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12 2 Handbook of Nanophysics Principles and Methods. equilibrium occurs at the saturating pressure Pv Psat For a. 5 m 20 m fi nite D if the surface tension sl of the wet solid surface see. 200 nm Insert A is lower than the one sv of the dry solid surface. the solid favors liquid condensation One should therefore. ask if the solid can successfully stabilize a liquid phase when. 4 m the vapor phase is stable in the bulk i e Pv Psat To answer. this question one must compare the grand canonical poten. tial see Insert A of two configurations the liquid fi lled. interstice which we shall call the condensed state and the. vapor fi lled interstice i e the non condensed state with. sat the chemical potential of the reservoir Figure. 12 2 Outside of coexistence i e if 0 the pressure in the. 20 m two phases is different and is given by the thermodynamic rela. tion P l Pv l v with l v the number of molecules, per unit volume in each phase As the liquid is usually much. more dense and incompressible than the vapor the pressure. difference reduces to Pv P l l l k BT ln Psat Pv if. the vapor can be considered as an ideal gas Thus the condensed. state is favored if the confi nement is smaller than the critical. b c distance Dc, FIGURE 12 1 Stiction effect due to drying in the nanofabrication of a nano. mirror array c The spacing between the lamellae is 200 nm as described in l Dc 2 sv sl 12 1. a and imaged in b If the aspect ratio is larger than a critical value the. stiffness of the lamellae becomes too small to withstand the attractive action. of the capillary forces induced by the meniscus in the drying process We The left hand side of Equation 12 1 represents the free energy. remark that in drying processes the meniscus curvature and capillary pres required to condense the unfavorable liquid state and the right. sure may be quite smaller than the equilibrium values cf Insert B but they hand side the gain in surface energy Dc is thus the criti. still have a great impact After Heilmann R et al SPIE Newsroom 2008 cal distance that balances the surface interactions and the. Insert A Surface Tension,and Contact Angle, he surface tension of a fluid interface is defined in is the potential energy for an open system Its variation is. terms of the work required to increase its area equal to the work done on the system during a transforma. tion and its value is minimal at equilibrium, F On the diagram of Insert B let us consider a horizon. Alv N l Vl N v Vv T tal translation dx of the meniscus At equilibrium the. grand canonical potential is minimum d P ldVl, Here F is the free energy of a liquid vapor system T its tem PvdVv sl sv dAsl 0 Thus Pv P l sv sl D But.

perature Alv the interface area and Nl Vl Nv Vv the num according to Laplace s law of capillarity the pressure dif. ber of molecules and the volume of each phase respectively ference Pv P l is also related to the curvature of the menis. Rowlinson and Widom 1982 For a solid surface one can cus Pv Pl lv r 2 lv cos D where is the contact angle. likewise define the difference of surface tension sl sv for We deduce the Young Dupr law of partial wetting. wet and dry surfaces in terms of the work dF required to wet. a fraction dAsl of the surface initially in the dry state lv cos sv sl valid if S sv sl lv 0 12 2. It is shown in thermodynamics that the surface tension. is a grand canonical excess potential per unit area The total The parameter S is the wetting parameter de Gennes et al. grand canonical potential of a multiphase system 2003 The situation S 0 corresponds to perfect wetting. In this case a thin liquid layer covers the solid surface see. s s lv Alv sl sv Asl Insert C,75403 C012 indd 2 4 27 2010 4 37 23 PM. Capillary Condensation in Confi ned Media 12 3, 104 mol m3 and assume a contact angle 30 In ambient. conditions with a relative humidity of Pv Psat 40 one has. r K 0 6 nm and Dc 1 nm The scale is in the nanometer range. D and increases quickly with humidity it reaches 4 nm at 80 and. 18 nm at 95 relative humidity Therefore capillary conden. sates are ubiquitous in ambient conditions in high confinement. a b situations, We see from the Laplace Kelvin equation that the pres. FIGURE 12 2 a non condensed 2A sv DAPv b condensed. sure in capillary condensates is usually very low taking the. 2A sl DAP l, example of water in ambient conditions with relative humidity. bulk interactions to determine the phase diagram of the fluid Pv Psat 40 the pressure in the condensates is P l 120 MPa. Israelachvili 1992 From the above equation it is clear that i e 1200 bar With these severe negative pressures conden. capillary condensation can occur only if the liquid wets at least sates exert strong attractive capillary forces on the surfaces to. partially the solid surfaces which they are adsorbed Thus capillary condensation is usually. In the case of partial wetting the difference between the dry associated to important mechanical aspects such as cohesion. and the wet surface tension is related to the contact angle of friction elastic instabilities and micro structures destruction. the liquid onto the solid surface see Insert A and the critical Furthermore if the liquid phase wets totally the solid surfaces. distance reduces to see Insert A the surfaces may be covered by a liquid fi lm. even in a nonconfi ned geometry see Inserts C and D In this. case the critical distance for capillary condensation can be sig. Dc 2rK cos 12 3 nificantly enhanced at low humidity In the case of water the. condensation of a liquid fi lm has important consequences on. surface chemistry as surface species can be dissolved in the liq. where r K is the Kelvin s radius associated to the undersaturation uid phase and the capillary condensation at the level of contact. see Insert B between surfaces increases solute transport and is responsible. For an estimation of the order of magnitude of the con for dissolution recrystallization processes which lead to slow. finement at which capillary condensation occurs consider temporal evolution of mechanical properties of the materials. the case of water at room temperature lv 72 mJ m2 l 5 5 cf Section 12 4. Insert B Laplace Kelvin Equation, nother way to address capillary condensation is to con lv P.

