3.1. Influence of Biot number

The Biot number $Bi={{r}_{d}}/\dot {\gamma }$ is an important indicator of the surfactant mass transfer rate, and its influence on the droplet deformation is studied for $Bi$ ranging from 0 to 10, which are achieved by varying the desorption coefficient ${{r}_{d}}$ and adsorption coefficient ${{r}_{a}}$ at the same time but holding ${{r}_{d}}/{{r}_{a}}$ unchanged. The adsorption number is taken as $k={{x}_{in}}/ (1-{{x}_{in}})=0.429$, which leads to ${{j}^{*}}=0$ at $\tau =0$ ($\tau$ is the dimensionless time defined as $\tau =\dot {\gamma }t$), meaning an initial zero mass transfer of surfactants across the interface. The other parameters are chosen as $P{{e}_{b}}=10$ and $h=0.5$.

Figure 2(a) plots ${{D}_{f}}$ as a function of $Bi$, in which $Bi=0$ corresponds to the insoluble case. As shown in this figure, ${{D}_{f}}$ first increases and then decreases as $Bi$ increases, with the maximum ${{D}_{f}}$ obtained at $Bi=0.1$. To understand the role of the surfactant mass transfer on the average interface surfactant concentration, we present the dimensionless total mass of the interface surfactants $M_{s}^{*}$ for different $Bi$ also in figure 2(a). It is seen that for $Bi=0$, $M_{s}^{*}$ remains almost at 1, indicating no mass transfer of surfactants across the interface; whereas for $Bi>0$, $M_{s}^{*}$ is always larger than 1. This suggests that the adsorption process dominates the dynamics of soluble surfactants as the droplet is elongated, and some interface surfactants are swept to the poles, thus increasing the amount of interface surfactants. In addition, like ${{D}_{f}}$, $M_{s}^{*}$ first increases and then decreases as $Bi$ increases.

Figure 2.

(a) The deformation parameter $D_f$ and the dimensionless total mass of interface surfactants $M_{s}^{*}$ as functions of $Bi$; the distributions of (b) the dimensionless source term ${{j}^{*}}$ and (c) the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length $s$ under various $Bi$; (d) the dimensionless bulk surfactant concentration ${{\phi }^{*}}$ at different $Bi$.

To show the influence of the surfactant solubility on non-uniform effects, when the systems reach the steady state, we present the distributions of the dimensionless source term ${{j}^{*}}$ along the arc length $s$ in figure 2(b), the distributions of the dimensionless interface surfactant concentration ${{\psi }^{*}}$ in figure 2(c) and the distributions of the dimensionless bulk surfactant concentration ${{\phi }^{*}}$ in figure 2(d). Herein, the arc length $s$, measured counterclockwise from the right intersection point (i.e. the point ’$A$’ in figure 1a) between $y=0$ and the droplet interface, is normalized by the initial droplet radius $R$. As shown in figure 2(b), for $Bi=0$, no mass transfer occurs across the interface ($\,j=0$). For $Bi>0$, ${{j}^{*}}$ is negative and the desorption dominates the mass transfer process at the droplet tips or poles; while away from the tips, ${{j}^{*}}$ is positive and the adsorption becomes dominant. In addition, the amplitude of ${{j}^{*}}$ is enhanced with the increase of $Bi$, corresponding to an increased mass transfer rate, which further leads to decreased non-uniform effects of the interface surfactants (see figure 2c) and an increased non-uniformity of the bulk surfactant distribution (figure 2d). It is also seen in figure 2(c) that at low values of $Bi$ ($Bi<0.1$), the non-uniform effects keep nearly the same, but as $Bi$ further increases above 0.1, the non-uniform effects decrease sharply.

As mentioned above, at low values of $Bi$ ($Bi<0.1$), the non-uniform effects change slowly, so the increased $M_{s}^{*}$ dominates the droplet deformation and leads to an increase in ${{D}_{f}}$. As $Bi$ further increases above 0.1, $M_{s}^{*}$ and non-uniform effects both decrease (typically with the minimum ${{\psi }^{*}}$ increasing but maximum ${{\psi }^{*}}$ decreasing), which results in a decrease of ${{D}_{f}}$.

3.2. Influence of bulk Péclet number

The bulk Péclet number $P{{e}_{b}}={{R}^{2}}\dot {\gamma }/{{D}_{b}}$ represents the relative importance of convection to diffusion of the bulk surfactants, and its influence on the droplet deformation is investigated for $P{{e}_{b}}$ ranging from 0.1 to 100, which are achieved by solely varying ${{D}_{b}}$. The other parameters are fixed as $Bi=1$, $h=0.5$ and $k={{x}_{in}}/(1-{{x}_{in}})=0.429$.

Figure 3 shows the deformation parameter ${{D}_{f}}$, the dimensionless total mass of the interface surfactants $M_{s}^{*}$, the distributions of the dimensionless source term ${{j}^{*}}$ and the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along $s$, and the dimensionless bulk surfactant concentration ${{\phi }^{*}}$ for different $P{{e}_{b}}$; the results of insoluble surfactants are also shown for the purpose of comparison. As expected, ${{D}_{f}}$ is larger in the soluble cases than in the insoluble case and, as $P{{e}_{b}}$ increases, ${{D}_{f}}$ decreases continuously. This is because as $P{{e}_{b}}$ increases, the bulk surfactant distribution gets more non-uniform due to weaker bulk diffusion, which prevents the surfactant adsorption from the bulk to the interface at the droplet surface away from the poles, thus producing a decreased average interface surfactant concentration (represented by $M_{s}^{*}$ and shown in figure 3a). On the other hand, as $P{{e}_{b}}$ increases, the amplitude of ${{j}^{*}}$ decreases (see figure 3b), corresponding to a decreased surfactant mass transfer rate and increased non-uniform effects of interface surfactants, as shown in figure 3(c). In this case, $M_{s}^{*}$ dominates the droplet dynamics, and thus, the droplet deformation ${{D}_{f}}$ decreases.

