Correction: Modeling pore wetting in direct contact membrane distillation—effect of interfacial capillary pressure

Membrane distillation (MD) is a thermally driven separation process utilizing microporous membranes and operating on the principle of liquid-vapor equilibrium. In this process, only the volatile component (typically water) of the feed solution evaporates at the pore inlet, transfers through the pore, and exits from the pore outlet in either vapor or condensed form. The membrane material must be hydrophobic to prevent liquid water from entering the pore.MD finds applications in the desalination of seawater and brackish water and treating concentrated brine from the reverse osmosis (RO) process (Rácz et al., 2014;Ibrar et al., 2022). Despite its impressive performance, commercialization faces challenges due to pore wetting, causing a significant decrease in MD flux and selectivity (Peña et al., 1993;Alklaibi and Lior, 2005;Gryta, 2005;2007;Karakulski and Gryta, 2005;Peng et al., 2005;Tun et al., 2005;He et al., 2008;Qtaishat et al., 2009;Pangarkar et al., 2011;Camacho et al., 2013;Guillen-Burrieza et al., 2013;Peng et al., 2013;Saffarini et al., 2013;Rezaei and Samhaber, 2016).Efforts have been made to mitigate MD pore wetting, including methods like liquid entry pressure (LEP) evaluation, introducing air bubbles into the feed solution, and dewetting the pores for regeneration and reuse (Baghbanzadeh et al., 2016;Warsinger et al., 2017;Ibrar et al., 2022;Hou et al., 2023). One of the most useful methods to evaluate the membrane's resistance against pore wetting is LEP, which is related to the contact angle and pore geometry (Rácz et al., 2014;Yazgan-Birgi et al., 2018). New devices have been designed and constructed to introduce air bubbles into the feed solution (Rezaei et al., 2018), and the pores have been dewetted to regenerate and reuse the membrane (Shin et al., 2016;Warsinger et al., 2017).The principle of LEP is based on the following Laplace equation:Δp 2σ cos θ r , (1a)where Δp is the pressure required to make liquid (usually water) enter into a cylindrical pore of radius r, σ is the surface tension of water, and θ is the contact angle (CA). Note that θ is usually measured on a flat surface of the material of which the membrane is made and considered intrinsic to the material. The contact angle in MD is integral to understanding surface properties and wetting behavior and is closely linked to thermodynamics. Surface energy, a key thermodynamic concept, delineates the energy at interfaces between phases. In MD, the contact angle, a representation of equilibrium between cohesive and adhesive forces, is mathematically expressed by the Young-Laplace equation, connecting the contact angle (θ) with surface tensions (γSL, γSG, and γLG). Hydrophobic behavior, characterized by contact angles exceeding 90 °, implies reduced wetting, while contact angles below 90 °indicate hydrophilic behavior, signaling increased wetting. This alignment with thermodynamics underscores the tendency of systems to seek lower energy states. Hydrophobic surfaces minimize solid-liquid interfacial energy, while hydrophilic surfaces minimize liquid-gas interfacial energy. In DCMD, thermodynamics governs the vapor-liquid equilibrium. The contact angle influences membrane surface wetting, impacting mass transfer and overall MD performance. Designing and optimizing MD systems for efficiency hinges on thermodynamic principles.Consideration of small capillaries introduces confinement effects that alter water behavior. While thermodynamics still governs wetting, capillary size, roughness, and confinement modify equilibrium conditions. The meniscus formed in capillaries may deviate from the flat surface scenario due to these effects. The Young-Laplace equation (ΔP = γLG/R + γLG/R s -γSLG/R l ) elucidates the equilibrium of forces at a curved liquid interface. Regarding changes in the meniscus at high pressure in trapped air, an increase in pressure (ΔP) impacts the curvature of the liquid-gas interface (ΔR), potentially altering meniscus shape. The specific impact depends on factors like material interfaces, trapped air characteristics, and system geometry, emphasizing the need for experimental validation to comprehensively understand these interactions.In DCMD, a capillary is in contact with the feed and permeate water stream at the pore entrance and exit, respectively, and gas is trapped in between. When a capillary made of hydrophobic material is placed between two water phases, both at room temperature (see Figure 1A), the meniscus formed at the pore entrance is convex rightward, and the meniscus at the pore exit is convex leftward (Ashoor et al., 2016). The liquid phase