# How Dimensionality Affects the Structural Anomaly in a Core-Softened Colloid

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. Methods and Simulation Details

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Vilaseca, P.; Franzese, G. Isotropic soft-core potentials with two characteristic length scales and anomalous behaviour. J. Non-Cryst. Solids
**2011**, 357, 419–426. [Google Scholar] [CrossRef] - 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] - Shukla, A.; Mylonas, E.; Di Cola, E.; Finet, S.; Timmins, P.; Narayanan, T.; Svergun, D.I. Absence of equilibrium cluster phase in concentrated lysozyme solutions. Proc. Natl. Acad. Sci. USA
**2008**, 105, 5075–5080. [Google Scholar] [CrossRef] - Yethiraj, A. Tunable colloids: Control of colloidal phase transitions with tunable interactions. Soft Matter
**2007**, 3, 1099. [Google Scholar] [CrossRef] - Jagla, E.A. Phase behavior of a system of particles with core collapse. Phys. Rev. E
**1998**, 58, 1478. [Google Scholar] [CrossRef] - Jagla, E.A. Minimum energy configurations of repelling particles in two dimensions. J. Chem. Phys.
**1999**, 110, 451–456. [Google Scholar] [CrossRef] - Jagla, E.A. Core-softened potentials and the anomalous properties of water. J. Chem. Phys.
**1999**, 111, 8980–8986. [Google Scholar] [CrossRef] - de Oliveira, A.B.; Netz, P.A.; Barbosa, M.C. Interplay between structure and density anomaly for an isotropic core-softened ramp-like potential. Physica A
**2007**, 386, 744–747. [Google Scholar] [CrossRef] - Barbosa, M.A.; Salcedo, E.; Barbosa, M.C. Multiple liquid-liquid critical points and density anomaly in core-softened potentials. Phys. Rev. E
**2013**, 87, 032303. [Google Scholar] [CrossRef] - Fomin, Y.D.; Tsiok, E.N.; Ryzhov, V.N. Inversion of sequence of diffusion and density anomalies in core-softened systems. J. Chem. Phys.
**2011**, 135, 234502. [Google Scholar] [CrossRef] - Yan, Z.; Buldyrev, S.V.; Giovambattista, N.; Stanley, H.E. Structural Order for One-Scale and Two-Scale Potentials. Phys. Rev. Lett.
**2005**, 95, 130604. [Google Scholar] [CrossRef] - Fomin, D.Y.; Gribova, N.V.; Ryzhov, V.N.; Stishov, S.M.; Frenkel, D. Quasibinary amorphous phase in a three-dimensional system of particles with repulsive-shoulder interactions. J. Chem. Phys.
**2008**, 129, 064512. [Google Scholar] [CrossRef] - Lascaris, E.; Malescio, G.; Buldyrev, S.V.; Stanley, H.E. Cluster formation, water-like anomalies, and re-entrant melting for a family of bounded repulsive interaction potentials. Phys. Rev. E
**2010**, 81, 031201. [Google Scholar] [CrossRef] [PubMed] - Buldyrev, S.V.; Malescio, G.; Angell, C.A.; Giovambattista, N.; Prestipino, S.; Saija, F.; Stanley, H.E.; Xu, L. Unusual phase behavior of one-component systems with two-scale isotropic interactions. J. Phys. Condens. Matter
**2009**, 21, 504106. [Google Scholar] [CrossRef] - Saija, F.; Prestipino, S.; Malescio, G. Anomalous phase behavior of a soft-repulsive potential with a strictly monotonic force. Phys. Rev. E
**2009**, 80, 031502. [Google Scholar] [CrossRef] - Malescio, G.; Saija, F. A Criterion for Anomalous Melting in Systems with Isotropic Interactions. J. Phys. Chem. B
**2011**, 115, 14091–14098. [Google Scholar] [CrossRef] - Prestipino, S.; Saija, F.; Malescio, G. Anomalous phase behavior in a model fluid with only one type of local structure. J. Chem. Phys.
**2010**, 133, 144504. [Google Scholar] [CrossRef] - Prestipino, S.; Saija, F.; Giaquinta, P.V. Hexatic phase and water-like anomalies in a two-dimensional fluid of particles with a weakly softened core. J. Chem. Phys.