Figure 3.

(a) The deformation parameter ${{D}_{f}}$ and the dimensionless total mass of interface surfactants $M_{s}^{*}$ as functions of $P{{e}_{b}}$; the distributions of (b) the dimensionless source term ${{j}^{*}}$ and (c) the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length $s$ under various $P{{e}_{b}}$; (d) the dimensionless bulk concentration ${{\phi }^{*}}$ under various $P{{e}_{b}}$.

3.3. Influence of adsorption number

The adsorption number $k={{r}_{a}}{{\phi }_{0}}/{{r}_{d}}$ is a dimensionless parameter closely related to the initial adsorption and desorption states of surfactants. For $k<{{x}_{in}}/(1-{{x}_{in}})=0.429$, the initial source term ${{j}^{*}}<0$, indicating a dominant desorption process over the interface at the initial stage of the simulation; whereas for $k>0.429$, ${{j}^{*}}>0$ and the adsorption process dominates at $\tau =0$. In this section, $k$ varies from 0.1 to 1 by increasing ${{r}_{a}}$ alone, and the other parameters are taken as $Bi=1$, $P{{e}_{b}}=10$ and $h=0.5$.

Since $k$ affects the initial mass transfer of surfactants, in figure 4(a) we present the distributions of the dimensionless source term ${{j}^{*}}$ along the arc length at different time $\tau$ for $k= 0.1$, 0.429, 0.6 and 1. It is seen that, for $k=0.1$, initially, ${{j}^{*}}<0$ throughout the droplet interface, suggesting that the desorption dominates; as time passes by, the overall ${{j}^{*}}$ increases due to desorption, and an equilibrium state is finally achieved with ${{j}^{*}}<0$ at the droplet poles (desorption) and ${{j}^{*}}>0$ at the equator (adsorption). For $k=0.429$, ${{j}^{*}}=0$ at $\tau =0$, revealing an initially quasi-equilibrium configuration. In this case, the interface surfactant concentration increases at the droplet poles but decreases at the equator, and at equilibrium we also find that ${{j}^{*}}<0$ at the poles and ${{j}^{*}}>0$ at the equator. For $k=0.6$ and 1, ${{j}^{*}}>0$ at the beginning; as time passes by, ${{j}^{*}}$ decreases gradually, and finally we still get ${{j}^{*}}<0$ at the poles and ${{j}^{*}}>0$ at the equator. Interestingly, we notice that the difference between ${{j}^{*}}$ distributions at different values of $k$ reduces as the droplet evolves towards the equilibrium state, and eventually the influence of $k$ on non-uniform effects can be negligible.

Figure 4.

(a) The distributions of the dimensionless source term ${{j}^{*}}$ and (b) the dimensionless interface concentration ${{\psi }^{{*}}}$ along the arc length $s$ for various $k$ at different times.

In figures 4(b) and 5 we plot the evolution of the distribution of ${{\psi }^{*}}$ along the arc length $s$ and the distribution of ${{\phi }^{*}}$ in the computational domain for different $k$. It is observed that the interface and bulk surfactant distributions are initially of unity for all $k$, and then the high- and low-concentration regions are formed at the poles and equators, respectively, for both interface and bulk surfactants. In addition, the average interface and bulk surfactant concentrations vary significantly for different $k$. Specifically, for $k=0.1$, the surfactant desorption dominates; ${{\phi }^{*}}$ first increases in the neighbourhood of the interface, and then, the high surfactant concentration spreads to the whole domain due to diffusion, leading to an overall increase of ${{\phi }^{*}}$ and a decrease of ${{\psi }^{*}}$. For $k=0.429$, as the droplet deforms, the high- and low-concentration regions are formed at the droplet poles and equator, respectively, and a relatively small change of the average interface and bulk surfactant concentrations is observed. For $k=0.6$ and 1, the surfactants are initially absorbed onto the interface, and then a low bulk concentration region appears in the neighbourhood of the interface that later diffuses to the whole domain. As a result, the overall ${{\phi }^{*}}$ decreases and ${{\psi }^{*}}$ increases.

Figure 5.

The distributions of the bulk surfactant concentration ${{\phi }^{*}}$ at different times for (a) $k=0.1$, (b) $k=0.428$, (c) $k=0.6$ and (d) $k=1$.

In figure 6 we present ${{D}_{f}}$ and $M_{s}^{*}$ at different values of $k$ and find that as $k$ increases, $M_{s}^{*}$ increases, leading to an increase in ${{D}_{f}}$. It is also noticed that as $k$ changes from 0.1 to 1, ${{D}_{f}}$ increases from 0.310 to 0.376, with an increment up to $20.92\,\%$.

Figure 6.

