# Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

^{2}s

^{−1}, so ${\tau}_{D}\approx 10$ s. Last, the memory time was estimated from experimental results to be ${\tau}_{a}\approx 1$ s [21]. It is important to notice that, as long as the system obeys Fick’s law and Langmuir’s kinetic, it in principle can be investigated within the scope of this model.

## 3. Results and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Table of Symbols

Symbol | Definition |

$\rho (z,t)$ | Bulk density of diffusing particles. It is a function of space ( z ) and time ( t ) |

$\sigma $(t) | Density of adsorbed particles. It is a function of time (t) |

D | Diffusion coefficient |

L | Cell thickness |

${\rho}_{0}$ | Total density of particles |

${\kappa}_{a}$ | Rate of adsorption |

${\kappa}_{d}$ | Rate of desorption |

${\rho}_{R}=\rho /{\rho}_{0}$ | Reduced bulk density |

${\sigma}_{0}$ | Number of available sites |

${\sigma}_{R}=\sigma /{\sigma}_{0}$ | Reduced surface density |

${\tau}_{D}={d}^{2}/D$ | Diffusion time |

${\tau}_{\kappa}=d/2{\kappa}_{a}$ | Adsorption time |

$\tau =1/{\kappa}_{d}$ | Desorption time |

${\tau}_{a}$ | Memory time |

${t}^{*}=4t/{\tau}_{D}$ | Dimensionless time |

${\left(\mathsf{\Delta}Z\right)}^{2}$ | Mean Square Displacement |

$\beta ={\rho}_{0}L/{\sigma}_{0}$ | Ratio of particles in the bulk to the number of available sites |

Subscript i | May be l (left) or r (right). Indicates the substrate considered. |

