An integrated model for adsorption-induced strain in microporous solids

Chemical Engineering Science 64 (2009) 4744 -- 4753

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An integrated model for adsorption-induced strain in microporous solids Samir H. Mushrif, Alejandro D. Rey ∗ Department of Chemical Engineering, McGill University, 3610 University Street, Montréal, Québec, Canada H3A 2B2

A R T I C L E

I N F O

Article history: Received 18 August 2008 Received in revised form 6 April 2009 Accepted 7 April 2009 Available online 17 April 2009 Dedicated to Prof. Morton M. Denn on the occasion of his 70th birthday Keywords: Adsorption Chemical potential Porous media Solid mechanics Statistical thermodynamics Strain

A B S T R A C T

Deformation of porous materials during adsorption of gases, driven by physico- or chemo-mechanical couplings, is an experimentally observed phenomenon of importance to adsorption science and engineering. Experiments show that microporous adsorbents exhibit compression and dilation at different stages of the adsorption process. A new integrated model based on the thermodynamics of porous continua (assumed to be linear, isotropic and poroelastic) and statistical thermodynamics is developed to calculate the adsorption-induced strain in a microporous adsorbent. A relationship between the strain induced in the adsorbent and the equilibrium thermodynamic properties of the adsorbed gas is established. Experimental data of CO2 adsorption-induced strain in microporous activated carbon adsorbents (Yakovlev, V.Y., Fomkin, A.A., Tvardovskii, A.V., Sinitsyn, V.A., 2005. Carbon dioxide adsorption on the microporous ACC carbon adsorbent. Russian Chemical Bulletin, International Edition 54, 1373–1377) is used to fit the model parameters and to validate the model. Assuming that the initial contraction in a microporous adsorbent is caused due to an attractive interaction between the adsorbed gas and the adsorbent, we demonstrate that there also exists a repulsive interaction amongst the adsorbed gas molecules and that this repulsive interaction can be correlated to the adsorption-induced strain. The proposed correlation can be extended to take into account the adsorbate–adsorbent attractive interaction in order to offer an undisputed and complete explanation of the adsorption-induced strain in microporous adsorbents. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Adsorption is a phenomenon in which a solid surface with unbalanced forces, when exposed to a gas, bonds (physically in physisorption or chemically in chemisorption) to the gas molecules. The solid substrate is referred to as the adsorbent whereas the gas phase is the adsorbate. The solid adsorbent is usually a porous medium and the pores are classified as micropores ( < 20 Å), mesopores (20–50 Å), and macropores ( > 50 Å) (Everett, 1972). Adsorbents like alumina, silica, activated carbon, activated carbon fibers, zeolite, to name a few, are extensively used in separation and purification processes where physisorption takes place inside the pores and adsorption is analyzed as a volume phenomenon (Do, 1998). In the chemical industry, 90% of the chemicals are manufactured using catalytic processes (Masel, 1996) where adsorption is also studied as a 2D surface phenomenon. Depending upon the pore size, the type of adsorption, and the underlying physics either approach is used to describe adsorption equilibria and kinetics. Given the environmental, industrial, and chemical importance of adsorption, extensive experimental and theoretical research has

∗ Corresponding author. Tel.: +1 514 398 4196; fax: +1 514 398 6678. E-mail address: [email protected] (A.D. Rey). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.04.014

been done in the field adsorption science and the review article by DaR browski (2001) critically summarizes the key developments. Most of these studies consider that an adsorbent is thermodynamically inert and it only contributes in inducing an external force field on the adsorbate. This assumption reduces the adsorption process to a thermodynamic phenomenon without any mechanics. However, the adsorbent may not be an inert component during the process of adsorption and this was first shown by Meehan (1927) when a dimensional change was observed in charcoal due to CO2 adsorption. Wiig and Juhola (1949) and Haines and McIntosh (1947) also observed dimensional changes in carbonaceous adsorbents upon adsorption and more recently Levine (1996), Yakovlev et al. (2003, 2005), Day et al. (2008), and Cui et al. (2007) have also shown that dimensional changes take place in porous adsorbents during gas physisorption. Deformation and surface stresses are also caused during chemisorption and those have been experimentally observed by Packard and Webb (1988), Grossmann et al. (1994, 1996), and Gsell et al. (1998), to name a few. Since an adsorbent cannot be inert during the process of adsorption, adsorption properties are bound to be affected due to the dimensional changes occurring in the solid. The degree of deformation can be as large as 50% (Reichenauer and Scherer, 2000) or as small as 0.1% (Yakovlev et al., 2005), but even a relatively small deformation can cause a substantial impact on the experimentally

