Comparison of theory and experiment

Shearstressesmeasuredinnon-linearresponseofthedispersionunderstrongsteady

shear-ing, andfrequeny dependent shearmoduliarisingfromthermalshear stress utuations

inthequiesentdispersionweremeasured andttedwithresultsfromthe shematiF

( ˙ γ) 12

-model. Some results from the mirosopi MCT for the equilibriummoduliare inluded

also; see setion 3.2.5 for more details [230℄. In the following disussion, we rst start

with more general observations on typial uid and glass like data, and then proeed to

a more detailed analysis. Figures 3.12 and 3.13 show measurements in uid states, at

φ _eff = 0.540

^and

φ _eff = 0.567

^, respetively, while g. 3.14 was obtained in the glass at

φ eff = 0.627

^{. From the ts to all}

φ eff

^{, the glass transitionvalue}

φ ^c _eff = 0.58

^{was obtained,}

whihagreeswellwiththe measurementsonlassialhardsphereolloids[121,191,194℄.

Crystallization eets

Westartthe omparisonofexperimentaland theoretialresultsbyreallingthe

interpre-tation of time in the ITT approah. Outside the linear response regime, both

Φ(t)

^and

g(t)

^{desribe the} deorrelationof equilibrium, uid-likeutuations under shear and in-ternalmotion. Integrating through the transientsprovidesthe steady state averages,like

the stress. While theory nds that transient utuations always relax under shear, real

systems may either remain in metastable states if

γ ˙

^{is too small to shear melt them, or}

undergo transitionstoheterogeneous statesforsome parameters. In theseirumstanes,

the theory annot beapplied,and the rheologialresponse ofthe system, presumably, is

dominated by the heterogeneities. Thus, are needs tobe taken in experiments, inorder

to prevent phase transitions, and to shear melt arrested strutures, before data an be

reorded.

The smallsize polydispersity of the present partilesenables the system to grow

rystal-litesaordingtoitsequilibriumphasediagram. Fortunately,whenreordingowurves,

viz. stress as funtion of shear rate, data an be taken when dereasing the shear rate.

Wend that the resulting 'down' ow urves orrespond toamorphous states and reah

the expeted low-

γ ˙

^{asymptotes (}

σ = η ₀ γ ˙

^{, see g. 3.12), exept for very low}

γ ˙

^{, when}

an inrease in stress indiates the formation of rystallites. 'Up' ow urves, however,

obtained when moving upwards in shear rate during the measurement of the stress are

aeted by rystallites formed after the initialshearing at 100 s

−1

during the timesweep

and the rst frequeny sweep experiments. See the hysteresis between 'up' and 'down'

owurvesings.3.12and3.13,wheremeasurementsfortwouiddensitiesare reported.

Above aritialshear stress

γ ˙ c ∼ 4 ⁻¹

^{nohysteresis has been observed, whihproved that}

all the rystallites have been molten. In the present work we fous on the 'down' ow

urves, and onsider only data taken either for

γ > ˙ γ ˙ c

^{, or (for}

γ < ˙ γ ˙ c

^{) before the time}

rystallization sets in. This time was estimated fromtimesweep experimentsas the time

where rystallization aused a 10

%

^{deviation of the omplex modulus}

G ^∗

^{. Thus, only}

the portions of the ow urves unaeted by rystallization are taken into aount. In

g. 3.12,g. 3.13,g. 3.15 and g. 3.19 vertial bars denote the limits. We nd that the

eet of rystallization onthe owurves is maximalaround

φ = 0.55

^{and beomes}

pro-gressively smaller and shifts to lower shear rates for higher densities as disussed in the

hapter rystallization (see hap. 2.3 and g. 2.33). This agrees with the notion that

-6 -4 -2 0 log ₁₀ (Pe ₀ )

-2 -1 0 1

log 10 ( σ R 3 H /k B T)

down up

-4 -2 0 2

log ₁₀ ( ω R _H ² /D ₀ ) -1

0 1 2

log 10 (G’, G’’ R 3 H /k B T) G’’

G’

Figure3.12: The redued owurves andtheorresponding modulifor a uidstate at13.01wt%,

