# Entropy and Random Walk Trails Water Confinement and Non-Thermal Equilibrium in Photon-Induced Nanocavities

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

_{n}= 150 K, M

_{w}400 K) purchased from Sigma-Aldrich (St. Louis, MO, USA) used to prepare solution 5% w/w in water. Thin layers (426 ± 1 nm) on Si wafer substrates were made by spin-coating for 60 s at 2500 rpm, and finally, cured at 110 °C for 15 min at a temperature rate of 0.37 °C s

^{−1}and then left to cool at room temperature. WLRS measures the thickness of PAM films coated on Si wafers.

#### 2.2. 157 nm Laser

^{TM}200), Lambda Physik AG (Coherent), Göttingen, Germany), under continuous nitrogen flow (99.999%) at 10

^{5}Pa and room temperature. The layers were mounted into a computer-controlled X-Y-Z-θ translation-rotation motorized stage, placed inside a 316 stainless-steel chamber. The laser temporal pulse duration at FWHM, the energy of an unfocused laser beam per laser pulse, the photon fluence per laser pulse and laser repetition rate were set up at 15 ns, 28 mJ, 250 Jm

^{−2}and 10 Hz. For dipping the amount of oxygen inside the stainless-steel chamber, nitrogen purging of the chamber was applied for 10 min before the irradiating stage.

#### 2.3. AFM Imaging and AFM-NI

^{−1}and tip radius of 8 nm, operating at a resonance frequency of 300 kHz at ambient conditions. The surface parameters of the samples were also evaluated.

#### 2.4. Fractal Analysis

_{f}that describes the topology and the cavitation of a surface quantitatively. D

_{f}was derived from AFM images by four different algorithms, the cube counting, triangulation, variance and power spectrum methods, besides an algorithm provided by the AFM’s “lake pattern” software (diSPMLab Vr.5.01). A detailed description of the concept and the specific methodologies of the different algorithms can be found in [27]. The D

_{f}was calculated for the four different methods using “Gwyddion, SPM data visualisation and analysis tool” [73]. The D

_{f}calculated with the four different algorithms follow the same trend, despite small dimensionality divergences coming up from systematic errors, because of the different converging speed of the fractal analytical approaches.

#### 2.5. Water Contact Angle (CA)

#### 2.6. White Light Reflectance Spectroscopy (WLRS)

_{2}layer on the top with a thickness of 2–3 nm.

#### 2.7. Random Walk Model

^{2}times and the mean escape time was calculated. In addition, the mean-escape time distribution for different cavities and number of molecules was used to evaluate the thermodynamic state inside the cavity. The model was designed and run in MATLAB. 9.4.0.813654 (R2018a), The MathWorks Inc.; Natick, MA, USA.

## 3. Results

#### 3.1. Surface Analysis

_{a}(area roughness or roughness average) is the arithmetic mean of the height deviation from the image’s mean value,${R}_{a}=\frac{1}{n}\sum _{i=1}^{n}\left|{Z}^{i}-\overline{Z}\right|$. The area RMS (R

_{rms}) is the value defined as the square root of the mean value of the squares of the distance of the points from the image mean value: ${R}_{rms}=\sqrt{\frac{1}{N}\sum _{i=1}^{N}{\left(Z-{Z}_{i}\right)}^{2}}$. Finally, the maximum range of Z

_{max}is defined as the maximum value of z-heights. The surface parameter values (z-height, area roughness, area RMS, and maximum range) of photon exposed areas were more considerable compared to the non-irradiated ones. However, because surface parameters are area size-dependent (Figure 2d,e), they are utilized only for a comparative qualitative evaluation of area modification under 157 nm laser irradiation.

#### 3.2. Fractal Analysis of 157 nm Photon Processed PAM Polymeric Matrixes

_{i}values above a threshold Z height are known as “islands”, while those with Z

_{i’}s below the threshold height value are named as “lakes”. AFM “island-lake structure” of non-irradiated and VUV irradiated 2 μm × 2 μm areas are shown in Figure 3.

_{i}heights of non-irradiated and the irradiated regions (10

^{3}laser pulses) were set at 0.75 and 1.94 nm respectively, and the irradiated areas show a diverging surface topology, in agreement with previous results [13,16,56,58]. Following a standard procedure, two parameters, the fractal dimensionality D

_{f}(which is a dimensionless number) and the “periphery to the area ratio” (PAR) are used to describe a set of “islands” or “lakes”. Both parameters are linked to the surface roughness, cavitations and topological entropy [27,74]. PAR is the ratio of logarithms of the perimeter Π to the area A, where $\Pi =\alpha \left(1+{D}_{f}\right){A}^{\left(1-{D}_{f}\right)/2}$. For assessing the state of cavitations, the fractal dimensionality is calculated by the partitioning, the cube counting, the triangulation, and the power spectrum algorithms [58]. Results are compared with those derived directly from the AFM “lake” pattern software, Figure 4a.

^{3}laser pulses, Figure 4a. For a constant “lake” surface area the number of “lakes”, and thus the number of cavities, is a function of the laser pulses (laser fluence), Figure 4b. The number of “lakes” within a given surface area vs. the number of laser pulses is shown in Figure 5a. The number of “lake” areas rises almost exponentially with the number of laser pulses and small area “lakes” prevail over larger ones The fractal dimensionality vs. laser fluence has a non-monotonous complex structure. Small size features (1–10

^{2}nm

^{2}) are associated with nanocavity-like structures, Figure 5b. It is also confirmed that below 10

^{3}laser pulses small size features contribute to a high cavitation state because small size features have a higher dimensionality than large size structures, Figure 5c. On the contrary, large size features are prominent at 10

^{3}laser pulses, indicating the complexity of the associated processes. In addition, for the same number of laser pulses, small size cavitation prevails over larger ones, Figure 5a. The experimental results indicate that water confinement is rather associated with small cavitations, in agreement with WLRS measurements (vide infra).

