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Effect of Surface Elasticity on Ostwald Ripening in Emulsions Marcel B. J. Meinders,†,‡ William Kloek,†,‡,§ and Ton van Vliet*,†,| Wageningen Centre for Food Sciences, Diedenweg 20, Wageningen, The Netherlands, Agrotechnological Research Institute ATO, Bornsesteeg 59, Wageningen, The Netherlands, and Wageningen University, Food Physics Group, P.O. Box 8129, 6700 EV Wageningen, The Netherlands Received March 1, 2000. In Final Form: March 7, 2001 The coarsening of droplets in an emulsion with a size distribution that initially is given by the LifshitzSlyozov-Wagner (LSW) distribution is studied by means of numerical calculations taking into account elastic interfacial behavior. Droplets smaller than the critical radius will shrink while droplets larger than the critical radius will grow. For a zero interfacial elasticity the stationary LSW distribution is obtained and its coarsening rate matches theoretical values. The critical droplet radius, number-averaged droplet radius, and volume-surface-averaged droplet radius increase with time. Low interfacial elasticity with respect to initial interfacial tension causes the initial LSW distribution to become bimodal. The size distribution of the coarsening peak can still be described by the LSW distribution. The smaller peak that accumulates in time has an average radius that depends on the ratio between the interfacial elastic modulus and the interfacial tension. For large ratios (E/σ > 1), the system goes within a short time to a stable situation without changes in particle size with time.
Introduction Dispersions such as emulsions and foams are sensitive to coarsening phenomena like coalescence and Ostwald ripening, since their thermodynamically most stable state is the completely demixed one. Coalescence is often considered as the most important destabilization mechanism leading to coarsening of dispersions. However, coalescence can often be prevented by a careful choice of stabilizers and is mainly of interest during processing. On the other hand, Ostwald ripening will continuously occur as soon as curved interfaces are present. The curvature of particles causes higher solubilities of the dispersed phase at the particle boundary compared to in the bulk or near to larger particles. The concentration gradient of the dispersed phase in the continuous phase causes large particles to grow at the expense of smaller particles. The solubility of a particle with radius a is approximated (according to Kelvin):
R R Cs(a) ) C∞ exp ≈ C∞ 1 + a a
()
(2u/31- 1)
81eu2 exp
(
)
(1)
where C∞ is the bulk solubility and R is the so-called capillary length, a parameter that among others depends on the interfacial tension. The Ostwald ripening equations have been treated independently by Lifshitz and Slyozov1 and by Wagner2 for the case of infinite dilute dispersions. The result of the LSW analyses can be summarized as follows:3 (1) In the stationary region, the shape of the particle distribution function f∞(u) is time invariant and is given by * Corresponding author. E-mail:
[email protected]. † Wageningen Centre for Food Sciences. ‡ Agrotechnological Research Institute ATO. § Present affiliation: DMV-International, P.O. Box 13, 5460 BA Veghel, The Netherlands. | Wageningen University. (1) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35. (2) Wagner, C. Z. Electrochem. 1961, 65, 581.
f∞(u) )
, 0 e u e 1.5 and x3 32(u + 3)7/3(1.5 - u)11/3 f∞(u) ) 0, u > 1.5 (2)
where e is exp(1) and u is the ratio of the particle radius a(t) and the critical radius ac(t). Both radii are timedependent. Particles with a radius larger than ac will grow, particles smaller than ac will shrink, and if the radius equals ac the particle will neither grow nor shrink. A property of this distribution function is that the numberaveraged radius equals the critical radius. So during coarsening the critical radius also increases. (2) The cube of the number-averaged particle size increases linearly with time according to
ω)
d〈a3〉 4 ) RDC∞ dt 9
(3)
where D is the diffusion coefficient of dispersed phase molecules in the continuous phase. Note that following the current definition C∞ is in mole per mole while in the rest of the paper it is in number of molecules per unit volume. Much (theoretical) work on Ostwald ripening is focused on the effect of volume fraction. At finite volume fractions the diffusion field of one particle will interfere with the diffusion field around another particle. This local environment effect can be accounted for by using sink-source terms for every particle.4,5 Numerical simulations showed that the size distribution becomes broader and the change in the cube of the number-averaged radius with time is faster with increasing volume fraction. For the infinitely (3) Kabalnov, A. S.; Shchukin, E. D. Adv. Colloid Interface Sci. 1992, 389, 69. (4) Enomoto, Y.; Kawasaki, K.; Tokuyama, M. Acta Metall. 1987, 4, 907. (5) Voorhees, P. W.; Glicksman, M. E. Acta Metall. 1984, 32, 2013.
