J. Phys. Chem. 1990, 94, 8965-8968 compound. However, in the present investigation we have studied comparatively the behavior of the different oxygen species in V205 in oxidation reactions and have arrived at different conclusions. On one side, the stability of vacancies associated to the removal of one of the three crystallographic different oxygen atoms in the V 2 0 5structure (Le., O(l), 0 ( 2 ) , and O(3)atoms) was investigated by S C F H F C O calculations based on an INDO Hamiltonian. The calculation supports the view that the most stable vacancy corresponds to the removal of an O(3) center. The electronic structure calculation of a V205monolayer shows that the O(3) centers accumulate more electronic charge than the O(2)and O(1) atoms. Therefore, the formation of O(3) vacancies leaves the defect structure in a more reduced form than in case of removal of O(2)or O(1) centers. The larger degree of reduction of V atoms in the O(3) defect structure is the leading factor for the stabilization of this type of vacancy. Moreover, we have monitored the participation of lattice oxygen in V 2 0 s in the oxidation of DMSO by infrared spectroscopy. The evolution of the absorption bands associated to the stretching
8965
vibrations of V=O(l) bonds (band I) as well as V-0(2) and V-0(3) bonds (band 11) shows a sharp decrease in the intensity of the latter band as the oxidation reaction proceeds. However, the intensity of the V=O(l) absorption band varies to a much lower extent during the oxidation reaction. This result is in line with the investigations of refs 5 and 6, where it was shown that the isotopic exchange between I80 tracer and the O(3) and/or O(2) atoms in the V z 0 5lattice is much easier than the exchange of 0(1)centers. Summarizing, both theoretical and experimental results presented in this work point toward the identification of the O(3) oxygen vacancy as the most relevant one in oxidation processes involving the catalyst V2Os.
Acknowledgment. This work was partially-supported by the Comisi6n Interministerial de Ciencia y Tecnologia, Spain (CICYT, project PPA86-0433). We express our appreciation to Dr. C. P. Herrero for a critical reading of the manuscript. Registry No. DMSO, 67-68-5; V205, I 3 14-62-I .
Brownian Motion of Lipid Domains in Electrostatic Traps in Monolayers H. M. McConnell,* P. A. Rice, and D. J. Benvegnu Stauffer Laboratory for Physical Chemistry, Stanford University, Stanford, California 94305 (Received: April 2, 1990; In Final Form: June 27, 1990)
Fluorescence microscopy can be used to visualize lipid monolayers at the air-water interface. These monolayers frequently show coexisting lipid phases, with a rich variety of domain shapes. The sizes, shapes, and shape transitions of isotropic domains can be accounted for phenomenologically in terms of a competition between an interphase line tension, A, and p2,the square of the difference in the dipole densities in neighboring domains. Theoretically, domain sizes, shapes, and shape transitions all depend on the ratio X/p2. In the present work it is shown that p2 can be determined from an analysis of the amplitude of Brownian motion of lipid domains. By combining measurements of the equilibrium sizes and shapes of domains, and Brownian motion, one can in principle determine both X and p2 in lipid monolayers.
