J. Phys. Chem. 1982, 86, 421-428
However, in order to improve substantially the desription of the electrical double layer, one has to look for more refined models. By a careful combination of quantum chemical and statistical mechanical methods a full molecular description of the electrical double layer seems to be within reach.
T
absolute temperature permittivity in vacuum relative permittivity coordinate of one particle coordinates of all particles one-particle distribution function concentration of species i at zero potential two-particle distribution function two-particle correlation function
€0
Acknowledgment. We are indebted to Drs. B. Halle, B. Jonsson, N. Mazer, and H. Wennerstrom for valuable discussions and linguistic criticism.
gij
Nomenclature
8
U
ull
Urn EMC Epg
EM A” Sex ui uij
9
d k R
configurational integral configurational energy ion-ion configurational energy ion-micelle configurational energy
(u)
configurational energy defined according to eq 11 self-energy of micelle excess free energy excess entropy external potential pair potential mean electrostatic potential electrostatic potential Boltzmann’s constant universal gas constant
r ri
c2+ C-
P08m
Pmir
B
421
.- 1
unit charge valency charging parameter surface charge density ionic radius micellar radius cell radius radial distance radial distance of ion i distance between ions i and j aggregation number mean concentration of species i mean concentration of amphiphile mean concentration of neutral salt concentration of monovalent counterion concentration of divalent counterion concentration of monovalent co-ion osmotic pressure mixed pressure term degree of ion association
Nuclear and Electron Spin Relaxation in Copper and Nickel Tutton Salts J. M. McNally and R. W. Krelllck’ h p a r f m n t of Chemlsby, Unlversny of Rochester, Rochester, New York 14627 (Received: August 4, 1981; I n Finei Form: October 5, 1081)
We have measured the proton spin-lattice relaxation times of polycrystalline samples of zinc potassium Tutton salts which were doped with varying concentrations of either copper potassium Tutton salt or nickel potatsium Tutton salt. Rotating-frameexperimentsin which the nuclear spins were locked at the magic angle wem condudad to eliminate the first-order contribution of spin diffusion to nuclear relaxation. The dominant mechanism for nuclear relaxation is the direct dipolar interaction between the nuclear spins and the impurity electron spins, and one is able to determine the correlation time for electron spin relaxation through analysis of the data. In samples of higher concentrations, spin exchange was found to be the dominant mechanism for relaxation for both the nickel and the copper salts. The exchange interaction contribuh to the relaxation times out to relatively large metal-metal separations and was found to vary exponentially with this separation. In very dilute samplep the mechanism for isolated electron spin relaxation involves direct energy transfer to lattice phonone. The nickel Tutton salt was found to couple more efficiently to the lattice and has a shorter spin-lattice relaxation time than the copper sample.
Introduction Electron spin relaxation times of transition-metal complexes are generally very short and difficult or impossible to measure with electron spin resonance spectroscopy. A n alternative approach for determination of these relaxation times involves measurement of nuclear relaxation times in samples in which the dominant mechanism for nuclear relaxation is direct dipolar coupling between the nuclear and electron spins.’ When this is the case, experimental data from measurements of nuclear relaxation times can (1) J. M. McNally and R. W . Kreilick, Chem. Phys. Lett., 79, 534 (1981).
be directly analyzed to yield the electron spin relaxation time. Nuclear spin-lattice relaxation in a pure diamagnetic solid in which molecular motion is restricted is a very inefficient process, and extremely long relaxation times are observed. If the sample is doped with a low concentration of a paramagnetic species, one observes a dramatic d e crease in the relaxation times as nuclear spin energy can be efficiently coupled to the paramagnetic impurity ions and then to the surrounding lattice.2 The nuclear and electron spin systems are coupled by two mechanism^.^ The first mechanism involves n direct ( 2 ) N. Bloembergen, Physica, 15,386(1949).
