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J. Phys. Chem. 1980, 84, 1944-1950
CF3CH2NH2than for CH3CH20H and CH3CH2NH2.3’5 The barriers to internal rotation for the CF, groups in CF3CH2F (3.58 kcal/mol),15 gauche-CF3CH20H, and trans-CF3CHzNHz follow the same trend as the CH, barriers in CH3CHzF (3.33 kcal/mol),16 gaucheCH3CH20H (3.62 kcal/mol),15 and trans-CH3CH2NH2 (3.74 kcal/m01),~but the barriers for the CF, groups are somewhat higher. A dipolar interaction such as the one described above cannot exist for ethanol and ethylamine, and we must conclude that, while the positions of the C-F bonds are important for stabilizing certain conformations, the CF3 barriers are not appreciably affected by the conformational preferences that exist for the OH and NH2 groups. Acknowledgment. The authors thank the Mississippi State University Office of Research for partial support of this work. Acknowledgment is also extended to the Mississippi Imported Fire Ant Authority for the funds to purchase the Nicolet and Spex instruments. We also extend our thanks to the Langley Research center, Hampton, VA, for the loan of the microwave spectrometer and to Professor R. L. Cook for providing easy access to the instrument. Miniprint Material Available: Full-sized photocopies of Tables I, 111, IV, and V (9 pages). Ordering information
is given on any current masthead page.
References and Notes (1) Presented in part at the Sixth International Conference on Raman Spectroscopy, Bangalore, India, Sept 1978. (2) Taken in part from the thesls of H. V. Anjarla to be submitted to the Department of Chemistry. (3) J. R. Durig and Y. S. Li, J. Chem. Phys.,63,41 10 (1975); M. Tsubol, K. Tamagake, A. K. Hirakawa, J. Yamaguchi, H. Nakagawa, A. S. Manocha, E. C. Tuazon, and W. G. Fateley, ibid., 63, 5177 (1975). (4) Y. S. Li and V. W. Laurle, paper presented at the 24th Symposium on Molecular Structure and Spectroscopy, Columbus, OH, 1969. (5) J. R. Durig, W. E. Bucy, C. J. Wurrey, and L. A. Carrelra, J. Phys. Chem., 79,988 (1975); A. J. Barnes and H. E. Hallam, Trans. Farahy Soc., 66, 1932 (1970). (6) R. Kakar and P. J. Seibt, J. Chem. Phys., 57, 4060 (1972), and references therein. (7) I. D. Warren, and E. B. Wilson, J. Chem. Phys., 56, 2137 (1972). (8) H. Wolff, D. Horn, and H. G. Rollar, Spectrochim. Acta, Part A , 29, 1835 (1973). (9) P. J. Krueger and H. 3. Mettee, Can. J . Chem., 42, 340 (1964). (10) J. Travert and J. C. Lavalley, Spectrochim. Acta, Part A , 32, 637 (1976). (1 1) C. J. Wurrey, L. A. Carreira, and J. R. Durig in “Vibrational Spectra and Structure”, J. R. Durig, Ed., Elsevier, Amsterdam, 1976, Chapter 1. (12) H. Gillman and R. G. Jones, J. Am. Chem. Soc., 65, 1458 (1973). (13) F. A. Miller and B. M. Harney, Appl. Spectrosc., 24, 291 (1970). (14) K. 0. Hartman, G. L. Carison, R. E. Witkowski, and W. 0. Fateley, Spectrochim. Acta, Part A , 24, 157 (1968). (15) A. D. Lopata and J. R. Durlg, J . Raman Spectrosc., 6, 61 (1977). (16) G. Sage and W. Klemperer, J . Chem. Phys., 39, 371 (1963).
