Article pubs.acs.org/cm
Tuning Magnetic and Structural Transitions through Valence Electron Concentration in the Giant Magnetocaloric Gd5−xEuxGe4 Phases Jinlei Yao, Peng Wang, and Yurij Mozharivskyj* Department of Chemistry and Chemical Biology, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada S Supporting Information *
ABSTRACT: Valence electron concentration is a viable chemical tool to control the crystal structure and magnetism of Gd5Ge4. A decrease in the valence electron concentration achieved through the substitution of Eu2+ for Gd3+ leads to the formation of the interslab Ge−Ge dimers, phase transitions to the Gd5Si2Ge2- and Gd5Si4-type structures, and a ferromagnetic ordering in the Gd5−xEuxGe4 system. Gd4.75Eu0.25Ge4 and Gd4.50Eu0.50Ge4 undergo temperature-induced magnetostructural transformations accompanied by giant magnetocaloric effects. KEYWORDS: magnetocaloric effect, valence electron concentration, magnetostructural transition
stoichiometry were loaded into tantalum ampules, which were sealed by arc melting under an argon atmosphere. The ampules were heated in a high frequency induction furnace at 1450 °C for 4 h and thereafter tempered at 1200 °C for 15 h. For comparison, the binary Gd5Ge4 compound was prepared by arc melting of the constituted elements under an argon atmosphere, annealed at 1050 °C for 22 h, and then, quenched in cold water. Powder X-ray diffraction (XRD) data of the samples were collected on a PANalytical X’Pert Pro diffractometer with a linear X’Celerator detector using Co Kα radiation. Lattice parameters and phase analyses were carried out by Rietveld refinements using the Rietica software,13 and the refinement results are given in Table S1 and Figure S1 of the Supporting Information. Single crystal XRD data at room temperature were obtained on a STOE IPDS II diffractometer with Mo Kα radiation. Numerical absorption corrections were performed using the program X-Shape included with the STOE IPDS software package.14 The low-temperature (94−300 K) data for the Gd4.75Eu0.25Ge4 and Gd4.50Eu0.50Ge4 single crystals were also collected on a Bruker SMART Apex II CCD diffractometer (Mo Kα radiation) in a reciprocal hemisphere. Intensities were extracted and then corrected for the Lorentz and polarization effects through the SAINT program.15 Numerical absorption correction was based on the crystal shape obtained from the optical face indexing. The space groups were assigned on the basis of the systematic absences and the statistical analysis of the intensity distributions. Structure determination using direct methods and refinement using full-matrix least-squares on F2 were carried out with the SHELXL package.16,17 Due to the similar Xray scattering factors of Gd and Eu, the site occupancies for Gd and Eu could not be refined using our XRD data. According to the simple relationship between the ionic size and R-site occupation,18 the Eu atoms are predicted to preferentially occupy the R1 site with a large
1. INTRODUCTION The 1997 discovery of the giant magnetocaloric effect (MCE) in Gd5Si2Ge21 prompted extensive research on the Gd5SixGe4−x and related R5T4 systems (R = rare earth and T = pelement).2−12 From the practical aspect, motivations for such studies were and remain to be attempts to prepare new highperformance magnetocaloric materials that can be used for magnetic refrigeration. From the fundamental prospective, scientists are trying to develop understanding of the structure− property relationship in R5T4 and to establish factors that would allow tuning their properties. It has been shown that the structural and magnetic behavior of R5T4 can be controlled through external parameters/forces, such as temperature, magnetic field, and pressure,2−4 as well as the chemical substitution, for example, Si/Ge5 and Gd/La,6 which can be treated as an internal force. Among the chemical approaches, changes in the valence electron concentration (VEC) are particularly appealing as they allow fine tuning of the structural features of R5T4.7−11 However, there was no solid evidence on how the VEC can alter the nature of magnetic interactions in the R5T4 phases until now. In this work, we show that a small reduction in the VEC of Gd5Ge4, which is antiferromagnetic (AFM), induces the ferromagnetic (FM) ordering in Gd4.75Eu0.25Ge4. We also demonstrated that further reductions in the VEC promote two successive structural transitions. 