LETTER pubs.acs.org/JPCL
Hindered Diffusion of an Oil Drop Under Confinement and Surface Forces Hannah Lockie,†,‡ Scott McLean,† and Raymond R. Dagastine*,† †
Department of Chemical and Biomolecular Engineering and the Particulate Fluids Processing Centre, ‡School of Chemistry and the Particulate Fluids Processing Centre, The University of Melbourne, Parkville VIC 3010, Australia ABSTRACT: Theoretical modeling of coalescence and dropdrop collision stability relies on an accurate physical understanding of surface forces, hydrodynamic drainage behavior, deformation, and internal drop flow. Brownian diffusion of alkane drops emulsified in aqueous solutions above a silica plate is measured using the total internal reflection microscope (TIRM). The behavior of drops over a range of separations is consistent with an immobile oilwater interface in both clean and anionic surfactant systems, counter to the hydrodynamic boundary conditions commonly assumed for fluidfluid interfaces. SECTION: Surfaces, Interfaces, Catalysis
I
nterfacial mobility of interacting drops and bubbles diffusing in aqueous solutions produces substantial differences in hydrodynamic behavior compared with that of drops with immobile liquid interfaces.17 During dropdrop collisions, flow at the interfacial boundary provides substantial increase in the rate of thin film drainage and may allow for coalescence in an interaction that would otherwise be kinetically stable.79 Rising drops or bubbles, found in solvent extraction and air flotation applications, have substantially reduced drag coefficients when nonzero fluid velocity is allowed at the phase boundary.1,10,11 Experimental evidence of internal drop mobility has been presented through measurement of terminal velocities and drag coefficients for drops of millimeter dimensions that undergo significant shape change (to ellipsoidal or oscillating profiles) in both confined channel flow12 and free drop rise or fall.10,11,13,14 Particle tracers have been used to investigate mass-transfer induced flow in smaller drops on the order of 100 μm radius.15 Evidence of mobility without mass transfer has also been provided under certain conditions for the small clean bubbles of this size.1 However, to our knowledge, there has been no direct experimental verification that mobility occurs without mass transfer at the phase boundary for drops over the practically relevant size range of 0.5200 μm radii, where near-sphericity is maintained during drop-rise and colloidal interactions (i.e., for low Reynolds number systems). Nonetheless, scaling analyses of micrometer-sized drop coalescence commonly require a partially mobile boundary condition for empirically fitting data.9,16,17 Previous studies employed atomic force microscopy18,19 and focused on the drainage between micrometer-sized drops at collision velocities at or greater than comparable velocities from Brownian motion. The lubrication model theory used to describe these data required the condition of interface immobility to theoretically model both surfactant18,19 and surfactant-free19 alkane drop interactions. The total internal reflection microscope (TIRM) r 2011 American Chemical Society
is used here to measure directly the hindered diffusion of an emulsion drop undergoing Brownian motion near a solid wall, in a configuration relevant to flow in a microfluidic channel. The sensitivity of diffusion to flow at the interface provides insight into the mobility of the oilwater interface and the corresponding hydrodynamic boundary conditions often assumed in theoretical models. Theoretical formulations based on viscosity ratio and length scales have been commonly applied to distinguish cases of mobility, immobility, and partial mobility.5,20 The systems used here (and in previous work5,20) are of viscosity ratios, length scales, and diffusion rates that fall within the partially or fully mobile category of such classifications. Under the experimental conditions of this study, the viscosity ratio of bromododecane and water is 4.0. If, as an example, the theoretical analysis and classification of Davis et al.5 is applied to compare the length scale of the drop interaction, given as (radius/film thickness)1/2, with the viscosity ratio, then a scaling value, m, of between 0.8 and 4 is obtained for the range of drop sizes and film thicknesses investigated here. According to the assignments of Davis,5 m = O(1) corresponds to a partially mobile interface. Values of m .1 are classified as fully mobile, and only values of m , 1 are assigned the immobile boundary condition. Interfacial cleanliness at the air or oilwater interface has also been highlighted by many authors21,22 as being important in not arresting flow at the interface, and to this end we use a purified alkane in a clean surfactant-free aqueous solution. In addition, the oil drop was selected as one of many from an emulsion in contrast with previous AFM studies18,19 where the total interfacial area of the system was significantly smaller. Received: August 26, 2011 Accepted: September 13, 2011 Published: September 13, 2011 2472
