Langmuir 1985,1, 301-305
301
Mechanical Testing of Monolayers. 1. Fourier Transform Analysis Kevin C. O'Brient and Jerome B. Lando* Case Western Reserve University, Cleveland, Ohio 44106 Received October 4, 1984. I n Final Form: February 25, 1985
A new technique has been developed to perform mechanical testa on monolayer films. The viscoelastic properties of the films can be understood through use of linear system theory, where the viscoelastic parameters for the f i i can be thought of as the response function of the system. The tangent of the phase difference (6) between stress and strain and the modulus of the film (IE*I)are determined as a function of the frequency applied to the sample. The computation of 6 and IE*I is facilitated by computing the Fourier transforms of the stress and strain signals. 1.0. Introduction Numerous techniques have been developed to examine the mechanical properties of monolayers a t the gas-water interface. Canal torsion pendulums for m ~ n o l a y e r slongitudinal ,~~ waves,61o and capillary ripple methods"-13 and a unique isotropic compression method14 have been used to examine the viscoelastic coefficients of monomolecular films. The advantages and disadvantages of these techniques have been outlined by Jolyl5 and Gaines.lG A modification of Van den Tempel's longitudinal wave method6?' by this laboratory has been previously rep0rted.l' The technique which has been developed in the present study enables five factors to be examined: (1) the frequency applied to the sample, (2) tan 6 (where 6 is the phase difference between stress and strain) vs. frequency applied to the film, (3) the amplitude of the complex modulus as a function of frequency, (4) correction for the coupling between the underlying subphase and the film during the duration of the test, (5) monitoring of the movement of the water subphase during the experiment as a function of time and depth. 2.0. Mechanical Properties of Monolayer Films 2.1. Analysis Of Viscoelastic Materials. Bulk polymer systems have been shown to be viscoelastic in nature. The modulus for these systems can be thought of as a complex modulus which consists of a real and imaginary part. The following equation illustrates this concept: E* = E' + iE" E* = complex modulus E' = real modulus E" = imaginary modulus The real modulus describes the time-dependent response of the system, while the imaginary modulus describes the time-dependent response of the system to an applied stress. The ratio of the imaginary to the real modulus is defined as tan 6. tan 6 = E"/E'
2.2. Definitions of Gas-Water Interface Viscoelastic Coefficients. Viscoelastic coefficients for monolayers can be defined based on the constitutive relation of Mann and Hansen." These coefficients reflect the time-dependent and time-independent response of a monolayer that has flow patterns in one direction. The stress tensor component uxxscan be represented asll QXXS
= Y + kEUXX + k
4 X X
'Present address: Center for Energy Studies, University of Texas, Austin, TX 78712.
where y is the surface tension of the film, k , is the elastic coefficient, k, is the viscosity coefficient for the film, and u,, and ri,, are the strain tensor and its time derivative. The major assumptions of this relation are (1)the surface is isotropic, (2) the strain and time rate of strain are small so that k, and k, are constants, and (3) the surface is flat. The elastic and viscosity coefficients have been shown to be related to the shear and dilational properties of the film.'* For the elastic term,
k,=K+c where K is the dilational modulus of the film and c is the shear modulus for the film. The viscosity term can be represented as k = q + {
where q is the dilational viscosity coefficient and { is the shear viscosity coefficient. These coefficients are related to the moduli measured by the test outlined in the text. The real modulus is a combination of shear elasticity and surface dilational elasticity, while the imaginary modulus is a combination of surface shear viscosity and surface dilational viscosity.
