Langmuir 1997, 13, 4887-4891
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Lattice Gas Model for the Adsorption of Guanidinium Nitrate at the Mercury-Water Interface Abdel K. Belkasri and Dale A. Huckaby* Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129
Lesser Blum Department of Physics, University of Puerto Rico, Rio Piedras, Puerto Rico 00931-3343 Received March 24, 1997. In Final Form: June 24, 1997X A lattice gas of trigonal-planar cations and anions on the triangular lattice is constructed to model the adsorption of guanidinium nitrate at the mercury-water interface. For physically realistic intermolecular interactions and electrochemical potentials, the Pirogov-Sinai theory is used to rigorously prove the presence at low temperatures of a hexagonal hydrogen-bonded phase in the model. This phase is consistent with the monolayer structure recently proposed on the basis of experiments.
1. Introduction The interfacial properties of salts which form condensed monolayers at the mercury-water interface have been the focus of some recent experiments.1-10 The interfacial properties of one such salt, guanidinium nitrate4,9 (see Figure 1), led de Levie et al. to conjecture a plausible structure for the condensed monolayer which forms at the interface.9 In particular, capacitance measurements indicated the presence of a monolayer at the interface, and the concentration dependence of the capacitance indicated the presence of equal numbers of nitrate ions and guanidinium ions in that monolayer. The X-ray crystal structure of the salt showed the presence of planes of hydrogen-bonded structures. This led to the conjecture9 that a condensed monolayer is present at the mercury-water interface which consists of a planar hexagonal array of hydrogenbonded guanidinium and nitrate ions, each ion having a trigonal planar structure which supports six hydrogen bonds to neighboring ions (see Figure 8 of ref 9). A comparison of the inhibition of electron transfer through this condensed monolayer with that through a condensed monolayer of monomethylguanidinium nitrate supports the conjecture of an open hydrogen-bonded structure.10 A lattice gas can be used to model any phases possible in the continuum which can be realized on the lattice. In the present paper we construct a two-dimensional lattice gas model containing nitrate ions and guanidinium ions which can occupy the sites of a triangular lattice at the mercury-water interface. The triangular lattice was * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, August 15, 1997. (1) Frumkin, A. N.; Damaskin, B. B. Dokl. Akad. Nauk SSSR 1959, 129, 862. (2) Damaskin, B. B.; Nikolaeva-Fedorovich, N. V. Zh. Fiz. Khim. 1961, 35, 1279. (3) Mikhailik, Y. V.; Damaskin, B. B. Elektrokhimiya 1979, 15, 566. (4) Dyatkina, S. L.; Damaskin, B. B.; Vygotskaya, M. Z. Elektrokhimiya 1980, 16, 996. (5) Srinivasan, R.; de Levie, R. J. Electroanal. Chem. 1986, 206, 307. (6) Sridharan, R.; Saffarian, M. H.; de Levie, R. J. Electroanal. Chem. 1987, 236, 311. (7) Vetterl, V.; de Levie, R. J. Electroanal. Chem. 1991, 310, 305. (8) Wandlowski, T.; de Levie, R. J. Electroanal. Chem. 1992, 329, 103; 1993, 345, 413; 1993, 352, 279; 1995, 388, 199. (9) Wandlowski, T.; Jameson, G. B.; de Levie, R. J. Phys. Chem. 1993, 97, 10119. (10) Wandlowski, T.; Jameson, G. B.; de Levie, R. J. Electroanal. Chem. 1994, 379, 215.
S0743-7463(97)00309-0 CCC: $14.00
Figure 1. Illustration of the hydrogen bonding (dotted lines) between a nitrate anion, NO3-, and a guanidinium cation, C(NH2 )3+. Below is a simplified representation for the ions which is used in the present paper.
chosen to model this system, for not only does it allow close-packed configurations which are possible in a continuous interface but it can also embed the hydrogenbonded hexagonal phase9 proposed on the basis of experiments. A justification for the procedure of imposing a lattice to determine crystalline phases present in the continuum can be found in a recent investigation.11 There, the freezing transition on a smooth interface is obtained by first imposing a symmetry-breaking lattice of sticky sites on an otherwise continuous interface. The sites have a stickiness parameter which differs from that of the continuous interface. (The stickiness parameter is related to the chemical potential in an equivalent lattice gas model.11,12) Then, after a crystalline phase forms on the large lattice, the stickiness parameters are decreased to zero. The crystalline phase is then stable if it has the lowest free energy. The crystallization process is thus well-modeled by imposing a symmetry-breaking lattice of sticky sites which is removed after crystallization occurs. After introducing some physically realistic intermolecular interaction parameters for the model in section 2, we proceed in section 3 to determine the structure of the ground-state configurations for the lattice gas model. One type of ground state is consistent with the structure (11) Hernando, J. A.; Blum, L. Physica A 1996, 232, 74. (12) Huckaby, D. A.; Blum, L. Langmuir 1995, 11, 4583.
