Theory of Sorption Hysteresis in Nanoporous Solids: I. Snap-Through Instabilities
TTheory of Sorption Hysteresis in Nanoporous Solids:I. Snap-Through Instabilities
Zdenˇek P. Baˇzant and Martin Z. Bazant August 22, 2011
Abstract:
The sorption-desorption hysteresis observed in many nanoporous solids, at vapor pres-sures low enough for the the liquid (capillary) phase of the adsorbate to be absent, has long beenvaguely attributed to changes in the nanopore structure, but no mathematically consistent expla-nation has been presented. The present work takes an analytical approach to account for discretemolecular forces in the nanopore fluid and proposes two related mechanisms that can explain thehysteresis at low vapor pressure without assuming any change in the nanopore structure. The firstmechanism, presented in Part I, consists of a series of snap-through instabilities during the fillingor emptying of non-uniform nanopores or nanoscale asperities. The instabilities are caused by non-uniqueness in the misfit disjoining pressures engendered by a difference between the nanopore widthand an integer multiple of the thickness of a monomolecular adsorption layer. The second mecha-nism, presented in Part II, consists of molecular coalescence within a partially filled surface, nanoporeor nanopore network. This general thermodynamic instability is driven by attractive intermolecularforces within the adsorbate and forms the basis to develop a unified theory of both mechanisms. Theultimate goals of the theory are to predict the fluid transport in nanoporous solids from microscopicfirst principles, and to determine the pore size distribution and internal surface area from sorptiontests.
Introduction
The sorption isotherm, characterizing the isothermal dependence of the adsorbate mass con-tent on the relative vapor pressure at thermodynamic equilibrium, is a basic characteristic ofadsorbent porous solids. It is important for estimating the internal pore surface of hydratedPortland cement paste and other materials. It represents the essential input for solutions of thediffusion equation for drying and wetting of concrete, for calculations of the release of methanefrom coal deposits and rock masses, for the analysis of sequestration of carbon dioxide in rockformations, etc. Its measurements provide vital information for determining the internal surfaceof nanoporous solids [38, 2, 31, 3, 27, 47, e.g.].An important feature sorption experiments with water, nitrogen, alcohol, methane, carbondioxide, etc., has been a pronounced hysteresis, observed at both high and low vapor pressuresand illustrated by two classical experiments in Fig. 1c,d) [38, p. 277] and [28] (see also[29, 40, 3, 27, 47, e.g.]). For adsorbates that exist at room temperature in a liquid form, e.g.water, the room temperature hysteresis at high vapor pressures near saturation has easily beenexplained by non-uniqueness of the surfaces of capillary menisci of liquid adsorbate in largerpores (e.g., the ‘ink-bottle’ effect [21]). However, a liquid (capillary) water can exist in the McCormick Institute Professor and W.P. Murphy Professor of Civil Engineering and Materials Science,Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, Illinois 60208; [email protected](corresponding author). Associate Professor of Chemical Engineering and Mathematics, Massachusetts Institute of Technology,Cambridge MA 02139. a r X i v : . [ phy s i c s . f l u - dyn ] A ug a) (b)(c) (d)w adsorption capillarity h h0 0.5 1.0 012345 0.2 0.4 0.6 0.8 h y s t e r e s i s Γ c T
1Γ BET Isotherm, c T = 54 2yPowers &Brownyard1946 r e l . w e i g h t l o ss rel. vapor pressure (P/P S ) rel. vapor pressure (P/P S ) Fig. 1 r e l . w e i g h t l o ss i n % Feldman &Sereda19680 0.2 0.4 0.6 0.8 1.0 1.0
Figure 1: (a) Typical desorption and sorption isotherms; (b) BET isotherm; (c)-(d) Desorption andsorption isotherms measured on hardened Portland cement paste. pores only if the capillary tension under the meniscus (which is given by the Kelvin-Laplaceequation) does not exceed the tensile strength of liquid water, which is often thought to beexhausted at no less than 45% of the saturation pressure, if not much higher. Anyway, at vaporpressures less than about 80% of the saturation pressure, the liquid phase represents a smallfraction of the total evaporable water content of calcium silicate hydrates (C-S-H) [32, Fig. 3]However, the hysteresis at low vapor pressures (lower than 80% of saturation in the case ofC-S-H) has remained a perplexing and unexplained feature for over 60 years. In that case, mostor all of the adsorbate is held by surface adsorption. The gases and porous solids of interestgenerally form adsorption layers consisting of several monomolecular layers (Fig. 1b). Themulti-layer adsorption is described by BET isotherm [22] (Fig. 1a), which is reversible. Sorptionexperiments have generally been interpreted under the (tacit) hypothesis of free adsorption, i.e.,the adsorption in which the surface of the adsorption layer is exposed to gas.In nanoporous solids, though, most of the adsorbate is in the form of hindered adsorption2ayers, i.e., layers confined in the nanopores (which are sometimes defined as pores ≤ > δ between each pair. At h = 1, either all of the pore space can be filledby liquid, or an anticlastic (hyperbolic paraboloid) meniscus surface of zero total curvature r − = r − + r − = 0 and liquid pressure equal to p s can exist between each two spheres, with r = − r (where r , r = principal curvature radii). This can explain 100% differences amongequilibrium liquid contents w at h = 1 observed in some experiments. It can even be shown thatwhen both δ and r (with r = − r ) approach zero in a certain way, then also the liquid content(as a continuum) approaches 0. Thus, in theory, an arbitrarily small but nonzero equilibriumliquid content at h = 1 is possible, though extremely unlikely.This three-dimensional picture, for example, explains why (as shown in Fig. 1a by dashedlines) the non-uniqueness of sorption isotherm extends to h > h = p v /p s ( T ) = relativevapor pressure, or relative humidity in the case of water, p v = pressure of vapor or gas; p s ( T ) =saturation vapor pressure). For h >
