Article pubs.acs.org/Macromolecules
Determination of Concentration-Dependent Diffusion Coefficients in Polymer−Solvent Systems: Analysis of Concentration Profiles Measured by Raman Spectroscopy during Single Drying Experiments Excluding Boundary Conditions and Phase Equilibrium David Siebel,* Philip Scharfer, and Wilhelm Schabel Thin Film Technology, Institute of Process Engineering, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany S Supporting Information *
ABSTRACT: Diffusion coefficients in polymer solutions are strongly dependent on composition and temperature. In many cases sufficient data are not available due to time-consuming measurements and a high number of material systems of interest. In this work a new method is proposed that allows determination of concentration-dependent diffusion coefficients directly from spectroscopic data of single film drying experiments. The quantitative data of concentration profiles in the sample are used to calculate diffusion coefficients by Fick’s law. The results are independent of boundary conditions in the gas phase. Two polymer solutions of poly(vinyl acetate) (PVAc) and the solvents methanol and toluene are investigated. The results are compared to diffusion data from the literature. Experimental proof of the independence from boundary conditions is provided. Diffusion coefficients down to values of approximately 10−14 m2 s−1 are accessible. The diffusion data are in good agreement with literature data and suitable for modeling purposes.
1. INTRODUCTION Understanding mass transport in polymer−solvent solutions is important to many different branches of science and technology and therefore subject to ongoing research. Diffusion dynamics greatly influence drying procedures and residual solvent contents of polymeric materials. The diffusion coefficient is a function of shape and size of the penetrant molecule, nature of the polymer, and concentration of the solvent and temperature.1,2 An overview of various theories describing fundamental properties of polymer solvent systems concerning diffusion is provided by some reviews on the topic.3,4 For a given solvent the diffusion coefficient can change by several orders of magnitude depending on solvent concentration.5 In polymer solutions the diffusion coefficient is known to decrease with decreasing solvent content. This behavior is caused by a reduction of free volume available for diffusion steps.6 Predictive methods, such as the free-volume theory,6 and models based thereon7 have so far shown insufficient results in practical applications. For example, an estimation of freevolume parameters from viscosity data yields satisfying results in some systems (toluene−PVAC and toluene−polystyrene) but is not universally applicable (e.g., in the systems ethylbenzene−polystyrene and chloroform−PVAc).8 To overcome this problem, various works propose fitting of some of the free-volume parameters.8−12 This method leads to good results for process design applications but requires measurement data, which accordingly is still a necessity. A vast number © 2015 American Chemical Society
of different methods to determine diffusion coefficients in polymer solutions have been proposed and applied to different material systems. Gravimetric measurements,2,13 nuclear magnetic resonance,14 and fluorescence spectroscopy15,16 are among the most important of these techniques. Each exhibits specific advantages and disadvantages concerning measuring times, accuracy, and applicable concentration range. In contrast to the high practical interest in diffusion data, its availability is extremely low due to time-consuming measurement routines and a very high number of material systems of interest. Even though the number of solutes used in practical applications is limited, new products and processes constantly require usage of new, uncharacterized polymers. This applies especially to concentration-dependent data at low solvent concentration which is of crucial importance for reliable process design. A possible solution to the shortage of measurement data is the direct calculation of diffusion coefficients in polymer solvent systems from spectroscopic data. In mixtures of liquids this method has been used successfully.17,18 In such systems the diffusion coefficients are high, allowing sample dimension of several centimeters. In contrast, in polymeric systems the expected change in diffusion coefficients is much larger, and much lower diffusion coefficients have to be measured. Thick Received: September 30, 2015 Revised: November 10, 2015 Published: November 17, 2015 8608
DOI: 10.1021/acs.macromol.5b02144 Macromolecules 2015, 48, 8608−8614
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ment interval. In the beginning of an experiment large intervals are chosen (10 μm) because the drying process is very dynamic. At later stages of drying the time delay is negligible compared the overall measurement time, and the interval can be reduced to 1 μm. The implications of the time delay will be discussed later (see section Independence of Boundary Conditions). The overall experimental time is dependent on the material system, temperature, and film thickness and varies from hours to several days. By means of a calibration, the local composition is derived quantitatively in terms of solvent loading (defined as g (solvent) g (polymer)−1). The calibration procedure is described by Scharfer.26 Pure component spectra of all components are given in Figure 1S. For analysis only the region of Raman shifts between 2750 and 3150 cm−1 has been used. The calibration curves are given in Figure 2S. For both systems a linear relation between the ratio of intensities of solvent spectrum to polymer spectrum and solvent loading was found as predicted by theory.27 The slope of the curve is the calibration constant for the respective system. Analysis of mixture spectra at different times and positions, which are shown exemplarily at different solvent loadings for both systems in Figure 3S, yields concentration profiles as exemplary depicted in Figure 1. In a given time interval the solvent loading usually decreases
samples with sizes of several centimeters lead to long measurement times. Therefore, methods and experimental design cannot be transferred directly. The approach of calculating diffusion coefficients directly from spectroscopic data of film drying experiments is untested. . A suitable experimental technique for the investigation of thin polymeric films is confocal Raman spectroscopy, which has previously been employed successfully by several authors.10,11,19−21 This work proposes an approach of direct calculation of diffusion coefficients with respect to the specific challenges in polymer− solvent systems. Reduction of measurement times and availability of a wide concentration range are achieved by drying thin samples in the range of 20−200 μm. The method is tested in two different material systems and compared to existing literature data.