© 1997 American Chemical Society
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Figure 2. Illustration of the triangular lattice with several ions in different orientations.
proposed in ref 9. We then use the Pirogov-Sinai theory13 to show that phases exist at finite temperatures which are small perturbations of these ground-state configurations. 2. Model In order to model the adsorption of guanidinium cations and nitrate anions at the mercury-water interface, we construct a two-component lattice gas on a triangular lattice at the interface. In this lattice gas model each lattice site can be either vacant or occupied by a nitrate anion, NO3-, or a guanidinium cation, C(NH2 )3+. Both of the ions have a trigonal-planar structure with ONO and NCN bond angles of 120° (see Figure 1). The central N or C atom of an ion will occupy a lattice site, the ion being in either of the two orientations in which the three NO or CN bonds point toward neighboring lattice sites (see Figure 2). Hydrogen bonding between the nitrate oxygens and the amine hydrogens of the guanidinium cation is possible in the model only if the ions are in the relative configuration illustrated in Figure 1. (If the ions are in the relative configuration of ions 2 and 3 in Figure 2, the geometry of the orbitals will not result in sufficient overlap for forming hydrogen bonds.) The grand partition function for the model can be written as
Ξ)
∑ξ e-H(ξ)/kT
(1)
where the Hamiltonian for the configuration ξ ) {ξl} (l runs over the sites of the triangular lattice) has the form
H(ξ) )
∑ Ei,j(ξ) - µ+N+(ξ) - µ-N-(ξ)
(2)
(i,j)
Ei,j(ξ) ) E(ξi,ξj) is the intermolecular pair interaction energy between a configuration ξi at site i and a configuration ξj at site j, a configuration ξi specifying whether site i is vacant or is occupied by an ion and, if so, the type and orientation of the ion at the site. N+(ξ) and N-(ξ) are the total numbers of cations and anions in ξ, and µ+ and µ- are, respectively, the electrochemical potentials of a cation and an anion at a lattice site of the triangular lattice at the mercury-water interface. We will consider only pair interactions between firstneighbor ions. The interaction energy Ei,j between two ions on i and j incorporates the following: (i) an electrostatic interaction < 0 between a cation and an anion on neighboring sites, as illustrated with (13) Pirogov, S.; Sinai, Ya. G. Theor. Math. Phys. 1975, 25, 1185; 1976, 26, 39.
Figure 3. The twenty-five symmetrically different types of configurations about a triangle in a configuration of ions on the triangular lattice. For example, in the configuration of ions illustrated in Figure 2, ions 1 and 2 share two triangles which have configurations of type 5, whereas ions 3 and 4 share two triangles which have configurations of type 10.
ions 2 and 3 of Figure 2, or - for the case where the two neighboring ions are of the same type, as illustrated with ions 1 and 2 of Figure 2. (ii) a hydrogen-bonding energy 2H < 0, as illustrated with ions 3 and 4 of Figure 2. In this case E3,4 ) + 2H. (iii) a steric repulsion γ > 0 for the case in which the ions point bonds toward one another, as is illustrated with the pairs (4, 5) and (5, 6) of Figure 2. The interaction energies for these cases are E4,5 ) - + γ and E5,6 ) + γ. Neglecting boundary effects (or assuming periodic boundaries), we can write the Hamiltonian for a configuration ξ as a sum of restricted Hamiltonians to each elementary triangle τ in the lattice. Since the Hamiltonian in eq 2 is a sum of terms associated with each occupied site and with each occupied first-neighbor pair of sites, then each of these terms can be partitioned equally among the elementary triangles of the lattice. Thus, 1/2 of the pair interaction energy Ei,j will be associated with each of the two triangles which share the edge linking sites i and j, and 1/6 of the electrochemical potential of an ion at site i will be associated with each of the six triangles which have a vertex at site i. This partition defines the Hamiltonian restricted to a triangle τ, Hτ(ξ). The Hamiltonian can then be written as a sum of restricted Hamiltonians
H(ξ) )
∑τ Hτ(ξτ)
(3)
where the summation is over all the triangles of the triangular lattice and ξτ is the restricted configuration of ξ to the vertices of triangle τ. In Figure 3 are listed all the possible symmetrically different configurations which
Adsorption of Guanidinium Nitrate