2. EXPERIMENTAL SECTION The starting materials were pieces of gadolinium (99.99 wt.%, distilled grade, Metal Rare Earth Limited, China), europium (99.9+ wt.%, distilled grade, Rhone Poulenc), and germanium (99.9999 wt.%, Alfa Aesar). The samples were handled in an Ar-filled glovebox. The surface of Eu metal lumps was scrapped with a file before cutting into pieces. Elements based on the Gd5−xEuxGe4 (x = 0.25−1.0) © 2011 American Chemical Society
Received: October 20, 2011 Revised: December 19, 2011 Published: December 20, 2011 552
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Table 1. Crystallographic and Magnetic Data for Gd5−xEuxGe4 x
0
0.25
0.25
0.50
0.50
1.0
temp (K) str. type a (Å) b (Å) c (Å) γ (deg) vol (Å3) R1 [I > 2σ(I)] dGe1−Ge1 (Å)
293(2) Sm5Ge4 7.6760(4) 14.7819(8) 7.7738(3) 90 882.06(7) 0.0211 3.620(1)
293(2) Sm5Ge4 7.6781(6) 14.842(1) 7.7797(7) 90 886.5(1) 0.0386 3.560(3)
94(2) Gd5Si4 7.5434(2) 14.8260(3) 7.8284(2) 90 875.52(4) 0.0353 2.692(1)
100(2) Gd5Si4 7.5669(8) 14.905(2) 7.8648(8) 90 887.0(2) 0.0249 2.692(2)
293(2) Gd5Si4 7.5993(6) 15.256(2) 8.0218(6) 90 930.0(1) 0.0381 2.675(3)
TC/N (K) μeff(R) (μB)
128 8.14
97 8.06
293(2) Gd5Si2Ge2 7.633(1) 14.909(2) 7.8481(7) 92.98(1) 891.9(2) 0.0468 2.691(6) 3.487(5) 132 8.70
site volume due to the larger ionic radii of Eu ions, that is, Eu2+ (1.17 Å) > Eu3+ (0.947 Å) > Gd3+ (0.938 Å) for a six-coordinate environment.19 Our first-principles electronic structure calculations by the Stuttgart tight-binding, linear-muffin-tin-orbital program using the atomic sphere approximation (TB−LMTO−ASA)20,21 also show that the assignment of Eu to the R1 site gives the lowest energy when compared with other models. However, the site preference for an R atom does not exclude its presence on other sites, which is observed in many R-substituted R5T4 systems.18 Therefore, the same Gd/Eu statistical mixtures consistent with the loaded compositions were assumed on all R sites. Details of the single-crystal data collections and refinements are listed in Table S2 of the Supporting Information. Atomic position parameters and isotropic displacement parameters are given in Table S3 of the Supporting Information. Further details of the crystal structure investigated are available from the Fachinformationszentrum Karlsruhe, 76344 Eggenstein−Leopoldshafen, Germany, on quoting the depository number CSD 423095 (Gd5Ge4, 293 K), 423096 (Gd4.75Eu0.25Ge4, 293 K), 423097 (Gd4.75Eu0.25Ge4, 94 K), 423098 (Gd4.50Eu0.50Ge4, 293 K), 423099 (Gd4.50Eu0.50Ge4, 100 K), and 423100 (Gd4EuGe4, 293 K). Magnetization of Gd5−xEuxGe4 was measured on a Quantum Design SQUID magnetometer in a magnetic field up to 5 T. Maxima in the derivatives of the magnetization with respect to temperature were taken as Curie (TC) temperatures. The cusp temperature on the thermal magnetization curve was taken as the Neel (TN) temperature of Gd5Ge4. Weiss temperatures (θ) and the effective magnetic moments of Gd/Eu atoms (μeff) were obtained by fitting the paramagnetic data to the Curie−Weiss law. The isothermal magnetization curves of Gd4.75Eu0.25Ge4 and Gd4.50Eu0.50Ge4 were measured from 0 to 50 kOe with a step of 2 kOe (Supporting Information, Figure S2). The MCE in terms of the isothermal entropy change, ΔS, was derived from the magnetization data (Supporting Information, Figure S2) through a numerical integration of the Maxwell equation:
ΔS(T )ΔH =
170 7.93
orbitals for Gd and 4s, 4p, and 4d orbitals for Ge. The Gd 6p and Ge 4d orbitals were treated by the Löwdin downfolding technique. The half-filled Gd 4f orbitals were treated as core states. Crystal orbital Hamilton population (COHP) were calculated to determine the relative influence of various interatomic orbital interactions.26 The kspace integrations were performed by the tetrahedron method.27 The self-consistent charge density was obtained using 256 irreducible kpoints in the first Brillouin zone for the orthorhombic cells.
