11314
J. Phys. Chem. C 2007, 111, 11314-11319
In Situ Measurements of Lubricant Temperature and Pressure at a Sliding Contact C. U. Amanda Cheong and Peter C. Stair* Department of Chemistry, Northwestern UniVersity, 2145 Sheridan Road, EVanston, Illinois 60208 ReceiVed: February 2, 2007; In Final Form: May 21, 2007
The in situ temperature and pressure of silicone (polydimethylsiloxane) grease at a ball-on-flat sliding contact was successfully measured by UV Raman spectroscopy. The ratio of Stokes and anti-Stokes Raman peaks from the 490 cm-1 Si-O-Si bending mode were used to calculate the grease temperature at 10 and 20 cm/s sliding speeds. The peak shifts detected during sliding were used to calculate the lubricant pressure. The results show that the silicone temperature at the contact area increased with the sliding speed. Under a stationary contact, the silicone temperature was measured to be 22 °C. At 10 and 20 cm/s sliding speeds, the silicone temperatures were measured to be 45 and 62 °C, respectively. The pressure, on the other hand, did not depend on the sliding speed and remained at 0.6 GPa. These results were found to be consistent with calculations based on Hertzian contact theory.
Introduction The importance of temperature in determining the properties of sliding or rolling interfaces has led to a number of theoretical and experimental studies on this subject. The temperature of a sliding contact rises due to the work done against friction. The temperature distribution depends on the load, the rubbing speed, the surface topography, material properties, and the presence of lubricants at the interface. Under lubricated sliding, the temperature rise, due to shearing of the lubricant film, influences its chemical stability and viscosity. Both factors affect the performance of the lubricant. Under dry sliding, the frictional heat dissipation at the rubbing surface increases fatigue and wear of the sliding materials. During dry friction and boundary lubrication, the surfaces of asperity contacts experience high temperatures of short duration which is termed “flash temperature”. Due to its short duration and transient nature, flash temperatures are difficult to measure. In the late 1930s and early 1940s, Blok1 and Jaeger2 formulated a theoretical analysis of flash temperature for rubbing solids which is still being used in the present day. The flash temperature theory formulated by Blok does not apply to lubricated contacts because it assumes that the effect of the oil film can be neglected. Later, Archard,3 Crook,4 and Klaus5,6 developed theories to estimate lubricant film temperatures at rubbing and rolling interfaces. A number of experiments have been designed to probe, directly, the temperature at a contact interface. These experiments fall into two categories, spectroscopic methods employing infrared (IR) emission microscopy and electro-mechanical methods employing surface micro transducers or thermocouples. In the 1970s, Wymer and Macpherson,7 and Winer and co-workers8-10 used IR emission microscopy to determine the temperature of rubbing surfaces in rolling and sliding elastohydrodynamic (EHD) contacts. In Winer’s experiments, a sliding EHD contact was formed using a 31.8 mm diameter chrome steel ball rotated and loaded against a 1.6 mm thick sapphire flat. The infrared radiation emitted at this contact was measured with an infrared radiometric detector having a spot size * To whom correspondence should be addressed. E-mail: pstair@ northwestern.edu.
