# Bragg–Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom

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## Abstract

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## 1. Introduction

## 2. Theory

## 3. Results and Discussion

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Baoukina, S.; Monticelli, L.; Marrink, S.J.; Tieleman, D.P. Pressure- area isotherm of a lipid monolayer from molecular dynamics simulations. Langmuir
**2007**, 23, 12617–12623. [Google Scholar] [CrossRef] [Green Version] - Blanco, E.; Pineiro, A.; Miller, R.; Ruso, J.M.; Prieto, G.; Sarmiento, F. Langmuir monolayers of a hydrogenated/fluorinated catanionic surfactant: From the macroscopic to the nanoscopic size scale. Langmuir
**2009**, 25, 8075–8082. [Google Scholar] [CrossRef] [PubMed] - Javanainen, M.; Lamberg, A.; Cwiklik, L.; Vattulainen, I.; Ollila, O.S. Atomistic model for nearly quantitative simulations of Langmuir monolayers. Langmuir
**2018**, 34, 2565–2572. [Google Scholar] [CrossRef] [PubMed] - Ermakov, Y.A.; Asadchikov, V.; Roschin, B.; Volkov, Y.O.; Khomich, D.; Nesterenko, A.; Tikhonov, A. Comprehensive study of the liquid expanded–liquid condensed phase transition in 1, 2-dimyristoyl-sn-glycero-3-phospho-L-serine monolayers: Surface pressure, Volta potential, X-ray reflectivity, and molecular dynamics modeling. Langmuir
**2019**, 35, 12326–12338. [Google Scholar] [CrossRef] [PubMed] - Linse, P.; Bjoerling, M. Lattice theory for multicomponent mixtures of copolymers with internal degrees of freedom in heterogeneous systems. Macromolecules
**1991**, 24, 6700–6711. [Google Scholar] [CrossRef] - Karnieli, A.; Markovich, T.; Andelman, D. Surface pressure of charged colloids at the air/water interface. Langmuir
**2018**, 34, 13322–13332. [Google Scholar] [CrossRef] [Green Version] - Agudelo, J.; Bossa, G.V.; May, S. Incorporation of Molecular Reorientation into Modeling Surface Pressure-Area Isotherms of Langmuir Monolayers. Molecules
**2021**, 26, 4372. [Google Scholar] [CrossRef] - Walter, J.; Sehrt, J.; Vrabec, J.; Hasse, H. Molecular dynamics and experimental study of conformation change of poly (N-isopropylacrylamide) hydrogels in mixtures of water and methanol. J. Phys. Chem. B
**2012**, 116, 5251–5259. [Google Scholar] [CrossRef] - Culver, H.R.; Clegg, J.R.; Peppas, N.A. Analyte-responsive hydrogels: Intelligent materials for biosensing and drug delivery. Acc. Chem. Res.
**2017**, 50, 170–178. [Google Scholar] [CrossRef] - Cilla, S.; Floría, L. Internal degrees of freedom in a thermodynamical model for intracell biological transport. Phys. D
**1998**, 113, 157–161. [Google Scholar] [CrossRef] - Guigas, G.; Weiss, M. Size-dependent diffusion of membrane inclusions. Biophys. J.
**2006**, 91, 2393–2398. [Google Scholar] [CrossRef] [Green Version] - Ilie, I.M.; den Otter, W.K.; Briels, W.J. A coarse grained protein model with internal degrees of freedom. Application to α-synuclein aggregation. J. Chem. Phys.
**2016**, 144, 085103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kumaki, J.; Hashimoto, T. Conformational change in an isolated single synthetic polymer chain on a mica surface observed by atomic force microscopy. J. Am. Chem. Soc.
**2003**, 125, 4907–4917. [Google Scholar] [CrossRef] [PubMed] - Shen, X.; Viney, C.; Johnson, E.R.; Wang, C.; Lu, J.Q. Large negative thermal expansion of a polymer driven by a submolecular conformational change. Nat. Chem.
**2013**, 5, 1035–1041. [Google Scholar] [CrossRef] [PubMed] - Yefimov, S.; Van der Giessen, E.; Onck, P.R.; Marrink, S.J. Mechanosensitive membrane channels in action. Biophys. J.
**2008**, 94, 2994–3002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rasmussen, T.; Flegler, V.J.; Rasmussen, A.; Böttcher, B. Structure of the mechanosensitive channel MscS embedded in the membrane bilayer. J. Mol. Biol.
