Measurement of the Curvature-Dependent Surface Tension in Nucleating Colloidal Liquids

V. D. Nguyen, F. C. Schoemaker, E. M. Blokhuis, and P. Schall

Phys. Rev. Lett. 121, 246102 – Published 14 December 2018

Abstract

The curvature dependence of the surface tension is central to the nucleation of liquids, but remains difficult to access experimentally and predict theoretically. This curvature dependence arises from the curvature-dependent molecular structure, which, for small nuclei, can deviate significantly from that of the planar liquid interface. Simulations and density functional theory have been used to predict this curvature dependence, however with contradicting results. Here, we provide the first direct measurement of the curvature-dependent surface tension in nucleating colloidal liquids. We employ critical Casimir forces to finely adjust colloidal particle interactions and induce liquid nucleation, and image individual nuclei at the particle scale to measure their curvature-dependent surface tension directly from thermally excited surface distortions. Using continuum models, we elucidate the interplay between nucleus structure, particle pair potential, and surface tension. Our results reveal a 20% lower surface tension for nuclei of critical size compared to bulk liquids, leading to 3 orders of magnitude higher nucleation rates, thus highlighting the importance of surface tension curvature corrections for accurate prediction of nucleation rates.

Received 12 April 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.246102

Physics Subject Headings (PhySH)

Interface & surface thermodynamics Nucleation

Statistical Physics & ThermodynamicsPolymers & Soft MatterCondensed Matter, Materials & Applied Physics

Authors & Affiliations

V. D. Nguyen¹, F. C. Schoemaker¹, E. M. Blokhuis², and P. Schall¹

¹Van der Waals-Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
²Leiden Institute of Chemistry, Leiden University, Einsteinweg 55, 2333 CC Leiden, Netherlands

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Vol. 121, Iss. 24 — 14 December 2018

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Images

Figure 1
Light scattering measurement of colloidal liquid nucleation and growth. Mean cluster size as a function of time, determined from the time decay of correlations of the scattered intensity, see inset. The scattered intensity is recorded at a 90° angle with the incident beam after waiting times of $Δ t = 5$ , 10, 30, 100, 180, and 360 min; its autocorrelation $C (τ)$ (inset) shows double-exponential decays due to solvent fluctuations (at $τ \sim 10^{- 2} ms$ ) and colloidal particle clusters (at $τ \sim 10 ms$ ) growing in time, shifting the decay to larger $τ$ .
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Figure 2
Colloidal liquid nucleation at the particle scale. (a)–(d) Confocal microscope images of liquid nucleation. (a) Arrow indicates a typical nucleus that disappears after a minute. (b) Arrow indicates a typical nucleus that is stable and grows. (c) Several stable nuclei that have grown to full droplets. (d) Two nuclei merging at later stages, manifesting the well-known Ostwald ripening of molecular liquids. (e)–(g) Reconstructed images of nucleation corresponding to the confocal microscope images (a)–(c). Large red (small blue) spheres indicate particles with more than four (less than or equal to four) nearest neighbors, providing a good representation of liquid and gas particles [4]. Here, nearest neighbors are identified as particles separated by less than $2 r_{\min}$ , the minimum of the pair correlation function. (h) Time-averaged particle density as a function of distance $r$ from the nucleus center of mass, for the unstable nucleus in (e) (black squares), stable nucleus in (f) (red circles), and a fully grown droplet in (g) (blue triangles). Densities are determined by particle counting. From the images, particle centers are located in three dimensions using recent extensions of the original code by Crocker and Grier [26].
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Figure 3
Gibbs free energy of nucleation. (a) Three-dimensional reconstruction of liquid nuclei coexisting during nucleation equilibrium. (b) Gibbs free energy of nucleation as a function of nucleus radius determined from the relative frequency of cluster sizes (inset), where we have used the particle density $ρ = 3.3 μ m^{- 3}$ to convert $N$ to $R$ . Black squares show data for $Δ T = 0.3 ° C$ corresponding to $T / T_{c}^{liq} = 0.72$ [13], with $T_{c}^{liq}$ the colloidal liquid critical temperature, and red circles show $Δ T = 0.4 ° C$ corresponding to $T / T_{c}^{liq} = 0.83$ . Solid curves are fits by classical nucleation theory.
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Figure 4
(a) Distribution of excess areas of ellipsoidal nuclei over the spherical ground state. Symbols distinguish nucleus sizes, see legend. (b) Surface tension of nuclei as a function of size, determined from thermally induced surface shape fluctuations. (c) Same data as in (b), plotted as a function of the inverse nucleus radius. Solid line is a fit up to second order in terms of ( $1 / R$ ). (d) Particle pair potential (main panel) and liquid pair correlation function (inset) for $Δ T = 0.3 ° C$ (open symbols) and $Δ T = 0.4 ° C$ (closed symbols).
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