Interpretation of NMR Relaxation Data for Reorientation in Brittle and Plastic Phases of Organic Solids David W. Larsen” Chemistry Depaflment, University of Missouri, St. Louis, Missouri 63 12 1
and John H. Strange Physics Laboratory, University of Kent, Canterbury, Kent, CT2 7NR. United Kingdom (Received January 14, 1980)
NMR relaxation behavior which is not accounted for by simple BPP theory has been observed for many organic solids which exhibit high-temperature plastic phases. The motional processes in question involve anisotropic molecular reorientation in the brittle phase and endospherical reorientation in the plastic phase. Models for these motions are presented from which NMR relaxation times T1and T1,are calculated. The model for plastic-phase motion involves reorientation about a molecular axis with simultaneous independent changes of position between molecular centers. The model for brittle-phase motion involves reorientation about a molecular axis, characterized by wells of unequal depth. Both threefold and fourfold rotations are considered. The results of the calculations me in good agreement with experimental data for azabicyclononane. Calculations of this type could provide a basis for understanding the detailed nature of these motions.
Introduction A class of molecular solids with globular structures is known to form a plastic crystalline high-temperature phase.l In general, one observes a phase transition below which the substance exists in a brittle phase. Substances of this type have recently been the subject of NMR relaxation s t u d i e ~ . ~ -Three l~ types of motional processes have been consistently found in these studies. In the brittle phase at low temperature, anisotropic molecular reorientation is observed, while in the plastic phase isotropic molecular reorientation is observed at lower temperatures and translational diffusion is observed at higher 0022-3654/80/2084- 194450 1.OO/O
temperatures. Since these substances have molecules with almost spherical shape, they are solids for which the simplest theories for NMR relaxation should apply. Two features of the NMR data for reorientation cannot be explained15in terms of the usual Bloembergen-PurcellPound (BPP) type expressions.16 (i) Above the Tl minimum for isotropic reorientation in the plastic phase, Tl f Tlpis observed in some cases, and where T1 and T1,can be observed over a temperature range, the usual Arrhenius plot shows that the gradients of both Tl and Tlpare identical within experimental error. Departure from linearity, described as a kink, in the 0 1980 American Chemical Society
Brittle and Plastic Phases of Organic Solids
The Journal of Physical Chemistry, Vol. 84, No. 15, 1980 1945
high-temperature TLarm has also been observed in some cases. (ii) T1/Tlpnear and below the T1, minimum for anisotropic reorientation in the brittle phase depends on the compound studied. The usual BPP-type expressions predict that this ratio should depend only on wo, the Larmor frequency in the strong field, and on w 1 = rHl, the Larmor frequency in the rotating magnetic field, H1. The BPP-type expressions for the spin-lattice relaxation time Tl and for the irotating frame spin-lattice relaxation time T1, due to dipolar interactions between like spins are given by eq 1 and 2. The expressions assume that a
of unequal depth, and the molecular reorientation is characterized by equal potential wells. We have found that this model reproduces the essential features of NMR relaxation data for plastic-phase isotropic reorientation. The results of the calculation are presented below, and requirements for the parameters to give the observed NMR anomolies are discussed.
