Evolution and Disappearance of Solvent Drops on Miscible Polymer Subphases

Abstract

Traditionally, an interface is defined as a boundary between immiscible phases. However, previous work has shown that even when two fluids are completely miscible, they maintain a detectable “effective interface” for long times. Miscible interfaces have been studied in various systems of two fluids with a single boundary between them. However, this work has not extended to the three-phase system of a fluid droplet placed on top of a miscible pool. We show that these three-phase systems obey the same wetting conditions as immiscible systems, and that their drop shapes obey the Augmented Young-Laplace Equation. Over time, the miscible interface diffuses and the shape of the drop evolves.

We place 2-microliter drops of water atop miscible poly(acrylamide) solutions. The drop is completely wetted by the subphase, and then remains detectable beneath the surface for many minutes. An initial effective interfacial tension can be approximated to be on the order of 0.5 mN/m using the capillary number. Water and poly(acrylamide) are completely miscible in all concentrations, and yet, when viewed from the side, the drop maintains a capillary shape. Study of this behavior is important to the understanding of effective interfaces between miscible polymer phases, which are pervasive in nature.

Graphical abstract

Introduction

Generally, when one measures interfacial tension between fluids, these fluids are immiscible. The definition of interfacial tension involves a minimization of the free energy of the system including the interface, which requires that interface be in thermodynamic equilibrium. However, there are many cases in which two miscible fluids will maintain a distinct interface for long times despite the fact that they are not in equilibrium and will eventually become a homogeneous mixture. A familiar example of this is honey being poured into tea. The honey maintains a rounded drop shape due to its interfacial tension with the water, but then gradually dissolves into solution. This phenomenon is described in the literature as an “effective interface” with “effective interfacial tension (EIT)” (first proposed in [1], with many references since). The interfacial thickness and interfacial tension associated with this effective interface are controlled by the diffusive mixing of the two phases.

The theory behind these “effective interfaces” relies on the assumption that they are in a local equilibrium. So, rather than a full minimization of the interfacial free energy, the free energy is instead expanded in powers of concentration gradient to find the effective interfacial free energy. This non-equilibrium theory of miscible interfaces has been developed for both simple and complex miscible fluid interfaces (For references, see: [2–8]).

A number of research groups have measured the EIT between miscible fluids using various techniques. Smith and collaborators made early measurements of EIT using a Wilhelmy plate and Kojima and collaborators showed that the toroidal shapes achieved by miscible drops can be best explained with the addition of a small interfacial energy between the miscible fluids [9,10]. A number of groups studied the EIT of drops, bubbles, and plumes of glycerine in water [11–13]. May and collaborators measured the EIT using capillary wave relaxation [14,15], and Pojman and collaborators measured the EIT using spinning drop tensiometry [16]. For a review of diffuse interface models and Korteweg stresses across fully miscible interfaces, see [17].

Truzzolillo and collaborators examined the effective interfaces between colloidal suspensions or polymeric solutions and their solvents. They predict a quadratic relation between EIT and polymer concentration for long-chain polymers, an exponential behavior for compact colloidal particles, and something in between for short-chain polymers [18,19]. Polymer systems were studied by Zoltowski and collaborators using spinning drop tensiometry and were shown to obey the theoretically obtained relationship between their EIT and the concentration difference from bulk fluid to drop [4]. An informative review of experimental miscible interface work between complex phases can be found in Truzzolillo and Cipelletti’s paper from 2016 [3].

In the systems we will be studying, a drop of solvent is placed on a miscible polymer solution subphase. The drop is wetted by the subphase and sits just beneath the surface with only a thin wetting layer between the drop and the subphase-air interface. There is an interaction free energy per unit area between any two interfaces leading to a distance-dependent “disjoining pressure” between them. As an example, when a fluid drop and a more rigid surface are pushed together, repulsive disjoining pressures will act to flatten the drop (reduce the drop’s curvature near the interface) [20–23]. Detectable deformation of the drop will occur in the portion of the drop that is within the “interaction zone”, where the disjoining pressure is significant. Most observed disjoining films are only a few nanometers thick [24,25]. The thickest disjoining films in the literature, between near-critical fluids, have been measured on the order of hundreds of nanometers [26,27]; however, Chan and collaborators suggest that the range of the “interaction zone” could be hundreds of microns [20].

Most of the work done examining miscible interfaces has been in two-phase systems where a drop is suspended within the bulk fluid. In our work, we explore miscible interfaces between a dilute polymer solution and its solvent, where the solvent drop is placed at the air-solution surface. This allows us to observe the three-phase wetting condition of the drop / subphase / air contact line and compare it to the immiscible wetting condition. If the drop has been wetted by the subphase, our experimental methods allow us to examine the pressures on the drop via the drop curvature. Our system is unique in several ways. First, before dissolution, the solvent drops obey immiscible wetting conditions leading to their being fully wetted by the polymer subphase. Second, small gravitational driving forces (due to the small density difference between drop and subphase) yield a very sensitive probe of the pressures acting on the drop; therefore, we can examine how disjoining pressures affect the upper and lower drop interfaces at very large distances from the air-liquid interface. Third, we may observe the power law spreading, slow evolution, and eventual disappearance of solvent drops as their interfaces diffuse.

