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Aqueous Foam Drainage Characterized by Terahertz Spectroscopy Justin Heuser,* James Moller, Wolfgang Spendel, and Gilbert Pacey Department of Chemistry and Biochemistry, Hughes Hall, Miami UniVersity, Oxford, Ohio 45056 ReceiVed April 20, 2008. ReVised Manuscript ReceiVed June 30, 2008 Aqueous foam drainage has been studied using terahertz (THz) spectroscopy. Water is highly absorbing of THz radiation, allowing drainage to be determined based on water content at respective foam height. These drainage profiles were validated using a model constructed from published equations and tailored to this specific study. In addition, a slow-draining foam was scanned to produce a two-dimensional foam image.
Terahertz(THz) spectroscopy has surfaced as a relatively new field in materials analysis. Advances in the generation of THz radiation have allowed more frequent use of these systems. Lying between far-infrared and microwave regions of the electromagnetic spectrum, THz waves generally lacked reliable radiation sources until the development of femtosecond (fs) lasers, photoconductive antennae, backward wave oscillators, and pulsed photomixing.1-4 Unique spectra have been observed by passing THz waves through various materials, with special attention given to the ability of water to quench transmission. Gaseous water absorptions dominate the broad THz spectrum between 0.03 and 3.00 THz, as is evident by Figure 1. Bulk water properties, however, are substantially different and result in a broad absorption throughout the THz region. This phenomenon creates potential for analyzing water-containing materials having little or no absorptive characteristics (besides the water within) such as foams. Foam creation and longevity are important in many applications, both industrially and commercially.5-7 Structural properties of the foam in each application are critical to proper performance and stability.8 Due to the fragile and dynamic nature of foams, these properties and flow dynamics are difficult to measure. Methods that have been used to investigate foam properties provide insufficient information regarding drainage, as has been discussed previously.9 It is during real-time drainage that important data regarding foam behavior are extracted. Temperature fluctuations and mechanical stresses may cause foams to degrade or lose functionality.6 Therefore it is imperative to have an effective measurement and qualifying method that characterizes the foam, yielding drainage rate data. Foam properties such as rigidity,
structure, and mechanical strength can be extracted from drainage rate profiles.8,10,11 Therefore a method that can collect real-time data of a draining foam system is required to characterize the foam properties sufficiently. The work described here details the application of THz spectroscopy for studying foam drainage by way of transmissive radiation absorption. Foam Drainage. Drainage is defined as the liquid flow between fragile film membranes (lamellae) via plateau borders under the influence of gravity and capillary forces.5,12,13 Water drains from the thin films through the plateau borders, which are essentially junctions of multiple films, and distorts the bubbles into polyhedral shapes.14,15 As the water drains from these lamellar regions into plateau borders and on into vertices, the foam gas bubbles become less stable and are increasingly susceptible to bursting. The Gibbs-Marangoni stabilization, however, plays a role in maintaining bubble life.16 Here, surfactant molecules diffuse to thinned areas, providing additional support before the bubble bursts. This stabilization makes it difficult to predict foam behaviors using models and theory. Figure 2 labels interfaces of a draining region consisting of the lamellae, a plateau border, and gas regions.17 Immediately after a foam has been created, the bubbles are somewhat spherical.16 This shape does not remain long, however, as the vertices quickly drain and pull the lamella closer resulting in polyhedral bubbles. Opposing forces dictate the velocity and degree of drainage. Gravity and capillary suction act to pull the water down to the foam/liquid interface as steric repulsion forces, van der Waals interactions, and electrostatics make up the disjoining pressure keeping the films from contacting each other.16 These events make it more difficult to classify foams effectively. The primary objective of this study was to analyze the foam drainage rates of various solutions using THz spectroscopy and compare them against a simulation. A secondary objective was to attempt THz imaging of a slow-draining foam.
* To whom correspondence should be addressed. E-mail: heuserja@ gmail.com.
Experimental Section
Introduction
(1) Mittleman, D. M. Sensing with Terahertz Radiation; Spring-Verlag: Berlin, Heidelberg, 2003. (2) Siegel, P. H. IEEE Trans. MicrowaVe Theory Tech. 2002, 50, 910–928. (3) Beard, M. C.; Turner, G. M.; Schmuttenmaer, C. A. J. Phys. Chem. B 2002, 106, 7146–7159. (4) Schmuttenmaer, C. A. Chem. ReV. 2004, 104, 1759–1779. (5) Hutzler, S.; Cox, S. J.; Wang, G. Colloids Surf. A: Physicochem. Eng. Aspects 2005, 263, 178–183. (6) Prud’homme, R. K.; Khan, S. A. Foams: Theory, Measurements, and Applications; Marcel Dekker, Inc.: New York, 1996. (7) Koehler, S. A.; Stone, H. A.; Brenner, M. P.; Eggers, J. Phys. ReV. E 1998, 58, 2097–2106. (8) Weaire, D.; Hutzler, S.; Drenckhan, W.; Saugey, A.; Cox, S. J. Prog. Colloid Polym. Sci. 2006, 133, 100–105. (9) Heuser, J. A.; Taulbee, A. R.; Spendel, W. U.; Hughes, M. R.; Pacey, G. E. Am. Lab. 2008, 40, 20–23.
