Langmuir 1986, 2, 194-203
194
surface pressure; it is reasonable to anticipate that the changes in surface pressure can affect other properties. By suitable structural modification of the photoactive molecule or the environment it should be possible to “engineer” systems where much more drastic changes can occur.
Acknowledgment. We are grateful to the U S . Army Research Office (Contract DAAG29-84-K-0011)for support of this research. D. G. Whitten is grateful to the Alexander von Humboldt Foundation for a revisit to the MPIGottingen made possible by a Humboldt Prize “renewal”.
An Equilateral Triangle of Spin-’/, Nuclei in a Uniaxial System: NMR Spectra, Molecular Motions, Order Parameter, and Potential Energy Function for AsF3 Intercalated in C40A~F6*2CH3N02 Gerald Ray Miller,*+Michael J. Moran,T Henry A. Resing,* and Tung Tsangs Code 6120, Naval Research Laboratory, Washington, D.C. 20375, and Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742 Received July 16, 1985. I n Final Form: November 20, 1985 For AsF3 co-intercalated in a graphite intercalation compound (GIC) of original composition C,AsFg2CH3N02,the 19FNMR spectrum is a 1:2:1 triplet originating from mutual nuclear dipolar coupling among the three 19Fnuclei. The nonzero dipolar splitting indicates that the AsF3molecules have a preferred orientation in the intercalation galleries and are not tumbling isotropically. The triplet splitting follows a (3 cos2K - 1)/2 dependence where K is the angle between the magnetic field, B, and the graphite c axis of the GIC. The magnitude of the maximum triplet splitting yields the magnitude, but not the sign, of the order parameter, S = ((3 cos26 - 1)/2), where 0 is the angle between the molecular C3 axis and the c axis of the GIC. A least-squares fit of the temperature dependence of the splitting yields IS1 = 0.4359 - 0.000433T(K)with 1.2 = 0.9993. On the basis of Andrew and Bersohn’s calculations of the NMR spectrum of three spin-l/, nuclei at the vertices of an equilateral triangle, we show that the presence of a 1:2:1 triplet requires that reorientation about the molecular C3axis and molecular reorientation about the crystalline c axis must both be rapid compared to the magnitude of the dipolar interactions (7 < 10” s). The substantial temperature dependence of the triplet splitting is not consistent with a fixed angle 0. On the basis of the very narrow line widths observed ( 100 Hz, fwhm), we conclude that the motion over 6 is rapid on the NMR time scale as well and that intermolecular dipolar interactions are averaged by rapid translational diffusion. On the basis of motional averaging of the remaining dipolar interactions over a Boltzmann distribution in a potential energy well, V(0) = V2 cos 20 + V4cos 40, we find that the temperature dependence can be fit well for V, = +6380 J/mol and V4 = +(l/6)V2 = +lo63 J/mol. These parameters correspond to S = -0.306 at 300 K and yield a well 12.8 kJ/mol in depth and centered on 6 = 90” (preferred orientation of the molecular C3 axis is parallel to the graphite sheets). No simple potential energy functions have been found that give a positive order parameter and the observed temperature dependence.
Introduction Graphite intercalation compounds (GICs) exhibit twodimensional characteristics and their electrical conductivities are often highly anis~tropic.l-~Unusually large in-plane electrical conductivities have been found for GICs derived from graphite and AsF,. It has been proposed1g4 that the intercalation of AsF, into graphite involves the formation of the Asp,- and AsF, species according to eq 1 and that the concomitant oxidation of the graphite lattice 2xC
+ 3AsF5
-
2C,AsF6
+ AsF3
(1)
yields the charge carriers (holes in the valence band) which are responsible for the very high electrica! conductivity. The 19FNMR spectrum of AsF,-intercalated graphite is a single narrow which suggests that if more than one fluorine species is present, fluorine chemical exchange is rapid on the NMR time scale. We have been studying ___I
+Permanentaddress: Department of Chemistry & Biochemistry, University of Maryland, College Park, MD 20743. Permanent address: Department of Chemistry, West Chester University, West Chester, PA 19380. 8 Permanent address: Department of Physics and Astronomy, Howard Lrniversity, Washington, DC 20059.
0743-7463/86/2402-0194$01.50/0
these materials and in order to determine the NMR spectrum of the AsF6- ion in the absence of fluorine chemical exchange with other fluorine-containing species, we prepared’ by reaction 2 a GIC containing the AsF6-ion. NO~ASF,+ 40C + 2CH3NO2 C,~ASF~.~CH~ ( 2N) O ~ The graphite used was highly oriented pyrolytic graphite (HOPG), which is polycrystalline graphite with the c axes of the individual crystallites aligned ( ~ spread) 2 ~ but disordered in the a-b plane. AsF3 was subsequently intercalated into this second-stage GIC in order to study (1) Bartlett, N.; McQuillan, B. W. In Intercalation Chemistry; Whittingham, N. W., Jacobson, A. J., Eds.; Academic Press: New York, 1982; p 19. (2) Dresselhaus, M. S.; Dresselhaus, G. Adu. Phys. 1981, 30, 139. (3) Solin, S. A. Adu. Chem. Phys. 1982, 49, 455. (4) Bartlett, N.; Biagoni, R. N.; McQuillan, B. W.; Robertson, A. S. J. Chem. SOC.,Chem. Commun. 1978, 200. (5) Resing, H. A.; Miller, G. R.; Moran, M. J.; Reardon, J. P.; Dominguez, D.; Vogel, F. L.; Wu, T. C. Eztended Abstracts, 15th Biennial Conference on Carbon, 1981; p 369. (6) Miller, G. R.; Moran, M. J.; Resing, H. A.; Banks, L. Extended Abstracts, 15th Biennial Conference on Carbon, 1981; p 385. (7) Moran, M. J.; Miller, G. R.; DeMarco, R. A.; Resing, H. A. J. Phys. Chem. 1984. 88, 1580.
0 1986 American Chemical Society
Langmuir, Vol. 2, No. 2, 1986 195
Spin-’I2 Nuclei in a Uniaxial System
-I
..-t
’...
h & e
Figure 2. AsF3in the interlamellar region of a stage two graphite intercalation compound; definition of the angles 0 and 4.
