Coarse-Graining of Adsorption in Microporous Materials - American

Article pubs.acs.org/JPCC

Coarse-Graining of Adsorption in Microporous Materials: Relation between Occupancy Distributions and Local Partition Functions Federico G. Pazzona,* Pierfranco Demontis, and Giuseppe B. Suffritti Dipartimento di Chimica e Farmacia, Università degli Studi di Sassari, via Vienna, 2, I-07100 Sassari, Italy S Supporting Information *

ABSTRACT: We derive a recurrence equation that relates the effective interactions among the particles adsorbed in two adjacent cages of a microporous material to the local probability distributions of cage occupancies. We show that such a relation can be used to estimate the approximate values of local partition functions, which can in turn be used as input for coarsegrained models of adsorption. After validating this theory by reproducing with good approximation the local partition functions of a lattice gas, we apply this method to the problem of adsorption of methane in all-silica zeolites DAY and ITQ-29 at various temperatures. The adsorption isotherms we obtain from the coarse-grained models built on the basis of the estimated values of the local partition functions are in very good agreement with the adsorption isotherms calculated from atomistic simulations.

I. INTRODUCTION The thermodynamic properties of molecules confined within a host material are ruled by the energy landscape that arises from both host−guest and guest−guest interactions.1 On the one hand, confinement complicates the description of the properties of the confined molecular species, e.g., with respect to the bulk case.2 On the other hand, however, the regular structure of cages (or cavities) observed in many microporous materials (such as zeolites) immediately suggests a simplified description of the problem of adsorption. In this coarse-grained description, the main observable property is the number of molecules adsorbed within each cage rather than the specific molecule positions inside the crystal. Many coarse-grained models of adsorption of small molecules in microporous materials have been produced over the years.3−8 The primary goal of massively reducing the number of degrees of freedom in a given atomistic host−guest sample is to enlarge the space and time scales in numerical simulations. Obviously, such a reductionistic approach must preserve some carefully chosen properties of the original system in order to produce a coarse-grained model that actually resembles the original system on either the mesoscopic or the macroscopic scale. Therefore, a property must be identified that can be considered representative of the host−guest system under study. The most important sources of information about the global static properties of a host−guest system are probably the system’s adsorption isotherms, i.e., plots of the number of guest molecules adsorbed in the host material versus either the chemical potential (or fugacity) or the pressure at fixed temperature. Adsorption isotherms can be determined from both experimental data and atomistic simulations, and because they describe with immediacy how the host material reacts to different pressure conditions, their determination is one of the main goals in the study of the adsorption properties of microporous materials.3,9−38 © XXXX American Chemical Society

Adsorption isotherms are also the primary feature that a coarse-grained model of adsorption is required to reproduce. This introduces the problem of the correct parameterization of the interactions in coarse-grained models. The nature of the parameters representing the molecular interactions depends strictly on the observable properties sampled by the coarsegrained model. For example, if a coarse-grained model of adsorption is supposed to sample the cage occupancies rather than the particle positions (the cage occupancy being defined as the number of particles adsorbed within a given cage at every instant in time), then the effective interactions within a cage with a given occupancy and between neighboring cages with different occupancies are required for mimicking of the actual intra- and intercage average interactions of the corresponding atomistic system. The first problem is then how to determine the effective interactions that would enter the parameterization of the coarse-grained model. In some cases, accurate adsorption isotherms can be produced by neglecting the interactions between neighboring cages. The average distributions of occupancies7,8,39−50 produced when the system is subjected to different chemical potentials can then be related to each other in order to retrieve approximate values of intracage partition functions that produce isotherms in good agreement with those obtained through atomistic simulations.51 In other cases, however, neighboring interactions cannot be neglected. A description based merely on self-interacting cages thus is unsatisfactory, and the need emerges for an alternative method to retrieve effective interactions between the molecules adsorbed in different cages. Whereas expanded ensemble methods (EEMs)8,52−60 can be used for the calculation of Received: September 30, 2014 Revised: November 10, 2014

