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The electron spin relaxation and reorientation correlation times of63CU(II)-bis(dimethyldithiocarbamate) in Solution
JOURNAL
OF
MAGNETIC
RESONANCE
36, 31 l-323
(1979)
F.G. HERRING AND J. M. PARK Department of Chemistry, University of British Columbia, Vancower, British Columbia, Canada V6T 1 W5 Received November 28, 1978; revised March 16, 1979 The electron spin relaxation of 63Cu(II)-bis(dimethyIdithiocarbamate) has been studied as a function of temperature in benzene, toluene, carbon disulfide, carbon tetrachloride, chloroform, and dichloromethane. In order for the fast motional spin relaxation theory to be tested the anisotropies of the spin-Hamiltonian parameters were determined from spectral simulations of the EPR in frozen toluene: these are AA = - 370 MHz, Ag = 0.0672. With these values it is found that the spin relaxation of Cu(II)-bii(diethyldithiocarbamate) is explained by the existing fast motional relaxation theories. The reorientational correlation time (Q) and spin-rotational term (a”) have been determined as functions of temperature and solvent. It has been found that (a) for a given solvent the linear relationship rs = (V&KT) + r0 is obeyed, and (b) for various solvents the linear relationship rs = Cn+ T; is obeyed; these results support previous findings that the reorientational correlation time of a molecule such as the one studied can be treated using mod&d hydrodynamic theories. We have also found that the linear plots of 7s vs (n/T) for different solvents intersect at a common “isorotational point.” This behavior implies a linear correlation between V,, and rO, and is also found in the data of Kivelson and Huang. These results support Kivelson’s formulation of V,, in terms of the stickiness factor, provided that this increases with temperature. Finafly the (Y”measured is found to conform to simple theories of spin-rotational coupling and is consistent with isotropic rotational diffusion.
INTRODUCTION
In studies of molecular motion, the rotational correlation time 72 for second-rank tensors is one of the most useful parameters which can be measured. Theoretical relationships have been established (1-8) which enable one to determine 72 from the transverse relaxation rates of magnetic resonance spectra. This approach has been successfully applied to the analysis of the electron paramagnetic resonance linewidths of paramagnetic species in solution. Freed and co-workers (2-3) have studied nitroxide spin probes with a view to investigating the motion of biological molecules, while Kivelson ef al. (4-6) have used the paramagnetic complexes copper(I1) and vanadyl bis(acetylacetonate) in investigations of the properties of simple solvents of fairly low viscosity. An earlier study (7) in this laboratory established that Cu-bis(diethyldithiocarbamate (CU(DEDC)~) could be used successfully for the same purpose. 311
0022~2364/79/120311-13$02,00/O Copyrisbbt 0 1979 by Academic Pa@. Inc. All rights of reproduction in my form reerved. Printed in Great Britain
312
HERRING
AND
PARK
Here we report the results of a systematic investigation of the temperature dependence of the EPR linewidths of 63Cu-bis(dimethyldithiocarbamate) (Cu(DMDC)z) in benzene, toluene, chloroform, dichloromethane, carbon disulfide, and carbon tetrachloride. To check for the effect of unresolved superhyperfine structure on the EPR linewidths we have used the perdeutero analog of Cu(DMDC)z (Cu(&-DMDC )2) in most of our experiments. This work is one of a series of studies using various Cu dialkyldithiocarbamates as motional probes of liquid structure. THEORETICAL
Linewidth
Analysis
The total spin Hamiltonian
for CU(DMDC)~
can be written
where the terms successively represent the electronic Zeeman, hyperfine, nuclear Zeeman, and nuclear quadrupole interactions. For Cu, S = $ and I = $, and the other symbols have their usual meanings. In liquid solution, the spin Hamiltonian reduces to the isotropic form Xi, = g&B0 * 8 + a$ * i,
El
which gives the line positions of the EPR spectrum. It has been well established (5) that when conditions for motional narrowing obtain, TY1, the transverse relaxation rate for an EPR transition is given by a polynomial in MI, the nuclear magnetic quantum number corresponding to the given transition. For the Lorentzian lines observed in solution the peak-to-peak firstderivative linewidth AB is given by
Thus the peak-to-peak linewidth is given by a polynomial in MI, the coefficients of which are functions of the field Bo, of the microwave angular frequency, wo, and of the anisotorpic parameters in &pr. For these purposes the nuclear Zeeman and quadrupole interactions can be neglected, and A and g approximated by axial tensors. It can then be shown (8) that AB(MI)=&‘+cY’+@M~+~M;
+SM;.
r41
Here 2726
lY’ = (3gop)“2 -$(E-$)
+(~)1’2FDa(wp1+w;1j(
1 -$)]I,
[5a]
EPR OF COPPER-DITHIOCARBAMATE
272
6 = (3&p)“*
QD*a(LO-’ 15
+w;l)(
1 -$I)
-$(3’“$(
COMPLEX
1 +I)],
3 L3
[5d]
where
D = (A, -A3/(6h)“*, -1 w, =
(6b)
cl& (w7J2 + 1’
These equations were derived by Bruno et al. (8), using the solution of the stochastic Liouville equation developed by Freed and co-workers (3). Similar equations have been developed by Wilson and Kivelson (5) using a Van Vleck transformation, and by us using Redfield theory with first-order wavefunctians. The residual linewidth (Y”, which is independent of T* in this formulation, is interpreted in terms of a spin-rotational interaction (9); the other terms are functions of both r2 and B0 as well as of the static Hamiltonian parameters. Our procedure for analyzing the CU(DMDC)~ EPR linewidths using these equations differs from the other reported methods and is based on the fact that (Y’, 0, ‘y, and S are ail functions of 7*. Then, if the values of (Ai1- A,) and (gll- gi) (obtained from solid-state measurements described below) are treated as tied parameters, the only independent variables in the linewidth expressions [4] are a” and 72. The four experimental linewidths can then be least-squares-fitted to the expressions obtained from Eqs. [4] and [5], and best-fit values of r2 and (Y” obtained. The calculation is facilitated by noting that the linewidth expressions are linear in (Y”, and approximately so in r2 since the pseudosecular and nonsecular terms (in p and u) are fairly small. The rms deviation between measured and fitted linewidths is typically of the order of 0.05 G, corresponding to uncertainties in r2 of 2 to 5 % . The earlier method of Tapping and Herring (7) (and presumably also of Kivelson), in which a = 1~’+ (Y”, p, ‘y, and S were determined by simple manipulation of the linewidth expressions [4], obscures the field dependence of these four parameters
314
HERRING
AND
PARK
and gives an average of their values over the B. values of the four EPR lines. The different functional dependences of (Y,p, y, and S upon B0 mean that errors of 10 to 15% can be introduced by assuming in this way that they are measured at a single intermediate field. Our procedure is to use the fitted linewidths to derive the equivalent mean calculated values of (Y,0, ‘y, S, which can then be compared with those obtained from the measured linewidths. Correlation Times
Several workers (4) have found that rotational correlation times can be represented by the hydrodynamic Stokes-Einsteih-Debye equation, 72=-j-3
V&ll
[71
where k is the Boltzmann constant, T the absolute temperature, 17 the macroscopic viscosity of the solvent, and V,, the effective volume of the solute molecule regarded as a sphere. Kivelson introduced the empirical factor K which relates Vea to the molecular volume V (usually estimated from the crystallographic volume V,) v,fl=Kv.