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

determined equilibrium thermodynamic characteristics of the adsorption system (Yakovlev et al., 2003). For example, Kharitonov et al. (2001) showed that the differential molar isosteric heat of adsorption (CO2 -activated carbon system) increased in the initial adsorption region and the authors suggested that it was due to the contribution of energy from the adsorption deformation phenomenon. Yakovlev et al. (2005) further confirmed this fact by experimentally observing CO2 adsorption-induced strain in a similar adsorbent and they also estimated the correction to the isosteric heat of adsorption for the non-inertness of the adsorbent to be 10–15%, when the strain was not more than 0.1%. In case of CO2 and H2 S sequestration, even though the strain is small it has been reported that large stresses developed due to adsorption may significantly affect the geomechanical and permeability characteristics, thereby implicating sorption estimates (Cui et al., 2007; Day et al., 2008; Viete and Ranjith, 2006). Also in the case of chemisorption, it has been shown that not only the properties of the adsorbent but also the adsorbate properties are modified in a strained adsorbent (Packard and Webb, 1988; Gsell et al., 1998). Wu and Metiu (2000) also showed using atomistic level modeling of CO adsorption on Pd that the parameters controlling the thermodynamics of chemisorption, like binding energy and vibrational frequencies, were altered when the adsorbent was strained. Recognizing the non-inertness of an adsorbent and realizing the significance of the effect of adsorption-induced strain and stress on: (i) the adsorption characteristics, (ii) the adsorbent, and (iii) the adsorbate, theoretical studies have also been performed in this field. Dergunov et al. (2005) developed a model which took into account the adsorption deformation to calculate the potential energy of adsorption. They showed that the maximum contraction in the adsorbent corresponds to the minimum stress. Recently Pan and Connell (2007) combined Myers' solution thermodynamics approach for adsorption in micropores (Myers, 2002) and Scherer's strain model (Scherer, 1986) to calculate gas adsorption-induced swelling in coal. Serpinskii and Yakubov (1981) had developed an analytical expression between the strain and the amount of gas adsorbed using vacancy solution theory and Hooke's law, but their expression required the bulk modulus to be a function of the amount of gas adsorbed, if compression and dilation in the adsorbent were to be observed. A similar approach was also taken by Jakubov and Mainwaring (2002), where they expressed the strain as a function of the amount of gas adsorbed and the difference between the chemical potentials of the gas, adsorbed on a strained and on an unstrained adsorbent. However, prior information of strain was required to calculate the chemical potential of the adsorbate that would have been adsorbed if the solid was prevented from strain. Ravikovitch and Neimark (2006) developed a non-local density functional theory-based model to calculate the adsorption-induced strain for Kr and Xe adsorption on zeolite. They were successfully able to reproduce the experimentally observed contraction and expansion of the adsorbent. When adsorption is treated as a 2-D surface phenomenon, Ibach (2004) showed that a Maxwell type relation exists between the dependence of chemical potential of an adsorbate on surface strain and the de¨ pendence of surface stress on coverage. Weissmuller and Kramer (2005) studied the metal–electrolyte system, using a continuum description of a solid adsorbent, where they identified experimentally measurable state variables in the system and established a relation between the state variables of the surface and those of an ¨ adsorbate. Lemier and Weissmuller (2007) also calculated hydrogen adsorption-induced strain in nanocrystalline Pd using the theory of thermochemical equilibrium in solids, developed by Larche and Cahn (1973), and by expressing the adsorption-induced strain as a function of state variables, pressure and chemical potential of the ¨ adsorbate. Muller and Saúl (2004) reviewed some key theoretical

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contributions in adsorption-induced stress on a plane surface, including some of the well recognized work of Ibach (1997). In summary, the significant facts in the adsorption-induced stress and strain literature lead to the following observations: (i) experimental investigations clearly show the existence and implications of adsorption-induced strain and stress (Meehan, 1927; Haines and McIntosh, 1947; Wiig and Juhola, 1949; Levine, 1996; Kharitonov et al., 2001; Yakovlev et al., 2003, 2005; Viete and Ranjith, 2006; Cui et al., 2007; Day et al., 2008), (ii) theoretical studies are needed to better understand the phenomenon, (iii) some fundamentally significant research, using thermodynamic and atomistic approaches, has been done to study the effects of stress and strain on the physics of surface ¨ adsorption (Ibach, 1997, 2004; Wu and Metiu, 2000; Weissmuller ¨ 2007), and (iv) a betand Kramer, 2005; Lemier and Weissmuller, ter understanding of the adsorption-induced strain, particularly in microporous adsorbents like activated carbon and zeolites, where an adsorbent first undergoes contraction followed by an expansion, needs further research in this area. Hence, this paper focuses on the adsorption-induced strain in microporous adsorbents. Two previous significant and very valuable contributions in this problem are: (i) the recent work of Ravikovitch and Neimark (2006) using non-local density functional theory approach; however, in this work a simple relation between the adsorption-induced strain and an equilibrium adsorption property was not established and the strain was assumed to be entirely due to the deformation of pore space, thereby neglecting the change in the volume of the solid matrix (a solid adsorbent is composed of pore space and solid matrix, c.f. Fig. 2). (ii) Jakubov and Mainwaring (2002) developed a relation between the adsorptioninduced strain and the difference in the chemical potentials of an adsorbate, when adsorbed on a strained and on an unstrained adsorbent and they make a similar assumption of strain in pore space only. However, when they calculate the difference in the chemical potentials using the difference in adsorption isotherms, on a strained and on an unstrained adsorbent, the magnitude of the difference in the isotherms appears to be too large (up to 50%) for the observed strain (  0.05%). In the present paper: (i) we develop a simple relationship between the adsorption-induced strain and an equilibrium adsorption property by (ii) taking into account the strain in the solid matrix and in the pore space and (iii) we show that this relationship can be used to predict the adsorption-induced strain in microporous adsorbents, and (iv) can provide a molecular level explanation for the adsorption-induced strain which to the best of our knowledge has not been previously done. A novel feature of this model is the integration of adsorbent mechanics with statistical thermodynamics. The organization of this paper is as follows. Section 2.1 presents the key equations of the mechanics and thermodynamics of porous adsorbents (Coussy, 2004) and the method to calculate the difference between the chemical potentials of the gas, adsorbed on a strained and on an unstrained adsorbent. The statistical mechanical model for the chemical potential difference is described in Sections 2.2 and 2.3 describes a method to predict the adsorption-induced strain by combining the approaches in Sections 2.1 and 2.2. Section 3 discusses and validates the results with previously presented experimental data (Yakovlev et al., 2005). Section 4 presents the conclusions.