1, or p v > p s , the total curvature of the menisci is changedfrom positive to negative, the pores contain overpressurized vapor, and the hysteresis, or non-uniqueness continues [12, 13]. This non-uniqueness and hysteresis explains why the slope of theisotherm for h > h >
1. (In theory, this nonuniqueness can extend up to the criticalpoint of water). In cements these phenomena are complicated by the fact that the chemicalreactions of hydration withdraw some water from the pores, and create self-desiccation bubbles.As a result, one practically never has concrete devoid of any vapor, even for p > p s .The consequence of the non-uniqueness is that the sorption isotherm is not a function oflocal thermodynamic variables. Instead, it is a functional of the entire previous history ofadsorbate content. Here we will show that the same functional character extends to the rangeof hindered adsorption in nanopores, consistent with the extensive experimental data that3onsistently exhibits sorption hysteresis over the entire range of relative humidities.The recent advent of molecular dynamic (MD) simulations is advancing the knowledge ofnanoporous solids and gels or colloidal systems in a profound way [37, 23, 24, 25, 34, 35, 42,36, 46]. Particularly exciting have been the new results by Rolland Pellenq and co-workers atthe Concrete Sustainability Hub in MIT led by Franz-Josef Ulm [14, 15, 16]. These researchersused numerical MD simulations to study sorption and desorption in nanopores of coal andcalcium silicate hydrates. Their MD simulations [14, Fig. 3,4] demonstrated that the fillingand emptying of pores 1 and 2 nm wide by water molecules exhibits marked hysteresis.Especially revealing is the latest paper of Pellenq et al. from MIT [15]. Simulating achain of nanopores, they computed the distributions of disjoining (or transverse) pressure andfound that it can alternate between negative (compressive) and positive (tensile), depending onthe difference of pore width from an integer multiple of the natural thickness of an adsorbedmonomolecular layer (see Figs. 4 and 11 in [15]. This discrete aspect of disjoining pressure,which cannot be captured by continuum thermodynamics, was a crucial finding of Pellenqet al. which stimulated the mathematical formulation of snap-through instabilities in Part Iof this work. Oscillations between positive and negative disjoining pressures have also beenrevealed by density-functional-theory simulations of colloidal fluids or gels in [4] (where theexcess transverse stress is called the “solvation pressure” rather than the disjoining pressure).This work is organized as follows. In Part I, we begin by summarizing the classical theoryof multilayer adsorption on free surfaces by Brunauer, Emmett and Teller (BET) [22], which iswidely used to fit experimental data, but assumes reversible adsorption without any hysteresis.We then develop a general theory of hindered adsorption in nanopores which accounts for crucialand previously neglected effects of molecular discreteness as the pore width varies. This leadsus to the first general mechanism for sorption hysteresis, snap-through instability in nonuniformpores, which is the focus of this Part I.In Part II [1], we will show that attractive forces between discrete adsorbed molecules canalso lead to sorption hysteresis by molecular coalescence in arbitrary nanopore geometries,including perfectly flat surfaces and pores. This second mechanism for hysteresis is a generalthermodynamic instability of the homogeneous adsorbate that leads to stable high-densityand low-density phases below the critical temperature. The mathematical formulation of thesecond part is thus based on non-equilibrium statistical mechanics. Similar models have beendeveloped for surface wetting by nanoscale thin films [54, 55, 56], starting with Van der Waalsover a century ago [57]. Even more relevant models, accounting for nanoscale confinement,have been developed for ion intercalation in solid nanoparticles with applications to Li-ionbatteries [51, 52, 53, 50]. In that setting, analogous phenomena of hysteresis [49] (in the batteryvoltage vs. state of charge, in the limit of zero current) and nanoparticle size dependence [48]have now been observed in experiments. These connections, which convey the remarkablegenerality of hysteresis in adsorption phenomena, will be developed more in the second partin the context of a statistical physics approach. Here, in the first part, we begin to build thetheory using more familiar models from solid mechanics and continuum thermodynamics. Continuum Thermodynamics of Hindered Adsorption in Nanopores
