COLLOIDAL METALLURGYJMAK theory [Avrami, J. Chem. Phys. 1940] Volume of a crystallite that was...

$: COLLOIDAL METALLURGYJMAK theory [Avrami, J. Chem. Phys. 1940] Volume of a crystallite that was formed at time t and grew ‘till time t Extended volume fraction of crystallized matter$
COLLOIDAL METALLURGY

Laboratoire Charles Coulomb

CNRS-Université Montpellier 2

Montpellier, France

Neda Ghofraniha (Post-Doc)

Elisa Tamborini (PhD)

Ameur Louhichi (M2)

Julian Oberdisse

Luca Cipelletti

Laurence Ramos

Motivation

MECHANICAL PROPERTIES OF ATOMIC POLYCRYSTALS

[Kumar Acta Mater. 2003]

2 competiting processes to control deformation

• Grain-boundary (GB) sliding

• Dislocation slip

[Richeton Nature Materials2005]

DISLOCATION GB

J.

Weis

s, L

GG

E/C

NR

S

Extremely small grains

Unrealistically high strains

Numerical simulations

Experiments on metals

Difficulty of preparing samples with small grains

Difficulty of measurements

Motivation

COLLOIDAL CRYSTALS: Model systems to address a variety of

fundamental/applied questions

• Structure (direct visualization / light scattering)

• Freezing (crystallization kinetics…)

• Melting (role of defects…)

• Behavior under external forces (shear…)

• Pattern formation, surface structuring, photonic crystals…

A. Van Blaaderen’s group R. Dullens’s group A. Yodh’s group

OUR OBJECTIVE

Use colloidal crystals as analog of atomic crystals

to get time- and space-resolved data on the behavior of the

materials under mechanical stress

Outline

• Design of a Colloidal Analog of a Metal with Grain

Refiner

• Controlling and Modeling the Microstructure

• Plasticity and Dynamics of the Grain-Boundaries

under Shear

Pluronics F108

PEO-PPO-PEO

Design of a Colloidal Analog

fcc crystal lattice

a = 31.7 nm

SANS

110

-23

10-22

10-21

10-20

10-19

I/(

)²

(cm

3)

q (nm-1)

~ 30 nm

fcc lattice

BLOCK-COPOLYMER IN WATER

5 10 15 200.1

1

10

100

1000

10000

G'

G"

G', G

" (P

a)

T (°C)

THERMOSENSITIVITY OF F108 PEOx-PPOy-PEOx

temperature

~ 30 nm

fcc lattice


T

f

Rheology

viscous

5 10 15

1

10

100

1000

10000

/

0

T (°C)


T DEPENDENCE OF f : MAPPING TO HS

feff = a (T- T0)

0.0 0.2 0.4 0.6

1

10

100

1000

10000 block copolymer polycrystals

Hard-Spheres [Pusey PRL 1995]

Hard-Spheres [Chaikin PRE 2002]

/

0

f, feff

=a(T-T0)


T0 = 4.2 °C

a = 0.045 °C-1

T DEPENDENCE OF f : MAPPING TO HS

Similar mapping found from S(q) [Mortensen PRL 1992]

feff = a (T- T0)

Design of a colloidal analog of a metallic alloy

3D VISUALIZATION OF THE NETWORK OF GBs

• NPs confined in the grain-boundaries

• analogy with impurities

in atomic & molecular systems

[Lee Metall. Mater. Trans. A 2000] [Losert PNAS 1998]

F108 in water

+

nanoparticles

(~ 1% or less) Polystyrene, Silica

sNP = 35 nm

accelerated x170 times (duration ~1 h)

20 mm

10 mm

T = 0.007°C/min .


HIERARCHICAL STRUCTURE OF THE NPs Silica

sNP = 30 nm

fNP = 0.5 %

T = 0.02°C/min .

NPcc in the GB ~ 10% >> fNP

Outline


Refiner



under Shear

By analogy with atomic or molecular systems (« annealed » or « quenched » samples)

0.1 1

0.1

1

10

100

I (c

m-1)

q (nm-1)

2°C/min

0.02°C/min

0.007°C/min

0.001°C/min

-2

No modification of

the crystalline structure

Controlling & Modeling the Microstructure

ROLE OF THE HEATING RATE

0.02 °C/Min

T

0.007 °C/Min

0.0005°C/Min

0.00025°C/min

Fluorescent polystyrene NP

sNP = 36 nm

fNP=0.5 %

.

