs for oering me this interesting topic which allowed me to work both with analytical and numerical methods. I also want to thank Professor Peter Nielaba for his willing to be the co-examiner. The whole soft matter theory group group was very kind and I enjoyed working with them during the last year. Thanks to Oliver Henrich for the helpful discussions. Thanks to Dr. Thomas Voigtmann for the discussions on the numerical algorithm. Special thanks to Dr. Joseph Brader for reading the manuscript of this thesis and for his helpful suggestions. I also thank Professor Matthias Ballau, Jerome Crassous and Miriam Siebenbürger for providing experimental data and for the interesting discussions which motivated the last chapter of this thesis. Finally, I want to thank my parents for the nancial support and the helpful suggestions during my studies.

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Introduction

1.1 Motivation

Colloidal dispersions like hard spheres in a solvent behave, if they are suciently diluted, like liquids. By increasing the packing fraction above some critical value a transition to an amorphous solid, a glassy state, can be observed, provided that crystallization eects are suppressed. The mechanism for this behavior is the cage eect: Each colloidal particle is arrested in a cage formed by the next neighbor particles. The development of a self consistent microscopic theory describing many aspects of the dynamics of glass forming systems, the mode-coupling theory of the glass transition, goes back to W. Götze. An overview can be found in [1] and [2]. The mode-coupling theory considers time and wave vector dependent correlation functions of density uctuations. The equations of motion are coupled nonlinear integro-dierential equations which are regular in all control parameters and dene functions with stretched dynamics, hence they have to be considered on logarithmic time scales. The ideal glass transition is described by a bifurcation of the long time limit of the correlators. Predictions of the mode-coupling theory are compared with experimental results in [3]. The universal aspects of the theory can be discussed in the framework of schematic models as rst done in [4]. To study rheological properties of colloidal dispersions the mode-coupling theory was extended by M. Fuchs and M. E. Cates by introducing a stationary shear ow [5]. A detailed analysis can be found in [6]. In the following we study the universal aspects of this extended theory in the framework of a schematic model rst introduced in [7].

1.2 The schematic F

₁₂

model

A simplied model including the ideal glass transition and the stretched glassy dynamics considers one normalized correlator φ(t), with the initial condition φ(0) = 1, which obeys an integro- dierential equation:

1

Γφ˙(t) +φ(t) + Z _t

0 m^¡t−t⁰^¢φ˙^¡t⁰^¢dt⁰ = 0. (1.1) Because we assume colloidal particles with Brownian dynamics [1], only rst-order time derivatives occur in equation (1.1). The parameterΓ>0sets the time scale of the system. A vanishing memory kernel m(t) would lead to an exponential decay of the correlator. To model the cage eect we choose [2]:

m(t) ≡ v₁φ(t) +v₂φ²(t). (1.2) Increasing the particle caging is only modeled by increasing the coupling parameters v₁, v₂ ≥0.

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Because of ^R₀^tm(t−t⁰) ˙φ(t⁰)dt⁰ →0for t→0the correlator obeys the initial condition:

φ(t→0) = 1−Γt. (1.3)

1.2.1 The long time limit and some general properties

In a glass, because of the cage eect, enforced density uctuations for t = 0 can not decay to zero for t→ ∞ like in a liquid [1]. Hence we consider the glass form factor

f ≡ lim

t→∞φ(t) and dene

• the liquid state: f = 0.

• the glassy state: f >0.