A

dx.doi.org/10.1021/jp509890z | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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neighborhood of a cage with occupancy m, namely, pμ({nj}, m), which we are going to define below, are assumed to be the same as in the full lattice 3 described by eq 1. On the basis of such an assumption, we can define the joint probability of finding cage 0 occupied by m particles and the neighboring cages occupied by n1, n2,..., nν particles as

intercage partition functions, we will show herein that under certain conditions we can write down a relation between local occupancy distributions61 and local partition functions and use the connection between the two to deduce approximate values of the intercage partition functions from grand canonical Monte Carlo (GCMC) simulations in a systematic and very simple way.

pμ ({nj}, m) = ζμ−1 e βμm Q(0)(m)

II. METHOD An ideal host−guest system can be represented as N indistinguishable particles occupying L identical cages (with each cage denoted by i with i = 1,..., L) of a perfectly regular host material. Such cages constitute a structured lattice, which we call 3 . In many situations, the particles inside every cage interact mostly with each other and with the particles in the first ν neighboring cages; thus, the interactions with particles located in other cages can be neglected. The grand canonical partition function of such a system can be approximated by3,8,17

ν

×

nL

where {nj} = {n1,..., nν} and ζμ is the mean-field grand canonical partition function defined as ζμ =

∑ e βμm Q(0)(m) ∑ e βμn

1

m

Q(0)(n1) Z(m , n1)

n1

× ... ×

βμni

(0)

Q (ni)

∏

Z(ni , nj)

∑e

βμnν

(0)

Q (n ν ) Z ( m , n ν ) (5)

The drawback of dealing with a distribution such as pμ({nj}, m) is that such an object consists of (nmax + 1)ν+1 entries for each value of the chemical potential μ. It is certainly preferable to work with its two-cage counterpart, namely, the much simpler pμ(n, m) that is defined as the joint probability distribution of two neighboring cages (occupied by n and m particles). Therefore, as a further restriction, we assume that the occupancies of cages 1, 2,..., ν are always constrained in such a manner that they have equal values, namely, n1 = n2 =...= nν = n. This assumption drastically simplifies the resulting probability distributions and leads to a simple relation that can be used to estimate the intercage partition functions. Thus, the joint probability distribution defined above becomes

(1)

j ∈ 5i

i=1

where β = (kBT)−1 (where kB is the Boltzmann constant and T is the temperature), μ is the chemical potential, and cage occupancies n1,..., nL denote the number of particles within cage i (i = 1,..., L). Because we will deal with only isothermal− isochoric calculations, we omit the dependence on T and V in the notation. We represent the maximum cage occupancy as nmax, i.e., n = 0, 1,..., nmax. In eq 1, the quantity Q(0)(n) is the canonical partition function of a cage containing n particles when it is considered to be a closed system, i.e., neglecting the interactions with particles in the neighboring cages. For structureless particles we have ni 1 Q(0)(ni) = ... dr ni e−βU (r ) 3ni v v ni !Λ (2)

−1

pμ (n , m) = ζμ̃ e βμmQ(0)(m)[e βμnQ(0)(n) Z(m , n)]ν

∫ ∫

i

j

ζμ̃ =

∑ e βμmQ(0)(m) ∑ [e βμnQ(0)(n) Z(m , n)]ν m

(7)

n

The single-cage occupancy probability (pμ(n)) and the conditional probability of the central cage to have occupancy m given that the surrounding cages are constrained to occupancy n (g(m|n)) can be obtained from eqs 6 and 7, respectively, as

ni nj

∫v ... ∫v dr n ∫v ... ∫v dr n e−βϕ(r ,r )

(6)

where

where U(rni) is the potential energy of configuration rni of the ni particles in the cage i, v is the volume of the cage, and Λ = (h2/ 2πmkBT)1/2 is the de Broglie wavelength. The term Z(ni, nj) in eq 1 (satisfying the symmetry requirement Z(ni, nj) = Z(nj, ni)) is the intercage partition function; it represents the configurational integral of two cages, e.g., i and j, respectively, occupied by ni and nj particles, that interact through the potential ϕ(rni, rnj) as follows Z(ni , nj) =