3.0. Relation of Linear Analysis to Mechanical Properties The viscoelastic properties of a polymer can be determined through use of linear systems analysis. If the (1) Harkim, W.-D.;Kirkwood, J. G. J . Chem. Phys. 1938, 6, 53,298. (2) Joly, M. Kolloid-Z. 1939,89, 26. (3) Langmuir, I,; Schaefer, V. J. J . Am. Chem. SOC.1937, 59, 2400. (4) Inokuchi, K.Bull. Chem. SOC.Jpn. 1953, 26, 500. (5) Tshoegl, N. W.; Alexander, A. E. J. Colloid Sci. 1960, 25, 168. (6) Lucassen, J. Tram. Faraday SOC.1968,64, 2230. (7) Lucassen, J.; Van den Temple, M. Chem. Eng. Sci. 1972,27,1280. (8) Maru, H. C.; Wasan, D. T. Chem. Eng. Sci. 1979,34, 1295. (9) Crone,. H. M.; Snik, A. F. M.; Prulis, J. A.; Kruger, A. J.; Van den Tempel, M. J. Colloid Interface Sci. 1980, 74, 1. (10) De Feijter, J. A. J . Colloid Interface Sci. 1979, 69, 375. (11) Hansen, R. 5.; Mann, J. A. J . Appl. Phya. 1964, 35, 152. (12) Goodrich, F. C. Roc. R. SOC.London, Ser. A 1961, 260, 481. (13) Mann, J. A.; Hansen, R. S. J . Colloid Sci. 1963, 18, 757. (14) Abraham, B. M.; Miyano, K.; Ketterson, J. B. Znd. Eng. Chem., Prod. Rea. Dev. 1984,23, 245. (15) Joly, M. Surf. Colloid Sci. 1972,5. (16) Gaines, G. L. "Insoluble Monolayers at Liquid-Gas Interface"; Interscience Publisher: New York, 1966. (17) OBrien, K. C.; Rogers, C. E.; Lando, J. B. Thin Solid Film 1983, 102, 131. (18) Mann, J. A. Surf. Colloid Sci. 1984, 13, 145.
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polymer behaves as a linear system, then the mechanical properties of the plymer can be described as a transfer function for a linear system. This approach is the basis of conventional dynamic-mechanical testing equipment in bulk polymers (for example the Rheovibron). A system is described as being linear and time invariant under the following condition^:'^ (1)The response (output) of the system to a periodc input is a periodic output having the same frequency as the input. (2) If the system is excited with a single sine wave, the output of system is a single sine wave. (3) The phase difference (6) between excitation (input) and output for the system is independent of the amplitude of the input signal. (4) The amplitude of the output signal from the system depends on the frequency of the input signal but not the elapsed time of the experiment. The system is considered to be stable if the output of the system is zero when the input is zero or vanishingly small. The superposition principle holds for linear timeinvariant systems. This principle states that the response of a linear time-invariant system to a sum of several excitations is the s u m of the responses of the system to the individual e~citati0ns.l~ The response of a linear system as a function of time can be described in terms of the excitation through the following equation: y(t) = l r m h ( r ) x (t
7) d7
(la)
or y ( t ) = h.x(t)
Ob)
where y ( t ) = output as a function of time, x ( t - T ) = input as a function of time, and h(7) = transfer function for system and where eq l b is the convolution of r ( t )and h. The transform function is defined as the response of the system to a unit impulse excitation. If h(7) has been determined, then the response of the system to various periodic inputs can be predicted. The transfer function will be a function of the frequency of the periodic excitation. The transfer function can be examined in the frequency domain by performing Fourier transforms on eq l a or lb: Y(io) = H(iw)X(iw) (2) where Y(iw) = Fourier transform of response, X(iw) = Fourier transform of excitation, and H(io) = Fourier transform of transfer function. The transfer function can then be defined in terms of the ratio of the Fourier transforms of output to input signal: H(iw) = Y(iw)/X(iw) This approach suggests that H(iw) can be written as H(iw) = IH(iw)I exp(i6)
(3)
where IH(iw)) describes the amplitude of the transfer function and 6 contains the phase information. Three techniques can be used to determine the transfer function for a linear system.20 Impulse Excitation. Let x ( t ) = 6 ( t ) , where 6 ( t ) is defined as the impuse function that has the following characteristics:21 6 ( t ) = 0 for t = 0 and 6 ( t ) m as t 0 such that J?ZG(t) dt = 1. Inserting x ( t ) in eq 1 and solving for y ( t ) , one obtains y ( t ) = h(t). This implies that if a linear
- -
