Linear and non linear rheology

This setion presents the basis of the rheology of olloidal suspensions. In the rest of

the hapter the mode oupling theory is introdued to give a quantitative desription

of the visoelastiity of the system in the linear regime and of the ow behaviour in

the stationary regime. By linear visoelastiity we onsider a system experiening an

osillatory strain. The dependene of the stress to the amplitude

γ

^{and frequeny of}

the soliitationis used to estimate the elasti and loss ontributionusually represent by

the moduli

G ^′

^and

G ^′′

^{. If we x the frequeny, the linearity is onsidered per denition}

for strains

γ

^{where these two moduli are onstant. At this point the struture of the}

system is not disturbed and the system is in equilibrium. If the strain exeeds a ertain

value

γ _o

^{this ondition is not respeted anymore and the struture is disturbed leading}

10 ^-1 10 ⁰ 10 ¹ 10 ²

10 ^-5 10 ^-3 10 ^-1 10 ¹ 10 ³

dg /dt [s ^-1 ]

s [P a .s ]

10 ^-1 10 ⁰ 10 ¹

10 ^-1 10 ⁰ 10 ¹ 10 ²

0.01 Hz 0.1 Hz 1 Hz 10 Hz

g [%]

G ´, G ´´ [P a ]

B)

C) A)

D)

Figure3.8: A) step ow experiment performed at dierent shear rate with dierent time

inre-ments: 50

ms

^{(rosses), 100}

ms

^{(squares), 200}

ms

^{(irles). The lines present the}

ts obtained from equation 3.10. B) Comparison of the owurves measured

experi-mentally with inreasing and dereasing shear rates (full and dashed lines) with the

stresses measured for dierent waiting times from the step ow experiments: 50

ms

(down triangles), 100

ms

^{(heked boxes), 1}

s

^(up triangles), 10

s

^{(hollow squares),}

100

s

^{(hollow irles) and stationary regime (full irles). C) Strain sweepat}

dier-entfrequenies. D)Shear stressmeasured asfuntionof thestrain fromthe stepow

experiments at dierent shear rates (hollow symbols) and frequenies (full symbols).

The samesymbolsare used forthe samevalues ofthe frequeny andshear rates. The

dashed lines present the ts fromequation 3.10.

to the relaxation and ow of the system. The simplest representation of suh a system

is the model of Maxwell. In the rest of the experiments the theory was applied to the

frequeny dependent measurementof

G ^′

^and

G ^′′

^{inthe linearregime. For this reasonthe}

modulihave been measured as funtion of the strain for dierent frequenies in order to

ensure that the deformations used in the frequeny dependent experiment belong for all

the frequenies to the linear regime. An example is given by the gure 3.8 C). A dense

solutionwithan eetivevolume

φ ef f = 0.622

^issubjet deformationsweep experiment is presented for adense13.01

wt%

^{solution at14}

^o C

^{fordierent frequeny. Themoduliare}

onstant untila strain around4

%

^{. This solutionrepresents the highest eetive volume}

fration investigated in this work, for this reason a strain of 1

%

^{full the ondition of}

linearityin allthe frequeny sweep experimentspresented inthis hapter.

The ow urves have been measured in the stationary regime. This means pratially

that the solution is sheared at a onstant shear rate

γ ˙

^{long enough, to ensure a laminar}

ow resulting in a onstant stress that we will dene as

σ( ˙ γ)

^{, whih is the} prerequisite of the theory. The stationarity has been heked by step ow experiments also for the

highest eetive volume fration (13.01

wt%

^{solution at 14}

^o C

^,

φ ef f = 0.622

^{). Figure}

3.8 A) presents the experiments performed for dierent shear rates, with dierent time

inrements (50, 100, 200

ms

^{). No signiant variations ould be observed between the}

dierenttimes. Forthe lowest shear rates (

γ ˙ = 10 ⁻⁴ − 10 ⁻¹ s ⁻¹

^{). The stress}

σ( ˙ γ, t)