I31
Almost invariably K 5 1 for molecular motion, and in many cases K is close to the ratio of the slip-to-stick limits, E, developed by Hu and Zwanzig (10). Some workers (II, 12) have found that the following empirical modification of Eq. [7] gives a better fit to the data: %h+7 72= kT 0. 191 The quantity 7. has been found to have the same order of magnitude as the free rotation time for the molecule in question, but its real interpretation remains obscure. We interpreted our measured r2 values using both Eq. [S] and Eq. [9], the latter giving the better fit. The results are shown in Table 2. We also made Arrhenius-type plots of the temperature dependence of 72, 72 = 7m exp(AE/RT),
[a
the parameters for which are also listed in Table 2. EXPERIMENTAL
The 63C~(DMDC)2 used in this work was prepared from isotopically pure 63C~N03 and sodium dimethyldithiocarbamate, as described earlier. 63Cu(d6DMDC)2 was prepared from 63C~N03 and potassium perdeuterodimethyldithiocarbamate, which was synthesized by standard methods from potassium hydroxide, carbon disulfide, and dimethylamine-d6 (99 mole%, Merck Sharp and Dohm) (16).
EPR OF COPPER-DIT’HIOCARBAMATE
COMPLEX
315
Solvents were of spectrograde quality, stored over 4-A molecular sieves and purified by passage through an alumina column immediately before use. Solutions were made up in a glove bag under nitrogen, to concentrations of (l-6) x low4 M, and were thoroughly degassed on a vacuum line by several freeze-pump-thaw cycles before being sealed off under their own vapor pressure. The viscosity data required in this work were obtained from Ref. (13). EPR measurements were made using an X-band homodyne spectrometer employing a Varian 12-in. magnet with Mark II Fieldial control. Phase-sensitive detection at 100 kHz was by an Itahaco Dynatrac 391 A lock-in amplifier. Field calibration was by a proton magnetometer, the resonance frequency of which was monitored by an H-P 5246L frequency counter; the same instrument, fitted with a plug-in, Model 5256A, served to monitor the microwave frequency. Temperature control was by a Varian temperature control unit as described earlier (7). Typical experimental conditions were: microwave frequency, 9.03 GHz; power, 1.5 mW; scan time, 5 min; time constant, 0.4 sec.; modulation in the range 0.4 to 2.0 G, chosen to be 5 l/6 of the EPR linewidth. Linewidths were measured from individual scans of 25 or 50 G. We elected to measure the linewidths directly rather than determining them from relative line heights, as the latter method is particularly susceptible to systematic errors arising from any spurious broadening of the sharpest line. Many of the data were taken in digital form using a Fairchild F-8 microprocessor controlling a paper tape punch (14). The paper tapes containing the lineshape and calibration data were processed in an IBM 370/168 computer. The method of analysis was to fit the regions near the extrema of each line to cubic functions of the scan position, and then find the actual extrema analytically. The baseline crossovers were similarly found by performing a linear regression on the data near the center of the line.
RESULTS
Solid-State Measurements The hyperfine and g-tensor components were obtained from the spectra of polycrystalline samples with the aid of spectral simulations using the programs POLYFIELDS and POLYSPEC written by J. A. Hebden and J. C. Tait in these laboratories (15). Figure 1 compares the observed and simulated spectra of 63CU(d6DMDC)* in toluene at approximately 100 K, and Table 1 shows the principal values of the g and hyperfine tensors obtained from the simulated spectra of the Cu compIex in Ni(DMDC)2 at room temperature and 100 K, and in the toluene glass. The values are typical for 63Cu dithiocarbamates (16). We note that there are variations between the results obtained in Ni(DMDC)z and toluene glass. In accord with other workers (1-6) we have used the results from the toluene glass, but note that the results in Ni(DMDC)2 indicate only minimal temperature dependence of the spinHamiltonian parameters. In addition, for the purpose of linewidth analysis we have made the approximation that the spin-Hamiltonian parameters are axial, i.e., A i = %A + A,) and a. = $(gx + g,).
HERRING
316
AND PARK
h a
FIG. 1. (a) Observed and (b) simulated EPR spectra of Cu(DMDC)z in frozen toluene at -100 K. Spectrometer settings: microwave frequency, 9.0426 GHz; microwave power, 9 mW; scan width, 1000 G; scan time, 10 min; time constant, 0.4 set; modulation, 6.6 G at 100 kHz.
ANISOTROPIC
Matrix Ni(DMDC)2 Ni(DMDC)* Toluene
SPIN-HAMILTONIAN
TABLE 1 PARAMETERS 9.50 GHz
FOR
CU(DMDC)~
DETERMINED
Temperature W 300 100 100
2.0195 2.0197 2.0222
2.0140 2.0170 2.0175
2.0785 2.0795 2.0870
-117 -128 -126
-93.5 - 109 -117
-460 -487 -492
AT
EPR OF COPPER-DITHIOCARBAMATE
COMPLEX
317
Lineshapes The EPR lineshapes of CU(DMDC)~ in solution are essentially Lorentzian. For the sharpest line in toluene at approximately -20°C (the temperature of minimum linewidth for A& = $), analysis by the method of Goldman er al. (2) shows that the lineshape is consistent with a line of 75% Lorentzian character and 25% Gaussian (of the same peak-to-peak width). For broader lines the Lorentzian character is approximately 95%. There is a small but significant difference in linewidth between 63C~(DMDC)2 and 63Cu(d6-DMDC)z. Comparison with simulated lineshapes shows that this is consistent with the broadening due to a superhyperfine interaction with twelve equivalent protons each of splitting constant -0.3 G. For linewidths of 3 G this additional width is - 0.5 G, but decreases for broader lines; the simulations suggest that the p-p linewidths of the perdeutero compound are close to the true Lorentzian widths. It was possible to make empirical corrections to data obtained from the undeuterated complex which brought them into agreement with the “deuterated” values; the resulting change in r2 was typically < 5%, of the same order as the experimental error. The lineshapes of both the deuterated and undeuterated complexes show some deviation from Lorentzian character; it is possible that unresolved 14N superhyperfine structure is responsible for this, and we are considering measurement of the 14N hyperfine coupling by other experimental techniques. For the present we note that with our method of analysis these deviations from a Lorentzian lineshape represent only a small and largely systematic error in our measured r2 values, and hence will not affect the overall interpretation of our results. The concentrations of the solutions used were in the range (l-6) x 10e4 M; we established that there was no significant change in the width of the narrowest line over the concentration range low3 to 10e5 A4, so we are confident that electronic spin--spin interactions are negligible in our experiments. 15 r
o-f3o I
I
-60I
II -40
11 -20 Temp
FIG. topline
2. EPR linewidths corresponds to MI
AB for Cu(&-DMDQt = + 4.
0
0'
1
20"1
40
(‘C)
in carbon
disulfide
as functions
of temperature.
