Adsorption Thermodynamics with Multisite Occupancy at Criticality

Langmuir 2000, 16, 10167-10174

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Adsorption Thermodynamics with Multisite Occupancy at Criticality A. J. Ramirez-Pastor,* J. L. Riccardo, and V. D. Pereyra Departamento de Fı´sica and Centro Latinoamericano de Estudios Ilya Prigogine, Universidad Nacional de San Luis, CONICET, Chacabuco 917, 5700 San Luis, Argentina Received December 23, 1999. In Final Form: July 28, 2000 The thermodynamics of interacting homonuclear dimers adsorbed on homogeneous surfaces in a monolayer regime is addressed. Dimer adsorption is described in the framework of the lattice gas theory. Hypercubic lattices are used as a model of homogeneous substrate in one and two dimensions. The effect of lateral interactions on the behavior of the adlayer is considered. The adsorption isotherm, energy of adsorption, differential heat of adsorption, and configurational entropy are obtained by Monte Carlo simulations and compared with analytical calculations. Particular attention is devoted to the two-dimensional case with repulsive nearest neighbor lateral interactions in the critical and subcritical regime. Different ordered phases are observed and their relationship with the behavior of the thermodynamic functions of the adsorbed monolayer is shown.

1. Introduction The adsorption of gases on solid surfaces is a topic of fundamental interest for various applications.1 From the theoretical point of view, the process can be described in terms of the lattice gas models.1-6 Although most of the works on the subject are devoted to the adsorption of monatomic gases, there have been a few studies related to multisite occupancy adsorption.7-9 In many experimental cases, the molecules consist of a number of single k components or elementary units, the so-called kmers. Even such simple gases as oxygen, nitrogen, and carbon monoxide are polyatomic. Larger linear molecules such as CnH2(n-1) (n-alkanes) adsorbed on solid surfaces should be regarded under the light of a multisite adsorption model,10 in order to properly account for the effects of configurational entropy (kmer size and flexibility) on the thermodynamics of the adlayer. Although the leading early contributions of Flory11,12 and Di Marzio13 have provided the basis for qualitative predictions of adsorbed kmer properties, many aspects of the multisite adsorption thermodynamics on homogeneous surfaces remain unexplored. More recent reports indicate that the phase * To whom all correspondence should be addressed. E-mail: [email protected]. (1) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: New York, 1974. (2) Dash, J. G. Films on Solid Surfaces; Academic Press: New York, 1975. (3) Phase Transitions in Adsorbed Films; Dash, J. G., Ruvalds, J., Eds.; Plenum: New York, 1980. (4) Ordering in Two Dimensions; Shina, S. K., Ed.; Elsevier: New York, 1980. (5) Binder, K.; Landau, D. P. Surf. Sci. 1976, 61, 576. (6) Patrykiejew, A.; Sokolowski, S.; Binder, K. Surf. Sci. Rep. 2000, 37, 207. (7) Nitta, T.; Kuro-oka, M.; Katayama, T. J. Chem. Eng. Jpn. 1984, 17, 45; Nitta, T.; Yamaguchi, A. J. J. Chem. Eng. Jpn. 1992, 25, 420. (8) Marczewski, A. W.; Derylo-Marczewska, M.; Jaroniec, M. J. J. Colloid Interface Sci. 1986, 109, 310; Boro´wko, M.; Rz˘ ysko, W. J. Colloid Interface Sci. 1996, 182, 268. (9) Rudzin´ski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992. (10) Rudzin´ski, W.; Nieszporek, K.; Cases, J. M.; Michot, L. I.; Villeras, F. Langmuir 1996, 12, 170. (11) Flory, P. J. Chem. Phys. 1942, 10, 51. (12) Flory, P. J. Principles of Polymers Chemistry; Cornell University Press: Ithaca, NY, 1953. (13) DiMarzio, E. A. J. Chem. Phys. 1961, 35, 658.

