How Solutes Modify the Thermodynamics and Dynamics of Filling and Emptying in Extreme Ink-Bottle Pores

Abstract

We investigate the filling and emptying of extreme ink-bottle porous mediamicrometer-scale pores connected by nanometer-scale pores-when changing the pressure of the external vapor, in a case where the pore liquid contains solutes. These phenomena are relevant in diverse contexts, such as the weathering of building materials and artwork, aerosol formation in the atmosphere, and the hydration of soils and plants. Using model systems made of vein-shaped microcavities interconnected by a mesoporous matrix, we show experimentally that the presence of nonvolatile solute shifts the condensation and evaporation transitions and in a way that is consistent with a modified Kelvin-Laplace equation that takes into account the osmotic pressure of the solution. Emptying occurs far below saturation, when the Kelvin stress, mediated by the large curvature of the liquid-vapor interfaces in the nanopores, is large enough to 1 induce spontaneous bubble nucleation in the micro-veins. Filling, on the other hand, occurs close to equilibrium (i.e., at saturation, p sat for pure water and p s < p sat for a solution), driven by the weak capillary pressure of the liquid-vapor interface in the micro-veins. Interestingly, solutes allow the system to reach situations where the vapor is supersaturated with respect to the solution (p s < p < p sat). We show that in that latter situation, a condensation layer covers the outer surface of the porous system, preventing the generation of Kelvin stresses but inducing osmotic stresses and flows that are vapor-pressure-dependent. The timescales and dynamics reflect these different driving forces: emptying proceeds through discrete, stochastic nucleation events with very fast, unsteady bubble growth associated with a poroelastic relaxation process, while filling occurs collectively in all veins of the sample through a slower steady-state process driven by a combination of osmosis and capillarity. The dynamics can however be rendered symmetrical between filling and emptying if bubbles pre-exist during emptying, a case that we explore using cycling of the vapor pressure around equilibrium.

Adsorption, Desorption, and Crystallization of Aqueous Solutions in Nanopores

“…Equilibrium is established when where μ w sol ( P , T , x ) is the chemical potential of water in solution at pressure P , temperature T , and mole fraction x and μ w vap ( p , T ) is the chemical potential of water vapor at pressure p and temperature T . By integrating the isothermal Gibbs–Duhem equation with the saturation pressure of pure water, p sat ( T ), as the reference pressure and with the chemical potential of bulk pure water and vapor (μ o ) as the reference, the chemical potential of liquid water in solution and vapor can be written as and where is the osmotic pressure of the solution (assuming v w sol ≈ v w liq ), p sol ( x ) is the vapor pressure of solution with dissolved mole fraction of solute x , and v w sol (m 3 /mol) is the molar volume of the water component of the pore solution at temperature T (assuming v w sol to be independent of pressure, i.e., considering the liquid to be incompressible). On defining the activity of water vapor as a = p / p sat (also referred to as relative humidity) and the activity of solution with dissolved mole fraction of solute x as a sol ( x ) = p sol ( x )/ p sat at equilibrium, eqs , , and yield The pressure difference between the solution and the water vapor is again mechanically balanced by the capillary pressure at the liquid–vapor meniscus as given by Young–Laplace where γ sol (N/m) is the surface tension at the solution–vapor interface, θ sol , as seen in Figure d,j, is the equilibrium angle the solution makes with the pore wall, and r p is the pore radius.…”

“…We assume that the molar volume of water is independent of the concentration of solute (i.e., v w sol ≈ v w liq ), and eq yields …”

“…Using the Kelvin–Laplace relation for pure solvent (eq ) and solution (eq ) for the case when vapor activity a < a at on the adsorption branch and a < a dt on the desorption branch, we derive a relation independent of r p and with the same assumption as made previously (i.e, the molar volume of water is independent of the concentration of solute, v w sol ≈ v w liq ), eq yields …”

“…where μ w sol (P, T, x) is the chemical potential of water in solution at pressure P, temperature T, and mole fraction x and μ w vap (p, T) is the chemical potential of water vapor at pressure p and temperature T. By integrating the isothermal Gibbs− Duhem equation with the saturation pressure of pure water, p sat (T), as the reference pressure and with the chemical potential of bulk pure water and vapor (μ o ) as the reference, the chemical potential of liquid water in solution and vapor can be written as 27 μ μ μ ), 27 p sol (x) is the vapor pressure of solution with dissolved mole fraction of solute x, and v w sol (m 3 /mol) is the molar volume of the water component of the pore solution at temperature T (assuming v w sol to be independent of pressure, i.e., considering the liquid to be incompressible). On defining the activity of water vapor as a = p/p sat (also referred to as relative humidity) and the activity of solution with dissolved mole fraction of solute x as a sol (x) = p sol (x)/p sat at equilibrium, eqs 8, 9, and 10 yield…”

Condensation‐Controlled Toposelective Vapor Deposition in Nano‐ and Microcavities: Theory, Methods, Applications, and Related Technologies

“…[ 36 ] Therefore, the surface selectivity could be affected by many factors that need to be considered when designing an advanced condensation‐based VD process. This is easy to see from the Kelvin equation, which has been corrected and expanded in various ways, including the introduction of non‐ideal behavior of fluids, [ 37 ] the contact angle between the adsorbate and adsorbent in slit‐shaped pores [ 38 ] or with tilted pore walls, [ 39 ] binary mixture setups, [ 40 ] adsorbed statistical multilayer on surfaces before the capillary filling occurs, [ 41 ] condensation in cavities between non‐contacting convex surfaces, [ 42 ] osmotic pressure caused by solutes, [ 43 ] flexural properties of meniscus inside nanopores [ 44 ] and even electric fields as a control parameter. [ 45 ] The Kelvin equation also presumes a macroscopic concept of surface tension, and it does not address adsorption fluctuations.…”