Validity of James’ Assumptions

5.7.6.1. Validity of James’ Assumptions

As said above, from the scattering curves, three important parameters can be calculated as :

1. From the position of the maximum (Q m ) a rough estimate of the mean distance between the precipitates is obtained ( Φ = 2π/Q max ).

2. Hence the relative number density of the particles (i.e. the total number of particles per unit volume of the sample) is also obtained (N = Φ –3 ).

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3. The size of the particles is obtained from the Guinier plots on the higher Q side of the maxi- mum. Within the limit of validity QR G ≤ 1.2, the scattering function S Q is related to the radius

⎛⎞ 3 of gyration (R G ) according to S Q = exp ⎜ ⎜ − 2/3 ⎟ ⎟ and R G = ⎜⎟ R S , where R S is the radius

of the “spherical” particle. It has been mentioned before that a high density of precipitating nano particles is responsible for

an interference effect, which produces a maximum (Q m ) in the SANS spectrum. From this value of Q m ,

a characteristic wavelength Φ of density fluctuations has been calculated, which has been interpreted as the ‘mean distance between the nano precipitates’, and hence a relative density of nano precipitates is obtained. However, as pointed out by Guinier [47], this is only an approximate evaluation. For compari- son purposes between the different samples, this relative number density can still be used as a param- eter; this also gives an idea about the ‘order of magnitude’ of the number of nano particles [2].

The parameter Φ is not plotted, because it will show the same typer of behaviour as Q m . The parameter N plotted against crystallization time in Figures 5.24 for the samples nucleated at 634°C for

2, 4 and 8 h. It is seen from Figure 5.24 that N decreases rapidly upto 30 min, which can be considered due to larger nano particles or nuclei ‘eating’ the smaller ones before the growth process starts, i.e. a redissolution process for the smaller nano particles as the larger ones continue to grow at their expense. After 30 min, N does not change significantly upto about 60 min, and it then approaches a saturation value when there is a growth of ‘stable nuclei’. For the sample nucleated at 665°C, this change is less remarkable, since the saturation arrives at a lower time due to earlier growth process. This is obviously due to the need of a shorter time for the growth process owing to the higher nucleation temperature.

at 634°C, 4h 634°C, 8h

10 665°C, 4h

Time of crystallization (min)

665°C, 8h 0 10 20 30 40 50 60 70 80 90 100 110 120

Figure 5.24 : Variation of the number of nuclei with crystallization time for different nucleated samples.

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40 /cm Crystallization ei at

30 nucl 680° . of

20 No 10

710°C 0 Time of cystallization (min) 740°C

Figure 5.25 : The number of nuclei against crystallization time for the sample nucleated at 577°C for 19 h at three growth temperatures.

The above behaviour can be described according to the following model. The importance of two-stage heat-treatment for the study of nucleation of glass-ceramics has been emphasized by James [48], who made two assumptions as :

1. After nucleation, the glass contains an assembly of nuclei, some of which will have grown into small nano crystals, the large majority of which do not redissolve on heating to the second stage of heat-treatment.

2. The nucleation rate at the growth temperature is negligible. If the assumption (1) is correct, then by varying the growth temperature should not radically

change the number of nano crystals, which was observed after a given nucleation heat-treatment. As a matter of 'test', the sample nucleated at 577°C for a very long time of 19 h was heat-treated at three growth temperatures, i.e. 680°C, 710°C and 740°C respectively. These data are shown in Figure 5.25. Indeed, it is seen that N varies significantly at these crystallization temperatures, which ‘invalidates’ the assumption (1). For testing the assumption (2), it was observed that the blank glass showed a high level of nucleation at the growth temperature, as shown later. This ‘invalidates’ the assumption (2). There- fore, it can be said that for the study of nucleation and crystallization behaviour of 'nano particles' of magnetite, the assumptions of James [48] does not appear to be valid for a two-stage heat-treatment. In order to explain the above data, the situation is shown schematically in Figure 5.26.

According to classical theory, the size of critical nucleus increases with rising temperature. Con- sequently, a cluster of ‘critical size’ at the lower nucleation temperature (say at 550 or 577°C) will not constitute a ‘critical size’ at the growth temperature and naturally will redissolve, since they are not energetically favourable, or rather thermodynamically unstable nuclei. However, during the nucleation heat-treatment, many of the nuclei that reach the ‘critical size’ will continue to grow and will attain a size larger than the ‘corresponding critical size’ at the growth temperature, which will make them ‘sta- ble’ and also enable them to grow to larger particles at this growth temperature at longer time. Hence, the number of nano particles after a longer growth treatment would not be a good estimate of the number

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of nuclei formed at lower nucleation temperature, because of the ‘redissolution’ of many of these nuclei that were present in the ‘original’ nucleated sample.

G* Δ T 3

Figure 5.26.

A schematic diagram of the free energy against radius of the nuclei at three

different temperatures.

Now, let us say that T 1 is the nucleation temperature (T N ) and T 3 is the crystallization tempera- ture (T C ). Therefore, according to Figure 5.26, at T N the total number of nuclei (N*) with r 1 * ≥ r is stable. If we increase the temperature to T C and make them grow, the number of nuclei (N 1 *) with

r 1 *<r<r 3 * are not stable, but the number of nuclei (N 2 *), which are stable at T C , will grow at this temperature with r ≥r 3 *. This implies that at T C , i.e. say at 710°C, N* = N 1 *+N 2 * as t → 0, but as t → ∞,

N* = N 2 *, since N 1 * number of of nuclei have already redissolved during the initial time of crystallization at T C [3].

Here, it should be mentioned that the nucleation rate of magnetite will depend on the level of super-saturation at the nucleation temperature, which is related to the amount of iron remaining dis- solved in the glass matrix. As the precipitation occurs, the level of super-saturation will decrease with time, and hence the nucleation rate will itself decrease and gradually approaches zero. Thus, the number of nuclei (N) should reach a constant maximum or saturation value, as observed in this case of crystal- lization of nano particles of magnetite. Thereafter, Ostwald ripening effect may slowly take over. Thus, N should reach a ‘maximum’ simply due to all the available magnetite being eventually precipitated so that no new nuclei can be formed.

However, the saturation number of nuclei reached at a given temperature will depend on the rapidity of the overall precipitation process, which is governed by the nucleation and growth rates, and so will vary strongly with temperature. Hence, the use of two-stage heat-treatment might have resulted in the redissolution of the small nano particles of magnetite into the glass with the disappearance of certain number of particles and a partial dissolution of others. The process proposed here for the growth of nano particles is at least as ‘probable’ as the process of redissolution of sub-critical nuclei at the growth temperature. However, in practice, the situation might be more complex than the model pro- posed here, by considering the complex nature of the basalt glass [49].

It should be noted that a problems arises in applying a two-stage heat-treatment to the basalt glass for the study of nano crystallites. The two-stage method has been successfully applied to similar systems such as lithium disilicate, where the crystal and the liquid phases have the same composition, and the growth temperature used was not too high ; although not apparently applicable to a basalt glass,

221 this method is still valid for many other systems. However, in basalt glass, the two-stage heat-treatment

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was used to develop and then grow ‘large enough’ crystallites (still in the nano range) in order to give a SANS spectrum [45].