J . Phys. Chem. 1988, 92, 4160-4165
4160
Size Quantization in Layered HgI, Colloids M. W. Peterson, 0. I. Micic,+ and A. J. Nozik* Solar Energy Research Institute, Golden, Colorado 80401 (Received: December 8, 1987)
Colloidal particles of Hg12 with a particle size less than about 25 8, are formed by the reaction of HgC12with NaI in acetonitrile. The particles are charged with CI- ions that fill normally empty tetrahedral coordination sites at the edges; therefore they have an empirical formula Hg12Cb,t6. About 40% of the mercury in the colloidal solution is in the form of the ionic complex Hg12C1-. These conclusions are based on extensive spectroscopicstudies of various Hg-I-Cl species and experiments involving ultracentrifugation, ultrafiltration, and treatment of the colloids with ion-exchange resins. The absorption spectrum of the colloidal Hg12 consists of three peaks at 4.26, 4.94, and 6.04 eV, and is attributed to size quantization effects. This spectrum is consistent with the first three allowed transitions in a simple particle-in-a-box model with infinite potential barriers in which the colloidal particles have the usual tetrahedral, layered structure of red Hg12,with dimensions of 26.1 8, perpendicular to the layer plane (four layers thick) and 13.3 A in the direction parallel to the layer plane.
Introduction
In recent years there has been much interest in the nature of very small particle sized (20-100 h;) semiconductor colloids that exhibit quantization These effects are manifested as large (up to 3 eV) blue shifts of the fundamental absorption edge of the semiconductor,'-25 enhanced photoredox properties of illuminated colloidal SOIS,'~,~~ and structure in the optical absorption s p e ~ t r a . ~ , " , ' ~Also , ~ ' , ~several ~ theoretical models for size quantization effects in small semiconductor particles have been presented. 1 7 8 1 1 , 2 3 Two controversies currently exist regarding the reported results with quantized colloids. The first controversy concerns the proposed existence of preferred particle sizes in the colloid, termed "magic number^".^^^^" The second controversy concerns the question of whether the optical absorption spectra are due only to semiconductor particles, or whether interfering ionic and/or molecular species are present in the system that contribute to, or perhaps even dominate, the observed s p e ~ t r a . ~ ~ ? ~ ~ These controversies are most lively with respect to the layered semiconductor colloids Pb12,11324,2S Bi13,11*24325 and Hg12.21,25The preparation of these colloids from the reaction of the metal salts with iodide ions leads to blue-shifted optical spectra exhibiting several well-resolved peaks.11s2125 These peaks have been attributed to magic numbers that exhibit one transition per particle size." Another explanation2' for peaks or structure in the absorption spectra of quantized colloids invokes multiple transitions in a single particle size between higher quantum states in the quantum wells of the semiconductor particles. In this paper, we examine this model further for Hg12 colloids. We also examine in great detail the even more basic questions, raised in our previous publication on Hg12,,' of whether the spectra exhibited by our preparation is due entirely to quantized colloidal particles of Hg12,and to what extent are contributions present from ionic or molecular species in solution? Since we prepare our HgIz colloids from the reaction of HgCl, with a stoichiometric amount of NaI in acetonitrile, the following products are possible: colloidal Hg12, molecular Hg12, molecular HgCI2, HgICl, Hg13-, Hg12C1-, HgCIJ-, Hg12C12Z-, HgI>-, HgICI,,-, Hg13C12-, I-, 12, 13-, and C1-. We determine the importance of each of these possible species in our preparation of colloidal Hg12 by optical and various physical separation experiments. 8
3
Experimental Section
