4209
Langmuir 1994,10, 4209-4218
Monolayers of Perfluoropolyethers with a Hydrophilic Head Group Werner A. Goedel,**tJHong W U , Matthew ~ C. Friedenber + Gerald G. Fuller,? Mark Foster,g and Curtis W. Frank*$,
f
Department of Chemical Engineering, Stanford University, Stanford, California 94305, and Institute of Polymer Science, The University of Akron, Akron, Ohio 44325 Received April 25, 1994@ Perfluoropoly(oxypropylene), F((CFz)30),((CFz)z)X,with a hydrophilic head group (X = -COOHI has been investigated as a thin film spread onto the surfaceofwater and as films transferred to solid substrates via the Langmuir-Blodgett technique. Between the onset of the isotherm and collapse, the polymer forms a continuous film of uniform thickness. The density of this film is the same as in the bulk polymer, and the film thickness depends on the area per molecule. The theory of tethered polymers predicts an entropic contribution to the free energy of the system due to deformation of the polymer coils. We propose a thermodynamic analysis in which we mathematically separate the contributions to the surface pressure from the head groups and from the stretching of the polymer chains. In addition, we investigate the influence of the subphase composition (aqueous solutions of CaC12, poly(ethylenimine), HC1, FeCl3, N H 3 ) and compare the isotherm with a model system for the isolated head group. Based on the thermodynamic analysis and the comparison of isotherms, we conclude that-in the case of a chain length of approximately 100 o-bonds-there is no significant entropic contribution from the polymer chains.
Introduction Two different classes of polymers have predominantly been investigated as insoluble films on liquid surfaces and used as materials for the Langmuir-Blodgett (LB) technique: flexible polymers with surface active repeat units and stiffchain molecules. In the first case the flexible adsorbed polymer exists as an essentially two-dimensional rather than a three-dimensional random coil. The entropy loss associated with the transition from a three-dimensional to a two-dimensional conformation is compensated by the adsorption energy of the amphiphilic repeat units. The statistical mechanics of this special case have been treated by several author^.^,^ In the case of stiff chain polymers, the equilibrium conformation of the polymer backbone is essentially one-dimensional so that no changes of the backbone conformation are necessary during the adsorption process. Therefore, only a weak surface activity of the repeat units is necessary to assemble the polymer at the two-dimensional surface. A third class of surface bound polymeric materials, chains tethered to an interface of liquids, has been much less investigated experimentally. These polymers bind to the interface solely by the end groups and retain the conformation of a distorted three-dimensional coil. Therefore, the adsorbed layer is considerably thicker than in the two former cases. If a soluble polymer is tethered to an interface of a liquid, it forms a layer that is swollen with solvent (Figure la). This case has been investigated in detail theoreti~ally,4-~
* To whom correspondence should be addressed. t Stanford University.
*Present address: Max Planck Institute for Colloids and Interfaces, Hs. 9.9,Rudower Chaussee 5,12489Berlin, Germany. 5 The University of Akron. Abstract published in Advance ACS Abstracts, October 1,1994. (1)For an introduction into the characterization and the transfer of insoluble films see: Gaines, G. L., Jr. Insoluble Monolayers at LiquidGas Interfaces; Interscience: New York, 1966. Roberts, G. LangmuirBlodgett Films; Plenum Press: New York, 1990. Ulman, A. An Introduction to Ultrathin Organic Films; Academic Press: San Diego, @
CA, 1991. (2)Huggins, M.L.Makromol. Chem. 1966,87,119. (3)Huggins, M.L.Kolloid 2.2.Polym. 1973,251,449. (4)Alexander, S.J.Phys. 1977,38, 983.
b)
a)
air
polymer A
Figure 1. Schematic comparison of a thin film of polymer tethered to the surface of a solvent (a)and a nonsolvent (b)and micro-phase-separatedAB two-block copolymerin the lamellar regime (c).
