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Mesopores in porous solids can produce a pronounced sorption hysteresis at moderate and high reduced vapor pressures of the ambient gas that is condensed in the pores. Unlike to other conventional porous materials, cement pastes often behave exceptionally. The water sorption hysteresis frequently persists at very low humidity. This hysteresis is reflected in a corresponding hysteresis loop of the solid skeleton volume. We discuss a theoretical model based on the strong compression force exerted by a condensate on the walls of narrow slit pores embedded in an elastic solid. This compression force is shown to be capable of shifting walls of narrow slit pores. Humidity-dependent closing and reopening of slit pores can produce hysteresis loops even at low humidity.

Hardening of cement pastes results from a hydration reaction that forms the C-S-H gel, where the notation C = CaO, S = SiO_{2}, and H = H_{2}O is used [

A comparison of different cement pastes with low and high volume fractions of small pores suggests that sorption hysteresis at low humidity only appears in materials with a large amount of small pores. The diagrams in

Several arrangements of the C-S-H sheets in cement pastes have been proposed. Feldman and Serena [

gaps filled with one or a few layers of physically bound water [

Conventional capillary forces [

On the other hand, for reopening the pore, the elastic energy must be sufficiently high to overcome the interface energy at the solid-solid contact region of a closed slit pore. It can be shown that sorption hysteresis can appear even at low air humidity, if material parameters satisfy some conditions necessary for closing and reopening the slit pores. If these conditions are satisfied, the resulting sorption hysteresis is also reflected in a hysteresis of the volume change of the solid skeleton. In this paper we consider the possibility of sorption hysteresis resulting from closing and reopening of narrow slit pores in solid materials with relatively high Young modulus such as hardened cement pastes. Apart from the Young modulus, the tension exerted on pore walls is also high in hardened cement pastes. It will be shown that pore walls in cement pastes can be shifted in a similar way as in soft solids. We elucidate conditions for the appearance of low-humidity sorption hysteresis in cement-based materials.

However, as the elasticity theory is a macroscopic theory, and the solid is considered as a continuous medium without periodic lattice structure, the expression for the elastic energy of a closed pore configuration is also correct when b is not equal to a lattice constant. Thus, the elastic energy of the closed pore shown in _{0}), where elastic stresses diverge, is excluded, since the linear elasticity theory is not applicable there. Let E and v be the Young modulus and Poisson’s ratio, respectively. Then the elastic energy per dislocation line length L (length of the slit pore along the y-axis) embedded in a solid cylinder with radius R is [

The core radius r_{0} is generally assumed to be comparable with a lattice constant (r_{0} ; 0.3 nm). Unfortunately,

W_{el} (Equation (1)) increases logarithmically with increasing radius of a cylindrically shaped elastic solid. To avoid this divergence, we consider a circular slit pore with radius r and thickness b = H_{0}. The corresponding model in solid state physics is a closed circular dislocation line. Thus, using an approximate expression known from dislocation theory, the elastic energy required to close the pore is [

This energy may be compared with the adhesion energy of the solid-solid contact

We expect that the slit pore is open if W_{el} > W_{S}, and completely closed if W_{S} > W_{el}. Hence, equation W_{el} = W_{S} defines a threshold value b_{0} at which the pore is completely closing. Instead of b_{0}, we can also evaluate a threshold value E_{0} for Youngs’s modulus

where b = H_{0} is the width of the open pore without condensed water. If E < E_{0} the closed pore is stable, whereas for E > E_{0} this pore configuration is unstable. In the case w = 0.1 J/m^{2}, r = 10 nm, b = 1 nm, v = 0.3 and r_{0} = 0.3 nm, we obtain E_{0} = 40 GPa . If we assume that w results from van-der-Waals interactions, the value of w may be obtained from the Hamaker constant. In this case, w is estimated to vary in the range between 0.005 J/m^{2} and 0.01 J/m^{2} [_{0} is supposed to be much lower (E_{0} ; 2 GPa). Closing and reopening of narrow slit pores in hardened cement pastes could be driven by the dependence of Young’s modulus E on air humidity. The elastic modulus of Portland cement at 100% relative air humidity was found to be about twice times larger than its value at relative humidity smaller than 20%. Hence, narrow slit pores with appropriate aspect ratios b/2r could be open at high air humidity and closed at low air humidity.

Equation (2) can be generalized to the case of a slit pore that contains one or a few water layers. Micropores with pore wall distances H lower than one nanometer are ubiquitous in cement-based materials. Let H_{0} be the width of the force-free open slit pore which contains no condensed water. The wall distance adapts to the layered water which produces a compression force. When a pore is not completely closing and the slit still contains one or a few water layers, the wall shift b is smaller than H_{0}. Thus, inserting b = H_{0} − H into Equation (2), we obtain the elastic energy for the partially compressed circular slit pore

The evaluation of the threshold value of Young’s modulus E_{0} (Equation (3)) for closing and reopening a slit pore neglects the tension of condensates on pore walls. In a previous paper [

However, Monte-Carlo simulations suggest that the tension exerted by adsorbates on the walls of micropores could be much larger than predicted by the classical theory of capillarity. A theoretical description of the wall tension for slit pores can start from the grand potential W that includes contributions resulting from the interaction between adsorbate molecules and substrate-adsorbate interactions. In the present case, where the pore can change its width, we add the elastic energy contribution W_{el} produced by the deformation of the solid skeleton. Let us first consider a closed circular pore as a reference state. According to Equation (4), the elastic energy per unit area is

where the factor k is defined as

If the pore opens and the distance between the walls increases, work must be done to overcome the attractive forces between the pore walls. The work per unit area w(H) may be evaluated by Monte-Carlo simulations of a grand canonical ensemble [

Mechanical stability of the open pore configuration requires that

where p_{0} = H_{0}k is a constant and

is the tension of the condensate exerted on the pore walls. Obviously, the slit pore is completely closed if p(H) ≤ −p_{0}. Let us assume that the pore is partially open. The condition for a minimum of the grand potential necessary for the stability of the open pore,

If more than one minimum of

Let us first discuss the mechanism of pore closing and reopening in the framework of the conventional macroscopic theory of capillarity. This macroscopic theory could be applied to porous soft solids. In a previous paper we have shown that empty slit pores embedded in a soft elastic solid are not stable if the wall distance H_{0} is lower than a critical value [_{0} is sufficiently small, and the unstable empty state of the pore can be disregarded.