T

= 20 ^o C

^{, and}

φ _eff = 0.540

^{. Flow urves measured proeeding from higher to lower}

shear rates (alled 'down' ow urves) and dynami experiments were tted where

eets from rystallization an be negleted; the lower limitsof the unaeted-data

regions are marked by vertial bars. The red lines show the ts with the shemati

F ₁₂ ^{( ˙ γ)}

^{-model while the blue lines show the results from mirosopi MCT (solid}

G ^′

^,

broken

G ^′′

^{), with} parameters:

ε = − 0.05

^,

^D _D ^S

0 = 0.14

^{, and}

η ∞ = 0.3 k _B T /(D ₀ R _H )

^;

the moduli were saled up by a fator

c _y = 1.4

^.

the glass transitionslows down the kinetis of rystallization and auses the average size

of rystallites to shrink [191℄. For the highest densities, whih are in the glass without

shear, the hysteresis in the lowest

γ ˙

^{has been attributed to a non} stationarity of the up urve (see g. 3.14). This eet has been onrmed by step ow experiments, but does

not aet the bak urves(see setion3.2.3).

The linearresponse modulisimilarlyare aeted by the presene of small rystallites at

low frequenies.

G ^′ (ω)

^and

G ^′′ (ω)

^{inrease above the behavior expeted for a solution}

(

G ^′ (ω → 0) → η ₀ ω

^and

G ^′′ (ω → 0) → c ω ²

^{) even at low density, and exhibit elasti}

ontributions atlowfrequenies (apparent from

G ^′ (ω) > G ^′′ (ω)

^{)(see gs. 3.12 and3.13).}

This eet follows the rystallization of the system during the measurement after the

shearing at

γ ˙ = 100 s ⁻¹

^{. The data have only been onsidered before the} rystallization time. For higher eetive volume fration other eets suh as ageing and an ultra-slow

proess had to be taken into aount and will be disussed more in detail in the next

setion.

Shapes of ow urves and moduli and their relations

The ow urves and moduli exhibit a qualitative hange when inreasing the eetive

paking fration from around 50% to above 60%. For lower densities (see Fig. 3.12),

the ow urves exhibit a Newtonian visosity

η 0

^{for small shear rates, followed by a}

sublinear inrease of the stress with

γ ˙

^{; viz. a region of shear thinning behavior. For}

the same densities, the frequeny dependent spetra exhibit a broad peak or shoulder,

whihorresponds tothe nalor

α

-relaxationdisussedinsetion3.2.4. Itspeakposition (or alternatively the rossing of the moduli,

G ^′ = G ^′′

^{) is roughly given by}

ωτ = 1

^(see

g. 3.13). These properties haraterize a visoelasti uid. For higher density, see g.

3.14, the stress in the ow urve remainsabove a nite yield value even for the smallest

shear rates investigated. The orresponding storage modulus exhibits an elasti plateau

atlowfrequenies. The lossmodulus drops farbelowthe elastione. These observations

haraterizea softsolid. The lossmodulusrisesagain atverylowfrequenies, whihmay

indiatethattheolloidalsolidatthisdensityismetastableandmayhaveanitelifetime

(an ultra-slow proess is disussed insetion 3.2.5).

Simple relations,like the 'Cox-Merz rule', have sometimes been used in the past to

om-parethe shapesof theowurves

σ( ˙ γ)

^{with theshapesofthe}dissipativemodulus

G ^′′ (ω)

^.

Both quantities an beinterpreted interms of a (generalized) visosity, on the one hand

as funtion of shear rate

η( ˙ γ) = σ( ˙ γ)/ γ ˙

^{, and on the other hand asfuntion of frequeny}

η(ω) = G ^′′ (ω)/ω

^{. The Cox-Merz rule states that the funtional forms of both visosities}

oinide.

Figures3.12 to3.14 providea sensitive test of relationsinthe shapesof

σ( ˙ γ)

^and

G ^′′ (ω)

^.

Figure 3.13 shows most onlusively, that no simple relation between the far-from

equi-librium stress as funtion of external rate of shearing exists with the equilibrium stress

utuationsatthe orrespondingfrequeny. While

σ( ˙ γ)

^inreases monotonially,the dis-sipative modulus

G ^′′ (ω)

^{exhibits a minimum for uid states lose to the glass} transition.

It separates the low-lying nal relaxation proess in the uid from the higher-frequeny

relaxation.