Materials and Methods

Materials

All water was purified to 18MΩ·cm resistivity, with < 10 ppb total organic carbon (Milli-Q Academic Unit, Millipore Corporation, Billerica, MA). The purified water had a surface tension of 71.5 ± 0.4 mN/m measured using a Wilhelmy pin apparatus (Nima Technology Limited, Coventry, England).

Poly(acrylamide) (PA) of molecular mass 5 – 6 MDa (CAS# 9003-05-08) was purchased in powder form from Polysciences (Warrington, PA) and used as received. Aqueous solutions of 1% w/w PA were prepared in the pure water by adding the PA powder in increments of 2 g every 2 – 3 days in a 3/4 full 1 L bottle under nitrogen with continuous gentle mixing on a gyratory shaker (New Brunswick Scientific, Edison, NJ, model G79, speed “4”). After adding the final 2 g of powder, water was added to bring the total volume up to 1 L, and stirring continued for 2 – 3 more days or until the solution was homogeneous, whichever took longer. The observed entanglement concentration is 0.45 wt% [28]. The surface tension of the PA solution after pouring a fresh sample was 70.8 ± 0.5 mN/m, which typically decreased by ~ 1 – 2 mN/m over 5 minutes. Dilutions to 0.8 wt%, 0.6 wt%, 0.4 wt%, and 0.2 wt% were created from the original 1% stock.

Experimental Methods

Side-view videos were taken using a model K2 DistaMax™ long-distance microscope purchased from Infinity Photo-Optical Company (Boulder, CO). The microscope was outfitted with a STD objective and lit from behind using a variable light source with a diffusing lens.

For these side-view experiments (shown in Figure 1), we used a 3 cm glass cube cell (purchased from Hellma Analytics) filled to a depth of approximately 2 cm with polymeric subphase fluid. A pipette was then used to place a 2 µL drop of water atop the subphase at the center of the cell. This was done as gently as possible to avoid adding kinetic energy into the system and perturbing the effective interface between the two materials. The cell was then covered to keep the humidity high and prevent skinning of the polymer subphase. The cell was large enough that the lowest, center region of the interface was nearly flat and the drop could not be detected to rise along the subphase meniscus due to the curvature of the meniscus. During these trials, the microscope was focused at the center of the cell. In Figure 1, the meniscus is seen as a dark band at the top of the image above the drop. Water drops were observed on multiple concentrations of PA subphase and their evolving drop shapes were recorded.

Figure 1.

Still images from side-view experiments. A 2µL droplet of water was placed atop a 1% or 0.4% PA solution and a wetting layer of PA can be seen to spread over top of the drop. The drop gradually changes shape with time as the EIT between the miscible phases decreases. On the higher concentration subphase, the water drops remain rounder and disappear more slowly, giving evidence for a higher EIT.

Top-view videos were taken using a Nikon AZ100 microscope (Nikon Instruments Inc, Melville, NY) equipped with an AZ-Plan Apo 0.5 · (NA 0.05/ WD 54 mm) objective. The microscope camera (model DS-QiMc–Nikon) had 0.6 × magnification. Images were captured with NIS-Elements Basic Research software and analyzed using NIS-Elements Advanced Research software, both purchased from Nikon Instruments Inc, Melville, NY.

For these top-view videos (shown in Figure 2), we used a 7 cm radius Petri dish contained within a chamber that kept the relative humidity above 95%. The Petri dish was filled with subphase fluid to a depth of approximately 1 cm. Beneath the Petri dish, we placed a black-lined Ronchi Ruling slide (Edmund Optics, Barrington, NJ – 150 lines per inch), which enhanced our visualization of the drop (for example, see [29]). A 2 µL drop of water was then pipetted, gently, atop the subphase surface (see Figure 2). The Ronchi Ruling slide was necessary in these top-view experiments because the drop is so thin in the vertical dimension that it is difficult to visualize relying only on the index gradient through the sample. The dark bands on the Ronchi Ruling slide shift slightly due to the curvature along all bounding interfaces and index of refraction gradients along the optical path through the subphase and drop, thus allowing us to see the edges of the drop more clearly. After videos were taken, a bare image of the Ronchi Ruling through the sample (before the drop was deposited) was subtracted from each subsequent image to make any shift of the lines at the edges of the drop more apparent. This procedure was not needed in the side-view case because the drop is much wider in the horizontal dimension and, thus, allowed for more bending of the light rays passing through it.

Figure 2.

Still images from top-view experiments. A 2µL droplet of water was placed atop a 1% PA solution and is visualized using the Ronchi ruling technique.

Water drops were observed on multiple concentrations of PA subphase and their time-evolving radii as well as their final disappearance times were recorded. Disappearance of the drop in either view is a complex problem controlled by a combination of the drop curvature, the decreasing height of the drop, the broadening of the diffuse boundary between the drop and subphase, and the decreasing index of refraction difference between the drop and the subphase. Thus, the “disappearance time” depends on the particular optics used, so care was taken to make sure the optics were consistent from one experiment to the next. In addition to consistent optics, we also used consistent image enhancement to increase visibility of the shifted Ronchi Ruling lines. A number of image detection methods were tried in order to automate the process of determining a “disappearance time”, however, they were each confirmed to be less sensitive than the human eye.

