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plotted solvent properties on 6, the overall scatter of the activated carbon correlation seems larger, in energy terms, than the others. Most probably, this scatter is due to specific surface interactions of some of the adsorbates. Nevertheless, Figure 8 implicates solvation phenomena as a major determinant of partition coefficients. The region of highest partition coefficient, defined by the square and stars, is a remarkably thin wedge approximately bounded at the corners by 2-ethyl-1-hexanol (7), nitrobenzene (8), and benzene (5). Several of the compounds within the wedge are aromatics bearing electron withdrawing substituents (8, acetophenone, benzaldehyde, and benzoic acid) while aromatics bearing electron donating substituents are excluded (aniline, toluene, phenol, and ethylbenzene). It is not justified to conclude that specific interactions with electron-poor aromatics are solely responsible for strong adsorption, however, because nonaromatic compounds with similar solubility parameters are also strongly adsorbed (7 and 3,5,5-trimethyl-2-cyclohexene-1-one). Conclusion Stereographs of Hansen solubility parameter correlations convey more complete information than a series of twodimensional orthogonal projections. The shapes of solubility regions and correlations of complex phenomena such as adsorption equilibria can be considered in much greater detail than was heretofore possible. With the advent of flexible computer graphics software and high-resolution hardware, stereographs can be routinely and conveniently obtained without great programming effort. Acknowledgment The author is indebted to Dr. R. E. Overfield for initially
suggesting the stereographic method of display and for many helpful discussions. Literature Cited Abe, I., et al. Bull. Chem. SOC.Jpn. 1980, 53, 1199-1205. Abraham, M. H. J. Chem. Soc., ferkln Trans. 2 1972, 1343-1357. Barton, A. F. M. Chem. Rev. 1975, 75, 731-753. Beerbower, A.; Dickey, J. R. ASLE Trans. 1989, 72, 1-20. Blanks, R. F.; Prausnltz, J. M. Ind. Eng. Chem. Fundam. 1964, 3, 1-8. Cramer, R. D. J. Am. Chem. SOC. 1980, 102, 1837-49, 1849-1859. Crowley, J. D., et al. J. faint Technol. 1967, 39, 19-27. Deal, C. H.; Derr, E. L. Ind. Eng. Chem. Process Des. Dev. 1984, 3, 394-399. Eastman Chemicals “Eastman Cellulose Acetate for Coatings”, Publ. E-140B, 1981; p 7. Hansen, C. M. J. faint Technol. I987a, 39, 104-117. Hansen, C. M. J. faint Technol. 1987b, 39, 505-510. Hansen, C. M.; Skaarup, K. J. faint Techno/. 1987c, 39, 511-514. Hansen. C. M. Ind. Eng. Chem. Prod. Res. Dev. 1989, 8 , 2-11. Hansen, C. M.; Beerbower, A. I n Mark, H. F., et al., Ed., “Kirk-Othmer Encyclopedia of Chemical Technology”, 2nd ed., Supplement: Wiley: New York, 1971; pp 889-910. Hlldebrand, J. H.; Prausnltz, J. M.; Scott, R. L. “Regular and Related Solutlons”; Van Nostrand-Relnhold; New York, 1970. Julesz, B. “Foundatlons of Cyclopean Perception”; Unlverslty of Chicago Press: Chicago, 1971; pp 143-149. Kamlet, M. J.; Abboud, J. L. M.;Tan, R. W. in Taft, R. W., Ed. “Progress In Physical Organic Chemistry”, Vol. 13; Wlley: New York, 1981; pp 485-630. Noble, B.; Daniel, J. W. “Applied Llnear Algebra”, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1977; p 284. Teas, J. P. J. f a h t Technol. 1088, 40, 19-25. Wu. K.4.; Ware, W. R. J. Am. Chem. SOC. 1979, 701, 5906-5909.
Received for review July 27, 1983 Accepted November 1, 1983
Supplementary Material Available: Listing of the stereographic program, description of the algorithm, and experimeqtal data for Figure 3 (8 pages). Ordering information is given on any current masthead page.