The deformation parameter ${{D}_{f}}$ and the dimensionless total mass of interface surfactants $M_{s}^{*}$ versus the adsorption number $k$.

3.4. Influence of adsorption depth

In this section the effect of the adsorption depth $h={{\psi }_{0}}/({{\phi }_{0}}R)$ on droplet deformation is investigated for $h$ varying from 0.1 to 1. Different values of $h$ are achieved by varying ${{\phi }_{0}}$, and ${{r}_{a}}$ should be changed accordingly to keep a constant ${{\phi }_{0}}{{r}_{a}}$. The other parameters are chosen as $Bi=1$, $P{{e}_{b}}=10$ and $k=0.429$.

In figure 7 we plot ${{D}_{f}}$ and $M_{s}^{*}$ as functions of $h$, and the distributions of ${{j}^{*}}$, ${{\psi }^{*}}$ and ${{\phi }^{*}}$ at different values of $h$. As shown in figure 7(a), as $h$ increases, $M_{s}^{*}$ and ${{D}_{f}}$ decrease. This is because as the adsorption depth $h$ increases, ${{r}_{a}}$ increases but ${{\phi }_{0}}$ decreases, indicating a decreased bulk surfactant concentration but a faster adsorption rate. With a faster adsorption rate from the bulk phase, a decreased bulk concentration in the vicinity of the droplet equator, as observed in figure 7(d), could hinder the further adsorption from the bulk phase. As a result, $M_{s}^{*}$ decreases with the increase of $h$. In addition, it can be seen in figure 7(b) that with increasing $h$, the amplitude of ${{j}^{*}}$ decreases, leading to the decrease of the surfactant mass transfer rate and, thus, to the increase of the non-uniform effects of interface surfactants, as shown in figure 7(c). In this case, $M_{s}^{*}$ dominates the droplet behaviour, so the droplet deformation ${{D}_{f}}$ decreases as $h$ increases.

Figure 7.

(a) The deformation parameter ${{D}_{f}}$ and the dimensionless total mass of interface surfactants $M_{s}^{*}$ as functions of $h$; the distributions of (b) the dimensionless source term ${{j}^{*}}$ and (c) the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length $s$ under various $h$; (d) the dimensionless bulk surfactant concentration ${{\phi }^{*}}$ under various $h$.

3.5. Summary of the influences of bulk surfactant parameters

By investigating the influences of various bulk surfactant parameters on droplet deformation, we have shown that the dimensionless total mass of the interface surfactants $M_{s}^{*}$ and the dimensionless source term ${{j}^{*}}$ can be good indicators for the variation of the average interface surfactant concentration and non-uniform effects. The increase of $M_{s}^{*}$ enhances the average interface surfactant concentration, while the increase of the amplitude of ${{j}^{*}}$ promotes the surfactant mass transfer rate, thus weakening non-uniform effects of interface surfactants. In most cases, $M_{s}^{*}$ determines the change of ${{D}_{f}}$. In addition, we summarize the minimum ($D_{f}^{min}$) and maximum ($D_{f}^{max}$) deformation parameters obtained from the simulations and their relative change in table 2. It can be seen that all the bulk surfactant parameters, including $Bi$, $P{{e}_{b}}$, $k$ and $h$, influence the droplet deformation but their influences are different. Specifically, the influences of $Bi$, $P{{e}_{b}}$ and $h$ are relatively small with the relative ${{D}_{f}}$ changes below $3\,\%$, while $k$ has a pronounced effect on the deformation with the relative ${{D}_{f}}$ change as high as $20.92\,\%$. This indicates that ${{D}_{f}}$ can be effectively controlled by varying $k$, and thus, in the following 3-D simulations, $k$ is varied to represent different solubilities of surfactants.

Table 2.

The minimum and maximum deformation parameters and their relative change for different bulk surfactant parameters.

Finally, it is interesting to show the connection between the bulk surfactant parameters used in our investigation and those used in existing experiments. Recently, Kalogirou & Blyth (Reference Kalogirou and Blyth2020) summarized the typical values of the properties of fluids and surfactants and the ranges of dimensionless parameters in existing experiments, from which one can obtain the ranges of the bulk surfactant parameters involved in experiments: $P{{e}_{b}}$ in the range of $1\unicode{x2013}{{10}^{7}}$, $Bi$ in the range of ${{10}^{-9}}\unicode{x2013}{{10}^{1}}$, $k$ in the range of ${{10}^{-11}}\unicode{x2013}{{10}^{1}}$ and $h$ in the range of ${{10}^{-4}}\unicode{x2013}{{10}^{3}}$. In this section we have investigated the impact of dimensionless parameters on the droplet behaviour for $P{{e}_{b}}$ ranging from 0.1 to 100, $Bi$ ranging from 0.01 to 10, $k$ ranging from 0.1 to 1 and $h$ ranging from 0.1 to 1. It is clear that the bulk surfactant parameters used in our investigation all fall within the ranges of the previous experimental parameters except for $P{{e}_{b}}=0.1$, which is slightly lower than the lower bound of $P{{e}_{b}}=1$ in experiments. Although $P{{e}_{b}}=0.1$ may have rarely happened in realistic surfactant systems, it was still investigated numerically in the literature (Severino, Campana & Giavedoni Reference Severino, Campana and Giavedoni2005; Xu et al. Reference Xu, Li, Lowengrub and Zhao2006; Shang et al. Reference Shang, Luo, Bai, He and Hu2024), which can provide additional perspectives for understanding surfactant dynamics and their control mechanisms.

4.1. Droplet deformation

For a clean droplet in simple shear flow, Vananroye et al. (Reference Vananroye, Van Puyvelde and Moldenaers2007) found that in the Stokes flow regime, the droplet deformation can be accurately predicted by the MMSH model, which is a combination of the Maffettone & Minale (Reference Maffettone and Minale1998) and Shapira & Haber (Reference Shapira and Haber1990) models, when $Ca\le 0.3$. Liu et al. (Reference Liu, Ba, Wu, Li, Xi and Zhang2018) found that the presence of insoluble surfactants enhances the droplet deformation in a simple shear flow, and the contribution of surfactants increases with $Ca$. Later, Liu et al. (Reference Liu, Zhang, Ba, Wang and Wu2020) focused on the deformation and breakup of a 3-D droplet on a solid surface subject to shear flow, and showed that the addition of insoluble surfactants always promotes droplet breakup regardless of viscosity ratio and the promoting effect is more pronounced at low viscosity ratios. Here, we investigate how soluble surfactants influence the droplet behaviour at various $Ca$ and $\lambda$.

Figure 8(a) shows the evolution of the deformation parameter ${{D}_{f}}$ for $Ca=\{ 0.1, 0.2, 0.3 \}$ and $\lambda =1$ in the clean, insoluble-surfactant and soluble-surfactant ($k=0.429$ and 1) cases, and the steady state ${{D}_{f}}$ vs $Ca$ is plotted in figure 8(b), where the deformation parameters for the clean, insoluble-surfactant and soluble-surfactant cases are denoted as ${{D}_{f0}}$, ${{D}_{f1}}$ and ${{D}_{f2}}$ for description convenience. In addition, the corresponding steady-state droplet shapes in the surfactant cases are plotted in figure 9, where the droplet surface is coloured by the dimensionless interface surfactant concentration ${{\psi }^{*}}$. As shown in figure 8(a), ${{D}_{f}}$ first increases and then evolves to a steady value for all the cases except the soluble-surfactant case with $k=1$ and $Ca=0.3$, in which the droplet eventually breaks into two parts (see figure 9l) and, thus, we only show the evolution of ${{D}_{f}}$ before the breakup occurs. When the droplet can reach the steady state, the steady-state deformation value increases with $Ca$, and the surfactants prefer to accumulate at the tips of the droplet where the interface surfactant concentration is the highest (figure 9g). For a fixed $Ca$, the addition of surfactants always enhances the droplet deformation, but the enhancement of ${{D}_{f}}$ varies with the surfactant type and $k$. For the insoluble case and the soluble case with $k=0.429$, ${{D}_{f}}$ varies slightly. This is because the soluble case with $k=0.429$ corresponds to the state where the surfactant desorption and adsorption balance and, thus, the surfactant mass exchange between the interface and the bulk can be roughly ignored, like in the insoluble case. For the soluble case with $k=1$, ${{D}_{f}}$ needs a much longer time to achieve the steady state, and ${{D}_{f}}$ is greatly enhanced, compared with the insoluble case and the soluble case with $k=0.429$.

Figure 8.

(a) The evolution of ${{D}_{f}}$ at $Ca=\{ 0.1, 0.2, 0.3 \}$, (b) ${{D}_{f}}$ vs $Ca$ for $\lambda =1$ in the clean, insoluble and soluble ($k=0.429$ and 1) systems. For the evolution of ${{D}_{f}}$ at $Ca=0.3$ in the soluble system with $k=1$, we only show the results before droplet breakup. In (b) the predictions from the MMSH model are also shown for comparison, and an inset is included to show the increment of ${{D}_{f}}$ caused by insoluble surfactants (denoted as ${{\delta }_{1}}$) and the increment caused by solubility (denoted as ${{\delta }_{2}}$) under various $Ca$.

Figure 9.

(a) The steady droplet shapes or the droplet shape after the breakup for $Ca=\{ 0.1, 0.2, 0.3 \}$ and $\lambda =1$ in the clean, insoluble and soluble ($k=0.429$ and 1) systems. The droplet surface is coloured by the dimensionless interface surfactant concentration for systems with surfactants. In each graph, the maximum and minimum values of the interface surfactant concentration are labelled. Note that the droplet eventually breaks up into two parts in (l). Results are shown for (a) $Ca = 0.1$, clean; (b) $Ca = 0.2$, clean; (c) $Ca = 0.3$, clean; (d) $Ca = 0.1$, insoluble; (e) $Ca = 0.2$, insoluble; (f) $Ca = 0.3$, insoluble; (g) $Ca = 0.1, k = 0.429$; (h) $Ca = 0.2, k = 0.429$; (i) $Ca = 0.3, k = 0.429$; (j) $Ca = 0.1, k = 1$; (k) $Ca = 0.2, k = 1$; (l) $Ca = 0.3, k = 1$.