behavior of interacting kmers may be unusually rich. For instance, the phase diagrams of dimers with both attractive and repulsive nearest neighbor interactions on a square lattice have been reported recently.14 The phase behavior is particularly interesting for repulsive dimers; at θ ) 0.5 a c(4 × 2) ordered phase develops, while at θ ) 2/3 a novel ordered phase (called a zigzag phase) is reported. The thermodynamic implication of such structural ordering is major on the collective migration of the adparticles and its importance is demonstrated in the analysis of the collective diffusion coefficient of dimers with nearest neighbor repulsion.15 The structural ordering of interacting dimers has been also analyzed by A. J. Phares et al.,16 through approximate calculations of dimer entropy on a semi-infinite square lattice by using transfer matrix techniques. They concluded that there is a finite number of ordered structures for dimers with repulsive nearest neighbor interactions. However, some of the predicted structures are artifacts of the calculation technique as it arises from simulation analysis.14 The present contribution addresses the analysis of adsorption thermodynamics of interacting kmers at subcritical temperatures and kmer phase behavior. Thus, coverage and temperature dependence of configurational entropy, internal energy, differential heat of adsorption, and adsorption isotherms are reported for attractive and repulsive dimers. Furthermore, analytical solutions are developed in the framework of mean-field approximation, based upon recent exact formulation of kmer thermodynamics in onedimensional lattices presented in refs 17-19.17-19 A comprehensive comparison between Monte Carlo simulations in the grand canonical ensemble and analytical results is carried out. A special interest presents the results of the differential heat of adsorption and entropy for dimers (14) Ramirez-Pastor, A. J.; Riccardo J. L.; Pereyra, V. D. Surf. Sci. 1998, 411, 294. (15) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo J. L.; Pereyra, V. D. Surf. Sci. 1997, 391, 267. (16) Phares, A. J.; Wunderlich, F. J.; Grumbine, D. W.; Curley, J. D. Phys. Lett. A 1993, 173, 365, (and reference therein). (17) Ramirez-Pastor, A. J. Adsorcio´ n y difusio´ n con mu´ ltiple ocupacio´ n de sitios; Ph.D. Thesis, Universidad Nacional de San Luis, 1998. (18) Ramirez-Pastor A. J.; Eggarter T. P.; Pereyra V. D.; Riccardo J. L. Phys. Rev. B 1999, 59, 11 027. (19) Ramirez-Pastor, A. J.; Roma´, F.; Aligia A.; Riccardo, J. L. Langmuir 2000, 16, 5100.

10.1021/la991675m CCC: $19.00 © 2000 American Chemical Society Published on Web 12/02/2000

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Ramirez-Pastor et al.

in two dimensions. These observables are particularly affected by the presence of order-disorder phase coexistence. The outline of the present work is as follows: In Section 2, we present the mean-field approximation for adsorbed dimers in two dimensions, and derive the analytical form of entropy, adsorption isotherm, and differential heat of adsorption. Section 3 is devoted to describing the simulation scheme of kmers adsorption in the grand canonical ensemble. The analysis of the results and discussion are given in Section 4. Finally, the conclusions are drawn in Section 5. 2. Lattice Gas Model for Multisite Occupancy Adsorption: Mean-Field Approximation In this section we describe the lattice gas model for the adsorption of particles with multisite occupancy in the monolayer regime. We consider the adsorption of homonuclear linear kmers on a square lattice. The adsorbate molecules are assumed to be composed of k identical units in a linear array with constant bond length equal to the lattice constant a. The kmers can only adsorb flat on the surface occupying k lattice sites (each lattice site can only be single-occupied). The surface is represented as an array of M adsorptive sites. To describe a system of N kmers adsorbed on M sites at a given temperature T, let us introduce the occupation variable ci which can take the values ci ) 0 or 1, if the site i is empty or occupied by a kmer unit, respectively. The kmer retain its structure upon adsorption and desorption. The Hamiltonian of the system is given by

H)w

∑ cicj - N(k - 1)w + o ∑i ci - µ ∑i ci

(1)

(i,j)′

where w is the nearest neighbor interaction energy that corresponds to repulsive (attractive) interaction for w > 0 (w < 0); (i,j)′ represents pairs of NN sites, o is the adsorption energy of a kmer unit and µ is the chemical potential. The term N(k - 1)w is subtracted in eq 1 since the summation over all the pairs of NN sites overestimates the total energy by including N(k - 1) bonds belonging to the N adsorbed dimers. Finally, o is set equal to zero for simplicity, without any loss of generality. 2.1. Exact Results in One Dimension. Let us assume a system of N interacting kmers adsorbed on a onedimensional chain of M sites at a given chemical potential µ and temperature T. The exact form of the free energy per site, f, was obtained explicitely in refs 17-19, as a function of T and coverage θ ) kN/M,

F(M, N, T) ) M θ θ θ w - R - kBT ln + (1 - θ) ln (1 - θ) k k k θ θ 2R ln R + kBT - R ln - R + k k

f(θ,T) )

[

]

[

]

{[

] [

]