pressure is slightly higher than the gas phase pressure to counterbalance the capillary pressure. If the temperature of the feed water is gradually increased, the gas phase pressure near the pore entrance will increase due to the evaporation of water, and it may surpass that of the liquid pressure when the feed water temperature is high enough. Then, in order to counterbalance the pressure difference, the meniscus at the pore entrance should change to concave leftward (Figure 1B); otherwise, the gas would appear in the feed water as gas bubbles. Thus, it is possible for the meniscus to change from the convex right (Figure 1A) to the concave left (Figure 1B), particularly at the pore entrance (Biswas and Kartha, 2019). This effect is negligible at the pore exit because the permeate stream is maintained at room temperature.Based on this conceptual experiment, the discussions in this work use contact angles below 90 o in the pore, in most cases, which allows drawing water into the capillary pore at the feed side of the pore, even when the pore is made of hydrophobic material.Indeed, Gryta (2007) reported the possibility of partial pore wetting based on experiments conducted using hydrophobic MD membranes. Gryta's comprehensive investigation identified a spectrum of pore-wetting phenomena encompassing four distinct categories:1) Non-wetted: The entire membrane pore is filled with gas/vapor. 2) Surface-wetted: The pore is partially filled with liquid. A gas/ vapor layer remains between the liquid layers at the entrance and exit of the pore. 3) Partial-wetted: As pore wetting proceeds, some pores are completely filled with liquid. 4) Wetted: The pore is completely filled with liquid, and the feed solution leaks to the permeate.Confirmation of the concept of partial pore wetting is supported by Gryta's work, where SEM/EDX analysis revealed concentration profiles of magnesium and calcium within the membrane pore (Gryta, 2007). Zhu et al. (2015) also observed the partial pore wetting of the PVA/PVDF composite hollow fiber membrane used for DCMD by applying SEM/EDX.In wetting experiments involving PVDF, understanding channel geometry is paramount for grasping the interaction dynamics between liquids and PVDF membranes, as well as the influence of different geometrical features on wetting behavior. Research on PVDF hollow fiber membranes immersed in various solutions has yielded valuable insights into wetting behavior (Ritter, 2022). Moreover, investigations into the impact of channel wettability and geometry on water plug wetting underscore the importance of these factors in wetting phenomena (Pfeiffer et al., 2017). Consequently, examining the channel geometry of PVDF in wetting experiments becomes crucial for comprehending liquid-membrane interactions, understanding the influence of geometrical features on wetting behavior, and discerning how membrane properties are influenced by channel geometry.Parameter screening studies on PVDF/PVP multi-channel capillary membranes further highlight the significance of channel geometry in shaping membrane properties and performance. Essential factors such as PVDF content, PVP molecular weight, pore size, and surface roughness play pivotal roles in determining membrane characteristics and behavior in wetting experiments (Back et al., 2019). In summary, when discussing wetting phenomena, it is imperative to consider parameters such as feed salinity, feed cross velocity, and channel geometry, as they significantly impact how liquids spread on a solid substrate. is to that this introduces an the of pores as cylindrical This a representation observed in and in the et al. made a of pore wetting of membrane distillation also using and two pore wetting the of membrane s ) and 2) the of liquid in the pore pore and the to a PVDF and that the of partial pore the of pore wetting in hydrophobic membranes is to the of the pore This is by the of or hydrophilic the of a hydrophobic material to a hydrophilic through various This further the of a contact angle of than 90 the the of these a comprehensive of pore wetting based on mass and remains the work by et al. on membrane distillation a into the of pore wetting with a distinct on mass and This in to the in the of membrane and for et underscore the need for a understanding of pore-wetting and further in this crucial of membrane and of this work is to a for DCMD in which mass and transfer is particularly the influence of the capillary pressure at the liquid-gas interface. the the effects of pore contact angle in the pore, feed and permeate pressure on the of water in the pore, the temperature at the and the MD flux are The by the are further with the observed by the experimental and reported