QQ22-3654l82l2Q86-Q421$01.25IQ 0 1982 American Chemical Society
422
The Journal of Physical Chemistty, Vol. 86, No. 3, 1982
dipole-dipole interaction between the electron spin of the paramagnetic impurity ions and all of the nuclear spins in the surrounding diamagnetic matrix. This interaction varies with the separation of the nuclear spins from the Paramagnetic impurity site (1/ri:), and it is necessary to sum over all possible interactions to obtain an equation relating the time rate of change of the macroscopic nuclear moment and the correlation time for electron spin relaxation. This equation predicts that the In of the macroscopic magnetization will evolve as the square root of time after the equilibrium population has been altered by an rf pulse. The second mechanism for transfer of energy between the nuclear and electron spin systems involves a strong dipolar coupling between the electron spin and a nearby nuclear spin followed by equilibration of energy in the nuclear spin system via nuclear spin diffusion. When this mechanism is dominant, one predicts that the In of the magnetization will exhibit a linear dependence on time. In many samples both relaxation mechanisms contribute to the observed decay of magnetization in a normal 18Oo-7-9O0 or 9Oo-7-9O0 pulse experiment, and it is impossible to analyze the experimental data to obtain either the diffusion constant or the correlation time for electron spin relaxation. Nuclear spin diffusion arises from proton-proton dipole-dipole interactions which are proportional to p/rm3(l - 3 cos2 en). In this expression rn, is the proton-proton separation and 8, is the angle between this vector and the external magnetic field. If 0, can be set to 54.7O (magic angle) and held at this angle during decay of the nuclear magnetization, one can efficiently eliminate fmt-order spin diffusion as a mechanism for relaxati~n.~When this is the case, the dominant mechanism for nuclear relaxation is the direct electron-nuclear dipolar interaction and one can readily analyze nuclear relaxation data to obtain correlation times for electron spin relaxation. One can obtain diffusion-free relaxation experimentally by setting the magnetic field slightly off resonance, applying a 54.7O puke, waiting a period long enough for the spins to precess into the x’-z’ plane, and then applying a locking pulse which h collinear with the magnetization vector. The spins are held at the magic angle by the locking pulse, and decay of the magnetization while the locking pulse is on is due mainly to direct relaxation. If one monitors the intensity of the magnetization as a function of the length of the locking pulse, one can effectively obtain the time rate of change of the macroscopic nuclear magnetization due to direct relaxation. We have used this technique to determine correlation times for electron spin relaxation for a series of concentrations of copper and nickel Tutton salts which were doped into zinc Tutton salt. X-ray studies of these mat e r i a l ~show ~ that the copper and nickel complexes isomorphically substitute for the zinc complex. The known crystal structure of these materials allows one to calculate average separations of the paramagnetic ions as a function of concentration. In this paper we report experimental data on both nuclear and electron spin relaxation times for a series of concentrations of the two paramagnetic ions. The experimental data are analyzed to yield exchange energies in the more concentrated samples and electron (3) I. J. Lowe and D. Tse, Phys. Rev., 166, 279 (1968); D. Tse and I. J. Lowe, ibid., 166, 292 (1968). (4) D. Tse and S. R. Hartmann, Phys. Reo. Lett. 21,511 (1968); N. A Lin and S. R. Hartmann, Phys. Rev. B, 5, 4079 (1973). (5) M. Tardy and G. Pannetier, Bull. SOC.Chirn. Fr., 3651 (1968); J. E. Weidenborner,Acta Crystallogr., 14,63 (1961);H. E. Swaneon, H. F. McMurdie, M. C. Morris, and E. H. Evans, Natl. Bur. Stand. ( U S ) , Monogr., &5, 1 (1969).
McNally and Krelllck
spin-lattice relaxation times in the dilute samples. The magnitude of the exchange energy is found to vary exponentially with the average separation of the paramagnetic ions. Theoretical Section The basic equations for relaxation of a given nuclear spin by the odd electron of a paramagnetic impurity at a distance rivfrom the nucleus have been derived by Lowe and Tse3 for the case in which the nucleus relaxes in the external Zeeman field (l/Tu) and the case in which the spins relax in the presence of a 90° resonant spin-locking field 1/Tb*. This treatment assumes that spin diffusion has been eliminated as an effective relaxation mechanism. If one averages the angular terms in the dipolar equations and notes that the electron Zeeman frequency (up)is large compared to the nuclear Zeeman frequency (wo) which is normally large compared to the Zeeman frequency in the spin-locking field (we), one obtains r
