universiteit twente physics of complex fluids dispersion ... · universiteit twente. physics of...

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UNIVERSITEIT TWENTE. Physics of Complex Fluids

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Dispersion Rheology

Dirk van den Ende

Dept. of Science and Technology

University of Twente


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Outline Dispersions of non-interacting hard spheres •  Volume fraction dependence •  Brownian particles and Péclet number •  Shear induced diffusion

Soft particle dispersions Weakly aggregating dispersions

microstructure in relation to •  flowcurve •  linear viscoelasticity

Dispersions out of thermodynamic equilibrium

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3/50

Sphere in liquid •  goes with the flow •  has to rotate, additional friction

NVp

V φ =

Einstein calculated: η = ηo (1+2.5φ )


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At higher concentrations •  particles collide with each other •  excluded volume effects

Dispersions

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φ

η/ηs

η(φ)/ηs Phan and Russel; 1996


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Einstein’s result

In terms of number of added particles, ΔN:

Mean field approach: starting with N particles, call that your “solvent” and add again ΔN particles:

Vfree: volume available to the added particles

α slightly larger than 1 due to interstitial solvent.

[η]=2.5, φ = NVp/V

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7/50

rewriting this equation

or

Krieger Dougerhty equation:

α = 1/φm


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φ

η/ηs φm = 0.53

φm = 0.65

Einstein

5


9/50 Krieger, 1972

ηr

Pe = γ’a2/D0

colloidal particles competition between diffusion and convection

tflow ~ 1/γ’ tdiff ~ a2/D0

polystyrene particles a = 400 nm


10/50

Viscoelastic effects: HS dispersions show visco-elasticity. What is the origine of the elasticity?

J. van de Werf et al.; 1989

log(ωa2/D0)

silica particles, a = 50 nm

η’’ r

η’

r

Entropy and distorsion of the pair distribution fuction g(r)

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Non colloidal particles shear-induced self-diffusion in simple shear flow

D = a2’ D ()

a particle radius

’ rate of shear

volume fraction y

x

Dxx Dxy 0

Dyx Dyy 0

0 0 Dzz

Tensor character:


12/50 V. Breedveld et al. ; 2002

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Simple model •  Particles follow flow lines

if not prohibited by excluded volume •  While colliding they role over each other •  Collisions are not completed due

to interaction with a third particle

Δz

z

x


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Taking an average collision time tc(φ), we calculate the displacement vector s(θ,φ) per collision.

Diffusion tensor:

9


17/50 V. Breedveld, thesis UT ; 2000 J. Kromkamp et al. ; 2006

φ

Dij /γ’ a

2


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Soft particle rheology

P.A. Nommensen; 1999

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φeff

ηr, D’Haene, thesis Leuven; 1992


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The mystery of the missing factor 3

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Weakly aggregating colloidal dispersions


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depletion interaction

Parameters: •  particle concentration •  polymer concentration and size •  structure of the aggregates

Fint

12


23/50 Wolters et al. 1996

Flow curve

γ/γ#

η/η o

Silica particles (a = 38nm) in Polystyrene (5970 g/mole)

φ = 8%


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R volume fraction of aggregates:

Fint ≥ Fhyd :

mean field approximation

viscosity curve:

fractal aggregate

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25/50

γ/γ#

η/η o

df = 2.3±0.2


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measuring cylinder

and sampleholder

spring, K

air bearing, D

exciter

measuring elasticity in a shear flow: G' (',) G'' (',)

Steady rotation in the r,φ plane, Oscillation in the r,z plane

Zeegers et al. ; 1995

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27/50

10|-‐2 10|-‐1 10|0 10|1 10|2

‘ [rad/s]

10|-‐2

10|-‐1

10|0

10|1

10|2

G' [P

a]

10|-‐2 10|-‐1 10|0 10|1 10|2

‘ [rad/s]

10|-‐2

10|-‐1

10|0

10|1

10|2

G'' [Pa] Fmax

Fmax

R h

Ábreak = ‡/…

Áescape

.

tescape

increasing γ’

ω [rad/s]


28/50

Shear cell on Confocal Scanning Laser Microscope

Silica particles (1 m) and poly(ethylene glycol) in a methanol-bromoform mixture

100 x

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Different cross sections through an aggregate

