Probing the phase diagram of colloidal suspensions under high pressure by neutron and light scattering

Probing the phase diagram of colloidal suspensions under high pressure by neutron and light scattering

R. Vavrin ¹ , J. Kohlbrecher ¹ , A. Wilk ¹ , M.P. Lettinga ² , M. Ratajczyk ³ , J. Buitenhuis ² , G. Meier ²

(1) Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland (2) Forschungszentrum Jülich, D-52428 Jülich, Germany

(3) Institute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland

PAUL SCHERRER INSTITUT

Choice of systems:

• Particles with hairs of different length

• Characterization of phase diagram under pressure

• Percolation through control of interaction

Aims:

• How does structure change while phase transition occurs?

• How fast does structure change upon quenches?

• How correlates diffusion dynamics with structure?

Pressure Dependent Small-Angle Neutron Scattering

Abstract

We have applied small angle neutron scattering (SANS), diffusive wave spectroscopy (DWS) and static and dynamic light scattering (SLS and DLS) to investigate the phase diagram of a sterically stabilized colloidal system consisting of silica particles on which octadecyl-chains are grafted dissolved in toluene. The system exhibits a phase separation at about -3°C at ambient pressure in a volume fraction range between φ=0.05 and 0.39. We have determined by DLS the pressure dependence of the coexistence temperature and the spinodal to be dP/dT=77bar/K. Since this is a rather large value, the system properties can be nicely examined by using pressure variations. The curiosity of this system is that it shows a gel-line intersecting the coexistence line around the critical point being at a volume fraction of about φ=0.16. We have accessed the coexistence line by DLS under pressure for dilute solutions with volume fractions lower than the spinodal concentration and have determined the stability limits by kinetic SANS experiments performing a pressure jump into the metastable and unstable regimes. The gel line was measured by DWS under high pressure using the condition that the system became non-ergodic when crossing it and we determined the coexistence line at higher volume fractions from the DWS-limit of turbid samples.

R

stearyl alcohol coated silica particles (R~32nm)

Model System

TEM

Normalized raw data of the two-cell DWS setup for a volume fraction of 39.2% is shown on the left side. In the center, the same data is corrected for the decay of the second cell. With increasing pressure, the correlation functions decay at later lag times and eventually build up a plateau, which is a clear sign of a non-ergodic state. Temperature of measurement T=14°C.

Schematic phase diagram of adhesive hard sphere system. Shown is the action of pressure on the loci of coexistence and percolation lines.

0

2R

ε ε

T < T_crit, P > P_crit T > T_crit, P < P_crit

attractive interaction repulsive interaction

( , , ) V r T P

kT

r

0 5 10 15 20 25 30 35

-10 -8 -6 -4 -2 0

Volume fraction in %

phase diagram at ambient pressure

T

P^-1 or percolation line

coexistence line

stable

percolated

ƒ effective attraction:

depending on solvent quality

ƒ temperature/pressure

modifies interaction potential

ƒ reversible phase separation

5000 bar Pressure Cell for Combined SANS-DLS/DWS

The transparent sapphire windows allow a combined small angle neutron scattering (SANS) and dynamic light scattering (DLS) experiment to get simultaneous information about static and dynamic properties of the system.

0 10 20 30 40 50

600 700 800 900 1000

one phase bi-phasic

spinodal

elapsed time / min

P / bar

1a)

10^-3 10^-2 10^-1 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ 0.0

0.5 1.0 0.0 0.5 1.0

0.0 0.5 1.0 0.0

0.5 1.0

0.0 0.5 1.0

P= 940 bar

t /ms

G(2) (τ)-1 _{P= 795 bar}G(2) (τ)-1 L(τ)

G(2) (τ)-1 L(τ)^{P= 585 bar}

b)

2 4 6 8 10 12 14 16 18 20 22 400

600 800 1000 1200 1400 1600 1800 2000

bi - phasic spinodal

P / bar

T / ^oC

dP/dT = 77.5 bar/K

c)

Fig.a: Plot showing at which pressures measurements have been performed. Horizontal lines indicate loci of transition pressures.

Fig.b: The transition is determined by the pressure at which the correlation functions clearly deviate from a narrowly distributed CONTIN distribution. Here as an example measurements at T=8.3°C. The single exponential decay gives a narrow CONTIN peak, top at P=585bar, and broadening indicating crossing the coexistence, middle at P=795bar.

Correlation function at the bottom at P=940bar indicates transition to spinodal region. No CONTIN analysis possible.