and predictions of thermodynamic properties and of the concentration profile and coil conformation of the polymer have been made. These theoretical predictions have been verified experimentally in measurements of the concentration profile,'* adsorption kinetics, and compressibility perpendicular to" and in the plane of the interface.12 In the latter case it was possible to verify theoretical predictions of the shape of the surface pressure/area isotherm of a tethered soluble polymer. If a polymer chain is tethered to a nonsolvent, it forms a distinct phase (Figure lb). Because of the tethering this phase will assume a characteristic shape. The most prominent cases are polymer micelles and phase separated block copolymers, but it is equally possible that the polymer forms a continuous insoluble monolayer at the liquidgas interface. In this monolayer the polymer is tethered to the polymerlliquid interface and forms a thin asymmetrical film with a liquidpolymer and a polymer/ gas interface. This film resembles a single layer in the lamellar regime of a phase separated AB two-block copolymer (Figure IC),which has already been investigated (5)deGennes, P. G. Macromolecules 1980,13, 1069. (6)Milner, S. T.;Witten, T. A.; Cates, M. E. Macromolecules 1988, 21,2610. (7)Venema, Paul; Odijik, The0 J.Phys. Chem. 1992,96, 3922. (8)Wijmans, C. M.;Scheutjens, J. M. H. M.; Zhulina, E. B. Macromolecules l992,25,2657. (9)Review article: Halperin,A.;Tirrell,M.;Lodge,T.P.Adu. Polym. Sci. 1992,100, 31. (lo) Kent,M. S.;Lee, L.-T.;Farnoux, B.; Rondelez,F.Macromolecules 1992,25,6231. (11)Milner, S. T.Europhys. Lett. 1988,7,659. (12)Granick, S.;Herz, J. Macromolecules 1986,18, 460.
0743-746319412410-4209$04.5010 0 1994 American Chemical Society
Goedel et al.
4210 Langmuir, Vol. 10,No. 11, 1994 Table 1. Polymers Used in This Study Perfluoropoly(oxypropy1ene) F((CF~)~O),((CFZ)~)X abbreviated head group name trade name -X MJgmol-l M,IM,
1-OH 1-COOH
Demnum SA Demnum SH
-CH20H -COOH
4000 3800
1.4
theoretica1ly.l3-l6 In the present study we focus on this class of materials. It has been shown that hydrophobic polymers with high glass transition temperatures form hard disklike particles, when spread onto a water surface.17Js These particles do not change their shape upon compression of the surface. As a consequence, the isotherms depend on the spreading conditions and the method used to detect the surface pressure and change only slightly when the polymer chains are tethered to the ~ a t e r . l ~ ,Obviously, ~O these systems cannot be treated as a continuous film of polymer chains, that assume equilibrium conformations a t any surface concentration. We have recently shownz1 that the main chain of perfluoropolyethers with low glass transition temperatures is insoluble and non-surface-active at the water surface. However, a single hydrophilic group is sufficient to render them surface-active. The isotherms of these hydrophilic-terminated polymers are independent of compression speed, show no hysteresis below collapse, and, therefore, reflect states near equilibrium. At that point we speculated that the polymer forms a continuous film of uniform thickness resembling a layer of a phase separated block copolymer. According to Semenov, the morphology of phase separated block copolymers results from balancing the interfacial tensions and the entropy changes due to stretching of the polymer chains.14J5 Because of the close resemblance between phase separated block copolymers and a polymer film tethered to a liquid surface, we considered it possible that the entropy changes due to stretching of the polymer chains might influence the isotherm. Thus, the scope of the present paper is to investigate (i) whether the tethered perfluoropolyethers can indeed be regarded as a uniform amorphous film and (ii)whether the stretching of the polymer chains gives rise to a significant contribution to the surface pressure. To get insights into point i, we imaged the film of a tethered perfluoropolyether on the water surface with Brewster angle microscopy and investigated the X-ray reflectivity of films transferred to solid substrates. To get insights into point ii, we tried to estimate the relative contribution of the head groups and the polymer chain by (a) analyzing the isotherm with thermodynamic arguments, (b)varying the head groups and counterions to the head groups, and (c) investigating model systems for head groups that are adsorbed to a liquid interface but not bound to a polymeric chain.
Experimental Section Linear perfluoropoly(oxypropy1ene)with one CF3 group at one end and a COOH or CH2OH group on the other end of the chain (seeTable 1)were obtained from Daikin Ltd. and used as received. (13)Helfand, E.Macromolecules 1976,8, 552. (14)Semenov, A. N. Sou. Phys. JETP 1986,61,733. (15) Semenov, A. N. Macromolecules 1993,26,6617. (16)Ohta, T.;Kawasaki, K. Macromolecules 1986,19, 2621. (17)Kumaki, J. Macromolecules 1988,19, 2258. (18) Kumaki, J. Macromolecules 1988,21,749. (19)Niwa, M.; Hayashi, T.; Higashi, N. Langmuir 1990,6,263. (20)Yoshikawa, M.; Worsfold, D. J.; Matsuura, T.; Kimura, A.; Shimidzu, T. Polym. Commun. 1990,31, 414. (21)Goedel, W.A.;Xu, C.; Frank, C. W. Langmuir 1993,9, 1184.