2. Model development Microporous adsorbents like activated carbon and zeolites exhibit a typical deformation behavior where the adsorbent undergoes contraction in the initial stage of adsorption and later it expands. An illustration of this behavior is shown in Fig. 1, where the adsorptioninduced strain is plotted as a function of amount of gas adsorbed. As mentioned above, the model is based on the integration of the

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S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

dissipation related to the skeleton and (iii) the deformation in the skeleton is small, the differential Helmholtz free energy density can be written as (Coussy, 2004) ds = ij dij + P d − Ss dT

(1)

where P is the pressure,  is the porosity, T is the temperature, Ss is the entropy, ij are the linearized strain components and ij are Cauchy stress components. Porosity  is defined as the ratio of pore volume (V) of the connected pore space to the undeformed skeleton volume 0 . Using a Legendre transformation, the state variables can be changed if Eq. (1) is expressed in terms of free energy Gs as Gs = s − P 

(2)

Differentiating Eq. (2) and substituting Eq. (1) in it, we get dGs = ij dij −  dP − Ss dT Fig. 1. Schematic of adsorption-induced strain in microporous adsorbents. A typical trend observed in microporous adsorbents where the adsorbent first contracts and then expands.

Pore Space (containing adsorbate)

Solid Matrix

(3)

The Maxwell relations for Eq. (3) are as follows:

*ij * =− , *P *ij

*ij *Ss =− *T *ij

and

* *Ss = *T *P

(4)

The strain tensor ij can be split into a shape changing but volume conserving part (deviatoric) plus a volume changing but shape conserving part (dilation), giving: 1   kk 3 ij k=1    3

ij =

+

eij  shape changing

(5)

volume changing

Then we can introduce a trace E and a deviatoric eij strain: E = ii

eij = ij − 13 ij E

and

(6)

where ij is the 3D Kronecker delta. A similar decomposition of the stress tensor gives

Occluded Pore Space Skeleton = Empty Pore Space + Solid Matrix Fig. 2. A Porous adsorbent continuum consisting of: (i) the solid matrix and (ii) the pore space (adapted from Coussy, 2004). It is also referred to as skeleton at some places in the text.

theory of thermoelasticity of porous continua for the adsorbent and a statistical thermodynamics-based model for the chemical potential of the adsorbate. A necessary and sufficient description of both the models and their interrelationship is presented here. 2.1. Mechanics of porous adsorbent The thermodynamics-based theory of porous continua, developed by Maurice Biot and described recently by Coussy (2004), is used here to model the porous adsorbent. The adsorption system is assumed to be a superimposition of the skeleton continuum (i.e. the solid microporous adsorbent) and the adsorbate. The skeleton continuum consists of a solid matrix and a pore space (without the adsorbate) and the adsorbate gas particles are present in the pore space (in the connected pore space, not in the occluded pore space) of the skeleton (c.f. Fig. 2). Let s be the Helmholtz free energy of the skeleton per unit initial (undeformed) volume of the skeleton. 0 (cc/gm) is the initial (undeformed) volume of the skeleton and V0 (cc/gm) is the pore volume (of the connected pore space of an undeformed skeleton) associated with 0 . With the assumptions that: (i) the porous skeleton continuum is thermoelastic, (ii) there is no

 = 13 ii and sij = ij − 13 ij 

(7)

Introducing Eqs. (6) and (7), Eq. (3) now reads (Coussy, 2004): dGs =  dE + sij deij −  dP − Ss dT

(8)

The state equations for {sij , , , Ss }, with the free energy Gs = Gs (eij , E, P, T) formulation, are       *Gs *Gs *Gs , = , =− sij = *eij P,T,E *E P,T,e *P  ,T 

Ss = −

*Gs *T

ij



ij

(9) eij ,P,E

A material is isotropic when the energy functions only depend on the first invariant of the strain tensor or the trace of the strain tensor (Coussy, 2004). In other words, the energy functions only depend on the total volume of the material and are independent of the shape of the material. The elastic energy part of the total free energy of a linear, isotropic and thermoporoelastic material is only dependent on the first invariant of the strain tensor and on the second invariant of its deviatoric part. Differentiating the terms in Eq. (9), with an assumption that the skeleton is linear and isotropic, gives {d, dsij , d, dSs } as follows (Coussy, 2004): d = k dE − b dP − 3 K dT d = b dE +

dP − 3  dT N

and

dsij = 2eij

(10) (11)

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

dSs = 3 K dE − 3  dP + C

dT T

(12)

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Ln (P)

where 2

k=

* Gs *E 2

2

(bulk modulus),

2

b=−

* Gs , *E *P

K = −

2

1 * Gs , =− N *P2

* Gs , *T *E

2

C = −T

* Gs , *T 2 Psx

2

3  =

* Gs *P *T

and

ws

Px

2

* G  = 2s *eij

(shear modulus)

(13)

Eqs. (10)–(12) are the constitutive equations of an adsorbent that is linear, thermoporoelastic and isotropic and when an adsorbent undergoes any deformation during the process of adsorption. Since adsorption is an isothermal process we set dT = 0 in Eqs. (10)–(12). The interest of the present work is the adsorption-induced strain and stress. Eq. (11) can be used to calculate the change in strain due to change in porosity and pressure and the coefficient b (also referred as Biot's coefficient) can be viewed as the relationship between strain change and porosity change when the pressure and tempera-

mx Amt. of Gas Adsorbed Fig. 3. An illustration of adsorption isotherms when the adsorbent undergoes deformation, as shown in Fig. 1, (full line) and when the adsorbent is prevented from deformation (dashed line).