dx.doi.org/10.1021/jz201171n | J. Phys. Chem. Lett. 2011, 2, 2472–2477
The Journal of Physical Chemistry Letters
LETTER
TIRM is a technique for monitoring the effects of surface forces and hydrodynamic drainage on a spherical particle undergoing Brownian motion in a liquid near a solid transparent surface, as illustrated in the inset of Figure 1b. The elevations sampled by the particle are obtained as a function of time through measuring the intensity of scattered light from a diffusing particle illuminated by an evanescent wave created by a total internal reflection at the silicaaqueous interface. This light decays exponentially with distance from the surface, allowing measurement of changes in elevation to ∼1 nm.23 The probability distribution of elevations sampled can be converted to a potential energy (PE) profile via the Boltzmann relation, as described in detail by Prieve23,24 and demonstrated in Figure 1a. Analysis of the PE of a solid particle as a function of separation allows accurate measurement of a range surface forces.2429 The experimental system and PE analysis are equivalent for a nondeforming drop (i.e., for a rigid sphere); however, to date the only two studies that have reported drop-solid measurements using the TIRM have used surfactantstabilized systems without extensive oil purification steps.30,31 A laser of 640 nm wavelength is passed unreflected through an optically matched solid prism and silica slide at an angle suitable for total internal reflection at the solidwater interface, thereby creating an evanescent wave. Scattered light is collected by a photomultiplier tube (PMT) after passing through narrow bandpass filter. A laser of 532 nm is used as optical tweezers to restrict the particles lateral movement, allowing statistically sufficient data to be obtained with 100 000 data points taken at 0.01 s intervals. A small body force is applied by optical tweezers altering the apparent weight of the particle and is often subtracted from the PE measurements (as illustrated in the inset of Figure 1a). The resulting PE profile can then be compared with theoretical models for a range of colloidal forces. The optical tweezers force is linearly related to laser power; therefore, by varying power, we are able to determine the particle radius to within 0.01 μm. Details of the TIRM experimental setup and analysis techniques available are provided elsewhere.23,32,33 High purity bromododecane (>99%, Sigma Aldrich) was purified by passing through a Fluorosil column several times. Bromododecane in Milli-Q water emulsions (0.003% (v/v)) are prepared and injected into the fluid cell of the TIRM. Different aqueous phases are then investigated through exchanging the solution in the cell while holding a single drop in position. Salts are >99.9% purity and baked for 12 h at 50 °C less than the specified melting point. The elevation of the particle above the surface is often formulated relative to the minimum energy in the PE profile (or the elevation of maximum probability, hm) where the attractive and repulsive forces balance, as shown in Figure 1a. A number of methods have been developed for the TIRM to determine the absolute separation of the particle and flat interface.24,3336 The most commonly used, the optical method, finds the height of the particle (h) based on the intensity measurement (I(h)), the intensity of scattered light when the particle is at zero separation from the solid plate (I0), and the penetration of the evanescent wave (β) through the following IðhÞ ¼ I0 eβh
ð1Þ
The stuck intensity is obtained through the well-established method23 of measuring the intensity of the particle or drop resting on the silica plate (at zero separation). The “sticking out” of drop (or particle) is achieved through solution exchange with a
Figure 1. (a) PE profile of bromododecane drop (2.97 μm radius) above a silica plate over a range of electrolyte concentrations (with apparent gravity subtracted). Inset: 2 mM NaCl PE profile as a function of relative separation with (pink squares) apparent gravity and surface forces and (brown squares) apparent gravity subtracted. Debye lengths fitted for data are (purple squares) 0.2 mM k1 = 21.4 ( 2 nm; (blue circles) 0.5 mM k1 = 14.0 ( 1 nm; (green triangles) 1 mM k1 = 9.8 ( 1 nm; (orange squares) 2 mM k1 = 6.9 ( 1 nm; (red circles) 5 mM k1 = 9.4 ( 1; and (blue triangles) 10 mM k1 = 6.0 ( 1 nm. (b) Hydrodynamic separation versus optical separation between the drop with the solid interface using an immobile alkanewater interface hydrodynamic boundary condition for NaCl (purple squares), NaClO4 (green circles), and NaNO3 (orange circles) solutions. The solid line indicates perfect agreement, and the dotted lines reflect a 5% error in determining drop size. A completely mobile interface was also used to calculate the hydrodynamic separation for the NaCl solution data (halffilled purple squares). Inset: Sketch shows the scattering of an exponentially decaying evanescent wave for a particle at a separation, h.