^rst

dereases in the very small time, and then inreases again to reah the onstant value

σ( ˙ γ )

^{. The} experimental values of

σ( ˙ γ)

^{(solidirles) have been reported inthe gure 3.8}

B) and omparedto the owurves measuredwith inreasing and dereasingshear rates

(full and dashed urves) in the experimental onditions desribed above and with the

measurement for dierent time. No signiant dierene an be observed whih proves

that the ow urves are measured in the stationary regime. On the ontrary a marked

disrepany an be observed if the time of measurement is not long enough. This result

an be diretly ompared to the work of Heymann et al. [219, 220℄ where the authors

investigated the inuene of the measurement time on the ow urves of onentrated

suspensionsof spherial partiles.

Step owand strainsweep experimentsare very loseinthe sensethat they desribe the

evolution of the system as funtion of the time and strain and have been the objet of

reentstudies[208,221,222℄. Ifweonsider thestepowexperimentsofthegure3.8A)

theoriginoftherstdeayintheshorttimeisnotfullyunderstoodandmustberelatedto

shorttime relaxationproess. Nevertheless theexperimentalpointsinthe shorttimesfor

the lowshearrates superpose intoamaster urve,whihan be desribed asapowerlaw

withanexponent -0.4. Then thestressinreases againquasilinearlyfortheslowest shear

rate to nally relax and reah a onstant value. The two experimentsan be ompared,

if we onsider the evolution of the stress as a funtionof the strain. Forthis the moduli

of the strain sweep are rst transformed into omplex modulus

G ^∗ = ((G ^′ ) ² + (G ^′′ ) ² ) ^0.5

and thenonvertedinstress

σ ^∗ = G ^∗ /γ

^{. Conerningthe step ow}experimentsthe strain isobtained by multiplyingthe time

t

^{withthe shear rate}

γ ˙

^{. Tothis high volumefration}

the system is in the glassy state and the omplex modulus is almost onstant between

10

−2

and 10

Hz

^{and the value at the plateau}

G ^∗ _p

^is approximately equal to 23

P a

^{. The}

omparison is presented in the gure 3.8D).

σ ^∗

^{is inreasing linearly with the strain for}

the small strain and relax for the higher strain. This kind of resaling underlines the

orrespondene of the two experiments in the linear regime. For the non linear regime a

diret omparison is obtained if the stress is measured for both experiments are done at

the samestrainand shearrate. Forthis purpose we have todene thestrain-rate dened

as

γ ˙ 0 = γω

^{. The omparisonisdone onthe gure3.9, whereboth}experimentssuperpose eahother to desribe the seond relaxationof the stress.

A simple approah lose to the one proposed by Wyss et al. [208℄ was used to aount

of the linear visoelastiity dened by the omplex modulus

G ^∗ ( ˙ γ 0 = ˙ γ)

^{(in the linear}

regime) atthe lowerstrains and of the stationarity and ow at higherstrains dened by

the onstant stress

σ( ˙ γ )

^.

The model onsists on a simple Maxwell element, dened with its general equation of

movement:

dγ dt = 1

G dσ

dt + σ

η

^(3.9)

We onsider now a steady shear (

dγ/dt = ˙ γ

^),

G = G ^∗ ( ˙ γ ₀ = ˙ γ )

^{the linear visoelasti}

ontribution,

η = σ( ˙ γ)/ γ ˙

^with

σ( ˙ γ)

^{the stress inthe stationaryregime.} Consideringthat at

t = 0

^,

σ(t) = 0

^{, the resolution of this equation gives the following expression for the}

stress:

σ( ˙ γ, t) = σ( ˙ γ) (1 − exp( − G ^∗ ( ˙ γ ₀ = ˙ γ) ˙ γt/σ( ˙ γ)))

^(3.10)

Thisexpressionhavebeenusedforthestepowandstrainsweep experimentsonsidering

G ^∗ ( ˙ γ 0 = ˙ γ) = G ^∗ _p = 23 P a

^{(see g. 3.8 A) and D) and g. 3.9). This simple approah}

desribes very well the seond relaxation observed experimentally in the glassy regime

for both experiments. Of ourse this kind of experiment is muh more ompliated and

the orrelation between the time, frequeny, strain and shear rate has to be addressed

moreindepth. Nevertheless itshows learlythe onnetionbetweenlinearandnonlinear

rheology whih willbe presented inthe rest of the hapter.