r
(l/tc) = (l/te)
+ (l/Tle)
In these expressions te is the correlation time for the exchange interaction and Tl, is the electron spin-lattice relaxation time. The summation is carried out over all of the paramagnetic sites v. The angle between the effective field in the rotating frame and the Zeeman field (5) is given by 5 = tan-’ [rH,/(w - w0)l
H1= rf magnetic field
(2)
and an expression for the nuclear relaxation time at any given 5 can be found as
($)
cos2 f =-+-
Tli
sin2 5 Tlt
(3)
For nuclei spin-locked at the magic angle (5 = 54.74O), the expression for the relaxation time is c
In order to determine the time evolution of the macroscopic magnetization, one must average over all of the nuclei in the sample. The equation for the time evolution of the magnetization at position ri is given by (5)
The Journal of Physical Chemistty, Vol. 86, No. 3, 1982 423
Spln Relaxation In Copper and Nickel Tutton Salts
'i E
with solutions of the form
a
The exponential recovery of the magnetization will vary from site to site reflecting the dependence of individual Tf(riV)'son the proximity of a paramagnetic ion. The macroscopic moment which reflects the ensemble average of all of the individual spins is not generally expected to show an exponential recovery except when each nuclear site has an adjacent paramagnetic ion. A technique for averaging over all of the nuclear sites to obtain an analytic solution for the time evolution of the macroscopic moment has been developed by Silbernagel.e The sample is assumed to contain No sites for paramagnetic ions with a fraction c randomly occupied. If a given site k is occupied (probability c), the contribution to the total recovery will be exp(-t/TIP(rik), while if unoccupied (probability 1- c) the contribution is unity. If one takes a weighted average of these contributions, the time evolution of the macroscopic moment can be expressed as
M ( t ) = II((1- C) + c exp[-t/Tf(riV)II V
riv
> r~
- e-t/Tlp(rlv))]
riv > ro
L 1. 00
0. 50
SORT
1. 5 0
2. 00
2. 50
(TIME. S E C ) * l B 1
Figure 1. Tlme evolution of the magnetization calculated wlth the discrete product sum (eq 7, symbols) and the approximate analytical equatlons (eq Q, solid lines) for a serles of dlfferent excluded volumes.
(7)
In this expression, ro is the radius of the excluded volume element for nuclei in close proximity to a paramagnetic ion, where the nuclear resonance is shifted out of the range of experimental observation. In the limit of low concentrations of paramagnetic ions, the product sum of eq 7 can be expanded to yield
M ( t ) N exp[-cC(l
c = o 02
tlt
+
h
(8)
V
The summation in this expression can be replaced by a volume integration in the limit of low concentration to yield the following analytic expression: I
1
I 2. 0
1. 0 SORT
In the limit in which C t / r $ >> 1, eq 9 can be simplified, yielding the following long-term limiting solution:
M ( t ) 0: e~p[-(t/Tf))'/~]
(10)
where
(l/Tf)1/2 = Y3rs/2No~C1/2 and C is defined by eq 4. When the effects of spin diffusion are negligible, this equation predicts that In M will decay as the square root of time. The unknown in this equation are the concentration of paramagnetic ions and the correlation time for electron spin relaxation. For samples of known concentration, it can be used to evaluate correlation times over an extremely large range of values (10" - 10-11 8). The metal Tutton salts which have been studied have a known crystal structure, and it is therefore possible to carry out the exact product-sum calculation (eq 7) for the time dependence of M and compare these results to those predictad from the approximate analytical solution (eq 9). A program was written which randomly populated the metal sites and evaluated the product-sum equation for all protons within a radius of 70 A. The correlation time (6)
M.Ft. McHenry and B.G.Silbernagel,Phye. Rev. B, 6,2968 (1972).
I 3. 0
8
2
4. 0
(TIME. S E C > * 1 E 1
Flgurr 2. Plot of the time evolution of the magnetlratlon calculated with the dlscrete product sum (eq 7, symbols) and the kqptenn limitlng sdution (eq 10,solid Unes) for a serlee of dlfferent concentratlone (mole fraction): (A) c = 0.213,t , = 1.25 X 10" 8; (B) c = 0.101,t c = l.6Q X 10" 8; (C)c = 0.450,t c = 1.88 X 10" s; (D) c = 0.016, t , = 2.2,x IO-^ 8.