Free Adsorption:
When a multi-molecular adsorption layer on a solid adsorbent surface isin contact with the gaseous phase of the adsorbate, the effective thickness a of the layer is well4escribed by the BET equation [22, eq. 28]:Θ = as = Γ w Γ = 11 − h − − h + c T h , c T = c e ∆ Q a /RT (1)where T = absolute temperature; Γ w = mass of adsorbate per unit surface area; Γ = mass ofone full molecular monolayer per unit area; Θ = dimensionless surface coverage; h = relativepressure of the vapor in macropores with which the adsorbed water is in thermodynamic equi-librium; R = universal gas constant (8314 J kmole − ◦ K − ) ; c = constant depending on theentropy of adsorption; ∆ Q a = latent heat of adsorption minus latent heat of liquefaction; s =effective thickness of a monomolecular layer of the adsorbate; a = effective thickness of the freeadsorption layer (in contact with vapor; Fig. 1b). For the typical value of c T = 54, the BETisotherm is plotted in Fig. 1b, where the number of adsorbed monolayers approaches five atthe saturation pressure.Eq. 1 can be easily inverted: h = h ( a ) = A + √ A + B (2)where A = Bc T (cid:18) − s a (cid:19) , B = 1 c T − Hindered Adsorption:
Consider now a pore with planar rigid adsorbent walls parallelto coordinates x and z and a width 2 y that is smaller than the combined width 2 a of the freeadsorption layers at the opposite walls given by Eq. (1). Then the adsorbate has no surface incontact with the vapor and full free adsorption layers are prevented from building up at oppositepore walls, i.e., the adsorption is hindered and a transverse pressure, p d , called the disjoiningpressure [26], must develop. For water in highly hydrophillic C-S-H, the adsorption layers canbe up to 5 molecules thick, and so, in pores less than 10 molecules wide (2 y < . h . The adsorbent communicatesby diffusion of the adsorbate along the pore with the water vapor in an adjacent macropore.In a process in which thermodynamic equilibrium is maintained, the chemical potentials µ of the vapor and its adsorbate, representing the Gibbs’ free energy per unit mass, must remainequal. So, under isothermal conditions,d µ = ρ − a (d˜ p d + 2d p a ) / ρ − v d p v (4)Here ρ = mass density of the vapor and ρ a = average mass density of the adsorbate (whichprobably is, in the case of water, somewhere between the mass density ρ w of liquid water andice). The superior ˜ is attached to distinguish the disjoining pressure obtained by continuumanalysis from that obtained later by discrete molecular considerations (˜ p d = 0 if the nanopore isnot filled because the transverse pressure due to water vapor is negligible); p a = π a /y = in-planepressure in the adsorption layer averaged through the thickness of the hindered adsorption layer;it has the dimension of N/m , and (in contrast to stress) is taken positive for compression; π a = longitudinal spreading ’pressure’ in the adsorption half-layer of thickness y (here the term‘pressure’ is a historically rooted misnomer; its dimension is not pressure, N/m , but force perunit length, N/m); π a is superposed on the solid surface tension ga s which is generally larger inmagnitude, and so the total surface tension, γ = γ s − p a , is actually tensile [7, Fig. 2] (thus thedecrease of spreading pressure with decreasing h causes an increase of surface tension, which isone of the causes of shrinkage).Further note that if p d and p a were equal, the left-hand side would be d µ = ρ − a d p d , whichis the standard form for a bulk fluid. Also, in contrast to solid mechanics, the left-hand side of5q. (4) cannot be written as (cid:15) y d p d + 2 (cid:15) x d p a because strains (cid:15) x and (cid:15) y cannot be defined (sincethe molecules in adsorption layers migrate and the difference between p d and p a is caused bythe forces from solid adsorbent wall rather than by strains).Consider now that the ideal gas equation p v ρ − v = RT /M applies to the vapor ( M =molecular weight of the adsorbate; e.g., for water M = 18.02 kg/kmole). Upon substitutioninto Eq. (4), we have the differential equation:for h ≤ h f : ρ − a d p a = ( RT /M ) d p v /p v (5)for h > h f : ρ − a (d˜ p d + 2d p a ) / RT /M ) d p v /p v (6)where h f = value of h at which the nanopore of width 2 y gets filled, i.e., h f = h ( y ) based onEq. (2). Factors 2 and 3 do not appear for h < h f because the free adsorbed layer can expandfreely in the thickness direction. Integration of Eq. (6) under the assumption of constant ρ a yields: for h ≤ h f : p a = π a y = ρ a RTM ln h (7)for h > h f : ˜ p d + 2( p a − p af ) = 3 ρ a RTM ln hh f (8)where p af = p a ( h f ) = longitudinal pressure when the nanopore just gets filled, i.e., when a = y .It is now convenient to introduce the ratio of the increments of in-plane and disjoiningpressures, κ = d p a / d˜ p d (9)which we will call the disjoining ratio. If the adsorbate were a fluid, κ would equal 1. Sinceit is not, κ (cid:54) = 1. The role of κ is analogous to the Poisson ratio of elastic solids. A rigorouscalculation of κ would require introducing (aside from surface forces) the constitutive equationrelating p a and p d (this was done in [11], but led to a complex hypothetical model with toomany unknown parameters).We will consider κ as constant, partly for the sake of simplicity, partly because (as clarifiedlater) κ is determined by inclined forces between the pairs of adsorbate molecules (Fig. 3b,c); κ should