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Figure 4. Zero temperature phase diagram for case a: 0 < -2 < -2H < γ. Points A, B, and C are, respectively, (3 + 6H, -3), h, B h , and C h are reflections of the points A, B, and C with respect to the line µ+ ) µ-. In the region (6H, -6), and (12 - 6H, -6H). A containing the origin, the ground-state configurations consist entirely of triangles having the configuration of type 10. One of these six symmetry-related ground-state configurations is illustrated in Figure 6. In the region containing a drawing of a triangle with a configuration of type 24, the ground-state configurations consist entirely of triangles of type 24. One of these six ground-state configurations is illustrated in Figure 7. The structures of the ground-state configurations in other regions of the phase diagram are described in the text. Table 1. Values of the Restricted Hamiltonian Hτ(i) for the 25 Symmetrically Different Configurations Illustrated in Figure 3 i
Hτ(i)
i
Hτ(i)
1 2 3 4, 5 6 7, 8 9 10 11, 12 13
0 -µ+/6 -µ-/6 -/2 - µ+/3 -/2 + γ/2 - µ+/3 -/2 - µ-/3 -/2 + γ/2 - µ-/3 /2 + H - µ+/6 -µ-/6 /2 - µ+/6 - µ-/6 /2 + γ/2 - µ+/6 - µ-/6
14 15 16 17 18, 19 20, 21 22 23 24 25
-3/2 - µ+/2 -3/2 + γ/2 - µ+/2 -3/2 - µ-/2 -3/2 + γ/2 - µ-/2 /2 + H + γ/2 - µ+/3 - µ-/6 /2 + H + γ/2 - µ+/6 - µ-/3 /2 - µ+/3 - µ-/6 /2 + γ/2 - µ+/3 - µ-/6 /2 - µ+/6 - µ-/3 /2 + γ/2 - µ+/6 - µ-/3
can occur about a triangle in the triangular lattice. The possible values for the Hamiltonian restricted to a triangle are listed in Table 1. The index i in Table 1 refers to the index of the different configurations in Figure 3. In section 3 we will determine the minima of the restricted Hamiltonian Hτ(ξ) and construct the ground state configurations for two physically realistic domains of interaction parameters. We will then prove the existence at low temperatures of several types of ordered phases in the model system.
3. Low-Temperature Phase Diagram The Hamiltonian restricted to a triangle, Hτ(ξ), is said to constitute an “m-potential”14 in a domain of the parameter space (, H, γ, µ+, µ-) if, for every point in the domain, there exists a configuration ξG which is composed entirely of triangles with a minimal value of the restricted Hamiltonian,
H° ) min Hτ(ξ) ξ,τ
(4)
The configuration ξG is a ground-state configuration, and all the other ground-state configurations in the domain are also composed entirely of triangles which have restricted Hamiltonians with the minimum value H°. Moreover, if Hτ(ξ) is an m-potential and the groundstate configurations are finitely degenerate and related by symmetries of the Hamiltonian, the Pirogov-Sinai theory13-15 proves that, at sufficiently low temperature, phases exist in the model which have the structure of the (14) Slawny, J. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic: New York, 1987; Vol. 11, p 127.
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Figure 5. Zero temperature phase diagram for case b: 0 < -2 < γ < -2H. Points A, B, and D are, respectively, (3 + 6H, -3), h, B h , and D h are reflections of the points A, B, and D with respect to the line µ+ ) µ-. (6H, -6), and (12 + 6H + 6γ, 3γ). A
ground-state configurations together with a number of defects or excitations. This theory proves that the phase diagram at zero temperature remains stable for sufficiently low temperatures if the conditions mentioned above are satisfied. (See, for example, ref 12 for the essence of the method or ref 14 for more details.) In Table 1 are listed the 25 different possible values for the Hamiltonian restricted to a triangle. A consideration of Table 1 together with eq 4 allows us to determine the minima of Hτ(ξ) and to construct the different groundstate configurations in the plane (µ+, µ-). Since the intermolecular interactions are symmetric with respect to the interchange of ion type, the phase diagram is symmetric with respect to the line µ+ ) µ-. For the general case of the model in which < 0, H < 0, and γ > 0, a consideration of Table 1 indicates that, for a region of the (µ+, µ-) plane about the point (µ+ ) 0, µ) 0), triangles with a configuration of type 10 have the lowest value of the restricted Hamiltonian. The restricted Hamiltonian forms an m-potential in this region, and the ground-state configurations consist entirely of triangles of type 10. These six symmetry-related ground-state configurations, one of which is illustrated in Figure 6, contain a hexagonal array of hydrogen-bonded guani(15) Holsztynski, W.; Slawny, J. Commun. Math. Phys. 1978, 61, 177.