sider the coexistence of a liquid and its vapor across a Pv Pl l l kBT ln sat 12 5. curved interface Because of the Laplace law of capillar. ity the pressure in the two phases are not equal Pint Pext lv r. with r is the radius of mean curvature of the interface The pres. sure is always higher on the concave side Because of this pres. sure difference the chemical potential of coexistence is shifted. v Pv l Pl Pv lv sat, We have assumed here that the liquid is on the convex side. a configuration compatible with an undersaturation For an. ideal vapor and an uncompressible liquid We check that in a flat slit the critical confi nement and. the Kelvin s radius are related by Dc 2r Kcos The capil. P lary condensate is thus limited by a meniscus whose cur. kBT ln sat Pl l Pv 12 4, Pv vature is equal to the Kelvin s radius The Laplace Kelvin. law is however more general than Equation 12 3 and allows. from where we get the Laplace Kelvin equation for the equi to predict the critical confi nement in arbitrarily complex. librium curvature Thomson 1871 geometries,75403 C012 indd 3 4 27 2010 4 37 24 PM. 12 4 Handbook of Nanophysics Principles and Methods. Insert C Perfect Wetting,The Disjoining Pressure, hen the energy of the dry solid surface sv is thermodynamic properties of the liquid film It is minimum. larger than the sum sl lv of the solid liquid at equilibrium so that the pressure in the liquid is not the. and liquid vapor interfaces S 0 the affinity of same as in the vapor. the solid for the fluid is such that it can stabilize a liquid fi lm. of thickness e in equilibrium with an undersaturated vapor. without any confinement The existence of such wetting fi lms Pv Pl d 12 7. must be taken into account when determining the liquid. vapor equilibrium in a confined space, The pressure difference d is called the disjoining pressure.

