3.3.1. Formation of hot spots inside the droplet

The interaction between a droplet of diameter ${D_0}$ and radiation of wavelength ${\lambda _0}$ can be characterized based on the non-dimensional Mie size parameter (Park & Armstrong Reference Park and Armstrong1989)

(3.11)\begin{equation}{\alpha _m} = \frac{{{\rm \pi} {D_0}}}{{{\lambda _0}}}.\end{equation}

The Mie size parameter ${\lambda _0}$ dictates the evolution of the thermal energy field inside the droplet through scattering, absorption and extinction cross-sections of the incident radiation source. Homogeneous internal heat generation occurs for ${\alpha _m} < 1.5$, corresponding to a very small droplet size (diameter 2.25 μm). Figures 4(a) and 4(b) show the normalized droplet size (diameter) during the entire evaporation process for different irradiation intensities and initial polymer concentration. The initial droplet diameter ${D_0}$ of 850 μm regresses to approximately 300–400 μm during the evaporation phase A.

Comparing the droplet diameter with the equivalent length scale corresponding to the Mie size parameter, we observe that our experimental conditions result in non-uniform heat evolution inside the droplet. Although the essential mechanism of heat evolution is non-uniform for all our cases due to significantly large droplet size, we can still isolate two distinct heating regimes (fast heating vs slow heating) based on the ratio of evaporation time scale $(t)$ to the thermal diffusion time scale $({t_d})$. The thermal diffusion time scale ${t_d}$ provides insights into the time required for the temperature field inside the droplet to equilibrate. Therefore, depending on the evaporation time scale, the laser heating can either be slow or fast. Quantitatively, $t/{t_d} < 1$ leads to inhomogeneous heating forming hot spots (characterized as fast heating). However, for $t/{t_d} > 1$, homogeneous heating of the droplet is attained, causing a symmetric and approximately uniform temperature profile evolution inside the droplet (characterized by slow heating). It can be inferred from figure 4(a) that, for a fixed concentration $(c/{c^\ast } = 83.3)$, the ratio of $t/{t_d}$ varies with ${I^\ast }$. For ${I^\ast } = 0.7$, $t/{t_d} \sim 2.2$ suggests slow heating, leading to a symmetric expansion of the membrane. However, for ${I^\ast } \ge 1.5$, the ratio $t/{t_d} < 1$ corresponds to fast heating, resulting in localized hot spots and subsequent asymmetric membrane growth. Literature on droplet–laser interaction and boiling suggests that liquid droplets can become superheated and remain in a liquid metastable state above 373 K at atmospheric pressure (Park & Armstrong Reference Park and Armstrong1989). The maximum temperature at hot spots that the droplet can attain before spontaneous nucleation in pure liquids is (Prishivalko & Leiko Reference Prishivalko and Leiko1980)

(3.12)\begin{equation}{T_{su}} \sim 0.9{T_{cr}} \sim 582\ \textrm{K}. \end{equation}

Here, ${T_{cr}}$ represents the critical temperature of water and its value is 647 K. This study, however, deals with aqueous polymeric droplets, and as we have discussed in § 3.2, evaporation of the droplets leads to skin formation and regions of excessively high concentration of polymeric networks that can act as nucleation sites forming vapour bubbles at a temperature lower than the ${T_{su}}$ required by spontaneous nucleation.

The interaction between the laser and the droplet can be represented as a volumetric heat generation term in the energy equation. The actual mechanism of the heat generation is an electromagnetic and molecular phenomenon that occurs due to the interaction of the photons from the laser beam with the molecules of the polymeric droplet. The differential form of the energy equation is given as

(3.13)\begin{equation}{\rho _l}{c_l}\left( {\frac{{\partial T}}{{\partial t}} + \boldsymbol{u}\boldsymbol{\cdot }{{\boldsymbol{\mathsf{\nabla}}}}T} \right) = {k_l}{\nabla ^2}T + {q_s}.\end{equation}

Here, the left-hand side represents the unsteady and convective term inside the droplet and the right-hand side represents the diffusion term and source term (volumetric heat generation). Also, ${\rho _l}$, ${c_l}$, T, t, u, ${{\boldsymbol{\mathsf{\nabla}}} }$, ${k_l}$, ${q_s}$ represent the density of the droplet, the specific heat capacity of the droplet, temperature, time, flow velocity inside the droplet, gradient operator, thermal conductivity of the droplet and volumetric heat generation source term, respectively. The volumetric heat generation depends on the optical properties of the droplet. The properties relevant to the laser–polymeric droplet interaction, such as refractive index and absorption coefficient, are almost equal to those of water. Supplementary figure S5 compares the absorptivity of water and PAM solution of concentration $c/{c^\ast } = 83.3$ for a wavelength range of $2$ to $16\ {\rm \mu}\textrm{m}$. As seen in the figure, the absorptivity values are almost identical to water. Liquid water is opaque to radiation in the infrared region and has a substantial absorption coefficient of ${10^5}\ {\textrm{m}^{ - 1}}$.

We estimate the average temperature of the droplet by integrating equation (3.13) over the entire droplet volume V. Here, we assume flow inside the droplet to be incompressible and that density changes of liquid are negligible with temperature. Note we do not assume a uniform temperature distribution inside the droplet. We have

(3.14)\begin{equation}\int_V {{\rho _l}{c_l}} \left( {\frac{{\partial T}}{{\partial t}} + \boldsymbol{u}\boldsymbol{\cdot }{{\boldsymbol{\mathsf{\nabla}}} }T} \right)\,\textrm{d}V = \int_V {{k_l}{\nabla ^2}T\,\textrm{d}V} + \int_V {{q_s}\,\textrm{d}V} .\end{equation}

The first term on the left-hand side of (3.14) becomes

(3.15)\begin{equation}\int_V {{\rho _l}{c_l}\frac{{\partial T}}{{\partial t}}\,\textrm{d}V} = \frac{{4{\rm \pi} {R^3}{\rho _l}{c_l}}}{3}\frac{{\textrm{d}T}}{{\textrm{d}t}},\end{equation}

where R is the radius of the droplet. The second term on the left-hand side of (3.14) can be expressed in a modified form using the product rule of derivatives.