Densities and viscosities of the various poly(acrylamide) solution subphases were required for analysis. Densities were measured at 23°C using a 5000M Anton Paar Densimeter with µg/mL precision. Densities of the aqueous poly(acrylamide) solutions were as follows: 1% − 1.00035 g/mL, 0.8% − 0.99971 g/mL, 0.6% − 0.99919 g/mL, 0.4% − 0.99862 g/mL, 0.2% − 0.99806 g/mL, pure water − 0.99754 g/mL. Standard deviations were less than 50 µg/mL over 10 trials. Viscosities for the polymer solutions were measured using a Cannon Ubbelohde viscometer (size 450, Cannon Instrument Co. State College, PA).

Results and Discussion

Temporal Evolution of the System

Despite being placed gently, the kinetic energy of physically placing a water drop on a PA subphase results in an initially deformed drop shape, as can be seen in the earliest time panels of Figure 1. Significant care is taken to place the drop in the center of the dish or cell so that there will be negligible forces driving the drop up the meniscus, and the drop will be as close to axisymmetric as possible [30]. Within seconds, the drop is wetted by the subphase and sits just beneath the surface. Over the next 10–30 seconds, the drop recovers from the initial shape perturbation and the forces quasistatically balance so that the position of the drop is stable to the precision of our measurements.

As the drop evolves, the wetting film above it does not change in apparent thickness, measured from the top center of the drop to the bottom of the meniscus. The only change in the drop’s center-of-mass height is through shape change. The average apparent film thicknesses over time are 140 ± 40 µm for the drop on 1% PA and 110 ± 20 µm for the drop on 0.4% PA with no statistically significant variation. The thickness varies slightly from one experiment to the next. The drop’s unperturbed shape is elongated, rotationally symmetric, and nearly top-bottom symmetric (see Figure 1 and Figure 2 at late times).

Over the next 5–20 minutes (depending on subphase concentration), the drop interface gradually becomes more diffuse and the drop shape gradually becomes flatter and more elongated. Visually, the drop outline becomes lighter, broader, and more difficult to discern. This flattening and elongation gives evidence for a decreasing EIT and increasing dominance of gravitational forces in the system (i.e. increasing Bond number). Eventually, the drop interface diffuses to the point that it is no longer detectable by our optics and “disappears”.

These observations provide three ways in which we may examine the EIT between the solvent drop and the polymer solution subphase: 1) The solvent drops recover from the initial perturbation because their capillary forces work against viscous flow to reduce the curvature of the drop/subphase interface. 2) A film of subphase forms between the drops and the air due to the relative magnitudes of the drop/air, subphase/air, and drop/subphase interfacial tensions. 3) The post-recovery shapes of drops can be described by solutions to the Augmented Young-Laplace Equation (AYLE). We examine each of these in detail below.

Interface Shape Recovery

Using the recovery time of our drops from initial perturbation, we may make an approximation of the early-time EIT between PA and water. Recovery is slow, so inertia is not significant. The upper bound on the Reynolds number is 0.15 (using 0.1 mm/s for the velocity of the drop’s lower interface as it recovers, 1.7 mm for the length scale – the distance from the bottom of the initially perturbed drop shape to the bottom of the recovered drop shape, and the viscosity of the less viscous water phase). The recovery is governed by a visco-capillary time, T_v = µL/γ_DB, where γ_DB is the EIT, μ is the dynamic viscosity of the subphase and L is the drop size. The time it takes for the drop interface to recover from perturbation and reach a nearly symmetric shape is ~ 20 sec for the 0.4 wt% PA subphase, which has a viscosity of ~ 5 mPa·s, leading to an estimate for the EIT of order 0.5 mN/m at early times. All of our different concentration PA subphases yield the same order of magnitude early time EITs. Literature measurements of EITs between polymer suspensions and their monomer (poly(dodecyl acrylate) and dodecyl acrylate [4]) and between microgels and their solvents (poly-N-isopropylacrylamide and water [19]) are shown to be this same order of magnitude at early times. The EIT decreases significantly over the course of the experiment as the interface diffuses.

Wetting Condition

In the case of three immiscible phases in contact with one another, three spreading coefficients may be defined to determine the dominant surface tension [31,32]:

$$S_1={\gamma }_{\mathit{\text{BA}}}-({\gamma }_{\mathit{\text{DA}}}+{\gamma }_{\mathit{\text{DB}}}),$$

$$S_2={\gamma }_{\mathit{\text{DB}}}-({\gamma }_{\mathit{\text{BA}}}+{\gamma }_{\mathit{\text{DA}}}),\text{and}$$ (1)

$$S_3={\gamma }_{\mathit{\text{DA}}}-({\gamma }_{\mathit{\text{BA}}}+{\gamma }_{\mathit{\text{DB}}}),$$

where the γ’s are the interfacial tensions of the interfaces between the drop (D), bulk (B), and air (A). The signs of these spreading coefficients dictate whether the three phases will produce a contact line (if all three are negative), or whether one phase will be completely wetted by another (if one is positive). The spreading coefficient pertinent in this work is S₃. In this case, if the drop-air surface tension is greater than the sum of the other two (positive S₃), then the bulk fluid will form a thin wetting between the drop and the air, pinching the drop off from the interface.