(4)

nν

∑ ... ∑ ∏ e n1

j

j=1

M

Ξμ =

∏ e βμn Q(0)(nj) Z(m , nj)

(3)

pμ (n) =

where ϕ does not include the interactions among particles adsorbed in the same cage. The square root in eq 1 is introduced to compensate for counting every Z(ni, nj) term twice. The aim of this work is to retrieve a simple relation that links Z(ni, nj) to a properly chosen distribution function. Such a relation could then be used as a possible alternative to EEMs for the calculation of intercage partition functions.8,52−60 Let us consider a subsystem made of one cage, labeled 0, and a number (ν) of its neighbors, labeled 1, 2,..., ν. With the aid of some approximations, we can construct a simpler mean-field model of adsorption out of such a subsystem. First, we assume that cage 0 interacts solely with its neighboring cages and vice versa and that the subsystem does not interact with the rest of the system. Nevertheless, the occupancy distributions of a single cage, pμ(m) (with m = 0,..., nmax), and of the cages in the

∑ pμ (n , m)

and gμ(m|n) =

m

pμ (n , m) pμ (n)

(8)

The basic assumption of the mean-field approach is that pμ(·) and gμ(·|n) (with n = 0, 1,..., K) must reflect, respectively, the single-cage occupancy distribution and the conditional probability distribution of a cage in the neighborhood of an n-occupied cage that we obtain from simulations of the system described by eq 1. By calculating the ratio gμ(m + 1|n)/gμ(m|n), we obtain the recurrence relation for the mean-field intercage partition function, represented as Zμ̃ (m + 1, n)ν = e−βμZμ̃ (m , n)ν ×

B

Q(0)(m)

Q(0)(m + 1) gμ(m + 1|n) gμ(m|n)

(9)



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system; they do not make use of any mean-field assumptions about the interactions among the particles adsorbed in a cage and its surroundings. The recurrence relation, eq 9, deserves some comment. It can be formally reversed to give a formula for gμ(m + 1|n) in terms of gμ(m|n), μ, and the (supposedly known in this case) ratios Z̃ μ(m + 1, n)v/Z̃ μ(m, n)v and Q(0)(m + 1)/Q(0)(m). However, such a reversed relation is not useful in practice. This is because the condition Z̃ μ(n, 0) = Z̃ μ(0, n) = 1 serves as the starting point for the derivation of the entries of matrix Z̃ μ(·, ·) through eq 9, whereas in the reversed recurrence relation (aimed at obtaining the entries of matrix gμ(·|·)), an analogous starting point is missing. Therefore, eq 9 is limited to the derivation of the mean intercage partition functions and unfortunately cannot be extended to the mean-field derivation of the conditional occupancy distributions in a lattice-gas problem where all of the self- and intercage partition functions are known in advance.

where we changed the symbol used for the intercage partition function from Z to Z̃ μ to highlight the fact that when the involved quantities are extracted from a numerical simulation the statistical error in the evaluation of the distributions gμ(·|n) introduces a slight dependence of the calculated value of function Z̃ μ on μ as well as a slight asymmetry of the function Z̃ μ with respect to the exchange of the occupancies, i.e., the calculated value of Z̃ μ(n, m) is not exactly equal to that of Z̃ μ′(n,m) (with μ ≠ μ′), and the calculated value of Z̃ μ(n, m) (with n ≠ m) is not exactly equal to that of Z̃ μ(m, n). We can retrieve chemical-potential-independent symmetry entries for the matrix of intercage partition functions by defining the following mean values ωμ(n , m)[Zμ̃ (m , n) + Zμ̃ (n , m)]

∑

Z̅(m , n) =

(10)

μ ∈ {μ}

with weight matrices ωμ(·, ·) defined as ωμ(m , n) =

pμ (n , m) 2 Σ

p (n , m)