Foam studies have been performed on the following aqueous systems: sodium dodecyl sulfate (SDS), cetyl trimethylammonium (10) Grassia, P.; Neethling, S. J.; Cervantes, C.; Lee, H. T. Colloids Surf. A: Physicochem. Eng. Aspects 2006, 274, 110–124. (11) Pitois, O.; Fritz, C.; Vignes-Adler, M. J. Colloid Interface Sci. 2005, 282, 458–465. (12) Koehler, S. A.; Hilgenfeldt, S.; Stone, H. A. , In Foams and Films; Weaire, D., Banhart, J., Eds.: D. Verlag MIT: Berlin, 1999. (13) Stevenson, P. Chem. Eng. Sci. 2006, 61, 4503–4510. (14) Breward, C. J. W.; Howell, P. D. J. Fluid Mech. 2002, 458, 379–406. (15) Fortes, M. A.; Coughlan, S. J. Appl. Phys. 1994, 76, 4029–4035. (16) Pugh, R. J. In Handbook of Applied Surface and Colloid Chemistry; Krister, H., Ed.; John Wiley and Sons: Chichester, 2001. (17) Verbist, G.; Weaire, D.; Kraynik, A. M. J. Phys.: Condens. Matter 1996, 8, 3715–3731.
10.1021/la8012427 CCC: $40.75 2008 American Chemical Society Published on Web 09/24/2008
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Figure 1. Gaseous water absorption peaks in the THz region.
Figure 2. 3D depiction of plateau borders and a vertex (not showing thin films).
bromide (CTAB), Triton X-114, Ultra Dawn dish soap, and Guinness beer. These samples were chosen because they are representative of anionic surfactant, cationic surfactant, nonionic surfactant, mixed surfactants, and protein surfactant solutions, respectively. The solutions were made at 0.01 M except for Guinness beer which was used directly from the can and Ultra Dawn which was at 3 g/L. Data were collected at specific times and at five different locations with five scans averaged at each position. Information about drainage could therefore be extracted either at a fixed time (instantaneous absorption data) showing absorption at the five locations or at a fixed location showing absorption versus time (kinetic data). Graphs of both instantaneous and kinetic data could therefore be plotted for all surfactants investigated. The foam was generated from a nitrogen flow rate of 20 mL/min into a frit of pore sizes ranging from 4 to 8 µm. Once the foam reached the top of the cell the gas was stopped, a stopwatch was started, and an initial scan was collected. Each five-point scan had duration of ∼6 s. The experimental setup for studying foam drainage using THz radiation is shown in Figure 3. THz radiation was emitted from a Picometrix T-Ray 2000 instrument; a basic schematic of this system is shown in Figure 4, labeling the individual pieces of the system. After a laser pulse is emitted from the Vitesse box, it is directed off of a mirror and through a small optical bench which prohibits back reflection and splits off a portion of the signal to reduce the power. It then passes through a grating dispersion compensator (GDC) to correct for pulse broadening that occurs in the fiber optic cables. These cables direct the pulse into the control box and then into the
transmitter and receiver heads where emission and detection occur, respectively. High-density polyethylene (HDPE) focusing lenses reduce the beam to approximately one millimeter at the focal point where the foam sample is inserted. An x-y translational stage moved the foam drainage apparatus, a HDPE cell, to obtain spectra at five separate points between the solution level and top of the foam. In the THz region HDPE exhibits minimal absorption and dispersion, making it suitable as a construction material. This cell system was designed and tested specifically for these experiments and consisted of two HDPE walls fastened with screws and fitted internally with a large O-ring (Figure 5). A small hole milled into the side accommodated the dispersion tube with polymer sealant applied to prevent leakage. Simulations. The underlying equations for the initial foam profile after lofting and hold-up evolution during draining were obtained from refs 6 and 18, respectively. In the beginning of the model a foam was “created” by lofting a wet foam with a chosen bubble diameter. Based on the drainage equation by Verbist et al.,18 a random network of Plateau borders was solved for the case of a wet foam undergoing steady rise at superficial velocity G and in which liquid drainage is in the opposite direction. The governing equation is
(
)
∂R ∂ 2 √R ∂R R )0 + ∂τ ∂ξ 2 ∂ξ
(1)
where R, τ, and ξ are dimensionless plateau border area, time, and location in the foam, respectively. Specifically,
ξ)
τ)
z ) zo
z
t ) to
Cσ Fg t η*
R)
√CσFg aP x2o
(2)
(3)
(4)
where C ) (3 - π/2), σ is surface tension, F is density, η*is effective viscosity, and g is the gravitational constant. For the case of interest, the solution is given by
R(ξ) )
V [tanh(√V(ξ - ξa))]2
(5)
where V is the scaled superficial velocity or (18) Verbist, G.; Weaire, D.; Kraynik, A. M. J. Phys.: Condens. Matter 1996, 8, 3715–3731.