3. RESULTS AND DISCUSSION The single-crystal and powder XRD analyses showed that the Gd5−xEuxGe4 compounds with x = 0−0.25, 0.50, and 1.0 crystallize with the Sm5Ge4-, Gd5Si2Ge2-, and Gd5Si4-type structures at room temperature (RT), respectively, and their purity is larger than 94 wt % (Table 1, Figure 1, and Tables S1−3 and Figure S1 of the Supporting Information). All three
∑ (Mi + 1 − Mi)/(Ti + 1 − Ti)δH i
where δH is a magnetic field step and Mi and Mi+1 are the magnetization values at Ti and Ti+1 temperatures, respectively.22 Electronic structure calculations were carried out with the TB− LMTO−ASA method. The calculations were performed on the Sm5Ge4- (in a zero magnetic field) and Gd5Si4-type (in a 35 kOe field) structures of the Gd5Ge4 compound. The structural data were taken from ref 23. Exchange and correlation were treated by the local density approximation.24 All relativistic effects except spin−orbit coupling were taken into account using a scalar relativistic approximation.25 In the ASA method, space is filled with overlapping Wigner−Seitz (WS) atomic spheres whose radii were determined by an automatic procedure.26 To satisfy the overlap criteria of the atomic spheres in the TB−LMTO−ASA method, empty spheres were included in the unit cell employing automatic sphere generation for the calculations.26 The WS radii employed for the calculations are as follows: Gd = 1.76− 1.94 Å and Ge = 1.51−1.56 Å. The basis sets included 6s, 6p, and 5d
Figure 1. Three crystal structures of Gd5−xEuxGe4. R is a mixture of Gd and Eu. The interatomic distance of the interslab Ge1−Ge1 bond is indicated in the figure. 553
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structures consist of ∝2[R5Ge4] nanoslabs (∼7 Å thick) within which the Ge2 and Ge3 atoms form the intraslab Ge2−Ge3 dimers. A remarkable difference between the three structures lies in how the ∝2[R5Ge4] slabs are connected: in the Gd5Si4type structure, the slabs are connected through interslab Ge1− Ge1 dimers (dGe1−Ge1 < 2.7 Å); in the Gd5Si2Ge2-type one, intact and broken Ge1−Ge1 dimers alternate between the slabs; and in the Sm5Ge4-type one, all interslab dimers are broken (dGe1−Ge1 > 3.4 Å). Low-temperature (LT) single-crystal XRD analysis revealed that both Gd4.75Eu0.25Ge4 (Sm5Ge4-type) and Gd4.50Eu0.50Ge4 (Gd5Si2Ge2-type) undergo transitions to the Gd5Si4 structure (Table 1 and Figure 1). The structural transitions can be viewed as shear movement of the ∝2[R5Ge4] slabs primarily along the a direction, resulting in the reformation of the broken interslab Ge1−Ge1 dimers. The magnetization data (Figure 2a) show that both transformations are accompanied by FM
Figure 3. Temperature (T) dependence of magnetization (M) and lattice parameters of Gd5−xEuxGe4 (x = 0.25 and 0.50). The lattice parameters were obtained from single-crystal X-ray diffraction. The magnetization was measured on polycrystalline samples as cooling and heating in a 100 Oe field for three circles.
Figure 2. (a) Temperature (T) dependence of magnetization (M) of Gd5−xEuxGe4 in a magnetic field of 100 Oe. The M of Gd5Ge4 is enlarged by a factor of 10. (b) Entropy change (−ΔS) for the magnetic field changing from 0 to 5 T around the Curie temperatures.