}

(1 - θ - R) ln (1 - θ - R) (2) where R is given by

b)

{[

dF ) -SdT - Π dM + µ dN

[

]

]

(4)

where

∂F (∂T )

S)-

∂F (∂M )

Π)-

M,N

T,N

µ)

∂F (∂N )

(5)

M,T

Then the coverage dependence of the chemical potential, µ, and the entropy per site, s, are straightforward from eqs 2, 3, and 5

µ(θ,T)/kBT ) w/kBT + ln [k(b - 1 + θ) + θ] + (k - 1) θ + b + (k - 1) ln k (k - 1) ln 1 k ln [k(b + 1 - θ) - θ] (6)

[

]

θ θ ln + (1 - θ) ln (1 - θ) - 2R ln R k k θ θ - R ln - R - (1 - θ - R) ln (1 - θ - R) (7) k k

s(θ,T)/kB )

[

] [

]

The differential heat of adsorption qd is defined as20,21

(

)

∂µ/kBT ∂T

)

θ

qd

(8)

kBT2

which can be calculated explicitly from eq 6,

[

(

1 [k(b - 1 + θ) + θ] 1 1 + [k(b + 1 - θ) - θ] [k(b + 1 - θ) + θ]

qd(θ,T) ) -w + χ

)]

(9)

with

χ)

2wθ(1 - θ)e-w/kBT b

(10)

In addition, f ) u - Ts and the adsorption energy per site, u, can be calculated from eqs 2 and 7 as

(kθ - R)

u(θ,T) ) w

(11)

Hereby, we derive a mean-field approximation for interacting kmers on a two-dimensional lattice, on the grounds of eq 2. 2.2. Mean-Field Approximation for Adsorbed kmers. The Bragg-Williams approximation is the simplest mean-field treatment for interacting adsorbed particles, even in the case of multisite occupancy. In this context, the canonical partition function Q(M,N,T) for a system of N kmers adsorbed on M sites at a temperature T, considering nearest neighbor lateral interaction of magnitude w between adsorbed molecules is given by,

Q(M, N, T) )

2θ(1 - θ) (k-1) θ+b k 1k (k - 1) 2 4A θ (θ - θ2) 1k k

R)

adlayer can be deduced from eq 2 along with the differential form of F in the canonical ensemble

∑ ω(Ek)e-E (M,N)/k T k

B

(12)

{Ek}

}

where ω(Ek) is the number of configurations of N kmers on M sites with energy Ek. If a mean-field approximation

1/2

(3)

with A ) 1 - e-w/kBT. All the equilibrium properties of the

(20) Molecular Thermodynamics of Adsorption of Gas and Liquid Mixtures; Fundamentals of Adsorption, Myers, A. L., Liapis, A. I., Eds.; Engineering Foundation, New York, 1987. (21) Ramirez-Pastor, A. J.; Bulnes, F. Physica A 2000, 283, 198.

Adsorption Thermodynamics with Multisite Occupancy at Criticality

Accordingly, from eq 19 and the formalism presented in Section 2.1., s is given by

is introduced at this point

Q(M, N, T) ) e-Ek(M,N)/kBT

∑ ω(Ek) ) -Ek(M,N)/kBT

e

Ωk(M, N, c) (13)

where Ωk(M, N, c) is the number of configurations of N kmers on a lattice with M sites and connectivity c, and Ek(M,N) the mean total energy of the system assuming that the kN occupied sites of the lattice are randomly distributed over M sites. Ωk(M,N,c) is calculated considering that the molecules are distributed completely at random on the lattice and assuming the arguments given by different authors,7,12,18 to relate the configurational factor Ωk(M,N,c) for any c, with the same quantity in one dimension (c ) 2). Thus

Ωk(M, N, c) ) K(c,k)N Ωk(M, N, 2)

[M - (k - 1) N]! N! [M - kN]!

