- Email: [email protected]
Birefringence and stress growth in uniaxial extension of polymer solutions
J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Birefringence and stress growth in uniaxial extension of polymer solutions T. Sridhar a,∗ , D.A. Nguyen a , G.G. Fuller b a
Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
b
Received 6 August 1999; accepted 9 August 1999
Abstract Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule. Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted. Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Birefringence; Stress
1. Introduction The ability of polymer chains in solution to become extended and aligned in a flow field is primarily responsible for the complex spectrum of properties exhibited by these solutions. Flow fields having a ∗
Corresponding author. Tel.: +61-3-9905-3427; fax: +61-3-9905-5686.
0377-0257/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 9 ) 0 0 0 8 0 - 4
300
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
significant extensional component facilitate such molecular unraveling. The dynamics of the transformation of polymer chains from a random coil to an extended structure remains an outstanding problem in rheology. The problem has motivated several avenues of research, both theoretical and experimental, over the last three decades. Recent progress has been rapid primarily due to several new experimental techniques being developed. Amongst these the development of birefringence as a sensitive measure of segmental orientation [1] and the measurement of extensional stresses by the filament stretching rheometer [2,3] are pertinent to the current work. In addition, improvements in our ability to simulate the stochastic processes governing chain configuration [4] have informed our ideas. Using such techniques, bead-rod [5] and bead-spring models [6] of a polymer chain have been developed. These studies have exposed an amazing diversity of chain conformation resulting from the competition between initial conformation, Brownian motion and the effect of the flow field [6]. Optical birefringence is sensitive to local orientation of the polymer chain. The polymer stress, however, responds to the overall deformed length of the polymer chain. The stress optical law provides a convenient relationship between birefringence and stress. Significant improvements in the measurement of optical birefringence [7] now enable even transient flows to be conveniently studied. The noninvasive nature of the measurement has resulted in a vast literature on the use of birefringence to examine flow problems. While the stress optical rule has been verified in shear flows, there is some evidence that in strong flows the anticipated chain extension may lead to a break down of this rule. Keller and co-workers have pioneered the use of optical techniques to study extensional flow [8–11]. They used opposed nozzles, four-roll mills and cross-slot devices to create an extensional flow field and used birefringence to probe the degree of chain orientation. In these devices the flow field is non-uniform but the presence of a stagnation point permits molecules to achieve a large strain. The birefringence is highly localised. The existence of a coil-stretch transition at a critical strain rate equal to the universe of the longest chain relaxation time was demonstrated. These studies were concerned with measuring the conformational relaxation time and elaborating the various birefringence structures observed. The relationship between such a relaxation time and molecular weight and solvent quality was explored using simple models of the polymer chain. Fuller and Leal [12] and Dunlop and Leal [13] studied dilute polymer solutions in two-and four-roll mills. These devices permit the flow field to be varied from simple shear to a purely two-dimensional extensional flow field. The presence of the polymer modifies the flow field and except for very dilute solutions, the changes in polymer conformation and the resultant changes in flow field are intricately linked. Dunlop et al. [14] and Geoffrey and Leal [15] have used flow birefringence to indicate changes in conformation and light scattering to directly measure changes in velocity gradient. These studies show significant inhibition of chain extension even in dilute solutions. One major drawback with most of these studies is that the polymer stress is not directly measured. There are very few reported instances where both stress and birefringence have been simultaneously measured using polymer solutions in extensional flows. Talbot and Goddard [16,17] measured birefringence and stress in a semi dilute separan solution using a fibre spinning device. The birefringence was measured at various positions along the spinline and hence simultaneous values of birefringence and stress for a fluid element as it flows down the spinline was obtained. This work documented the breakdown of the stress optic coefficient when the stress exceeded 2 kPa, which was the lowest recorded stress. Cathey and Fuller [18] performed similar measurements in an opposing jet device and explored the effects of solvent quality and strain rate. The birefringence was shown to saturate at high strain rate, while the extensional viscosity exhibited a maximum with strain rate. The latter finding was attributed to the decrease in fluid residence time in the flow field as the strain rate increased.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
301
Both these devices are subject to uncertainties in the flow field. In the fibre spinning device, the initial conditions of a fluid element is not known and is determined by the shear flow in the tubes leading to the spinline. In the opposing jet device, the fluid residence time varies and hence an accurate measure of the strain is not available [19]. Furthermore these devices do not permit a study of the effect of strain and the development of stress and birefringence with time from a stress free state. Partly this arises from the difficulty of creating well-defined extensional flows in the laboratory and measuring the resultant stress. The recent development of the filament stretching rheometer [2,3,20] has effectively overcome this difficulty and it is now possible to measure the stress in a well-defined extensional flow field. Simultaneously, recent developments in optical rheometry permit the measurement of birefringence in time dependent flows [1,7]. Doyle et al. [5] reported simultaneous stress and birefringence measurements. In order to avoid scattering at the air-filament interface, the experiments were performed in a plateau tank with the filament immersed in a fluid of similar density. There are significant experimental difficulties associated with such a procedure including the limitation it places on the maximum strain that can be achieved. This implies that it is not easy to achieve steady states in stress and birefringence. Doyle et al. [5] report a maximum stress of about 40 kPa at a Weissenberg number of 2.84, which corresponds to a strain of around 3.5. Under these conditions several other experimental problems could arise. The flow field during the early stages is affected by the shear flow near the end plates [21] and the measured stresses are dominated by the solvent contribution. In spite of these difficulties, this work captured some of the intricacies of the dynamics of chain unraveling. In particular the stress and birefringence in start up and relaxation exhibited a hysteresis. The stress at the same average level of chain deformation (as measured by birefringence) could assume different values depending on the history of deformation. This finding was later explained by Doyle et al. [5] as arising from the different configurations obtained during stretching and relaxation. In this work, optical rheometry on the filament stretching rheometer is used to examine both steady and time dependent strain rates. We perform the experiments in air and are able to reach large strains. A technique developed by Talbott and Goddard [16] is adapted to mask out light, which passes around the shrinking filament.
2. Experimental procedure The fluid sample studied in this work consists of 0.05% by weight of monodisperse polystyrene of 2 million molecular weight dissolved in piccolastic (a low molecular weight polystyrene). The solution was prepared by Professor S. Muller. The shear viscosity of the solution at 21.5◦ C is 87.5 Pa s and the dominant Zimm relaxation time was estimated to be 8.4 s. Details of the filament stretching rheometer have been published [22]. A ‘master plot’ technique [23], is used to ensure that any desired strain rate or strain rate history can be imposed on the sample. All experiments were carried out at a constant temperature of 21.5 ± 0.5◦ C. The flow birefringence of the sample was measured using the optical train shown in Fig. 1(a). A polarised laser light with a wavelength of 632 nm is generated by a helium–neon (He–Ne) laser and is deflected at 90◦ by a simple right angle prism. The light then passes through a linear polariser (P) and a photoelastic modulator (PEM). The polariser is fixed at an angle of 45◦ relative to the photoelastic modulator. Before passing through the filament, the light is transmitted through a quarter wave plate (Q), which is oriented parallel to the linear polariser. The light then passes through a lens, a circular polariser
302
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 1. (a) Optical train used for flow birefringence experiments; (b) simple function of the lens and the mask on laser light.