3.2.4 Theory

Mirosopi approah

The next setions providea full desription of the theory developed as mentioned in the

introdutionbythe professorMatthiasFuhsand hisoworkers. Weonsider

N

^spherial

partiles with radius

R H

^{dispersed in a volume}

V

^{of solvent (visosity}

η s

^{) with imposed}

homogeneous, and onstant linear shear-ow. The ow veloity points along the

x

^-axis

and its gradient along the

y

^{-axis. The motion of the partiles (with positions}

r _i (t)

^for

i = 1, . . . , N

^{)is desribed by}

N

^{oupledLangevin equations [206℄}

ζ dr i

dt − v ^solv (r i )

= F _i + f _i .

^(3.11)

Solventfrition ismeasuredby theStokesfrition oeient

ζ = 6πη s R H

^{. The}

N

^vetors

F _i = − ∂/∂r _i U ( { r _j } )

^{denote the} interpartile fore on partile

i

^{deriving from potential}

Figure3.9: Shear stress measured as funtion of the strain from the strain sweep (full symbols)

andstepowexperiments (hollowsymbols). The samesymbolsareused forthe same

values of shear rates

γ ˙

^{and strainrates}

γ ˙ ₀ = ωγ

^.

interations with allother partiles;

U

^{is the potential energy whih depends onall}

par-tiles' positions. The solvent shear-ow is given by

v ^solv (r) = ˙ γ y x ˆ

^{, and the Gaussian}

white noise foresatises (with

α, β

^{denoting diretions)}

h f _i ^α (t) f _j ^β (t ^′ ) i = 2ζ k B T δ αβ δ ij δ(t − t ^′ ) ,

where

k _B T

^{is the thermal energy. Eah partile experienes} interpartile fores, solvent frition, and random kiks. Interation and frition fores on eah partile balane on

average,so thatthe partilesare atrest inthe solventonaverage. The Stokesianfrition

is proportional to the partile's motion relative to the solvent ow at its position; the

latter varies linearly with

y

^{. The random fore on the level of eah partile satises the}

utuation dissipation relation.

An important approximation in Eq. (3.11) is the neglet of hydrodynami interations,

whihwould arise fromthe propertreatmentof the solvent owaround movingpartiles

[33,206℄. Inthefollowingwewillarguethatsuheetsanbenegletedathighdensities

where interpartile fores hinder and/or prevent strutural rearrangements, and where

the system is lose to arrest into an amorphous, metastable solid. Another important

approximation in Eq. (3.11) is the assumption of a given, onstant shear rate

γ ˙

^{, whih}

does not vary throughout the (innite) system. We start with this assumption in the

philosophy that, rst, homogeneous states should be onsidered, before heterogeneities

and onnement eets are taken into aount. All diulties in Eq. (3.11) thus are

onneted to the many-body interations given by the fores

F _i

^{, whih ouple the}

N

Langevin equations. In the absene of interations,

F _i ≡ 0

^{, Eq. (3.11) leads to}

super-diusive partilemotion termed'Taylordispersion' [206℄.