An Instrument for Measuring the Static and Dynamic Response to Shear at the Air/Water Interface and of Insoluble Monolayers: Some New Insights into Two-Dimensional Phases 6. M. Abraham and K. Miyano” Argonne N8tIOn8lLaboratory, Argonne, Illinois 60439
J. 6. Ketterson Northwestern Universlw, Evanston, Illlnols 6020 1
An apparatus has been designed, assembled, and employed in experiments to determine, as a function of molecular areal density, the viscoelastic behavior of insoluble monolayers spread on water. The apparatus has several unique features: (1) surface tension can be monitored constantly without touching the monolayer; (2) compression of the film is isotropic in an annular space entirely free of obstructions; (3) the stress transducer, located at the geometric center, defines the inner wall of the annulus and the trough wall defines the outer wall of the annulus. A wide range of viscosity and shear modulus can be measured by controlling the stress or strain applied to the film. The apparatus can be used to study the airlwater interface of surfactant solutions as well as the surface of water covered by an insoluble monolayer.
Introduction The study of insoluble monolayers spread on an aqueous substrate has been an active field of research since its
* Research Institute of Electrical Communications, Tohoku University, Sendai, Japan. 0196-4321/84/1223-0245$01.50/0
inception by Langmuir (1917). Hundreds of compounds have since been spread on water and the surface pressure (a)determined as a function of the molecular area ( A )and the temperature (7‘). From the resultant diagram, which is accepted as the analogue of the PVT diagram in three dimensions, one may calculate thermodynamic parameters 0 1984 American Chemical Soclety
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such as a compressibility coefficient and a thermal expansion coefficient. Further, the distinct changes in the slope of the H-A diagram signal phase changes according to the Ehrenfest criteria. Using the compressibility along with shear viscosity measurements as indicators, Harkins and Copeland (1942) assigned phase labels to the various regions of the phase diagram of fatty alcohol monolayers. Similar phase assignments have been made for many other substances although no relevant physical parameters that unambiguously characterize the films (such as X-ray diffraction) have been made. There exist, therefore, great lacunae in the information necessary to gain a greater insight into these systems. In spite of the limited range of measurements that could be performed and the restricted information that was obtained from them, the persistence of the field attests to both its technical importance and scientific interest. The impetus to broaden the range of measurements comes from the growing interest among physicists in the theoretical problem of melting transitions in two dimensions, first put forward by Kosterlitz and Thouless (1973). An experimental attack required the ability to make a measurement that distinguishes a liquid from a solid and that would provide quantitative information about the strength of the transition. In three dimensions the static shear modulus is the unique feature that properly distinguishes a liquid from a solid even though an extremely viscous liquid can sometimes be confused with a soft solid. Measurement of the two-dimensional shear modulus would aid considerably in the verification of the phase assignments for different regions of the 7-A diagram. The shear modulus might also provide some insight into the structure of monolayers as well as into both the interaction between film and substrate and between surfactant molecules. The importance of such measurements was recognized very early during the emergence of the field (Langmuir and Schaefer, 1937). To the best of our knowledge, however, there was only one measurement of the shear modulus reported in the early literature (Mouquin and Rideal, 1927). A more recent set of measurements (Abraham et al., 1981) showed that the earlier measurements were made with contaminated water; however, the later ones suffered from the drawbacks of limited sensitivity and poor geometry. The parallel vane transducer used resembled a channel viscometer with the consequence that a highly viscous film never came to equilibrium in the channel. In contrast to the limited study of the shear modulus, the rheology of soluble surfactants at the air/water and oil/water interfaces have been studied extensively (Joly, 1972; Goodrich, 1973). A variety of rotational viscometers have been developed, with the deep channel traction viscometer being the most highly developed in practice and in theory. Unfortunately, the design of all these instruments makes them totally unsuitable for studying the rheology of insoluble monolayers as a function of molecular surface density, and inadequate as well for the study of shear moduli of solutions. We shall describe the design of a unique instrument which was assembled to measure the surface elastic properties as well as the surface shear viscosity of an insoluble monolayer as a function of (areal) density. The instrument is also suitable for similar measurements at the air/water interface for surfactant solutions. Three criteria were established that had to be met by the design: (1) measurements were to be made as a function of density and the film was to be compressed uniformly; (2) surface tension or surface pressure was to be measured at will without disturbing the film; and (3) the stress-strain re-
TORSION WIRE
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CAPILLARY WAVE GENERATING BLADE
,COIL
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&
,
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-
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Figure 1. Schematic drawing of the mechanical components of the surface shear instrument. The support rod for the capillary wave blade is driven up/down by a small linear stepping motor. The torsion wires (2 mil Pt-5% W alloy) are also the electrical leads for the coil.
