ARTICLE pubs.acs.org/Langmuir
Coadsorption of n Monomer Species on Terraces and Nanotubes Alain J. Phares Department of Physics, Mendel Science Center, Villanova University, Villanova, Pennsylvania 19085-1699, United States ABSTRACT: The knowledge of the partition function, Z, of a system of particles adsorbed on a surface is all that is required to determine the occupational characteristics of the adsorbates and the thermodynamic properties of the system. The surface considered is a terrace or a nanotube of arbitrary periodic lattice geometry, L atomic sites in length, and M0 sites in the width of the terrace or in the normal cross section of the nanotube. The matrix method introduced in 2007 to obtain Z for the adsorption study of one species of monomers is now generalized to the study of the coadsorption of any number, n, of monomer species. We provide proof that Z can be related to the eigenvalues of a real and non-negative matrix (T matrix) of rank (n þ 1)M, where M is an integer multiple of M0 . In the infinite-L limit, we also prove that Z is the largest eigenvalue of the T matrix, raised to the power of 1/M. Because the rank of this matrix increases exponentially with M, we develop a technique for its recursive construction applicable to any lattice geometry, which is easily programmed and efficiently adaptable for supercomputing and multiparallel processing. As examples, we consider the coadsorption on square, equilateral triangular, and honeycomb surfaces. This general formulation can now be applied to model a whole new set of experiments involving the coadsorption of two or more monomer species, on terrace or nanotube surfaces with various periodic lattice structures.
1. INTRODUCTION For almost a century now, lattice models have been applied to a number of physical, chemical, and biological systems and continue to be a very powerful tool in providing crucial insights into otherwise mathematically and theoretically intractable systems. The earliest models are the Langmuir adsorption theory18 and the Ising model applied to ferromagnetism.919 There are only very few exactly solved lattice models,20 and as the complexity of the physical system increases, it becomes necessary to rely on numerical solutions and on the use of supercomputing power and multiparallel processing. This work is a further application of lattice models to surface adsorption studies, which ultimately requires numerical computations. Depending on the manner in which a metal is cut, its exposed surface may be flat, with its atomic sites forming a lattice having a periodic geometrical pattern made of squares, rectangles, isosceles or equilateral triangles, honeycombs, and so forth.21 Rather than being flat, the exposed surface may exhibit parallel and finitewidth terraces separated by steps. Such nonflat surfaces are called stepped surfaces. All terraces are very long, have the same width and lattice geometry, and are observed to have the same adsorption properties. The steps are characterized not only by their lattice geometry, which may be different from that of the terraces, but also by the number of atomic sites in their height and by their angular orientation relative to the terraces. Atoms (and molecules in general) have been assembled to form long tubes with diameters on the order of a few nanometers, or nanotubes. The most renowned of such assemblies are the carbon nanotubes.22 This article considers adsorption on very long terrace or nanotube surfaces. r 2011 American Chemical Society
The particles adsorbed (the adsorbates) on a terrace or a nanotube surface (the substrate) of a given lattice geometry come from a medium to which the substrate is exposed. The medium could be a gas or a solution. In this article, we consider n distinct species of particles, which, when adsorbed on the surface, bond to a single atomic site, thus behaving as monomers. Therefore, a lattice site could be vacant or occupied by one of the monomer species and could therefore be in one of (n þ 1) occupational states. The system is at thermodynamic equilibrium and absolute temperature T. The size of the substrate is determined by the number of atomic sites in its horizontal width, if a terrace, or in its horizontal circular cross section, if a nanotube, and by the number of sites in its vertical length. In the horizontal direction, let M0 be the minimal number of sites, forming a chain that may be linear or zigzag and whose pattern repeats L times in the vertical direction. Thus, the total number of lattice sites is LM0 . The chemical potential energy of the monomers of the ith species, μi0 , depends on their partial pressure in the gas or their partial concentration in the solution. The bonding to the lattice, or the adsorbatesubstrate interaction energy, depends on the monomer species. In addition, if the substrate is a terrace, then for a given monomer species the adsorbatesubstrate interaction energy depends on whether adsorption takes place on a site at one of the edges of the terrace or in the bulk. Finally, one has to account for pairwise adsorbateadsorbate interaction energies between nearest neighbors (first neighbors), next-nearest neighbors (second neighbors), and higher-order neighbors if necessary. In general, consider a Received: February 14, 2011 Published: June 08, 2011 8105
