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Publication Date (Web): January 5, 2015 ... One can view Young's law and classical Neumann's triangle conditions as two extreme limits of droplet shap...
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Polymeric Droplets on Soft Surfaces: From Neumann’s Triangle to Young’s Law Zhen Cao and Andrey V. Dobrynin* Polymer Program, Institute of Materials Science, University of Connecticut, Storrs, Connecticut 06269-3136, United States S Supporting Information *

ABSTRACT: Shape deformation of polymeric droplets on gel-like surfaces is studied by using a combination of the molecular dynamics simulations and theoretical calculations. On the basis of the results of molecular dynamics simulations, we have developed a theoretical model of droplet shape deformation on elastic surfaces which takes into account surface and elastic energy contributions. Analysis of simulation results in the framework of this model shows that the equilibrium droplet shape is controlled by the dimensionless parameter γSL/GSa, where γSL is the surface tension of the substrate/liquid interface, GS is the shear modulus of the substrate, and a is the contact radius of polymeric droplet. This parameter describes crossover between Neumann’s triangle conditions for liquid droplets on liquid substrates and Young’s law for droplets on rigid substrates. In the limit when the elastocapillary length is much larger than the contact radius, γSL/GS ≫ a, we recover the Neumann’s triangle conditions for liquid droplets floating on liquid substrates. However, in the opposite limit, γSL/GS ≪ a, our model reproduces Young’s law. The model predictions are in a very good agreement with simulation results and experimental data.

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INTRODUCTION Spreading of liquid on solid or liquid substrates plays an important role in understanding processes governing oil recovery, printing and painting, surface waterproofing, locomotion of insects on surface of water, and cornea lubrication.1−4 For a small droplet placed on the solid surface the equilibrium is attained when the contact angle θ1 satisfies Young’s equation1,2,5 which provides the relationship between surface tensions of the liquid, γL, substrate, γS, and substrate/ liquid γSL interfaces and the contact angle θ1 (see Figure 1a). This condition corresponds to the minimum of the system free energy6 and can be visualized as in plane balance of the surface tensions. For a liquid droplet floating on another liquid (liquid substrate) three surface tensions at the contact line should balance (see Figure 1b). This balance results in the Neumann’s triangle conditions1,2,7 that represent balance of the vertical and horizontal components of the surface tensions at the contact line. Note that as in the case of the Young’s equation the Neumann’s conditions can also be derived from minimization of the system free energy at constant droplet volume (see discussion below). Comparing conditions describing equilibrium of the liquid droplet on the solid substrate and of the liquid droplet floating on the liquid substrate, it appears from Figure 1a that the vertical component of liquid surface tension γL sin θ1 acting on the substrate at the contact line is not balanced. In reality this is not the case, and this vertical component of the force is balanced by the stress generated in the substrate due to wetting ridge formation near the triple phase (substrate/liquid/air) contact line (see Figure 1c).8−10 For a displacement d of the contact line region the generated © XXXX American Chemical Society

elastic stress per unit length of the contact line is on the order of GSd, where GS is the shear modulus of the substrate. For water (γL ≈ 70 mN/m) spreading over the glass (GS ≈ 50 GPa) the substrate displacement d ≈ γL sin θ1/GS ≈ 1.4 pm is impossible to observe. This substrate deformation becomes on the order of tens of nanometers for elastomeric substrates with shear modulus on the order of 0.1 MPa. Therefore, for deformable (gel-like) surfaces the local force balance at the contact line in addition to surface tensions of corresponding interfaces should also include contributions arising from local elastic stresses generated by the substrate deformation.11 These additional contributions can be taken into account by balancing surface stresses, ϒi, at the triple phase contact line (see Figure 1c). This surface stress balance is an extension of the classical Neumann’s triangle condition1,2,7 describing equilibrium of small droplet floating on a liquid (see Figure 1b). Note that in the elastic solids γi and ϒi are related by the Shuttleworth equation12 and for liquids γi = ϒi. One can view Young’s law and classical Neumann’s triangle conditions as two extreme limits of droplet shape deformation on extremely hard and soft substrates, respectively. Recent experimental,8−10,13−15 theoretical,11,16−19 and simulation20 studies were focused on understanding mechanism of droplet shape deformation on soft substrates with particular emphases on the substrate deformation near the triple phase line (the wetting ridge) and measurement of the substrate Received: August 14, 2014 Revised: December 19, 2014