2

ture are kept constant (b = −(* Gs / *E *P) = (*/ *E)(−*Gs / *P) = */ *E). All the previous adsorption-induced strain modeling efforts have assumed that the strain is completely due to the change in porosity (b ≈ 1) which may not be a completely valid assumption for all the adsorbents. However, if the occluded pore space is absent (c.f. Fig. 2) it can be a valid assumption. Parameter N (also referred as Biot's modulus) relates the change in porosity variation to change in pressure variation when strain and temperature are kept constant 2

(1/N = −(* Gs / *P 2 ) = −(*/ *P)(*Gs / *P) = −(*/ *P)). Eq. (10) can then be used to calculate the stress change due to the changes in strain and pressure. From the Maxwell relations, the coefficient b can also be viewed as the relationship between stress change and pressure change (*/ *P =−(*/ *E)) when the strain and temperature are held constant. In the case of adsorption, the adsorbent skeleton is an open thermodynamic system which can exchange mass with the surroundings. The adsorbate gas gets adsorbed in the connected pore space of the solid adsorbent. If ma is the amount of gas adsorbed per unit initial volume of the adsorbent skeleton (0 ) and a is the density of the adsorbed gas in the porous space then, ma = a 

(14)

Differentiating Eq. (14) with respect to pressure (P) and rearranging, we get 1 dma ma d a d = − 2

a dP dP

a dP

(15)

The term dma /dP and ma can be calculated form the adsorption isotherm and if a model describing the density of the adsorbate in the pores is available, the change in porosity can be calculated using Eq. (15). The porosity change can further be used to calculate the total strain and stress in the adsorbent using Eqs. (10) and (11) and the poroelastic properties of the adsorbent. However, it has to be noted that Eq. (15) calculates the change in porosity based on the difference between the amount of gas adsorbed in a strained adsorbent (experimental adsorption isotherm) and the amount of gas that would have been adsorbed if the adsorbent is prevented from strain (using a model for adsorbate density in pores). Given the extremely small magnitude of strain in adsorbents, an exceptionally accurate model for the density of the adsorbate in the pores, a , is needed and any simplistic analytical model cannot be used. Molecular level density computation techniques may provide the required accuracy

in this case. Hence, instead of correlating the adsorption-induced strain (via porosity) to the amount of gas adsorbed and to its density in the pore space (as in Eq. (15)), we propose another method where we correlate the adsorption strain (via porosity) to the chemical potential of the adsorbate. This alternative approach is based on the following sequence where we calculate the difference between the chemical potentials of the gas adsorbed on a deformed and on an undeformed adsorbent using the experimental strain data: From Eq. (11) the change of porosity is related to the strain by d = dP  unknown

b  Biot’s coefficient

dE dP  Experimental strain data

+

1 N 

(16)

Biot’s modulus

Given the material properties (b,N), using Eq. (16), we calculate d/dP from available experimental strain data. We calculate (P) by numerically integrating d/dP in step (II) and use experimental adsorption isotherm (ma (P)) to calculate the density of the adsorbate ( a (P)), using Eq. (14). At this stage we have density as a function of pressure. a = a (P). If the adsorbent would have been prevented from strain and if ma is the amount of gas in an unstrained adsorbent at pressure P then

V ma (P) = 0 a (P), 0 = 0 (17)

0

Using the above mentioned steps (I–III) we can calculate the adsorption isotherm for the adsorbent if it would have been prevented from strain. Fig. 3 shows an illustration of the adsorption isotherms with strain (ma (P)) and without strain (ma (P)), for an adsorbent which exhibits an adsorption-induced strain as shown in Fig. 1. The two adsorption isotherms can be used to calculate the difference in the chemical potentials of the gas, adsorbed on a strained and on an unstrained adsorbent. If Pxs is the pressure required get mx amount of gas adsorbed when the adsorbent undergoes deformation and if Pxws is the pressure to get the same amount of gas adsorbed (c.f. Fig. 3 for illustration) when the adsorbent is prevented from strain then  s Px s ws diff =  −  = k T ln (18) B x x x Pxws where sx and ws x are the chemical potentials of the adsorbed gas (J/molecule) when mx amount of gas is adsorbed, with strain and without strain, respectively; kB is the Boltzmann's constant. In