AsshowninFig. 3.11, thefrequenydependene of

G ^′′

^{intheminimumregionisgiven by}

-6 -4 -2 0 log ₁₀ (Pe ₀ )

-1 0 1

log 10 ( σ R 3 H /k B T) down

up

-4 -2 0 2

log ₁₀ ( ω R ² _H /D ₀ ) 0

1 2

log 10 (G’, G’’ R 3 H /k B T) G’’

G’

G’’

Figure3.13: The redued owurves andtheorresponding modulifor a uidstate at13.01wt%,

T

= 18 ^o C

^{, and}

φ _eff = 0.567

^{. The vertial bars mark the minimal Pelet number or}

resaled frequeny for whih the inuene of rystallization an be negleted.

Mi-rosopi parameters:

ε = − 0.01

^,

^D _D ^S ₀ = 0.14

^{, and}

η ∞ = 0.3 k _B T /(D ₀ R _H )

^{; moduli}

sale fator

c _y = 1.4

^.

thesalingfuntion

G

^{ofsetion3.2.4, whihdesribestheminimumasrossoverbetween}

two powerlaws. The approximation for the modulus aroundthe minimum

G ^′′ (ω) ≈ G _min

relaxation time

τ

^{is large, viz. time sale separation holds for small}

| ε |

^{[162℄. The}

parameters in this approximation follow from Eqs. (3.27,3.28) whih give

G min ∝ √

− ε

and

ω min ∝ ( − ε) ^1/2a

^{. For paking frations too far below the glass} transition, the nal relaxationproessisnot learly separatedfromthehigh frequeny relaxation. Thisholds

ing. 3.12, where the nal strutural deay proess onlyforms ashoulder. Closer tothe

transition, in g. 3.13, it is separated, but rystallization eets prevent us from tting

Eq. (3.39) tothe data.

Asymptoti power-law expansions of

σ( ˙ γ)

^{exist lose to the glass} transition, whih were dedued from the stability analysis in setion 3.2.4 [45, 231, 232℄; yet we refrain from

entering their detailed disussion and desribe the qualitative behavior in the following.

For the same parameters in the uid, where the minimum in

G ^′′ (ω)

^{appears, the ow}

urvesfollowaS-shapeinadouble logarithmiplot, rossingoverfrom alinear behavior

σ = η 0 γ ˙

^{atlow shearrates toa downward urved piee, followed by a pointof inetion,}

and anupward urved piee,whih nallygoesoverintoaseondlinearbehavior atvery

large shear rates, where

σ = η _∞ ^{γ ˙} γ ˙

^{. This S-shape an be reognizedin gs. 3.12 and 3.13.}

Beause of the nite slope of

log ₁₀ σ

^versus

log ₁₀ γ ˙

^{at the point of inetion, one may}

speulate about aneetive power-law

log ₁₀ σ ≈ c + c ^′ log ₁₀ γ ˙

^{. In Fig. 3.12 this happens}

atPe

0 ≈ 10 ⁻²

^{. Yet, the power-lawisonlyapparentbeausethe pointofinetion moves,}

theslopehangeswithdistanetotheglasstransition,andthe linearbitinthe owurve

never extends overan appreiablewindowin

γ ˙

^[232℄.

A qualitative dierene of the glass ow urves to the uid S-shaped ones, is that the

shape of

σ( ˙ γ)

^{onstantly has an upward urvature in} double-logarithmi representation.

The yield stress an be read o by extrapolating the ow urve to vanishing shear rate.

InFig.3.14thisleadstoavalue

σ ⁺ ≈ 0.24 k B T /R _H ³

^at

φ eff = 0.622

^{,whihisinagreement}

with previous measurements in this system over a muh redued window of shear rates

[39℄. Whilethis agreementsupports the predition ofan dynamiyieldstress in the ITT

approah, and demonstrates the usefulness of this onept, smalldeviations in the ow

urve at low

γ ˙

^{are present in g. 3.14. We postpone to setion 3.2.5 the disussion of}

these deviations,whihindiatethe existene ofan additionalslowdissipative proess in

the glass. Its signature is seen most prominently inthe loss modulus

G ^′′ (ω)

^{in g. 3.14.}

On the ontrary the storage modulus of the glass shows striking elasti behavior.

G ^′ (ω)

exhibits a near plateau over more than three deades infrequeny, whih allows to read

o the elasti onstant

G ∞

^easily.

Mirosopi MCT results

Inludedingures3.12to3.14are alulationsusingthe mirosopiMCT given byEqs.

