Diffusive and capillary instabilities of viscous fluid threads in microchannels

We experimentally examine the formation of viscous fluid threads in hydrodynamic focusing sections using both miscible and immiscible fluid pairs at relatively low flow rates. A systematic comparative study is conducted between diffusive and capillary regimes using a viscous oil and a variety of polar organic solvents in a simple microflow geometry. Silicone oil and various low molecular weight alcohols are used as model fluids to investigate the dynamics of viscous multiphase flows at ultralow interfacial tension and with partially miscible systems. An original methodology based on analysis of thread width and detachment length in the viscous regime is developed to quantify various interfacial destabilization processes over a wide range of injection flow rates. For miscible fluid pairs, we investigate several regimes of thread swelling and, for immiscible fluid pairs, we discuss diverse modes of droplet formation and wetting dynamics. This work provides a comprehensive general classification of immiscible and miscible fluid dynamics with large viscosity contrasts in microchannels together with a unifying phenomenological description of thread behavior based on simple functional relationships to better delineate the role of flow parameters and fluid properties on viscous microflow processes.

Figure 1

Fluid properties. (a) Evolution of interfacial tension ${\gamma }_{12}$ between silicone oil and alcohol as a function of alcohol viscosity ${\eta }_2$. (b) Characteristic capillary velocity $V^{*}={\gamma }_{12}/{\eta }_2$ of fluid pairs. Inset: solvent viscosity ${\eta }_2$ as a function of carbon number. (c) Schematics of square hydrodynamic focusing section where a viscous fluid $L1$ is centrally injected at $Q_1$ and a solvent $L2$ is symmetrically introduced at $Q_2$ resulting in various regimes, including (i) dripping droplet, and (ii) viscous thread.

Figure 2

Maps of flow regimes for miscible and immiscible fluid pairs with characteristic micrographs. Flow rates ($Q_1$, $Q_2$) in μl/min. (a) Miscible fluid pair $C\mathrm{3-2}$ ($L2$: 2-propanol). Structural flow regimes: bulging (5, 2), tubing (10, 10), threading (20, 40), and engulfment (1, 90). Unstable flow regimes: winged (0.5, 1.2), diffusive (0.5, 5), diffusive buckling (0.5, 20), and fragmentation (0.5, 200). (b) Immiscible fluid pair $C\mathrm{7-1}$ ($L2$: 1-heptanol). Structural flow regimes: bulging (5, 1), tubing (5, 3), threading (5, 40), and engulfment (0.5, 45). Destabilizing flow regimes: dripping (0.2, 0.5), dripping tail (0.2, 9), jetting (0.3, 7), and jetting beads (0.3, 9). Base flow regime transitions at constant φ: (i) bulging-tubing, (ii) tubing-threading, (iii) threading-engulfment.

Figure 3

Flow maps of miscible fluid pairs, $C\mathrm{3-2}$ (2-propanol), $C\mathrm{4-1}$ (1-butanol), $C\mathrm{4-2}$ (2-butanol), and $C\mathrm{5-1}$ (1-pentanol), including structural regimes, such as threading ($•$), tubing ($\blacksquare$), engulfment ($✦$), and bulging ($◆$), and unstable regimes, such as diffusive ($◂$), diffusive buckling ($▾$), winged ($▴$), fragmentation (), and limiting (). See main text for subregime transition lines (i)–(iv).

Figure 4

Diameter of miscible fluid threads. (a) Evolution of stable diameter ${\varepsilon }_0/h$ as a function of flow rate ratio φ across structural flow regimes. Lower solid line: Eq. (2), upper solid line: Eq. (3). (b) Variation of minimum diameter ${\varepsilon }_{\mathrm{M}}/h$ as a function of φ showing the role of $Q_{\mathrm{T}}$ on iso-φ curves for fluid pair $C\mathrm{3-2}$. (c) Evolution of ${\varepsilon }_{\mathrm{M}}/h$ with $Q_{\mathrm{T}}$ along iso-φ curves for miscible fluid pairs, fitting parameter $Q_{\mathrm{C}}=4.5$ (2-propanol), 3.2 (1-butanol), 2.1 (2-butanol), and 1.6 μl/min (1-pentanol). (d) Comparison of measured ${\varepsilon }_{\mathrm{M}}/h$ with Eq. (4). (e) Evolution of $Q_{\mathrm{C}}$ with solvent viscosity ${\eta }_2$.