(15)

In the particular case of rigid straight kmers it follows that K(c,k) ) c/2. On the other hand,

1 kN Ek(N, M) ) kNo + λN w 2 M

( )

(16)

where the first and second terms in the RHS of eq 16 account for the kmer-lattice and kmer-kmer interactions, respectively; and λ ) [2(c - 1) + (k - 2)(c - 2)] is the number of nearest neighbor sites of an adsorbed kmer. Hence, the canonical partition function Q(M,N,T) can be written as

Q(M, N, T) ) [M - (k - 1)N]! -(kNo+λkN2w/2M)/kBT (17) e K(c,k)N N![M - kN]! The Helmholtz free energy F(M,N,T) is given by:

F(M, N, T) ) kBT ln Q(M, N, T) ) 2

N 1 kBT ln Ωk(M, N, c) - kNo - wλk ) 2 M kBT {ln[M - (k - 1)N]! - ln N! - ln[M - kN]! + N2 1 (18) N ln K(c,k)} - kNo - wλk 2 M The Helmholtz free energy per site can be obtained as a function of coverage and temperature,

{[

k-1 k-1 θ ln 1 θ + k k θ θ θ ln + (1 - θ) ln (1 - θ) - ln K(c,k) + θo + k k k 1 θ2 λw (19) 2 k

] [

] }

] [

]

In addition, the isotherm equation takes the form

y(θ,T) ) Ck K(c,k)e(µ-ko)/kBT )

[

]

(k-1) θ k (1 - θ)k

θ 1-

(k-1)

eλwθ/kBT (21)

where Ck ) k. Finally, the differential heat of adsorption, qd, and total adsorption energy per site u result

qd(θ) ) -λwθ - ko

(14)

where, K(c,k) is, in general, a function of the connectivity and the size of the molecules and Ωk(M,N,2) can be readily calculated18 giving

Ωk (M, N, 2) )

s(θ) k-1 k-1 θ θ θ ln 1 θ - ln ) 1kB k k k k θ (1 - θ) ln (1 - θ) + ln K(c,k) (20) k

[

{Ek}

f(θ,T) ) kBT - 1 -

Langmuir, Vol. 16, No. 26, 2000 10169

(22)

and

u(θ) )

1 θ2 λ w + θo 2 k

(23)

For w < 0, subcritical isotherm present the typical van der Waals loops characteristic of the first-order phase transitions. The critical temperature and coverage, Tkc and θkc, are obtained from the conditions:22

( ) ∂2µ ∂θ2

) 0;

θ)θkc

(

)

∂Π/kBT ∂θ

)0

(24)

θ)θkc

where Π (eq 5) is the so-called spreading pressure. Consequently,

θkc )

(2k - 1) - xk2 - k + 1 3(k - 1)

(25)

with the limits θc1 ) 0.5 for k ) 1, and θc∞ ) 1/3 for k f ∞. The critical temperature for finite value of k follows from the expression,

{

) ]}

k-1 k λw ) - θkc (1 - θkc ) 1 θc k k kBTc

[ (

-1

(26)

3. Monte Carlo Simulation The thermodynamic properties of the present model have been investigated following a generalized Monte Carlo procedure (proposed in ref 23) for kmers adsorption in the grand canonical ensemble. Square lattices of size M ) L × L with periodic boundary conditions were used and relaxation toward equilibrium relied upon Glauber dynamics and the Metropolis scheme of transition probabilities. For repulsive lateral interactions, where ordered phases are expected to develop, the first 105 Monte Carlo steps (MCS) of each run were discarded to allow for equilibrium and typically the next 105 MCS were used to compute averages. At temperatures far from the critical point 5 × 103 MCS were found to be enough to obtain (22) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley: Reading, MA, 1960. (23) Ramirez-Pastor, A. J.; Nazzarro, M. S.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1995, 341, 249; Ramirez-Pastor, A. J.; Pereyra, V. D.; Riccardo J. L. Langmuir 1999, 15, 5707; Ramirez-Pastor, A. J.; Riccardo J. L.; Pereyra, V. D. Langmuir 2000, 16, 682.

10170



sufficient accuracy. In the vicinity of critical points, up to 106 MCS had to be used in order to let the system relax from metastable states. It should be noted that displacement (diffusional relaxation) of adparticles to nearest neighbor positions, by either jumps along the kmer axis or reptation by rotation around the kmer end, must be allowed in order to reach equilibrium in a reasonable time. The effect of forbidding displacement moves in such a way on the calculation of adsorption isotherms are shown by the open squares in Figure 5 in comparison with solid squares for w/kBT ) 7.5. Although data shown in open symbols are averages over typically 106-107 MCS, they are affected by a large systematic error for high coverages. Thermodynamic quantities, such as mean coverage θ and mean adsorption energy per site u are obtained as simple averages:

θ)

1 M

u)

M

∑i 〈ci〉

(27)

1 〈H〉 M

(28)

The differential heat of adsorption can be obtained from our simulation as,24,25

∂u qd ) ∂〈N〉

Figure 1. (a) Lattice coverage versus chemical potential (in units of kBT) for attractive interacting dimers; curves from right to left correspond to: w/kBT ) 0; w/kBT ) -2 and w/kBT ) -5. (b) Same as (a) for attractive interacting kmers with w/kBT ) -5 and different sizes of the adsorbate; curves from right to left correspond to: k ) 1,2,3,4. (c) Same as (a) for repulsive interacting dimers; curves from left to right correspond to: w/kBT ) 0; w/kBT ) 2 and w/kBT ) 5. (d) Same as (a) for repulsive interacting kmers with w/kBT ) 5 and different sizes of the adsorbate; curves from left to right correspond to: k ) 1,2,3,4.

