# Structural and Dynamical Behaviour of Colloids with Competing Interactions Confined in Slit Pores

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## Abstract

**:**

## 1. Introduction

## 2. The Model and the Simulation Method

## 3. Results

#### 3.1. Equilibrium Properties

#### 3.1.1. Low Density: The Cluster-Crystal

#### 3.1.2. Intermediate Density: The Hexagonal Phase

#### 3.1.3. High Density: The Lamellar Phase

#### 3.2. Dynamic Properties

#### 3.2.1. Low Density: Cluster-Crystal

#### 3.2.2. Intermediate Density: Cylindrical Phase

#### 3.2.3. High Density: Lamellar Phase

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ruiz-Franco, J.; Zaccarelli, E. On the role of competing interactions in charged colloids with short-range attraction. Annu. Rev. Condens. Matter Phys.
**2021**, 12, 51–70. [Google Scholar] [CrossRef] - Ciach, A.; Pękalski, J.; Góźdź, W. Origin of similarity of phase diagrams in amphiphilic and colloidal systems with competing interactions. Soft Matter
**2013**, 9, 6301–6308. [Google Scholar] [CrossRef] [Green Version] - Ciach, A. Universal sequence of ordered structures obtained from mesoscopic description of self-assembly. Phys. Rev. E
**2008**, 78, 061505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pini, D.; Parola, A. Pattern formation and self-assembly driven by competing interactions. Soft Matter
**2017**, 13, 9259–9272. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhuang, Y.; Zhang, K.; Charbonneau, P. Equilibrium phase behavior of a continuous-space microphase former. Phys. Rev. Lett.
**2016**, 116, 098301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhuang, Y.; Charbonneau, P. Equilibrium phase behavior of the square-well linear microphase-forming model. J. Phys. Chem. B
**2016**, 120, 6178–6188. [Google Scholar] [CrossRef] [Green Version] - Royall, C.P. Hunting mermaids in real space: Known knowns, known unknowns and unknown unknowns. Soft Matter
**2018**, 14, 4020–4028. [Google Scholar] [CrossRef] [Green Version] - Zhang, T.H.; Kuipers, B.W.M.; de Tian, W.; Groenewold, J.; Kegel, W.K. Polydispersity and Gelation in Concentrated Colloids with Competing Interactions. Soft Matter
**2015**, 11, 297–302. [Google Scholar] [CrossRef] [Green Version] - Toledano, J.C.F.; Sciortino, F.; Zaccarelli, E. Colloidal systems with competing interactions: From an arrested repulsive cluster phase to a gel. Soft Matter
**2009**, 5, 2390–2398. [Google Scholar] [CrossRef] - Klix, C.L.; Royall, C.P.; Tanaka, H. Structural and dynamical features of multiple metastable glassy states in a colloidal system with competing interactions. Phys. Rev. Lett.
**2010**, 104, 165702. [Google Scholar] [CrossRef] [Green Version] - Campbell, A.I.; Anderson, V.J.; van Duijneveldt, J.S.; Bartlett, P. Dynamical arrest in attractive colloids: The effect of long-range repulsion. Phys. Rev. Lett.
**2005**, 94, 208301. [Google Scholar] [CrossRef] [Green Version] - Zhuang, Y.; Charbonneau, P. Recent advances in the theory and simulation of model colloidal microphase formers. J. Phys. Chem. B
**2016**, 120, 7775–7782. [Google Scholar] [CrossRef] [Green Version] - Khandpur, A.K.; Foerster, S.; Bates, F.S.; Hamley, I.W.; Ryan, A.J.; Bras, W.; Almdal, K.; Mortensen, K. Polyisoprene-polystyrene diblock copolymer phase diagram near the order-disorder transition. Macromolecules
**1995**, 28, 8796–8806. [Google Scholar] [CrossRef] - Hu, H.; Gopinadhan, M.; Osuji, C.O. Directed self-assembly of block copolymers: A tutorial review of strategies for enabling nanotechnology with soft matter. Soft Matter
**2014**, 10, 3867–3889. [Google Scholar] [CrossRef] - Doerk, G.S.; Yager, K.G. Beyond native block copolymer morphologies. Mol. Syst. Des. Eng.
**2017**, 2, 518–538. [Google Scholar] [CrossRef] - Guo, Y.; van Ravensteijn, B.G.P.; Kegel, W.K. Self-assembly of isotropic colloids into colloidal strings, Bernal spiral-like, and tubular clusters. Chem. Commun.
**2020**, 56, 6309–6312. [Google Scholar] [CrossRef] [PubMed] - Serna, H.; Díaz Pozuelo, A.; Noya, E.G.; Góźdź, W.T. Formation and internal ordering of periodic microphases in colloidal models with competing interactions. Soft Matter
**2021**, 17, 4957. [Google Scholar] [CrossRef] - Míguez, H.; Yang, S.M.; Ozin, G.A. Optical properties of colloidal photonic crystals confined in rectangular microchannels. Langmuir
**2003**, 19, 3479–3485. [Google Scholar] [CrossRef] - Pękalski, J.; Rzadkowski, W.; Panagiotopoulos, A.Z. Shear-induced ordering in systems with competing interactions: A machine learning study. J. Chem. Phys.