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V)G
to xo
Heuser et al.
(6)
and where ξa is selected such that R equals a known constant at the bottom of the foam. Plateau border area and holdup profile are related by
ε(aP) )
ap ν + ap nPl
(7)
where ν is the bubble volume, nP is the number of plateau borders per bubble, and l is the plateau border length.
The model for holdup profile evolution in a standing foam due to drainage comes from the work of Narsimhan and Ruckenstein. The governing equation is
{[ [
]( )
}
(8)
where t* and z* are time and location from the foam top surface nondimensionalized according to
Figure 3. Photograph of experimental setup for investigating foam drainage with a THz spectrometer.
Figure 4. Basic schematic of Picometrix T-Ray 2000 Spectrometer.
]
(2(y*)2 + 1)β ((y*)2 + 1) ∂β ∂y* + (y*)4 2(y*)3 ∂y* ∂z* (2(y*)2 + 1) ∂β ((y*)2 + 1) ∂y* 2 - * CP β (y*)4 ∂y 2(y*)3 ∂z* ((y*)2 + 1) ∂2y* CPβ 2(y*)3 ∂z*2
∂y* ) -8.1 ∂t*
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t t* ) ) τ
t nPl µR 1145 ν Fg
(9)
The boundary conditions are
∂y* ∂z*
and
z* )
z R
(10)
where R is bubble radius, µ is viscosity, and z is distance from the foam top. The dependent variable is a mapping of holdup according to
y* )
1 -ε ε
(11)
The dimensionless group CP relates the capillary pressure inside a bubble to the variation of pressure due to gravity over a distance equal to the bubble radius
CP )
σ FgRR
npl ν
(12)
and the term β is a velocity coefficient which includes the effect of surface viscosity.
|
)z*)0
1 CP
(13)
and
y*| z*)L* ) 1.68705
(14)
Surface viscosity using 3 g/L SDS in water has been determined to have a range from 0.02 to 2.2 µPa · m · s in the literature.19 In the presented model, 2.0 µPa · m · s was selected as an appropriate value. Bubble size was difficult to predict because it is not consistent throughout drainage, bursting occurs, and gas from smaller bubbles diffuses into larger ones. Values from 10 to 2000 µm were used because the frit size was slightly below 10 µm, and the bubble diameters after 60 min were typically near 0.2 cm. The bubble diameter used for the comparison SDS model was 0.06 cm, just under halfway between the two extremes. Note that the model assumed spherical shape of bubbles, which is incorrect during drainage, but made the model much simpler to compute and was sufficient for this study. Literature values for CTAB are not as abundant, but based on lower-concentration studies, CTAB has a
Figure 5. HDPE cell designed and constructed for the foam experiments; left is a front view of the face with screws fastened and right is a top view with frit at bottom.
Figure 6. HDPE cell adjacent to representative data indicating relative position of beam.
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Figure 7. Kinetic data for Guinness beer foam; absorbance versus time excluding 0.150 fraction line.
Figure 8. Kinetic drainage comparison of five aqueous surfactant systems at foam height fraction 0.300.
slightly higher surface viscosity than SDS.20 The final bubble size for CTAB foams was marginally different from those in the SDS experiment and therefore left unchanged. Therefore, 3.0 µPa · m · s and 0.06 cm were used for the surface viscosity and bubble diameter for the CTAB model, respectively. Values for Triton X-114 could not be found, but ref 20 listed Triton X-100 results which provided a rough estimate. The final bubbles in the Triton X-114 experiment were appreciably larger than the previous two systems, and the model was adjusted accordingly. A surface viscosity value of 1.0 µPa · m · s and bubble diameter of 0.20 cm were used as parameters for the nonionic surfactant.