10.1021/la000304z CCC: $20.00 © 2001 American Chemical Society Published on Web 05/30/2001
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concentrated case the square of the number-averaged particle radius instead of the cube increases linearly with time and also a stationary state is obtained in which the distribution function can be described with an analytical function similar to the LSW distribution. These results of the effect of volume fraction on coarsening rates and the form of the distribution function were also confirmed by simple numerical simulations of the coarsening process.6 Another factor that is hardly considered in emulsion coarsening literature is the effect of the interfacial rheology. The interfacial tension σ is incorporated in the coarsening equations in the capillary length parameter R. The latter is given by
R)
2vdσ kbT
(4)
where vd is the molecular volume of the dispersed phase, kd is Boltzmann’s constant, and T is temperature. A lowering of the interfacial tension will lead to a smaller capillary length and therefore a lower solubility at the droplet boundary. This will cause a lower coarsening rate, as is also clear from eq 3, which shows that the coarsening rate is proportional to the capillary length. Almost all studies on Ostwald ripening assume the interfacial tension to be constant, but this may not be true, especially when considering ripening of emulsions. From theoretical calculations on Ostwald ripening in a solid dispersion, it appeared that mechanical stresses around a particle can strongly influence the ripening rates.7 The mechanical stresses around a solid particle can be compared with interfacial stresses at an emulsion droplet boundary. Emulsion droplets are normally stabilized by a surfactant or amphifilic polymer. The adsorbing surfactant causes a lowering of the interfacial tension which makes it more easy to disrupt droplets, but it also provides the droplet steric or electrostatic repulsion to stabilize it against coalescence. The interfacial tension is closely related to the amount of adsorbed stabilizer; the interfacial tension decreases with an increase in surface load. The surface load is directly related to the bulk concentration of the stabilizer although, dependent on the type of stabilizer, many other effects such as history are of importance. If the interfacial area is increased and the stabilizer does not desorb, the surface load will decrease and the interfacial tension will become higher. In this case the interface is said to be elastic and its elastic behavior can be characterized by an interfacial dilation elasticity modulus E. This modulus is defined as the change in interfacial tension divided by the relative change in interfacial area:
dσ dσ ) E) dA/A d ln A
(5)
which is defined by the difference between the interfacial tension at equilibrium (σe) and the interfacial tension during steady-state expansion/compression (σ) divided by the surface deformation rate:
ηd )
σ - σe d ln A with d ln A˙ ) d ln A˙ dt
(6)
Elastic interfaces can be formed by proteins that adsorb in a train-loop-like manner at the interface. It is not likely that several “trains” from one protein molecule desorb cooperatively. However, there will always be some viscous behavior due to intra- or intermolecular rearrangements. The interface will become more elastic if the proteins will form extra (intra-) molecular interactions such as S-Sbridges. An example of such a stabilizer is egg-white protein. Another way to make an elastic interface is by particles that have a certain affinity for the continuous phase/dispersed phase interface and therefore adsorb at the interface and stabilize it by Pickering stabilization.8 According to Lucassen8 this would be an efficient way to stabilize emulsion droplets against Ostwald ripening. The aim of this paper is to describe the effect of surface elasticity on the coarsening of emulsions. Simulation Simulation Model. The simulations are based on a method described by De Smet et al.9 Consider N dispersed particles having radii ai with i ) 1, ..., N, which are initially (t ) 0) distributed according to the LSW-distribution (eq 2). Each particle consists of ni ) 4/3πa3i /νd molecules. The number of molecules dni that particle i will loose or gain in time dt can be described by a combination of Fick’s law and Kelvin’s equation (eq 1)