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S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

formulating Eq. (18) we make use of the fact that the adsorbed molecules are in equilibrium with the surrounding, unadsorbed gas molecules {(gas) = (adsorbate)}. However, it has to be noted that the usage of Eqs. (15)–(17) imply that the constitutive Eq. (11) of the solid adsorbent (the skeleton in Fig. 2) is independent of the type and characteristics of the adsorbate. In partial summary, this section: (i) provides a relationship between the adsorption-induced strain and adsorption-induced porosity change (Eq. (11)) and (ii) illustrates a method to calculate the difference between the chemical potentials of the adsorbate addiff sorbed on a strained and on an unstrained adsorbent (x ), using the relationship in (i) and the experimental adsorption isotherm. Section 2.2 describes a statistical thermodynamics-based model for diff and Section 2.3 illustrates a method to predict the adsorptionx diff induced strain by comparing x calculated using the method presented in this section with that using the method presented in Section 2.2. It has to be noted that Eqs. (10) and (12) are not used in the present work; however, they can be used to calculate the adsorption-induced stress and entropy once the adsorption-induced strain is known and this paper presents a method to calculate the adsorption-induced strain. 2.2. Chemical potential of the adsorbate A summary of different approaches used to model the chemical potential of the adsorbed molecules is given by Kevan (1998). Adsorption in micropores has been previously modeled using Lattice gas models (Masel, 1996; Kreuzer et al., 1999, 2000; Aranovich et al., 2004; Do and Do, 2005; Tovbin and Rabinovich, 2006; Hocker et al., 1999). According to the lattice gas model, the adsorbed gas is considered as a layer where the molecules are free to move around but are not allowed to leave the surface. When the chemical structure of the adsorbent surface does not change, the adsorbate does not undergo any chemical change upon adsorption (like molecular dissociation) and there is a complete absence of adsorbate–adsorbate interaction then the chemical potential of the adsorbed gas is given as (Kreuzer et al., 2000; Do and Do, 2005)

a = − 0 + kB T ln



1−



− ln(q3 qint )

(19)

where 0 is the binding energy (positive) of an isolated adsorbate molecule on the surface relative to the molecule far away from the surface with zero kinetic energy, is the coverage, q3 is the vibrational partition function of the adsorbed molecule, and qint is the internal partition function of the molecule. The chemical potential calculated using Eq. (19) is only applicable for adsorbate molecules that are not interacting with each other. However, as mentioned before, densely adsorbed molecules in micropores do exhibit repulsive interactions. The effect of lateral interactions between the adsorbed molecules can be modeled using the popular analytical quasichemical approximation (Hill, 1960; Masel, 1996). The quasichemical approximation has previously been used to model the lateral interactions between the adsorbed molecules in micro and mesoporous materials (Kreuzer et al., 1997, 1999, 2000; Tovbin and Rabinovich, 2006; Tovbin and Votyakov, 2001; Tovbin, 1999; ¨ Votyakov et al., 1999; Lemier and Weissmuller, 2007; Salazar and Gelb, 2005) and if the chemical potential is split into two parts (Kreuzer et al., 1999, 2000), i.e. a term due to adsorbate–adsorbent interaction (Eq. (19)) and a term due to adsorbate–adsorbate interaction, then the later according to the Quasichemical approximation becomes (Hill, 1960; Kreuzer et al., 1997, 2000) 1 2

int a = c nn + ckB T ln



 − 1 + 2 1 −  + 1 − 2

 (20)

where 

 = 1 − 4 (1 − ) 1 − exp



− nn kB T

 (21)

c is the number of nearest neighboring adsorbate particles (site coordination number) of the molecule whose chemical potential is a + int a and nn is the strength of interaction between the nearest neighbor adsorbate molecules. For a repulsive interaction between the adsorbate molecules, nn > 0 and for an attractive interaction, nn < 0. It has been suggested in the literature that the adsorptioninduced contraction strain in microporous adsorbents is caused due to the strong attractive forces between the gas molecules and pore walls and as a result of these attractive forces the gas molecules reach very high densities when adsorbed in these microporous adsorbents (Ravikovitch and Neimark, 2006; Aranovich et al., 2005; Aranovich and Donohue, 2003). But it has also been shown that due to this dense packing of the gas molecules, repulsive interactions amongst the gas molecules are developed since the intermolecular distance is less than the minimum in their potential curve (Ravikovitch and Neimark, 2006; Aranovich et al., 2005; Aranovich and Donohue, 2003; Ibach, 1999). It has to be noted that though there exist repulsive interactions between the adsorbate molecules, adsorption continues since the decrease in the free energy of the adsorbed molecule due to the molecule–adsorbent interaction is larger in magnitude than the increase in the free energy due to its repulsive interactions with the neighboring adsorbate molecules (Aranovich et al., 2005). Applying the lattice gas model and Quasichemical approximation (Eqs. (19) and (20)) to calculate chemical potentials sx and ws x we get

ws x = − 0 + kB T ln

+





1−

− ln(q3 qint ) + c ws nn

ws

− 1 + 2 1 ckB T ln ws 2  + 1 − 2



ws = 1 − 4 (1 − ) 1 − exp





1−



− ws nn kB T

 (22)

and

sx = − 0 + kB T ln

+ 





1−

s

− ln(q3 qint ) + c snn

− 1 + 2 1 ckB T ln s 2  + 1 − 2

s = 1 − 4 (1 − ) 1 − exp





1−



− snn kB T

 (23)

