Local and global force balance for diffusiophoretic transport S. Marbach, H. Yoshida, L. Bocquet To cite this version: S. Marbach, H. Yoshida, L. Bocquet. Local and global force balance for diffusiophoretic transport. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2020, 892, 10.1017/jfm.2020.137. hal-02998824 HAL Id: hal-02998824 https://hal.archives-ouvertes.fr/hal-02998824 Submitted on 10 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Local and global force balance for diffusiophoretic transport S. Marbach12, H. Yoshida3 and L. Bocquet1† 1Ecole Normale Supérieure, PSL Research University, CNRS, 24 rue Lhomond, Paris, France 2Courant Institute of Mathematical Sciences, NYU, 251 Mercer Street, New York, NY, USA 3Toyota Central R&D Labs., Inc., Bunkyo-ku, Tokyo 112-0004, Japan (Received xx; revised xx; accepted xx) Electro- and diffusio- phoresis of particles correspond respectively to the transport of particles under electric field and solute concentration gradients. Such interfacial transport phenomena take their origin in a diffuse layer close to the particle surface, and the motion of the particle is force-free. In the case of electrophoresis, it is further expected that the stress acting on the moving particle vanishes locally as a consequence of local electroneutrality. But the argument does not apply to diffusiophoresis, which takes its origin in solute concentration gradients. In this paper we investigate further the local and global force balance on a particle undergoing diffusiophoresis. We calculate the local tension applied on the particle surface and show that, counter-intuitively, the local force on the particle does not vanish for diffusiophoresis, in spite of the global force being zero as expected. Incidentally, our description allows to clarify the osmotic balance in diffusiophoresis, which has been a source of debates in the recent years. We explore various cases, including hard and soft interactions, as well as porous particles, and provide analytic predictions for the local force balance in these various systems. The existence of local stresses may induce deformation of soft particles undergoing diffusiophoresis, hence suggesting applications in terms of particle separation based on capillary diffusiophoresis. 1. Introduction Phoresis corresponds to the motion of a particle induced by an external field, say Θ∞: typically an electric potential for electrophoresis, a solute concentration gradient for diffusiophoresis, or a temperature gradient for thermophoresis (Anderson 1989; Marbach & Bocquet 2019). The particle velocity is accordingly proportional to the gradient of the applied field, writing in the general form arXiv:1910.13901v1 [cond-mat.soft] 30 Oct 2019 vP = µP × (−∇Θ∞) (1.1) with Θ∞ the applied field infinitely far from the particle. Phoretic motion has several key characteristics. First the motion takes its origin within the interfacial diffuse layer close to the particle: typically the electric double layer for charged particles, but any other surface interaction characterized by a diffuse interface of finite thickness. Within this layer the fluid is displaced relatively to the particle due e.g. to electro-osmotic or diffusio-osmotic transport; see Fig. 1 for an illustration (Derjaguin 1987; Anderson 1989). Second, motion of the particle is force-free, i.e. the global force on the particle is zero, the particle moves at a steady velocity. This can be understood in simple terms for example for electrophoresis: the cloud of counter-ions around the particle experiences a force due to the electric field which is opposite to that applied directly to the particle, so that the † Email address for correspondence: [email protected] 2 S. Marbach, H. Yoshida and L. 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A surface (gray) is in contact with a gradient of solute (red particles). Here the particles absorb on the surface creating a pressure in the fluid (represented by yellow arrows). This pressure build-up is stronger where the concentration is highest, and induces a hydrodynamic flow vDO from the high concentration side to the low concentration side. (b) If this phenomenon occurs at the surface of a particle, the diffusio-osmotic flow will induce motion of the particle at a certain speed vDP in the opposite direction. This is called diffusiophoresis. total force acting on the system of the particle and its ionic diffuse layer experiences a vanishing total force. Both electro- and diffusio- phoresis and correspondingly electro- and diffusio-osmosis can all be interpreted as a single osmotic phenomena, since the two are related via a unique driving field, the electro-chemical potential (Marbach & Bocquet 2019). Interestingly these phenomena have gained renewed interest over the last two decades, in particular thanks to the development of microfluidic technologies which allow for an exquisite control of the physical conditions of the experiments, electric fields or concentration gradients. However, in contrast to electrophoresis, diffusiophoresis has been much less investigated since the pioneering work of Anderson and Prieve. Its amazing consequences in a broad variety of fields have only started to emerge, see Marbach & Bocquet (2019) for a review and Abécassis et al. (2008); Palacci et al. (2010, 2012); Velegol et al. (2016); Möller et al. (2017); Shin et al. (2018) as a few examples of applications. The diffusiophoretic velocity of a particle under a (dilute) solute gradient writes vDP = µDP × (−kBT rc∞) (1.2) where µDP is the diffusiophoretic mobility, rc∞ is the solute gradient far from the sphere, kB Boltzmann’s constant and T temperature. For example, for a solute interacting with a spherical particle via a potential U(z), where z is the distance to the particle surface, the diffusiophoretic mobility writes (Anderson