The interface potential Wslv e and the wetting parameter. sv d sl lv are Legendre transforms of each other,Wslv e sv d sl lv e d e 12 8. In the theory of wetting liquid films are described by the For instance in the case of van der Waals forces the inter. concept of interface potential Derjaguin 1944 de Gennes face potential results from dipolar interactions going as 1 r6. 1985 The excess potential per unit area of a solid surface cov between molecules and varies as 1 e2. ered by a wetting film does not reduce to the sum sl lv of the. surface tensions a further excess must be taken into account. corresponding to the fact that the molecular interactions which Aslv Aslv. Wslv e d e, generate the surface tension do not operate over a thickness of 12 e 2 6 e 3. liquid that can be considered infinite The excess grand canoni 1 3 12 9. cal potential of the humid solid surface of area A is then sv sl lv 9 Aslv 2 3. sv sl lv Wslv e e Pl Pv 12 6 The Hamaker constant Aslv has the dimension of an energy. Israelachvili 1992 It lies typically between 10 21 and 10 18 J. where the interface potential Wslv e vanishes for a mac and has negative sign when the liquid wets the solid i e if the. roscopic fi lm The excess potential sv A describes the interface potential is positive. adsorption corresponds to capillary condensation and the. 12 1 3 Mesoporous Systems, porous volume is completely fi lled by liquid nitrogen before. Capillary condensation has been extensively studied in relation the saturating pressure is reached Th is adsorption branch. to sorption isotherms in mesoporous media i e nanomate shows usually a strong hysteresis and the capillary desorption. rials with pore sizes between 2 and 50 nm in the prospect is obtained at a lower vapor pressure than the condensation. of using those isotherms for the determination of porosity Th is feature underlines the fi rst order nature of capillary. characteristics such as the specific area and the pore size condensation It is shown in the next paragraph that for suf. distribution Figure 12 3 for instance shows a typical adsorp ficiently simple pore shapes the desorption branch is the stable. tion isotherm of nitrogen in a mesoporous silica at 77 K In a one and corresponds to the liquid vapor equilibrium through. fi rst domain of low vapor pressure the adsorption is a func curved menisci The desorption branch may be used to deter. tion of the relative vapor saturation only and corresponds mine the pore size distribution of the medium through the. to the mono and polylayer accumulation of nitrogen on the Laplace Kelvin relation using appropriate models Barret. solid walls Th is regime allows the determination of the spe Joyner Halenda Barrett et al 1951 More can be found on the. cific area for instance through the Brunauer Emmett Teller physics of phase separation in confi ned media in the review of. model Brunauer et al 1938 At a higher pressure a massive Gelb et al 1999. 75403 C012 indd 4 4 27 2010 4 37 24 PM,Capillary Condensation in Confi ned Media 12 5. Insert D Perfect Wetting,The Prewetting Transition.

and Capillary Condensation, n a situation of perfect wetting a liquid fi lm condenses d l kBT ln Psat Pv 16 S 3 9 Aslv with S the wetting. on a flat isolated solid surface if the humid solid surface parameter 12 2. tension sv is lower than the dry one In a confined geometry such as sketched in Figure 12 2. the grand canonical potential of the noncondensed state. sv sl lv Wslv e e d sv 12 10, is shifted above the prewetting transition because the solid. surface tension sv has to be replaced by the humid value sv. If the fi lm exists its thickness at equilibrium with the vapor is The modified Equation 12 1 and the Laplace Kelvin relation. implicitly determined by the analogue of the Laplace Kelvin lv r K d l give the critical distance Derjaguin and. equation 12 5 Churaev 1976,Wslv e P Wslv e,d e Pv Pl l kBT ln sat 12 11 Dc 2rK 2e 2 12 12. The difference with the partial wetting case is not simply to. The thickness e realizing the equality in relation 12 10 is. decrease the available interstice by twice the fi lm thickness. a minimum thickness for the wetting fi lm and the associ. In the case of van der Waals forces for example, ated chemical potential and vapor pressure Pv corre. spond to a prewetting transition Above the transition the Dc 2rK 3e with e Aslv 6 l 1 3 12 13. thickness of the adsorbed fi lm increases with the vapor. pressure until it reaches a macroscopic value at satura The effect of adsorbed fi lms becomes quantitatively impor. tion In the case of van der Waals wetting for instance tant for determining the critical thickness at which capillary. the vapor pressure at the prewetting transition is given by condensation occurs in situations of perfect wetting. 9 large forces represent a valuable tool to study the thermodynamic. and mechanical properties of the condensates Experimentally. the ideal geometry involves a contact with at least one curved. surface either a sphere on a plane two spheres or two crossed. A Q mmol g, cylinders so that locally the topology resumes to a sphere of.

5 radius R close to a flat Surface force apparatus SFA use mac. roscopic radius R in order to take advantage of the powerful. Derjaguin approximation which relates the interaction force. F D at distance D to the free energy per unit area or other appro. eff ect of drying aft er the nanofabrication of a system of lamellae of width and spacing of 200 nm and variable height Separating the two surfaces is oft en complicated due to the fragile nature of the microstructures In divided media capillary forces not only control the cohe sion of the media but also have dramatic infl uence on the aging properties of materials Since the condensation

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