IO k H z Figure 1. 19FNMR spectrum of AsF3 (1:2:1 triplet) and AsF, (broadened singlet or partially resolved doublet) for K = Oo (c axis )I B ) and for K = 90° (c axis I B). possible fluorine chemical exchange between the AsFf, the AsF,, and any AsF5 formed by the reverse reaction of eq 1. The 19FNMR spectrum found7 for this doubly intercalated material is shown in Figure 1. A sharp 1:2:1 triplet due to AsF3 is superimposed on the broad doublet spectrum of the AsF,- ion. There is no evidence for the presence of AsF5 or for fluorine chemical exchange between AsF, and AsF,-. The principal aims of this paper are to determine the temperature dependence of this triplet splitting, to determine what kinds of molecular reorientation are necessary for the spectrum to be a simple 1:2:1 triplet, and to determine what potential energy well orientation and shape could account for the observed temperature dependence of the triplet splitting. The splitting of the 1:2:1 triplet follows the 3 cos2 K 1dependence7 characteristic of dipolar interactions. Here K is the angle between the magnetic field, B, and the director where we define the director as the normal to the average plane of the microcrystalline graphite’s a-b planes. If each AsF, molecule were tumbling isotropically about its center of mass, the dipole-dipole interactions between the nuclear magnetic moments of the 19Fnuclei present in the molecule would be averaged to zero (as they are in the liquid or gaseous state). The presence of this splitting requires that the tumbling of the AsF, molecule be anisotropic in the 8 A high intercalation gallery established by the size of the AsF,- ion (Figure 2). The absence of any further splitting indicates that 75As(I= 3/2)-19Fdipolar and indirect spin-spin interactions are averaged to zero, presumably by fast quadrupolar relaxation of 75As. We consider below the case of an AsF, molecule partially oriented by a planar surface. We examine the effects on the NMR spectrum of various combinations of fast and slow molecular reorientations about several axes passing through the molecular center of mass. By “fast” or “rapid” we mean fast compared to some splitting due to an interaction between magnetic nuclei; in the present case, the maximum splittings possible are tens of kilohertz so the reorientation rate must be on the order of a hundred kilohertz or greater, corresponding to a correlation time of = s or less. When the motion is “fast”, the resulting spectrum contains the nucleus-nucleus interaction averaged over the motion. In contrast, when the motion is “slow” compared to the splitting interaction, the resulting spectrum is the weighted average of the spectra for the possible orientations of the molecule. We show that the
presence of a 1:2:1 triplet requires that reorientation be fast both about the molecular C3 axis and about the director (angle $I in Figure 2). The partial orientation of such a molecule can be described by the order parameter. S = (1/2)(3 cos2 8 - 1) (3) where 8 is the angle between the molecular symmetry axis and the director (Figure 2). A given value of S could result from a unique value of 8, determined by eq 3, or by fast sampling by the molecule of a variety of B’s, in which case S = ((1/2)(3cos2 8 - l)),the weighted average of (1/2)(3 cos28 - 1)over the molecular motions. In either case, the magnitude of S is directly related to the experimentally observed triplet splitting. IS( = 0.306 for AsF, a t 300 K in our GIC.7 For IS1 < the sign of S in not determined in the NMR experiment but the sign is necessary to determine the preferred orientation of the molecule in the crystal lattice. The triplet splitting, and hence, /Si, is a decreasing linear function of the temperature. Writing the potential energy function of the AsF, molecule oriented by a plane as V = V2cos 28 V4 cos 48, we examine the behavior of ((1/2)(3 cos2 8 - 1))as a function of R T / V z ,for several values of V4/ V2. We show that the temperature dependence of S is consistent with the order parameter being negative (corresponding to the preferred orientation of the AsF, molecule with its C3 axis parallel to the graphite planes) and not consistent with S being positive. The general method developed should prove useful for determining the preferred orientation of other symmetric top molecules on planar surfaces.
+
Experimental Section The sample used in this study was the same one reported on earlier.7 It was prepared by the reaction of HOPG with NO2AsF6 following the dissolved in CH3N02to form C40AsF6.2CH3N02, methods developed by Forsman and Mertwoy* and by Billaud et al.’ Subsequent intercalation of AsF, was accomplished by exposing the material to AsF3 vapor. The details of the preparation have been given.I The 19FNMR spectra were obtained with a Bruker SXP-60 spectrometer/Nicolet 1080 data system, operating at 56.4 MHz. The temperature dependence of the splitting between the outermost components of the triplet is shown in Figure 3. It is essentially linear from 260 to 335 K. The absolute value of the order parameter can be derived by dividing the observed splitting by the value calculated for perfect alignment of the C3 axis of the AsF3 molecule along the director. Assuming that the shape of the AsF3 molecule is not distorted upon intercalation, the theoretical splitting may be calculated to be 19464 Hz from the known (8)Forsman, W. C.; Mertwoy, H.E. Synth. Metals 1980, 2, 171. (9) Billaud, D.; Pron, A.; Vogel, F. L. Synth. Metals 1980, 2, 177.
196 Langmuir, Vol. 2, No. 2, 1986
Miller et al.
6400
0.32
6200 c L
O 3 ' 1st 0.30
0.29
5600 240
270 300 T, 'K
330
Figure 3. Temperature dependence of the splitting between the outer lines of the AsF, triplet (left) and of the magnitude of the order parameter, IS1 (right). geometry in the gas statelQ(rAsF = 1.706 A, LFAsF = 96.2O) and the value of the magnetic moment of the 19Fnucleus. With this value for the maximum splitting, the right-hand ordinate of Figure S1 as a function of temperature. A least-squares fit of 3 gives I the d a t a yields I S1 = 0.4359 - 0.0004337'with r2 = 0.9993.
Orientational Probability Specification of the instantaneous orientation of a molecule with respect to a surface on which it is bound can be made by means of the three Euler angles." For AsF, in graphite, the Euler angles are the polar angle, 8, of the molecular C3 axis with the director, the azimuthal angle, 4, of the molecular C, axis about the director, and the phase angle, 0,which fixes the orientation of the equilateral triangle of 19Fatoms in its plane. There will be, in general, a distribution in each coordinate due to molecular reorientation and vibration. We would like to know the probability distribution P(0) dQ. We make the simplifying assumption that we can write this distribution as a product of probability functions of the individual Euler angles: P ( 0 ) dQ = [P@) dPl[p&@sin 8 d01[P4(4)d+l (4) Halle and Wennerstrom12 have recently discussed the evaluation of orientational probabilities and potential energy functions from NMR data and the limitations inherent in these methods. For a molecule of arbitrary symmetry, a second-rank tensor (the order tensor13J4)can be defined which has, in general, five independent elements (Le., a symmetric tensor of zero trace). For a molecule of 3-fold or higher symmetry on a featureless surface, the number of elements is reduced to one, the order parameter, S, defined in eq 3. S represents a single term in the expansion of P ( 0 ) in a set of orthogonal functions. The usefulness of S derives from its ready availability from splittings or shifts of lines in the NMR ~ p e c t r u m . ' The ~ limitations are that higher order terms in 8 may not be negligible in the expansion, that the NMR spectrum yields directly only IS1 and, therefore, not the sign of S when -1/2 < S 6 +l/*, and finally, that S clearly cannot contain information on P&3) or P$(+).To proceed further in describing P ( Q ) ,one could find another, non-NMR experi(10) Clippard, F. B.; Bartell, L. S. Inorg. Chem. 1970, 9, 805. (11) Goldstein, H. Classical Mechanics, 2nd ed.; Addison-Wesley: Reading, MA, 1980; pp 143-148. (12) Halle, B.; Wennerstrom, H. J . Chem. Phys. 1981, 75, 1928. (13) Saupe, A. Angew. Chem., Int. Ed. Engl. 1968, 7, 97. (14) Zannoni, C. In The Molecular Physics of Liquid Crystals; Luckhurst, G. R., Gray, B. W., Eds.; Academic Press: New York, 1979; Chapter 3.
ment sensitive to other terms in the e~pansion,'~ substitute another nucleus in the molecule with different nuclear interaction^,'^ model the data by using probability functions of proper symmetry,', or impose a condition in which motional averaging does not occur in the NMR spectrum so that other features of P ( Q )beyond S may manifest themselves.16 In the following section, we calculate the spectrum for three equivalent nuclei for several combinations of molecular motions.