2. EXPERIMENTAL SECTION Materials. For the experiments poly(vinyl acetate) (Carl Roth GmbH + Co. KG, Karlsruhe, Germany, catalogue number 9154.1) with a molecular weight of 55 000−70 000 g mol−1 was used. Prior to sample preparation the polymer was dried under vacuum at 25 °C for at least 48 h. Toluene (Merck KGaA, Darmstadt, Germany, catalogue number 1.08326.1001) and methanol (Carl Roth GmbH + Co. KG, Karlsruhe, Germany, catalogue number 4627.2) were used as solvents. The material systems were selected on the basis of good data availability on the diffusional behavior and analyzability of the Raman signals.12,22,23 For the analysis of measurement data densities and refractive indices of the materials are necessary. An overview over this data is given in Table 1S. Polymer solutions were prepared by dissolving polymer pellets in the respective solvent at the desired concentration at room temperature. The samples were stirred for 48 h. Inverse Micro-Raman Spectroscopy. All spectroscopic data in this work have been obtained during drying experiments of thin polymer films using inverse micro-Raman spectroscopy (IMRS). The setup allows drying of polymer solutions under defined boundary conditions in a drying channel with adjustable airflows and simultaneously measuring Raman signals in the drying sample. The experimental setup has been presented including schematic illustrations and discussed at length in previous works.21,24 Thin glass slides (size: 0.12 by 0.12 m2; thickness: 150 μm) were used as substrates and placed on a temperature-controlled metal plate. The objective lens is located in the center of the sample below the substrate in order to avoid edge effects. Optic accessibility is provided by a notch in the heated metal plate. Before the coating step, the glass substrates were cleaned with acetone, isopropanol, and water. The temperatures of the substrate, drying air, and drying channel were constant and uniform. Polymer solutions consisting of PVAc and one of the solvents with an initial solvent loading of 1.5 g (solvent) g (polymer)−1 (this corresponds to a solvent mass fraction of 0.6) were cast by an automated device onto the substrate by knife coating and subsequently dried. The film thickness has been found to vary about 1 μm for different experiments with the same polymer solution. For the experiments an air velocity of 0.2 m s−1 and a temperature of 20 °C were used. In the system methanol−PVAc the air velocity was additionally increased up to 1 m s−1. This corresponds to gas phase mass transfer coefficients of 3.7 × 10−3−8.4 × 10−3 m s−1 as calculated according to Ameel.25 The dry film thickness of the samples has been adapted to the measurement range of the experimental setup and was in the range of 28−74 μm. During the drying process, Raman spectra at different positions in the sample were measured. The measurements start in the glass substrate below the film and proceed step by step through the film up to its solvent−air interface. This procedure will be referred to as “depth scan” and yields a concentration profile through the sample. This routine was repeated during the drying process until no further changes in composition were detected. Shifting of the focus and data processing takes 1 s per measured spectrum. Measuring a whole profile accordingly takes about 5−30 s, depending on the spatial measure-
Figure 1. Exemplary results of a measurement by IMRS in the system toluene−PVAc at 20 °C and an air velocity of 0.2 m s−1. from the bottom to the top of the film where evaporation takes place. The shrinkage of the film due to evaporation is clearly visible by a decrease of the maximum film height over time. After a fast decrease of solvent loading (constant rate period) in the early stages of drying, the mean solvent loading remains almost constant after longer times (falling rate period). Figure 1 contains all data necessary for analysis of the diffusion coefficient by the proposed method. From each data point a single value of the diffusion coefficient can be calculated at the respective solvent loading. Since the sample changes its solvent loading from the initial solvent loading X0 to almost dry polymer, a wide range is covered in a single experiment.