#### 3.3. AFM-NI

^{3}laser pulses was 2.0 ± 0.8 and 1.6 ± 0.42 and 2.55 ± 1.29 GPa, respectively, Figure 7a. The significant errors of Young’s moduli at different points in the same sample are credited to various morphological heterogeneities and a progressive phase transformation to a relatively high carbonized state. Young’s moduli follow a similar trend with fractal dimensionality vs. laser fluence, Figure 4a and Figure 5b.

^{3}laser pulses.

#### 3.4. Water Contact Angle (CA)

_{f}and area RMS indicate a secure interconnection between surface morphology and D

_{f}, Figure 8b,c.

#### 3.5. White Light Reflectance Spectroscopy (WLRS)

^{−14}m

^{3}, defined by the cross-sectional diameter of the white light beam of 3.5 × 10

^{−4}m and the thickness of the polymeric layer of 426 nm.

#### 3.6. Random Walk Model

^{−5}m

^{2}s

^{−1}) and the current non-interactive random walk model for 10

^{3}runs is shown in Table A1 and Table A2. There is a noticeable difference between the two models for small size nanocavities because the diffusion constant for small size nanocavities is undetermined. The mean escape time from random walk models with the interactive model for the different number of confined molecules, cavity and the entrance-escape hole size is given in Figure 10 and Appendix A, Table A3, Table A4, Table A5 and Table A6.

## 4. Discussion

#### 4.1. 157 nm Molecular Photodissociation of PAM Polymeric Chains

^{3}nm) and cavitations are shreds of evidence of significant photochemical topological matrix alterations (Figure 2c–e). Similar structures were previously observed on PAM hydrogel surfaces by cross-link concentration variations [78].

#### 4.2. Trapping of Water Molecules in Nanocavities

^{−25}m

^{3}, the volume stress exerted on the walls of the cavity from the collisions of a molecule with the walls of a cavity should be of the order of $~\frac{kT}{V}=7.9\times {10}^{4}\text{}\mathrm{Pa}$, a value that almost matches the atmospheric pressure outside the cavity. By increasing the number of molecules inside a small cavity, the volume stress should be increased proportionally to the number of molecules because of mechanical collisions with the cavity walls. Consequently, extremely high pressures should be developed inside small cavities, in agreement with [88]. In addition, for small size cavities, there is rather an entropic than an energy barrier that balances the flow kinetics of molecules in and out the cavity [94,95]. Previous studies indicated that in the case of elastic collisions in the cavity, the molecular dynamics depends on the number of molecules inside the cavity and is either frictionless (inertial dynamics), moderately frictional (Langevin dynamics), or strongly frictional (Brownian dynamics) [96], where the noise term should be properly taken into account. For small entrance-escape holes, the number correlation function generally decays exponentially with time. The transition rate in the frictionless limit is given by a microcanonical ensemble. As the strength of the friction is increased, the rate of collisions approaches the diffusive limit without a Kramers turnover. In this work, random-walk calculations of non-interactive and interactive molecules in the cavity for 10

^{3}and 10

^{2}runs, point to variable escape times of water molecules from different size nanocavities (1–10

^{3}nm) and entrance-escape holes (0.3 – 5 × 10

^{2}nm), Figure 10 and Appendix A, Table A1 and Table A2. For the same cavity size the mean escape time falls with large entrance-escape hole size, extended over a wide dynamic range of 10 orders of magnitude. The mean escape time for the interactive model is independent of the number of molecules inside small cavities and interestingly, the mean escape time fluctuates a great deal inside tiny cavities, Figure 12a–d and Table A3, Table A4, Table A5 and Table A6; suggesting that the system is in non-thermal equilibrium, a state that dominates the statistics and the dynamics of molecules inside small cavities.

^{3}nm), while the ratio deviates for small ones, suggesting again a non-equilibrium thermal state and large fluctuations inside small size cavities, Figure 14. Most interesting, the local fluctuations of mean molecular escape time are prominent for small size cavities and small number of molecules, while the mean molecular escape time remains steadier for a larger number of molecules, Figure 12a–d, in agreement with molecular dynamic results [6,85,89,90,91,92,96] and general nanothermodynamic considerations [31]. In addition, the mean escape time distribution of molecules for both the non-interactive and interactive models (1 and 150 molecules) inside different size small cavities reveals a rather non-thermal distribution and the absence of a thermal equilibrium state inside the cavities, Figure 13 and Figure 15.

#### 4.3. Stress-Strain Response in Polymeric Matrixes-A Relation between Physics and Mechanics

^{th}component of a traction force$d\overrightarrow{F}$ along the i-axis, along a unit vector $\overrightarrow{n}$ perpendicular on an infinitesimal surface area $d\overrightarrow{A}$,${\sigma}_{ij}$is the $\left(i,j\right)$ component of the stress tensor, and${n}_{j}$ is the j

_{th}component of the $\overrightarrow{n}$ vector that is perpendicular to the surface area $d\overrightarrow{A}$.