(CP) and the detector. Two lock in amplifiers are used to measure the output at the principal frequency and at the first harmonic. The purpose of the lens and the mask (a cylindrical rod, painted black, with a diameter of 1.7 mm) is to prevent part of the laser light that passes around the filament from reaching the detector. Talbott [17] has given a simple analysis of the action of the lens and the mask as shown in Fig. 1(b). The lens is located such that parts of the laser light passing around the filament remain collimated and are brought to focus at the focal point of the lens where the mask is located. Light rays passing through the filament, which acts as a cylindrical lens, is focused at nearly the front focal point of the lens. Hence these light rays are collimated by the lens, and pass through the circular polariser and then to the detector. The analysis by Talbott [17] shows that the error in assuming that the path length is equal to the filament diameter is negligible. The signal from the detector was then fed into a signal conditioner, which produced a low pass filter output and the intensity of the signal. The latter signal was sent to two lock-in amplifiers. The low pass filter signal and the two signals from the lock-in amplifiers were collected by the PC via the DAC card. Further details of the procedure and the data analysis are available in Fuller [1]. To increase the birefringence signal and make it convenient to focus the light on the filament, an initial sample size of 6 mm in diameter and 1.7 mm in length was used. One disadvantage of using such a small initial aspect ratio is that the initial flow field is dominated by shear near the end plates. The curvature of the filament also causes problems in focussing the optics and does lead to rather unreliable results at small time. A diameter-measuring device (ZUMBACH ODAC) measured the mid-point filament diameter
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
303
Fig. 2. Flow birefringences and tensile stresses of sample SM1 at ε = 4.9 s−1 for two identical runs.
while a force transducer measured the force on the end plates. The signals from the diameter measuring device and force transducer were sent to a PC via a digital-to-analog converter (DAC) card and several signal conditioners are incorporated to improve the quality of the data collected. 3. Results and discussion 3.1. Steady strain rates Fig. 2 shows the transient growth of birefringence and stress at a constant strain rate. At small time the stress and birefringence are dominated by the solvent contribution. At 0.4 s corresponding to a strain of 2 units, the polymer chain begins to orient with the flow and unravel and as a consequence both the stress and birefringence increase. Two identical experiments are shown in Fig. 2 demonstrating good reproducibility.
304
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 3. Plot of Trouton ratio of sample SM1 for various strain rates (2−10 s−1 ) against strain at a temperature of 21.5◦ C.
Fig. 3 shows that the Trouton ratio depends mainly on the strain and the effect of strain rate is confined to a region near the steady state. The behaviour of this fluid is typical of similar Boger fluids [3,21,23,24] The steady state extensional viscosity decreases with increasing strain rate and Gupta et al. [23] have suggested several possible reasons for such a behaviour. The stress saturates after about 5 strain units and the saturation Trouton ratio is around 800 at low strain rates. On the assumption that the polymer chain is in a fully extended configuration, one can estimate the Trouton ratio using Batchelor’s formula for rigid rods in suspension [25]. Tr =
π ηs cNA l 3 3 ηo MW ln(π/φ)
For polystyrene the monomer length is 2.5 × 10−8 cm [25], hence the length of a fully extended polymer is l = 4.5 × 10−4 cm. The equivalent diameter is 6.3 × 10−8 cm. φ is the volume fraction of polymer, c is the polymer concentration in g/ml. The equation predicts a Tr of 1500 whereas the measured value is around 800. This suggests that under these conditions the chain is not in the fully extended configuration.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
305
Fig. 4. Plot of flow birefringence of sample SM1 for various strain rates (2–10 s−1 ) against strain at a temperature of 21.5◦ C.
It is possible that the chain has attained a steady configuration but further extension is inhibited due to the formation of folds or entanglements [25]. Fig. 4 shows the behaviour of the solution birefringence at different strain rates. At low strain rates the birefringence measurements are influenced by two factors. One is the low birefringence value under these conditions. The other more important influence is the strong curvature of the filament during the initial period. The experimental difficulties of getting an adequate signal due to surface scattering and the uncertainty in the diameter leads to a poor resolution of the birefringence. At larger strain the filament assumes a more cylindrical shape and the birefringence is also much larger. Above a strain of 2, the birefringence increases rapidly. A saturation of the birefringence is observed above a strain of 5. The saturation birefringence appears to be independent of strain rate whereas the birefringence at intermediate strain depends on strain rate. The data in Fig. 4 reflect the birefringence of both the polymer and the solvent. In order to remove the effect of the solvent, the birefringence of the polymer chain was calculated by subtracting the solvent
306
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 5. Polymer flow birefringence for various strain rates.
birefringence. The solvent birefringence was estimated from the solvent stress (3 ηS ε˙ ) and the stress optic coefficient for the solvent. There is sufficient evidence in the literature [26] that these oligomeric solvents obey the stress optic rule. Fig. 5 shows the polymer contribution to the birefringence as a function of the strain. Surprisingly, the strain rate has a minor influence on the growth of the birefringence. In Fig. 3, we have already shown that the Trouton ratio is independent of strain rate and is predominantly dependent on strain. Hence Figs. 3 and 5 suggest a relationship between Trouton ratio and polymer birefringence. This directly contradicts the stress optical rule, which postulates a relationship between birefringence and stress. Note that the solvent contribution to the stress has been ignored because it is insignificant at strains larger than 2. The saturation birefringence of 4 × 10−5 can be compared with the birefringence for a fully extended polymer chain [27] 2 2π np n2o + 2 1n = N (α1 − α2 ) , n 3 1
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
307
where, np is the number density of polymer in solution, n and no are the refractive indices of solution and solvent, respectively, N is the number of Kuhn segments per molecule, and α 1 and α 2 are the polarizability of a subunit parallel and perpendicular to the chain axis. For a solution of high molecular weight PS in low molecular weight PS, no = n = 1.6 and (α 1 − α 2 ) = −145 × 10−25 cm3 . The number of Kuhn segments is estimated assuming that there are 15 C–C bonds per segment. Hence the calculated value of birefringence of a fully extended polystyrene chain for SM1 solution is −1n1 = 4.8 × 10−5 . This estimate is subject to significant uncertainities. Hence the birefringence saturates at about the expected value for full extension. We have shown previously that the Trouton ratio saturates at 50% of the expected value for a fully extended chain. Theoretical simulations are examined to further probe this issue. Larson et al. [6] have examined the chain extension using bead spring models. A variety of chain configurations such as dumbbells, folds, kinks, coils and half dumbbells are shown to arise. The dynamics of each of these configurations is quite distinct. Folded configurations show a rapid extension with strain, but appear to ‘hesitate’ at about 50% extension before extending further. These folded configurations
Fig. 6. Two levels of the steady state of Trouton ratio of SM1 for a strain rate of 0.5 s−1 at a temperature of 19.5◦ C.