Resonance Line Shapes The spectrum of three interacting spin-'/, nuclei at the vertices of an equilateral triangle has been calculated by Andrew and Bersohn" (hereinafter referred to as AB). They calculated the spectra for a stationary triangle and for a rapidly reorienting triangle, both for the case of a single crystal and for the case of a powder of randomly oriented crystallites. We base our calculations of the 19F NMR spectra of AsF, molecules undergoing various motions on the results of AB. We only consider the case A,, not the case A,X (X:I = 3 / 2 ) , so all calculations (including the rigid lattice) assume that the 75Asspin is decoupled from the 19Fnuclei. We begin by assuming that 0 is uniquely defined by eq 3 for an observed order parameter, S. Later, we will consider distributions of molecular orientation and average over such distributions. Thus for AsF, in our GIC at 300 K, the angle 8 may be either 43' or 6 9 O , corresponding to S = +0.306 or -0.306, respectively. In a single crystal of such uniaxial assemblies, the distribution of the azimuthal angle, +, reflects the symmetry of the potential energy well in which the molecule moves. The effects we discuss require this symmetry to be 3-fold or higher. Because the orientation of the crystallites in HOPG is random in the plane perpendicular to the director, the distribution of 4 is random. This results in spectra that are two-dimensional powder patterns. We calculate the NMR spectrum for each of four different states of molecular motion: case A, stationary AsF, molecules (the rigid lattice); case B, fast reorientation of each AsF3 molecule about its C3 axis (motional averaging over the phase angle p); case C, fast reorientation of the AsF, about the director (motional averaging over 4); case D, fast reorientation about both the C,-axis and the director (motional averaging over both /3 and 4). These spectra are quite different and the molecular motions of the AsF, molecule can be determined without ambiguity from the experimental spectra. We do not consider the transitional regimes lying between one state of motional narrowing and another although such consideration can be useful; even for randomly distributed 4 the symmetry of the &dependence for the individual crystals may emerge in the transitional regime18 of, for example, case B going to case D. Case A: The Rigid Lattice. For an arbitrary angle $ between the magnetic field, B , and the C, axis of the molecule, AB showed that the spectrum consists of a central (unshifted) line and three pairs of satellites disposed symmetrically about the center line. For their calculation of the intensity of the spectrum of a powder as a function of the field measured from the center line, F(h),they multiplied the transition probability p ( $ ) of each (15) Chapoy, L. L.; DuPre, D. B.; Samulski, E. T. In Liquid Crystals and Ordered Fluids; Johnson, J. F., Porter, R. S., Eds.; Plenum Press: New York, 1978; Vol. 3. (16) Resing, H. A,; Slotfeldt-Ellingsen,D. E. J . Mugn. Reson. 1980,38, 401. (17) Andrew, E. R.; Bersohn, R. J . Chem. Phys. 1950, 18, 159. (18) Kooser, R. G.; Resing, H. A. J . Phys. Chem. 1983, 87, 2564.
Langmuir, Vol. 2, No. 2, 1986 197
Spin-'/, Nuclei in a Uniaxial System S = - 0 . 3 06
S=0.306 3 I
2
1 1
Table I. Second Moments" for S = f0.306 and K = 90°
M ,GZ
3
I '
'2
I
1
3
1 ?
I 32
1 1 1
1
case
s = +0.306 (e = 430)
A
6.22 0.66 0.67 0.19
B
C D
s = -0.306
(e = 690)
5.62 1.90 1.89 0.19
Assuming that all 76As-'9F dipolar and indirect spin-spin interactions are averaged to zero (cf. text).
u -4
-2
0
2
4
, -4
-2
0
2
4
h v/a
Figure 4. Calculated "F NMR spectrum for AsF, (assuming 75As is effectively decoupled from the 19F):case A, rigid lattice; case B, fast reorientation about the C3 axis; case C, fast reorientation about the director; case D, fast reorientation about both the C3 axis and the director.
satellite by d(cos $)/dh, which accounts for the fraction of triangular groups for which the angle $ lies in the interval dll. when one has a powder randomly oriented in three dimensions. In the present case, the GIC has randomly oriented crystallites in the a-b plane. Therefore the intensity as a function of field, G(h), measured from the spectrum center, is p ( $ ) multiplied by d@/dh,where the latter assumes equal probability for aJl values of 4. The details of the calculations are given in the Appendix. The effect of having the molecular C3-axesevenly distributed on a cone of half-angle 0 is to smear the intensity of each satellite over a range of frequency (magnetic field). This is illustrated in Figure 4A which shows spectra calculated at K = 90' for both S = +0.306 (0 = 43') and for S = -0.306 (0 = 69'). In this orientation each of the six satellites has two singularities; the two singularities associated with each of the three low-field satellites are indicated in the figure with l's, 2'9, and 3's. Spectra for other values of K are similar in form with each satellite having two or three singularities. Clearly, the experimentally observed spectra are much simpler than these rigid lattice spectra. If one could freeze out the molecular motion and observe the spectrum, however, the difference between the calculated spectra for S > 0 and S C 0 may be great enough, even with additional broadening due to intermolecular dipolar broadening, to determine the sign of S. Case B: Fast Reorientation about the C3Axis (but not about the director). In this calculation, the averaging over all values of the phase angle /3 leads to zero intensities for two of the three pairs of satellites. For a single AsF3 molecule, the remaining pair of satellites form a 1:2:1 triplet with the central line. The even distribution of molecular C3 axes on the cone of half-angle 0 gives rise to an anisotropy pattern for each member of the satellite pair. Figure 4B shows the simple symmetric form of one satellite anisotropy pattern (dashed line) at K = 90' as well as the full spectrum. There is a substantial difference between the spectrum calculated for S > 0 and that for which S C 0, but neither is the simple triplet observed experimentally. Case C: Fast Reorientation about the Director (but not about the C3 axis). In this calculation, each of the three pairs of satellites has nonzero intensity a t all K . Due to the motional averaging over the azimuthal angle, +, each of the satellite lines is sharp (in contrast to cases A and B, above). The splittings follow the expected 3 cos2 K 1behavior. Figure 4C shows the spectra for K = 90'. Again
the spectra are very different for S > 0 and S C 0, but neither duplicates the triplet observed experimentally. Case D: Fast Reorientation about Both the Director and the C3Axis. In this calculation it is necessary to average over both @ and P. Two satellites symmetrically disposed about the center line result, yielding a 1:2:l triplet (Figure 4D). Because of the averaging over 6, the pair of satellites are sharp. The motional averagings of the lines in case C into the lines of case D are also shown in Figure 4. The splitting, as in case C, is proportional to 3 cos2 K - 1. Since the identical triplet results for both signs of S , one cannot determine the sign of S directly from the spectrum. Comparison of the experimental spectra (Figure 1)with the calculated spectra (Figure 4) clearly demonstrates that the observed spectra result from rapid reorientation of the AsF3 molecule about its C3 symmetry axis and about the director. In comparing the spectra calculated for S = +0.306 with those for S = -0.306 (left side and right side of Figure 4, respectively), one finds identical spectra only for case D. For case A, the spectra are qualitatively similar and intermolecular broadening (not accounted for in Figure 4) may obscure many of the detailed differences. In both cases B and C, on the other hand, the spectra are very different for positive and negative S, being much narrower for positive S. Qualitatively, this is due to the fact that the cone half-angle is much smaller for positive S (43') than it is for negative S (69'). The differences in the spectra for positive and negative S are also reflected in the second moments of the spectra. The general expressions for the second moment, M, are derived in the Appendix. For the spectra shown in Figure 4,the values of the second moments are given in Table I. Note that for cases B and C, the narrower spectra for S > 0 have second moments about 3 times smaller than their S 0 counterparts. For case A, the difference is only 10%. Physically, case A may be attained by lowering the temperature sufficiently to achieve the rigid lattice. If the reorientation about the director became slow (in the NMR sense defined earlier) upon cooling, well before the temperature at which reorientation about the C3 axis became slow, we would expect a spectrum characteristic of case B to be an intermediate spectrum before we cooled enough to obtain the rigid lattice spectrum. Conversely, if the C3-reorientationbecame slow first upon cooling, we would expect a spectrum of case C to be an intermediate spectrum. For pyramidal molecules such as AsF3 and NH3, case C (fast reorientation about the director and slow reorientation about C3) would seem to be physically unlikely in most quasi-two-dimensional systems. Case C requires that the potential energy well in which the molecule resides restrict severely C3 reorientation while allowing +-reorientation to occur easily.