(
( ))
Ri dni ) 4πDai(Cm - Cs) ) 4πDai Cm - C∞ 1 + dt ai
(7)
where Cm is the concentration in the medium. This equation may be rewritten by introducing the critical drop radius ac for a droplet that is in equilibrium with the medium (ac(Cm - C∞) ) RC∞), giving
(
)
Rc R i dni ) 4πDC∞ai dt ac ai
(8)
The number of molecules ni,j+1 in particle i at time j + 1 is given by
ni,j+1 ) ni,j +
dni,j ∆tj dt
(9)
If the stabilizer desorbs on compression of the interface within the time scale of deformation, the surface load will remain the same and therefore the interfacial tension will remain approximately the same. This ideal behavior is often observed for low molecular surfactants. Whether or not a stabilizer will desorb on compression of the interface depends on the time scale of deformation and the time scale of adsorption/desorption. The dependence of the interfacial tension on the time scale of deformation can be described by the interfacial dilatational viscosity ηd,
where ∆tj is the time step. Assuming the total number of exchanged particles during each time step to be equal to zero (Σi dni,j /dt ) 0) allows us to calculate the unknown concentration in the medium Cm or the critical radius ac. This corresponds to the situation where the total number of molecules and the number of dissolved molecules in the continuous phase are constant. De Smet et al.9 considered the case for a constant interfacial tension σ0, independent of the dispersed particle radii. This allows eq 9 to be written as
(6) Yarranton, H. W.; Masliyah, J. H. J. Colloid Interface Sci. 1997, 196, 157. (7) Kawasaki, M.; Enomoto, Y. Physica A 1988, 150, 463.
(8) Lucassen, J. In Anionic Surfactants: physical chemistry of surfactant action; Dekker: New York, 1981. (9) De Smet, Y.; Deriemaeker, L.; Finsy, R. Langmuir 1997, 13, 6884.
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ni,j+1 ) ni,j + MjPi,j with
Mj ) 4πDC∞R0∆tj Pi,j )
(10)
ai,j -1 ac,j
(11)
Σiai,j Nj
(12)
and
ac,j )
with Nj the total number of droplets present at time j. The critical radius depends on the time step, since it is directly related to the drop size distribution. The real time can be converted to a time parameter T by summation of all Mj over all time steps. This time is proportional to the number of exchanged molecules.
T ) 4πDC∞R0t
ai,j σi,j ) σ0 + 2E ln ai,0
(14)
This equation and eq 5 show that the interfacial tension of an elastic interface will decrease on shrinkage of a droplet. This will cause a decrease of the capillary length R and therefore reduce the coarsening rate. The effect of surface elasticity can be incorporated in the ripening equations through Ri,j, given by
(
)
2vd ai,j σ + 2E ln kbT 0 ai,0
(15)
Preservation of mass during each time step yields for the concentration in the medium Cm
Σi(ai,j + Ri,j) Cm,j ) C∞ Σiai,j
2. Calculation of the various types of average droplet radii a10, a21, and a32, where akl is given by
(16)
Simulation Approach. Calculations are carried out for 5000 droplets (Nmax ) 5000) with a mean radius of ac,0 ) 20, a molecular volume of νd ) 0.664, and ac,0/R0 ) 3. The calculation involved the following steps: 1. Generation of Nmax droplets according to a certain normalized distribution function f(u), where u is the ratio of the droplet radius ai and the number-averaged radius a10. The radius of droplet i is determined by solving
i ∫0uf(u) ∂u ) 1 - Nmax for u by taking i to range from 1 to Nmax with a step size of 1. Note that u is a continuous parameter but that i has an integer value. This results in an array of Nmax droplets in which the droplets are sorted from large (index i ) 1) to small (i ) Nmax). N, which is the number of droplets that have not obtained their final size, is set to Nmax.
()
1/(k-l)
Nmax
(13)
To incorporate the effect of surface elasticity, a function needs to be derived that describes the interfacial tension of a droplet as a function of its radius. This can be achieved by rewriting eq 5 in terms of particle radius a. Substitution of A ) 4πa2 and dA ) 8πa and using the condition that, at t ) 0, σ ) σ0 and a ) a0 yield the following relation:
Ri,j )
Figure 1. Evolution of the droplet size distribution of an emulsion with time due to Ostwald ripening for a low-viscous interface (b ) 0). The initial distribution is the LSW-distribution.