While formulation of the onsidered mirosopi model handily uses Langevin

equa-tions, theoretial analysis proeeds more easily from the reformulation of Eq. (3.11) as

Smoluhowski equation. It desribes the temporal evolution of the distribution funtion

Ψ( { r _i } , t)

^{of the partilepositions}

∂ _t Ψ( { r _i } , t) = Ω Ψ( { r _i } , t) ,

^(3.12)

employingthe Smoluhowski operator [33, 206℄,

Ω =

the system relaxes intoa unique stationarystate atlong times, so that

Ψ(t → ∞ ) = Ψ s

holds. Homogeneous, amorphous systems are studied so that the stationary distribution

funtion

Ψ s

^is translationally invariant but anisotropi. Negleting ageing, the formal solutionof the Smoluhowskiequation withinthe ITT approah an be broughtintothe

form[41, 44℄

where the adjoint Smoluhowski

Ω ^†

^{operator arises from partial} integrations. It ats on the quantities tobeaveraged with

Ψ s

^.

Ψ e

^{denotes the equilibrumanonial} distribution funtion,

Ψ _e ∝ e ^{−U/(k B T )}

^{, whihis the} time-independentsolution of Eq. (3.12) for

γ ˙ = 0

^;

in Eq. (3.14), it gives the initial distribution at the start of shearing (at

t = 0

^{). The}

potential part of the stress tensor

σ xy = − P N

i=1 F _i ^x y i

^{entered via}

ΩΨ e = ˙ γ σ xy Ψ e

^{. The}

simple, exat result Eq. (3.14) is entral to the ITT approah as it onnets steady

state propertiesto time integrals formed with the shear-dependent dynamis. The latter

ontains slowintrinsipartile motion.

In ITT, the evolution towards the stationary distribution at innite times is

approxi-mated by following the slowstrutural rearrangements,enoded in the transient density

orrelator

Φ q (t)

^{. It isdened by [41, 44℄}

Φ q (t) = 1

NS q h δ̺ ^∗ _q e ^{Ω † t} δ̺ q(t) i ^{( ˙ γ=0)} .

^(3.15)

It desribesthe fate ofan equilibriumdensity utuationwith wavevetor

q

^{, where}

̺ q = P N

j=1 e ^{iq·r j}

^{, under the ombined eet of internal fores, Brownian motionand shearing.}

Note that beause of the appearane of

Ψ e

^{in Eq. (3.14), the average in Eq. (3.15) an}

be evaluated with the equilibrium anonial distribution funtion, while the dynamial

evolution ontains Brownianmotion and shear advetion. The normalizationis given by

S _q

^theequilibriumstruturefator[33,206℄forwavevetormodulus

q = | q |

^{. Theadveted}

wavevetor enters inEq. (3.15)

q(t) = q − γt q ˙ _x y ˆ ,

^(3.16)

q (0)

t>0 t=0

q (t)

Figure3.10: Shear advetion of a utuation with initial wavevetor in

x

^-diretion,

q(t = 0) = q (1, 0, 0) ^T

^{, and adveted wavevetor at later time}

q(t>0) = q (1, − γt, ˙ 0) ^T

^{. At all}

times,

q(t)

^is perpendiular to the planes of onstant utuation amplitude. Note that the magnitude

q(t) = q p

1 + ( ˙ γt) ²

^{inreases with time. Brownian motion,}

negleted in this sketh,would smear out the utuation.

where unit-vetor

y ˆ

^{points in}

y

^{-diretion) The} time-dependene in

q(t)

^{results from the}

ane partile motion with the shear ow of the solvent. Translational invariane

un-der shear ditates that at a time

t

^{later, the} equilibrium density utuation

δ̺ ^∗ _q

^{has a}

nonvanishing overlap only with the adveted utuation

δ̺ q(t)

^{(see g. 3.10), where a}

non-deorrelatingutuation is skethed under shear. In the ase of vanishing Brownian

motion, viz.

