Cantilevered-Capillary Force Apparatus for Measuring Multiphase

Apr 1, 2013 - Two cameras with orthogonal views are used to record videos of the ...... Cuvelier , D.; Nassoy , P. Hidden Dynamics of Vesicle Adhesion...
0 downloads 0 Views 12MB Size
Download PDF
Article pubs.acs.org/Langmuir

Cantilevered-Capillary Force Apparatus for Measuring Multiphase Fluid Interactions John M. Frostad,*,† Martha C. Collins,‡ and L. Gary Leal† †

Department of Chemical Engineering, University of California, Santa Barbara, California 93016-5080, United States University College Cork, Kane Building, College Road, Cork, Ireland

‡

ABSTRACT: A new instrument is presented for investigating interactions between individual colloidal particles, emulsion droplets, foam bubbles, and other particle−particle or particle−surface interactions. Measurement capabilities are demonstrated by measuring interfacial tension, coalescence time for emulsion droplets, adhesion between giant multilamellar vesicles, and adhesion between model food emulsion particles. The magnitude of the interaction force that can be measured or imposed, ranges from 1 nN to 1 mN for particles ranging in size from 10 μm to 1 mm in diameter.

1. INTRODUCTION Interfacial phenomena are present in countless consumer products, from everyday items like shampoo and milk, to specialty products like sporting equipment and automotive parts. The processing and production of these products requires combining ingredients of different and/or immiscible phases. The desired result in many cases is a suspension, emulsion, or foam that is stable against phase separation. During the production and shelf life of a given product, however, the drops, bubbles, and/or particles may aggregate, coalesce, break apart, sediment, react, exchange mass, or phase separate, among other processes, all of which change the properties of the final product. To make these types of consumer products, one would like to have a solid scientific understanding of all of these potential processes and how they will improve or damage the product. For products involving deformable particles or fluid−fluid interfaces, many fundamental questions are still under investigation. For example, an active area of research involves predicting how long it takes for two emulsion droplets in a fluid−fluid system to coalesce under different conditions.1−8 This is, in fact, a very complicated problem from both a theoretical and experimental point of view. Another active area of research9−16 involves trying to understand the physics that govern interactions such as adhesion between vesicles in suspension. In approaching such fundamental questions it is often useful to study a simplified system of just two droplets or vesicles. In this paper, a new instrument called a cantilevered© 2013 American Chemical Society

capillary force apparatus (CCFA) is presented that is capable of making direct measurements of the coalescence time for two drops as well as the force of adhesion between two vesicles along with several other similar measurements. The CCFA consists of two opposing capillaries that can be submerged in a liquid, in which one of the capillaries acts as a force transducer. The force transducer is the key feature of the CCFA because it provides the ability to dynamically measure and/or apply forces of interaction between two deformable particles between 10 μm and 1 mm in diameter or between a particle in the same size range, and a surface. The CCFA is particularly well-suited to working with deformable particles because they are held by suction pressure at the tips of the capillaries (shown in Figure 1) as opposed to deposited on a surface or freely suspended in a fluid. The design of the CCFA is similar in concept to two existing instruments that have been used for similar research: a micropipette technique developed by Moran et al.17 and the atomic force microscope18 (AFM). However, for experiments involving deformable particles in the indicated size range, the CCFA has some significant advantages over both instruments. It should be noted that several other techniques exist for measuring forces smaller than 1 nN in similar systems, such as the biomembrane force probe,19 optical tweezers, and magnetic Received: October 17, 2012 Revised: December 28, 2012 Published: April 1, 2013 4715

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 1. (a) Droplets of silicone oil in a polybutadiene matrix held at the tips of glass capillaries with an inner diameter of 50 μm. (b) Giant multilamellar vesicles in aqueous solution held by capillaries with an inner diameter of 20 μm.

Figure 2. (a) Schematic representation of the cantilever-capillary force apparatus (CCFA). The cantilevered capillary is flexible when a force is applied to the tip. (b) Close-up photograph of the capillaries in the CCFA. In this photograph, the rigid capillary has an outer diameter of 360 μm and the cantilevered capillary has an outer diameter of 150 μm.

tweezers.20 These other techniques are not discussed here, however, because they are generally used for making extremely sensitive biological measurements (e.g. single molecules or receptor−ligand bonds), while the CCFA is designed to investigate micrometer-sized deformable particles. The specifics of the design of the CCFA are presented in section 2 with direct comparisons to the AFM and micropipette technique. Section 3 provides the details of the equipment used and the validation of the device, and section 4 is devoted to providing several demonstrations of the measurement capabilities.

In both the AFM and the micropipette technique, a cantilever that obeys eq 1 is also used as a force transducer. In the AFM, the cantilever is made from a thin piece of metal or silicon with a reflective surface, and in the micropipette technique a glass pipette is heated and drawn down to have a small diameter and then bent into a shape similar to the cantilevered capillary in the CCFA. At this point it is necessary to make a distinction in terminology between a micropipette and a capillary. To some extent it is possible to use these terms interchangeably, but for the purposes of this paper the term micropipette will be used to refer to thin glass tubes made from standard glass pipettes by this heating and drawing method. The term capillary, on the other hand, will be used to refer to the small glass tubes with uniform cross section that are typically used as columns for gas chromatography. The distinction may seem trifling, but it is of key importance to the fabrication and practicality of the CCFA, as will be made clear shortly. Use of micropipettes requires routinely producing newones, because they are so fragile that they break very easily. The capillaries, however, are quite resistant to breaking, and in principle, a single cantilever can be cleaned and reused indefinitely. In addition, using capillaries made from gas chromatography columns is very convenient because they can be purchased in standard sizes, which is advantageous for calibration. Also, a full line of products and adaptors is available (from IDEX) for connecting the capillary tubing to other standard tubing at a relatively low cost. One feature of the AFM that makes it a powerful measurement tool is the ability to track the motion of the cantilever by reflecting a laser off of the surface to provide very high spatial and time resolution. A laser can also be used to track the motion of the cantilever in the CCFA because it is made from a

