The statistical thermodynamics of polyatomic species mixtures adsorbed on
two-dimensional substrates was developed on a generalization in the spirit of
the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In
this scheme, the coverage and temperature dependence of the Helmholtz free
energy and chemical potential are given. The formalism leads to the exact
statistical thermodynamics of binary mixtures adsorbed in one dimension,
provides a close approximation for two-dimensional systems accounting multisite
occupancy and allows to discuss the dimensionality and lattice structure
effects on the known phenomenon of adsorption preference reversal.Comment: 13 pages, 4 figure

Matoz-Fernandez, D. A.

Ramirez-Pastor, A. J.

Chemical Physics Letters

English

arXiv

The statistical thermodynamics of polyatomic species mixtures adsorbed on two-dimensional substrates was developed on a generalization in the spirit of the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In this scheme, the coverage and temperature dependence of the Helmholtz free energy and chemical potential are given. The formalism leads to the exact statistical thermodynamics of binary mixtures adsorbed in one dimension, provides a close approximation for two-dimensional systems accounting multisite occupancy and allows to discuss the dimensionality and lattice structure effects on the known phenomenon of adsorption preference reversal.Fil: Matoz Fernandez, Daniel Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Física Aplicada ; ArgentinaFil: Ramirez Pastor, Antonio Jose. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Física Aplicada ; Argentin

Matoz Fernandez, Daniel Alejandro

Ramirez Pastor, Antonio Jose

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:1406.4809v1  [cond-mat.stat-mech]  18 Jun 2014Adsorption preference reversal phenomenonfrom multisite-occupancy theory fortwo-dimensional latticesD. A. Matoz-Fernandez and A. J. Ramirez-Pastor 1Departamento de F́ısica, Instituto de F́ısica Aplicada, Universidad Nacional deSan Luis-CONICET, Ejército de los Andes 950, D5700BWS, San Luis, ArgentinaAbstractThe statistical thermodynamics of polyatomic species mixtures adsorbed on two-dimensional substrates was developed on a generalization in the spirit of the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In this scheme,the coverage and temperature dependence of the Helmholtz free energy and chemicalpotential are given. The formalism leads to the exact statistical thermodynamicsof binary mixtures adsorbed in one dimension, provides a close approximation fortwo-dimensional systems accounting multisite occupancy and allows to discuss thedimensionality and lattice structure effects on the known phenomenon of adsorptionpreference reversal.1 IntroductionTwo previous articles [1,2] referred to as papers I and II, respectively, weredevoted to the study of binary mixtures adsorption. In paper I, the multicom-ponent adsorption of polyatomic species was described as a fractional statisticsproblem, based on Haldanes statistics [3,4]. The thermodynamic functions cal-culated for a monomer-dimer mixture were applied to describe the adsorptionof methane-ethane mixtures in zeolites. The obtained results revealed that,at low pressure, the adsorbed phase is almost entirely ethane, but at high-pressure methane displaces ethane reproducing the adsorption preference re-versal (APR) phenomenon observed in Monte Carlo simulations, mean-fieldtheory, and exact calculations [5,6,7,8] of the system under study.1 Corresponding author. E-mail: antorami@unsl.edu.arPreprint submitted to Chemical Physics Letters 19 June 2014In Refs. [7,8], the authors had shown how the competition between two speciesin presence of repulsive mutual interactions can be responsible of the displace-ment of one species by the other. The results in paper I contributed to theunderstanding of the phenomenon, showing that if realistic single species ad-sorption energy values are used when studying mixtures, the APR will appearas a result of the difference of size (or number of occupied sites) between theadsorbed species. Thus, to introduce repulsive lateral interactions in the ad-sorbate, as done in Refs. [7,8], is a fictitious or effective way of taking intoaccount geometric or steric effects by means of energetic arguments. A realdescription of the displacement of flat ethane