## References

- Li, J.R.; Kuppler, R.J.; Zhou, H.C. Selective gas adsorption and separation in metal–organic frameworks. Chem. Soc. Rev.
**2009**, 38, 1477–1504. [Google Scholar] [CrossRef] [PubMed] - Adamson, A.; Gast, A. Physical Chemistry of Surfaces; Wiley: Hoboken, NJ, USA, 1997. [Google Scholar]
- Barbero, G.; Evangelista, L.R. Adsorption Phenomena and Anchoring Energy in Nematic Liquid Crystals; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Perlmutter, S.H.; Doroski, D.; Moddel, G. Degradation of liquid crystal device performance due to selective adsorption of ions. Appl. Phys. Lett.
**1996**, 69, 1182–1184. [Google Scholar] [CrossRef] - Zhang, M.; Soto-Rodríguez, J.; Chen, I.C.; Akbulut, M. Adsorption and removal dynamics of polymeric micellar nanocarriers loaded with a therapeutic agent on silica surfaces. Soft Matter
**2013**, 9, 10155–10164. [Google Scholar] [CrossRef] - Yaseen, M.; Salacinski, H.J.; Seifalian, A.M.; Lu, J.R. Dynamic protein adsorption at the polyurethane copolymer/water interface. Biomed. Mater.
**2008**, 3, 034123. [Google Scholar] [CrossRef] [PubMed] - Patrick, N.; Youssef, B.; Burd, S.D.; Cairns, A.J.; Ryan, L.; Katherine, F.; Tony, P.; Shengqian, M.; Brian, S.; Lukasz, W.; et al. Porous materials with optimal adsorption thermodynamics and kinetics for CO
_{2}separation. Nature**2013**, 495, 80. [Google Scholar] [CrossRef] - Langmuir, I. The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum. J. Am. Chem. Soc.
**1918**, 40, 1361–1403. [Google Scholar] [CrossRef] [Green Version] - Kuan, W.H.; Lo, S.L.; Chang, C.M.; Wang, M.K. A geometric approach to determine adsorption and desorption kinetic constants. Chemosphere
**2000**, 41, 1741–1747. [Google Scholar] [CrossRef] [PubMed] - Swenson, H.; Stadie, N.P. Langmuir’s Theory of Adsorption: A Centennial Review. Langmuir
**2019**, 35, 5409–5426. [Google Scholar] [CrossRef] [Green Version] - Arfken, G.; Weber, H.; Harris, F. Mathematical Methods for Physicists: A Comprehensive Guide; Elsevier Science: Amsterdam, The Netherlands, 2013. [Google Scholar]
- Bénichou, O.; Grebenkov, D.; Levitz, P.; Loverdo, C.; Voituriez, R. Optimal Reaction Time for Surface-Mediated Diffusion. Phys. Rev. Lett.
**2010**, 105, 150606. [Google Scholar] [CrossRef] - Levesque, M.; Bénichou, O.; Voituriez, R.; Rotenberg, B. Taylor dispersion with adsorption and desorption. Phys. Rev. E
**2012**, 86, 036316. [Google Scholar] [CrossRef] - Simonin, J.P. On the comparison of pseudo-first order and pseudo-second order rate laws in the modeling of adsorption kinetics. Chem. Eng. J.
**2016**, 300, 254–263. [Google Scholar] [CrossRef] [Green Version] - Azizian, S. Kinetic models of sorption: A theoretical analysis. J. Colloid Interface Sci.
**2004**, 276, 47–52. [Google Scholar] [CrossRef] [PubMed] - Ho, Y.; McKay, G. Pseudo-second order model for sorption processes. Process Biochem.
**1999**, 34, 451–465. [Google Scholar] [CrossRef] - Liu, Y.; Shen, L. From Langmuir Kinetics to First- and Second-Order Rate Equations for Adsorption. Langmuir
**2008**, 24, 11625–11630. [Google Scholar] [CrossRef] - Salvestrini, S. A modification of the Langmuir rate equation for diffusion-controlled adsorption kinetics. React. Kinet. Mech. Catal.
**2019**, 128, 571–586. [Google Scholar] [CrossRef] - Barbero, G.; Evangelista, L.R. Adsorption phenomenon of neutral particles and a kinetic equation at the interface. Phys. Rev. E
**2004**, 70, 031605. [Google Scholar] [CrossRef] [Green Version] - Levesque, M.; Bénichou, O.; Rotenberg, B. Molecular diffusion between walls with adsorption and desorption. J. Chem. Phys.
**2013**, 138, 034107. [Google Scholar] [CrossRef] [Green Version] - Zola, R.S.; Lenzi, E.K.; Evangelista, L.R.; Barbero, G. Memory effect in the adsorption phenomena of neutral particles. Phys. Rev. E
**2007**, 75, 042601. [Google Scholar] [CrossRef] - Zola, R.; Freire, F.; Lenzi, E.; Evangelista, L.; Barbero, G. Kinetic equation with memory effect for adsorption–desorption phenomena. Chem. Phys. Lett.
