Melting of colloidal crystal in a two-dimensional periodic substrate: Switch from a single crossover to two-stage melting

TL;DR

The paper investigates the melting of a $2D$ colloidal lattice on a square-periodic substrate at filling $n=1$. It uses Metropolis Monte Carlo simulations to compute translational and bond-orientational order, structure factors, susceptibilities, and defect configurations as functions of the substrate parameter $d$ and strength $A_s$. For $d \lesssim 9\lambda$, two continuous transitions separate a low-temperature crystal from a hexatic phase and then a modulated liquid, with $g_6(r)$ decaying algebraically and $\eta_6(T) \approx 1/4$ near the hexatic–liquid boundary, indicating a non-KTHNY-type two-stage melting. For larger $d$, a single crossover to a modulated liquid occurs, and sufficiently strong substrates can stabilize a square-ground state with no melting. These results show that substrate periodicity can qualitatively alter 2D melting and provide design principles for tunable colloidal and related 2D systems.

Abstract

The melting transitions of a colloidal lattice confined to a two-dimensional ($2D$) periodic substrate of square symmetry are studied using Monte Carlo simulations. When the strengths of interparticle and particle-substrate interactions are comparable, the incommensurate nature of square and triangular ordering leads to the formation of a partially pinned solid with only one of the smallest {\bf G} vectors of the substrate present. This low-temperature phase has true long-range order. By varying the lattice parameter of the substrate while keeping the filling fraction constant, it is seen that the transition from this low-temperature solid to a high-temperature modulated liquid phase can happen via either a single crossover transition or by a two-stage melting process. The transitions are found to be second-order in nature when the lattice parameter is $d \lesssim 9 λ$, as confirmed by the finite-size scaling behavior of the specific heat. For the two-stage melting scenario, the intermediate phase is found to be hexatic. The transitions observed in this work are different from the predictions of the KTHNY theory. The study reveals how constraints from substrate periodicity can fundamentally alter melting dynamics, offering insights into the design of tunable colloidal systems and advancing the understanding of phase transitions in two-dimensional particle systems.

Figures (10)

• Figure 1: (a) A schematic of the partially pinned structure of the colloidal lattice with the underlying substrate. The black circles represent the colloidal particles, and the red lines indicate the underlying square symmetric substrate with spacing $d$. The pinned colloidal particles are located at the corners of the squares (b) The potential landscape of the substrate formed by the pinned particles which interact with the interstitial ones via screened Coulomb interaction.
• Figure 2: Main plot: The variation of specific heat $C_v$ (in units of $k_B$) as a function of $k_BT$ (in units of $U_1=U_0\frac{e^{-5}}{5\lambda}$) at $d=5\lambda$ for $N_p=900$. The $C_v$ has two peaks, one at $k_BT_{c1}=0.07\:U_1$ and the other at $k_BT_{c2}=0.25\:U_1$. These two peaks correspond to the phase transitions from solid to hexatic and hexatic to modulated liquid phases. The inset plot (a) shows the variation of $C_v$ for the other two smaller system sizes with $N_p=144$ and $400$. For $N_p=144$, the solid to hexatic transition happens at $k_BT_{c1}=0.075\:U_1$ and the hexatic to liquid transition at $k_BT_{c2}=0.28\:U_1$ Similarly, for $N_p=400$, the solid to hexatic transition occurs at $k_BT_{c1}=0.07\:U_1$ and the hexatic to liquid transition at $k_BT_{c2}=0.26\:U_1$. Inset plot (b) shows the enlarged portion of the hexatic-liquid transition for $N_p=144$ and $N_p=400$. There is a clear shift in both the transition temperatures with system size as well as an increase in the $C_v$ values at both transitions. Here, the error bars indicate the standard deviation of the mean , and the dotted lines shown are guides to the eye.
• Figure 3: (i) Variation of translational order parameter $\Psi_{T1}$ corresponding to ${\bf G_1}$ as a function of $k_BT$ (in units of $U_1=U_0\:\frac{e^{-5}}{5\lambda}$) for $N_p=900$ at $d=5\lambda$. The corresponding structure factor $S_1$ for ${\bf G_1}$ is shown in the inset plot (a). Here, both $\Psi_{T1}$ and $S_1$ show a continuous drop around the transition temperature $k_BT_{c1}$. Inset plot (b) shows the variation of the orientational order parameter $\Psi_6$ with $k_BT$. It has a continuous drop around the second transition temperature $k_BT_{c2}$. (ii) Variation of translational order parameter $\Psi_{T2}$ as a function of $k_BT$ for $N_p=900$ at $d=5\lambda$. The corresponding variation of $S_2$ for the reciprocal lattice vector ${\bf G_2}$ is shown in the inset plot. $\Psi_{T2}$ and $S_2$ also drop continuously around the transition temperature $k_BT_{c1}$. The dotted line indicates the guide to the eye, and the error bars are the standard deviation of the mean.
• Figure 4: Main plot: Variation of the translational susceptibilities $\chi_{T1}$ and $\chi_{T2}$ as a function of $k_BT$ (in units of $U_1=U_0\:\frac{e^{-5}}{5\lambda}$) for $N_p=900$ at $d=5\lambda$. The inset plot shows the variation of orientational susceptibility $\chi_6$ as a function of $k_BT$. Both the plots accurately locate the transition temperatures $k_BT_{c1}=0.07\:U_1$ and $k_BT_{c2}=0.25\:U_1$ marked by a solid blue line, using the peaks of $\chi_T$'s and $\chi_6$. Error bars represent the standard deviation of the mean, and the dotted lines indicate guides to the eye.
• Figure 5: Structure factor in different phases for the case $d=5\lambda$. (a) The partially pinned solid phase with true long-range positional and orientational orders at $k_BT=0.01\:U_1$. (b) The intermediate hexatic phase with short-range positional order and quasi-long-range orientational order at $k_BT=0.20\:U_1$. (c) The liquid phase with short-range positional and orientational orders at $k_BT=0.35\:U_1$.