by methane and upright ethaneshould include multisite-occupancy adsorption.Paper II was a step further, addressing the rigorous statistical thermodynam-ics of s-mer (particle occupying s lattice sites)k-mer (particle occupying k lat-tice sites) mixtures adsorbed on one-dimensional substrates. The formalismpresented in paper II was the first exact model of adsorption of polyatomicmixtures in zeolites and allows to demonstrate that the APR phenomenonis the result of the difference of size (or number of occupied sites) betweenthe adsorbed species. In other words, the results in paper II revealed that areal description of the phenomenon of APR may be severely misunderstood,if the polyatomic character of the adsorbate is not properly incorporated inthe thermodynamic functions from which experiments are interpreted.In contrast to the statistics for simple particles, where the arrangement of theadsorption sites in space is immaterial, the structure of lattice space playssuch a fundamental role in determining the statistics of k-mers. Then, it is ofinterest and of value to inquire how a specific lattice structure influences themain thermodynamic properties of adsorbed polyatomics. In this sense, theaim of the present work is to extend the study in paper II to two-dimensionallattices. The problem is not only of theoretical interest, but also has practicalimportance (most of the experiments in surface science are carried out in twodimensions).2 Theory: lattice model for adsorption of alkane binary mixturesIn this paper, the adsorption of a binary mixture of straight rigid k-mers andstraight rigid l-mers on two-dimensional lattices is considered. The k(l)-mersare assumed to be composed by k(l) identical units in a linear array withconstant bond length equal to the lattice constant a. Without any loss of gen-erality, we assume that l < k. The substrate is modeled by a two-dimensionalarray of M sites (M → ∞) and connectivity γ, where periodic boundaryconditions apply. Under this condition, all lattice sites are equivalent, henceborder effects will not enter our derivation. The k(l)-mers can only adsorb flat2on the surface occupying k(l) contiguous lattice sites. In addition, double siteoccupancy is not allowed as to represent properties in the monolayer regime.Since different particles do not interact with each other, all configurations ofNk k-mers and Nl l-mers on M sites are equally probable. Then, the canonicalpartition function Q(M,Nk, Nl, T ) equals the total number of configurations,Ω(M,Nk, Nl), times a Boltzmann factor including the total interaction energybetween adparticles and substrate ǫ(Nk, Nl),Q(M,Nk, Nl, T ) = Ω(M,Nk, Nl) exp[−ǫ(Nk, Nl)kBT]. (1)In order to calculate Ω(M,Nk, Nl), the DiMarzio’s lattice theory can be used[9]. Let’s start calculating the number of distinct ways to pack Nk rigid rodsonto a lattice with d allowed orientations (directions),Ωk(M, {Nk}d) =∏di=1 [M − (k − 1)Nk,i]!(M − k Nk)! (M !)d−1∏di=1 (Nk,i)!, (2)where Nk,i is the number of k-mers lying in the direction i and Nk =∑di=1Nk,iis the total number of k -mers on the surface.Now, using the DiMarzio counting scheme, the number of ways to place the(j1 + 1)th l type molecule onto the lattice (the subscript reminds us that weare discussing the orientation 1), given that j1 l molecules have already beenplaced in the direction 1 and Nk type k molecules have already been placed,is seen to be [10],νj1+1 = (M − kNk − l j1)[M − k Nk − l j1M − (k − 1)Nk,1 − (l − 1)j1]l−1. (3)The total number of ways to place Nl,1 indistinguishable molecules onto thelattice in this orientation is,∏N1−1j1=0 νj1+1(Nl,1)!=(M − k Nk)! [M − (k − 1)Nk,1 − (l − 1)Nl,1]!(M − k Nk − l Nl,1)! [M − (k − 1)Nk,1]!Nl,1!. (4)Similar expressions can be obtained for the other orientations and total num-bers of ways to place Nl hard rod molecules type l when Nk type k have beenplace in the surface is,Ωl(M, {Nl}d) =(M − k Nk)!∏di=1 [M − (k − 1)Nk,i − (l − 1)Nl,i]!(M − k Nk − l Nl)!∏di=1 [M − (k − 1)Nk,i]! (Nl,i)!. (5)Them the product obtained from Eqs. (2) and (5) gives the total number of3ways to pack the molecules in the mixture:Ω(M, {Nk}d, {Nl}d) =∏di=1 [M − (k − 1)Nk,i − (l − 1)Nl,i]!(M − k Nk − l Nl)! (M !)d−1∏di=1 (Nk,i)! (Nl,i)!. (6)Equation (6) is exact when all molecules live in one direction [2]. For the caseof an isotropic distribution of molecules, i.e., Nk(l),i = (2/γ)Nk(l), then theappropriate generalization of Eq. (6) is,Ω(M,Nk, Nl) =M !