(3.16)\begin{equation}\int_V {\boldsymbol{u}\boldsymbol{\cdot }{{\boldsymbol{\mathsf{\nabla}}} }T\,\textrm{d}V} = \int_V {{{\boldsymbol{\mathsf{\nabla}}} }\boldsymbol{\cdot }(\boldsymbol{u}T)\,\textrm{d}V} - \int_V {T{{\boldsymbol{\mathsf{\nabla}}} }\boldsymbol{\cdot }\boldsymbol{u}} .\end{equation}

As the flow inside the droplet is essentially incompressible in the liquid phase it hence satisfies the continuity equation

(3.17)\begin{equation}{{\boldsymbol{\mathsf{\nabla}}} }\boldsymbol{\cdot }\boldsymbol{u} = 0.\end{equation}

Note that we are still in the liquid phase and bubbles have not formed during temperature evolution. Substituting equation (3.17) in (3.16) and using the Gauss divergence theorem (considering the droplet surface as the Gaussian surface $S$), we have

(3.18)\begin{equation}\int_V {\boldsymbol{u}\boldsymbol{\cdot }{{\boldsymbol{\mathsf{\nabla}}} }T\,\textrm{d}V} = \int_V {{{\boldsymbol{\mathsf{\nabla}}} }\boldsymbol{\cdot }(\boldsymbol{u}T)\,\textrm{d}V} = \int_S {T\boldsymbol{u} \boldsymbol{\cdot} \hat{n}\,\textrm{d}S} = 0.\end{equation}

The first term on the right-hand side of (3.14) represents heat transfer due to conduction and second term represents heat generation due to radiation.

In general, the thermal diffusion time scales are comparatively higher than the radiative heating time scale. For our experimental condition the thermal diffusion time scale $({t_d})$ is of the order of seconds whereas the radiative heating time scales are smaller than a fraction of a microsecond

(3.19)\begin{equation}\int_V {{q_s}\,\textrm{d}V} \gg \int_V {{k_l}{\nabla ^2}T\,\textrm{d}V} .\end{equation}

Therefore, the dominant mechanism of temperature evolution inside the droplet is through radiative heating.

The last term of (3.14) is the total internal heat generation within the droplet volume represented as ${Q_s}$

(3.20)\begin{equation}{Q_s} = \int {{q_s}\,\textrm{d}V} = \int_0^{2R} {\alpha I(r,z)\,\textrm{d}A\,\textrm{d}r} ,\end{equation}

where $\alpha $ is the absorption coefficient, $I(r,z)$ is the beam profile, $\textrm{d}V = \textrm{d}A\,\textrm{d}r$ is the differential volume and $\textrm{d}A$ is the differential area perpendicular to the direction of propagation of the laser (see supplementary figure S6a). Note that r is in the radial direction along the laser beam and z is a direction perpendicular to the beam. In general, the beam intensity has a Gaussian profile, i.e. intensity is a function of z. However, the incident beam has a diameter of 3.5 mm. The beam diameter is approximately four times larger than the droplet diameter (850 μm). Hence, to obtain an approximate estimate of the temperature, we use a constant beam profile along the z direction (see supplementary figure S7), i.e.

(3.21)\begin{equation}I(r,z) = I(r).\end{equation}

As the laser beam travels through the droplet, the energy in the beam attenuates according to the Beer–Lambert law given as

(3.22)\begin{equation}I(r) = {I_0}\,{\textrm{e}^{ - \mu r}},\end{equation}

where $\mu $ is the extinction coefficient

(3.23)\begin{equation}\mu = \alpha + \kappa ,\end{equation}

and $\kappa $ is the scattering coefficient. However, $\alpha \gg k$, which implies $\mu \sim \alpha $. Using (3.21), (3.22) in (3.20), ${Q_s}$ becomes

(3.24)\begin{equation}{Q_s} = \int_0^{2R} {\alpha {I_0}\,{\textrm{e}^{ - \mu r}}\,\textrm{d}A\,\textrm{d}r} ,\end{equation}

where

(3.25)\begin{equation}\textrm{d}A = {\rm \pi}[R\sin \theta {\textrm{]}^2},\end{equation}

and

(3.26)\begin{equation}\sin \theta = \frac{{\sqrt {{R^2} - {{(R - r)}^2}} }}{R} = \frac{{\sqrt {2Rr - {r^2}} }}{R}.\end{equation}

The differential circular area perpendicular to the laser beam is

(3.27)\begin{equation}\textrm{d}A = {\rm \pi}(2Rr - {r^2}).\end{equation}

Using (3.27) in (3.24), we have

(3.28)\begin{equation}{Q_s} = \alpha {\rm \pi}{I_0}\int_0^{2R} {{\textrm{e}^{ - \mu r}}(2Rr - {r^2})\,\textrm{d}r} .\end{equation}

Using (3.15), (3.18), (3.19), (3.28) in (3.14) we have

(3.29)\begin{equation}\frac{{{\rho _l}{c_l}{R^3}}}{3}\frac{{\textrm{d}T}}{{\textrm{d}t}} = \frac{\alpha }{4}{I_0}\int_0^{2R} {{\textrm{e}^{ - \mu r}}(2Rr - {r^2})\,\textrm{d}r} .\end{equation}

Simplifying the definite integral in (3.29), we have

(3.30)\begin{equation}\frac{{{\rho _l}{c_l}{R^3}}}{3}\frac{{\textrm{d}T}}{{\textrm{d}t}} = \frac{{\alpha {I_0}}}{{2{\mu ^3}}}(\mu R + (\mu R - 1)\,{\textrm{e}^{2\mu R}} + 1)\,{\textrm{e}^{ - 2\mu R}}.\end{equation}

Simplifying further the rate of change in droplet temperature, we have

(3.31)\begin{equation}\frac{{\textrm{d}T}}{{\textrm{d}t}} = \frac{{3\alpha {I_0}}}{{2{\mu ^3}{\rho _l}{c_l}}}\frac{{(\mu R + (\mu R - 1)\,{\textrm{e}^{2\mu R}} + 1)\,{\textrm{e}^{ - 2\mu R}}}}{{{R^3}}}.\end{equation}

We can write (3.31) in a condensed form by incorporating a constant A and a change of variable $G(R(t),\mu )$

