Spreading of oil-in-water emulsions on water surface - Langmuir (ACS

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Spreading of oil-in-water emulsions on water surface Neda Sanatkaran, Valery G Kulichikhin, Alexander Ya. Malkin, and Reza Foudazi Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01435 • Publication Date (Web): 24 Aug 2018 Downloaded from http://pubs.acs.org on August 28, 2018

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Spreading of oil-in-water emulsions on water surface Neda Sanatkaran, † Valery G. Kulichikhin, ‡ Alexander Ya. Malkin, ‡ Reza Foudazi † * †

Department of Chemical and Materials Engineering, New Mexico State University, Las Cruces,

NM 88003, U.S.A. ‡

A.V. Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Moscow

119991, Leninskii pr., 29

*

Corresponding Author. Email: [email protected], Tel.: (575) 646-3691, Fax: (575) 6467706. Address: MSC 3805, P. O. Box 30001, Las Cruces, New Mexico 88003-3805

ACS Paragon Plus Environment

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ABSTRACT

This work presents the spreading behavior of oil-in-water (o/w) emulsions on the water surface recorded using high-speed photography method. We study a series of o/w emulsions with two different droplet sizes of 4.50 and 0.75 µm and volume fractions of the oil phase in the 20-80% range. Results show that for all the emulsions a rapid spreading occurs upon the collision with the water surface, which then forms a thin film expanding with time. Appearance of a dry spot in the center of collision is observed in the spreading of the emulsions in mid-volume fraction range that induces a bursting-like spreading. For the highly concentrated emulsions, the deliberation of decompression energy from the deformed oil phase droplets inhibits the bursting, increase the equilibrium propagation radius, and reduce the dissipation time. The role of viscoplasticity (existing of the yield stress) is considered and a model describing the propagation step of the emulsion spreading is presented. The model shows that the peculiarities of the spreading are determined by the competition between yielding, plastic viscosity, and interfacial tension. By comparing the model prediction and experimental results, it is suggested that the spreading behavior of the emulsions is not only a consequence of the surface tension gradient, but also the coalescence of the oil droplets during spreading.

KEYWORDS. Emulsions, Concentrated Emulsions, Marangoni Effect, Spreading, Viscoplastic


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INTRODUCTION Spreading of a liquid on another liquid or solid surface can be found in many applications such as coating, inkjet printing, medical treatments, and filtration processes. The dynamics of spreading of various liquids such as surfactant,1–3 polymer solutions,4 and suspensions5,6 on a solid or liquid surface have been investigated. Spreading of a surface-active compound (surfactant) solution on a liquid is mainly driven by the surface tension gradient (as a consequence of the concentration gradient) across the liquid-air interface, which can be thermodynamically determined by the spreading coefficient as follows:

= Υ − (Υ + Υ )

(1)

where Υ being the interfacial tensions between the and phases in contact, and 1, 2, and 3

represent air, water, and surfactant solution, respectively. The spreading happens if ≥ 0. The concentration gradient induces stress on the surface, known as “Marangoni effect”,7 leading to

the expansion and instability of the surfactant solution. The scaling relationships and boundary conditions of Marangoni-driven spreading depend on different factors such as the thickness of substrate, geometry of spreading, and rheological properties of the surface and injected liquid.8,9 The spreading behavior of surfactant solutions on solid and liquid substrates is rather well understood and several theoretical models have been suggested to predict their spreding.1,2,10,11 In addition, the role of oil spreading as an antifoam, in contact with surfactant solutions and air, have been discussed.12–14 For example, the impact of oil droplet coalescence on the mode of spreading13 as well the size of the oil droplets on the instability of the film12 have been reported. Spreading of emulsions has been reported on both solid and liquid surfaces in dilute regime.15–17 However, the mechanism of spreading of an emulsion in the range of semi-dilute and


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concentrated regime with respect to the internal phase droplet size, has not fully understood.18 The presence of internal phase alters the rheological properties of the emulsion to nonNewtonian behavior with a noticeable elasticity originated from interdroplet interactions and in some cases a yield stress appears due to jamming of droplets.19 In addition, the spreading process can be influenced by the coalescence instability of the oil droplets due to the drainage of the surfactant solution from the interdroplet layer.18 In this work, we study the spreading behavior of o/w emulsions – stabilized by sodium dodecyl sulfate (SDS) – with different droplet sizes (from sub-micron to micrometer) and volume fractions (from the dilute to the highly concentrated regime) on the surface of the pure water. The experimental results are linked to the rheological behavior of the emulsions to better understand the mechanism of the emulsion spreading and its instability dynamic.