Even though our system contains a miscible interface, it obeys the same wetting conditions as an immiscible system using the EIT for γ_DB. The values for our surface tensions are water’s γ_DA = 71.5 ± 0.4 mN/m, and PA’s γ_BA = 70.8 ± 0.5 mN/m; note that PA’s surface tension tends to drift downwards by 1–2 mN/m over the first 5 minutes. If the EIT between these two miscible fluids is less than (γ_DA − γ_BA), the spreading coefficient S₃ will be positive and a wetting film of PA will occur. Although, accounting for error, it looks like S₃ could initially be negative or positive, with the downward drift of the PA surface tension (γ_BA), it is likely that S₃ will be positive. Our experimental results verify that this analysis of the spreading coefficient correctly predicts the interfacial behavior. As shown in the experimental work, images of drops of water atop PA do pinch off from the interface and a layer of PA forms on top of them even though the PA is the denser fluid (see Figure 1.)

Additionally, reflection confocal microscopy was used to characterize the macroscopic flatness of the PA interface after a drop of water was placed. These experiments were carried out by collaborators at the Max Planck Institute for Polymer Physics in Mainz, Germany using experimental procedures described in [33,34]. For comparison to a known low-contact-angle drop, confocal images were taken of a phase separated 1:1 mixture of butanol and heavy water (D₂O). Drops of the butanol rich phase atop the butanol poor phase have a contact angle of ~ 3°. This contact line was clearly observed, showing that this method can detect extremely low contact angles. When a drop of water is placed atop a PA subphase, no contact angle is observed and the interface is macroscopically flat to less than a micron over ~ 1.5 mm of interface. This suggests that the water drop is, indeed, beneath the surface and that it does not have a strong enough buoyancy force to deform the PA interface.

Pressures Governing Capillary Drop Shape

Pressures governing drop shape

The third observed result of effective interfacial tension cited above was that our drops have capillary shapes. In order to examine those shapes further, we will use numerical techniques to calculate and fit the drop shape.

We use the Augmented Young-Laplace Equation (AYLE) for an immiscible drop (for examples, see [20,35–38]) to analyze our drop shapes:

$$\gamma \kappa =P_L+\mathrm{\Delta }\rho \mathit{\text{gz}}+P(z)$$ (2)

where κ is two times the mean curvature, P_L is the constant Laplace pressure in the absence of gravity (as used to define the Bond number below in equation 10), Δρ is the difference in density from the outer phase to the drop phase, g is the gravitational acceleration, z is the height change from a reference height, and P(z) is the pressure balancing the buoyancy of the drop and stopping its vertical motion. As shown in the Supporting Information Figure S1, in order for the solution to the AYLE to form a closed, continuously smooth drop shape, it must include an additional pressure opposing the buoyancy. In the system under examination here, this pressure either arises from hydrodynamic forces (due to slow draining of the solution film above the drop or the motion of drop shape change), or from the disjoining pressure between interfaces in the system.

As mentioned earlier, to the detectability limit of our optics, the film between the drop and air interface does not change thickness over the course of our experiments. However, the buoyant force applied to the film is extremely small due to the very small density difference between the fluids. Assuming the drop is a disk with the same radius and volume as the drop, the buoyant force results in a pressure of 0.02 Pa applied to the film. With pressures this small and kinematic viscosities between 5 and 45 mm²/s for the various polymer solutions, the film may be draining too slowly for us to detect significant motion before the drop is no longer visible. Examples of slow film drainage occurring over hundreds of seconds do appear in the literature in systems such as an air bubble in an electrolyte pressed against a quartz plate [20].

When we use a submerged syringe to deposit water drops from beneath the interface, the drops rise to the interface and immediately form a film; they never break the interface and they do not “dimple” on approach. “Dimples” occur when the fluid cannot drain quickly enough and gets trapped against the interface [20]. To contrast, in the case of deposition from the top, the film is first formed when the drop pinches off of the interface, and then it may drain. Both cases produce the same apparent film thickness after initial transients. The absence of both “dimple” deformation and detectable film drainage suggests that the hydrodynamics of film thinning does not contribute significant pressures in the system [20].

While the film thickness is not changing significantly, the shape of the drop is changing due to the temporal evolution of the EIT. Using the characteristic velocity measured for the outward motion of the edge of the drop V ~ 0.01 mm/sec, the capillary number (Ca = μV/γ where μ is the dynamic viscosity of the PA) is ~ 0.5 for our system. This Ca is higher than one might expect at these low speeds, due to the very low surface tension. Thus, capillary forces are somewhat greater than hydrodynamic forces in the system at this point. This, again, shows that viscous hydrodynamic forces are not dominant in balancing the buoyant force and halting the vertical motion of the drop.

The second possibility for the source of the additional pressure P(z) in the AYLE is disjoining pressure. Traditionally, disjoining pressures have been investigated only for much smaller film thicknesses than observed here, on the order of the nanometer thickness for molecular interfaces. The thickness of the film above our drops is on the order of 100 µm. Disjoining film thicknesses in literature are at most 600 nm for cases involving critical fluids [26,27] and 120nm for those involving polymer solutions [39,40]. However, their corresponding disjoining pressures are measured to be on the order of kilopascals, whereas our estimated pressure is 20 millipascals. Thus we are in a position to probe the asymptotic, large-distance tail of the disjoining pressure because of the incredibly low buoyancy force of the drop.