**2012**, 137, 104503. [Google Scholar] [CrossRef] [PubMed] - Coslovich, D.; Ikeda, A. Cluster and reentrant anomalies of nearly Gaussian core particles. Soft Matter
**2013**, 9, 6786–6795. [Google Scholar] [CrossRef] - Quesada-Perez, M.; Moncho-Jorda, A.; Martinez-Lopez, F.; Hidalgo-Álvarez, R. Probing interaction forces in colloidal monolayers: Inversion of structural data. J. Chem. Phys.
**2001**, 115, 10897–10902. [Google Scholar] [CrossRef] - Contreras-Aburto, C.; Méndez-Alcaraz, J.M.; Castañeda-Priego, R. Structure and effective interactions in parallel monolayers of charged spherical colloids. J. Chem. Phys.
**2010**, 132, 174111. [Google Scholar] [CrossRef] [PubMed] - Haddadi, S.; Skepö, M.; Jannasch, P.; Manner, S.; Forsman, J. Building polymer-like clusters from colloidal particles with isotropic interactions, in aqueous solution. J. Colloid Interface Sci.
**2020**, 581, 669–681. [Google Scholar] [CrossRef] [PubMed] - Grillo, F.; Fernandez-Rodriguez, M.A.; Antonopoulou, M.N.; Gerber, D.; Isa, L. Self-templating assembly of soft microparticles into complex tessellations. Nature
**2020**, 582, 219–224. [Google Scholar] [CrossRef] [PubMed] - Ranganathan, V.T.; Bazmi, S.; Wallin, S.; Liu, Y.; Yethiraj, A. Is Ficoll a Colloid or Polymer? A Multitechnique Study of a Prototypical Excluded-Volume Macromolecular Crowder. Macromolecules
**2022**, 55, 9103–9112. [Google Scholar] [CrossRef] - Marques, M.S.; Nogueira, T.P.O.; Dillenburg, R.F.; Barbosa, M.C.; Bordin, J.R. Waterlike anomalies in hard core–soft shell nanoparticles using an effective potential approach: Pinned vs adsorbed polymers. J. Appl. Phys.
**2020**, 127, 054701. [Google Scholar] [CrossRef] - Lafitte, T.; Kumar, S.K.; Panagiotoulos, A.Z. Self-assembly of polymer-grafted nanoparticles in thin films. Soft Matter
**2014**, 10, 786–794. [Google Scholar] [CrossRef] - Bos, I.; van der Scheer, P.; Ellenbroek, W.G.; Sprakel, J. Two-dimensional crystals of star polymers: A tale of tails. Soft Matter
**2019**, 15, 615–622. [Google Scholar] [CrossRef] - Angell, C.A. Two phases? Nat. Mater.
**2014**, 13, 673–675. [Google Scholar] [CrossRef] - Gallo, P.; Amann-Winkel, K.; Angell, C.A.; Anisimov, M.A.; Caupin, F.; Chakravarty, C.; Lascaris, E.; Loerting, T.; Panagiotopoulos, A.Z.; Russo, J.; et al. Water: A Tale of Two Liquids. Chem. Rev.
**2016**, 116, 7463–7500. [Google Scholar] [CrossRef] - Gallo, P.; Bachler, J.; Bove, L.E.; Böhmer, R.; Camisasca, G.; Coronas, L.E.; Corti, H.R.; de Almeida Ribeiro, I.; de Koning, M.; Franzese, G.; et al. Advances in the study of supercooled water. Eur. Phys. J. E
**2021**, 44, 143. [Google Scholar] [CrossRef] - Chaplin, M. Anomalous Properties of Water. 2023. Available online: http://www.lsbu.ac.uk/water/anmlies.html (accessed on 16 February 2023).
- Kellu, G.S. Density, thermal expansivity, and compressibility of liquid water from 0.deg. to 150.deg. Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. J. Chem. Eng. Data
**1975**, 20, 97–105. [Google Scholar] [CrossRef] - Errington, J.R.; Debenedetti, P.D. Relationship between structural order and the anomalies of liquid water. Nature
**2001**, 409, 318–321. [Google Scholar] [CrossRef] [PubMed] - Yan, Z.; Buldyrev, S.V.; Giovambattista, N.; Debenedetti, P.G.; Stanley, H.E. Family of tunable spherically symmetric potentials that span the range from hard spheres to water-like behavior. Phys. Rev. E
**2006**, 73, 051204. [Google Scholar] [CrossRef] [PubMed] - Gibson, H.M.; Wilding, N.B. Metastable liquid-liquid coexistence and density anomalies in a core-softened fluid. Phys. Rev. E
**2006**, 73, 061507. [Google Scholar] [CrossRef] - Vilaseca, P.; Franzese, G. Softness dependence of the anomalies for the continuous shouldered well potential. J. Chem. Phys.
**2010**, 133, 084507. [Google Scholar] [CrossRef] [PubMed] - Xu, L.; Giovambattista, N.; Buldyrev, S.V.; Debenedetti, P.G.; Stanley, H.E. Waterlike glass polyamorphism in a monoatomic isotropic Jagla model. J. Chem. Phys.