be constant in multi-molecular layers because the orientation distribution of these forcesis probably independent of the nanopore width. Note that κ would equal 0 only if all theseintermolecular forces were either in-plane or orthogonal (Fig. 3a, as in a rectangular grid).For constant disjoining ratio κ , we may substitute p a = κ ˜ p d in Eq. (8), and we get˜ p d = ρ a κ RTM ln hh f (10)For κ = 0, this equation coincides with equation 29 in [7] but, in view of Fig. 3, a zero κ mustbe an oversimplification.According to this continuum model of hindered adsorption, which represents a minor exten-sion of [7], the sorption isotherm of the adsorbate mass as a function of vapor pressure wouldhave to be reversible. However, many classical and recent experiments [38, 28, 40, 27, 47,e.g.] as well as recent molecular simulations [14, 15, 16] show it is not. Two mutually relatedmechanisms that must cause sorption irreversibility in nanopores with fixed rigid walls will bepresented, one here in Part I, and one in Part II which follows.6 Φ dΦds0a)b) c)ssrepulsion attraction un s t r e ss e d Fig. 2F= Δs ΔsC=1 F FF<0 F=0 F>0s s s C Figure 2: (a) Interatomic pair potential; (b) the corresponding interatomic force and secant stiffness;(c) interatomic forces between opposite pore walls visualized by springs. a) b)c) d)Tension CompressionFig. 3n=1 n=2 2y2yn=2 n=3 F y Pd=F y /δ F x F x F y F x s s s s F y F y F x s Figure 3:
Various simple idealized molecular arrangements between the walls of a nanopore. echanism I: Snap-Through Instability The local transverse (or disjoining) pressure p d can be determined from the transverse stiffness C n , defined as C n = ∆ F/ ∆ s where ∆ F transverse resisting force per molecule and ∆ s = changeof spacing (or distance) between the adjacent monomolecular layers in a nanopore containing n monomolecular layers of the adsorbate. Since large changes of molecular separation areconsidered, C n varies with s and should be interpreted as the secant modulus in the force-displacement diagram (Fig. 2b). For this reason, and also because many bond forces areinclined (“lateral interactions” [17, 18, 19])rather than orthogonal with respect to the adsorption layer (shown by the bars in Fig. 3), C n is generally not the same as the second derivative d Φ / d r of interatomic potential nor thefirst derivative d F/ d s of force F (Fig. 2a,b).To estimate C n , one could consider various idealized arrangements of the adsorbate molecules(as depicted two-dimensionally for two different pore widths 2 y in Fig. 3) and thus obtainanalytical expressions for C n based on the classical mechanics of statically indeterminate elastictrusses. However, in view of all the approximations and idealizations it makes no sense to delveinto these details. Diverging Nanopore:
Consider now a wedge-shaped nanopore between two divergingplanar walls of the adsorbent (Fig. 4a), having the width of 2 y where y = kx . Here x =longitudinal coordinate (Fig. 4), k = constant (wedge inclination) and s = effective spacingof adsorbate molecules at no stress. In the third dimension, the width is considered to bealso s . The adsorbate molecules are mobile and at the wide end (or mouth) of the pore theycommunicate with an atmosphere of relative vapor pressure h in the macropores.We assume the hindered adsorbed layer to be in thermodynamic equilibrium with the vaporin an adjacent macropore. This requires equality of the chemical potentials ¯ µ per molecule (¯ µ = µ/M , the overbar being used to label a quantity per molecule). At the front of theportion of the nanopore filled by adsorbate, henceforth called the ‘filling front’ (marked bycircled 2, 3 or 4 in Fig. 4), Eq. (10) of continuum thermodynamics gives a zero transversepressure, ˜ p d = 0, and so ¯ µ = ¯ µ a = ¯ µ v .However, in the discrete treatment of individual molecules, the chemical potential can bealtered by transverse tension or compression ∆ p d (Fig. 4), which can develop at the fillingfront and act across the monomolecular layers unless the nanopore width 2 y at the filling fronthappens to be an integer multiple of the unstrained molecular spacing s . We will call ∆ p d the’misfit’ (part of) disjoining (or transverse) pressure, by analogy with the misfit strain energyfor a dislocation core in the Peierls-Nabarro model [20].The misfit pressure, which, at the filling front, represents the total transverse pressure (orstress), is determined by the average change ∆ s of spacing s between adjacent monomolecularlayers, which is ∆ s = 2 kx/n − s ( n = 1 , , ... ) (11)where n is the number of monomolecular layers across the nanopore width, and s is the naturalspacing between the adjacent monomolecular layers in free adsorption, i.e., when the transversestress vanishes (note that for the triangular arrangements in Fig. 2b,c, s is obviously less thanthe natural spacing of unstressed adsorbate molecules, shown as s in Fig. 2a). So, the forcebetween the molecules of the adjacent layers is F = C ∆ s and the strain energy of the imaginedsprings connecting the molecules is F ∆ s/ C (∆ s ) / )0 1 2xb)c)d)e) tension+ a b c d-tensiontensioncompresssion1 layer 2 layers 3 layers 4 layersfree adsorptionhindered adsorption vaporFig. 4ΔP d ΔP d ΔP d ΔP d P d =0~P d ~ extra pressure due tointeger number of layers at frontP d x x x x x 12s s
12+ - b+ - 3 4 5
Figure 4:
Filling of a continuously diverging nanopore and disjoining pressures.