308
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 7. Plot of stress-optic coefficient against strain; data points: experiment with a strain rate of around 2 s−1 (Wi = 16.8), lines: bead rod simulations (Wi = 10.65).
can require 8 to 9 strain units before reaching full extension. Dumbbell shaped configurations reach full extension at a lower strain. On an extensional viscosity versus strain plot this translates into a plateau corresponding to a strain of about 5 units and a slow growth from there up to a steady state corresponding to full extension. Experiments at such high strains are dogged by experimental difficulties. The measured forces become very small and inertia becomes increasingly important. The most important issue is filament breakage and the difficulty of maintaining a constant strain rate. For the fluid used in this work, it has been possible to overcome these difficulties and obtain data up to a strain of 7.8. Fig. 6 shows that the Trouton ratio increases rapidly up to a plateau of around 800 before increasing slowly up to a value of 1500. This shows similarity with the behaviour of the folded structure discussed above and the final steady state is also consistent with the estimates of the Batchelor theory. Birefringence depends on the orientation at the Kuhn length scale and is not expected to be able to differentiate between folded and fully extended structures. The saturation of birefringence under these conditions is therefore not surprising. From the simultaneous measurements of stress and birefringence, the stress optic coefficient can be calculated. Fig. 7 shows that the stress optical coefficient decreases rapidly with increasing strain. This
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
309
Fig. 8. Comparison between shear stress relaxation with extensional stress birefringence and extensional relaxation.
dramatic breakdown of the stress optic rule is consistent with the bead rod simulations of Doyle et al. [5]. Fig. 7 also shows the simulations of Doyle et al. [5] for two different values of N, the number of links in the chain. While the values of N used do not correspond to the polymer used in this work, the agreement is qualitatively good. The stress optic law is premised on a linear force law for the spring. The fact that the stress optic coefficient breaks down at even low strains suggests that, even under these conditions, the stress arises predominantly from portions of the chain that are extended beyond the linear region. The filament stretching rheometer makes it possible to study relaxation of stress and birefringence. Orr and Sridhar [24] have examined the relaxation of stresses and showed that the initial relaxation is very rapid and is followed by a slower relaxation. This finding has generated significant interest and several theoretical studies address this issue [28]. In this work simultaneous stress and birefringence measurements were made during relaxation after extensional flow for a certain strain. The data are shown in Fig. 8. The relaxation of birefringence is similar at the two strains and is significantly slower than the rate at which the tensile stress relaxes. The relaxation of shear stress, at a shear rate of 8.5 s−1 ,
310
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 9. Extensional flow start-up and subsequence relaxation for stress versus birefringence of sample SM1; data points: asterisk: Wi = 41.7, circle: Wi = 16.8, square [1]: Wi = 2.84, line: [1]: conformation dependent FENE model with ζ max /ζ o = 8 (Wi = 2.84). −1 obtained in a Rheometrics rheometer is also √ shown on Fig. 8. The shear rate of 8.5 s is chosen so that the extensional strain rate is ε˙ = γ˙ / 3. Under these conditions of equivalent deformation, the relaxation of birefringence and shear stress is similar. The relaxation data for birefringence in shear and extension are in qualitative agreement with the bead-rod simulations of Doyle et al. [5]. The simulations also demonstrate that the evolution of stress and birefringence during start-up and relaxation exhibits a pronounced hysteresis. This hysteresis arises due to the skewed distribution of dumbbell lengths in a FENE model. The stress results predominantly from dumbbells near their full extension due to the highly non-linear restoring force. On the other hand, the contribution to birefringence scales linearly with length. In a bead-rod model, the simulations show that the chain configuration during start-up and relaxation is significantly different. During relaxation, the chain essentially relaxes from the free ends. Fig. 9 shows a plot of stress versus birefringence for two different strain rates and demonstrates a pronounced hysteresis. The stress is much larger, for a given deformation, during start-up than during relaxation. Doyle et al.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
311
Fig. 10. Stress growth of sample SM1 during extensional flow start-up then stop and then re-start the flow. The flow stops at point A for 0.2 s then re-starts at point B; after stopping for 2 s the flow re-starts at point C. Position D shows the same level of stress as that at point B and point E has the same stress as point C.
[5] first documented such a hysteresis and their data is also shown in Fig. 9. Fig. 9 shows that the stress-birefringence relationship during stretching depends on strain rate. During relaxation, the stress drops rapidly whereas the birefringence changes slowly. After this initial period, the stress birefringence profile is independent of strain rate. The simulations of Doyle et al. [5] and the earlier study of Grassia and Hinch [29] suggests an universal curve for a fully extended chain onto which can be superimposed the relaxation of all other chains by a shift in time. The process of relaxation of these chains is similar, in that they all relax from the chain ends in the so-called stem and flower configuration. As a result, a universal curve for stress birefringence during relaxation is not surprising and relaxation of partially extended chains eventually follows this curve. The substantial differences in the chain configuration during start-up and relaxation have significant ramifications for the modelling of time dependent flows. Orr and Sridhar [22] demonstrated one consequence of this hysteresis. In a filament stretching rheometer, filaments at the same stress levels can be
312
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 11. Flow birefringence of sample SM1 during extensional flow start-up then stop and then re-start the flow. The flow stops at point A for 0.2 s then re-starts at point B; after stopping for 2 s the flow re-starts at point C. Position D shows the same level birefringence as that at position B and point E has the same birefringence as point C.
created by a combination of stretching and relaxation for different periods. The response of such filaments to further deformation was shown to be significantly different. The difference arises due to the different chain conformations obtained — open chains respond rapidly to the imposed deformation whereas compact chains respond more slowly. These findings cast doubt on the validity of pre-averaging procedures adopted for development of closed form constitutive equations. We report here similar experiments where both stress and birefringence are measured. Fig. 10 shows the stress in an experiment where the fluid was stretched at a constant strain rate for a specified period, followed by relaxation for a given time followed by further stretching at the same strain rate. During relaxation the stress decays by nearly two orders of magnitude. Note that the stress growth on re-imposition of flow is very rapid. The chain conformation at points A and B shown on the Fig. 10 are not very dissimilar and the stress at points B and D are identical. Yet the rate of stress growth at point B is much larger than at point D and is, in fact similar to the rate of change of stress at point A [22].
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
313
Fig. 12. Flow birefringence of sample SM1 during extensional flow start-up then stop and then re-start the flow. Data points have been shifted to the left for comparison.