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model that takes into account pairwise adsorbateadsorbate interaction energies up to and including Ith order neighbors. We then define a sequence of sites in the horizontal direction of the lattice substrate as made up of r consecutive chains of M0 sites such that the Ith neighbor of any site in that sequence is found in either the same sequence or the two neighboring sequences, just above and just below but not beyond. Because terraces and nanotubes are usually very long, for_ convenience we assume the lattice to have an integer number, L , of sequences in its _vertical length so that the total number of lattice sites is LM0 = L M. The occupational state of the surface and its various adsorption properties are derived using statistical thermodynamics, which relies on the construction of the partition function of the mediumsubstrate system. The partition function, Z, depends on the substrate's size and geometry, the chemical potential energies of the individual species, and the adsorbatesubstrate and pairwise adsorbateadsorbate _ interaction energies. The wrapping of a vertical L M terrace of a given lattice geometry onto a vertical cylinder of an appropriate _ diameter, without altering the lattice periodicity, produces an L M tube. This refers to setting a periodic boundary in the horizontal _ M direction of the terrace. Another wrapping of the vertical L M tube onto a horizontal cylinder of appropriate _ diameter, without altering the lattice periodicity, results in an L M toroid. This double wrapping refers to setting periodic _ _ boundary conditions in both the L and M directions of the L M terrace. Finite-size scaling2327 refers to the method of obtaining, by numerical extrapolation, the properties of adsorption on an infinitely long nanotube or an infinitely planar 2D surface. For an infinitely long nanotube with a given number of sites M in its circular cross section, one extrapolates to infinity the_ numerical results obtained for increasing values of its length, _ _ L . For an infinite planar substrate, the starting point is an L L toroidal surface, _ and the numerical results obtained for increasing values of L are extrapolated to infinity. Other methods are combined with finite-size scaling, such as Monte Carlo samplings and simulations2831 and group renormalization techniques.3235 Our method of approach does not involve any sampling, simulation, approximation, or finite-size scaling. The_general statistical formulation of the adsorption problem on an L M terrace or nanotube surface is presented in section 2. Section 3 shows how the partition function, Z, of the adsorption mediumsubstrate system can be related to a real non-negative matrix, the T matrix, of rank (n þ 1)M, which depends on the monomers’ chemical potentials and the adsorbatesubstrate and adsorbateadsorbate interaction energies. As follows from the PerronFrobenius theorem,3639 the eigenvalue of largest modulus, R, of the T matrix is real and positive. Section 3 establishes a one-to-one correspondence between the T-matrix elements and the occupational states of two consecutive sequences of M sites. It also shows that Z is formally obtained in terms of all of the eigenvalues of the T matrix; however, in the infinite-L limit, it is given exactly as the largest eigenvalue, R, raised to the power of 1/M: Z ¼ R 1=M , M ¼ rM 0
ð1Þ
This exact expression of the partition function is subsequently used to determine the adsorption properties of the system numerically, without relying on approximation methods such as those mentioned above. Because of the number of parameters
involved and the exponential increase in the rank of the T matrix, the problem requires supercomputing power and multiparallel processing, which is also the case when using alternative methods. Thus, for computational purposes, it is necessary to develop an efficient and easily programmable method of constructing and numerically storing the T matrix. The objective of section 4 is to provide such a method, which recursively constructs the T matrix for lattices of arbitrary geometry, such that the knowledge of the T matrix associated with a surface of a given horizontal size (M) is obtained from the knowledge of the T matrices associated with smaller surfaces. In increasing complexity, sections 57 apply the general recursive construction formalism of section 4 to square, zigzag and armchair equilateral triangle, and zigzag and armchair honeycomb surfaces, respectively, whether the surfaces are terraces or nanotubes. Once the T matrix is iteratively constructed and stored, an eigenvalue solver package adapted to multiparallel processing, such as ScaLAPACK, can then be used to carry out the numerical computation of the largest eigenvalue R, leading to the partition function from which all of the pertinent thermodynamic properties of the system can be obtained. Section 8 is the summary and discussion.
