# Chasing the Critical Wetting Transition. An Effective Interface Potential Method

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. The Effective Interface Potential Method

## 4. Other Methods of Determination of the Critical Surface Field

#### 4.1. Determination of the Critical Surface Field by Thermodynamic Integration

#### 4.2. BLK Method for Symmetric Surface Fields Revisited

## 5. Discussion

## 6. Conclusions

- The effective interface potential method can be used to determine the location of the critical wetting transition. The limitation of this method is that its accuracy decreases if the bulk fluctuations become important.
- The thermodynamic integration method can be used to estimate the location of the critical wetting transition. Extrapolation to the thermodynamic limit is non-trivial.
- The Binder–Landau–Kroll method of determination of the critical wetting transition also leads to reasonable results if sufficiently big system sizes are considered.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The effective interface potentials calculated for the 2D non-symmetric system at $J/{k}_{B}T=1.0$ and for $L=210,D=210$. The potentials are calculated for the surface fields ${H}_{1}/J=-0.915$, −0.92, −0.925, −0.93, −0.935, −0.94, −0.942, −0.945, −0.948, −0.950, and −0.955. The thick line corresponding to ${H}_{1}/J=-0.955$ denotes the effective interface potential with no detectable local minimum.

**Figure 2.**The effective interface potentials calculated for the 2D non-symmetric system at $J/{k}_{B}T=1.0$ and for $L=630,D=210$. The potentials are calculated for the surface fields ${H}_{1}/J=-0.910$, −0.912, −0.915, −0.918, −0.920, −0.922, −0.925, −0.928, −0.930, −0.932, −0.935, −0.938, and −0.942. The thick line corresponding to ${H}_{1}/J=-0.942$ denotes the effective interface potential with no detectable local minimum.

**Figure 3.**The effective interface potentials calculated for the 2D non-symmetric system at $J/{k}_{B}T=1.0$ and for $L=1260,D=210$. The potentials are calculated for the surface fields ${H}_{1}/J=-0.910$, −0.912, −0.915, −0.918, −0.920, −0.925, −0.928, −0.930, −0.932, −0.935, and −0.938. The thick line corresponding to ${H}_{1}/J=-0.938$ denotes the effective interface potential with no detectable local minimum.

**Figure 4.**The estimate of the wetting surface field for the 2D system at $J/{k}_{B}T=1.0$ Circles correspond to the surface critical fields obtained from simulations. The straight line denotes a regression fit. The black diamond denotes the exact result [7].

**Figure 5.**The estimate of the wetting surface field for the 2D system at $J/{k}_{B}T=0.625$ Circles correspond to the surface critical fields obtained from simulations. The straight line denotes a regression fit. The black diamond denotes the exact result [7].

**Figure 6.**The effective interface potentials calculated for the 3D non-symmetric system at $J/{k}_{B}T=0.35$ and for $L=63,D=60$. The potentials are calculated for surface fields from ${H}_{1}/J=-0.89$, −0.891, −0.892, −0.893, −0.894, −0.895, −0.896, −0.898, −0.90, −0.902, −0.904, −0.906, and −0.908. The thick line corresponding to ${H}_{1}/J=-0.908$ denotes the effective interface potential with no detectable local minimum.

**Figure 7.**The effective interface potentials calculated for the 3D non-symmetric system at $J/{k}_{B}T=0.35$ and for $L=126,D=60$. The potentials are calculated for surface fields ${H}_{1}/J=-0.89$, −0.891, −0.892, −0.893, −0.894, −0.895, −0.896, −0.898, −0.90, −0.902, −0.904, −0.906, and −0.908. The thick line corresponding to ${H}_{1}/J=-0.908$ denotes the effective interface potential with no detectable local minimum.

**Figure 8.**The effective interface potentials calculated for the 3D non-symmetric system at $J/{k}_{B}T=0.35$ and for $L=252,D=60$. The potentials are calculated for surface fields ${H}_{1}/J=-0.89$, −0.891, −0.892, −0.893, −0.894, −0.895, −0.896, −0.898, −0.90, −0.902, −0.904, −0.906, and −0.908. The thick line corresponding to ${H}_{1}/J=-0.908$ denotes the effective interface potential with no detectable local minimum.

**Figure 9.**The effective interface potentials calculated for the non-symmetric system at $J/{k}_{B}T=0.25$ and for $L=252,D=80$. The potentials are calculated for surface fields from top ${H}_{1}/J=-0.5565$, −0.557, −0.558, −0.559, −0.560, −0.561, −0.562, −0.563, −0.564, −0.565, −0.568, −0.570, −0.575, −0.582, −0.588, −0.592, −0.596, −0.600, −0.604, and −0.608. The thick line corresponding to ${H}_{1}/J=-0.608$ denotes the effective interface potential with no detectable local minimum.

**Figure 10.**The integrand of Equation (16) vs. the surface field for $J/{k}_{B}T=0.25$ and for three system sizes L listed in Figure. The inset shows zoom-out of the main Figure.

**Figure 11.**The integrand of Equation (16) vs. the surface field for $J/{k}_{B}T=0.35$ and for three system sizes L listed in Figure. The inset shows zoom-out of the main Figure.

**Figure 12.**$\partial cos\left(\theta \right)/\partial {H}_{1}$ vs. the surface field for $J/{k}_{B}T=0.25$ and for three system sizes L listed in Figure. The inset shows full dependence of $cos\left(\theta \right)$ vs. ${H}_{1}/J$.

**Figure 13.**$\partial cos\left(\theta \right)/\partial {H}_{1}$ vs. the surface field for $J/{k}_{B}T=0.35$ and for three system sizes L listed in Figure. The inset shows full dependence of $cos\left(\theta \right)$ vs. ${H}_{1}/J$.

**Figure 14.**Symbols denote the estimate of the surface field for which $\partial cos\left(\theta \right)/\partial {H}_{1}=0$ at a given L. Thick lines denote the linear regression extrapolating $L\to \infty $. Panel (

**a**) is for $J/{k}_{B}T=0.35$ while panel (

**b**) is for $J/{k}_{B}T=0.25$.

**Figure 15.**${\chi}_{1}$ for five different system sizes L, evaluated at $J/{k}_{B}T=0.35$ for the symmetric surface fields, ${H}_{1}={H}_{D}$. The big black circle with error bars denotes the simulational result of Binder et al. [15].

**Figure 16.**Estimation of ${H}_{1c}(L=\infty )$ from simulational data presented in Figure 15, evaluated at $J/{k}_{B}T=0.35$ for the symmetric surface fields.

**Figure 17.**${\chi}_{1}$ for different system sizes, evaluated at $J/{k}_{B}T=0.25$ for the symmetric surface fields.

**Figure 18.**Estimation of ${H}_{1c}(L=\infty )$ from simulational data presented in Figure 17, evaluated at $J/{k}_{B}T=0.25$ for the symmetric surface fields.

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Bryk, P.; Terzyk, A.P.
Chasing the Critical Wetting Transition. An Effective Interface Potential Method. *Materials* **2021**, *14*, 7138.
https://doi.org/10.3390/ma14237138

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Bryk P, Terzyk AP.
Chasing the Critical Wetting Transition. An Effective Interface Potential Method. *Materials*. 2021; 14(23):7138.
https://doi.org/10.3390/ma14237138

**Chicago/Turabian Style**

Bryk, Paweł, and Artur P. Terzyk.
2021. "Chasing the Critical Wetting Transition. An Effective Interface Potential Method" *Materials* 14, no. 23: 7138.
https://doi.org/10.3390/ma14237138