_{i}are the body forces (e.g., the weight of the volume element dV). From Equation (4) and the Gauss theorem, the surface integral of the components of the traction forces is transformed into a volume integral

#### 4.4. Internal Energy Variation during Molecular Water Confinement

^{−12}Fm

^{−1}, ${\epsilon}_{1}~80$ is the relative electric permittivity of the polymer-water system, ${k}_{B}=1.38\times {10}^{-23}{\mathrm{J}\text{}\mathrm{K}}^{-1}$ is Boltzmann’s constant and T = 300 K is the absolute temperature. Because the energy of each laser pulse at 157 nm is 28 mJ, the number of photons carried in one laser pulse is $n=2.26\times {10}^{16}$ photons/laser pulse, and this number equals to the number of photon-activated dipole binding sites. Each VUV photon at 157 nm dissociates one molecular bond and creates one active site on the polymeric matrix, Figure 11. For a 1.12 × 10

^{−4}m

^{2}cross-sectional area of the 157 nm laser beam and 426 nm layer thickness, it is found that 4.73 × 10

^{26}photon-activated dipole binding sites are generated within 1 m

^{3}per laser pulse. For a cross-section area of the WLRS beam of 4.90 × 10

^{−8}m

^{2}and 426 nm matrix thickness, the volume $dV$ of the polymeric matrix occupied by the white light beam is 4.09 × 10

^{−14}m

^{3}and thus the total number of active binding sites per laser pulse within the volume occupied by the white beam is N

_{b}= 2.31 × 10

^{13}. From Equation (21) $<\Phi >\approx 1.51\mathrm{x}{10}^{-23}\mathrm{J}$ for $\lambda l=0.05$ (vide infra) and finally $\delta {U}_{d}=1.43\times {10}^{-11}\mathrm{J}.$

#### 4.5. Entropic Energy Variation during Molecular Confinement

_{b}(n), N

_{c}(n), Ε

_{α}) per unit time is specified by the frequency of water molecules confined in the nanocavities. The rate of visits is regulated by the mean escape time of water molecules. n and ${N}_{c}\left(n\right)$is the number of laser pulses and nanocavities, respectively, N

_{a}is the number of water molecules outside the nanocavities with energy Ε

_{α}and N

_{c}(n) is the number of nanocavities. The number of microstates is equal to the number of indistinguishable permutations $\left\{{N}_{a}\left(n\right),{N}_{b}\left(n\right)+{N}_{c}\left(n\right)\right\}$between the number of water molecules N

_{a}and the number of nanocavities ${N}_{c}\left(n\right)$ and the photon-induced dipole binding sites ${N}_{b}\left(n\right)$

_{b}= 2.31 × 10

^{13}, $y\left(n\right)=2,$the entropic energy at 300 K is ${k}_{B}T\delta S=1.31\times {10}^{-8}\mathrm{J}$, which is almost three orders of magnitude larger than $\delta {U}_{d}.$ Equations (32), (33) properly reflect the extensive variable character of the entropy as it should be.

#### 4.6. Surface Strain from the Confinement of Water Molecules

_{c}(n), y(n) is a measure of the surface carbonization. By using a linear functional for both y(n) and E(n), the best fit of Equation (36) to the experimental data of Figure 17 for relative humidity 80% is for $\beta \left(n\right)=0.2$ and $0\le \lambda l<0.05$. The above fitting values suggest a small and large contribution from the electric dipole interactions and the entropic variation in surface strain, respectively. From Equation (36), the surface strain is proportional to the square root of the number of nanocavities and the concentration of the water molecules (RH) and inversely proportional to the square root of Young’s modulus of the surface, in agreement with the experimental results of Figure 17. Finally, the entropic jump, probed by WLRS, trails the confinement of water molecules in nanocavities, while the deep physical root of surface entropy variation originates from the different “time flow and scales” and the validity and invalidity of thermal equilibrium outside and inside the nanocavities, respectively, Figure 15, Figure 16, Figure 17.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Mean escape time (τ) of a given molecule, errors and total traveling distance of the confined molecule in a spherical cavity (cavity diameter D of 1 nm and 10 nm) for different entrance–escape hole diameters (h) calculated by the diffusion and the non-interactive random walk models.

h (nm) | Cavity Diameter D = 1 nm | Cavity Diameter D = 10 nm | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Diffusion | Random Walk | Diffusion | Random Walk | |||||||

τ (s) | τ (s) | Error (s) | Distance (nm) | Error (nm) | τ(s) | τ (s) | Error (s) | Distance (nm) | Error (nm) | |

0.3 | 3.98 × 10^{−14} | 5.51 × 10^{−11} | 1.7 × 10^{−12} | 35.5 | 1.1 | 3.98 × 10^{−11} | 5.57 × 10^{−8} | 1.7 × 10^{−9} | 35,931 | 1133 |

0.5 | 2.39 × 10^{−14} | 1.91 × 10^{−11} | 5 × 10^{−13} | 12.4 | 0.3 | 2.39 × 10^{-11} | 2.19 × 10^{−8} | 7 × 10^{−10} | 14,151 | 465 |

1 | 1.19 × 10^{-11} | 4.93 × 10^{−9} | 1.6 × 10^{−10} | 3182 | 107 | |||||

2 | 5.98 × 10^{-12} | 1.28 × 10^{−9} | 4 × 10^{−11} | 822 | 26 | |||||

5 | 2.39 × 10^{−12} | 1.90 × 10^{−10} | 5 × 10^{−12} | 128 | 4 |

**Table A2.**Mean escape time (τ) of a given molecule, errors and total traveling distance of the confined molecule in a spherical cavity (cavity diameter 10

^{2}nm and 10

^{3}nm) for different entrance–escape hole diameters (h) calculated by the diffusion and the non-interactive random walk models.

h (nm) | Cavity Diameter D = 10^{2} nm | Cavity Diameter D = 10^{3} nm | ||||||||
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Diffusion | Random Walk | Diffusion | Random Walk | |||||||

τ (s) | τ (s) | Error (s) | Distance (nm) | Error (nm) | τ (s) | τ (s) | Error (s) | Distance (nm) | Error (nm) | |

