Introduction

The interfacial tension is a crucial property of multiphase liquid systems such as classical emulsions and polymer blends. It directly characterizes the interactions at the interface between the components of the mixture. Various techniques are available for the determination of interfacial tension. Xing et al. (2000) compared five experimental techniques for the determination of interfacial tension in polymer blends and showed that the retraction of deformed drop method (DDRM) is of high simplicity and efficiency. Such a method can be used in the retraction of pre-sheared drops (Luciani et al. 1997; Guido and Villone 1999) or for elongated drops produced by the breakup of a long thread (Mo et al. 2000). DDRM method is based on a fundamental assumption that the drop is slightly deformed and it adopts an ellipsoidal shape with the three semiaxes L(t), B(t), W(t) and L(t)≥W(t)≥B(t), where t denotes time.

The DDRM used by Luciani et al. (1997) is based on the small deformation (SD) theory of Taylor (1934) and Rallison (1984). The crucial quantity to be measured for the determination of interfacial tension, Γ, is the variation in time of the deformation parameter D 1(t)=L(t)−B(t)/L(t)+B(t) during the retraction process of the deformed drop. According to the small deformation theory, the evolution of D 1(t) during the drop retraction can be written as

$$D_{1} {\left( t \right)} = D_{0} \exp {\left( { - {f_{1} t} \mathord{\left/ {\vphantom {{f_{1} t} \tau }} \right. \kern-\nulldelimiterspace} \tau } \right)}$$
(1)

with

$$f_{1} = \frac{{40{\left( {p + 1} \right)}}} {{{\left( {2p + 3} \right)}{\left( {19p + 16} \right)}}},\;\;\;\tau = \frac{{\eta _{m} R}} {\Gamma }$$
(2)

where p=η d /η m is the ratio between the viscosity of the drop, η d , and the matrix, η m , R is the radius of an equivalent sphere having the same volume as the deformed drop (R 3=LBW). D 0 is the initial deformation of the drop. The determination of the interfacial tension by Eq. (1) comes from the slope of the plot lnD 1(t)/D 0 vs time, t. To use Eq. (1), Luciani et al. (1997) assumed that the drop shape is axisymmetric (B=W). However, such assumption is not inherent to the small deformation theory (Mo et al. 2000) and one can in principle measure the three axes of the drop during the retraction process. Nevertheless for very small deformation, the difference between B and W and the axisymmetric assumption remains valid within the range of experimental errors of the technique. However, when the droplet shape departs largely from sphericity, the axisymmetric assumption becomes invalid. This is crucial because during experiments it is very difficult to apply very small deformations and at the same time visualize the shape of the deformed drop. It is also not easy to decide experimentally whether the deformation is small or not.

Another DDRM method was suggested by Mo et al. (2000), who used of the model of Maffettone and Minale (MM) (Maffettone and Minale 1998). In the MM model, the retraction of the ellipsoidal drop can be written as (Mo et al. 2000)

$$D_{2} (t) = L^{2} {\left( t \right)} - B^{2} {\left( t \right)} = {\left( {L^{2} - B^{2} } \right)}_{0} \exp {\left( { - {f_{1} t} \mathord{\left/ {\vphantom {{f_{1} t} \tau }} \right. \kern-\nulldelimiterspace} \tau } \right)}$$
(3)

The MM model was proved to be accurate only to the zeroth order of drop deformation (Yu and Bousmina 2003; Yu et al. 2003), which means that Eqs. (1) and (3) are equivalent in describing the drop retraction. This is not surprising since the parameters of the MM model (parameter f 1 for instance) were determined from the SD theory. Nevertheless, it was shown that Eq. (3) works better than Eq. (1) when the deformation of drop increases (Mo et al. 2000). However even the MM model is not able to describe the drop deformation and retraction when the drop is subjected to large deformations (Yu and Bousmina 2003). Under large deformation neither the plot of lnD 1(t) nor that of lnD 2(t) vary linearly with time.

In this paper, we propose a new method that relaxes the small deformation assumption. The results of the new technique will be compared both to some experimental results of the literature and to the predictions of the SD and MM models.

Theory and Method

We describe the ellipsoidal drop by a second rank tensor G (Wetzel and Tucker 2001; Jackson and Tucker 2003; Yu and Bousmina 2003). In the principal axes of the drop, G can be expressed as

$${\user2{G}}{\left( t \right)} = {\left( {\begin{array}{*{20}c} {{1/L^{2} {\left( t \right)}}} & {0} & {0} \\ {0} & {{1/B^{2} {\left( t \right)}}} & {0} \\ {0} & {0} & {{1/W^{2} {\left( t \right)}}} \\ \end{array} } \right)}$$
(4)

where L, B, and W are the three semi-axes of the ellipsoidal drop. The time evolution of G is given by (Wetzel and Tucker 2001)

$$\frac{{DG_{{ij}} }} {{Dt}} + L_{{ki}} G_{{kj}} + G_{{ik}} L_{{kj}} = 0$$
(5)

where D/Dt is the material derivate and L ij is the velocity gradient tensor taken at the surface of the drop. During the retraction of the drop, L ij possesses only diagonal components and therefore, Eq. (5) can be reduced to

$$\frac{{d\ln L{\left( t \right)}}} {{dt}} = L_{{11}} {\left( t \right)}$$
(6)

Similar expressions can be obtained for B(t) and W(t). Since the retraction of the drop is driven by interfacial tension, Eq. (6) can be used to determine its value during the retraction process. We will show later that L ij can expressed in the following form:

$$L_{{ij}} {\left( t \right)} = {{F}'_{{ij}} {\left( t \right)}} \mathord{\left/ {\vphantom {{{F}'_{{ij}} {\left( t \right)}} \tau }} \right. \kern-\nulldelimiterspace} \tau $$
(7)

where τ is a characteristic time that is related to the interfacial tension (τ=η m R/Γ, R being the radius of the initial spherical droplet) and \( {F}\ifmmode{'}\else$'$\fi_{{ij}} {\left( t \right)} \) is a given function of time. The expression of \({F}'_{{ij}} {\left( t \right)}\) depends on the model under consideration and will be given later for the two models (JT and YB models). Equation (6) can be integrated to obtain

$$\ln \frac{{L{\left( t \right)}}} {{L_{0} }} = \frac{1} {\tau }{\int_0^t {{F}'_{{11}} {\left( {{t}'} \right)}d{t}'} } = \frac{{F_{{11}} {\left( t \right)}}} {\tau }$$
(8)

where L 0 is the initial length of the drop. The interfacial tension can then be determined from the slope of lnL(t)/L 0 vs F 11(t). B(t) and W(t) are given by equations similar to Eq. (8) and can also be used to determine the interfacial tension. It is, however, preferable to use L(t) instead of B(t) and W(t) since the variation of L(t) is larger and therefore the experimental errors are less important than those obtained with the other axes.

Let us now derive the analytical expression for L ij as a function F ij (t) and interfacial tension, Γ using two approaches. The first approach is the use of the YB model that gives an analytical expression of L ij at the surface of the drop using the boundary integral formalism. In the YB model, the velocity gradient due to the interfacial tension is expressed as

$${\user2{L}} = {\user2{L}}^{\alpha } + {\user2{L}}^{\beta } $$
(9)

where \({\user2{L}}^{\alpha } \) is given by

$${\user2{L}}^{\alpha } = {\user2{\bar{L}}}^{\alpha } - tr{\left( {{\user2{\bar{L}}}^{\alpha } } \right)}{\user2{P}}$$
(10)

where P is a constant diagonal matrix and takes the form

$$P_{{ij}} = {\left( {\begin{array}{*{20}c} {0} & {0} & {0} \\ {0} & {{\tfrac{1} {2}}} & {0} \\ {0} & {0} & {{\tfrac{1} {2}}} \\ \end{array} } \right)}\;\;\;{\text{for}}\;\;\;p \geqslant 1\;\;\;{\text{and}}\;\;\;P_{{ij}} = {\left( {\begin{array}{*{20}c} {{ - 1}} & {0} & {0} \\ {0} & {{\tfrac{3} {2}}} & {0} \\ {0} & {0} & {{\tfrac{1} {2}}} \\ \end{array} } \right)}\;\;\;{\text{for}}\;\;\;p < 1$$
(11)

The components of the tensor \({\user2{L}}^{\alpha } \) are obtained from the velocity at the surface of the ellipsoid:

$$\bar{L}^{\alpha }_{{11}} = {{u}'^{\alpha }_{1} {\left( {{\left[ {L,0,0} \right]}} \right)}} \mathord{\left/ {\vphantom {{{u}'^{\alpha }_{1} {\left( {{\left[ {L,0,0} \right]}} \right)}} {\tau L}}} \right. \kern-\nulldelimiterspace} {\tau L},\;\bar{L}^{\alpha }_{{22}} = {{u}'^{\alpha }_{2} {\left( {{\left[ {0,B,0} \right]}} \right)}} \mathord{\left/ {\vphantom {{{u}'^{\alpha }_{2} {\left( {{\left[ {0,B,0} \right]}} \right)}} {\tau B}}} \right. \kern-\nulldelimiterspace} {\tau B},\;\bar{L}^{\alpha }_{{33}} = {{u}'^{\alpha }_{3} {\left( {{\left[ {0,0,W} \right]}} \right)}} \mathord{\left/ {\vphantom {{{u}'^{\alpha }_{3} {\left( {{\left[ {0,0,W} \right]}} \right)}} {\tau W}}} \right. \kern-\nulldelimiterspace} {\tau W}$$
(12)

\( {u}\ifmmode{'}\else$'$\fi^{\alpha }_{i} {\left( {i = 1,2,3} \right)} \) is calculated by the boundary integral equation:

$${u}'^{\alpha }_{i} {\left( {\user2{x}} \right)} = \frac{R} {{2\pi {\left( {p + 1} \right)}}}{\int\limits_{S_{y} } {\frac{{{\user2{r}} \cdot {\left[ {{\user2{n}}{\left( {\user2{x}} \right)} + {\user2{n}}{\left( {\user2{y}} \right)}} \right]}}} {{r^{3} }}{\left\{ {1 - \frac{{3{\left[ {{\user2{r}} \cdot {\user2{n}}{\left( {\user2{x}} \right)}} \right]}{\left[ {{\user2{r}} \cdot {\user2{n}}{\left( {\user2{y}} \right)}} \right]}}} {{r^{2} }}} \right\}}dS_{y} } }$$
(13)

\({\user2{L}}^{\beta } \) is related to \({\user2{L}}^{\alpha } \) by

$$diag{\left( {{\user2{L}}^{\beta } } \right)} = {\user2{A}}^{{ - 1}} \cdot {\user2{B}} \cdot diag{\left( {{\user2{L}}^{\alpha } } \right)}$$
(14)

where diag( ) mean the vector composed by the diagonal elements of the tensor between parenthesis. The tensors A and B are function of the viscosity ratio and the semiaxes of ellipsoids (Yu and Bousmina 2003):

$$ {\mathbf{A}} = 2{\left( {\begin{array}{*{20}c} {{1 + {\left( {p - 1} \right)}S_{{11}} }} & {{{\left( {p - 1} \right)}S_{{12}} }} & {{{\left( {p - 1} \right)}S_{{13}} }} \\ {{{\left( {p - 1} \right)}S_{{21}} }} & {{1 + {\left( {p - 1} \right)}S_{{22}} }} & {{{\left( {p - 1} \right)}S_{{23}} }} \\ {{{\left( {p - 1} \right)}S_{{31}} }} & {{{\left( {p - 1} \right)}S_{{32}} }} & {{1 + {\left( {p - 1} \right)}S_{{33}} }} \\ \end{array} } \right)} $$
(15)

and

$$ {\mathbf{B}} = {\left( {p - 1} \right)}{\left( {\begin{array}{*{20}c} {{1 - 2S_{{11}} }} & {{ - 2S_{{12}} }} & {{ - 2S_{{13}} }} \\ {{ - 2S_{{21}} }} & {{1 - 2S_{{22}} }} & {{ - 2S_{{23}} }} \\ {{ - 2S_{{31}} }} & {{ - 2S_{{32}} }} & {{1 - 2S_{{33}} }} \\ \end{array} } \right)} $$
(16)