Experiments designed to sort out the composition of the synthesized raw colloid are based on preparing each of the individual possible products listed above separately, comparing optical absorption spectra of the various species, and performing physical separations of colloidal particles from small ionic or molecular species using such techniques as ultracentrifugation, ultrafiltration, and treatment of sols with ion-exchange resins. 'Boris Kidric Institute, Vinca, Belgrade, Yugoslavia 1 1001
0022-3654/88/2092-4160$01.50/0
( a ) Preparation of Colloids and Ionic or Molecular Species. Colloidal Hg12. Typically 25 mL of a 2 X lo4 M solution of NaI (Bakers Analyzed) in acetonitrile (Aldrich Spectro grade) was slowly added to a 1 X lo4 M solution of HgClz (Aldrich,
(1) Brus, L. E. J. Chem. Phys. 1983, 79, 5566; J . Chem. Phys. 1984,80, 4403; J. Phys. Chem. 1986, 90, 2555. (2) Rossetti, R.; Ellison, J. L.; Gibson, J. M.; Brus, L. E. J . Chem. Phys. 1984,80, 4464. (3) Fojtik, A.; Weller, H.; Koch, U.; Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1984,88, 969. Weller, H.; Koch, U.; Gutierrez, M.; Henglein, A. Ibid. 1984, 88, 649. (4) Nozik, A. J.; Williams, F.; Nenadovic, M. T.; Rajh, T.; Micic, 0. I. J . Phys. Chem. 1985, 89, 397. Williams, F.; Nozik, A. J. Nature 1984, 311, 21. Peterson, M. W.; Nenadovic, M. T.; Rajh, T.; Herak, R.; Micic, 0. I.; Goral, J. P.; Nozik, A. J. J. Phys. Chem. 1988, 92, 1400. Rajh, T.; Vucemilovic, M. I.; Dimitrijevic, N. M.; Micic, 0. I.; Nozik, A. J. Chem. Phys. Lett. 1988, 143, 305. (5) Rossetti, R.; Hull, R.; Gibson, J. M.; Brus, L. E. J . Chem. Phys. 1985, 82, 552; J . Chem. Phys. 1985.83, 1406. (6) Fischer, Ch.-H.; Weller, H.; Fojtik, A.; Lume-Pereira, C.; Janata, E.; Henglein, A. Ber. Bunsen-Ges. Phys. Chem. 1986, 90, 46. (7) Schmidt, H. M.; Weller, H. Chem. Phys. Lett. 1986, 129, 615. Weller, H.; Schmidt, H. M.; Koch, U.; Fojtik, A,; Baral, S.;Henglein, A,; Kunath, W.; Weiss, K.; Dieman, E. Chem. Phys. Lett. 1986, 124, 557. (8) Ekimov, A. I.; Onushchenko, A. A. JETP Lett. 1984,40, 1136. Ekimov, A. I.; Efros, A. L.; Onushchenko, A. A. Solid State Commun. 1985, 56, 921. Efros, AI.; Efros, A. Sou. Phys. Semicond. 1982, 16, 772. (9) Heinglein, A,; Kumar, A,; Janata, E.; Weller, H. Chem. Phys. Lett. 1986, 132, 133. (10) Chestnoy, N.; Harris, T. D.; Brus, L. E. J . Phys. Chem. 1986, 90, 3393. Chestnoy, N.; Hull, R.; Brus, L. E. J. Chem. Phys. 1986, 85, 2237. (11) Sandroff, C. J.; Hwang, D. M.; Chung, W. M. Phys. Rev. B 1986, 33, 5953. Sandroff, C. J.; Chung, W. M. J . Colloid Interface Sci. 1987, 115, 593. Sandroff, C. J.; Farrow, L. A. J. Chem. Phys. Lett. 1986, 130, 458. Sandroff, C. J.; Kelty, S. P.; Hwang, D. M. J. Chem. Phys. 1986,85, 5337. (12) Nedeljkovic, J. M.; Nenadovic, M. T.; Micic, 0.;I.; Nozik, A. J. J. Phys. Chem. 1986, 90, 12. (13) Dannhauser, T.; O'Neil, M.; Johansson, K.; Whitten, D.; McLendon, G. J . Phys. Chem. 1986, 90, 6074. (14) Youn, H.-C.; Tricot, Y.-M.; Fendler, J. H. J . Phys. Chem. 1987, 91, 581. Watzke, H. J.; Fendler, J. H. J. Phys. Chem. 1987, 91, 854. ( 1 5 ) Kamat, P. V.; Dimitrijevic, N. M.; Fessenden, R. W. J . Phys. Chem. 1987, 91, 396. (16) Wang, Y.; Herron, N. J. Phys. Chem. 1987, 91, 257. (17) Bahnemann, D. W.; Kormann, C.; Hoffman, M. R. J. Phys. Chem. 1987, 91, 3789. (18) Rajh, T.; Peterson, M. W.; Turner, J. A,; Nozik, A. J. J . Electroanal. Chem. 19878, 228, 55. (19) Papavassilious, G. C. J. Solid State Chem. 1981, 40, 330. (20) Hanglein, A. Prog. Colloid Polym. Sci. 1987, 73, 1 . Henglein, A.; Fojtik, A.; Weller, H. Ber. Bunsen-Ges. Phys. Chem. 1987, 91, 441. (21) Micic, 0. I.; Nenadovic, M. T.; Peterson, M. W.; Nozik, A. J. J . Chem. Phys. 1987, 91, 1295. (22) Anpo, M.; Shima, T.; Kodama, S.; Kubokawa, Y . J. Phys. Chem. 1987, 91,4305. (23) Wang, Y.; Suna, A,; Mahler, W.; Kasowski, R. J. Chem. Phys., in press. (24) Wang, Y . ;Herron, N. J . Phys. Chem. 1987, 91, 5005. (25) Micic, 0. I.; Zongguan, L.; Mills, G.; Sullivan, J. C.; Meisel, D. J. P h j s . Chem. 1987, 91, 6221.