resolution of 38 µm. The contact temperatures deduced from these readings were time-averaged values since a large number of surface asperities would pass through the field of view during one sampling interval. Most temperature measurements using IR emission radiometry have been done on the ball surface instead of the lubricant. In 1976, Winer and co-workers8 reported a temperature mapping of naphthenic mineral oil at an EHD contact. Since the emission spectrum of the oil and the steel ball surface overlap, extensive data filtration was performed to isolate the lubricant emission signals. In their subsequent investigations, only ball surface temperatures were reported, most likely due to the masking of the lubricant emission by the black body emission from the ball surface. The shear stress profile at an EHD contact provides information on how lubricants respond to changing conditions across an EHD contact. Cann and Spikes11-13 have used the ball surface temperature, measured by IR radiometry, to deduce lubricant shear stress in an EHD contact. They argued that, since heat input into both surfaces results from the heat generated by lubricant shear and thus the shear stress at that position, the shear stress profile can be obtained. Three assumptions were made in their calculations: (1) All of the heat generation was due to fluid shear, not due to compression or asperity contact. (2) All of the heat generated from fluid shear was lost by conduction to the solids, not by convection. (3) The temperatures of the two contacting surfaces were the same at the corresponding contact points. Provided that these assumptions were valid, the heat generated by fluid shear could be equated to the heat entering the sliding surfaces, and thence, the shear stress could be determined. Clearly, a much more accurate shear stress profile would be obtained if the temperature of the lubricant at the shear plane was measured directly. Nonspectroscopic methods of measuring surface temperature in the concentrated contact involved subsurface thermocouples and surface micro transducers. Nonspectroscopic methods have an advantage over spectroscopic methods that nontransparent materials can be used for both surfaces. However, again, only the solid surface temperature could be studied. The lubricant film temperature was not measured directly, and therefore, the
10.1021/jp070928e CCC: $37.00 © 2007 American Chemical Society Published on Web 07/11/2007
Measurements at a Sliding Contact influence of temperature rise on lubricant rheology could not be studied directly. Besides temperature, the pressure on a lubricant at a rubbing interface may influence its effectiveness through changes in chemistry and viscosity. Raman spectroscopy using visible wavelength excitation has been used to study the effect of high pressure on some lubricants and liquids. It was found that the Raman peaks shift as a function of pressure14-16 and that the lubricant pressure distribution under EHD conditions is consistent with theory.17,18 As discussed by Jonas and others, the frequency shift observed for a band may be negative or positive depending on whether the dominant intermolecular force for the corresponding oscillator is attractive or repulsive.19-23 The range of lubricants that could be studied with visible Raman spectroscopy has been limited by their inherently low Raman scattering cross section and by fluorescence interference. In this paper, the application of ultraviolet Raman spectroscopy to obtain the temperature and pressure of silicone grease at a ball-on-flat sliding contact is described. With UV Raman spectroscopy, it is possible to both increase the scattering intensity and avoid fluorescence.24-27 In so doing, lubricants that could not be studied using visible Raman spectroscopy are detectable. Experimental Section The UV Raman spectrometer used to measure Raman spectra has been described previously in refs 24 and 25. Briefly, a Lexel Second Harmonic Generation (SHG) laser was used to provide continuous UV radiation at 244 nm focused, vertically downward, to a diameter of ∼50 µm onto the sample position. Light scattered up from the sample is collected with a high quality ellipsoidal reflector and focused into a Spex 1877 Triplemate triple monochromator. The triple monochromator filters out Rayleigh scattering and disperses the Raman scattered light onto an imaging photomultiplier tube. The laser power at the sample position during the experiments was typically 7mW. Dow Corning DC-976 high vacuum silicone grease was employed to carry out the temperature and pressure measurements. Silicone grease is stable from -40 °C to 204 °C with a high viscosity, 2 × 106 cSt. The silicone grease is a polydimethylsiloxane (PDMS) having the molecular structure shown.
Figure 1 is a Raman spectrum of the silicone grease. The Raman band at 490 cm-1, used for the temperature and pressure measurements, is assigned to the Si-O-Si out-of-plane skeletal bending vibration.26,27 This band is used for the temperature and pressure measurements because its relatively low Raman shift results in a more precise measurement of temperature. Its high intensity and absence of overlap with bands due to the sapphire window material are also advantages. Calibration experiments were performed under controlled temperature conditions to verify that grease is a suitable lubricant and that temperature rises can indeed be measured with the UV Raman spectrometer. A layer of silicone grease about 0.5 mm thick was spread on the flat surface of an aluminum block. A K-type thermocouple was embedded into the grease. The temperature of the unheated grease was recorded to be 27 °C, and the Stokes and anti-Stokes Raman spectra were recorded. Then the grease was heated by placing the aluminum block on
J. Phys. Chem. C, Vol. 111, No. 30, 2007 11315
Figure 1. Stokes Raman spectrum of silicone high vacuum grease.
