Colloids and Surfaces A: Physicochemical and Engineering Aspects
Colloid-interface interactions initiate osmotic flow dynamics
Graphical abstract
Introduction
The understanding of the transport of colloids at, or across, interfaces is still a scientific challenge meeting applications in many processes. For example, flow through semi-permeable membranes is a common process in living bodies (kidneys, membrane cells, etc.) and in industrial applications (filtration, desalting, etc.). Beyond these applications, the recent development of microfluidic experiments and of nano-scale engineered interfaces has revived the question of the role played by colloid-surface interactions on the transport at, or across, interfaces [1]. When considering the colloid transport across interfaces, the classical model for osmotic flow derives from the semi-empirical formulation of Kedem and Katchalsky [2] that considers non-equilibrium thermodynamics with the assumption of linearity between the fluxes and the driving forces. The mechanical approach (adapted from Darcy law) and the thermodynamic formulation converges to the writing of the velocity of the solvent, uw, and the colloids, uc, through the membrane [3]:Where p is the pressure, Πcc is the osmotic pressure due to the colloid-colloid interactions, σ, kw and P are respectively the reflection coefficient, water and the solute permeabilities. In this model, the membrane is considered as a discrete transition region (an active layer assumed to be completely uniform across its thickness) between two homogeneous solutions. It is therefore assumed that flows through the membrane are caused by the differences in potentials occurring across the membrane.
These equations describe the fluxes between two compartments by assuming that the drive is the difference in chemical potentials. However, one has to note that the effective forces leading to the transport are not accounted by this formalism. Early works on osmosis underline the importance played by the interaction between colloids or molecules and membrane interface on the osmosis flows. Van’t Hoff explained osmosis in terms of the work done by the rebounding molecules of a solute on a selective semipermeable membrane: “The mechanism by which, according to our present conceptions, the elastic pressure of gases is produced is essentially the same as that which gives rise to osmotic pressure in solutions. It depends, in the first case, upon the impact of the gas molecules against the wall of the vessel; in the latter, upon the impact of the molecules of the dissolved substance against the semipermeable membrane, since the molecules of the solvent, being present upon both sides of the membrane through which they pass, do not enter into consideration [4]”. Einstein [5] considered that the colloids exerted a pressure on the material at the origin of the partition: “We must assume that the suspended particles perform an irregular move (even if a very slow one) in the liquid, on account of the molecular movement of the liquid; if they are prevented from leaving the volume V* by the partition, they will exert a pressure on the partition just like molecules in solution”. Fermi [6] stated that the pressure on the side of the membrane facing the solution is increased by the impacts of the molecules of the dissolved substances, which cannot pass through the membrane.
This interest for considering the effects of interactions with the membrane has been recently discussed [7], [8], [9], [10] but there is still little knowledge on how the colloid-interface interactions play a role on the dynamic of osmotic flow and how these interactions are related to the properties of the fluid and the membrane. The aim of this paper is to put forward a model that implements the role of the colloid-membrane interactions on the dynamics of osmotic flow.
Section snippets
Theoretical background
A new model has been recently established from the momentum balance for the fluid and the colloid phase on an energy landscape [11]. The concept of energy landscapes [12] allows the mapping of the colloid-membrane interaction energy (related to the Gibbs free energy that can also be expressed per unit of volume as a pressure, Πi named interfacial pressure in the paper) for all of the spatial positions of the colloids in the vicinity or inside the membrane. This map represents the overall
Transient description of osmotic flows
The ability of the model to describe the dynamics of the osmotic flow is illustrated for two different and complementary case studies (Fig. 3):
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The Reverse Osmosis at a constant flow rate: the model dynamically describes the local concentration through the membrane, the selectivity of the membrane and the increase of the counter pressure
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The 1748 Abbé Nollet osmosis experiment [26] where two compartments with two different concentrations induce an osmotic flow that change the height of the
Discussion
The Two Fluids on Energy Landscape (TFEL) model allows the description of the transfer of solute having an interaction with its surrounding environment. The model unravels the effect of the combination of interactions between the three bodies (Fig. 1): the fluid, the solute and its environment (a membrane interface in this paper). In the case of the flow of a mixture of fluid and colloid through a membrane, the model describes the flow without applying the Kedem-Katchalsky approach as
Conclusions
A model describing the flow of a colloidal dispersion through a membrane has been developed with a two-fluid approach (to account for the colloid and the fluid transport) on an energy landscape (to represent the interfacial barrier). The model takes into account the interaction between the colloids and the membrane via an interfacial pressure. These interactions allow the description of the membrane selectivity for the colloids. Moreover, the colloid-membrane interactions enable, with a totally
Acknowledgements
The author thanks Yannick Hallez, Fabien Chauvet, Leo Garcia, Martine Meireles and Pierre Aimar for the fruitful discussions.
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