0 1988 American Chemical Society
Size Quantization in Layered Hg12 Colloids
-
99.995%) in acetonitrile. The presumed reaction is
HgC12
+ 2NaI
colloidal Hg12 + 2NaC14
(1)
However, as we shall show in this paper, this preparation also produces Hg12CI-; the resulting solution is herein referred to as "raw colloid". In some experiments, the raw colloidal solution is passed through a column of ion-exchange resin (Amberlite monobed resin MB-1) to remove ionic species that are present in the solution. In addition to HgC12, reactions were also carried out with Hg(N03)2 (Fisher Scientific, ACS certified). Molecular Hg12. Molecular Hg12 was formed by dissolving crystalline red HgI, (Aldrich 99.995%) in acetonitrile. Previous based on optical absorption, colligative properties, and conductivity data has shown that the dissolved species consists of linear molecules of Hg12, perhaps in the form of dimers or trimer^.^',^^ It should be noted that the dissolution of crystalline HgI, does not lead to the same species of Hg12 as produced by eq 1 with the same mercury concentrations. This difference is explained in the Discussion section. Ionic H$+ Species. The ionic complexes Hg12CI-, HgC121-, M solutions of and Hg13- were formed by reacting 2 X molecular Hg12 or HgCI, in acetonitrile with either NaCl (in wet acetonitrile) or NaI. The corresponding reactions are HgCl2
+ I-
+ C1HgI2 + I-
HgIz
+
+
+
HgC12I-
(2)
HgI2C1-
(3)
HgI3-
(4)
The reactions were performed by stepwise addition of the halide ion in excess of the stoichiometric amount required to form the complex, and they were followed by absorption spectroscopy and conductivity measurements. The formation of the above three ionic complexes were found to be quantitative as evidenced by linear Lambert-Beer behavior with added I- or C1- concentrations up to a total ha1ide:Hg ratio of 3, followed by a break in the curve at higher halide concentrations; the conductivity measurements also showed a break at a ha1ide:Hg ratio of 3. The formation of the doubly charged species HgI:-, Hg12C1?-, HgIC13", and Hg13C12-was attempted according to the reactions
+ I- e Hg142HgC121- + I- e HgCl2I2,Hg12C1- + I- e HgI3CI2HgIClT + C1- F HgIC13,Hg1,-
'
The Journal of Physical Chemistry, VoZ. 92, No. 14, 1988 4161
(5)
(6) (7)
(8)
The formation of Hg14,- was not quantitative but rather showed an equilibrium with Hg13- and I- that was primarily shifted toward the latter two species. HgI2CI2*-,Hg13C12-,and HgIC132- also did not form in the concentration range studied as evidenced by the absence of spectral changes upon addition of the halide ions to the singly charged complexes (eq 6-8). The reaction of HgC12 and Hg12 resulted in a spectrum that showed a simple additive effect and indicated that HgICl does not form in acetonitrile. (b) Physical Measurements. UV spectra were recorded on a HP-8450A UV-visible spectrophotometer; the resulting data were then downloaded into an Apple MacIntosh for analysis. Both colloid and molecular or ionic spectra were deconvoluted assuming either constant or variable peak widths. The deconvolution routine consists of two parts; a Simplex routine is first used to refine the initial guess of parameters to the point where a Marquardt routine can then be used to obtain the final converged parameter values. (26) Griffiths, T.R.; Anderson, R. A. J . Chem. SOC.,Faraday Trans. 2 1979, 75, 957.
(27) Ellendt, G.; Cruse, K. 2. Phys. Chem. 1952, 201, 130. (28) Griffiths, T. R.; Symons, M. C. R. Trans. Faraday SOC.1960, 56, 1752. (29) Eliezer, I.: Algavish, G. Inorg. Chim. Acta 1974, 9, 257 and references therein. (30) Gaizer, F.; Beck, M. T.J . Inorg. Nucl. Chem. 1967, 29, 21.
n (a'
3 2
52
42
6 2
Energy (eV)
Figure 1. Optical absorption spectra of molecular HgI, formed by dissolving crystalline red Hg12 in acetonitrile: (a) 1 X lo4 M HgI,; (b) 5 X M HgI,.
Y Y
32
52
42
62
Energy (eV)
Figure 2. Absorption spectra of HgI,CI-, HgCI,I-, HgIC, and "raw" colloid formed from the reaction of HgCI, with NaI.
Ultracentrifugation of the colloid was done on three 3-mL samples at 60000 rpm in a Beckman L5-65 ultracentrifuge. The resulting samples were divided into three fractions for spectroscopic analysis. Ultrafiltration experiments were performed with Spectra/Por Type F (polyfluoride) filters with molecular weight cutoffs (MWCO) of 5000 (average pore size 15 A) and 10000 (average pore size -25 A). The filters were first washed with -50 mL of acetonitrile to remove the wetting agent and then allowed to dry. The resulting filter was then wetted with the colloidal suspension and -2-5 mL was passed through it at a pressure of -30 psi of argon.
