# Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. The Ground States

#### 3.1. The Ground States of Open Systems

#### 3.2. The Ground States for Fixed Number of Particles

## 4. Thermodynamics of the System at $\mathbf{T}>\mathbf{0}$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Vasudevan, S.A.; Rauh, A.; Barbera, L.; Karg, M.; Isa, L. Stable in bulk and aggregating at the interface: Comparing core–shell nanoparticles in suspension and at fluid interfaces. Langmuir
**2018**, 34, 886. [Google Scholar] [CrossRef] [PubMed] - Isa, L.; Buttinoni, I.; Fernandez-Rodriguez, M.A.; Vasudevan, S.A. Two-dimensional assemblies of soft repulsive colloids confined at fluid interfaces. Europhys. Lett.
**2017**, 119, 26001. [Google Scholar] [CrossRef] - Rauh, A.; Rey, M.; Barbera, L.; Zanini, M.; Karg, M.; Isa, L. Compression of hard core–soft shell nanoparticles at liquid–liquid interfaces: Influence of the shell thickness. Soft Matter
**2017**, 13, 158. [Google Scholar] [CrossRef] [PubMed] - Rey, M.; Elnathan, R.; Ditcovski, R.; Geisel, K.; Zanini, M.; Fernandez-Rodriguez, M.A.; Naik, V.V.; Frutiger, A.; Richtering, W.; Ellenbogen, T.; et al. Fully tunable silicon nanowire arrays fabricated by soft nanoparticle templating. Nano Lett.
**2016**, 16, 157. [Google Scholar] [CrossRef] [PubMed] - Riest, J.; Mohanty, P.; Schurtenberger, P.; Likos, C.N. Coarse-graining of ionic microgels: Theory and experiment. Z. Phys. Chem.
**2012**, 226, 711. [Google Scholar] [CrossRef] - Vogel, N.; Fernandez-Lopez, C.; Perez-Juste, J.; Liz-Marzan, L.M.; Landfester, K.; Weiss, C.K. Ordered arrays of gold nanostructures from interfacially assembled [email protected] PNIPAM hybrid nanoparticles. Langmuir
**2012**, 28, 8985. [Google Scholar] [CrossRef] [PubMed] - Nazli, K.O.; Pester, C.; Konradi, A.; Boker, A.; Van Rijn, P. Cross-Linking Density and Temperature Effects on the Self-Assembly of SiO
_{2}—PNIPAAm Core–Shell Particles at Interfaces. Chemistry**2013**, 19, 5586. [Google Scholar] [CrossRef] [PubMed] - Volk, K.; Fitzgerald, J.P.; Retsch, M.; Karg, M. Time-Controlled Colloidal Superstructures: Long-Range Plasmon Resonance Coupling in Particle Monolayers. Adv. Mater.
**2015**, 27, 7332. [Google Scholar] [CrossRef] [PubMed] - Honold, T.; Volk, K.; Rauh, A.; Fitzgerald, J.P.S.; Karg, M. Tunable plasmonic surfaces via colloid assembly. J. Mater. Chem. C
**2015**, 3, 11449. [Google Scholar] [CrossRef][Green Version] - Geisel, K.; Rudov, A.A.; Potemkin, I.I.; Richtering, W. Hollow and core–shell microgels at oil–water interfaces: Spreading of soft particles reduces the compressibility of the monolayer. Langmuir
**2015**, 31, 13145. [Google Scholar] [CrossRef] [PubMed] - Karg, M. Functional materials design through hydrogel encapsulation of inorganic nanoparticles: Recent developments and challenges. Macromol. Chem. Phys.
**2016**, 217, 242. [Google Scholar] [CrossRef] - Sheverdin, A.; Valagiannopoulos, C. Core-shell nanospheres under visible light: Optimal absorption, scattering, and cloaking. Phys. Rev. B
**2019**, 99, 075305. [Google Scholar] [CrossRef] - Ciach, A.; Pekalski, J. Exactly solvable model for self-assembly of hard core–soft shell particles at interfaces. Soft Matter
**2017**, 13, 2603. [Google Scholar] [CrossRef] [PubMed][Green Version] - Groda, Y.G.; Grishina, V.; Ciach, A.; Vikhrenko, V.S. Phase diagram of the lattice fluid with SRLA-potential on the plane triangular lattice. J. Belarusian State Univ. Phys.
**2019**, 3, 81. [Google Scholar] [CrossRef] - Grishina, V.; Vikhrenko, V.; Ciach, A. Triangular lattice models for pattern formation by core-shell particles with different shell thicknesses. J. Phys. Condens. Matter
**2020**, 32, 405102. [Google Scholar] [CrossRef] [PubMed] - Somerville, W.R.C.; Law, A.D.; Rey, M.; Vogel, N.; Archer, A.J.; Buzza, D.M.A. Pattern formation in two-dimensional hard-core/soft-shell systems with variable soft shell profiles. Soft Matter
**2020**, 16, 3564. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ciarella, S.; Rey, M.; Harrer, J.; Holstein, N.; Ickler, M.; Löwen, H.; Vogel, N.; Janssen, L.M.C. Soft particles at liquid interfaces: From molecular particle architecture to collective phase behavior. arXiv
**2020**, arXiv:2008.13695v1. [Google Scholar] - Almarza, N.G.; Pȩkalski, J.; Ciach, A. Periodic ordering of clusters and stripes in a two-dimensional lattice model. II. Results of Monte Carlo simulation. J. Chem. Phys.
**2014**, 140, 164708. [Google Scholar] [CrossRef] [PubMed][Green Version]

**Figure 1.**The interaction potential as a function of the distance between the particle centers in units of the first-neighbor repulsion J. The symbols denote the interaction between the cores occupying the lattice sites, and the line is to guide the eye. The shown interactions ${J}_{2}={J}_{5}=1/3$ correspond to a crossover between different patterns formed by the particles, as described in Section 3.1.