where S ij is the Eshelby tensor expressed with contracted notation (Eshelby 1957). Equation (12) shows that \( {\user2{\ifmmode\expandafter\bar\else\expandafter\=\fi{L}}}^{\alpha } \) is proportional to the interfacial tension, Γ, and thus \({\user2{L}}^{\beta } \) is also proportional to Γ. Therefore, L is directly proportional to Γ, and can be expressed in the form of Eq. (7). \({F}'_{{ij}} {\left( t \right)}\) can be obtained by multiplying Eq. (9) with τ. F 11 (t) is calculated by numerical integration over time period 0∼t by using Eqs. (9), (10), (11), (12), and (13).

Another approach is suggested by Jackson and Tucker (2003) (the JT model). According to their model, the velocity gradient due to the interface is a blend of the Eshelby model and the slender-body model:

$$ {\user2{L}} = \left\{ {\begin{array}{*{20}l} {{f{\user2{L}}_{{{\text{Eshelby}}}} + {\left( {1 - f} \right)}{\user2{L}}_{{{\text{slender}}}} } \hfill} & {{p < {\text{0}}{\text{.1}}} \hfill} \\ {{{\user2{L}}_{{{\text{Eshelby}}}} } \hfill} & {{p \geqslant {\text{0}}{\text{.1}}} \hfill} \\ \end{array} } \right. $$
(17)

where f is a mixing parameter which depends on the dimensionless length L/R of the ellipsoidal drop (Jackson and Tucker 2003). L Eshelby is calculated by

$${\user2{L}}_{{{\text{Eshelby}}}} = - \frac{{2q}} {{\pi \tau }}{\user2{B}}:{\user2{S}}:{\user2{\bar{P}}}$$
(18)

with

$${\user2{\bar{P}}} = {\user2{P}} - \tfrac{1} {3}tr{\left( {\user2{P}} \right)}\delta ,\;\;{\user2{P}} = diag{\left[ {\frac{R} {B}E{\left( {1 - \frac{{B^{2} }} {{W^{2} }}} \right)},\frac{R} {W}E{\left( {1 - \frac{{W^{2} }} {{L^{2} }}} \right)},\frac{R} {L}E{\left( {1 - \frac{{L^{2} }} {{B^{2} }}} \right)}} \right]}$$
(19)

where q is a function of the viscosity ratio (Jackson and Tucker 2003). B and S are fourth-order tensors which are functions of viscosity ratio and the three semiaxes of ellipsoid (Wetzel and Tucker 2001).

L slender is calculated as follows in the absence of flow:

$${\user2{L}}_{{{\text{slender}}}} = - \frac{{{\sqrt {L \mathord{\left/ {\vphantom {L R}} \right. \kern-\nulldelimiterspace} R} }}} {{2{\sqrt 5 }\tau {\left( {1 + 0.8p{\left( {L \mathord{\left/ {\vphantom {L R}} \right. \kern-\nulldelimiterspace} R} \right)}^{3} } \right)}}}{\left[ {\begin{array}{*{20}c} {1} & {0} & {0} \\ {0} & {{ - \tfrac{1} {2}}} & {0} \\ {0} & {0} & {{ - \tfrac{1} {2}}} \\ \end{array} } \right]} + \frac{4} {{3\tau {\left( {1 + p} \right)}{r_{0} } \mathord{\left/ {\vphantom {{r_{0} } R}} \right. \kern-\nulldelimiterspace} R}}{\left( {1 - \frac{{r_{0} }} {W}} \right)}{\left[ {\begin{array}{*{20}c} {0} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {{ - 1}} \\ \end{array} } \right]}$$
(20)