than a larger unit cell. These structural facts support the +2 oxidation state for Eu. Third, the fit to the PM data of Gd5−xEuxGe4 using the Curie−Weiss law yields an effective magnetic moment for Eu of μeff(Eu) = 7.89−13.78 μB, assuming Gd takes the theoretical moment of 7.94 μB. Since μ(Eu3+) = 0 and μ(Eu2+) = 7.94 μB theoretically, the magnetic data advocate Eu2+ in Gd5−xEuxGe4. The overestimation of μeff(Eu), especially in the low Eu-content samples, results from the underestimation of the actual moments of Gd, which frequently shows enhanced magnetic moments due to the contribution from the 5d electrons.30,31 Fourthly, since only the rare-earth atoms are magnetically active in the R5T4 phases, any rare-earth substitution with a nonmagnetic element will lead to weaker magnetic interactions or even destruction of the longrange magnetic ordering. Such negative dilution effects have been observed in several R5T4 systems, such as Gd5−xLuxGe4, Gd5−xLaxGe4,6 R5−xMgxGe4,9 and Gd5−xZrxSi4,32 no matter how the VEC changes. Thus, the Gd5−xEuxGe4 phase should be AFM with a lower TN or even PM if the Eu ion is Eu3+. However, our samples display the FM properties and the TC increases with increasing the Eu content (Figure 2a), indicating the substitution enforces the FM interactions. Considering the fact that Eu2+ is magnetic and Eu3+ is diamagnetic, the FM properties induced by the Eu substitution suggest the presence of Eu2+ too. In summary, Eu in Gd5−xEuxGe4 is testified divalent based both on the structural and magnetization data. Now, we can write electronic formulas for Gd5−xEuxGe4, using the Zintl−Klemm electron counting method.7,33 Within this formalism, the intraslab Ge2−Ge3 and interslab Ge1−Ge1 dimers, both denoted as Ge2, are isoelectronic with halogen dimers and carry the negative charge of 6−. The Ge1
orderings. Presence of temperature hystereses for the samples with x = 0.25 and x = 0.50 during cooling and heating indicates their first-order nature (Figure 3b). The first-order structural transitions are confirmed by the crystallographic data at various temperatures (Figure 3a). Both the samples undergo a phase transition to the Gd5Si4-type structure when the paramagnetic (PM) state turns into the FM ordering at low temperatures, accompanied by an abrupt change in the a parameter. The magnetostructural Sm5Ge4 to Gd5Si4 and Gd5Si2Ge2 to Gd5Si4 temperature-driven transitions achieved through chemical modification of the Gd5Ge4 binary have been observed before, for example, in Gd5SixGe4−x5 and Gd5GaxGe4−x.12 However, until now, inducing these types of transitions was accomplished through the substitution of Ge and besides required relatively large substitution levels (>17 at.% for Gd5GaxGe4−x and >25 at. % for Gd5SixGe4−x). In Gd5−xEuxGe4, the two transformations are attained through the substitution of Gd and at much lower levels of 5 and 10 at.%. To understand effects of the Eu substitution, the oxidation state of Eu in Gd5−xEuxGe4 will be addressed first. A significant expansion of the unit cell upon the Eu substitution (Table 1) implies that Eu is primarily divalent. The ionic radii of Eu3+ (0.947 Å, six-coordinated) and Gd3+ (0.938 Å) are similar but much smaller than that of Eu2+ (1.17 Å).19 In addition, when compared to Gd3La2Ge4 (V = 929.5 Å3),28 Gd3Eu2Ge4 displays a larger unit cell (V = 947.7 Å3).29 Since La3+ (1.032 Å) is the biggest trivalent rare-earth ion, the presence of Eu3+ in Gd3Eu2Ge4 should result in a smaller rather 554
