The effect of anisometry of dispersed droplets on their coalescence during annealing of polymer blends
Ivan Fortelný Josef Jůza Taťana Vacková Miroslav Šlouf
Institute of Macromolecular Chemistry AS CR, v. v. i., Heyrovského nám. 2, 162 06 Prague 6,CzechRepublic
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Derivation of the equation for time evolution of average inter-droplet distance in a system of coalescing anisometric droplets
It is assumed that the change in the average distance between a droplet and its nearest neighbor consists of contributions of the droplet shape relaxation and of van der Waals force induced coalescence. For a system of randomly oriented droplets, the average inter-droplet distance, h, is given by the equation (Eq. 5 in the paper)
(S1)
For a system of parallelly oriented droplets with a shorter inter-droplet distance in perpendicular direction to their orientation, h can be expressed as (Eq. 19 in the paper)
(S2)
Time dependence of the ratio of droplet axes, c, at the droplet shape relaxation can be expressed as (Eq. 7 in the paper)
(S3)
where is the relaxation time. The rate of the change in the droplet distance induced by their shape relaxation, (dh/dt)R, can be obtained by substitution from Eq. S3 into Eq. S1 or S2 and their derivative with respect to t. It is considered that hr does not change with time due to the droplet shape relaxation. The following equation is obtained in the both cases
(S4)
In our preceding paper [23], theory of the coalescence induced by van der Waals force was derived. It was assumed that the minor phase consist of the system of monodisperse spheres randomly distributed in the matrix. Interaction of a droplet only with its nearest neighbor localized in the average distance was considered. It was assumed that droplets kept spherical shape and their system was monodisperse through the whole coalescence process. The approach of spherical droplets is described by the equation [1,2]
(S5)
where F is the driving force of coalescence and a function g(m) is given by
(S6)
where g1k = 1.711, g2k = 0.461, gmk = 0.402
and (S7)
For van der Waals force induced coalescence, the force F can be expressed as [1]
(S8)
where A is the Hamaker constant [3]. After substitution from Eq. S8 into Eq. S5, the following equation is derived for the time dependence of the average distance between a droplet and its nearest neighbor caused by van der Waals force
(S9)
If it is assumed that changes in inter-droplet distance caused by the droplet shape relaxation and van der Waals forces induced coalescence are additive, the time dependence of the average inter-droplet distance is described by the equation
(S10)
Substitution to Eq. S10 from Eqs. S4 and S9 leads to
(S11)
Equation S11 is Eq. 9 in the paper.
Solution of Equation S9
Equation S9, describing coalescence of spherical droplets can be solved analytically. Its integration from 0 to tc and from h0 to hc leads to
(S12)
where
(S13)
For systems with h0hc, hc was supposed to be zero. In this case, also Eq. 14 (in the paper) has analytical solution for ttN
(S14)
where RN is the equivalent drop radius at the time tN.
References
- Fortelný I, Živný A (1998) Film drainage between droplets during their coalescence in quiescent polymer blends. Polymer 39:2669-2675
- Zhang X, Davis RH (1991) The collision of small drops due to Brownian and gravitational motion. J Fluid Mech 230:479-504
- Elmendorp JJ (1991) Dispersive mixing in liquid systems. In: Rauwendaal C (ed) Mixing in polymer processing. Marcel Dekker, New York, pp 17-100
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