Since L Eshelby and L slender are proportional to 1/τ, Eq. (17) can then be written in the form of Eq. (7). \({F}'_{{ij}} {\left( {{t}'} \right)}\) for the JT model can be written as

$$ \begin{array}{*{20}l} {{{F}'_{{ij}} {\left( {{t}'} \right)} = } \hfill} & {{ - f\frac{{2q}} {\pi }{\user2{B}}{\left( {{t}'} \right)}:{\user2{S}}{\left( {{t}'} \right)}:{\user2{\bar{P}}}{\left( {{t}'} \right)}} \hfill} \\ {{} \hfill} & {{ + {\left( {1 - f} \right)}{\left\{ { - \frac{{{\sqrt {{L{\left( {{t}'} \right)}} \mathord{\left/ {\vphantom {{L{\left( {{t}'} \right)}} R}} \right. \kern-\nulldelimiterspace} R} }}} {{2{\sqrt 5 }{\left( {1 + 0.8p{\left( {{L{\left( {{t}'} \right)}} \mathord{\left/ {\vphantom {{L{\left( {{t}'} \right)}} R}} \right. \kern-\nulldelimiterspace} R} \right)}^{3} } \right)}}}{\left[ {\begin{array}{*{20}c} {1} & {0} & {0} \\ {0} & {{ - \tfrac{1} {2}}} & {0} \\ {0} & {0} & {{ - \tfrac{1} {2}}} \\ \end{array} } \right]} + \frac{4} {{3{\left( {1 + p} \right)}{r_{0} } \mathord{\left/ {\vphantom {{r_{0} } R}} \right. \kern-\nulldelimiterspace} R}}{\left( {1 - \frac{{r_{0} }} {{W{\left( {{t}'} \right)}}}} \right)}{\left[ {\begin{array}{*{20}c} {0} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {{ - 1}} \\ \end{array} } \right]}} \right\}}} \hfill} \\ \end{array} $$
(21)

To sum-up, in both YB and JT models, the velocity gradient tensor can be expressed in the form of Eq. (7) and therefore the interfacial tension can be calculated from the slope of the plot ln L(t)/L 0 vs F 11(t) (see Eq. 8). F 11(t) is related to the function \( {F}\ifmmode{'}\else$'$\fi_{{11}} {\left( {{t}\ifmmode{'}\else$'$\fi} \right)} \) through (Eq. 8). The expression of \({F}'_{{11}} {\left( {{t}'} \right)}\) is given explicitly for the JT model (Eq. 21). For the YB model, \( {F}\ifmmode{'}\else$'$\fi_{{11}} {\left( {{t}\ifmmode{'}\else$'$\fi} \right)} \) is given implicitly by Eqs. (9), (10), (11), (12), (13), and (14). In fact both \({\user2{L}}^{\alpha } \) and \({\user2{L}}^{\beta } \) (\({\user2{L}}^{\beta } \) is related to \({\user2{L}}^{\alpha } \) by Eq. 14) can be written if the form given by Eq. (12), where the velocity has to be integrated numerically at the apex of the ellipsoid. This means that the velocity gradient given by Eq. (9) can also be expressed in the form given by Eq. (12) and therefore by Eq. (7) that has to be used to extract the interfacial tension from the characteristic time τ=η m R/Γ.

To use the method more simply, an approximate procedure is suggested in the Appendix. The values of the function \({F}'_{{11}} {\left( t \right)}\) appearing in Eq. (8) can be directly calculated from the values supplied in Table A1.

Results and discussions

It has been shown that the YB and the JT models can describe the deformation and relaxation of the drop in a quite satisfied manner (Jackson and Tucker 2003; Yu and Bousmina 2003), including the case of large deformation. Hence, the determination of interfacial tension by drop retraction from YB model or JT model relaxes the restriction about the small deformation assumption. We thus examined this approach for systems with various viscosity ratios and various initial deformations of the drop. The interfacial tension obtained from the new approach is compared to that determined experimentally using the small deformation models.