As shown in figure 8(b), the simulated ${{D}_{f}}$ values all agree well with the predictions from the MMSH model in the clean case. In the presence of surfactants, ${{D}_{f}}$ is enhanced for all the $Ca$ values considered, and the increased deformation $\delta ={{D}_{f2}}-{{D}_{f0}}$ consists of two parts: (1) the insoluble part ${{\delta }_{1}}={{D}_{f1}}-{{D}_{f0}}$, which is attributed to the decrease of the interfacial tension induced by the average interface surfactant concentration, and non-uniform effects (Liu et al. Reference Liu, Ba, Wu, Li, Xi and Zhang2018) caused by the non-uniform capillary forces and Marangoni stresses; and (2) the soluble part ${{\delta }_{2}}={{D}_{f2}}-{{D}_{f1}}$, which is attributed to the change of the average interface surfactant concentration and surfactant distribution caused by the adsorption and desorption at the interface. For the insoluble case, ${{\delta }_{1}}$ increases monotonously with $Ca$, consistent with the previous results of Liu et al. (Reference Liu, Ba, Wu, Li, Xi and Zhang2018). For the soluble case with $k=0.429$, ${{\delta }_{2}}$ is nearly zero owing to the balance between the surfactant desorption and adsorption, whereas for $k=1$, ${{\delta }_{2}}$ is greatly enhanced, because a higher $k$ favours the adsorption process, leading to a much higher average interface surfactant concentration.

In figure 10 we present the dimensionless total mass of interface surfactants $M_{s}^{*}$ and the minimum and maximum values of interface surfactant concentration (i.e. $\psi _{min}^{*}$ and $\psi _{max}^{*}$) in the steady state. It is seen in figure 10(a) that in the insoluble case and the soluble case with $k=0.429$, $M_{s}^{*}$ is equal or close to 1, indicating negligible surfactant mass transfer between the bulk and the interface. As $k$ changes from 0.429 to 1, $M_{s}^{*}$ is increased by nearly $50\,\%$ due to a dominant adsorption at the interface, leading to the greatly enhanced ${{D}_{f}}$ in figure 8(b). In addition, we note that $M_{s}^{*}$ increases with $Ca$, since the dilution of interface surfactants at large $Ca$ favours the surfactant adsorption and suppresses the desorption at the interface. In figure 10(b) it is observed that as surfactants change from the insoluble case to the soluble case with $k=0.429$, the minimum interface surfactant concentration $\psi _{min }^{*}$ increases but the maximum interface surfactant concentration $\psi _{max}^{*}$ decreases; while as $k$ increases from 0.429 to 1, both $\psi _{min }^{*}$ and $\psi _{max}^{*}$ increase. From these three cases, the non-uniform effects, represented by $\psi _{max}^{*}-\psi _{min }^{*}$, is the highest in the insoluble case, and it changes little as $k$ changes from 0.429 to 1.

Figure 10.

(a) The dimensionless total mass of interface surfactants $M_{s}^{*}$, and (b) the minimum and maximum values of interface surfactant concentration ($\psi _{min}^{*}$ and $\psi _{max}^{*}$) for various $Ca$ in the insoluble and soluble ($k=0.429$ and 1) systems at $\lambda =1$.

In figures 11 and 12 we plot ${{D}_{f}}$, $M_{s}^{*}$, $\psi _{min}^{*}$ and $\psi _{max}^{*}$ as functions of $Ca$ for the viscosity ratios of 0.3 and 5. In figure 11 the predictions from the MMSH model at $\lambda =0.3$ and 5 are also shown for comparison, and it is seen that the simulated ${{D}_{f}}$ values in the clean case match perfectly with the MMSH predictions. As shown in figure 11(a), at $\lambda = 0.3$, the droplet deformation is higher in the insoluble system (${{D}_{f1}}$) than in the soluble system with $k=0.429$ (${{D}_{f2}}$). The increment caused by the solubility ${{\delta }_{2}}={{D}_{f2}}-{{D}_{f1}}<0$ and $| {{\delta }_{2}} |$ increases with $Ca$. This is because although $M_{s}^{*}$ is larger in the soluble system with $k=0.429$, non-uniform effects are more pronounced in the insoluble system (figure 12a,b), which dominate the droplet deformation, thereby resulting in a negative ${{\delta }_{2}}$; and as $Ca$ increases, non-uniform effects increase rapidly in the insoluble system (figure 12b), leading to the increase of $| {{\delta }_{2}} |$ with $Ca$. In the soluble system with $k=1$, $M_{s}^{*}$ is much larger than in the other two systems, and thus, ${{D}_{f}}$ is higher. In figure 11(b) we find that at $\lambda =5$ the ${{D}_{f}}$ values obtained in the insoluble system and in the soluble system with $k=0.429$ are nearly the same for all $Ca$, which is attributed to their similar $M_{s}^{*}$ and the relatively weak non-uniform effects, as shown in figure 12(c,d); while in the soluble system with $k=1$, ${{D}_{f}}$ becomes much higher due to the remarkably enhanced $M_{s}^{*}$.

Figure 11.

Plots of ${{D}_{f}}$ vs $Ca$ in the clean, insoluble and soluble ($k=0.429$ and 1) systems for (a) $\lambda =0.3$ and (b) $\lambda =5$, where the predictions from the MMSH model are also shown for comparison. In (a,b) an inset is included to show the increment of ${{D}_{f}}$ caused by the insoluble surfactant (denoted as ${{\delta }_{1}}$) and the increment caused by the solubility (denoted as ${{\delta }_{2}}$) under various $Ca$.

Figure 12.