The
318
Linewidth
HERRING
AND
PARK
Studies
Figure 2 shows the EPR linewidths of 63Cu(d6-DMDC)2 in carbon disulfide as functions of temperature. In general we find that the correlation times r2 obtained from our linewidth data fit Eq. [9] over most of the range of measurement. The intercept r. is numerically of the same order as the free rotation time for 63Cu(DMDC)2, as noted by other workers, and the effective volume V,, is less than the crystallographic volume (2.16 x 1O-22 cm3) (10). In general, too, the residual linewidth LY” gives a linear plot against T/T as expected for a simple spin-rotational interaction induced by diffusional motion (9, 17). Interestingly, all the plots had a nonzero positive (Y” intercept in the range 0.3 to 0.8 G. This effect is qualitatively similar to the results of McClung and Kivelson (18) for C102, but larger than their analysis would predict. The presence of such a residual linewidth has also been noted in nitroxide spectra (1). We tried to estimate the anisotropy of the motion using the values of LY”.As a p-p linewidth (Y” is given by (20)
2 a”=--[L( WY
h go0
720
[ill
(sP),2+(sP):+(sp):]+3(6g):[~-~]],
where x, y, z are the diffusion axes and Sg = g, -g,. The correction factor < f12> /(l - h2) (20) for finite rotational jumps is expected to be of order unity (21). The 72M are related to the principal values of the (axial) rotational diffusion tensor R by T& = 6R, +M2(R,, - R,),
M=0,2
WI
(and 72 = r20), where 720 is equivalent to 72 used in Eq. [5]. If we define N = RI//R and take rzo = V,,&kT
then [ll]
becomes
II131 TABLE
PROPERTIES OF THE
c.52
CH2C12
CHC13 C7b Cd36
cc14
(cm’) 0.33 0.41 0.53 0.53 0.59 0.88
11.3 13.2 17.5 16.5 15.0 26.5
1.41 1.34 1.35 1.30 1.07 1.23
a From Eq. [7]. b From Eq. [9], with V, = 2.16 x lo-” ’ From Eq. [lo].
STUDIED,
AT
300K
b
v,~x1o22"
Solvent
2
SOLVENTS
Ice 0.65 0.62 0.63 0.60 0.50 0.57
cm3.
0.48 0.83 0.72 0.68 0.42 0.54
3.0 -4.3 -1.4 -1.9 2.4 1.5
-219.8 -222.0 -221.5 -221.5 -222.3 - 220.5
2.0453 2.0453 2.0456 2.0453 2.0453 2.0462
7oJc
AE’
(psec)
(kJ mole-‘)
0.34 0.17 0.27
0.11 0.22 0.13
8.8 10.9 10.5 12.8 10.6 13.4
EPR OF COPPER-DITHIOCARBAMATE
COMPLEX
319
20r
16G E 1.28
OB-
0.4 -
o.ol -4.0
-32
-2.4
-1.6 Beto (G)
-0.0
0.0
FIG. 3. Experimental (points) and calculated (line) values of the linewidth parameters y and /3 for CU(&-DMDC)~ in CS2.
Comparison of the slope of (Y”vs T/v from this equation with the measured slope implies that N - 1, a result which is consistent with diffusion by relatively large-angle jumps (21). Table 2 summarizes our main results for the six solvents studied. The a0 and go values were obtained by a least-squares analysis of the isotropic Hamiltonian ff]; they are relatively independent of solvent, and some of the observed variation in their values may be attributed to the fact that effective values are quoted, uncorrected for dynamic frequency shifts (8) which would alter the line positions by - 0.5 G. As shown by Freed et al. (I ), a plot of y vs p is a useful diagnostic for the validity of the theory. Figure 3 shows the observed and calculated -y-/3 plots for 63Cu(DMDC)2 in carbon disulfide. The agreement between theory and experiment is good, as was found for all the solvents studied; the deviations at large values of 72 (slow m&on) are attributable to experimental errors in the linewidths, and in some instances, to the incipient breakdown of the fast-motional tumbling approximation. In general we were able to obtain good linear plots for (Y’ vs n/T, In r2 vs l/T, and 72 vs q/T (in all cases with correlation coefficients 0.99), as the simple theories of rotational diffusion would predict. In addition to these correlations for individual solvents we have plotted 72 vs n at 300 K for all the solvents studied; this is shown in Fig. 4. As has been found by Braumann et al. (Ilb, c) there is a good correlation between n and 72, with the exception of benzene, which shows anomalous behavior in other correlations, as mentioned below. For the range of viscosities studied the correlation achieved compares favorably with that found by Kowert and Kivelson using vanadyl acetonacetonate (22). Since K as well as 7 varies with each solvent, the exact significance of this interesting correlation is not clear, but the agreement between the results for CU(DMDC)~ and CU(DEDC)~ is encouraging, and the overall trend seems to show that the motion of the probe is essentially similar in all the solvents studied, though modified in detail by the varying solute-solvent interactions.
HERRING
320 35
O----O O-.-O
AND PARK
63CuEt,dtc %Me#c
.I ,'
30 I’
,’
caq,o ,’ I’
, &,
,*-
CHC13 O,,”
25rps1
,/&CH3 cH2a2 ,o’
20-
csz,o* I’ *I’
15lo,' 5,
,’
./
.’
r’
,’
,I’
.’
a-to, .:‘&H&, .I %%-
/4H2C12 9
/’
.’
I’
,./ &6H6
(32
.’
I
0’
0.2
0.4
I T)ccp,ds
b8
FIG. 4. Values of 72 vs solvent viscosity at 300 K for Cu(DMD(&
and Cu(DEDQ.
DISCUSSION
In agreement with earlier results we find that our measured r2 values are quite well estimated by the simple hydrodynamic model of molecular reorientation, in which the effective molecular volume is less than the crystallographic volume; in this context it is encouraging to find that there is good agreement between the K values obtained from CU(DEDC)~ and CU(DMDC)~ (cf. Fig. 4). There does not, however, appear to be any clear correlation between K and solvent properties such as dipole moment, and a comparison with the results of Ref. (4) shows that even the relative K values are functions of the solute as much as of the solvent. The detailed interpretation of K is not clear. Kivelson et al. have noted that K can be estimated using the Hu and Zwanzig formulation for viscous motion with “slipping,” and Fury and Jonas (11~) have obtained good approximations to K by taking weighted means of the Hu and Zwanzig factors 8 for the three axial ratios of a completely anisotropic molecule. Using their approach, in which motion about the magnetic symmetry axis is discounted as causing no relaxation, we estimate K = 0.45 for CU(DMDC)~, in reasonable agreement with our experimental results. When 72 vs (n/T) is plotted for CU(DMDC)~ in all solvents studied, it is apparent that all the lines pass through a common point, 72 = 11 f 1 psec, and (n/T) = (1.15 hO.1) x 1O-3 cp K-‘. The implication of this is that there is an “isorotational point” (so-called by analogy with isokinetic points), which requires a linear relationship between K’ (or V,K'/R) and ro; that this is so can be seen from the plot of K’ vs 7. in Fig. 5. (This plot was chosen for clarity: the plots of r2 vs n/T can be reconstructed using the parameters in Table 2.) The linear relationship between K’ and 7. (in picoseconds) is K’=
with a correlation
-0.053~+0.61,
coefficient of 0.97.