-
Results ( a ) Spectral Studies. The optical absorption spectra of molecular Hg12 dissolved in acetonitrile at concentrations of l X lo4 and 5 X M are shown in Figure 1. The spectra of the solution resulting from our colloidal preparation (raw colloid) is shown in Figure 2 together with the spectra of Hg12C1-, HgC121-, and Hg13-. All of these spectra are highly reproducible. Although there are similarities among them, there are also real and significant differences. Although the results shown in Figures 1 and 2 show that the raw colloid preparation does not consist of pure molecular HgI,, HgI,Cl-, HgC121-, or Hg13-, and that the spectrum of the raw colloid is unique, many experiments and analyses are required to determine its composition. One important result is that crystalline Hg12 dissolved in acetonitrile does not yield the same spectrum as that produced when HgC1, is reacted with NaI to produce raw HgI, colloid with
4162
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988
Peterson et al.
Raw Colloid
-s-
B
-5 2 0 0
i? K al C
eio 51 n Q:
004 00
’
01
02
03
04
I
05
00
32
for several wavelengths showing linear Lambert-Beer behavior. the same Hg concentrations. The dissolution process produces molecular Hg12 [as a monomer, dimer, or trimer27-29]that exhibits the following equilibrium:28
HgClZI-
+ HgI3-
+
95
(a) Ultracentrifugatlon 0 4
0 ‘5
(b) Ultrafiltration
(c) Amberlite Treatment
I
(10)
Hg1,Cl- in acetonitrile does not show this equilibrium as evidenced by linear Lambert-Beer behavior as HgI,Cl- is diluted. We therefore eliminate HgC1,I- as a species in the raw colloid. Hg13is also eliminated on basis of the absence of the equilibria of eq 9 and 10. Additional results show that NaCl and HgCl, in acetonitrile do not absorb at energies below 6.2 and 5.2 eV, respectively, and there is no evidence for I-, which absorbs at 5.1 eV. Also, there is no evidence of Iz or 13-(the equilibrium I, I- e I< is shifted to I, in acetonitrile) as indicated by spectral data (see Figure 4) and a negative starch-iodine test of partially evaporated sol redissolved in water. Therefore, based on the spectral data of the various possible ionic and molecular species, we conclude that the essential components of the raw colloid are Hg12Cl- and colloidal particles of Hg12. Attempts to synthesize the colloid by reacting Hg(NO& with NaI resulted in the formation of molecular HgIz (as indicated by the absorption spectrum), and not of the raw Hg12 colloid. Thus, it appears that the presence of chloride is essential to the formation of colloidal Hg12 from Hg2+ salts and iodide; a proposed mechanism for this behavior is presented in the Discussion section. Finally, the titration of HgC12 with I- results in nonlinear Lambert-Beer behavior, with complex equilibria, resulting ultimately in colloidal Hg12 and Hg12Cl- a t [Hg]/[I-] = 1/2. ( b ) Physical Studies. To support the conclusion that the raw colloid consists of colloidal Hg12 and Hg12C1-, and also to obtain more quantitative information, we subjected the raw colloid to separation techniques such as ultracentrifugation, ultrafiltration, and ionic selective adsorption. Ultracentrifugation of the raw
+
Figure 4. Comparison of absorption spectra of “raw” colloid and I, I-.
(9)
Thus, at high Hg12 concentrations (1 X lo4 M) the solution consists primarily of molecular HgI,; at lower concentrations (5 X M) it contains the dissociated species Hg1,- and HgI+ (Figure 1). On the other hand, the spectrum of the synthesized raw colloid does not change over the HgI, concentration range of 5 X lod to 5 X M, but rather shows Lambert-Beer behavior (see Figure 3). We interpret this result as strong evidence that the concentration of molecular HgI, in the raw colloid is very low. We find that the reaction of wet NaCl in acetonitrile with molecular HgIz results in the quantitative formation of Hg12C1at [Hg2’] = [Cl-1; no further spectral changes are observed after further additions of Cl- up to [Hg2+]= 0.5 [Cl-1. In order to obtain HgC121- in our system it would be necessary to have the following equilibrium present and shifted to the right: 2HgI2C1-
6 2
Energy (eV)
Figure 3. Absorption of the “raw” colloid as a function of concentration
2HgIz e Hg13- + HgI+
52
42
Mercuric Concentration (~JM)
Enerav (eV)
Figure 5. Spectra of colloidal HgIz obtained from (a) difference spectra from ultrafiltration experiments, (b) difference spectra from ultrafiltration experiments, and (c) spectra after “raw” colloid is treated with Amberlite ion-exchange resin. The three deconvoluted peaks are shown for each spectrum.