a hot plate. The temperature of the grease was monitored with the thermocouple. The Stokes and anti-Stokes Raman spectra were accumulated for 60 s at each of 6 different temperatures, 32, 37, 47, 53, 63, and 81 °C. In situ temperature and pressure measurements of the silicone grease at a ball-on-flat sliding contact were performed by coupling the UV Raman spectrometer to a modified Falex ISC200PC Ball-On-Disk tribotester as described in ref 24. The geometry of the spectrometer optics and contact are diagramed in ref 25. Briefly, the test ball is fixed on one end of a precisionbalanced cantilever arm to simulate pure sliding against the under side of a horizontal, spinning sapphire window. The pressure between the ball and sapphire window is applied by loading weights on the opposite end of the cantilever. Laser excitation and Raman scattered light collection are performed through the sapphire window looking down on the top of the ball. After spreading a thin layer of grease onto the lower side of the sapphire disk, a 6.35 mm diameter chrome steel ball was loaded against the sapphire window to create the ball-on-flat contact. The load applied to the chrome steel ball was 0.61 N. Two levels of sliding speed were applied to the sapphire window, 10 and 20 cm/s. Alignment of the sapphire-ball contact with the Raman instrument was performed by translation of the tribotester to a position where the signal from the lubricant was minimized, corresponding to the minimum lubricant film thickness. The accuracy of contact location was limited to approximately 25 µm as determined by the intensity of the Raman signal and the precision of the translation stage. Stokes and anti-Stokes Raman spectra of the silicone grease at the contact center under stationary conditions and during sliding were recorded. The spectra were integrated for 600 s under stationary conditions. At a 10 cm/s sliding speed, the spectra were integrated for 1800 s and for 2400 s at a 20 cm/s sliding speed. Results and Discussion A. Temperature Measurements. The temperature of a sample can be measured from the ratio of Stokes and anti-Stokes Raman intensities. For a nonabsorbing sample in thermal equilibrium, the ratio of intensities, expressed as radiant power, is related to the temperature according to eq 1
11316 J. Phys. Chem. C, Vol. 111, No. 30, 2007
( ) ( )( )
exp
-hνk Ia[νk] νo - νk ) kT Is[νk] νo + νk
Cheong and Stair
4
(1)
where T ) temperature; h,k ) Planck’s and Boltzmann’s constants; νo ) laser frequency; νk ) vibrational frequency; and (νo ( νk) ) the frequencies of the Stokes (-) and antiStokes (+) scattered light. As discussed by Rassat and Davis,28 eq 1 can be expressed in terms of photon count rates according to eq 2
exp
( ) ( )( ) -hνk na[νk] νo - νk ) kT ns[νk] νo - νk
3
(2)
where n refers to the photon count rate. Rassat and Davis also reported that eq 2 can give erroneous temperature results due to changes in the spectrometer efficiency with wavelength. By comparing temperatures measured using Raman intensities, Tmeas, to known values, Tabs, obtained by, for example, a thermocouple, a correction factor can be obtained according to eq 3
CF ) exp
[ (
hνk 1 1 k Tabs Tmeas
)]
(3)
The Raman peak areas obtained in the temperature calibration experiments were determined by peak fitting software which accounts for the baseline and provides a fitted peak area for a Gaussian form.29 Equation 3 was applied to the 490 cm-1 peak of unheated grease to obtain a correction factor of 0.905. With this correction factor, eq 2 was applied to calculate the temperature obtained from the other 6 sets of Stokes and antiStokes Raman spectra of the calibration experiment. As shown in Figure 2, the Raman and thermocouple measured temperatures agree very well with a maximum discrepancy of 4 °C. This agreement demonstrates the absence of significant laser-induced heating of the silicone grease. Figure 3 presents the Stokes and anti-Stokes Raman spectra of the 490 cm-1 grease peak before sliding and during 10 cm/s sliding. The Raman peak areas were obtained using the same procedure as the temperature calibration experiments. The temperature of the grease before sliding was determined to be 22 °C. At 10 cm/s sliding, the temperature increased to 44 °C; at 20 cm/s sliding, the temperature increased to 62 °C. These results indicate that the temperature increase, ∆T, is approximately proportional to sliding speed, consistent with conductive heat dissipation and a coefficient of thermal conductivity that is independent of sliding speed. As with the calibration measurements, the maximum error is estimated to be 4 °C. The measured lubricant temperatures are averages over the area of the incident laser beam and the time of spectral integration. Since the Raman spectra were taken over 1800 and 2400 s during sliding, the temperature values were average temperatures over that period of time. Similarly, since the laser spot size is about 50 µm, the temperature results were spatial averages over that dimension. Because of the temporal and spatial averaging, the measured temperatures are not “flash” temperatures. The fluid temperature at the contact interface depends on the roughness of the rubbing surfaces, the amount of load exerted, the sliding speed, the physical properties of the lubricant, the thickness of the lubricant film, and the rate of lubricant flow through the contact. All of these parameters need to be included in a mechanical model to obtain a theoretical estimate of the
Figure 2. Comparison of grease temperatures measured by Raman and thermocouple.
