Modeling of interface mobility in the description of flow-induced coalescence in immiscible polymer blends

Ivan Fortelný  Josef Jůza

Institute of Macromolecular Chemistry AS CR, Heyrovského nám. 2, 162 06 Praha 6, Czech Republic

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Derivation of equations for dependences of h on rotation angles

For shear flow with unperturbed velocity u0 = (˙y, 0, 0), F is given by the equation [1]

(S1)

where is polar angle, and  is azimuth (see Fig. 1 in [1]). K is a function of p defined as [2]

(S2)

whereK is the value of K for p, very weakly dependent on inter-droplet distance. K = 12 for force on rigid spheres in unperturbed flow and K = 12.24 for force on doublet of touching spheres [1,3].

Rotation of the droplets is described by the following equations [1,4]

(S3)

(S4)

whereβ is a function of the ratio of distance between centers of spheres and 2R. β = 0.075 was recognized as satisfactory approximation [1].

The following equation was derived [1,5] for the rate of matrix drainage between spherical droplets

(S5)

where

(S6)

andm is defined as

(S7)

Substitution of F with F + mdF/dtin Eq. (S5), using Eq. (S1) for F and division of Eq. (S5) by Eq. (S3) leads to the following equation (Eq. (10) in the paper) for spherical droplets

(S8)

The same procedure applied to flattened droplets, where time dependence of h is given by Eq. (7) in the paper, leads to the equation (Eq. (11) in the paper)

(S9)

where

(S10)

Angles  and  are bound by the equation [1]

(S11)

Equation (S11) is utilized at calculation of maximum initial azimuth, M, for which the droplet collision is followed by their fusion.

For the extensional flow with unperturbed velocity u0 = ε˙(-x, -y, 2z), where ε˙ is the deformation rate, rotation of the spheres is described by the equation [4,6,7]

(S12)

For the driving force of the coalescence in extensional flow, the following equation is valid [6,7]

(S13)

The equation for the dependence of the inter-droplet distance on rotation angle can by derived by the same procedure as for the shear flow [6]

(S14)

where

(S15)

(S16)

(S17)

For flattened droplets, where drainage of the matrix is described by Eq. (7) in the paper, the following equation is valid for the dependence of inter-droplet distance on the rotation angle

(S18)

where capillary number, Cae, for extensional flow is defined as Cae = mε˙R/.

Collision of the droplets in extensional flow is followed by their fusion if their distance decreases below hc until the polar angle * is achieved. During calculation of h(*), Eq. (S18) for flattened droplets is combined with Eq. (S14) for spherical droplets in the same way as for shear flow. Expression for * is determined from the condition dh/dt = 0 [6]

(S19)

For systems with a Newtonian matrix, *= arctg(2). Minimum initial polar angle, 0(m), at which h(*) = hc should be determined. Probability of coalescence, Pc, for extensional flow can be calculated from the Eq. (15) in the paper.

References

  1. Fortelný I, Jůza J (2012) The Effect of Matrix Elasticity on the Shear Flow Induced Coalescence of Dispersed Droplets. J Rheol 56: 1393-1411
  2. Jaeger PhT, Janssen JJM, Groeneweg F, AgterofWGM (1994) Coalescence in emulsions containing inviscid drops with high interfacial mobility. Colloids Surfaces A: Physicochem Eng Aspects 85: 255-264
  3. Nir A, Acrivos A (1973) On creeping motion of two arbitrary-sized touching spheres in a linear shear fields. J Fluid Mech 59: 209-223
  4. Wang H, Zinchenko AZ, Davis RH (1994) The collision rate of small drops in linear flow fields. J Fluid Mech 265: 161 – 188
  5. Zhang X, Davis RH (1991) The collision of small drops due to Brownian and gravitational motion. J Fluid Mech 230: 479 – 504
  6. Jůza J, Fortelný I (2013) Flow induced coalescence in polymer blends. ChemChemTechnol 7:53-60
  7. Fortelný I, Živný A. (2003) Extensional flow induced coalescence in polymer blends. RheolActa 42: 454 – 461