system is excited with a unit impulse, the resulting response is the transfer function for the system. The advantage of this technique is that theoretically the transfer function is determined over all frequencies. The reason behind this can be understood by examining the Fourier transform of the impulse function.22 The Fourier transform has a unit value over all frequencies, thus an impulse excitation theoretically has frequencies components ranging from +a to As a result, the complete transfer function over a wide range of frequencies can be quickly determined in one experiment. The disadvantage of the impulse technique is that the system may behave in a nonlinear fashion a t high frequencies. This would result in the breakdown of the technique since the analysis previously outlined is based on the assumption that the system is linear over the frequency range of the experiment. Another possible problem is that many systems are unable to sustain an impulse excitation. Harmonic Excitation. Let x ( t ) = A sin (ut)in eq 1. This implies a periodic excitation with amplitude A and frequency w. The response of a linear system to a periodic input should be a periodic output of the same frequency. The response of the system can be represented by y ( t ) B sin (ut a), where 6 is the phase difference between input and output signal. The transfer function can be thought of as containing phase and amplitude information as shown in eq 3. The phase information is obtained by examining 6, while the amplitude of the transfer function is contained in the ratio of the amplitudes of output to input signal. The period of the input signal is varied in order to determine the transfer function over a range of frequencies. This approach is the basis for the Rheovibron analysis of bulk polymer systems. The major advantage of harmonic excitation is the relative simplicity of the electronic equipment needed for the experiment, the probability of introducing nonlinear effects is much smaller than in the impulse experiment, and that most systems are more likely to withstand periodic as opposed t~ impulse excitation. The disadvantage of harmonic excitation is that the input frequency must be varied over the frequency range of interest. This could prove to be very time consuming if the frequency range is quite large. Ratio of Transforms. Equation 2 shows that the transfer function can be defined as the ratio of the Fourier transform of the output to the transform of the input signal. This suggests another possible method to determine H(iw). The requirement for this technique is that over the frequency range of interest X(io) does not equal zero. One advantage to this method is that the excitation is not required to be a specific waveform. A major drawback to this approach is that the transform ratio is very sensitive to noise. The second technique was chosen to examine the mechanical properties of monolayers. The test frequency was regulated by adjusting the barrier’s speed. The phase difference between stress and strain was determined by examining the Fourier transform of the barrier movement and the surface pressure of the film. The mathematical model developed to calculate the drag stress of the water subphase was based on a periodic strain input to the sample.
--.
+
4.0. Experimental Section (19) Stewart, J. L. “Fundamentals of Signal Theory“; McGraw-Hill: New York, 1960. (20) Faure, P.; Depeyrot, M. “Elements of System Theory”; NorthHolland Publishing Co.: Amsterdam, 1977. (21) Aseltine, J. A. “Transform Methods in Linear Systems Analysis“: McGraw-Hill: New York, 1960.
4.1. Materials and Procedures. The synthesis, water substrate, and procedure for spreading monolayer films of vinyl (22) Brigham, 0. E. “The Fast Fourier Transform”; Prentice-Hall Inc.: Englewoods Cliffs, NJ, 1974.
Langmuir, Vol. 1, No. 3, 1985 303
Mechanical Testing of Monolayers
7 3.54 3.5 -
INPUT
-1
9.0
I
stmorn. F I x
I I
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7
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-
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1
1
0
I i
I
4
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Figure 1. Flow chart for analysis of pressure and strain data. I
I
X - AXIS
I
2
I
5
6
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Figure 3. Power spectra for bulk poly(viny1 stearate) aged for 3 h. Major and minor harmonics are designated.
XVSTORE
0
I
9'
8
I
8-9'.