The contractive strain in the adsorbent also results in contraction of the pore space (Eq. (16)) and as a result it may bring the gas molecules further closer thereby increasing the repulsive interaction amongst them. However, when the adsorbent undergoes dilation, the repulsive interactions between the adsorbate molecules are reduced since the increase in pore space will take the molecules apart. Hence s ws in Eqs. (22) and (23), snn > ws nn for a contractive strain and nn < nn for an expansive strain. Applying Eqs. (22) and (23) to Eq. (18) we get 1 2

s ws diff x = c( nn − nn )+ ckB T ln



s −1+2 s +1 − 2



ws +1−2 ws −1+2

 (24)

It has also been shown by Wu and Metiu (2000) that the nearest neighbor interaction nn (CO adsorption on Pd) is larger for a contractive strain and is smaller for an expansive strain. Based on

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

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Wu and Metiu (2000) and on the fact that in case of adsorption in micropores, the contraction or expansion of the pore space takes the adsorbed molecules closer or farther, respectively, we can write

snn − ws nn = K(0 − ) Eq. (24) can then be written as

 s  ws   − 1+2  +1−2 1 diff = cK(  −  )+ T ln ck B x 0 2 s +1−2 ws − 1+2

(25)

(26)

Eq. (26) expresses the difference between the chemical potentials of the adsorbate adsorbed on a strained and on an unstrained adsorbent as a function of the change in the pore space of an adsorbent (0 − ) (and consequently the adsorption strain via Eq. (16)), coverage and temperature T. 2.3. Calculating adsorption-induced strain The model equation (26) correlates diff to the adsorptioninduced porosity change, temperature and coverage. However, diff can also be calculated for a given porosity () using the experimental adsorption isotherm data, as shown in Section 2.1. The difference or residual,  = (diff )Eq. (26) − (diff )Eq. (18)

(27)

can thus be minimized to yield the sought after adsorption-induced strain. Using the model equation (26) and experimental adsorption isotherm, an iterative procedure can be set-up to predict the adsorption-induced porosity change and hence the adsorptioninduced strain. However, the dependence of diff on  is larger than on T and (Eq. (26)) and hence instead of directly comparing diff calculated using Eq. (26) to that calculated using the method in Section 2.1, the procedure is slightly modified so as to cancel out the error introduced in the model during the fitting procedure. A flowchart of the procedure is shown in Fig. 4 and a detailed explanation is given in Appendix A. 3. Results and discussion In this section the relationship of adsorption-induced strain in a linear, isotropic, and poroelastic microporous adsorbent with the chemical potential of the adsorbate, as developed in Section 2 is validated using the experimental strain data of Yakovlev et al. (2005). The adsorption-induced strain in microporous activated carbon adsorbent was measured by Yakovlev et al. (2005) for CO2 adsorption. Figs. 5a and b show the experimental adsorption isotherms and the experimental adsorption-induced strain data, adapted from Yakovlev et al. (2005). The strain data shows the typical trend observed in microporous adsorbents where the adsorbent first contracts and then expands. The data at 243 K is used in the present study for validation and for fitting parameters, where as the data at 273 and 293 K is used to validate the calculated results. Experimental adsorption isotherms and the experimental strain data are interpolated and a numerical derivative of strain with respect to pressure (dE/dP) is calculated. Material parameters b and N are needed to calculate d/dP (from Eq. (16)), however, their values are unknown for the microporous activated carbon adsorbent. Bouteca and Sarda (1995) and Coussy (2004) have reported orders of magnitude of these properties for different materials and based on those values, we assume b = 0.5 and N = 100 GPa. The porosity of the undeformed activated carbon adsorbent is reported by Yakovlev et al. (2005) as 0 = 0.52875. Fig. 6 shows the porosity change calculated from the experimental strain data (in Fig. 5b), employing Eq. (16). Using the porosity data, we calculate the adsorption isotherm if the adsorbent is prevented from deformation (using

Fig. 4. Flowchart of the procedure for calculating the adsorption-induced strain. The corresponding explanatory text is given in Appendix A.

Eq. (17)). However, since the change in porosity is of the order of magnitude of 10−3 , it is not possible to visually differentiate the isotherms, with and without strain, hence we do not show the plots of the calculated isotherms (An exaggerated illustration is shown in Fig. 3). Using the two isotherms we calculate diff (Eq. (18), Section 2.1). Fig. 7 shows the diff as a function of the amount of gas adsorbed at 243, 273 and 293 K. The chemical potential difference (diff ) data in Fig. 7 and the porosity data in Fig. 6, at 243 K, are used to fit the parameter K in Eq. (26). The strength of interaction between the nearest neighbor adsorbate molecules in an undeformed adsorbent ( ws nn ) is taken equal to kB T ( > 0, hence repulsive interaction). is calculated as the ratio of the amount of gas adsorbed at a pressure to the maximum adsorption capacity of the adsorbent at that temperature, as determined from the adsorption isotherm. “c”, the number of nearest neighboring adsorbate particles (site coordination number), is taken as 6. Fig. 8 compares the chemical potential difference (diff ) calculated using the model equation (26) and fitted parameter K with the chemical potential difference calculated using the procedure in Section 2.1 (as in Fig. 7). Though a quantitative agreement is obtained, it can be noticed that the model Eq. (26) underestimates diff at larger strains and it is attributed to the assumption of linear dependence of the nearest neighbor interaction on the porosity since the potential curve for the adsorbate molecules may exhibit a steep variation after a particular intermolecular distance.