fluid temperature. Currently, there is no direct comparison available between our results and any temperature results reported in the literature. However, it is helpful to examine some literature results to see if our values are reasonable. Cheng and Lacey,30 using a mechanical model, calculated the asperity temperature of an AISI #52100 ball surface to be 410 °C during the oxidation of a paraffin oil. Their sliding speed was 23 cm/s compared to our 10 cm/s, and their load was 392 N compared to our 0.61 N. Cann and Spikes11 used IR radiometry to measure the ball surface temperature rise above the lubricant oil temperature in the fully flooded EHD lubrication regime. The lubricants used were commercial base stocks and were held at a temperature of 40 °C. The sliding speed ranged from 50 to 200 cm/s and the load ranged from 28 to 72 N. At 50 cm/s and 28 N, the chrome steel ball surface was found to be about 50 °C. When the sliding speed was doubled and the load increased to 44 N, the ball surface temperature increased to about 85 °C. These measurements indicate that our temperature values are reasonable. B. Pressure Measurements. It is observed from the Raman spectra presented in Figure 3 that an asymmetric broadening of the peak appeared during sliding. To examine the broadening more closely, the Raman peaks from the O-Si-O bending mode during sliding are superimposed on those before and after sliding and compared in Figure 4. An asymmetric broadening on the high wavenumber side is evident. This asymmetric broadening is observable at both sliding speeds and for both Stokes and anti-Stokes peaks. The possibility that the broadening is an artifact of the rotating sapphire window was evaluated by removing the chrome steel ball and measuring the silicone Raman spectrum while the sapphire window was rotated at 10 cm/s. No peak broadening was observed. Chemical degradation of the grease is also ruled out since the peak broadening is reversible; that is, the peak returned to its original width and position after sliding. Thus, it is clear that the peak broadening is caused by the sliding action. The asymmetric peak broadening of the 490 cm-1 Si-O-Si bending mode can be modeled as a pressure induced peak shift to higher frequency, indicating that the dominant intermolecular force between Si-O-Si units is repulsive,20-23 which is consistent with weak dipole-dipole interactions and a lack of short range order in the fluid. We note that the load on the contact results in pressure on the lubricant only during sliding
Measurements at a Sliding Contact
J. Phys. Chem. C, Vol. 111, No. 30, 2007 11317
Figure 3. Stokes and anti-Stokes Raman spectra of silicone grease before and during a sliding experiment at 10 cm/s. The arrow marks the peak corresponding to the Si-O-Si bending vibration. The other prominent peaks are due to the sapphire window.
Figure 4. Comparison of the Si-O-Si Raman peak from the silicone under stationary (filled symbols) and sliding (open circles) conditions.
not under stationary conditions. Under stationary conditions when the solid surfaces are relatively smooth, the surface contacts, rather than the lubricant, bear most of the load. Under these conditions the Raman spectrum represents primarily lubricant at atmospheric pressure. Under sliding, the contact surfaces are separated, and the lubricant film supports the applied load. The shift in the Stokes 490 cm-1 peak was determined by peak fitting, and the results are presented in Figure 5. First, the
Figure 5. Fitting results for stationary and sliding conditions. For stationary conditions the silicone peak has been fit to a single asymmetric Gaussian peaks. For sliding conditions the peak has been fit to two peaks with the same shape as used for the stationary conditions. The separation between the two peaks is 22 cm-1.