colloid at 60000 rpm for 4 h resulted in a supernatant solution with a spectrum that precisely matched that of Hg12Cl- with 34% of the initial Hg concentration. A rough calculation of the smallest HgIz particle diameter that would be removed by this process is about 18 A, assuming that the hydrodynamic density of the Hg12 particle is equal to its bulk solid-state density. The difference spectrum between the initial raw colloid and the spun supernatant solution (containing Hg12Cl-) represents pure colloidal Hg12, and this spectrum is presented in Figure 5a. Ultrafiltration of either molecular Hg12 or Hg12C1- through 5000 MWCO (15 8,pore size) or 10000 MWCO (25 8, pore size) filters results in no loss of material as evidenced by the difference spectra. On the other hand, when the raw colloid is passed through a 10000 MWCO filter (25 8, pores), a small absorbance loss is observed in the filtrate (-5% relative to the raw material). Filtration of the raw colloid through the 5000 MWCO filter (1 5 8,pores) results in a 27% absorbance loss. The resulting difference spectrum between the initial raw colloid and the filtrate is shown
Size Quantization in Layered HgI, Colloids
08
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988 4163 Hg12C1- or Hg12 to begin cluster growth that ultimately leads to the unit cell configuration. A proposed mechanism for colloid formation is as follows:
1
HgCl2 HgC1,IHgC1212,-
+ I+ I-+
HgC12I-
(2)
HgC12122-
(6)
HgI2C1- + C1-
(1 1)
-
-C
The tetrahedral HgC12122-unit can react with Hg12C1- to form the Hg12 clusters with C1- filling the tetrahedral voids at the crystallite edges: HgI2C122-
+ xHgIzC1-
-+
(HgI2),+,ClY’+ + (2 + x
+ y)C1(12)
32
52
42
62
Energy (eV)
Figure 6. Absorption spectra of “raw” colloid after treatment with Amberlite ion-exchange resin: (a) original “raw” colloid; (b) after a single
treatment; (c) after two treatments. in Figure 5b. It is seen that this difference spectrum, representing colloidal Hg12, agrees quite well with the ultracentrifugation result in Figure 5a. Attempts to isolate the colloid from the filter have largely failed; the spectra from such experiments show three peaks on a strong gray background that may be filter decomposition or possibly initially isolated NaCl that has been redissolved. Treatment of the raw colloid with ion-exchange resin (Amberlite) to remove ionic species results in a loss of about 40% of the initial Hg concentration. The spectrum of the treated colloid is shown in Figure 5c. It is comparable to the difference spectra shown in parts a and b of Figure 5 for the ultracentrifugation and ultrafiltration experiments, strongly indicating that treatment of the raw colloid with Amberlite leaves behind only colloidal HgI, particles. Repeated treatment of the raw colloid with Amberlite eventually results in total loss of material (Figure 6 ) . This implies that the Hg12particles are charged, presumably with Cl- ions. Quantitative analysis of the initially treated raw colloid shows it has the composition Hg12Clo,6compared to the initial raw colloid which has a composition Hg:I:Cl = 1:2:2. The treated raw colloid does not react with additional added C1-, as does molecular Hg12. This provides further support for the presence of stable particles of Hg12, and not a composite of molecular HgI, and Hg12Cl-. The observation that colloidal HgI, particles are charged is not surprising since the structure of HgI, is tetrahedral, and the coordination spheres at the edges of particles made from stoichiometric amounts of Hg2+ and I- are incomplete and could only be filled in our preparation by coordinating C1-. The smaller the HgI, particle size, the higher the Cl-/I- ratio is expected to be.
Discussion
On the basis of spectral studies of the possible molecular and ionic species in raw Hg12 colloid, and of various separation techniques involving ultracentrifugation, ultrafiltration, and treatment with ion-exchange resins, we conclude that the raw colloid, prepared by reacting HgCl, with N a I in the ratio of 1:2, consists of a mixture of about 60% colloidal Hg12 and 40% Hg12Cl-. Although the colloidal Hg12 clusters also contain coordinated C1- ions, we refer to them in this paper as HgI, colloids since we believe they maintain the crystal structure of red HgI,. Attempts to produce the colloid from Hg(N03), were for the most part unsuccessful; the resulting spectrum showed only a small distortion from that of molecular Hg12. The magnitude of the distortion was related to the order of addition (i.e., Hg2+to I- or I- to Hg2+). This result may be understood from the nature of the Hg coordination in solution. In the edge-sharing tetrahedral structure of colloidal red HgI,, voids will exist in the coordination sphere that need to be filled. As a result, the initial unit necessary to start particle growth would be tetrahedral Hg12X22-,where X could be either I- or C1-. This tetrahedral unit can then react with