Cl41
EPR
OF
IO t
06
COPPER-DITHIOCARBAMATE
COMPLEX
32 1
l CHC13
t
K’
t 04
t
01 -5
’
’
-3
FIG. 5. Values of 7;) vs K~ for Cu(DMDC)z and chloroform (20) (solid symbols).
’
’ -'
’
’
’
qpsec)'
in various
solvents
’ 3
(open
circles),
and Ni(MNT);
in butanol
To check whether this relation could be an artifact of our analysis we have also plotted in Fig. 5 the published data of Huang and Kivetson (20) for the compound NiS.&(CN)41-, which has a structure very similar to that of Cu(DMCb)*. If we assume with Huang and Kivelson that K - 1 .O for their chloroform data (a reasonable assumption since V,a - 250 A3) then their data give (K, rO) values of (1.0, - 4.0 psec) for chloroform and (0.37, 3.4 psec) for n-butanol, and these values are in good agreement with Eq. [14]. Thus to within experimental error these data, too, pass through the isorotational point for CU(DMDC)~. This observation also accounts for the apparent temperature dependence of K noted in Ref. (22). These findings establish the validity of Eq. [9] as a basis for analyzing reorientational correlation times; unfortunately most of the relevant data in the literature have been analyzed in terms of Eq. [7] so we are unable to make further cotmpzrrison which might illuminate the relationship between K’ and ro. One possibility which our data cannot eliminate is that the plots of Q vs q/T for different solvents become nonlinear and converge at low viscosities (rz < 10 psec), perhaps asymptotically approaching the free rotation time of the probe as 77+ 0. However, on the basis of the data presented above we offer the following interpretation of our observations. To take into account deviations from the hard sphere or slip model of solutesolvent interactions, Hoe1 and Kivelson (17) introduced the stickiness factor $ defined by
where g is the Hu and Zwanzig “slip” factor; s is zero only if K = 3 (i.e., when slip boundary conditions apply) and thus is an approximate measure of the “stickiness” of a molecular collision.
322
HERRING
AND
PARK
If we assume that 72 is given by the simple hydrodynamic given by Eq. [ 151, we have
equation [7], but with
K
Vrl rl=-P+g=s(l-E). kT We now propose that s is temperature dependent; in particular if s varies as [kT/v with [ constant, Eq. [16] has the form of Eq. [9] and we can make the identifications, K = .F, in fair agreement with experiment and r. = Vg(l - K’). This latter relation can be compared with Eq. [14] found experimentally. Equating slopes and using the crystallographic volume of CU(DMDC)~ we find 5 = 8.2 x lOlo sec/cm3, which gives s = rkT/q in the range 0.17 to 1.2 for the range of (T/q) studied, and thus is reasonably consistent with the original dbfinition of s requiring -s”/( 1 - 3) I s I 1. The constant term 0.61 in Eq. [ 141 is to be compared with the value 1 .O obtained by the above analysis; this seems to be fair agreement given that the form of Eq. [ 151 is somewhat arbitrary in the absence of a good theory to describe the stickiness of molecular collisions. The requirement of the above model that s decrease at low temperatures suggests that structuring of the solvent may influence the motion of the probe. Hoe1 and Kivelson (17) have suggested that such a mechanism might explain the nondiffusive motion they observed at high viscosities. The same process may account for the deviation from linearity of some of our 72 vs (v/T) plots at large r2, although the overall behavior of our 72 results and our (Y” vs (T/v) plots are consistent with essentially diffusive motion. In this context we note that our plots of y vs p show satisfactory agreement with theory, without requiring the spectral densities to be modified for nondiffusive reorientation in the manner found necessary by Goldman et al. (2) for peroxylaminedisulfonate. SUMMARY
AND
CONCLUSIONS
We have determined the static spin-Hamiltonian parameters for CU(DMDC)~ and show that this compound can be used as a paramagnetic motional probe of the liquid state. There is evidence for some proton superhyperfine broadening of the EPR lines, but this can be accounted for by a small empirical correction. Our results show that the modified hydrodynamic model (9) is a valid means of interpreting reorientational correlation times, while use of the simple equation [7] may obscure useful information. The linear correlation we find between K’ and r. supports Kivelson’s formulation of r in terms of a stickiness factor, provided that the latter decreases with decreasing temperature; but clearly more theoretical work is needed to put this model on a firm foundation. The Cu dithiocarbamates have advantages over the acetylacetonate probes used in earlier work (5,6) in that they have smaller linewidths, enabling greater precision to be attained and a wider range of r2 values to be studied (since r2 and (Y” may be extracted from the two smallest linewidths even when the broader lines are no longer described by simple motional narrowing theory). The reduced linewidths are a consequence of the smaller value of gll-gl in dithiocarbamates, which reduces (Y’ (and a”) and makes the linewidth variation more evident.
EPR
OF
COPPER-DITHIOCARBAMATE
32:s
COMPLEX
It is also relatively simple to modify the structure of dithiocarbamates by adding appropriate side chains, and thus to tailor a molecule to a selected geometry, without greatly altering its magnetic or chemical properties. These characteristics lay the foundation for further studies using other Cu dithiocarbamates. In particular one would hope that comparison of the motional behaviors of a series of related spin probes would shed some light on the physical significance of hitherto purely empirical quantities such as V,, and Q (23). ACKNOWLEDGMENTS The authors wish to thank from the National Research
P. Phillips for help and discussions. Council of Canada.
This
work
was funded
by a research
grant
REFERENCES
1. S. A. GOLDMAN, (1972). 2 3. 4. 5, 6. 7. 8. 9. 10.
Il.
13.
14. 1.5,
16. 17.
18. 19. 20.
21. 22. 23.
B. V. BRUNO,
C. F. POLNASZEIC,
AND
J. H. FREED,
.I.