(29)

where the thermal average 〈...〉, means the time average over the Monte Carlo simulation run. Additionally, the configurational entropy of the adlayer, as a function of θ and T, has been calculated by means of the thermodynamic integration method.26 Accordingly,

s(θ,T) s(θ ) 0,T) u(θ, T) - u(θ ) 0, T) ) kB kB kBT θ 1 µ(θ′)dθ′ (30) kBT 0

∫

The calculation of the entropy and the energy in the reference state, s(θ ) 0,T) and u(θ ) 0,T), respectively, is trivial [s(θ ) 0,T) ) u(θ ) 0,T) ) 0]. 4. Results and Discussion First we address the comparison between the analytical one-dimensional adsorption isotherm from eq 6 and the MC simulation. The simulations have been performed for chains of M ) 1000 sites and periodic boundary conditions. Adsorption isotherms for dimers k ) 2 with different attractive lateral interactions (w/kBT ) 0, -2, -5) are shown in Figure 1a. As expected, isotherms shift to lower values of µ/kBT and their slope increases as the ratio w/kBT increases. Figure 1b shows adsorption isotherms for fixed a value of w/kBT ) -5 and different adsorbate size (k ) 1,2,3,4). The shape of the isotherms is fairly independent of the size of the molecules and curves shift to the left for increasing k, because larger molecules with attractive units facilitate the adsorption. On the other hand, repulsive dimers present an interesting behavior as shown in Figure 1c. The isotherms develop a neat step at θ ) 2/3 for low temperatures. Dimers (24) Soto, J. L. Statistical Thermodynamics of Sorption in Molecular Sieves; Ph.D. Thesis, University of Pennsylvania, Philadelphia, 1979. (25) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (26) Binder, K. Applications of the Monte Carlo Method in Statistical Physics. Springer: Berlin, 1984.

Figure 2. (a) Differential heat of adsorption versus lattice coverage for attractive interacting dimers; curves from top to bottom correspond to: w/kBT ) -5; w/kBT ) -2 and w /kBT ) -1. (b) Same as (a) for attractive interacting kmers with w/kBT ) -5 and different sizes of the adsorbate; curves from top to bottom (for θ < 0.5) correspond to: k ) 4,3,2,1. (c) Same as (a) for repulsive interacting dimers; curves from top to bottom correspond to: w/kBT ) 1; w/kBT ) 2 and w/kBT ) 5. (d) Same as (a) for repulsive interacting kmers with w/kBT ) 5 and different sizes of the adsorbate; curves from left to right correspond to: k ) 1,2,3,4.

avoiding configurations with nearest neighbor repulsive heads order in a structure of alternating dimers separated by an empty site at θ ) 2/3. The width of the step is directly proportional to the energy per dimer necessary to alter such an ordered structure. As the size of adparticles increases, the step shifts to higher coverages. Based on analogous arguments to the ones given for dimers, it is straightforward that steps are located at θ ) k/(k + 1) (see Figure 1d). The differential heat of adsorption as a function of coverage has been plotted in Figure 2. qd for attractive dimers is monotonically increasing on coverage with a plateau at qd ≈ w/kBT (Figure 2a). As the size of the kmer


Langmuir, Vol. 16, No. 26, 2000 10171

Figure 3. (a) Phase diagram for attractive dimers adsorbed on a square lattice. (b) Phase diagram for repulsive dimers adsorbed on a square lattice.