9s the sum of the strain energies of the strain components. Since continuum thermodynamicsgives zero disjoining (transverse) pressure p d at the filling front, it suffices to add to C n (∆ s ) / µ a per molecule at the filling front due to longitudinal pressure p a only.So, in view of Eq. (10), the chemical potential per molecule at the filling front x fn with n monomolecular layers is ¯ µ f,n = C n (cid:32) kx f n − s (cid:33) + ¯ µ a (12)where the overbar is a label for the quantities per molecule. Since ˜ p d = 0 at the filling front x ∗ ,the only source of ¯ µ n is the longitudinal spreading pressure p a in the adsorption layer.Let us now check whether at some filling front coordinate x ∗ (Fig. 4) the diverging nanoporeis able to contain either n or n + 1 monomolecular layers with the same chemical potential permolecule. For n + 1 layers, ¯ µ f,n +1 = C n (cid:32) kxn + 1 − s (cid:33) + ¯ µ a (13)Setting ¯ µ n = ¯ µ n +1 , we may solve for x . This yields the critical coordinate and critical porewidth for which the molecules in n and n + 1 monomolecular layers have the same chemicalpotential per molecule: x ∗ f,n = √ C n + √ C n +11 n √ C n + n +1 √ C n +1 s k , y ∗ f,n = 2 kx ∗ f,n (14)So the critical relative pore width 2 y ∗ f /s at the filling front is a weighted harmonic mean of n and n + 1 (and a simple harmonic mean if C n = C n +1 ).Equality of the chemical potentials per molecule at the filling front for n and n +1 monomolec-ular layers in the same nanopore, which occurs for the pore width given by Eq. (14), impliesthat no energy needs to be supplied and none to be withdrawn when the number of monomolec-ular layers is changed between n and n + 1. So the equilibrium content of hindered adsorbate inthe nanopore for a given chemical potential of vapor is non-unique. Similar to non-uniquenessof capillary surfaces, this non-uniqueness underlies the sorption-desorption hysteresis in thenanopores. Misfit Disjoining Pressure:
In view of Eq. (12), its value corresponding to ¯ µ n for n monomolecular layers in the nanopore is p d,n = C n (cid:32) s − kx ∗ n (cid:33) + ˜ p d ( x n ) (15)where ˜ p d ( x n ), based on continuum thermodynamics, is non-zero if x n (cid:54) = x f,n . In contrast tostress, the pressure is considered as positive when compressive. Replacing n with n + 1, wefind that the disjoining pressure makes a jump when the number of monomolecular layers inthe nanopore changes from n to n + 1;∆ p d,n = p d,n +1 − p d,n = 2 kx n (cid:18) C n n − C n +1 n + 1 (cid:19) + s ( C n +1 − C n ) (16)(see Fig. 4). At the filling front, the jump is from transverse tension to compression (Fig.5c). The sudden jumps ∆ p d,n of the misfit pressures from tension to compression diminish with10ncreasing n ( n = 1 , , , ... ) as the wedge-shaped nanopore is getting wider; see Figs. 5c and4d. For n >
10, these jumps become insignificant.Note that, since the changes ∆ s of molecular distance are large, the C values depend on F or ∆ p d (Fig. 2b). So Eq. 16 is actually a nonlinear equation for ∆ p d and its numericalsolution would require iterations. But here we are aiming at conceptual explanation ratherthan numerical results. Misfit Chemical Potentials and Their Effect on Sorption Isotherm:
The variationof chemical potential at the filling front x f is shown in Fig. 5d. Since transverse tension at thefilling front gives the same chemical potential as transverse compression of equal magnitude,the misfit chemical potential, defined as the part of chemical potential due to p d at the fillingfront, varies continuously, provided the pore width varies continuously, too; see Fig. 5d. Thisis because transverse tension gives the same chemical potential as transverse compression ofequal magnitude.The total chemical potential at the filling front is obtained by adding the chemical potential¯ µ a ( x f ) obtained from continuum thermodynamics, which yields the potential variation in Fig.5e. Considering the relation of filling front coordinate x f to the adsorbate mass w shown (in asmoothed form) in Fig. 5b, and the relation h = e ( M/RT ) µ f , one can deduce the solid curve inFig. 5e representing the diagram of equilibrium states of mass content w versus relative vaporpressure h in the macropore.Why are the segments of the pressure variation in Fig. 5c linear, and why are the segmentsof the chemical potential variation in Fig. 5d,e,f parabolic? The reason is that the variationof nanopore width has been idealized as linear (and that the plots are made for constant C ).These segments take different shapes for other width variations. Sequential Snap-Throughs of Adsorbate Content:
In sorption testing and most prac-tical problems, the relative vapor pressure h is the variable that is controlled, and the adsorbatemass w is the response. Consequently, the states at the reversal points 1, 3 5, 7 of the equilib-rium diagram in Fig. 5 for the diverging nanopore are unstable. Likewise the states at points1, 3, 5, 7 in Fig. 6d for the nanopore of step-wise variable width. The loss of stability canbe evidenced by checking that the molecular potential loses positive definiteness. Fundamentalthough such checks may be, it is simpler and more intuitive to argue in terms of infinitely smalldeviations d h from the equilibrium state.Consider, e.g., that, in Fig. 5f or 6d, a sufficiently slow gradual increase of h has moved theequilibrium state from point 2 to point 3, which is a local maximum of h as a function of w . Fora further infinitesimal increase d h there is on the equilibrium diagram no longer any point closeto point 3. So, borrowing a term from structural mechanics [10], we realize that the adsorbatemass content w must dynamically ‘snap through’ at constant h along vertical line 34 to point4. After dissipating the energy released along segment 34 (the rate of which depends on thelingering times of adsorbed molecules and diffusion along the hindered adsorbed layer [11]),thermodynamic equilibrium is recovered at point 4. It is stable because a further infinitesimalincrement of d h finds, next to point 4, an equilibrium state with adsorbate content incrementedby d w .If h is increased slowly enough further, the equilibrium system will move from point 4 topoint 5 at which a local maximum of h is reached again and the loss of stability gets repeated,since a further increase d h can find equilibrium only after a dynamic snap-through to point6. Each snap-through will release some energy which must be damped and dissipated by thesystem. So the local maxima of h at points 1, 3, 5 and 7 are the critical states giving rise tothe so-called ‘snap-through instability’ [10]. The equilibrium states on curved segments 1 e )b)c)d)e)f) x f Filling frontmassw0 xat filling front x f
1f 32de c b a 6 8snap-throughs 1e3c5a7=equilibriumstateshysteresisunstableequilibrium0 754 x0 xx s 2n = 1 n = 2 n = 3contentΔμ d μ f μ a (x) by continuumthermodynamics(no misfits)Fig. 5 ΔP d (x f ), n=3 ΔP d (x) x x x at filling front compr . tension w w(h) by equilibriumthermodynamics(no misfits) ef d 12 34 b 5a 6 7c hystheresisunstable equilibrium1e3c5a7 = equilibriumstates Figure 5:
Misfit disjoining pressures and chemical potentials in a continuously diverging nanopore,with dynamic snap-throughs of adsorbate content. )b)c)d)e) w k h ge f 2dj 3 ci b 6 87a541w(h) by equilibriumthermodynamics hysteresis hh12fc45cb78=equilibrium pathunstableequilibrium(no misfits)1 2 hc4 5 6ab g3defj i12ed45b6 = equilibrium path0 2y < 5
2y = 2s x x x xxx f
2y > 4s ΔPd = 0ΔPd < 0ΔPd = 0ΔPd > 0ΔPd(x)Δμ d μ f at filling front μ a (x) by continuumthermodynamicssnap-throughscompressiontensionx 2y > 3s
2y Pa
Fig. 6
Figure 6:
Step-wise diverging nanopore and misfit disjoining pressures. c a w ( h ) is followed when h is decreased. To showit, consider point 7 in Fig. 5f or 6d as the starting point. During a slow enough decrease h , thesystem will follow the stable states along segment 76 a until a local minimum of h is reachedat point a , which is the stability limit. Indeed, if h is further decremented by d h , there is noequilibrium state near point a . So the equilibrium state a is unstable and the system will ‘snapthrough’ dynamically at constant h along path ab . At point b stable equilibrium is regainedafter sufficient time. When h is decreased further slowly enough, the equilibrium states movethrough segment b c until again a local minimum of h is reached and stability is lost at point c .Thereafter, the system ‘snaps through’ along line cd to point d , where equilibrium is regained,etc.In the diverging pore in Fig. 5, the snap-through means that when the equilibrium fillingfront reaches the critical points, x , x , or x , it will advance forward a certain distance atconstant h , as fast as diffusion along the micropore, controlled by the lingering times of theadsorbate molecules, will permit. The cross-hatched areas in between the sorption and desorp-tion isotherms, such as area 34 cd Sequential Snap-Throughs for Step-Wise Nanopore Width Variation:
The dia-grams in Fig. 5d,e,f are valid only for a micropore with continuously diverging rigid planarwalls (Fig. 5a). This is, of course, an idealization. Because of the atomistic structure of porewalls, the pore width in reality varies discontinuously, as exemplified in Fig. 6a. The chance ofa width exactly equal to an integer multiple of s is small.Consider that the jumps of nanopore width (Fig. 6a) occur at x , x , x , ... , and that at x is narrower than s , at x exactly equal to 2 s , and at x wider than 3 s . Thus the fillingfront in pore segment ( x , x ) is in transverse compression, in segment ( x , x ) at zero trans-verse pressure, and in segment ( x , x ) in transverse tension; see Fig. 6b. The correspondingstrain energies, representing the misfit chemical potential ∆¯ µ d per molecule, have a pulse-likevariation as shown in Fig. 6c. Continuum thermodynamics, which ignores the misfits, givesa monotonically rising staircase variation of the chemical potential ˜ µ a ( x ) (per unit mass) as afunction of the filling front coordinate x f , represented by path jiedhgba (Fig. 6d). Superpos-ing on this staircase the misfit chemical potential ∆¯ µ d (converted to unit mass), one gets thenon-monotonic step-wise path of equilibrium states, shown by the bold line 12ed455b6 in Fig.6d. Taking into account the dependence of the adsorbate mass w in the nanopore on the fillingfront coordinate x f , one can covert the diagram in Fig. 6d into the sorption isotherm in Fig.6e, usually plotted as w versus h . The monotonic staircase hkf ejicba would represents theequilibrium path if the misfit disjoining pressures were ignored.When the rise of h , and thus µ f , is controlled, the segments 23 and 56 in Fig. 6e are unstableand