Broad slits of soft solids can be closed by the compression force exerted by a condensate. In this case, the macroscopic classical theory of capillarity is appropriate to describe hysteresis. Starting from Equation (7), the contribution w(H) to the grand potential is [

where

is the condensate pressure. Here, r denotes the number density of the condensate, k the Boltzmann constant, T the temperature and m the relative humidity of the water vapor, which is in thermodynamic equilibrium with the condensate in the slit pore. Obviously, as long as the pore is completely filled with condensate, the condensate

pressure p is independent of H , i.e. we have

where

In _{0} is plotted versus the reduced condensate tension p/p_{0}. The pore is open and filled with condensate if

For considering the hysteresis in more detail, it is useful to introduce the notation

configuration is absolutely stable

satisfied, the pore is open. Increasing the magnitude of pore pressure P along the path a ® b the reduced pore thickness gradually decreases to zero (Equation (8)). The condition of mechanical stability (Equation (10)) is always satisfied. At point b, where H is equal to zero and the pore is closed, the contribution of the surface energy term w (first constant term 1 in Equation (13)) disappears in the thermodynamic potential

pore remains closed even after a slight increase of P. If the pore partially reopened, the reduction of elastic energy would be lower than the expenditure of surface energy w. Thus, the pore remains closed along the line e ® c. However, if P is larger than

Thus, the reduced pore width H/H_{0} switches from zero to

Equation (11) for potential w(H) provides us only a crude macroscopic description. Broad pores embedded in rigid materials with Young’s modulus larger than a few GPa do not close when the relatively weak pore tension evaluated by Equation (12) is exerted. In rigid solids such as hardened cement pastes broad slits with wall distances exceeding a nanometer cannot close. The evaluation of the strong condensate tension p(H) in narrow pores requires the application of a microscopic theory. A recent neutron scattering study confirmed that the distance H between hydrated calcium silica layers of the lamellar C-S-H structure increases with increasing water content [

Unfortunately, in [_{1} and H = H_{2} the slopes _{1} and H > H_{2} . A pore width in the range H_{1} < H < H_{2} is mechanically unstable _{1} and H = H_{2} of stable pore width regions are crossed, the pore wall distance H must jump towards a stable state. Usually, discontinuous transitions are accompanied with hysteresis. Thus, we expect hysteresis if the pore tension p(H) varies and crosses the stability limits p(H_{1}) and p(H_{2}) . A complete elucidation of the hysteresis behavior of a pore requires a detailed knowledge of the function p(H) in dependence on relative humidity m. Let us estimate the range of the hysteresis region for the condensate tension data p(H) simulated by Bonnaud et al. [_{0} = 1 nm . Assuming reasonable values for the Young modulus (E = 40 GPa ), aspect ratio of the slit

(2r/H_{0} = 10 ), cut-off radius ( r_{0} = 0.3 nm), and Poisson’s ratio (0.3), we obtain k = 5 GPa/nm . The condition

Sorption hysteresis at moderate and low humidity, which is often observed in cement based materials, could be explained by strong cohesion forces which attract the walls of narrow slit pores. The theoretical model presented here suggests a possible mechanism that can lead to a sorption hysteresis loop accompanied with a related loop of the pore volume change. Evaluating the elastic energy of closed slit pores, the attraction force between pore walls was found to be sufficiently large to close narrow slit pores. Monte-Carlo simulations and experimental results suggest that the attraction force between pore walls is substantially stronger than the conventional capillary tension predicted by the macroscopic theory of capillarity [

According to the elasticity theory, closing and reopening of slit pores depends on the Young modulus E, the slit thickness H_{0}, the aspect ratio 2r/H_{0} of the pore and the adhesion energy w of the solid surfaces. If a parameter

However, apart from changing P , a closed slit pore can also reopen when the stiffness of the solid skeleton is increasing. Actually, the modulus of elasticity at the saturation point of water vapor (m = 1 ) was found to be about twice as high as its value at m = 0.2 . Thus, micropores can reopen when the stiffness of the solid skeleton increases with increasing relative humidity. Hardened cement pastes have a relatively large Young modulus. We expect that only narrow slit pores of these rigid materials can deform remarkable and produce hysteresis loops at low humidity. According to the microscopic description of the condensate-induced pore compression, a necessary condition for hysteresis is that the condensate tension p(H) varies and crosses stability limits H_{1} and H_{2} (

It should be noted, however, that there are also other attempts to explain low-humidity sorption hysteresis without changing the pore volume in cement pastes [

Financial support by the German Research Foundation DFG (grant MO 600/9-1) is gratefully acknowledged.

PeterSchiller,MircoWahab,ThomasBier,SandraWaida,Hans-JörgMögel, (2016) Sorption Hysteresis of Hardened Cement Pastes. Journal of Materials Science and Chemical Engineering,04,40-48. doi: 10.4236/msce.2016.42005