**2020**, 152, 204905. [Google Scholar] [CrossRef] [PubMed] - Imperio, A.; Reatto, L. Microphase morphology in two-dimensional fluids under lateral confinement. Phys. Rev. E
**2007**, 76, 040402(R). [Google Scholar] [CrossRef] - Alba-Simionesco, A.; Coasne, B.; Dosseh, G.; Dudziak, G.; Gubbins, K.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Effects of confinement on freezing and melting. J. Phys. Condens. Matter
**2006**, 18, R15. [Google Scholar] [CrossRef] - Nygård, K. Colloidal diffusion in confined geometries. Phys. Chem. Chem. Phys.
**2017**, 19, 23632. [Google Scholar] [CrossRef] - Zangi, R. Water confined to a slab geometry: A review of recent computer simulation studies. J. Phys. Condens. Matter
**2004**, 16, S5371. [Google Scholar] [CrossRef] - Martí, J.; Calero, C.; Franzese, G. Structure and dynamics of water at carbon-based interfaces. Entropy
**2017**, 19, 135. [Google Scholar] [CrossRef] [Green Version] - Serna, H.; Noya, E.G.; Góźdź, W.T. Assembly of Helical Structures in Systems with Competing Interactions under Cylindrical Confinement. Langmuir
**2018**, 35, 702–708. [Google Scholar] [CrossRef] - Serna, H.; Noya, E.G.; Góźdź, W.T. The influence of confinement on the structure of colloidal systems with competing interactions. Soft Matter
**2020**, 16, 718–727. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Serna, H.; Noya, E.G.; Góźdź, W.T. Confinement of Colloids with Competing Interactions in Ordered Porous Materials. J. Phys. Chem. B
**2020**, 124, 10567–10577. [Google Scholar] [CrossRef] [PubMed] - Pękalski, J.; Almarza, N.G.; Ciach, A. Effects of rigid or adaptive confinement on colloidal self-assembly. Fixed vs. fluctuating number of confined particles. J. Chem. Phys.
**2015**, 142, 204904. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Almarza, N.G.; Pękalski, J.; Ciach, A. Effects of confinement on pattern formation in two dimensional systems with competing interactions. Soft Matter
**2016**, 12, 7551. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schwanzer, D.F.; Coslovich, D.; Kahl, G. Two-dimensional systems with competing interactions: Dynamic properties of single particles and of clusters. J. Phys. Condens. Matter
**2016**, 28, 414015. [Google Scholar] [CrossRef] [PubMed] - Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids; Oxford University Press: Oxford, UK, 2017. [Google Scholar]
- Faken, D.; Jónsson, H. Systematic analysis of local atomic structure combined with 3D computer graphics. Comput. Mater. Sci.
**1994**, 2, 279–286. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool. Model. Simul. Mater. Sci. Eng.
**2010**, 18. [Google Scholar] [CrossRef] - Bores, C.; Almarza, N.G.; Lomba, E.; Kahl, G. Inclusions of a two dimensional fluid with competing interactions in a disordered, porous matrix. J. Ournal. Phys. Condens. Matter
**2015**, 27, 194127. [Google Scholar] [CrossRef] [Green Version] - Qiao, C.; Zhao, S.; Liu, H.; Dong, W. Connect the Thermodynamics of Bulk and Confined Fluids: Confinement-Adsorption Scaling. Langmuir
**2019**, 35, 3840–3847. [Google Scholar] [CrossRef] [PubMed] - Lechner, W.; Dellago, C. Accurate determination of crystal structures based on averaged local bond order parameters. J. Chem. Phys.
**2008**, 129, 114707. [Google Scholar] [CrossRef] [PubMed] - Yu, B.; Li, B.; Jin, Q.; Ding, D.; Shi, A.C. Confined self-assembly of cylinder-forming diblock copolymers: Effects of confining geometries. Soft Matter
**2011**, 7, 10227–10240. [Google Scholar] [CrossRef] - Mittal, J.; Truskett, T.M.; Errington, J.R.; Hummer, G. Layering and position-dependent diffusive dynamics of confined fluids. Phys. Rev. Lett.
**2008**, 100, 145901. [Google Scholar] [CrossRef] [Green Version] - Litniewski, L.; Ciach, A. Effect of aggregation on adsorption phenomena. J. Chem. Phys.
**2019**, 150, 234702. [Google Scholar] [CrossRef] - Liu, Y.; Zhao, W.; Zheng, X.; King, A.; Singh, A.; Rafailovich, M.; Sokolov, J.; Dai, K.; Kramer, E. Surface-induced ordering in asymmetric block copolymers. Macromolecules
**1994**, 27, 4000–4010. [Google Scholar] [CrossRef]

