Brownian motion near a soft surface

Yilin Ye; Yacine Amarouchene; Raphaël Sarfati; David S. Dean; Thomas Salez

doi:10.5802/crphys.263

Research article

Brownian motion near a soft surface

Yilin Ye ^1,²; Yacine Amarouchene ¹; Raphaël Sarfati ³; David S. Dean ¹; Thomas Salez ¹

¹ Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33400 Talence, France
² Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS, École Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
³ Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY, USA

Comptes Rendus. Physique, Volume 26 (2025), pp. 619-630

Abstract (Original language)
Résumé

Brownian motion near soft surfaces is a situation widely encountered in nanoscale and biological physics. However, a complete theoretical description is lacking to date. Here, we theoretically investigate the dynamics of a two-dimensional colloid in an arbitrary external potential and near a soft surface. The latter is minimally modelled by a Winkler’s foundation, and we restrict the study to the colloidal motion in the direction perpendicular to the surface. We start from deterministic hydrodynamic considerations, by invoking the already-established leading-order soft-lubrication forces acting on the particle. Importantly, a negative softness-induced and position-dependent added mass is identified. We then incorporate thermal fluctuations in the description. In particular, an effective Hamiltonian formulation is introduced and a temperature-dependent generalized potential is constructed in order to ensure equilibrium properties for the colloidal position. From these considerations and the Fokker–Planck equation, we then derive the relevant Langevin equation, which self-consistently allows to recover the deterministic equation of motion at zero temperature. Interestingly, besides an expected multiplicative-noise feature, the noise correlator appears to be modified by the surface softness. Moreover, a softness-induced temperature-dependent spurious drift term has to be incorporated within the Ito prescription. Finally, using numerical simulations with various initial conditions and parameter values, we statistically analyze the trajectories of the particle when placed within a harmonic trap and in presence of the soft surface. This allows us to: (i) quantify further the influence of surface softness, through the added mass, which enhances the velocity fluctuations; and (ii) show that intermediate-time diffusion is unaffected by softness, within the assumptions of the model.

Le mouvement brownien proche de surfaces déformables est une situation fréquemment rencontrée en physique à l’échelle nanométrique et en biologie. Cependant, une description théorique complète fait encore défaut à ce jour. Dans cet article, nous étudions théoriquement la dynamique d’un colloïde bidimensionnel soumis à un potentiel extérieur arbitraire, au voisinage d’une surface élastique. Cette dernière est modélisée de manière minimale par une réponse de type Winkler, et nous restreignons l’étude au mouvement du colloïde dans la direction perpendiculaire à la surface. Nous commençons par des considérations hydrodynamiques déterministes, en invoquant les forces de lubrification élastique, au premier ordre en déformabilité, déjà établies. Il est important de noter qu’une masse ajoutée, négative et dépendante de la position, induite par la déformation de la surface, est identifiée. Nous incorporons ensuite les fluctuations thermiques dans la description. En particulier, une formulation hamiltonienne effective est introduite et un potentiel généralisé dépendant de la température est construit afin d’assurer les propriétés d’équilibre pour la position du colloïde. À partir de ces considérations et de l’équation de Fokker–Planck, nous dérivons ensuite l’équation de Langevin pertinente, qui permet de retrouver de manière auto-consistante l’équation déterministe du mouvement à température nulle. Il est intéressant de noter qu’en plus du bruit multiplicatif attendu, le corrélateur du bruit semble être modifié par la déformabilité de la surface. De plus, un terme de dérive spurieuse dépendant de la température, induit par l’élasticité, doit être incorporé dans la prescription d’Ito. Enfin, en utilisant des simulations numériques avec diverses conditions initiales et valeurs de paramètres, nous analysons statistiquement les trajectoires de la particule lorsqu’elle est placée dans un piège harmonique en présence de la surface élastique. Cela nous permet de : (i) quantifier l’influence de l’élasticité de la surface, à travers la masse ajoutée, qui renforce les fluctuations de vitesse ; et (ii) montrer que la diffusion à temps intermédiaire est indépendante de l’élasticité, dans les hypothèses du modèle.

Received: 2025-07-03
Revised: 2025-09-04
Accepted: 2025-09-10
Published online: 2025-10-02

DOI: 10.5802/crphys.263

Keywords: Brownian motion, soft lubrication, elasticity
Mots-clés : Mouvement brownien, lubrification élastique, élasticité

Author's affiliations:

Yilin Ye ^{1, 2}; Yacine Amarouchene ¹; Raphaël Sarfati ³; David S. Dean ¹; Thomas Salez ¹

¹ Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33400 Talence, France
² Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS, École Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
³ Department of Civil and Environmental Engineering, Cornell University, Ithaca, NY, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Yilin Ye and Yacine Amarouchene and Rapha\"el Sarfati and David S. Dean and Thomas Salez},
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Yilin Ye; Yacine Amarouchene; Raphaël Sarfati; David S. Dean; Thomas Salez. Brownian motion near a soft surface. Comptes Rendus. Physique, Volume 26 (2025), pp. 619-630. doi: 10.5802/crphys.263

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