r r

r

Alterations in Cage Structure with Addition of Polymer

I. Adding small amounts of polymers to dense suspensions suppress local colloid “cage” coherence

II. Adding large amounts of polymers compresses cage (locally densify)

Corresponds to Glass Melting GelFluid

I II

1.6

2

2.4

2.8

0 0.05 0.1 0.15

S(q*)

cp/c

p*

c

0.400.400.450.42

Rg/R

0.0530.0780.0530.06

PRISMExpt

cp/cp*

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

=0.33=0.29=0.24=0.207=0.161=0.108

S(q

)

qD0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

cp/c

p

*=0.0

cp/c

p

*=0.1 : Gel

cp/c

p

*=0.15 : Gel

cp/c

p

*=0.2 : Gel

S(q

)

qD

• Local structure arrested

• Intermediate fluctuations suppressed

• Enhanced low angle scattering

• Insensitivity of gel structure at ALL measurable length scales to cp/cp*, c

q*

Microstructure of GelsRg/R=0.078

cp/cp*= 0.6 c= 0.4

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8

S(q

)

qD

=0.42

=0.3

=0.19

Hard Spheres

No added polymer

Lines : PY

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8

00.020.030.040.05

S(q

)

qD

cp/c

p*

Traversing vertically at = 0.4

Comparisons of Equilibrium Structure with PRISM

LINES: PRISM

Rg/R=0.078

Excellent agreement with theory

Fluid Structure

r

Confocal microscopy

(r-Rij)/Rij

gij(r)

1

1

# of nearest neighbors

Fourier Transform

Ultra small angle x-ray scattering

neutron scattering

coherence of local cage

S(0) = dimensionless osmotic compressibility

Sij(q)

qRij

1

q* ~ 2/D

q*

2

I (q, )~ V P q S qs c P

0

0

I(q, )S(q)I(q, )

0LimS(q) 1

4 nsq sin2

Is

Ii

102

103

104

105

106

10-3 10-2 10-1

I(q)

a.u

.

q (Å-1)

P(q)

c = 0.42 ± 0.01

Scattering Theory

Liquid State Integral Equation Approach

p = monomer; s = sphere

Cpp

CpsCss

gpsgss

gpp

D =2R

site-site density

pair correlation function

polymer self-structure factor

• Hard core colloid-colloid interaction

• p-p interaction depends on solvent quality

• Repulsive polymer segment-sphere interaction

T

T

11

gij(r)

Sij(q)

Fluid Structure and

Thermodynamics

PRISM Schweizer and Curro (1987)

p

“polymer random coil”

u(r)Athermal good

Ideal

Summary Equilibrium

• Theory and experimental measurements of S(q) agree.

• Non monotonic variation of “cage” coherence with increasing polymer concentration.

Non-Equilibrium

• Scattering experiments suggest formation of heterogeneous regions as gel forms

• Ensemble-averaged structure is independent of strength and range of attraction after gelation.

The UNICAT facility at the Advanced Photon Source (APS) is supported by the Univ of Illinois at Urbana-Champaign, Materials Research Laboratory (U.S. DOE, the State of Illinois-IBHE-HECA, and the NSF), the Oak Ridge National Laboratory (U.S. DOE under contract with UT-Battelle LLC), the National Institute of Standards and Technology (U.S. Department of Commerce) and UOP LLC. The APS is supported by the U.S. DOE, Basic Energy Sciences, Office of Science under contract No. W-31-109-ENG-38.

Funding: DOE-BES via UIUC MRL

NSF-NSEC (RPI-UIUC-LANL)

S(q) ~ (qD) ~ -3.5 at high c

• Unphysical fractal exponent at high c

• Fluctuations insensitive to cp/cp* and Rg/R

• Porod-like exponents suggesting compact clusters with fractally rough surfaces?

• Departure from Porod-like decay occurs at intermediate length scale qD ~ 1.5-2.