Results and Discussion Guinness beer represents protein-based foams and was expected to have a slow drainage rate due to the “stickiness” of proteins. The experimental data support this claim, although other foaming additives and protein concentration were not known. As the water drained from the foam, gas from smaller bubbles diffused into larger ones, creating bubbles bigger than were originally present. Also, bursting of bubbles (in effect creating larger bubbles) was (19) Stevenson, P. J. Colloid Interface Sci. 2005, 290, 603–606. (20) Poskanzer, A. M.; Goodrich, F. C. J. Phys. Chem. 1975, 79, 2122–2126.
observed as the lamella thinned to the point of rupture due to external forces and weak points in the films. In order to properly interpret the presented data, Figure 6 gives a typical height versus absorption plot placed adjacent to the foam cell. Each bar on the graph represents absorption at the corresponding foam height fractions. The foam cell next to the plot has been roughly scaled to match the foam heights if the bars were extended to the left and onto the cell (shown by arrows). This depiction is meant as a representation of where the beam approximately passes through the HDPE cell. The beam was determined to be about one millimeter in diameter and its position was found by blocking the beam with a piece of metal and monitoring the signal loss. From the initial position, subsequent positions were chosen to cover the majority of the foam height. HDPE absorbs little in the THz region and therefore was sufficient for a cell in these experiments. Absorption values were determined from the total area under the power curve between 0.03 and 2.00 THz. The time trace data were entered into MathCad for Fourier transform (FT) analysis and subsequent area calculations. Kinetic foam drainage consists of water draining from the foam as time passes. The kinetic data graphically represent absorption of THz radiation (by water) decreasing at consecutive
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Figure 9. SDS experimental drainage curve with normalized and adjusted SDS-based model after 60 min drainage.
Figure 10. CTAB experimental drainage curve with normalized and adjusted CTAB-based model after 60 min drainage.
time increments. Less water in the foam is directly related to less THz absorption. Kinetic data for Guinness beer are shown in Figure 7. Here, the data for the lowest fraction, i.e., directly above the liquid interface, have been omitted due to lack of information. Complete absorption was observed near the foamliquid interface preventing valuable absorbance data from being collected. As for the other four data curves, the decreasing absorbance values are due to water draining from the foam. The higher fraction (near the top of the cell) is expected to contain less water and reach lamellar steady-state faster because less net water volume drains into the film at higher elevation above the solution surface. This is evident in the graph by the gray trace and shows least absorption of all the curves. All the foam systems exhibit water content dependence on foam height due to the interplay between foam film strength and hydrostatic pressure. The foam reaches steady-state when the water pressure and film strength balance as observed from the decreased water content with increasing foam height. This steady-state is maintained until the bubble ruptures due to lamellar film weakening. Because the foam water volume increases closer to the solution surface, the absorption follows that trend.
The error ranges for the Guinness data are small relative to the other surfactants. This is likely due to the slower drainage rate and longer-lasting Guinness foam. It is more difficult to reproduce foams that drain quickly. Kinetic data for the remaining surfactants Dawn Ultra, SDS, CTAB, and Triton X-114 at foam height fraction 0.300, are shown graphically versus Guinness beer in Figure 8. These data were all obtained using a frit with a pore range of 4-8 µm and a nitrogen flow rate of 20 mL/min. It is clear that the Guinness foam drains more slowly than the others and the Triton X-114 drains most quickly. However, at this fraction, it is difficult to differentiate between Dawn Ultra, SDS, and CTAB. Contrary to the stability of Guinness foam, Triton X-114 drains and decomposes quickly such that foam no longer remains in the upper part of the cell after 60 min. Triton X-114 is a nonionic surfactant and the stabilization forces, such as Van der Waals and charge repulsion, are not nearly as strong. The protein-based foam has many areas of charge localization and steric hindrances, showing resistance to drainage and degradation. Despite Dawn Ultra, SDS, and CTAB being similar at this fraction, overall they
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Figure 11. Triton X-114 experimental drainage curve with normalized and adjusted Triton X-114-based model after 60 min drainage.