Theory Reorientation in the Plastic Phase. “Isotropic” reorientation in the plastic phase is thought to involve random r motion between a number of equilibrium positions separated by potential barriers.18 Correlation functions may be evaluated by group theoretical methods17p21>22 which account for symmetry properties of the crystal structure. 1 These methods are most readily adapted to the intramo_ lecular contributions to relaxation. We find that the inT1, termolecular contributions must be included for a satis27, 57, factory fit of the experimental data. A two-dimensional -model which comprises two three-spin units (molecules) 1 + h127,2 1 + C O O ~ ~ , 1~ + ~ C O O ~ ~ : is used. The orientation of each molecule and the sepa(2) ration between molecules are allowed to vary with time. powder average may be taken and MZm&is that portion We were unable to obtain the desired behavior without of the second moment that is modulated by the motion. including two motions of this type in the calculation. These equations predict Tl = T1, for oar, 2, or for isotropic reorientation. The inunits (molecules or molecular segments) which undergo termolecular contribution introduces terms” in r C / 2 ,and threefold random reorientation and simultaneous change these terms have the effect of shifting the minima slightly, of distance between their respective centers, as indicated: but they do not account for the observed behavior. Thus a more detailed model is required to fit the observed data. Models for reorieritation in molecular crystals have been discussed recently.IR Two features of the reorientation appear to provide a basis for understanding the observed NMR behavior. First, since the molecules themselves are Molecule B is assumed to jump relative to molecule A, and not spherically symmetric or indeed may not contain even the process is described by two potential wells of depths an n-fold symmetry axis, reorientation should be characE2and E{, for the pair together and separated, respecterized by potential wells of unequal depth. Second, the tively. The threefold reorientations are assumed to occur positions of neighbors may be time dependent because of independently with time constant r3,whether A and B are the effects of “bumps” in the molecular structure as the close or separated. Thus molecules reorient. We have considered various simple 7 3 = 703 exp(E,/RT) models that incorporate these two features of reorientations in globular molecular solids. Using these models, we 7 2 = 702 exp(Ez/RT) have derived expressions for Tland Tlp,and we have compared the exprlessions with experimental data. Q‘ = r0{ exp(Ei/RT) (3) For brittle-phase anisotropic reorientation, we have Correlation functions for the spherical harmonics are found that a planar model in which the molecule jumps calculated as described by Andersonmfor the pairwise spin between potential wells of unequal depth is a useful apinteractions proach. Both threefold and fourfold cases were considered. Following the approach presented p r e v i o ~ s l ywe , ~ have ~~~~ (Fm(OF,(t) ) = Cp#’irnCpj(tli)Fjrn (4) obtained expressions that reproduce the qualitative feai I tures of brittle-phase anisotropic reorientation relaxation where Pi is the probability that the spin pair has configdata and, in some cases, a quantitative fit can be obtained. uration i at t = 0, Pj(tli) is the probability that the spin The results of these calculations are presented below, and pair has configuration j at time t given that it had conthey are compared with experimental data for a typical figuration i at t = 0, and Firnand Fjmare the spherical compound that exhibits brittle-phase anisotropic reorienharmonics characterizing the nuclear dipole-dipole intation. t e r a c t i o n ~ .The ~ ~ sums over i and j include all possible For plastic-phase isotropic reorientation, we have found configurations. that a simple motion between n-fold unequal potential For the threefold reorientation, there are three sites (a, wells will not fit the experimental data. We attribute this b, c) for to the neglect of tirne dependence of neighbor positions. Inclusion of this feature involves a complex calculation in Pa = Pb = P, = 1 / 3 three dimensions. For simplicity we have used a planar model in which time-dependent intermolecular separation P,(tla) = 7311 + 2 exp(-t/r3)l is superimposed upon molecular reorientation. The intermolecular separation is characterized by potential wells Pdtla) = Y3[1 - exp(-t/r3)1 (5)
+
+
1
1946
The Journal of Physical Chemistry, Vol. 84, No. 15, 1980
Larsen and Strange
For the molecular separations, there are two sites (2 and 2’) for P2
= Qz
P2) =
lo2,
Qy
+ Qz’ e x p ( - t / ~ , ~ ) P2412) = Q2U- exp(-t/~,~)l
P2(t12) =
Q2
P2(t12’) = Qz[l - exp(-t/~,~)l P24t12’) = Qzt
Qzf = 72!/(72 72’) (6) where Q2 and Q2, are the equilibrium fractions for molecules together and separately, respectively, and the relative jump frequency rC2-l= ( T Z ) - ~ + ( T ~ , ) - ~ . The correlation functions are calculated by forming the required compound probabilities in eq 4. For the intermolecular contribution Q2
=
+ Q2 exp(-t/rC2)
72/(72
721)
5
3
4
10%
Figure 1. TI and TI, vs. reciprocal temperature for Isotropic reorientation model. Calculated from eq 8-11 with E, = 12 kcalhol, E ’ = 10 kcai/mol, T,,, = ~ ~ =2 51X s,E3 = 8 kcal/mol, 703= S.
