7250
J. Chem. Phys., Vol. 112, No. 16, 22 April 2000
O’Connor, Ginsburg, and Blum
With this modification, H (t) must be multiplied by a func-
tion, h(t;c ,), which is given in the Appendix for both T1
and T1Q . For T1 ,h(t;c ,) is essentially 1 for all values of
jumps, the middle will decay roughly twice as slow as the
horns or edges. Thus, without any further complications such
as axis motion, the mechanisms should be distinguishable in
both the fast (T1) and slow (T1Q) regimes.
t in which M (t) is observable and hence it can be ignored.
Often, methyl-d3 powder patterns show -dependent re-
laxation behavior between that of rotational diffusion and
jumping.13,29 For reasons discussed later, we combine the
mechanisms in a serial or homogeneous manner where all
methyls switch between jumping and diffusing often during
the experiment, but do not change their rate or c , as defined
above. With this model each deuteron has the same average
environment and an effective rate, 1/Tx(c ,, f ), results
from the weighted sum of the individual rates,
For T1Q , the h(t;c ,) term should be included. The exact
form of H (t) with this modification will be shown below.
For these two models, is the only relaxation rate pa-
rameter dependent on orientation. Thus, for simplicity, T1
and T1Q will now be subscripted by instead of to denote
the relaxation rate of a resonance. The values of range from
0° to 90° and are related to through Eq. ͑9͒. Figure 1
illustrates the T1 differences between the two models for the
ϭ0° ͑edge͒ and 90° ͑maxima͒ resonances. Because the
ϭ0° and 90° frequencies have the smallest and largest T1
values, the curves in Fig. 1 and the corresponding ratios of
T1,90 to T1,0 demonstrate the maximum T1- dependence of
1/T ,, f ͒ϭf/T ,͒ϩ 1Ϫf ͒/T ,͒, ͑16͒
͑
͑
͑
͑
x
c
xj
c
xd c
where Tx refers to either T1 or T1Q ; the subscripts d and j
refer to the rotational diffusion and jump mechanisms, re-
spectively; and f represents the fraction of time the deuteron
jumps and is equivalent to the parameter used by Torchia
and Szabo3 to generalize their models. Txd(c ,) and
Txj(c ,) are derived with the substitutions of Eqs. ͑7͒ and
͑9͒, respectively, into Eq. ͑1͒ or ͑2͒.
the models. For slow rotation ( cϾ1), the models have
0
similar dependencies and are difficult to distinguish with
T1 data. In this region, the T1,90 /T1,0 ratios equal 1.5 and 1.4
for the rotational diffusion and jump models, respectively.
As the reorientation rate increases and passes through the T1
minimum ( cϭ1), the T1,90 /T1,0 ratio for rotational dif-
0
For this homogeneous combination model, H (t) be-
fusion first increases from 1.5 to 2.2 and then decreases to 1
comes
in the fast motion regime ( cϽ1). For the jump model
0
the ratio goes smoothly from 1.4 ͑slow͒ to 2 ͑fast͒. Again,
the models appear too similar around the minimum to dis-
cern by T1 anisotropy alone. However, when the motion is
fast, the entire powder pattern relaxes at the same rate ͑no
dependence͒ for the rotational diffusion model in contrast to
the jump model which has a T1,90 /T1,0 ratio of 2. This dif-
ference can be observed experimentally.
H
t; , f ͒ϭw h t•f; , ͒exp Ϫt/T , , f ͒
͑
͑
͓
͑
͔
c
ϩ
c
ϩ
x
c
ϩ
ϩ 1Ϫw ͒h t•f; , ͒
͑
͑
ϩ
c
Ϫ
ϫexp Ϫt/T , , f ͒ ,
͑17͒
͓
͑
͔
x
c
Ϫ
where the Ϯ’s are the angles from the ‘‘Ϯ’’ transitions cor-
responding to ͓determined from Eq. ͑9͔͒; wϩ is the frac-
tion of the intensity from the ‘‘ϩ’’ transition at which can
be calculated if the line shape is known; and, h(t•f;c ,) is
the function incorporating the dependence with t multi-
plied by f. Once again, h(t•f;c ,) is essentially 1 for T1
A plot of T1Q vs would look similar to that of T1 vs
c
in that the rates of the two mechanisms are relatively
c
similar, but the -dependence is different. For T1Q , the
middle region of the powder pattern, with ϭ54.7°, also
becomes a distinguishing factor. The jump mechanism has
and given by Eqs. ͑A5͒ and ͑A6͒ of the Appendix for T1Q
.
Finally, for this model of methyl reorientation, including
a distribution of c’s results in
T
1Q,0 :T1Q,54.7 :T1Q,90 ratios of 6:9:4, independent of c . For
rotational diffusion, these ratios vary with and are
c
12:27:16 in the fast region. With slow diffusive motion, the
T1Q,54.7 lies between ϭ0° and 90° values, as with T1 ;
however, the -dependence is reversed, as compared to T1 ,
with the ϭ0° resonance relaxing 4.5 times slower than the
ϭ90° resonance. The difference in these ratios makes T1Q
experiments most suitable for distinguishing between the
mechanisms in the slow motion regime.
H
t; ,, f ͒ϭ d G ; ,͒H t; , f ͒, ͑18͒
͑ ͑
m c c m c
͑
͵
where H (t;c , f ) is the sum of only two angles, Eq. ͑17͒.
Relative to Fig. 1, the distribution flattens and raises the
curves as its width increases. This effect extends both the
time scales associated with the minimum and the at which
c
the models have significantly different T1 and T1Q behavior.
If the model applies, Eqs. ͑4͒–͑5͒ using either Eq. ͑17͒ with
two parameters or Eq. ͑18͒ with three parameters should fit
the relaxation behavior of the entire methyl-d3 powder pat-
tern. These fits are in contrast to using Eq. ͑3͒, which would
have 3 or 4 parameters for each . It should be noted that
successful fits only suggest the validity of the model and do
not prove it.
The T1 and T1Q behavior of the two mechanisms can be
divided into that of fast ( cϽ1) and slow ( cϾ1) me-
0
0
thyl reorientation and summarized in relation to the changes
of the powder pattern with an increasing delay time, t. In the
fast regime, T1Q relaxation is similar for both mechanisms
with the middle region (ϭ54.7°) decaying the slowest. For
T1 in the fast regime, the rotational diffusion mechanism
predicts that the entire powder pattern relaxes at the same
rate and the jump mechanism predicts that the horns (
ϭ90°) will decay twice as slow as the edges (ϭ0°). For
slow reorientation, T1 relaxation is similar for both mecha-
nisms with the horns decaying roughly 50% slower than the
edges. For T1Q in the slow regime, with rotational diffusion
the edges will decay 4.5 times slower than the horns and, for
EXPERIMENT
The synthesis and characterization of PMPS-d3 was de-
scribed previously.30 For PAMS-d3 , trideuteromethylstyrene
(AMS-d3) was synthesized by reacting trideuteromethylphe-
164.107.254.56 On: Mon, 08 Dec 2014 18:29:27
O'Connor, Robert D.