Concurrent birefringence measurements are shown on Fig. 11. Note the good reproducibility of the three experiments during the initial period prior to the relaxation. The birefringence decays by about a factor of 5 during the 2 s relaxation (point A to point C). During this period the stress has decayed by two orders of magnitude (Fig. 10). The birefringence at points B and D and at points C and E are similar. The rate of increase of birefringence at points B and D and at points C and E are also similar. This is more clearly demonstrated in Fig. 12, where the curves in Fig. 11 have been shifted left so that the points B and D and the points C and E coincide. The three experiments superimpose suggesting that at constant strain rate the current value of birefringence determine the rate of change of birefringence. In time dependent extensional flows, which can be broken down into a series of extensions and relaxation, the current state of polymer chain and its ability to respond to imposed deformation depends not on the stress but on the chain configuration. As such the history of deformation is contained within the chain configuration. Attempts to capture this information by just the second moment of the end to end distribution function is likely to lead to problems in predicting time dependent flows.
314
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
4. Conclusions This work presents simultaneous stress and birefringence measurements using the filament stretching rheometer. The data shows that both stress and birefringence saturate at a strain of 5. The saturation values for stress and birefringence are lower than that inferred on the assumption of complete unravelling of the polymer chain. It is speculated that the formation of folded structures is responsible for this behaviour. Evidence is presented that suggests these folded structures can be further unraveled at higher strain. The stress at a strain of 7.8 units is in agreement with estimates from the Batchelor theory. Such folded structures have been predicted by Brownian simulations and have been observed for DNA solutions [6]. The relaxation of birefringence is shown to be much slower than the relaxation of stress leading to the stress- birefringence hysteresis and a breakdown of the stress optical rule. In time dependent flows simulated by a sequence of stretching and relaxation and further stretching, the dynamics of stress and relaxation are shown to be different. Stress growth is seen to depend on both the configuration and the imposed strain rate. The current state of stress does not by itself uniquely represent the current configuration. On the other hand, the dynamics of the growth of birefringence depends only on the current value of birefringence and the imposed strain rate.
Acknowledgements This work was supported by a grant from the Australian Research Council. It is dedicated to Professor D.V. Boger in acknowledgement of his many contributions to rheology. References [1] G.G.Fuller, Optical rheometry of complex fluids, Oxford University Press, London, 1995. [2] T. Sridhar, V. Tirtaatmadja, D.A. Nguyen, R.K. Gupta, Measurement of extensional viscosity of polymer solutions, J. Non-Newtononian Fluid Mech. 40 (1991) 271–280. [3] V. Tirtaatmadja, T. Sridhar, A filament stretching device for measurement of extensional viscosity, J. Rheol. 37 (1993) 1081–1102. [4] H.C. Ottinger, Stochastic Processes in Polymeric Fluids, Springer, New York, 1996. [5] P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110. [6] R.G. Larson, H. Hua, D.E. Smith, S. Chu, Brownian dynamics simulation of a DNA molecule in an extensional flow field, J. Rheol. 43 (1999) 267–304. [7] P.L. Frattini, G.G. Fuller, A note on phase-modulated flow birefringence, J. Rheol. 28 (1984) 61–70. [8] A.A. Keller, A.J. Muller, J.A. Odell, Entanglements in semi-dilute solutions as revealed by elongational flow studies, Progr. Colloid Polym. Sci. 75 (1987) 179–200. [9] A.J. Muller, J.A. Odell, A. Keller, Elongational flow and rheology of monodisperse polymer in solution, J. Non-Newtonian Fluid Mech. 30 (1988) 99–118. [10] D.P. Pope, A. Keller, Alignment of macromolecules in solution by elongational flow: a study of the effect of pure shear in a four-roll mill, Colloid Polym. Sci. 255 (1977) 633–643. [11] D.P. Pope, A. Keller, A study of chain extending effect of elongational flow in polymer solutions, Colloid Polym. Sci. 256 (1978) 751–756. [12] G.G. Fuller, L.G. Leal, Flow birefringence of dilute polymer solutions in two-dimensional flows, Rheol. Acta 19 (1980) 580–600.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
315
[13] P.N. Dunlap, L.G. Leal, Dilute polystyrene solutions in extensional flow: birefringence and flow modification, J. Non-Newtonian Fluid Mech. 23 (1987) 5–48. [14] P.N. Dunlap, C.H. Wang, L.G. Leal, An experimental study of dilute polyelectrolyte solutions in strong flows, J. Polym. Sci. Part B: Polym. Phys. 25 (1987) 2211–2238. [15] E. Geffroy, L.G. Leal, Flow birefringence studies in transient flows of a two-roll mill for the test fluid M1, J. Non-Newtonian Fluid Mech. 35 (1990) 361–400. [16] W.H. Talbott, J.D. Goddard, Streaming birefringence in extensional flows of polymer solutions, Rheol. Acta 18 (1979) 505–517. [17] W.H.Talbott, Ph.D. thesis, University of Michigan , Ann Arbor, 1978. [18] C.A. Cathey, G.G. Fuller, The optical and mechanical response of flexible polymer solutions to extensional flow, J. Non-Newtonian Fluid Mech. 34 (1990) 63–88. [19] R.K. Gupta, T. Sridhar, in: A.A. Collyer, D.W. Clegg (Eds), Elongational rheometers in Rheological Mesurements, Elsevier, New York, 1988. [20] S.H. Spiegelberg, G.H. McKinley, Stress relaxation and elastic decohesion of viscoelastic polymer solutions in extensional flow, J. Non-Newtonian Fluid Mech. 67 (1996) 49–76. [21] S.H. Spiegelberg, D.C. Ables, G.H. McKinley, The role of end-effects on measurements of extensional viscosity in filament stretching rheometers, J. Non-Newtonian Fluid Mech. 64 (1996) 229–267. [22] N. Orr, T. Sridhar, Probing the dynamics of polymer solutions in extensional flow using step strain rate experiments, J. Non-Newtonian Fluid Mech. 82 (1999) 203–232. [23] R.K. Gupta, D.A. Nguyen, T. Sridhar, Extensional viscosity of dilute polystyrene solutions: effect of concentration and molecular weight, 1999, in press. [24] N. Orr, T. Sridhar, Stress relaxation in uniaxial extension, J. Non-Newtonian Fluid Mech. 67 (1996) 77–103. [25] D.F. James, T. Sridhar, Molecular conformation during steady-state measurements of extensional viscosity, J. Rheol. 39 (1995) 713–724. [26] R.C. Ng, L.G. Leal, Concentration effects on birefringence and flow modification of semi-dilute polymer solutions in extensional flows, J. Rheol. 37 (1993) 443–468. [27] A. Peterlin, Streaming birefringence of soft linear macromoleules with finite chain length, Polymer 2 (1961) 257–264. [28] J. Remmelgas, L. Gary Leal, N.V. Orr, T. Sridhar, Viscous and elastic stresses in extensional rheometry, J. Non-Newtonian Fluid Mech. 76 (1998) 111–135. [29] P.S. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 208 (1996) 255–288.
Birefringence and stress growth in uniaxial extension of polymer solutions T. Sridhar a,∗ , D.A. Nguyen a , G.G. Fuller b a
Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
b
Received 6 August 1999; accepted 9 August 1999
Abstract Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule. Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted. Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations. ©2000 Elsevier Science B.V. All rights reserved. Keywords: Birefringence; Stress
1. Introduction The ability of polymer chains in solution to become extended and aligned in a flow field is primarily responsible for the complex spectrum of properties exhibited by these solutions. Flow fields having a ∗
Corresponding author. Tel.: +61-3-9905-3427; fax: +61-3-9905-5686.
0377-0257/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 9 ) 0 0 0 8 0 - 4
300
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
significant extensional component facilitate such molecular unraveling. The dynamics of the transformation of polymer chains from a random coil to an extended structure remains an outstanding problem in rheology. The problem has motivated several avenues of research, both theoretical and experimental, over the last three decades. Recent progress has been rapid primarily due to several new experimental techniques being developed. Amongst these the development of birefringence as a sensitive measure of segmental orientation [1] and the measurement of extensional stresses by the filament stretching rheometer [2,3] are pertinent to the current work. In addition, improvements in our ability to simulate the stochastic processes governing chain configuration [4] have informed our ideas. Using such techniques, bead-rod [5] and bead-spring models [6] of a polymer chain have been developed. These studies have exposed an amazing diversity of chain conformation resulting from the competition between initial conformation, Brownian motion and the effect of the flow field [6]. Optical birefringence is sensitive to local orientation of the polymer chain. The polymer stress, however, responds to the overall deformed length of the polymer chain. The stress optical law provides a convenient relationship between birefringence and stress. Significant improvements in the measurement of optical birefringence [7] now enable even transient flows to be conveniently studied. The noninvasive nature of the measurement has resulted in a vast literature on the use of birefringence to examine flow problems. While the stress optical rule has been verified in shear flows, there is some evidence that in strong flows the anticipated chain extension may lead to a break down of this rule. Keller and co-workers have pioneered the use of optical techniques to study extensional flow [8–11]. They used opposed nozzles, four-roll mills and cross-slot devices to create an extensional flow field and used birefringence to probe the degree of chain orientation. In these devices the flow field is non-uniform but the presence of a stagnation point permits molecules to achieve a large strain. The birefringence is highly localised. The existence of a coil-stretch transition at a critical strain rate equal to the universe of the longest chain relaxation time was demonstrated. These studies were concerned with measuring the conformational relaxation time and elaborating the various birefringence structures observed. The relationship between such a relaxation time and molecular weight and solvent quality was explored using simple models of the polymer chain. Fuller and Leal [12] and Dunlop and Leal [13] studied dilute polymer solutions in two-and four-roll mills. These devices permit the flow field to be varied from simple shear to a purely two-dimensional extensional flow field. The presence of the polymer modifies the flow field and except for very dilute solutions, the changes in polymer conformation and the resultant changes in flow field are intricately linked. Dunlop et al. [14] and Geoffrey and Leal [15] have used flow birefringence to indicate changes in conformation and light scattering to directly measure changes in velocity gradient. These studies show significant inhibition of chain extension even in dilute solutions. One major drawback with most of these studies is that the polymer stress is not directly measured. There are very few reported instances where both stress and birefringence have been simultaneously measured using polymer solutions in extensional flows. Talbot and Goddard [16,17] measured birefringence and stress in a semi dilute separan solution using a fibre spinning device. The birefringence was measured at various positions along the spinline and hence simultaneous values of birefringence and stress for a fluid element as it flows down the spinline was obtained. This work documented the breakdown of the stress optic coefficient when the stress exceeded 2 kPa, which was the lowest recorded stress. Cathey and Fuller [18] performed similar measurements in an opposing jet device and explored the effects of solvent quality and strain rate. The birefringence was shown to saturate at high strain rate, while the extensional viscosity exhibited a maximum with strain rate. The latter finding was attributed to the decrease in fluid residence time in the flow field as the strain rate increased.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
301
Both these devices are subject to uncertainties in the flow field. In the fibre spinning device, the initial conditions of a fluid element is not known and is determined by the shear flow in the tubes leading to the spinline. In the opposing jet device, the fluid residence time varies and hence an accurate measure of the strain is not available [19]. Furthermore these devices do not permit a study of the effect of strain and the development of stress and birefringence with time from a stress free state. Partly this arises from the difficulty of creating well-defined extensional flows in the laboratory and measuring the resultant stress. The recent development of the filament stretching rheometer [2,3,20] has effectively overcome this difficulty and it is now possible to measure the stress in a well-defined extensional flow field. Simultaneously, recent developments in optical rheometry permit the measurement of birefringence in time dependent flows [1,7]. Doyle et al. [5] reported simultaneous stress and birefringence measurements. In order to avoid scattering at the air-filament interface, the experiments were performed in a plateau tank with the filament immersed in a fluid of similar density. There are significant experimental difficulties associated with such a procedure including the limitation it places on the maximum strain that can be achieved. This implies that it is not easy to achieve steady states in stress and birefringence. Doyle et al. [5] report a maximum stress of about 40 kPa at a Weissenberg number of 2.84, which corresponds to a strain of around 3.5. Under these conditions several other experimental problems could arise. The flow field during the early stages is affected by the shear flow near the end plates [21] and the measured stresses are dominated by the solvent contribution. In spite of these difficulties, this work captured some of the intricacies of the dynamics of chain unraveling. In particular the stress and birefringence in start up and relaxation exhibited a hysteresis. The stress at the same average level of chain deformation (as measured by birefringence) could assume different values depending on the history of deformation. This finding was later explained by Doyle et al. [5] as arising from the different configurations obtained during stretching and relaxation. In this work, optical rheometry on the filament stretching rheometer is used to examine both steady and time dependent strain rates. We perform the experiments in air and are able to reach large strains. A technique developed by Talbott and Goddard [16] is adapted to mask out light, which passes around the shrinking filament.
2. Experimental procedure The fluid sample studied in this work consists of 0.05% by weight of monodisperse polystyrene of 2 million molecular weight dissolved in piccolastic (a low molecular weight polystyrene). The solution was prepared by Professor S. Muller. The shear viscosity of the solution at 21.5◦ C is 87.5 Pa s and the dominant Zimm relaxation time was estimated to be 8.4 s. Details of the filament stretching rheometer have been published [22]. A ‘master plot’ technique [23], is used to ensure that any desired strain rate or strain rate history can be imposed on the sample. All experiments were carried out at a constant temperature of 21.5 ± 0.5◦ C. The flow birefringence of the sample was measured using the optical train shown in Fig. 1(a). A polarised laser light with a wavelength of 632 nm is generated by a helium–neon (He–Ne) laser and is deflected at 90◦ by a simple right angle prism. The light then passes through a linear polariser (P) and a photoelastic modulator (PEM). The polariser is fixed at an angle of 45◦ relative to the photoelastic modulator. Before passing through the filament, the light is transmitted through a quarter wave plate (Q), which is oriented parallel to the linear polariser. The light then passes through a lens, a circular polariser
302
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 1. (a) Optical train used for flow birefringence experiments; (b) simple function of the lens and the mask on laser light.