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transitions around TC = 97 and 132 K, respectively, which yield FM ground states with the Gd5Si4 structure. For comparison, the ground state of Gd5Ge4 is an AFM Sm5Ge4type structure with TN = 128 K. As demonstrated previously,34 the FM coupling between the adjacent slabs is mediated through the long-range Ruderman−Kittel−Kasuya−Yosida (RKKY) indirect interactions via the R−Ge−Ge−R pathway. Presence of the interslab Ge−Ge dimers promotes FM interactions, and their absence leads to the AFM ground state, as in Gd5Ge4. The AFM interactions in Gd5Ge4 can be tuned into the FM one, concomitant with a transition to the Gd5Si4 structure, through applying external magnetic fields, for example, Happ = ∼18 kOe at 4 K,35 or imposing chemical or hydrostatic pressure.3,4 However, the Eu substitution leads to the lattice expansion, implying that the chemical pressure is unlikely to be responsible for the onset of FM interactions in Gd5−xEuxGe4. To uncover factors stabilizing FM interactions in Gd5−xEuxGe4, we assume a FM ground state with the Gd5Si4 structure for Gd5Ge4 and compare it to the ground states of the Eu−substituted phases. For such ground states, the nce decreases from 3 e− in Gd5Ge4 to 2.75 e− for the Eu0.25 and 2.5 e− for Eu0.50 substitutions. A reduction in nce likely establishes the FM coupling between the slabs through the R− Ge−Ge−R pathway in Gd4.25Eu0.25Ge4 and Gd4.50Eu0.50Ge4; however, the FM coupling is not so strong and is destroyed at high temperatures, leading to a transition to the energetically favorable PM Sm5Ge4- or Gd5Si2Ge2-type phases. A further decrease in nce enforces the Ge1−Ge1 bonding, stabilizing the Gd5Si4 structure and the interslab FM coupling in Gd4EuGe4. The VEC-stabilized FM interactions are also supported by the enhanced Curie temperature of Gd4EuGe4 upon further Eu substitution (Table 1). The magnetostructural transitions in Gd4.75Eu0.25Ge4 and Gd4.50Eu0.50Ge4 yield giant MCEs (Figure 2b). The ΔS values may change after the synthetic conditions are optimized. Still, it is clear that the maximum ΔS value of −40 J/kg K for Gd4.75Eu0.25Ge4 is significantly larger than −27 J/kg K for Gd4.50Eu0.50Ge4. This difference results from a more global structural transformation in Gd4.75Eu0.25Ge4 as compared to that in Gd4.50Eu0.50Ge4; all interslab Ge1−Ge1 dimers are reformed in the former, but only half of those in the latter (Figure 1). The difference of 13 J/kg K provides an estimate of the structural contribution to the total entropy change in Gd4.50Eu0.50Ge4. Obviously, in Gd4.75Eu0.25Ge4, the lattice contribution will be roughly doubled. The lattice contribution of 13 J/kg K in Gd4.50Eu0.50Ge4 is close to the value of 9.8 J/kg K derived for Gd5Si2Ge2,36 which undergoes a magnetostructural transition analogous to that in Gd4.50Eu0.50Ge4.
monomers, present when Ge1−Ge1 dimers are broken, adopt the closed shell configuration and carry the negative charge of 4−. Thus, the electronic formulas for the Sm5Ge4- (x = 0− 0.25), Gd5Si2Ge2- (x = 0.5), and Gd5Si4-type (x = 1.0) phases can be written as (Gd3+)5−4.75(Eu2+)0−0.25(Ge26−)(Ge4−)2(1− 0.75e− ), (Gd 3+ ) 4.5 (Eu 2+ ) 0.5 (Ge 2 6− ) 1.5 (Ge 4− )(1.5e− ), and (Gd3+)4(Eu2+)(Ge26−)2(2e−), respectively. Clearly, different structures permit different numbers of conduction electrons (nce), which in turn implies that their stability might depend on the VEC. Elucidation of this dependence comes from the electronic structure calculations performed on Gd5Ge4 with the Gd5Si4and Sm5Ge4-type structures using the TB−LMTO−ASA method (Figure 4). In the case of the Gd5Si4 structure,20 the Fermi level falls into the σp* region of both the interslab and
Figure 4. Crystal orbital Hamilton population (COHP) curves for the interslab Ge1−Ge1 interactions of Gd5Ge4 with the Gd5Si4- and Sm5Ge4-type structures. The vertical dash line with x = 1 is the Fermi level for the Gd4EuGe4 within a rigid band approximation. The COHP data for the Sm5Ge4-type structure are magnified by 10. Interactions with −COHP > 0 and −COHP < 0 are bonding and antibonding, respectively. The σs and σs* states are at lower energies and are fully occupied.
