^{1,a)}, J. Kohlbrecher

^{1}, A. Wilk

^{1,2}, M. Ratajczyk

^{2}, M. P. Lettinga

^{3}, J. Buitenhuis

^{3}and G. Meier

^{3}

### Abstract

We have applied small angle neutron scattering (SANS), diffusing wave spectroscopy (DWS), and dynamic light scattering (DLS) to investigate the phase diagram of a sterically stabilized colloidal system consisting of octadecyl grafted silica particles dispersed in toluene. This system is known to exhibit gas-liquid phase separation and percolation, depending on temperature , pressure, and concentration . We have determined by DLS the pressure dependence of the coexistence temperature and the spinodal temperature to be . The gel line or percolation limit was measured by DWS under high pressure using the condition that the system became nonergodic when crossing it and we determined the coexistence line at higher volume fractions from the DWS limit of turbid samples. From SANS measurements we determined the stickiness parameter of the Baxter model, characterizing a polydisperse adhesive hard sphere, using a global fit routine on all curves in the homogenous regime at various temperatures, pressures, and concentrations. The phase coexistence and percolation line as predicted from correspond with the determinations by DWS and were used to construct an experimental phase diagram for a polydisperse sticky hard sphere model system. A comparison with theory shows good agreement especially concerning the predictions for the percolation threshold. From the analysis of the forward scattering we find a critical scaling law for the susceptibility corresponding to mean field behavior. This finding is also supported by the critical scaling properties of the collective diffusion.

The SANS data are based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. We thank A. Giacometti for sending numerical data for Fig. 2. E. Zaccarelli, A. Giacometti, and Jan Dhont are thanked for stimulating discussions. A. Wilk and M. Ratajczyk acknowledge support within the European Network of Excellence SoftComp.

I. INTRODUCTION

II. THEORETICAL:PRESSURE AND TEMPERATURE DEPENDENCE OF INTERACTIONS IN STICKY SPHERE SYSTEMS

III. EXPERIMENTAL

A. Materials

B. Light scattering

C. Diffusing wave spectroscopy

D. Small angle neutron scattering

IV. RESULTS AND DISCUSSION

A. Determination of and spinodal-binodal using DLS

B. Characterization of the gel line using DWS results

1. Ergodic to nonergodic transition

2. Internal dynamics

C. Global fit of SANS data

D. Critical scaling law

E. Phase diagram

V. CONCLUSIONS

### Key Topics

- Percolation
- 43.0
- Phase diagrams
- 34.0
- High pressure
- 22.0
- Correlation functions
- 21.0
- Light scattering
- 21.0

## Figures

Schematic phase diagram of adhesive hard sphere system. Shown is the action of pressure on the loci of coexistence and percolation lines. The observed temperature shift under the action of 1 kbar pressure amounts to about .

Schematic phase diagram of adhesive hard sphere system. Shown is the action of pressure on the loci of coexistence and percolation lines. The observed temperature shift under the action of 1 kbar pressure amounts to about .

Theoretical phase diagram of an adhesive hard sphere model taken from Fantoni *et al.* (Ref. 35. Their new data (dotted lines) is denoted as PY and is in this plot compared to MC simulations from Miller and Frenkel (Ref. 7) which are shown as full lines and filled squares. Dashed lines denote the model C1 according to Fantoni *et al.* (Ref. 35). In this model, the value for is to a good approximation given by Eq. (7).

Theoretical phase diagram of an adhesive hard sphere model taken from Fantoni *et al.* (Ref. 35. Their new data (dotted lines) is denoted as PY and is in this plot compared to MC simulations from Miller and Frenkel (Ref. 7) which are shown as full lines and filled squares. Dashed lines denote the model C1 according to Fantoni *et al.* (Ref. 35). In this model, the value for is to a good approximation given by Eq. (7).

(a) 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 . The single exponential decay gives a narrow CONTIN peak, top at , and broadening indicating crossing the coexistence, middle at . Correlation function at the bottom at indicates a transition to the spinodal region. No CONTIN analysis possible. (b) Phase transition pressures at a volume fraction of as a function of temperature. The two straight lines in the figure have a slope of as determined by linear fits to the data.

(a) 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 . The single exponential decay gives a narrow CONTIN peak, top at , and broadening indicating crossing the coexistence, middle at . Correlation function at the bottom at indicates a transition to the spinodal region. No CONTIN analysis possible. (b) Phase transition pressures at a volume fraction of as a function of temperature. The two straight lines in the figure have a slope of as determined by linear fits to the data.

Normalized raw data of the two-cell DWS setup for a volume fraction of 16% is shown on the left side. On the right side, the same data are 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 nonergodic state. Temperature of measurement is .

Normalized raw data of the two-cell DWS setup for a volume fraction of 16% is shown on the left side. On the right side, the same data are 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 nonergodic state. Temperature of measurement is .