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Figure 1. Droplet wetting of solid (Young’s law: γS = γSL + γL cos θ1) (a), liquid (Neumann’s triangle conditions: γS = γSL cos θ2 + γL cos θ1; γSL sin θ2 = γL sin θ1) (b), and gel-like (Neumann’s triangle conditions: ϒS cos θS = ϒSL cos θ2 + γL cos θ1; ϒSL sin θ2 + ϒS sin θS = γL sin θ1) (c) substrates.

Figure 2. Snapshots of the polymeric droplets’ wetting of elastic substrates.

Table 1. System Parameters droplet radius Rd [σ] substrate dimensions Lx = Ly [σ] droplets ρdσ3 ρcσ3 GL [kBT/σ3]

0.953 0.000 0.000

10.92 45.2 ρsσ3 ρcσ3 GS [kBT/σ3]

15.81 65.6 0.953 0.000 0.000

19.78 82.1 substrates

0.962 0.044 0.024

0.970 0.095 0.072

25.74 106.9 0.979 0.159 0.162

30.64 127.3 0.986 0.209 0.252

34.62 143.9 0.999 0.309 0.498

1.013 0.412 0.833

⎧ ⎡⎛ ⎞12 ⎛ ⎞6 ⎛ ⎞12 ⎛ ⎞6 ⎤ ⎪ σ σ σ ⎥ ⎢ σ ⎪ 4εLJ⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ − ⎜ ⎟ + ⎜ ⎟ ⎥ rij ≤ rcut r r r r ⎝ ⎠ ⎝ ULJ(r) = ⎨ ⎝ ij ⎠ cut cut ⎠ ⎦ ⎣⎝ ij ⎠ ⎪ ⎪0 rij > rcut ⎩

surface stresses21 from substrate deformation near the contact line. In this paper we use molecular dynamics simulations to study macroscopic and microscopic features of the contact between soft (gel-like) substrates and polymeric droplets. The range of studied system parameters allowed us to cover crossover between Neumann and Young-like behavior of polymeric droplets on elastic substrates with different values of the substrate shear modulus. We focus on the substrate deformation under the droplet and show that this deformation determines the apparent macroscopic values of the contact angles and results in renormalization of the substrate/liquid surface stresses. These simulation results are used to develop a model which accounts for both capillary effects and elastic deformation of the substrate. For soft substrates our model recovers Neumann’s triangle conditions for the surface tension balance for a liquid droplet floating on a liquid substrate. However, in the limit of a rigid substrate it reduces to Young’s expression. The crossover between different droplet and substrate deformation regimes is controlled by the ratio of the elastocapillary length γSL/GS and the contact radius a of a droplet.

(1)

where rij is the distance between the ith and jth beads, σ is the bead diameter, and the cutoff distance rcut was set to 2.5σ. The Lennard-Jones interaction parameter εLJ was equal to 1.5kBT for interactions between beads of the same type, and was equal to 0.25, 0.40, 0.75, or 1.20 kBT for interactions between beads of different types (kB is the Boltzmann constant and T is the absolute temperature). Connectivity of beads into polymer chains and cross-links bridging chains together were modeled by the finite extension nonlinear elastic (FENE) potential22 ⎛ 1 r2 ⎞ ⎟ UFENE(r ) = − kspringR max 2 ln⎜1 − 2 R max 2 ⎠ ⎝

(2)

with the spring constant kspring = 30kBT/σ2 and the maximum bond length Rmax = 1.5σ. The repulsive part of the bond potential was modeled by the LJ potential with rcut = 21/6σ and εLJ = 1.5kBT. The gel-like substrates were placed on a rigid wall represented by the external potential acting on all beads in a system

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MODEL AND SIMULATION DETAILS We performed coarse-grained molecular dynamics simulations of polymeric droplets on soft gel-like substrates (see Figure 2). In our simulations we used bead−spring representation of polymer chains and gels.22 Both polymeric droplets with size Rd and gel-like substrates were made of chains with the number of monomers N = 32. The substrate elastic properties were controlled by varying the degree of cross-linking between polymer chains, ρc. The interactions between all beads in a system were modeled by the truncated-shifted Lennard-Jones (LJ) potential23