D 0 = 0

^{in Eq. (3.13), we nd}

Φ q (t) ≡ 1

^{, beause the adveted wavevetor}

takes aount of simple ane partile motion [223℄. The relaxation of

Φ q (t)

^thus

her-alds deay of strutural orrelations. Within ITT, the time integralover suh strutural

deorrelationsprovides anapproximation to the stationarystate:

Ψ s ≈ Ψ e + γ ˙ 2k _B T

∞

Z

0

dt X

k

k x k y S _k ^′

k NS _k(t) ² Φ ² _k (t) Ψ e ̺ ^∗ _k(t) ̺ k(t)

,

^(3.17)

with

S _k ^′ = ∂S _k /∂k

^{[224℄. Thelastterminbrakets inEq. (3.17)expresses, thatthe}

expe-tation value of a general utuation

A

ⁱⁿ ITT-approximation ontains the (equilibrium) overlap with the loal struture,

h ̺ ^∗ _k ̺ k A i ^{( ˙ γ=0)}

^{. The dierene between the} equilibrium andstationarydistributionfuntionsthen followsfromintegratingovertimethespatially

resolved (viz. wavevetor dependent) density variations.

Thegeneralresultsfor

Ψ _s

^{,the exat oneof Eq. (3.14)and the}approximationEq. (3.17), an beapplied toompute stationaryexpetationvalues like for example the

thermody-nami transverse stress,

σ( ˙ γ) = h σ xy i /V

^{. Equation (3.14) leads to an exat non-linear}

Green-Kuborelation:

σ( ˙ γ) = ˙ γ

∞

Z

0

dt g(t, γ) ˙ ,

^(3.18)

where the generalizedshear modulus

g(t, γ) ˙

^{depends on shear rate viathe} Smoluhowski operator fromEq. (3.13)

g(t, γ ˙ ) = 1

k _B T V h σ xy e ^{Ω † t} σ xy i ^{( ˙ γ=0)} .

^(3.19)

In ITT, the slow stress utuations in

g (t, γ) ˙

^are approximated by following the slow strutural rearrangements, enoded in the transient density orrelators. The generalized

modulus beomes, using the approximation Eq. (3.17), or, equivalently, performing a

mode oupling approximation [44, 45,216℄:

g(t, γ) = ˙ k B T

Summation over wavevetors has been turned into integration in Eq. (3.20) onsidering

aninnite system.

The familiar shear modulus of linear response theory desribes thermodynami stress

utuations in equilibrium, and is obtained from Eqs. (3.19,3.20) by setting

γ ˙ = 0

^[33,

158,225℄. WhileEq. (3.19)then givesthe exatGreen-Kuborelation,the approximation

Eq. (3.20) turns into the well-studied MCT formula. For nite shear rates, Eq. (3.20)

desribes how ane partile motion auses stress utuations to explore shorter and

shorterlengthsales. Therethe eetivefores, asmeasuredby thegradientofthediret

orrelationfuntion,

S _k ^′ /S _k ² = nc ^′ _k = n∂c k /∂k

^{,beomesmaller,andvanish}asymptotially,

c ^′ _k→∞ → 0

^{; the diret orrelation funtion}

c k

^{is onneted tothe struture fator via the}

Ornstein-Zernike equation

S k = 1/(1 − n c k )

^{, where}

n = N/V

^{is the partile density.}

Note, that the equilibrium struture sues to quantify the eetive interations, while

shear just pushesthe utuationsaround onthe 'equilibriumenergy landsape'.

Strutural rearrangements of the dispersion aeted by Brownian motion is enoded in

the transient density orrelator. Shear indued ane motion, viz. the ase

D 0 = 0

^,

is not suient to ause

Φ k (t)

^{to deay. Brownian motion of the quiesent orrelator}

Φ ^{( ˙} _k ^γ=0) (t)

^{leads at high densities to a slow strutural proess whih arrests at long times}

in (metastable) glass states. Thus the ombination of strutural relaxation and shear

is interesting. The interplay between intrinsi strutural motion and shearing in

Φ k (t)

is aptured by

(i)