(3.32)\begin{equation}\frac{{\textrm{d}T}}{{\textrm{d}t}} = AG(R(t),\mu ),\end{equation}

where

(3.33)\begin{equation}A = \frac{{3\alpha {I_0}}}{{2{\mu ^3}{\rho _l}{c_l}}},\end{equation}

and

(3.34)\begin{equation}G(R(t),\mu ) = \frac{{(\mu R + (\mu R - 1)\,{\textrm{e}^{2\mu R}} + 1)\,{\textrm{e}^{ - 2\mu R}}}}{{{R^3}}}.\end{equation}

Note from (3.32) that the rate of temperature change is not a constant and changes nonlinearly with time. Integrating equation (3.32) using experimental regression curves for $R(t)$ we have

(3.35)\begin{equation}\int_{{T_0}}^T {\textrm{d}T} = A\int_0^t {G(R(t),\mu )\,\textrm{d}t} ,\end{equation}

where ${T_0}$ is the droplet initial temperature at $t = 0$ and T is the droplet temperature at any time $t$

(3.36)\begin{equation}T = {T_0} + A\int_0^t {G(R(t),\mu )\,\textrm{d}t} .\end{equation}

The temporal variation of volume-averaged temperature of the droplet for low and high irradiation intensities is shown in figure 5(b). Note that the above analysis becomes increasingly accurate for higher irradiation intensities under our stated assumptions. It can be seen that the interface temperature of the droplet (figure 5a) obtained during the evaporation phase for different irradiation intensities during the evaporation phase is less than the volume-averaged temperature of the droplet for different irradiation intensities. This essentially implies that the laser--droplet interaction occurs through volumetric heat generation inside the droplet.

3.3.2. Bubble growth

The bubble formed in the superheated aqueous polymeric droplet evolves under the influence of liquid inertia, the surface tension, pressure difference between the vapour bubble and the ambient liquid phase of the droplet. The bubble growth generally occurs in two phases. Initially, the bubble growth is slow due to the equilibrium of forces. However, it is accelerated with the increase in bubble size as a consequence of reduced surface tension due to an increase in temperature at the hot spots. After an initial exponential growth phase, the temperature and pressure within the bubble reduce, and the growth rate slows down. The temperature reduction within the bubble is due to the latent heat requirement of evaporation that happens at the liquid–vapour interface during the bubble growth. The amount of heat required for evaporation at the bubble boundary depends on the growth rate of the bubble. The standard bubble growth equation (Rayleigh–Plesset equation) coupled to the heat diffusion equation results in a modified Rayleigh–Plesset equation. The bubble growth takes place on a significantly short time scale (${O}(\mu s)$ compared with the evaporation time scales $({O}(1s))$ over which the laser heating takes place. Therefore, the liquid and vapour are assumed to be in equilibrium during bubble growth.

The bubble radius $({R_b})$ in the asymptotic limit of sufficiently large values relative to the nucleus size takes the form of the Plesset–Zwick scaling given as (Plesset & Zwick Reference Plesset and Zwick1954)

(3.37)\begin{equation}{R_b} \sim {R_{b0}}\frac{2}{{{\rm \pi} \gamma }}{\left( {\frac{{\beta {t_b}}}{3}} \right)^{1/2}},\end{equation}

where ${R_{b0}}$ is given by

(3.38)\begin{equation}{R_{b0}} = \frac{{2{\sigma _L}}}{{{P_V}(T) - {P_0}}}.\end{equation}

Note that, initially during bubble growth phase, ${R_b} \ll {R_{onset}}$. Here, ${\mathrm{\sigma }_L}$ is the air–liquid surface tension, ${P_V}(T)$ is the saturated vapour pressure at a temperature $T > {T_b}$, ${P_0}$ is the saturation pressure corresponding to ${T_b}$ and ${P_V}(T)$ can be calculated from the Clausius–Clapeyron equation

(3.39)\begin{equation}{P_V}(T) = {P_0}\,{\textrm{e}^{{L^\ast }(T - {T_b})/{R_g}T{T_b}}},\end{equation}

where ${L^\ast }$ is the latent heat of vaporization and ${R_g}$ is the specific gas constant of water vapour. The temperature scale T in (3.39) can be estimated from (3.36). Higher laser irradiation intensity causes a larger pressure difference across the bubble interface, causing faster bubble-induced membrane growth, as is observed for high ${I^\ast }$ compared with low ${I^\ast }$ (see figures 6 and 7).

Figure 6. High-speed image sequence for $c/{c^\ast } = 83.3$ at ${I^\ast } = 0.7$. It depicts symmetric membrane growth and consequent buckling of the polymeric shell structure. Here, ${t_1}$ represents reference time scale starting from the end of droplet evaporation (phase A). The scale bar represents 1 mm.

Figure 7. High-speed image sequence for $c/{c^\ast } = 83.3$ at ${I^\ast } = 2.2$ depicting asymmetric membrane growth, membrane rupture and the subsequent catastrophic breakup of parent droplet. The scale bar represents 1 mm.

In (3.37), $\gamma $ is a non-dimensional number that controls the bubble growth rate and is given by

(3.40)\begin{equation}\gamma = \frac{{AL\rho ^{\prime}}}{{3k{R_{b0}}\beta }}{\left( {\frac{{{\alpha_l}}}{{{\rm \pi} \beta }}} \right)^{1/2}},\end{equation}

where

(3.41)\begin{gather}A = \frac{{{P_V}(T) - {P_0}}}{{\rho (T - {T_b})}},\end{gather}

(3.42)\begin{gather}\beta = {\left( {\frac{{2{\sigma_L}}}{{\rho R_{b0}^3}}} \right)^{1/2}},\end{gather}

where k is the conductivity of the liquid, A is a proportionality constant relating pressure difference to temperature difference, $\rho ^{\prime}$ is the vapour density, ${\rho _l}$ is the liquid density and ${\alpha _l}$ is the thermal diffusivity of the liquid.

The bubble radius in a super-heated liquid medium grows as $t_b^{1/2}$. The coefficient of $t_b^{1/2}$ depends on the degree of super-heat (refer to § 3.3.1 and (3.36)).