2. INSTRUMENT The conceptual design of the CCFA is fairly simple. As shown in Figure 2a, one capillary referred to as the “rigid capillary” is positioned directly opposite a second capillary that has been bent to a 90° angle near the tip. The second capillary is referred to as the “cantilevered capillary”. The bend in the cantilevered capillary causes it to be flexible when a force is applied to the tip of the capillary. For small deflections (relative to the length), the motion x(t) of the cantilevered capillary is governed by the equation for a driven harmonic oscillator: meff x ̈ + bdx ̇ + kx = F(t )

(1)

Here, meff is the effective mass, bd is the drag coefficient, k is the spring constant, and F(t) is the driving force which, in principle, is equivalent to whatever interaction forces are present and/or applied during an experiment. The motion of the cantilevered capillary can be monitored digitally by using a sensor to track the position of a laser that has been reflected off a mirror that is attached to the capillary. A close-up picture of the capillaries in the device itself is shown in Figure 2b. 4716

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

of the capillaries. In order to provide a flow through the capillaries they are connected using standard gas chromatograph connections from IDEX to a reservoir of fluid or a syringe pump. This allows capability for producing flows with a specific flow rate by programming the syringe pump or with a constant pressure by adjusting the height of the reservoir, relative to the height of the chamber. Some additional details regarding the fabrication of the CCFA are provided in the Appendix and in the Ph.D. dissertation of John Frostad.21

capillary that is strong enough to allow a mirror to be glued on near the end (see details in the Appendix). This is not practical with a micropipette because it is so fragile. In both the CCFA and the AFM it is possible to measure deflections with nanometer resolution at thousands of hertz. For the micropipette technique, however, the deflection must be monitored optically with the use of a microscope and video camera. This limits the spatial resolution to hundreds of nanometers, and the time resolution depends on the frame rate of the camera and requires individual image analysis, which can be very time/resource intensive. The major disadvantage of using the AFM to study deformable particles is that the cantilever is essentially a flat surface or sharpened tip and interacts with objects on a planar surface. Therefore, a lot of effort must be made to attach a particle to the cantilever, because it generally requires cleanroom preparation and it is difficult to vary the size of the particle or study a population of particles. The micropipette technique does not suffer from this limitation, since particles can easily be placed on the tip of the micropipet using a suction pressure. However, the technique is not commercially available and little work has been published using the micropipette technique in the past 10 years, presumably because of the great difficulty involved in replicating and operating the device.

3.2. Cantilever Calibration. As mentioned previously, the cantilevered capillary can be treated as a harmonic oscillator. This allows the cantilever to be calibrated by measuring the resonance frequency in air as a function of the length. The length of the capillary can be varied easily by clamping it at different points. The absolute length and spring constant of the capillary is found by fitting the data to the following equation which gives the resonance frequency M of a harmonic oscillator: M=

1 2π

k(L) meff (L)

(2)

Both the spring constant k and the effective mass meff are functions of the length of the cantilever as follows: k=

3Eπ (R o 4 − R i 4) 4L3

3. MATERIALS AND METHODS meff = M +

3.1. Equipment. Capillary tubes with outer diameters of 360 and 150 μm, and various inner diameters were purchased from New Objective (e.g. TT360-100-50-N-5). The deflection of the cantilever is measured by bouncing a 4.5 mW laser (ThorLabs) off of the small mirror made from a piece of polished silicon wafer glued to the cantilever, and detecting the position of the beam on a position sensitive photodetector (Thorlabs PDP90A). The detector reading is calibrated before every experiment by placing the rigid capillary into contact with the cantilever and measuring the signal at known displacements. The range of deflection that can be detected depends on the path length from the cantilever to the sensor and can be changed as needed. Under typical operating conditions a maximum range of 5−10 μm is used. The motion of the rigid capillary is controlled using a piezoelectric stage with a range of 500 μm and a resolution of 5 nm (Physik Instrumente P-625.1 with E-625.CR). The piezoelectric stage can be programmed to perform essentially any type of motion needed for an experiment. Data acquisition is performed using MATLAB via a USB data acquisition card from National Instruments (USB 6009). All experiments are automated, and the hardware is controlled using MATLAB. Two cameras with orthogonal views are used to record videos of the experiment and verify alignment

=

α L3

(3)

33ρc L 140

(4)

In eqs 3 and 4 E is the elastic modulus of the cantilever, Ro and Ri are the outer and inner radii of the capillary, M is mass at the end of the cantilever, and ρc is the density per unit length of the capillary. The factor of 33/140 comes from the moment of inertia calculation for an annular cantilever. Rather than calculate the parameter α from the Young’s modulus and dimensions of the capillary, α is measured experimentally using a length of straight capillary material to remove uncertainty due to lack of information about the elastic modulus for the specific material. It is not possible to determine the absolute length of the cantilever as a result of gluing the mirror on the end, and mechanically “deactivating” a portion at the end. Therefore, the length L is defined according to L = L0 + ΔL

(5)

Here, L0 is some unknown reference length and ΔL is the measured change in length of the cantilever. The parameters M and L0 are determined by fitting the data; typically only 6−10

Figure 3. Calibration curves for the cantilevered capillary. (a) The spring constant of the cantilever is determined by fitting the resonance frequency as a function of the length. (b) The effective mass and drag coefficient of the cantilever are determined by fitting the response to an impulse force. The data in panel b was taken in water where the drag is relatively small. 4717