μ ′∈ {μ} μ ′

III. NUMERICAL SIMULATIONS First, we assess the validity of our mean-field theory by calculating the cage and the intercage partition functions from numerical simulations of a lattice-gas model where both the functions Q(0)(n) and Z(n, m) are known in advance. The obtained estimations of Q̅ (0)(n) and Z̅ (n, m) are then used as input for new numerical simulations that we expect to produce a good approximation of the asdorption isotherm of the original system. Our lattice-gas model was set as a square lattice of L2 (with L = 16) nodes with the connectivity set as ν = 4. The maximum occupancy was set to nmax = 16. The closed-cage partition functions were assumed to have the following form ⎛ nmax ⎞ −βH(n) ⎟e Q(0)(n) = ⎜ ⎝ n ⎠ (15)

(11)

for every μ, n, and m. Equation 11 assigns greater importance to the more frequently sampled values of neighboring occupancies n and m and arithmetically averages the values of Z̃ μ with reversed occupancies. The set of chemical potentials {μ} must be carefully chosen in order to cover in a proper manner the full range of loadings allowed in the system. For example, by defining the loading of the system as the average occupancy of a single cage, i.e., ⟨n⟩μ = Σnnpμ(n), such a range is 0 < ⟨n⟩μ < nmax. We choose the set {μ} = {μ1, μ2,..., μk,...} in such a way that ⟨n⟩μk+1 − ⟨n⟩μk ≈ 1/2. The recurrence relation, eq 9, starts from Z̃ μ(n, 0) = Z̃ μ(0, n) = 1 for every occupancy n and chemical potential μ; it makes use of the closed-cage partition functions Q(0)(0), Q(0)(1),..., Q(0)(nmax) that can be retrieved if unknown (because they usually are in off-lattice microscopic simulations of host−guest systems) with high accuracy from simulations of the system without intercage interactions. We shall refer to this as the reference system through51 Q̅ (0)(n) =

where the occupancy-dependent interaction potential H(n) was defined arbitrarily as a potential with a weak interaction component (1 < n < 4) followed first by a stronger attraction (4 < n < 13) and then by a repulsion component (13 < n < 16). Such a potential is reported in Table 1. The intercage partition function is represented as

(0)

ωμ(0)(n) Q̃ μ (n)

∑

(12)

μ ∈ {μ}

Z(n , m) = e−βϕ(n , m)

In this case, the Q̅ (·) values computed by means of eq 12 replace the Q(0)(·) values in eq 9. The chemical-potentialdependent closed-cage partition functions appearing on the right-hand side of eq 12 are given by the recurrence relation (0)

(0) (0) Q̃ μ (n + 1) = e−βμQ̃ μ (n)

with ϕ simply defined as ϕ(n, m) = ( /2)[H(n) + H(m)]. The temperature was arbitrarily set to T = 300 K. We will refer to the system working with known local partition functions as the original system. We carried out simulations of the original system in the canonical ensemble and used the method described in our previous work for the calculation of the adsorption isotherm.62 We used 107 MC displacement steps for every value of the loading.

pμ(0) (n + 1) pμ(0) (n)

(13)

that starts with (0) = 1 for every μ and where is the single-cage occupancy probability for such a reference system. The weights in eq 12 are represented by Q̅ (0) μ

ωμ(0)(n)

=

p(0) μ

Table 1. Effective Interaction Parameters Used for the Simulation of the Lattice-Gas Modela

pμ(0) (n) Σ p(0) (n) μ ′∈ {μ} μ ′

(16) 1

(14)

which assigns more importance to the occupancy values that are more frequently sampled in the reference system. The difference between the set of eqs 13 and 14 and the set of corresponding equations reported in our previous work51 is due to the fact that the set of eqs 13 and 14 refer to the reference

a

C

n

H(n)b

n

H(n)b

n

H(n)b

0 1 2 3 4 5

0.0 0.0 0.0 −0.1 −0.2 −0.4

6 7 8 9 10 11

−0.8 −1.6 −3.2 −3.6 −3.4 −3.0

13 14 15 16

0.0 4.0 6.0 7.0

See eq 15. bRepresented in kJ mol−1. dx.doi.org/10.1021/jp509890z | J. Phys. Chem. C XXXX, XXX, XXX−XXX