2. GENERAL FORMULATION OF THE ADSORPTION PROBLEM The adsorbed monomers are found on sites forming an adsorbate lattice, which may or may not be commensurate with the substrate lattice. For example, if the atoms of the substrate surface form a honeycomb lattice and adsorption takes place exactly on top of an atomic site or is slightly shifted but maintains the same lattice geometry, then one has a commensurate adsorbate lattice. However, if adsorption on a honeycomb surface takes place in the hollows of the hexagonal structures, then the adsorbate lattice is not commensurate because it has an equilateral triangle geometry. From here on, the lattice surface, or the surface geometry, refers to the adsorbate lattice or the adsorbate lattice geometry. There are three categories of lattice sites on a terrace. There are those found in the bulk of the terrace, those found on the sites at the top of a step, or step-up sites, and those found at the bottom of a step, or step-down sites. There is no such distinction on nanotube surfaces; all sites are bulk sites, and the adsorbate substrate interaction energy of a monomer of the ith species is denoted as VB,i. To account for the effect due to the edges of a terrace, one has to consider three types of adsorbatesubstrate interaction energies of the ith species: (a) at bulk sites, VB,i, (b) at step-up sites, VS1,i, and (c) at step-down sites, VS2,i. In most cases, it is sufficient to consider nearest-neighbor (first-neighbor) and next-nearest-neighbor (second-neighbor) adsorbateadsorbate interactions. For convenience, only first- and second-neighbor interactions are taken into account, although the present work can be extended to include higher-order interactions. First- and second-neighbor interaction energies between a monomer of the ith species and a monomer of the jth species are denoted as Vij and Wij, respectively. We use the convention that positive energies refer to attractive forces and negative energies to repulsive forces. The chemical potential energy of the monomer of the ith species, μi0 , mentioned in section 1 always appears in combination with the corresponding bulk adsorbatesubstrate interaction energy, and it is convenient to introduce its shifted value, μi (simply referred to as the chemical potential), as μi ¼ μi 0 þ VB, i 8106
ð2Þ
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Unlike nanotubes, terraces have edge sites, and the differential adsorbatesubstrate energies between monomers on step-up (step-down) sites and those on bulk sites have to be considered, namely, U1i (U2i):
and for terraces, it is X X X ðμi θi Þ þ ðVij θij þ Wij βij Þ þ ðU1i γ1i þ U2i γ2i Þ ε¼ i
iej
i
ð3Þ
ð10Þ
Let kB stand for Boltzmann’s constant. With the sign convention adopted for attractive and repulsive forces, the activities associated with μi, Vij, Wij, U1i, and U2i are Vij Wij μ xi ¼ exp i , yij ¼ exp , zij ¼ exp ð4Þ kB T kB T kB T
The entropy per site divided by Boltzmann’s constant is then obtained from a knowledge of the previously computed quantities, namely, ε S ¼ ln Z ð11Þ kB T
U1i ¼ VS1, i VB, i , U2i ¼ VS2, i VB, i
s1i ¼ exp
U1i U2i , s2i ¼ exp kB T kB T
ð5Þ
It is convenient to think of a vacant site as occupied by a monomer of the zeroth species, which has zero interaction energy with the substrate and zero interaction energy with any other occupied site, so that the convention used throughout this article is x0 ¼ y0i ¼ z0i ¼ s10 ¼ s20 ¼ 1
ð6Þ
The partition function, Z, for nanotubes, depends on the first three sets of activities, eq 4, because there are no edges, and for terraces, the partition function depends on all five sets of activities, eqs 4 and 5. As follows from statistical thermodynamics, all of the pertinent properties of the mediumsubstrate system are derived from the knowledge of the partition function. The only ones that we will be discussing are the statistical occupational averages of the respective species: (a) the coverage θi of the lattice by the monomers of the ith species; (b) the numbers per site, θij and βij, of first- and second-neighbor monomers of the ith species with those of the jth species, respectively; and (c) for terraces, the numbers per site of monomers of the ith species found on step-up sites, γ1i, and on step-down sites, γ2i: θ i ¼ xi
Dðln ZÞ Dðln ZÞ Dðln ZÞ , θij ¼ yij , βij ¼ zij Dxi Dyij Dzij γ1i ¼ s1i
Dðln ZÞ Dðln ZÞ , γ2i ¼ s2i Ds1i Ds2i
ð7Þ
ð8Þ