unreachable. Indeed, when h or µ f is infinitesimally increased above point 2, there is nonearby equilibrium state, and so the system will ‘snap through’ dynamically to point 3. At thatpoint, equilibrium is regained, and h and µ f can be raised again, slowly enough to maintainequilibrium, along path 345. A similar dynamic snap-through is repeated along segment 56,after which the stable segment 678 can be followed. Likewise, in the diagram of µ f versusthe filling front coordinate x f (Fig. 6d), forward snap-throughs at increasing µ f (which is amonotonically increasing function of h ) occur along segments 23 and 56.When h or µ f is decreased slowly enough from point 8, the stable equilibrium path 876 bc isfollowed until stability is lost at point c (Fig. 6e). Then the system snaps through dynamically14rom c to d , follows equilibrium path def , and snaps dynamically from f to g . Likewise, in Fig.6d, backward snap-throughs at decreasing µ f occur along segments bc and ef .Obviously, the states on segments c f e and 5 e in Fig. 6d, can neverbe reached. They represent unstable equilibrium. The shaded areas g eg and d bd representhysteresis, which leads to energy dissipation. Snap-Throughs in a System of Nanopores:
The diverging nanopore (Fig. 4, 5aand 6a) is not the only pore geometry producing sorption hysteresis. There are infinitelymany such geometries. In the simple model of discrete monolayers pursued in Part I, the onlygeometry avoiding hysteresis due to sequential snap-throughs is hypothetical—the widths of allthe nanopores would have to be exactly equal to the integer multiples of the natural spacing s of monomolecular layers in free adsorption, so as to annul the misfit pressures. Below, we willshow that if molecular coalescence is allowed in the lateral direction, then even these specialpore geometries will exhibit sorption hysteresis, and so the effect is extremely general.An essential feature of nanoporosity is that there are nanopores of many different thicknesses2 y densely distributed as shown in Fig. 7. At a given vapor pressure, all the nanopores that arenarrower than a certain width 2 y are filled by adsorbed water and the wider ones are empty,containing only vapor; see Fig. 7a,c,e.As the relative pore pressure h is increased, larger and larger pores fill up. A critical state(or a local maximum of h ) is reached for a pore width at which the misfit chemical potential∆ µ d due to misfit disjoining pressure is for n monomolecular layers equal to or larger than themisfit chemical potential for n + 1 layers. After that state, the system loses stability and regainsit only when all the nanopores up to a certain larger width get filled without increasing h . Fordecreasing h , the stability loss would occur for a different pore width.The distribution of nanopore thicknesses 2 y may be characterized by a continuous cumu-lative frequency distribution function ϕ ( y ) that represents the combined volume of all thenanopores with thicknesses < y . This case, though, is not qualitatively different from thediverging nanopore studied previously. For ϕ ( y ) ∝ ky , the nanopore system in Fig. 7 becomesmathematically equivalent to the linearly diverging nanopore studied before.The way the hysteresis in the individual nanopores gets superposed to produce a pronouncedhysteresis on the macroscale is schematically illustrated in Fig. 8. Analogy with Snap-Through Buckling of Flat Arch:
There is an instructive analogywith the snap-through buckling of elastic arches or shells under controlled load (Fig. 9 [10,p.231]. If the arch is flat enough and flexible enough not to fracture, the equilibrium diagram oftotal load p versus midspan deflection u follows the diagram sketched in Fig. 9. The segments051 and 432 consist of stable states at which the potential energy is positive definite (i.e., hasa strict local minimum). But this is not true for the equilibrium states on the segment 12, atwhich the potential energy does not have a strict local minimum.Consider that load p is increased from point 0 up to the local maximum at critical point1 (Fig. 9). If load p is increased further by an infinitesimal amount ddp , there is no nearbyequilibrium state. The arch must follow at constant load the dynamic snap-through path14, during which there is accelerated motion, with the load difference from the equilibriumcurve below being equal to the inertial force, which provides rightward acceleration. The archgains kinetic energy up to point 3, swings over (along a horizontal line), and then vibrates atconstant load about point 3 until the kinetic energy is dissipated by damping (without damping,it would vibrate indefinitely). Then, if the load is increased further, the arch moves throughstable equilibrium states on the segment 34.When the load is decreased, starting at point 4, the arch will follow the stable equilibrium15 ) 1 2 3 4 5 6b)c)d) ΔPdPd compression ΔPdΔPdΔPd = 0 ΔPd = 0 ΔPdΔPd extra pressure due tointeger number of layersn=2 layersn=3 layersn=2 layersn=1 layers P tensiontension filled part cannot1 layer1 layer 2 layers exist in equilibriumsnap-through vapor 2y
Fig. 7
Pd compression(continuum thermodynamics1 2 3 4 5 6 7 8P snap-through
Figure 7:
System of nanopores of different widths communicating through vapor phase . ig. 8 w hpore 1pore 2pore 3 Figure 8:
Superposition of hysteretic loops from different nanopores.