higher electrolyte concentration (typically 50100 mM). This screens the repulsive electric double-layer forces that are present at lower concentrations. The hydrodynamic separation is an alternate method that can be performed concurrently during any experiment. The separation is determined by matching the apparent diffusion constant with theoretical hindered diffusion constant. The method is detailed by Bevan and Prieve,34 with a short overview provided herein. The apparent diffusion constant of the particle above the plate is found by measuring the autocorrelation function, R(τ), and using the initial slope of the function, R0 (τ), as a function of delay time, τ, given by Dapparent ¼ β2 2473
R 0 ð0Þ Rð0Þ
ð2Þ
dx.doi.org/10.1021/jz201171n |J. Phys. Chem. Lett. 2011, 2, 2472–2477
The Journal of Physical Chemistry Letters
LETTER
The calculation of the theoretical diffusion constant has two elements. The first accounts for the reduced hydrodynamic mobility of the particle near the wall according to the relation D(h) = Dunhindered f(h), where f(h) is a hydrodynamic drag function discussed below and Dunhindered is the StokesEinstein diffusion constant for a particle far from a wall. The second element accounts for the separation dependent probability, p(h), of the particle due to surface forces. The probability, p(h), is related to the PE, ϕ(h), via the Boltzmann relation, p(h) = A exp(ϕ(h)/kT), where A is a constant, T is temperature, and k is the Boltzmann constant. The apparent diffusion coefficient is the weighted average of these two elements given by Z ∞ f ðhÞDunhindered ðIÞ2 pðhÞ dh ð3Þ Dtheoretical ¼ ∞ Z ∞ I 2 ðhÞpðhÞ dh ∞
For the solid particlesolid plate system, the selection of the hydrodynamic drag function f(h) is straightforward, and good agreement has been shown between hydrodynamic and optical separations.34 For both a solid sphere and a drop with an immobile interface, the full form of the drag function is given by Bart,37 where a convenient approximation for a particle of radius R above a plate at is given by1 f ðhÞ =
x þ x2 þ x3 , 1 þ 2:34x þ ð17=8Þx2 þ x3
x ¼ h=R
ð4Þ
Should the drop have a completely mobile interface, the drag function is given by Brenner.2 The hydrodynamic drag is significantly reduced, with a nonzero tangential velocity and the interface of the moving drop, and a convenient approximation for of the full drag function is given by1 f ðhÞ =
ð8=3Þx þ 7:44x2 , 1 þ 8:11x þ 7:44x2
x ¼ h=R
ð5Þ
A partially mobile interface would lead to a drag function intermediate between eqs 4 and 5. An accurate method to deduce whether the interface is immobile or fully mobile is to compare the mean hydrodynamic separation, under the assumption of immobility or mobility, with the independently measured mean optical separation. Figure 1a shows PE profiles of a bromododecane drop of 2.97 μm radius undergoing Brownian motion above a silica plate. The Debye lengths closely match the expected values based on ionic strength over a range of NaCl concentrations (0.2 to 2 mM), where the closest approach of the drop is limited at least 6 D lengths due to electrical double-layer forces. Figure 1a shows that the dropsilica interaction remains sufficiently repulsive to remain levitated up to 10 mM electrolyte concentration, which has not been possible in solidsolid systems because of the strength of the van der Waals interaction at close separations. The negative surface charge on oil drops in surfactant free aqueous solutions has been demonstrated here and elsewhere with both electrokinetic38,39 and surface force measurements;19,40 although the origin of this charge remains a subject of ongoing debate.41,42 As the drop approaches the surface at higher electrolyte concentrations (5 and 10 mM), the van der Waals attraction becomes significant, illustrated by the negative potential over the range of the secondary minimum. There is also an increased