⎡ 2 ⎛ σ ⎞ 9 ⎛ σ ⎞3 ⎤ U (z) = εw ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ⎝z⎠ ⎦ ⎣ 15 ⎝ z ⎠

(3)

where εw was set to 1.0kBT. Simulations were carried out in a constant number of particles and temperature ensemble. The constant temperature was maintained by coupling the system to a Langevin B

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Macromolecules Table 2. Surface Tension γL [kBT/σ2] GS [kBT/σ3] γS [kBT/σ2] εLJ [kBT] 1.20 0.75 0.40 0.25

1.812 0.000 1.812 1.309 2.674 3.385

0.024 1.843

0.072 1.879

1.339 2.700 3.414 3.575

1.356 2.723 3.445

0.162 1.967 γSL [kBT/σ2] 1.440 2.798 3.529

0.252 2.027

0.498 2.213

0.833 2.395

1.480 2.847 3.585

1.631 3.028 3.760

1.746 3.177 3.936

Figure 3. Average shape of polymeric droplets with initial size Rd = 25.74σ and angles θ1 and θ2 produced by wetting of elastic substrates. G̃ = GSσ3/ kBT is reduced shear modulus of the substrate, and ε̃LJ = εLJ/kBT is reduced value of the LJ interaction parameter between beads forming droplets and substrates.

thermostat23 implemented in LAMMPS.24 In this case, the equation of motion of the ith bead is m

dvi⃗(t ) R = Fi ⃗(t ) − ξvi⃗(t ) + Fi⃗ (t ) dt

Polymeric Droplets. The polymer droplets were made by confining polymer chains into a spherical cavity (see for details ref 25). After removing the confining cavity, droplets were equilibrated by performing MD simulation runs lasting 104τLJ. The size of equilibrium droplets Rd was obtained by calculating their radius of gyration. Elastic Substrates. The soft gel-like substrates were made by confining polymer chains into a slab with thickness H0; then the chains were randomly cross-linked by the FENE bonds (see for details ref 26). After cross-linking, the confining slab was removed and substrates were equilibrated in the external potential given by eq 3 by performing MD simulation runs lasting 104τLJ. The equilibrium substrate thickness was varied between 20σ and 64σ depending on droplet size (see Supporting Information for particular substrate thicknesses). It was calculated by using the height distribution function of the

(4)

where m is the bead mass set to unity for all beads in a system, vi⃗ (t) is the ith bead velocity, and F⃗i(t) denoted the net deterministic force acting on the ith bead. The stochastic force F⃗Ri (t) had a zero average value and δ-functional correlations ⟨F⃗Ri (t)F⃗Ri (t′)⟩ = 6kBTξ(t − t′). The friction coefficient ξ was set to ξ = m/τLJ, where τLJ is the standard LJ-time τLJ = σ(m/ εLJ)1/2. The velocity-Verlet algorithm with a time step Δt = 0.01τLJ was used for integration of the equations of motion. In our simulations we used periodic boundary conditions in x and y directions. System sizes are summarized in Table 1. All simulations were performed using LAMMPS.24 C

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polymer-phobic. However, the polymer phobicity of the substrate increases with increasing the substrate shear modulus GS. This is manifested in decrease of the indentation depth produced by a droplet in a substrate and decrease the value of the angle θ2. This should not be surprising since with increasing substrate shear modulus the elastic energy of indentation produced by a droplet increases as well favoring shallow indentations. In the case of a droplet floating on liquid substrate the Neumann’s triangle (see inset in Figure 4) represents a balance

substrate. The shear modulus of the substrate was obtained from 3-D simulations of a polymeric gel with the same degree of cross-linking and values of interaction parameters as described in ref 25. Droplet Spreading. Simulations of droplets spreading on gel-like substrates started with placing a droplet at a distance of 2.0σ from a substrate. A harmonic potential with the spring constant Ksp = 100kBT/σ2 was applied to the droplet center of mass for 100τLJ to bring it in contact with a substrate then the potential was removed. The system was equilibrated for 4 × 104τLJ followed by the production run lasting 104τLJ. Work of Adhesion. We have calculated the potential of mean force between two films to evaluate the work of adhesion between a droplet and a substrate. In these simulations, two films having structures of gel-like substrate and polymeric droplet with initial dimensions 10σ × 10σ × 10σ were pushed toward each other. In these simulations z-component of the center of mass of the polymeric film was fixed at z = 0.0σ. The center of mass of gel-like film, zcm, was tethered at z* by a harmonic potential U (zcm , z*) =