**2007**, 438, 144–147. [Google Scholar] [CrossRef] - Guimarães, V.G.; Ribeiro, H.V.; Li, Q.; Evangelista, L.R.; Lenzi, E.K.; Zola, R.S. Unusual diffusing regimes caused by different adsorbing surfaces. Soft Matter
**2015**, 11, 1658–1666. [Google Scholar] [CrossRef] - Recanello, M.; Lenzi, E.; Martins, A.; Li, Q.; Zola, R. Extended adsorbing surface reach and memory effects on the diffusive behavior of particles in confined systems. Int. J. Heat Mass Transf.
**2020**, 151, 119433. [Google Scholar] [CrossRef] - Fernandes, M.; Lenzi, E.; Evangelista, L.; Li, Q.; Zola, R.; de Souza, R. Diffusion and adsorption-desorption phenomena in confined systems with periodically varying medium. Chem. Eng. Sci.
**2021**, 233, 116386. [Google Scholar] [CrossRef] - Guimarães, V.G.; Lenzi, E.K.; Evangelista, L.R.; Zola, F.C.; de Souza, R.T.; Zola, R.S. Symmetry breaking in an electrolytic cell under AC field and non-identical adsorbing electrodes. J. Electroanal. Chem.
**2017**, 789, 44–49. [Google Scholar] [CrossRef] - Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific Publishing Company: Singapore, 2000. [Google Scholar]
- Glöckle, W.G.; Nonnenmacher, T.F. Fractional relaxation and the time-temperature superposition principle. Rheol. Acta
**1994**, 33, 337–343. [Google Scholar] [CrossRef] - Li, X.; Shaw, R.; Evans, G.M.; Stevenson, P. A simple numerical solution to the Ward–Tordai equation for the adsorption of non-ionic surfactants. Comput. Chem. Eng.
**2010**, 34, 146–153. [Google Scholar] [CrossRef] - Liggieri, L.; Ravera, F.; Passerone, A. A diffusion-based approach to mixed adsorption kinetics. Colloids Surf. A Physicochem. Eng. Asp.
**1996**, 114, 351–359. [Google Scholar] [CrossRef] - Mazumder, S. Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods; Elsevier Science: Amsterdam, The Netherlands, 2016. [Google Scholar]
- Galassi, M.; Gough, B. GNU Scientific Library: Reference Manual; GNU Manual; Network Theory: London, UK, 2009. [Google Scholar]
- Maximus, B.; Ley, E.D.; Meyere, A.D.; Pauwels, H. Ion transport in SSFLCD’s. Ferroelectrics
**1991**, 121, 103–112. [Google Scholar] [CrossRef] - López-González, B.; Dector, A.; Cuevas-Muñiz, F.; Arjona, N.; Cruz-Madrid, C.; Arana-Cuenca, A.; Guerra-Balcázar, M.; Arriaga, L.; Ledesma-García, J. Hybrid microfluidic fuel cell based on Laccase/C and AuAg/C electrodes. Biosens. Bioelectron.
**2014**, 62, 221–226. [Google Scholar] [CrossRef] - Sparavigna, A.; Lavrentovich, O.D.; Strigazzi, A. Periodic stripe domains and hybrid-alignment regime in nematic liquid crystals: Threshold analysis. Phys. Rev. E
**1994**, 49, 1344–1352. [Google Scholar] [CrossRef] [Green Version] - Zola, R.S.; Evangelista, L.R.; Yang, Y.C.; Yang, D.K. Surface Induced Phase Separation and Pattern Formation at the Isotropic Interface in Chiral Nematic Liquid Crystals. Phys. Rev. Lett.
**2013**, 110, 057801. [Google Scholar] [CrossRef] - Kuksenok, O.V.; Shiyanovskii, S.V. Surface Control of Dye Adsorption in Liquid Crystals. Mol. Cryst. Liq. Cryst. Sci. Technol. Sect. A Mol. Cryst. Liq. Cryst.
**2001**, 359, 107–118. [Google Scholar] [CrossRef] - Linse, P. Effect of solvent quality on the polymer adsorption from bulk solution onto planar surfaces. Soft Matter
**2012**, 8, 5140–5150. [Google Scholar] [CrossRef] - Kotono, M.; Takahiro, F.; Yasuhiro, U.; Kenichi, S.; Ryota, I.; Hidetoshi, Y.; Mihoko, S.; Hideji, M.; Ken, R.; Akihiro, K. Ultrafine Membrane Compartments for Molecular Diffusion as Revealed by Single Molecule Techniques. Biophys. J.
**2004**, 86, 4075–4093. [Google Scholar] [CrossRef] [Green Version] - Chipot, C.; Comer, J. Subdiffusion in Membrane Permeation of Small Molecules. Sci. Rep.
**2016**, 6, 35913. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Spiechowicz, J.; Łuczka, J. Subdiffusion via dynamical localization induced by thermal equilibrium fluctuations. Sci. Rep.
**2017**, 7, 16451. [Google Scholar] [CrossRef] [Green Version] - Höfling, F.; Franosch, T. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys.
**2013**, 76, 046602. [Google Scholar] [CrossRef]