Temperature Dependence of S All of the calculations in the foregoing section were predicated on 6 having a fixed value, either 43' (S = +0.306) or 69' ( S = -0.306). To a first approximation, no
198 Langmuir, Vol. 2, No. 2, 1986
Miller et al. Table 11. Value of the Order Parameter, S, vs. R T / V , S
RTIV,
00
RT/IV,I 0.2 0.4 0.6 0.8
1.0
1.2
1.4
1.6
1.8
2.0
Figure 5. Order parameter, S, vs. RT/JV,(.Dotted lines show approximate method fits. Curves a, b, c, d, and e represent V4/V2 = I/,, 0, and -I/,, respectively. temperature dependence is expected for S if 8 is fixed since the spectrum should remain the same as long as the rates of reorientation are fast compared to the dipolar interactions being averaged by the motions. In fact, we observe a 10% change in IS1 between 260 and 335 K. Physically, a fixed Euler angle 8 appears unreasonable in a system enjoying rapid reorientation about the other two Euler angles, 4 and @, unless there were a very strong attachment to the plane (like a covalent bond). In fact, the extremely narrow line widths of the triplet ( N 100 Hz, fwhm) indicate a near absence of intermolecular dipolar broadening near room temperature which, in turn, suggests that the AsF, molecules are also undergoing rapid translational diffusion in the intercalation galleries as well. Under these circumstances, we introduce a more realistic model which utilizes a distribution in the polar angle, 8, for the AsF, molecules in the interlamellar region. The order parameter becomes S = where indicates the weighted average over all molecular orientations. For motion of AsF, in a potential energy well defined by the function V , J T ( 1 / 2 ) ( 3 cos2 8 - l)e-"iRT sin 8 d8
S=
(5)
&'e-V/RT sin 6 d8 where R is the ideal gas constant. Since S is only a function of 8, we only need the 8-dependence of V. Because of the symmetry of the +c and the -c directions in the GIC lattice, we can write a Fourier expansion for V(8): V(8)= V , cos 26 + V , cos 48 + ... (6) The first term is expected to be dominant. For I V4/V,l 6 l/,, the second term flattens or sharpens the potential energy well, depending on signs, but introduces no new minima. The total depth of the well is 21V21. This expansion is equivalent to an expansion in terms of Legendre polynomials of cos 8.
4.0000 3.0000 2.0000 1.3333 1.0000 0.8000 0.6667 0.5714 0.5000 0.4000 0.3333 0.2857 0.2500 0.2000 0.1667 0.1250 0.1000 0.0667 0.0500 0.0400 -4.0000 -3.0000 -2.0000 -1.3333 -1.0000 -0.8000 -0.6667 -0.5714 -0.5000 -0.4000 -0.3333 -0.2857 -0.2500 -0.2000 -0.1667 -0.1250 -0.1000 -0.0667 -0.0500 -0.0400
v, =
v, =
-V9J4 -0.07384 -0.09713 -0.14123 -0.20036 -0.25054 -0.29199 -0.32557 4.35245 -0.37382 -0.40436 -0.42408 -0.43733 -0.44665 -0.45878 -0.46633 -0.47529 -0.48046 -0.48716 -0.49044 -0.49238 0.07748 0.10355 0.15528 0.23014 0.29949 0.36173 0.41625 0.46321 0.50324 0.56613 0.61197 0.64634 0.67300 0.71200 0.73962 0.77713 0.80207 0.84000 0.86219 0.87718
-V9J6 -0.07033 -0.09237 -0.13400 -0.18980 -0.23739 -0.27713 -0.30982 -0.33645 -0.35806 -0.38986 -0.41120 -0.42599 -0.43666 -0.45081 -0.45975 -0.47042 -0.47660 -0.48461 -0.48854 -0.49087 0.07487 0.10038 0.15155 0.22709 0.29885 0.36486 0.42400 0.47593 0.52091 0.59265 0.64544 0.68498 0.71547 0.75948 0.79010 0.83077 0.85710 0.89565 0.91707 0.93090
Vd
=0
-0.06331 -0.08284 -0.11944 -0.16818 -0.20985 -0.24508 -0.27468 -0.29949 -0.32029 -0.35255 -0.37586 -0.39315 -0.40635 -0.42501 -0.43750 -0.45313 -0.46250 -0.47500 -0.48125 -0.48500 0.06960 0.09393 0.14385 0.22408 0.29690 0.37054 0.43928 0.50165 0.55694 0.64601 0.71156 0.75862 0.79310 0.83909 0.86807 0.90270 0.92283 0.94909 0.96200 0.96968
v, =
v, =
V9/6 -0.05632 -0.07333 -0.10482 -0.14610 -0.18099 -0.21042 -0.23528 -0.25634 -0.27430 -0.30305 -0.32486 -0.34192 -0.35563 -0.37637 -0.39140 -0.41197 -0.42556 -0.44570 -0.45696 -0.46423 0.06425 0.08735 0.13582 0.21320 0.29400 0.37521 0.45364 0.52653 0.59189 0.69694 0.77015 0.81929 0.85235 0.89174 0.91385 0.93820 0.95164 0.96860 0.97674 0.98152
V?/4 -0.05283 -0.06856 -0.09750 -0.13495 -0.16623 -0.19238 -0.21434 -0.23288 -0.24867 -0.27396 -0.29325 -0.30845 -0.32078 -0.33968 -0.35365 -0.37325 -0.38664 -0.40744 -0.41984 -0.42830 0.06155 0.08401 0.13169 0.20931 0.29218 0.37707 0.46025 0.53820 0.60822 0.71965 0.79495 0.84333 0.87439 0.90943 0.92822 0.94854 0.95973 0.97384 0.98062 0.98460
Equation 5 was integrated numerically for values of R T I V , ranging from -4 to +4 and for ratios of V4 to V 2 of -1/4, -lI6, 0, lI6,and lI4. Simpson's rule was employed for the integrations with a step size of lo.Values of the order parameter, S, are given in Table I1 for these ratios of V4to V , and selected values of RT/V,. In Figure 5, the value of S calculated from eq 5 is plotted against the dimensionless parameter RT/IV,I for both V , < 0 (upper portion of the figure, 0 d S 6 1) and V , > 0 (lower portion of the figure --'I2d S < 0); curves a, b, c, d, and e represent V 4 / V 2= 'I4, 0, -lI6,and -I/,, respectively. Assuming the well depth 21 V21is not itself a function of temperature for a particular system, Figure 5 is a plot of the order parameter us. temperature scaled by R/lV21. In general, the temperature dependence of S is nonlinear. It will appear to be linear if the second derivative of the appropriate curve is sufficiently small in the temperature region studied or the temperature region itself is sufficiently small. The tabulated values of S in Table I1 were used to find the values of V, and V , which best fit the data. For a particular ratio V 4 / V 2 a, value of V 2 was found which yielded the correct value of (SIat 300 K. This value of V2was then used with entries in the same column of Table I1 to determine other pairs of S and T (from R T / V , values). In Figure 6, we plot the temperature dependence thus determined for each of the tabulated values of V4/ V , for V , > 0. The fit is quite sensitive to the potential energy parameters. The best fit shows a slight curvature, as ex-
Langmuir, Vol. 2, No. 2, 1986 199
Spin-'I2 Nuclei in a Uniaxial System I
\
0
0.29
250
270
290
310
T'K
Figure 6. Temperature dependence of IS( for V , > 0; curves a, b, c, d, and e correspond to V4/V2 = lj4,l j 8 , 0, -lI6, and respectively. The temperature dependence of IS1 for V, < 0 lies in the shaded area for lj4> V, > -lj4. The experimental points are shown; the best fit occurs for V = +6380 cos 28 + 1063 cos 48. l4
l2 IO
c,
I
4t
\..,,,
,
2
0 0
10
20
30
40
50 6 ,d e 9
60
70
80
10
20
30
40
50
60
70
80
90
8 , deg Figure 8. Probability function e-VIRTsin 0 for T = 265 and 335
330
90
