diffusion coefficient in terms of the known concentrations in tables 10, 19, 20 and 21 and the heat treatment time (t2-t1). ies-
a2 Lj-L,
D = --- • ... 5.3
4 (t2“t 1) LjL2
where = In (1_ci/co)
This procedure allows the approximate calculation of the minimum value of the diffusion coefficient required to have led to the reduction in core potassium observed between the as drawn and heat treated fibres.
For the fibre treated at 550°C for 8 hours using the data in tables 12, 19 and 20; C0=18%, Cj=4.2% and C2=0.2% if a constant background level of 2% potassium is assumed.
Inserting these values into equation 5.3 gives a minimum value for the diffusion coefficient of 4.8x10-14 m 2s-1. This is about two orders of magnitude less than would be expected for the interdiffusion of sodium and potassium in a silicate glass.
The high proportions of other elements in the glass would be expected to change the diffusion coefficient. The mixed-ion
effect alone is known to reduce the conductivity of glass by up to' two orders of magnitude and calcium also reduces diffusion
coefficients of monovalent cations. These results also show the minimum diffusion coefficient required to have reached the observed position. The values of C2 especially are rather
unsubstantiated and the fibres analysed may have been fully equilibrated within minutes of the start of the heat
treatment. The equation used assumes an infinite medium
which, in the context of the mobility of potassium, the fibre is not. The general background level of potassium is
also a complication.
Of more interest than the measurement of diffusion
coefficients during heat treatment would be some understanding of the diffusive processes prevalent during fibre drawing.
The time-temperature cycle during fibre drawing is represented by a complicated integral [25,26], as is the equation governing preform neck-down [59]. To provide a
mathematical model of diffusion processes during fibre drawing it would be necessary to combine Ficks equation in a
co-ordinate system shrinking according to one equation and with a time-temperature cycle governed by another. As all three of the equations usually require solution by numerical methods it seems unlikely that an analytical solution could be found to such an assembly and an empirical equation might be of
The temperature/time regime during fibre drawing is in effect measured by the diffusion behaviour of the elements in the
fibre. The temperature cycle is encoded into the effective value of the diffusion coefficient and the associated time cycle is encoded into the time term. It is impossible to
deconvolve the two terms but their sum, Dt characterises their combined effect on the diffusion behaviour of the potassium.
Using equation 2.8:-
C = C 0 (l-e-a V 4 D t)
it can be shown that:-
Dt = -a2 (5.4)
4 In (l-C/C0)
Using this equation and data from the core glass analysis (Table 12) and the as drawn fibre core analysis (Table 19) gives a value for Dt of:-
Dt = 9.4x10-11 m 2
This value is an empirical measure of the time-temperature cycle during fibre drawing and if this value of Dt were to be multiplied by the ratio of the diffusion coefficients of
potassium and another element it could be substituted back into the diffusion equations in order to estimate the
diffusion behaviour of that element during fibre drawing, at least under the drawing conditions prevailing during the manufacture of this fibre.
5.5 Investigation of Preform Draw-Down Region
The data acquired in the earlier parts of the investigation seemed to indicate, as expected, that the shape of the
preform draw down stage. To further investigate this stage of fibre manufacture, and also to check that the ratio of core size to fibre size remained the same, a section of preform end from which fibre had been drawn was supplied, along with a section of the fibre. The fibre used was an erbium doped fibre 3115.01, but the investigation was limited to the germanium doping concentration profile.
To see if the germanium concentration profile varied at all during fibre drawing a number of profiles were acquired at a variety of magnifications. These are shown in figure 55. As can be seen the shape and gradient of the outer edge of the fibre does not seem to vary during the process, though in the smaller sections the depth of the central depletion region
seems to be reduced. This latter effect is probably due to a
resolution limitation as these profiles of the preform
sections were acquired from bulk samples. In the case of the smallest sample the diameter of the sample was 2.7mm,
/
indicating a core diameter of around 30/im. The normally stated
resolution of analysis in a bulk sample is around 2/un, but this assumes a low electron energy around 20keV. These
analyses were performed in the TEM at lOOkeV in order to have directly comparable results between the fibre and bulk x-ray data. This would be expected to increase the interaction volume and hence degrade the resolution of the analysis.
To investigate any change in effective core diameter with respect to fibre diameter the data in figure 56 were acquired.
examined, but the scaling errors in the 20Ox and 30Ox
magnifications precluded their use. Both curves have a very similar shape and are very close in size. As the scale of the preform section relies for its accuracy on a measurement taken from a micrograph of a feature 6mm long the experimental
measurement errors are too large to interpret the slight size disparity as meaningful.
This reinforces the conclusion reached from examination of the Raman fibres that the preform refractive index profile persists unchanged into the fibre. It also indicates that the ratio of preform core diameter to overall diameter is the same as that of the fibre and fibre core. This means that
fibre core diameters can be determined accurately from preform data and fibre diameter measurements.
Chapter 6