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 4. Static and dynamic validation of the CCFA. (a) Interfacial tension between air and water is measured. The stretching force as a function of stretching distance is fit by numerically solving the Young−Laplace equation to give a value of 72.1 ± 0.2 mN/m. (b) The lubrication force between two hard spheres (radius = 0.515 mm) as a function of the separation velocity in two fluids of different viscosity.

points are sufficient to produce a good fit, as shown in Figure 3a. Typical values of the spring constant are between 0.01 and 10 N/m depending on the diameter and length of the cantilever used. The resulting force that can be measured ranges from about 5 nN to 5 mN, and the best resolution that can currently be achieved is ∼1 nN. A major advantage of this new cantilever is that the spring constant can easily be calibrated to within 1−2% error using this method, compared to the 10−20% error that is typical for AFM cantilever calibration. The reason for the relatively larger error in the case of the AFM is because it relies on calibration methods like the added mass method or the thermal fluctuation method that are complicated due to the small and varying dimensions of the cantilever.18 No error estimate is reported by Moran et al. for the micropipette technique. Because the capillary tubing is uniform in diameter, the cantilever can be adjusted to different lengths. This means that the spring constant of a single cantilever can be changed by more than a factor of 10 just by adjusting the free length. A typical cantilever is between 3 and 10 cm long. Once the spring constant is known, it is also necessary to measure the effective mass and drag coefficient of the cantilever. These properties will depend on the fluid in which the capillaries are submerged and so should be measured prior to each experiment. This is easily done by providing an impulse force to the cantilever, measuring the response of the cantilever, and then fitting the data with the effective mass and drag coefficient as adjustable parameters. The response of a typical cantilever to an initial impulse force in water is shown in Figure 3b. It is interesting to note that, in this case, although the Reynolds number calculated for the motion of the fluid is small, and therefore the fluid is essentially free of inertia, the cantilever itself has a significant inertial contribution to its motion. This is because the cantilever is relatively large (compared to that in an AFM), but in most experiments the inertial contribution to the motion of the cantilever is negligible compared to the force on the cantilever at an instantaneous position. In experiments where acceleration of the cantilever is not negligible, (e.g. due to very fast motion of the particles) it is completely accounted for by using eq 1 when interpreting the data. 3.3. Validation. The CCFA was validated under both static and dynamic measurement conditions. For the static case, the interfacial tension of an air−water system was measured. This is done by holding a bubble of air with constant volume between

the tips of the two capillaries using suction pressure (see inset to Figure 4a). The bubble is stretched by moving the capillaries apart and then held stationary at different stretching lengths to wait for equilibration. The force required to maintain the stretched shape is then measured at each stretching distance. In Figure 4, F is the stretching force, γ is the surface tension, rc is the inner radius of the capillary, and b − b0 is the stretched distance where b0 is the reference distance between the capillaries when the shape of the drop corresponds to a segment of a sphere. The data can be fit using the same method as outlined in Moran et al. to calculate the interfacial tension. The result for this experiment was a value of 72.1 ± 0.2 mN/m which is in good agreement with the literature value. To validate the CCFA under dynamic conditions, the maximum attractive force was measured between two glass beads being separated from contact. The measured force during separation depends on the details of the driving force F(t) in eq 1. In this case, the driving force is made up of the lubrication force FL, the DLVO force FDLVO (electrostatic and van der Waals), and the force due to hard contact FC between the glass beads: F(t ) = FL + FDLVO + FC FL =

3πμR2d ̇ ; 2d

FDLVO =

(6)

A R ZκR −bκ e + H2; 2 12d

FC = kc(dc − d) → d < dc

(7)

Here, μ is the viscosity, d is the minimum distance between the beads assuming smooth surfaces, R is the radius of the beads, Z is the electrostatic interaction constant (see Israelachvili22), κ is the Debye length, AH is the Hamaker constant, and kc is an effective spring constant that resists compressing the beads closer than the separation distance at which they are in contact due to surface roughness, dc. The form of the hard contact force, FC, given here is not derived from a theory. Instead it is simply an approximation that is convenient for solving the equation for the driven harmonic oscillator (eqs 1 and 6) and that prevents the solution from allowing values of d that are much less than the value of the surface roughness. The exact value of the surface roughness, dc, was unknown and therefore used as an adjustable parameter to fit the data. The DLVO force can be neglected in the calculations because of the relatively large initial separation distance due to surface 4718

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 5. Exaggerated picture of misalignment that occurs due to deflection of the cantilever. Figure 7. Critical value of the compressed distance at which a buckling instability occurs as a function of the drop radius. The data is taken from Bradley and Weaire (ref 25).

roughness. For a constant velocity separation of the beads from contact, the relationship between d and x is given by x(t ) = x0 + Vt + dc − d(t )

As the cantilever deflects, the angle θ (see Figure 5) of the tip of the cantilever changes. For small deflections, the angle at the tip is approximately equal to the derivative of Δx with respect to L1. Using this approximation along with eq 3, it can be shown that the maximum force that can be measured for a cantilever of dimensions L1 and L2 (see Figure 5) for a given constraint on Δy is

(8)