4750

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

1

16

x 10-3

0.5

14

0

0 - 

In (P)

12

10

8

-0.5 -1 -1.5

6

-2

4

-2.5 0

2

4 6 8 10 Amt. of Gas Adsorbed (mmoles/gm)

0

12

1

2

3

4

5

6

7

8

9

10

11

12

Amount of Gas Adsorbed (mmoles/gm) Fig. 6. Porosity change in the deformed adsorbent calculated using Eq. (16) and experimental strain data (Yakovlev et al., 2005) at 243 K (•), 273 K (䊏), and 293 K ().

5 4 3 Strain x 103

0.5

2

x 10-22

0 -0.5

1

x 10-23 1

-1

diff

-1

-1.5

0

-2

-1 diff

0

-2.5

-2 0

1

2

3 4 5 6 7 8 9 10 Amt. of Gas Adsorbed (mmoles/gm)

11

12

-2

-3 -3

-3.5 Fig. 5. (a) CO2 Adsorption isotherms and (b) CO2 adsorption-induced strain data (adapted from Yakovlev et al., 2005) at 243 K (•), 273 K (䊏), and 293 K (). The adsorbent is microporous activated carbon material.

-4

-4

0

2 4 6 8 Amt. of Gas Adsorbed (mmoles/gm)

-4.5 0 After validating model Eq. (26) (from Fig. 8), Fig. 9 shows the predicted adsorption-induced strain (using the procedure described in Section 2.3 and Fig. 4) in comparison with the experimental strain data of Yakovlev et al. (2005) at 243, 273, and 293 K. The predicted adsorption-induced strain data matches well with the experimentally observed strain data. It has to be noticed that the above procedure utilizes adsorption isotherm as the only experimental signal to predict the strain, once the model parameter K and material parameters b and N are known. We note that the model is not restricted to adsorption in subcritical conditions (for CO2 , T crit = 304.1 K) and is equally applicable to model adsorption-induced strain in supercritical conditions. It is also independent of the geometry of pores; cylindrical, spherical or slit shaped. Exact material parameters b and N were not available for the present work and though approximate values of these properties have proven adequate for the “proof of concept”, experimentally determined values are needed to correlate the skeleton properties with the solid matrix properties and porosity thereby availing more accurate understanding of the different stresses developed in an

2

10

4 6 8 10 Amt. of Gas Adsorbed (mmoles/gm)

12

Fig. 7. diff (calculated using Eq. (18), Section 2.1) as a function of the amount of gas adsorbed; at 243 K (•), 273 K (䊏), and 293 K ().

adsorbent and the structural characteristics of the adsorbent. A correct analysis of the structural characteristics of the adsorbent is also needed to determine the effect of adsorption-induced strain on equilibrium sorption properties. A competition between the adsorbate–adsorbent attractive interaction and the adsorbate–adsorbate repulsive interaction is believed to govern the type of strain developed in an adsorbent (Yakovlev et al., 2005; Ravikovitch and Neimark, 2006; Aranovich et al., 2005). When adsorption takes place in micropores that are of the size of a few molecular diameters, the adsorbate molecule attracts the opposite pore walls due to dispersive interaction and a contractive stress is developed due to these attractive linkages between the adsorbate and the adsorbent framework. At lower coverage, as more molecules are adsorbed the total attractive interaction increases and causes

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

adsorption deformation mechanism. A more detailed analysis including: (i) experimentally determined material parameters and (ii) a more comprehensive chemical potential model taking into account adsorbate–adsorbent attractive interaction (Eq. (23)), in addition to the adsorbate–adsorbate repulsion, can further confirm this mechanism and it can also explain why different adsorbate gases induce different strains in an adsorbent.

x 10-22

0.5 0 -0.5

0

-1.5

-0.5 -1

-2 diff

diff

0.5

-1

4. Conclusions

-1.5



-2.5

-2 -2.5

-3

-3

-3.5

-3.5

-4

-4

0

2 4 6 8 Amt. of Gas Adsorbed (mmoles/gm)

-4.5 0

2

10

4 6 8 10 Amt. of Gas Adsorbed (mmoles/gm)

12

Fig. 8. diff calculated using Eq. (26) as a function of the amount of gas adsorbed at 273 K () and at 293 K (). The filled symbols indicate calculated diff using Eq. (18) and experimental adsorption isotherm (as described in Section 2.1).

6

x 10-3

5 4

Strain x 103

3 2 1 0 -1 -2 0

1

4751

2

3

4

5

6

7

8

9

10

11

12

Amt. of Gas Adsorbed (mmoles/gm) Fig. 9. Predicted CO2 adsorption-induced strain in microporous activated carbon adsorbent at 243 K (), 273 K (), and 293 K (). Filled symbols indicate the experimental data from Yakovlev et al. (2005).