The cumulant fit coefficient is corrected for multiple scattering and plotted against pressure at a temperature of measurement of . The arrows indicate the first nonergodic file measured as described in Sec. III C. The samples with a volume fractions of 1% and 5% remain ergodic.

The cumulant fit coefficient is corrected for multiple scattering and plotted against pressure at a temperature of measurement of . The arrows indicate the first nonergodic file measured as described in Sec. III C. The samples with a volume fractions of 1% and 5% remain ergodic.

vs for different volume fractions (top: 5%, 11.2%, and 39.2%; bottom: 16%) and temperatures as given in the figures (plotted with an offset). In each data set the pressure is varied and values are indicated in the figures. Full lines are calculated according to Eq. (13) with according to Ref. 18. The was calculated on the basis of the Robertus model with sticky hard sphere interactions. Deviations of fit from data at intermediate are due to not taking the experimental resolution into account. This was proven not to influence the analysis of the stickiness at small . For details of fit see text.

vs for different volume fractions (top: 5%, 11.2%, and 39.2%; bottom: 16%) and temperatures as given in the figures (plotted with an offset). In each data set the pressure is varied and values are indicated in the figures. Full lines are calculated according to Eq. (13) with according to Ref. 18. The was calculated on the basis of the Robertus model with sticky hard sphere interactions. Deviations of fit from data at intermediate are due to not taking the experimental resolution into account. This was proven not to influence the analysis of the stickiness at small . For details of fit see text.

Plot of inverse stickiness vs pressure calculated from parameters of the global fit for volume fraction 16% using Eq. (6).

Plot of inverse stickiness vs pressure calculated from parameters of the global fit for volume fraction 16% using Eq. (6).

The forward intensity plotted vs the reduced pressure according to Eq. (16). We used for the SANS data (filled circles) and LS (filled triangles) at . SANS data (filled squares) at and with [pressure value from Fig. 3(b)]. The straight line through the data points has the slope of −1 suggesting a mean field type of behavior. All curves are vertically shifted to the SANS data to show the general critical behavior irrespective of the chosen temperature of measurement. The other, steeper straight line shown in the figure has a slope of −1.24, which would correspond to a scaling behavior of the 3D Ising case. Clearly, our data are not in agreement with this expectation.

The forward intensity plotted vs the reduced pressure according to Eq. (16). We used for the SANS data (filled circles) and LS (filled triangles) at . SANS data (filled squares) at and with [pressure value from Fig. 3(b)]. The straight line through the data points has the slope of −1 suggesting a mean field type of behavior. All curves are vertically shifted to the SANS data to show the general critical behavior irrespective of the chosen temperature of measurement. The other, steeper straight line shown in the figure has a slope of −1.24, which would correspond to a scaling behavior of the 3D Ising case. Clearly, our data are not in agreement with this expectation.

The relaxation rates divided by vs the reduced pressure for the 5, 11, and 16% samples at . The straight line has the slope of 1 suggesting mean field type of behavior. We have used the value for the critical pressure of from , as deduced from Fig. 3(b). The same was used in Fig. 8. From the measured transmissions of our samples we have then determined the transmission at the sample at that pressure and have used this value to determine the for the sample.

The relaxation rates divided by vs the reduced pressure for the 5, 11, and 16% samples at . The straight line has the slope of 1 suggesting mean field type of behavior. We have used the value for the critical pressure of from , as deduced from Fig. 3(b). The same was used in Fig. 8. From the measured transmissions of our samples we have then determined the transmission at the sample at that pressure and have used this value to determine the for the sample.

Final experimental phase diagram. Lines are guides to the eye. Dotted line: DWS coexistence line. Solid line: DWS percolation line. Dashed line: SANS percolation line. The phrase “visual inspection” in the inset refers to the phase diagram given in Ref. 18. All other symbols are explained in the inset.

Final experimental phase diagram. Lines are guides to the eye. Dotted line: DWS coexistence line. Solid line: DWS percolation line. Dashed line: SANS percolation line. The phrase “visual inspection” in the inset refers to the phase diagram given in Ref. 18. All other symbols are explained in the inset.

Comparison between theoretical phase diagram and our data (triangles). Data are converted into temperature using our global fit parameters and Eq. (6). Also shown is simulated percolation data from Kranendonk *et al.* (Ref. 6) and Seaton and Glandt (Ref. 5). MC is data from Miller and Frenkel (Ref. 7) which is similar to the data of Fantoni *et al.* (Ref. 35, see Fig. 2). The PY model is given by Eqs. (7) and (8), respectively.

Comparison between theoretical phase diagram and our data (triangles). Data are converted into temperature using our global fit parameters and Eq. (6). Also shown is simulated percolation data from Kranendonk *et al.* (Ref. 6) and Seaton and Glandt (Ref. 5). MC is data from Miller and Frenkel (Ref. 7) which is similar to the data of Fantoni *et al.* (Ref. 35, see Fig. 2). The PY model is given by Eqs. (7) and (8), respectively.

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