Figure 12. Foam image 10 × 10 mm2 of 50/50 v/v water/glycerin with 0.01 M SDS and 10-4 M 1-dodecanol.
exhibit intricate differences when the complete plots are closely inspected. Using the models that were previously discussed, the experimental data were plotted against the simulation data in Figures 9-11. After 60 min have passed, the SDS curve was similar in shape to the experimental profile, but did not overlap. This was expected because the model did not account for environmental influences, gas diffusion (affecting bubble diameter), foam cell shape, or bubbles bursting. Also, it is impossible to experimentally begin collecting data at time zero, whereas the model begins simulation immediately. Therefore, the model was shifted left to overlay with the experimental data for trend comparison, shown in Figure 5. After this adjustment the liquid hold-ups are quite close; this indicates that the predicted behavior was seen. Because the influencing factors are not included in the model, there is an expected deviation near steady-state (60 min) where the discounted effects are expected to dominate. In the experiment bubbles change shape and sometimes burst, reducing water volume in the films and plateau borders. This accounts for
the offset of the curves and validates correlation between model and experiment. The CTAB plot in Figure 10 also aligns well after normalization and x-axis adjustment, and the trends are similar. However, it appears the model predicted less water directly above the liquid layer, yet the experimental data suggests otherwise. Perhaps additional data points in the region of deviation could help to better interpret the drainage phenomenon in this case. Both curves are still quite close in foam profile character after 60 min of drainage. Triton X-114 foams are essentially nonexistent after 60 min, with little foam remaining atop the liquid layer. For this case, the foam cell rupture rate is competitive with the water drainage rate, making model correlation problematic. However, in the absence of further model development, THz can be used to differentiate and quantitatively distinguish foam properties as a function of composition. The model predicted near-zero liquid hold-up until about 0.5 relative foam height. Although the data overlay the model almost perfectly in Figure 11 after normalizing and adjusting, only three data points from the experiment are effectively present and could collectively show zero absorption when including error bars. The model did align well with the data regardless and additional data points would certainly better characterize the foam at the lower region. Further model development, including the parameters determining steady-state, could lead to comprehensive models capable of differentiating foam dynamic and static foam properties as a function of foam composition. However, this work demonstrates the suitability and advantages of using THz radiation to measure foam characteristics and extract mechanistic data. Imaging. One way to observe bubble size is to obtain a foam image, difficult to do unless the foam is stabilized for the time necessary to scan the image area. To accommodate this requirement, special solutions were made using SDS, glycerin, and 1-dodecanol. Glycerin serves to increase bulk viscosity whereas 1-dodecanol increased surface viscosity by serving as a stabilizing cosurfactant. The solution used consisted of 50:50 v/v water/glycerin with an SDS concentration of 0.01 M and 1-dodecanol concentration of 10-4 M. A 0.6 cm-wide-gap cut into an HDPE cup was filled about 2 cm from the bottom with this solution, and a Pasteur pipet with the tip cut just before the inflection was affixed inside the cup. Nitrogen flowed in at 50 mL/min to create the foam and the cup was imaged immediately after air flow was stopped using a similar setup discussed previously. Bubbles produced with these parameters
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were significantly larger than those in the aforementioned foams, as expected by the larger gas dispensing hole size. The resulting image, using a resolution of 0.3 mm, scanning window of 10 × 10 mm2, and five averages per acquisition, is shown in Figure 12. The lighter colored areas represent high transmission, whereas the darker colored areas indicate signal loss due to water absorption. Bubbles can be vaguely made out; however, because there was some overlap of bubbles and plateau borders, the true shapes and sizes are uncertain. The experimental setup did not isolate a single layer of bubbles and, therefore, had some overlap through the foam thickness, yet the image still reveals information into the foam structure at the submillimeter scale. Plateau borders can possibly be identified by the triangular-shaped darker regions between the highly transmissive bubbles. Based on the image window and assuming the bubble in the middle of the left side is a whole bubble, the bubbles are approximately 5 mm in diameter.
Conclusions Overall the application of THz spectroscopy and imaging in studying aqueous foams has been demonstrated successfully as a proof-of-concept. Real-time data were collected as water drained from foams and provide differentiation between solution properties. This method, along with further model development, can be used to investigate and characterize foams quantitatively.
From the data presented it is clearly shown that THz spectroscopy is suitable for profiling drainage in aqueous foam systems. Information can be presented showing time-based water loss via draining as well as instantaneous plots portraying liquid hold-up at selected time slices. These data are helpful in both characterizing foam properties based on surfactant used and comparing one system to another. Because foam drainage is sensitive to liquid environment, these plots provide means to compare the effect of solution additives on the behavior of a foam. Imaging a foam, while difficult, provides yet another dimension of characterization. Although only special systems will maintain structure long enough for a THz image, the data obtained provide a unique view of the foam. Contrary to higher-powered imaging systems, the THz source introduces a negligible amount of heat making it a nondestructive analysis method. This is important when collecting real-time data of any material. The image would be optimal had only a single layer of bubbles been present and future work could likely provide that opportunity. Acknowledgment. The authors thank Barry Landrum for helping to both design and fabricate the HDPE cell. This project was funded by National Science Foundation ECS-0304297 and DoD DURIP 2004. LA8012427