As discussed below, our results indicate that x + 1 = 36 and y = 24. The failure of Hg(N03)2 to form Hg12 colloids can be understood in terms of the inability of the large nitrate ions to fill the coordination sphere in a tetrahedral configuration; hence, the necessary precursor for particle growth is not present in these solutions. Further support for this explanation is found in the results of changing the order of addition. That is, adding I- to a solution of Hg(N03), results in a reasonably pure spectrum of molecular HgI,, while the addition of Hg(N03), to a solution of I-, where HgI2- in low concentration is expected to be formed initially, results in a distorted spectrum. This distorted spectrum is consistent with minor formation of colloidal Hg12. The inability of Hg(N03), to produce colloidal Hg12 is also consistent with the lack of colloid formation upon reacting Hg12 with C1-. When we titrate Hg12 with C1-, we find formation of Hg12C1- that shows no further changes after [HgI,] = [Cl-1. On the other hand, titration of HgC12 with I- results in an initial formation of HgC1,I- that continues to react after [HgC12] = [I-], ultimately resulting in the formation of both colloidal HgI, and Hg12Cl-. This result may readily be interpreted on the basis of the stability of three ionic complexes and colloidal HgI,. Our results imply that the relative stability of these ionic complexes in acetonitrile goes as Hg12Cl- > Hg12C122-> HgICl,-, and we assume that colloidal Hg12 is comparable in stability to Hg12CI-. Hence, addition of C1- to a solution of Hg12C1- will result in no further reaction while addition of I- to a solution of HgIC1,- will result in formation of the necessary Hg12C1$- precursor that may then react to form colloidal Hg12 or Hg12Cl-. Our results indicating that the dissolution of crystalline HgI, does not produce clusters, while the reaction of HgCl, with NaI can produce clusters (at the same Hg concentration), are also explained by the mechanism presented above. In the former case, the empty coordination sites at the crystallite edges cannot be filled, and therefore clusters cannot be stabilized. The dissolution thus continues until monomeric, dimeric, or trimeric Hg12 molecules are formed with a nontetrahedral structure. Addition of I- to HgI, leads to HgIC ions, and not to Hg12 clusters stabilized with excess I-. The only possible way to form stable tetrahedrally bonded Hg12Xyrclusters is via reactions 2, 6, 11, and 12. Deconvolution of the spectra of the colloidal, ionic, and molecular species were performed for accurate determination of peak positions in order to quantify spectral differences. Both Lorentzian and Gaussian line shapes with constant and variable line widths were investigated. It was found that, in order to successfully fit the ionic spectra (Le., minimize x2 without severe oscillations in the residual sum of squares), it was necessary to use Gaussian line shapes and variable line widths. The spectra of colloidal HgI, were taken as the difference spectra from ultrafiltration and ultracentrifugation experiments and from the Amberlite treated material (see Figure 5 ) . A total of six such spectra were analyzed with quite consistent results; the high-energy transition at about 6 eV showed some variations since this part of the spectrum is quite noisy. The results of these deconvolutions,summarizing peak positions, line widths, and intensities, are presented in Table I and shown in Figure 5 for three experiments in which colloidal HgI, is isolated.
4164
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988
TABLE I: konvolution Parameters for Colloidal HgIl and Ionic ComDlexes ~
~~
colloidal HgIz 4.26 4.94 6.04
Hg12C1HgI3Peak Positions, eV 3.75 4.19 4.92 5.51 5.14
HgClZI-
0.78 0.70 0.67 0.54 0.35
TABLE 11: Charge Confinement Calculations for Colloidal HgI, Considering One Transition per Particle Size (Infinite Potential Barriers)
peak
3.78 4.15 4.94 5.48 5.74
3.85 4.39 5.05 5.44
band gap
positn, eV increase, eV 4.26 2.14
4.94
Peak Widths, eV 0.83 0.67 1 .o
Peterson et al.