akl )
∑ i)1
aki
Nmax
∑ i)1
(17)
ali
3. Calculation of the concentration in the medium Cm/ C∞ by using eq 16. 4. Calculation of dni/dT using eqs 7 and 13. 5. Calculation of the time step ∆Τj. The number of exchanged molecules for each particle ∆ni,j ) (dni/dT)∆Tj cannot be larger than ni,j - ni,min, where ni,j - ni,min ) 4/ π a 3 3 i,min/νd is the minimum number of molecules in droplet i with a minimum size ai,j,min ) ai,0 exp[-σ0/(2E)]. The minimal size follows from eq 15 and the fact that when Ri ) 0, dni/dT g 0 (eq 8). The time step ∆Tj is now taken as the minimum value so that ni,j+1 g ni,min, for all i. If ∆Tj is larger than 104, the latter is taken to be the time step. 6. Calculation of ni,j+1, ai,j+1, and Tj+1 ) Tj + ∆Tj. 7. Steps 2-6 are repeated until the number of particles equals 500 (E ) 0 case) or until a steady situation is reached Simulation Results Simulations have been carried out for initial distributions (T ) 0) which are described by the LSW-distribution. The interfacial elasticity/interfacial tension ratio (b ) E/σ0) was set to 0 for the case of no surface elasticity and was chosen to be larger than 0 for the presence of surface elasticity. Both cases will be treated separately. No Surface Elasticity. Figure 1 shows the evolution of the droplet size distribution as a function of dimensionless time for b ) 0. It shows that the radius distribution broadens with time and that the surface area under the distribution curve decreases with time due to the fact that for b ) 0 (the viscous case) droplets may dissolve completely. Figure 2 shows the corresponding normalized frequency distributions for b ) 0, in which the size-axis is made dimensionless by dividing the droplet size by the critical radius (eq 12). The normalization procedure consists of dividing the frequency of every size class by its integral z ) ∫u0 f(u) ∂u. Figure 2 shows that the curves at various coarsening times overlap each other and correspond to the initial LSW-distribution.
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Meinders et al. Table 1. Theoretical and Simulated Scaling Behavior of the Cubed Averaged Radii with Dimensionless Time for b ) 0 radius, akl
slope ckla in 3 + cklT a3kl ) akl,0
simulated (ckl/cc)1/3
theoretical
a0 a10 a21 a32
0.03529 0.03529 0.04036 0.04432
1 1 1.046 1.079
1 1 1.046 1.080
a The parameter c is the slope of the line between the cube of kl the kl-averaged particle radius and T, whereas cc is the slope of the line between the cube of ac and T.
Figure 2. Normalized size distribution of an emulsion undergoing Ostwald ripening plotted as a function of the dimensionless radius a/ac at various dimensionless times. It shows the stationary LSW-distribution (see text).
Figure 3. Evolution of the various droplet radii (a) and cubed radii with dimensionless time during coarsening of a LSWdistribution for a low-viscous interface (b ) 0).
Figure 3 shows the corresponding evolution of the various average droplet radii and cubed radii for b ) 0. All average radii increased with time, and the critical radii are equal to the number-averaged droplet radii during the coarsening process. All cubed radii changed linearly with time, as can be seen in Figure 3b. The simulated and theoretical slopes are compiled in Table 1. The theoretical slopes were calculated from the LSWdistribution function f(u), since this shape is maintained during coarsening. The kl-averaged dimensionless droplet radius ukl ) akl/ac is given by the following expression:
∫01.5f(u)uk ∂u ukl ) 1.5 ∫0 f(u)ul ∂u
(18)
The theoretical slope equals (ckl/cc)1/3 because in this case a10 equals ac. The LSW-theory predicts a coarsening rate given by eq 3. Multiplying this with dt/dT (eq 13) results in a dimensionless theoretical coarsening rate d[a310]/dT of 1/(9π) ) 0.03537. This is very close to the calculated coarsening rate. Also, the calculated and theoretical slopes in the scaling relation between the cubed radii and dimensionless time are very close to each other. With Surface Elasticity. Figure 4 shows the evolution of the frequency size distribution for different b ) E/σ0 ratios, ranging from b ) 0.25 to b ) 5 (the values are indicated in the figure). The initial distributions are the same as the initial distribution shown in Figure 1. Also shown in the figure (dotted lines) are the size distributions that correspond to the situation were all particles have their minimum sizes ai,j,min ) ai,0 exp[1/(2b)]. In this case the σi ) Ri ) 0 (eqs 14 and 15), and from eq 8 it follows that dni/dt g 0. It is noted that in reality a zero interfacial tension will not be reached. If the interfacial tension approaches zero, inherently also E will go to zero. Furthermore, on strong compression of a monolayer, collapse of the adsorbed layer will occur, resulting in an effective lower E. Moreover, for very low droplet radii, also bending elasticity has to be taken into