0.3 | 3.98 × 10^{−8} | 5.71 × 10^{−5} | 1.8 × 10^{−6} | 3.68 × 10^{7} | 1.2 × 10^{6} | 3.98 × 10^{−5} | 5.3 × 10^{−2} | 1.7 × 10^{−3} | 3.42 × 10^{10} | 1.1 × 10^{9} |

0.5 | 2.39 × 10^{−8} | 2.00 × 10^{−5} | 6 × 10^{−7} | 1.29 × 10^{7} | 4.1 × 10^{5} | 2.39 × 10^{−5} | 2.01 × 10^{−2} | 6.3 × 10^{−4} | 1.3 × 10^{10} | 4.1 × 10^{8} |

1 | 1.19 × 10^{−8} | 4.63 × 10^{−6} | 1.4 × 10^{−7} | 2.99 × 10^{6} | 9 × 10^{4} | 1.19 × 10^{−5} | 5.43 × 10^{−3} | 1.6 × 10^{−4} | 3.5 × 10^{9} | 1.1 × 10^{8} |

2 | 5.98 × 10^{−9} | 1.22 × 10^{−6} | 4 × 10^{−8} | 7.89 × 10^{5} | 2.5 × 10^{4} | 5.98 × 10^{−6} | 1.33 × 10^{−3} | 4.1 × 10^{−5} | 8.56 × 10^{8} | 2.7 × 10^{7} |

5 | 2.39 × 10^{−9} | 1.95 × 10^{−7} | 6 × 10^{−9} | 1.26 × 10^{5} | 4 × 10^{3} | 2.39 × 10^{−6} | 1.92 × 10^{−4} | 6.7 × 10^{−6} | 1.24 × 10^{8} | 4.3 × 10^{6} |

10 | 1.19 × 10^{−9} | 5.18 × 10^{−8} | 1.6 × 10^{−9} | 3.34 × 10^{4} | 1060 | 1.19 × 10^{−6} | 4.99 × 10^{−5} | 1.6 × 10^{−6} | 3.22 × 10^{7} | 1.1 × 10^{6} |

20 | 5.97 × 10^{−10} | 1.21 × 10^{−8} | 3 × 10^{−10} | 7821 | 244 | 5.97 × 10^{−7} | 1.23 × 10^{−5} | 4.1 × 10^{−7} | 7.93 × 10^{7} | 2.6 × 10^{5} |

50 | 2.39 × 10^{−10} | 1.87 × 10^{−9} | 5 × 10^{−11} | 1204 | 35 | 2.39 × 10^{−7} | 1.99 × 10^{−6} | 6.4 × 10^{−8} | 1.28 × 10^{6} | 4.1 × 10^{4} |

100 | 1.19 × 10^{−7} | 5.07 × 10^{−7} | 1.5 × 10^{−8} | 3.27 × 10^{5} | 9.8 × 10^{3} | |||||

200 | 5.97 × 10^{−8} | 1.25 × 10^{−7} | 3.7 × 10^{−9} | 8.05 × 10^{4} | 2.4 × 10^{3} | |||||

500 | 2.39 × 10^{−8} | 1.88 × 10^{−8} | 5.6 × 10^{−10} | 1.21 × 10^{4} | 3.6 × 10^{2} |

**Table A3.**Mean escape time (τ) and associated errors of different number molecules confined in a 5-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 10

^{2}runs.

Cavity Diameter = 5 nm | ||||
---|---|---|---|---|

N (Number of Molecules) | h = 1 nm | h = 2 nm | ||

τ (s) | Error (s) | τ (s) | Error (s) | |

1 | 1.41 × 10^{−9} | 1.4 × 10^{−10} | 2.02 × 10^{−10} | 2.0 × 10^{−11} |

2 | 1.38 × 10^{−9} | 1.4 × 10^{−10} | 2.04 × 10^{−10} | 2.2 × 10^{−11} |

3 | 1.57 × 10^{−9} | 1.6 × 10^{−10} | 2.00 × 10^{−10} | 2.1 × 10^{−11} |

4 | 1.50 × 10^{−9} | 1.4 × 10^{−10} | 2.05 × 10^{−10} | 1.9 × 10^{−11} |

5 | 1.39 × 10^{−9} | 1.4 × 10^{−10} | 1.95 × 10^{−10} | 1.8 × 10^{−11} |

6 | 1.46 × 10^{−9} | 1.3 × 10^{−10} | 2.01 × 10^{−10} | 2.2 × 10^{−11} |

7 | 1.27 × 10^{−9} | 1.2 × 10^{−10} | 2.06 × 10^{−10} | 2.1 × 10^{−11} |

8 | 1.30 × 10^{−9} | 1.4 × 10^{−10} | 2.25 × 10^{−10} | 2.4 × 10^{−11} |

9 | 1.82 × 10^{−9} | 1.8 × 10^{−10} | 2.37 × 10^{−10} | 2.1 × 10^{−11} |

10 | 1.41 × 10^{−9} | 1.4 × 10^{−10} | 2.35 × 10^{−10} | 2.4 × 10^{−11} |

12 | 1.23 × 10^{−9} | 1.5 × 10^{−10} | 2.20 × 10^{−10} | 2.1 × 10^{−11} |

15 | 1.53 × 10^{−9} | 1.5 × 10^{−10} | 2.18 × 10^{−10} | 2.0 × 10^{−11} |

20 | 1.47 × 10^{−9} | 1.3 × 10^{−10} | 1.87 × 10^{−10} | 1.9 × 10^{−11} |

**Table A4.**Mean escape time (τ) and associated errors of different number molecules confined in a 10-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 10

^{2}runs.