1 K sp(zcm − z*)2 2

(5)

with the value of the spring constant Ksp varying between 200 and 500 kBT/σ2. We have moved location of the film’s tethering point with an increment Δz* = 0.1σ. For each location of the tethering point, we have performed a simulation run lasting 5 × 103τLJ, during which we have calculated distribution of the center of mass of the film. The weighted histogram analysis method (WHAM)27 was applied to calculate potential of the mean force between two films from distribution functions of the film center of mass. The potential of the mean force was used to calculate work of adhesion, W = ΔF/A, as a function of the interaction parameter and the cross-linking density of the gel-like substrates. Surface Tension. The surface tension of the polymeric droplet and gel substrates was evaluated by integrating the difference of the normal PN(z) and tangential PT(z) to the interface components of the pressure tensor. Note that in our simulations, the z direction was normal to the interface.

Figure 4. Dependence of the magnitude of the vertical component of the liquid surface tension, γL sin θ1, on the magnitude of the vertical component of the substrate/liquid surface tension, γSL sin θ2, for droplets of different sizes on substrates with shear modulus varying between 0.024kBT/σ3 and 0.833kBT/σ3 and different values of the LJ interaction parameters for polymer−substrate pairs: εLJ = 0.25kBT (filled pink pentagons), εLJ = 0.40kBT (filled black squares), εLJ = 0.75kBT (filled red circles), and εLJ = 1.20kBT (filled blue triangles). Open symbols correspond to droplets floating on liquid (polymeric melt) substrates. Dashed line corresponds to Neumann’s condition given by eq 7. Inset shows Neumann’s triangle for the droplet on liquid substrate.

of surface tensions at the contact line resulting in the following expression for its vertical components

ξ

γ=

∫−ξ (PN(z) − PT(z)) dz

γL sin θ1 = γSL sin θ2

(6)

(7)

In Figure 4, we test validity of the Neumann’s triangle condition by plotting dependence of the vertical components of the surface tensions as a function of the angles θ1 and θ2 and surface tensions for different strengths of the LJ interaction parameter, shear modulus of the substrate, and droplet sizes. We used macroscopic values of the angles that were obtained by fitting averaged droplet shape by two spherical caps and using obtained values of the caps’ heights and contact (droplet footprint) radius a to calculate angles θ1 and θ2 (see eq 13). The values of surface tensions for this plot were obtained from separate simulations as discussed above (see Table 2). Neumann’s triangle condition is shown by the dashed line. Note that a deviation from this line indicates violation of the Neumann’s triangle condition for a liquid substrate as described by eq 7. The following conclusions can be drawn from this figure. The values of the angles obtained for polymeric droplets on melt-like polymeric substrates (open symbols) are described by Neumann’s triangle condition, eq 7. This should be expected since substrate deformation lacks any elastic contribution. The second set of data which closely follows the Neumann’s triangle

where 2ξ is the thickness of the interface that was determined from the monomer density profile as an interval within which the monomer density changes from zero to its bulk value. The results of surface tension calculations are presented in Table 2. The surface tension of the substrate/polymer interface was calculated from the work of adhesion, γSL = γS + γL − W.

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SIMULATION RESULTS Figure 3 illustrates the variation of the droplet shape and angles as a function of the strength of the polymer/substrate interactions and substrate shear modulus, GS. For comparison, we also show polymeric droplets on the surface of polymeric film (see first column in Figure 3). It follows from this figure that with increasing the affinity between droplet and substrate (increasing the strength of the LJ interactions) the depth of indentation produced by a droplet in a substrate monotonically increases. This increase is accompanied by increase in the value of the angle θ2 and decrease of the angle θ1. Thus, improving affinity between droplet and substrate makes the substrate less D

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Figure 5. Superimposed droplet and surface profiles at triple phase contact line for droplets of different initial sizes Rd = 10.92σ (black squares), Rd = 15.81σ (red circles), Rd = 19.78σ (blue triangles), Rd = 25.74σ (green triangles), Rd = 30.64σ (pink triangles), and Rd = 34.62σ (dark blue triangles) having value of the LJ interaction parameter between polymer−substrate pairs εLJ = 1.20kBT interacting with substrates having values of shear modulus: GS = 0.833kBT/σ3 (a), GS = 0.162kBT/σ3 (b), and GS = 0.0kBT/σ3 (c).