3. RESULTS AND DISCUSSION Derivation of Governing Equations and Data Processing. Diffusion in binary polymer solutions above glass transition temperature can be described by Fick’s law.28 jiV = −DiV
∂ρiV (1)
∂z
where is the flux of the solvent i, is Fick’s diffusion coefficient, ρVi is the concentration of the solvent, and z is the spatial coordinate. The diffusion coefficient at any position and the corresponding local solvent content can be calculated jVi
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DVi
DOI: 10.1021/acs.macromol.5b02144 Macromolecules 2015, 48, 8608−8614
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Macromolecules directly from this equation if the local gradient and mass flux are known. In systems of constant volume Fick’s second law of diffusion can be derived from this equation which has been used to calculate diffusivities in liquid mixtures.18 In polymeric systems the usage of very thin polymer films of several 10 μm is convenient to reduce experimentation time. Since keeping the film volume constant and simultaneously covering a wide range of concentration are experimentally very challenging, drying experiments are used. However, in this case shrinkage of the sample has to be taken into account. Therefore, it is convenient to express Fick’s law in an alternative frame of reference that is fixed to the polymer mass as proposed by Crank referred to as “polymer coordinates”.29
jiP = −DiP
1 ∂Xi VP̂ ∂ζ
superscript refers to the number of the depth scan, indicating that a central difference was used. From the boundary condition at the bottom of the film it is known that the flux is given by
jiP |z = 0 = 0
(8)
jPi (z)
The shape of the function is unknown. Therefore, a model-based simulation, described by Schabel, has been performed to calculate local mass fluxes using independently determined diffusion coefficients and a full description of phase equilibrium and mass transport.30,31 The results for the system toluene−PVAc are given in Figure 2.
(2)
Here jPi is the flux of solvent i in polymer coordinates, V̂ P is the specific volume of the polymer, Xi is the solvent loading in solvent mass per polymer mass, and ζ is the local coordinate in polymer coordinates. This description of Fick’s law also contains the diffusion coefficient DPi in the alternative frame of reference, which is not suitable for comparison. Therefore, the diffusion coefficient can be replaced by DiP = DiV φP 2
(3)
where φP is the local volume fraction of polymer. Since the local coordinate in the polymer mass based frame of reference cannot be measured directly, it is rewritten in terms of Cartesian coordinates: ∂ζ = φP ∂z
(4)
Figure 2. Calculated local mass flux during the drying of a toluene− PVAc sample at 20 °C. The diffusion coefficients are taken from VDI Heat Atlas.31 The position in the film is normalized by the timedependent film height.