**Figure 1.**System studied in this work. It consists of a liquid with diluted neutral particles (diffusing particles) that diffuse in the $z-$direction and may be adsorbed (desorbed) following Langmuir’s equation. Notice that the left surface is located at $z=-L/2$, while the right surface is at $Z=L/2$. Gray spheres represent occupied sites on the surfaces, while green spheres represent free adsorption sites.

**Figure 2.**Role played by the parameter $\beta ={\rho}_{0}d/{\sigma}_{0}$ on the surface dynamics. (

**a**) shows the left surface ($Z=-1$) vs. ${t}^{*}$ for ${\tau}_{D}/{\tau}_{l}=100$, ${\tau}_{\kappa l}/{\tau}_{l}=10$ and for two values of ${\tau}_{al}/{\tau}_{l}$, that is, 0.1 or 20 and $\beta =0.2$ and $\beta =2$. The inset shows the bulk density ${\rho}_{R}$ vs. Z when ${t}^{*}=0.6$ for $\beta =2$ (in red) and $\beta =0.2$ (black). (

**b**) shows the left surface ($Z=-1$) vs. ${t}^{*}$ for ${\tau}_{D}/{\tau}_{l}=100$, ${\tau}_{\kappa l}/{\tau}_{l}=1$, and ${\tau}_{al}/{\tau}_{l}$=0.1 for several values of the parameter $\beta $.

**Figure 3.**Effect of $\beta $ on non-identical surfaces. For both figures, the left surface uses ${\beta}_{l}=1.0$, ${\tau}_{\kappa l}/{\tau}_{l}=1$ and ${\tau}_{al}/{\tau}_{l}=0.01$, and the diffusion time is ${\tau}_{al}/{\tau}_{l}=0.01$. In (

**a**), all the characteristic times of the surface at $Z=1$ are the same as the surface at $Z=-1$, except the parameter ${\beta}_{r}$. The main figure shows the time evolution of ${\sigma}_{Rl}$, while the inset shows ${\sigma}_{Rr}$. In (

**b**), the left surface uses the same parameters as in (

**a**), but the right surfaces uses ${\tau}_{\kappa r}/{\tau}_{r}=10$, ${\tau}_{ar}/{\tau}_{r}=20$ for ${\beta}_{r}=0.2$ and ${\beta}_{r}=2.0$. Both surface dynamics are plotted against ${t}^{*}$ in the main figure, while the inset shows the bulk distribution for three different values of ${t}^{*}$ and both values of ${\beta}_{r}$.

**Figure 4.**${\sigma}_{Ri}/{\beta}_{i}$ for two different values of ${\beta}_{r}$. The left surface uses the same parameters as used in Figure 3, while the surface at $Z=1$ uses ${\tau}_{\kappa r}/{\tau}_{r}=0.01$ and ${\tau}_{ar}/{\tau}_{r}=0.01$. From this figure, it becomes clear that increasing the $\beta $ of a single surface changes the behavior of the opposite surface and that larger values of $\beta $ mean higher coverage but a smaller overall number of adsorbed particles. The inset shows the bulk distribution for three different values of ${t}^{*}$ and for ${\beta}_{r}=0.2$ and ${\beta}_{r}=2.0$.

**Figure 5.**Mean square displacement (MSD) vs. ${t}^{*}$ for several values of $\beta $. The solid curves represent the case in which both surfaces are equal, while the dashed curves represent the MSD for the $Z=1$ surface in the case of non-equal surfaces. (

**a**) shows the case in which ${\tau}_{\kappa i}/{\tau}_{i}=1$, while (

**b**) shows the case in which ${\tau}_{\kappa i}/{\tau}_{i}=0.01$ (identical surfaces and right surface for the non-identical case). The dotted lines in (

**a**) show examples of the exponent (${t}^{a}$) of the MSD, indicating the subdiffusive behavior of the system.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

de Souza, R.F.; de Almeida, R.R.R.; Omori, E.K.; de Souza, R.T.; Lenzi, E.K.; Evangelista, L.R.; Zola, R.S.
Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics. *Physchem* **2023**, *3*, 1-12.
https://doi.org/10.3390/physchem3010001

**AMA Style**

de Souza RF, de Almeida RRR, Omori EK, de Souza RT, Lenzi EK, Evangelista LR, Zola RS.
Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics. *Physchem*. 2023; 3(1):1-12.
https://doi.org/10.3390/physchem3010001

**Chicago/Turabian Style**

de Souza, Renato F., Roberta R. Ribeiro de Almeida, Eric K. Omori, Rodolfo T. de Souza, Ervin K. Lenzi, Luiz R. Evangelista, and Rafael S. Zola.
2023. "Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics" *Physchem* 3, no. 1: 1-12.
https://doi.org/10.3390/physchem3010001