(M − kNk − lNl)![M − (k − 1)γ2Nk − (l − 1)γ2Nl]!M !Nk!Nl!γ2.(7)In the canonical ensemble, the Helmholtz free energy F (M,Nk, Nl, T ) relatesto Ω(M,Nk, Nl) through,βF (M,Nk, Nl, T ) = − lnQ(M,Nk, Nl, T )= − ln Ω(M,Nk, Nl) + βǫ(Nk, Nl),(8)where β = 1/kBT , ǫ(Nk, Nl) = ǫk Nk + ǫl Nl and ǫi represents the interactionenergy of a i-mer (i = k, l) adsorbed on the substrate.The chemical potential of the adsorbed species i, µi,ads, can be calculated as[11],µi,ads =(∂F∂Ni)Nj{i, j = k, l}. (9)From Eqs. (7-9) it follows that,µk,ads =(k − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θkk)− k ln (1− θk − θl)− k ln(γ2)+ β ǫk,(10)andµl,ads =(l − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θll)− l ln (1− θk − θl)− l ln(γ2)+ β ǫl,(11)where θi = iNi/M represents the partial coverage of the species i {i = k, l}. Atequilibrium, the chemical potential of the adsorbed and gas phase are equal.Then,µk,ads = µk,gas, (12)andµl,ads = µl,gas, (13)4where µk,gas (µl,gas) corresponds to k-mers (l-mers) in gas phase.The chemical potential of each kind of molecule in an ideal gas mixture, attemperature T and pressure P , isβµk,gas = βµ0k + lnXP, (14)andβµl,gas = βµ0l + ln(1−X)P, (15)where µ0k and µ0l (X and 1 − X) are the standard chemical potentials (molefractions) of k-mers and l-mers, respectively. In addition,βµ0i = − ln(2πmikBTh2)3/2kBT {i = k, l}. (16)3 Adsorption preference reversal phenomenon in two dimensionsOur model can be used for predict the behavior of APR phenomenon forn-alkanes mixtures. As in Refs.[2,7], we start defining the Φi’s parameters,βΦk ≡ βǫk − βµ0k − lnXP, (17)andβΦl ≡ βǫl − βµ0l − ln(1−X)P. (18)Then, from Eqs. (10-18), it results that(k − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θkk)− k ln (1− θk − θl)− k ln(γ2)+ βΦk = 0,(19)and(l − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θll)− l ln (1− θk − θl)− l ln(γ2)+ βΦl = 0.(20)Now, following the line of Refs.[2,7], we define the quantity A ≡ exp [β(Φk − Φl)],5which is obtained from the equilibrium Eqs. (19) and (20),A =1−XXexp[β(ǫk − ǫl)− β(µ0k − µ0l )]=kθllθk(1− θk − θl)k−l[γ2− (k−1)kθk −(l−1)lθl]k−l(γ2)k−l.(21)As shown in Fig. 1, if the APR phenomenon occurs, then there exists somevalue of coverage, θ∗ (0 < θ∗ < 1), at which the partial isotherms coincide(θk = θl = θ∗). The crossing point (µ∗, θ∗) separates two adsorption regimes.Thus, for µ < µ∗, θk is larger than θl. This tendency is reverted for µ > µ∗,where θl is larger than θk. Then, the existence of the point θ∗ is directly relatedto the displacement of the species k by the species s, and, consequently, tothe presence of APR phenomenon. The value of θ∗ can be obtained from Eq.(21),θ∗ =1−(lAk)1/(k−l)2−(k−1k+ l−1l) (2γ) (lAk)1/(k−l), (22)with the condition that 0 < A < 1. This condition is satisfied given that theadsorption energies corresponding to linear alkanes are attractive and increase,in absolute value, linearly with the chain length [12,13,14,15,16].In the following, the dependence of θ∗ on the adsorption energies ǫk and ǫl,the geometry γ and the mole fraction X will be investigated. As is common inthe literature [15,16], we adopt a “bead segment” model of the alkane chains,in which each methyl (bead) group occupies one adsorption site on the surfacewith adsorption energy ǫ0. Under this consideration, ǫk = kǫ0 and ǫl = lǫ0. Inaddition, the values of βµ0k and βµ0l were obtained by using Eq. (16) with miequal to the molar mass of an alkane of size i, CiH(2i+2) [18], and T=300 K.Figure 2 shows the dependence of θ∗ on k for a (l − k) mixture with l =1 2 , γ = 4 (square lattice), X = 0.5 and different values of the adsorptionenergy βǫ0(= −3,−6,−12,−24). The curves increase monotonically with thesize k and converge asymptotically to a finite value as k → ∞. This finitevalue tends to 0.5 for high values of βǫ0. In the case of the figure, θ∗(∞) ≈0.375, 0.465, 0.497, and 0.5 for βǫ0 = −3,−6,−12, and −24, respectively.In order to analyze the effect of lattice geometry, the coverage θ∗ was calculatedas a function of k for a typical case (βǫ0 = −3, l = 1 and X = 0.5) and one-dimensional (γ = 2), honeycomb (γ = 3), square (γ = 4) and triangular(γ = 6) lattices. The results are shown in Fig.3. The general form of the2 The (l− k) mixture with l = 1 and k ≥ 2 is the simplest case of an alkane binarymixture