Figure 2. Overhead view of the Lauda film balance. Periodic strain is applied by oscillation of barrier from B to B'.
stearate have been reported previously." Films were polymerized at the gas-water interface as described in previous reports.24 Infrared spectra of irradiated films demonstrated that the conversion of monomer to polymer in monolayers irradiated at the gas-water interface under these conditions was 75%. 4.2. Hardware. A Lauda Film balance interfaced to a DEC MINC-11 was used to prepare the monolayer films. Details of the computer interface have been described previou~ly.~~ Figure 1is an overhead view of the Lauda. A periodic strain is applied to the sample by oscillating a barrier between B and B'. The frequency applied to the sample can be varied by adjusting the speed or amplitude of the barrier's oscillation. Typically amplutides of oscillation were 2 mm (less than 1% of the length of the film). 4.3. Software. Figure 2 is a flow chart for the analysis of data once it has been loaded onto a VAX 11/780, operating under VMS version 3.2. All programs were written in VAX-11 FORTRAN 1977. DYNMEC. Main program called on to analyze pressure and strain data. The user enters the number of data files for each sample ( n ) and the length of the film. The following programs are subroutines called on by DYNMEC. INPUT. Time domain data (i.e., surface pressure vs. time) is input through use for this routine. SMOOTH. Digital filter developed to eliminate high-frequency components that may be present in input data through use of a Hanning function. The bandwidth of filter was determined to be 9.8 Hz.= (23) OBrien, K. C.; Lando, J. B., manuscript in preparation. (24) OBrien, K. C.; Lando, J. B., manuscript in preparation. (25) OBrien, K. C. Rev. Sci. Instrum., submitted for publication.
FIX. This program removes spikes that might be present in input data. Points greater than two standard deviations from the mean are removed from the input and replaced with a value that is an average of the two neighboring points. o m . Computes fast Fourier transform of pressure or position data by using the relation N-l
Xk+l= j - 0 Zj+le2*iiklN k = 0, ..., N-1 i = (-1)'l2 where the real input vector is 2,and the output vector is XI (complex). The complex output vector X I consists of a real component A, and an imaginary component B,. The length of the input array, N , is a power of two. The algorithm is from the International Mathematics and Statistics Library (IMSL). OPOWER. Computes power spectra of Fourier transform data. Power is computed through the relation power = A: + BIZ
where A, is the real coefficient and B, is the imaginary coefficient of the Fourier transform of the input data. The major harmonic of the signal is determined by searching for the frequency that has the maximum power in the power spectra. XCALC. The position vs. time data is generated by this program. The barrier movement as a function of time can be represented by a triangular wave with the same major harmonic as the pressure data. MODULUS. This program computes the complex, real, and imaginary moduli for the film. The phase difference, 6, between pressure and position is also calculated using the following relations assuming A + iB, represents the Fourier transform of the movement of the karrier vs. time and C, + iD,represents the Fourier transform of the surface pressure of the film as a function of time: Q, = (A, + iB,)/(C, + iD,) tan 6 = Im(Q,)/Re(Q,) where IE*l = amplitude of complex modulus and 1 = length of the film in cm. The correction for the drag of the water substrate can be applied, if necessary, by using the relation E", = E': + K" E', = E'u + K' where K'and K''are corrections to the real and imaginary moduli to account for the drag of the underlying subphase. XYSTORE. This program stores the moduli and tan 6 data as a function of frequency for each sample. (26) Beauchamp, K. G. "Signal Processing using Analog and Digital Techniques"; Wiley: New York, 1973.
304 Langmuir, Vol. 1, No. 3, 1985
O'Brien and Lando
" I l
L 1
a
xxxx
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3.5 3.0
'oI(
xx
L 2.5
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0
0
2.
1
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A
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Figure 5. Modulus poly(viny1stearate).
a1 0
I 20
I 0
_80
I
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Figure 4. (a) Percent error in 6 vs. frequency (Hz) for a mechanical test at the gas-water interface. (b) Percent error vs. 6 (deg) for a mechanical test at the gas-water interface. Test frequency was 0.05 Hz.