increases the plateau flattens over a wide range of coverage since adsorbate islands are more compact for larger particles (Figure 2b). The behavior of repulsive interacting kmers is completely different. qd is nearly zero for coverages θ < k/(k + 1), since the adsorbed particles can rest separated by more than, or by at least one, empty site in this regime. Adsorption of a kmer for θ > k/(k + 1) requires the creation of a chain of k empty sites, therefore k - 1 particles must be forced to new positions becoming nearest neighbors. The resulting energy change for this process is given by the factor (k - 1)w (because of the referred rearrangement of the k - 1 molecules), plus 2w coming from the interaction of the adsorbed kmer with its neighbors. The value of the differential heat at coverages θ > k/(k + 1) is roughly qd ≈ 2w + (k - 1)w. The dependence of qd on the strengh of lateral interactions and size of particles are depicted in Figure 2 c and d, respectively. Hereafter, we present the analysis of the adsorption of interacting dimers on a square lattice and also the thermodynamic implications. The computational simulations were developed for squares L × L lattices, with L ) 192, and periodic boundary conditions; with this lattice size we verified that finite size effects are negligible. The phase diagrams for both attractive and repulsive lateral interactions have been obtained by A. J. RamirezPastor et al.14 In this work, we stress the study of adsorption isotherms, differential heat of adsorption, and configurational entropy, emphasizing their behavior at critical and subcritical temperatures. The phase diagram for attractive dimers (Figure 3a) is similar to that of a simple lattice gas of monomers, with critical temperature, kBTc|w| ) 0.689, larger than the one for monomers by a factor of about 1.2 (for monomers the value of kBTc|w|, is 0.567). In the phase diagram for repulsive nearest neighbor lateral interactions (Figure 3b), the two well-defined ordered phases, as a c(4 × 2) -like structure at θ ) 0.5, and a zigzag (ZZ) structure at θ ) 2/3 are observed. The c(4 × 2) ordered structure is characterized by a repetition of alternating files of adsorbed dimers separated

Figure 4. Snapshot of the ordered phases corresponding to the repulsive phase diagram for homonuclear dimers. (a) Different c(4 × 2) structures and (b) a zigzag (ZZ) order.

by two empty sites (see Figure 4a). There are two possible orientations of such a structure with the same energy. The equilibration of the adsorbed system at this temperature required approximately 1.5 × 106 MCS. The ordered phase at θ ) 2/3 is characterized by domains of parallel ZZ strips oriented at (45° from the lattice symmetry axes, separated from each other by strips of single empty sites (see Figure 4b). The periodicity of the ZZ varies from 1 to L; the largest value corresponding to diagonal strips spanning throughout the lattice. The internal energy per site in this phase is u ) w/3. The ZZ ordered phase presents different degenerate structures with the same energy, corresponding to all the possible arrangements of the diagonal rows (columns). However, the entropy per lattice site in the thermodynamic limit tends to zero in this case, as suggested by the local minimun of s(θ,T) at θ ) 2/3 in Figure 8. In the following, we address the study of the thermodynamic quantities in different regions of the phase diagram to disclose the influence of the ordered phases in the form of adsorption isotherm, differential heat of adsorption, and configurational entropy. Typical adsorption isotherms obtained by MC simulations in the grand canonical ensemble (symbols) and comparison with MFA (lines) are shown in Figure 5. In both the attractive and the repulsive case, disagreement between MC and MFA is found at critical and subcritical

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Figure 5. Adsorption isotherms for homonuclear dimers adsorbed on a square lattice with nearest neighbor interactions. The symbols represent the results from MC simulations and the solid lines represent the results from MFA. The dotted lines are included in the figure as a guide for the eyes. (a) Attractive case: solid circles, w/kBT ) 0; open up triangles, w/kBT ) -0.50; open diamonds, w/kBT ) -1.02; open down triangles, w/kBT ) -1.43; and open squares, w/kBT ) -2.04. (b) Repulsive case: solid circles, w/kBT ) 0; solid up triangles, w/kBT ) 2; solid diamonds, w/kBT ) 4 [path (A-B) in Figure 3b]; solid down triangles, w/kBT ) 5 [path (C-D) in Figure 3b] and solid squares, w/kBT ) 7.5 [path (E-F) in Figure 3b]. The open squares correspond to MC simulations without diffusional relaxation and w/kBT ) 7.5.