**Figure 1.**The Lennard-Jones plus Yukawa potential used to model the interactions between colloidal particles and the Lennard-Jones potential truncated and shifted at the energy minimum used to model the interactions between the slit walls and the particles.

**Figure 2.**Sketch of the bulk phase diagram, using data from Ref. [17]. The state points studied for each pore size ${W}^{*}=$ 5, 7, 9 and 11 are marked with different symbols, and their colours indicate the structure adopted by the confined fluid in each thermodynamic state, as provided in the legend.

**Figure 3.**Local density isosurfaces ${\rho}_{iso}^{*}=0.30$ for all the ordered microphases obtained at different slit widths, ${W}^{*}$. Note that the density chosen for the isosurfaces is somewhat lower than that in our previous work on SALR systems modelled with the square-well linear model (in which we chose ${\rho}_{iso}^{*}=0.40$) [25,26,27]. The reason for this new choice is that the clusters obtained with the Lennard-Jones plus Yukawa model used in this work are appreciably smaller [17]. Two views are presented for cluster-crystal and hexagonal phases and one for the lamellar phase. The number densities in reduced units, ${\rho}^{*}$, are specified and the temperature is ${T}^{*}=0.30$.

**Figure 4.**Cluster-size distributions of the cluster-crystal in bulk and in the slit pores of width ${W}^{*}$ at ${T}^{*}=0.3$ and ${T}^{*}=0.4$.

**Figure 6.**Top panel:the fraction of particles with local icosahedral environments as a function of temperature for the bulk and the confined systems at a low density at which the cluster-crystal phase is stable. Central panel: the fraction of particles with local icosahedral environments as a function of temperature for the bulk and the confined systems at an intermediate density at which the cylindrical phase is stable. Bottom panel: the fraction of particles within a hexagonal local environment as a function of temperature for the bulk and the confined systems at a high density at which the lamellar phase is stable. Note that the classification of particles in these plots is based solely on analysis of the local structure around each particle; the dynamics of the particles was not taken into account in this analysis.

**Figure 7.**Particle mean squared displacement (MSD) of the fluid confined in slit pores of different width ${W}^{*}$ at different temperatures at densities at which the cluster-crystal (

**top**row), the cylindrical (

**middle**row) and the lamellar (

**bottom**row) phases are stable. For comparison, the MSD for the bulk system under similar thermodynamic conditions are also included. The dashed black and blue lines show the expected behaviour for diffusive (MSD$\propto {t}^{\beta}$, $\beta =1$) and ballistic ($\beta =2$) behaviour.

**Figure 8.**Diffusion coefficient as a function of pore width, for low (

**left**panel), medium (

**middle**panel) and high (

**right**panel) densities, at temperatures around those at which the periodic microphases start to spontaneously form.

**Table 1.**Average number of particles confined in the slit pores at which the fluid organises into ordered structures at ${T}^{*}=0.3$ at densities at which the bulk fluid assembles into a cluster-crystal, a cylindrical and a lamellar phase. Note that the chemical potential for the more dense lamellar phase might not be reliable due to the low acceptance probability of the insertion/deletion MC moves. In any case, we only used these simulations to generate the initial configurations for the NVT MD runs.

Phase | ${\mathit{W}}^{*}=5.0$ | ${\mathit{W}}^{*}=7.0$ | ${\mathit{W}}^{*}=9.0$ | ${\mathit{W}}^{*}=11.0$ |

Cluster-Crystal | $\begin{array}{c}N=1030\hfill \\ {\mu}^{*}=-1.20\hfill \end{array}$ | $\begin{array}{c}N=1748\hfill \\ {\mu}^{*}=-1.00\hfill \end{array}$ | $\begin{array}{c}N=2403\hfill \\ {\mu}^{*}=-1.00\hfill \end{array}$ | $\begin{array}{c}N=2807\hfill \\ {\mu}^{*}=-1.00\hfill \end{array}$ |