(3.20)to (3.24)evaluatedfor hardspheres [230℄. This is presentlypossiblewithoutshear

only (

γ ˙ = 0

^{), beause of the} ompliations arising fromanisotropy and time dependene inEq. (3.22). Theonlyaprioriunknown,adjustableparameteristhemathingtimesale

-6 -4 -2 0 log ₁₀ (Pe ₀ )

-0.5 0 0.5 1

log 10 ( σ R 3 H /k B T) down

up

-4 -2 0 2

log ₁₀ (ωR ² _H /D ₀ ) 0

1 2

log 10 (G’, G’’ R 3 H /k B T) G’’

G’

Figure3.14: The redued owurves andthe orresponding moduli foraglass stateat13.01wt%,

T

= 14 ^o C

^{, and}

φ _eff = 0.622

^{. See gure 3.12 for further} explanations. Mirosopi parameters:

ε = 0.03

^,

^D _D ^S

0 = 0.08

^{, and}

η _∞ = 0.3 k _B T /(D ₀ R _H )

^{; moduli sale fator}

c _y = 1.4

^{(blue). Curvesfromtheshemati F}

^{( ˙} ₁₂ ^γ)

^{-modelwithan additional}dissipative

proessinluded (Eq.3.38)areshown asdashedlines;

δ = 10 ⁻⁷ Γ

^{(longdashes,light}

green) and

δ = 10 ⁻⁸ Γ

^{(short dashes, dark green). Here}

Γ = 88 D ₀ /R ² _H

^{. The red}

urves give the shemati model alulations for idential parameters but without

additional dissipative proess (viz.

δ = 0

^).

t 0

^{,whihweadjustedbyvaryingtheshorttimediusionoeientappearingintheinitial}

deay ratein Eq. (3.21). The omputationswere performed with

Γ q (t) ≡ Γ q = D s q ² /S q

^,

and values for

D s /D 0

^{are reported inthe aptions of Figs. 3.12 to3.14, and in table 3.2.}

Gratifyingly, the stress values omputed from the mirosopi approah are lose to the

measured ones; they are too small by 40% only, whih may arise from the approximate

struture fators entering the MCT alulation; the Perus-Yevik approximation was

used here [33℄. In order toompare the shapes of the modulithe MCT alulations were

saled up by a fator

c _y = 1.4

^{in gs. 3.12 to 3.14. Mirosopi MCT also does not hit}

the orret value for the glass transition point [162, 164℄. It nds

φ ^MCT _c = 0.516

^{, while}

our experiments give

φ ^exp _c ≈ 0.58

^{. Thus, when omparing, the relative separation from}

the respetive transition point needs to be adjusted as, obviously, the spetra depend

sensitively on the distane to the glass transition; the tted values of the separation

parameter

ε

^{are inluded in g. 3.16.}

Considering the low frequeny spetra in

G ^′ (ω)

^and

G ^′′ (ω)

^{, mirosopi MCT and}

shemati model provide ompletely equivalent desriptions of the measured data.

Dif-ferenes in the ts in Figs. 3.12 to 3.14 for

ωR ² _H /D ₀ ≤ 1

^{onlyremain beause of slightly}

dierenthoiesof thetparameterswhihwere nottuned tobelose. These dierenes

servetoprovidesome estimateof unertainties inthe ttingproedures. Mainonlusion

oftheomparisonsistheagreementofthemodulifrommirosopiMCT,shematiITT

model, and from the measurements. This observation strongly supports the universality

of the glass transitionsenario whih is a entral line of reasoning in the ITT approah

tothe non-linear rheology.

Atlarge

γ ˙

^andlarge

ω

hydrodynamiinterations beome important. In the owurves,

η ^γ _∞ ^˙

^{, and, in the loss modulus,}

η ^ω _∞

^{beome relevant} parameters, and the strutural relax-ationaptured in ITTandMCT is notsuient aloneto desribethe rheology.

