Equation of state for coarse-grained nanoparticle dispersion model. (a) The phase diagram for pure NPs. The inset is a blowup of the region close to the critical point to emphasize the shapes of the coexistence (heavy solid black line) and spinodal (heavy dashed black line) lines. We show sample data for (symbols) that were used to determine the isotherms (in solid lines), and the coexistence (red dashed line) and spinodal lines (dashed blue line). See text for the fitting procedure used to obtain these lines. (b) phase diagram showing the coexistence line (solid line) and the spinodal line (dashed line), from which we get the critical point , , and .
Snapshots illustrating the particle dispersion under different thermodynamic conditions: (a) a gas phase system at , ; (b) a gas-liquid phase system at , ; and (c) a liquid phase system at , .
Isotherms of vs for the modified NP with the Yukawa potential. The lack of a “loop” indicatives that the long-ranged repulsion suppresses phase separation down to at least . Each isotherm is separated by 0.2 in temperature, and curves are progressively shifted by a factor of 1.5 on the pressure axis for clarity. The lines are intended as a guide for the eye.
The pair distribution function between the centers of NP. We separate into the contributions between pairs of small or large NP, and also the cross correlations between small and large NPs. We define particles to be nearest neighbors if , where is the location of the minimum of . (Inset) An expanded view of near . The arrows show the locations of the for the three possible pairs.
Fraction of clustered NP as a function of . At low we observe that most NPs are in a clustered state. As we increase , the system becomes dispersed, except for large density (low ) where clusters are unavoidable. The lines are only intended as a guide to the eye, and are the results of spline fits to the data.
(a) Average cluster mass as a function of along isochores. (b) The same data with a reduced temperature scale, where is normalized by , show that the data collapse to a single master curve.
A comparison between the particle clustering transition temperatures and estimated from and , respectively. The nonmonotonic behavior of suggests that is not a good measure of the assembly transition temperature for this system. The inset shows that conforms reasonably well to an Arrhenius relation, as expected from simple theoretical models of equilibrium particle assembly. The black line indicates the best fit of Eq. (7); the red dotted line is only a guide for the eye.
The average cluster mass as a function of density for various . Note that for our system, similar to the behavior of wormlike micelles, colloidal fluids, protein suspensions, and dipolar fluids (Refs. 47, 54, and 55). The lines simply connect data points.
Potential energy as a function of density, showing a double-well structure in the potential energy. If there were phase separation, the free energy would have a double-well dependence on density; thus the density dependence of the entropy must be responsible for the lack of phase separation. The lines simply linearly interpolate between points.
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