where M 3 (9/20)(T2h2/R6)= (360/R6)(G2);R is the radius of the molecule in 8. The intramolecular contributions are evaluated as
1/T1 = 2.65 X 107M[j3(wo) + 4j3(2~00)]
(10)
l/TIp = 2.65 X lo7M[(3/2)jd2wl) + (5/2)jdu0) + j3(2q)l (11) where
The indices k and 1 each run over a, b, c, corresponding to the three orientations of the molecules A and B, and the functions Fk,Jare Fa,, Fab,.. , where the subscripts indicate molecule A in configuration k and molecule B in configuration 1. Fk.1 and Fk,{refer to molecules together and separated, respectively. In the above expressions, all Fki are set equal to zero for simplicity, and it is assumed that 7,2 >> 73. All Fk,l are then adjusted so that (F,(O)) = 0, and the correlation functions are evaluated numerically. Powder averages over angles are taken and spectral densities are determined as Fourier transforms of correlation functions. Finally T1 and T1, are calculated by using standard method^.^^^^^^^^ The intermolecular contributions are
where j J w ) 5 T J ( ~ + ~ ~ 7 , ~ ) . We wish to illustrate the behavior of the equations derived above by use of a plot. We note that while Tl # Tlp in the high-temperature arm of the T1minimum for isotropic reorientation has been reported in most of the references cited above, the temperature range over which this can be observed is relatively small, between the brittle-plastic phase transition and the onset of domination by translational diffusion. For example, Tl/Tlpx 2 has been observed15for 3-azabicyclo[3.2.2]nonane(ABN) at -300 K, a temperature at which one expects T1= Tlpfor simple isotropic motion. In general, the minima associated with the motion are not observed. For this reason, we choose to present a plot made by using parameter values that are typical of those for isotropic reorientation in the plastic phases of organic solids. With the present model for reorientation, the parameters characterizing the motion are E3,ro3,Ez, 702,E;, r o i ,and M. Figure 1shows a plot of Tl and Tlpvs. 103/Tcalculated s, and M from eq 8-11, with E3 = 8 kcal/mol, 703 = = 1 G2 (R = 2.7 A), which are reasonable values for these parameters. The Larmor frequency vo = 30 MHz and the rf field H1= 10 C. The other parameters were chosen to give the observed type of behavior but with the constraints E2,E; > E3 and rO2= ro2/> 703; they are E2 = 12 kcal/mol, =5 X E2’ = 10 kcal/mol, and ro2= s. Figure 1 shows that Tl/Tl, = 2.5 with identical gradients for TIand TIPon the high-temperature side of the Tl minimum. This behavior is observed for a range in which both relaxation times change by a factor of 30. At the highest temperature, both T1 and TIPdeviate from linear behavior, but in a plastic crystal, this region is obscured by the effects of translational diffusion so that the contributions due to reorientation are not observed. The range of parameters that show this behavior is discussed below. Reorientation in the Brittle Phase. Threefold Case. Molecular reorientation in the brittle phase is thought to be highly anisotropic, usually about a symmetry axis.