in the following are made to the The feed only in the liquid The pore is and The of the membrane material is that only the transfer in the pore is This and the following two are made to the effects of mass and transfer in the pore on the MD mass flux the effects of The into the water the pore only from the pore The enter from the pore layer resistance of the feed liquid is The liquid mass transfer the pore the The vapor mass transfer the pore the The meniscus of the liquid-vapor interface vapor pressure. transfer in the is The liquid is MD in a pore is for the at which the liquid-vapor phase remains the pore (Figure DCMD, both of the pore are in contact with liquid, and it possible that liquid from both for the in the it is that the liquid only from the pore entrance that is in contact with the feed is some to the water entry from the feed Gryta (2007) reported that water partially filled the pore from the feed side to the of in the of membrane by the and in the of the pore by DCMD by membrane using water as a and also the of on the feed side by when DCMD by membrane et al., et al., 2017). on this the is also for air membrane distillation with some changes in the the liquid phase l , the in a pore is to the l l Δp ) is the of the liquid, is the of the liquid, l is the of the liquid phase in the pore, and Δp l is the for the liquid l + is the pressure of the liquid at the pore is the pressure of the gas phase by trapped between two liquid phases, and is the capillary pressure at the interface et al., 2σ cos θ r σ and θ o ) are the liquid surface tension and contact angle, the the temperature of the liquid properties is ) ) , σ r ) r , r + and are the temperature at the pore entrance and at the liquid-gas interface. Thus, + is the liquid phase this work, the membrane pore in a of are considered for the of water at an pressure and is and the for such pores are the is used for the of water vapor in air that is trapped between two liquid , the mass flux in the gas phase is r + a r and are the gas and the temperature of the gas phase is the of the gas is the molecular of water and a are the pressure of the gas phase and the partial pressure of the air in the gas and and are the vapor pressure of water at the interface and at the pore exit, following are used in a + , is the pore and and are the feed and permeate pressure s + , s is the vapor pressure of water in the gas behavior in a + + in Membrane and and are the at the liquid-gas interface and at the pore exit for the of water vapor in air is the at an which is by + + l , is the mass flux of water through the liquid and gas l + for the vapor pressure of water, used at the liquid-gas interface and at the pore exit, the transfer in the liquid the following equation is used at the + a is the temperature in the liquid phase and is the from the pore in the l l and are and specific of the liquid, solution of the equation + and are the the at l at is the of evaporation of liquid and is the transfer of the gas phase and be from the in the specific solution of the + l l l ) because at l + l l the temperature of is considered r + r r r r ) r r l and l be by and following are used in using and the is in Figure to the properties of water, as in the following parameters used in the that l and are and of are used at both parameters on the The transfer using through the transfer , required in by , using = which is the of changes with the change of temperature and it that it the because the transfer at the liquid-gas interface by the evaporation of the liquid is than the transfer by l ) in how the pore radius l , and , while = = = = θ = °, and = The pore r, from to Figure the influence of pore radius on wetting characteristics, when the pore radius r = l that of the pore is by the liquid phase due to the capillary in such a small pore. As r to l to , by a significant in from to at the radius = the in Figure This underscores the impact of the r on Despite lower than due to significant temperature by the liquid phase in the pore, remains with This is to the effect of the decrease in the increase in l In summary, the between pore size, capillary forces, and temperature dynamics the wetting behavior within the impact of feed temperature on l , and , while = = = r = θ = °, and = Figure the effect of the impact of feed temperature ) on wetting characteristics l with , pore of the entire pore with liquid. in Figure from the increase to vapor pressure with feed temperature This behavior is to temperature at higher In Figure ) the dynamics of temperature a decrease to when is a significant to when is This the observed in the impact of temperature on the between and vapor pressure as the effect of on l , and when , , , r, and are