This yields
DiV = −jiP
VP̂ φP ∂z φP2 ∂Xi
(5)
Figure 2 shows that the mass flux increases almost linearly from the bottom of the film to the surface at drying times higher than 6 s. This result is consistent for all calculations and both material systems under investigation. Various exemplary simulations proved that the deviation from linear behavior increases with stronger concentration dependence of the diffusion coefficient. The simulated results imply that the assumption of a linearly increasing mass flux is acceptable, even for strongly concentration dependent system (as in the case of the system toluene−PVAc). The deviation from a linear behavior is highest at the bottom of the film and does not exceed 30%. In most cases it is lower than 15%. Since the diffusion coefficients found in the literature often vary by more than 1 order of magnitude for the same material system,32 this inaccuracy is acceptable. At very short times a linearization is not suitable due to the fact that evaporation has not yet affected the bottom of the film. An approximation of the time needed until a linear approximation can be made is given by the short time asymptotic solution (STAS) as shown by Middleman.33
Using eq 5, the diffusion coefficient can be calculated directly from experimental results as depicted in Figure 1. According to eq 5, the gradient and the local mass flux have to be known in order to calculate the diffusion coefficient. The gradient can in principle be taken directly from the measurement data. However, prior to analysis, data smoothing proved to be beneficial to the results. In this work a parabolic fit of the profiles has been made. The following equation was used: X(z) = az 2 + b
(6)
A parabola with its vertex at the bottom of the film is in accordance with the boundary condition at the bottom of the film (no concentration gradient). The implications of this smoothing procedure will be discussed later. The local mass flux cannot be measured directly. From the change in mean solvent loading between two depth scans the flux through the film surface can be calculated by X̅in + 1 − X̅in − 1 hend (7) t ̅ n+1 − t ̅ n−1 P In eq 7, ji |z=zmax is the mass flux through the film/air interface, ρP is the polymer density, and hend is the film thickness of the completely dry film. X̅ i is the mean solvent loading of a depth scan at a time t ̅ (since the measurement points are not taken at exactly the same time this is also an averaged value). The jiP |z = z max = ρP
tSTAS =
s0 2 16DiV
(9)
In this equation, tSTAS specifies the upper time limit for the validity of the short time asymptotic solution (meaning that linearization is not possible) and s0 is the film thickness (which 8610
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circles). For solvent mass fractions above 0.15 no data for comparison is available. In case of toluene the calculated data (gray triangles) fit very well to the literature data (black diamonds) down to solvent mass fractions of 0.05 as shown in Figure 4. The variance of the
is assumed to be constant in the derivation of eq 9). Assuming high solvent mass fractions in the beginning of the drying process, the diffusion coefficient can be estimated to be higher than 10−10. For example, the diffusion coefficients of toluene in PVAc34 and benzene in polystyrene35 are reported to be 8.4 × 10−10 m2 s−1 and 6 × 10−10 m2 s−1 at 25 °C and a solvent mass fraction of 0.6. Smaller solvents are expected to diffusive even faster. The calculated values for tSTAS are in good agreement with the simulated results. An initial film thickness of 200 μm results in an upper estimate of tSTAS = 25 s. This leads to the conclusion that at typical initial film thicknesses and solvent mass fractions the first two depth scans cannot be used for the aforementioned procedure. Results and Comparison to Literature Data. From the results of IMRS measurements diffusion coefficients of toluene and methanol in PVAc at 20 °C have been calculated. For comparison, diffusion data from gravimetric sorption experiments measured in a magnetic sorption balance have been taken from Mamaliga (2004) for both material systems.22 In the system methanol−PVAc additionally data calculated from sorption and desorption experiments reported by Kishimoto (1964) are used.36 The results for the material system methanol−PVAc in comparison to literature data are given in Figure 3. The results
Figure 4. Diffusion coefficients calculated directly from Raman profiles in the system toluene−PVAc at 20 °C as a function of solvent mass fraction in comparison to gravimetric sorption data by Mamaliga (2004).22 The data were calculated under the assumption of a linearly increasing mass flux toward the film surface and parabolic smoothing of the solvent loading profiles.
data calculated from Raman profiles appears to be higher than that of the data reported by Mamaliga et al. (2004).22 On the other hand, a much wider concentration range (0.05−0.6) is accessible by calculation from Raman profiles. At very low solvent mass fractions no diffusion coefficients could be calculated, roughly at the concentration at which no gravimetric sorption data are reported. The very low diffusion coefficients in the system toluene−PVAc lead to mass fluxes below the resolution limit of the setup, which renders the calculation of diffusion coefficients impossible. In this concentration range only very sensible methods of measurement are able to determine diffusion coefficients in acceptable measurement times. The experimentation time in this system is longer than in the system methanol−PVAc as a result of the lower partial pressure and diffusion coefficients. After 2.5 h no further change in solvent loading could be detected. In both cases it can be observed that the diffusion coefficients calculated from the same scan through the film are almost equal, even though local solvent loading differs significantly. Theory and literature suggest decreasing diffusion coefficients with decreasing solvent loadings (i.e., toward the free surface of the film). The reason for this behavior can be found in eq 5 and the employed method of data smoothing. The diffusion coefficient is antiproportional to the gradient of the solvent loading. The employed parabolic smoothing is very good at fitting the gradient at the bottom of the film where the gradient is low. However, near the surface of the film real gradients are much steeper than those of a least-squares fitted parabola. This behavior is shown in Figure 5. It depicts measurement data (symbols), a least-squares fit of a parabola to the measurement
Figure 3. Diffusion coefficients calculated directly from Raman profiles in the system methanol−PVAc at 20 °C as a function of solvent mass fraction in comparison to gravimetric sorption data by Mamaliga et al. (2004) and Kishimoto (1964).22,36 The data were calculated under assumption of a linearly increasing mass flux toward the film surface and parabolic smoothing of the solvent loading profiles.