lation for the film was to be measured over a wide dynamic range. Sensitivity to very small stresses and the ability to apply small strains were essential to be certain that one was in the linear stress/strain regime. The manner in which each of these conditions was met will now be presented, followed by a discussion of the type of measurements made and of the analysis with examples.
Instrument Description A detailed description of the first version of this instrument has been published (Abraham et al., 1983), but several improvements have been made which were essential to extend the dynamic range and which warrant the extended remarks. A Teflon cup with a sloping inner wall holds the subphase on which the film is spread (Figure 1). The film is compressed by withdrawing the subphase through a drain in the bottom. The subphase level, however, is always brought to the original elevation by raising the cup. The fiducial mark for the level is also one electrode of the capillary wave generating system with which the surface tension is measured. The stress transducer, a Teflon rotor, is located at the center of the cup. A strain can be applied to the film either by rotating the cup or the rotor. The resultant stress is measured by recording the motion of the rotor with an optical lever. A more complete description of each component follows. Trough and Mechanical Ancillaries. The cup or trough was machined from a Teflon pancake 6 in. (15.24 cm) in diameter by 2.25 in. (5.72 cm) thick by cutting a concial hole, 90" apex angle, through the pancake. A water-tight cup was formed by bolting the machined pancake to a flat copper heat-exchange plate with eight copper bolts. The seal between the plate and the pancake was made with a 5-mil Teflon sheet. A flanged Teflon drain tube, sealed by the same sheet, was inserted through the heat-exchange plate and bolted on the underside t o make a rigid tight seal. The resulting assembly can be heated or cooled by circulating water through the heat exchanger. The cup was then secured to a stepping motor actuated rotary optical table. Because of the water inlet and outlet tubes to the heat exchanger, the maximum angle of rotation is limited to 90". The minimum angle corresponds to one step of the motor, 7.3 X radians. The speed can be pre-set over wide limits. This assembled unit was then rigidly attached to a vertical linear translation stage which is also actuated by a stepping motor. Vertical motion can be resolved to cm which corresponds to one step of the motor. The stage can be moved 5 cm but
Ind. Eng. Chem. Prod. Res. Dev., Vol.
I
=--
I
--
Figure 2. Schematic drawing of the data acquisition system.
in practice the limit is 2.5 cm because the span of the capillary wave generator sets a limit on the minimum radius. The compression ratio of the film is, as a consequence, limited to 2.5:l. The drain tube was connected to a rigidly mounted buret to form a “U” tube with the Teflon cup. A glass capillary tube was inserted through a closure at the top of the buret to a point just above a stopcock at the bottom. The outer end of the capillary tube was connected to an aspirator through a valve. A glass rod that had been drawn to a point was sealed to the capillary, then bent to point down. The glass point (water level fiducial mark) and the capillary tube from the point upward were platinized to form a conducting path to the outside. As the water is always buffered or acidified, the conduction path is continuous through the drain to the water surface in the cup when the water is in contact with the point. Thus, when the stopcock is closed and an aliquot of water is withdrawn from the buret, electrical contact to the fiducial mark is broken. When the stopcock is opened, water flows out of the cup into the buret but contact, both physical and electrical, are re-established only on raising the cup. The signal that contact has been made is the re-appearance of the capillary wave (see below). The contact is reproducible to 25 counts or 25 pm. The drop in water level in the cup is then calculated from the number of pulses (counts) applied to the stepping motor. Surface Pressure. The capillary wave generator/detector system used to measure the surface pressure was first described by Soh1 et al. (1978). In brief, a platinum foil, the generator blade, is held parallel to the water surface and about 0.5 mm above it. The blade is at ground potential but 170 V at about 100 Hz is applied to the fiducial mark. When the water is in contact with the fiducial mark a ripple approximately 100 A in amplitude is generated on the water