The theoretical bubble growth time scale according to (3.37) is approximately $O({10^{ - 7}})s$ to $O({10^{ - 6}})s$ for the bubble to reach the onset size of the droplet for ${I^\ast } = 2.2$ and ${I^\ast } = 0.7$, respectively. The bubble observed in our experiments forms near the surface of the droplet and interacts with the skin layer formed during the evaporation phase. Once the bubble interacts with the skin layer, it evolves as a membrane (see supplementary figure S6b). The nature of the membrane is different for different irradiation intensities. Also, the bubble growth time scale is smaller than the membrane growth time scale by an order of magnitude.

3.3.3. Membrane growth mechanics

The membrane growth for ${I^\ast } = 0.7$ is slower compared with ${I^\ast } = 2.2$ (see figures 6 and 7). We model the membrane growth using a spring, mass, damper model with external forcing from the pressure difference across the membrane interface coupled to the maximum pressure inside the droplet through the Clausius–Clapeyron equation (3.39). Our modelling approach is similar to various rheological/solid models based on spring, mass and damper systems like the Kelvin–Voigt model and standard linear solid model (Maxwell model), to name a few (Eldred, Baker & Palazotto Reference Eldred, Baker and Palazotto1995; Renaud et al. Reference Renaud, Dion, Chevallier, Tawfiq and Lemaire2011).

The governing dynamical law for the viscoelastic membrane is given as

(3.43)\begin{equation}{\ddot{R}_m} + 2\zeta {\omega _n}{\dot{R}_m} + \omega _n^2{R_m} = \frac{{{F_{ext}}}}{m},\end{equation}

where ${R_m} = {R_m}(t)$ (figure 8a) represents the radial coordinate of the envelope of the membrane, ${\dot{R}_m}$ represents the velocity of the membrane, ${\ddot{R}_m}$ represents the membrane acceleration, ${\omega _n}$ is the natural frequency of the membrane characterizing the energy storage capacity, $\zeta $ is the damping coefficient of the membrane characterizing the losses due to viscous action, ${F_{ext}}$ represents the external force on the membrane due to the pressure and m is the mass of the membrane characterizing the inertia of the system. Equation (3.43) is essentially Newton's second law of motion for the membrane. Assuming approximate spherical symmetry, the membrane mass can be given as

(3.44)\begin{equation}m = {\rho _m}4{\rm \pi} R_{onset}^2{h_0},\end{equation}

where ${\rho _m}$ is the membrane density, ${R_{onset}}$ is the onset radius and ${h_0}$ is the initial membrane thickness (skin-layer thickness) formed during the evaporation phase (refer to phase A). The sudden expansion is due to the pressure developed inside the bubble and ${F_{ext}}$ is the external pressure force on the membrane

(3.45)\begin{equation}{F_{ext}}\sim \Delta P{\rm \pi} R_m^2.\end{equation}

The pressure build-up inside the bubble is due to the sudden phase change of liquid to vapour at hot spots

(3.46)\begin{equation}\Delta P = {m_{in}}{R_g}T/v,\end{equation}

where $\Delta P = {P_V}(T) - {P_0}$, ${m_{in}}$ is the initial mass of the vapour inside the bubble, $v = 4{\rm \pi} R_{onset}^3/3$. The scale of ${m_{in}}$ can be evaluated by equating equation (3.39) with (3.46)

(3.47)\begin{equation}{m_{in}} \sim \frac{{4{\rm \pi} R_{onset}^3{P_0}}}{{3{R_g}T}}({\textrm{e}^{{L^\ast }(T - {T_S})/{R_g}T{T_S}}} - 1).\end{equation}

Therefore, the governing dynamical law becomes

(3.48)\begin{equation}{\ddot{R}_m} + 2\zeta {\omega _n}{\dot{R}_m} + \omega _n^2{R_m} = \frac{C}{{{R_m}}},\end{equation}

where

(3.49)\begin{equation}C = \frac{{3{m_{in}}{R_g}T}}{{16{\rm \pi} {\rho _m}{h_0}R_{onset}^2}}.\end{equation}

Notice that the external forcing term on the right-hand side makes the equation nonlinear. Further, note that the external force is inversely proportional to the radius ${R_m}$, which indicates that, as ${R_m}$ increases, the pressure decreases due to a fixed amount of vapour occupying an increasingly larger volume. The initial conditions are

(3.50)\begin{gather}{R_m}(0) = {R_{onset}},\end{gather}

(3.51)\begin{gather}{\dot{R}_m}(0) = 0.\end{gather}

The physical properties of the membrane are captured in ${\omega _n}$ and $\zeta $; ${\omega _n}$ is related to the energy stored in the membrane and the storage modulus E whereas $\zeta $ is a parameter related to the losses occurring in the membrane. The storage modulus E is defined as the ratio of potential energy stored in the system per unit volume of the membrane. We have used a linear spring model to represent the stored potential energy in the membrane. The energy stored in the membrane is ${k_s}R_m^2/2$, where ${k_s}$ is the equivalent spring constant of the membrane

(3.52)\begin{equation}E\sim \frac{1}{2}\frac{{{k_s}R_m^2}}{{4{\rm \pi} R_m^2{h_0}}}.\end{equation}

Solving for ${k_s}$ we have

(3.53)\begin{equation}{k_s}\sim 8{\rm \pi} {h_0}E.\end{equation}

Dividing both sides of the above equation by the mass of the membrane $m = {\rho _m}4{\rm \pi} R_{onset}^2{h_0}$ and taking the square root, we have

(3.54)\begin{equation}{\omega _n}\sim \sqrt {\frac{{2E}}{{{\rho _m}R_{onset}^2}}} .\end{equation}

The numerical value of E used to calculate ${\omega _n}$ is $O({10^6}{-} {10^7})\ \textrm{Pa}$ (Stafford et al. Reference Stafford2004; O'Connell et al. Reference O'Connell, Wang, Ishola and McKenna2012; Li & McKenna Reference Li and McKenna2015). The second-order nonlinear differential equation is solved numerically, and the temporal evolution of the membrane radius ${R_m}$ is plotted in figure 8(b) along with the experimental curve for ${I^\ast } = 0.7$ and ${I^\ast } = 2.2$ for $c/{c^\ast } = 83.3$. The theoretical scales of the membrane growth agree with the experimental observations within the experimental uncertainty The membrane growth for ${I^\ast } = 2.2$ is comparatively faster than for ${I^\ast } = 0.7$.