Using eqs 6−8, eq 1 is solved numerically for individual separation velocities over the range used in these experiments (10−70 μm/s). The minimum value of x in the solution is proportional to the maximum attractive force that is predicted to be measured during separation. A value of ∼180 nm for the surface roughness produced the fit to the data shown in Figure 4b. The fact that good agreement with the data is found for two fluids of different viscosity suggests that the value is correct. 3.4. Limitations. As with all measurement techniques there are some limitations to the capabilities of the CCFA. The most obvious is that the resolution of the force is on the order of 1 nN. This means that it would not be suitable for making measurements requiring piconewton resolution like many of the single-molecule force measurements performed in the AFM and other instruments.18 It is because of the much smaller dimensions of the cantilever in the AFM (previously cited as a liability during calibration) that make it possible to produce cantilevers with much smaller spring constants than what is possible with a cantilevered capillary. A more subtle limitation is due to the loss of alignment that can occur as the cantilever deflects. Figure 5 shows an exaggerated picture of this effect. Another force measurement technique called a surface forces apparatus22,23 avoids this issue by using a doublecantilever force transducer, but the AFM and micropipet technique suffer from the same problem. The amount of misalignment Δy that can be tolerated in a particular experiment limits the maximum force that can be reasonably measured. From solid mechanics, the displacement Δx due to a force F acting on the cantilever at a distance L1 from the fixed end, is given by

Δx =

FL13 α

Fmax =

kL1 Δy 3L 2

(10)

When fabricating the cantilever for the CCFA, care is taken to make L2 as short as possible. A typical cantilever in the CCFA can measure up to 50 μN with only a 100 nm misalignment. On the other hand, a typical cantilever used in an AFM to study droplet interactions24 can measure only 30 nN (a factor of 1000 less) with a 100 nm misalignment. Another limitation can occur when forces are measured between two deformable particles, such as drops. This is depicted in Figure 6. In a typical experiment with the CCFA, one expects the geometry to be axisymmetric. As two drops are compressed, the axisymmetric configuration is stable to perturbations until a critical compressive force is reached (Figure 6b). Above this critical force, the axisymmetric configuration becomes unstable and the drops will adopt a nonaxisymmetric configuration (Figure 6c). If an axisymmetric configuration is important to the interpretation of the measurements then this adds an additional limit to the maximum force that can be applied with the CCFA on deformable particles. Bradley and Weaire25 performed a numerical analysis of this instability using a “surface evolver” technique to find the onset of this instability for a small range of drop sizes. Their results are replotted here in Figure 7 in terms

(9)

Figure 6. Two drops held by capillaries interacting under a compressive force: (a) before contact; (b) under symmetric compression; (c) after the critical force is reached and an asymmetric configuration is more stable. 4719

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

of the compressed distance (b0 − b)/D (see Figure 6) and the radius of the drops R/D nondimensionalized by the capillary diameter. To date, there is no explicit formula to predict the maximum force that can be applied for this geometry. Analytical treatment of the problem is rather unwieldy, but Elfring and Lauga26 preformed an analysis for a similar geometry that could provide a reasonable starting point.

4. MEASUREMENT CAPABILITIES The CCFA is able to make many different types of measurements that are relevant for characterizing multiphase fluids and for testing theoretical predictions. As was demonstrated in the Validation section, the interfacial tension between two fluids can be measured, and additional details are given in section 4.1. Two other types of measurements are demonstrated qualitatively in sections 4.2 and 4.3 to be suggestive of the range of experiments that can be performed with the CCFA. Finally in section 4.4, a more quantitative demonstration is made of the measurement capabilities. To the knowledge of the authors, the latter two measurements demonstrated here have not been previously performed in any other device. For each of the measurements in the latter three sections, a complete study has been performed and will be published separately in papers that are currently under preparation. 4.1. Interfacial Tension. To measure interfacial tension, the suspending fluid must be transparent so that the drop fluid can be observed through the microscope. The suspending fluid also needs to wet the capillaries better than the drop fluid so that the drop does not spread onto the capillaries. Surface treatment of the capillaries can be done in some cases to accomplish this, though in many cases it is unnecessary since the drop fluid and suspending fluid can easily be switched. Moran et al.17 solved the Young−Laplace equation numerically to fit the stretching data, but a simpler method for computing the interfacial tension from this measurement is also available. Kusumaatmaja and Lipowsky27 derived a linear approximation of the Young−Laplace equation for stretching or compressing a capillary bridge (same geometry as in Figure 4a) near zero force. The result is an effective spring constant for the drop. The use of this method is demonstrated using a liquid−liquid system of a single droplet of 1000 cS poly(dimethylsiloxane) in 650 cP polybutadiene (1530−2070 Mn) for which the interfacial tension is 4.58 ± 0.01 mN/m. The effective spring constant is calculated from the slope of the linear portion of the force curve for a range of drop radii R divided by the mean capillary diameter D = (D1 + D2)/2. The data are compared to the theoretical prediction of Kusumaatmaja and Lipowsky for the symmetrical case when D1 = D2, repeated here for reference (eq 45 in ref 27): kdrop γ

=

Figure 8. Effective spring constant of the drop divided by the measured interfacial tension as a function of the drop radius divided by the capillary diameter.

tension from the slope of the force versus stretching distance that is no less accurate than the exact solution. 4.2. Droplet−Droplet Coalescence. The first demonstration is related to the interaction of two emulsion droplets. A particular parameter of interest for the stability of emulsions is the time scale over which two droplets in contact will coalesce. This time scale is often referred to as the “drainage time” when the rate-limiting step is the thinning of the fluid film separating the drops by hydrodynamic drainage of fluid out of the film. In the CCFA, two droplets can be submerged in another liquid and then brought into contact by moving the rigid capillary closer to the cantilevered capillary. In the present demonstration, two drops with a radius of 71.0 ± 0.1 μm are brought together with a constant velocity (100 μm/s) and then held in contact with an approximately constant force (77 ± 1 nN) until they coalesce. Data for this experiment with 1000 cS poly(dimethylsiloxane) (PDMS) as the drop fluid and 650 cP polybutadiene (1530− 2070 Mn PBD) as the suspending fluid are presented in Figure 9.