Figure 7. Shape of half the potential energy well (well is symmetric about 0 = 90') compared to a pure cos 28 well (dotted line) of the same barrier ( V shifted so that the bottom of the well corresponds to zero energy). pected from Figure 5, but the curvature is much smaller than we could evaluate considering the experimental errors. For values of V2 < 0 (well centered on 9 = Oo) all fits for IV41V21< 1/4 have a much higher slope than the experimental data; the range of slopes is indicated by the shaded area in Figure 6. The lack of sensitivity to the shape of the well results from the convergence of the family of curves near S -0.3 for V, < 0 in Figure 5. On the basis of the temperature dependence of ISJ,we conclude that V2 > 0 (the well is centered on 9 = 90') and that the order parameter is negative. The best fit of the experimental data occurs for V2 = +6380 J/mol, V4 = +(l/6)V2 = +lo63 J/mol. The shape of this well is shown in Figure 7 where the potential energy scale has been shifted to make the bottom of the well zero. The well is flatter than that of the pure V2 cos 2%potential energy function shown for comparison. The barrier height is 2V2 or 12760 J/mol. The changes in the order parameter with temperature are due to changes in the function e-(Vs COB 28 + V, cos 4 8 ) / R T sin g which is the probability function for the orientation of the C3 axis of the AsF, molecule, the Po(%) sin B of eq 4. This
K.
probability function is shown in Figure 8, calculated at 265 and 335 K for the best fit potential energy function. The increased probability of greater deviations from 0 = 90' at 335 K corresponds to the smaller magnitude observed for the order parameter a t the upper limit of the temperature region studied. This approach to determining the location, depth, and shape of the potential energy function for a molecule on a planar surface and, thereby, the sign of the order parameter is based on the assumption that V itself is not a function of temperature. The assumption is a reasonable one for this GIC where the intercalation gallery height is determined by another, much larger intercalant, the AsFc ion. The method is not limited to cases such as this where the combination of the temperature span and the magnitude of the second derivative of the appropriate curve in Figure 5 is such that the temperature dependence of IS1 is almost perfectly linear. This approach should work well over extended temperature ranges for any of the curves of Figure 5 as long as V is not a function of temperature. The extension to ratios of V4 to V 2other than those tabulated in Table I1 is straightforward. There is a faster, approximate method of determining V when the temperature dependence of (SIis nearly linear. Neglecting V4 and higher order terms, one makes a series expansion about the potential energy minimum. For V2 > 0, we define the parameter X = 21V21/RTand change the variable from 9 to u = cos 0. If X >> 1, the integration limits for u may be changed from (-1,l) to (-m,m) and the order parameter is, approximately,
S = -72
+ (3RT/8VJ + ...
(7)
The next term is 3(7~X)-'/~e-~, which is quite small for X >> 1. For V, < 0, we change the variable from 0 to u = sin 9 since the major contributions to the integrals come from small u. Using Taylor's expansion, we get
S = 1 - 3RT/41V,I
+ ...
(8)
where the next term, -(3/4X2), is also quite small when X >> 1. When we plot S vs. T, the intercept of eq 7 for T = 0 is -'I2 while the intercept for eq 8 is +l. The linear least-squares fit of our experimental data, IS1 = 0.4359 0.000433T, has an intercept of S = -0.4359 (if V, > 0, S C 0) or S = +0.4359 (if V2 < 0, S > 0). The agreement between -1/2 and -0.4359 strongly suggests that V, > 0, S < 0, and the potential minimum lies at 0 = 90" (as we found by the general method, above). For this choice, X 6, making the higher order correction terms small in eq 7. Alternatively, assuming only the linear term resulting from an expansion of V, cos 4%in a fashion similar to our
200 Langmuir, Vol. 2, No. 2, 1986
Miller et al.
Conclusions Experimentally, we have observed the temperature dependence of the 19FNMR triplet spectrum of AsF, intercalated in a graphite intercalation compound containing AsF,-. From a consideration of the effects of motions about the three Euler angles p, 4, and 8 on the NMR spectrum of three spin-l/, nuclei a t the vertices of an equilateral triangle and a comparison of calculated spectra with the observed spectra, we conclude that motional averagings occur over all three angles. From the observed temperature dependence of the triplet splitting, we deduce that the potential energy well has its minimum at 8 = 90° with a well depth of 12.8 kJ/mol. Thus the sign of the order parameter, S , is negative. The preferred orientation of the AsF, molecule in this graphite intercalation compound is with its C3 axis parallel to the graphite planes. H -4
-2
0
2
4
A v/a
Figure 9. Calculated 19FNMR spectrum for h F 3 (assuming75As
is effectively decoupled from ' V ) :case E, fast reorientation about C3 axis, slow reorientation about director, slow change in 8; case F, fast reorientation about C, axis, slow reorientation about director, fast change in 8; case G, fast reorientation about C3 axis and director, slow change in 8 ; case H, fast reorientation about C3axis and director, fast change in 8. Dotted lines show anisotropy of one component in cases E and F. treatment of V , cos 28 is important, one can draw a straight line from the vertical intercept, S = *0.4359, to the experimental value of S in the center of the observed range. The straight line should be tangent to the curve with the parameters V , and V, descriptive of the potential energy well. The dotted lines in Figure 5 indicate again that agreement cannot be obtained for S > 0 (the upper part of Figure 5) but is obtained for S < 0. In the case of AsF3 in this GIC, the experimental evidence for fast translational diffusion in addition to fast reorientation about P and 4 strongly suggested the successful approach taken above which involves averaging over the motion in 8 (i.e., we assumed changes in 8 were fast on the NMR time scale). It is worth generalizing the consideration of motion of a symmetric top on a planar surface to include the possibility of either fast or slow changes in 8. Fast changes lead to an averaged spectrum. Slow changes in 8 lead to a spectrum that is the sum of the spectra for various B's, each spectrum weighted by the probability of 8. Figure 9 shows the spectra calculated for S = -0.306, K = 90°, at 300 K for the four combinations of fast and slow motions in p and 8. The details of the calculations are given in the Appendix. Case E: Fast C,-axis reorientation, slow reorientation about the director, slow change in 8. Case F: Fast C,-axis reorientation, slow reorientation about the director, fast change in 8. Case G: Fast C,-axis reorientation, fast reorientation about the director, slow change in 8. Case H: Fast C,-axis reorientation, fast reorientation about the director, fast change in 8. Each of these spectra is quite different from the others. Comparisons of spectrum G and spectrum H with our experimental spectrum indicate that 8 changes fast for AsF3in our sample. Spectrum H is identical with spectrum D (Figure 4D), a 1:2:1 triplet with narrow lines. This is why one cannot distinguish between the case of fixed 6 and the case of fast changes in 8 on the basis of the experimental spectrum at one temperature. It is the temperature dependence of the triplet splitting that enables a distinction to be made between case D and case H.