**2011**, 134, 064507. [Google Scholar] [CrossRef] [PubMed] - Reisman, S.; Giovambattista, N. Glass and liquid phase diagram of a polyamorphic monatomic system. J. Chem. Phys.
**2013**, 138, 064509. [Google Scholar] [CrossRef] [PubMed] - Das, G.; Gnan, N.; Sciortino, F.; Zaccarelli, E. Unveiling the complex glassy dynamics of square shoulder systems: Simulations and theory. J. Chem. Phys.
**2013**, 138, 134501. [Google Scholar] [CrossRef] - Krott, L.B.; Bordin, J.R.; Barraz, N.M.; Barbosa, M.C. Effects of confinement on anomalies and phase transitions of core-softened fluids. J. Chem. Phys.
**2015**, 142, 134502. [Google Scholar] [CrossRef] - Luo, J.; Xu, L.; Angell, C.A.; Stanley, H.E.; Buldyrev, S.V. Physics of the Jagla model as the liquid-liquid coexistence line slope varies. J. Chem. Phys.
**2015**, 142, 224501. [Google Scholar] [CrossRef] - Bordin, J. Waterlike features, liquid-crystal phase and self-assembly in Janus dumbbells. Phys. A Stat. Mech. Appl.
**2016**, 459, 1–8. [Google Scholar] [CrossRef] - Pinheiro, L.; Furlan, A.; Krott, L.; Diehl, A.; Barbosa, M. Critical points, phase transitions and water-like anomalies for an isotropic two length scale potential with increasing attractive well. Phys. A Stat. Mech. Appl.
**2017**, 468, 866–879. [Google Scholar] [CrossRef] - de Haro, M.L.; Rodríguez-Rivas, A.; Yuste, S.B.; Santos, A. Structural properties of the Jagla fluid. Phys. Rev. E
**2018**, 98, 012138. [Google Scholar] [CrossRef] [PubMed] - Higuchi, S.; Kato, D.; Awaji, D.; Kim, K. Connecting thermodynamic and dynamical anomalies of water-like liquid-liquid phase transition in the Fermi–Jagla model. J. Chem. Phys.
**2018**, 148, 094507. [Google Scholar] [CrossRef] - Ryzhov, V.N.; Tareyeva, E.E.; Fomin, Y.D.; Tsiok, E.N. Complex phase diagrams of systems with isotropic potentials: Results of computer simulations. Physics-Uspekhi
**2020**, 63, 417. [Google Scholar] [CrossRef] - Martín-Roca, J.; Martinez, R.; Martínez-Pedrero, F.; Ramírez, J.; Valeriani, C. Dynamical anomalies and structural features of active Brownian particles characterized by two repulsive length scales. J. Chem. Phys.
**2022**, 156, 164502. [Google Scholar] [CrossRef] - Bretonnet, J.L.; Bomont, J.M. Analytical treatment of the structure for systems interacting via core-softened potentials. Chem. Phys.
**2022**, 555, 111445. [Google Scholar] [CrossRef] - Nogueira, T.; Bordin, J.R. Patterns in 2D core-softened systems: From sphere to dumbbell colloids. Phys. A Stat. Mech. Appl.
**2022**, 605, 128048. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B.; Barbosa, M.C. Self-Assembly and Water-like Anomalies in Janus Nanoparticles. Langmuir
**2015**, 31, 8577–8582. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B. How Competitive Interactions Affect the Self-Assembly of Confined Janus Dumbbells. J. Phys. Chem. B
**2017**, 121, 4308–4317. [Google Scholar] [CrossRef] - Jiménez-Millán, S.; García-Alcántara, C.; Ramírez-Hernández, A.; Sambriski, E.; Hernández, S. Self-Aassembly of core-corona colloids under cylindrical confinement: A Monte Carlo study. J. Mol. Liq.
**2021**, 335, 116219. [Google Scholar] [CrossRef] - Pérez-Figueroa, S.E.; Gallegos-Lozano, A.; Mendoza, C.I. Packing core–corona particles on a spherical surface. Soft Matter
**2022**, 18, 6812–6824. [Google Scholar] [CrossRef] - Isa, L.; Buttinoni, I.; Fernandez-Rodriguez, M.A.; Vasudevan, S.A. Two-dimensional assemblies of soft repulsive colloids confined at fluid interfaces(a). Europhys. Lett.