Figure 8. (a) Snapshots indicating hole nucleation and rupture of membrane. (b) Experimental and theoretical comparison of membrane radius evolving with time for different irradiation intensities $({I^\ast })$ at $c/{c^\ast } = 83.3$. (c) Temporal evolution of membrane thickness for different irradiation intensities $({I^\ast })$ at $c/{c^\ast } = 83.3$. Data are represented as mean values ± SD from four individual droplets. ${h_{crit}}\sim 1\ {\rm \mu}\textrm{m}$ denotes the membrane thickness just before the rupture initiation.

The membrane for ${I^\ast } = 0.7$ is qualitatively different from that of ${I^\ast } = 2.2$ (see figures 6 and 7 and supplementary material and movies). A low irradiation intensity forms a membrane with a shell-like structure. On the contrary, the membrane resembles a viscoelastic membrane for higher irradiation intensities. Figure 8(a) shows a close-up view of the viscoelastic membrane formed for ${I^\ast } = 2.2$. Further inference from the high-speed images (refer to figure 6 for ${t_1} > 1.4\ \textrm{ms}$) and videos (see supplementary material and movies) reveal textures/cracks on the shell-like structure for low irradiation intensities. The textures/cracks near the hole are dominantly visible as thin dark lines that radiate out from the central hole location. In contrast, no textures/cracks are observed for high irradiation intensity (see figure 8a), and multiple holes appear due to membrane thinning (refer to § 3.3.4) and the location where hole nucleates is caused due to thin-film instability (Sharma & Reiter Reference Sharma and Reiter1996). The nature of the membrane can be explained based on the ratio $t/{t_d}$. For $t/{t_d} > 1$ (refer to § 3.3.1), the droplet heating by the laser is slow, and the laser exposure time is significant, forming a shell structure that undergoes a rupture forming holes at the poles. The slow heating produces a structure that is more solid like. On the contrary, for $t/{t_d} < 1$, heating by the laser is fast and the exposure time short, leading to a viscoelastic membrane that thins out and ruptures through multiple holes due to the high pressures inside the bubble. The losses in the shell structure are large compared with the viscoelastic membrane. Solving the membrane inflation model quantitatively, we get $\zeta $ for slow heating is approximately 7 times larger than fast heating, depicting that more losses occur in the membrane for slow heating than fast heating. This is a numerical validation of the fact that the membranes formed because of slow heating and fast heating have different properties.

3.3.4. Thinning of the membrane causes hole formation

As the membrane inflates due to the bubble growth, the membrane envelope radius ${R_m}$ increases, however, the thickness of the membrane h (figure 8a) has to decrease as a result of the conservation of mass. Applying conservation of mass for the membrane during the growth phase, we have

(3.55)\begin{equation}{\rho _m}4{\rm \pi} R_m^2(t)h = {\rho _m}4{\rm \pi} R_{onset}^2{h_0} = \textrm{const}\textrm{.},\end{equation}

where ${\rho _m}$ is the membrane density, ${R_{onset}}$ is the onset radius of membrane growth and ${h_0}$ is the membrane thickness at the end of evaporation phase A

(3.56)\begin{equation}h(t) = {h_0}{\left( {\frac{{{R_{onset}}}}{{{R_m}}}} \right)^2}.\end{equation}

This shows that, as the membrane grows, the thickness of the membrane decreases. Equation (3.56) is plotted for ${I^\ast } = 0.7$ and ${I^\ast } = 2.2$ in figure 8(c). The initial thickness ${h_0}$ for low irradiation intensity is larger $({h_0} \sim 30\,{\rm \mu} \textrm{m)}$ than the high irradiation intensity $({h_0} \sim 10\ {\rm \mu}\textrm{m)}$. The membrane thickness reduces with time faster for ${I^\ast } = 2.2$ compared with ${I^\ast } = 0.7$.

For low irradiation intensity, the membrane resembles a shell that grows until it ruptures by forming holes at the poles (weakest points) (see figure 6). Further several texture/crack patterns are also observed. However, for high irradiation intensity, the membrane ruptures through multiple holes due to thinning of the film in a very short time scale of a fraction of a millisecond. The rupture time scale can be evaluated by applying the Reynolds thin-film equation (Scheludko, Platikanov & Manev Reference Scheludko, Platikanov and Manev1965) to the viscoelastic membrane. The Reynolds equation is valid for the thinning of the viscoelastic membrane as the membrane thickness is less than $30\ {\rm \mu}\textrm{m}$, which is an order of magnitude less than the droplet/membrane radius

(3.57)\begin{equation}\frac{{\textrm{d}{h^{ - 2}}}}{{\textrm{d}t}} = \frac{{4\Delta P}}{{3\eta r_t^2}},\end{equation}

where h is the membrane thickness, $\eta $ is the viscosity of the polymer solution, $\Delta P$ is the total pressure difference applied across the membrane and ${r_t}$ is a length scale (see supplementary figure S4) characterizing the initial hole size on the surface of the membrane (see figure 8a).

Equation (3.57) is based on the assumption that, for small values of ${r_t}({r_t} \ll {R_{onset}})$, the membrane is a plane parallel, and the fluid flow inside the membrane during the thinning process mimics the flow behaviour between parallel plates getting squeezed.

