6 � b2 Deff, the time taken to diffuse to the nearest neighbour pore, although it is less pronounced than the peak for a regular lattice As 6 increases the diffraction
7.5. ESR EXPERIMENTS 105 Observation Parameter
time 6.
I
Deff
10.0 J-LS 1 3 p,m 3.2 x 1 0-5 m2 s-1 14.0 p,s 20 j.Lffi 2.2 X 10-5 m2 s-1 16.5 p,s 26 p,m 2 . 1 x 1 0-5 m2 s-1 1 9.5 p,s 26 j.Lffi 2 . 1 x 1 0-5 m2 s-1 best fit 25 p,m 2 . 1 X 10-5 m2 s-1Table 7 . 6 : Parameters used in fits to the data shown in Figures 7.15 and 7 . 1 6 using the permeable
wall model of eqn (7.4 1 ) .
Inverting the data
All the models presented thus far depend specifically on the assumed exponential distribution of well sizes. In principle it is possible to transform the echo attenu ation function to reveal the appropriate weight distribution function,
p(
l), given a knowledge of the specific nature of the electron-barrier interaction. The simplest model to use is the time independent diffraction model where the averaged propaga tor is given by a sum of one-dimensional well autocorrelation functions. The inverse Fourier transform of the6.. =
19 p,s data is shown in Figure 7. 1 7 a. This longest time was used as the model inherently assumes the long time limit approximation applies to all wells. The data is analysed to reveal a corresponding distribution functionPetr(l),
shown in Figure 7. 1 7b, using both a non-negative least squares (NNLS) and a least distance (LDP) algorithm in which the kernel is taken to be the expected triangular autocorrelation function. The exponential weight function,p( l)
is also using the valueI =
25J.Lm.The difference in the results of the NNLS and LDP algorithms serves to empha sise the difficulties inherent in an inversion of the averaged propagator in order to find the distribution of autocorrelation functions. The direct inversion also assumes perfectly reflecting walls, no boundary relaxation and the long time limit for the trapped electrons and therefore more weight is placed on the echo attenuation data results shown above. However the analysis shown here shows the possibility of such an approach in other more well defined systems.
Two population model
Finally we have attempted to fit the data presented here with a model comprising two populations of electrons, one of weight x which is free to diffuse without restrictions with self diffusion coefficient D and one of weight 1 - x which is completely confined. The best fit to D and x is shown in Figure 7.18 and clearly gives a poor representation of the data.
(a)
1
•0.6
•N'
'-" •IQ..
....,0.4
• •0.2
0
0
20
40
60
80
100
z(!-!m)
(b)
0.6
0.06
0.5
0.05
0.4
0.04
,..-._ ,..-._ - '-"0.3
� -0.03 �
0.2
0.02
0.1
0.01
0
0
0 20 40 60 80 1 00 l(�-Lm)
Figure 7 . 1 7 : (a) The spatial Fourier transform of the E(q) data for � = 19.5 J-LS and 6 = 2.6
J-LS along with (b), the corresponding distribution function p(l) obtained using the NNLS (filled
squares) , left hand scale) and LDP (open circles, right hand scale) algorithms of Lawson and Hanson in which the kernel is taken to be the expected triangular autocorrelation function. The solid curve (right hand scale) shows the exponential weight function p(l), calculated using the value
7.6. S UMMARY
0 1 0000
q
300001
0.01
1 07
Figure 7 . 1 8 : Fit to the E(q) data of Figure 7.16 (� = 10, 14, 16.5 and 1 9 . 5 Jl-S as filled circles, open
circles, filled squares and open squares respectively) using a simple two site model in which one fraction of electrons is able to diffuse freely (D = 3 x 10-5 m2 s- 1 , x = 0.9 ) while the remaining
electrons are confined. Descending amplitude corresponds to ascending 6.. The data are poorly represented by such a model.
7.6
Summary
The experiments on the polystyrene sphere array clearly demonstrate the versatil ity and usefulness of taking a diffractive approach to PGSE NMR. Experiments performed on the polystyrene sphere array showed the coherence peak predicted at
q = b-1 . These results were the first published[38] PGSE data showing an increase in echo signal strength with an increase in gradient strength. A similar interference effect has recently been reported in an oil-water emulsion system by Balinov et. al[75] . The shape of the attenuation curve was consistent with predictions of the pore glass, pore hopping model. Structural parameters pertaining to the sample were revealed[67] . Through non-linear least square fitting, the analytic theory was fitted to echo attenuation data for each time 6. . The parameters were consistent with expected results but with short times
6..
giving smaller fitted values for b.The shape of the pore voids formed by the polystyrene spheres were also analysed to test the use of the long time limit spherical pore structure function to model
IS0(q) l2 • The results showed a best-fit average spherical pore size of a "' 0.43b,
broadly consistent with the PGSE data results which found a "' 0.33b[70] .
I also demonstrated the usefulness of the pore hopping model outlined in this chapter, and the parallel barrier model outlined in Chapter 6, by applying them to the analysis of PGSE ESR data obtained from the conduction electrons in a
(FA)2PF6 sample where the electron diffusion data clearly involved restricted diffu sion. I fitted the data to two models, one based on the parallel relaxing wall model and the second on pore hopping. The pore hopping approach proved useful in find ing simple, analytic expressions for E( q ). I wrote computer programs to fit these models to the ESR data in order to test both the consistency of the mechanism of each model in simulating the observed data, and to obtain structural parameters relevant to the system under investigation. The pore hopping model proved better at modelling the data although neither method was conclusive. The ESR data was also inverted to reveal the average propagator. The corresponding distribution func-
tion
Peff( l)
was found using two different algorithms in which the kernel was taken to be the expected triangular autocorrelation function.In Chapter 8 the restricted diffusion exhibited by entangled polymer chains will be investigated.