V. Tolpekin, thesis UT; 2004


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0.4 1 10 40

r/a

1

10

100

N(r)

df = 2.0 ± 0.1

z = +5 um z = 0 um z =-5 um

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Aggregate growth

t [s] 0 1000 2000 3000 0

5

10

15

20

25 <D

>/a


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Modeling the aggregate growth

adjustable: Ko(' ), Q

dni /dt = ½ Ai-j,j ni-j nj - Aij ni nj + Bj pji nj - Bi ni

Aggregation: Aij =4/3 ' (Ri+ Rj)3

Break-up: Bi = Ko(' ) (Ri /a)Q

d Rdf /dt = C [R3 +3 R2 R ] - Ko(' ) RQ Rdf

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33/50 (input: df = 2.0, tagg= 460 s)

Results of the modeling

t [s] 0 1000 2000 3000 0

5

10

15

20

25 <D

>/a


34/50

Aging of soft colloidal suspensions studied by macro- and micro-Rheology

40 °C

20 °C

E.H. Purnomo et al.; 2008

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35/50

(micro-) rheology probes the aging

Structural relaxation time: ts ~ tc

log w

G’

G” increasing Age

log G

Aging in soft glassy materials

* relaxation processes slow down with age of the sample… * equilibrium is never reached…

tc

log t

log <x2>

increasing Age


36/50

Thermosensitive polyNipam-polyNipmam particles

40 °C

20 °C

10 20 30 40 50 60

50

75

100

T [oC]

R g [n

m]

light scattering data

liquid glass

40 oC

20 oC

with fluorescent ( ) tracer particles eff = V0 n

V0 (T)

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37/50

Experiment To obtain reproducible results… …rejuvenate the sample

quench > yield 0 < yield

stre

ss

qu

ench

tw t

1: macro-rheology (HAAKE RS 600)

2: particle tracking (Confocal Scanning Laser Microscopy)

quench > yield stre

ss

qu

ench

tw t <

x2>


38/50

0 100 200 300 400 500 600 700 800

time [s]

0

1

2

…(t) [-‐]

0

2

4

Â(t) / Â

y

t w

t

/y

t [s]

tw [s]

Creep measurements

T = 24 oC c = 7 %w/w

J(t-tw,tw) = ( (t)- (tw))/0

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39/50

10 - 2 10 - 1 10 0 10 1 10 2 10 1

10 2

10 3

G'

G"

10 - 2 10 - 1 10 0 10 1 10 2 10 1

10 2

10 3

10 - 2 10 - 1 10 0 10 1 10 2 10 1

10 2

10 3

G’

G"

(rad/s)

G’’,

G’

(Pa)

tw= 30, 600, 3000, 10000s

•  Brownian motion (~√) •  local viscosity (~ ) (Mason, Weitz 1995)

age dependence at low frequencies

particle-liquid interaction at high frequencies

Linear rheology, age dependence

T = 25 oC c = 7 %w/w


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Soft Glassy Rheology model

P. Sollich et al., Phys. Rev. Lett. 78, 2020 (1997); S.M. Fielding et al., J. Rheol. 44, 323 (2000)

•  Dissipation due to internal yielding of the particles •  Activated rate process with effective noise temperature x x > 1: liquid, G* = fnc (), J = fnc (t-tw) x < 1: glass, G* = fnc (t), J = fnc ((t-tw)/tw)

102

G',

G" (

Pa)

101

102 104

t Elastic response

10-1 100 10+1

ω [1/s]

G’’ tw

Microscopic picture

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41/50

glass aging

liquid non aging

100

101

102

G',

G" (

Pa) (a)

T=35 oCx=0.67±0.03

100 101 102 10310 0

10 1

ωtG

', G

" (Pa

) (b)

T=37 oCx=0.87±0.03

100 10110-4

10-3

10-2

10-1

100

ω(rad/s)

G',G

" (Pa

) (d)

T=40 oCx>3

10-1 100 10110-2

10-1

100 101 102 103ωt

(c)

T=38 oCx=2.2±0.3

ω(rad/s)

G',G

" (Pa

)

100

100

101

102

G',

G" (

Pa) (a)