^{rst a formally exat} Zwanzig-Moritype equation of motion, and

(ii)

seondamodeouplingfatorisationinthememoryfuntionbuiltwithlongitudinalstress

utuations [41, 44℄. The equation of motionfor the transient density orrelatorsis

∂ t Φ q (t) + Γ q (t)

wherethe initialdeay rate

Γ q (t) = D 0 q ² (t)/S q(t)

generalizesthe familiarresultfrom lin-ear response theory to adveted wavevetors; it ontains Taylor dispersion. The memory

equationontainsutuatingstressesand similarlylike

g(t, γ) ˙

^{inEq. (3.17),isalulated}

in mode ouplingapproximation

where we abbreviated

p = q − k

^{. The vertex} generalizes the expression in the quiesent ase [44℄:

V qkp (t, t ^′ ) = S q(t) S k(t ^′ ) S p(t ^′ )

q ² (t) q ² (t ^′ ) V ^qkp (t) V ^qkp (t ^′ ) , V ^qkp (t) = q(t) · k(t) nc k(t) + p(t) nc p(t)

.

^(3.23)

Withshear, wavevetors inEq. (3.23) are adveted aording to Eq. (3.16).

Equations (3.17,3.21,3.22), with the spei example of the generalized shear modulus

Eq. (3.20),forma losedset of equationsdetermining rheologialproperties ofa sheared

dispersion from equilibrium strutural input [41, 44℄. Only the stati struture fator

S _q

^{is required to predit}

(i)

^{the time dependent shear modulus within linear response,}

g ^lr (t) = g (t, γ ˙ = 0)

^{, and}

(ii)

^{the stationary stress}

σ( ˙ γ)

^{from Eq. (3.18). The loss and}

storage moduli of small amplitude osillatory shear measurements [33, 158℄ follow from

Eq. (3.19) inthe linear response ase

(i) G ^′ (ω) + i G ^′′ (ω) = iω

∞

Z

0

dt e ^{−i ω t} g(t, γ ˙ = 0) .

^(3.24)

While,inthelinearresponseregime,modulusanddensityorrelatoraremeasurable

quan-tities, outsidethe linearregime, both quantities serve as toolsin the ITT approahonly.

The transient orrelator and shear modulus provide a route to the stationary averages,

beause they desribe the deay of equilibrium utuations under external shear, and

their time integral provides an approximation for the stationary distribution funtion,

see Eq. (3.17). Determination of the frequeny dependent moduliunder large amplitude

osillatoryshearhas beomepossiblereentlyonly[226℄,and requiresanextensionofthe

present approah totime dependent shear rates inEq. (3.13) [221℄.

Universal aspets

The summarized mirosopi ITT equations ontain a bifuration in the long-time

be-havior of

Φ q (t)

^{, whih orresponds to a} non-equilibrium transition between a uid and a shear-molten glassy state. Close to the transition, (rather) universal preditions an

be made about the non-linear dispersion rheology and the steady state properties. The

entral preditions are introdued in this setion and summarized in the overview given

by g. 3.11. It is obtained from the shemati model whih is also used to analyse the

data, and whihis introdued insetion 3.2.4.

A dimensionless separation parameter

ε

^{measures the distane to the transition whih}

is situated at

ε = 0

^{. A uid state (}

ε < 0

^{) possesses a} (Newtonian) visosity,

η 0 (ε <

0) = lim γ→0 ˙ σ( ˙ γ)/ γ ˙

^{, and shows} shear-thinning upon inreasing

γ ˙

^{. Via the relation}

η 0 = lim _ω→0 G ^′′ (ω)/ω

^{, the Newtonian visosity an also be taken from the loss modulus at}

low frequenies, where

G ^′′ (ω)

^{dominates over the storage modulus. The latter varies like}

G ^′ (ω → 0) ∼ ω ²

^{. Aglass(}

ε ≥ 0

^{),intheabsene ofow, possessesanelastionstant}

G ∞

^,

whihanbemeasuredinthe elastishear modulus

G ^′ (ω)