−π ln(csc θ + cot θ) −

cos θ 1 + cos2 θ

(11) Figure 9. Representative data for a single coalescence experiment between two drops. The dashed blue line shows the position (read as micrometers on the y-axis; i.e., the maximum position is ∼95 μm) of the rigid capillary as a function of time. The measured force is shown by the solid black line, and the sharp decrease at around 37 s indicates coalescence.

Here, kdrop is the effective spring constant of the drop and θ is the equilibrium contact angle of the drops defined as follows, assuming that R > D/2:

θ=

⎛D⎞ π + arccos⎜ ⎟ ⎝ 2R ⎠ 2

(12)

The comparison is plotted in Figure 8 and shows good agreement without any fitting parameters. The approximation given by Kusumaatmaja and Lipowsky therefore provides an explicit expression that can be used to calculate the interfacial

In Figure 9, the position of the rigid capillary as a function of time is shown by the dashed blue line. The force measured by the cantilever is shown by the solid black curve. Initially, the force is zero because there is no motion in the fluid and 4720

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 10. Adhesion of giant multilamellar vesicles. (a) Sequence of images showing the adhesion and separation of the vesicles. (b) Sample force curve of an adhesion measurement performed at 30 μm/s with a 10 s equilibration time. The dashed blue line shows the position (read as 10 μm units on the y-axis; i.e., the maximum position is ∼120 μm) of the rigid capillary as a function of time. The measured force is shown by the solid black line, and the minimum in the curve corresponds to the force of adhesion.

Figure 11. Sequence of photographs showing arrested coalescence, compression, and separation of model fat globules.

is important for many consumer products, among other applications. One aspect of this interaction that is of interest is the strength of adhesion under varying conditions. Here, data are presented from a measurement of the force of adhesion between two giant multilamellar vesicles as the result of a depletion attraction between the vesicles due to the presence of a polymer in solution. Figure 10a shows some images taken of two charged vesicles, made from di(tallow ethyl ester)dimethyl ammonium chloride (diC18:1 DEEDMAC) surfactant, in the presence of 2 wt % of 15 000 MW poly(allylamine) in aqueous solution. Because the vesicles are multilamellar, the interior of the vesicles in these images contain additional surfactant bilayer that can be seen in the images. In the experiment the vesicles are pressed together, held in contact for a short time, and then pulled apart. Figure 10b shows details of the measurement. The position of the rigid capillary as a function of time is indicated in Figure 10b by the dashed blue line, and the measured force is indicated by the solid black line. When the vesicles are brought into contact there is a compressive (positive) force, and then when they are separated an attractive (negative) force is measured that decreases until the force applied overcomes the attractive interaction and the vesicles separate. In this case there is no measurable drag force on the cantilevered capillary due to motion of the rigid capillary because the suspending fluid has a much lower viscosity (close to that of water) than the suspending fluid in the previous demonstration. The global minimum in the force curve in Figure 10b corresponds to the force of adhesion for this experiment. The magnitude of the force of adhesion can be measured as a function of the

the drops are not close together. Then as the rigid capillary is moved forward, the force becomes increasingly repulsive due to a drag force on the cantilever from the motion of the rigid capillary. It is the relatively high viscosity of the suspending fluid that results in a significant drag force during the motion of the rigid capillary. Once the rigid capillary stops moving, the force equilibrates to the force resulting from global compression of the droplets and becomes constant to within experimental error. The drainage time is measured either from the point in time when the motion of the rigid capillary begins, or from when it stops, and until coalescence occurs. The choice of starting time depends on the particular interpretation of the data that is intended. Coalescence is signaled by a sharp decrease in the force caused by the rapid minimization of surface area after the two drops are joined. The measured force at this point is negative because capillary forces act to pull the tips of the capillaries together. The drainage time can be measured as a function of the drop radii, the contact force, the viscosity, interfacial tension, Hamaker constant, and approach velocity. In this case the drainage time was 33.36 ± 0.01 s (measured from when the capillary stops moving), and values of the drainage time are found to be reproducible to within 5% over 50 trials. It is also possible to impose a time-dependent contact force, interaction velocity, or drop radius. For example, in another study21 the drainage time is measured when the drops come into contact by increasing in size with a constant volumetric growth rate. 4.3. Vesicle−Vesicle Adhesion. The second demonstration involves the interaction of vesicles in suspension, which 4721

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 12. (a) Force as a function of the change in position of the rigid capillary for two sequential compression cycles on the same two droplets. (b) Fit of the data during the compression with the theory for compression of spheres in contact.

Figure 13. Sequence of photographs showing arrested coalescence and compression to the point where yielding occurs in the droplets (image 3), after which the suction pressure on the droplets was not strong enough to separate them.

forward in 0.5 μm steps with a 0.5 s equilibration between each step so that the compression and separation of the drops is quasi-static. The measured force as a function of the change in position of the rigid capillary is shown in Figure 12a, and the reference position is such that the value is zero when the drops come into contact. Here, the data labeled “approach 1” and “separation 1” coincide with the images in Figure 11. The data labeled “approach 2” and “separation 2” are from a second compression cycle (shown in Figure 13) of the same two droplets in which the yield stress has been exceeded. A theoretical expression for the force as a function of compression of two identical elastic spheres was originally developed by Herz28 and can also be found in the work of Walton:29