T=35 oCx=0.67±0.03

100 101 102 10310 0

10 1

ωtG

', G

" (Pa

) (b)

T=37 oCx=0.87±0.03

100 10110-4

10-3

10-2

10-1

100

ω(rad/s)

G',G

" (Pa

) (d)

T=40 oCx>3

100 10110-4

10-3

10-2

10-1

100

ω(rad/s)

G',G

" (Pa

) (d)

T=40 oCx>3

10-1 100 10110-2

10-1

100 101 102 103ωt

(c)

T=38 oCx=2.2±0.3

ω(rad/s)

G',G

" (Pa

)

100

10-1 100 10110-2

10-1

100 101 102 103ωt

(c)

T=38 oCx=2.2±0.3

ω(rad/s)

G',G

" (Pa

)

100

Macro-rheology (curves: SGR model)

mass concentr. c: 7 %w/w tw: 3, 30, 300 s

T= 35oC

T= 38oC

T= 37oC

T= 40oC

t

t


42/50

+: micro-rheology 0 5 10 15

0.50

1.00

1.50

2.00

φ

G p [Pa]

li quid

glas s

hard s pher e limit

(b)

liquid

glass

eff = V0 n

φeff

E.H. Purnomo et al., 2008

3 5 7 9

20

30

40

c [%]

li quid

glas s

T [

o C]

(a)

Phase diagrams

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43/50

10 1 10 2 10 3 10 4 10 5 10 6 10 1

10 2

10 3

G'

G"

10 1 10 2 10 3 10 4 10 5 10 6 10 1

10 2

10 3

10 1 10 2 10 3 10 4 10 5 10 6 10 1

10 2

10 3

t

G'

G"

The structural relaxation time is not observable How to measure the structural relaxation time ?

T = 25 oC c = 7 %w/w


44/50

5 m

rn(t)

rn(t+s)

particle tracking µ-rheology

<>: ensemble averaging

Stokes Einstein Relation (Newtonian fluid):

Generalized Stokes Einstein Relation:

: fluoresent tracer observed by CSLM

rn = (xn,yn)

Bursac et al; Nature materials 2005

J(t): retardation function

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45/50

<r2> polyNipam, 4%

liquid

glass

32oC

30oC

27oC

detection limit

eff = 1.2

eff = 1.3

eff = 1.5

eff = 1.4

31oC


46/50

Glass transition at c = 4%

tw

300 s (o) 3000 s (●)

eff = 1.3

eff = 1.4 30 <Tg< 31 oC 1.3 <g< 1.4

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47/50

tw [s]

300 800

1300 1800 3000 3500 4000 4500

10|0 10|1 10 |2 10 |3 10 |4

10 |-‐3

10 |-‐2

t-tw [s]

<r2 >

[m

2 ]

polyNipam-polyNipmam, 4%, 27 °C Age dependence

10|-‐2 10 |-‐1 10 |0 10 |1

10 |0

10 |1

(t-tw)/

<r2 >

/<

r2 >p

0.0 2.5 5.00.00

0.25

0.50

[ks]

tw [ks]

= (0.09 ± 0.01) tw

polyNipam-polyNipmam, 4%, 27 °C scaling with tw


48/50 10 |-‐1 10 |0 10 |1 10 |2

10 |-‐1

10 |0

G’/G’

G’’/G’

micro-rheology viscoelastic moduli

T.G. Mason, 2000

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49/50

Macro- vs µ-rheology

10|-‐3 10 |-‐2 10|-‐1 10 |0 10|110|-‐1

10|0

10|1

10|2

ω [1/s]

G',G'' [Pa]

Experimental observation:

G'macro / G'micro 2

tcmacro / tcmicro 5

T = 27 °C c = 4 % w/w tw = 1300 s

Mean field calculation of the frequency dependent drag on a

stationary particle in a low viscous cell surrounded by a viscoelastic bulk

= [G”/] =


50/50 [s-1]

G’

G’’

[P

a]

micro- vs macro T = 27 °C c = 4 % w/w tw = 1300 s

10 |-‐3 10|-‐2 10|-‐1 10 |0 10 |110|-‐1

10 |0

10 |1

b/a =1.05

G*

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