EXPERIMENTAL SECTION

Emulsion preparation. Silicone oil with the kinematic viscosity of 5 cSt (Sigma-Aldrich) as the oil phase and 80 mM sodium dodecyl sulfate (Sigma-Aldrich) solution (about 10 times higher than its CMC=8×10-3 mM)20 as the aqueous phase are used to prepare the emulsions. The silicone oil is gradually added to the SDS solution to attain coarse emulsions with the 30% oil volume fraction (). The coarse emulsions are mixed further using an ultrasonic apparatus (Sonics & Materials, Inc., US) with different mixing times. The samples are transferred to the centrifuge for 60 min at 11,000 rpm to obtain highly concentrated emulsions through creaming effect.


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The concentration of oil phase in the creams is measured (both above 80%) through evaporation method. The creams are diluted down with the same aqueous phase to produce emulsions with a range of volume fractions from 80% to 10% while keeping the surfactant concentration constant. Droplet size measurement. The droplet size of creams (. ) is measured using a Zetasizer (Malvern Instruments Ltd., UK). A drop of emulsions is diluted with the same continuous phase and transferred to a Zetasizer cell for droplet size measurement. Emulsions with average droplet size of 4.50 and 0.75 µm are obtained corresponding to the ultrasonication time of 30 and 60 s, respectively. The polydispersity index, PDI = (peak width/peak mean)2, of the emulsions is about 0.4, as obtained from Zetasizer. Rheological characterization. The flow behavior of the samples is measured via a Discovery HR-3 rheometer (TA Instruments, US) using a cross-hatched parallel plate geometry (to avoid the wall slip) with 20 mm diameter and 1 mm gap. The flow curves are measured from 100 to 0.1 s-1 shear rate. Visualization procedure. The experimental set-up includes a transparent round Petri dish with dimensions of 64 mm diameter and 13 mm height, a blunt needle connected to a syringe pump (Harvard Apparatus PHD2000), a 500-kW lamp as light source, a high-speed MotionPro X5Plus camera (DEL Imaging Systems, LLC), and a Nikon lens with 105 mm ring to enlarge the images. The experiments are performed in an isolated area to eliminate the effect of temperature variation and air streams on the region of the emulsion contact with the water surface. Ultrapure water (EMD Millipore, US) is chosen as the substrate. The thickness of the substrate layer in Petri dish is 6 mm. A drop of the emulsion with constant volume of 33 mm3 is injected by an L shape needle using the syringe pump. The needle was positioned vertically and very close to the


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substrate surface to avoid any form of vortex or splash on the water surface. The spreading is recorded with the high-speed camera with 100 frames per second and images are analyzed using an open access imaging software (ImageJ).

RESULTS

Since the flow properties of emulsions are different from those of surfactant solutions, a different spreading behavior is expected. To confirm whether the initial spreading of the emulsions is triggered by the Marangoni effect, the substrate is replaced with an SDS solution at its CMC concentration. No spreading is observed in the absence of the concentration gradient, indicating the spreading of the emulsions is initiated by the Marangoni stresses. On the pure water substrate, however, three types of spreading behaviors are observed depending on the internal phase volume fraction and the internal phase droplet size, labeled as type I, II, and III. These three types are discussed below by illustrating the schematic of each spreading type for the emulsions with . = 4.50 in Figure 1. Type I. At low concentrations of dispersed phase, as the emulsion drop collides with the water surface, the outer layer of the emulsion spreads rapidly, causing instability in the form of capillary waves on the water surface (Figure 1-Type I, ≈ 0.01 ). To make sure that the collision is not cause of the initial rapid spreading of emulsion, we replaced DI water with water containing SDS slightly below its CMC and ran the test under the same conditions. No rapid spreading of the emulsion outer layer was observed. Thus, the effect of initial collision of the emulsion drop with the substrate can be ignored.


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In type I spreading, the frontier film rapidly propagates on water surface along with capillary waves induced by the rapid spreading of the emulsion’s outer layer. While the capillary waves fade away, the thin film still propagates towards the periphery of the Petri dish but slows down further due to the reduction of the Marangoni stresses (Figure 1-Type I, 0.01 < < 0.08 ). Later, the surfactant depletion from the contact point increases the surface tension in the center, which causes fingering instability at the edge of the film disk (Figure 1-Type I , ≈ 0.96 ) and

the retraction of the thin film (Figure 1-Type I, > 0.96 ).22 Simultaneously, the gradual

dissipation of the remained emulsion (cap) is observed (0.01 < < 0.96 ), which inhibits the full retraction. As soon as the emulsion fully dissipates, some holes appear on the surface - and the retraction rate of the thin film accelerates. The experimental images are presented in Supporting Information, Figure S1.


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Figure 1. Schematic of various stages of different spreading types of o/w emulsion on water surface: (a) Type I and III from side and top view and (b) Type II from side and top view. Indicators: (1) substrate, (2) emulsion, (3) thin layer, (4) emulsion cap, (5) holes, (6) thinning of the film, (7) dry spot (film break-up) and, (8) 2nd thin layer. Stages: (A) emulsion deposition, (B) initial rapid spreading followed by film propagation, (C) maximum expansion, (C-1) dry spot formation, (C-2) dry spot expansion and bursting, (C-3) expansion of the 2nd film, (D) fingering, and (E) hole formation.


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Type II. As the concentration of dispersed phase increases, bursting of the emulsion cap is observed. First, the emulsion outer layer spreads rapidly at the time of the collision, similar to the dilute emulsion spreading (Figure 1-Type II, 0.01 < < 0.02 ), followed by the propagation of

the film. However, the film becomes thinner around the emulsion cap (Figure 1-Type II, ≈ 0.09 ) and later, ruptures from the center and leaves a dry spot behind- (Figure 1-Type II, ≈ 0.22 ). The dry spot expands by time which can be seen as a blue circle with a pronounced

boundary around emulsion cap in Figure 1-Type II, t ≈ 0.32 s. This phenomenon has been reported during the spreading of other fluids due to instability and thinning of the spreading layer.22,23 As the dry spot expands, the emulsion cap bursts (Figure 1-Type II, ≈ 0.38 ), which creates a second thin film propagating to the edge of Petri dish (Figure 1-Type II, 0.38 <
0.38 ). The experimental images are shown Supporting Information, Figure S2. Type III. For highly concentrated samples, the initial rapid spreading upon the emulsion

collision to the water surface occurs similar to the previous types. While no dry spot is formed for this type of spreading, a fast dissipation (instead of bursting) of the emulsion cap was observed (Figure 1-Type III, 0.02 < < 0.4 ), followed by fingering, hole formation, and the

retraction of the film (Figure 1-Type III, ≥ 0.4 ). The stages of Type I and III are similar, but the kinetic of latter is faster. The change in spreading behavior shifts to lower volume fractions for the emulsions with smaller droplet size of the internal phase, as presented in Table 1. The appearance of the dry spot in Type II spreading is a factor of droplet size and volume fraction of the internal phase of the emulsions.


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However, Type III spreading occurs after = 60 % regardless of the droplet size in the studied range. Table 1. Spreading behavior of the emulsions with different droplet sizes and volume fractions of internal phase.

Spreading Type Droplet size (. , µm)

Type I

Type III

Type II

Volume fraction (, %)

4.50

10,20,30, 40

50,60

70,80

0.75

10,20,30

40,50, 60

70,80

While the spreading behavior of the emulsions with . = 0.75 is similar to the one

shown in Figure 1, the timing is slightly different from the emulsions with . = 4.50 .

Table 2 present the time (* ) and radius (+* ) at which the film spreads to its maximum as well as the time of dry spot formation (, ) and its maximum radius (+, ) upon the bursting for Type II

spreading emulsions. Table 2. Time and radius of the maximum film spreading and time and maximum radius of dry spot prior to the bursting of Type II emulsions.

. (µm) 4.5

0.75

(%) 50 60

+* (mm) 26.8 25.8

+, (mm) 18.3 16.7

* (s) 0.08 0.08

, (s) 0.09 0.07

40 50

27.2 26.0

12.6 12.3

0.06 0.07

0.20 0.15

60

25.4

12.1

0.09

0.12


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DISCUSSION

When the emulsion contacts the water surface, the surface tension gradient between the emulsion and the substrate is at its maximum and the spreading is mainly a Marangoni-driven process. A power-law model has been suggested for spreading of soluble surfactant solutions on the water surface with different power values in two different spreading regions.1,2 The model describes the time dependency of the thin film spreading +() as follows:

+() = +- + . *

(2)

where ω and m are the coefficients of the model. The +() has been obtained using the area measurement module of ImageJ software from the beginning of spreading to the fingering step and retraction. Due to the polydispersity of droplets of the internal phase, the propagation and retraction of the thin film are not necessarily symmetrical, thus, we considered an average diameter for the area. We measured the area of the film and approximated the average radius of a circle with equivalent surface area as the radius of the film. Figure 2 illustrates the time evolution of the emulsions frontier up to its equilibrium point for the emulsion with droplet size . = 4.50 and different spreading types. The same trend is observed for the other droplet size.


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(a)

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(b)

Figure 2. Propagation step of the frontier layer with time for emulsions with . = 4.50 and different volume fractions in (a) logarithmic scale and (b) linear scale (symbols). The dash lines in (b) correspond to the predicted propagation radius from eq. (A-14). The arrow shows when the spreading radius of highly concentrated emulsions takes over the lower volume fraction samples during the propagation.

The time evolution of the thin film is divided into two regions. Region I is when the acceleration of the frontier occurs. As the emulsions’ thin film propagates further, the rate of propagation decays (region II- deceleration). A similar trend has been reported for the spreading of surfactant solutions on the water surface.1 Lee and Starov related the appearance of two propagation regions to the disintegration of surfactant micelles and their diffusion to the bulk of substrate in the first step of spreading.1 In our experiments, the concentration of the SDS is about 10 times higher than its critical micelle concentration (CMC~8 mM). However, the values obtained

through +()~. * scaling for emulsion propagation is lower than the surfactant solution


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spreading exponents in both regions ( : acceleration, , : deceleration), as presented in Table 3. Table 3. Power law exponents obtained by fitting eq. (2) on propagation regions of the emulsions

. () 4.5

0.75

(%) 20 50 80 20 50 80

0.96 1.12 1.16 1.10 1.26 1.67

, 0.52 0.60 0.32 0.88 0.81 0.67

The value of m obtained by Lee and Starov for the SDS solution spreading above its CMC was about 0.55 for the acceleration region and 0.18 for the deceleration region.1 Vernay et al.16 reported that the propagation radius of very dilute emulsions scales with ~ /1 , while others showed various conditions such as change of surfactant type or geometry could result in different scaling exponents.8 The deviation of the values in our work, compared to the values reported by others,1,8,16 can be attributed to the viscoplastic behavior of the emulsions.24 Therefore, we try to capture this deviation through modeling of emulsion spreading. Upon the collision of emulsion on water surface, the Marangoni stress must initially overcome the viscous stresses (23 ) present between the emulsion and water surface:

23 =

Υ 4

(3)

where 4 is the radial coordinate of the spreading, and Υ is the surface tension of the film as a function of the surfactant concentration at the surface. Unlike Newtonian surfactant solutions, emulsions are non-Newtonian fluids with viscoplastic behavior (i.e., having yield stress). Figures


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3 illustrates the typical flow curve of emulsions under investigation in the range of 0.1-100

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shear rate. The viscoplastic behavior of emulsions implies that their spreading dynamic is nonlinear.

(a)

(b)

(c)

(d)

Figure 3. Flow curves of the emulsions with (a,b) . = 4.50 (symbols), fitted by (a) H-B model (dash lines)

and (b) Bingham model (dash lines). Flow curves of the emulsions with (c,d) . = 0.75 (symbols), fitted by (c) H-B model (dash lines) and (d) Bingham model (dash lines).


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As seen in Figure 3, the emulsions with volume fraction higher than 0.5 exhibit yield stress even below the random close packing of spheres, * = 0.64.25 Kim et al. have reported the

viscoplastic behavior below * for ionic stabilized emulsions with the dispersed droplet size in

the range of few micrometers to nanometers and attributed their finding to the effect of interdroplets interactions such as electrostatic repulsion.25 The flow curve of the emulsions under investigation can be well described by Herschel-Bulkley (H-B) model with a yield stress, 2- , in a one-dimensional flow field:26

23 = 2- + 6(78 )9

(4)

where 6 and : are the model parameters with the : value changing from 0.5 for highly concentrated to 1 for semi-dilute emulsions. The Bingham plastic model is a special case of H-B model with n=1 and 6 = ; . We consider the emulsion drop as a cylinder to model its spreading on a liquid surface. The schematic of an emulsion spreading in the propagation stage is shown in Figure 4. The equations of motions in cylindrical coordinates are considered to find the governing scaling law for emulsion spreading. The details of modeling can be found in Appendix.


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Figure 4. A sketch of an emulsion spreading on the water surface at (a) the collision, (b) propagation, and (c) radius of emulsion spreading at equilibrium.

To calculate the propagation from eq. (A.14), a rough estimation of 33 mN/m is considered for the surface tension gradient across the substrate surface affected by the presence of SDS in the emulsion drop and 2- as well as ; are obtained from fitting of Bingham model on emulsions flow curves (see Figure 3b and 3d). Table 4 presents the plastic viscosity as well as the yield stress for the emulsions under investigation fitted with Bingham plastic equation.


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Table 4. Plastic viscosity and yield stress of the emulsions with different droplet sizes and volume fractions of internal phase obtained from fitting of Bingham Plastic model.

Droplet size (. , µm) Volume fraction (, %) 20 30

4.50

0.75

Plastic viscosity (; , Pa.s) 0.17 0.19 0.2 0.2

4.50

0.75

Yield stress (2- , Pa) 0.025 0.04

0.08 0.14

40

0.2

0.2

0.15

0.2

50

0.32

0.2

0.5

0.35

60

0.3

0.3

0.7

1.5

70

0.3

0.8

2

6

80

1

5

21

28

The experiment values of propagation of the frontier layers for the emulsion with droplet size . = 4.5 and different spreading types versus the model suggested for Bingham plastic are shown in Figure 2b next to experimental results. The model, eq. (a.14), predicts the trend of spreading dimeter with respect to volume fraction of emulsions with values in the range of experimental observations. The observed deviation of the model is not due to the neglecting of viscous forces in water substrate, because they will decrease the predicted values of spreading radius from model. Eq. (A.14) indicates that the spreading of the thin frontier layer is affected by yield stress, surface tension gradient, and the plastic viscosity of the initial emulsion drops. For all emulsions the surface tension gradient is the same, thus, the difference in spreading radius is expected to arise from the change in the yield stresses and the viscosities of the emulsions with respect to the droplet size and volume fraction of the internal phase.


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Unlike surfactant solution, emulsions consist of internal phase in the form of droplets which are stabilized within the continuous phase by surfactants. When the emulsion touches the water surface, the surfactant molecules migrate across the surface due to the Marangoni stress, leading to the interdroplet film drainage inside emulsions.20,27 The presence of electrostatic layer between oil droplets makes the kinetic of drainage complicated. Experimental studies have shown that for the ionic stabilized emulsions, the drainage speed of the interdroplet film due to the Marangoni effect is significant, even in a weak Marangoni regime.10,27 As a result, the oil droplets gradually coalesce and release the energy stored during the emulsification process (∆=> = Υ∆?′, where ?A is the total surface area of the oil phase). The deviation of eq. (A.14) from experimental data could be due to the steady-state assumption in the modeling and/or an additional driving force exerted by the release of coalescence energy (increases the spreading radius of the thin layer). The velocity of the frontier layers can be calculated from the time differentiation of spreading radius. The average rate of the propagation decreases with increasing the volume fraction of emulsions. The experimental velocity of the thin film frontiers (+8 ) as a function of spreading

radius for the emulsions with droplet size . = 4.5 and different spreading types are shown in Figure 6.


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Figure 6. Velocity of the frontier layer versus spreading radius during spreading for emulsions with . = 4.50 and different volume fractions.

In the beginning of the spreading, a big jump in velocity profile represents the initial rapid spreading of the emulsion drops upon collision. As the spreading continues, the surface tension gradient becomes smaller and the radius of frontier layer becomes bigger leading to the slowdown of velocity. For the highly concentrated emulsions (type III spreading), a non-uniform velocity profile is observed. As the internal phase volume fraction exceeds the maximum random packing fraction (* ~0.64), the oil droplets are no longer spherical and packed together in a polyhedron configuration with a very thin interdroplets film keeping them apart.28 This structure can be obtained through exerting an external pressure to compress the repulsive oil droplets and store an interfacial elastic energy.29 Unlike the emulsions with the spherical oil droplets, the spreading of highly concentrated emulsions allows compressed droplets to gradually revert to the spherical shape as the emulsion spreads on the water surface and releases the stored elastic


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energy of compression. The compression energy is a function of volume fraction and droplet size:26

=~ where B′

3B A − * C D 2 *

(5)

is the elastic modulus of the concentrated emulsion.1 For highly concentrated

emulsion, compression energy must be considered in the equations of motion; therefore, the spreading equations of these samples have an additional positive term. In contrast to the developed model, thus, the experimental results for spreading radius of highly concentrated emulsions takes over the lower volume fraction samples during the propagation (the arrow in Figure 2). The surfactant concentration gradient (as the initial body force driving the spreading) is about 5 Pa at its maximum considering the surface tension of SDS equal to 39 mN/m and water 72 mN/m at 25 ºC,30 while the decompression energy for the highly concentration emulsion is about 10 times higher. Finally, the frontier layer reaches an equilibrium point (+8 = 0), meaning that the surfactant concentration gradient and the viscous term across the liquid-air interface are at equilibrium. At this point, due to the opposite surface tension gradient formed behind the frontier layer, the thin film retracts back to the center of the collision.15,31 Although this phenomenon can be observed for all the emulsion samples (Figure 6), the frontier layer retraction (negative velocity) happens at different time scales for the different types of spreading. During the thin film retraction, some holes spontaneously form in several spots in the thin film which are advected toward the periphery of the film.


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During the spreading, the presence of surfactant molecules across the surface exerts a strong electrostatic repulsion in the thin film, which acts as a barrier against hole formation.21 However, when soluble surfactants such as SDS are present in the system, the bulk diffusion of surfactant molecules occurs right after the equilibrium point.1 The migration of surfactant molecules to the substrate creates local inhomogeneity in the thin film. Different mechanisms of perforation of the film has been suggested, depending on the film thickness, the ratio between the oil droplets and film thickness, the magnitude of Péclet number as well as the magnitude of spreading coefficient at the surface.12,16,21,32 Vernay et al. have demonstrated that the perforation of film in dilute oil-in-water emulsions spreading occurs when the entering of oil droplet to the surface is thermodynamically favorable and the Marangoni stresses exist across the surface.16,21 While in their report, the size of oil droplets in comparable to the film thickness, in our experiments the size of oil droplets are smaller than the film. Thus, bridging or dewetting are not possible mechanism for the perforation. Neel and Villermau32 related the perforation to the instability of the film caused by Marangoni stresses, for spreading films with the thickness range of 1 −

100 and with the Péclet number (Pe) larger than unity. In our experiments, the Péclet

number (Pe) could be estimated as EGF +* /HI , where +* is the equilibrium point of the spreading

(zero velocity), EGF is the average velocity of the spreading layer, and HI is the surfactant

diffusion coefficient equal to 6 × 105K / for SDS.27 The Péclet number of the emulsions

under investigation is around 101 , thus, self-sustained Marangoni-induced instability of the film is expected to occur.

During the propagation period below < 0.7, the film rupture takes place as the volume fraction of the emulsions increases, generating a dry zone at the center. The film rupture produces a new concentration gradient around the emulsion cap which induces the bursting of the cap and creates


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Page 22 of 37

another layer of thin film. Here, the Péclet number is about 4 and we can safely ignore the diffusion of surfactant to the sublayers of water during the time scale (~0.1 s) of the propagation of the frontier layer until it reaches to the equilibrium point. The film rupturing is dependent on the magnitude of the Bond number (L) which is the ratio between gravitational to surface tension gradient forces:33

L=

MNO ∆ΥK

(6)

where M is the substrate density and N is the gravitational acceleration. When L ≪ 1, the gravitational forces can be neglected and the outflow of the spreading film under Marangoni stresses causes an extreme film thinning which can lead to film rupturing.33 The initial volume of the emulsion drop is 33 mm3 and the final thickness of the thin layer (O) at the equilibrium point (+* ) is an order of 10-2 mm for all samples. Thus, the L ≈ 7 × 105Q and L ≪ 1 condition exists. The possibility of film rupturing in the spreading of surfactant solutions has been related to the velocity gradient of frontier layer and the area near the collision center as well as the duration of the propagation step.1 For the studied emulsions, the migration of the SDS molecules to the water surface not only induces the spreading of the emulsion outer layer, but also involves the outflow of continuous phase from the emulsion cap, resulting in water drainage from interdroplets film inside the emulsion cap. Since the oil droplets are apart in dilute emulsions, the emulsion cap gradually dissipates as the frontier layer is propagating, which maintains the thickness and spreading velocity of the thin film and prevents it from film rupturing. As the volume fraction of the internal phase increases


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and/or the oil phase droplet size becomes smaller, the dissipation of the emulsion cap does not occur during the propagation of the emulsion thin layer. The drainage of the water from interdroplet layer in semi-dilute emulsions brings the oil droplets closer, thus, the interdroplet interactions temporarily increase the yield stress within the remained emulsion.34 The increase in the yield stress prevents the emulsion cap from dissipation during the frontier layer propagation. Therefore, the thickness of the spreading layer decreases, which destabilizes the spreading layer and forms a dry spot around the emulsion cap (i.e., leads to rupturing). As the frontier layer propagates further, the dry spot expands and forms a surface tension gradient across the dry zone, resulting in the bursting of the emulsion cap.

Figure 8. Dissipation time for the emulsions under investigation. Solid lines correspond to median of the experimental points from = 20 to 60 %.

Figure 8 shows the time between collision and complete dissipation of the emulsion (, ) in each type of spreading. The dissipation of type II emulsions is slightly faster than type I emulsions due to the bursting. The release of compressed energy - in highly concentrated emulsions


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Page 24 of 37

( = 0.7 − 0.8) escalates the dissipation of the emulsion cap, hinders the formation of dry spot, and inhibits the emulsion cap from the bursting.

CONCLUSION

The spreading behavior of emulsions on the water substrate is investigated for a series of o/w emulsions with different droplet sizes and volume fractions of the oil phase. Depending on the concentration of the emulsions, three types of spreading are observed. Spreading of the dilute emulsions goes through the rapid spreading of emulsion outer layer, the formation of a disk-like thin film, propagation of the film, and slow dissipation of the remained emulsion drop followed by fingering instability and retraction of the thin film. In the emulsions with mid-volume fraction of the oil, the thin film ruptures during the propagation step and forms a dry spot around the remained emulsion causing a bursting of the emulsion cap. Theoretical study for the emulsion spreading suggests that the viscoplasticity of the emulsions controls the spreading in addition to the Marangoni effect. While the effect of viscoplasticity of the emulsions on their spreading behavior is modeled, it is suggested that the interdroplet interactions between dispersed phase droplets of the emulsions should be considered as a controlling factor of their coalescence in the prediction of spreading behavior of such systems. The higher propagation radius in experimental results compared to the model prediction is attributed to the interfacial energy release upon the coalescence of the oil phase droplets. In addition, the gradual release of compression energy in


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highly concentrated emulsions enhances the dissipation of the emulsion cap without formation of the dry spot, increases the equilibrium propagation radius, and reduces the dissipation time.


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ASSOCIATED CONTENT

Supporting Information Experimental images of the different types of spreading (PDF). AUTHOR INFORMATION

Corresponding Author * Email: [email protected] Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT

N.S. and R.F. thank ACS PRF for funding (55725-DNI7), Dr. Fangjun Shu (MAE, NMSU) for providing high-speed camera and MotionPro software and NSF award #1438584 that made the purchase of the rheometer possible. V.G.K. and A.Y.M. are grateful to Russian Foundation for Basic Research (grant #16-03-00259) for financial support.

ABBREVIATIONS

SDS, sodium dodecyl sulfate; o/w, oil-in-water; CMC, critical micelles concentration.


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Hotrum, N. E.; van Vliet, T.; Cohen Stuart, M. A.; van Aken, G. A. Monitoring Entering and Spreading of Emulsion Droplets at an Expanding Air/Water Interface: A Novel Technique. J. Colloid Interface Sci. 2002, 247 (1), 125–131.

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Semakov, A. V.; Postnov, E. M.; Kulichikhin, V. G.; Malkin, A. Y. Explosive Spreading of a Concentrated Emulsion over a Liquid Surface. Colloid J. 2017, 79 (3), 414–417.

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Foudazi, R.; Masalova, I.; Malkin, A. Y. The Rheology of Binary Mixtures of Highly Concentrated Emulsions: Effect of Droplet Size Ratio. J. Rheol. 2012, 56 (5), 1299–1314.

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Néel, B.; Villermaux, E. The Spontaneous Puncture of Thick Liquid Films. J. Fluid Mech. 2018, 838, 192–221.

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Gaver, D. P.; Grotberg, J. B. Droplet Spreading On A Thin Viscous Film. J. Fluid Mech. 1992, 235, 399–414.

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Kim, H. S.; Scheffold, F.; Mason, T. G. Entropic, Electrostatic, and Interfacial Regimes in Concentrated Disordered Ionic Emulsions. Rheol. Acta 2016, 55 (8), 683–697.

(35)

Nikolov, A. D.; Wasan, D. T.; Chengara, A.; Koczo, K.; Policello, G. A.; Kolossvary, I. Superspreading Driven by Marangoni Flow. Adv. Colloid Interface Sci. 2002, 96 (1–3), 325–338.

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Traykov, T. T.; Ivanov, I. B. Hydrodynamics of Thin Liquid Films. Effect of Surfactants on The Velocity of Thinning of Emulsion Films. Int. Z Multiph. Flow 1977, 3 (1957), 471–483.

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Nikolov, A.; Wasan, D. Superspreading Mechanisms: An Overview. Phys J Spec Top. 2011, 197 (1), 325–341.


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APPENDIX In line with previous studies for modeling the propagation step,8,35,36 we consider the emulsion drop as a cylinder which symmetrically expands in r and shrinks in z directions under the Marangoni effect, independent of the azimuth, T. Thus, the velocity components in 4 and U

directions are non-zero, which are noted as EF (4, U) and E (4, U), respectively. In contrast to previous works,1,2 we consider the viscous dissipation in water substrate is negligible since the viscosity of emulsions are at least two orders of magnitude higher than that of substrate. Therefore, the continuity and momentum equations can be described for the propagation of emulsion film as follows:

WM3 1 W W (M3 4EF ) + (M3 E ) + W 4 W4 WU =0

M3 C

WEF WEF + EF D W W4

=−

WX W4

1W W2F (42FF ) − − 4 W4 W4 +

(A.1)

(A.2)

WΥ W?


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M3 C

Page 32 of 37

WE WE WE + EF + E D W W4 WU =− −

WX WU

1W W2 (42F ) − 4 W4 WU

(A.3)

+ M3 N

where M3 is the density of the thin film,

YZ Y[

is the body force exerted on the thin film caused by

the Marangoni stresses, ? is the area of the thin film expanding across the substrate surface with

time, and N is the gravitational acceleration in z direction. To consider the non-Newtonian behavior of the emulsions, we first need to find the magnitude of rate-of-strain, 78 , (also known as rate of deformation) in cylindrical coordinates. The 78 is the second invariant of the rate-of-strain

tensor, 78 ̿ = ∇E̅ + (∇E̅ )_ , where ∇E̅ is the velocity gradient and (∇E̅ )_ is its transpose. The rate of deformation for the spreading of drop in cylindrical coordinates can be obtained as:

1 78 = `78 ̿ ` = a 78 ̿ : 78 ̿ 2 = a2 cC

WEF EF WE WEF WE D +d e +C D f+g + h W4 4 WU WU W4

(A.5)

The stress for Herschel-Bulkley fluid in a multidimensional incompressible flow filed is as follows:

i

2 = −jkl 78 , 2 > 2- 2 = 0, 2 ≤ 2-


(A.6)

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where jkl =

no p8

+ 6(78 )95, and 2 and 78 are the ij-component of stress tensor and rate-of-

strain tensor, respectively. By placing the Herschel-Bulkley model in the equation of motion, the eq. (A.2) and (A.3) become nonlinear and cannot be solved analytically. Therefore, some simplifying assumptions have to be introduced as follows:

i.

The maximum Reynold number (+q = a steady state (

ii.

∆w

∆u

Y

Y_

rs tu v

) is small (on the order of 100) and therefore,

= 0) of the thin film flow can be considered.2,37

≪ 1, where + and ℎ are the thin film radius and thickness, respectively. Thus, to

study the dynamic of the emulsion spreading, the lubrication theory can be applied,23,38 therefore, we have

Yy Y

=

Yz Y

= 0.9

iii.

The gravity effect is negligible and can be ignored, M3 N = 0.

iv.

The film is in contact with air, so the pressure gradient across the film can be considered equal to zero.

The boundary conditions are as follows:

Υ(0, +) = ΥK

WΥ 2{+ℎ 2FF (+) = ℎ ( | ) W4 F}u

(A.7)

By applying assumption ii, the continuity equation can be solved to obtain the EF =

uu8 F

, where +8

is the velocity of frontier layer. Thus, the rate of deformation is:


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++8 ++8 WE 78 = a2 ~− + + g h 4 4 W4 ++8 WE a = 4 +g h 4 W4

To prove that the

Yz YF

Page 34 of 37

(A.8)

term can be neglected in this equation, we solve the eq. (A.3) for a

Newtonian fluid with viscosity µ (to be able to analytically evaluate the order of magnitude of terms). By considering the preceding assumptions, eq. (A.3) simplifies as:

M3

++8 WE 1W WE =+ C4 D 4 W4 4 W4 W4

(A.9)

The solution to this differential equation is as follows:

E (4) = 4

rs

uu8 v

+

(A.10)

where and are constants. Since r and EF are in the order of mm and mm/s, the E is also in the order of mm/s, thus, the is in the order of unity. Therefore, the

insignificant compared to the

uu8 F

Yz YF

term in eq. (A.8) is

term according to the obtained experimental values.

Consequently, the magnitude of the rate-of-strain tensor is approximately (A.3), we need to define 2FF :


uu8 F

. To solve the eq.

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2WEF 2FF = −2 C + 6(78 )95 D 78 W4

9 2++8 = 2- + 6 4

(A.11)

Now, the boundary condition, eq. (A.7) can be used to determine the spreading conditions: 9 2+8 WΥ 2{+ 2- + 6 = ( | ) + W4 F}u YZ

where ( YF

F}u

(A.12)

) is proposed to increase linearly with time, /+,39 where is a constant and is

time. The final equation is as follows:

29 {6+8 9 + 2{+ 9 2- − + 95 =0

(A.13)

If 2- = 0, we have the equation for spreading of a power-law liquid. The solution to this

equation for a Bingham plastic liquid (: = 1 and 6 = ; known as plastic viscosity) is:

; −2vo += + +0 + q 2{2{2- n

−

; {2-

(A.14)

; −2vo 2 +8 = − +0 + q 2{2- 2; {2- n

In these equations, we assumed that at time zero, the radius of droplet cylinder is +K . For a

Newtonian liquid (2- = 0, : = 1, and 6 = ), we have:


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+=a + +0 2 4{ +8 =

4{

Page 36 of 37

(A.15)

2 4{ + +0

The solution for cases with : ≠ 1 and non-zero yield stress (H-B model) cannot analytically be obtained, but one may consider a general spreading equation according to eq. (A.14) as follows:

+ = + q

−

_ >

+ 1


(A.16)

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Equilibrium Radius (mm)

Page 37 of 37

Yield Stress (Pa)


37

Spreading of oil-in-water emulsions on water surface - Langmuir (ACS

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