peak measured under stationary conditions was fit to an asymmetric Gaussian form to determine the width and asymmetry parameters.31 Each of the broadened Stokes peaks at 10 and 20 cm/s sliding speeds was then fit with a pair of peaks having shape parameters constrained to same values as the fit to the peak under stationary conditions. The correlation coefficient of the two-component fit was 0.97 for the 10 cm/s sliding
11318 J. Phys. Chem. C, Vol. 111, No. 30, 2007
Cheong and Stair
speed and 0.98 for the 20 cm/s sliding speed. The results produce a peak shift of 22 cm-1 for the 10 cm/s sliding speed and 25 cm-1 for the 20 cm/s sliding speed; (3 cm-1 with a 95% confidence limit. To estimate the pressure responsible for the observed peak shift, we consider a simple model for the relationship between the Si-O-Si bending frequency, the volume occupied by PDMS, and its compressibility. To establish this relationship, we examine the results from vibrational spectroscopy of zeolites. In zeolites silicon has a tetrahedral configuration similar to PDMS. Extensive experimental and theoretical studies have identified a strong Raman band near 500 cm-1 to be due to the out-of-plane bending vibration of Si-O-Si units, the same vibrational motion that is responsible for the 490 cm-1 band in PDMS. Dutta and co-workers32 have found a linear correlation between the Si-O-Si bond angle in a series of zeolites and the Raman shift of the Si-O-Si bending mode. From this correlation, a Raman band at 490 cm-1 corresponds to a SiO-Si angle of 148°, whereas a Raman shift of 515 cm-1 (a change of 25 cm-1) corresponds to a Si-O-Si angle of 139°. The measured ∠Si-O-Si bond angle for PDMS is reported to be approximately 143°,33 in excellent agreement with the values obtained from the correlation. The radius of gyration, S, can be related to the bond angle, θ, along the polymer chain (see below).34
S is defined as the root-mean-squared distance of the collection of atoms or groups from their common center of gravity.34 S is proportional to the ratio of cosine functions given in eq 4
S∝
cos θ (11 -+ cos θ)
1/2
(4)
The relationship between S and pressure for the polymer is determined by the coefficient of coil compressibility, R, given by eq 535
R)-
3 dS S dP
( )
(5)
A finite pressure change can then be expressed as
∆P ) -
()
S2 3 ln R S1
(6)
Combining eqs 4 and 6, P2 can be expressed in terms of the bond angle of the polymer chain as
P 2 ) P1 -
(
)
1 + cos θ2/1 - cos θ2 3 ln 2R 1 + cos θ1/1 - cos θ1
(7)
In the sliding experiments, P1 is the pressure of the silicone grease before sliding, i.e., atmospheric pressure (0.1 MPa). Based on Dutta’s work, when the Raman shift is 490 cm-1, the Si-O-Si bond angle, β, is 148° which gives θ a value of 32°. During sliding, the Raman peak shifted to 515 cm-1 which implies a decrease of β to 139° and a value of θ equal to 41°. We were not able to obtain data on the coefficient of coil
compressibility for PDMS grease. However, in an experimental comparison of compressibilities for polymer blends, the coefficient of coil compressibility was observed to be 40-300% of the volume compressibility, depending on the temperature and the blend composition.35 Assuming that the coil compressibility is equal to the volume compressibility, 1.3 GPa-1 at 40 °C, the pressure of the grease during sliding, P2, is estimated to be 0.6 GPa. The maximum pressure in the contact based on the Hertzian theory is 0.38 GPa, which is in reasonable agreement with our estimation given the uncertainty in the value of the coil compressibility. It is observed from the pressure experiments that doubling the sliding speed from 10 to 20 cm/s did not increase the measured lubricant pressure. This result is consistent with calculations using the elastohydrodynamic theory. The calculations confirm that merely changing the sliding speed without changing the load does not increase the lubricant pressure at the contact zone. Conclusions The in