1-COOH and 1-OH contained approximately 30% of pertluoropoly(oxypropy1ene) chains with CF3 groups on both ends. Perfluorodecanoic acid (PCR, 98% pure), perfluoroalkane mixture FC72 (3M), 1,1,2-trichloro-1,2,2-trifluoroethane (Aldrich), iron(II1) chloride (Aldrich, 98%), tridecafluoro-1,1,2,2-tetrahydrooctyl-1-trichlorosilane(Hiils), and poly(ethylenimine1(Aldrich) were used as received. Water (resistivity 18.2 x lo6 P cm-l, total dissolved organic carbon '5 ppm) was purified with an ion exchangdfilter system (Millipore). A saturated calcium chloride solution was prepared by dissolving a n excess of calcium chloride (Baker, reagent grade) in purified water. This solution was filtered hot. The filtrate was stirred for 2 h with activated charcoal and then filtered again. Overnight crystals of calcium chloride precipitated from this solution. The solution was stored together with these crystals and decanted prior to each use. Isotherms on the surface of pure water and aqueous solutions were recorded using a 11 cm x 70 cm (7 cm x 25 cm for concentrated calcium chloride solutions) rectangular Langmuir trough made of poly(tetrafluoroethylene), equipped with two symmetrical compressionbarriers (KSV, Helsinki). The number average molecular weight was used to calculate the mean area per molecule of the polymers. To obtain isotherms a t the interface between a saturated aqueous calcium chloride solution and a perfluoroalkane mixture by the method of stepwise addition, perfluorodecanoic acid was first dissolved in a mixture of one part ethanol and three parts of 1,1,2-trichloro-1,2,2-trifluoroethane (mass fraction 0.1 mg/g). Then the needle of a syringe that contained this solution was brought in contact with the interface and raised a millimeter while maintaining adherance to the interface, and the contents of the syringe were ejected over a 5-min period. The solvent mixture has a density between the density ofthe perfluoroalkane mixture and the CaCl2 solution. The ethanol is insoluble in the perfluoroalkanes,while the trichlorotrifluoroethane is insoluble in the aqueous phase; thus, the mixture is neither completely miscible with the aqueous nor with the perfluorinated phase. The mixture, however, rapidly spreads a t the liquidiquid interface even without a dissolved amphiphile, resulting in a temporary rise of the surface pressure, which completely decays within 30 min. Spreading of a solution of perfluorodecanoicacid in this mixture gave rise to a temporary overshoot in surface pressure, which reached a stable value usually within 30 min, and a new addition was carried out 1h later. The mean molecular area was calculated under the assumption that all the amphiphiles adsorb to the interface. Isotherms at the liquidiquid interface obtained by the compression method were measured using a quartz trough similar t o the design given by Brooks and Pethica.22 Hydrophobization of the quartz surface was carried out by immersing it for 4 h in a solution of tridecafluoro-1,1,2,2tetrahydrooctyl-1-trichlorosilanein perfluoroalkane mixture FC72 (1 drop per 5 mL), followed by wiping with FC72 and 2-propanol, washing with copious amounts of purified water, and blow drying with nitrogen. All isotherms were recorded a t a temperature of 25 "C. The surface pressures were measured via the Wilhelmy method, using ash-free filter paper plates. Hydrophilic glass and silicon substrates were first cleaned with 2-propanol and then treated with a hot mixture of 7 parts concentrated sulfuric acid and 3 parts hydrogen peroxide (30%). The Brewster angle microscope (BAM) was constructed following the design of Honig, Overbeck, and M o b i ~ s . 2 3 , ~ ~ p-polarized light from a 632.8-nm, 8-mW HeNe laser (Melles Griot) is reflected off the aidwater interface at the Brewster angle (53.1"). The reflected beam passes through a 50 mm focal length lens (Newport) into a n analyzer at known angle to the incident polarization and finally to a CCTV camera (VDC3825, Sanyo, 0.05 lux sensitivity). Rotation of the analyzer allows the image contrast to be adjusted by varying the reflected polarization that is passed to the camera. The images were recorded on videotape with a VCR (Magnavox VHS) and subsequently digitized with a DT2862 frame grabber (Data Translation). Digitized images were enhanced in contrast and corrected for (22)Brooks, J.H.; Pethica, B. A. Trans.Faraday SOC.1984,60,208. (23)Honig, D.;Overbeck,G. A.; Mabius, D.Adu.Muter. 1992,4,419. (24)Friedenberg, M. C.;Fuller, G . G . ;Frank, C. W.; Robertson, C. R. Langmuir 1994,10,1251.