The Journal of Physical Chemistry, Vol. 84, No. 15, 1980 1947
Brittle and Plastic Phases; of Organic Solids
Expressions for T1and T1, for threefold reorientation with one potential well different from the other two have been given by Anderson.20 The internuclear vector is assumed to be perpendicular t o the rotation axis, and the expressions apply only to the intramolecular contribution. The intermolecular contribution can be calculated by using the method of the previous section; however, because of the assumption of unequal wells, the calculation involves very many terms. The time constants in t,he correlation functions are r2, r2/2, r3, r3/2, and (rL1f T ~ - ~ ) - 'where , the ri are defined by Hoffman.% For simplicity,we consider only the intramolecular clontributions, since they agree qualitatively with the observed behavior. The expressions of Anderson20are
-4 lo
u 4.5
5.0
40
3.5
10%
(13) The depth of the single well, E A , differs from the depth of the other two wells, E g . In this case, there are two decay modes,24with time constants, 7 2 9 ( k +~ and 7 3 (3kB)-l,in which kA and kB are rate constants for particles leaving sites A and E3, respectively. The kinetic behavior of the motion is different for the two cases: (I) E A > EB and (11)E A < E B . In case I, kB > k~ and r2 373, whereas in case 11, k B .< k A and r2 (2kA)-l and 7 3 = (3kB)-l. We illustrate these functions in a plot as in the previous section. The temperature range over which anisotropic reorientation in the brittle phase can be observed in NMR relaxation data is in general relatively large, and the T1, minimum can often be seen. For this reason, it is appropriate to compare our derived expressions directly with experimental data. In Figure 2, relaxation datal5 for ABN are presented. A comparison is made with the unequal wells functions in the two separate threefold cases given above. The parameters used in the calculation are given in the figure caption. In constructing the plots, the various sets of activation parameters that fit tlhe Tlpvalues across the minimum (for 103/T < 4) were determined, and from these the sets of parameters that gave the shortest T, values were used in Figure 2. BPP values from eq 1 and 2 are also shown for comparison. I t can be seen that both cases I and 11, and BPP, can be fitted to the experimental Tlpvalues near the minimum. The functions diverge at lower temperatures, and the low-temperature arm of Tl, exhibits a kink for case 11. T1 values for both cases I and I1 fall below corresponding BPI' values. The effect is larger for case 11, with Tl values shorter than BPP Tl values by a factor of 2. This is the largest factor that can be obtained for the threefold case with two barriers EA and E B , and this is insufficient to fit the data for ABN. The two decay modes both depend on kg. In case I, r2 3r3,Le., the two decay modes have essentially the same time constant, and the decrease of TI values by proper choice of parameters is relatively small. In case 11, r:! and r3 = (3kg)-', i.e., the two decay modes have independent constants, and the decrease of T1 values by proper choice of parameters is larger. Thus the largest effects will be found for kinetic schemes in which decay modes have independent time This is illustrated in the next section, in which
-
-
-
Figure 2. T , and T i , vs. reciprocal temperature for threefold and fourfold anisotropic reorientation models: (solid Ilne) from eq 1 and 2 ( E = 9.7 kcal/mol, r0 = 3.3 X s, M,, = 13.3G2);(dashed line) form eq 12 and 13 (case I: EA = 10.7kcallmol, EB= 9.9kcal/md, ro = 4 X s, M,, = 19.0G2);(solid dot line) from eq 12 and 13 (case 11: E A = 8.1 kcallmol, E, = 9.9 kcal/mol, r o= 7 X s, M m = 18.1 G2);(dotted line) from eq 17 and 18 (E$ = 9.9 kcal/mol, EB = 8.1 kcal/mol, r o = 2.4 X lobi4s, M, = 14.0G , A = 7~14).H, = 15 G in all calculatlons. Open circles represent experiment data points for ABN (see text).
+
we consider the fourfold case. Reorientation in the Brittle Phase. Fourfold Case. In this section we extend the calculations of Andersonz0to the case of fourfold reorientation described by two barriers E A = Ec and EB = ED. We assume two pairs of equivalent sites at 0 , r and at a/2,3a/2. In this case, there are three decay modes with time constants, r 2 (2kA 2kg)-', r3 5 (2k~)-', and r 4 = (2kB)-l. The conditional probabilities (analogous to eq 5 and 6) are
+
PA(tl.4) = QA -k QBe-t/'z Pc(tlA) =
&A
+ 1/3e-t/78
+ QBe-t/'2 - 1/e-t/73
PB(tlA) =
&B -
&Be-t/rz
+ he-
Pg(tlB) = Q B
+
PD(tlB) =
+ &Ae-t/'z - l/e-t/'4
QB
PA(tlB) =
QA -
t/r4
QA~-~/~~
Pc(tlC) = f'~(tlA) = PB(~~B) PD(~ID)
PD(tlA) = Pg(tlC) = P,(tlC) = Pg(tlA) Pc(tlB) = P A ( ~ ~=DPc(tlD) )
PA(tlB)
(14)
Substitution of these quantities into eq 4 gives
-
in which QA = QC = 1 / 2 k ~ / (+k k~g ) and QB = QD = 'I2 QA. The expressions for Fj given by W ~ e s s n e rare ~~ substituted into eq 15, and powder averages are taken.
