where μ is the dynamic viscosity, and dv/dx is the velocity gradient. For the velocity
gradient dv is given by the maximum injector velocity, dvi. It is more difficult to estimate
dx, which depends on the thickness of the boundary layer around the bubble. Tramper et
al. (1986) used the diameter of a cell for dx. For normal medium with a dynamic
viscosity equal to that of water and a cell diameter of 10 μm a shear stress of 8.8 N·m−^2 is
obtained, which is of the same order of magnitude as the shear stresses reported to
damage cells. However, in these reports exposure times are of the order of minutes to
hours while here a bubble is formed within 0.1 seconds, in which case much higher shear
stresses may be required to damage cells.
Another possible mechanism of cell death may be through direct contact between cells
and the bubble surface. During bubble formation the bubble surface is rapidly expanding.
Absorption of surfactants on to the surface as well as distribution of these surfactants
over the surface requires time. As a consequence, less surfactant molecules are present at
the newly formed surface and the surface tension of a rapidly expanding surface (the
dynamic surface tension) is higher than the static surface tension of a surface that is in
equilibrium with the bulk liquid. Thus, depending on the speed of bubble formation and
surfactant adsorption, cells may come in contact with a bubble surface that is not or only
partly covered with surfactant molecules. As shown by Jordan et al. (1994) this may lead
to direct cell death or adsorption of the cells to the bubble. Michaels et al. (1995b) and
Jordan et al. (1994) looked at the speed of Pluronic F68 adsorption onto newly formed
bubbles as a function of concentration. Both authors showed that at a concentration
higher than 0.01% adsorption took place very rapidly (<0.1 ms) at the source. This would
mean that in order for the surface creation to be faster than the adsorption process, bubble
formation should be very rapid.
Murhammer and Goochee (1990b) showed that the type of sparger influenced cell
death by air-bubbles. They used two air-lift reactors with different sparger designs, a
membrane gas distributor and a porous stainless-steel gas distributor. For the membrane
distributor cells could be grown in the presence of 0.2% Pluronic F68, while for the
stainless-steel distributor no growth occurred at 0.2% Pluronic F68. However, using 0.2%
Pluronic L35, which has a lower molecular weight, or higher concentrations of Pluronic
F68 (0.5%), growth was possible. They proposed that the increased pressure drop present
for the stainless-steel distributor, which results in a higher gas entrance rate, was
responsible for increased turbulence in the sparger region causing an increase in cell
damage. The protective effect of higher Pluronic concentrations could then be due to
either extra dampening of the hydrodynamic forces liberated at bubble formation, or
more adsorption of Pluronic to the cells. However, the concentration at which saturation
of adsorption to cells occurs is much lower than the 0.2% used here. Consequently,
addition of extra Pluronic would be useless. Dey and Emery (1999) showed that the
effect of Pluronic on the severity of bubble break-up was minimal after a concentration of
0.025%. Although this can not be directly translated to the situation of bubble formation,
it suggests that the hypothesis of extra dampening is also unlikely. Possibly, in the
stainless-steel distributor Pluronic adsorption cannot keep track with the very rapid
formation of new bubble surface causing attachment of cells to the surface leading either
to direct cell death or cell-bubble attachment and cell death later at the surface. Higher
Pluronic concentrations lead to higher speeds of adsorption, which may explain the
protection offered when more Pluronic F68 is added. Finally, another explanation is
Lethal effects of bubbles in animal-cell culture 467