account, which leads to an apparent higher surface tension. For the low b ) E/σ0 ) 0.25 ratio, it nicely shows the development of a bimodal size distribution. These types of size distributions are often observed in daily practice. With time, the initial distribution peak, with its center around 23, becomes broader, its integral becomes smaller, and it moves to larger radius. At low radius, a new peak is formed which increases in time toward the minimal size distributions. Initially the small peak builds up at low radius due to the fact that the smallest particles will reach their minimal size first, and it gradually evolves toward larger radii. Contrary to the nonelastic case, the droplets do not disappear but approximate their minimal radii. In principle, the case E ) 0 can also be considered as bimodal, consisting of the normal coarsening peak at high radius and a δ-peak at zero radius in which the particles are accumulated that have reached their minimal size (a ) 0) and consequently have disappeared. Introduction of approximately zero interfacial tension will cause the δ-peak to appear near a ) 0, because the particles do not disappear anymore. Now slightly increasing the interfacial tension will result in a broadening of the peak and in a shift toward larger radius. For low b ) E/σ0 ratios, the final size distribution will approximate the minimal size distribution. In this case Ri ) Rc ) 0 for every droplet, and according to eq 8, the system is stable. For medium b ) E/σ0 ratios on the order of 0.5, a similar trend can be observed: part of the initial peak coarsens to larger radii and part of the peak accumulates at smaller
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Figure 5. Evolution of various average droplet sizes as a function of the dimensionless time for various surface elasticitysurface tension ratios (indicated). The dotted line equals the mean droplet radius for b ) 0 and is the same as that shown in Figure 3. The lowest curve, indicated as b ) 0 (corr), is the mean radius divided by the initial number of droplets present.
Figure 4. Evolution of the droplet size distribution of an emulsion with an elastic interface due to Ostwald ripening for b ) 0.25 (a), b ) 0.5 (b), b ) 1 (c), b ) 2 (d), and b ) 5 (e). The initial distribution is the LSW-distribution. The dimensional times corresponding to the distributions are indicated. Also shown are the size distributions that correspond to the situation where all particles have their minimum sizes ai,j,min ) ai,0 exp[1/(2b)] (dotted lines).
radii. The system evolves toward a stable size distribution that deviates from the minimal size distribution, with peak position and peak width between the initial and final size distributions. For large b ) E/σ0 ratios (b > 1) the system evolves quickly toward a stable situation with a size distribution similar to the initial one. For infinitely large E it follows that Rc/ac ) R/a is a constant, and from eq 8 it follows that the system is stable. For disproportionation between two equal sized droplets, Lucassen8 derived that it stops if E > σ/2. Here it is found that for a LSW droplet size distribution roughly E > σ is sufficient. Figure 5a shows the number-averaged radii (a10) as a function of dimensionless time, for various E/σ0 ratios. The number-averaged radius a10 increases with time for E/σ0 ) 0 (dotted line) but decreases in time for E/σ0 > 0. This difference between a nonelastic and an elastic interface can be explained by the disappearance of the droplets for E/σ0 ) 0 while for E/σ0 > 0 the droplets do not dissolve but accumulate at low droplet radii. To correct for this effect, we also plotted the mean radius for b ) 0 where we included the disappeared particles that have zero radius. Then a10 ) Σa/N0, with N0 ) 5000, is the initial number of particles present, and the calculation is similar to that performed for E/σ0 > 0, which makes comparison possible. The result corresponds to the lowest curve in the figure and is indicated as b ) 0 (corr). The number-averaged radius is mainly determined by the number of droplets in a certain size class and not by the low number of droplets that will grow continuously. The final number-averaged radius increases with increasing elasticity. This is because with increasing elasticity the droplets that shrink will finally obtain a larger radius with increasing E/σ0. For large surface elasticity (b ) 1, 2, 5) the system is stable and no significant change in the averaged radius is observed. The volume-surface-averaged droplet radius (a32) is the most relevant average droplet radius. It is the radius that
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Figure 7. Relative droplet radius at stabilization as a function of the ratio elastic modulus-interfacial tension, assuming a constant elastic modulus or assuming the elastic layer to be a Langmuir monolayer (see text).
