Determination of interfacial tension by the retraction method of highly deformed drop

Abstract

The traditional retraction of the deformed drop method (DDRM) to determine the interfacial tension is reformulated to relax the limit the small deformation assumption. The kernel of the new formalism is the calculation of the velocity gradient on the vertex of the ellipsoidal drop. Two models were used for such calculations: the Jackson and Tucker model [J Rheol 47:659–682] and the Yu and Bousmina model [J Rheol 47:1011–1039]. The method can be used either in the retraction of shear deformed drop, or in the retraction of elongated drops produced by the breakup of a long thread. Comparison with experimental results of the literature showed that conversely to the classical DDRM, good accuracy is obtained when the new modeling for the determination of interfacial tension is used both under small and large deformations.

Appendix

To make easy use of the method proposed is the present paper we provide here an approximate method for the function ${F}'_{{11}} {\left( t \right)}$ appearing in the YB model for various viscosity ratios and various aspect ratios. The function ${F}'_{{11}} {\left( t \right)}$ can be fitted using the following polynomials:

$${F}'_{{11}} {\left( t \right)} = \frac{{{\sum\limits_{j = 1}^n {{\sum\limits_{i = 1}^m {k_{{ij}} c^{{i - 1}} {\left( t \right)}d^{{j - 1}} {\left( t \right)}} }} }}} {{L{\left( t \right)}}}$$

(A1)

where c(t)=B(t)/L(t) and d(t)=B(t)/W(t) are two aspect ratios. The coefficients k _ij for a given range of viscosity ratio, p, are listed in Table A1. F ₁₁(t) can be easily calculated from Eq. (8) together with Eq. (A1). For other values of p not listed in the table, a linear interpolation can be made from F ₁₁(t) between the nearest values of p. The coefficients for p out of the range 0.01~10 are not listed in Table A1. Direct calculation using the model is recommended for these viscosity ratios.

Table A1 Polynomial coefficients k _ij in Eq. (<equationcite>A1</equationcite>) for a range of viscosity ratio

The fitting procedure used here is somewhat similar to the one adopted by Tjahjadi et al. (1992). The estimation of interfacial tension from their method needs first the knowledge of a theoretical curve L(t)/R∼t, which is determined by double linear interpolations from the fitting results of numerical simulations. Then interfacial tension is determined by taking two shapes during retraction for the aspect ratios and non-dimensional time from the curve L(t)/R∼t. The advantage of the method suggested by Tjahjadi et al. is its ability to describe some complex shapes, such as dumbbell and other non-ellipsoidal shapes. However, there are two concerns about such a method. (i) If the viscosity ratio and the initial aspect ratio is not included in the table of polynomial coefficients, which are fitted from the numerical simulations, double linear interpolations are needed. Additional assumptions had been made in such a procedure such that the evolution of aspect ratios is a bilinear function of viscosity ratio and the initial aspect ratio. In contrast, our approach only (implicitly) assumes that F ₁₁(t) is a linear function of viscosity ratio. (ii) Usually, only two shapes of the drop during the retraction are needed to determine the interfacial tension from the method suggested by Tjahjadi et al. (1992). Therefore, this poses the problem of the choice of the best images to be selected. The authors did not supply any information about this issue. If a series of two different images are used, it is then expected that one will obtain a series of interfacial tensions from the method. Such a problem is not encountered in our method. The interfacial tension is obtained only from the fitting of the straight line F ₁₁(t)∼lnL(t)/L ₀. Therefore, there is no need to find the two best images or the way of making averages.