(CP) and the detector. Two lock in amplifiers are used to measure the output at the principal frequency and at the first harmonic. The purpose of the lens and the mask (a cylindrical rod, painted black, with a diameter of 1.7 mm) is to prevent part of the laser light that passes around the filament from reaching the detector. Talbott [17] has given a simple analysis of the action of the lens and the mask as shown in Fig. 1(b). The lens is located such that parts of the laser light passing around the filament remain collimated and are brought to focus at the focal point of the lens where the mask is located. Light rays passing through the filament, which acts as a cylindrical lens, is focused at nearly the front focal point of the lens. Hence these light rays are collimated by the lens, and pass through the circular polariser and then to the detector. The analysis by Talbott [17] shows that the error in assuming that the path length is equal to the filament diameter is negligible. The signal from the detector was then fed into a signal conditioner, which produced a low pass filter output and the intensity of the signal. The latter signal was sent to two lock-in amplifiers. The low pass filter signal and the two signals from the lock-in amplifiers were collected by the PC via the DAC card. Further details of the procedure and the data analysis are available in Fuller [1]. To increase the birefringence signal and make it convenient to focus the light on the filament, an initial sample size of 6 mm in diameter and 1.7 mm in length was used. One disadvantage of using such a small initial aspect ratio is that the initial flow field is dominated by shear near the end plates. The curvature of the filament also causes problems in focussing the optics and does lead to rather unreliable results at small time. A diameter-measuring device (ZUMBACH ODAC) measured the mid-point filament diameter
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
303
Fig. 2. Flow birefringences and tensile stresses of sample SM1 at ε = 4.9 s−1 for two identical runs.
while a force transducer measured the force on the end plates. The signals from the diameter measuring device and force transducer were sent to a PC via a digital-to-analog converter (DAC) card and several signal conditioners are incorporated to improve the quality of the data collected. 3. Results and discussion 3.1. Steady strain rates Fig. 2 shows the transient growth of birefringence and stress at a constant strain rate. At small time the stress and birefringence are dominated by the solvent contribution. At 0.4 s corresponding to a strain of 2 units, the polymer chain begins to orient with the flow and unravel and as a consequence both the stress and birefringence increase. Two identical experiments are shown in Fig. 2 demonstrating good reproducibility.
304
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 3. Plot of Trouton ratio of sample SM1 for various strain rates (2−10 s−1 ) against strain at a temperature of 21.5◦ C.
Fig. 3 shows that the Trouton ratio depends mainly on the strain and the effect of strain rate is confined to a region near the steady state. The behaviour of this fluid is typical of similar Boger fluids [3,21,23,24] The steady state extensional viscosity decreases with increasing strain rate and Gupta et al. [23] have suggested several possible reasons for such a behaviour. The stress saturates after about 5 strain units and the saturation Trouton ratio is around 800 at low strain rates. On the assumption that the polymer chain is in a fully extended configuration, one can estimate the Trouton ratio using Batchelor’s formula for rigid rods in suspension [25]. Tr =
π ηs cNA l 3 3 ηo MW ln(π/φ)
For polystyrene the monomer length is 2.5 × 10−8 cm [25], hence the length of a fully extended polymer is l = 4.5 × 10−4 cm. The equivalent diameter is 6.3 × 10−8 cm. φ is the volume fraction of polymer, c is the polymer concentration in g/ml. The equation predicts a Tr of 1500 whereas the measured value is around 800. This suggests that under these conditions the chain is not in the fully extended configuration.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
305
Fig. 4. Plot of flow birefringence of sample SM1 for various strain rates (2–10 s−1 ) against strain at a temperature of 21.5◦ C.
It is possible that the chain has attained a steady configuration but further extension is inhibited due to the formation of folds or entanglements [25]. Fig. 4 shows the behaviour of the solution birefringence at different strain rates. At low strain rates the birefringence measurements are influenced by two factors. One is the low birefringence value under these conditions. The other more important influence is the strong curvature of the filament during the initial period. The experimental difficulties of getting an adequate signal due to surface scattering and the uncertainty in the diameter leads to a poor resolution of the birefringence. At larger strain the filament assumes a more cylindrical shape and the birefringence is also much larger. Above a strain of 2, the birefringence increases rapidly. A saturation of the birefringence is observed above a strain of 5. The saturation birefringence appears to be independent of strain rate whereas the birefringence at intermediate strain depends on strain rate. The data in Fig. 4 reflect the birefringence of both the polymer and the solvent. In order to remove the effect of the solvent, the birefringence of the polymer chain was calculated by subtracting the solvent
306
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 5. Polymer flow birefringence for various strain rates.
birefringence. The solvent birefringence was estimated from the solvent stress (3 ηS ε˙ ) and the stress optic coefficient for the solvent. There is sufficient evidence in the literature [26] that these oligomeric solvents obey the stress optic rule. Fig. 5 shows the polymer contribution to the birefringence as a function of the strain. Surprisingly, the strain rate has a minor influence on the growth of the birefringence. In Fig. 3, we have already shown that the Trouton ratio is independent of strain rate and is predominantly dependent on strain. Hence Figs. 3 and 5 suggest a relationship between Trouton ratio and polymer birefringence. This directly contradicts the stress optical rule, which postulates a relationship between birefringence and stress. Note that the solvent contribution to the stress has been ignored because it is insignificant at strains larger than 2. The saturation birefringence of 4 × 10−5 can be compared with the birefringence for a fully extended polymer chain [27] 2 2π np n2o + 2 1n = N (α1 − α2 ) , n 3 1
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
307
where, np is the number density of polymer in solution, n and no are the refractive indices of solution and solvent, respectively, N is the number of Kuhn segments per molecule, and α 1 and α 2 are the polarizability of a subunit parallel and perpendicular to the chain axis. For a solution of high molecular weight PS in low molecular weight PS, no = n = 1.6 and (α 1 − α 2 ) = −145 × 10−25 cm3 . The number of Kuhn segments is estimated assuming that there are 15 C–C bonds per segment. Hence the calculated value of birefringence of a fully extended polystyrene chain for SM1 solution is −1n1 = 4.8 × 10−5 . This estimate is subject to significant uncertainities. Hence the birefringence saturates at about the expected value for full extension. We have shown previously that the Trouton ratio saturates at 50% of the expected value for a fully extended chain. Theoretical simulations are examined to further probe this issue. Larson et al. [6] have examined the chain extension using bead spring models. A variety of chain configurations such as dumbbells, folds, kinks, coils and half dumbbells are shown to arise. The dynamics of each of these configurations is quite distinct. Folded configurations show a rapid extension with strain, but appear to ‘hesitate’ at about 50% extension before extending further. These folded configurations
Fig. 6. Two levels of the steady state of Trouton ratio of SM1 for a strain rate of 0.5 s−1 at a temperature of 19.5◦ C.