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missing at first sight, namely, the values of Z̅ (m, n) where n is very different from m. This is due to an accuracy problem that naturally arises in the computation of the conditional distributions gμ(·|·). More specifically, by thoroughly examining eq 9, we can immediately argue that the accuracy of the calculated value of Z̃ μ(m + 1, n) depends on the accuracy of the values of Z̃ μ(m, n), Q̅ (0)(m), Q̅ (0)(m + 1), gμ(m|n), and gμ(m + 1|n) that were previously calculated. Now, let us suppose that the accuracy of Q̅ (0)(·) is acceptable (as it actually is). In such a case, it is the accuracy of the conditional distribution gμ(·|n) that determines the accuracy of the Z̃ μ(·, n) values. If the chemical potential μ corresponds to a low loading value, e.g., to ⟨n⟩μ = 1, then a pair of neighboring occupancies such as (n, m) = (1, 12) will be rarely, if ever, sampled during the simulation. To ensure the highest accuracy possible in the calculation of the entries of Z̃ μ(·,·), we chose to restrict eq 9 to the cases in which the values of both gμ(m|n) and gμ(m + 1|n) are greater than 10−5. The missing entries in the mean values of Z̅ (·,·) are a consequence of this restriction. Another consequence appears at high loadings: neighboring occupancy pairs where n is much larger than 1, such as (m, n) = (1, n), are very rarely sampled. In such a case, eq 9 cannot start from the initial condition of Z̃ μ(0, n) = 1 but must instead start from some pair (m, n) for which gμ(m|n) > 10−5. Even if both gμ(m|n) and gμ(m + 1|n) are known with acceptable accuracies, the calculation of Z̃ μ(m + 1, n) seems to be impossible because it requires a known value of Z̃ μ(m, n). When this problem occurs, we circumvent it by simply replacing Z̃ μ(m, n) in eq 9 with a known entry, Z̃ μ′(m, n), that was previously calculated at a lower chemical potential, i.e., where μ′ < μ. Therefore, the discrepancies between Q̅ (0)(·) and Q(0)(·) (Figure 1a) are related to the accuracy of the determination of the occupancy distributions p(0) μ (·) in a manner similar to how the discrepancies between Z̅ (·, ·) and Z(·, ·) (Figure 1b−q) are related to the accuracy of the conditional occupancy distributions gμ(·|·). After the evaluation of the approximate array Q̅ (0)(·) and the matrix Z̅ (·,·), we carried out GCMC simulations of the approximate system, i.e., the system working with local partition functions Q̅ (0)(·) and Z̅ (·, ·) rather than Q(0)(·) and Z(·, ·), again using 107 MC steps for every value of the chemical potential with displacements, insertions, and removals selected with equal probability. (Details about the lattice-gas GCMC simulations can be found in the Supporting Information.) During the numerical simulations of the approximate system, the missing entries of the matrix Z̅ (·,·) were treated as null entries. This is not expected to introduce a significant error because such entries are related to very different occupancies of neighboring pairs that were very rarely, if ever, sampled in simulations of the original system. The resulting adsorption isotherm is shown in Figure 2 (●) with that of the original isotherm (○) for comparison. To show that the introduction of lateral interactions has significant effects on the shape of the curve, we also included a plot of the adsorption isotherm evaluated in the absence of lateral interactions, i.e., by using ϕ = 0 (Figure 2, ×). For very low loadings, namely, ⟨n⟩μ < 4, the agreement among the data is almost perfect. This is due to the fact that for low occupancies both the closed-node partition functions (Figure 1a) and the internode partition functions (Figure 1b− g) were determined with very good accuracy. At intermediate to high loadings, the approximate isotherm is slightly lower than the original one. This can be attributed to the fact that for intermediate to high occupancies the approximate internode

First, the node occupancy distributions pμ(0) (·) were calculated from simulations of the system without lateral interactions; these distributions were used together with the adsorption isotherm (by means of eqs 12−14) to calculate the approximate values of the closed-node partition functions, namely, the array Q̅ (0)(·). In Figure 1a, we compare their values

Figure 1. (a) Values of the original closed-node partition function, Q(0)(·): (○) as compared to the corresponding approximate values, Q̅ (0)(·); (●) as estimated through eqs 12−14. (b−q) Values of the original internode partition function, Z(·, ·): (○) as compared to the corresponding approximate values, Z̅ (·, ·); (●) as estimated through eqs 9−11.