If we assume that f ≥0exists, then because of equation (1.2) the limit g ≡ lim

t→∞m(t)

also exists, hence we can assume φ(t) = f +f₀(t) and m(t) = g+g₀(t) with f₀(t) → 0 and g₀(t)→0fort→ ∞. Using this, equation (1.1) and^R₀^tf˙₀(t⁰)dt⁰ =f₀(t)−f₀(0) =f₀(t) +f−1 leads to:

1

Γf˙₀(t) +f+f₀(t) +g(f₀(t) +f−1) + Z _t

0 g₀^¡t−t⁰^¢f˙₀^¡t⁰^¢dt⁰ = 0. (1.4) We further assume that φ(t) is completely monotone. This means that for all t ≥ 0 and n = 0,1,2, . . . the relation

(−1)ⁿ ∂ⁿ

∂tⁿφ(t) ≥ 0

is satised. Then the limit f˙₀(t) → 0 for t → ∞ is obvious. First we dene g₀(t) for negative arguments: g₀(−t) ≡ g₀(t). Because g₀(t) is bounded, |g₀(t)| ≤ g_max, we can consider R_∞

0

¯¯

¯g₀(t−t⁰) ˙f₀(t⁰)^¯^¯¯dt⁰ ≤g_max^R₀^∞^¯^¯¯f˙₀(t⁰)^¯^¯¯dt⁰ =g_max^¯^¯¯R_∞

0 f˙₀(t⁰)dt⁰^¯^¯¯=g_max|f −1|<∞. With this we can show that the integral in equation (1.4) vanishes fort→ ∞: ^¯^¯^¯^R₀^tg₀(t−t⁰) ˙f₀(t⁰)dt⁰

¯¯

¯≤ R_∞

0

¯¯

¯g₀(t−t⁰) ˙f₀(t⁰)^¯^¯¯dt⁰ →^R₀^∞^¯^¯¯0·f˙₀(t⁰)^¯^¯¯dt⁰ = 0for t→ ∞. We can conclude: If φ(t) is a completely monotone solution of the F₁₂ model with long time limit f, then the following equation is satised:

f+g(f −1) = 0. (1.5)

Because of equation (1.2) we conclude g=v₁f+v₂f². Obviouslyf = 0is a special solution of equation (1.5). The question, which solution of equation (1.5) represents the correct glass form factor, can be answered by the following theorems proven by W. Götze and L. Sjögren in a more general framework [8]:

• TheF₁₂model has a unique solution. It is dened for allt≥0and is completely monotone.

• The long time limit of the solution is given by the maximum solution of equation (1.5).

In the following we analyze under which conditions equation (1.5) has nontrivial solutions.

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1.2.2 The ideal glass transition

To nd out for which values of v₁ andv₂ equation (1.5) has positive solutions, we dene:

F(f) ≡ v₂f²+ (v₁−v₂)f + (1−v₁). With this and the assumption f 6= 0 we can rewrite equation (1.5):

F(f) = 0. (1.6)

Values of v₁, v₂ lying on a bifurcation line we call critical points v₁^c, v^c₂ and the corresponding solutionsf_c critical glass form factors. First we are interested in the type B transition line with f_c > 0. By crossing this line the two complex solutions of equation (1.6) merge into two real solutions, so that F(f_c) = 0 and _∂f^∂F(f)

¯¯

¯f=fc

= 0, or written explicitly,

v₂^cf_c²+ (v₁^c−v₂^c)f_c+ (1−v₁^c) = 0 (1.7) and

2v₂^cf_c+ (v^c₁−v^c₂) = 0 (1.8) have to be satised. Equation (1.8) leads to f_c= ^v^c²_2v^−vc¹^c

2 . Obviously f_c>0 requiresv^c₂ > v₁^c. By using these results and equation (1.7) we obtain:

v₁^c = v₂^c Ãs4

v₂^c −1

!

, (1.9)

f_c = 1− 1

pv^c₂. (1.10)

Equation (1.9) and equation (1.10) are valid for 1< v^c₂≤4 so that we can conclude0< f_c≤ ¹₂, f_c= ¹₂ for v^c₂= 4 and f_c→0 for v^c₂ →1. The type B transition line is shown in gure 1.1. We now analyze the type A transition line withf_c= 0. By crossing this line one of the two negative nontrivial solutions of equation (1.7) merges into a positive solution, so that −4v₂^c(1−v^c₁) = 0 has to be satised. We conclude:

v₁^c = 1, (1.11)

f_c = 0. (1.12)