Appendix In this appendix, we undertake the calculation of the NMR spectrum of three identical spin-l/, nuclei at the vertices of an equilateral triangle which is undergoing motions about various combinations of the Euler angles p, 8, and 4 while on the surface of a plane. We use this as a simplified model for AsF3 molecules intercalated between graphite layers. While we have applied these calculations in the main text of this paper to the motions of AsF, molecules in our graphite intercalation compound, the calculations are generally applicable to two-dimensional systems of symmetric top species containing an equilateral triangle of spin-'/* nuclei on the principal rotation axis when these nuclei are decoupled from any other magnetic nuclei. These calculations are an extension of the classic work of Andrew and Bersohnl' (AB) from the three-dimensional case to the two-dimensional case. Case A: The Rigid Lattice. For an equilateral triangle of spin-1/2nuclei rigidly fixed in space, AB have given the spectrum as a central line a t frequency y B with relative intensity (1 + 3 ~ ~ y - ~flanked ) / S by three pairs of satellites shifted by Au = &2y, f(32 + y) and f(32 - y) in frequency (Hz) with relative intensities = 3(1 - x 2 s,-2 1/16 (All p = (1 - 2 ~ - ' ) / 8 p = (1 + xy-')/8 where x = a ( l - 3 cos2 +)/a
(A21
and y2 = x 2
+ a2 + bz = a2(27 cos4 + - 42 cos2 I,L + 19)/4
(A31 a and b are functions of the angles defined by AB and is the angle between the magnetic field B and the molecular C3 axis. These parameters are independent of the phase angle p. We have followed the "energy of transition" convention (Table I and Figure 1 of AB) which is equivalent to setting their magnetic moment K = (the spin of 19F). Our a = 3h2y2r-3/8(where r is the distance between vertices of the equilateral triangle) is equivalent to r(~a of AB. (From the magnetic moment of 19F and the known geometry of the AsF, molecule,'0 r = 1.706 X cm and F-As-F angle 96.Z0, we get r = 2.540 X lo-' cm, y = 4005 Hz/G, and a = 2433 Hz.) It is also convenient to introduce the dimensionless frequency shift h = Au/2a. Our h is equivalent to the h / a of AB. AB found the detailed line shape F ( h ) for a random three-dimensional powder of equilateral triangles by considering the uniform distribution of the molecular C, axis in space:
+
Langmuir, Vol. 2, No. 2, 1986 201
Spin-'I2 Nuclei in a Uniaxial System
F ( h ) = p($)ldh/d(cos
$)I-'
(A4)
Our p ( $ ) is the same transition probability listed in Table I of AB. Let us now consider the two-dimensional system with angle K between the magnetic field, B , and the director, the normal to the plane. The distribution of the molecular C3 axes is "conelike", that is, they are all inclined a t an angle 0 with respect to the director but the azimuthal angle 4 is random between 0 and 360'. In general, the randomness of 4 corresponds to an assumption of a smooth, featureless planar surface. (In our case of AsF, in the GIC, the randomness of the graphite a axis in the plane perpendicular to the director, characteristic of HOPG, assures a random angle 4. If the &dependence of a potential energy well is of %fold symmetry or higher, the results to follow apply to this case as well.) We now write $ in terms of the other angles as cos $ = cos
K
cos 0
+ sin
K
sin 0 cos 4
(A5)
If we choose 0 and K in the range 0-90°, then the angle $ is between IK - 01 and K + 0. The line shape G ( h )for the two-dimensional system is quite similar to the polycrystalline system.l6Jg Since the distribution is random with respect to 4 rather than cos $, we have in place of eq A4
G ( h ) = p($)Idh/d4l-l
That is, G ( h ) of the two-dimensional system is the polycrystalline result, eq A4 as given by AB, multiplied by the factor A-'l2, where the function A is defined by
(A8)
Using eq A5 to eliminate the angle 4, we get
- cos2 0 + 2 cos K cos 0 cos $ - cos2$
K
h = f(3x
COS'
$ = (7/9) *(2/9)(3h2 - 2)'i'
+
2(3~ - y)'[(l - X Y - ~ ) / ~ ]= 6x2 + 2y2 (A12)
where x and y are given in terms of cos $ by eq A2 and A3 and cos $ is given by eq A5. For a GIC, it is necessary to sum over the angle 4: (cosn 4 ) = ( 1 / 2 7 r ) ~ ' ~ c o s4n d$ = 1, 0,
y2,0, and y8 (A13)
may be rewritten in terms of the Legendre polynomials P 2 ( u )= (3u2 - 1 ) / 2 and P 4 ( u )= (35u4 - 30 u2 + 3)/8: (cos' $) = (2/3)P2(c0s K)P~(COS 0) + 1 / 3 (cos4 $) = (8/35)P4(~0sK)P,(COS 0)+ (4/7)P2(C0S 0)
+ 1/5
(A14)
Substitution into eq A2, A3, and A5 gives (A9)
The dependences of h on cos $ have been given by AB for the various satellite components. The inverse functions, expressing cos $ in terms of h, have also been given by AB. Thus cos $ is really a dummy variable and may be eliminated from eq A7 and A9. We note that the spectrum exists only when A is positive. Also, infinities or singularities may be introduced into the spectrum when A goes to 0 (at $ = IK - 01 and K + 0) and when F(h)goes to infinity (at $ = 90°, possible only when K + 0 2 90') in the range of h. These singularities also correspond to dh/d$ = 0, that is, when the splittings are maximum or minimum as functions of 4. These characteristics have been previously noted16J9for two-dimensional systems in general. From (A2) and (A3), we get
h = fy/a:
M = 8y2[3(1- ~ ~ y - ~ ) / 1+62(3x ] + ~ ) ~ +[ (x y1- ' ) / 8 ]
for n = 0, 1,2,3, and 4, respectively. The results for ( cos2 $) and (cos4$) are somewhat complicated. However, they
G ( h ) = F(h))d(cos$)/d@I-' = F(h)IA(h)[-'/' (A7)
A = sin'
+
(-46).