with the exact partition functions provided by eq 15. Figure 1 illustrates that the values of the approximate partition functions are very close to those of the original functions. Thereafter, the conditional distributions gμ(·|·) are evaluated from simulations of the original system, i.e., the system with lateral interactions, and then introduced in eqs 9−11 with the approximate values of the closed-node partition function Q̅ (0)(·) to retrieve values for the internode partition function, namely, the matrix Z̅ (·, ·). In Figure 1b−q, we compare the approximate values with the original values of the entries of such a matrix. As in the previous case, the approximate values are quite close to the original ones. However, some entries are D



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29, 64 cages in both cases) was simulated at temperatures of 200, 300, and 400 K at various values of the chemical potential chosen in order to cover the full range of allowed loadings. The evaluation of each adsorption isotherm started with a chemical potential corresponding to ⟨n⟩μ ≈ 0.5 molecules per cage and proceeded by increasing the chemical potential to saturation. (We assumed nmax = 17 for the DAY case and 15 for the ITQ29 case.) Each point of the isotherm was sampled using the normal GCMC method discussed by Snurr et al.76 by means of 106 MC equilibration steps, with displacements, insertion, and removals invoked with equal probability (during which the average number of sorbate molecules per unit cell, N̅ uc, was calculated), and followed by the use of 106 N̅ uc MC steps (leading to a maximum of about 2.1 × 108 steps) for the sampling of the equilibrium properties. Access to the sodalite cages and to the double six-ringed cages was forbidden without introducing a bias, i.e., by simply rejecting insertions and displacements into the forbidden regions.76 According to the approach followed by Snurr et al., our GCMC simulations worked with fugacities rather than chemical potentials, whereas our coarse-grained model works directly with chemical potentials. Therefore, fugacities were converted to chemical potentials by means of thermodynamic data provided by the NIST database.77 Three adsorption isotherms were produced for each system: (i) one in which the interaction potential between sorbate molecules residing in different cages was set to zero (aimed at the computation of the closed-cage partition functions Q̅ (0)(·) through eqs 12−14), (ii) one in which the interactions were extended to include the first neighboring cages (aimed at the calculation of the matrix Z̅ (·, ·) through eqs 9−11), and (iii) one in which all of the possible interactions within the cutoff were taken into account (to check the validity of the locality approximation, eq 1). Comparison of the data represented by the + symbols (all interactions within the cutoff) with that represented by the ○ symbols (only intracage and neighboringcage interactions) in Figures 4 and 5 illustrates that the latter two types of isotherm were almost exactly coincident in all of the systems studied. Therefore, we can reasonably assume that in such cases the shape of the adsorption isotherms is essentially determined by the intracage and neighboring-cage interactions, i.e., the condition that makes eq 1 a valid approximation. For ITQ-29 at 200 K, the isotherms computed by taking into account only the intracage interactions do not differ much from the isotherms that include neighbor interactions (Figure 5a) but do differ in all of the other cases (Figures 4a−c and 5b,c), thus justifying the need to parameterize in a proper manner the lateral interactions in the coarse-grained model. First, we derived the approximate partition functions by following the procedure described in section II. Our estimations of the local partition functions are reported in the Supporting Information. Herein, we briefly describe their behavior. The qualitative trends of the closed-cage partition function Q̅ (0)(·) are the same as those shown in Figure 1: the lattice-gas case is the same, the case of methane in DAY zeolite is the same with a maximum, i.e., a free-energy minimum, between n = 12 and 13, and the case of methane in ITQ-29 zeolite is the same with a maximum between n = 12 and 14. The value of Q̅ (0)(·) becomes less than unity, i.e., the free energy becomes positive, in the DAY case with occupancies greater than a threshold value of between 21 and 22 and in the ITQ-29 case with occupancies with a threshold value of 18.