for electron relaxation was obtained from experimental data through use of the analytical expression. The calculations were carried out to determine the effect of the magnitude of the excluded volume and concentration on the time rate of decay of the magnetization. Plots of log M vs. the square root of time for a series of different excluded volumes and concentrations are given in Figures 1and 2. The symbols in Figure 1designate the discrete product-sum calculation (eq 7) while the solid line represents the low-concentration analytic solution (eq 9). Excluded volumes with radii from 1.59 to 12.56 A gave linear plots at longer times with excellent agreement between the two calculations. There is some deviation from linearity at short times with ro = 12.56 A, and the analytical solution yields values which are slightly smaller than those of the product s u m at longer times. The difference in the values at longer t i m e probably reflects the limited size of the data set used in the calculation. The actual excluded volume in the sample is determined by the magnitude of dipolar shifts of signals from nuclei which are near the metal atom. In the case of the metal Tutton salts, only water molecules in the first coordination sphere can experience a dipolar shift large enough to remove the signals from the central
4U The Journal of Physical Chemistry, Vol. 86, No. 3, 1982
resonance line. These protons are all within 6 A of the metal ion where excellent agreement is obtained between ths product-sum and analytical solutions. Figure 2 shows plota of log M vs. the square root of time for a series of different metal ion concentrations (mole fraction 0.016-0.213). The discrete product-sum calculation is again denoted by the symbols while the solid lines were calcuktd with the long-term limiting solution (eq 10). Excellent agreement between the exact and approximate d y t i c a l solutions is obtained in all cases with linear evolution of In M with the square root of time. The d y t i c a l eolution was derived with the assumption of low concentration, but this plot indicates that this equation can be used to evaluate correlation times for samples with m o d ~ a t e l yhigh concentrations without serious error. The intensity of the free induction decay at the termination of an infinitely long locking pulse ( M ( w ) )depends on the angle at which the spins are locked ([), the magnitude of the nuclear Zeeman frequency, and the correlation time producta (wite) which are found in the relaxation equations. In the presence of a strong radio frequency field, the magnetization precesses around the effective field in the rotating frame and the projection of the magnetid o n along the effective field position determines M (a). The general equation for M ( w ) for any locking field angle (€1 is given by M(O)C, cos [ M(w) = (11) C, cos2 [ + C, sin2 5
McNaliy and Kreilick
l
j
l
-:I.
/
l
l
0
LOG [CORRELATION
(b)
l
I
l
TIME
l
l
~
-7.0
-9. 0
-5. 0
(Tt)l
I si
"
0
8 0
while the expression for the magic-angle spin lock is
2. 0
4. 0 THETA
6. 0
8. 0
(DEGREES) * 1 8 - '
Figure 3. (a) Plot of eq 12 for a series of proton Zeeman frequencies. (b) Plot of eq 11 for a series of correlation times for electron spin relaxation.
when wpte >> 1is assumed. In these expressions M(0) is the mrrlmum value of the magnetization which is observed after a single x / 2 pulse. The ratio M ( w )f M ( 0 ) vs. the correlation time for electron spin relaxation is plotted in Figure 3a for three different proton Zeeman frequencies. When wet, s1D2
Flgurr 5. Plot of the in of magnetization vs. time from a 90°-~-900 pulse sequence for a mde fractbn of 0.0423 nickel Tutton salt in zinc Tutton salt at 77 K.
I 18. 0
LOG
I 19.0
I 20. 0
Q
21. 0
CONCENTRATION (IONWCC)
Flgurr 6. Plot of log T , vs. log concentration for the copper (0)and nickel (0)Tutton salts diluted in a Zn Tutton matrix. These results were obtained from 18Oo-~-9O0and 90°-~-900 experiments.
field. Figure 5 shows a plot of the time dependence of the decay of the magnetization for a sample of Ni Tutton salt in Zn Tutton salt. The data from these experiments reflect the combined effect of the direct dipolar interaction and spin diffusion on nuclear relaxation. At short times the plots are nonlinear, indicating that direct relaxation is the dominant mechanism, while at longer times the plots become linear as spin diffusion becomes the chief mechanism for relaxation. These curves can be computer simulated (solid line in Figure 5) by appropriately mixing decay constants for the two relaxation mechanisms: MJt) = A exp(-t/TJdiffusion + B[-(t/7'Jdi,~'21
(13) This type of calculation gave good results for the diffusional contribution which dominates at long times, but there is a large error in the direct contribution as the mixing coefficient B is very small for samples with lower concentrations. The diffusional relaxation times agreed well with times obtained by neglecting the initial points in the decay curve and determination of slopes from the portion of the curve which evolved linearly with time. Figure 6 is a log-log plot of the concentration dependence of the diffusional component of TIfor both the copper and nickel Tutton salts. The diffusional contribution changes linearly with concentration for both types of samples with
426
McNaliy and Kreilick
The Journal of Physical Chemistfy, Vol. 86,No. 3, 1982
I
P
E J t
e
-
\c
\D L
l
8
l
.QO
1
.
.
t
1
"\.
1 3. 08
SORT
-
_ u l -
5 00
7 . 00
2 ( - 1 M E . S E C ) +10
Flgure 7. Plot of In of magnetlratiin vs. the square root of time obtained from magicangle spin-lock experlments for varlous mole fractions of nickel Tutton in zinc Tutton: (A) c = 0.0186, (B) c = 0.0423, (C) c = 0.0961, (D) c = 0.186.
L O G CONCENTRATION