separation velocity, vesicle radii, polymer concentration, vesicle composition, and membrane tension. 4.4. Fat Globule Characteristics. Apart from the more common deformable particles, like emulsion droplets and vesicles, there are other interesting particles that can be interrogated using the CCFA. In this final demonstration, hexadecane oil droplets in water, containing 70 wt % of smaller, solid wax particles (see Figure 11), are used as a model for a fat globule such as those contained within milk and cheese. The solid wax particles within the hexadecane cause the drop fluid to behave as a yield-stress fluid. Oscillatory shear measurements in a rheometer, of the bulk fluid used to make the droplets, show that the fluid has an elastic modulus of approximately 100 kPa and yields under strains greater than approximately 0.5%. When these particles are brought into contact they undergo arrested coalescence, meaning that the liquid surrounding the solid wax particles in each droplet coalesces, but the internal structure of the droplets prevents the drops from completely merging to form a single drop as shown in Figure 11. At high concentrations of solid particles, the internal structure also enables the droplets to be separated after partial coalescence. This can result in the (nearly) reversible formation of larger multiparticle structures that are important for food applications. Here, the CCFA is used to study the interaction of two of these droplets as well as make an in situ measurement of the rheology of the droplets. The droplets are held at the tips of the capillaries, and it can be seen in Figure 11 that some of the oil from the droplet has been sucked into the capillaries in order to hold it on the capillary. In these experiments the spring constant of the cantilevered capillary was 0.773 ± 0.015 N/m. The droplets are brought together by moving the rigid capillary

F=

4ER1/2 ⎛⎜ ΔL ⎞⎟ 3(1 − ν 2) ⎝ 2 ⎠

3/2

(13)

Here, F is the force, E is the elastic modulus, and ΔL is the distance that the rigid capillary moves to compress the spheres. For the purposes of the present analysis it has been assumed that the Poisson’s ratio ν of the material is 0.5 (perfectly elastic). Equation 13 has been used to fit the data during the compression of the droplets with the elastic modulus as an adjustable parameter as shown in Figure 12b. Using the droplet radius of 47 μm, the elastic modulus is ∼7.7 kPa. This is smaller by about a factor of 10 relative to the oscillatory shear measurements of the bulk material. Pawar et al.30 studied the same set of materials and found that the elastic modulus is very sensitive to the concentration of solid wax. If the difference in elastic modulus was simply a result of the droplets having a lower 4722

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

Figure 14. (a) Photograph showing how a cantilevered capillary can be made using a propane torch and bending the heated tip. (b) Photograph showing a small mirror held by vacuum suction on a plastic pipet tip being glued to a cantilevered capillary with a diameter of 360 μm. Inset: photograph of a cantilevered capillary tied in a knot to demonstrate the mechanical strength.

Figure 15. Photograph of clamp, clamp holder, and flexible port for the cantilevered capillary in the fluid chamber.

to find a way to relate the properties of the bulk material to the properties of the droplets. Additional in situ measurements using the CCFA could be used to provide the data for determining this relationship. Apart from the rheology of the droplets, the force of adhesion between them can be measured with the CCFA. In Figure 12a, the minimum in the measured force during the first separation of the droplets corresponds to the maximum force that must be applied to separate the drops from contact. The magnitude of this force of adhesion provides a quantitative measure of the stability of the multiparticle aggregates. It is clear from the images in Figure 11 that a capillary bridge forms between the droplets that resists their separation. The measured force of adhesion for these droplets can be compared to a theoretical prediction for the force of adhesion between two rigid spheres connected by a capillary bridge. Rabinovich et al. present a simple expression based on an equilibrium theory for two spheres at zero separation distance that does not depend on the volume of fluid in the capillary bridge:31

concentration of solid particles than the starting bulk material, then according to the measurements of Pawar et al. the concentration would be close to 50 wt % in the droplets. During the second compression cycle on the droplets, yielding of the internal structure was observed by a decrease in the measured force upon further compression (see Figure 12) and visually by the development of a flattened region between the droplets as shown in Figure 13. In this particular case, the increased contact area between the droplets resulted in a stronger adhesion and the suction pressure holding the droplets on the capillaries was not quite strong enough to separate them. The strain at which yielding occurs in this experiment can be estimated, by dividing the compression distance by the initial droplet diameter, to be ∼8%. This is larger than the strain at which yielding occurs in the oscillatory shear measurements by about a factor of 10. The difference in the elastic modulus and the yield-strain, between the bulk material and the droplets, raises an interesting point. If the rheological properties of the droplets in a system such as this are needed, then it is necessary 4723

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir Fcapillary ≈ 4πγR cos θ

Article

second part of the holder is a metal cylinder that screws onto the clamp on one end and has a thin cylindrical tube coming out of the other end for the capillary to pass through. This second part of the holder can be freely rotated in a round channel of the third, outer part of the holder and locked into position with a thumb screw. The outer part of the holder is a metal arm connected to 3-D positioning stages. The result is that the capillary can be rotated and positioned in 3-D without unclamping the capillary. A second design challenge was that the capillaries need to be submergible in a small volume of fluid in a chamber that can be sealed, but still allow movement of the capillaries as described below. A silicone nipple (see Figure 15) was used to provide a flexible port to the chamber for the cantilevered capillary to allow between 1/4 and 1/2 inch of travel in three orthogonal directions for alignment of the capillary tips. The thin tube coming out of the middle part of the holder is designed to pass through a small hole in the nipple and be slightly larger than the opening so that the silicone seals around it. For the rigid capillary, a glass tube with an inner diameter a couple thousandths of an inch wider than the capillaries was used as the port. The rigid capillary fits loose enough in the glass tube to allow free motion (up to 1/2 in. of coarse motion and 500 μm of fine motion) into and out of the chamber so that the distance between the tips of the capillaries can be adjusted. At the same time, the tight clearance of the capillary in the glass tube provides a large resistance to leakage of fluid out of the chamber. A third design challenge was that, because the intention is to measure very small forces of interaction, the device had to be isolated as well as possible from external vibrations and other sources of mechanical noise. Several commercially available vibration-isolating tables were tested, but to achieve the performance required for the very low noise needed for these experiments it was necessary to construct a custom vibrationisolating table. Fortunately, a simple and inexpensive design, in which the breadboard from an optical table was suspended by 2 m of bungee cord from a steel frame, provided sufficient isolation from mechanical noise. One last design challenge worth mentioning was the need for complete automation of the experiments and data collection. This was necessary in part to avoid additional noise due to touching the apparatus during an experiment. Automating the experiments also allows the experiments to be performed in exactly the same way (within the precision of the equipment) from one trial to the next for consistency and repeatability. All hardware communication and control was performed using a custom MATLAB program, and each type of experiment or calibration step of the instrument is done via a simple script.