**2017**, 119, 26001. [Google Scholar] [CrossRef] - Wang, J.; Mbah, C.F.; Przybilla, T.; Englisch, S.; Spiecker, E.; Engel, M.; Vogel, N. Free Energy Landscape of Colloidal Clusters in Spherical Confinement. ACS Nano
**2019**, 13, 9005–9015. [Google Scholar] [CrossRef] [PubMed] - Osterman, N.; Babič, D.; Poberaj, I.; Dobnikar, J.; Ziherl, P. Observation of Condensed Phases of Quasiplanar Core-Softened Colloids. Phys. Rev. Lett.
**2007**, 99, 248301. [Google Scholar] [CrossRef] - Villada-Balbuena, A.; Jung, G.; Zuccolotto-Bernez, A.B.; Franosch, T.; Egelhaaf, S.U. Layering and packing in confined colloidal suspensions. Soft Matter
**2022**, 18, 4699–4714. [Google Scholar] [CrossRef] - Kang, C.; Honciuc, A. Self-Assembly of Janus Nanoparticles into Transformable Suprastructures. J. Phys. Chem. Lett.
**2018**, 9, 1415–1421. [Google Scholar] [CrossRef] - Wu, D.; Honciuc, A. Design of Janus Nanoparticles with pH-Triggered Switchable Amphiphilicity for Interfacial Applications. ACS Appl. Nano Mater.
**2018**, 1, 471–482. [Google Scholar] [CrossRef] - Rafael Bordin, J. Distinct self-assembly aggregation patters of nanorods with decorated ends: A simple model study. Fluid Phase Equilibria
**2019**, 499, 112251. [Google Scholar] [CrossRef] - Bordin, J.R. Distinct aggregation patterns and fluid porous phase in a 2D model for colloids with competitive interactions. Phys. A Stat. Mech. Appl.
**2018**, 495, 215–224. [Google Scholar] [CrossRef] - Cardoso, D.S.; Hernandes, V.F.; Nogueira, T.; Bordin, J.R. Structural behavior of a two length scale core-softened fluid in two dimensions. Phys. A Stat. Mech. Appl.
**2021**, 566, 125628. [Google Scholar] [CrossRef] - Fonseca, E.R.; Mendoza, C.I. Self-assembly of core-corona particles confined in a circular box. J. Phys. Condens. Matter
**2019**, 32, 015101. [Google Scholar] [CrossRef] - Marques, M.S.; Bordin, J.R. Interplay between adsorption, aggregation and diffusion in confined core-softened colloids. JCIS Open
**2021**, 4, 100029. [Google Scholar] [CrossRef] - Pekalski, J.; Bildanau, E.; Ciach, A. Self-assembly of spiral patterns in confined systems with competing interactions. Soft Matter
**2019**, 15, 7715–7721. [Google Scholar] [CrossRef] - 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] - Pant, S.; Ghorai, P.K. Structural anomaly of core-softened fluid confined in single walled carbon nanotube: A molecular dynamics simulation investigation. Mol. Phys.
**2016**, 114, 1771–1777. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B. Confinement effects on the properties of Janus dimers. Phys. Chem. Chem. Phys.
**2016**, 18, 28740–28746. [Google Scholar] [CrossRef] - Dobnikar, J.; Fornleitner, J.; Kahl, G. Ground states of model core-softened colloids. J. Phys. Condens. Matter
**2008**, 20, 494220. [Google Scholar] [CrossRef] - Bildanau, E.; Pękalski, J.; Vikhrenko, V.; Ciach, A. Adsorption anomalies in a two-dimensional model of cluster-forming systems. Phys. Rev. E
**2020**, 101, 012801. [Google Scholar] [CrossRef] - Tsiok, E.; Fomin, Y.D.; Ryzhov, V. The effect of confinement on the solid–liquid transition in a core-softened potential system. Phys. A Stat. Mech. Appl.
**2020**, 550, 124521. [Google Scholar] [CrossRef] - Fomin, Y.D.; Ryzhov, V.N.; Tsiok, E.N. Interplay between freezing and density anomaly in a confined core-softened fluid. Mol. Phys.
**2020**, 118, e1718792. [Google Scholar] [CrossRef] - Fomin, Y.D. Between two and three dimensions: Crystal structures in a slit pore. J. Colloid Interface Sci.
**2020**, 580, 135–145. [Google Scholar] [CrossRef] - Barros de Oliveira, A.; Netz, P.A.; Colla, T.; Barbosa, M.C. Thermodynamic and dynamic anomalies for a three-dimensional isotropic core-softened potential. J. Chem. Phys.
**2006**, 124, 084505. [Google Scholar] [CrossRef] - de Oliveira, A.B.; Netz, P.A.; Colla, T.; Barbosa, M.C. Structural anomalies for a three dimensional isotropic core-softened potential. J. Chem. Phys.