**2019**, 431, 3081–3090. [Google Scholar] [CrossRef] - Zhang, Y.; Daday, C.; Gu, R.X.; Cox, C.D.; Martinac, B.; de Groot, B.L.; Walz, T. Visualization of the mechanosensitive ion channel MscS under membrane tension. Nature
**2021**, 590, 509–514. [Google Scholar] [CrossRef] - Thorne, J.B.; Vine, G.J.; Snowden, M.J. Microgel applications and commercial considerations. Colloid Polym. Sci.
**2011**, 289, 625–646. [Google Scholar] [CrossRef] - Urich, M.; Denton, A.R. Swelling, structure, and phase stability of compressible microgels. Soft Matter
**2016**, 12, 9086–9094. [Google Scholar] [CrossRef] [Green Version] - Marcisz, K.; Mackiewicz, M.; Romanski, J.; Stojek, Z.; Karbarz, M. Significant, reversible change in microgel size using electrochemically induced volume phase transition. Appl. Mater. Today
**2018**, 13, 182–189. [Google Scholar] [CrossRef] - Butler, M.D.; Montenegro-Johnson, T.D. The swelling and shrinking of spherical thermo-responsive hydrogels. J. Fluid Mech.
**2022**, 947, A11. [Google Scholar] [CrossRef] - Reese, C.E.; Mikhonin, A.V.; Kamenjicki, M.; Tikhonov, A.; Asher, S.A. Nanogel nanosecond photonic crystal optical switching. J. Am. Chem. Soc.
**2004**, 126, 1493–1496. [Google Scholar] [CrossRef] [PubMed] - Giner-Casares, J.J.; Brezesinski, G.; Möhwald, H. Langmuir monolayers as unique physical models. Curr. Opin. Colloid Interface Sci.
**2014**, 19, 176–182. [Google Scholar] [CrossRef] - Stefaniu, C.; Brezesinski, G.; Möhwald, H. Langmuir monolayers as models to study processes at membrane surfaces. Adv. Colloid Interface Sci.
**2014**, 208, 197–213. [Google Scholar] [CrossRef] [PubMed] - Oliveira, O.N., Jr.; Caseli, L.; Ariga, K. The past and the future of Langmuir and Langmuir–Blodgett films. Chem. Rev.
**2022**, 122, 6459–6513. [Google Scholar] [CrossRef] [PubMed] - McConnell, H.M.; Radhakrishnan, A. Condensed complexes of cholesterol and phospholipids. Biochim. Biophys. Acta Biomembr.
**2003**, 1610, 159–173. [Google Scholar] [CrossRef] [Green Version] - Jurak, M. Thermodynamic aspects of cholesterol effect on properties of phospholipid monolayers: Langmuir and Langmuir–Blodgett monolayer study. J. Phys. Chem. B
**2013**, 117, 3496–3502. [Google Scholar] [CrossRef] - Janich, C.; Hädicke, A.; Bakowsky, U.; Brezesinski, G.; Wölk, C. Interaction of DNA with Cationic Lipid Mixtures: Investigation at Langmuir Lipid Monolayers. Langmuir
**2017**, 33, 10172–10183. [Google Scholar] [CrossRef] - Luque-Caballero, G.; Maldonado-Valderrama, J.; Quesada-Pérez, M.; Martín-Molina, A. Interaction of DNA with likely-charged lipid monolayers: An experimental study. Colloids Surf. B
**2019**, 178, 170–176. [Google Scholar] [CrossRef] - Nobre, T.M.; Pavinatto, F.J.; Caseli, L.; Barros-Timmons, A.; Dynarowicz-Łątka, P.; Oliveira, O.N., Jr. Interactions of bioactive molecules & nanomaterials with Langmuir monolayers as cell membrane models. Thin Solid Films
**2015**, 593, 158–188. [Google Scholar] - Rojewska, M.; Smułek, W.; Kaczorek, E.; Prochaska, K. Langmuir Monolayer Techniques for the Investigation of Model Bacterial Membranes and Antibiotic Biodegradation Mechanisms. Membranes
**2021**, 11, 707. [Google Scholar] [CrossRef] - Salay, L.C.; Ferreira, M.; Oliveira, O.N., Jr.; Nakaie, C.R.; Schreier, S. Headgroup specificity for the interaction of the antimicrobial peptide tritrpticin with phospholipid Langmuir monolayers. Colloids Surf. B
**2012**, 100, 95–102. [Google Scholar] [CrossRef] - Martynowycz, M.W.; Rice, A.; Andreev, K.; Nobre, T.M.; Kuzmenko, I.; Wereszczynski, J.; Gidalevitz, D. Salmonella membrane structural remodeling increases resistance to antimicrobial peptide LL-37. ACS Infect. Dis.