Qualita-tive dierenes appear in the moduli, espeially in

G ^′ (ω)

^{, between the shemati model}

and the mirosopi MCT. Whilethe storage modulus of the F

( ˙ γ)

12

^{-modelrosses over to}

a high-

ω

^{plateau already at ratherlow}

ω

^{, the mirosopi modulus ontinues to inrease}

for inreasing frequeny, espeially at lower densities; see the region

ω > ≈ 10 ² D 0 /R ² _H

ⁱⁿ

Figs. 3.12 to 3.13. The latter aspet is onneted to the high-frequeny divergene of

the shear modulus of partiles with hard sphere potential [161℄, as aptured within the

MCT approximation[225,230℄,Asarefullydisussed byLionbergerandRussel,

lubria-tion fores maysuppress this divergene and its observation thus depends on the surfae

propertiesof the olloidalpartiles[187℄. Clearly,the regionof (rather)universal

proper-ties arisingfromthenon-equilibriumtransitionbetween shear-thinninguid andyielding

glass is left here,and partilespei eets beomeimportant.

Parameters

In the mirosopiITT approah fromsetion 3.2.4 the rheology is determinedfrom the

equilibrium struture fator

S q

^{alone. This holds at low enough frequenies and shear}

rates, and exludes the time sale parameter

t ₀

^{of Eq. (3.27), whih needs to be found}

by mathing to the short time dynamis. This predition has as onsequene that the

ow urves and moduli should be a funtion only of the thermodynami parameters

haraterizing the present system, viz. itsstruture fator.

-6 -4 -2 0 log ₁₀ (Pe ₀ )

-2 0

log 10 ( σ R 3 H /k B T)

-2 0 2

log 10 (G’R 3 H /k B T)

-4 -2 0 2

log ₁₀ ( ω R ² _H /D ₀ ) 0

2

log 10 (G’’ R 3 H /k B T)

Figure3.15: Theplotsdemonstratethattheredued owurvesandthereduedmoduliareunique

funtions only depending on

φ _eff

^{. All ow urves are down urves. The ts using}

the shemati F

( ˙ γ)

12

^{-model were performed with the data points at 13.01wt% taken}

before the onset of rystallization (data to the right of the vertial bars). Blak

diamonds: 12.10wt% and

φ _eff = 0.527

^{. Blak irles: 13.01wt% and}

φ _eff = 0.527

^.

Reddiamonds: 12.10wt%and

φ _eff = 0.578

^{. Red irles: 13.01wt%and}

φ _eff = 0.580

^.

Green diamonds: 13.01wt% and

φ _eff = 0.608

^{. Green irles: 13.58wt% and}

φ _eff =

0.606

^.

Figure3.15supportsthislaimbyprovingthattherheologialpropertiesofthedispersion

onlydependonthe eetive paking fration,ifpartilesize istaken aountof properly.

Figure3.15 olletsow urves and modulimeasured for dierentonentrations of

par-tilesaording toweight, andfordierentradii

R H

^adjustedbytemperature. Whenever the eetive paking fration,

φ eff = (4π/3)nR ³ _H

^{, is lose, the rheologial data overlap}

in the window of strutural dynamis. Obviously, appropriatesales for frequeny, shear

rateand stressmagnitudes needtobehosen toobservethis. Thedependene ofthe

ver-ties on

S q

^(Eqs. (3.20,3.23)) suggests that

k B T

^{sets the energysale aslong asrepulsive}

interations dominate the loal paking. The length sale is set by the average partile

separation, whih an be taken to sale with

R H

^{in the present system. The time sale}

of the glassy rheology withinITT is given by

t 0

^{from Eq. (3.27), whih we take to sale}

with the measured dilute diusion oeient

D ₀

^{. Thus the resaling of the rheologial}

data an be done with measured parameters alone. Figure3.15 shows quitesatisfatory

saling. Whether thepartilesare truely hard spheres isnot ofentralimportanetothe

data ollapse in Fig. 3.15 as long as the stati struture fator agrees for the

φ _eff

^used.

Fitswith theF

( ˙ γ)

12

^{-modeltoalldata arepossible,andare ofomparablequalitytothe ts}

shown in Figs.3.12 to3.14.

The tted parameters used in the shemati F

( ˙ γ)

12

^{-model are summarized in Fig. 3.16.}

Parameters orresponding to idential onentrations by weight are marked by idential

olours. Within the satter of the data one may onlude that all tparameters depend

onthe eetive pakingfration only. This again supports the mentioned dependene of

the glassy rheologyon the equilibriumstruture fator. The initialrate

Γ

^{,whih sets}

t 0

^,

appearsauniquefuntion of

φ _eff

^{,also; an}observationwhihisnot overed by thepresent ITT approah. It suggests that hydrodynami interations appear determined by

φ eff

ⁱⁿ

the present system also.