Chem. Phys. 56, 716
S. A. GOLDMAN, G. V. BRUNO, AND J. H. FREED, J. Chem. Phys. 59,307l (1973). J. H. FREED, G. V. BRUNO AND C. F. POLNASZEK, J. Phys. Chem. 75,3385 (1971). J. HWANG, D. KIVELSON, AND W. PLACHY, J. Chem. Phys. S&l753 (1973). R. WILSON AND D. KIVELSON, J. Chem. Phys. 44,154 (1966). R. WILSON AND D. KIVELSON, J: Chem. Phys. 44,4445 (1966). F. G. HERRING AND R. L. TAPPING, Can. J. Chem. 52,4016 (1974). G. V. BRUNO, J. K. HARRINGTON, AND M. P. EASTMAN, J. Phys. Chem. 81,llll (1977). P. W. ATKINS AND K. KIVELSON, J. Chem. Phys. 44,169 (1966). G. M. Hu AND R. ZWANZIG, J. Chem. Phys. 60,4354 (1974). (a) M. FURY AND J. JONAS, J. Chem. Phys. 65,2206 (1976). (b) D. R. BAUER, J. I. BRAUMAN, AND R. PECORA, J. Amer. Chem. Sot. %,6840 (1974). (c)J. I. BRAUMAN, R. PECORA, AND D. R. BAUER, J. Chem. Phys. 59,531O (1973). N. B. VARGAFTIK, “Tables on the Thermophysical Properties of Liquids and Gases” 2nd ed., Wiley, New York, 1975; “Selected Values of Physical and Thermodynamic Properties of Hydtocatbons and Related Compounds”, Am. Petrol. Inst. Res. Proj. No. 44, 1953; T. W. PHILLIPS AND K. P. MURPHY, J. Chem. and Eng. Data 15,304 (1970), and Hydrocarbon Processing 4, 136 (1970). F. G. HERRING, J. R. MAYO, AND P. S. PHILLIPS, J. Mugn. Reson. 34,413 (1979). F. G. HERRING, C. A. MCDOWELL, AND J. C. TAIT, J. Chem. Phys. 57,4564 (1972). M. J. WEEKS AND J. P. FACKLER, Znorg. Chem. 7,2548 (1968). D. HOEL AND D. KIVELSON, .J. Chem. Phys. 62,4535 (1975). R. E. D. MCCLUNG AND D. KIVELSON, J. Chem. Phys. 49,338O (1968). F. W. B. EINSTEIN AND J. S. FIELD, Actu Crystallogr. B 30,2928 (1974). R. HUANG AND D. KIVELSON, 1. Appl. Chem. 207 (1972). D. E. O’REILLY, J. Chem. Phys. 57,885 (1972). B. KOWERT AND D. KIVELSON, J. Chem. Phys. &I,5206 (1976). F. G. HERRING AND P. S. PHILLIPS, to be published.
OF
MAGNETIC
RESONANCE
36, 31 l-323
(1979)
F.G. HERRING AND J. M. PARK Department of Chemistry, University of British Columbia, Vancower, British Columbia, Canada V6T 1 W5 Received November 28, 1978; revised March 16, 1979 The electron spin relaxation of 63Cu(II)-bis(dimethyIdithiocarbamate) has been studied as a function of temperature in benzene, toluene, carbon disulfide, carbon tetrachloride, chloroform, and dichloromethane. In order for the fast motional spin relaxation theory to be tested the anisotropies of the spin-Hamiltonian parameters were determined from spectral simulations of the EPR in frozen toluene: these are AA = - 370 MHz, Ag = 0.0672. With these values it is found that the spin relaxation of Cu(II)-bii(diethyldithiocarbamate) is explained by the existing fast motional relaxation theories. The reorientational correlation time (Q) and spin-rotational term (a”) have been determined as functions of temperature and solvent. It has been found that (a) for a given solvent the linear relationship rs = (V&KT) + r0 is obeyed, and (b) for various solvents the linear relationship rs = Cn+ T; is obeyed; these results support previous findings that the reorientational correlation time of a molecule such as the one studied can be treated using mod&d hydrodynamic theories. We have also found that the linear plots of 7s vs (n/T) for different solvents intersect at a common “isorotational point.” This behavior implies a linear correlation between V,, and rO, and is also found in the data of Kivelson and Huang. These results support Kivelson’s formulation of V,, in terms of the stickiness factor, provided that this increases with temperature. Finafly the (Y”measured is found to conform to simple theories of spin-rotational coupling and is consistent with isotropic rotational diffusion.
INTRODUCTION
In studies of molecular motion, the rotational correlation time 72 for second-rank tensors is one of the most useful parameters which can be measured. Theoretical relationships have been established (1-8) which enable one to determine 72 from the transverse relaxation rates of magnetic resonance spectra. This approach has been successfully applied to the analysis of the electron paramagnetic resonance linewidths of paramagnetic species in solution. Freed and co-workers (2-3) have studied nitroxide spin probes with a view to investigating the motion of biological molecules, while Kivelson ef al. (4-6) have used the paramagnetic complexes copper(I1) and vanadyl bis(acetylacetonate) in investigations of the properties of simple solvents of fairly low viscosity. An earlier study (7) in this laboratory established that Cu-bis(diethyldithiocarbamate (CU(DEDC)~) could be used successfully for the same purpose. 311
0022~2364/79/120311-13$02,00/O Copyrisbbt 0 1979 by Academic Pa@. Inc. All rights of reproduction in my form reerved. Printed in Great Britain
312
HERRING
AND
PARK
Here we report the results of a systematic investigation of the temperature dependence of the EPR linewidths of 63Cu-bis(dimethyldithiocarbamate) (Cu(DMDC)z) in benzene, toluene, chloroform, dichloromethane, carbon disulfide, and carbon tetrachloride. To check for the effect of unresolved superhyperfine structure on the EPR linewidths we have used the perdeutero analog of Cu(DMDC)z (Cu(&-DMDC )2) in most of our experiments. This work is one of a series of studies using various Cu dialkyldithiocarbamates as motional probes of liquid structure. THEORETICAL
Linewidth
Analysis
The total spin Hamiltonian
for CU(DMDC)~
can be written
where the terms successively represent the electronic Zeeman, hyperfine, nuclear Zeeman, and nuclear quadrupole interactions. For Cu, S = $ and I = $, and the other symbols have their usual meanings. In liquid solution, the spin Hamiltonian reduces to the isotropic form Xi, = g&B0 * 8 + a$ * i,
El
which gives the line positions of the EPR spectrum. It has been well established (5) that when conditions for motional narrowing obtain, TY1, the transverse relaxation rate for an EPR transition is given by a polynomial in MI, the nuclear magnetic quantum number corresponding to the given transition. For the Lorentzian lines observed in solution the peak-to-peak firstderivative linewidth AB is given by
Thus the peak-to-peak linewidth is given by a polynomial in MI, the coefficients of which are functions of the field Bo, of the microwave angular frequency, wo, and of the anisotorpic parameters in &pr. For these purposes the nuclear Zeeman and quadrupole interactions can be neglected, and A and g approximated by axial tensors. It can then be shown (8) that AB(MI)=&‘+cY’+@M~+~M;
+SM;.