Cavity Diameter = 10 nm | ||||||
---|---|---|---|---|---|---|

N (Number of Molecules) | h = 1 nm | h = 2 nm | h = 5 nm | |||

τ (s) | Error (s) | τ (s) | Error (s) | τ (s) | Error (s) | |

1 | 1.35 × 10^{−8} | 1.5 × 10^{−9} | 2.59 × 10^{−9} | 4.6 × 10^{−10} | 2.42 × 10^{−10} | 2.7 × 10^{−11} |

2 | 1.33 × 10^{−8} | 1.3 × 10^{−9} | 2.07 × 10^{−9} | 2.3 × 10^{−10} | 2.97 × 10^{−10} | 4.7 × 10^{−11} |

3 | 1.09 × 10^{−8} | 1.3 × 10^{−9} | 1.74 × 10^{−9} | 1.7 × 10^{−10} | 2.06 × 10^{−10} | 2.5 × 10^{−11} |

4 | 1.35 × 10^{−8} | 1.5 × 10^{−9} | 2.21 × 10^{−9} | 3.2 × 10^{−10} | 2.00 × 10^{−10} | 1.8 × 10^{−11} |

5 | 1.10 × 10^{−8} | 1.1 × 10^{−9} | 1.57 × 10^{−9} | 1.7 × 10^{−10} | 2.00 × 10^{−10} | 2.4 × 10^{−11} |

6 | 1.43 × 10^{−8} | 1.4 × 10^{−9} | 1.78 × 10^{−9} | 1.7 × 10^{−10} | 2.14 × 10^{−10} | 2.5 × 10^{−11} |

7 | 1.55 × 10^{−8} | 1.6 × 10^{−9} | 1.85 × 10^{−9} | 1.9 × 10^{−10} | 2.51 × 10^{−10} | 3.1 × 10^{−11} |

8 | 1.21 × 10^{−8} | 1.1 × 10^{−9} | 2.10 × 10^{−9} | 2.2 × 10^{−10} | 2.03 × 10^{−10} | 2.1 × 10^{−11} |

9 | 1.23 × 10^{−8} | 1.1 × 10^{−9} | 2.07 × 10^{−9} | 2.0 × 10^{−10} | 2.54 × 10^{−10} | 2.4 × 10^{−11} |

10 | 1.17 × 10^{−8} | 1.3 × 10^{−9} | 1.73 × 10^{−9} | 1.5 × 10^{−10} | 2.31 × 10^{−10} | 3.1 × 10^{−11} |

12 | 1.28 × 10^{−8} | 1.2 × 10^{−9} | 1.82 × 10^{−9} | 1.7 × 10^{−10} | 1.88 × 10^{−10} | 2.4 × 10^{−11} |

15 | 1.29 × 10^{−8} | 1.2 × 10^{−9} | 1.66 × 10^{−9} | 1.6 × 10^{−10} | 2.21 × 10^{−10} | 2.3 × 10^{−11} |

20 | 1.13 × 10^{−8} | 1.0 × 10^{−9} | 1.74 × 10^{−9} | 1.5 × 10^{−10} | 2.40 × 10^{−10} | 2.5 × 10^{−11} |

**Table A5.**Mean escape time (τ) and associated errors of different number molecules confined in a 15-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 10

^{2}runs.

Cavity Diameter = 15 nm | ||||||||
---|---|---|---|---|---|---|---|---|

N (Number of Molecules) | h = 1 nm | h = 2 nm | H = 5 nm | h = 7.5 nm | ||||

τ | Error (s) | τ (s) | Error (s) | τ (s) | Error (s) | τ (s) | Error (s) | |

1 | 1.41 × 10^{−9} | 1.4 × 10^{−10} | 6.17 × 10^{−9} | 7.0 × 10^{−10} | 7.61 × 10^{−10} | 1.1 × 10^{−10} | 2.95 × 10^{−10} | 3.6 × 10^{−11} |

2 | 1.38 × 10^{−9} | 1.4 × 10^{−10} | 7.26 × 10^{−9} | 8.3 × 10^{−10} | 8.49 × 10^{−10} | 8.1 × 10^{−11} | 2.88 × 10^{−10} | 2.7 × 10^{−11} |

3 | 1.57 × 10^{−9} | 1.6 × 10^{−10} | 6.39 × 10^{−9} | 7.5 × 10^{−10} | 7.52 × 10^{−10} | 7.7 × 10^{−11} | 3.88 × 10^{−10} | 5.9 × 10^{−11} |

4 | 1.50 × 10^{−9} | 1.4 × 10^{−10} | 7.29 × 10^{−9} | 6.8 × 10^{−10} | 8.47 × 10^{−10} | 8.8 × 10^{−11} | 3.93 × 10^{−10} | 5.2 × 10^{−11} |

5 | 1.39 × 10^{−9} | 1.4 × 10^{−10} | 7.09 × 10^{−9} | 6.8 × 10^{−10} | 8.47 × 10^{−10} | 9.2 × 10^{−11} | 3.77 × 10^{−10} | 4.3 × 10^{−11} |

6 | 1.46 × 10^{−9} | 1.3 × 10^{−10} | 6.13 × 10^{−9} | 7.7 × 10^{−10} | 8.37 × 10^{−10} | 1.1 × 10^{−10} | 3.21 × 10^{−10} | 4.2 × 10^{−11} |

7 | 1.27 × 10^{−9} | 1.2 × 10^{−10} | 7.26 × 10^{−9} | 7.4 × 10^{−10} | 6.41 × 10^{−10} | 7.0 × 10^{−11} | 3.72 × 10^{−10} | 4.3 × 10^{−11} |