Integrating equation (3.57) with respect to time form $t = 0$ to $t = \tau $ we have

(3.58)\begin{equation}\int_0^\tau {\frac{{\textrm{d}{h^{ - 2}}}}{{\textrm{d}t}}\,\textrm{d}t} = \int_0^\tau {\frac{{4\Delta P}}{{3\eta r_t^2}}\,\textrm{d}t} .\end{equation}

The rupture time scale $\tau $ therefore becomes

(3.59)\begin{equation}\tau \sim \frac{{3\eta r_t^2}}{{4\Delta P{h^2}}},\end{equation}

where the total pressure $\Delta P$ is composed of the capillary pressure $({\sim} {10^2}\ \textrm{Pa})$, pressure gradient due to the bubble expansion $({\sim} {10^5}\ \textrm{Pa})$ and disjoining pressure $({\sim} {10^{ - 2}}\ \textrm{Pa})$. The dominant component of the pressure is the pressure due to the bubble expansion, with capillary and disjoining pressures being insignificant. The theoretical time scale of rupture $(\tau \sim 0.1\ \textrm{ms})$ for ${I^\ast } = 2.2$ agrees with the experimental value of 0.1 ms. The blue dotted line in figure 8(c) denotes the critical membrane thickness of $({h_{crit}}\sim 1\ {\rm \mu}\textrm{m)}$. Note that for ${I^\ast } = 2.2$ the membrane thickness h reaches the critical thickness ${h_{crit}}$ within approximately 0.1 ms, leading to the rupture of the membrane. The temporal scale of hole formation on the membrane during growth is a direct consequence of film thinning.

3.4.1. Catastrophic breakup of droplet

After the membrane ruptures, bubbles develop again inside the droplet (see § 3.3). The pressure exerted by these bubbles during their breakup catastrophically fragments the droplet with the ejection of secondary droplets. The Weber number of the droplet defining the catastrophic breakup is calculated in the following way. The pressure force due to bubble breakup scales as

(3.60)\begin{equation}{F_P}\sim ({P_{int}} - {P_{amb}}){\rm \pi} r_b^2,\end{equation}

where ${P_{int}}$ is the pressure inside the bubble, ${P_{amb}}$ is ambient atmospheric pressure and ${r_b}$ is the radius of the bubble. Similarly, the surface tension force scales as

(3.61)\begin{equation}{F_S}\sim {\sigma _L}{\rm \pi} {D_{onset}}.\end{equation}

The Weber numbers (We) for the catastrophic breakup are $O({10^3})$. The secondary droplet diameter is in the range of (0.01–0.16 mm), whereas the velocities are in the range of (1–6 m s⁻¹) (See supplementary figure S3). It is observed that the secondary droplet diameter $({\alpha _s})$ and velocity $({\beta _s})$ are negatively correlated (a Pearson correlation coefficient of ~−0.5) with higher diameter secondary droplets corresponding to lower velocity and vice versa (see supplementary figure S3). Here, the correlation coefficient is used to indicate the dependence between ${\alpha _s}$ and ${\beta _s}$. The empirical Pearson coefficient (Swinscow and Campbell Reference Swinscow and Campbell1997) is given as

(3.62)\begin{equation}Corr({\alpha _s},{\beta _s}) = \frac{{\sum ({{\alpha_s} - \overline {{\alpha_s}} } )({{\beta_s} - \overline {{\beta_s}} } )}}{{\sqrt {\sum {{({{\alpha_s} - \overline {{\alpha_s}} } )}^2}\sum {{({{\beta_s} - \overline {{\beta_s}} } )}^2}} }},\end{equation}

where $\overline {{\alpha _s}} $ and $\overline {{\beta _s}} $ represent the average secondary droplet diameter and average secondary droplet velocity, respectively.

3.4.2. Fragmentation of droplets through the formation of a stable sheet

Figure 9 shows the high-speed images depicting the time evolution of the polymeric droplet into a stable sheet and its subsequent collapse. After the skin-layer formation, the key mechanism driving the drop deformation and propulsion is the local asymmetric boiling of the liquid induced by the absorption of laser energy. The local asymmetric boiling leads to the formation of small-sized vapour bubbles, which subsequently undergo breakup and induce a thrust force on the droplet. It is to be noted that, during propulsion of the liquid sheet, the laser radiation pressure and thermal radiation pressure caused by the droplet surface heating are negligible (Delville et al. Reference Delville, de Saint Vincent, Schroll, Chraibi, Issenmann, Wunenburger, Lasseux, Zhang and Brasselet2009; Klein et al. Reference Klein, Bouwhuis, Visser, Lhuissier, Sun, Snoeijer, Villermaux, Lohse and Gelderblom2015). The induced thrust force deforms and breaks the droplet with a primary rim detachment $({t_1}\sim 0.04\ \textrm{ms})$, forming a thin sheet. This deformation of the droplet occurs within the inertial time scale ${\tau _i}\sim {D_{onset}}/U\sim {10^{ - 4}}\ \textrm{s}$, where ${D_{onset}}$ represents the diameter of the polymeric droplet at the pre-breakup instant and U represents the maximum sheet velocity (1–2 m s⁻¹). Concurrently, the expanding sheet is subjected to very high accelerations of $O({10^4})\ \textrm{m}\ {\textrm{s}^{ - 2}}$, which could lead to Rayleigh–Taylor instability. The instability growth, ${t_{RT}}$ can be estimated as follows (Villermaux and Clanet Reference Villermaux and Clanet2002):

(3.63)\begin{equation}{t_{RT}}\sim {\left( {\frac{{{\sigma_L}}}{{{\rho_L}a_s^3}}} \right)^{1/4}},\end{equation}

where ${\sigma _L}$ represents the surface tension of the liquid, ${\rho _L}$ represents the density of the liquid and ${a_s}$ represents the radial acceleration of the sheet. The theoretical growth rate $({t_{RT}})$ is$\sim 0.09\ \textrm{ms}$. The experimental inter-frame time interval is 0.04 ms, which is of the order of ${t_{RT}}$. The developed liquid sheet gradually stretches out and accumulates liquid at the edge, forming a thin rim and ligaments $({t_1}\sim 0.36\ \textrm{ms)}$. At this instant, the centre of the sheet is thick compared with the outer region. As the sheet further expands, its thickness decreases while the rim diameter increases. Further, the sheet evolves to form a stable sheet $({t_1}\sim 0.68\ \textrm{ms)}$ with a thick rim and ligaments $({t_1}\sim 1.00\ \textrm{ms)}$. Initially, corrugations develop on the rim, which appear as mere noise in the experiments. Later they form apparent perturbations with a characteristic wavenumber, ${k_r}$ from which ligaments evolve. The wavelength corresponding to the corrugations gives the experimental growth rate of Rayleigh–Taylor instability on the rim. The fastest-growing Rayleigh–Taylor mode on the rim is given as (Klein et al. Reference Klein, Kurilovich, Lhuissier, Versolato, Lohse, Villermaux and Gelderblom2020)