Importantly, allt parameters exhibit smooth and monotonous drifts as funtion of the

externalthermodynamiontrolparameter,viz.

φ eff

^here. Nevertheless,themoduliatlow frequenies (e.g.

G ^′ (ω)

^at

ωR ² _H /D 0 = 0.01

^{), or the stresses at low shear rates (e.g.}

σ( ˙ γ )

at

γR ˙ ² _H /D 0 = 10 ⁻⁴

^{) hange by more than an order in magnitude in Figs. 3.12 to 3.14.}

Even larger hanges may be obtained from taking experimental data not shown, whose

tparametersare inluded inFig.3.16. Itisthis sensitivedependene of the rheologyon

smallhanges of the external ontrolparameters that ITTaddresses.

When omparing the parameters from the shemati model to the ones obtained from

the mirosopi MCT alulation of the moduli, one observes qualitative and

semi-quantitative agreement (see the aptions to Figs. 3.12 to 3.14, table 3.2, and the upper

inset of Fig. 3.16). For example, the inrease of the prefator

v σ

^{of stress utuations}

is aptured in the mirosopi vertex where

S q

^{enters (this follows beause the resaling}

fator

c y

^{is density} independent). Also the hydrodynami visosity

η ∞ = η _∞ ^ω

^roughly

agrees and may be taken

φ eff

-independent in the ts with the mirosopi moduli. On loser inspetion, one may notie that the separation parameter of the mirosopi hard

sphere alulation obtains larger positive values than

ε

^{tted with the shemati model.}

Moreover, it follows an almost linear dependene on the eetive paking fration as

asymptotiallypredited by MCT,

ε ≈ 0.65 (φ eff − φ ^c _eff )/φ ^c _eff

^{withglass transitiondensity}

φ c = 0.587

^{slightly higher than from the shemati model ts. The diering behavior}

of the separation parameter from the ts with the F

( ˙ γ)

12

^{-model in the glass is not}

under-stood presently. The mirosopi alulation signals glassy arrest more learly than the

-0.4 -0.3 -0.2 -0.1 0

ε

20 40 60 80

Γ , v σ

Γ v _σ

0.45 0.5 0.55 0.6

φ _eff 0.2

0.4 0.6 0.8

η o o ω , γ .

η _oo ^ω η _oo ^{γ .}

0.4 0.5 φ _eff 0.6 0.08

0.1 0.12

0.56 φ _eff 0.6 -0.04

0

ε 0.04

γ _c

Figure3.16: The tted parameters of the F

( ˙ γ)

12

^{-model (open symbols). Blak symbols: 10.75wt%,}

red symbols: 12.10wt%, green symbols: 13.01wt%, blue symbols: 13.58 wt%.

ε

^and

γ _c

^are dimensionless. Filled magenta symbols, inluded in the upper inset, give the

ε

^{values tted in the mirosopi MCT alulations for 13.01wt%. The unit of}

v _σ

is

k _B T /R ³ _H

^while

Γ

^{isgiven in units of}

D ₀ /R ² _H

^{. The high frequenyand high shear}

visosities

η ∞ ^{ω,˙ γ}

^{are given in units of}

k _B T /(D ₀ R _H )

^.

Figure3.17: Newtonian visosity

η 0

^{(diamonds, left axis), elasti onstant}

G ∞

^{(squares), and}

yield stress

σ ⁺

^{(irles; data resaled by a fator 30; both}

G ∞

^and

σ ⁺

^{right axis),}

as funtions of the eetive paking fration

φ _eff

^{as obtained from the ts}

per-formed with the F

( ˙ γ)

12

^{-model. Filled symbols indiate where diret} measurements

of

η ₀

^{were possible. Blak symbols: 10.75wt%, red symbols: 12.10wt%, green}

symbols: 13.01wt%, blue symbols: 13.58 wt%. The line gives a power-law t to

the visosity-date over the full range using the known

γ = 2.34

^{exponent from}

MCT,

log ₁₀ η ₀ = A − γ · log ₁₀ (φ ^c _eff − φ _eff )

^{; the ritial paking fration is found}

as

φ ^c _eff = 0.580

^{. The horizontal bar denotesthe ritial elasti onstant}

G ^c _∞

^.