r41
Here 2726
lY’ = (3gop)“2 -$(E-$)
+(~)1’2FDa(wp1+w;1j(
1 -$)]I,
[5a]
EPR OF COPPER-DITHIOCARBAMATE
272
6 = (3&p)“*
QD*a(LO-’ 15
+w;l)(
1 -$I)
-$(3’“$(
COMPLEX
1 +I)],
3 L3
[5d]
where
D = (A, -A3/(6h)“*, -1 w, =
(6b)
cl& (w7J2 + 1’
These equations were derived by Bruno et al. (8), using the solution of the stochastic Liouville equation developed by Freed and co-workers (3). Similar equations have been developed by Wilson and Kivelson (5) using a Van Vleck transformation, and by us using Redfield theory with first-order wavefunctians. The residual linewidth (Y”, which is independent of T* in this formulation, is interpreted in terms of a spin-rotational interaction (9); the other terms are functions of both r2 and B0 as well as of the static Hamiltonian parameters. Our procedure for analyzing the CU(DMDC)~ EPR linewidths using these equations differs from the other reported methods and is based on the fact that (Y’, 0, ‘y, and S are ail functions of 7*. Then, if the values of (Ai1- A,) and (gll- gi) (obtained from solid-state measurements described below) are treated as tied parameters, the only independent variables in the linewidth expressions [4] are a” and 72. The four experimental linewidths can then be least-squares-fitted to the expressions obtained from Eqs. [4] and [5], and best-fit values of r2 and (Y” obtained. The calculation is facilitated by noting that the linewidth expressions are linear in (Y”, and approximately so in r2 since the pseudosecular and nonsecular terms (in p and u) are fairly small. The rms deviation between measured and fitted linewidths is typically of the order of 0.05 G, corresponding to uncertainties in r2 of 2 to 5 % . The earlier method of Tapping and Herring (7) (and presumably also of Kivelson), in which a = 1~’+ (Y”, p, ‘y, and S were determined by simple manipulation of the linewidth expressions [4], obscures the field dependence of these four parameters
314
HERRING
AND
PARK
and gives an average of their values over the B. values of the four EPR lines. The different functional dependences of (Y,p, y, and S upon B0 mean that errors of 10 to 15% can be introduced by assuming in this way that they are measured at a single intermediate field. Our procedure is to use the fitted linewidths to derive the equivalent mean calculated values of (Y,0, ‘y, S, which can then be compared with those obtained from the measured linewidths. Correlation Times
Several workers (4) have found that rotational correlation times can be represented by the hydrodynamic Stokes-Einsteih-Debye equation, 72=-j-3
V&ll
[71
where k is the Boltzmann constant, T the absolute temperature, 17 the macroscopic viscosity of the solvent, and V,, the effective volume of the solute molecule regarded as a sphere. Kivelson introduced the empirical factor K which relates Vea to the molecular volume V (usually estimated from the crystallographic volume V,) v,fl=Kv.
I31
Almost invariably K 5 1 for molecular motion, and in many cases K is close to the ratio of the slip-to-stick limits, E, developed by Hu and Zwanzig (10). Some workers (II, 12) have found that the following empirical modification of Eq. [7] gives a better fit to the data: %h+7 72= kT 0. 191 The quantity 7. has been found to have the same order of magnitude as the free rotation time for the molecule in question, but its real interpretation remains obscure. We interpreted our measured r2 values using both Eq. [S] and Eq. [9], the latter giving the better fit. The results are shown in Table 2. We also made Arrhenius-type plots of the temperature dependence of 72, 72 = 7m exp(AE/RT),
[a
the parameters for which are also listed in Table 2. EXPERIMENTAL
The 63C~(DMDC)2 used in this work was prepared from isotopically pure 63C~N03 and sodium dimethyldithiocarbamate, as described earlier. 63Cu(d6DMDC)2 was prepared from 63C~N03 and potassium perdeuterodimethyldithiocarbamate, which was synthesized by standard methods from potassium hydroxide, carbon disulfide, and dimethylamine-d6 (99 mole%, Merck Sharp and Dohm) (16).
EPR OF COPPER-DIT’HIOCARBAMATE
COMPLEX
315
Solvents were of spectrograde quality, stored over 4-A molecular sieves and purified by passage through an alumina column immediately before use. Solutions were made up in a glove bag under nitrogen, to concentrations of (l-6) x low4 M, and were thoroughly degassed on a vacuum line by several freeze-pump-thaw cycles before being sealed off under their own vapor pressure. The viscosity data required in this work were obtained from Ref. (13). EPR measurements were made using an X-band homodyne spectrometer employing a Varian 12-in. magnet with Mark II Fieldial control. Phase-sensitive detection at 100 kHz was by an Itahaco Dynatrac 391 A lock-in amplifier. Field calibration was by a proton magnetometer, the resonance frequency of which was monitored by an H-P 5246L frequency counter; the same instrument, fitted with a plug-in, Model 5256A, served to monitor the microwave frequency. Temperature control was by a Varian temperature control unit as described earlier (7). Typical experimental conditions were: microwave frequency, 9.03 GHz; power, 1.5 mW; scan time, 5 min; time constant, 0.4 sec.; modulation in the range 0.4 to 2.0 G, chosen to be 5 l/6 of the EPR linewidth. Linewidths were measured from individual scans of 25 or 50 G. We elected to measure the linewidths directly rather than determining them from relative line heights, as the latter method is particularly susceptible to systematic errors arising from any spurious broadening of the sharpest line. Many of the data were taken in digital form using a Fairchild F-8 microprocessor controlling a paper tape punch (14). The paper tapes containing the lineshape and calibration data were processed in an IBM 370/168 computer. The method of analysis was to fit the regions near the extrema of each line to cubic functions of the scan position, and then find the actual extrema analytically. The baseline crossovers were similarly found by performing a linear regression on the data near the center of the line.
RESULTS
Solid-State Measurements The hyperfine and g-tensor components were obtained from the spectra of polycrystalline samples with the aid of spectral simulations using the programs POLYFIELDS and POLYSPEC written by J. A. Hebden and J. C. Tait in these laboratories (15). Figure 1 compares the observed and simulated spectra of 63CU(d6DMDC)* in toluene at approximately 100 K, and Table 1 shows the principal values of the g and hyperfine tensors obtained from the simulated spectra of the Cu compIex in Ni(DMDC)2 at room temperature and 100 K, and in the toluene glass. The values are typical for 63Cu dithiocarbamates (16). We note that there are variations between the results obtained in Ni(DMDC)z and toluene glass. In accord with other workers (1-6) we have used the results from the toluene glass, but note that the results in Ni(DMDC)2 indicate only minimal temperature dependence of the spinHamiltonian parameters. In addition, for the purpose of linewidth analysis we have made the approximation that the spin-Hamiltonian parameters are axial, i.e., A i = %A + A,) and a. = $(gx + g,).
HERRING
316
AND PARK
h a
FIG. 1. (a) Observed and (b) simulated EPR spectra of Cu(DMDC)z in frozen toluene at -100 K. Spectrometer settings: microwave frequency, 9.0426 GHz; microwave power, 9 mW; scan width, 1000 G; scan time, 10 min; time constant, 0.4 set; modulation, 6.6 G at 100 kHz.
