Phoretic effects at the nanoscale
- Jure Dobnikar (Institute of Physics, Chinese Academy of Sciences and University of Cambridge, Department of Chemistry, United Kingdom)
- Benjamin Rotenberg (CNRS and Sorbonne Université, France)
- Daan Frenkel (University of Cambridge, United Kingdom)
- Emmanuel Trizac (Laboratoire de Physique Theorique et Modeles Statistiques, France)
Electrophoretic phenomena are relatively well understood. However, progress in numerical modeling of other phoretic transport phenomena is hampered by the lack of a well-defined statistical mechanical formulation. The standard description of transport induced by thermodynamic gradients is based on Onsager's macroscopic thermodynamics of irreversible processes. In the linear response regime, the matrix which couples the chemical potential gradients to the fluxes of each species involves coefficients that are related to equilibrium fluctuations of microscopic quantities via Kubo relations . This connection between equilibrium and non-equilibrium situations is exploited in simulations at various levels to determine transport coefficients of all kinds, such as viscosities or electrical conductivities .
Thermal conductivities can be determined in molecular simulations from both equilibrium simulations or under non-equilibrium conditions [3,4]. Classical Green-Kubo expressions for the thermal conductivity exist for systems interacting through pair-additive forces. However, similar expressions are lacking in the case of systems with many-body interactions (see, however 
Care must be taken when using polarizable force fields to correctly account for the contribution of the "electronic" degrees of freedom . Similarly, with coarse-grained and implicit-solvent models one should properly account for the contribution of the degrees of freedom that have been averaged out to the transport coefficient . In addition, great care must be taken to account for finite size effects . The non-equilibrium approach seems in principle simpler to deal with complex systems, in particular to determine Soret coefficients. Several strategies have been developed to simulate such situations, including simulations in which the system explicitly includes two regions associated with two different thermostats  or in which a constant heat-flux is imposed throughout the system . Some technical difficulties associated with such simulations include heat exchange with the thermostat or the possibility to use periodic boundary conditions or, in the case of confined fluids, dissipation at the solid boundaries . The Soret coefficients for mixtures have also been investigated by molecular simulations, using the heat exchange method [11,12], by which it was possible to identify a pure chemical contribution to this effect . The equilibrium or non-equilibrium approaches have already been exploited to determine the thermal conductivity in a variety of systems such as simple fluids , alkane mixtures , electrolyte solutions , molten salts ...
The microscopic simulation of osmotic pressure and osmotic flows has also attracted little attention compared to the high importance of this phenomenon. The osmotic coefficients of solutions, e.g. electrolytes, can be obtained from direct molecular simulations  or, within a coarse-graining strategy whereby effective potentials are first determined from molecular simulations [18,19]. Several studies of osmosis or reverse osmosis through membranes have been reported, such as model membranes , ZIFs  or functionalized carbon nanotubes [22,23]. Such simulations involve systems with explicit reservoirs of different compositions and under non-equilibrium conditions. The choice of proper boundary conditions in the direction of the flow is not without its difficulties, it is not always possible to maintain a steady state flow. The continuous description of osmotic flow through nanopores or of diffusiophoresis involves the pressure gradient along a wall induced by a change in the solute profile in the direction perpendicular to the wall. Microscopic simulations should be able to confirm this mechanism, using improved algorithms for the computation of local pressure .
Most of the modelling of thermophoresis and diffusio-osmosis of colloidal suspensions, as well as the interpretation of experiments of osmosis and reverse osmosis through nanopores is based on continuous approaches (e.g. recent experiments have measured electrolyte solution transport in single nanotubes under the effect of pressure, potential and salt concentration gradients , finding that osmotically induced electric currents exceeded by two order of magnitude their pressure driven counterpart, and these results were discussed in the framework of continuous theories). Diffusio-osmosis can also be modelled at this level, e.g. to account for the effect of hydrodynamic slip [26,27]. Recently, mesoscale simulations (SRD) were used to model and quantify the thermophoretic behaviour of a colloid  and a Lattice-Boltzmann scheme has been proposed to capture osmosis in addition to electro-osmotic effects . The use of molecular simulation to improve simple models has been much less explored in the context of thermal or osmotic transport .
