[Paper Review] Bond-orientational Order in Melting of Colloidal Crystals

Xin Qi, Yong Chen+2 more|arXiv (Cornell University)|Mar 9, 2006

Material Dynamics and Properties3 citations

TL;DR

This study uses Brownian dynamics simulations to investigate melting in two-dimensional soft Yukawa colloidal crystals, demonstrating a two-stage transition—solid to hexatic to liquid—consistent with the KTHNY theory. The bond-orientational order parameter $\Phi_6$ and single-particle order parameter distribution $P(\phi_6)$ reveal that local order breaks down gradually in 2D due to disclinations, whereas in 3D, melting occurs via a single first-order transition with global loss of order.

ABSTRACT

Using Brownian dynamics simulation, we study the orientational order in melting transition of colloidal systems with $'$soft$'$ Yukawa potential. The bond-orientational order parameter $Φ_{6}$ and the bond-orientational order function $g_B(r)$ are calculated in two-dimensional systems. It is found that a two-stage transition and the hexatic phase are indeed existent in two-dimensional melitng, which is consistent with the prediction of the Kosterlitz-Thouless-Halperin-Nelson-Young theory. For comparing with the melting process in three-dimensional systems, the probability distribution of single-particle local order parameter is introduced. Based on the extensive simulations, it is qualitatively suggested that the breakdown of local order only occurs on the fractional part of the colloidal systems for the two-dimensional melting, but in three-dimensional melting, this breakdown takes place on the whole systems at the same time.

Motivation & Objective

• To investigate the melting mechanism of two-dimensional soft Yukawa colloidal crystals using large-scale simulations.
• To verify the existence of a hexatic phase and two-stage melting as predicted by the KTHNY theory.
• To compare the nature of melting in 2D and 3D systems, particularly the breakdown of local order.
• To evaluate the effectiveness of bond-orientational order parameters and single-particle order distributions in detecting phase transitions.
• To clarify whether finite-size effects obscure the hexatic phase in soft-core systems.

Proposed method

• Brownian dynamics simulations were performed using reduced units with $\sigma = 1$, $U_0 = 1$, and $\rho = 1$ for $N = 2500$ particles in a 2D rectangular box with periodic boundary conditions.
• The inter-particle potential was modeled using a soft Yukawa potential with screening parameter $\lambda = 8$, representing a 'soft' core interaction.
• The bond-orientational order parameter $\Phi_6$ was calculated to quantify long-range orientational order, with $|\Phi_6|^2$ tracking the degree of order across temperatures.
• The orientational correlation function $g_B(r)$ was computed to analyze the decay of orientational correlations, testing for algebraic decay expected in the hexatic phase.
• The probability distribution $P(\phi_6)$ of the single-particle local order parameter was used to detect structural heterogeneity and disclination formation.
• Simulations were run over a wide range of reduced temperatures $T^*$ from 0.050 to 1.200 and densities $\rho$ from 0.6 to 1.2 to ensure statistical convergence.

Experimental results

Research questions

• RQ1 Does a two-stage melting transition occur in two-dimensional soft Yukawa colloidal crystals, as predicted by the KTHNY theory?
• RQ2 What is the nature of the breakdown of local orientational order in 2D versus 3D systems during melting?
• RQ3 How do the bond-orientational order parameter $\Phi_6$ and the single-particle distribution $P(\phi_6)$ distinguish between solid, hexatic, and liquid phases?
• RQ4 Is the hexatic phase characterized by algebraic decay of $g_B(r)$ with an exponent of $-2/3$, as expected in the KTHNY theory?
• RQ5 Does the melting transition in 3D soft Yukawa systems exhibit a first-order character with global loss of order, contrasting with 2D?

Key findings

• A two-stage melting transition was clearly observed in 2D soft Yukawa colloidal crystals, with a solid-to-hexatic transition at $T^* = 0.600$ and a hexatic-to-liquid transition at $T^* = 0.608$.
• The orientational correlation function $g_B(r)$ exhibited algebraic decay with an exponent of $-2/3$ at $T^* = 0.609$, consistent with the hexatic phase.
• The probability distribution $P(\phi_6)$ in 2D changed from Gaussian-like at low and high temperatures to a broad, orderless distribution at intermediate $T^*$, indicating the presence of isolated disclinations characteristic of the hexatic phase.
• In contrast, $P(q_6)$ in 3D systems remained Gaussian-like throughout the melting process, indicating uniform thermal disruption without intermediate phases.
• The single-particle order parameter distribution $P(\phi_6)$ was found to be more intuitive and informative than $g_B(r)$ for visualizing structural changes during melting.
• The variance of $P(\phi_6)$ in 2D was significantly larger than in 3D, reflecting greater structural heterogeneity and localized order fluctuations in 2D systems.