(gray triangles) show the typical behavior of diffusion coefficients in polymer−solvent systems. The diffusion coefficient varies little at high solvent mass fractions (>0.3) and shows a significant drop in the limit of low solvent mass fractions (0.3) the calculated values vary by a factor of about 4, showing no clear trend with increasing air velocity. At low solvent loadings the different experiments are not distinguishable. This indicates that the measurement results are indeed independent of boundary conditions. Moreover, the time required to measure a profile appears not to have a significant influence on the results. If this was the case, faster drying (i.e., higher air velocity) should lead to significantly steeper profiles as the film keeps drying during
⎛ A + Bi Xi ⎞ = exp⎜ − i ⎟ (m /s) ⎝ 1 + CiXi ⎠ DiV 2
(10)
This expression has three independent parameters. Since no measurement data are available at solvent loadings higher than 1.5 g g−1, fitting without any further conditions usually leads to unphysical values in the limit of high solvent loadings. In the limit of infinite dilution of the polymer eq 10 can be rewritten as ⎛ D V (X ) ⎞ ⎛ B⎞ lim ⎜ i 2 i ⎟ = exp⎜ − i ⎟ X i →∞⎝ (m /s) ⎠ ⎝ Ci ⎠
(11)
This expression can be equaled to the diffusion coefficient of the pure solvent to overcome this problem. One of the parameters Bi and Ci is eliminated, resulting in two independent fit parameters. For the fit in this work, NMR data for the pure solvents reported by Pickup et al.34 (toluene) and Woolf38 (methanol) have been used. Figure 7 shows the results of the fit in comparison to measurement data exemplarily for methanol− PVAc (left) at the lowest air velocity and for toluene−PVAc (right). The fit shows very good agreement with the measurement data at solvent mass fractions below 0.2. At higher solvent mass fractions the agreement is slightly worse in the system methanol−PVAc, since the value at a solvent mass fraction was fixed during the fitting procedure. This behavior could not be observed in the system toluene−PVAc. Table 1 gives an overview of the parameters Ai, Bi, and Ci for both systems and all process parameters tested in this study. As expected, the values vary little for the system methanol−PVAC 8612
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Figure 7. Comparison of fit to eq 10 (lines) and measurement data (symbols) of concentration-dependent diffusion coefficients in the system methanol−PVAc (left) and toluene−PVAc (right) at 20 °C and an air velocity of u = 0.2 m s−1. The value at solvent mass fraction 1 was fixed to the diffusion coefficient of the pure solvent.
Table 1. Parameters of a Fit to Eq 10 for All Experiments in This Work at 20 °C system
air velocity [m s−1]
A
B
C
range of validity
toluene−PVAc methanol−PVAc methanol−PVAc methanol−PVAc
0.2 0.2 0.6 1.0
43.34 30.39 30.13 30.39
162.75 111.17 118.93 123.83
8.2 5.57 5.96 6.21
solvent mass fractions of 0.05−1 whole concentration range whole concentration range whole concentration range
at different air velocities, proving independence of boundary conditions. Additionally to the comparison to literature data, these diffusion coefficients have been used in the aforementioned model-based simulation by Schabel for the prediction of film drying processes to test their practical applicability.30 As an expression for the diffusion coefficient, eq 10 was used with the parameters given in Table 1. An exemplary comparison of predicted concentration profiles (lines) and measurement data in the system toluene−PVAc from Raman spectroscopy (symbols) is given in Figure 8 at various times during the drying experiment in both the constant and the falling rate period. The predicted profiles are in good agreement with the experimental data. As a consequence of the abundance of data points for the fit at low solvent loadings, the predicted profiles are very accurate at long drying times, correctly describing the measurement data up to 78 h after the experiment started.