(14)

Here, γ is the interfacial tension between the oil and aqueous phase, R is the radius of the spheres, and θ is the contact angle of the liquid forming the bridge, on the solid spheres. The interfacial tension was measured for the suspending fluid in hexadecane to have a value of 21.4 mN/m at 21.5 °C. For this value of γ, a radius of 47 μm, and contact angle of zero, eq 14 predicts a force of ∼12.6 μN. This estimate is the same order of magnitude as the measured value of 3 μN. An interesting feature of the droplet adhesion is that the droplets do not separate after reaching the maximum adhesion force. Instead, the capillary bridge continues to hold the droplets together with a lesser force until some critical separation distance is reached and the capillary bridge ruptures. The same behavior is observed for rigid spheres.31 Because the measurements demonstrated in this section are the first of their kind, it is clear that there are many interesting questions one could pursue. Possibilities for additional measurements include varying the particle size and composition, the suspending fluid composition, the interaction velocity, compressive load, and equilibration time. In a separate paper that is currently in preparation, some of the dynamic aspects of the interaction between two similar droplets are investigated.

5. CONCLUSION A new instrument, called a cantilevered-capillary force apparatus (CCFA), for characterizing multiphase fluid interactions has been presented with demonstrations of the range of measurement capabilities. This technique provides the ability to easily and accurately measure phenomena that would not be practical, or in some cases even possible, using existing instruments. Measurements of systems and phenomena, beyond those demonstrated here, that have not yet been tested, but that are possible, include dynamic interfacial tension, surface rheology, dynamic interactions between elastic spheres (to complement the well-known JKR theory22), interactions between capsules, and many variations of the measurements already tested, including particles interacting with a surface.

■

APPENDIX To put the 90° angle near the tip of the cantilevered capillary a straight piece of capillary is heated in a propane torch and bent by hand. The polymer coating on the outside of the capillaries that adds protection against breaking can be preserved by shielding the capillary with foil as shown in Figure 14a. This is much easier than the process of making a cantilever from a micropipette which requires the use of a commercial instrument to heat and draw the pipettes first, and then a special “microforge” to shape the micropipette into a capillary. To give an indication of the mechanical robustness of the capillaries used in the CCFA, the inset to Figure 14b shows a length of capillary that had been tied in a knot. In developing the CCFA a few design challenges had to be overcome. First, the cantilevered capillary needs to be clamped rigidly at the entry point to the chamber so that the spring constant does not change during an experiment but still allow for 3-D alignment and rotation of the capillary. To achieve this, a three-part holder is used (refer to Figure 15). The innermost part is simply a threaded fitting and nut that tightens a ferrule around a plastic sleeve holding the capillary (purchased from IDEX). This firmly grips the capillary at a fixed length when it is tightened down and is labeled as the “clamp” in Figure 15. The

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the Institute for Multiscale-Materials Studies and the National Science Foundation for funding this project, Alexandra Paul for assistance with the interfacial tension measurements, Sebastian Bernasek for assistance with the vesicle measurements, Andy Weinberg and Nicole Holstrom 4724

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725

Langmuir

Article

(20) Neuman, K. C.; Nagy, A. Single-Molecule Force Spectroscopy: Optical Tweezers, Magnetic Tweezers and Atomic Force Microscopy. Nat. Methods 2008, 5, 491−505. (21) Frostad, J. M. Fundamental Investigations of Phase Separation in Multiphase Fluids. Ph.D. Dissertation, University of California, Santa Barbara, December 2012. (22) Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: Waltham, MA, 2010. (23) Israelachvili, J. N.; Min, Y.; Akbulut, M.; Alig, A.; Carver, G.; Greene, W.; Kristiansen, K.; Meter, E.; Pesika, N.; Rosenberg, K.; Zeng, H. Recent Advances in the Surface Forces Apparatus (SFA) Technique. Rep. Prog. Phys. 2010, 73 (3), 036601. (24) Lockie, H. J.; Manica, R.; Stevens, G. W.; Grieser, F.; Chan, D. Y. C.; Dagastine, R. R. Precision AFM Measurements of Dynamic Interactions between Deformable Drops in Aqueous Surfactant and Surfactant-Free Solutions. Langmuir 2011, 27 (6), 2676−2685. (25) Bradley, G.; Weaire, D. Instabilities of Two Liquid Drops in Contact. Comput. Sci. Eng. 2001, 3 (5), 16−21. (26) Elfring, G. J.; Lauga, E. Buckling Instability of Squeezed Droplets. Phys. Fluids 2012, 24 (7), 072102−072102−15. (27) Kusumaatmaja, H.; Lipowsky, R. Equilibrium Morphologies and Effective Spring Constants of Capillary Bridges. Langmuir 2010, 26, 18734−18741. (28) Landau, L. D.; Pitaevskii, L. P.; Lifshitz, E. M.; Kosevich, A. M. Theory of Elasticity, 3rd ed.; Butterworth-Heinemann: Oxford, U.K., 1986; Vol. 7. (29) Walton, K. The Effective Elastic Moduli of a Random Packing of Spheres. J. Mech. Phys. Solids 1987, 35 (2), 213−226. (30) Pawar, A. B.; Caggioni, M.; Hartel, R. W.; Spicer, P. T. Arrested Coalescence of Viscoelastic Droplets with Internal Microstructure. Faraday Discuss. 2012, 158, 341−350. (31) Rabinovich, Y. I.; Esayanur, M. S.; Moudgil, B. M. Capillary Forces Between Two Spheres with a Fixed Volume Liquid Bridge: Theory and Experiment. Langmuir 2005, 21 (24), 10992−10997.