Figure 2. (a) PE profiles of bromododecane drop (2.97 μm radius) above a silica plate in NaCl solutions with and without surfactant (SDS). Debye lengths fitted for data in NaCl are (purple squares) 0.2 mM k1 = 20.8 ( 2 nm; (red circles) 0.3 mM k1 = 17.3 ( 2 nm; (green diamonds) 0.5 mM k1 = 12.3 ( 1 nm; and (orange triangles) 1 mM k1 = 9.5 ( 1 nm. Debye lengths fitted at equivalent concentration with 0.1 mM SDS and NaCl as required to make equivalent concentration are unfilled (purple squares) 0.2 mM k1 = 19.27 ( 2 nm; (red circles) 0.3 mM k1 = 17.0 ( 1 nm; and (green diamonds) 0.5 mM k1 = 11.9 ( 1 nm. Debye length for SDS (orange triangles) 1 mM k1 = 9.3 ( 1 nm. Measurements are all taken for the same drop with solution exchange to SDS systems performed following the clean interface measurements. (b) PE profile versus relative separation of bromododecane drop (5.48 μm radius) above a silica plate in NaCl solutions. Debye lengths fitted for data in NaCl are (green triangles) 0.2 mM k1 = 21.2 ( 2 nm; (orange circles) 0.5 mM k1 = 11.3 ( 1 nm; (purple squares) 1 mM k1 = 7.4 nm (1 nm; and (red diamonds) 2 mM k1 = 4.6 ( 1 nm. Schematic (inset) shows the motion of a large and small drop. The dotted lines denote where deformation may occur during the motion.
extent of double-layer overlap, which has been shown previously to cause deviations in apparent Debye lengths.43 This suggests that a more sophisticated method (beyond the scope of the current work) is required to quantify precisely the length scale and surface charge of the electrical double force in the calculation of PE. The hydrodynamic and optical separations at the most probable separation, hm, are compared for the data presented Figure 1a and for data using other electrolytes (NaNO3 and NaClO4) in Figure 1b. For all concentrations and all electrolyte types, the two methods agree to within (2 nm when the interface is assumed to be immobile, as indicated by the correlations in Figure 1b. This is an excellent correspondence given that experimental error in the optical and hydrodynamic separation 2474
dx.doi.org/10.1021/jz201171n |J. Phys. Chem. Lett. 2011, 2, 2472–2477
The Journal of Physical Chemistry Letters
Figure 3. Hydrodynamic drop-plate separation versus optical separation under the assumption of interface immobility for a 2.97 μm drop with NaCl (purple circles) and with NaCl + 0.1 mM SDS (green diamonds) and for a 5.48 μm drop with NaCl (red squares). For comparison, hydrodynamic separation versus optical separation for comparison with the assumption of mobility for 2.97 μm drop with NaCl + 0.1 mM SDS (half-filled green squares) and 5.48 μm drop with NaCl a 5.48 μm drop (half-filled red squares). The 1:1 line indicates perfect agreement between optical and hydrodynamic separation, and dotted lines show agreement given 5% error in determining drop size. Schematic (top left corner): organic drop under confined diffusion with an immobile oilwater interface. Schematic (bottom right corner): internal flow and confined diffusion of organic drop with a mobile interface.