^{inthelimitoflow}frequenies,

-2 0 2 4 6 8

Figure3.11: Overviewofthepropertiesofthe F

( ˙ γ)

12

^-modelharateristiforthe transitionbetween uid andyielding glass. The upper panel shows numerially obtained transient

or-relators

Φ(t)

^for

ε = 0.01

^{(blak urves),}

ε = 0

^(red),

ε = − 0.005

^{(green), and}

ε = − 0.01

^{(blue). The shear rates are}

| γ/Γ ˙ | = 0

^{(thik solid lines),}

| γ/Γ ˙ | = 10 ⁻⁶

(dotted lines), and

| γ/Γ ˙ | = 10 ⁻²

^{(dashed lines). For the glass state at}

ε = 0.01

(blak),

| γ/Γ ˙ | = 10 ⁻⁸

^(dashed-dotted-line) is also inluded. All urves were alu-lated with

γ _c = 0.1

^and

η ∞ = 0

^{. The thin solid lines give the} fatorization result Eq. (3.25) with saling funtions

G

^for

| γ/Γ ˙ | = 10 ⁻⁶

^{; label}

a

^{marks the ritial}

law (3.27), and label

b

^{marks the von} Shweidler-law (3.28). The ritial glass form fator

f c

^{is indiated. The inset shows the ow urves for the same values}

for

ε

^{. The thin blak bar shows the yield stress}

σ ⁺ _c

^for

ε = 0

^{. The lower panel}

showsthe visoelasti storage (solidline) andloss(broken line)moduli forthe same

values of

ε

^{. The thin green lines are the} Fourier-transformed fatorization result Eq. (3.25) with saling funtion

G

^{taken from the upper panel for}

ε = − 0.005

^.

The dashed-dotted lines show the t formula Eq. (3.39) for the spetrum in the

minimum-region with

G min /v σ = 0.0262

^,

ω min /Γ = 0.000457

^at

ε = − 0.005

^(green)

and

G _min /v _σ = 0.0370

^,

ω _min /Γ = 0.00105

^at

ε = − 0.01

^{(blue). The elastionstant}

at the transition

G ^c _∞

^{ismarked also.}

G ^′ (ω → 0, ε ≥ 0) → G ∞ (ε)

^{. Here thestoragemodulusdominatesoverthelossone,whih}

drops like

G ^′′ (ω → 0) ∼ ω

^{. Enforing steady shear ow melts the glass. The stationary}

stress of the shear-molten glass always exeeds a (dynami) yield stress. For dereasing

shear rate, the visosity inreases like

1/ γ ˙

^{, and the stress levels o onto the} yield-stress plateau,

σ( ˙ γ → 0, ε ≥ 0) → σ ⁺ (ε)

^.

Closetothe transition,the zero-shearvisosity

η 0

^{,the elastionstant}

G ∞

^{,and the yield}

stress

σ ⁺

^{show universal anomalies as funtions of the distane to the} transition: the visositydivergesinapower-law

η ₀ (ε → 0 − ) ∼ ( − ε) ^−γ

^{withmaterialdependentexponent}

γ

^around

2 − 3

^{,the elastionstantinreaseslikea}square-root

G ∞ (ε → 0+) − G ^c _∞ ∼ √

ε

^,

andthe dynamiyieldstress

σ ⁺ (ε → 0+)

^{alsoinreases withinniteslopeaboveitsvalue}

σ _c ⁺

^{at the} bifuration. The quantities

G ^c _∞

^and

σ _c ⁺

^{denote the respetive values at the}

transition point

ε = 0

^{, and measure the jump in the elasti onstant and in the yield}

stress atthe glass transition; inthe uid state,

G ∞ (ε < 0) = 0

^and

σ ⁺ (ε < 0) = 0

^hold.