8 | 1.30 × 10^{−9} | 1.4 × 10^{−10} | 6.37 × 10^{−9} | 6.8 × 10^{−10} | 6.60 × 10^{−10} | 6.8 × 10^{−11} | 2.65 × 10^{−10} | 3.3 × 10^{−11} |

9 | 1.82 × 10^{−9} | 1.8 × 10^{−10} | 6.11 × 10^{−9} | 6.1 × 10^{−10} | 7.77 × 10^{−10} | 7.6 × 10^{−11} | 3.88 × 10^{−10} | 4.1 × 10^{−11} |

10 | 1.41 × 10^{−9} | 1.4 × 10^{−10} | 6.49 × 10^{−9} | 6.9 × 10^{−10} | 8.88 × 10^{−10} | 1.0 × 10^{−10} | 2.67 × 10^{−10} | 3.3 × 10^{−11} |

12 | 1.23 × 10^{−9} | 1.5 × 10^{−10} | 6.90 × 10^{−9} | 7.8 × 10^{−10} | 7.38 × 10^{−10} | 8.1 × 10^{−11} | 3.21 × 10^{−10} | 4.0 × 10^{−11} |

15 | 1.53 × 10^{−9} | 1.5 × 10^{−10} | 6.52 × 10^{−9} | 6.9 × 10^{−10} | 7.58 × 10^{−10} | 7.7 × 10^{−11} | 2.96 × 10^{−10} | 3.2 × 10^{−11} |

20 | 1.47 × 10^{−9} | 1.3 × 10^{−10} | 6.19 × 10^{−9} | 5.3 × 10^{−10} | 9.07 × 10^{−10} | 8.9 × 10^{−11} | 3.39 × 10^{−10} | 4.5 × 10^{−11} |

**Table A6.**Mean escape time (τ) and associated errors of different number molecules confined in a 20-nm spherical cavity for different entrance–escape hole diameters (h) calculated by the interactive random walk models for 10

^{2}runs.

Cavity Diameter = 20 nm | ||||||||
---|---|---|---|---|---|---|---|---|

N (Number of Molecules) | h = 1 nm | h = 2 nm | h = 5 nm | h = 10 nm | ||||

τ (s) | Error (s) | τ (s) | Error (s) | τ (s) | Error (s) | τ (s) | Error (s) | |

1 | 1.26 × 10^{−7} | 1.6 × 10^{−8} | 1.74 × 10^{−8} | 2.0 × 10^{−9} | 2.27 × 10^{−9} | 3.85 × 10^{−10} | 4.89 × 10^{−10} | 7.03 × 10^{−11} |

2 | 7.59 × 10^{−8} | 7.5 × 10^{−9} | 1.69 × 10^{−8} | 1.8 × 10^{−9} | 1.82 × 10^{−9} | 2.24 × 10^{−10} | 4.07 × 10^{−10} | 4.78 × 10^{−11} |

3 | 9.88 × 10^{−8} | 9.5 × 10^{−9} | 1.63 × 10^{−8} | 1.7 × 10^{−9} | 1.91 × 10^{−9} | 2.04 × 10^{−10} | 3.46 × 10^{−10} | 4.19 × 10^{−11} |

4 | 9.80 × 10^{−8} | 8.5 × 10^{−9} | 1.51 × 10^{−8} | 1.7 × 10^{−9} | 1.70 × 10^{−9} | 1.85 × 10^{−10} | 4.17 × 10^{−10} | 5.17 × 10^{−11} |

5 | 9.22 × 10^{−8} | 9.6 × 10^{−9} | 1.75 × 10^{−8} | 1.8 × 10^{−9} | 1.96 × 10^{−9} | 2.24 × 10^{−10} | 4.54 × 10^{−10} | 4.66 × 10^{−11} |

6 | 8.20 × 10^{−8} | 7.7 × 10^{−9} | 1.32 × 10^{−8} | 1.3 × 10^{−9} | 1.87 × 10^{−9} | 2.11 × 10^{−10} | 3.51 × 10^{−10} | 4.63 × 10^{−11} |

7 | 8.51 × 10^{−8} | 8.3 × 10^{−9} | 1.51 × 10^{−8} | 1.2 × 10^{−9} | 1.87 × 10^{−9} | 1.87 × 10^{−10} | 4.48 × 10^{−10} | 5.82 × 10^{−11} |

8 | 9.64 × 10^{−8} | 8.5 × 10^{−9} | 1.43 × 10^{−8} | 1.5 × 10^{−9} | 1.90 × 10^{−9} | 2.05 × 10^{−10} | 4.10 × 10^{−10} | 4.13 × 10^{−11} |

9 | 6.85 × 10^{−8} | 5.9 × 10^{−9} | 1.66 × 10^{−8} | 1.6 × 10^{−9} | 1.88 × 10^{−9} | 2.15 × 10^{−10} | 3.56 × 10^{−10} | 3.69 × 10^{−11} |

10 | 9.94 × 10^{−8} | 9.6 × 10^{−9} | 1.56 × 10^{−8} | 1.5 × 10^{−9} | 2.05 × 10^{−9} | 2.18 × 10^{−10} | 4.32 × 10^{−10} | 4.07 × 10^{−11} |

12 | 8.36 × 10^{−8} | 9.8 × 10^{−9} | 1.55 × 10^{−8} | 1.5 × 10^{−9} | 1.77 × 10^{−9} | 1.81 × 10^{−10} | 4.77 × 10^{−10} | 5.87 × 10^{−11} |

15 | 8.01 × 10^{−8} | 7.7 × 10^{−9} | 1.40 × 10^{−8} | 1.4 × 10^{−9} | 2.23 × 10^{−9} | 2.01 × 10^{−10} | 3.88 × 10^{−10} | 4.80 × 10^{−11} |

20 | 7.68 × 10^{−8} | 7.8 × 10^{−9} | 1.49 × 10^{−8} | 1.6 × 10^{−9} | 1.69 × 10^{−9} | 1.89 × 10^{−10} | 4.26 × 10^{−10} | 4.87 × 10^{−11} |

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**Figure 1.**Surface parameters of irradiated polyacrylamide (PAM) layers for a 2 μm × 2 μm area: (

**a**) Mean z-height; (

**b**) area roughness (R

_{a}); (

**c**) area RMS; (

**d**) maximum range. The area roughness and area RMS parameters show an increment with laser pulses up to ~ 200 lp followed by a dip at 10

^{3}lp.