Fig. 9
5P P1 2 3 4deflection
Figure 9:
Analogy with snap-through of an arch.
Energy dissipated by sorption hysteresis on a full h -cycle → → (shaded area),and dissipation during mid-range reversals (a,b). states along segment 432 until a local minimum is reached at point 2. If the load is decreasedfurther by an infinitesimal amount d P , there is no equilibrium state near point 2. So the archmust snap through dynamically to point 5, the load being again balanced by inertia forces whichprovide leftward acceleration. During this snap-through the arch gains kinetic energy, swingsover ot the left of point 5 and vibrates about point 5 until the kinetic energy is dissipated bydamping. Then the load can be decreased further following the stable equilibrium states belowpoint 5.Note that even though the arch is elastic and the structure-load system is conservative,hysteresis is inevitable. During the cycle, the arch dissipates an energy equal to the cross-hatched area 51325 in Fig. 4. Energy Dissipated by Hysteresis and Material Damage:
The Gibbs free energydissipated per unit mass of the nanoporous material is d G = w d µ where d µ = RTM d ln h , whichhas in thermodynamic equilibrium the value for the adsorbate species in the vapor and for theadsorbed phases. Therefore, the free energy dissipated per unit volume of material due to thehysteresis during a complete cycle, e.g., a drying-wetting cycle of hardened cement paste, is∆ G = RTM (cid:73) w ( h ) h d h (17)Since h is in the denominator, integrability, i.e., the finiteness of ∆ G , requires that lim h → w/h =1 /h n where 0 ≤ n <
1. Graphically, ∆ G is proportional to the area between the sorption anddesorption isotherms in the diagram of w/h versus h (Fig. 10).The energy could be dissipated in two ways:1) By internal friction in the adsorbed fluid during the dynamic snap-throughs (or the18olecular coalescence phenomena discussed in Part II [1]), or2) by fracturing or plastic damage to the nanopore surfaces.However, the latter seems unlikely since it could be associated with every disjoining pressurechange and not particularly with the snap-through. The existence of the former is undeniable,and the point here is to show that the hysteresis is perfectly explicable without postulating anydamage to the nanopore surface.Anyway, the degree of material damage due to a drying-wetting cycle, if any, could bechecked by measuring the strength or the fracture energy, or both, of the material before andafter the cycle. This would have to be done slowly enough on thin enough specimens havingdrying half-times less than 1 hour ( < h in the capillary pores can be changed without creating a significant gradient of h across the specimen wall (in thicker samples, most of the material damage is done by non-uniform shrinkage stresses engendered by non-uniformity of h across the wall thickness [45]).Shrinkage and creep experiments on such specimens have been performed at Northwestern [44],but no cycles were performed and strength changes were not checked. It could also be checkedwhether the snap-throughs might be associated with the acceleration of concrete creep due tosimultaneous drying, called the drying creep (or Pickett effect). Sorption Potential:
Note that, based on the derivation of Eq. (17), it further followsthat β = RTM wh = ∂G∂h (18)In other words, the Gibbs’s free energy per unit mass of adsorbate as a function of h is apotential for the adsorbate content parameter β during a one-way change of h . Conclusions of Part I
We can summarize the findings of the first part as follows:1. One mechanism that must be causing sorption hysteresis at low vapor pressure is a seriesof snap-through instabilities causing path-dependent non-uniqueness of adsorbate contentand dynamic jumps of water content of nanopores at constant vapor pressure.2. The snap-through instabilities are a consequence of the discreteness of the adsorbate,which leads to non-uniqueness of mass content and to misfit disjoining (transverse) pres-sures due to a difference between the pore width and an integer multiple of the thicknessof a transversely unstressed monomolecular layer of the adsorbate.3. The hysteresis is explained by the fact that the snap-through instabilities for sorption anddesorption follow different paths.4. The snap-through instabilities are analogous to snap-through buckling of arches and shells,long known in structural mechanics. They cause hysteresis and energy dissipation evenwhen the arch or shell is perfectly elastic.If a quantitative version of this theory were developed, it might be possible to infer from thehysteresis the surface area and the size distribution of the nanopores filled by hindered adsor-bate. Our preliminary analysis of snap-through instabilities suggests that the key to making thisconnection is to account for inclined forces, or “lateral interactions”, in the statistical thermo-dynamics of hindered adsorption. In the Part II, we will show that attractive lateral interactions19enerally lead to sorption hysteresis in any pore geometry due to molecular coalescence of theadsorbate.
Acknowledgment:
The research was funded partly by the U.S. National Science Foundation under GrantCMS-0556323 to Northwestern University (ZPB) and Grant DMS-0948071 to MIT (MZB) and partly bythe U.S. Department of Transportation under Grant 27323 provided through the Infrastructure TechnologyInstitute of Northwestern University (ZPB). Thanks are due to Franz-Josef Ulm and Rolland J.-M. Pellenqof MIT for stimulating discussions of disjoining pressure based on MD simulations, and to Laurent Brochardand Hamlin M. Jennings for further valuable discourse.
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