Langmuir, Vol. 10,No. 11, 1994 4211
Monolayers of Perfluoropolyethers
50 c)
0
100 50
Area per Molecule
150 100
150
[A2]
Figure 2. Isotherm of 1-COOHon water. Isotherms of groups b and c are offset horizontally for clarity. (a) Variation of compression speed: (i) 1 A2 s-l(repeat unit)-', (ii) lo-' & s-l(repeat unit)-', (iii) k s-l(repeat unit)-'. (b) Two consecutive cycles of compression and expansion up to a maximum pressure of 40 mN/m. (c) Compression down to 30 k and subsequent re-expansion.
geometric distortion on a SPARCstation 2 using Sun VISION image processing tools (Sun Microsystems). Angle-dependent X-ray reflectivity was measured with a wavelength1 = 0.154nm on a rotating anodebased reflectometer with a pyrolyticgraphite primary beam monochromator yielding a relative wavelength resolution (dud) of 0.015(61= 0.0022 nm). The incident beam had an anglular divergenceof 0.00022 radian. Measurementstypically lasted severalhours and were performed at room temperature in air.
Results Figure 2 shows the isotherm of 1-COOH on pure water. Up to a surface pressure of 40 mN/m the isotherm is independent of compression speed. The isotherms of 1-COOH shown in Figure 2a, for example, have been measured within a time frame of approximately 2 min (i), 2 h (ii), and 2 days (iii). In addition, we detect no hysteresis when the film is compressed to 40 mN/m and re-expanded (Figure 2b). If the film is compressed beyond this pressure, however, the isotherm of the expansion is shifted to lower areas per molecule (Figure 2c). In the following sets of experiments we investigate whether the system can be treated as a fluid amorphous layer of uniform thickness. Brewster Angle Microscopy of 1-COOHon Pure Water. In Figure 3 a film of 1-COOH adsorbed a t the surface of pure water is visualized using a Brewster angle microscope. Before proceeding, we note that the illumination is nonuniform due to the Gaussian intensity profile of the laser used for illumination. Moreover, the pictures are out of focus right and left of the center due to the viewing angle of the camera. The fingerprint-like pattern is a result of laser interference and imperfection in the optics. Finally, a ring of white spots and a dark spot at the top left center results from damaged regions within the camera. The contrasts of parts a-f and j of Figure 3 have been increased by a factor of 2 compared to the images in parts g-i in Figure 3. In Figure 3a,b we spread the appropriate amount of material to obtain the desired area per molecule and recorded the image 30 min after spreading. In Figure 3c-j we compressed or expanded a film of originally 250 Azper molecule stepwise to the desired area per molecule and kept it at this area per molecule for 5 min before adjusting to the next area per molecule. If not otherwise stated, each image is taken at the end of this 5-min period.