**2019**, 5, 1214–1222. [Google Scholar] [CrossRef] [PubMed] - Fainerman, V.; Vollhardt, D. Surface pressure isotherm for the fluid state of Langmuir monolayers. J. Phys. Chem. B
**2006**, 110, 10436–10440. [Google Scholar] [CrossRef] [PubMed] - Klug, J.; Masone, D.; Del Pópolo, M.G. Molecular-level insight into the binding of arginine to a zwitterionic Langmuir monolayer. RSC Adv.
**2017**, 7, 30862–30869. [Google Scholar] [CrossRef] [Green Version] - Levental, I.; Janmey, P.; Cēbers, A. Electrostatic contribution to the surface pressure of charged monolayers containing polyphosphoinositides. Biophys. J.
**2008**, 95, 1199–1205. [Google Scholar] [CrossRef] [Green Version] - Chachaj-Brekiesz, A.; Kobierski, J.; Wnętrzak, A.; Dynarowicz-Łatka, P. Electrical properties of membrane phospholipids in Langmuir monolayers. Membranes
**2021**, 11, 53. [Google Scholar] [CrossRef] - Miñones, J.; Yebra-Pimentel, E.; Iribarnegaray, E.; Conde, O.; Casas, M. Compression—expansion curves of cyclosporin A monolayers on substrates of various ionic strengths. Colloids Surf. A
**1993**, 76, 227–232. [Google Scholar] [CrossRef] - Hąc-Wydro, K.; Dynarowicz-Łątka, P. Nystatin in Langmuir monolayers at the air/water interface. Colloids Surf. B
**2006**, 53, 64–71. [Google Scholar] [CrossRef] [PubMed] - Wnętrzak, A.; Chachaj-Brekiesz, A.; Janikowska-Sagan, M.; Rodriguez, J.L.F.; Conde, J.M.; Dynarowicz-Łatka, P. Crucial role of the hydroxyl group orientation in Langmuir monolayers organization–The case of 7-hydroxycholesterol epimers. Colloids Surf. A
**2019**, 563, 330–339. [Google Scholar] [CrossRef] - Strzalka, J.; Chen, X.; Moser, C.C.; Dutton, P.L.; Ocko, B.M.; Blasie, J.K. X-ray scattering studies of maquette peptide monolayers. 1. Reflectivity and grazing incidence diffraction at the air/water interface. Langmuir
**2000**, 16, 10404–10418. [Google Scholar] [CrossRef] - Davis, H.T. Statistical Mechanics of Phases, Interfaces, and Thin Films; Wiley: Hoboken, NJ, USA, 1996. [Google Scholar]
- Han, Y.; Huang, S.; Yan, T. A mean-field theory on the differential capacitance of asymmetric ionic liquid electrolytes. J. Phys. Condens. Matter
**2014**, 26, 284103. [Google Scholar] [CrossRef] [PubMed] - Jones, R.A. Soft Condensed Matter; Oxford University Press: Oxford, UK, 2002; Volume 6. [Google Scholar]
- Andelman, D.; Brochard, F.; Knobler, C.; Rondelez, F. Structures and phase transitions in Langmuir monolayers. In Micelles, Membranes, Microemulsions and Monolayers; Springer: New York, NY, USA, 1994; pp. 559–602. [Google Scholar]
- Bossa, G.V.; Gunderson, S.; Downing, R.; May, S. Role of transmembrane proteins for phase separation and domain registration in asymmetric lipid bilayers. Biomolecules
**2019**, 9, 303. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Akasaka, R. Calculation of the critical point for mixtures using mixture models based on Helmholtz energy equations of state. Fluid Phase Equilib.
**2008**, 263, 102–108. [Google Scholar] [CrossRef] - Bell, I.H.; Jäger, A. Calculation of critical points from Helmholtz-energy-explicit mixture models. Fluid Phase Equilib.
**2017**, 433, 159–173. [Google Scholar] [CrossRef] [Green Version] - Knobler, C.M.; Desai, R.C. Phase transitions in monolayers. Annu. Rev. Phys. Chem.
**1992**, 43, 207–236. [Google Scholar] [CrossRef] - Dynarowicz-Łatka, P.; Dhanabalan, A.; Oliveira, O.N. A study on two-dimensional phase transitions in langmuir monolayers of a carboxylic acid with a symmetrical triphenylbenzene ring system. J. Phys. Chem. B
**1999**, 103, 5992–6000. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of a system composed of molecules present in two states: small (S, light red cubes) and large (L, dark red cuboids). From diagrams (