Equation (1.11) and equation (1.12) are valid for 0 ≤ v₂^c ≤ 1. The type A transition line is also shown in gure 1.1. We notice that the type A transition line continuously merges into the type B transition line. For the following we are only interested in type B transitions. Close to the transition line we can write v₁ =v₁^c+δv₁, v₂ =v^c₂+δv₂ and f =f_c+δf with some small quantitiesδv₁,δv₂ andδf. We follow [7] and dene the exponent parameter

λ ≡ 1−f_c

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and the separation parameter measuring the distance from the critical point:

² ≡ f_cδv₁+f_c²δv₂ 1−f_c .

Obviously, ² <0 leads to a liquid while ² >0 leads to a glassy state. With these denitions we can rewrite equation (1.9) and equation (1.10):

v₁^c = 1

λ² (2λ−1), (1.13)

v₂^c = 1

λ². (1.14)

Equation (1.13) and equation (1.14) are valid for ¹₂ ≤λ <1. Finally we analyze δf to leading order in ². We substitute v₁ =v₁^c+δv₁, v₂ = v^c₂+δv₂ and f =f_c+δf in equation (1.6), use equation (1.7) and solve the resulting quadratic equation for δf. For the limit ²→0 we obtain to leading order:

f(²→0) = f_c+ (1−f_c)² r ²

1−λ. (1.15)

0 0.5 1 1.5 2

v₁ 0

1 2 3 4

v2

f_c>0 f_c=0

glass

liquid

Figure 1.1: The phase diagram of the schematic F₁₂ model. By crossing the type B transition line the glass form factor f jumps from zero (liquid) tof_c>0(glass). A path crossing the type A transition line leads to a continuous variation off.

1.2.3 The numerical algorithm

I. Hofacker [9], supported by M. Fuchs, W. Götze and A. Latz [10], has developed an ecient multi-grid algorithm for solving a set of coupled nonlinear integro-dierential equations on logarithmic time scales. The algorithm was later extended and improved by A. P. Singh [11], Th.

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Voigtmann [12], M. Fuchs and others [13]. We follow [13] and specialize the algorithm for solving equation (1.1). The algorithm makes use of the fact that for long times the solutions of equation (1.1) are slowly varying functions (the solutions are completely monotone, hence the time derivatives of all orders monotonically decay to zero). First we choose some ¯t < t and consider the convolution integral

I(t) ≡ Z _t

0 m^¡t−t⁰^¢φ˙^¡t⁰^¢dt⁰

in equation (1.1): I(t) = ^R₀^¯^tm(t−t⁰) ˙φ(t⁰)dt⁰ +^R_¯_t^tm(t−t⁰) ˙φ(t⁰)dt⁰. After partial integration, substituting t⁰⁰ = t−t⁰ in the second integral and renaming t⁰⁰ to t⁰ we obtain I(t) = m(t−¯t)φ(¯t)−m(t)φ(0) +I₁(t,¯t) +I₂(t,¯t)withI₁(t,¯t)≡^R₀^¯^tm˙ (t−t⁰)φ(t⁰)dt⁰ and I₂(t,¯t)≡ R_t−_¯_t

0 φ˙(t−t⁰)m(t⁰)dt⁰. The time derivatives in the integrands are slowly varying functions in the considered integration sets if we assume for example ¯t ≈ ₂^t. Now we consider a system of time grid points

t^d_i ≡ i·h^d with0≤i≤N−1,

h^d ≡ h⁰·2^d

withh⁰ >0, the grid indexd= 0,1,2, . . .and we assume thatN ≥4is a power of2. To switch form grid d−1 to grid dmeans to double the step size. We dene the function values on the grid points:

φ^d_i ≡ φ^³t^d_i^´,

φ˙^d_i ≡ φ˙^³t^d_i^´,

m^d_i ≡ m^³t^d_i^´,

I_i^d ≡ I^³t^d_i^´. Because of equation (1.2) we conclude:

m^d_i ^hφ^d_iⁱ = v₁φ^d_i +v₂^³φ^d_i^´². (1.16) Now for ¯ι < i we can writeI₁^³t^d_i, t_¯^d_ι^´=^P^¯^ι_k=1^R_t^td^d^k

k−1

˙

m^³t^d_i −t⁰^´φ(t⁰)dt⁰ and I₂^³t^d_i, t_¯^d_ι^´=^P^i−¯_k=1^ι ^R_t^td^d^k

k−1

φ˙^³t^d_i −t⁰^´m(t⁰)dt⁰. Because of the complete monotony ofφ(t)andm(t) we can apply the generalized mean value theorem: I₁^³t^d_i, t^d_¯_ι^´=^P^¯^ι_k=1m˙ ^³t^d_i −t^#d_k ^{´ R}_t^td^d^k

k−1

φ(t⁰)dt⁰ and I₂^³t^d_i, t^d_¯_ι^´ = ^P^i−¯_k=1^ι φ˙^³t^d_i −t^∗d_k ^{´ R}_t^t_d^d^k

k−1

m(t⁰)dt⁰ with some t^#d_k , t^∗d_k ∈ ^ht^d_k−1, t^d_kⁱ. For 1 ≤ k ≤ N −1 we dene the moments:

dΦ^d_k ≡ 1 h^d

Z _td k

t^d_k−1φ^¡t⁰^¢dt⁰,

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dM_k^d ≡ 1 h^d

Z _td k

t^d_k−1m^¡t⁰^¢dt⁰.

The slowly varying time derivatives can be approximated bym˙ ^³t^d_i −t^#d_k ^´≈ _h¹d

³

m^d_i−k+1−m^d_i−k^´ and φ˙^³t^d_i −t^∗d_k ^´≈ _h¹d

³

φ^d_i−k+1−φ^d_i−k^´. With this we can approximate the convolution integral:

I_i^d ≈ m^d_i−¯_ιφ^d_¯_ι −m^d_iφ^d₀+

¯

Xι

k=1

³

m^d_i−k+1−m^d_i−k^´dΦ^d_k+

i−¯ι

X

k=1

³

φ^d_i−k+1−φ^d_i−k^´dM_k^d. (1.17) To approximate the time derivative in equation (1.1) we use a well known formula from the dierentiation of interpolation polynomials:

1

Γφ˙^d_i ≈ D¯^d_i ^hφ^d_j<iⁱ+D^dφ^d_i, (1.18)

D¯_i^d^hφ^d_j<iⁱ ≡ 1 Γh^d

µ1

2φ^d_i−2−2φ^d_i−1

¶ ,

D^d ≡ 3 2

1 Γh^d. At last we dene:

A^d ≡ 1 +dM₁^d+D^d,

B^d ≡ φ^d₀−dΦ^d₁,

Σ^d_i ≡

¯

Xι

k=2

³

m^d_i−k+1−m^d_i−k^´dΦ^d_k+

i−¯ι

X

k=2

³

φ^d_i−k+1−φ^d_i−k^´dM_k^d,

C_i^d ≡ Σ^d_i +m^d_i−¯_ιφ_¯^d_ι −m^d_i−1dΦ^d₁−φ^d_i−1dM₁^d+ ¯D_i^d^hφ^d_j<iⁱ.

By using these denitions, equation (1.17) and equation (1.18) we obtain the discretized version of equation (1.1):

A^dφ^d_i = B^dm^d_i ^hφ^d_iⁱ−C_i^d. (1.19) With the choice

¯ι ≡

¹i 2 º

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only moments with 1 ≤ k ≤ ^N₂ occur in equation (1.19) for each grid. The rst step of the numerical algorithm is the initialization: We use the initial condition (1.3) to initialize the rst grid by