-
1948
The Journal of Physical Chemistry, Vol. 84, No. 15, 1080
Larsen and Strange
TABLE I: Hypothetical Set of Activation Barriers for the Various Contributions t o TIIllustrated in Figure 3e barriers in contributiona high temp low temp
contribution
apparent barrierb high temp low temp
I(E, - 2 E , ' ) I C = 8d EZC= 12d 8.8d 11.8d I-E,'l = 10 { E 2 '= 10 I-E,'I = 12 l(E2'- 2E,)I = 14 (TI I-E,I = 8 8.2 7.8 l(E,' - E , - E,)I = 10 :d,=+8E,, - E,) = 6 (T,),intrn I-E,I = 8 E, = 8 8 8 a These are the Arrhenius activation barriers for the various terms in each of the three contributions to T,. Values are given in both high-temperature and low-temperature limits. These are the Arrhenius activation barriers obtained from the approximately linear regions in Figure 3. Terms are listed in order of decreasing importance. kcal/mol. e These values are taken as typical for molecular reorientation in the plastic phase. This table illustrates that apparent energy barriers may differ greatly from fundamental barriers when complex reorientation occurs among unequal potential wells.
(T112
{
{
*
The averaged correlation function for m = 1 is then obtained
&A
cos2 Ae-t/73 + QB cos2 Ae-t/74] (16)
in which A is the angle between the rotation axis and 7, the vector joining the proton pair. Decay modes three and four transfer molecules from A to C and from B to D, respectively. Thus these modes make no contribution in , these modes simply change 7 by eq 16 if A = ~ / 2 since 180' in that case. The spectral densities are determined and 1'2 and T1,are calculated as above. The resulting expressions are
W
lo2
T
c Z
0
2
2 10 -1
4 j 3 ( 2 ~ 0 ) -!] cos2 AU4bO) + 4j4(200)11 (17)
QA COS2 AVs(W0)
QB
1/Tl, = 2y2M2 sin2 A(2QAQBsin2 A[3j2(2wl) + 5 j 2 b O ) + 2j2(2~0)1+ QA cos2 A[3j3(201) + 5j3(Wo) 2j&Wo)l
W
e
(L W
1/T1 = 4Y2WSin2 A ( ~ & A & B Sin2 A u 2 ( ~ o ) 4j2(200)1 -!-
c: u
QB COS2 A[3j4(2Wi)
1
+
5
+ 5j4(00) + 2j4(200)]) (18)
in which M2 is the "rigid lattice" second moment.26 Values of T1and TlPcalculated from eq 17 and 18 are also shown in Figure 2. The results for the fourfold reorientations are similar to those of the threefold reorientation, case 11, except that a greater decrease in T I is obtained in the former case. At the Tlp minimum, the fourfold case T1is smaller than the BPP T1by a factor of 3, and this is sufficient to fit the data for ABN.
Discussion Equations 8 and 10 are composed of an intramolecular an intermolecular contribution due contribution, ( T1)gintra, and in intermolecular contrito reorientation, (T1)ginter, bution due to change in molecular separation, (T1)2,with analogous contributions to T1,in eq 9 and 11. The various contributions are shown in Figure 3. Approximately linear behavior is found on both sides of all minima, but the gradients for the various contributions are quite different. Inspection of eq 8 shows that in the high-temperature limit (W07,2