(a) The dimensionless total mass of interface surfactants and (b) the minimum and maximum values of interface surfactant concentration ($\psi _{min }^{*}$ and $\psi _{max}^{*}$) versus $Ca$ in the insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =0.3$; (c) $M_{s}^{*}$ and (d) $\psi _{min }^{*}$ and $\psi _{max}^{*}$ vs $Ca$ in the insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =5$.

Figure 13 presents the steady droplet shapes in the clean, insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =\{ 0.3, 1, 5 \}$. As shown in this figure, as $\lambda$ changes from 1 to 0.3, the steady droplet shape and ${{D}_{f}}$ change slightly, but non-uniform effects become more pronounced for all the cases investigated (see figures 10b and 12b). Specifically, in the insoluble system the droplet deformation is larger at $\lambda =0.3$ than at $\lambda =1$ due to the enhanced non-uniform effects. In the soluble system, non-uniform effects are still more pronounced at $\lambda =0.3$, but the corresponding $M_{s}^{*}$ is lower (see figures 10a and 12a); under the combined influence of non-uniform effects and $M_{s}^{*}$, at $\lambda = 0.3$, we find a slightly smaller ${{D}_{f}}$ for $k=0.429$ but a slightly larger ${{D}_{f}}$ for $k=1$. Note that an exception occurs in the case of $Ca=0.3$, $k=1$ and $\lambda = 1$ where the droplet cannot reach a steady deformation but eventually break into two daughter droplets, and this will be discussed in the next subsection. In addition, it can be observed in figures 10, 12 and 13 that compared with the results at $\lambda =1$ and $\lambda = 0.3$, at $\lambda = 5$, ${{D}_{f}}$, $M_{s}^{*}$ and non-uniform effects are overall smaller or weaker, and these values increase slowly with the increase of $Ca$, which leads to very small ${{D}_{f}}$ for the whole range of $Ca$.

Figure 13.

The final droplet shapes for $Ca=\{ 0.1, 0.2, 0.3 \}$ in the clean, insoluble and soluble ($k=0.429$ and 1) systems at $\lambda =\{ 0.3, 1, 5 \}$, in which the red solid lines, the blue dashed lines and the green dash-dotted lines represent the droplet interfaces at $\lambda =1$, $\lambda =0.3$ and $\lambda =5$, respectively. Results are shown for (a) $Ca = 0.1$, clean; (b) $Ca = 0.2$, clean; (c) $Ca = 0.3$, clean; (d) $Ca = 0.1$, insoluble; (e) $Ca = 0.2$, insoluble; (f) $Ca = 0.3$, insoluble; (g) $Ca = 0.1, k = 0.429$; (h) $Ca = 0.2, k = 0.429$; (i) $Ca = 0.3, k = 0.429$; (j) $Ca = 0.1, k = 1$; (k) $Ca = 0.2, k = 1$; (l) $Ca = 0.3, k = 1$.

To sum up, in the insoluble system, as the viscosity ratio $\lambda$ increases, non-uniform effects weaken, leading to a monotonic decrease of ${{D}_{f}}$, unlike a droplet in the clean system where ${{D}_{f}}$ first increases slightly and then decreases with $\lambda$ (Maffettone & Minale Reference Maffettone and Minale1998; Vananroye et al. Reference Vananroye, Van Puyvelde and Moldenaers2007). On the other hand, in the soluble system, under the combined action of $M_{s}^{*}$ and non-uniform effects, with the increase of $\lambda$, ${{D}_{f}}$ first increases slightly and then decreases for $k=0.429$, but decreases monotonously for $k=1$.

To predict the deformation parameter $D_f$ in the shear system with soluble surfactants, a simple way is to replace the capillary number $Ca$ with the effective capillary number $C{{a}_{e}}$ in the MMSH model, where the effective capillary number is defined as $C{{a}_{e}}=Ca/[1+{{E}_{0}}\ln (1-{{x}_{in}}{{M}_{s}})]$. The prediction model obtained in such a way is called the modified MMSH model and, clearly, the modified MMSH model has taken into account the influence of ${{M}_{s}}$ that mainly affects the effective interfacial tension. Figure 14 shows the comparison between the simulated deformation parameters ($D_f$) and those predicted from the modified MMSH model ($D_{f,M}$) for three different viscosity ratios. It is seen that for $\lambda =0.3$, the difference between $D_f$ and $D_{f,M}$ is the largest, which is followed by $\lambda =1$, and when $\lambda$ is increased to $5$, $D_{f,m}$ agrees well with the simulated $D_{f}$ with a maximum relative error of $4.71\,\%$. This suggests that the modified MMSH model is only suitable for predicting droplet deformation at high viscosity ratios, which is easily understandable because the modified MMSH model has not incorporated the influence of non-uniform effects and non-uniform effects can be ignored only at high viscosity ratios (see figure 12b,d). To predict the droplet deformation at low and moderate viscosity ratios, non-uniform effects should be properly quantified, which we would like to do in future work.

Figure 14.

The steady-state deformation parameter as a function of $Ca_e$ in the insoluble and soluble ($k=0.429$ and 1) systems for (a) $\lambda =0.3$, (b) $\lambda =1$ and (c) $\lambda =5$. The filled symbols represent the simulation results ($D_f$) while the hollow symbols represent the predictions from the modified MMSH model ($D_{f,M}$).