dinium and nitrate ions, the centers of the hexagons in the array being vacant. Therefore, at sufficiently low temperatures the equilibrium states for the model in this region will have the same basic structure as in Figure 6, except for some defects or excitations. A condensed monolayer with this structure was indeed proposed on the basis of experiments of the adsorption of guanidinium nitrate at the mercury-water interface.9 In addition to the ground states having the structure illustrated in Figure 6, the model also exhibits other types of ground-state structures. We shall determine the ground-state configurations and illustrate the zero temperature phase diagram in the entire (µ+, µ-) plane for the model in two physically realistic domains of interaction parameters (, H, γ), case a, for which 0 < -2 < -2H < γ, and case b, for which 0 < -2 < γ < -2H. In these two cases, the electrostatic interaction energy is of smaller magnitude than either the hydrogen bond energy or the steric repulsion energy. This is physically reasonable, for screening effects should cause the electrostatic interaction to be small. Regions of the zero temperature (µ+, µ-) plane which correspond to different ground-state configurations are illustrated in Figure 4 for case a and in Figure 5 for case b. For both cases the restricted Hamiltonian, Hτ(ξ), constitutes an m-potential in the entire (µ+, µ-) plane. In
Adsorption of Guanidinium Nitrate
Figure 6. Illustration of a ground-state configuration composed of triangles of type 10. The ions form a hydrogen-bonded array. This phase exists in both cases a and b around the origin of the plane (µ+, µ-).
each region of the plane are illustrated the configurations about a triangle which have the minimum value of the restricted Hamiltonian, the ground states in the region consisting entirely of triangles of these types. As illustrated for case a in Figure 4 and for case b in Figure 5, in a region of the (µ+, µ-) plane containing the origin, the ground-state configurations are sixfold degenerate and have the structure illustrated in Figure 6. The low-temperature phases for the two-dimensional lattice gas in this region thus consist of a hexagonal array of hydrogen-bonded guanidinium and nitrate ions. For larger magnitudes of the electrochemical potentials, µ+ and µ-, the model exhibits other types of phases. Both case a and case b present qualitatively the same phase diagram except for the region about the diagonal µ+ ) µ-. In case a the ground-state configurations in this region are composed of triangles of one type, either type 24 or type 22. These symmetry-related ground states are sixfold degenerate, and thus there are low-temperature phases in the model which have the structure of these groundstate configurations, together with some excitations. One of these ground-state configuration is illustrated in Figure 7. For case b in the region near µ+ ) µ-, the ground-state configurations are infinitely degenerate. In some of these configurations (as in the configuration of Figure 6), guanidinium and nitrate ions form a hexagonal hydrogenbonded array. But unlike the configuration of Figure 6, the centers of the hexagons are occupied by ions of a single type, each being in either of the two possible orientations. Since these configurations are infinitely degenerate, the Pirogov-Sinai theory cannot be used to determine the structure of the low-temperature phases in this region. The other ground-state configurations, common to both cases a and b, can also be constructed. The vacant configuration, composed of triangles of type 1, occurs if
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Figure 7. Illustration of a ground-state configuration composed of triangles of type 24. This phase is possible only for case a, in which the steric repulsion interaction γ is strong enough to prevent neighboring ions from both pointing bonds toward each other.
the electrochemical potentials are sufficiently negative. If the ground-state configurations are composed of triangles of type 14 (or triangles of type 16), these ground states have every site occupied by an ion of a single type, all in the same orientation. Since such configurations are twofold degenerate, there are thus low-temperature phases having the structure of these ground-state configurations, together with some defects or excitations. In regions in which the ground-state configurations are composed of triangles of type 2, 3, 4, 5, 7, or 8, the PirogovSinai theory cannot be used to ascertain the structure of the low-temperature phases because these ground-state configurations are infinitely degenerate. In conclusion, we have introduced a lattice gas model containing guanidinium cations and nitrate anions on a triangular lattice at the mercury-water interface. The Pirogov-Sinai theory was used to prove the existence of equilibrium phases. Independent of the relative amplitudes of the interaction parameters in the model, a phase in which the guanidinium ions hydrogen bond to the nitrate ions to form an interfacial monolayer is always present, in agreement with experiments.9 The model also exhibits other interesting phases at low temperatures, and it will be interesting to see if experiments can detect such phases at the mercury-water interface. Acknowledgment. L.B. is grateful for the hospitality of Georgetown University and for conversations with Robert de Levie which introduced the authors to this interesting interfacial system. L.B. was supported by the EPSCoR program, L.B. and D.H. were supported by the Office of Naval Research, and D.H. was supported by the Robert A. Welch Foundation, Grant P-0446. A.B. was a Robert A. Welch Foundation Postdoctoral Fellow. LA9703095