Figure 6. Dependence of the difference between surface stress and surface tension of the substrate (a) and substrate/droplet interface (b) on the ratio of the elastocapillary length, γL/GS, to the bead size σ for different values of the LJ interaction parameter between polymer−substrate pairs εLJ = 0.40kBT (black squares), εLJ = 0.75kBT (red circles), and εLJ = 1.2kBT (blue triangles). The values of the surface stress for these plots were obtained from Neumann’s triangle conditions shown in Figure 1c.

condition corresponds to a weak polymer/substrate interactions characteristic feature of which is shallow substrate indentations (see Figure 3). However, the majority of our simulation data demonstrate a significant deviation from Neumann’s line. Furthermore, the further away simulations data are located from the dashed line (see Figure 4) than stronger substrate elasticity influences shape of the droplet. It was shown theoretically16 and confirmed experimentally21 that close to the triple phase contact line the droplet/substrate profile is universal and is independent of the droplet size. It is controlled by interfacial tensions and by the stress generated in the deformed substrate. To show that this is indeed the case in our simulations as well Figure 5 shows deformation of the substrates and droplets at the triple phase contact line for rigid, soft, and liquid (polymeric melt) substrates. For these figures we have superimposed averaged images of the droplets and substrates at the triple phase contact point for droplets with sizes varying between 10.92σ and 34.62σ. As follows from these figures for all our droplets the droplet and substrate profiles close to the contact line are universal and do not depend on the droplet size. This should be expected since at the length scales l much smaller than the radius of the contact line a, l ≪ a, the curvature of the contact line can be neglected in calculating the elastic response of the substrate. However, these profiles depend on the substrate shear modulus which is manifested in variation of the values of the contact angles between different

phases as seen in Figure 5a−c. Note that for polymeric droplets on liquid (polymeric melt) substrates the values of the microscopic contact angles calculated at the triple phase contact line and those obtained from the droplet shape (see discussion above) are close to each other (see Table SI3). Knowing values of the angles and surface tension of the droplet γS we can use Neumann’s triangle conditions (see Figure 1c) to calculate surface stresses at the triple phase contact line. The results of these calculations are summarized in Figure 6a,b (see Table SI3 for data points for these plots and values of the corresponding contact angles). It follows from these figures that the difference between surface stresses and surface tensions calculated for the flat undeformed gel-like interface decreases with increasing value of the elastocapillary length γL/GS. For polymeric droplets on a very soft substrates with γL/GS > 50σ our data sets converge. In this range of system parameters the surface stress ϒi approaches surface tension γi which is expected for a liquid droplet on a liquid substrate. Analysis of our simulation data presented above demonstrates that substrate deformation influences contact properties of droplets on both macroscopic and microscopic length scales. It follows from this analysis that elastic response of the substrate could be taken into account by introducing surface stress which accounts for renormalization of the surface tension due to substrate deformation at the triple phase contact line. In the next section we present a variational approach which in a E

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Figure 7. Approximation of the deformed droplets and elastic substrate indentations by two spherical caps for snapshots of droplets shown in Figure 2. Original droplet shapes are shown by blue lines, and spherical cap approximations are shown by the dashed red lines.

self-consistent way accounts for elastic substrate deformation and provides equilibrium conditions for macroscopic angles as determined by the droplet shape deformation and its coupling with the substrate deformation.