temperatures. For attractive interactions (Figure 5a), the system undergoes a first-order phase transition that appears as the discontinuity in the simulated isotherms. The isotherms in the repulsive case shown in Figure 5b have more features because of the complexity of the phase diagram. At T > Tc, the isotherms do not present any peculiar behavior (solid circles and solid up triangles in the figure). For T < Tc, the shape of the isotherms is very dependent on temperature. Following the path (A-B) in the phase diagram of Figure 4b, the corresponding isotherm (full diamonds) presents a unique pronounced step at θ ) 0.5 that can be traced to the formation of a c(4 × 2) order of dimers. At lower temperatures (path E-F), the isotherm (solid squares) develops a step at θ ) 2/3 (less pronounced than the one at θ ) 0.5), corresponding to the ZZ phase. Even though equilibrium can be achieved for θ < 0.5, a strong metastability develops if no diffusional relaxation is included and the order-disorder phase transition at θ ) 2/3 (zigzag phase) is completely smeared out as observed in the enhanced plateau and following step corresponding to open squares in Figure 5b. Since the symmetry of particle-vacancy (valid for monatomic particles), is broken for dimers, the isotherms and phase diagram are asymmetric with respect to θ ) 0.5. This effect is also observed in the dependence of the entropy as a function of coverage where the deep minima correspond to the referred order structures (Figure 8). The energy of adsorption per site, u, versus coverage, for both attractive and repulsive lateral interactions is depicted in Figure 6 a and b. In the attractive case, u grows monotonically with coverage and the agreement


Figure 6. Energy per site (in units of w) versus lattice coverage for homonuclear dimers adsorbed on a square lattice with nearest neighbor interactions. The symbols represent the results from MC simulations and the solid lines represent the results from MFA. (a) Attractive case: open up triangles, w/kBT ) -0.50; open diamonds, w/kBT ) -1.02; open down triangles, w/kBT ) -1.43. (b) Repulsive case: solid up triangles, w/kBT ) 2; solid diamonds, w/kBT ) 4; solid down triangles, w/kBT ) 5 and solid squares, w/kBT ) 7.5.

Figure 7. Energy per site (in units of w) versus lattice coverage for homonuclear dimers adsorbed on a square lattice with repulsive nearest neighbor interactions. The solid squares (connected by the line) represent the results from MC simulations (w/kBT ) 7.5) and the solid lines represent the results from MFA.

between MFA and simulations is qualitatively good. Interesting results arise for repulsive interactions where three typical regimes appear for T , Tc (see also Figure 7). (i) For θ < 0.5, the dimers do not interact with each other; for low coverage they are isolated, as the coverage increases they form c(4 × 2)-like islands; and finally a perfect c(4 × 2) ordered phase is reached at θ ) 0.5; (ii) for 0.5 < θ < 2/3, a new dimer added to the surface increases the coverage by a factor of 2/M, while u increases linearly from u/w ) 0 to u/w ) 1/3; (ii) finally, for 2/3 < θ < 1, the situation is slightly more complicated, due to the fact that in order to adsorb a new dimer it is necessary to displace at least one of the previously adsorbed dimers, as a result the energy increases linearly with coverage. The slope of


Figure 8. Configurational entropy of the adlayer (in units of kB) versus lattice coverage for the cases represented in Figure 6. The dotted lines are included in (b) as a guide for the eyes.

u versus θ changes discontinuously at θ < 0.5 and θ < 2/3. For T > Tc, u has a continuous derivative over all the range of θ. The configurational entropy s(θ,T) for attractive and repulsive dimers is depicted in Figure 8. The curves have been obtained from eq 30, where the dependences µ(θ,T) and u(θ,T) have been taken from the simulated adsorption isotherm and energy of adsorption, respectively. On the other hand, MFA comes from eq 20. In the attractive case (Figure 8a), s(θ,T) is asymmetric with respect to θ ) 0.5 and decreases for all θ as the strengh of the lateral interactions increases. All curves shown correspond in this case to T > Tc. The coverage dependence of s(θ,T) is very interesting in the repulsive case (Figure 8b) where at T > Tc the entropy only has a maximun for θ > 0.5. However for T j Tc a minimum appears at θ ) 0.5 corresponding to the c(4 × 2) order. If temperature is further lowered, a second minimum develops related to the ZZ phase. Both minima tend to zero as T f 0. This result represents a qualitative and quantitative improvement about the knowledge of the critical behavior of dimers, for which transfer-matrix calculations16 of Phares et al. predict more than two minima for s(θ,T f 0) and a finite value for s(θ ) 2/3,T f 0) in contrast to s(θ ) 2/3,T f 0) f 0 found in this work. (It should be noted that Phares et al. argued correctly that although they obtained more than two minima for s(θ,T f 0), only the ones at θ ) 0.5 and θ ) 2/3 may be genuine; the others being artifacts of the calculation technique.) MFA agrees with simulation data only for T f ∞. The differential heat of adsorption is monotonically increasing with θ for attractive dimers and T > Tc as expected (see Figure 9a). As the temperature decreases, the difference between MFA and MC simulation gets more important. Below Tc, the calculation of qd is rather cumbersome due to coverage fluctuations. The coverage dependence of qd for repulsive dimers changes qualitatively from T < Tc (see Figures 9b and 10). For 0 < θ < 0.5, the dimers do not interact with each other giving qd(θ) ) 0. However, near the critical coverage, a small maximum appears in qd(θ), when one approaches

Langmuir, Vol. 16, No. 26, 2000 10173

Figure 9. Differential heat of adsorption versus lattice coverage for the cases represented in Figure 6. The dotted lines are included in (b) as a guide for the eyes.