The disjoining pressure may also be significant over a larger distance in this particular system because of the broad, diffuse interface between PA and water. Little work has been done studying disjoining pressures in entangled polymer solutions with these large film thicknesses. The disjoining pressure is an interaction between the concentration gradients that form the air-PA and the PA-water interfaces, but the PA-water interface is quite a bit wider than an immiscible interface would be. So the added pressure could act over a larger distance than in an immiscible system.

We may gain some information about the form of this disjoining pressure by taking a polynomial expansion of P(z) about the vertical position z = 0, which we will set to be the point of peak curvature in our drops, and then assessing the significance of each term. This can be done for any given function P(z) so long as it is smooth and differentiable in the region of the drop. Equation 2 then becomes

$$\gamma \kappa =P_L+\mathrm{\Delta }\rho \mathit{\text{gz}}+(P_0+P_{1}z+P_2{z}^2+P_3{z}^3+P_4{z}^4+\cdots ).$$ (3)

Given this description of the disjoining pressure, the polynomial coefficients can be probed using the observed curvature of the drop. This is done by analyzing and fitting Equation 3 to the shapes of the drops seen in Figure 1. However, we cannot separate the contributions of P_L from P₀ or Δρg from P₁ without independent information on the form of P(z) or the value of the EIT. The AYLE, therefore, does not permit the determination of EIT by drop shape analysis alone.

Drop Shape Analysis

From our side-view images, we created a pixel-by-pixel outline for each drop at multiple time steps, giving a profile from which the shape may be analyzed. The reproducibility of this tracing process is shown in Figure 3. Residual distances were taken as the Euclidean distance between points with the same arc length. This residual distance was on the order of 0.01 mm between three hand-drawn outlines of the same drop. In order to examine the symmetry of these drops, we reflected the drops about their midline and calculated the residuals between the drop and its inversion. These residuals were less than or equal to 0.01 mm, slightly smaller on average than the residuals between three outlines.

Figure 3.

A1 and B1 show a typical flat and a round drop reflected about their horizontal axes (about our z=0 line). The mean residual distances between the top of the drop and the flipped lower half are 0.0013 mm for the flat drop and 0.0095 mm for the round drop. A2 and B2 show the same flat and round drops outlined three independent times by eye. The mean residual distances between these three outlines are 0.0209 mm for the flat drop and 0.0083 mm for the round drop.

We next compute a parametric representation of each of the pixel-by-pixel outlines. These data sets are discrete, but may be cast into continuous form by calculating their Fourier series representation, providing an easily differentiable interpolation of the data [41]. There is no parameter describing position around the circumference of the drop image intrinsic to the data, so one is defined and appended to the list so that each list element contains (x, z, s). This allows the data to be separated into two curves, x(s) and z(s). The appended parameter is an approximate arc length defined using the Euclidean distance between neighboring points.

Using these equations and the values for x, z, and s at each point, we are able to calculate as many Fourier coefficients as we wish and then sum the series in order to get functions for x(s) and z(s). For each data set, the number of Fourier components is adjusted to approximate the data well, while not taking so many components that ringing appears in the curvature, which will be calculated in Equation 3 below (for Fourier fits, see Supporting Information S3).

We assume that the drop images are two-dimensional cross sections of three-dimensional objects that are rotationally symmetric about their central axis (see Figure 2 where the drops are circular when viewed from above). Assuming axisymmetry, the full drop surface can be reconstructed by rotating the profile around its vertical axis:

$$\{x(s),0,z(s)\}\to \{x(s)\text{ cos }\theta ,x(s)\text{ sin }\theta ,z(s)\}$$

where θ is the angle around the circle as viewed from the top and x becomes our radial coordinate. For the axisymmetric shape described, the mean curvature, κ, can be written [42,43]:

$$\mathrm{\kappa }=\frac{x(x\prime z″-z\prime x″)+z\prime ({(x\prime )}^2+{(z\prime )}^2)}{x{({(x\prime )}^2+{(z\prime )}^2)}^{3/2}}.$$ (4)

where the primes are derivatives with respect to s, introduced in the parametric form of the profile. This form of the curvature does not assume arc length parametrization, but choosing this parametrization simplifies the equation. Arc length parametrization bounds s between zero and the total arc length around the drop (the perimeter of the profile). Because our drops will have different sizes, we have chosen our parameter to be arc length parametrized and scaled by total arc length when comparing outlines. The Fourier series representation of our data facilitates taking the derivatives needed to extract the curvature from our data.

Supporting information S3 shows the curvature data for two typical drops, one early in the drop shape evolution and one later. As expected, this curvature is nearly zero along the top and bottom of the drop, with some ringing due to the limitations of the finite number of terms in the discrete Fourier series, and then peaks at the rounded ends of the drop. We choose z = 0 to be the location of maximum curvature, which, in the case of our water drops, is also the approximate mid-plane of the drop. By choosing z = 0 at the maximum in the curvature, the net linear term in equation 3 is identically zero. Additionally, the drops, and, thus, their corresponding curvatures, are very symmetric in z (see Figure 3), which suggests that higher order odd terms in the curvature are also small.