Figure 5

Viscous detachment length $L_{\mathrm{D}}$ of miscible fluid threads. (a) Micrographs of fluid contactor in primary flow regimes, fluid pair $C\mathrm{3-2}$, from top to bottom: (i) $\phi =2.5\times {10}^{\text{–}3},1\times {10}^{\text{–}2},2.5\times {10}^{\text{–}2}$, (ii) $\phi =5\times {10}^{\text{–}2},1\times {10}^{\text{–}1},2.5\times {10}^{\text{–}1}$, and (iii) $\phi =3.3\times {10}^{\text{–}1},5\times {10}^{\text{–}1},1\times {10}^0$. (b) Evolution of stable detachment length $L_{\mathrm{D}0}/h$ as a function of φ. Solid line: Eq. (6). Inset: schematics of measurement of $L_{\mathrm{D}}$. (c) Variation of initial and final viscous detachment locations, $x_0$ and $x_1$, as a function of φ for fluid pair $C\mathrm{4-1}$. Solid line (i): $x/h=3.84\phi$. (d) Chart of micrographs showing a reduction of $L_{\mathrm{D}}$ for diffusive regimes at low $Q_{\mathrm{T}}$. Fluid pair $C\mathrm{3-2}$, flow rates ($Q_1$, $Q_2$) in μl/min, from top to bottom, $\phi =0.25$ (5, 20), (1, 4), (0.5, 2), and $\phi =0.5$ (5, 10), (1, 2), and (0.5, 1). (e) Evolution of $L_{\mathrm{D}}/h$ as a function of $Q_{\mathrm{T}}$ along iso-φ curves for fluid pair $C\mathrm{3-2}$. (f) Evolution of $L_{\mathrm{D}}/h$ as a function of φ along iso-φ curves for fluid pair $C\mathrm{3-2}$. Solid line: Eq. (6). (g) Evolution of $L_{\mathrm{D}}/L_{\mathrm{D}0}$ as a function of $Q_{\mathrm{T}}$ with fluid pair $C\mathrm{3-2}$. Solid line: $L_{\mathrm{D}}/L_{\mathrm{D}0}={\left(1+Q_{\mathrm{D}}/Q_{\mathrm{T}}\right)}^{\text{–}1}$ with $Q_{\mathrm{D}}=1.7\mu \mathrm{l}/\min$. (h) Comparison of $L_{\mathrm{D}}/h$ and $L_{\mathrm{D},\mathrm{M}}^{*}/h$ calculated based on Eq. (7). Inset: evolution of $Q_{\mathrm{D}}/Q_{\mathrm{C}}$ for miscible fluid pairs. Solid line: $Q_{\mathrm{D}}/Q_{\mathrm{C}}=0.5$.

Figure 6

Winged thread regime. (a) Micrographs showing lateral span $S$ of diffusive pools for fluid pair $C\mathrm{3-2}$, flow rates in μl/min, $Q_1=0.5$ and $Q_2=$ (i) 0.5, (ii) 0.7, (iii) 0.9, (iv) 1.1, and (v) 1.3. (b) Evolution of normalized span $S$/$h$ as a function of φ for various iso-$Q_1$ curves for fluid pair $C\mathrm{3-2}$. Solid lines: $S/h=1+1.5\phi /Q_1$. (c) Span $S$/$h$ versus $Q_2$ for data corresponding to (b). Solid line: $S/h=1+1.5/Q_2$. (d) Variation of normalized pool width $w$/$h$ versus reciprocal flow rate $2/Q_2$ for $Q_1\le 1\mu \mathrm{l}/\min$. Solid lines: $w/h=2Q_{\mathrm{W}}/Q_2$. (e) Evolution of $Q_{\mathrm{W}}$ with solvent viscosity ${\eta }_2$; solid line: $Q_{\mathrm{W}}=4{{\eta }_2}^{\text{–}1/3}$. (f) Ratio $Q_{\mathrm{W}}/Q_{\mathrm{C}}$; solid line: $Q_{\mathrm{W}}/Q_{\mathrm{C}}=5\times {10}^{\text{–}2}$. (g) Micrographs showing saturated circulating regions with fluid pair $C\mathrm{5-1}$ at large $Q_1=5$ and $Q_2=$ (i) 0.2, (ii) 0.3, (iii) 0.4, (iv) 0.5, and (v) 0.6. (h) Evolution of normalized island size $D$/$h$ versus $Q_2/2$ for $Q_1\ge 2\mu \mathrm{l}/\min$. Solid line: $D/h=2.5\times {10}^{\text{–}1}{\left(Q_2/2\right)}^{\text{–}1/2}$. (i) Evolution of $D$/$h$ with $Q_1$ for fixed $Q_2=0.5$. (j) Measurement of normalized subdroplet size $d_{\mathrm{D}}/h$ with $Q_2/2$. Solid line: $d_{\mathrm{D}}/h=0.1$.

Figure 7

Flow maps of immiscible fluid pairs, $C\mathrm{3-1}$ (1-propanol), $C\mathrm{6-1}$ (1-hexanol), $C\mathrm{7-1}$ (1-heptanol), and $C\mathrm{8-1}$(1-octanol), including threading ($•$), tubing ($\blacksquare$), engulfment ($✦$), bulging ($◆$), jetting (▸), dripping ($◂$), and wetting () regimes. Dashed lines represent regime extrapolations drawn from analogy between maps. See main text for solid transition lines.

Figure 8

Diameter of immiscible fluid threads. (a) Evolution of stable diameter ${\varepsilon }_0/h$ as a function of flow rate ratio φ in the viscous regime. Lower solid line: Eq. (2); upper solid line: Eq. (3). Inset: micrographs of stable regimes with fluid pair $C\mathrm{8-1}$. (b) Evolution of width $\varepsilon /h$ in the capillary regime with partial lubrication due to thread wetting. Solid line: Eq. (2); dashed line: Eq. (8). (c) Micrographs of wetting threads, flow rates ($Q_1$, $Q_2$) in μl/min, (i) fluid pair $C\mathrm{6-1}$ (0.2, 0.2), (ii) (0.2, 0.8), (iii) fluid pair $C\mathrm{7-1}$ (2, 2), fluid pair $C\mathrm{8-1}$ (0.8, 15), (v) fluid pair $C\mathrm{6-1}$ (2,10), and (vi) (0.5, 30).

Figure 9

Detachment length $L_{\mathrm{D}}$ of immiscible fluid threads. (a) Micrographs of fluid contactor in primary flow regimes, fluid pair $C\mathrm{7-1}$, from top to bottom: (i) $\phi =1\times {10}^{\text{–}2},2\times {10}^{\text{–}2},2.5\times {10}^{\text{–}2}$; (ii) $\phi =5\times {10}^{\text{–}2},1\times {10}^{\text{–}1},2.5\times {10}^{\text{–}1}$; and (iii) $\phi =3.3\times {10}^{\text{–}1},5\times {10}^{\text{–}1},1\times {10}^0$. (b) Evolution of stable detachment length $L_{\mathrm{D}0}/h$ as a function of φ in the viscous regime. Solid line: Eq. (6). (c) Evolution of $L_{\mathrm{D}}/h$ as a function of $Q_{\mathrm{T}}$ and φ along iso-$Q_1$ curves, fluid pair $C\mathrm{7-1}$. Solid lines: $L_{\mathrm{D}}/h=0.87{Q_{\mathrm{T}}}^{\text{–}0.2}$ (top) and $L_{\mathrm{D}}/h=1.9{\phi }^{1/2}$ (bottom). (d) Chart of micrographs showing an increase of $L_{\mathrm{D}}$ for capillary regimes at low $Q_{\mathrm{T}}$. Fluid pair $C\mathrm{8-1}$, flow rates ($Q_1$, $Q_2$) in μl/min, from top to bottom, $\phi =0.02$ (2, 100), (0.8, 30), (0.2, 10), and $\phi =0.2$ (20, 100), (0.8, 4), and (0.2, 1). (e) Evolution of $L_{\mathrm{D}}/h$ as a function of $Q_{\mathrm{T}}$ along iso-φ curves for fluid pair $C\mathrm{8-1}$. Solid line: $L_{\mathrm{D}}/h={Q_{\mathrm{T}}}^{\text{–}0.2}$. (f) Evolution of $L_{\mathrm{D}}/h$ as a function of φ along iso-φ curves for fluid pair $C\mathrm{8-1}$. Solid line: Eq. (6). (g) Scaled length $L_{\mathrm{D}}/L_{\mathrm{D}0}$ versus $Q_{\mathrm{T}}$. Solid lines: $L_{\mathrm{D}}/L_{\mathrm{D}0}=1+k{Q}_{\mathrm{T}}^{-1/2}$. Inset: coefficient $k$ as a function of φ. Solid line $k=k_0{\phi }^{\text{–}1}$ with $k_0=0.17{(\mu \mathrm{l}/\min )}^{1/2}$. (h) Comparison of $L_{\mathrm{D}}/h$ with Eq. (9).