When the initial deformation of the drop is very large, neither the deformation parameter D nor the parameter L 2−B 2 can be described by a single exponential decrease like Eqs. (1) or (3). This is shown more explicitly in Fig. 1, where the scatter symbols represent the experimental data of Almusallam et al. (2000) for PBd/PDMS system with p=1 and the strain of pre-shear γ 0=5.0, and the solid lines represent the predictions obtained from Eqs (1) and (3). Clearly, the small deformation models are unable to describe the relaxation of the drop after imposition and cessation of a large amplitude of deformation. One can argue that the fit can be carried out at longer times in the linear region. This of course gives better results for the determination of interfacial tension by the SD models, but the linear zone is not always well identified when the drop is deformed under large amplitude of deformation. Of course, if the technique has to be used for viscoelastic components, it is better to consider a time interval starting beyond the longer time of elastic relaxation of the pure components, where the drop shape recovery is only driven by interfacial tension. Examination of various systems revealed in fact that even in this case the variation of the deformation parameter in time (Eqs. 1 and 3) is not clearly linear.

Fig. 1
figure 1

Determination of the interfacial tension by the SD model and the MM model

Full size image

The calculations of the interfacial tension based on YB and JT models are shown in Fig. 2, which illustrates excellent agreement between the experiments and the models fit. The interfacial tension determined by YB and JT models are 1.9 mN/m and 2.4 mN/m respectively.

Fig. 2
figure 2

Determination of the interfacial tension by JT model and YB model

Full size image

In Table 1, we compare the calculation results by different methods for different systems. χ 2 in the table denotes the correlation coefficient of the linear fit. It is shown that SD and MM models give acceptable results only for very small deformation (Nos. 1, 5, 6), i.e., the maximum stretch ratio is of about 3. MM model seems to do a better job than SD model due to its good linear fit with experiments. The new approach gives better linear fit with the experimental data for the retraction of slightly deformed drop than the traditional DDRM, and thus the determined interfacial tension is slightly larger than the traditional methods. However, the difference between the interfacial tension measured by the traditional method and the new approach lies within the acceptable experimental errors. The advantages of the new approach become significant when the initial deformation of drop is large. It is seen from Table 1 that SD and MM models fail to fit the experimental results for large drop deformation (Nos. 2–4 and 7–8), while YB and the JT models can still fit the experimental results quite well (except No. 8).

Table 1 Comparisons of different methods in different systems
Full size table

It is also noticed that the interfacial tension determined by YB model starts to decrease when the stretch ratio L 0/B 0 of drop is larger than 10. This might be due to the over-simplified determination of the velocity gradient tensor by Eq. (11). The interfacial tension determined by JT model for PIB/PDMS system is larger than all other models. This is probably because the viscosity ratio (p=0.0667) is near the transition zone between the Eshelby result and the slender-body result in the JT model.

The new approach can be applied to drop retraction after imposition of both small and large deformations. One restriction of this method is that the shape of the drop should remain ellipsoidal. This condition is satisfied under most cases except for systems with very small viscosity ratio and large initial deformation (No. 8). When p≪1 and L 0/B 0≫1, the shape of the drop changes into rod or dumbbell during the retraction. For systems with p≪1, it is better to measure the interfacial tension by the drop retraction with limited initial drop deformation (L 0/B 0<10).

Another approach for the determination of interfacial tension for largely deformed drop has been suggested by Tjahjadi et al. (1992). The technique is also available for shapes other than ellipsoid. However, the Tjahjadi et al. (1992) technique is based on a large number of previous numerical simulations by using boundary integral method and the determination of the interfacial tension was obtained with some curve-fittings and interpolations from their tabulated results. Moreover, the results of these add-hoc curve fittings provide values of interfacial tension only for p=0.01~10. If the technique has to be applied to viscosity ratio other than this range, further numerical simulations and curve fittings have to be performed.

The approach proposed in the present paper does not require any curve fittings and can be applied to arbitrary viscosity ratio both under small and large deformations.

Conclusions

The traditional DDRM to determine the interfacial tension by SD model or MM model is updated with extension to large drop initial deformation by using YB model or JT model. This approach requires accurate measurement of the length of three axes during the drop retraction. The method shows good description of the experimental results and thus the error on the determination of interfacial tension is very small compared to the classical DDRM method that fails when the drop is deformed under large amplitude of deformation.