1

10

100

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

=0.329; 3.4

=0.292; 3.12

=0.244, 3.04

=0.207, 3.17

=0.161, 2.55

=0.108, 2.49

S(q

)

qD

0.01

0.1

1

10

100

1000

0.10.150.2

0.1 1 10

S(q

)

cp/c

p*

Rg/R = 0.078

s = 0.40 ± 0.01

qD

-3.45-3.53-3.43

cp/cp*= 0.6

c= 0.4Rg/R=0.078

clusters of size

c ~ 3-5 D

Scattering At Small Angles

0

0.1

0.2

0.3

0.4

0 0.05 0.1 0.15 0.2 0.25

Rg/R = 0.053

Rg/R = 0.078

Rg/R = 0.053

S(q

D=

2)

cp/c

p*

s = 0.40

s = 0.40

s = 0.45

6.6

6.8

7

7.2

7.4

7.6

7.8

0 0.05 0.1 0.15 0.2 0.25

q*

cp/c

p*

c

0.40.40.450.42

Rg/R

0.0530.0780.0530.06

Expt PRISM

• Ensemble-averaged inter-particle spacing is “frozen” after gelation

• Experiments fall out of agreement with theory due to non-equilibrium transition

local structural arrestsuppression of intermediate scattering

Signature of the Gel

Not near spinodal --Why the upturn in S(q) at small q’s ?

Clusters and Voids

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

=0.329; v/D=4.1

=0.292; v/D=3.6

=0.244; v/D=3.3

=0.207; v/D=3.6

=0.161; v/D=2.3

=0.108; v/D=2.2

[S(q

)]-1

/2

(qD)2

0

0.2

0.4

0.6

0.8

1

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5

[S(q

)]-1

/2

cp/c

p*

/D

5.8

5.4

5.5

Rg/R = 0.078

c = 0.40 ± 0.01

(qD)2

Debye-Bueche Analysis:

Total Scattering Intensity :

S(q)=SD-B(q)+Sc(q)

(inter-cluster) (intra-cluster))exp(~)(

vBD

rrS

222

BDBD )q1(

)0(S)q(S

qq*

Gel

c

Voids

Clusters

eff >bulk

v

Experimental System

100 nm

Colloid:

D = 100 ± 5 nm Sterically Stabilized Silica (Hard Spheres)

Polymer:

Polystyrene (near theta state) in decalin

0

10

20

30

40

50

0 0.1 0.2 0.3 0.4 0.5

Carnahan Starling

(1/k

T)d

/d

c

Gel

Fluid-Crystal

Fluid-Fluid

No added polymer

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5

c p/c

p

*

c

Gel Boundary

Phase Diagram

Aggregated ClustersGel

Rg/R=0.078 The arrows indicate the path taken in X-ray scattering measurements.

Introduction

Colloid-Polymer mixtures are used in a number of industrial and scientific applications to achieve desired material properties. Understanding the effects of microscopic interactions provides tunable paramaters to manipulate the macroscopic properties such as the flow behavior governed by the underlying phase transitions.

Suspensions can undergo both equilibrium and non-equilibrium phase transitions depending on the strength and range of these interactions. Strong short-ranged attractions often give rise to gelation whereas weaker longer ranged attractions result in equilibrium fluid-fluid or fluid-crystal transitions.

Here we present a comprehensive study of the microstructure of the gel phase of a well characterized colloidal suspension by Small Angle X-ray Scattering as the gel boundary is traversed. The location of the boundary is controlled by carefully tuning the strength and range of interparticle interactions by adding non-adsorbing polymer.

Fumed Silica GelWeak Strong

Hair Gel (Cosmetics)

Toothpaste (Hygiene)

Microstructure of Dense Colloid Suspensions and GelsS. Ramakrishnan, S.A. Shah, Y.-L. Chen, K.S. Schweizer, and C.F. Zukoski, Department of Chemical and Biomolecular Engineering, University of Illionois at Urbana-Champaign, IL 61801,

Jan Ilavsky Department of Chemical Engineering Purdue University, IN 47097, Pete R. Jemian, Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, IL 61801