At an area of 1000 k per molecule we detect patches of two-dimensional foam that move around on the surface due to convection and air currents and have a brighter reflectivity than the background (Figure 3a,b). Some of these patches are smaller, others larger than the 1 x 2 mm field of view of the microscope, and there is a large variation in the size of the dark circular regions inside the patches. The perimeters of the patches are circular, elliptical, or elongated. Coalescence of two adjacent dark regions gives rise to transient concave structures, which relax on a time scale of several seconds. Figure 3b, for example, is taken 0.2 s after Figure 3a. The concave part of the right perimeter relaxes to some extent within that time. When more polymer is spread or the film is compressed, the relative area and the size of the dark regions decrease (Figure 3c at 500 Azand Figure 3d a t 125 k)until they vanish at an area of 115 Az (Figure 3e), which coincides with the onset of the pressure rise of the isotherm. During further compression no features can be detected with the Brewster angle microscope (Figure 30 even if the analyzer is rotated, until we exceed the collapse pressure of 40-45 mN/m. Coinciding with the collapse, we observe a grainy structure containing bright circular patches (Figure 3g). When the film is re-expanded to an area greater than the area a t collapse, the patches graduallyvanish even at pressures above 0 mN/m. Figure 3h, for example, was recorded immediately after reexpansion to 50 k,and Figure 3i was recorded 5 min later at the same area per molecule. After complete reexpansion to 115 Az and waiting for 15 min only a few faint patches can be detected (Figure 3j); the main appearance is again featureless as in Figure 3e. Transfer. A film of 1-COOHcould be transferred from water to a variety of substrates by vertical deposition. Transfer took place during the down stroke of a glass substrate that was made hydrophobic by treatment with (tridecafluoro-l,l,2,2-tetrahydrooctyl)trichlorosilane.In the upstroke 80%ofthe transferred material was deposited back to the water surface. A transfer in the upstroke with a transfer ratio of 1.1f0.2 was found for hydrophilic substrates like glass, silicon, and platinum at surface pressures from 2 to 40 mN/m. The film was displaced back to the water surface when the substrate was dipped down again. For X-ray reflectivity measurements we transferred monolayers of 1-COOHfrom the water surface to silicon substrates a t surface pressures of 10,20, and 30 mN/m and obtained a transfer ratio of 1.20, 1.13, and 1.11, respectively. X-ray Reflectivity. The q dependent X-ray reflectivities of silicon substrates to which the monolayers of 1-COOH were transferred at different surface pressures are depictedin Figure 4a (filled crossed, and open circles). q was calculated from the angle of reflection 0 (angle of I according incidence) and the wavelength of the X-rays , to q = (4dA)sin 0. The reflectivity of the sample is sensitive to variations normal to the substrate in the scattering length density blv, which is proportional to the electron density ee ((bl v)ee-l= 2.82 x m). To calculate the fitted reflectivity, we used a five layer model (air, polymer main region, head group or intermediate region, silicon dioxide, silicon).The thickness of the layers, blv, and the roughness of the interfaces between layers were varied to yield the optimum matching between the fitted and the experimental ref l e ~ t i v i t y . ~The ~ , ~fits ~ to the experimental data are included in Figure 4a (solid, dashed, and dotted lines) (25) Russel, T.P.Mater. Sci. Rep. 1990,5, 171. (26) Foster, M.D.Crit. Reu. Anal. Chem. 1993,24, 179-241.
4212 Langmuir, Vol. 10,No.11, 1994
and the resulting scattering length density profiles normal to the substrate are given in Figure 4b. When we compare the profiles of the films transferred a t 10 and 20 mN/m, we observe two trends: (i)The scattering length density b/u of the intermediate layer increases with increasing transfer pressure, while its thickness is nearly constant. (ii)The film thickness of the main region increases with
Goedel et al.
increasing transfer pressure, while the scattering length density is not affected. From the chemical composition and the bulk density (1.8g/cm3)of 1-COOH we calculate a bulk value of b/u = 0.1467 x cm A-l, which is close to the b/u of the main region obtained in the fit. The film transferred a t 30 mN/m, however, has properties in between those of the two other films. When the films
Langmuir, Vol. 10, No. 11, 1994 4213
Monolayers of Perfluoropolyethers
oh
I
.-; r: 1
I
0.0
-fit
n,,=lOmNm"
exp.;
&=20"m"
exp.:----- fit
I
(
0.2
0.4
0
.
1
I
I
0.6
0.8
9
In his case, polymer chains are tethered to both interfaces of a lamellar domain (Figure IC).In our case, however, the polymer forms a thin film with two different interfaces and the polymer is tethered to only one interface (Figure lb). In the Semenov theory, polymer coils, tethered to opposite interfaces, may in principle penetrate each other. In the limit of strong stretching, however, the free ends of the polymer chain are mainly concentrated in the center region. We, therefore, assume that his expressions for the change in entropy due to the stretching of the polymer chains would still be valid if-in a thought experiment-the lamella was split in the center to form an asymmetrical layer. However, we have to take into account the free energy of the new interface. We measure an apparent surface pressure, which is given by the difference between the derivatives of the free energy with respect to area of the pure water surface and the adsorbed polymer layer
n = -(E) aA polymer film + (E) aA waterlair
(1)
We postulate that the following contributions to the free energy are additive: (i)a contribution to the surface pressure due to an elastic stretching of the polymer chains, (ii)the interfacial tension at the polymertwaterinterface, and (iii) the interfacial tension at the polymer/air interface. We thus obtain
5 0.05
-
...........