0.91 0.55 0.83 0.44 0.33
0.74 0.95 0.72 0.5 1
6.04
no. of Hg12 L,,
layers
A
1 2 3 4 1 2 3 4 1 2 3 4
7.6 13.8 19.9 26.1 7.6 13.8 19.9 26.1 7.6 13.8 19.9 26.1
2.84
4.94
Intensities (Hg), lo4 cm-' M-' 0.62 1.4 2.0
0.24 0.74 1.57 0.78 0.49
0.52 0.68 1.84 0.8 1 0.98
0.04 0.32 1.17 0.13
It is noted that in general the fit of the ionic complexes is somewhat better than that of the colloidal spectra. This is not surprising since the colloidal spectrum should be a convolution of a line-shape function onto the size distribution function; the resulting total convolution will thus not necessarily be a simple Gaussian reflecting the line width of the colloid. To model the observed optical transitions arising from charge confinement, we have chosen a simple of a particle confined to a three-dimensional box with dimensions L, = L, = L,, and L,, and with an infinite potential barrier. L.y is the dimension of the particle in the layer plane, and L, is the dimension perpendicular to the layers. Our previously publishedz1charge-confinement calculations for colloidal Hg12 were based on three peaks at 4.17, 4.88, and 5.71 eV that were derived from fitting spectral data of the raw colloid. However, the deconvolution of the spectra of pure colloidal HgIz derived in the present more detailed and accurate experiments yields peaks at 4.26,4.94, and 6.04 eV. Given these more accurate values for the transition energies of colloidal Hg12, we have reevaluated our initial calculations of particle dimensions based on charge confinement. We have also added an additional consideration in these calculations, which is to predict the possible multiple excited-state transitions for a given particle size applying the selection rules: An, = 0, An, = 0, and An, = 0. Under these conditions for Hg12 the increase in the transition energy ( A E ) compared to the band gap energy is
AE = h2r2(n2 + n,2)/2pXyL,,2
+ h2r2n>/2p,L,Z
and Eobd =
Ebg
+ AE
-
where n,, ny, and n, represent changes between electron and hole states of the same quantum number (i.e., n, nxh, where nxe = nxh),EoW is the observed transitions energy, p,, = 0.22mo,1 , = 0.26 mo,and Ebg = 2.1 eV. The results of our calculations are summarized in Tables I1 and 111. It is apparent that, within the limits of the simple theory used here and our reported TEM results, there may be either magic numbers or multiple transitions or both. That one particle size will account for all observed transitions based on the lowest energy calculation (4.26 eV) is striking. Furthermore, the spectrum obtained from the ultrafiltration experiments (removal of some of the colloid) shows peak ratios that are consistent with the ultracentrifugation experiment (removal of all of the colloid). If we had magic numbers present, one would expect that the low-energy peak (largest particle size) should decrease in intensity at an accelerated rate relative to the high-energy peaks. Since this is not observed, the implication is that of a single particle size distribution. Our original conclusions that the multiple peaks could be due to either magic numbers or multiple transitions may now
L,,
A
no. of HgIz
molecules
15.8 13.9 13.3
25 29 36
12.9 11.8 11.5 15.8 10.4 9.8 9.6
16 21 27 12 11
14 19
TABLE III: Charge Confinement Calculations for Colloidal HgI, Allowing for Multiple Transitions obsd peak positn, e~ 4.26
4.94 6.04
particle dimensions L,,
A
13.8 19.9 26.1 (26.1)a 13.8 19.9 26.1 7.6 13.8 19.9 26.1
L,,
A
15.8 13.9 13.3 (ll.l)n 12.9 11.8 11.5 15.8 10.4 9.8 9.6
predicted transitns, eV 4.26 4.26 4.26 (4.32 4.94 4.94 4.94 6.04 6.04 6.04 6.04
5.31 4.91 4.90
5.99 5.98)"
6.05 5.58
'Calculated by using finite potential barriers of 4 eV for electron and hole wells. Other values are for infinite potential barriers. be restricted to multiple transitions from a single size. From our calculations for multiple transition from a single particle size the particle should consist of 36 molecules in four layers. This then leads to four layers of nine molecules in each layer (3 X 3) with 24 empty coordination sites at the edges. If all these sites were filled with C1-, the empirical formula for the cluster would be Hg12Clo,660.6&. Our analysis of the Amberlitetreated colloid showed the cluster formula to be Hg12Cb,60,6,which is in good agreement with the expected composition if all the empty tetrahedral sites are filled. To check the validity of our assumption in our simple model calculation that the use of infinite potential barriers for the particles is reasonable, we also performed a calculation using a barrier height of 4 eV for both the electron and hole wells. The value of 4 eV is based on considering that acetonitrile is a wide gap s e m i c o n d ~ c t o rwith ~ ~ an effective band gap of about 9.5 eV and that the HgI, band gap falls in the middle of that for acetonitrile. For the case of a single particle size with allowed multiple excited-state transitions, the first three calculated transitions energies are at 4.32, 4.90, and 5.98 eV for an HgIz cluster with 4 layers (26.1 8,)and a lateral dimension of 1 1 . 1 8, (compared to 13.3 8, for the infinite potential); the experimental values are 4.26, 4.94, and 6.04 eV (see Table 111). This cluster size contains 25 HgI, molecules and would also have an expected composition of approximately HgIzClo,70~7-, in agreement with the experimentally determined formula. Thus, introduction of a finite potential of 4 eV in place of an infinite potential in our model calculations results in a slight decrease of the calculated particle size that would produce theoretical excited-state transitions that closely match the experimental values. We conclude that the simplicity of our model does not warrant consideration of finite potentials because (1) there is great uncertainty in the actual values of the effective electron (31) Williams,