shematimodelt. Theshorttimediusionoeient

D s /D 0

^{inthemirosopi}

alula-tion dereases as expeted fromonsiderations ofhydrodynamiinterations. The initial

rate

Γ

^{, however, of the shematimodelinreases with pakingfration. The ad ho}

in-terpretatationof

Γ

^{asmirosopiinitialdeay rateevaluatedforsometypialwavevetor}

q ∗

^{, viz.the ansatz}

Γ = D s q _∗ ² /S q ∗

^{, thusapparently does not hold.}

While the model parameters adjusted in the tting proedure only drift smoothly with

density,therheologialpropertiesofthedispersionhangedramatially. Figure3.17shows

the Newtonian visosity as obtained from extrapolations of the ts in the F

( ˙ γ)

12

^{-model. It}

hanges by 6 ordersin magnitude. From the ombinationof

G ^′′ (ω)

^{- and ow urve data}

we an follow this divergene over more than one deade in diret measurement. From

the divergene of

η 0

^{the estimate of the ritial paking fration an be obtained using}

the power-law Eq. (3.28), beause the exponent

γ

^{is known. We nd}

φ ^c _eff = 0.580

ⁱⁿ

nie agreementwith the value expeted for olloidalhard spheres. On the glass side, the

elasti onstant and yield stress jump disontinuously into existene. Reasonable values

are obtained from the F

( ˙ γ)

12

^{-model ts ompared to data from omparable systems. The}

strong inrease of the elasti quantities upon smallinreases of the density isapparent.

Additional dissipative proess in glass

One of the major preditions of the ITT approah onerns the existene of glass states,

whih exhibit an elasti response for low frequenies under quiesent onditions, and

whih ow only beause of shear and exhibit a dynami yield stress under stationary

shear. Figure 3.14 shows suh glassy behavior, as is revealed by the analysis using the

Table 3.2:Parameters of the ts with the mirosopi MCTto the linear-response moduli

G ^′ (ω)

and

G ^′′ (ω)

^{. The rsttwo olumns of separation parameter}

ε

^{andshort-time diusion}

oeient ratio

D _s /D ₀

^{orrespondtothetsshowninFigs.3.12to3.14andFig.3.19}

(solid lines),whiletheseondolumnsof

ε ^′

^and

D _s ^′ /D ₀

^{orrespondtothe} dashed-lines in Fig. 3.19; when no value is given, the values fromthe rst two olumns apply. In

all ases

c _y = 1.4

^and

η ∞ = 0.3 k _B T /(D ₀ R _H )

^{are used.}

φ eff ε D s /D 0 ε ^′ D ^′ _s /D 0

0.527 - 0.08 0.15

0.540 - 0.05 0.15

0.567 - 0.01 0.15

0.580 0.005 0.13 - 0.01 0.15

0.608 0.02 0.11 -0.003 0.15

0.622 0.03 0.08 -0.003 0.15

-0.5 0 0.5 1.0 1.5

-3 -2 -1 0 1 2

10s 60s 600s 3600s 8200s 21600s 86400s G''

G'

t _w

t _w

log ₁₀ w [rads ^-1 ] lo g 1 0 (G ', G '') [P a ]

-0.5 0 0.5 1.0 1.5

1 2 3 4 5 6 7

t _w

log ₁₀ (w t) [rad]

lo g 1 0 (G ', G '') [P a ]

Figure3.18: Ageing experimenton adenseoreshellsuspensioninthe glassystate(

φ _eff = 0.622

^,

T = 14 ^o C

^{). The storage}

G ^′ (ω)

^{and loss}

G ^′′ (ω)

^{moduli for dierent waiting times}

t _w

^{. Thedatahavebeenalsoplottedasfuntionof}

ωt

^{assuggestedreently[205,233℄.}

See text for further explanation.

-1 0 1

log 10 (G’ R 3 H /k B T)

-4 -3 -2 -1

log 10 ( ω R ²

H /D

0 ) -1

0

log 10 (G’’ R 3 H /k B T)

Figure3.19: Fits with mirosopi MCT to the linear-response moduli

G ^′ (ω)

^{(upper panel) and}

Figure3.19: Fits with mirosopi MCT to the linear-response moduli

G ^′ (ω)

^{(upper panel) and}

Im Dokument Structure, Dynamics and Association of Thermosensitive Core-Shell Particles (Seite 88-104)