4.2. Droplet breakup

As $Ca$ increases above a critical value, the droplet no longer reaches a steady deformation but breaks up, and the critical value is called the critical capillary number of droplet breakup, denoted as $C{{a}_{cr}}$. In simple shear flow, $C{{a}_{cr}}$ is often quantified as the function of the viscosity ratio and confinement ratio (Janssen et al. Reference Janssen, Vananroye, Van Puyvelde, Moldenaers and Anderson2010). Liu et al. (Reference Liu, Ba, Wu, Li, Xi and Zhang2018) showed that the presence of insoluble surfactants tends to reduce $C{{a}_{cr}}$. Complementing the previous works, we study the influence of soluble surfactants on $C{{a}_{cr}}$ under various viscosity ratios and Reynolds numbers. In the simulations the height of the computational domain is set as $H=4R$ with $R=25$ so that the confinement ratio is $2R/H=0.5$, and the domain length and width are set as $10R$ and $6R$, respectively. As previously done in Liu et al. (Reference Liu, Ba, Wu, Li, Xi and Zhang2018), we increase $Ca$ by 0.01 at a time, and the lowest $Ca$ at which the droplet breaks up is considered as $C{{a}_{cr}}$.

Figure 15 presents $C{{a}_{cr}}$ for different viscosity ratios $\lambda$ in the clean, insoluble and soluble ($k=0.429$ and 1) systems, where insets are included to show snapshots of the droplet shapes just before and after breakup. It is seen that in each case, $C{{a}_{cr}}$ first decreases and then increases as $\lambda$ increases. The addition of surfactants always promotes the droplet breakup and for a fixed $\lambda$, $C{{a}_{cr}}$ is slightly smaller in the soluble case with $k=0.429$ than in the insoluble case, which is because in the soluble case the interface surfactant concentration in the necking region is increased, accelerating the necking process (see the ${{\psi }^{*}}$ distributions at the droplet surface before breakup in figure 15). In addition, increasing $k$ from 0.429 to 1 favours the droplet breakup, consistent with the droplet deformation trend shown previously in § 4.1.

Figure 15.

Plot of $C{{a}_{cr}}$ for different viscosity ratios in the clean, insoluble and soluble ($k=0.429$ and 1) systems. The insets are included to show the droplet shapes immediately before and after droplet breakup, and the droplet surface is coloured by ${{\psi }^{*}}$ in both insoluble and soluble systems.

In figure 16 we plot the snapshots of the droplet breakup process in the clean, insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =1$, and for the surfactant cases, we present the distributions of the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length in the $x$–$z$ mid-plane at different dimensionless time $\tau$ in figure 17. As shown in figure 16, we can see that the droplet experiences the stretching and necking stages before breakup. For insoluble surfactants, we can see in figure 17(a) that in the initial stretching stage ($\tau \le 10$), $\psi _{max}^{*}$ occurs near the droplet poles while $\psi _{min }^{*}$ at the equator of the droplet; the maximum interface surfactant concentration $\psi _{max}^{*}$ first increases to 2.056 then decreases slightly to 2.017, and the minimum interface surfactant concentration $\psi _{min }^{*}$ decreases. After the stretching stage, $\psi _{min }^{*}$ starts to migrate from the equator towards the poles at $\tau =19$, and a neck gradually forms in the middle of the droplet, where a local maximum interface surfactant concentration $\psi _{neck}^{*}$ appears, which promotes the thinning of the neck. At the same time, $\psi _{max}^{*}$ decreases severely from 2.017 at $\tau =19$ to 1.508 at $\tau =54.4$ (with a decrease of about $25\,\%$) while $\psi _{min }^{*}$ almost keeps at 0.45, exhibiting a severe dilution of the interface surfactants. The necking stage almost ends at $\tau =54.4$, when $\psi _{neck}^{*}$ increases to 0.917; finally, the droplet breaks up into two daughter droplets at $\tau =55$. For the soluble surfactants with $k=0.429$ (figure 17b), in the stretching stage ($\tau \le 10$), $\psi _{max}^{*}$ increases to 1.621, and $\psi _{min }^{*}$ decreases to around 0.59; $\psi _{max}^{*}$ is lower than in the insoluble case, but $\psi _{min }^{*}$ is higher, which is caused by the surfactant adsorption and desorption at the interface. In the necking stage, $\psi _{max}^{*}$ decreases from 1.621 to 1.473, with a decrease of about $10\,\%$, which is much smaller than that in the insoluble case, while $\psi _{min }^{*}$ holds almost constant. This means that in the soluble case an increased $M_{s}^{*}$ and a reduced non-uniformity of surfactants are observed compared with the insoluble case. In addition, before the droplet breaks up, $\psi _{neck}^{*}$ increases to 1.127 at $\tau =62.4$, much higher than in the insoluble case. Therefore, the surfactant dilution is effectively supplemented by the bulk surfactants, and the increase of $\psi _{neck}^{*}$ reduces the interfacial tension in the neck region, which both contribute to the droplet breakup. As $k$ increases to 1 (see figure 17c), $\psi _{max}^{*}$ increases to 2.108 and $\psi _{min }^{*}$ holds at around 0.9 in the stretching stage. In the necking stage, $\psi _{max}^{*}$ decreases with a decrement within $10\,\%$ while $\psi _{min }^{*}$ remains roughly a constant, and $\psi _{neck}^{*}$ increases to 1.524, much higher than the other two cases. Thus, with the higher average interface surfactant concentration and $\psi _{neck}^{*}$, $C{{a}_{cr}}$ decreases to 0.3.