**Figure 2.**Atomic force microscopy (AFM) surface image of polyacrylamide (PAM) layers. Scan area 2 × 2 μm

^{2}, laser fluence 250 J m

^{−2}per pulse: (

**a**) non-irradiated PAM layer; (

**b**) irradiated PAM layer with 100 laser pulses (lp), 25 kJ m

^{−2}; (

**c**) 200 lp, 50 kJ m

^{−2}; (

**d**) 10

^{3}lp, 250 kJ m

^{−2}; (

**e**) scan area 2.3 × 2.3 μm

^{2}, 10

^{3}lp, 250 kJ m

^{−2}. The surface morphology is area size-dependent.

**Figure 3.**Atomic force microscopy (AFM) image of ‘‘lake’’ (grey) and ‘‘island’’ (orange) for a fractal area of 2 × 2 μm

^{2}: (

**a**) Non-irradiated area; (

**b**) irradiated area with 10

^{3}laser pulses.

**Figure 4.**Fractal dimensionality, “lake” surface area and lake number vs. laser fluence. (

**a**) Surface -fractal dimensionality calculated with four different fractal analytical methodologies. Colors, symbols and lines to assist the eye: blue squares for partitioning (PA), purple spheres for power spectrum (PS), dark yellow stars for cube counting (CC), green triangles for triangulation (TR), and brown pentagons for atomic force microscopy (AFM) “lake” pattern (LA). The five methods show a similar fractality trend vs. laser fluence; (

**b**) “lake” surface area vs. the number of lakes at different number of laser pulses (laser fluence).

**Figure 5.**Fractal parameters of polyacrylamide (PAM) at different laser fluence: (

**a**) Number of ‘‘lakes’’ for different fractal size vs. the number of laser pulses (n); (

**b**) fractal dimensionality vs. the number of laser pulses (n) for different fractal size. The concentration of small size nanocavities increases at higher laser fluence; (

**c**) fractal dimension vs. fractal size at a different number of laser pulses (lp).

**Figure 6.**Typical force-distance (F-D) curves of polyacrylamide (PAM) thin layer surfaces (426 nm) irradiated with a different number of laser pulses (lp) (250 J m

^{−2}per laser pulse): (

**a**) F-D curves of the non-irradiated layer; (

**b**) F-D curves with 200 lp; (

**c**) F-D curves with 300 lp; (

**d**) F-D curves with 400 lp.

**Figure 7.**Young’s modulus and adhesion force of irradiated polyacrylamide (PAM) surfaces showing enhanced carbonization at higher laser fluence. (

**a**) Young’s modulus; (

**b**) adhesion force of a PAM surface irradiated at different laser fluence up to 250 J m

^{−2}; (

**c**) Young’s modulus and adhesion force column charts of PAM vs. laser fluence.

**Figure 8.**Water contact angle (CA) vs. the number of laser pulses: (

**a**) Water CA vs. the number of laser pulses. (

**b**) column chart diagram of CA and fractal dimensionality vs. laser fluence. The mean correlation factor is −0.833; (

**c**) column chart diagram of CA and area RMS of PAM vs. laser fluence. The mean correlation factor is 0.768 pointing to a strong correlation between fractal dimensionality, CA, and area RMS over a wide range of laser fluence; (

**d**) column chart diagram of water CA at different time intervals. The almost similar slopes point to a uniform surface response at different VUV photon fluence. From Figure 4a, Figure 5a,b, Figure 7a,b and Figure 8a the surface chemical modification is saturated at ~500 laser pulses, because of the low penetrating depth of the 157 nm laser photons, indicating the strong correlation between fractal dimensionality, CA, area RMS, Young’s modulus and surface modification.

**Figure 9.**Principle of operation of white light reflectance spectroscopy (WLRS). (

**a**) White light beam reflection in PAM surfaces. (

**b**) experimental details and geometry of the reflected beams (

**c**) surface strain response during water confinement in nanocavities. The contribution of the SiO

_{2}layer at the interference pattern is negligible.

**Figure 10.**Non-interactive random walk of one water molecule in a nanocavity. (

**a**) The water molecule enters the cavity (yellow arrow) and then it collides with the inside walls of the spherical cavity (10 nm) several times (A–I points and blue lines) before escaping from the entrance-escape hole (3 nm, red line); (

**b**) mean escape time for 10

^{3}different random walk runs in 1 nm (green), 10 nm (red), 10

^{2}nm (blue), and 10

^{3}nm (magenta) spherical cavities for different entrance-escape hole diameters (0.3 nm–500 nm). The y-axis represents a logarithmic time scale.

