Growth kinetics

Kapur (1978) presented an extensive review of the kinetics of wet granulation processes. On the supposition that the growth is controlled by a single mechanism, mathematical models for the changes in size and size distribution were presented. Such models are highly useful in analysing agglomeration processes, especially in the understanding of the agglomeration mechanisms and the resulting size distributions of a particular process. If, in the course of the agglomeration, successive granule size distributions exhibit similarity characteristics when plotted against an appropriate dimensionless size, it is reasonable to assume that there has been no change in the growth mechan isms governing the process. The cumulative granule size distribution is usually normalized by plotting it against d/dso, where di0 is the median granule size. When the series of normalized distributions coincide, the distribution is said to be self-similar or self-preserving.

Published investigations on growth kinetics relate to low shear granulators such as rotating drums and there appear to be no studies in the pharmaceutical literature to establish growth kinetics. Leuenberger etal. (1990) presented a graph showing that wet granulation of lactose in a Diosna V10 high shear mixer produced a self-similar granule size distribution. In experiments with lactose-corn starch mixtures, they found that the resulting size distributions were similar to either the distribution obtained with lactose or the distribution for corn starch which means that increasing concentrations of corn starch at some point produce a change in the growth mechanism.

In agglomerate growth by nucleation it seems reasonable to postulate that the particles are well mixed, and that the collision frequency and probability of coalescence are independent of particle size. The agglomerate growth is then described by random coalescence kinetics. Kapur (1978) showed that these growth kinetics imply that the resulting size distributions are self-similar and that the rate of growth is described by the following relation:

where V(t) and V{ denote the mean agglomerate size at time t and the initial agglomerate size, respectively; X is the specific coalescence rate which, when time-invariant, implies a straight-line relationship between t and log mean granule size.

Figure 14 shows the granule size distributions obtained by granulating a 8:2 mixture of lactose and corn starch with a 4°/o gelatine solution in a Glatt WSG 15 fluidized bed. The graph demonstrates self-similar granule size distributions which fit very well to a log-normal distribution with a geometric standard deviation of about 1.9. The addition of binder solution gave rise to a granule growth rate described by a straight line correlation between In d50 and the added amount of binder solution, which is proportional to time t.

In their studies on fluidized bed granulation, Schaefer and Woerts (1978b) showed that the resulting granule size distributions are of lognormal type. The geometric standard deviation is almost constant in the course of the process but reduces when the added amount of liquid becomes relatively high. Other researchers, for example Ormos etal. (1975), have presented similar results. The growth mechanism is likely to change when a high proportion of granulating liquid has been added and all the primary particles are nucleated. In the course of the process the solvent evaporates turning the binder solution into a highly viscous, immobile liquid which

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DIMENSIONLESS GRANULE SIZE

Fig. 14 Self-similar size distributions obtained by fluidized bed granulation of a lactose-corn starch mixture 8:2 with a 4% gelatine solution. Mass of added binder solution relative to mass of solids: 0.2, O; 0.3, •; 0.35, V; 0.45, Produced from data supplied from Schaefer and

Woerts (1978b).

may facilitate further growth by coalescence of agglomerates and even balling because shear forces are absent.

Wet granulation in a high shear mixer proceeds by nucleation and coalescence mechanisms, the latter being dominant when the liquid saturation has been increased beyond a certain limit. In the case of calcium hydrogen phosphate, growth by nucleation is supposed to dominate until the liquid saturation has been increased to about 80% and above, c.f. Fig. 12.

Figure 15 demonstrates self-similarity of the size distributions achieved in the wet massing phase of granulating calcium hydrogen phosphate (dgw = 8.5 ¿on) in a high shear mixer. It is apparent that in the wet massing phase there is a change in size distribution from 3 to 6 min. The difference between the two distributions can be seen in context with the changing liquid saturation in the range of 55-80% at 0 and 3 min and above 80% at 6 and 8 min wet massing. The graph reflects, therefore, the change in growth mechanism from nucleation (0 to 3 min) into coalescence (6 to 8 min). Experiments with binder solutions of Kollidon 90, 25 and VA64 in varying concentrations show self-similar size distributions identical with the distributions shown in Fig. 15 and with the same distinction between

LL tI C3

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 DIMENSIONLESS GRANULE SIZE

Fig. 15 Self-similar size distributions obtained by wet granulation in a Fielder PMAT 25 of calcium hydrogen phosphate with 10% m/m solution of hydrolysed gelatine. Wet massing times: 0 and 3 min, O; 6 and 8 min, •. Solid lines: log-normal distributions with sg = 2.01 and 1.48. Data from Holm eta/. (1993).