Figure 2. Adsorption isotherm of the lattice-gas system used to assess the validity of the method: (○) results from the original system (with known local partition functions), (●) results from a system working with partition functions recalculated from the occupancy distributions of the original system, (×) isotherm data when neighbor interactions are absent, and (inset) the effective occupancy-dependent interaction potential H(n). Chemical potentials given in kJ mol−1.

partition functions are slightly less accurate (Figure 1i−q, n > 7). However, despite the inevitable accuracy issues in the determination of the internode partition functions, the method described above produced a very acceptable approximation of the original system, as illustrated in Figure 2. We applied the method described above to the case of methane adsorbed in the all-silica zeolites DAY (dealuminated zeolite Y63−65) and ITQ-29 (all-silica zeolite A46,63,64,66−72). The framework of DAY consists of a tetrahedral array of sodalite units interconnected through six-membered oxygen bridges. Ten sodalite units enclose one α cage (each of diameter ∼11.8 Å), and every unit cell contains eight α cages. Every α cage is connected to 4 neighboring cages (according to a diamond-lattice topology) through 12-membered oxygen rings (each of diameter ∼7.5 Å, Figure 3). The structure of

Figure 3. Unit cells: (a) DAY and (b) ITQ-29. Oxygen and silica atoms are in red and yellow, respectively. Orange clouds enclose the available space points around which a methane molecule can be centered. (Access to the sodalite cages and to the double six-ringed cages was forbidden in the GCMC simulations.)

ITQ-29 consists of a simple cubic network of α cages (each of diameter ∼11.4 Å) with eight α cages per unit cell, interconnected through eight-ringed windows (each of diameter ∼4.5 Å, Figure 3). The methane−zeolite and methane−methane force fields used for parameterization of the atomistic simulations were taken from Dubbeldam et al.73 with a cutoff of 13 Å. In our atomistic GCMC simulations, the zeolite framework was kept rigid (at high loadings, the framework flexibility is expected to affect the sorption properties of molecules larger than methane).74 This allowed the pretabulation of the methane−zeolite potential energy on a 3D 124 × 124 × 124 grid with ∼0.2 Å spacing over the entire unit cell of each of the two zeolites under study, thus reducing the CPU time significantly.75 An array of 2 × 2 × 2 unit cells of each atomistic system (methane in DAY and methane in ITQE



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Figure 4. Adsorption isotherms for methane in DAY zeolite at different temperatures: (a) 200, (b) 300, and (c) 400 K. The chemical potential is given in kJ mol−1. (×) Only intracage interactions are considered; (○) and (+) include neighboring-cage interactions and the extension to all of the possible interactions within the cutoff, respectively; and (●) isotherms derived from GCMC simulations of the coarse-grained model.

Figure 5. Adsorption isotherms for methane in ITQ-29 zeolite at different temperatures: (a) 200, (b) 300, and (c) 400 K. The chemical potential is given in kJ mol−1. (×) Only intracage interactions are considered; (○) and (+) include neighboring-cage interactions and the extension to all of the possible interactions within the cutoff, respectively; and (●) isotherms derived from GCMC simulations of the coarse-grained model.

This is due to the fact that each cage in the system that lacks intercage interactions is allowed to be more crowded than those in the fully interacting system. This in turn causes higher occupancies to be reached. At such occupancies, repulsive effects prevail. As a consequence, the closed-cage partition functions become less than unity. The intercage partition functions of all systems show a weakly attractive trend (Z̅ (n, m) > 1, negative free energy) at low occupancies, whereas around n, m = 9, repulsive effects cause 0 < Z̅ (n, m) < 1 and the trend in the function Z̅ (n, ·) to decrease at fixed (high) values of n (corresponding to positive and increasing free energy). Our estimated local partition functions were used as input for coarse-grained GCMC simulations, which were carried out over systems of the same size as those in the atomistic simulations, i.e., 64 cages arranged as a diamond lattice for the simulation of the methane−DAY system and 64 cages arranged cubically for the methane−ITQ system. Each isotherm was computed over 108 GCMC steps (after 107 steps of equilibration). Filled circles in Figures 4 and 5 illustrate the coarse-grained isotherms obtained. The agreement of these data with that of the reference atomistic simulation is satisfactory; thus, we can conclude that our parameterization method works properly for the cases studied in this article and is very likely to provide accurate coarse-grained descriptions of host−guest systems where the host structure is made of approximately spherical pores connected through narrow windows and the guest is a small molecular species.