Fig.c: Phase transition pressures at a volume fraction of φ=0.10 as a function of temperature. The two straight lines in the figure have a slope of dP/dT_trans=77.5±2.5 bar/K

Determination of dP/dT _trans

g₁

( )

t ^{= exp⎛}_⎝⎜^{−K1t + 1}₂ ^K2t2⎞_⎠⎟

cumulant fit

14° C

DWS-Data Analysis: Second Cell Set-Up

normalised raw data divided by ScndCell, cleaned P

P

second cell:

ensemble averaging ergodic Þ nonergodic

( )

^{P 0 d}_dP^Θ

(

^{P P0}

)

Θ = Θ + −

( )

2

ln 12 for 2 2

2 2

0

B B B

r R

V r k T R r R

R r R

τ

∞

⎧ <

⎪ ⎡ Δ ⎤

⎪ ⎛ ⎞

= ⎨⎪ ⎢⎣ ⎜⎝ + Δ ⎟⎠⎥⎦ >≤ ≤+ Δ + Δ

⎪⎩

Pressure and Temperature Dependence of Interactions in Sticky Sphere Systems

( )

2

2 2

0 2

SW

r R

V r R r R

r R

ε

∞ <

⎧⎪

= −⎨ ≤ ≤ + Δ

⎪ > + Δ

⎩

Baxter’s model for sticky hard spheres: Square Well Potential:

( ) ( )

0

L T kB T

P T

ε = ^⎧⎨ ^{Θ − < Θ}

⎩ ≥ Θ

1 12 [ ] 12 ( )

exp exp 1

2 2

B

L P

R ε R T

τ

Δ Δ ⎡ ⎛ Θ ⎞ ⎤

= + Δ = + Δ ⎢ ⎣ ⎜ ⎝ − ⎟ ⎠ ⎥ ⎦

16%, T=2C 16%, T=8.3C

39.2%, T=15.1C 5%, T=4.3C 11%, T=3C 11%, T=15.1C

16%, T=15.1C

16%,T=4.2C 5%, T=15.1C

Simultaneous fit of SANS-data for different P, T, and vol%

The SANS data has been described by the model of Robertus for polydisperse spherical particles with a sticky hard sphere interaction potential:

( ) ( ) ( )

( ) ( )

,perc ,perc

, for , (vol%)

, (vol%) otherwise

with

, 2 exp 1

12

B B B

B

B

B

P T P T

P T

R P

P T L

T

τ τ τ

τ τ

τ

⎧ >

= ⎪⎨

⎪⎩

⎡ ⎛Θ ⎞⎤

= + ΔΔ ⎢⎣− ⎜⎝ − ⎟⎠⎥⎦

It is assumed that the scattering curve will not change anymore with pressure as soon as one reaches the percolation line. The percolation occurs for a specific stickiness, which depends only on the volume fraction. Therefore the stickiness parameter, determined by SANS, stays constant for pressures above the percolation transition.

34.2 nm 2.3 nm 0.124 Rm

R σ

= Δ =

=

( )

( )

B,min

B,min

5% 0.08 11.2% 0.122 τ

τ

=

= Fit results:

0

107.2 9.7 C L =

Θ = °

( )

( )

B,min

B,min

16% 0.131 39.2% 3.05 τ

τ

=

=

Summary

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.0

0.1 0.2 0.3 0.4 0.5

τ B

Volume fraction Percolation

Spinodal Percolation

( )( )

2

1 / 2

1 for 0.12

1 3

1 4 14

for 0.12 12 2 1 1

B

ϕ ϕ ϕ

ϕ ϕ

τ ϕ ϕ ϕ

ϕ ϕ

⎧ ⎛ + − ⎞ <

⎪ − ⎜⎜ ⎟⎟

⎪ ⎝ ⎠

=⎨

+ −

⎪ ≥

⎪ + −

⎩

( )

2

2

19 2 1

B 12 1ϕ ϕ

τ ϕ

− +

= −

We have shown that a combination of light scattering techniques (static and dynamic light scattering and diffusive wave spectroscopy) with small angle neutron scattering can be utilized to measure the phase diagram of a polydisperse sticky hard sphere colloidal system including the percolation transition which we have determined by the ergodic to non-ergodic transition from DWS under high pressures. We have used for the first time pressure as the leading variable also for the dynamic measurements. We could verify the large value for dP/dT_trans utilizing high pressure dynamic light scattering. Through that dynamic experiment we found characteristic signatures for binodal and spinodal points. We were able to fit all SANS data with a model taking the structure factor of a polydisperse sticky hard sphere system into account using the Robertus model. The resulting parameters in terms of the Baxter model fit well to results from our light scattering and enabled us to construct a phase diagram also including the percolation. The agreement with theoretical phase diagrams is good concerning the absolute values for the stickiness and shape. Our data provide for the first time a full account of the realistically modeled scattering curves.

Theoretical models assume mono-disperse systems using a simplified expression for the structure factor of which we know that it gives wrong results at low q due to neglecting polydispersity effect.

Conclusion

References:

• J. Kohlbrecher, J. Buitenhuis, G. Meier and M. L. Lettinga, J. Chem. Phys. 125, 044715 (2006)

• G. Meier, R. Vavrin, J. Kohlbrecher, J. Buitenhuis, M. L. Lettinga and M. Ratajczyk, Meas. Sci. Technol. 19, 034017 (2008)

• J. Kohlbrecher, A. Bollhalder, R. Vavrin and G. Meier, Rev. Sci. Instrum. 78, 125101 (2007) Final experimental phase diagram. Lines are guides to the eye.

Dashed line: coexistence line. Solid line: percolation line.

Comparison between theoretical phase diagram and our data. Additional experimental points from the figure above (τB(vol%)) are also plotted. Data is converted into temperature using our global fit parameters

1 1

K/bar 77.5

trans

d dP

dP dT

Θ ⎛ ⎞−

= ⎜⎝ ⎟⎠ = Parameter taken

From DWS/DLS: