Electric field deformation of protein-coated droplets in thin channels

Greg Randall. Langmuir , Just Accepted Manuscript. DOI: 10.1021/acs.langmuir.8b01713. Publication Date (Web): July 30, 2018. Copyright © 2018 America...
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Electric Field Deformation of Protein-Coated Droplets in Thin Channels Greg C. Randall* General Atomics, San Diego, California 92121, United States

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S Supporting Information *

ABSTRACT: High-strength droplet interfaces are attractive for many applications, specifically in cases where droplets are channeled through fluidic devices and manipulated by electromagnetic fields. Using models and experiments, we study the deformation of droplets and capsules with protein interfaces in an electric field in thin and wide electrode gaps. Proteins are chosen from candidates expected to display qualitatively different interfacial interactions and strengths: a globular protein (bovine serum albumin), a reversible crosslinking peptide (AFD4), and a hydrophobin (cerato ulmin). Dilute protein additives can lead to over 1 order of magnitude stronger oil−water interfaces than those stabilized by small surfactants. We develop small deformation models to evaluate a protein membrane’s interfacial elasticity, notably accounting for the electric field perturbation encountered in a gap and a careful treatment of a generalized elastic interface with both surface tension and interfacial elasticity. Results indicate that globular proteins, which typically have comparable surface tension and interfacial elasticity, can be modeled well by this generalized elastic interface. We further find that when in a gap, droplets and capsules migrate toward one electrode, deform asymmetrically, exhibit polar spreading on the electrode, and predictably stretch more than in the infinite gap scenario at constant field strength.

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INTRODUCTION Like surfactants, proteins are known to self-assemble on liquid interfaces to alter fluid−fluid interfacial properties and even form quasi-2D viscoelastic films.1−6 As opposed to small molecule surfactant interfaces that primarily exhibit a 2D liquid-like behavior, in typical laboratory or process conditions (e.g., strain rates of 10−3−1 s−1), equilibrated protein interfaces can often behave like a 2D elastic solid. For example, for a small concentration (O ≈ 10−5 M) of bovine serum albumin in a water droplet placed in oil, the protein molecules selfassemble into a concentrated interfacial layer on the oil−water interface, the interfacial energy lowers, and over ∼10 h, a thin viscoelastic interfacial membrane forms.7−12 Substantial interfacial elasticity (>10 mN/m) typical of a capsule has been measured despite the O[10 nm] thickness of a selfassembled protein membrane.2 Capsules, that is, droplets with an elastic membrane, have shown interesting deformation behavior not seen in simple surface tension droplets like increased strength, tank-treading, wrinkling, and instabilities.13 Strengthening or engineering droplet interfaces can help maintain stable emulsions, create robust drug delivery vehicles, and enable microfluidic droplet reactors14 and artificial blood cells.15 Many applications are specifically concerned with droplet and capsule deformation in electric fields.16 This work is motivated by fabrication of millimeter-scale perfectly spherical polymer shells for nuclear fusion laser compression experiments made from droplet-within-droplet emulsion precursors.17 In this process, the shell fluid is polymerized in an ideally concentric droplet configuration so that the resulting © XXXX American Chemical Society

shell has a uniform wall thickness. This is often not the case, and the process yield can be very low and limited to thin walls. However, an E ∼ kV/cm AC electric field (∼10 MHz) can actively force droplets-within-droplets to adopt a perfect concentric arrangement.18,19 To employ active droplet centering in shell production, it is simultaneously crucial to limit droplet stretching in the strong centering electric field, as even small 1% sphericity deformations are problematic in fusion shell targets.17 On the other hand, precision control of deformation can offer a fabrication route for alternate inertial fusion shell designs with custom perturbations expected to counteract inertial fusion drive nonuniformities.20 Motivated by the desire to control droplet deformation in fluidic devices, we report experimental measurements and models of the small deformation of surfactant- and proteinladen water droplets in an electric field in both wide and thin oil gaps. An electric field induces a nonuniform Maxwell stress on the droplet interface and in turn stretches it into an ellipsoid.18,21,22 Feng and Scott23 and Lac and Homsy16 provide detailed reviews of the extensive past work on this fundamental problem for the case of a simple surface tension droplet deforming in a uniform electric field. Electric fieldinduced deformation has also been studied in droplets with complex interfaces like surfactants,24 vesicles,25 interfacial polymerized capsules,26−31 and particles.32 Protein-covered Received: May 23, 2018 Revised: July 19, 2018 Published: July 30, 2018 A

DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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Langmuir droplets are unique because they can adopt the qualitatively different extreme structures (2D fluid or 2D elastic solid) depending on the interface preparation procedure.33,34 Furthermore, as we observe here, surface tension can have non-negligible influence on deformation even after a protein forms a capsule membrane. Analogous to a previous microfluidic extensional flow study,35 we investigate representatives from different protein families with qualitatively large differences in oil−water interfacial strength: (1) Silwet L-77 as a small molecule anionic surfactant, (2) bovine serum albumin (BSA) as a globular protein, (3) AFD4 as a small tunable interfacial crosslinking peptide,36,37 and (4) cerato ulmin (CU) as a hydrophobin, regarded as the most surface active family of proteins found in nature.38 In particular for BSA, we explore denaturing39 to probe differences in interfacial strength on the interfacial solid formation method. Figure 1a,b shows schematics of the study. We detail the small-deformation solution for a generalized capsule with both interfacial tension

(i.e., uniform elastic prestress) and interfacial Hookean elasticity. In contrast to thicker and stronger interfacially polymerized membranes, we argue that surface tension is essential to include in the deformation model for some globular protein membranes. For the first time, the examination of electric field deformation in a thin channel is performed to more directly mimic the walls of a fluidic device. In thin gaps, we find dielectrophoretic motion toward an electrode, spreading near the electrode, and asymmetric deformation larger than in the wide gap case at constant field strength (i.e., constant voltage/gap height). We further extract approximate interfacial elasticity parameters and suggest conditions to improve measurement resolution.

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EXPERIMENTAL SECTION

Materials. Fluids. We use ultrapure deionized (DI) water (Thermo Scientific, Barnstead Nanopure system) filtered through 0.45 μm filters (Aerodisc, Pall Life Sciences, AP-4424) with conductivity ∼10−5 S/m measured in air in contact with process vessels (VWR, 23226-505). The oil solution is a mixture of 20% (v/v) tetrachloroethylene (TECE, Sigma-Aldrich, 270393) in mineral oil (Aldrich, 161403) so that the density is matched to the aqueous droplet. Fluid densities were measured at 21 °C on a densitometer (Anton Paar, DMA 4500) resulting in 0.998 for deionized water, 0.849 g/mL for mineral oil, and 1.599 g/mL for TECE, with errors of ±0.001 g/mL. The dielectric constant ε2 = 2.26ε0 (where ε0 = 8.85 × 10−12 F/m) for the density-matched oil solution is assumed to be a linear combination with volume % prefactors of the mineral oil (ε = 2.2ε040) and TECE (ε = 2.5ε041) and constant at the ≪GHz frequencies used here. The dielectric constant of water42 is ε1 = 80.1ε0, giving a dielectric constant ratio S = ε1/ε2 = 35.4. Surfactants and Proteins. We are studying the behavior of the following surfactants and proteins, with each serving as a representative of a different interfacial behavior: Silwet L-77 (Silwet, Helena Chem. Co., Collierville, TN, KC8L1297, Mw = 600 g/mol43), bovine serum albumin (BSA, a globular protein,7−12,44 >96%, Sigma, A9418-5G, 66 kDa, 4 × 4 × 14 nm heart-shaped structure at moderate pH44,45), AFD4 (a reversible interfacial cross-linking peptide,36,37 GenScript, Piscataway, NJ, Mw = 2435, peptide sequence: Ac-MKQLADS LHQLAHK VSHLEHA-CONH2), and cerato ulmin (CU, a hydrophobin, donated from Paul Russo at Georgia Tech, 7.6 kDa, molecular diameter ∼3 nm46,47). A schematic of each is depicted in Figure 1 (inset). All proteins were used in as-received state without further purification. Aqueous solution concentrations were large enough to highly populate the interface, specifically: 0.1% Silwet, 10−4 M BSA (either native or denatured), 5 × 10−5 M AFD4/2 × 10−4 M ZnSO4, and 10−4 M CU. Solutions were visibly clear with the exception of the somewhat cloudy Silwet solutions or any milky white BSA solutions that were intentionally denatured. More details on each emulsifier are provided in the Supporting Information. Charge Effects. In modeling deformation, we do not consider surface charge effects; however, protein molecules often carry a net surface charge in water. A strong surface charge would potentially alter the electric field around the droplet, potentially inducing electrohydrodynamic flow and altering the deformation. We did not observe electrohydrodynamic flow in any of these stretching experiments, which would be seen in tests with tracer microbubbles or small contaminants for a flow of >O[0.1 mm/s]. Past simulation work at characteristic surface charge values suggests interfacial charge has a 1) or by interfacial elasticity (γ(5 + νs)/Es < 1). Equation 3 reduces to the solution for a surface tension droplet16 as γ(5 + νs)/Es → ∞:

where Ts is the constitutive interfacial stress tensor, Tbulk,i is the volumetric stress tensor of the bulk fluid i evaluated at the interface, and ||Tbulk|| is the jump in stress between the two bulk phases across the interface, or specifically ||Tbulk|| = Tbulk,2 − Tbulk,1 by convention.53 The normal and tangential components of eq 1 simplify for an axisymmetric surface (derived in the Supporting Information). A small deformation solution considers how the bulk stress imbalance on the reference shape perturbs the interface,22,56 so that eq 1 becomes ∇s·Ts ≃ −||N·Tbulk||. As in previous droplet electric deformation problems in a uniform electric field,16 we find ||N· Tbulk|| for pure dielectrics by solving the electrostatics problem using the spherical reference geometry in Appendix 1. Interfacial Constitutive Model. With the bulk driving force known, an interfacial constitutive relation Ts must be chosen to solve for protein-coated droplet deformation. Given that we are studying steady-state deformation without any bulk electrohydrodynamic flow, interfacial viscosity has no role in the mechanical balance. Deformation will be governed by interfacial tension and elastic stresses. The various elastic membrane and shell models that have been used to model capsule deformation all reduce to a Hookean model at low deformation.13,57 An interfacial Hookean model requires two parameters and are typically chosen among these four: interfacial Young’s modulus Es, interfacial shear modulus Gs, interfacial dilatational modulus Ks, and surface Poisson ratio νs. Relations exist to write each parameter in terms of two others (see the Supporting Information). In terms of the interfacial Young’s modulus Es and the interfacial Poisson ratio νs, a generalized elastic interface with surface tension (or uniform prestress in its reference state) follows the interfacial constitutive equation: Ts =

9ε2E 2R (S − 1)2 (5 + νs) 16Es (S + 2)2 γ 1 + E (5 + νs) s

D≃

9ε2E2R (S − 1)2 16γ (S + 2)2

surface tension droplet

(4)

It reduces to the solution for an elastic capsule26−29 with a stress-free reference state as γ(5 + νs)/Es → 0: D≃

9ε2E2R (S − 1)2 (5 + νs) 16Es (S + 2)2

capsule

(5)

28

As previously noted, a capsule’s deformation depends on both Es and νs, but small deformation would only vary ±20% with νs because it can only range from −1 to 1. However, νs has been fit in or extracted from past work to a narrower range for both thick elastic membranes (νs ≈ 0)13,59 and thin protein membranes (νs = 0.6−0.8).50,60 We will simply assume νs = 0.6 to fit our experiments on protein-coated droplets for Es. Consequently, a capsule will deform ∼5.6× more than a hypothetical droplet with γ equivalent to the capsule’s Es.28 However, capsules made from globular proteins like BSA are unique because Es ≈ γ. In fact, due to the ∼5.6× factor, surface tension can still dominate BSA membrane interfacial elasticity. For this generalized capsule case, the electrostatic deformation (eq 3) depends on three constitutive parameters (Es, νs, and γ), so even with the νs = 0.6 assumption, the interfacial elastic parameters cannot be independently extracted from D versus E2R small deformation experiments. A value for the surface tension must be also determined. Thin Gap Solution. A thin electrode gap perturbs the otherwise uniform electric field just discussed, which in turn changes the bulk driving traction ||N·Tbulk||. As in a previous work on surface tension droplets, we consider the simplest nonuniform perturbation from a uniform field.23,61 Instead of the infinite gap electric potential ϕ(ρ → ∞) = −EρP1(cos θ), a

(2)

where Is is the surface identity or projection tensor (Is = I − nn), e is the infinitesimal surface strain tensor,13 and Tr indicates trace. With physical components of e indicated by εθ and εϕ, Tr[e] = εθ + εϕ. We are using the thin membrane approximation to ignore any out-of-plane stresses in the D

DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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simple nonuniform potential can be approximated by ϕ(ρ → ∞) = −EρP1(cos θ) − Λρ2P2(cos θ). In this work, Λ is fit to the fields generated in COMSOL Multiphysics around a droplet centered in a gap (e.g., similar to the method for a pin electrode of Mhatre and Thaokar62). Droplet shape in the nonuniform field is then characterized by two symmetric (s2,s4) and one asymmetric (s3) perturbation: (6)

23

RESULTS AND DISCUSSION

Interfacial Tension and Membrane Preparation Methods. To extract Es from a protein-coated droplet stretching experiment, we seek evidence that a 2D solid interfacial membrane can form for a given aging procedure, and, in cases where we expect γ(5 + νs)/Es ≈ O[1], a value for γ. Pendant drop results and analysis are included in the Supporting Information. Equilibrated γ values in mineral oil or TECE and parameters used in modeling density matched mineral oil/TECE are indicated in Table 1. From dynamic pendant drop studies, we observed (1) fast surface equilibration (O[s]) of the small molecule surfactant Silwet as compared to the longer (O[100 + s]) times for the larger proteins, (2) two-step equilibration for native BSA solutions typically interpreted8 as a O[100 s] interfacial population and a O[103−104 s] reorganization with an additional ∼20% surface tension drop, and (3) a comparably accelerated (O[100 s]) equilibration for denatured BSA. We tested aging procedures to form 2D solid elastic membranes by slowly suctioning fluid out of the pendant drop apparatus to observe for the presence of membrane wrinkles.37,50 The absence of wrinkles for 12 h room-temperature equilibration and Es = 47 mN/m for 70−75 °C denaturing. We only report an approximate Es value for the >12 h room-temperature equilibration because, due to its low magnitude, the experimental fit appears highly sensitive to γ. Although the absolute values of Es are sensitive to γ and Λ parameters, the trend supports the assumption that denaturing the BSA allows for progressively more intermolecular binding interactions and a stronger surface nanogel. However, the effect of γ on BSAcapsule deformation is more evident. Comparing the native BSA layers at different equilibration times, it may be initially surprising that the long-time equilibrated droplet (elastic membrane with wrinkling observed in tests) appears to deform more than the short-time case (interfacial liquid without wrinkling observed). However, recall that the interfacial tension drops ∼20% as the interfacial elastic membrane forms over 12 h. The observed higher deformation for the long-time room temperature membrane formation further supports the notion that, despite the additional elastic layer, γ governs the globular protein-coated droplet’s small deformation. Uncertainty Analysis. We have reported error estimates for the individual measurements in the Experimental Section; however, we will comment on the potential uncertainty in determination of Es using small-deformation models for the various protein-coated capsules. First note we already account for the expected accuracy of a linear fit to the small deformation model with adjustments of the reported values. CU and AFD4 data were fit to D < 0.02 so Es values were not adjusted. BSA experiments were fit to D < 0.05, so the best-fit Es was reported 12% higher as expected from previous work.16 Nevertheless, this is a source for uncertainty. There are still several potentially larger sources for uncertainty from K(ΛR/ E) (and the associated approximation for gap deformation near an electrode, ±15% for BSA and ±30% for AFD4, CU), ε2 (±10%), νs = 0.6 assumption (±7%), PDMS effect on E2 (+6%), linear fit to D/(E2R) (±5% for BSA and ±10% for AFD4 and CU), and specifically for the generalized capsule model of BSA, the value of γ (±20%). On the basis of the possibility of charge effects studied in simulations,48 another ∼ −10% uncertainty exists for AFD4. A root-mean-square of all of these errors yields ∼30% uncertainty in the reported Es values (±31% for BSA and 36% for AFD4/Zn2+ and 34% for CU). The largest sources for uncertainty are the γ value for BSA and, generally, the gap constant K (i.e., Λ) used to match

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CONCLUSION

Using different emulsifier representatives, we studied the small deformation of surfactant and protein-coated droplets in electric fields in both wide and thin gaps. Although there are generally many applications for droplets with strong interfaces, we were motived by a need to engineer droplets that resist deformation in a microfluidic droplet-within-droplet shell fabrication process that incorporates a strong electric field to center the droplets. A generalized small deformation model that incorporates both the effects of surface tension and the Hookean elasticity was presented. Furthermore, the effect of a thin gap was studied and observed. In an electric field in a thin gap, we found that droplets moved by dielectrophoresis until they collided into a PDMS-coated electrode. The small deformation was modeled in a nonuniform field parametrized by Λ, and the experimental results were compared to models by best-fitting Λ to the experimental gaps using finite element electrostatic solutions. Experiments and models indicated ∼2× higher deformation in gaps of R/h ≃ 0.3 over R/h ≃ 0 at the same E. Asymmetry in the deformation was also found in approximate agreement with the gap model; however, the droplet pole near the electrode was observed to spread, an effect that would dominate deformation in much smaller gaps. Although testing would require higher voltage sources than available, models pointed a stronger influence of the field nonuniformity on pure capsules compared to surface tension droplets. Experiments indicate protein-coated droplets are significantly stronger than a surfactant-coated droplet. Strength increased from a globular protein membrane with higher γ and a moderate Es, to a cross-linked peptide membrane and hydrophobin membrane with very high Es. In these latter cases, despite the presumably O[10 nm] thin protein membranes, the droplets showed extreme resistance to deformation. We used an approximation for νs and a bounded approximation for γ to extract the interfacial elasticity Es for each protein, and suggested improvements needed to refine the 30% uncertainty in measurements. Despite the formation of a visible elastic membrane for globular proteins like BSA, we argue that surface tension governs its dilatational small deformation behavior, an important consideration for dilatational interfacial experiments in general. This conclusion is supported by the observation of small but measurable shifts in experimental deformation results as the BSA interface preparation changes. In conclusion, dilute protein additives can greatly strengthen droplets in fluidic devices, and we are already working with a formulation that resists stretching in an electric field droplet-within-droplet centering device.

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APPENDIX 1: BULK ELECTROSTATIC INTERFACIAL STRESS Determination of the driving force for electric field deformation, that is, the jump in traction across the droplet interface ||N·Tbulk|| in its reference sphere shape, is outlined here with details available elsewhere.16,26−29 The bulk stress Tbulk is composed of hydrostatic pressure (−piI) and the H

DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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Langmuir

s i y ρ(θ ) ≃ R(1 + s2P2(cos θ )) = R jjj1 + 2 (3 cos2 θ − 1)zzz 2 k {

1

Maxwell stress T e = εi EiEi − 2 εiEi2 I, where I is the identity and i = 1,2 corresponds to either the droplet (i = 1) or the solvent (i = 2). Consequently, the jump in bulk stress is ||N· Tbulk|| = ΔpN + ||N·Te||, where Δp = p1 − p2. As discussed in previous work on droplets deforming in electric fields16 and shown in practice,63 an oil/water system behaves as a pure dielectric system at RF AC frequency. We must further assume negligible surface charge density (discussed above) or membrane potential (an O ≈ [0.05 V] effect in aqueous/ aqueous biological membranes). The solution for the electric field around (E2) and inside (E1) a dielectric sphere of radius R and dielectric constant ε1 in a dielectric fluid of dielectric constant ε2 is a textbook problem in electrostatics found by solving the Laplace equation for the electric potential in each phase i, ϕi, by separation of variables and using a series of Legendre polynomials; for example, see p 72 of Lim.64 For an infinitely wide electrode gap with ϕ(ρ → ∞) = −EρP1(cos θ), evaluating the Maxwell stress at the interface (ρ = R) and decomposing into normal and tangential stresses, we find N ·||N ·Tbulk || = Δp +

(11)

where s2 ≪ 1 is a general small parameter. In the following, we will conduct a perturbation analysis to O[s2] and will drop any terms of higher order. The solution assumes εθ and εϕ are both O[s2]. Using eq 11, the leading order differential geometry expressions (see the Supporting Information) of the perturbed axisymmetric shape are a11 ≃ R2(1 + s2(3 cos2 θ − 1)) a 22 ≃ R2 sin 2 θ(1 + s2(3 cos2 θ − 1))

7s 15s2 i y cos2 θ zzz b11 ≃ −R jjj1 − 2 + 2 2 k { 9s s i y b22 ≃ −R sin 2 θ jjj1 − 2 + 2 cos2 θ zzz 2 2 k {

9ε2E2 (S − 1) ((S − 1) cos2 θ 2 (S + 2)2

Tr[b] = a11b11 + a 22b22 ≃

+ 1)

a 22

A α ·||N ·Tbulk || = 0

−2 (1 − s2 + 3s2 cos2 θ ) R

∂a 22 ≃ cot θ(1 − 3s2 sin 2 θ ) ∂θ

(8)

(12)

For an electric potential nonuniformity parametrized by Λ so that ϕ(ρ → ∞) = −EρP1(cos θ) − Λρ2P2(cos θ), the normal and tangential stresses are61

The differential geometry expressions of eqs 12 can be inserted into the normal and tangential mechanical balances of eqs 10. To leading order, they simplify to

N ·||N · Tbulk ||

= Δp +

Es 2γ 2γ (εθ + εϕ) − (s2 − 3s2 cos2 θ ) + R(1 − νs) R R 9ε2E2 (S − 1) = Δp + ((S − 1) cos2 θ + 1) 2 (S + 2)2

ε2 (S − 1) [S(5ΛR(S + 2) 2 (S + 2)2 (2S + 3)2

− 3 cos θ(E(2S + 3) + 5ΛR(S + 2) cos θ ))2 + 9(E(2S + 3) + 5ΛR(S + 2) cos θ)2 sin 2 θ ]

Es

A α ·||N · Tbulk || = 0

■

1−

(9)

νs2

Es ν E ∂εϕ ∂εθ + s s2 + (εθ − εϕ) cot θ = 0 ∂θ 1 + νs 1 − νs ∂θ (13)

APPENDIX 2: SMALL DEFORMATION SOLUTION FOR A GENERALIZED HOOKEAN CAPSULE IN A UNIFORM FIELD

Simplifying notation using easily found expressions A and B, rearranging the normal balance gives εϕ =A cos2 θ + B − εθ. This is used to substitute out εϕ terms in the tangential balance, and a non-homogeneous ordinary differential equation solution is used to find εθ = (A + B)/2 − (A sin2 θ/4)(1 − 3νs)/(1 − νs). Next, as used previously for axisymmetric shells,65 a displacement vector strain analysis provides two additional equations with additional unknowns uρ and uθ, the displacements of the surface elements in the ρ and θ directions.

A2.1. Interfacial Balance

For the interfacial constitutive relation of a generalized Hookean membrane with surface tension (eq 2), the axisymmetric (see simplifications in Supporting Information) normal and tangential balances of eq 1 become: νE Es ji zy Tr[e·b] + jjjγ + s s 2 Tr[e]zzzTr[b] = −Δp − fe j z 1 + νs 1 − νs k { 22 ∂ ε νE Es ∂εθ a ∂a 22 Es ϕ + s s2 + (εθ − εϕ) 2 ∂ θ ∂ θ 2 ∂θ 1 + νs 1 − νs 1 − νs =0

y 1 ij duθ jj + uρzzz R k dθ { 1 εϕ = (uθ cot θ + uρ) R εθ =

(14)

The solutions for s2 and Δp can be found by matching the solution for uρ from eqs 14 to uρ = Rs2P2(cos θ) as assumed from the initial problem statement. After extensive algebra, we find (recall D = 3s2/4):

(10)

where εθ and εϕ are the physical components of the interfacial strain tensor e, and fe = 9(S−1)ε2E2[(S−1) cos2 θ + 1]/(2(S + 2)2) for the wide gap case. A2.2. Small Deformation from a Reference Sphere

s2 =

Because the perturbing Maxwell stress has a P2(cos θ) form, we assume the deformed droplet has the shape: I

3ε2E 2R (S − 1)2 (5 + νs) 4Es (S + 2)2 γ 1 + E (5 + νs) s

(15) DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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Dimensionless electric capillary numbers16 can also be constructed to show the relative importance of the Maxwell stress to either the stabilizing surface tension (Caγ = ε2E2R/γ) or the interfacial elasticity (CaEs = ε2E2R/Es). In terms of these electric capillary numbers, we find s2 =

3Ca Es (S − 1)2 (5 4 (S + 2)2 Ca Es

1+

Ca γ

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.8b01713.

(5 + νs)

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n=1

*Phone: (858) 455-3000. Fax: (858) 455-3181. E-mail: [email protected]. ORCID

Greg C. Randall: 0000-0002-8375-9041 Notes

The author declares no competing financial interest.

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yz

∑ snPn(cos θ)zzzzz 4

{

s2 =

3ε2E2R (S − 1)2 25ε2 Λ2R3 (S − 1)(4S + 3) + 2 4γ (S + 2) 28γ (2S + 3)2

s3 =

9ε2E ΛR2(S − 1)2 5γ(S + 2)(2S + 3)

ACKNOWLEDGMENTS This work was funded by General Atomics Internal Research and Development. I thank Tom Jones, Shirley Bei, and David Harding (University of Rochester, Laboratory for Laser Energetics) and Reny Paguio, Neil Alexander, Fred Elsner, and Brent Blue (General Atomics) for helpful comments, Paul Russo (Georgia Tech) for providing the hydrophobin samples and for discussions of proteins at interfaces, Wade Bittle (Laboratory for Laser Energetics) for design and fabrication of the RF impedance matching network, and Wendi Sweet (General Atomics) and Emma Willard (UC Davis) for pendant drop assistance.

(17)

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10ε2 Λ2R3(S − 1)2 7γ(2S + 3)2

Deformation for the Hookean capsule case in this nonuniform field is solved as above using Ts from eq 2 with γ = 0. We find

s4 =

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(1) Fuller, G. G.; Vermant, J. Complex Fluid-Fluid Interfaces: Rheology and Structure. Annu. Rev. Chem. Biomol. Eng. 2012, 3, 519− 543. (2) Sagis, L. M. Dynamic Properties of Interfaces in Soft Matter: Experiments and Theory. Rev. Mod. Phys. 2011, 83, 1367−1403. (3) Miller, R.; Ferri, J. K.; Javadi, A.; Krägel, J.; Mucic, N.; Wüstneck, R. Rheology of Interfacial Layers. Colloid Polym. Sci. 2010, 288, 937− 950. (4) Lyklema, J. Fundamentals of Interface and Colloid Science V: Soft Colloids; Academic Press: London, 2005. (5) Bos, M. A.; van Vliet, T. Interfacial Rheological Properties of Adsorbed Protein Layers and Surfactants: A Review. Adv. Colloid Interface Sci. 2001, 91, 437−471. (6) Mitropoulos, V.; Mütze, A.; Fischer, P. Mechanical Properties of Protein Adsorption Layers at the Air/Water and Oil/Water Interface: A Comparison in Light of the Thermodynamical Stability of Proteins. Adv. Colloid Interface Sci. 2014, 206, 195−206. (7) Graham, D. E.; Phillips, M. C. Proteins at Liquid Interfaces V. Shear Properties. J. Colloid Interface Sci. 1980, 76, 240−250. (8) Beverung, C.; Radke, C.; Blanch, H. Protein Adsorption at the Oil/Water Interface: Characterization of Adsorption Kinetics by Dynamic Interfacial Tension Measurements. Biophys. Chem. 1999, 81, 59−80. (9) Lu, J.; Su, T.; Thomas, R. Structural Conformation of Bovine Serum Albumin Layers at the Air−Water Interface Studied by Neutron Reflection. J. Colloid Interface Sci. 1999, 213, 426−437. (10) Alexandrov, N.; Marinova, K. G.; Danov, K. D.; Ivanov, I. B. Surface Dilatational Rheology Measurements for Oil/Water Systems with Viscous Oils. J. Colloid Interface Sci. 2009, 339, 545−550.

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s3 =

AUTHOR INFORMATION

Corresponding Author

where sn ≪ 1 is a general small parameter. Appendix 1 eq 9 indicates the bulk traction for the nonuniform field with potential ϕ∞ = −rEP1(cos θ) − Λr2P2(cos θ), where Λ parametrizes the nonuniformity (in units of V/m2). We recalculate the pure dielectric deformation extracted from Feng and Scott23 and Thaokar61 for a surface tension droplet (Ts = γIs) in this nonuniform electric field. Note important notation differences with Thaokar,61 for example, S = 1/SThaokar, E = E Thaokar / 2 , and Λ = Λ Thaokar / 2 . We find

s2 =

(1) More information on each emulsifier, (2) details on interfacial tension measurements, (3) relations between parameters used in a Hookean interfacial elastic model for Ts, and (4) derivation of normal and tangential interfacial balances for an axisymmetric shape (PDF)

+ νs)

APPENDIX 3: SMALL DEFORMATION SOLUTIONS IN A NONUNIFORM FIELD We calculate the pure dielectric deformation in a nonuniform electric field for two conditions: a pure surface tension droplet23,61 and a Hookean elastic capsule. These nonuniform field solutions were generated as in Thaokar,61 for example, applying a Legendre polynomial integration identity to the normal and tangential interfacial balances, using Mathematica v11 for the extensive algebra. We assume the deformed droplet has the shape:

s4 =

ASSOCIATED CONTENT

S Supporting Information *

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ij ρ(θ ) ≃ R jjjj1 + j k

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3ε2E2R(5 + νs) (S − 1)2 25ε2 Λ2R3(5 + νs) + 4Es 28Es (S + 2)2 (S − 1)(4S + 3) (2S + 3)2 3ε2E ΛR2(S − 1)(1777S − 1905 + νs(115S − 243)) 277Es(S + 2)(2S + 3) 10ε2 Λ2R3(S − 1)2 (19 + νs) 7Es(2S + 3)2 (19) J

DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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Langmuir

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DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.langmuir.8b01713 Langmuir XXXX, XXX, XXX−XXX