On the basis of eqs 7 and 8, it is justified to call xi the coverage, or occupational, activity of the monomers of the ith species, and there are n such activities, one for each species. Similarly, yij and zij are called the first-neighbor and second-neighbor interaction activities that are associated with a pair of adsorbateadsorbate first-neighbors and second-neighbors, respectively, where one member of the pair is the ith species and the other member is the jth species. Because activity yij (or zij) is the same as activity yji (or zji), there are n(n þ 1)/2 such interaction activities. When the lattice is a terrace, s1i and s2i are the step occupational activities of the adsorbates of the ith species found on step-up sites and step-down sites, respectively, for an added number of 2n activities. Consequently, for a nanotube, the partition function, Z, depends on n(n þ 2) activities, and for a terrace, on n(n þ 4) activities. For nanotubes, the statistical average of the energy per site, ε, is X X ðμi θi Þ þ ðVij θij þ Wij βij Þ ð9Þ ε¼ i
iej
As follows from statistical thermodynamics, the partition function, Z, is obtained from the configurational grand canonical partition function, Δ1, which requires _ the knowledge of the number of arrangements, A1, on the L M lattice for a given set of numbers of monomers of the different species, {Ni}, such that there are so many pairs, with one being the ith species and the other being the jth species, which are first neighbors, {Nij}, or second neighbors, {pij}. If the lattice is a terrace, one has to account for the monomers of a given species found on step-up or stepdown sites corresponding _ to the sets {q1i} and {q2i}. This number of arrangements, A1(L ;{Ni},{Nij},{pij},{q1i},{q2i}) for a terrace or _ A1(L ;{Ni},{Nij},{pij}) for a nanotube, is called the degeneracy because all of these arrangements correspond to states of the system having the same energy. There is no reference to the size M of the lattice in the argument of A1 because the number of sites in the width of the terrace, or the circumference of the nanotube, is maintained at a fixed value, allowing the length of the lattice to vary. The analyses, presented in this and the following section, refer to terraces but are easily adapted to nanotubes. Nanotubes do not have steps and therefore involve fewer parameters. The probability for a monomer of the ith species to be adsorbed is proportional to its occupational activity xi; its probability to be found specifically on a step-up or step-down site is proportional to s1i or s2i; and finally, its probability to be first or second neighbor with an adsorbate of the jth species is proportional to yij or zij. Thus, quantity A1 is multiplied by the Boltzmann’s statistical weight factors or activities, eqs 4 and 5, with each activity having as an exponent the number belonging to one of those found in the corresponding set of numbers: {Ni}, {Nij}, {pij},_ {q1i}, or {q2i}. The grand canonical partition function, Δ1(L ;{xi},{yij},{zij}, {s1i},{s2i}), is then obtained by summing the weighted degeneracy over all possible ways that the lattice may be occupied, from empty to full coverage: Δ1 ¼
X
_ Y N p q q A1 ðL ; fNi g, fNij g, fpij g, fq1i g, fq2i gÞ xNi i yij ij zijij s1i1i s22i i, j
ð12Þ Here, symbol Σ stands for the summation over all of the possible }, values that the numbers in each of the sets, {Ni}, {Nij}, {pij}, {q1i_ and {q2i}, may take on for a lattice that has, in its vertical length, L sequences of M sites found in its horizontal width. The partition function Z follows from raising Δ1 to_a power that is the reciprocal of the total number of lattice sites, L M = LM0 , namely, _ 0 ð13Þ Z ¼ ½Δ1 ðL ; fxi g, fyij g, fzij g, fs1i g, fs2i gÞ1=LM The thermodynamic limit in the length of the lattice is reached _ when L becomes infinite. Initially, effort was spent on computing the degeneracy, A1, from which the grand canonical partition function, Δ1, is obtained, leading to the partition function, Z, that 8107
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Langmuir in turn provides the statistical averages of the various thermodynamic quantities. In a number of very special cases, a method for computing the degeneracy relied on solving a number of linearly coupled recursion relations.4044 Subsequently, we have shown that it is possible, in these special cases, to bypass the degeneracy problem and go directly to the partition function, replacing the problem of solving linearly coupled recursion relations by that of solving an equal number of linear equations among a set of generating functions.4549 The linear equations are then solved using the standard matrix method, exhibiting the T matrix, whose eigenvalues provide the partition function, thus completely circumventing the degeneracy problem. Section 3 presents the generating function technique, which is now generalized to the coadsorption of n monomer species on terrace or nanotube surfaces of periodic geometry. As mentioned earlier, the following derivation assumes the surface to be a terrace. The derivation for a nanotube involves fewer parameters but is otherwise identical.