and contains all the properties of the multisite occupancy adsorption.6curves is similar to that reported in Fig. 2. In this case, the limit value θ∗(∞)strongly depends on the connectivity γ, being an increasing function of γ.To conclude the analysis of Eq. (22), the dependence of θ∗ on the molar fractionX is analyzed in Fig. 4. The parameters chosen for this study were βǫ0 = −3,l = 1 and γ = 4. Several conclusions can be drawn from the figure.For values of X between X = 0[X ≈ 0.448] and X ≈ 0.448[X = 1] and kvarying between k = 0 and k ≈ 20, θ∗ is a decreasing[increasing] function ofthe parameter k. For k > 20, all curves merge into one single curve whichconverges to a constant value as k → ∞. This constant value (near 0.38 in thefigure) depends on the lattice connectivity and the value of A.For small values of k, APR phenomenon disappears as the molar fraction isabove a certain critical value X = X∗. This behavior determines the phasediagram shown in the inset of Fig. 4, which does not change with the latticegeometry, and only depends on the differences between the relative standardchemical potentials and the adsorption energies.In summary, the statistical thermodynamics of polyatomic species mixturesadsorbed on two-dimensional substrates was developed on a generalizationin the spirit of the lattice-gas model and the classical Guggenheim-DiMarzioapproximation. The theoretical formalism allows to study the problem of ad-sorption of alkane binary mixtures, leads to the exact solution in one dimen-sion, provides a close approximation for two-dimensional systems accountingmultisite occupancy and represents a contribution to the understanding of thephenomenon of adsorption preference reversal.AcknowledgmentsThis work was supported in part by CONICET (Argentina) under projectnumber PIP 112-201101-00615; Universidad Nacional de San Luis (Argentina)under project 322000 and the National Agency of Scientific and TechnologicalPromotion (Argentina) under project PICT-2010-1466.7References[1] M. Dávila, J.L. Riccardo, A.J. Ramirez-Pastor, J. Chem. Phys. 130 (2009)174715.[2] M. Dávila, J.L. Riccardo, A.J. Ramirez-Pastor, Chem. Phys. Lett. 477 (2009)402.[3] F.D.M. Haldane, Phys. Rev. Lett. 67 (1991) 937.[4] Y.S. Wu, Phys. Rev. Lett. 73 (1994) 922.[5] Z. Du, G. Manos, T.J.H. Vlugt, B. Smit, AIChE J. 44 (1998) 1756.[6] A. Khettar, S.E. Jalili, L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 362(2002) 414.[7] K. Ayache, S.E. Jalili, L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 362(2002) 414.[8] L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 377 (2003) 551.[9] E.A. DiMarzio, J. Chem. Phys. 35 (1961) 658.[10] H.T. Peterson, D.E. Martire, M.A. Cotter, J. Chem. Phys. 61 (1974) 3547.[11] T.L. Hill, An Introduction to Statistical Thermodynamics, Addison WesleyPublishing Company, Reading, MA, 1960.[12] M.A. Chaer Nascimento (Ed.), Theoretical Aspects of Heterogeneous Catalysis,Kluwer Academic Publishers, Boston, MA, 2002; and references therein.[13] P. Ungerer, B. Tavitian, A. Boutin, Applications of Molecular Simulation in theOil and Gas Industry: Monte Carlo Methods, Editions Technip, Paris, 2005; andreferences therein.[14] J.A.C. Silva, A.E. Rodrigues, Ind. Eng. Chem. Res. 38 (1999) 2434.[15] F. Romá, J.L. Riccardo, A.J. Ramirez-Pastor, Langmuir 21 (2005) 2454.[16] F. Romá, J. L. Riccardo, A. J. Ramirez-Pastor, Ind. Eng. Chem. Res. 45 (2006)2046.[17] K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics. AnIntroduction, Springer, Berlin, 2010.[18] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 88th edn., CRCPress/ Taylor and Francis, Boca Raton, FL, 2008 (Internet Version 2008).84 Figure CaptionsFigure 1: Typical partial adsorption isotherms for a k-mer-l-mer mixture in the pres-ence of APR phenomenon. The crossing point (µ∗, θ∗) separates two adsorp-tion regimes. Thus, for µ < µ∗, θk is larger than θl. This tendency is revertedfor µ > µ∗, where θl is larger than θk. Then, the existence of the point θ∗ isdirectly related to the displacement of the species k by the species l, and,consequently, to the occurrence of APR phenomenon.Figure 2: Dependence of θ∗ on k for a (l−k) mixture with l = 1, X = 0.5 and differentvalues of the adsorption energy βǫ0 as indicated.Figure 3: θ∗ as a function of k for a typical case (βǫ0 = −3, l = 1 and X = 0.5) anddifferent geometries as indicated.Figure 4: θ∗ as a function of k for a typical case (βǫ0 = −3, l = 1 and γ = 4) anddifferent values of the molar fraction X as indicated.90 10 20 300.00.20.40.60.81.0Fig. 1: D. A. Matoz-Fernandez et al. species k species l  i, fractional coverage, chemical potential*0 10 20 30 40 50 600.20.30.40.5Fig. 2: D. A. Matoz-Fernandez et al.  *k 0 = - 3.0 0 = - 6.0 0 = - 12.0 0 = - 24.0 = 4; l = 1; X = 0.50 10 20 30 40 50 600.320.360.40Fig. 3: D. A. Matoz-Fernandez et al.  k  = 2  = 3  = 4  = 60 = - 3.0; l = 1; X = 0.5*0 20 40 60 80 1000.00.10.20.30.40.50.000.250.500.751.000 2 4 6 8 10 12 14 16 18Fig. 4: D. A. Matoz-Fernandez et al.k X= 0.1 X= 0.3 X= 0.5 X= 0.7 X= 0.9*0 = - 3.0; ; l = 1kX* Region of occurrenceof APR phenomenon

Adsorption preference reversal phenomenon from multisite-occupancy theory for two-dimensional lattices

CONICET Digital

arXiv:1406.4809v1  [cond-mat.stat-mech]  18 Jun 2014Adsorption preference reversal phenomenonfrom multisite-occupancy theory fortwo-dimensional latticesD. A. Matoz-Fernandez and A. J. Ramirez-Pastor 1Departamento de F´ısica, Instituto de F´ısica Aplicada, Universidad Nacional deSan Luis-CONICET, Eje´rcito de los Andes 950, D5700BWS, San Luis, ArgentinaAbstractThe statistical thermodynamics of polyatomic species mixtures adsorbed on two-dimensional substrates was developed on a generalization in the spirit of the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In this scheme,the coverage and temperature dependence of the Helmholtz free energy and chemicalpotential are given. The formalism leads to the exact statistical thermodynamicsof binary mixtures adsorbed in one dimension, provides a close approximation fortwo-dimensional systems accounting multisite occupancy and allows to discuss thedimensionality and lattice structure effects on the known phenomenon of adsorptionpreference reversal.1 IntroductionTwo previous articles [1,2] referred to as papers I and II, respectively, weredevoted to the study of binary mixtures adsorption. In paper I, the multicom-ponent adsorption of polyatomic species was described as a fractional statisticsproblem, based on Haldanes statistics [3,4]. The thermodynamic functions cal-culated for a monomer-dimer mixture were applied to describe the adsorptionof methane-ethane mixtures in zeolites. The obtained results revealed that,at low pressure, the adsorbed phase is almost entirely ethane, but at high-pressure methane displaces ethane reproducing the adsorption preference re-versal (APR) phenomenon observed in Monte Carlo simulations, mean-fieldtheory, and exact calculations [5,6,7,8] of the system under study.1 Corresponding author. E-mail: antorami@unsl.edu.arPreprint submitted to Chemical Physics Letters 19 June 2014In Refs. [7,8], the authors had shown how the competition between two speciesin presence of repulsive mutual interactions can be responsible of the displace-ment of one species by the other. The results in paper I contributed to theunderstanding of the phenomenon, showing that if realistic single species ad-sorption energy values are used when studying mixtures, the APR will appearas a result of the difference of size (or number of occupied sites) between theadsorbed species. Thus, to introduce repulsive lateral interactions in the ad-sorbate, as done in Refs. [7,8], is a fictitious or effective way of taking intoaccount geometric or steric effects by means of energetic arguments. A realdescription of the displacement of flat ethane by methane and upright ethaneshould include multisite-occupancy adsorption.Paper II was a step further, addressing the rigorous statistical thermodynam-ics of s-mer (particle occupying s lattice sites)k-mer (particle occupying k lat-tice sites) mixtures adsorbed on one-dimensional substrates. The formalismpresented in paper II was the first exact model of adsorption of polyatomicmixtures in zeolites and allows to demonstrate that the APR phenomenonis the result of the difference of size (or number of occupied sites) betweenthe adsorbed species. In other words, the results in paper II revealed that areal description of the phenomenon of APR may be severely misunderstood,if the polyatomic character of the adsorbate is not properly incorporated inthe thermodynamic functions from which experiments are interpreted.In contrast to the statistics for simple particles, where the arrangement of theadsorption sites