Technical Implications of Organic Syntheses in Ball Mills - Ball Milling Towards Green Synthesis: Applications, Projects, Challenges - page 255

and, therefore, more energy is provided, increasing the bulk temperature
and along with that the rate constant of the reaction. The fact that the re-
lation between n MB and yield is not similar for all reactions is due to the
different requirements regarding activation energy. Accordingly, a discrete
amount of energy has to be provided before conversion starts. For reactions
like the cross- 72 and homo-coupling procedure 62 higher activation energies
are needed compared to the two oxidation procedures.

10.3.4 Size of Milling Balls
Another important feature that is related to n MB is the diameter of the
grinding bodies d MB . Generally, the milling ball diameter depends on the
volume of the milling vessel and milling balls for materials listed in
Table 10.1 are available with sizes from 0.25 to 30 mm. The stress energy
transferred to the milling feed (E stress.feed ) showed a cubic proportionality to
d MB for stirred media mills operated under wet conditions, as indicated in
Eq. (10.1). 65,75,76 However, it is questionable if this relation is transferrable
to the conditions of organic synthesis in ball mills. The most significant
differences are the mill type and the type of processing: wet milling for
particle refinement and dry milling for organic syntheses. Thus, a more
general consideration has to be made regarding the influence of d MB on E kin
or E stress.feed . In general the stress energy transferred to the milling feed
(E stress.feed ) is proportional to the impact energy of the milling balls (E impact )
and the collision frequency (n collision ) (Eq. 10.5): 55
E stress,feed pE impact v collision

(10.5)

where E stress,feed (J) is the stress energy transferred to the feed material,
E impact (J) is the impact energy of colliding grinding bodies and n collision (Hz)
is the collision frequency.
Regarding E impact the direct proportionality to E kin and therefore to the
mass of the moving bodies in the milling beaker is obvious (Eq. 10.4). This
part of E stress.feed is mainly determined by the density and the number of the
milling balls. The collision frequency is expressed as the number of col-
lisions in a given time range (Eq. 10.6). Decreasing n MB , the ball-to-powder
ratio decreases, alongside with a decreased collision probability and thus
decreasing yield (Figure 10.7): 54
n collision / number of collisions
Dt

(10 : 6)

In general, the stress energy (E) is considered as an energy distribution
summing up all stress energies, SE, from all single stress events (Eq. 10.7):
E ¼ X

N

SE i ¼ SE SF t reaction (10 : 7)

i ¼ 1

where E (J) is the stress energy, SE (J) is the stress energy for a single event
and SF (Hz) is the stress frequency n collision .