3. PARTITION FUNCTION AND T MATRIX The features of adsorption on long, finite-width terraces making up a stepped surface are the same for all terraces because they all have the same lattice geometry. Thus, it is sufficient to consider the adsorption taking place on one of the terraces, making the distinction between the sites in the bulk of the terrace and those on its edges step-up and step-down sites. To make this section selfcontained, I summarize the basic terminology introduced in the previous sections. A chain of M0 sites, in the horizontal width of a terrace or in the circumference of a nanotube, has a pattern that repeats throughout its vertical length. There are r consecutive chains of M0 sites forming a sequence of M = rM0 sites, as required by the highest-order pairwise adsorbateadsorbate interaction energies being considered in the modeling of the _ adsorption _ system. We assume L/r to be an integer number, L , so that L is the number of sequences of M sites_in the length L of the lattice, and the total number of sites is L M. The study begins with the problem of establishing recursion relations that lead to the construction of the degeneracy. For a terrace, the degeneracy is _ A1(L ;{Ni},{Nij},{pij},{q1i},{q2i}). 3.1. Recursive Construction of the Degeneracy. The degeneracy A1 refers to the arrangements of monomers on the lattice corresponding to a given set of numbers, {Ni}, {Nij}, {pij}, {q1i}, and {q2i}. Among these arrangements, there are those for which the first sequence of M sites at the bottom of the vertical length of the lattice is in one of the (n þ 1)M occupational states whereas _ the remaining monomers are arranged on the remaining (L 1) sequences. We use the same letter t to indicate the restricted occupational state of the sequence at the bottom of the lattice and also to indicate its ranking. In the t state of ranking order t, the sets of occupational numbers of the first sequence of M sites are designated as {Nti}, {Ntij}, {ptij}, {qt1i}, and {qt2i}. For these sets of numbers, there are (Ni Nti ) monomers _ of the ith species still to be distributed on the remaining (L 1) sequences, of which (q1i qt1i) are on step-up sites and (q2i qt2i) are on step-down sites and form (Nij Ntij) first-neighbor pairs and (pij ptij) second-neighbor pairs. These numbers of pairs must include those occurring between one member of the pair in the lower first sequence and the other member of the pair in the upper second sequence. The _ arrange_ corresponding restricted ment on the remaining (L 1) sequences is At(L 1;{Ni
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Nti },{Nij Ntij},{pij ptij},{q1i qt1i},{q2i qt2i}). Consequently, the degeneracy A1 is the sum over index t of the arrangements, At, ranging from 1 to (n þ 1)M. For record-keeping purposes, it is essential to specify the ordering of the occupational states of a sequence of M sites. This is discussed in the forthcoming section. At this point, it is sufficient to choose as the first state t = 1, the one for which all of the M sites are vacant. This is consistent with the label given to the degeneracy A1 because there are no sites below the first sequence and adding at this location a sequence of M sites restricted to be vacant does not change the counting problem. As we repeat the same analysis for the remaining types of arrangements, we finally generate a set of (n þ 1)M linearly coupled recursion relations of the form _ At ðL ; fNi g, fNij g, fpij g, fq1i g, fq2i gÞ ¼
ðnX þ 1ÞM t0 ¼ 1
_ 0 0 0 0 0 At0 ðL 1; fNi Nit g, fNij Nijt g, fpij ptij g, fq1i qt1i g, fq2i qt2i gÞ
ð14Þ _ In the above equation, L has become any number of sequences of M sites above a lower sequence whose state of occupation and ranking order are both labeled t, whereas t0 is the state or the ranking order of the upper sequence0 just above0 the lower se0 quence. The occupational numbers, Nti , qt1i, and qt2i, refer to the counting found in the upper sequence only. The numbers of 0 0 first- and second-neighbor pairs, Ntij and ptij , refer to the counting of the pairs found in the upper sequence and of the pairs found between the lower and upper sequences but not those found in the lower sequence. _ There is only one way _ to cover L sequences of sites with no monomers, including L = 0, which corresponds to At = 1 for all values _ of t. These are the initial conditions required to compute A1 (L ;{N i},{Nij},{pij},{q1i},{q2i}), whose values allow the numerical computation of the grand canonical partition function, which ultimately provides the partition function.3842 It is a long path to follow, and