φ⁰_i ≈ 1−Γt⁰_i and

m⁰_i ≈ m⁰_i ^hφ⁰_iⁱ

for 0≤i≤N−1, approximate the moments by using the trapezoid rule, dΦ⁰_k ≈ 1

2

³

φ⁰_k−1+φ⁰_k^´

and

dM_k⁰ ≈ 1 2

³

m⁰_k−1+m⁰_k^´

for 1≤k≤ ^N₂, and consider grid0 as solved. The second step is the time-domain decimation:

We assume that gridd−1is solved. Then we initialize grid dby φ^d_i ≡ φ^d−1_2i

and

m^d_i ≡ m^d−1_2i

for 0≤i≤ ^N₂ −1. Using the denitions of the moments we easily verify dΦ^d_k = 1

2

³

dΦ^d−1_2k−1+dΦ^d−1_2k ^´ (1.20) and

dM_k^d = 1 2

³

dM_2k−1^d−1 +dM_2k^d−1^´ (1.21) for 1≤ k≤ ^N₂ −1. Because we assume that we already know the moments for 1≤ k≤ ^N₂ on gridd−1, we can use them directly to calculate the moments for1≤k≤ ^N₄ −1on gridd. Using equation (1.20), equation (1.21) and the trapezoid rule on grid d−1 lead us to

dΦ^d_k ≈ 1 4

³

φ^d−1_2k−2+ 2φ^d−1_2k−1+φ^d−1_2k ^´

and

dM_k^d ≈ 1 4

³

m^d−1_2k−2+ 2m^d−1_2k−1+m^d−1_2k ^´

for ^N₄ ≤k≤ ^N₂ −1. The last moments we easily approximate by dΦ^dN

2 ≈ φ^d−1_N−1 and

dM^dN

2 ≈ m^d−1_N−1.

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Now we can consider equation (1.19). It is an implicit equation for φ^d_i depending on all values φ^d_j<i. Starting withi= ^N₂ we can calculateφ^d_i for ^N₂ ≤i≤N−1by the iteration

³ φ^d_i^´

n ≡ 1

A^d µ

B^dm^d_i ^·³φ^d_i^´

n−1

¸

−C_i^d

¶

(1.22) with the initial value

³φ^d_i^´

0 ≡ φ^d_i−1.

Each new decimation step leads to ^N₂ new data points with the double step size compared with the previous decimation step. Obviously, for the initialization step the condition(N −1) Γh⁰ ¿1 has to be satised. The iteration of equation (1.22) has to be continued until the desired accuracy for φ^d_i is reached.

1.2.4 Examples

To obtain universal results, we introduce a rescaled time:

˘t ≡ Γt.

With the denitions

φ˘^³˘t^´ ≡ φ(t),

˘

m^³˘t^´ ≡ v₁φ˘^³˘t^´+v₂^³φ˘^³˘t^´´², (1.23) equation (1.1), equation (1.2) and equation (1.3) we easily obtain:

∂

∂˘tφ˘^³˘t^´+ ˘φ^³˘t^´+ Z _˘_t

0 m˘ ^³˘t−˘t⁰^´ ∂

∂˘t⁰φ˘^³˘t⁰^´dt˘⁰ = 0, (1.24)

φ˘^³t˘→0^´ = 1−˘t. (1.25)

With this the explicit dependency on Γ is eliminated. This can also easily be done in the numerical algorithm. Hence in all numerical calculations we will measure time in units of Γt. We choose N = 256, Γh⁰ = 10⁻⁶ and we iterate equation (1.22) until

¯¯

¯¯(φ^d_i)_n−(φ^d_i)_n−1 (φ^d_i)_n

¯¯

¯¯≤ 10⁻¹⁶ or n = 100000 is reached. We follow [7] and choose for all numerical examples v₂ = v₂^c and v₁ =v₁^c+^1−f_f ^c

c ². Figure 1.2 shows numerical solutions of the F₁₂ model forv₂^c= 2and dierent values for ². After the ²-independent short time dynamics determined by Γ the correlators for

² < 0 decay to zero with time scales determined by ². For ² → 0⁻ an increasingly stretched plateau is developed before the nal decay to zero. For ²≥0 the correlators arrest at positive plateau values determined to leading order by equation (1.15).