ROLE OF THE HEATING RATE


0.02 °C/Min

0.007 °C/Min

0.0005°C/Min

0.00025°C/min

fNP=0.5 % (v/v)

s = 36 nm


fNP

1% v/v

0.5% v/v

0.1% v/v

0.05% v/v

T=0.007°C/Min .

Analogy to grain refinement

in metallic alloys

ROLE OF THE NP CONCENTRATION


0.05% v/v

0.5% v/v

1% v/v

0.1% v/v

ROLE OF THE NP CONCENTRATION


vs heating rate vs NP content

0.0001 0.001 0.01

10

R (

mm

)

fNP

0.001 0.01

10

R (

mm

)

T (°C min-1)

.

AVERAGE CRYSTALLITE SIZE


JOHNSON-MEHL-AVRAMI-KOLMOGOROV (JMAK) THEORY

1. Nucleation occurs randomly and homogeneously over the entire

untransformed portion of the material (I (T))

2. The growth rate does not depend on the extent of

transformation (vg(T) )

3. Growth occurs at the same rate in all directions

(until crystals occupy all the available volume)


JMAK theory

[Avrami, J. Chem. Phys. 1940]

Volume of a crystallite that

was formed at time t and

grew ‘till time t

Extended volume

fraction of

crystallized

matter

Actual volume fraction that has crystallized

Average grain size

where

[Farjas PRB 2007, 2008]

JOHNSON-MEHL-AVRAMI-KOLMOGOROV (JMAK) THEORY


Ds diffusion coefficient

l length scale over which particles diffuse <~ s

|µ| chemical potential difference crystal/fluid

Adapated from Wikipedia

WILSON-FRENKEL LAW

CLASSICAL NUCLEATION THEORY

I = Gexp (- G* / kBT)

G* =

nucleation barrier to

form a stable nucleus

2

3

3

16

m

s


Assumption: NP impurities affect

only interfacial energy (thanks to P. Olmsted!)

NP less soluble

in crystal than fluid

NP decreases surface energy cost: GS = 4r2 - ENP

ENP ~ fNPr3

CRYSTAL

FLUID

CRYSTAL

FLUID

EFFECTS OF NPs

GS = 4r2 - 2 FpfNP

sNP

r3

3/2 FP fNP / sNP


0.0001 0.001 0.01

10

R (

mm

)

f

0.001 0.01

10

R (

mm

)

T(°C min-1)

without NPs with NPs

.

FITS OF EXPERIMENTAL RESULTS

4 fitting parameters

f = a (T-T0) (as determined by rheology)

FP ; µ /kBT = AMU x (f - fc) ; /kBT = AGAMMA / s 2 ; l = LAMBDA x s

NP


COMPARISON WITH SIMULATIONS/EXPERIMENTS ON HS

Fits Simul.

HS

Exp.

HS

AMU 17 ± 3

~ 14.7 ?

AGAMMA 0.75 ± 0.2

~ 0.64 ~ 0.5

LAMBDA 0.14±0.05

0.2-0.5 2.8-17

FP 126 ± 24

- -

0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.6110

-25

10-21

10-17

10-13

10-9

10-5

I*

HS (Exp.)

HS (Simulation)

Model

f

Auer & Frenkel, Adv Polym Sci 2005

Schätzel & Ackerson,PRE 1993

Harland & Van Megen, PRE 1997

Sinn, Stipp & Palberg, Prog Coll Polym Sci 2001

Cheng et al., PRL 2002


12 14 16 180.1

1

10

100

1000

10000

G', G

" (P

a)

T (°C)

G', G'' 0.5°C/min

G', G'' 0.007°C/min

G', G'' 0.1°C/min

CROSS-CHECK – CRYSTALLIZATION TEMPERATURE vs T

1E-4 1E-3 0.01 0.1 115

16

17

18

19

20

Rheology

DSC

Model

Tc (

°C)

T (°C/min)

.

.


Outline


Refiner



under Shear

Plasticity & Dynamics of the Grain-Boundaries

laser spring

motor

moving slide

fixed slide

CONFOCAL MICROSCOPY

Observation by confocal microscopy

t

= 3.6 %

t = 1 t = 2 t = 3 =0

50 µm

t = 1

t = 2617

Overlay of 2 images taken at

~ 5000 cycles

Deformation of the

crystalline grains

PROTOCOL (analogy to fatigue test in material science)


Shear-cell coupled to Mid-Angle Light Scattering set-up

SHEAR CELL

LASER

L1a L1b

PDT

L2a L2b L3a L3b

M

LPDT

CCD

PC

S

PDM

OF

BS

Z

COLLIMATOR

10 µm

10-6

10-5

10-4

10-3

10-2

10-1

10-2

10-1

100

101

102

103

104

105

106

107

108

SANS

SLS

USALS

MALS

q (Å

-1)

I (a

rb.

un

.)