in space is immaterial, the structure of lattice space playssuch a fundamental role in determining the statistics of k-mers. Then, it is ofinterest and of value to inquire how a specific lattice structure influences themain thermodynamic properties of adsorbed polyatomics. In this sense, theaim of the present work is to extend the study in paper II to two-dimensionallattices. The problem is not only of theoretical interest, but also has practicalimportance (most of the experiments in surface science are carried out in twodimensions).2 Theory: lattice model for adsorption of alkane binary mixturesIn this paper, the adsorption of a binary mixture of straight rigid k-mers andstraight rigid l-mers on two-dimensional lattices is considered. The k(l)-mersare assumed to be composed by k(l) identical units in a linear array withconstant bond length equal to the lattice constant a. Without any loss of gen-erality, we assume that l < k. The substrate is modeled by a two-dimensionalarray of M sites (M → ∞) and connectivity γ, where periodic boundaryconditions apply. Under this condition, all lattice sites are equivalent, henceborder effects will not enter our derivation. The k(l)-mers can only adsorb flat2on the surface occupying k(l) contiguous lattice sites. In addition, double siteoccupancy is not allowed as to represent properties in the monolayer regime.Since different particles do not interact with each other, all configurations ofNk k-mers and Nl l-mers onM sites are equally probable. Then, the canonicalpartition function Q(M,Nk, Nl, T ) equals the total number of configurations,Ω(M,Nk, Nl), times a Boltzmann factor including the total interaction energybetween adparticles and substrate ǫ(Nk, Nl),Q(M,Nk, Nl, T ) = Ω(M,Nk, Nl) exp[−ǫ(Nk, Nl)kBT]. (1)In order to calculate Ω(M,Nk, Nl), the DiMarzio’s lattice theory can be used[9]. Let’s start calculating the number of distinct ways to pack Nk rigid rodsonto a lattice with d allowed orientations (directions),Ωk(M, {Nk}d) =∏di=1 [M − (k − 1)Nk,i]!(M − k Nk)! (M !)d−1∏di=1 (Nk,i)!, (2)where Nk,i is the number of k-mers lying in the direction i and Nk =∑di=1Nk,iis the total number of k -mers on the surface.Now, using the DiMarzio counting scheme, the number of ways to place the(j1 + 1)th l type molecule onto the lattice (the subscript reminds us that weare discussing the orientation 1), given that j1 l molecules have already beenplaced in the direction 1 and Nk type k molecules have already been placed,is seen to be [10],νj1+1 = (M − kNk − l j1)[M − k Nk − l j1M − (k − 1)Nk,1 − (l − 1)j1]l−1. (3)The total number of ways to place Nl,1 indistinguishable molecules onto thelattice in this orientation is,∏N1−1j1=0 νj1+1(Nl,1)!=(M − k Nk)! [M − (k − 1)Nk,1 − (l − 1)Nl,1]!(M − k Nk − l Nl,1)! [M − (k − 1)Nk,1]!Nl,1!. (4)Similar expressions can be obtained for the other orientations and total num-bers of ways to place Nl hard rod molecules type l when Nk type k have beenplace in the surface is,Ωl(M, {Nl}d) =(M − k Nk)!∏di=1 [M − (k − 1)Nk,i − (l − 1)Nl,i]!(M − k Nk − l Nl)!∏di=1 [M − (k − 1)Nk,i]! (Nl,i)!. (5)Them the product obtained from Eqs. (2) and (5) gives the total number of3ways to pack the molecules in the mixture:Ω(M, {Nk}d, {Nl}d) =∏di=1 [M − (k − 1)Nk,i − (l − 1)Nl,i]!(M − k Nk − l Nl)! (M !)d−1∏di=1 (Nk,i)! (Nl,i)!. (6)Equation (6) is exact when all molecules live in one direction [2]. For the caseof an isotropic distribution of molecules, i.e., Nk(l),i = (2/γ)Nk(l), then theappropriate generalization of Eq. (6) is,Ω(M,Nk, Nl) =M !(M − kNk − lNl)![M − (k − 1)γ2Nk − (l − 1)γ2Nl]!M !Nk!Nl!γ2.(7)In the canonical ensemble, the Helmholtz free energy F (M,Nk, Nl, T ) relatesto Ω(M,Nk, Nl) through,βF (M,Nk, Nl, T ) = − lnQ(M,Nk, Nl, T )= − ln Ω(M,Nk, Nl) + βǫ(Nk, Nl),(8)where β = 1/kBT , ǫ(Nk, Nl) = ǫkNk + ǫlNl and ǫi represents the interactionenergy of a i-mer (i = k, l) adsorbed on the substrate.The chemical potential of the adsorbed species i, µi,ads, can be calculated as[11],µi,ads =(∂F∂Ni)Nj{i, j = k, l}. (9)From Eqs. (7-9) it follows that,µk,ads =(k − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θkk)− k ln (1− θk − θl)− k ln(γ2)+ β ǫk,(10)andµl,ads =(l − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θll)− l ln (1− θk − θl)− l ln(γ2)+ β ǫl,(11)where θi = iNi/M represents the partial coverage of the species i {i = k, l}. Atequilibrium, the chemical