However, the method is limited to the coarse-graining of systems whose adsorption isotherms do not illustrate singularities; adsorption isotherms of type I (IUPAC classification) satisfy such a requirement. Again, the reason is that accuracy is required in the determination of the conditional occupancy distribution functions gμ(·|·). More specifically, in order for eq 9 to work properly, the gμ(·|·) matrices (one matrix for every chemical potential μ) must be sampled with reasonable accuracy; this is not possible if, for example, the reference adsorption isotherm has one or more sharp steps. As a consequence, a portion of the loadings might be sampled inefficiently or not at all so that eq 9 simply does not have enough data to work. This is the case for the methane−DAY system at 100 K (shown in Supporting Information). When we move in such a system from a fugacity value of 2.2 × 10−3 to 2.200 1 × 10−3 bar (corresponding to the values of chemical potential of −5.122 88 and −5.122 85 kJ mol−1, respectively), the loading increases abruptly from ∼1 to ∼13 molecules per cage. All of the loadings in between cannot be sampled with GCMC, causing the distribution data from ⟨n⟩μ ≈ 1.5 to 12.5 molecules per cage to be missing. With such a gap, eq 9 cannot work, and a coarse-grained model should instead be produced through EEMs. F



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IV. CONCLUSIONS A simplified mean-field model enabled us to create a recurrence equation (eq 9) that connects local occupancy distributions of small molecules adsorbed in neighboring cages of a microporous material61 to local partition functions.8 In this article, we explored some of the possibilities and limitations of such a mean-field relation in the coarse-grained reproduction of adsorption isotherms. Combined with another relation aimed at the derivation of the intracage effective interactions,51 our recurrence equation enabled us to calculate effective interaction parameters through GCMC atomistic simulations of methane in DAY or ITQ-29 at various temperatures. Once introduced into coarse-grained simulations of the corresponding systems, such parameters produced adsorption isotherms in very good agreement with atomistic data. To use the recurrence relation, eq 9, it is necessary to sample the original adsorption isotherm along the whole spectrum of loadings (from infinite dilution to saturation) with sufficient detail, e.g., ⟨n⟩μk+1 − ⟨n⟩μk ≈ 0.5 molecules per cage, and with and without intercage interactions. As a consequence, isotherms of systems that have abrupt increments in their loading, e.g., methane in DAY at 100 K, cannot be reproduced through our method. In the absence of singularities in the reference adsorption isotherm (more specifically, when the adsorption isotherm is of type I), however, our method can be used as an alternative to EEMs for the calculation of intercage partition functions to be introduced into large-scale coarse-grained simulations. The possible developments of the method discussed in this article are multifarious. On the one hand, a line of work is represented by its application to systems where the interactions are long-ranged ones and the host systems are heterogeneous or nonspherical cages, e.g., host structures made of interconnected channels rather than cages such as silicalite. If the repeating unit of the host material contains cages (or channels) differing in size and shape, then in order to retrieve a relation between occupancy distributions and local partition functions the modifications to the approximated mean-field model must be carried out in such a way that eq 6 is replaced by a set of equations (one for every type of neighborhood). On the other hand, exploring the extent to which the present method can be applied in the coarse-grained simulation of bulk systems might be of a certain level of interest in the field of multiscale modeling.

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Rimozione e l′Immagazzinamento della CO2 developed at Università degli Studi di Sassari by means of the contract Ricercatore a Tempo Determinato financed through the resources of POR Sardegna FSE 2007/2013−Obiettivo Competitività Regionale e Occupazione, Asse IV Capitale Umano, Linea di Attività l.3.1.

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ASSOCIATED CONTENT

S Supporting Information *

Details regarding the lattice-gas Monte Carlo simulations, plots of local partition functions for the molecular systems investigated, and the adsorption isotherm for methane in DAY at 100 K. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Phone: +39 079 22 94 96. Fax: +39 079 22 95 59. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The present work was produced within the research project Progettazione su Calcolatore di Materiali Avanzati per la G



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