time. The volume-surface-averaged radius increases more rapidly in time for a lower surface elasticity. A lower surface elasticity allows the droplet to shrink further. However, the total volume should remain the same so less drops have to yield the remaining volume, which leads to larger droplets. Again, for large surface elasticity (b ) 1, 2, 5) the system is stable and no significant change in a32 is observed. The evolution in time of individual droplets with varying initial radii a0 for different b ) E/σ0 ratios is given in Figure 6. For every E/σ0 ratio it is clear that the smallest droplets directly start to shrink. For small E/σ0 ratios, larger droplets may initially grow, but after a certain time they will also start to shrink. A droplet obtains its maximum size at the moment the radius equals the critical radius. Dependent on the E/σ0 ratio, the droplet will be stabilized at a certain degree of shrinkage. With increasing E/σ0 ratio stabilization occurs sooner. For large E/σ0 ratios, the growing particles do not shrink but reach a stable value for the radius when Rc/ac ) R/a. The effect of surface elasticity on the evolution of the droplet size distribution may become even more pronounced, since the surface elastic modulus will not be constant with either time or deformation of the interfacial area. Especially when proteins are used as the interfacial emulsifier, the elastic modulus may increase with time due to interfacial network formation or other lateral attractive interactions between the stabilizers. The interfacial elastic modulus may also increase due to compaction of the interfacial layer or due to local bending of the interface. Assuming that E is proportional to the surface load Γ and that no collapse of the adsorbed layer occurs, for a Langmuir monolayer the modulus can be written as a function of the initial modulus E0 by E(t) ) E0(a02/a(t)2), resulting in the following radius-dependency of the interfacial tension:
( )
σi,j ) σ0 + E0 1 Figure 6. Evolution of droplet sizes with different initial sizes as a function of dimensionless time for several surface elasticity/ surface tension ratios; b ) 0 (a), b ) 0.25 (b), b ) 1 (c), and b ) 5 (d).
can be determined experimentally most easily and accurately. Furthermore, it is often a better measure for the “average” radius of a distribution and gives an indication of what happens with the bulk of an emulsion. Figure 5 shows that this average radius shows continuous coarsening of the emulsion whereas the number-averaged radius suggests that the emulsion would become finer in
2 ai,0
a2i,j
(19)
In this case a droplet will be stabilized as soon as ai,j ) ai,0[E0/(E0 + σ0)]1/2. Figure 7 shows that for a small interfacial elastic modulus-interfacial tension ratios a Langmuir monolayer will lead to less droplet shrinkage before stabilization is achieved compared to that for a constant interfacial elastic modulus. It is assumed that E is constant on dissolution or growth of the droplet and does not depend on the interfacial tension. The situation becomes more complicated as E is dependent on time, as
Ostwald Ripening in Emulsion
normally found for adsorbed protein layers. For a Langmuir monolayer E will first increase on shrinkage of the droplet due to the increase in adsorbed amount per unit of surface area and decreases after collapse of the adsorbed layer. Conclusions The effect of surface elasticity on coarsening of emulsions is studied by means of numerical calculations. Low interfacial elasticity (b ) E/σ0 < 1, with b the ratio between the surface elastic modulus E and the initial surface tension σ0) causes the initial continuous LSW size distribution to become bimodal. A peak occurs at a small droplet size that results from droplets that shrink to a
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minimum radius which is exp[-1/(2b)] of their initial size. The other peak at large radius results from coarsening of the original size distribution, and it becomes smaller with time. The rate of increase of the critical radius with time increases with increasing surface elasticity while the rate of increase of the volume-surface-averaged radius decreases with increasing surface elasticity. For large interfacial elasticity (b > 1) the system is stable. Acknowledgment. The authors would like to thank the reviewer whose suggestion lead to an adaptation and correction of the model and to a significantly improved manuscript. LA000304Z