4. CONCLUSIONS For both systems the calculated diffusion coefficients show good agreement with literature data from gravimetric sorption experiments. This suggests that data produced by this method can be used to characterize diffusional behavior in polymer solvent systems. A wider concentration range is available by analysis of Raman data of single drying experiments than by the reported literature data. Tests in the system methanol−PVAc show that the results of this method are independent of mass transfer in the gas phase as suggested by theory. This aspect is important, since a perfect control of conditions in the gas phase, as it is e.g. needed for gravimetric experiments, is experimentally very cumbersome. However, the results also show that at diffusion coefficients lower than 10−14 m2 s−1 the
Figure 8. Simulated (lines) and measured concentration profiles (symbols) of the drying of a toluene−PVAc film at 20 °C and an air velocity of u = 0.2 m s−1. The initial solvent loading of toluene was 1.5 g g−1. In the simulation the diffusion coefficient was implemented in form of eq 10 with parameters given in Table 1.
mass flux cannot be calculated from the measurement data, and accordingly determination of diffusion coefficients is impossible. For both tested systems a fit of the diffusion coefficient is offered which has been used to predict profiles in a drying film by a model-based simulation. The prediction accurately describes measurement data. The diffusion coefficients 8613
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(22) Mamaliga, I.; Schabel, W.; Kind, M. Chem. Eng. Process. 2004, 43 (6), 753−763. (23) Müller, M. Zum Stofftransport schwer flüchtiger Additive in Polymerbeschichtungen - Untersuchungen mit Hilfe der konfokalen MikroRaman-Spektroskopie; KIT Scientific Publishing: Karlsruhe, 2013. (24) Ludwig, I.; Schabel, W.; Kind, M.; Castaing, J. C.; Ferlin, P. AIChE J. 2007, 53 (3), 549−560. (25) Ameel, T. A. Int. Commun. Heat Mass Transfer 1997, 24 (8), 1113−1120. (26) Scharfer, P.; Schabel, W.; Kind, M. J. Membr. Sci. 2007, 303 (1− 2), 37−42. (27) Alsmeyer, F.; Koß, H.-J.; Marquardt, W. Appl. Spectrosc. 2004, 58 (8), 975−985. (28) Fick, A. Ann. Phys. 1855, 170 (1), 59−86. (29) Hartley, G.; Crank, J. Trans. Faraday Soc. 1949, 45, 801−818. (30) Schabel, W.; Scharfer, P.; Müller, M.; Ludwig, I.; Kind, M. Chem. Ing. Technol. 2003, 75 (9), 1336−1344. (31) Schabel, W. Section D5.2 - Polymer Solutions: Vapor-Liquid Equilibrium and Diffusion Coefficients. In VDI Heat Atlas; SpringerVerlag: Berlin, 2010. (32) Krüger, K.-M.; Pfohl, O.; Dohrn, R.; Sadowski, G. Fluid Phase Equilib. 2006, 241 (1−2), 138−146. (33) Middleman, S. An Introduction to Mass and Heat Transfer: Principles of Analysis and Design; Wiley: New York, 1998. (34) Pickup, S.; Blum, F. D. Macromolecules 1989, 22 (10), 3961− 3968. (35) Kosfeld, R.; Zumkley, L. Ber. Bunsen Phys. Chem. 1979, 83 (4), 392−396. (36) Kishimoto, A. E. J. Polym. Sci., Part A: Gen. Pap. 1964, 2 (3pa), 1421−1439. (37) Jeck, S.; Scharfer, P.; Schabel, W.; Kind, M. Chem. Eng. Process. 2011, 50 (5−6), 543−550. (38) Woolf, L. A. Pure Appl. Chem. 1985, 57 (8), 1083−1090.
determined with the proposed method are suitable for modeling purposes. In conclusion, this new method shows very interesting features, yielding reliable results over a wide concentration range for different systems. The relatively fast measurements and easy handling are definite advantages of this method. Limitations of this method are to be expected at very low diffusion coefficients.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02144. Figures 1S−3S and Table 1S (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail
[email protected] (D.S.). Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank the students Lifu Liu and Sarah Armbruster for their work in this project. REFERENCES
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