Cylindrical | $\begin{array}{c}N=1868\hfill \\ {\mu}^{*}=-0.60\hfill \end{array}$ | $\begin{array}{c}N=2923\hfill \\ {\mu}^{*}=-0.40\hfill \end{array}$ | $\begin{array}{c}N=3845\hfill \\ {\mu}^{*}=-0.40\hfill \end{array}$ | $\begin{array}{c}N=4065\hfill \\ {\mu}^{*}=-0.60\hfill \end{array}$ |

Lamellar | $\begin{array}{c}N=3432\hfill \\ {\mu}^{*}=0.50\hfill \end{array}$ | $\begin{array}{c}N=5399\hfill \\ {\mu}^{*}=0.20\hfill \end{array}$ | $\begin{array}{c}N=6124\hfill \\ {\mu}^{*}=0.50\hfill \end{array}$ | $\begin{array}{c}N=8058\hfill \\ {\mu}^{*}=0.40\hfill \end{array}$ |

**Table 2.**Estimation of the distance between clusters (${l}_{0}^{*}={l}_{0}/\sigma $) and the average cluster size (${d}_{0}^{*}={d}_{0}/\sigma $) in bulk and in the confined systems. The cluster size ${d}_{0}^{*}$ corresponds to the average cluster diameter in spherical and cylindrical clusters, and to the width of the lamellae in the lamellar phase.

${\mathit{W}}^{*}$ | $5.0$ | $7.0$ | $9.0$ | $11.0$ | Bulk |
---|---|---|---|---|---|

Cluster-crystal | $\begin{array}{c}{l}_{0}^{*}=5.9\hfill \\ {d}_{0}^{*}=3.3\hfill \\ {\rho}^{*}=0.1287\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=5.8\hfill \\ {d}_{0}^{*}=3.1\hfill \\ {\rho}^{*}=0.1561\hfill \end{array}$ | $\begin{array}{c}--\hfill \\ --\hfill \\ {\rho}^{*}=0.1668\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=6.0\hfill \\ {d}_{0}^{*}=3.3\hfill \\ {\rho}^{*}=0.1594\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=5.6\hfill \\ {d}_{0}^{*}=3.3\hfill \\ {\rho}^{*}=0.155\hfill \end{array}$ |

Hexagonal | $\begin{array}{c}{l}_{0}^{*}=5.3\hfill \\ {d}_{0}^{*}=3.0\hfill \\ {\rho}^{*}=0.2335\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=5.5\hfill \\ {d}_{0}^{*}=3.0\hfill \\ {\rho}^{*}=0.2610\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=5.8\hfill \\ {d}_{0}^{*}=3.2\hfill \\ {\rho}^{*}=0.2670\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=6.2\hfill \\ {d}_{0}^{*}=3.0\hfill \\ {\rho}^{*}=0.2309\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=5.6\hfill \\ {d}_{0}^{*}=2.8\hfill \\ {\rho}^{*}=0.252\hfill \end{array}$ |

Lamellar | $\begin{array}{c}--\hfill \\ {d}_{0}^{*}=2.9\hfill \\ {\rho}^{*}=0.4290\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=3.7\hfill \\ {d}_{0}^{*}=1.5\hfill \\ {\rho}^{*}=0.4820\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=4.7\hfill \\ {d}_{0}^{*}=2.2\hfill \\ {\rho}^{*}=0.4252\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=3.9\hfill \\ {d}_{0}^{*}=1.7\hfill \\ {\rho}^{*}=0.4578\hfill \end{array}$ | $\begin{array}{c}{l}_{0}^{*}=4.6\hfill \\ {d}_{0}^{*}=2.2\hfill \\ {\rho}^{*}=0.407\hfill \end{array}$ |

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**MDPI and ACS Style**

Serna, H.; Góźdź, W.T.; Noya, E.G.
Structural and Dynamical Behaviour of Colloids with Competing Interactions Confined in Slit Pores. *Int. J. Mol. Sci.* **2021**, *22*, 11050.
https://doi.org/10.3390/ijms222011050

**AMA Style**

Serna H, Góźdź WT, Noya EG.
Structural and Dynamical Behaviour of Colloids with Competing Interactions Confined in Slit Pores. *International Journal of Molecular Sciences*. 2021; 22(20):11050.
https://doi.org/10.3390/ijms222011050

**Chicago/Turabian Style**

Serna, Horacio, Wojciech T. Góźdź, and Eva G. Noya.
2021. "Structural and Dynamical Behaviour of Colloids with Competing Interactions Confined in Slit Pores" *International Journal of Molecular Sciences* 22, no. 20: 11050.
https://doi.org/10.3390/ijms222011050