From the Fourier representation plots of curvature versus height (Supporting Information S3B and D), we can see that the overall trend is for the curvature to be an even function of z that opens downward. There are asymmetric components in many individual plots; but, over all experiments as well as over multiple alternative tracings of the same drop, these asymmetries are not consistently toward the top or bottom of the drop, suggesting they are most likely due to error in our chosen outlines of the diffuse interfaces, possibly amplified by the interpolation and multiple derivatives performed on the Fourier representation of the data.

This symmetric model may be used to numerically generate drop shapes, which may be fit directly to the outline of each drop. Due to the drop’s symmetry, we keep only the second and fourth order terms in P(z) and our AYLE becomes:

$$\gamma \kappa =(P_L+P_0)+P_2{z}^2+P_4{z}^4$$ (5)

Numerical Solutions to the Augmented Young-Laplace Equation

We fit the drop shape data (the raw map of pixel coordinates) directly by generating solutions to the AYLE with the truncated expansion of P(z) in Equation 5, and adjusting the parameters to minimize the sum of the squared residuals between the solution and the drop outline. For this method, we express the curvature in terms of the angle of the tangent vector with respect to the surface (for example, see [20]). The derivatives of x and z are the components of the unit tangent vector to the surface and are therefore defined in terms of the tangent angle to the surface, ψ (see Figure 4). Here, the dotted derivatives are all with respect to the arc length (rather than the primed derivatives with respect to a general parameter).

$$\dot{x}=\text{cos }\psi \ddot{x}=-\dot{\psi }\text{ sin }\psi$$

$$\dot{z}=\text{sin }\psi \ddot{z}=\dot{\psi }\text{ cos }\psi$$

$${\dot{x}}^2+{\dot{z}}^2=1$$

The resulting equation for the curvature is:

$$\kappa =\dot{\psi }+\frac{\text{sin}\psi }{x}$$ (6)

We may now re-write the AYLE with this curvature:

$$\gamma (\dot{\psi }+\frac{\text{sin}\psi }{x})=\mathrm{\Delta }\rho \mathit{\text{gz}}+P_L+P(z)$$ (7)

To non-dimensionalize this, we choose the length scale R₀ = 2γ/P_L, which is the radius of a drop in the absence of external forces (g = 0 and P(z) = 0). This gives us dimensionless lengths as follows:

$$x=\frac{2\mathrm{\gamma }}{{\mathrm{P}}_{\mathrm{L}}}\stackrel{\sim }{x}z=\frac{2\mathrm{\gamma }}{{\mathrm{P}}_{\mathrm{L}}}\stackrel{\sim }{z}s=\frac{2\mathrm{\gamma }}{{\mathrm{P}}_{\mathrm{L}}}\stackrel{\sim }{s}$$

Then equation 7 becomes

$$(\dot{\stackrel{\sim }{\psi }}+\frac{\text{sin}\stackrel{\sim }{\psi }}{\stackrel{\sim }{x}})=\frac{4\mathrm{\Delta }\rho g\gamma }{P_L^2}\stackrel{\sim }{z}+2+\frac{2P(\stackrel{\sim }{z})}{P_L}$$ (8)

Without the added pressures P(z), a single parameter controls the dimensionless shape of the drop:

$$\frac{4\mathrm{\Delta }\rho g\gamma }{P_L^2}$$ (9)

This single parameter can be expressed as a more traditional Bond number

$$\mathit{\text{Bo}}=\frac{\mathrm{\Delta }\rho g{R}_0^2}{\gamma },$$ (10)

where γ/ρg = a², with a being the capillary length. This traditional Bond number does not take into account the effects of the additional pressure P(z), which would introduce a third length scale in the problem.

Figure 4.

Schematic of coordinate system used in analysis.

With our tangent-angle formulation, the differential equations governing curvature are first order (i.e. solving a boundary value problem). The drop shape data is now fit using numerically generated solutions to the differential equations:

$$\dot{\stackrel{\sim }{\psi }}=2+A{\stackrel{\sim }{z}}^2+B{\stackrel{\sim }{z}}^4-\frac{\text{sin}\stackrel{\sim }{\psi }}{\stackrel{\sim }{x}}$$

$$\dot{\stackrel{\sim }{x}}=\text{cos}\stackrel{\sim }{\psi }$$

$$\dot{\stackrel{\sim }{z}}=\text{sin}\stackrel{\sim }{\psi }$$

The three parameters (A, B, and an overall scaling factor on the solutions) are fit to the drop shape data by minimizing the sum of the residuals, or the Euclidean distance, between points with the same arc length value in the model solution and in the data (for details, see Supporting Information section Method for numerically fitting drop shapes).