ΔFsurf (a , δs , h) = Fsurf (a , δs , h) − AγS = −πa 2S + π (γLh2 + γSLδs 2)

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(9)

A droplet produces an indentation with depth δs and radius of curvature Rs in the elastic substrate (see Figure 8). The elastic energy of this indentation is28

MODEL OF A DROPLET ON AN ELASTIC SUBSTRATE Analysis of the droplet shape deformation shows that we can approximate resultant droplet shape with high accuracy by two spherical caps (see Figure 7). Therefore, to account for substrate elastic response to droplet shape deformations, we approximate the shape of the deformed droplet by two spherical caps with radii Rs and R1 that intersect along the circle with radius a and have angles θ1 and θ2 at the triple phase contact line as shown in Figure 8. The height of the spherical

3 ⎡ 2 δsa 1 a5 ⎤ ⎥ Uel(δs , a) = K ⎢δs 2a − + 3 Rs 5 R s2 ⎦ ⎣

(10)

where K = 2GS/(1 − v) is the substrate rigidity, GS is the substrate shear modulus, and v is the Poisson ratio. Note that for deformation of polymeric gels with Poisson ratio v = 0.5, the substrate rigidity K = 4GS. For small deformations δs/2Rs ≪ 1, we can approximate a2 ≈ 2Rsδs and rewrite the elastic energy as follows: Uel(δs , a) =

7K 2 28 δs a = GSδs 2a = CGSδs 2a 15 15

(11)

where numerical constant C = 28/15. Adding together surface free energy and elastic energy contributions, we obtain ΔF(a , δs , h) = −πa 2S + π (γLh2 + γSLδs 2) + CGSδs 2a (12)

It is convenient to express change in the system free energy in terms of angles θ1 and θ2 (see Figure 8) using the following geometric relations

Figure 8. Schematic representation of droplet and substrate deformation.

cap formed by the droplet above the substrate is equal to h, the indentation depth produced by droplet in a substrate is equal to δs. The change in the free energy of the system includes the surface free energy change due to change in the areas of liquid, substrate, and substrate/liquid interfaces and elastic energy contribution due to indentation produced by the droplet in the elastic substrate. Below we outline the main steps of derivation of the free energy expression with derivation details given in the Supporting Information. The surface free energy of a substrate with surface area A wetted by a droplet is equal to Fsurf (a , δs , h) = AγS − πa 2S + πδs 2γSL + πh2γL

h=

1 − cos θ1 θ a = tan 1 a sin θ1 2

δs =

1 − cos θ2 θ a = tan 2 a sin θ2 2

(13)

Substituting expressions for h and δs into eq 12, we obtain ⎡ ⎛θ ⎞ ΔF(a , θ1 , θ2) = −πa 2S + πa 2⎢γL tan 2⎜ 1 ⎟ ⎝2⎠ ⎣ ⎛ θ ⎞⎤ ⎛θ ⎞ 28 GSa3 tan 2⎜ 2 ⎟ + γSL tan 2⎜ 2 ⎟⎥ + ⎝ 2 ⎠⎦ 15 ⎝2⎠

(8)

where S = γS − γSL − γL is the spreading coefficient. The change of the system surface free energy upon contact of a droplet with a substrate can be written as follows:

(14)

Deformation of polymeric droplet occurs at constant volume F

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Macromolecules ⎛ θ1 ⎞ ⎛ θ2 ⎞ 4π 3 πa3 ⎡ 3⎛ θ1 ⎞ Rd = ⎢3 tan⎜ ⎟ + tan ⎜ ⎟ + 3 tan⎜ ⎟ ⎝2⎠ ⎝2⎠ ⎝2⎠ 3 6 ⎣ ⎛ θ ⎞⎤ + tan 3⎜ 2 ⎟⎥ ≡ Vd(a , θ1 , θ2) ⎝ 2 ⎠⎦ (15)

V0 ≡

In this case the optimal droplet shape is obtained by minimizing the Lagrange function L(a , θ1 , θ2 , λ) = ΔF(a , θ1 , θ2) − λ[Vd(a , θ1 , θ2) − V0] (16)

with respect to a, θ1, θ2, and λ (Lagrange multiplier). After some algebra (see Supporting Information for derivation details) we can rewrite the Neumann’s triangle condition eq 7 for soft deformable substrates as follows: ⎛ ⎞ 28 GSa⎟ sin θ2 γL sin θ1 = ⎜γSL + ⎝ ⎠ 15π