**2006**, 125, 124503. [Google Scholar] [CrossRef] [PubMed] - Bordin, J.R.; de Oliveira, A.B.; Diehl, A.; Barbosa, M.C. Diffusion enhancement in core-softened fluid confined in nanotubes. J. Chem. Phys.
**2012**, 137, 084504. [Google Scholar] [CrossRef] - Krott, L.B.; Barbosa, M.C. Anomalies in a water-like model confined between plates. J. Chem. Phys.
**2013**, 138, 084505. [Google Scholar] [CrossRef] - Bordin, J.R.; Diehl, A.; Barbosa, M.C. Relation Between Flow Enhancement Factor and Structure for Core-Softened Fluids Inside Nanotubes. J. Phys. Chem. B
**2013**, 117, 7047–7056. [Google Scholar] [CrossRef] [PubMed] - Krott, L.B.; Barbosa, M.C. Model of water-like fluid under confinement for hydrophobic and hydrophilic particle-plate interaction potentials. Phys. Rev. E
**2014**, 89, 012110. [Google Scholar] [CrossRef] [PubMed] - Krott, L.B.; Bordin, J.R. Distinct dynamical and structural properties of a core-softened fluid when confined between fluctuating and fixed walls. J. Chem. Phys.
**2013**, 139, 154502. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B.; Barbosa, M.C. High pressure induced phase transition and superdiffusion in anomalous fluid confined in flexible nanopores. J. Chem. Phys.
**2014**, 141, 144502. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B.; Barbosa, M.C. Surface Phase Transition in Anomalous Fluid in Nanoconfinement. J. Phys. Chem. C
**2014**, 118, 9497–9506. [Google Scholar] [CrossRef] - Krott, L.B.; Bordin, J.R.; Barbosa, M.C. New Structural Anomaly Induced by Nanoconfinement. J. Phys. Chem. B
**2015**, 119, 291–300. [Google Scholar] [CrossRef] [PubMed] - Bordin, J.R.; Barbosa, M.C. Waterlike anomalies in a two-dimensional core-softened potential. Phys. Rev. E
**2018**, 97, 022604. [Google Scholar] [CrossRef] [PubMed] - Dudalov, D.E.; Fomin, Y.D.; Tsiok, E.N.; Ryzhov, V.N. How dimensionality changes the anomalous behavior and melting scenario of a core-softened potential system? Soft Matter
**2014**, 10, 4966–4976. [Google Scholar] [CrossRef] - Bordin, J.R.; Krott, L.B. Solid-amorphous transition is related to the water-like anomalies in a fluid without liquid–liquid phase transition. J. Chem. Phys.
**2023**, 158, 134501. [Google Scholar] [CrossRef] [PubMed] - Allen, P.; Tildesley, D.J. Computer Simulation of Liquids; Oxford University Press: Oxford, UK, 1987. [Google Scholar]
- Dzugutov, M. Universal scaling law for atomic diffusion in condensed matter. Nature
**1996**, 381, 137–139. [Google Scholar] [CrossRef] - Dyre, J.C. Perspective: Excess-entropy scaling. J. Chem. Phys.
**2018**, 149, 210901. [Google Scholar] [CrossRef] - Bell, I.; Dyre, J.; Ingebrigtsen, T. Excess-entropy scaling in supercooled binary mixtures. Nat. Commun.
**2020**, 2020, 015012. [Google Scholar] [CrossRef] - Galliero, G.; Boned, C.; Fernández, J. Scaling of the viscosity of the Lennard-Jones chain fluid model, argon, and some normal alkanes. J. Chem. Phys.
**2011**, 134, 064505. [Google Scholar] [CrossRef] - Raveché, H.J. Entropy and Molecular Correlation Functions in Open Systems. I. Derivation. J. Chem. Phys.
**1971**, 55, 2242–2250. [Google Scholar] [CrossRef] - Baranyai, A.; Evans, D.J. Direct entropy calculation from computer simulation of liquids. Phys. Rev. A
**1989**, 40, 3817–3822. [Google Scholar] [CrossRef] [PubMed] - Klumov, B.A.; Khrapak, S.A. Two-body entropy of two-dimensional fluids. Results Phys.
**2020**, 17, 103020. [Google Scholar] [CrossRef] - Shell, M.S.; Debenedetti, P.G.; Panagiotopoulos, A.Z. Molecular structural order and anomalies in liquid silica. Phys. Rev. E
**2002**, 66, 011202. [Google Scholar] [CrossRef] - Errington, J.E.; Debenedetti, P.G.; Torquato, S. Quantification of Order in the Lennard-Jones System. J. Chem. Phys.
**2003**, 118, 2256–2263. [Google Scholar] [CrossRef]