this is why Monte Carlo samplings and simulations are so useful in providing an approximation to such an otherwise intractable problem, even in the simplest case of the adsorption of a single monomer species. We follow an alternate path that does not rely on any approximation beyond that of the precision in conducting the numerical computations. This path is based on introducing generating functions.4547 _ 3.2. Generating Functions and the T Matrix. With At(L ; {Ni},{Nij},{pij},{q1i},{q2i}), we associate _ a restricted t type of grand canonical partition function Δt(L ;{xi},{yij},{zij},{s1i},{s2i}) defined in exactly the same manner as the one corresponding to t = 1, or eq 11, in which we now replace index 1 by index t. Then, we introduce the (n þ 1)M generating functions associated with the restricted grand canonical partition functions, Δt, namely,4345 Gt ðη; fxi g, fyij g, fzij g, fs1i g, fs2i gÞ ¥ _ X Δt ðL ; fxi g, fyij g, fzij g, fs1i g, fs2i gÞηL ¼
ð15Þ
L ¼0
As mentioned earlier, there is _only one way of _keeping a lattice unoccupied or vacant for any L and including L = 0; therefore, Δt(0;{xi},{yij},{zij},{s1i},{s2i}) = 1. Consequently, for all values of t in eq 15, the first term in the series expansion, corresponding 8108
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_ to L = 0, is 1. We then use eq 12 with its added index t and the recursive relation among the restricted t types of arrangements, given by eq 14, to obtain Δt in a form that includes several summations, one of them being a summation over t0 running from 1 to (n þ 1)M. This expression of Δt is then used in eq 15, and we exchange the order of the summations, leaving the t0 summation for last. In the process, we split the product of activities in eq 12 as follows: Y i, j
N
p q
q
xNi i yij ij zijij s1i1i s22i
0
¼@
1
Y
0 0 0 Y N N t0 Nij Nijt pij ptij q1i qt0 q2i qt0 t t N t N p qt qt 2i xi i yij ij zijij s1i1i s22i A xi i i yij zij s1i 1i s2 i, j i, j 0
0
0
0
ð16Þ The first grouping corresponds to the activities associated with the occupational states of two consecutive sequences of M sites, with the lower sequence in the t-type occupational states and the upper sequence in the t0 -type occupational states, which we now refer to as Ttt 0 ¼
Y
0 t Nt N
0
0
pt qt0 qt 0
xi i yij ij zijij s1i1i s22i
ð17Þ
i, j
In the above expressions, the exponents are restricted as specified in the construction of the recursive relationships among the various types of arrangements, which yield three selection rules: 0 Rule 1: Exponent Nti is the number of monomers of the ith species found in the occupational state of the upper sequence. This excludes the monomers that occupy the sites in the lower sequence. Thus, the sum of these numbers cannot exceed the total number of sites 0M in a 0sequence. Rule 2: Exponents Ntij and ptij are, respectively, the numbers of first- and second-neighbor pairs of monomers, with one monomer in the pair being of the ith species and the other being of the jth species, such that the members of a pair are either both in the upper sequence or one is in the upper sequence and the other is in the lower sequence. This excludes the pairs found in the lower sequence. 0 0 Rule 3: For a terrace, exponents qt1i and qt2i are the numbers of monomers of the ith species on step-up and step-down sites, respectively, found only in the upper sequence. After all of these manipulations are performed, eq 15 is equivalently written as
set of (n þ 1)M linear equations among the generating functions. A final rearrangement of eq 18 yields ðnX þ 1ÞM 1 1 ½Ttt 0 ¼ δtt 0 Gt0 η η 0 t ¼1
Here, δtt0 is 0 for t 6¼ t0 and 1 for t = t0 , which is the usual notation for the Kronecker delta. We group the (n þ 1)M types of generating functions in a column matrix, G, ordered sequentially from 1 to (n þ 1)M to reach a matrix representation leading to 1 1 1 ð20Þ ½T IG ¼ Q ; G ¼ ½T I Q η η Matrices I and T are of rank (n þ 1)M. I is the identityt matrix, and t t t t i Nij pij q1i q2i T is a matrix whose matrix elements are Ttt0 = Πi,jxN i yij zij s1i s2 . This matrix is the T matrix. Matrix Q is a column matrix of (n þ 1)M rows, and all of its elements are 1/η. 3.3. Partition Function and T Matrix. We are interested in obtaining only G1, the generating function _ of the unrestricted grand canonical partition function, Δ1(L ;{xi},{yij},{zij},{s1i}, {s2i}), leading to the partition function Z. Let Cj1 be the cofactor of the element in the jth row and first column of matrix [T (1/η)I], and let |T (1/η)I| be the determinant of that matrix. Then eq 17 provides the solution to G1 as 0
¼1þη
t0
¼1
8