**a**–

**f**): illustration of how the system may phase-separate upon increasing ${\varphi}_{0}$. The phases, here colored in white and blue, can assume different sizes and compositions depending on the values of ${\varphi}_{0}$, $\lambda $, $\chi $, and $\xi $. The gray arrows below the diagrams indicate the increase in ${\varphi}_{0}$, with darker gray for larger ${\varphi}_{0}$. Results illustrate a two-dimensional lattice for $\xi =4$.

**Figure 2.**Diagrams (

**a**,

**b**): free energy $f(\xi ({\varphi}_{0}-{\varphi}_{S}),{\varphi}_{S})$ according to Equation (3), calculated as a function of ${\varphi}_{S}$ for ${\varphi}_{0}=0.23$ and ${\varphi}_{0}=0.24$; blue bullets mark the global minimum. Diagram (

**c**): ${\varphi}_{S}^{opt}$ as a function of ${\varphi}_{0}$. Diagram (

**d**): $f\left({\varphi}_{0}\right)$ as a function of ${\varphi}_{0}$ with red bullets marking the points that connect through a common tangent. In all four diagrams, $\xi =4,\lambda =1.5$, and $\chi =1.2$.

**Figure 3.**Left and right diagrams display the values of ${\varphi}_{S}^{opt}$ and $f\left({\varphi}_{0}\right)$, respectively, versus ${\varphi}_{0}$ for different $\lambda $ and fixed $\xi =4$. Each row corresponds to a specific value of $\chi $: $\chi =0.2$ (top), $\chi =1.2$ (middle), and $\chi =2.2$ (bottom). In all diagrams, the values of $\lambda $ are color-coded according to the side legend on the right. The ${\varphi}_{S}^{opt}$ values are solutions of Equation (4), i.e., the value of ${\varphi}_{S}$ that globally minimizes the free energy $f(\xi ({\varphi}_{0}-{\varphi}_{S}),{\varphi}_{S})$. The shaded gray areas mark the region where the ${\varphi}_{S}^{opt}$ values change discontinuously, and the color-matching dotted lines indicate discontinuity of ${\varphi}_{S}^{opt}$. Insets display the results over the entire range $0\le {\varphi}_{0}\le 1$.

**Figure 4.**Slope from the free energy expansion around ${\varphi}_{0}=0$ as a function of the interaction parameter $\chi $, calculated according to Equation (6). The values of $\lambda $ are color-coded according to the side legend. The black dotted lines mark the $\chi $ values used in the diagrams of Figure 3, i.e., $\chi =0.2$, $1.2$, and $2.2$. The insets display illustrations of two systems, one with particles mostly in the L-state (bottom left) and another with all particles in the S-state (top right).

**Figure 5.**Phase diagrams, $1/\chi $ versus ${\varphi}_{0}$, for $\xi =4$ and different values of $\lambda $: $\lambda =0$, 1, $1.5$, and 2 in panels (

**a**–

**d**), respectively. The critical value of $\chi $ is marked by colored bullets: blue ($\lambda =0$), green ($\lambda =1$), yellow ($\lambda =1.5$), and green ($\lambda =2$). The dashed line shown in all diagrams is the line on which all critical $\chi $ values lie. The “heat maps” in the background display the value of ${\varphi}_{S}^{opt}/{\varphi}_{0}$ color-coded according to the side legend.

**Figure 6.**Phase diagrams for $\lambda =10$ with $\xi =4$ (

**a**) and $\xi =1$ (

**b**). The dashed line in (

**a**) is the line along which the critical values of $\chi $ lie; the heat maps in (

**a**) and (

**b**) display the value of ${\varphi}_{S}^{opt}/{\varphi}_{0}$, color-coded according to the side legend in (

**b**). Diagram (

**c**): location of critical points, where each set of dashed lines and symbols correspond to $\xi =2$, 3, 4, from bottom to top lines; values of $\lambda $ are color-coded as specified in the side panel, and the gray bullets mark the limits $\lambda \to \pm \infty $. For $\lambda \to -\infty $, all critical points merge into one location at ${\chi}^{*}=2$ and ${\varphi}_{0}=0.5$, irrespective of $\xi $.

**Figure 7.**Scaled pressure ${\nu}_{0}P$ versus $\nu /{\nu}_{0}=1/{\varphi}_{0}$ for $\chi =0.2$ (

**a**), $\chi =1.2$ (

**b**), and $\chi =2.2$ (

**c**). Different colors correspond to different values of $\lambda $, as indicated in the side legend. The dotted line marks the point $\nu /{\nu}_{0}=\xi $. Results calculated for $\xi =4$.

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**MDPI and ACS Style**

Bossa, G.V.; May, S.
Bragg–Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom. *Molecules* **2023**, *28*, 5060.
https://doi.org/10.3390/molecules28135060

**AMA Style**

Bossa GV, May S.
Bragg–Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom. *Molecules*. 2023; 28(13):5060.
https://doi.org/10.3390/molecules28135060

**Chicago/Turabian Style**

Bossa, Guilherme Volpe, and Sylvio May.
2023. "Bragg–Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom" *Molecules* 28, no. 13: 5060.
https://doi.org/10.3390/molecules28135060