**Figure 11.**Nanocavitation by 157 nm laser photodissociation of polyacrylamide (PAM) matrixes: (

**a**) Molecular photodissociation at 157 nm. Vertical arrows indicate photon transitions between two vibrational levels of the ground (A) and an excited electronic state (Β). A transition from the excited to a repulsive electronic state (red curve) through an avoided crossing via a vibration state at the point (Γ) is very fast (< 1 ps) and breaks a molecular bond in the polymeric chain; (

**b**) PAM surface irradiation with 157 nm photons; (

**c**) (DE): a bond break is followed by molecular decomposition and nanocavitation; (

**d**) (CL): possible recombination of carbon dissociative products, and cluster formation in the gas phase; (

**e**) carbon cluster deposition on the polymeric matrix and possible structure of carbon nanocavitation.

**Figure 12.**Mean escape time for a different number of non-interactive molecules calculated for 10

^{3}runs. (

**a**) Fluctuations of mean molecular escape time are prominent for small size cavities, while remains constant for a large number of molecules; (

**b**–

**d**) mean escape time for 5, 10, 15 and 20 nm spherical cavities with different entrance-escape holes and number of molecules in the cavity. The mean escape time is independent of the number of molecules and is a function only of the cavity geometry (diameter and entrance-escape hole).

**Figure 13.**Distribution of mean escape times for 10

^{2}random walk runs of a water molecule (non-interactive model) for different cavity geometries (cavity size D and entrance-escape hole size h). (

**a**) D = 15 nm, h = 1 nm; (

**b**) D = 10 nm, h = 1 nm; (

**c**) D = 5 nm, h = 1 nm; (

**d**) D = 15 nm, h = 2 nm; (

**e**) D = 10 nm, h = 2 nm (

**f**) D = 5 nm, h = 1 nm. Distributions are non-normal and besides that skewness and long tails indicate non-equilibrium processes inside the cavities.

**Figure 14.**Mean escape time and mean travelling distance of a molecule within cavities of different geometries. (

**a**,

**b**) Gradient of the mean escape time and the mean distance that a molecule travels in the cavity before it escapes with different entrance-escape holes is diverging for very small size cavities (1, 10 nm); (

**c**,

**d**) mean escape time and the travelling distance gradients are constant for large size cavities (10

^{2}, 10

^{3}nm), suggesting a non-thermal equilibrium state and large fluctuations for small size cavities.

**Figure 15.**(

**a**–

**c**) Mean escape time distribution of 150 interactive molecules for different cavity and entrance-escape hole size for 10

^{2}runs; (

**d**) best-fitting of Figure 15a is for a log-normal distribution; (

**e**) mean escape time vs. the cavity ratio $\frac{h}{D}$. The time differentiation of molecular movements inside and outside cavities is provided by the dependence of the ratio $\frac{\mathrm{h}}{\mathrm{D}}$ on the waiting time τ, which for D=10

^{2}and 10

^{3}nm cavities goes as a power law of the waiting time with exponent −0.5.

**Figure 16.**Schematic layout of the interphase between the photon processed polyacrylamide PAM surface and the water vapor domain. (

**a**) Thermal equilibrium domain. Reference time and space scales are determined by the mean collision time t

_{col}between the water and air molecules and the entropy of the ideal gases and the mean collision distance. The entropy S

_{1}is given by the Sackur–Tetrode equation for the ideal gases [99]; (

**b**) local fluctuations domain. Nanocavitations on the surface with confined molecules. The time scale is determined by the mean escape time τ of water molecules. The entropy S

_{2}in this domain is determined by the number of microstates Ω(Ν

_{b}(n), N

_{c}(n), Ε

_{α}), which specify a state of ordered arrangements between nanocavities in one hand and molecular water ensembles of fixed molecular length near the surface on the other; (

**c**) volume matrix domain.

**Figure 17.**Relative surface deformation (strain) of the 426 nm polyacrylamide (PAM) layers measured with white light reflectance spectroscopy (WLRS) at different 157 nm irradiating conditions of the PAM matrix and relative humidity (RH). The solid lines at different RH represent the best fit of Equation (36) to the experimental data. The black lines at 80 % RH are the fittings for different λl values of 0, 0.05, 0.1 and 0.2. The best fit (orange line) respectively, is for $0\le \mathsf{\lambda}\mathrm{l}<0.05\text{}$ suggesting a small contribution to the relative surface deformation from electric dipole attachment of water molecules to active binding sites in the PAM matrix and a substantial contribution from the confinement of water molecules in nanocavities, Equation (36).

© 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/).

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**MDPI and ACS Style**

Gavriil, V.; Chatzichristidi, M.; Christofilos, D.; Kourouklis, G.A.; Kollia, Z.; Bakalis, E.; Cefalas, A.-C.; Sarantopoulou, E.
Entropy and Random Walk Trails Water Confinement and Non-Thermal Equilibrium in Photon-Induced Nanocavities. *Nanomaterials* **2020**, *10*, 1101.
https://doi.org/10.3390/nano10061101

**AMA Style**

Gavriil V, Chatzichristidi M, Christofilos D, Kourouklis GA, Kollia Z, Bakalis E, Cefalas A-C, Sarantopoulou E.
Entropy and Random Walk Trails Water Confinement and Non-Thermal Equilibrium in Photon-Induced Nanocavities. *Nanomaterials*. 2020; 10(6):1101.
https://doi.org/10.3390/nano10061101

**Chicago/Turabian Style**

Gavriil, Vassilios, Margarita Chatzichristidi, Dimitrios Christofilos, Gerasimos A. Kourouklis, Zoe Kollia, Evangelos Bakalis, Alkiviadis-Constantinos Cefalas, and Evangelia Sarantopoulou.
2020. "Entropy and Random Walk Trails Water Confinement and Non-Thermal Equilibrium in Photon-Induced Nanocavities" *Nanomaterials* 10, no. 6: 1101.
https://doi.org/10.3390/nano10061101