308
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 7. Plot of stress-optic coefficient against strain; data points: experiment with a strain rate of around 2 s−1 (Wi = 16.8), lines: bead rod simulations (Wi = 10.65).
can require 8 to 9 strain units before reaching full extension. Dumbbell shaped configurations reach full extension at a lower strain. On an extensional viscosity versus strain plot this translates into a plateau corresponding to a strain of about 5 units and a slow growth from there up to a steady state corresponding to full extension. Experiments at such high strains are dogged by experimental difficulties. The measured forces become very small and inertia becomes increasingly important. The most important issue is filament breakage and the difficulty of maintaining a constant strain rate. For the fluid used in this work, it has been possible to overcome these difficulties and obtain data up to a strain of 7.8. Fig. 6 shows that the Trouton ratio increases rapidly up to a plateau of around 800 before increasing slowly up to a value of 1500. This shows similarity with the behaviour of the folded structure discussed above and the final steady state is also consistent with the estimates of the Batchelor theory. Birefringence depends on the orientation at the Kuhn length scale and is not expected to be able to differentiate between folded and fully extended structures. The saturation of birefringence under these conditions is therefore not surprising. From the simultaneous measurements of stress and birefringence, the stress optic coefficient can be calculated. Fig. 7 shows that the stress optical coefficient decreases rapidly with increasing strain. This
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
309
Fig. 8. Comparison between shear stress relaxation with extensional stress birefringence and extensional relaxation.
dramatic breakdown of the stress optic rule is consistent with the bead rod simulations of Doyle et al. [5]. Fig. 7 also shows the simulations of Doyle et al. [5] for two different values of N, the number of links in the chain. While the values of N used do not correspond to the polymer used in this work, the agreement is qualitatively good. The stress optic law is premised on a linear force law for the spring. The fact that the stress optic coefficient breaks down at even low strains suggests that, even under these conditions, the stress arises predominantly from portions of the chain that are extended beyond the linear region. The filament stretching rheometer makes it possible to study relaxation of stress and birefringence. Orr and Sridhar [24] have examined the relaxation of stresses and showed that the initial relaxation is very rapid and is followed by a slower relaxation. This finding has generated significant interest and several theoretical studies address this issue [28]. In this work simultaneous stress and birefringence measurements were made during relaxation after extensional flow for a certain strain. The data are shown in Fig. 8. The relaxation of birefringence is similar at the two strains and is significantly slower than the rate at which the tensile stress relaxes. The relaxation of shear stress, at a shear rate of 8.5 s−1 ,
310
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 9. Extensional flow start-up and subsequence relaxation for stress versus birefringence of sample SM1; data points: asterisk: Wi = 41.7, circle: Wi = 16.8, square [1]: Wi = 2.84, line: [1]: conformation dependent FENE model with ζ max /ζ o = 8 (Wi = 2.84). −1 obtained in a Rheometrics rheometer is also √ shown on Fig. 8. The shear rate of 8.5 s is chosen so that the extensional strain rate is ε˙ = γ˙ / 3. Under these conditions of equivalent deformation, the relaxation of birefringence and shear stress is similar. The relaxation data for birefringence in shear and extension are in qualitative agreement with the bead-rod simulations of Doyle et al. [5]. The simulations also demonstrate that the evolution of stress and birefringence during start-up and relaxation exhibits a pronounced hysteresis. This hysteresis arises due to the skewed distribution of dumbbell lengths in a FENE model. The stress results predominantly from dumbbells near their full extension due to the highly non-linear restoring force. On the other hand, the contribution to birefringence scales linearly with length. In a bead-rod model, the simulations show that the chain configuration during start-up and relaxation is significantly different. During relaxation, the chain essentially relaxes from the free ends. Fig. 9 shows a plot of stress versus birefringence for two different strain rates and demonstrates a pronounced hysteresis. The stress is much larger, for a given deformation, during start-up than during relaxation. Doyle et al.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
311
Fig. 10. Stress growth of sample SM1 during extensional flow start-up then stop and then re-start the flow. The flow stops at point A for 0.2 s then re-starts at point B; after stopping for 2 s the flow re-starts at point C. Position D shows the same level of stress as that at point B and point E has the same stress as point C.
[5] first documented such a hysteresis and their data is also shown in Fig. 9. Fig. 9 shows that the stress-birefringence relationship during stretching depends on strain rate. During relaxation, the stress drops rapidly whereas the birefringence changes slowly. After this initial period, the stress birefringence profile is independent of strain rate. The simulations of Doyle et al. [5] and the earlier study of Grassia and Hinch [29] suggests an universal curve for a fully extended chain onto which can be superimposed the relaxation of all other chains by a shift in time. The process of relaxation of these chains is similar, in that they all relax from the chain ends in the so-called stem and flower configuration. As a result, a universal curve for stress birefringence during relaxation is not surprising and relaxation of partially extended chains eventually follows this curve. The substantial differences in the chain configuration during start-up and relaxation have significant ramifications for the modelling of time dependent flows. Orr and Sridhar [22] demonstrated one consequence of this hysteresis. In a filament stretching rheometer, filaments at the same stress levels can be
312
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
Fig. 11. Flow birefringence of sample SM1 during extensional flow start-up then stop and then re-start the flow. The flow stops at point A for 0.2 s then re-starts at point B; after stopping for 2 s the flow re-starts at point C. Position D shows the same level birefringence as that at position B and point E has the same birefringence as point C.
created by a combination of stretching and relaxation for different periods. The response of such filaments to further deformation was shown to be significantly different. The difference arises due to the different chain conformations obtained — open chains respond rapidly to the imposed deformation whereas compact chains respond more slowly. These findings cast doubt on the validity of pre-averaging procedures adopted for development of closed form constitutive equations. We report here similar experiments where both stress and birefringence are measured. Fig. 10 shows the stress in an experiment where the fluid was stretched at a constant strain rate for a specified period, followed by relaxation for a given time followed by further stretching at the same strain rate. During relaxation the stress decays by nearly two orders of magnitude. Note that the stress growth on re-imposition of flow is very rapid. The chain conformation at points A and B shown on the Fig. 10 are not very dissimilar and the stress at points B and D are identical. Yet the rate of stress growth at point B is much larger than at point D and is, in fact similar to the rate of change of stress at point A [22].
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
313
Fig. 12. Flow birefringence of sample SM1 during extensional flow start-up then stop and then re-start the flow. Data points have been shifted to the left for comparison.