When we perform these fits to the numerical AYLE solutions, we see that the experimental drops are not consistently deformed asymmetrically upward or downward compared to their fits (47% upward to 53% downward out of 38 samples), confirming our results shown in Figure 3. Any significant asymmetry would suggest that one or more of the higher odd-order terms played a significant role in determining the shape (see Supporting Information Figure S2). A significant positive or negative cubic term, for instance, would deform the drop upward or downward respectively. In addition, the residual distances between these fits and their corresponding drop outlines are nearly an order of magnitude smaller than the residuals between three alternative outlines of the same drop (see residuals in Figure 3 and Figure 5). This suggests that we do not have the experimental precision to measure any small cubic term that might be needed to describe the drop shapes. Thus, our symmetric model is well motivated. Our lack of experimental precision likely arises from the complex optics that led to the images we observe.

Figure 5.

Representative numerical drop shape fits. A) shows a round, early time water drop in 1.0% PA. B) shows a flat, late time water drop in 0.4% PA. The original drop outlines are in solid black. The numerical solutions are overlaid in dashed gray. The mean residual distance between the drops and their outlines is ~ 0.005 mm.

Analysis of drop curvature and pressures

Analysis of the drop images and the results of the numeric fitting discussed above show that the drops have apparent mirror symmetry despite the fundamental lack of mirror symmetry in the physical problem. This lack of detectable asymmetry arises first because the drop is flattened and sampling only a small range of P(z), and second because this small range is centered on a peak in the curvature. In the more common case (e.g. an air bubble in water pressing up against a wetted interface), the Bond number is much lower and the bubble covers a greater range of P(z). The bubble curvature is peaked, but the larger range of z reveals the curvature’s asymmetry (see Figure 6).

Figure 6.

Schematic diagram of pressure components acting on buoyant, wetted drops. The air-liquid interface is located at the top of each plot. The dotted lines show the buoyancy pressure, which goes linearly with depth. The dashed lines show an schematic functional form for P(z) (e.g. a decreasing exponential in z is pictured), which is largest near the interface and then decreases to zero far away. The solid black line shows the sum of these two pressures, which, but for scaling factors, is the curvature of the drop. Panel A shows the case of a highly symmetric drop spanning a highly symmetric piece of the curvature. In panel B, the z-axis has been expanded to show both the highly symmetric drop and an asymmetric drop that spans a greater height and, therefore, a more asymmetric portion of the curvatures. The gravitational pressure does not pass through the origin due to the inclusion of the constant P_L term.

Figure 6 contrasts the roles of additional pressure in the AYLE and the resulting total curvature for drops of different shapes. This figure shows the two opposing pressures on a drop near an interface: the buoyancy pressure and P(z). The interface is located at the top of the figure (note that, in order to show the physical orientation of the interface, our independent variable, z, is on the vertical axis). The buoyancy pressure is positive and linear with z. The additional pressure from the interface is a decreasing function of z, the form of which is not fully defined. Here we show it as a decaying exponential, for illustration’s sake. Both of these pressure functions are smooth functions of z, but only physically exist along the drop interface. Up to scaling factors, the total mean curvature as a function of height is given by the sum of these two pressures, as can be seen from AYLE. If the total curvature is a symmetric function, then the drop will be symmetric about a horizontal mirror plane. As we can see in the figure, the curvature is not symmetric everywhere, but it is nearly symmetric very close to the point of maximum curvature. Because our drops have very high Bond number and, therefore, have very small heights (are flattened), they exist close enough to the peak in the curvature that their total curvature function is approximately symmetric, as shown in Figure 6A.

Another case showing an asymmetric drop is shown in Figure 6B, which contains the same pressure fields extended to higher z values. For example, if a drop with the same pressure fields were to be significantly larger or have a significantly lower Bond number, it would span a greater height. This means the total curvature would be significantly more asymmetric. The portion of the drop at high negative z values would be less affected by P(z) and the curvature would be primarily determined by the buoyancy pressure. This is exactly the case for a typical sessile drop. These drops are highly asymmetric, bulging away from the surface. In this case, it would be possible to measure the EIT of the drop. The P(z) term would be negligible except very close to the surface because the height range over which P(z) acted would be smaller than the total height of the drop. The EIT could then be determined directly by neglecting the region near the interface in the shape analysis; ρg/γ would become the linear fitting parameter. However, in our case of a water drop on PA, because P(z) acts over the whole drop, its effect cannot be neglected. This means that, without prior knowledge of the form of P(z), we cannot directly measure the EIT using this method.

Drop Spreading and Dissolution

The diameter of the drops of water under the PA surface can be determined from either the top or side views. The drops’ diameters increase as power laws with time, as shown in Figure 7. Statistically significant variation in the exponent with polymer concentration cannot be detected in our data. Our data show a power law exponent of 0.25 ± 0.05.

Figure 7.

Diameter versus time for a water drop atop multiple concentrations of 5 MDa aqueous poly(acrylamide) solution subphases.

As mentioned above, the capillary number for this spreading is less than 1. The Reynolds number (Re = ρVL/μ) is also less than 1. Therefore the spreading is likely quasistatically following the slow decrease in the EIT. We note that immiscible sessile drops on a liquid pool with constant surface tensions often also exhibit power law spreading [44,45].

The drops eventually disappear in the optical measurements. This is due to some combination of the decreasing height of the drop (caused by the lowering EIT and concomitant spreading of the drop), the broadening of the diffuse boundary between the drop and subphase, and the decreasing index of refraction difference between the drop and the subphase. Thus, the disappearance is indicative of some later stage of dissolution of the drop into the subphase. Figure 8 shows that the disappearance time increases with increasing polymer concentration.