potential of the adsorbed and gas phase are equal.Then,µk,ads = µk,gas, (12)andµl,ads = µl,gas, (13)4where µk,gas (µl,gas) corresponds to k-mers (l-mers) in gas phase.The chemical potential of each kind of molecule in an ideal gas mixture, attemperature T and pressure P , isβµk,gas = βµ0k + lnXP, (14)andβµl,gas = βµ0l + ln(1−X)P, (15)where µ0k and µ0l (X and 1 − X) are the standard chemical potentials (molefractions) of k-mers and l-mers, respectively. In addition,βµ0i = − ln(2πmikBTh2)3/2kBT {i = k, l}. (16)3 Adsorption preference reversal phenomenon in two dimensionsOur model can be used for predict the behavior of APR phenomenon forn-alkanes mixtures. As in Refs.[2,7], we start defining the Φi’s parameters,βΦk ≡ βǫk − βµ0k − lnXP, (17)andβΦl ≡ βǫl − βµ0l − ln(1−X)P. (18)Then, from Eqs. (10-18), it results that(k − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θkk)− k ln (1− θk − θl)− k ln(γ2)+ βΦk = 0,(19)and(l − 1) ln[γ2−(k − 1)kθk −(l − 1)lθl]+ ln(θll)− l ln (1− θk − θl)− l ln(γ2)+ βΦl = 0.(20)Now, following the line of Refs.[2,7], we define the quantity A ≡ exp [β(Φk − Φl)],5which is obtained from the equilibrium Eqs. (19) and (20),A =1−XXexp[β(ǫk − ǫl)− β(µ0k − µ0l )]=kθllθk(1− θk − θl)k−l[γ2− (k−1)kθk −(l−1)lθl]k−l(γ2)k−l.(21)As shown in Fig. 1, if the APR phenomenon occurs, then there exists somevalue of coverage, θ∗ (0 < θ∗ < 1), at which the partial isotherms coincide(θk = θl = θ∗). The crossing point (µ∗, θ∗) separates two adsorption regimes.Thus, for µ < µ∗, θk is larger than θl. This tendency is reverted for µ > µ∗,where θl is larger than θk. Then, the existence of the point θ∗ is directly relatedto the displacement of the species k by the species s, and, consequently, tothe presence of APR phenomenon. The value of θ∗ can be obtained from Eq.(21),θ∗ =1−(lAk)1/(k−l)2−(k−1k+ l−1l) (2γ) (lAk)1/(k−l) , (22)with the condition that 0 < A < 1. This condition is satisfied given that theadsorption energies corresponding to linear alkanes are attractive and increase,in absolute value, linearly with the chain length [12,13,14,15,16].In the following, the dependence of θ∗ on the adsorption energies ǫk and ǫl,the geometry γ and the mole fraction X will be investigated. As is common inthe literature [15,16], we adopt a “bead segment” model of the alkane chains,in which each methyl (bead) group occupies one adsorption site on the surfacewith adsorption energy ǫ0. Under this consideration, ǫk = kǫ0 and ǫl = lǫ0. Inaddition, the values of βµ0k and βµ0l were obtained by using Eq. (16) with miequal to the molar mass of an alkane of size i, CiH(2i+2) [18], and T=300 K.Figure 2 shows the dependence of θ∗ on k for a (l − k) mixture with l =1 2 , γ = 4 (square lattice), X = 0.5 and different values of the adsorptionenergy βǫ0(= −3,−6,−12,−24). The curves increase monotonically with thesize k and converge asymptotically to a finite value as k → ∞. This finitevalue tends to 0.5 for high values of βǫ0. In the case of the figure, θ∗(∞) ≈0.375, 0.465, 0.497, and 0.5 for βǫ0 = −3,−6,−12, and −24, respectively.In order to analyze the effect of lattice geometry, the coverage θ∗ was calculatedas a function of k for a typical case (βǫ0 = −3, l = 1 and X = 0.5) and one-dimensional (γ = 2), honeycomb (γ = 3), square (γ = 4) and triangular(γ = 6) lattices. The results are shown in Fig.3. The general form of the2 The (l− k) mixture with l = 1 and k ≥ 2 is the simplest case of an alkane binarymixture and contains all the properties of the multisite occupancy adsorption.6curves is similar to that reported in Fig. 2. In this case, the limit value θ∗(∞)strongly depends on the connectivity γ, being an increasing function of γ.To conclude the analysis of Eq. (22), the dependence of θ∗ on the molar fractionX is analyzed in Fig. 4. The parameters chosen for this study were βǫ0 = −3,l = 1 and γ = 4. Several conclusions can be drawn from the figure.For values of X between X = 0[X ≈ 0.448] and X ≈ 0.448[X = 1] and kvarying between k = 0 and k ≈ 20, θ∗ is a decreasing[increasing] function ofthe parameter k. For k > 20, all curves merge into one single curve whichconverges to a constant value as k →∞. This constant value (near 0.38 in thefigure) depends on the lattice connectivity and the value of A.For small values of k, APR phenomenon disappears as the molar fraction isabove a certain critical value X = X∗. This behavior determines