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 DIMENSIONLESS GRANULE SIZE

Fig. 15 Self-similar size distributions obtained by wet granulation in a Fielder PMAT 25 of calcium hydrogen phosphate with 10% m/m solution of hydrolysed gelatine. Wet massing times: 0 and 3 min, O; 6 and 8 min, •. Solid lines: log-normal distributions with sg = 2.01 and 1.48. Data from Holm eta/. (1993).

the two stages according to the level of the liquid saturation.

The solid lines drawn in Fig. 15 are the estimated log-normal distributions. The geometric standard deviation sg of the agglomerates in the nucleation stage of the process is about 2. The size distributions develop in parallel in the log-probability plot. When the growth mechanism changes into coalescence of agglomerates, the geometric standard deviation should change to value 1.5 and, at the same time, the growth rate should increase. In experimental work, higher values of sg will normally be observed because, as shown in Fig. 16, the experimental size distributions over-represent larger agglomerates.

Nucleation of particles and coalescence of agglomerates involve similar mechanisms, as depicted in Fig. 3. Figure 15 demonstrates, however, that the size distribution achieved by the nucleation mechanism is wider than that achieved by coalescence of agglomerates. Growth by nucleation is by the kinetics of random coalescence which means that the collision frequency and the probability of coalescence are independent of size (Kapur, 1978). In contrast, growth by coalescence of agglomerates proceeds by the kinetics xi <a

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70 SO 30

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400 7001000 2000

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Granule size (jam)

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IDS'

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200 400 700 1000

Granule size (nm)

Fig. 16 Log-probability plot of the cumulative weight distribution during wet massing in Lodige FM 50. Granulation of calcium hydrogen phosphate with a 15% solution of Kollidon VA64 in water, a, Impeller speed 200 r.p.m., chopper speed 3000 r.p.m. b, Impeller speed 200 r.p.m., no chopper action. Wet massing times: 0 min, (O); 1 min, (x); 3 min, A; 6 min, □. Reproduced with permission from Schaefer eta/. (1987), Pharm. Ind. 49, 297-304. Editio Cantor, Denmark.

of non-random coalescence where the probability of coalescence is dependent on size as described earlier. It is a consequence of the non-random kinetics that a graph of the mean granule size against time is a straight line in a log-log plot (Kapur, 1978).

Figure 16 shows the granule size distributions obtained by granulating calcium hydrogen phosphate (dgw = 21 /¿m) in a Lodige FM 50 mixer, which is a high shear mixer of horizontal type. It appears that the action of the chopper has a great influence on the granule size distribution. With the chopper inactive, the resulting size distributions are log-normal in distribution. The size changes are characterized by self-similar distributions and a geometric standard deviation of about 2.8. Schaefer etal. (1987) showed that the intragranular porosity remained almost constant (about 29%) during wet massing. This means that the liquid saturation was also constant. With the chopper active, the intragranular porosity was reduced from 29 to 19% in the course of the process. The agglomerates must, therefore, have been saturated with liquid, which gives rise to growth by coalescence of agglomerates. Accordingly, Fig. 16 shows a change at 3 and 6min into narrower size distributions. The straight-line part of the two distributions is described by a geometric standard deviation of about 1.7. The sg of the entire distribution is greater (2.18 and 2.02, respectively) because the distributions are tailed upwards.

It is the author's experience that the granule size distributions shown in Fig. 16 are typical for wet granulation of cohesive powders in high shear mixers. When the nucleation mechanism dominates, the resulting granules have a wide size distribution, and the growth rate is described by a straight-line relationship between time and log mean granule size. When growth by coalescence of agglomerates becomes the dominating mechanism, the size distribution of the agglomerates is reduced because the probability of coalescence becomes size dependent (high for small agglomerates). The growth rate is then described by a straight-line relationship between time and mean granule size in a log-log plot.

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