Figure 8.

Disappearance time versus concentration squared for 2 µL water drops in 5 MDa aqueous poly(acrylamide) solution. The line is a linear fit to the data, which reveals the quadratic behavior of concentration with disappearance time. Error bars represent the largest confidence margins for the three individuals reporting disappearance time.

In order to determine whether evolution and disappearance of the drops is controlled by the increasing viscosity of the polymer with concentration, we performed experiments using a lower molecular weight PA solution (~1 MDa) in a similar viscosity range as our 5 MDa PA. When disappearance time was plotted versus viscosity for both the 1 MDa and the 5 MDa PA samples, the data lay on two independent curves. If viscosity were determining disappearance, we would expect these to collapse onto a single curve. An independent experiment was also performed on a subphase of 75% glycerol/water solution that had approximately the same viscosity and wetting condition as the 1% 5 MDa solution (noting that the indices of refraction may be more different in this case than between two molecular weights of the same polymer, and thus the disappearance conditions may differ). The water drop on glycerol/water flattened and fully disappeared within 5 minutes, much faster than on 1% 5MDa PA. We believe the combination of these results yields reasonable certainty that viscosity is not the dominant factor in time evolution.

We observe no optical evidence of mixing patterns in our side views, thus disappearance is controlled by diffusive, not by convective, processes. Being diffusive, the EIT is expected to decrease with the square root of time [46]. Dissolution of the polymer into the drop is controlled by (1) water being driven in to the surrounding network by osmotic pressure, (2) reptation of polymer chains out of the network, and (3) diffusion of escaped chains away from the interface [47]. Flow of water out of the drop driven by osmotic pressure should be slow due to the already low molar concentration of the polymer network; only a small decrease in the volume of the drop, ~ 5 – 10% over two minutes, is in fact measured. The balance of this diffusion and the unknown reptation time for chains leaving the network will determine if the dissolution process is transport limited. In our case of a drop of water atop PA, the small osmotic driving force likely decreases as polymer chains reptate into the drop. In the inverse problem of a drop of PA solution placed atop water, the PA drop disappears almost instantaneously, suggesting that the osmotic driving force is maintained due to the effectively infinite reservoir of solvent outside of the drop.

For miscible interfaces between long-chain polymer solutions and their solvents, [3]

$$\gamma =\frac{\mathrm{K}}{\delta }{(\mathrm{\Delta }\phi )}^2$$ (11)

where γ is the EIT, K is a material constant for the polymer, δ is the breadth of the diffuse interface, and Δφ is the difference in concentration between the two liquid phases. Equaiton 10 arises from a free energy minimization for a system in local equilibrium. For our slow dissolution, we make this same assumption. Δφ is the difference in bulk liquid concentrations assuming a linear concentration profile between the two. This value will be constant for each experiment. γ and δ will both be time dependent (γ decreasing with time and δ increasing), but this equation suggests that their product will remain the same. We do not know the exact time-dependence of γor δ; however, it is not unreasonable that disappearance time will be dependent on both, since the visibility of the drop will depend on the breadth of the interface and the curvature at the ends of the drop. As shown in Figure 8, the disappearance time data is consistent with prediction in equation (10).

Conclusions

In this work, we see clear evidence of an EIT between a polymer solution and its solvent. Drops of solvent atop a polymer subphase recover from perturbation, follow proper wetting conditions, and maintain a rounded, capillary shapes for long times.

We are able to examine the shape of these droplets in order to probe the pressures involved. In the case of a solvent drop on a polymer subphase, the disjoining pressure is comparable in magnitude to the buoyancy pressure over the full height of the solvent drops such that these drops take on flattened, approximately symmetric forms. We may model these drops by numerically solving an expansion of the AYLE.

The particular experimental system we explore is unique in a number of ways. The fact that our additional disjoining pressure is significant across the full height of the drops is uncommon for sessile drops. Most measurements of disjoining pressures use a probe to push a bubble or drop against an interface with pressures on the order of kilopascals, resulting in film thicknesses on the order of a hundred nanometers. In the case of this polymer/solvent system, drops are pushed against the air-polymer surface by buoyancy pressures on the order of millipascals, resulting in film thicknesses on the order of a hundred microns. The small density difference between the two fluids in our system drives this unusually sensitive way of measuring disjoining pressure. This is a unique class of systems that reveal the disjoining pressure far from the interface.

In addition to examining the drop shapes at a given time, we may also examine these shapes as the interface evolves in time. With time, the interface becomes more diffuse and the EIT between the miscible fluids decreases. This changes the capillary pressures on the drop and, therefore, changes the drop curvature.

Future work in this area has extensive application. A very low buoyancy pressure allows us to probe the tail end of the disjoining pressure where it is the same order of magnitude as the buoyancy. If we were to use another method to measure the EIT, as suggested by others in the literature, we would then be able to characterize P(z). Similarly, if we had access to the full form of P(z) theoretically, we could use this to determine the EIT. In this way, we believe that this is a powerful method for future study of systems of closed drops and low interfacial tensions.