Comparison with eq A4 gives
A = Id(cos $)/d& = sin2 K sin2 0 sin2 4
associated with each of the six satellite lines. The positions of these singularities are at Av = f2y, f(3x + y),and f(3x - y), calculated from (A2) and (A3) and from the values of $ given above. Typical examples are shown in Figure 4A for K = 90°. From eq A5 we have cos $ = sin 0 cos 4. For each satellite component, there are two singularities corresponding to cos $ = f sin 0 (when 4 = 0 or 180') and cos $ = 0 (when 4 = 90'). There are 1 2 singularities a t Av = f(27 cos4 0 - 12 COS' 0 4)'/', f(1/2)[9 COS' 0 - 6 f(27 cos4 0 - 1 2 cos2 0 + 4)1/2],f ( 3 f 19ll2)/2,and f(19'l2)/2. The satellite pattern is somewhat "powderlike" because of the conelike distribution of the molecular C3 axis and the associated singularities. As a simpler alternative to line shapes, we may also calculate the second moment, M , of the complete NMR spectrum. For an isolated equilateral triangle of spin-1/2 nuclei, we have
(A10)
+ y)/Ba
and f(3x - y)/2oc: cos2 $ = [(l- 6h)/9] f (2/9)(3h2 + 6h + 4)l/' ( A l l )
Thus the function A(h) may be obtained by substituting (A10) and ( A l l ) into (A9). By use of this A(h) in (A7), the calculated satellite spectra, G(h)vs. h, would be rather broad but with rather sharp singularities as noted above. Thus the main features of the NMR spectrum are the central component and several (two or three) singularities (19) Resing, H. A.; Garroway, A. N.; Weber, D. C.; Ferraris, J.; Slotfeldt-Ellingsen, D. C. Pure Appl. Chem. 1982,54, 595.
M/Ci2 = (216/35)P,(COs
K)Pd(COS 0) (32/7)P,(cos K)P~(COS 0) + 32/5 (A15)
Besides the constant term, this particular form for M involves only the two pairs of the second- and fourth-order Legendre polynomials and is in accordance with the group theoretical requirement.'O We also note that P2(cos 0) is identical with the order parameter S, and P4(cos0) = (35S2 - 1 0 s - 7)/18: Thus the second moment may also be expressed in terms of the order parameter as
M / a 2 = (12/35) X (358' - 1 0 s - 7)P,(COS
K)
-(32s/7)P,(COS
K)
+ 32/5 (A161
Case B: Fast Reorientation of the Equilateral Triangles about Their C3 Axes. The averaging over phase angle 0implies that c i = 6 = 0, where the bar denotes motional averaging. Hence we have 9 = IZl from eq A2 and A3, but f is still given by eq A2. With the transition probabilities given in eq A1 for the three satellite pairs in case A above, the intensities of the two pairs at 2 9 and f(32 - 9 ) would become zero whereas the frequency shifts are Av = f(3f + 9) = f 4 f for the other pair. For a single equilateral triangle, the spectrum would be three sharp (20) O'Reilly, D. E.; Tsang, T. Phys. Reu. 1962, 128, 2639. (21) Slichter, C. P.Principles of Magnetic Resonance; Springer Verlag: Berlin, 1978; p 76. (22) Jackson, J . D. Classical Electrodynamics; Wiley: New York, 1962; p 4.
202 Langmuir, Vol. 2, No. 2, 1986
Miller et al.
resonance lines with intensity ratios 1:2:1, where the two satellites are shifted by Av = f 4 f or h = f 2 f / a . Again, each satellite is broadened into a "powderlike" pattern by the random distribution of the angle 4. The shifts are
h = f(1 - 3 COS' $)
(A171
and the intensity is G(h) = Idh/d$l-' = 16 cos
+ sin
K
sin 01-' = 16A-'(cos
$)-'I ( A B )
where A is given by eq A9. For one of the components, we have cos $ = [(I - h)/3]'/' and the intensity is G(h) = (1/2)(1 - h)-1/2[3(sin2K - cos2 8) +2(3)'/2(1 h)'j2 COS K COS 0 - 1 + h]-'i2 (A19) As before, the main features are several (two or three) singularities at h = 1 - 3 cos' ( K - e), when 4 = 0; h = 1 - 3 cos2 ( K B ) , when $ = 180"; and h = 1, possible only when K 0 290'. The intensity of the other component is obtained by replacing h with -h. These results are quite similar to the spectra previously derived for anisotropic chemical shifts.16 Again, these spectra are powderlike because of the conelike distribution of the molecular C3 axes. A simple example is for K = 90" (magnetic field, B, perpendicular to the director). Equation A19 may be written as
+
M = 6Zw2+ 29-' = a'(3 cos2 K - 1)'(3S2 - 2s
(A24) The spectra for K = 90" are shown in Figure 4C. Here, the central component intensity is rather weak, thus the second moment given by eq A24 for case C is comparable to eq A21 for case B, although the spectra for case C appear to be more compact. Case D: Fast Reorientation about Both the Director and the C3Axis. It is necessary to average over both 4 and 0. However, we now have h = 6 = 0 and hence 9 = 1x1. Among the three pairs of satellite lines in case C, only the pair at Av = f 4 f may be observed since the other two pairs have zero intensity. The NMR spectrum is a pattern of three sharp resonance lines with an intensity ratio of 1:2:1. The satellites are shifted by
AV = f 4 f = f ( 1 - 3 COS2 d ) ( l
+
G(h) = (1/2)[(1
-
h)(3 sin' 6 - 1 + h)]-1/2 (A20)
That is, the satellite intensity is powderlike between the two singularities at h = 1- 3 sin2 0 (corresponding to both 4 = 0 and 180") and at h = 1. Similarly, the intensity for the other satellite may be obtained by replacing h with -h. In addition to the sharp central peak, the complete spectrum has four singularities at h = f l , f ( 1 - 3 sin2 0) or in frequency units Av = f2a, *2a(l- 3 sin28). The general features may be seen in Figure 4B. The second moment, M , may be calculated readily for any angle K . Instead of eq A12, we now have M = 8x2, with x given by eq A2, due to the motional averaging. Using eq A13, we get
M / a 2 = (144/35)P4(cos K ) P ~ ( C O S8) 4( 1 6 / 7 ) P 2 ( ~ 0K)Pz(COS ~ 0) + "15 = (6/35)(35 8' - 1 0 s ~)P,(cosK) 4- (16/7)SP,(cos K ) + "/j (A21) As expected,21the constant term in eq A21 is a factor of 4 smaller due to motional averaging. Case C: Fast Reorientation about the Director (but not about the C3 axis). There will be motional averaging over the azimuthal angle 4. From AB, eq A2 and A3 are averaged to
x
= (a/4)(1 - 3
COS'
~ ) (-13
COS'
0)
(A22)
9 = (a/4)(1 - 3 cos2 ~ ) ( 2 7sin4 0 -12 sin' 0+ 4)'I' (A23) The central line, with relative intensity (1+ 3T2ji-')/8, is flanked by three pairs of satellites shifted by Av = f29, f ( 3 f + j i ) , and &(3f - j i ) with relative intensities 3(1 T2jj-2)/16,(1 + f j i - ' ) / S , and (1- f9-')/8. Because of the motional averaging over the angle 4, the resonance lines are all sharp. On varying the angle K , the magnitudes of the satellite splittings vary as 1- 3 cos' K while the relative intensities and ratios of the splittings remain unchanged since they are functions of the angle 0 only and are independent of K . The second moment is
+ 1)
h = fS(3
- 3 COS'
COS' K
= =t2S(3
K)
COS'
- 1)
K
- 1)
(A251
The second moment is
M = ( A V ) ~= / ~2a2S2(3COS'
K
- 1)2
(A26)
For 8 = 43" (S = +0.306) and 0 = 69' ( S = -0.3061, the spectra for K = 90" are shown in Figure 4A-D. For the four cases, the spectra are quite different because of the various types of motional averaging. The spectra of cases A and B are powderlike because of the conelike distribution of the C3 axis without motional averaging over the azimuthal angle $. With averaging over 4, satellite lines are sharp in both cases C and D, but there are more lines in the former case due to the absence of averaging over the phase angle p. The three resonance line pattern of 1:2:1 intensity ratio is only possible in case D after motional averaging over both 4 and p. The motional averagings of the lines in case C into case D are also indicated in Figure 4. So far, we have dealt only with the cases for fixed 0. We now consider averaging over motion in 0. We consider the following combination of molecular motions at 300 K for S = -0.306. Case E: Fast Reorientation about the C 3Axis, Slow Reorientation about the Director, and Slow Change in 0. The spectrum, G(h), is obtained by averaging G(h,R) in case B with fixed 0 given by eq A19 with the distribution function Po(@sin 0 = e-"/RT sin 8 of the angle 8 (cf. eq 4): sin 0 d0 G(h) = $2TG(h,0)Po(0) 0 sin 0 d8/$2TPo(0) 0 (-427) A simple example is for K = 90". From eq A20, G(h,0) has two singularities a t h = f l and f ( 3 sin2 0 - 1). On averaging over Po(@sin 0, the singularities a t h = f l remain sharp because their positions are independent of 0. The singularities at f ( 3 sin2 0 - I) are broadened out. Case F: Fast Reorientation about the C 3Axis, Slow Reorientation about the Director, and Fast Change in 8. This is identical with case B as described previously. In eq A19, we now replace cos 0 and cos' 0 by (cos 0 ) and (COS'
e).