to o , and in Figure the liquid l as the permeate temperature This be to the wettability of the membrane higher permeate the membrane becomes to reduced liquid into the the between the feed temperature and the temperature at the interface where the vapor-liquid As the temperature , This decrease in temperature is of r on l , and with parameters = = = = θ = o , and = in Membrane and and be linked to the transfer dynamics within the on mass flux temperature is also a decrease in mass flux with it is to that the impact of on mass flux is than the influence of feed temperature a increase in in a measured decrease in of In a increase in to a increase in between the effects of and that the system is to in The between temperature parameters and mass flux underscores the of the system In summary, understanding the between and mass is crucial for optimizing investigations will the these observed the effect of on l , and when , , , r, and are to o , and of on l , and with parameters = = = r = θ = o , and = of on l , and with parameters = = = r = θ = o , and = in Membrane and Figure a the l a as feed pressure ) This from various Despite higher feed pressure liquid into the pore, a in temperature at the liquid-gas interface ) capillary pressure in the gas phase the liquid the pore The in Figure underscore that the influence of on l that of This is also in the of flux of water through liquid and gas as (Figure decrease in l with becomes examining the of capillary pressure As liquid the pore the pivotal is the impact of , by the between , its increase with , to an vapor pressure The in the of in l capillary forces play a in determining l , and a is observed for an increase in , causing an in , in behavior and in a decrease in interaction the observed phenomena in the between capillary pressure in the gas phase ) and feed pressure The of liquid entry into the pore the influence of capillary forces, particularly by , the behavior of the liquid and are in comprehending this the of surface capillary and the it becomes that the increase in ) the impact of , on the in l and This is in by the where an in , , and a in l decrease in l is particularly when as this a in wetting factors related to wettability properties into play as the temperature a in within to and water wettability properties et al., in the of membrane the of membrane wetting and pore This is at due to an temperature and increased transfer the in temperature the surface tension and contact angle, further to membrane wetting (Gryta, In summary, the impact of these factors at in a in wetting characteristics l elucidates the impact of feed pressure ) on key l , and , while specific , , , r, and are to and distinct contact θ = and θ = are the pore radius has been from to in Figure to in Figure Despite the different pressure for and both contact angles A is the in the of in Figure to Figure as in an in l (Figure In the the with in both the contact angle of used in studies to different wetting properties of A of on l , and where = = r = θ = o , and = in Membrane and contact angle 90 where the surface water, while an angle below 90 wetting by water specific angles are to these distinct wetting for experimental and the of and for are to the impact of on wetting characteristics and interfacial tension in using different pressure within this how changes in pressure contact angles and interfacial insights into the behavior of with surfaces different conditions. specific pressure are to a of and understand the of comprehensively et al., 2016). This in behavior underscores the to in , a that be influenced by parameters such as pore radius and contact The between and experimental factors in Figure the dynamics the observed in l , and to a understanding of how the system to changes in feed pressure ) the effect of on l , and when parameters , , , and are to and θ is either o or o and r is it is that the decrease in has the effect as the increase in As by either an increase in or a decrease in , l an while and a the significance of a pressure ) is in the between flux and permeate pressure As permeate pressure the flux also a between these two parameters et al., This in pressure and flux which are crucial for optimizing system and efficiency in various applications like reverse gas and membrane this pressure difference, the flux of through membranes, to separation efficiency and overall system the effect of on l , and when , , , r, and are to o , and In this specific an in to a decrease in l , a in and an in this while a the between liquid l feed pressure mass flux and liquid-gas interface temperature ) is crucial in various As a decrease in l transfer with The increase in mass flux a in or mass the in that the temperature with , which is for and optimizing like in of on l , and , with parameters = = = r = and = contact angles are θ = and θ = devices et al., In this must the of transfer and against like increased energy or system et al., the effect of θ on l , and , when , , , r, and are to and Figure l as θ because water is driven into the pore as the of the membrane material In Figure the that as θ because of the decrease in temperature with the decrease in the of the liquid A decrease in l the of the liquid phase and as a θ = o , l and and it is in the θ may increase it becomes o , where l becomes and the pore is filled only with the gas becomes as high as This scenario is in Figure where l and and increase with higher θ a key is the use of θ in Figure This is by the higher required in the membrane material to the feed pressure ) and the interface even with θ exceeding 90 °, in the pore, is partial liquid to Figure the θ is this where l and are reported as and is that θ may further increase it °, to l and as revealed in the Figure is against l for two different feed = and = the for both feed and is a increase in as l a increase when l is below = of θ on l , with parameters = = = r = and = of θ on l , and , with parameters = = = r = and = in Membrane and when l = This a scenario where is liquid in the pore, and in the vapor in this work some factors that would the DCMD performance, such as the transfer of the feed and permeate layer and the of the membrane the of the with the experimental in be the of the be by the it with observed by The that the DCMD flux as r (Figure 4) 2) as (Figure 3) as (Figure Figure 4) as (Figure as θ (Figure Figure and are and have been by experiments and The by 3) and 4) because only a studies have the effect of and on the MD of are as and observed a decrease in the flux with an increase in the feed pressure and it to membrane also a increase in the flux at a high pressure by the pore et al. observed a increase in the flux of DCMD when the feed pressure ) from to for a PVDF have this by the of the molecular and the in the pore and 2) to the increase in the transfer of the layer at the feed side and the in by the change of the meniscus from convex to summary, the observed by the experiments and also observed a decrease in the flux with an increase in the feed pressure and it to membrane also a increase in the flux at a high pressure by the pore The DCMD flux of the hollow fiber membrane change with an increase in et al., The DCMD flux increased with a decrease in is the pore when the feed pressure et al., for the effect of the increase in , the the flux decrease Thus, the with experimental with the experimental et experiments et al., pressure to the hollow fiber on both that the and the in the DCMD the pressure only to the feed side the side of the hollow fiber to et al., Then, it is possible that the hollow fiber in an increase in DCMD If such an the flux decrease with an increase in , as the The a increase in flux as feed pressure a where is maintained as in Figure The concept of the liquid l ) as an of partial pore wetting. of l pore wetting. has been of l various DCMD operating to it is to that the of wetting the entire pore as l becomes on the that l with an increase in , when the pore is as as (Figure Thus, the with of the experimental l to decrease with an increase in the feed pressure when the pore is as small as l , with parameters = = = = r = and = in Membrane and (Figure would be to this l with an increase in (Figure This with 2) of the experimental a to flux and partial pore wetting in DCMD while the influence of capillary pressure at the liquid-gas interface. In this that the contact angle the pore than 90 to the high pressure in the gas phase or in at the pore The that the with an increase in the pore with higher feed with an increase in permeate and with an increase in the contact have been by the effect of feed pressure the a decrease in the as feed pressure while the permeate pressure This with experimental in the of hollow which are considered the a decrease in with feed pressure while a pressure between the feed and a that experimental these of be to the of the pore DCMD the that partial pore wetting is with an increase in feed pressure when the pore is as as which is with experimental from membranes with an pore of the an when the pore is as small as This further experimental to its to the partial pore wetting as permeate pressure a supported by experimental of pore wetting. valuable insights into the of feed and permeate pressure in the of DCMD and its insights of the dynamics within DCMD, on should experimental validation and the to membrane characteristics and operating its

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