Two-Dimensional Melting of Colloidal Hard Spheres

Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens

Phys. Rev. Lett. 118, 158001 – Published 10 April 2017

Abstract

We study the melting of quasi-two-dimensional colloidal hard spheres by considering a tilted monolayer of particles in sedimentation-diffusion equilibrium. In particular, we measure the equation of state from the density profiles and use time-dependent and height-resolved correlation functions to identify the liquid, hexatic, and crystal phases. We find that the liquid-hexatic transition is first order and that the hexatic-crystal transition is continuous. Furthermore, we directly measure the width of the liquid-hexatic coexistence gap from the fluctuations of the corresponding interface, and thereby experimentally establish the full phase behavior of hard disks.

Received 9 December 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.158001

Physics Subject Headings (PhySH)

Equations of state Phase diagrams Phase transitions

2-dimensional systems Colloids Hard sphere colloids

Optical microscopy

Polymers & Soft MatterCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens^*

Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom

^*roel.dullens@chem.ox.ac.uk

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Vol. 118, Iss. 15 — 14 April 2017

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Images

Figure 1
The sedimentation-diffusion equilibrium for 2D colloidal hard spheres. (a) A typical experimental image of the system in sedimentation-diffusion equilibrium for a tilt angle of $α = 0.5 6 °$ . Inset, a schematic diagram of the experimental geometry showing the effect of tilting the sample by a small angle, $α$ , and the resultant in-plane component of gravity. (b) The density profile, $ϕ (z)$ , as a function of the height, $z$ , rescaled by the in-plane gravitational height, $h_{g ∥}$ . The data shown are averaged over samples at six different values of $α$ , with error bars, arising from the standard deviation of this average, smaller than the symbol size. The solid red line indicates the prediction for the density profile in the liquid phase from scaled particle theory [41]. (c) The equation of state $Π / ρ k_{B} T$ , averaged over six different values of $α$ , with error bars as in (b). The inset shows an expanded view of the behavior of the equation of state in the region of the discontinuity. The solid red line gives the prediction of scaled particle theory for the range of area fractions characteristic of a liquid, $Π / ρ k_{B} T = 1 / (1 - ϕ)^{2}$ . As a guide to the eye, the solid blue line shows a semiempirical fit to the behavior at high $ϕ$ of $Π / ρ k_{B} T = a / (ϕ_{c p} - ϕ)$ [42], where $ϕ_{cp}$ is the area fraction at close packing ( $ϕ_{cp} \approx 0.91$ ).
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Figure 2
Characterization of the liquid, hexatic, and crystal phases. (a) The height-resolved bond-orientational correlation function in time, $g_{6} (t)$ , for the sample at $α = 0.2 5 °$ for all bins across the whole range of area fractions. Note that a legend is only provided for the data between $0.65 < ϕ < 0.76$ . (b) The corresponding height-resolved modified Lindemann parameter, $γ_{L} (t)$ , for the same sample. The legend in panel (a) also applies to panel (b). (c) The variation in the exponents $η_{6} / 2$ and $β$ with $ϕ$ in the region of the transitions. Error bars indicate the standard deviation of the average over samples at six different tilt angles.
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Figure 3
Analysis of the liquid-hexatic interface and coexistence region. (a)–(d) Typical Voronoi plots with cells colored by $θ_{i}$ for systems at (a) $α = 0.5 6 °$ , (b) $α = 0.3 5 °$ , (c) $α = 0.2 5 °$ , and (d) $α = 0.06 7 °$ . (e) The mean height-height correlation function, $g_{h} (x)$ , rescaled by $g_{h} (x = 0)$ , for the six samples considered. Note that the $x$ axis is rescaled by the capillary length, $L_{c}$ . The solid red line shows an exponential decay, according to Eq. (2) and error bars indicate the standard deviation of the average over samples at six different tilt angles. (f) and (g) Plots used to determine the coexistence gap $Δ ϕ$ exhibiting the linear dependence of (f) $g_{h} (x = 0)$ with $\sqrt{1 / Γ \sin α}$ and (g) $L_{c}$ with $\sqrt{Γ / \sin α}$ .
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