from the Engineering Machine Shop at UCSB for assistance in fabricating the instrument, Professor Paul Hansma and Professor Michael Gordon for helpful advice, and Jon Caldwell, DMD for proofreading the manuscript. Martha Collins was partially supported by the CISEI internship program through the UCSB Materials Research Laboratory and International Center for Materials Research under Grants NSF DMR 1121053 and NSF DMR 0843934.

■

REFERENCES

(1) Lai, A.; Bremond, N.; Stone, H. A. Separation-Driven Coalescence of Droplets: An Analytical Criterion for the Approach to Contact. J. Fluid Mech. 2009, 632, 97−107. (2) Baret, J.; Kleinschmidt, F.; El Harrak, A.; Griffiths, A. D. Kinetic Aspects of Emulsion Stabilization by Surfactants: A Microfluidic Analysis. Langmuir 2009, 25 (11), 6088−6093. (3) Chen, D.; Cardinaels, R.; Moldenaers, P. Effect of Confinement on Droplet Coalescence in Shear Flow. Langmuir 2009, 25 (22), 12885−12893. (4) Xianhua, F.; Mussone, P.; Gao, S.; Wang, S.; Wu, S.; Masliyah, J. H.; Xu, Z. Mechanistic Study on Demulsification of Water-in-Diluted Bitumen Emulsions by Ethylcellulose. Langmuir 2010, 26 (5), 3050− 3057. (5) Toro-Mendoza, J.; Petsev, D. N. Brownian Dynamics of Emulsion Film Formation and Droplet Coalescence. Phys. Rev. E 2010, 81 (5), 051404. (6) Chan, D. Y. C.; Klaseboer, E.; Manica, R. Film Drainage and Coalescence Between Deformable Drops and Bubbles. Soft Matter 2011, 7 (6), 2235. (7) Janssen, P. J. A.; Anderson, P. D. Modeling Film Drainage and Coalescence of Drops in a Viscous Fluid. Macromol. Mater. Eng. 2011, 296 (3−4), 238−248. (8) Gong, Y.; Leal, L. G. Role of Symmetric Grafting Copolymer on Suppression of Drop Coalescence. J. Rheol. 2012, 56 (2), 397−433. (9) Cuvelier, D.; Nassoy, P. Hidden Dynamics of Vesicle Adhesion Induced by Specific Stickers. Phys. Rev. Lett. 2004, 93 (22), 228101. (10) Shen, Y.; Hoffmann, H.; Hao, J. Phase Transition of Precipitation Cream with Densely Packed Multilamellar Vesicles by the Replacement of Solvent. Langmuir 2009, 25 (18), 10540−10547. (11) Colbert, M.-J.; Raegen, A. N.; Fradin, C.; Dalnoki-Veress, K. Adhesion and Membrane Tension of Single Vesicles and Living Cells Using a Micropipette-Based Technique. Eur. Phys. J. E: Soft Matter 2009, 30 (2), 117−121. (12) Chatkaew, S.; Georgelin, M.; Jaeger, M.; Leonetti, M. Dynamics of Vesicle Unbinding Under Axisymmetric Flow. Phys. Rev. Lett. 2009, 103 (24), 248103. (13) Sengupta, K.; Limozin, L. Adhesion of Soft Membranes Controlled by Tension and Interfacial Polymers. Phys. Rev. Lett. 2010, 104 (8), 088101. (14) Bulut, S.; Oskolkova, M. Z.; Schweins, R.; Wennerstrom, H.; Olsson, U. Fusion of Nonionic Vesicles. Langmuir 2010, 26 (8), 5421−5427. (15) Ramachandran, A.; Anderson, T. H.; Leal, L. G.; Israelachvili, J. N. Adhesive Interactions Between Vesicles in the Strong Adhesion Limit. Langmuir 2011, 27 (1), 59−73. (16) Anderson, T. H.; Donaldson, S. H.; Zeng, H. B.; Israelachvili, J. N. Direct Measurement of Double-Layer, van der Waals, and Polymer Depletion Attraction Forces Between Supported Cationic Bilayers. Langmuir 2010, 26 (18), 14458−14465. (17) Moran, K.; Yeung, A.; Masliyah, J. Measuring Interfacial Tensions of Micrometer-sized Droplets: A Novel Micromechanical Technique. Langmuir 1999, 15 (24), 8497−8504. (18) Butt, H. J.; Cappella, B.; Kappl, M. Force Measurements with the Atomic Force Microscope: Technique, Interpretation and Applications. Surf. Sci. Rep. 2005, 59 (1−6), 1−152. (19) Evans, E.; Ritchie, K.; Merkel, R. Sensitive Force Technique to Probe Molecular Adhesion and Structural Linkages at Biological Interfaces. Biophys. J. 1995, 68 (6), 2580−2587. 4725

dx.doi.org/10.1021/la304115k | Langmuir 2013, 29, 4715−4725