Figure 10

Droplet regimes with dripping (open symbols) and jetting (solid symbols) patterns. (a) Evolution of normalized dripping droplet size $d$/$h$ as a function of ${\alpha }_2{\mathrm{Ca}}_2$. Dashed line: $d/h=2.2\times {10}^{\text{–}3}{\left({\alpha }_2{\mathrm{Ca}}_2\right)}^{\text{–}1}$, solid line: $d/h=0.5{\left({\alpha }_2{\mathrm{Ca}}_2\right)}^{\text{–}0.17}$. Inset: droplet size $d$/$h$ versus capillary number ${\mathrm{Ca}}_2$, solid line: $d/h=0.3{{\mathrm{Ca}}_2}^{\text{–}1/3}$. (b) Micrographs of dripping droplets, flow rates in μl/min, fluid pair $C\mathrm{8-1}$, $Q_1=0.1$, $Q_2=$ (i) 0.2, (ii) 0.6, (iii) 1, (iv) 10, and (v) time series of dripping tail breakup with $Q_1=0.5$ and $Q_2=10$. (c) Jetting droplet size with flow rate ratio φ, solid line: $d/h=3.1{\phi }^{1/2}$. (d) Micrographs of jetting droplets with flow rates ($Q_1$, $Q_2$); (i) fluid pair $C$-71 (0.5, 6); (ii) (0.3, 9); (iii) fluid pair $C$-81 (0.8, 1); (iv) (0.5, 2); (v) fluid pair $C$-31 (0.5, 5), (0.5, 7), (0.5, 10), (0.5, 16); and fluid pair $C\mathrm{8-1}$ (0.3, 30) and (0.3, 35). (e) Evolution of normalized droplet velocity $V_{\mathrm{D}}/J_{\mathrm{T}}$ as a function of $\left(d/h\right)C{a}_{\mathrm{T}}^{-1}$. Solid line: $V_{\mathrm{D}}/J_{\mathrm{T}}=1+0.5{\left[d/\left(h{\mathrm{Ca}}_{\mathrm{T}}\right)\right]}^{\text{–}1/3}$. Inset: droplet velocity $V_{\mathrm{D}}$ as a function of $J_{\mathrm{T}}$; solid line: $V_{\mathrm{D}}=J_{\mathrm{T}}$. (f) Evolution of segmented flow aspect ratio $d$/$L$ with flow rate ratio φ. Solid line for dripping: $d/L=2\phi$. Dashed line for jetting: $d/L=0.1+2\phi$. (g) Normalized emission frequency of dripping droplet $f/{\tau }_1$ as a function of ${\mathrm{Ca}}_{\mathrm{T}}{\phi }^{\text{–}1}$. Solid line: $f/{\tau }_1=10{\left({\mathrm{Ca}}_{\mathrm{T}}{\phi }^{\text{–}1}\right)}^{1/4}$. (h) Normalized emission frequency of jetting droplet $f/{\tau }_1$ as a function of ${\mathrm{Ca}}_{\mathrm{T}}{\phi }^{\text{–}1}$. Solid line: $f/{\tau }_1={10}^2{\left({\mathrm{Ca}}_{\mathrm{T}}{\phi }^{\text{–}1}\right)}^{1/2}$.