**Figure 2.**The system of unit cells with particles belonging to nine sublattices and the lattice vectors ${\mathrm{e}}_{i},i=1,2,3$. The particles 1 and 2 or 1 and 4 are the nearest neighbors, the particles 1 and 5 or 2 and 7 are the next nearest neighbors, the particles 1 and 3 or 1 and 7 are the third neighbors, the particles 1 and 6 are the fourth neighbors. The particles with the same texture in nearest unit cells with the separation $3a$ are the fifth neighbors.

**Figure 3.**The possible distributions of particles over the unit cell for the concentrations $c=n/9$ with $2\le n\le 4$. For the concentrations with $5\le n\le 7$, the particles and vacancies have to be interchanged. For the concentrations with $n=1$ or 8, the particle or vacancy can occupy any lattice site of the unit cell. There are several equivalent distributions of particles over the unit cell for the concentrations with $2\le n\le 4$ or vacancies for the concentrations with $5\le n\le 7$.

**Figure 4.**The dimensionless thermodynamic Hamiltonian per lattice site versus the chemical potential for the concentrations $c=n/9,n=0,1,2,...9$ at ${J}_{2,5}=1/4$.

**Figure 5.**Cartoons of the distribution of particles over the lattice sites and the corresponding chemical potential intervals for the system with ${J}_{2,5}=1/4$.

**Figure 6.**The phase diagram of the system. The vertical lines show the chemical potential intervals for the system states at ${J}_{2,5}=1/4$, ${J}_{2,5}=1/3$ and ${J}_{2,5}=1/2$.

**Figure 7.**The dimensionless thermodynamic Hamiltonian per lattice site versus the chemical potential for the concentrations $c=n/9,n=0,1,2,...,9$ at ${J}_{2,5}=1/2$.

**Figure 8.**Cartoons of the distribution of particles over the lattice sites and the corresponding chemical potential intervals for the system with ${J}_{2,5}>1/3$.

**Figure 9.**Cartoon of the interface between the $c=1/9$ and 2/9 phases for the system at ${J}_{2,5}<1/3$.

**Figure 10.**The snapshot of the system of 37 particles on the lattice of $36\times 36$ lattice sites ($c\simeq 0.029$) after 9000 Monte Carlo simulation steps (MCS) at ${J}_{2,5}=1/2$. The interface lines are parallel to the lattice vectors ${\mathrm{e}}_{i}$.

**Figure 11.**The chemical potential as a function of the concentration at ${J}_{2,5}=0.5$ and several temperatures. The isotherms are shifted in the vertical direction by 3 from each other for clarity. The isotherm at $T=0.1$ is not shifted.

**Figure 12.**Snapshot of the system at $T=0.2$, $\mu =2.6$, ${J}_{2,5}=1/4$ after 8000 MCS. The extra particle (defect) is shown in red. This structure corresponds to the ground state configuration shown in Figure 5.

**Figure 13.**Snapshot of the system at $T=0.3$, $\mu =2.6$, ${J}_{2,5}=1/2$ after 8000 MCS. The additional vacancies (defects) are shown in yellow. This structure corresponds to the ground state configuration shown in the bottom row of Figure 8.

**Figure 14.**Snapshot of the system at $T=0.3$, $\mu =2.6$, ${J}_{2,5}=1/2$ after 8000 MCS. The extra particles and additional vacancies (defects) are shown in red and yellow, respectively. This structure corresponds to the ground state configuration shown in the upper row of Figure 8.

**Figure 15.**The inverse thermodynamic factor as a function of the concentration at ${J}_{2,5}=0.5$ and several temperatures. The curves are shifted in the vertical direction by ${3}^{3n}$ from the lowest one for clarity.

**Figure 16.**The dimensionless specific heat as a function of the concentration at ${J}_{2,5}=0.5$ and several temperatures. The curves are shifted in the vertical direction by ${3}^{3n}$ from the lowest one for clarity.

**Figure 17.**The fine structure of the inverse thermodynamic factor (

**a**) and dimensionless specific heat (

**b**) at $T=0.3$ and ${J}_{2,5}=0.5$ simulated with the reduced chemical potential step $\Delta \mu =0.002$. The scatter of the results characterizes the precision of the simulation.

**Figure 18.**The chemical potential as a function of the concentration at ${J}_{2,5}=0.25$ and several temperatures. The isotherms are shifted in the vertical direction by 3 from each other for clarity. The isotherm at $T=0.1$ is not shifted.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Grishina, V.; Vikhrenko, V.; Ciach, A. Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models. *Entropy* **2020**, *22*, 1215.
https://doi.org/10.3390/e22111215

**AMA Style**

Grishina V, Vikhrenko V, Ciach A. Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models. *Entropy*. 2020; 22(11):1215.
https://doi.org/10.3390/e22111215

**Chicago/Turabian Style**

Grishina, Vera, Vyacheslav Vikhrenko, and Alina Ciach. 2020. "Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models" *Entropy* 22, no. 11: 1215.
https://doi.org/10.3390/e22111215