Adsorption is widely employed in separation, purification and catalytic applications and adsorption-induced strain is an experimentally observed phenomenon that affects the adsorbent and the equilibria and kinetics of adsorption (Yakovlev et al., 2003, 2005; Kharitonov et al., 2001; Cui et al., 2007; Day et al., 2008; Viete and Ranjith, 2006). Microporous adsorbents typically undergo an initial contraction followed by expansion and though experimental investigations provide a useful insight, theoretical studies are required to better understand the phenomenon. The present work assumes the solid adsorbent to be linear, isotropic and a continuous poroelastic medium and thermodynamics-based constitutive equations for the adsorbent are coupled with the equilibrium adsorbate chemical potential, modeled using a lattice gas model and the quasichemical approximation. The proposed model correlates the difference between the chemical potential of the gas adsorbed on a deformed adsorbent and that of a gas adsorbed on an undeformed adsorbent to the adsorption-induced strain. The correlation is validated using the experimental CO2 adsorption-induced strain data on activated carbon adsorbent (Yakovlev et al., 2005). Based on this correlation, a method which utilizes the experimental adsorption isotherm data is proposed and is able to successfully predict the adsorption-induced strain in the activated carbon adsorbent at 243, 273 and 293 K. The correlation is equally applicable to subcritical and supercritical gas adsorption, is independent of the pore geometry and is also in accord with the opinion that the initial compressive strain in microporous adsorbent is caused due to attractive interaction between the pore wall and adsorbate and the increased adsorbate–adsorbate repulsion at higher adsorption causes the adsorbent to expand. We show that there exists a repulsive interaction between adsorbed molecules in micropores and we compute the adsorption-induced strain on the basis of the change in repulsive interaction between the adsorbate molecules due to the adsorption strain, using approximate material parameters. The present work can also be extended to a more comprehensive model that takes into account the adsorbate–adsorbent attractive interaction and uses measured material parameters.

Notation b

the adsorbent to contract. Though the contraction in the adsorbent increases the repulsive interaction (due to the adsorbate molecules coming closer), the contribution of the dispersive interaction is dominating and thus the adsorbent continues to contract. However, the adsorbent ceases to contract when the total repulsive interaction due to the packing of the adsorbate molecules balances the attractive interactions. Further densification of adsorbate molecules into the pore space causes the repulsive interaction to dominate, thereby causing the adsorbent to expand. If the above mechanism is to be believed, the repulsive interaction is larger in a contracted adsorbent and smaller in a dilated adsorbent than that of an undeformed adsorbent when the same amount of gas is adsorbed in both the cases. Model Eq. (26) calculates diff based on this assumption and its agreement with the diff calculated using poromechanics and experimental adsorption isotherm (c.f. Fig. 8) further validates the

c d E Gs kB K ma N P q3

material parameter defined in Eq. (13); also referred as Biot's coefficient number of nearest neighboring adsorbate particles (site coordination number) distance between the trial and experimental strain curve (c.f. Fig. A1) trace of the strain tensor free energy of the skeleton defined in Eq. (2), J/cc Boltzmann's constant, J/mol K parameter defined in Eq. (25), J/mol amount of gas adsorbed, mmol/g m material parameter defined in Eq. (13); also referred as Biot's modulus, Pa pressure, Pa vibrational partition function of the adsorbed molecule

4752

qint Ss T V

S.H. Mushrif, A.D. Rey / Chemical Engineering Science 64 (2009) 4744 -- 4753

internal partition function of the molecule entropy of the skeleton, J/K temperature, K pore volume of the skeleton

Greek letters

 ij 

 0 nn

a  s 

parameter defined in Eq. (21) Kronecker delta bulk stress in the adsorbent skeleton, Pa coverage chemical potential, J/mol binding energy of an isolated adsorbate molecule, J/mol strength of interaction between the nearest neighboring adsorbate molecules, J/mol density of the adsorbed molecules in pore space porosity Helmholtz free energy density of the adsorbent skeleton, J/cc volume of the adsorbent skeleton, cc/gm

 diff 2 Fig. A1. The objective function ([(diff − ˜ )/ diff ] ) plotted against the distance between the trial and experimental strain curves at 273 K (as illustrated in the inset).

calculate the strain at a specific value separately because: (i) porosity and strain are related by a differential equation (16) with an initial condition, E = 0 and  = 0 and = P = 0 and (ii) it is not possible to diff

Subscripts and superscripts i,j int s ws x 0

indices interacting part of chemical potential with strain (superscript only) without strain amount of gas adsorbed without deformation

Acknowledgments Financial support for this work was provided by the National Science Foundation under Award Number EEC-9731689 (ADR). Samir Mushrif is the recipient of post-graduate scholarships from Natural Sciences and Engineering Research Council of Canada (NSERC) and the Eugenie Ulmer Lamothe fund of the Department of Chemical Engineering of McGill University. Appendix A The purpose of this appendix is to elaborate the procedure used to predict the adsorption-induced strain, as shown in Fig. 4. The adsorption-induced strain is expressed as a polynomial function of the coverage, and an initial guess of the unknown strain is calculated using an initial guess for the polynomial coefficients. For the guessed strain function E( ), the porosity function ( ) is calculated using Eq. (16). It has to be noted that though the strain and porosity are expressed as a function of , they can also be expressed as a function of P using the experimental adsorption isotherm. Given the porosity function ( ), the adsorption isotherm of an undeformed adsorbent is calculated (P ws ( )), as explained in Section 2.1. diff ( ) is computed by using ( ) in Eq. (26) and is then used in Eq. (18) to calculate P s ( ). Thus, having the two adsorption isotherms, with strain and without strain (P s ( ) and P ws ( ), respectively), the poros˜ ( ). The chemical potential difference ity function is recalculated  ˜ diff ( ) is then calculated by using ˜ ( ) in Eq. (26). A sum of the relative error [(diff − ˜ diff )/ diff ]2 at nine different values (in the range 0.1–0.99) is then minimized by changing the polynomial coefficients. This method calculates the adsorption-induced strain function, E( ), in the entire range (0.1   0.99) at once. It is not possible to

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