surface. The ripple is caused by the alternating strong electric field set up between the blade and the water; since the water response is quadratic in the applied electric field, the frequency of the ripple is twice the frequency of the excitation. The ripple is detected with a laser beam. The split beam from a laser, Figure 2, is directed vertically downward onto the water surface about 3 cm in front of the blade. A portion of the beam is reflected from the water which acts as a rocking mirror. The reflected beam is directed by mirrors onto a position sensing photodiode (PD 1). The signal from the photodiode is amplified, displayed on an oscilloscope, and fed to a phase-sensitive detector. From an initial calibration with clean water, subsequent readings of the phase enable one to calculate the surface tension of film covered water. Surface pressure is calculated from
where go, Q, p, and Tare the surface tensions of clean and film covered water, respectively, the density of water, and
23, No. 2, 1984 247
the period of the audiofrequency oscillator. The path length D and number of wavelengths No are obtained from calibration with pure water. The number of wavelengths N on film-coveredwater is calculated from the phase angle obtained from the lock-in amplifier. As seen from eq 1the capillary wave method using a fixed path, as done in this apparatus, is a relative measurement. The surface tension of clean water is required. For this purpose a single initial measurement is made with a Wilhelmy plate. Though the capillary wave and the Wilhelmy plate are of approximately equal precision, 0.1 dyn/cm, both are necessary. (The Wilmelmy plate is not necessary if a variable path is used. In this apparatus, it was convenient to use the plate.) The diameter of the water surface, thus the area, is obtained from the vertical rise of the trough, and the surface pressure is calculated from eq 1. The P A diagram is then completely determined, noninvasively, during the entire course of the experiment. The limitation to the accuracy of the density arises from two sources. The error in delivery of surfactant solution from the transfer pipet, which can be as much as lo%, and the extra area associated with the meniscus height of the water which is a function of the diameter of the water surface. The latter is systematic and constant from run to run; the former causes slight shifts in the relative position on the area axis from run to run. Stress Transducer. In the first version of the instrument the Teflon rotor, which contacted the water, was suspended by a long torsion wire. The instrument was severely limited in the range of viscosity and shear modulus that could be studied with a given wire. It is not uncommon for the viscosity or shear modulus of the film to vary by several orders of magnitude during the compression. Under these circumstances measurements can be made only at certain portions of the s-A diagram. A major modification to the instrument which circumvented the problem was to mount the rotor on a fiber glass frame which was also the support for a 2lI2turn coil that became part of a galvanometer. In this manner, a variable stress over a very wide range can be applied to the rotor which is simply determined by the magnitude of the current through the coil. The extended range is achieved at the expense of a slight reduction in sensitivity. Two short wires, instead of a single long one, are required for mounting the coil. These wires also serve as the current leads. Data Acquisition and Analysis Most real substances do not behave as ideal (Newtonian) liquids or as ideal elastic (Hookean) solids; monolayers are no exception. The variable and complex interactions possible in monolayer systems or at the interface of a surfactant solution show a response to strain that is quite complex. The data collected from all modes of operation of this instrument are the time-dependent stress developed in response to strain. In order to deduce the physical parameters relevant to the system, the data are first fitted to a solution to the differential equation of motion of the rotor. The parameters from this fit are then related to the solution to the differential equation of an appropriate viscoelastic model. The differential equations of motion for a Maxwell liquid and a Kelvin solid have been solved for conditions of constant strain, constant stress, and oscillatory damping (DeFeijter, 1979; Ferry, 1970; Goodrich, 1973; Joly, 1972). The parameters obtained from the curve fitting are substituted into the appropriate solution to obtain the physically relevant parameters.