Concurrent birefringence measurements are shown on Fig. 11. Note the good reproducibility of the three experiments during the initial period prior to the relaxation. The birefringence decays by about a factor of 5 during the 2 s relaxation (point A to point C). During this period the stress has decayed by two orders of magnitude (Fig. 10). The birefringence at points B and D and at points C and E are similar. The rate of increase of birefringence at points B and D and at points C and E are also similar. This is more clearly demonstrated in Fig. 12, where the curves in Fig. 11 have been shifted left so that the points B and D and the points C and E coincide. The three experiments superimpose suggesting that at constant strain rate the current value of birefringence determine the rate of change of birefringence. In time dependent extensional flows, which can be broken down into a series of extensions and relaxation, the current state of polymer chain and its ability to respond to imposed deformation depends not on the stress but on the chain configuration. As such the history of deformation is contained within the chain configuration. Attempts to capture this information by just the second moment of the end to end distribution function is likely to lead to problems in predicting time dependent flows.
314
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
4. Conclusions This work presents simultaneous stress and birefringence measurements using the filament stretching rheometer. The data shows that both stress and birefringence saturate at a strain of 5. The saturation values for stress and birefringence are lower than that inferred on the assumption of complete unravelling of the polymer chain. It is speculated that the formation of folded structures is responsible for this behaviour. Evidence is presented that suggests these folded structures can be further unraveled at higher strain. The stress at a strain of 7.8 units is in agreement with estimates from the Batchelor theory. Such folded structures have been predicted by Brownian simulations and have been observed for DNA solutions [6]. The relaxation of birefringence is shown to be much slower than the relaxation of stress leading to the stress- birefringence hysteresis and a breakdown of the stress optical rule. In time dependent flows simulated by a sequence of stretching and relaxation and further stretching, the dynamics of stress and relaxation are shown to be different. Stress growth is seen to depend on both the configuration and the imposed strain rate. The current state of stress does not by itself uniquely represent the current configuration. On the other hand, the dynamics of the growth of birefringence depends only on the current value of birefringence and the imposed strain rate.
Acknowledgements This work was supported by a grant from the Australian Research Council. It is dedicated to Professor D.V. Boger in acknowledgement of his many contributions to rheology. References [1] G.G.Fuller, Optical rheometry of complex fluids, Oxford University Press, London, 1995. [2] T. Sridhar, V. Tirtaatmadja, D.A. Nguyen, R.K. Gupta, Measurement of extensional viscosity of polymer solutions, J. Non-Newtononian Fluid Mech. 40 (1991) 271–280. [3] V. Tirtaatmadja, T. Sridhar, A filament stretching device for measurement of extensional viscosity, J. Rheol. 37 (1993) 1081–1102. [4] H.C. Ottinger, Stochastic Processes in Polymeric Fluids, Springer, New York, 1996. [5] P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110. [6] R.G. Larson, H. Hua, D.E. Smith, S. Chu, Brownian dynamics simulation of a DNA molecule in an extensional flow field, J. Rheol. 43 (1999) 267–304. [7] P.L. Frattini, G.G. Fuller, A note on phase-modulated flow birefringence, J. Rheol. 28 (1984) 61–70. [8] A.A. Keller, A.J. Muller, J.A. Odell, Entanglements in semi-dilute solutions as revealed by elongational flow studies, Progr. Colloid Polym. Sci. 75 (1987) 179–200. [9] A.J. Muller, J.A. Odell, A. Keller, Elongational flow and rheology of monodisperse polymer in solution, J. Non-Newtonian Fluid Mech. 30 (1988) 99–118. [10] D.P. Pope, A. Keller, Alignment of macromolecules in solution by elongational flow: a study of the effect of pure shear in a four-roll mill, Colloid Polym. Sci. 255 (1977) 633–643. [11] D.P. Pope, A. Keller, A study of chain extending effect of elongational flow in polymer solutions, Colloid Polym. Sci. 256 (1978) 751–756. [12] G.G. Fuller, L.G. Leal, Flow birefringence of dilute polymer solutions in two-dimensional flows, Rheol. Acta 19 (1980) 580–600.
T. Sridhar et al. / J. Non-Newtonian Fluid Mech. 90 (2000) 299–315
315
[13] P.N. Dunlap, L.G. Leal, Dilute polystyrene solutions in extensional flow: birefringence and flow modification, J. Non-Newtonian Fluid Mech. 23 (1987) 5–48. [14] P.N. Dunlap, C.H. Wang, L.G. Leal, An experimental study of dilute polyelectrolyte solutions in strong flows, J. Polym. Sci. Part B: Polym. Phys. 25 (1987) 2211–2238. [15] E. Geffroy, L.G. Leal, Flow birefringence studies in transient flows of a two-roll mill for the test fluid M1, J. Non-Newtonian Fluid Mech. 35 (1990) 361–400. [16] W.H. Talbott, J.D. Goddard, Streaming birefringence in extensional flows of polymer solutions, Rheol. Acta 18 (1979) 505–517. [17] W.H.Talbott, Ph.D. thesis, University of Michigan , Ann Arbor, 1978. [18] C.A. Cathey, G.G. Fuller, The optical and mechanical response of flexible polymer solutions to extensional flow, J. Non-Newtonian Fluid Mech. 34 (1990) 63–88. [19] R.K. Gupta, T. Sridhar, in: A.A. Collyer, D.W. Clegg (Eds), Elongational rheometers in Rheological Mesurements, Elsevier, New York, 1988. [20] S.H. Spiegelberg, G.H. McKinley, Stress relaxation and elastic decohesion of viscoelastic polymer solutions in extensional flow, J. Non-Newtonian Fluid Mech. 67 (1996) 49–76. [21] S.H. Spiegelberg, D.C. Ables, G.H. McKinley, The role of end-effects on measurements of extensional viscosity in filament stretching rheometers, J. Non-Newtonian Fluid Mech. 64 (1996) 229–267. [22] N. Orr, T. Sridhar, Probing the dynamics of polymer solutions in extensional flow using step strain rate experiments, J. Non-Newtonian Fluid Mech. 82 (1999) 203–232. [23] R.K. Gupta, D.A. Nguyen, T. Sridhar, Extensional viscosity of dilute polystyrene solutions: effect of concentration and molecular weight, 1999, in press. [24] N. Orr, T. Sridhar, Stress relaxation in uniaxial extension, J. Non-Newtonian Fluid Mech. 67 (1996) 77–103. [25] D.F. James, T. Sridhar, Molecular conformation during steady-state measurements of extensional viscosity, J. Rheol. 39 (1995) 713–724. [26] R.C. Ng, L.G. Leal, Concentration effects on birefringence and flow modification of semi-dilute polymer solutions in extensional flows, J. Rheol. 37 (1993) 443–468. [27] A. Peterlin, Streaming birefringence of soft linear macromoleules with finite chain length, Polymer 2 (1961) 257–264. [28] J. Remmelgas, L. Gary Leal, N.V. Orr, T. Sridhar, Viscous and elastic stresses in extensional rheometry, J. Non-Newtonian Fluid Mech. 76 (1998) 111–135. [29] P.S. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 208 (1996) 255–288.