the phasediagram shown in the inset of Fig. 4, which does not change with the latticegeometry, and only depends on the differences between the relative standardchemical potentials and the adsorption energies.In summary, the statistical thermodynamics of polyatomic species mixturesadsorbed on two-dimensional substrates was developed on a generalizationin the spirit of the lattice-gas model and the classical Guggenheim-DiMarzioapproximation. The theoretical formalism allows to study the problem of ad-sorption of alkane binary mixtures, leads to the exact solution in one dimen-sion, provides a close approximation for two-dimensional systems accountingmultisite occupancy and represents a contribution to the understanding of thephenomenon of adsorption preference reversal.AcknowledgmentsThis work was supported in part by CONICET (Argentina) under projectnumber PIP 112-201101-00615; Universidad Nacional de San Luis (Argentina)under project 322000 and the National Agency of Scientific and TechnologicalPromotion (Argentina) under project PICT-2010-1466.7References[1] M. Da´vila, J.L. Riccardo, A.J. Ramirez-Pastor, J. Chem. Phys. 130 (2009)174715.[2] M. Da´vila, J.L. Riccardo, A.J. Ramirez-Pastor, Chem. Phys. Lett. 477 (2009)402.[3] F.D.M. Haldane, Phys. Rev. Lett. 67 (1991) 937.[4] Y.S. Wu, Phys. Rev. Lett. 73 (1994) 922.[5] Z. Du, G. Manos, T.J.H. Vlugt, B. Smit, AIChE J. 44 (1998) 1756.[6] A. Khettar, S.E. Jalili, L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 362(2002) 414.[7] K. Ayache, S.E. Jalili, L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 362(2002) 414.[8] L.J. Dunne, G. Manos, Z. Du, Chem. Phys. Lett. 377 (2003) 551.[9] E.A. DiMarzio, J. Chem. Phys. 35 (1961) 658.[10] H.T. Peterson, D.E. Martire, M.A. Cotter, J. Chem. Phys. 61 (1974) 3547.[11] T.L. Hill, An Introduction to Statistical Thermodynamics, Addison WesleyPublishing Company, Reading, MA, 1960.[12] M.A. Chaer Nascimento (Ed.), Theoretical Aspects of Heterogeneous Catalysis,Kluwer Academic Publishers, Boston, MA, 2002; and references therein.[13] P. Ungerer, B. Tavitian, A. Boutin, Applications of Molecular Simulation in theOil and Gas Industry: Monte Carlo Methods, Editions Technip, Paris, 2005; andreferences therein.[14] J.A.C. Silva, A.E. Rodrigues, Ind. Eng. Chem. Res. 38 (1999) 2434.[15] F. Roma´, J.L. Riccardo, A.J. Ramirez-Pastor, Langmuir 21 (2005) 2454.[16] F. Roma´, J. L. Riccardo, A. J. Ramirez-Pastor, Ind. Eng. Chem. Res. 45 (2006)2046.[17] K. Binder, D.W. Heermann, Monte Carlo Simulation in Statistical Physics. AnIntroduction, Springer, Berlin, 2010.[18] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 88th edn., CRCPress/ Taylor and Francis, Boca Raton, FL, 2008 (Internet Version 2008).84 Figure CaptionsFigure 1: Typical partial adsorption isotherms for a k-mer-l-mer mixture in the pres-ence of APR phenomenon. The crossing point (µ∗, θ∗) separates two adsorp-tion regimes. Thus, for µ < µ∗, θk is larger than θl. This tendency is revertedfor µ > µ∗, where θl is larger than θk. Then, the existence of the point θ∗ isdirectly related to the displacement of the species k by the species l, and,consequently, to the occurrence of APR phenomenon.Figure 2: Dependence of θ∗ on k for a (l−k) mixture with l = 1, X = 0.5 and differentvalues of the adsorption energy βǫ0 as indicated.Figure 3: θ∗ as a function of k for a typical case (βǫ0 = −3, l = 1 and X = 0.5) anddifferent geometries as indicated.Figure 4: θ∗ as a function of k for a typical case (βǫ0 = −3, l = 1 and γ = 4) anddifferent values of the molar fraction X as indicated.90 10 20 300.00.20.40.60.81.0Fig. 1: D. A. Matoz-Fernandez et al. species k species l  i, fractional coverage, chemical potential*0 10 20 30 40 50 600.20.30.40.5Fig. 2: D. A. Matoz-Fernandez et al.  *k 0 = - 3.0 0 = - 6.0 0 = - 12.0 0 = - 24.0 = 4; l = 1; X = 0.50 10 20 30 40 50 600.320.360.40Fig. 3: D. A. Matoz-Fernandez et al.  k  = 2  = 3  = 4  = 60 = - 3.0; l = 1; X = 0.5*0 20 40 60 80 1000.00.10.20.30.40.50.000.250.500.751.000 2 4 6 8 10 12 14 16 18Fig. 4: D. A. Matoz-Fernandez et al.k X= 0.1 X= 0.3 X= 0.5 X= 0.7 X= 0.9*0 = - 3.0; ; l = 1kX* Region of occurrenceof APR phenomenon

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Adsorption preference reversal phenomenon from multisite-occupancy theory fortwo-dimensional lattices

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