The desribed results follow from the stability analysis of Eqs. (3.21,3.22) around an

arrested, glassy struture

f q

^{of the transient orrelator [44, 45℄.} Considering the time window where

Φ q (t)

^{is metastable and lose to arrest at}

f _q

^{, and taking all ontrol}

pa-rameters like density, temperature, et. to be lose to the values at the transition, the

stabilityanalysis yields the 'fatorization' between spatial and temporaldependenies

Φ q (t) = f _q ^c + h q G ( t/t 0 , ε, γt ˙ 0 ) + . . . ,

^(3.25)

where the (isotropi) glass form fator

f _q ^c

^{and ritial amplitude}

h q

^{desribe the spatial}

propertiesof the metastableglassystate. The ritialglassform fator

f _q ^c

^givesthe

long-lived omponent of density utuations, and

h q

^{aptures loal partile} rearrangements.

Both an betaken as onstants independent on shear rate and density, asthey are

eval-uated from the verties in Eq. (3.23) at the transition point. All time-dependene and

(sensitive)dependene onthe externalontrolparametersis ontainedinthe funtion

G

^,

whihoften isalled '

β

-orrelator' and obeys the non-linear stability equation

ε − c ^{( ˙ γ)} ( ˙ γt) ² + λ G ² (t) = d

The two parameters

λ

^and

c ^{( ˙ γ)}

^{in Eq. (3.26) are determined by the stati struture}

fator at the transitionpoint,and take values around

λ ≈ 0.73

^and

c ^{( ˙ γ)} ≈ 3

^for

S q

^taken

from Perus-Yevik approximation [33℄ for hard sphere interations [44, 45, 227℄. The

transition point then lies at paking fration

φ _c ≈ 0.52

^(index

c

^{for ritial), and the}

separation parameter measures the relative distane,

ε = C (φ − φ c )/φ c

^with

C ≈ 1.3

^.

The'ritial'exponent

a

^{isgivenbytheexponentparameter}

λ

^via

λ = Γ(1 − a) ² /Γ(1 − 2a)

[162, 164℄.

The time sale

t 0

^{in Eq. (3.27) provides the means to math the funtion}

G (t)

^{to the}

mirosopi, short-time dynamis. The Eqs. (3.21,3.22) ontain a simplieddesription

of the short time dynamisin olloidaldispersions via the initialdeay rate

Γ q (t)

^{. From}

this model for the short-time dynamis, the time sale

t 0 ≈ 1.6 10 ⁻² R ² _H /D 0

^{is obtained.}

Solvent mediated eets on the short time dynamis are well known and are negleted

in

Γ q (t)

^{in Eq. (3.21). If} hydrodynami interations were inluded in Eq. (3.21), all of the mentioned universal preditions of the ITT approah would remain true. Only

the value of

t 0

^{will be shifted and depend on the short time} hydrodynami interations.

This statementremainsvalid,as long asthe hydrodynami interations donot aet the

mode oupling vertex in Eq. (3.23). In this sense, hydrodynami interations an be

inorporated into the theory of the glass transition, and amount to a resaling of the

mathingtime

t 0

^{, only.}

Obviously,the mathing time

t ₀

^{alsoprovidesanupperut-o forthetime windowofthe}

strutural relaxation. At times shorterthan

t 0

^{the spei short-time dynamis matters.}

The ondition

γt ˙ 0 ≪ 1

^{follows and translates intoarestrition for the aessible rangeof}

shear rates,

γ ˙ ≪ γ ˙ ∗

^{, where the upper-ut o shear rate}

γ ˙ ∗

^{is onneted to the mathing}

time.

The parameters

ε

^,

λ

^and

c ^{( ˙ γ)}

^{in Eq. (3.26) an be determined from the} equilibrium struture fator

S q

^{at orlose tothe} transition, and, together with

t 0

^{and the shear rate}

˙

γ

^{they apture the essene ofthe rheologial anomaliesindense} dispersions. A divergent

γ

^{they apture the essene ofthe rheologial anomaliesindense} dispersions. A divergent

Im Dokument Structure, Dynamics and Association of Thermosensitive Core-Shell Particles (Seite 73-88)