Figure 10. Differential heat of adsorption versus lattice coverage for homonuclear dimers adsorbed on a square lattice with replusive nearest neighbor interactions. The solid squares represent the results from MC simulations (w/kBT ) 7.5) and the solid lines represent the results from MFA.

θ ) 0.5 from the left. A scaling study of this maximum allows the calculation of the critical coverage as is shown in ref 21 for adsorption of monomers. For 0.5 < θ < 2/3 the system changes from the c(4 × 2) to the ZZ phase, diminishing its free energy. This regime corresponds to the plateau in the differential heat of adsorption. The value of the plateau is 4w and it can be obtained from the slope of u versus θ. 4w equals the energy needed to change from the c(4 × 2) to the ZZ structure. Finally, for θ > 2/3, the energy cost upon adsorption of a dimer can be estimated by considering the adsorption of an incoming dimer on a perfect ZZ structure. To adsorb a new dimer it is necessary first to create a pair of empty sites moving one adsorbed dimer. This process involves an energy variation w as shown in Figure 11, where the dimer denoted A changes from interacting with 2 NN to interacting with 3 NN (intermediate state in Figure 11, where the NN interactions are highlighted as dashed lines). Then, the incoming dimer (denoted B) can be allocated on the lattice at an additional energy cost of 6w (see final state in Figure 11). Ultimately, the total energy

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The temperature and coverage dependence of the chemical potential (adsorption isotherm), adsorption energy, configurational entropy of the adlayer, and differential heat of adsorption of interacting dimers have been calculated through MC simulations in the grand canonical ensemble and analytical methods. From the study of thermodynamics of attractive dimers it follows that isotherms, differential heat of adsorption, and entropy are qualitatively similar to the monomer case, except for the fact that the mentioned thermodynamic quantities are nonsymmetrical around θ ) 0.5, since the symmetry of particle-vacancy is broken for dimers and larger particles. On the other hand, the repulsive case presents a qualitative behavior completely different to the one of monatomic particles. At low temperatures the adsorption isotherms present either one or two plateaus corresponding to order-disorder phase transitions at θ ) 0.5 and θ ) 2/3. A c(4 × 2) ordered phase appears at θ ) 0.5 and a ZZ phase at θ ) 2/3.

Figure 11. Snapshots of the adsorbate at θ > 2/3 showing the elementary step necessary to adsorb a new dimer: (a) initial state (ordered ZZ phase); (b) intermediate state in which one dimer (denoted A) has been shifted to the left, leaving a pair (i,j) of empty sites; (c) final state where a new dimer (denoted B) adsorbed on the pair (i,j) of sites.

variation is 7w, corresponding to the plateau of qd for θ > 2/3 in Figures 9b and 10. On the other hand, the MFA does not predict the existence of an ordered phase in the adsorbate, and it is not possible to distinguish the different adsorption regimes. 5. Conclusions A comprehensive study of adsorption thermodynamics of dimers has been presented emphasizing the adlayer properties at critical and subcritical temperatures.

The adsorption energy, differential heat of adsorption, and entropy are very sensitive probes of the appearance of these ordered phases. qd reveals the presence of the ordered phases through marked jumps. In the coverage dependence of entropy, the appearance of ordered phases is traced to the developing of minima in s(θ,T) as the temperature decreases. The critical temperature of the phase c(4 × 2) at θ ) 0.5 is lower than the one corresponding to the analogous c(2 × 2) phase appearing for repulsive monomers on a square lattice. The present study shows that the adsorption thermodynamics of the simplest polyatomic particles has many new features compared to the standard lattice gas of monomers. This encourages the study of critical behavior and thermodynamic functions of larger adsorbed particles.

Acknowledgment. This work is partially supported by the CONICET (Argentina). LA991675M

Adsorption Thermodynamics with Multisite Occupancy at Criticality

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