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In the present setup, shown schematically in Figure 2, the voltage output from a position sensing photodiode (PD 2), which is a direct measure of the angular deflection of the rotor, is fed to a programmable digital voltmeter which in turn interfaces with a dedicated computer. A voltage and corresponding time measurement constitute a data point and are stored on a flexible magnetic disk for analysis at a later time. The DVM and the computer replace, at a much lower cost, a signal averager and a computer. Depending on the strength of the film,we employ three different modes of operations. 1. When the film viscosity (7) and the stiffness (d are small, the rotor undergoes a damped oscillatory motion after a step function stress is applied. A cosine function with an exponential envelope is fitted to the data to obtain three parameters, frequency, damping, and mean position (a fourth parameter, the phase, is also determined in the course of the fit). From these and the viscoelastic model relevant to the experiment, one can calculate q and I* at the frequency of the rotor oscillation. 2. When 7 becomes too great for the oscillatory mode to be used, the trough is rotated through a small angle and the resulting time-dependent stress is obtained from the time-dependent rotation of the rotor. The mode can also be used when the shear modulus becomes very large. Again, the data are fitted to an equation describing the motion and the resulting parameters inserted into the appropriate model to obtain the physical quantities, shear viscosity, instantaneous (infinite frequency) shear modulus, and zero frequency (static) shear modulus. 3. Finally, when CL >> 7 a known stress is applied to the rotor by passing current through the coil. The time-dependent strain is followed as indicated by the rotor and the analysis is performed as before. The same parameters are extracted as in the immediately preceding case. Examples. The performance of the instrument will be illustrated by presenting examples of data from three different systems, a homologous series of fatty alcohols, a biological surfactant, and two soluble surfactants. As stated in the Introduction, the impetus for this new approach came from an interest in phase transitions in two dimensions. Our experience with a linear trough in which the film compression was uniaxial (Abraham et al., 1980) led us to believe that conventional phase assignments to certain regions of the a-A diagram could be incorrect. We therefore decided to examine the normal fatty alcohols C18 to CZzinclusive with the exception of C19, which was not available (Miyano et al., 1983). In Figure 3A are illustrated the a-A diagrams for the alcohols. Except for the slight shoulder displayed by Czo the shapes of the curves are similar. The pressure at the kink is higher as one progressively increases the number of carbon atoms; otherwise there is no dramatic difference. However, on examining Figure 3B, one sees a drastic difference. The kink, which is supposed to signal the transition from liquid to solid, can apparently signal the transition from solid to liquid as well. One sees that the Czz alcohol becomes increasingly more rigid with compression to develop finally a substantial static shear modulus. The Czl alcohol, on the other hand, displays a most unusual phenomonon, re-entrant melting. At densities both below and above the kink the film is solid (or nearly so) but at the kink there is an unambiguous melting. The C ,, alcohol becomes liquid when compressed beyond the kink in contrast with the CI8 which becomes solid. When compared with this rich display, the T-A diagram is very sterile indeed. Before there can be any hope of introducing order into this apparently chaotic condition,
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Figure 3. (A) n-A diagrams for a homologous series of long-chain fatty alcohols (CISOH,C,,OH, CZ10H,and CzzOH). (B)Stress development as a function of time after applying a shearing Strain to each of the alcohols. Labeled points correspond to equivalent labels on the x-A diagram.
the temperature plane of each compound must be mapped. Figure 4A illustrates the 7r-A diagram for a-dipalmitoyl lecithin, a biological surfactant. The compound is a zwitterion and is, therefore, sensitive to its environment. Although the film appears to be compressed with Ca2+ions in the subphase, the general form of the diagram with Na+ and Ca2+in the subphase is the same. However, on comparing Figures 4B and 4C we see that the viscosity of the film with Na+ in the subphase is much greater than with Ca2+,whereas the latter has a nonzero shear modulus (solid). Research with this compound is continuing with various buffers in the subphase. The fiial examples that will be presented are viscoelastic measurements on two surfactant solutions, a 0.1 wt % solution of hydrolyzed methyl cellulose (Dow Methocel A) and 0.5 wt % polyvinyl alcohol (PVA). The former solution had a bulk viscosity of 1.4 cP, and surface tension of 59 dyn/cm. The latter had a bulk viscosity of 1.2 CPand surface tension of 46 dyn/cm. Again here is a situation
Ind. Eng. Chem. Prod. Res. Dev., Vol. 23, No. 2, 1984
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