The program XCALC is necessary because of the low signal to noise ratio for the channel monitoring the barrier movement vs. time data. The high noise level preventa the direct analysis of the position of the barrier as a function of time over the small amplitudes of oscillation (2 mm) employed in the test. In order to assure accuracy in the analysis, at least 16 cycles of data were collected before a Fourier transform was computed. Data collection rates ranged from 10 to 1pointa/s. These were well above the Nyquist criteriaz2for the frequencies typically applied to the films (0.1-0.01 Hz). Figure 3 is the power spectra of the surface pressure variation for a monolayer of vinyl stearate that was polymerized at the gas-water interface. The relative power, as determined from eq 1, is plotted vs. the frequency in hertz. The frequency applied to the film can be determined from the maximum in the spectra, which occurs at 0.16 Hz. The minor harmonics (Le., k X major harmonic,where k = 3,5,...) were approximately lllW the strength of the major harmonic, thus they were not used in the Fourier transform analysis. The estimation of the error in the power spectra can be determined through use of a x2 error analysis.26 The bandwidth of the Hanning filter along with the rate of data collection is used to compute the degrees of freedom for the system. The error in the calculation of IE*I was determined to be i7% for a 90% confidence interval and 4~12%in a 99% confidence interval. An estimation of the error in the computation of 6 from the Fourier transform analysis listed above is plotted in Figure 4a,b. This estimation was determined by examining two triangular waves that have a known phase. difference. The Fourier transform analysis listed above was then performed on the two signals to compute 6. The percent error was determined by comparing the actual to the computed values for 6. Figure 4a is a plot of the percent error in 6 vs. frequency (in Hertz). Figure 4b is a plot of the percent error in 6 vs. 6 (in degrees). The frequency of the two waves was O.Ok Hz. Figure 4a suggests that over the range of test frequencies examined, the percent error in 6 should range from 3% t o less than 1% . The percent error in 6 ranges from
and tan delta (0)spectra for bulk
I
I m
(X)
1
'=1
R
w D
E
io
L
A
5.
a -2.0
-1.e
-1.6
-1.4
-1.2
-1.0
-a.e
0.6
-I
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Figure 6. Modulus (X) and tan 6 (0) spectra for bulk poly(viny1 stearate) sample retested from low to high frequency.
less than 1% for 6 less than loo to 4% for 6 close to 90°.
5.0. tan 6 and IE*I Spectra Figure 5 depicts the tan 6 and modulus spectra for a film of bulk polymerized poly(viny1 stearate). Films were tested at frequencies ranging from 0.4 to 0.01 hertz. Coupling between the water subphase and the film has been shown to be negligible over this frequency range.27 The modulus increases with a decrease in the test frequency. A large peak occurs in the tan 6 spectra at 0.071 Hz. A smaller and broader peak occurs at 0.025Hz. Figure 6 illustrates the effect of retesting the bulk poly(viny1 stearate) monolayer. The height of the tan 6 peak at 0.071 Hz is reduced by retesting the sample from low to high frequencies. The modulus appears to be independent of frequency over the (27) OBrien, K. C.; Mann, J. A.; Lando, J. B., manuscript in preparation.
Langmuir 1985,1, 305-312 entire range of test frequencies. The tan 6 peak originally located at 0.025 Hz has been split into two peaks a t 0.016 and 0.04 Hz. The irreversibility of the spectra suggest that the sample is being altered by the periodic strain. The peaks are also extremely narrow compared to most peaks observed in tan 6 spectra. A structural change in the monolayer appears to be occurring during the course of the experiment. Electron microscopy studies by Ries and WalkeP.29 along with optical studies by Day and Lando30 suggest that crystallite islands are present in the film. The size of these islands increased with increasing surface pressure (increasing strain applied to the film). This was attributed to the merging of many small neighboring islands to form less islands that were larger in size. The peaks in the tan 6 spectra could be due to the repacking of these islands during the course of the experiment. The periodic strain applied to the sample causes the islands to pack in a more efficient manner. This explanation is reinforced by the results of hysteresis studies on bulk poly(viny1 stearate) monolayer^.^' These studies have shown that a decrease in the areajmolecule in the isotherms of bulk polymer films can be accomplished by successive expansions and compressions of the monolayer. (28)Ries, H. E.; Walker, D. C. J. Colloid Sci. 1961, 16, 361. (29)Ries, H.E.; Kimball, W. A. Nature (London) 1958, 181, 901. (30)Day, D.; Lando, J. B. Macromolecules 1980, 13, 1478. (31)Uitenham, L. Ph.D. Dissertation, Case Western Reserve University, Cleveland, 1984.