ANISOTROPIC
Matrix Ni(DMDC)2 Ni(DMDC)* Toluene
SPIN-HAMILTONIAN
TABLE 1 PARAMETERS 9.50 GHz
FOR
CU(DMDC)~
DETERMINED
Temperature W 300 100 100
2.0195 2.0197 2.0222
2.0140 2.0170 2.0175
2.0785 2.0795 2.0870
-117 -128 -126
-93.5 - 109 -117
-460 -487 -492
AT
EPR OF COPPER-DITHIOCARBAMATE
COMPLEX
317
Lineshapes The EPR lineshapes of CU(DMDC)~ in solution are essentially Lorentzian. For the sharpest line in toluene at approximately -20°C (the temperature of minimum linewidth for A& = $), analysis by the method of Goldman er al. (2) shows that the lineshape is consistent with a line of 75% Lorentzian character and 25% Gaussian (of the same peak-to-peak width). For broader lines the Lorentzian character is approximately 95%. There is a small but significant difference in linewidth between 63C~(DMDC)2 and 63Cu(d6-DMDC)z. Comparison with simulated lineshapes shows that this is consistent with the broadening due to a superhyperfine interaction with twelve equivalent protons each of splitting constant -0.3 G. For linewidths of 3 G this additional width is - 0.5 G, but decreases for broader lines; the simulations suggest that the p-p linewidths of the perdeutero compound are close to the true Lorentzian widths. It was possible to make empirical corrections to data obtained from the undeuterated complex which brought them into agreement with the “deuterated” values; the resulting change in r2 was typically < 5%, of the same order as the experimental error. The lineshapes of both the deuterated and undeuterated complexes show some deviation from Lorentzian character; it is possible that unresolved 14N superhyperfine structure is responsible for this, and we are considering measurement of the 14N hyperfine coupling by other experimental techniques. For the present we note that with our method of analysis these deviations from a Lorentzian lineshape represent only a small and largely systematic error in our measured r2 values, and hence will not affect the overall interpretation of our results. The concentrations of the solutions used were in the range (l-6) x 10e4 M; we established that there was no significant change in the width of the narrowest line over the concentration range low3 to 10e5 A4, so we are confident that electronic spin--spin interactions are negligible in our experiments. 15 r
o-f3o I
I
-60I
II -40
11 -20 Temp
FIG. topline
2. EPR linewidths corresponds to MI
AB for Cu(&-DMDQt = + 4.
0
0'
1
20"1
40
(‘C)
in carbon
disulfide
as functions
of temperature.
The
318
Linewidth
HERRING
AND
PARK
Studies
Figure 2 shows the EPR linewidths of 63Cu(d6-DMDC)2 in carbon disulfide as functions of temperature. In general we find that the correlation times r2 obtained from our linewidth data fit Eq. [9] over most of the range of measurement. The intercept r. is numerically of the same order as the free rotation time for 63Cu(DMDC)2, as noted by other workers, and the effective volume V,, is less than the crystallographic volume (2.16 x 1O-22 cm3) (10). In general, too, the residual linewidth LY” gives a linear plot against T/T as expected for a simple spin-rotational interaction induced by diffusional motion (9, 17). Interestingly, all the plots had a nonzero positive (Y” intercept in the range 0.3 to 0.8 G. This effect is qualitatively similar to the results of McClung and Kivelson (18) for C102, but larger than their analysis would predict. The presence of such a residual linewidth has also been noted in nitroxide spectra (1). We tried to estimate the anisotropy of the motion using the values of LY”.As a p-p linewidth (Y” is given by (20)
2 a”=--[L( WY
h go0
720
[ill
(sP),2+(sP):+(sp):]+3(6g):[~-~]],
where x, y, z are the diffusion axes and Sg = g, -g,. The correction factor < f12> /(l - h2) (20) for finite rotational jumps is expected to be of order unity (21). The 72M are related to the principal values of the (axial) rotational diffusion tensor R by T& = 6R, +M2(R,, - R,),
M=0,2
WI
(and 72 = r20), where 720 is equivalent to 72 used in Eq. [5]. If we define N = RI//R and take rzo = V,,&kT
then [ll]
becomes
II131 TABLE
PROPERTIES OF THE
c.52
CH2C12
CHC13 C7b Cd36
cc14
(cm’) 0.33 0.41 0.53 0.53 0.59 0.88
11.3 13.2 17.5 16.5 15.0 26.5
1.41 1.34 1.35 1.30 1.07 1.23
a From Eq. [7]. b From Eq. [9], with V, = 2.16 x lo-” ’ From Eq. [lo].
STUDIED,
AT
300K
b
v,~x1o22"
Solvent
2
SOLVENTS
Ice 0.65 0.62 0.63 0.60 0.50 0.57
cm3.
0.48 0.83 0.72 0.68 0.42 0.54
3.0 -4.3 -1.4 -1.9 2.4 1.5
-219.8 -222.0 -221.5 -221.5 -222.3 - 220.5
2.0453 2.0453 2.0456 2.0453 2.0453 2.0462
7oJc
AE’
(psec)
(kJ mole-‘)
0.34 0.17 0.27
0.11 0.22 0.13
8.8 10.9 10.5 12.8 10.6 13.4
EPR OF COPPER-DITHIOCARBAMATE
COMPLEX
319
20r
16G E 1.28
OB-
0.4 -
o.ol -4.0
-32
-2.4
-1.6 Beto (G)
-0.0
0.0
FIG. 3. Experimental (points) and calculated (line) values of the linewidth parameters y and /3 for CU(&-DMDC)~ in CS2.
Comparison of the slope of (Y”vs T/v from this equation with the measured slope implies that N - 1, a result which is consistent with diffusion by relatively large-angle jumps (21). Table 2 summarizes our main results for the six solvents studied. The a0 and go values were obtained by a least-squares analysis of the isotropic Hamiltonian ff]; they are relatively independent of solvent, and some of the observed variation in their values may be attributed to the fact that effective values are quoted, uncorrected for dynamic frequency shifts (8) which would alter the line positions by - 0.5 G. As shown by Freed et al. (I ), a plot of y vs p is a useful diagnostic for the validity of the theory. Figure 3 shows the observed and calculated -y-/3 plots for 63Cu(DMDC)2 in carbon disulfide. The agreement between theory and experiment is good, as was found for all the solvents studied; the deviations at large values of 72 (slow m&on) are attributable to experimental errors in the linewidths, and in some instances, to the incipient breakdown of the fast-motional tumbling approximation. In general we were able to obtain good linear plots for (Y’ vs n/T, In r2 vs l/T, and 72 vs q/T (in all cases with correlation coefficients 0.99), as the simple theories of rotational diffusion would predict. In addition to these correlations for individual solvents we have plotted 72 vs n at 300 K for all the solvents studied; this is shown in Fig. 4. As has been found by Braumann et al. (Ilb, c) there is a good correlation between n and 72, with the exception of benzene, which shows anomalous behavior in other correlations, as mentioned below. For the range of viscosities studied the correlation achieved compares favorably with that found by Kowert and Kivelson using vanadyl acetonacetonate (22). Since K as well as 7 varies with each solvent, the exact significance of this interesting correlation is not clear, but the agreement between the results for CU(DMDC)~ and CU(DEDC)~ is encouraging, and the overall trend seems to show that the motion of the probe is essentially similar in all the solvents studied, though modified in detail by the varying solute-solvent interactions.
HERRING
320 35
O----O O-.-O
AND PARK
63CuEt,dtc %Me#c
.I ,'
30 I’
,’
caq,o ,’ I’
, &,
,*-
CHC13 O,,”
25rps1
,/&CH3 cH2a2 ,o’
20-
csz,o* I’ *I’
15lo,' 5,
,’
./
.’
r’
,’
,I’
.’
a-to, .:‘&H&, .I %%-
/4H2C12 9
/’
.’
I’
,./ &6H6
(32
.’
I
0’
0.2
0.4
I T)ccp,ds
b8
FIG. 4. Values of 72 vs solvent viscosity at 300 K for Cu(DMD(&
and Cu(DEDQ.