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-2 0 2 4 6 8 10 12 log₁₀(Γt)

0 0.2 0.4 0.6 0.8 1

φ

ε=0 ε=-10^-4 ε=-10^-3 ε=-10^-2 ε=10^-4 ε=10^-3 ε=10^-2

Figure 1.2: Numerical solutions of theF₁₂model forv^c₂= 2. The short time dynamics determined byΓis not dependent on². For² <0the correlators decay to zero with²-dependent time scales.

For ² → 0⁻ an increasingly stretched plateau is developed before the nal decay to zero. For

²≥0the correlators arrest at positive plateau values.

1.3 A system under shear: The F

₁₂^{( ˙}^γ)

model

To take account of the shear-induced decorrelation we, following [7], extend the F₁₂ model and replace equation (1.2) by:

m(t) ≡ 1

1 + ( ˙γt)²

³

v₁φ(t) +v₂φ²(t)^´. (1.26) The parameterγ˙ is the shear rate. The only eect of shearing is to cause a time dependent decay of the memory kernel. Obviously the system is symmetric in γ˙, hence it only depends on |γ|˙ . 1.3.1 The long time limit and some comments

As discussed above, the model without shearing has a unique for all t ≥ 0 dened completely monotone solution. Until now, this has been proven only for memory kernels without explicit time-dependency [8]. First we consider the limit|γ˙| → ∞. In this case the solution isφ(t) =e^−Γt which obviously has all properties discussed above. The case |γ˙|= 0 reproduces the F₁₂ model without shearing. Because of these two limits it is reasonable to assume that theF₁₂^{( ˙}^γ)model with 0<|γ|˙ <∞also has a unique for all t≥0dened completely monotone solution. Then we can conclude that the long time limit of the solution of theF₁₂^{( ˙}^γ) model is a solution of equation (1.5) and the numerical algorithm described above can be extended for solving the F₁₂^{( ˙}^γ) model. For

|γ| 6= 0˙ because of equation (1.26) we obtain g= 0. With this and equation (1.5) we conclude:

• For a nite shear rate the long time limit of the solution of theF₁₂^{( ˙}^γ) model is zero.

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We emphasize that the discussed properties of the F₁₂^{( ˙}^γ) model remain unproven. The main justication of our assumption is given by numerical results.

1.3.2 The numerical algorithm

To extend the numerical algorithm for solving the F₁₂^{( ˙}^γ) model we replace m^d_i ^hφ^d_iⁱ dened by equation (1.16) by:

m^d_i ^ht^d_i, φ^d_iⁱ = 1 1 +^¡γt˙ ^d_i^¢²

µ

v₁φ^d_i +v₂^³φ^d_i^´²

¶ .

Everything else remains unchanged.

1.3.3 Examples

We introduce a rescaled shear rate

˘˙

γ ≡ γ˙ Γ

and replace equation (1.23) by:

˘

m^³˘t^´ ≡ 1 1 +^³γ˘˙t˘^´²

µ

v₁φ˘^³˘t^´+v₂^³φ˘^³˘t^´´²

¶ .

Then we easily verify that equation (1.24) and equation (1.25) remain valid. Again the explicit dependency onΓis eliminated and the system only depends on^¯^¯¯γ˘˙^¯^¯¯, hence in all following numerical calculations we will measure the shear rate in units of ^¯^¯^¯^γ_Γ^˙^¯^¯^¯and time in units ofΓt. All other numerical parameters discussed above remain unchanged. Figure 1.3 shows numerical solutions of the F₁₂^{( ˙}^γ) model for v^c₂ = 2, ² = 10⁻³ and dierent shear rates. The short time dynamics is shear-independent. After developing a plateau the correlators decay to zero with time scales determined by the shear rates. Figure 1.4 shows numerical solutions of theF₁₂^{( ˙}^γ)model forv₂^c= 2,

²=−10⁻³ and various shear rates. Again, the short time dynamics is shear-independent. Small shear rates have only little eect on the correlators, however, if the time scale for the shear- induced decorrelation becomes larger then the time scale for the nal decay determined by ², then the long time dynamics is shear-dominated.