F.;Nozik, A. J. Nature 1978, 271, 137
J. Phys. Chem. 1988, 92, 4165-4171 and hole barriers in the Hg12-acetonitrile system, and (2) the barriers are expected to be large and the infinite barrier approximation is not expected to produce gross differences in the calculated results, as borne out by the above described case. The similarity between the spectra of ionic Hg12Cl- and the colloid spectra is consistent with multiple transitions from the colloidal system that approach the behavior of molecular or ionic species. This observation is also consistent with that of Wang and H e r r 0 n ' ~ 3in~ their study of PdS and CdS encapsulated in zeolite; clusters containing 12-14 molecules were observed to have properties that approach those of molecular species. Our estimated colloid size of 36 Hg12 molecules may well fall into this regime. Finally, we note that our results do not prove conclusively that the clusters in the Hg12colloidal sol are truly crystalline. Mercuric halides are known to form clusters that can involve solvent molecule^.^^^^^*^^ Further work is in progress to establish the detailed nature of the particles in more direct and unequivocal experiments, such as small-angle X-ray scattering and EXAFS. Con cI usion
We conclude from a series of optical measurements together with physical separation techniques involving ultracentrifugation, ultrafiltration, and treatment with ion-exchange resins that the reaction of 1 X lo4 M HgClz with 2 X lo-" M NaI in acetonitrile produces colloidal particles of Hg12and the ionic complex Hg12Cl-. About 60% of the mercury is present as HgI, clusters. These clusters are charged and contain C1- ions in normally unfilled tetrahedral coordination sites at the edges of the cluster; the empirical formula is Hg12Clo,60~6.Other possible molecular and ionic species, such as molecular Hg12, HgC12, HgICl, HgClzI-, Hg13-, Hg12C122-,HgC131-, HgI,Cl-, 13-, and I2 have been elim-
4165
inated as components of the initial colloidal solution. Absorption difference spectra from the ultracentrifugation and ultrafiltration of the initial colloidal solution, as well as the resultant spectrum obtained by treating the initial colloidal solution with the ionexchange resin Amberlite, are identical and this spectrum is attributed to small HgIz particles that are less than about 25 A. The spectrum of colloidal Hg12 exhibits three peaks at 4.26, 4.94, and 6.04 eV, which are attributed to quantization effects in the small Hg12 particles. Using a simple particle-in-a-box model with infinite potential barriers and the appropriate selection rules for optical transitions, the three peaks are consistent with either (1) the first three transitions in a Hg12 particle with the usual tetrahedral structure containing 36 Hg12 molecules arranged in four layers (total thickness of 26.1 A) with nine molecules per layer (total lateral dimension of 13.3 A), or (2) particles with magic numbers containing from one to four layers (7.6-26.1 A), depending upon the s ecific optical transition, and lateral dimensions from 9.6 to 15.8 The results of our physical separation experiments favor the former model of multiple transitions from a single dominant particle size. Model calculations using finite potential barriers of 4 eV did not seriously affect the results or our conclusions. Additional work is required to unequivocally establish that the 25-A colloidal particles are crystalline with the same hexagonal, layered crystal structure as red HgI,.
1.
Acknowledgment. This work was funded by the U S . Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, and the SERI Director's Development Fund. O.I.M. was supported by the US.-Yugoslavia Joint Research Fund. Registry No. Hg12, 7774-29-0.
Second Virial Coefficients of Molecules Absorbed in Spherical Cavities and Slltlike Micropores Perla B. Balbuenat and Donald A. McQuarrie* Department of Chemistry, University of California, Davis, California 95616 (Received: August 3, 1987; I n Final Form: January 20, 1988)
Virial expansions for the properties of a dilute gas within a spherical cavity and a parallel-plate micropore are developed, and calculations are presented for the second virial coefficients of a Lennard-Jones gas that interacts with the walls of a spherical cavity and a slitlike pore by a Lennard-Jones potential. The second virial coefficients of molecules within spherical cavities and parallel-plate micropores are expressed as single integrals involving Fourier transforms.
We start with the grand partition function, E , which is given
Introduction
Recently Rowlinson' has developed a formalism for the virial expansion of inhomogeneous systems. This formalism is based upon the grand potential, Q, which is taken to be a function of the temperature, T, the chemical potential, K , and a function of an external one-body potential, $, which replaces the conventional fixed geometric boundaries of a system. The types of systems for which this formalism is directly applicable are fluids within a system of capillary pores, or in the interstices of a chlathrate structure or a zeolite. In this paper we shall model a zeolite as a regular structure consisting of more or less separate spherical cavities and shall present a formal analysis of the absorption properties of a model zeolite and then present explicit calculations of a virial expansion of the isosteric heat of absorption. 'Permanent address: INTEC,Giiemes 3450,3000-Santa
Fe, Argentina.
0022-3654/88/2092-4165$01.50/0
by
where the activity X = ePikT, the deBroglie wavelength A = (h2/2amkT)1/2and
vN =
s ...
sdr,
... drN exp{-pU(r, ,...,rN) - @$(rl,...,rN)] (2)
where 8 = l / k T is the configuration integral. Following Rowlinson, we write
= VN/vIN (1) Rowlinson, J. S. Proc. R . Soc. London 1985, A402, 67.
0 1988 American Chemical Society
(3)