(3.64)\begin{equation}{k_r}\sim {({a_s}{\rho _L}/{\sigma _L})^{1/2}}.\end{equation}

The theoretical $(10\ \textrm{m}{\textrm{m}^{ - 1}})$ and experimental $(3.225 \pm 0.12\ \textrm{m}{\textrm{m}^{ - 1}})$ growth rates are approximately of the same order. The ligaments become Rayleigh–Plateau unstable and pinch-off into secondary droplets. The time taken between the formation of the ligament to pinch-off of the first droplet from the ligament gives the breakup time of the ligament. The experimental ligament breakup time is compared with the capillary time scale of the ligament $({t_{cl}})$, which is given as (Rao et al. Reference Rao, Karmakar and Basu2018)

(3.65)\begin{equation}{t_{cl}}\sim \sqrt {\frac{{{\rho _L}\varepsilon _b^3}}{{{\sigma _L}}},} \end{equation}

where ${\varepsilon _b}$ represents ligament diameter. The theoretical capillary breakup time is 0.14 ms, whereas the experimental breakup time is 0.33 ± 0.12 ms. Following the breakup of ligaments, the sheet collapses under the influence of surface tension $({t_1}\sim 1.16\ \textrm{ms)}$. The experimental sheet collapse time scale is compared with the capillary time scale of the sheet, which is given as (Avila & Ohl Reference Avila and Ohl2016)

(3.66)\begin{equation}{t_c}\sim \sqrt {\frac{{{\rho _L}D_{onset}^3}}{{8{\sigma _L}}}} .\end{equation}

Here, ${D_{onset}}$ represents the diameter of the polymeric droplet at the pre-breakup instant. The theoretical capillary time scale $({t_c})$ is 0.95 ms, whereas the experimental sheet collapse time is 0.71 ± 0.07 ms, indicating a similar order of magnitude.

Figure 9. High-speed image sequence depicting the evolution of a polymeric droplet into an expanding stable sheet and subsequent collapse. This mode of breakup is predominant at $c/{c^{\ast }} = 83.3$ and ${I^{\ast }} = 3,3.5$. Here, ${t_1}$ represents reference time scale starting from the end of droplet evaporation (phase A). The scale bar represents 1 mm.

Figure 10 shows the experimental evolution of the stable sheet diameter with time and its theoretical comparison. The inset shows the variation of rim diameter with time. As seen in figure 10, the experimental temporal evolution of the sheet length scale closely matches the theoretically predicted sheet evolution. The continuous accumulation of liquid at the edge drawn from the centre of the sheet increases the diameter of the rim. As shown in the inset of figure 10, the rim diameter increases linearly with time. We use the approach used by Klein et al. (Reference Klein, Kurilovich, Lhuissier, Versolato, Lohse, Villermaux and Gelderblom2020) to explain the stable sheet evolution dynamics in our case.

Figure 10. Experimental and theoretical comparison of temporal variation of normalized sheet length scale (L_s/D_onset). Inset represents the temporal variation of rim diameter corresponding to a stable sheet. The dotted line represents a linear fit of the data.

The evolving stable sheet can be modelled as

(3.67)\begin{equation}\frac{{{L_S}({t_1})}}{{{D_{onset}}}} = 1 + \sqrt {3W{e_d}} \frac{{{t_1}}}{{{t_c}}}{(1 - \sqrt 3 {t_1}/2{t_c})^2},\end{equation}

where

(3.68)\begin{equation}W{e_d} = \frac{{{E_{k,d}}}}{{{E_{k,cm}}}}We.\end{equation}

Here, ${L_S}$ represents approximate length scale of the sheet (refer to figure 9) and ${E_{k,d}}/{E_{k,cm}}$ is the ratio of deformation to propulsion kinetic energies. The Weber number defining the sheet evolution is calculated in the following way. As the nucleated bubbles are not visible at higher irradiation intensities, the Weber number of the droplet characterizing sheet fragmentation is modified as the ratio of drop displacement kinetic energy to its surface energy which is defined as

(3.69)\begin{equation}We\sim \frac{{{\rho _L}{D_{onset}}{U^2}}}{{{\sigma _L}}},\end{equation}

where U represents the axial velocity of the sheet. However, $We$ is rescaled to $W{e_d}$ to take into account the kinetic energy that is utilized for deformation. The ratio of deformation to propulsion kinetic energies can be written as

(3.70)\begin{equation}\frac{{{E_{k,d}}}}{{{E_{k,cm}}}}\sim \frac{{\int_0^R {u_r^2r\,\textrm{d}r} }}{{{U^2}\int_0^R {r\,\textrm{d}r} }},\end{equation}

where ${u_r}$ represents radial velocity inside the sheet. The radial velocity can be expressed as, ${u_r}\sim r/{t_{max}}$. Here, ${t_{max}}$ represents the time taken for the sheet to reach its maximum extension. Integrating equation (3.70), the ratio of deformation to propulsion kinetic energies can be written as

(3.71)\begin{equation}\frac{{{E_{k,d}}}}{{{E_{k,cm}}}}\sim \frac{{R_{SM}^2}}{{2{U^2}t_{max}^2}}.\end{equation}

Here, ${R_{SM}}$ represents the maximum radius of the stable sheet. Therefore, $W{e_d}$ can be obtained using (3.68) and (3.71).

3.4.3. Fragmentation of droplets through the formation of unstable sheet

Figure 11 represents the time evolution of the breakup of the polymer droplet into an unstable sheet. Here, the bubble breakup induced thrust force deforms the droplet into a sheet $({t_1}\sim \, 0.48\ \textrm{ms)}$ which evolves in an unstable manner. The unstable sheet develops pre-holes on it $({t_1}\sim 0.56\ \textrm{ms)}$, which subsequently evolve into holes $({t_1}\sim \, 0.72\ \textrm{ms)}$. Through hole growth, the unstable sheet finally ruptures $({t_1}\sim \, 0.76\ \textrm{ms)}$. The formation of a hole is always preceded by a localized thinning of the sheet, referred to as a pre-hole. The pre-hole to hole transition is evident through the formation of a darker rim surrounding the patch and a vanishing thickness of the inner zone of the patch (see figure 12a).