Figure 16.

The snapshots of droplet breakup for $\lambda =1$ in (a) the clean system with $C{{a}_{cr}}=0.392$, (b) the insoluble system with $C{{a}_{cr}}=0.34$, (c) the soluble system ($k=0.429$) with $C{{a}_{cr}}=0.33$ and (d) the soluble system ($k=1$) with $C{{a}_{cr}}=0.3$. Note that in (b–d) the droplet surface is coloured by the dimensionless interface surfactant concentration ${{\psi }^{*}}$.

Figure 17.

The distributions of the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length $s$ in the $x$–$z$ mid-plane at different dimensionless time $\tau$ in (a) the insoluble case, (b) the soluble case with $k=0.429$ and (c) the soluble case with $k=1$. The viscosity ratios are all taken as $\lambda =1$, and the critical capillary numbers are exactly the same as those in figure 16(b–d). Note that the pink and blue filled circles are to highlight the droplet tips and the necking regions, respectively.

In figure 18 we present the distributions of ${{\psi }^{*}}$ along the arc length $s$ at the $x$–$z$ mid-plane in the insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =0.3$ and $\lambda =5$. At $\lambda =0.3$, the breakup process is similar to that at $\lambda =1$, both experiencing the stretching and necking stages, but the necking stage lasts for a shorter period; the surfactant distributions are also similar to those at $\lambda =1$ for each case, but non-uniform effects are generally more pronounced than at $\lambda =1$, which hinders the increase of $\psi _{neck}^{*}$, and thus, increases $C{{a}_{cr}}$ for each case.

Figure 18.

The distributions of the dimensionless interface surfactant concentration ${{\psi }^{*}}$ along the arc length in the $x$–$z$ mid-plane at different dimensionless time $\tau$ for (a) $\lambda =0.3$ and (b) $\lambda =5$. Note that the pink and blue filled circles are to highlight the droplet tips and the necking regions, respectively.

At $\lambda =5$, the droplet breakup exhibits a significant difference. We take the soluble case with $k=1$, $\lambda =5$ and $C{{a}_{cr}}=0.58$ as an example, and the corresponding snapshots of the droplet shape and ${{D}_{f}}$ evolution are shown in figure 19. As seen in this figure, ${{D}_{f}}$ first increases rapidly to around 0.5 and then increases at a much lower speed until the breakup occurs, and the necking stage lasts very long since the droplet is exposed to weak shear flow due to a low inclination angle in this case. In figure 18(b) we can clearly observe that non-uniform effects are relatively weak at $C{{a}_{cr}}$ for all cases. Specifically, in the insoluble case the non-uniformity of interface surfactants is small, with $\psi _{max}^{*}-\psi _{min }^{*}=0.88$ at $\tau =5$, and decreases significantly in the necking stage (e.g. $\psi _{max}^{*}-\psi _{min }^{*}=0.33$ at $\tau =50$); when approaching the end of the necking stage ($\tau =80.75$), $\psi _{max}^{*}-\psi _{min }^{*}$ reduces to around 0.27 under the effect of dilution, and $\psi _{neck}^{*}$ is close to $\psi _{min }^{*}$, much lower than $\psi _{neck}^{*}$ at $\lambda =1$ and $0.3$. In the soluble case with $k=0.429$, $\psi _{max}^{*}-\psi _{min }^{*}$ is almost the same as that in the insoluble case before breakup, but due to the surfactant adsorption from the bulk phase, the average interface surfactant concentration ${{\psi }^{*}}$ is higher, quickening the breakup process. Upon increasing $k$ to 1, a similar distribution of ${{\psi }^{*}}$ is also observed, but the average interface surfactant concentration is higher and the surfactants become more non-uniform before droplet breakup. As a result, the droplet breakup is promoted, corresponding to a lower $C{{a}_{cr}}$.

Figure 19.

(a) Snapshots of the droplet shape and (b) the time evolution of deformation parameter ${{D}_{f}}$ in the soluble case with $k=1$ at $\lambda =5$ and $C{{a}_{cr}}=0.58$.

Another important dimensionless parameter governing droplet breakup is the Reynolds number, and its influence on $Ca_{cr}$ is investigated in the clean, insoluble and soluble ($k=0.429$ and 1) systems for the viscosity ratio of 1, and the results are displayed in figure 20. It is seen that in each of the systems considered, as $Re$ increases, the critical capillary number $C{{a}_{cr}}$ decreases monotonically, and the droplet breakup undergoes the transition from binary breakup ($Re\le 4$) to ternary breakup ($Re=10$). In addition, we observe that regardless of the value of $Re$, the presence of surfactants always reduces $C{{a}_{cr}}$, and the greater the solubility of surfactants, the more the reduction of $C{{a}_{cr}}$. As $Re$ increases, especially when increasing from 4 to 10, we interestingly find that the influence of solubility on $C{{a}_{cr}}$ weakens since inertia gradually dominates the breakup mechanism, and the influence of surfactants also shows a weakening trend.

Figure 20.

Critical capillary number of droplet breakup as a function of Reynolds number in the clean, insoluble and soluble ($k=0.429$ and 1) systems for $\lambda =1$. Binary breakup and ternary breakup are represented by the discrete symbols without ‘$+$’ and the discrete symbols with ‘$+$’, respectively. Each inset plots the droplet shapes before and after breakup.