**Figure 1.**Particle–particle (P-P) interaction potential (solid blue line) and wall–particle (W-P) interaction potential (dashed red line). Insert shows the schematic depiction of a hard core–soft shell polymer-coated nanoparticle as a core-softened effective colloid [25].

**Figure 2.**(

**a**) Transversal density profile for some systems that present the formation of two (green solid line), two-to-three (orange dashed and dotted lines), and three layers (violet solid line) of particles between plates. Examples of snapshots of systems with (

**b**) two layers for plates separated by ${L}_{z}=4.8$ and (

**c**) three layers for plates separated by ${L}_{z}=6.0$. The fluid particles are in blue and the confining walls are in red.

**Figure 3.**(

**a**) Translational order parameter $\tau $ and (

**b**) two-body entropy ${s}_{2}$ as functions of layer density ${\rho}_{l}$ at different separations of plates (${L}_{z}$) and $T=0.150$. Both quantities are related to contact layers.

**Figure 4.**Quantities related to ${L}_{z}=4.8$ at $T=0.150$: (

**a**) transversal density profile, (

**b**) lateral radial distribution function, and (

**c**) cumulative two-body entropy.

**Figure 5.**Quantities related to ${L}_{z}=6.0$ at $T=0.150$: (

**a**) transversal density profile, (

**b**) lateral radial distribution function, and (

**c**) cumulative two-body entropy.

**Figure 6.**Quantities related to ${L}_{z}=5.5$ at $T=0.150$: (

**a**) transversal density profile, (

**b**) lateral radial distribution function, and (

**c**) cumulative two-body entropy.

**Figure 7.**(

**a**) Translational order parameter and (

**b**) two-body entropy as function of central layer density. The central layers were analyzed at $T=0.150$.

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**MDPI and ACS Style**

Krott, L.B.; Bordin, J.R.
How Dimensionality Affects the Structural Anomaly in a Core-Softened Colloid. *Colloids Interfaces* **2023**, *7*, 33.
https://doi.org/10.3390/colloids7020033

**AMA Style**

Krott LB, Bordin JR.
How Dimensionality Affects the Structural Anomaly in a Core-Softened Colloid. *Colloids and Interfaces*. 2023; 7(2):33.
https://doi.org/10.3390/colloids7020033

**Chicago/Turabian Style**

Krott, Leandro B., and José Rafael Bordin.
2023. "How Dimensionality Affects the Structural Anomaly in a Core-Softened Colloid" *Colloids and Interfaces* 7, no. 2: 33.
https://doi.org/10.3390/colloids7020033