Case G: Fast Reorientations about the C 3Axis and the Director and Slow Change in 0. Again, the spectrum G ( h ) is obtained from eq A27. We now use G(h,0) from case D. From eq A25, these satellites are sharp and the line shapes have the form of the 8 functions: G(h,B) = 6[h f (1/2)(3
COS'
K
- 1)(3 COS' R -l)]
(A28)
Standard methodsz2of integrating eq A27 indicate that G(h) is proportional to eviRT/cos0, and eq A25 may be
Langmuir 1986, 2, 203-210 used to eliminate the angle 8. Singularities in G ( h )occur a t 8 = 90° or h = f(3 cos2 K -1)/2. Case H: Fast Reorientations about the C3Axis and the Director and Fast Change in 0. This is identical with case D as described previously. The complete NMR
203
spectrum is a pattern of three sharp resonance lines with an intensity ratios of 1:2:1. The satellite shifts are given by eq A25, where cos2 8 is replaced by (cos2 8). For these four cases, E-H,the calculated spectra are shown in Figure 8 for K = 90°.
“In Situ” Attenuated Total Reflection Fourier Transform Infrared Studies of the Goethite (a-FeO0H)-Aqueous Solution Interface M. Isabel Tejedor-Tejedor and Marc A. Anderson* Water Chemistry Program, University of Wisconsin-Madison, Madison, Wisconsin 53706 Received J u l y 30, 1985. I n Final Form: November 13, 1985 This paper examines the use of attenuated total reflection Fourier transform infrared spectroscopy (ATR-FTIR) as an “in situ” technique with which to study the goethite (a-FeO0H)-aqueous solution interface. Using a cylindrical internal reflection (CIR) cell with a ZnSe crystal we show that ATR infrared goethite spectra are similar to spectra recorded via transmission for bands arising from the bulk of the solid. These results are interpreted by using a thin film model. Interfacial spectra as a function of pH or pD, ionic strength, and nature of the “inert” electrolyte anion were also obtained. By employing this method we demonstrate that the positive and negative goethite particles induce structure on adjacent water layers. ”Inert” electrolytes affect this highly structured water in the order NO; > Cl- > Clod-. The signal intensity in the ATR-FTIR technique is largely dependent upon the dispersivity (state of aggregation) of the suspension; therefore, it is presently semiquantitative since an increase in goethite concentration (g/L) does not necessarily produce a proportional increase in signal intensity (unless the samples have identical degrees of dispersivity). Consequently, one must use internal standards, such as bulk OH or FeO groups, to compare peak heights for quantification purposes. Phosphate, which is known to form inner-sphere complexes with iron oxides, has also been examined both in solution and at the goethite interface by using ATR-FTIR. This technique shows that phosphate greatly perturbs the structuring of the water at the interface (more so than the “inert” electrolytes NO3-, C1-, and C104-) and it substitutes for surface OH groups. As such, ATR-FTIR should prove useful for studying surface complexation reactions in aqueous suspensions.
Introduction Our research is addressed to understanding those heterogeneous surface reactions that control the distribution of solute species between aqueous solutions and solid surfaces. Previous research on these interfacial reactions has largely resulted in model postulates concerning the molecularity of surface reactions since until now only indirect methods (e.g., adsorption isotherms) were available for these studies. Our own effort, therefore, has been directed toward finding a technique that identifies the chemical structure of (1)the adsorbent surface, (2) the adsorbate, and (3)any new species which may be formed by chemisorption. This information can be obtained from the interpretation of vibrational spectra of these chemical groups, and infrared spectroscopy is the most common method of studying vibrational modes. While IR studies have been successfully used for interpreting catalytic reaction mechanisms,l the use of IR in this research has largely been confined to identify reactions occurring at a gas-solid interface. In our studies, we are interested in characterizing the solid-water interfacial region. However, water has always been an anathema to the transmission mode of IR analysis. Water is such a strong absorber in the mid-infrared that extremely short-path-length cells are required in order to transmit enough energy to make useful measurements. Although such cells can be constructed with (1) Sheppard, N. NATO Adu. Study Inst. Ser., Ser. C 1980, 67.
some difficulty, they are almost impossible to fill or empty and are subject to clogging when used to study suspensions. Internal reflection spectroscopy (IRS) offers a solution to this problem. Since beam penetration into the solution is extremely small with this technique, effective path lengths of the correct magnitude for most aqueous solution studies (15 pm) can easily be obtained. Although internal reflection cells made with flat plates have been used previously in the analysis of aqueous sol u t i o n ~this , ~ ~crystal ~ design is not efficient with respect to energy throughput in the circular beam of the FTIR spectrometers. To avoid beam vignetting and a consequent increase in the energy throughput of the system, Wilks, in 1982, proposed a new internal reflection element (IRE) design employing a polished cylindrical rod with coneshaped ends.4 This cylindrical internal reflectance (CIR) cell has been shown to perform very well for qualitative and quantitative analysis of aqueous solution^.^^^ On the basis of these results, we decided to utilize this design for IR studies of aqueous colloidal suspensions. This paper evaluates the CIR technique as an “in situ” tool with which to study the surface chemistry of aqueous (2) Yang, R. T.; Low,M. J. D. Anal. Chem. 1973,45,2014. (3) Mattson, J. S.; Jones, T. T. Anal. Chem. 1976,48, 2164. (4)Wilks, P., Jr. Industrial Research and Development 1982, Sept, 132. (5) Messerchmidt, R. G. Scan Time 1983,2,3. (6)Wong, J. S.; Rein, A. J.; Wilks, D.; Wilks, P., Jr. Appl. Spectrosc. 1984,38,32.
0143-7463/86/2402-0203$01.50/0 0 1986 American Chemical Society