The present proposal focuses on thermal and osmotic effects, but these are not the only types of thermodynamic gradients relevant and challenging to simulate. Recent simulation schemes have been proposed to model chemically reacting systems such as cement . In that case, the reactions play an import role in the final structure of the material and consequently govern the mechanical properties. The possibility of temperature induced phase separation also raises interesting questions in the context of thermal gradients . Another important class of systems which has recently received a lot of attention from the modelling community is active systems, where the underlying fluid generates motion without external forces, as a result of internal driving mechanisms with free energy taken up and consumed at the level of each particle . Finally, it will be interesting to discuss the connection with similar driving mechanisms , with emphasis on multiscale strategies that have been successfully developed in recent years [35,36].
The statistical mechanical framework that has been developed to describe bulk transport phenomena cannot be simply extended to phoretic and related interfacial transport phenomena. Phoretic motion is generated under the gradient of thermodynamic forces that cannot induce bulk flow: electro-, thermo-, diffusio-, osmo-phoresis, etc. We use the term ‘osmosis’ to denote the corresponding induced interfacial flows: electro-, thermo-, diffusion-osmosis, etc. All these phenomena originate in forces that act on the fluid in the first few nanometres adjacent to a solid surface and play a key role in nano-scale transport because the hydrodynamic mass transport scales with the fourth power of the tube diameter, whilst phoretic fluxes scale with the square. Hence, for narrow channels, phoresis tends to be the dominant mass-transport mechanism.
As nano-scale transport becomes experimentally more important, it clearly is crucial to be able to predict phoretic transport. It is only in recent years that the need to develop the theoretical background and numerical tools to model phoresis has been recognized. Modelling challenges are related to the fact that the non-equilibrium physics at the nanoscale is different from its macroscopic counterpart: gradients are larger, confinement plays a dominant role and fluctuations are enhanced. At the molecular level, flow properties in strongly confined fluids deviate from continuum hydrodynamics predictions due to the granularity of the fluid components (e.g. structuring of water at surfaces).
For many years, the experimental tools to probe phoretic phenomena on the nano-scale were limited and phoretic effects, although known for over a century, remained poorly understood. We are now entering an era when, finally, experimental methods allow probing the behaviour and transport of fluids down to the nanoscale and provide a truly molecular view on phoretic transport. There have also been rapid advances in numerical and analytical tools. In short: after more than a century, a quantitative atomistic understanding of phoretic phenomena is within reach.
We propose to organize a CECAM flagship workshop focused on understanding the physical origins of the phoretic phenomena and on predicting the flow in nanoconfinement in diverse experimental applications such as water filtration, blue energy, oil recovery and protein characterization. We will build upon previous successful workshops, which explored the fundamentals of transport at the nanoscale and the issue of finite size scaling in simulations (Zaragoza 2015, Paris 2017). In this workshop, we aim at bringing together leading scientists working on the fluid flow in nanoconfinement and in thermodynamic gradients with theoretical, computational and experimental background, as well as industry representatives. In such multidisciplinary setting, we intend to discuss both, the open fundamental issues and the novel emerging technologies based on exploiting phoretic phenomena.
Objectives: The first objective of the meeting is to continue the discussion initiated in previous workshops on the challenges related to the simulation of nano-confined systems under thermodynamic gradients. Due to the rapid progress in the field, such a workshop is timely and necessary. Moreover, we wish to expand the scope to discuss both, the progress in fundamental understanding and designing novel technologies based on phoretic effects.
Knowledge transfer: As argued above, numerical modeling of phoretic phenomena has direct relevance for applications. In fact, a recently funded EU FET (`NANOPHLOW’) focuses specifically on the practical applications of phoretic phenomena. In this collaboration, numerical modeling will play a key role. For this reason, the NANOPHLOW FET has direct interest in the proposed workshop and will offer financial support to allow the members of its team to attend.
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