Figure 9. Dependence of the ratio γSL sin θ2/γL sin θ1 on the dimensionless parameter γSL/GSa for droplets of different sizes on substrates with shear moduli varying between 0.024 and 0.833 kBT/σ3 and different values of LJ interaction parameters for polymer− substrate pairs: εLJ = 0.25kBT (pink filled pentagons), εLJ = 0.40kBT (black filled squares), εLJ = 0.75kBT (red filled circles), and εLJ = 1.20kBT (blue filled triangles). The solid line corresponds to the following expression: (γSL sin θ2)/(γL sin θ1) = [1 + (28/15π)(GSa/ γSL)]−1. Experimental data for sessile droplets of ionic liquid on PDMS gels are shown by green stars.14

(17)

In our approximation for the elastic free energy of the substrate deformation the analogue of Neumann’s expression for the horizontal components of the surface tension has the following form: γS = γL cos θ1 + γSL cos θ2 − ⎛θ ⎞ × tan 2⎜ 2 ⎟ ⎝2⎠

14 GSa(1 + 2 cos θ2) 15π

sin θ1 ≪ 1, which occurs for elastocapillary lengths γL/GS ≪ σ. These values of γL/GS are smaller than those corresponding to a crossover to the Young regime, γL/GS ≪ a. In the opposite limit, γSL/GS ≫ a, simulation data approach the saturation limit. This corresponds to the Neumann regime, where angles measured at the triple phase contact line and those obtained from a droplet shape are almost identical (see Table SI3). The agreement between simulation data and eq 17 indicates that we can apply this equation to obtain values of the surface tension of polymer/gel interface by analyzing the shapes of liquid droplets. We tested this procedure analyzing data for sessile droplets of ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate doped with fluorophore Nile Red with surface tension γL = 48.8 mN/m on PDMS substrates (see Supporting Information for data analysis details).14 This analysis produced a value of the PDMS/ionic liquid surface tension to be γSL = 57.6 mN/m. Experimental data points are shown as green stars in Figure 9. In our description of the droplet shape deformation on the elastic surfaces we have neglected the wetting ridge which is formed at the contact line (see Figure 1c). For studied range of interaction parameters the height of the ridge, d, is comparable with the bead size σ in almost all our simulations except those with the largest value of the LJ interaction parameter, εLJ = 1.2kBT, between droplet−substrate pairs. These data are shown in Figure 10. The reduced ridge height shows nonmonotonic dependence on the value of parameter, γSL/GSa. It first increases with increasing the value of parameter γSL/GSa, and then it begins to decrease. The increase of the ratio d/a in the interval of the small values of the dimensionless parameter γSL/ GSa follows from the expression for the ridge height d ∝ γL/ GS8−10 taking into account that for our systems γS is on the order of γSL, d/a ∝ γL/GSa ∝ γSL/GSa. In the limit of large values of the parameter γSL/GSa the ratio d/a ∝(γL/GSa)−1 ∝ (γSL/GSa)−1.17 Therefore, the ratio d/a is small when angles satisfy the Young’s law or Neumann’s triangle condition for liquid substrate. The ridge height d has a maximum when

(18)

It is important to point out that a particular form of the last term in eq 18, which describes contribution due to elastic stress generated by substrate deformation, depends on a particular shape of the droplet in contact with substrate. Analysis of the different asymptotic regimes of eqs 17 and 18 shows that these equations can be reduced to the classical Neumann’s triangle conditions derived for liquid substrates when the elastocapillary length, γSL/GS, is much larger than the contact radius a, γSL/GS ≫ a (Neumann regime). In this case the terms that include elastic energy contributions can be neglected. However, in the opposite limit, γSL/GS ≪ a, the vertical component of the tension force (see eq 17) at the contact line is balanced by the substrate elastic force. Note that the value of the angle θ2 approaches zero, θ2 ∝ (γL/GSa) sin θ1, when both γL/GSa ≪ 1 and γSL/GSa ≪ 1. In this case eq 18 reduces to the Young’s law (Young regime). We want to point out that in the crossover between Neumann and Young regimes eqs 17 and 18 should not be viewed as equations describing the surface stress balance at the triple phase contact line since these equations result in different renormalizations of the corresponding surface stresses due to substrate deformation. We can use eq 17 and plot simulation data for droplets of different sizes on soft gel-like substrates. Figure 9 shows our data in these new reduced variables. As one can see, the agreement between simulation results and model prediction is excellent. Note that at small values of the parameter γSL/GSa ≪ 1 the simulation data follow a linear scaling dependence. In this interval of parameters the value of the angle θ2 ∝ (γL/GSa) sin θ1. From geometric relationship between angle θ2 and the spherical cap thickness (see eq 13) the indentation depth, δs, produced by a droplet in a substrate is on the order of δs ∝ aθ2 ∝ (γL/GS) sin θ1. Therefore, in the Young regime the indentation depth δs could be larger than the characteristic molecular scale (bead size) σ. The indentation depth δs becomes smaller than the bead size σ when δs/σ ∝ (γL/GSσ) G