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-2 0 2 4 6 8 10 12 log₁₀(Γt)

0 0.2 0.4 0.6 0.8 1

φ

|γ.

/Γ|=10^-10

|γ. /Γ|=10^-9

|γ. /Γ|=10^-8

|γ. /Γ|=10^-7

|γ. /Γ|=10^-6

|γ. /Γ|=10^-5

|γ. /Γ|=10^-4

Figure 1.3: Numerical solutions of the F₁₂^{( ˙}^γ) model for v^c₂ = 2 and ² = 10⁻³. The short time dynamics is shear-independent. After developing a plateau the correlators decay to zero with time scales determined by the shear rates.

-2 0 2 4 6 8 10

log₁₀(Γt) 0

0.2 0.4 0.6 0.8 1

φ

|γ.

/Γ|=10^-10

|γ. /Γ|=10^-9

|γ. /Γ|=10^-8

|γ. /Γ|=10^-7

|γ. /Γ|=10^-6

|γ. /Γ|=10^-5

|γ. /Γ|=10^-4

Figure 1.4: Numerical solutions of the F₁₂^{( ˙}^γ) model for v₂^c = 2 and ²= −10⁻³. The short time dynamics is shear-independent. Small shear rates have only little eect on the correlators. If the time scale for the shear-induced decorrelation becoming larger then the time scale for the nal decay determined by ², then the long time dynamics is shear-dominated.

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Chapter 2

The β -relaxation process

2.1 The β -scaling equation

As our numerical examples show, for small separation parameters and shear rates the correlators develop a stretched dynamics located around the critical plateau value f_c for some time window. This dynamics we call the β-process. In the following we determine the equation of motion for the time-dependent correction to f_c to leading order in ² and γ˙. First we consider the convolution integral in equation (1.1). After partial integration and using φ(0) = 1 we obtain: ^R₀^tm(t−t⁰) ˙φ(t⁰)dt⁰ = m(0)φ(t) −m(t) +^R₀^t_∂t^∂ (m(t−t⁰)φ(t⁰))dt⁰ =

−m(t) + _dt^d ^R₀^tm(t−t⁰)φ(t⁰)dt⁰. Using this we can rewrite equation (1.1):

1

Γφ˙(t) +φ(t)−m(t) + d dt

Z _t

0 m^¡t−t⁰^¢φ^¡t⁰^¢dt⁰ = 0. (2.1) Equation (1.15) motivates the ansatz

φ(t) ≈ f_c+ (1−f_c)²G(t) (2.2)

with a function G(t) =O^³²¹²^´. Following [7] we dene:

c^{( ˙}^γ) ≡ f_cv^c₁+f_c²v₂^c 1−f_c .

Using equation (1.13) and equation (1.14) lead us to:

c^{( ˙}^γ) = 1−λ

λ² . (2.3)

We choose v₁ =v₁^c+δv₁ and v₂=v^c₂+δv₂ like above. Then we can writeδv₁, δv₁=O(²). For suciently small shear rates we can approximate _{1+( ˙}¹_γt)² ≈1−( ˙γt)². By using this, equation (2.2), equation (1.26), f_c = ^v^c²_2v^−vc^c¹

2 , equation (1.14), and neglecting terms of the order ²^x with x >1 and ²^y( ˙γt)² withy >0we obtain:

m(t) ≈ λ µ

²−c^{( ˙}^γ)( ˙γt)²+c^{( ˙}^γ)+ 1

λG(t) +λG²(t)

¶

. (2.4)

Using equation (2.4), equation (2.2) and again neglecting terms of the order ²^x withx > 1and

²^y( ˙γt)² withy >0 lead us to:

Scaling laws in the rheology of colloidal dispersions