DLS under shear strain

GBs dynamics


DATA ANALYSIS

g2(t,t)-1 =

t t = i t = i+1 t = i+2

=0

t time

t delay between shear cycle

t =1

t =2 q//

ROI10 ROI9

dynamic

structure

factor

q10 q9


100

101

102

103

104

0.0

0.2

0.4

0.6

0.8

1.0

g2-1

t tr

ELASTICITY vs PLASTICITY

Elastic sample

(PDMS)

Plastic sample

(polycrystal)

1 10 100 1000 100000.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 60.0

0.5

1.0

t

g2-1

g2-1

t

CHOICE OF THE STRAIN AMPLITUDES

0.01 0.1 1 1010

100

1000

10000

storage modulus

loss modulus

G',

G"

(Pa)

(%)

0.025°C/min

f = 0.5 Hz

Elastic Plastic Viscous

= 1.6 %

= 2.5 %

= 4.6 %

= 5.2 %

= 3.5 %


1 10 100 1000 100000.1

1

10

100

1000

tr (

dec

orr

elat

ion

tim

e)

t (# shear cycles)

= 4.6 %

AGING law

DECORRELATION TIME vs # of SHEAR CYCLES

1 10 100 1000 100000.0

0.2

0.4

0.6

0.8

1.0

2

3

4

7

10

25

50

100

150

250

500

1500

g2-1

t

# shear cycles

t


tr

q=1.58 mm-1

1 10 100 1000 100000.1

1

10

100

1000

tr

t

q1

q2

q3

q4

q5

q6

q7

q8

q9

q10

1 10 100 1000 100000.0

0.2

0.4

0.6

0.8

1.0

g2-1

t

0.13

0.20

0.24

0.39

0.78

1.16

1.58

2.20

2.83

3.72

q (mm-1

)

q AGING laws

= 4.6 %


DECORRELATION TIME vs # of SHEAR CYCLES & q-VECTORS

10-2

10-1

100

101

102

103

104

10-4

10-3

10-2

10-1

100

= 1.5%

= 2.5%

= 3.5%

= 4.6%

= 5.2%

tr/t

t/tc


SCALING

0.1 110

2

103

104

= 1.5%

= 2.5%

= 3.5%

= 4.6%

= 5.2%

t

q (mm-1

)

STEADY STATE RELAXATION TIME

-1

ballistic motion

Length-scale dependent

plastic remodeling of the

network of grain-boundaries


2 /grain size)

0.1 1

100

101

102

q (mm

-1)

= 1.5%

= 2.5%

= 3.5%

= 4.6%

= 5.2%

t c


CROSSOVER TIME FROM AGING TO STEADY

Length-scale dependent

plastic remodeling of the

network of grain-boundaries

= 0

A PHYSICAL PICTURE

L

0

Stationary state

« reshuffling » length scale


0.1 1

100

101

102

q (mm

-1)

= 1.5%

= 2.5%

= 3.5%

= 4.6%

= 5.2%

t c

)1(

2

==L

ctq

CROSSOVER TIME FROM AGING TO STEADY

RESHUFFLING LENGTH SCALE

tc=1

1 2 3 4 5 60

10

20

30

40

50

60

70

L (

mm

)

(%)


Open Questions

1 10

10

100

L (

mm

)

(%)

?

?

Grain size

Scaling of the “reshuffling” length scale when approaching the elastic and flow

regimes?

Role of the microstructure ?

Analogy with the plasticity of other disordered materials?

Ballistic motion of the GB

Conclusions

Ghofraniha et al., Soft Matter (2012)

Tamborini et al., Langmuir (2012)

Tamborini et al. Rev. Sci. Inst. (2012)

Louhichi et al., PRE (2013)

Tamborini et al. ArXiv. (2013)

• Design of an original analog of atomic polycrystal

• Control & Modeling of the microstructure

• Time and length-scale dependent of the plasticity

due to the GB network re-modeling

Acknowledgements

F. Caton & D. Roux

(Lab. Rhéologie, Grenoble)

COLLOIDAL METALLURGYJMAK theory [Avrami, J. Chem. Phys. 1940] Volume of a crystallite that was...

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