305
A decrease in the areajmolecule suggesta an improvement in the packing order in the film. Further details on the interpretation of the tan 6 spectra of bulk poly(viny1 stearate) and vinyl stearate films are discussed in a later publications?2 Major differences occur in the spectra depending on whether the surface-active material is a monomer or a bulk polymer. Studies on the spectra of films polymerized a t the gas-water interface indicate that the tan 6 spectra was sensitive to the presence of monomer in the polymerized film.32
6.0. Conclusions Principles of system theory have been applied to the analysis of the viscoelastic properties of monolayer films. A technique has been developed to determine tan 6 and modulus spectra for monolayer films. The phase difference between stress and strain (6) and the moduli for the film can be determined by computing the Fourier transform of the stress and strain as a function of time. The coupling between the water subphase and film during the course of the test is negligible. The displacement of the water subphase as a function of time and depth can be computed on the basis of a transport analysis of the mechanical test.
Acknowledgment. We acknowledge J. A. Mann for fruitful discussions on the viscoelastic properties of monolayers. The financial support of NSF Grant DMR-811441 is also gratefully acknowledged. (32)OBrien, K. C.;Lando, J. B., manuscript in preparation.
Application of ESCA and Contact-Angle Measurements to Studies of an Adsorbed Fluorosurfactant L. J. Gerenser,* J. M. Pochan, M. G. Mason, and J. F. Elman Research Laboratories, Eastman Kodak Company, Rochester, New York 14650 Received October 5, 1984 ESCA and surface energy measurements were made on clean SiOzand poly(ethy1ene terephthalate) (PET) fiis coated with various amounts of the cationic fluorosurfactant Zonyl FSC. On SiOz,ESCA measurements show that the fluorosurfactant coverage is uniform and continuous at all coating thicknesses and the surfactant molecules reorient within certain coating regimes. Good agreement was found between calculated surfactant coverages, based on proposed molecular orientations, and actual values determined from ESCA measurements in a region where only a monolayer of surfactant is coated. On PET, ESCA measurements indicate incomplete surface coverage with possible aggregation even at coverages where the average thickness is considerably greater than that of a monolayer. Models and correlations are derived to relate dispersion energy measurements to measured ESCA surface coverage. of the surface of a material, but contact angle is assumed Introduction to be a function only of the first chemical layer on the Contact-angle measurements have been used to describe surface. Under certain circumstances, angular-dependent wettability and surface energetics of surfaces.lpZ Electron ESCA can provide information on the uniformity, thickspectroscopy for chemical analysis (ESCA) is a spectroscopic tool that can ascertain surface c o m p ~ s i t i o n . ~ ? ~ ness, and orientation of overlayers with thicknesses 150 A. The composition of the surface of a material can affect ESCA can detect compositions on the upper 50 8,or less such quantities as adhesion,’V2 photogeneration? and triboelectrification.6 Since the surface of the material is (1)Wu, S.“Polymer Interface and Adhesion”; Marcel Dekker: New significant in many applications, it is important to unYork, 1982. derstand how surface concentration can affect measurable (2)Osipow, L. I. “Surface Chemistry”; Reinhold New York, 1962. (3)Siegbahn, K.; Nordling, C.; Fahlman, A.; Nordberg, R.; Hamrin, properties. With this in mind, we have undertaken a study K.; Hedman, J.; Johansson, G.; Bergmark, T.; Karlsson, S.; Lindgren, I.; Lindberg, B. ‘ESCA Atomic, Molecular and Solid State Structure Studied by Means of Electron Spectroscopy”; Almquest and Wiksells: Uppsala, Sweden, 1967. (4)Clark, D.T.In ‘Advances in Polymer Science”;Springer-Verlag: New York, 1977;pp 126-187.
( 5 ) Pochan, J. M.; Gibson, H. W. J. Polym. Sci., Polym. Phys. Ed. 1982,20, 2059. (6)Gibson, H.W.; Pochan, J. M.; Bailey, F. C. Anal. Chem. 1979,51,
483.
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