Introduction
Fluorescence microscopy has now been used for almost a decade to observe lipid monolayers at the air-water interface.I4 A variety of coexisting phases have been observed taking advantage of the differential partitioning of fluorescent lipid probes.'-5 These coexisting phases include liquid and gas, liquid and solid, and immiscible l i q ~ i d s . ~ The , ~ domains observed in these systems exhibit a wide variety of sizes, shapes, and shape transitions. The term "shape transition" is used here to denote shape changes for isolated finite as well as infinite two-dimensional arrays of domains.l&l3 For coexisting domains that are isotropic in two ( I ) von Tscharner, V.; McConnell, H. M. Biophys. J . 1981,36,409-419. (2)Peters, R.: Beck, K. Proc. N a t l . Acad. Sci. U.S.A. 1983, 80, 7 183-71 87. ( 3 ) LGsche, M.; Sackmann, E.; Mbhwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983,87,848-852. (4)McConnell, H. M.; Tamm, L. K.; Weis, R. M. Proc. Nutl. Acad. Sci. U.S.A. 1984.81, 3249-3253. ( 5 ) Weis. R. M.; McConnell, H. M. Nature (London) 1984.310.47-49. (6)Subramaniam, S.;McConnell, H. M. J . Phys. Chem. 1987, 91, I7 15-1 718. (7)Rice. P. A.; McConnell, H. M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86,64454448. (8) Keller, D.;Korb, J.-P.; McConnell, H. M.J . Phys. Chem. 1987,9/, 6417-6422. (9)McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1988.92.4520-4525,
0022-3654/90/2094-8965$02.50/0
dimensions (such as coexisting immiscible fluid phases6v7),the domain shapes and shape transitions can be described as a problem in variational calculus, the minimization of a free energy F = FA Fel (1)
+
where FA is the free energy of the domain interfacial line tension, and FeIis the electrostatic energy of the system due to long-range electrostatic dipole-dipole repulsions. The minimization of the free energy is carried out keeping the areas of the two phases constant and varying their shapes. For isotropic fluids, FA is simply proportional to A, and Fel is proportional to p2, the square of the difference in the dipole density in the two phases. From (1) it can be seen that the results of such variational calculations depend only on the ratio Alpz. Unfortunately, no practical molecular theory exists for either h or p2. Nonetheless, potentially useful new chemical information might be derived from monolayer studies if X and p2 could be measured independently. One approach to the measurement of dipole densities and dipole density (IO) Andelman, D.; Brochard, F.; deGennes, P. G.;Joanny, J.-F.C. R. Acad. Sci. Paris Ser. C 1985,301,675-678.
( I I ) Andelman. D.;Brochard. F.; Joanny. J.-F. J . Chem. Phys. 1987,86, 3673-368I. (12)Keller. D.J.: McConnell. H. M.: Mov. V. T. J. Phvs. Chem. 1986, 90,'23i 1-23 15. (13)McConnell, H. M. Proc. Null. Acad. Sci. U.S.A. 1989, 86, 3452-3455.
0 1990 American Chemical Society
8966 The Journal of Physical Chemistry, Vol. 94, No. 26, 1990
McConnell et al. 7 =
a2p2/A3
(8)
Since the Boltzmann average of l$;)is kT for motion with two degrees of freedom, we have q = a2p2/A3 = 2kT/3p2r2
(9)
Thus, a measurement of q provides a measurement of p2, under the assumptions of our calculations. The probability of observing , a value of v / q between v / q and v / q + dq/q is P ( T / Q )where P(v/ri) = eXP(-v/v) (10) When experimental data are limited, it is convenient to calculate the expected standard deviation, (r. Using P ( v / q ) in eq 10, one obtains (r
Figure 1. A small domain of radius a is electrostatically trapped within a large domain of radius A. The trapped domain is displaced from the potential minimum at the center of the large domain by a distance p due to thermal fluctuations. The variable r is used in eq 2.
Model Calculations-Electrostatic Trap Our calculations use the model of the electrostatically trapped domain sketched in Figure 1. A small circular domain of radius a is composed of light liquid and is surrounded by a dark liquid circular domain of radius A , and this is in turn surrounded by light liquid. Here white and dark refer to domains that are strongly and weakly fluorescent. We calculate the electrostatic energy Fel(p)as a function of p , the displacement of the small inner circular domain from the center. As shown previously, the electrostatic energy can be calculated from the expression
where r is the distance between a point on the circumference of the inner circle and a point on the circumference of the outer c i r ~ l e . ~One * ~ line integral is around the inner circle, and one around the outer circle. These line integrals are in opposite directions, and the negative sign in eq 2 ensures that the resultant energy is positive. By using a vector potential (3)
where the integration is around the outer circle, and assuming a