Figure 11. High-speed image sequence depicting the evolution of a polymer droplet into an expanding unstable sheet followed by pre-hole/hole formation and subsequent rupture of the sheet. This mode of breakup is predominant at $c/{c^\ast } = 16.6,\,33.3$ and ${I^\ast } = 2.2,\,3,\ \textrm{and}\ 3.5$. The scale bar represents 1 mm.

Figure 12. (a) Image sequence depicting the growth of a single pre-hole/hole in an unstable sheet. The scale bar represents 1 mm, (b) Temporal evolution of pre-hole and hole diameter corresponding to several pre-holes and holes. The solid lines in (a) and (b) indicate a linear fit.

Figure 12(a) depicts how a single patch (pre-hole and hole) develops on an unstable sheet. It can be seen that the diameter of the patch $({D_{patch}})$ increases with time. However, the rate of increase of ${D_{patch}}$ in the pre-hole regime is comparatively slow compared with the hole regime. To investigate the evolution of pre-holes and holes in a more detailed way, we have tracked the number of pre-holes and holes on the unstable sheet. Figure 12(b) shows the temporal evolution of several pre-holes and holes on the sheet. Both ${D_{prehole}}$ and ${D_{hole}}$ vary linearly with time for all the tracked pre-holes and holes on the unstable sheet. As seen in the figure, the slope of the line for ${D_{prehole}}$ is comparatively smaller compared with that for ${D_{hole}}$, indicating that, even for a broader range of pre-holes and holes, the hole evolution is significantly faster compared with pre-hole evolution. The experimental growth rate of holes is in the range of 0.7–2 m s⁻¹. The growth rate can also be predicted using the Taylor–Culick law (Culick Reference Culick1960), assuming uniform and constant sheet thickness,

(3.72)\begin{equation}{V_H}\sim \sqrt {\frac{{2{\sigma _L}}}{{{\rho _L}{h_s}}}} ,\end{equation}

where ${h_s}$ represents the thickness of the sheet. The theoretical hole growth rate is $0({10^0})\ \textrm{m}\ {\textrm{s}^{ - 1}}$, which agrees well with the experimental values.

Figure 13 describes distinct droplet breakup modes observed in the current work, ranging from relatively low strength stable sheet breakup to high strength catastrophic breakup. For sheet type breakup, it is interesting to observe that higher ${D_{onset}}/{D_0}$ (lower $We$) corresponds to stable sheet breakup, whereas lower ${D_{onset}}/{D_0}$ (higher $We$) corresponds to unstable sheet breakup. As we have seen, the impulse pressure applied on the drop surface due to bubble breakup sets the fluid motion inside the entire drop. The axial velocity $(U)$ of the sheet follows from global mass conservation as

(3.73)\begin{equation}{P_e}{\tau _e}D_{onset}^2\sim {\rho _L}D_{onset}^3U,\end{equation}

where ${P_e}$ represents impulse pressure on the droplet due to bubble break up, ${\tau _e}$ represents the time scale on which the impulse acts

(3.74)\begin{equation}U\sim {P_e} {\tau _e}/{\rho _L}{D_{onset}},\end{equation}

In our experimental study, throughout the ranges of $c/{c^\ast }$ and ${I^\ast }$, ${P_e}$, ${\tau _e}$ and ${\rho _L}$ remain constant. Therefore, (3.74) suggests that axial velocity scales as

(3.75)\begin{equation}U\sim 1/{D_{onset}}.\end{equation}

Finally, using (3.69) and (3.75), the dependence of Weber number on ${D_{onset}}$ can be expressed as

(3.76)\begin{equation}We\sim 1/{D_{onset}}. \end{equation}

Equation (3.76) indicates that higher ${D_{onset}}/{D_0}$ corresponds to a lower Weber number and a stable sheet breakup, whereas a lower ${D_{onset}}/{D_0}$ corresponds to a higher Weber number and an unstable sheet breakup. As already seen, ${D_{onset}}/{D_0}$ is a function of both $c/{c^\ast }$ and ${I^\ast }$ (see § 3.3.1). At higher $c/{c^\ast }$ and ${I^\ast }$, sheet fragmentation of droplets occurs earlier in droplet's lifetime (${D_{onset}}/{D_0}$ is large) compared with lower $c/{c^\ast }$ and ${I^\ast }$ (${D_{onset}}/{D_0}$ is small). This indicates that high $c/{c^\ast }$ and ${I^\ast}\ (c/{c^\ast } = 83.3,\,{I^\ast } = 3,3.5)$ corresponds to predominantly stable sheet fragmentation, whereas at low $c/{c^\ast }$ and ${I^\ast}\ (c/{c^\ast } = 33.3,\,{I^\ast }\, = 3,\, 3.5\ \textrm{and}\,c/{c^\ast } = 16.6,\,{I^\ast } = 2.2,\,3,\,3.5)$ corresponds to unstable sheet fragmentation mechanism dominates. The characteristic ratio of rupture time scale to the capillary time scale in the context of sheet breakup is given as (Klein et al. Reference Klein, Kurilovich, Lhuissier, Versolato, Lohse, Villermaux and Gelderblom2020)

(3.77)\begin{equation}\frac{{{t_r}}}{{{t_c}}}\sim {\left( {\frac{{{n_0}}}{{{D_{onset}}}}} \right)^{ - 1}}{(We)^{ - 1}}{\left( {\frac{{{E_{k,d}}}}{{{E_{k,cm}}}}} \right)^{ - 1/4}},\end{equation}

where ${t_r}$ represents rupture time scale of the sheet, ${n_0}/{D_{onset}}$ represents the initial noise level from which the instability grows. The value of ${n_0}/{D_{onset}}$ is calculated considering ${t_r}/{t_c}\sim 1$, which is found to be of $O({10^{ - 2}})$. The ratio of ${t_r}/{t_c} > 1$ indicates stable sheet fragmentation of droplets, whereas ${t_r}/{t_c} < 1$ represents unstable sheet fragmentation. It shows that the stable sheet collapses before the occurrence of sheet rupture, whereas the unstable sheet ruptures before it collapses, as experimentally depicted in figures 9 and 10.

Figure 13. Regime map representing distinct droplet breakup modes observed in the present work.