DISCUSSION
In agreement with earlier results we find that our measured r2 values are quite well estimated by the simple hydrodynamic model of molecular reorientation, in which the effective molecular volume is less than the crystallographic volume; in this context it is encouraging to find that there is good agreement between the K values obtained from CU(DEDC)~ and CU(DMDC)~ (cf. Fig. 4). There does not, however, appear to be any clear correlation between K and solvent properties such as dipole moment, and a comparison with the results of Ref. (4) shows that even the relative K values are functions of the solute as much as of the solvent. The detailed interpretation of K is not clear. Kivelson et al. have noted that K can be estimated using the Hu and Zwanzig formulation for viscous motion with “slipping,” and Fury and Jonas (11~) have obtained good approximations to K by taking weighted means of the Hu and Zwanzig factors 8 for the three axial ratios of a completely anisotropic molecule. Using their approach, in which motion about the magnetic symmetry axis is discounted as causing no relaxation, we estimate K = 0.45 for CU(DMDC)~, in reasonable agreement with our experimental results. When 72 vs (n/T) is plotted for CU(DMDC)~ in all solvents studied, it is apparent that all the lines pass through a common point, 72 = 11 f 1 psec, and (n/T) = (1.15 hO.1) x 1O-3 cp K-‘. The implication of this is that there is an “isorotational point” (so-called by analogy with isokinetic points), which requires a linear relationship between K’ (or V,K'/R) and ro; that this is so can be seen from the plot of K’ vs 7. in Fig. 5. (This plot was chosen for clarity: the plots of r2 vs n/T can be reconstructed using the parameters in Table 2.) The linear relationship between K’ and 7. (in picoseconds) is K’=
with a correlation
-0.053~+0.61,
coefficient of 0.97.
Cl41
EPR
OF
IO t
06
COPPER-DITHIOCARBAMATE
COMPLEX
32 1
l CHC13
t
K’
t 04
t
01 -5
’
’
-3
FIG. 5. Values of 7;) vs K~ for Cu(DMDC)z and chloroform (20) (solid symbols).
’
’ -'
’
’
’
qpsec)'
in various
solvents
’ 3
(open
circles),
and Ni(MNT);
in butanol
To check whether this relation could be an artifact of our analysis we have also plotted in Fig. 5 the published data of Huang and Kivetson (20) for the compound NiS.&(CN)41-, which has a structure very similar to that of Cu(DMCb)*. If we assume with Huang and Kivelson that K - 1 .O for their chloroform data (a reasonable assumption since V,a - 250 A3) then their data give (K, rO) values of (1.0, - 4.0 psec) for chloroform and (0.37, 3.4 psec) for n-butanol, and these values are in good agreement with Eq. [14]. Thus to within experimental error these data, too, pass through the isorotational point for CU(DMDC)~. This observation also accounts for the apparent temperature dependence of K noted in Ref. (22). These findings establish the validity of Eq. [9] as a basis for analyzing reorientational correlation times; unfortunately most of the relevant data in the literature have been analyzed in terms of Eq. [7] so we are unable to make further cotmpzrrison which might illuminate the relationship between K’ and ro. One possibility which our data cannot eliminate is that the plots of Q vs q/T for different solvents become nonlinear and converge at low viscosities (rz < 10 psec), perhaps asymptotically approaching the free rotation time of the probe as 77+ 0. However, on the basis of the data presented above we offer the following interpretation of our observations. To take into account deviations from the hard sphere or slip model of solutesolvent interactions, Hoe1 and Kivelson (17) introduced the stickiness factor $ defined by
where g is the Hu and Zwanzig “slip” factor; s is zero only if K = 3 (i.e., when slip boundary conditions apply) and thus is an approximate measure of the “stickiness” of a molecular collision.
322
HERRING
AND
PARK
If we assume that 72 is given by the simple hydrodynamic given by Eq. [ 151, we have
equation [7], but with
K
Vrl rl=-P+g=s(l-E). kT We now propose that s is temperature dependent; in particular if s varies as [kT/v with [ constant, Eq. [16] has the form of Eq. [9] and we can make the identifications, K = .F, in fair agreement with experiment and r. = Vg(l - K’). This latter relation can be compared with Eq. [14] found experimentally. Equating slopes and using the crystallographic volume of CU(DMDC)~ we find 5 = 8.2 x lOlo sec/cm3, which gives s = rkT/q in the range 0.17 to 1.2 for the range of (T/q) studied, and thus is reasonably consistent with the original dbfinition of s requiring -s”/( 1 - 3) I s I 1. The constant term 0.61 in Eq. [ 141 is to be compared with the value 1 .O obtained by the above analysis; this seems to be fair agreement given that the form of Eq. [ 151 is somewhat arbitrary in the absence of a good theory to describe the stickiness of molecular collisions. The requirement of the above model that s decrease at low temperatures suggests that structuring of the solvent may influence the motion of the probe. Hoe1 and Kivelson (17) have suggested that such a mechanism might explain the nondiffusive motion they observed at high viscosities. The same process may account for the deviation from linearity of some of our 72 vs (v/T) plots at large r2, although the overall behavior of our 72 results and our (Y” vs (T/v) plots are consistent with essentially diffusive motion. In this context we note that our plots of y vs p show satisfactory agreement with theory, without requiring the spectral densities to be modified for nondiffusive reorientation in the manner found necessary by Goldman et al. (2) for peroxylaminedisulfonate. SUMMARY
AND
CONCLUSIONS
We have determined the static spin-Hamiltonian parameters for CU(DMDC)~ and show that this compound can be used as a paramagnetic motional probe of the liquid state. There is evidence for some proton superhyperfine broadening of the EPR lines, but this can be accounted for by a small empirical correction. Our results show that the modified hydrodynamic model (9) is a valid means of interpreting reorientational correlation times, while use of the simple equation [7] may obscure useful information. The linear correlation we find between K’ and r. supports Kivelson’s formulation of r in terms of a stickiness factor, provided that the latter decreases with decreasing temperature; but clearly more theoretical work is needed to put this model on a firm foundation. The Cu dithiocarbamates have advantages over the acetylacetonate probes used in earlier work (5,6) in that they have smaller linewidths, enabling greater precision to be attained and a wider range of r2 values to be studied (since r2 and (Y” may be extracted from the two smallest linewidths even when the broader lines are no longer described by simple motional narrowing theory). The reduced linewidths are a consequence of the smaller value of gll-gl in dithiocarbamates, which reduces (Y’ (and a”) and makes the linewidth variation more evident.
EPR
OF
COPPER-DITHIOCARBAMATE
32:s
COMPLEX
It is also relatively simple to modify the structure of dithiocarbamates by adding appropriate side chains, and thus to tailor a molecule to a selected geometry, without greatly altering its magnetic or chemical properties. These characteristics lay the foundation for further studies using other Cu dithiocarbamates. In particular one would hope that comparison of the motional behaviors of a series of related spin probes would shed some light on the physical significance of hitherto purely empirical quantities such as V,, and Q (23). ACKNOWLEDGMENTS The authors wish to thank from the National Research
P. Phillips for help and discussions. Council of Canada.
This
work
was funded
by a research
grant
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