. .
_----- &=30 0.00
0
20
40
60
80
100
120
mNm" 140
160
r [AI Figure 4. (a) X-ray reflectivity of silicon substrates covered with films of 1-COOH transferred from pure water at surface pressures of 10, 20, 30, and 40 mN/m. (b) Scattering length densities obtained from the fit to the experimental points. were stored at room temperature for several weeks, the reflectivity characteristics changed irreversibly and all films showed a reflectivity comparable to the film transferred at 10 mNtm. The following data analysis and experiments are intended to estimate the relative contributions to the surface pressure of the head groups and the polymer chains. "hemodynamic Analysis. We use two different thermodynamic descriptions ofthe isotherm in the region between the onset of the pressure rise and the collapse. In the first description,we take into account the stretching of the polymer chain. In the second treatment, we take into account the cross-sectional area of the amphiphile, but neglect the polymer chain. In both treatments, we fit a theoretical description to the experimental data using two fit parameters. In the first approach, we treat the adsorbed polymer film as a three-dimensional system. We regard the polymer as a continuous layer of uniform thickness that is free of solvent and that has one interface with the aqueous phase and one interface with the air (Figure lb). Semenov describes the lamellar regime of phase separated AB two-block copolymers in which two incompatible blocks form parallel lamellae of uniform thickness. Assuming an incompressible polymer melt, he obtains an expression for the conformationalfree energy of the lamellae and the surface tension of the interfaces as a function of lamella thickness.14J5 We assume that his theory is applicable, although we have to take into account several differences between his model and our system:
(%)polymer/air-inteflace +
(E)
waterlair-interface
(2)
Semenovgives an expression for the elastic free energy, of a layer of tethered polymer per area, A, as a function of the effective radius of gyration of a single link, a , the thickness of the film, R , and the number of links per chain, N . In his expression kBT and the volume of the link, Y , are taken as unity.
Feiastic,
Felastic
2
-4
A
R3
(3)
332a2@
To have expressions in SI units we substitute FelmtiJ kBT for Felastic, alv113for a, Rlv1I3for R , and AW3for A to yield Felastic
-4
(4)
We take the polymer as incompressible and obtain an expression for the layer thickness, R , as a function of the number of molecules, n, and area, A
RA = nNv
(5)
We combine eqs 4 and 5
4
Felastic
=5
@3N~2A-2 32a2
and take the first derivative ofthe elastic energy to obtain the first term of eq 2
-aFelastic _-- - 2 aA
12 'BT
7(3
Nv2 A
-3
(7)
In the system investigated by Semenov, the polymer is tethered to an interface between two weakly interacting
Goedel et al.
4214 Langmuir, Vol. 10,No.11, 1994 polymers (Flory parameter x 0.5). The interfacial thickness can be expected to be in the order of the dimensions of a repeat unit and independent of the area per molecule. We therefore use a simple ideal gas equation to express surface tension as a function of the area per head group. In addition, we assume that the head groups (a fluorinated strong carbonic acids) are completely dissociated and therefore contribute two particles per head group to the ideal gas. We thus postulate eq 8 in which yopolymerlwakrrepresents the interfacial energy of the water/ polymer interface in the absence of the head groups:
(x
L
Y
20
10
0
60
40
80
100
120
Area per Molecule [A2] By doing so, we neglect steric and energetic interactions between the head groups. This procedure has the advantage of limiting the number of necessary fit parameters, but might introduce a considerable error, which we will discuss below. The assumption of complete dissociation of the head groups might be disputed. We note however that we obtain similar results in the fitting procedures if we use lkBT instead of 2 k ~ Tin eq 8 (not included here for clarity). We assume that the free -CF3 ends are not preferentially adsorbed to the polymerfair interface. Thus, the composition and the surface tension of this interface are independent of the area per molecule.
Figure 5. Least-squares fits of thermodynamic equations to the experimental isotherm of 1-COOHon pure water: (. * -) Semenov theory; (-) Gaines theory. Table 2. Fitted Values and Independent Estimates of the Parameters of the T w o Thermodynamic Descriptions uarameter fitted value independent estimate c1 1.8 x N m6 7.4 x N m6 C2 -8.8 x N m-l