intraslab Ge dimers. This is in agreement with the Zintl− Klemm approach that predicts that three conduction electrons in the Gd5Si4 structure of (Gd3+)5(Ge26−)2(3e−) will occupy the antibonding levels within the Ge−Ge dimers. To minimize these energetically unfavorable antibonding contributions, Gd5Ge4 adopts the Sm5Ge4-type structure, in which the interslab dimers are broken and the nce is reduced to 1: (Gd3+)5(Ge26−)(Ge4−)2(1e−). However, the Gd5Si4 structure can be stabilized if the population of the antibonding Ge−Ge states is minimized. This goal is achieved in Gd4EuGe4 through the Eu substitution, which brings the total VEC down to 30 e−/ f.u. and the nce to 2. The bond strength of the interslab Ge1− Ge1 dimer is slightly enhanced by the VEC decrease, which is demonstrated by the change in the integrated COHP (−ICOHP), that is, from 1.81 eV/cell for x = 0 to 1.86 eV/ cell for x = 1.0. Interestingly, a lower Eu substitution level as in Gd4.5Eu0.5Ge4 (VEC = 30.5) yields the RT Gd5Si2Ge2-type structure with half of the interslab dimers broken. Most likely, this VEC and the resulting nce are just right to stabilize the RT Gd5Si2Ge2 structure. The relationship between the VEC and structure established for Gd5−xEuxGe4 is similar to those observed in other R5T4 phases.7−11 The LT structural and magnetic behavior of the electronically modified Gd5−xEuxGe4 phases is rather unique. Gd4.25Eu0.25Ge4 and Gd4.5Eu0.5Ge4 undergo first-order magnetostructural
4. CONCLUSION In conclusion, the VEC is demonstrated to be a powerful tool to tune the crystal structure and magnetism of Gd5Ge4, which may be applied to other R5T4 phases. A decrease in the VEC of Gd5Ge4 achieved through the Eu substitution results in the reformation of the interslab Ge1−Ge1 dimers, transitions from the Sm5Ge4- to Gd5Si2Ge2-, and Gd5Si4-type structures and the FM coupling between the ∝2[R5Ge4] slabs. Gd4.75Eu0.25Ge4 and Gd4.50Eu0.50Ge4 undergo temperature-induced magnetostructural transformations, accompanied by giant MCEs. 555
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(26) Jepsen, O.; Andersen, O. Z. Phys. B 1995, 97, 35. (27) Blöchl, P. E.; Jepsen, O.; Andersen, O. K. Phys. Rev. B 1994, 49, 16223. (28) Yang, H. F.; Rao, G. H.; Liu, G. Y.; Ouyang, Z. W.; Liu, W. F.; Feng, X. M.; Chu, W. G.; Liang, J. K. J. Alloys Compd. 2003, 361, 113. (29) Gd3Eu2Ge4, Gd5Si4-type, Space group Pnma, a = 7.6186(5) Å, b = 15.418(1) Å, c = 8.0683(6) Å, and V = 947.7(1) Å3. (30) Harmon, B. N.; Freeman, A. J. Phys. Rev. B 1974, 10, 1979. (31) Roeland, L. W.; Cock, G. J.; Muller, F. A.; Moleman, A. C.; McEwen, K. A.; Jordan, R. G.; Jones, D. W. J. Phys. F 1975, 5, L233. (32) Yao, J.; Lyutyy, P.; Mozharivskyj, Y. Z. Dalton Trans. 2011, 40, 4275. (33) Choe, W.; Pecharsky, V. K.; Pecharsky, A. O.; Gschneidner, K. A.; Young, V. G.; Miller, G. J. Phys. Rev. Lett. 2000, 84, 4617. (34) Haskel, D.; Lee, Y. B.; Harmon, B. N.; Islam, Z.; Lang, J. C.; Srajer, G.; Mudryk, Y.; Gschneidner, K. A.; Pecharsky, V. K. Phys. Rev. Lett. 2007, 98, 247205. (35) Levin, E. M.; Pecharsky, V. K.; Gschneidner, K. A.; Miller, G. J. Phys. Rev. B 2001, 64, 235103. (36) Planes, A.; Mañosa, L.; Saxena, A. Magnetism and Structure in Functional Materials; Springer: Berlin, Germany, 2005.
ASSOCIATED CONTENT
S Supporting Information *
Single-crystal crystallographic files in CIF format, tables of crystallographic data, Rietveld refinement using X-ray powder diffraction data and plot of magnetization versus magnetic field for Gd5−xEuxGe4 (pdf). This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +1 905 525 9140 × 27796; Fax: +1 905 521 2773; Email:
[email protected].
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ACKNOWLEDGMENTS This work was supported by Discovery Grant from the Natural Sciences and Engineering Research Council of Canada and by a grant from the ACS Petroleum Research Fund.
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REFERENCES
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