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CONCLUSIONS We study effect of the substrate elasticity on the equilibrium shape of polymeric droplets. Our analysis confirms that the equilibrium shape of a droplet is a result of a fine interplay between capillary and elastic forces. We have derived a generalized expression for the droplet free energy on the elastic substrate which accounts for these effects. Derived expressions for equilibrium angles (see eqs 17 and 18) describe crossover between Neumann’s triangle conditions for liquid droplets floating on a liquid and Young’s law describing droplets on infinitely rigid substrates. The crossover between these two regimes is governed by the dimensionless parameter, γSL/GSa, which is a ratio of the elastocapillary length γSL/GS and contact radius a. In the limit when the elastocapillary length γSL/GS is larger than the contact radius a the capillary forces control droplet−substrate interactions, and we recover the Neumann’s triangle condition for equilibrium macroscopic angles. However, in the opposite limit γSL/GS ≪ a, the deformation of the substrate is negligible in the interval of parameters γL/GS ≪ a and equilibrium conditions for droplet shape deformation on elastic substrate reduce to the Young’s law corresponding to balance of horizontal components of the surface tensions. It is interesting to point out similarity between behavior of a liquid droplet on elastic substrate and adhesion-wetting crossover behavior of elastic particles on elastic surfaces.21,25,26,29 In both cases the ratio of the elastocapillary length to the contact radius determines crossover between elastic and capillary forces dominated regimes. The presented here model can also be extended to describe deformation of the polymeric brush by a liquid droplet.30 This will be a subject of future study.

Figure 10. Dependence of the reduced ridge height d/a on the dimensionless parameter γSL/GSa for droplets of different initial sizes: Rd = 10.92σ (black squares), Rd = 15.81σ (red circles), Rd = 19.78σ (blue triangles), Rd = 25.74σ (green triangles), Rd = 30.64σ (pink triangles), and Rd = 34.62σ (dark blue triangles) and for value of the LJ interaction parameter between polymer−substrate pairs εLJ = 1.20kBT. Experimental data from ref 14 are shown by green stars.

elastocapillary length γSL/GS becomes comparable with the contact radius a. However, even in this range of parameters its value is below 7% of the value of the contact radius, a. The small value of the ridge height in comparison with the contact radius provides additional confirmation that the elastic energy generated by substrate deformation under the droplet provides dominant contribution to the total substrate elastic energy. Formation of the ridge at the triple phase contact line results in rotation of Neumann’s triangle by angle θs (see inset in Figure 11). Figure 11 shows variation of this angle as a function

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ASSOCIATED CONTENT

S Supporting Information *

Derivation of the free energy of a droplet in contact with elastic surface and comparison with experiments. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.V.D.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grants DMR-1004576 and DMR-1409710.

Figure 11. Dependence of the Neumann’s triangle rotation angle θs (see inset for angle definition) on the dimensionless parameter γSL/ GSa for droplets of different initial sizes: Rd = 10.92σ (black squares), Rd = 15.81σ (red circles), Rd = 19.78σ (blue triangles), Rd = 25.74σ (green triangles), Rd = 30.64σ (pink triangles), and Rd = 34.62σ (dark blue triangles) and for value of the LJ interaction parameter between polymer−substrate pairs εLJ = 1.20kBT.

REFERENCES

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DOI: 10.1021/ma501672p Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/ma501672p Macromolecules XXXX, XXX, XXX−XXX