situ temperature and pressure of PDMS grease have been successfully measured using UV Raman spectroscopy. The Stokes and anti-Stokes Raman peaks of the 490 cm-1 Si-OSi bending mode were used to determine the fluid temperature at 10 and 20 cm/s sliding speeds. Peak shifts detected during sliding were used to calculate the lubricant pressure. The results show that the PDMS temperature at the contact area increased with the sliding speed. Pressure, on the other hand, did not depend on the sliding speed. These results are consistent with the calculations based on Hertzian contact theory. Acknowledgment. We acknowledge the financial support from the Tribology Research Center at Northwestern University. We are highly grateful to Professor Herbert Cheng and Professor Jane Wang for stimulating discussions. References and Notes (1) Blok, H. IIme Congr. mondial petrole 1937, 3 (Sect. 4), 471-86. (2) Jaeger, J. C. J. Proc. R. Soc. New South Wales 1942, 56, 203. (3) Archard, J. F. Wear 1959, 2 (6), 438-455. (4) Crook, A. W. Philos. Trans. R. Soc. London, Ser. A 1958, 250 (981), 387-409. (5) Clark, D. B.; Klaus, E. E.; Hsu, S. M. Lubr. Eng. 1985, 41 (5), 280-287. (6) Naidu, S. K.; Klaus, E. E.; Duda, J. L. Ind. Eng. Chem. Prod. Res. DeV. 1986, 25 (12), 596-603. (7) Wymer, D. G.; Macpherson, P. B. Asle Trans. 1975, 18 (4), 229238. (8) Ausherman, V. K.; Nagaraj, H. S.; Sanborn, D. M.; Winer, W. O. J. Lubr. Technol.-Trans. Asme 1976, 98 (2), 236-243. (9) Nagaraj, H. S.; Sanborn, D. M.; Winer, W. O. J. Lubr. Technol.Trans. Asme 1977, 99 (2), 254-263. (10) Nagaraj, H. S.; Sanborn, D. M.; Winer, W. O. Wear 1978, 49 (1), 43-59. (11) Cann, P. M.; Spikes, H. A. Tribol. Trans. 1989, 32 (3), 414-422. (12) Spikes, H. A.; Cann, P. M. Tribol. Trans. 1990, 33 (3), 355-362. (13) Cann, P. M.; Spikes, H. A. Tribol. Trans. 1991, 34 (2), 248-256. (14) Gardiner, D. J.; Bowden, M.; Daymond, J.; Gorvin, A. C.; Dareedwards, M. P. Appl. Spectrosc. 1984, 38 (2), 282-284. (15) Gardiner, D. J.; Baird, E.; Gorvin, A. C.; Marshall, W. E.; Dareedwards, M. P. Wear 1983, 91 (1), 111-114. (16) Baird, E. M. The Application of Raman Spectroscopy to Studies of Elastohydrodynamic Contacts; Newcastle upon Tyne Polytechnic: Newcastle, U.K., 1988. (17) Jubault, I.; Mansot, J. L.; Vergne, P.; Mazuyer, D. J. Tribol.-Trans. Asme 2002, 124 (1), 114-120. (18) Gardiner, D. J.; Baird, E. M.; Craggs, C.; Dare-Edwards, M. P.; Bell, J. C. Raman microspectroscopy of a working elastohydrodynamic contact; 1989; Vol. 1, pp 301-313. (19) Schindler, W.; Sharko, P. T.; Jonas, J. J. Chem. Phys. 1982, 76 (7), 3493-3496. (20) Thomas, H. D.; Jonas, J. J. Chem. Phys. 1989, 90 (8), 4144-4149.
Measurements at a Sliding Contact (21) Cansell, F.; Petitet, J. P.; Fabre, D. J. Appl. Phys. 1989, 65 (8), 3280-3282. (22) Bradley, M. S.; Krech, J. H. J. Phys. Chem. 1992, 96 (1), 75-79. (23) Walrafen, G. E.; Hokmabadi, M. S.; Yang, W. H.; Piermarini, G. J. J. Phys. Chem. 1988, 92 (15), 4540-4542. (24) Cheong, C. U. A.; Stair, P. C. Tribol. Lett. 2001, 10 (1-2), 117126. (25) Cheong, C. U.; Stair, P. C. Tribol. Lett. 1998, 4 (2), 163-170. (26) Stair, P. C.; Li, C. J. Vac. Sci. Technol., A 1997, 15 (3, Pt. 2), 1679-1684. (27) Li, C.; Stair, P. C. Stud. Surf. Sci. Catal. 1996, 101, 881-890 (Pt. B, 11th International Congress on Catalysis-40th Anniversary, 1996, Pt. B).
J. Phys. Chem. C, Vol. 111, No. 30, 2007 11319 (28) Rassat, S. D.; Davis, E. J. Appl. Spectrosc. 1994, 48 (12), 1498505. (29) PeakFit Version 4, SPSS Inc. (30) Hsu, S. M.; Shen, M. C.; Klaus, E. E.; Cheng, H. S.; Lacey, P. I. Wear 1994, 175 (1-2), 209-218. (31) Half Gaussian Modified Gaussian form. (32) Dutta, P. K.; Shieh, D. C.; Puri, M. Zeolites 1988, 8 (4), 306-9. (33) Mark, J. E.; Flory, P. J. J. Am. Chem. Soc. 1964, 86 (2), 138-41. (34) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience Publishers: New York, 1969; p 432. (35) Janssen, S.; Schwahn, D.; Springer, T.; Mortensen, K. Macromolecules 1995, 28 (7), 2555-60.