determination may be up to (5 nm. In addition, all values fall within an experimental error of (5% in determining drop size. It should be noted that the same drop (and therefore drop size) is used for a range of concentrations for each type of electrolyte. This produces consistency in data acquired over a range of separations. The calculated hydrodynamic separation employing a mobile alkanewater boundary condition is shown for the NaCl solution data in Figure 1b as well. This demonstrates a very large disagreement between optical and hydrodynamic separation, with an average difference in separation of 70%. Furthermore, the data indicates that an assumption of even partial mobility would introduce disagreement between analyses that is not present under the immobile assumption. The application of a slip length model to either the liquid solid or liquidliquid interface would also result in deviations of the hydrodynamic separation from the optical separation. In this instance, we have chosen to not employ this model for two reasons. First, the aqueoussilica interface has been shown to behave as a no-slip interface from previous AFM studies in simple electrolyte solutions.44 Second, the interfacial mobility model for the liquidliquid interface reflects the physical situation in a more accurate manner. In the context of a slip length as a physical descriptor of film drainage, the mobile and immobile interface would correspond to slip lengths of infinity and zero, respectively. Figure 2a shows the PE profiles of the same 2.82 μm drop above a silica plate in both NaCl solutions and NaCl solutions with an anionic surfactant, sodium dodecyl sulfate (SDS). The clean interface electrolyte measurements are taken prior to surfactant measurements to ensure no contamination at the interface. Drops in the presence of a 0.1 mM concentration of SDS exhibit a greater magnitude of double-layer repulsive force due to the increased charge at the interface (also measured by Malvern 2000 Zeta-Sizer 2000 as 60 ( 10 mV in a 0.2 mM NaCl solution and 92 ( 10 mV with 0.1 mM SDS and 0.1 mM NaNO3).
LETTER
Therefore, the most probable separation of the drop is greater for each of the equivalent concentrations investigated, as shown systematically in Figure 2a. As expected, the Debye lengths fitted from data show reasonable agreement with theoretical values at each concentration, regardless of whether electrolyte concentration is with pure NaCl or anionic surfactant. The reduction in interfacial tension from SDS adsorption at this concentration leads to a decrease in the drop’s Laplace pressure of ∼10%. However, the agreement in both the Debye length and the agreement between the hydrodynamic and the optical separation (shown in Figure 3a) both indicate that the deformation of the drop of this radius is not significant. This is consistent with the single SDS condition measurement by Haughey and Earnshaw.30 In Figure 2b, a larger drop is used (in this case equivalent to a 40% reduction in Laplace pressure), and the drop PE profile is measured in surfactant-free NaCl solutions. The PE profiles are plotted in Figure 2b as a function of relative, not absolute separation, with the zero set as the most probable height. This allows comparison of the surface forces over a range of electrolyte concentrations at which the absolute separation varies over a range of ∼200 nm (as illustrated in other Figures 1a and 2a). The Debye length is reasonably matched to the expected value at 0.2 mM NaCl concentration, where the decay length of the repulsive force is large, at separations of 150200 nm. At higher ionic strengths where the decay length of the repulsive force is shorter, the apparent Debye length was found (using the fitting method that provided good agreement for drops of higher Laplace pressure) to be consistently lower than expected. The systematic deviation in Debye lengths suggests that approach of the drop toward the surface is being retarded to a greater extent than would be expected based on surface forces. This observation is consistent across many data sets for large drops (not shown here) and does not occur for smaller drops. It is proposed that this is due to deformation of the drop, allowing PE to be absorbed through shape change when the thin film pressure is sufficiently strong and thereby reducing sampling of closer separations. The deformation of the drop may affect the PE profile, the hydrodynamic behavior of the drop (as discussed below), and the scattering relationship with drop separation. A complex approach would be required to quantify precisely changes in the scattering relationship due to small deformations. Previous calculations for the scattering of a spheroid in an evanescent wave for similar sized particles show significant differences in the near-field scattering behavior for an aspect ratio of 1.05.45 For the drops in this study, this would correspond to ∼50 to 100 nm of deformation. An approximate calculation of the drop deformation using the force model of Chan et al.46 has been performed for the larger drop (5.48 μm radius) with the calculated surface forces disjoining pressure and Laplace pressure over the range of system conditions investigated. Using this approach, the model predicts deformations of 24 nm at concentrations of 0.5 to 2 mM. Therefore, deformation effects on the scattering relationship are expected to be small compared with the effect on PE as a function of separation. Furthermore, the deviations observed in the apparent Debye lengths are of the correct order of magnitude to be attributed to slight drop deformation. Figure 3 shows the relationship between hydrodynamic and optical separation for the NaCl solutions without surfactant with a drop of
