Rheologica Acta

, Volume 51, Issue 5, pp 385–394

Stress and neutron scattering measurements on linear polymer melts undergoing steady elongational flow

Authors

    • Department of Chemical and Biochemical EngineeringTechnical University of Denmark
  • Kell Mortensen
    • Department of Basic Sciences and Environment, Faculty of Life SciencesUniversity of Copenhagen
  • Anders Bach
    • Department of Chemical and Biochemical EngineeringTechnical University of Denmark
  • Kristoffer Almdal
    • Department of NanotechnologyTechnical University of Denmark
  • Henrik Koblitz Rasmussen
    • Department of Mechanical EngineeringTechnical University of Denmark
  • Wim Pyckhout-Hintzen
    • Jülich Centre for Neutron Science-1 & Institute for Complex SystemsForschungszentrum Jülich
Original Contribution

DOI: 10.1007/s00397-012-0622-1

Cite this article as:
Hassager, O., Mortensen, K., Bach, A. et al. Rheol Acta (2012) 51: 385. doi:10.1007/s00397-012-0622-1

Abstract

We use small-angle neutron scattering to measure the molecular stretching in polystyrene melts undergoing steady elongational flow at large stretch rates. The radius of gyration of the central segment of a partly deuterated polystyrene molecule is, in the stretching direction, increasing with the steady stretch rate to a power of about 0.25. This value is about half of the exponent observed for the increase in stress value σ, in agreement with Gaussian behavior. Thus, finite chain extensibility does not seem to play an important role in the strongly non-linear extensional stress behavior exhibited by the linear polystyrene melt.

Keywords

ScatteringPolymer meltUniaxial extensionPolystyrene

Introduction

Elongational flow has long been and continues to be a topic of significant rheological interest (Petrie 1979; McKinley and Sridhar 2002). For dilute polymer solutions, it is known from visual observations (Perkins et al. 1997) that the polymer molecules may become fully extended in strong elongational flows. Our understanding of the fully extended state continues to improve as a result of accurate theoretical modeling (Underhill and Doyle 2007). Moreover, the transition from near equilibrium to the fully stretched state is a rather sharp transition that occurs at a specific elongation rate (De Gennes 1974).

For entangled polymer melts, our understanding of the molecular behavior in strong elongational flow is less satisfactory. Simple continuum models such as the Oldroyd B model or the Lodge rubberlike liquid (Bird et al. 1987a) predict a stress singularity at a critical elongation rate. This singularity may be suppressed by invoking finite extensibility in the associated structural models (Bird et al. 1987b), but for most models, finite extensibility merely replaces the singularity with a sharp transition to an asymptotic behavior that may be associated with a fully stretched state.

While one would not expect such simple models such as the Oldroyd B model to provide more than just a first-order approximation for non-linear dynamics, it is more serious that the situation is not much better for most current structural models. This is true in particular for the tube theory framework for modeling the dynamics of entangled polymer systems. Going back to the pioneering development by Doi and Edwards (1986), the basic model predicts a limiting stress in strong extensional flow. The molecular origin of the limiting stress is the assumption of instantaneous chain retraction so that the stress is due to orientation only. Perfect orientation of the tubes but with no chain stretching gives a stress that is five times the plateau modulus. The corresponding steady elongational viscosity then decreases with elongation rate \(\dot{\epsilon}\) as \(\dot{\epsilon}^{-1}\). However, it was recognized early that chain stretching will occur at deformation rates faster than the inverse Rouse time (Marrucci and Grizzutti 1988; Pearson et al. 1989). The concept of chain stretching was consequently included in several models based on the Doi and Edwards framework (Fang et al. 2000; Mead et al. 1998; Schieber et al. 2003). The result of including chain stretching alone is to introduce a stress singularity at an elongation rate of the order the inverse Rouse time. This singulary may again be relieved by invoking finite extensibility. The resulting models exhibit four flow regimes: a linear viscoelastic regime dominated by reptation wherein the elongational viscosity is constant, a second regime wherein the elongational viscosity decreases as \(\dot{\epsilon}^{-1}\), and a third regime with a sharp increase in the elongational viscosity to an asymptotic behavior in the fourth regime. There was at the time no data for entangled monodisperse polymer melts to compare with (Mead et al. 1998). However, subsequent data on steady elongational viscosity of narrow molar mass distribution (NMMD) polystyrene (Bach et al. 2003a; Luap et al. 2005) demonstrated a rather different nonlinear behavior with an elongational viscosity scaling approximately as \(\dot{\epsilon}^{-0.5}\). The conflict was recognized immediately by Marrucci and coworkers who introduced the concept of interchain pressure in polymer dynamics (Marrucci and Ianniruberto 2004) whereby chain stretching is balanced by lateral forces between the chain and the tube wall. In this way, they were able to explain the \(\dot{\epsilon}^{-0.5}\) scaling. Moreover, they predicted a maximum in the steady elongational viscosity for smaller molar masses that was subsequently observed (Nielsen et al. 2006). Wagner and coworkers introduced the interchain pressure concept into their molecular stress function theory whereby they obtained a constitutive equation with the correct \(\dot{\epsilon}^{-0.5}\) scaling (Wagner et al. 2005). Later refinements have allowed quantitative predictions of available non-linear rheological measurements from linear viscoelastic data alone (Wagner and Rolón-Garrido 2009, 2010). The interchain pressure concept has also been introduced into the Rolie–Poly (Likhtman and Graham 2003) constitutive equation, whereby likewise a quantitative description of the elongational viscosity is obtained (Kabanemi and Hétu 2009). While there are currently two tube-based models capable of describing the non-linear stress behavior in elongational flow, the situation is not entirely satisfactory for two reasons. First the number of modifications needed on the basic model is in itself worrying. Second the very fact that two different tube-based models both describe the stress data quantitatively does not leave a clear picture.

To gain insight into the physics of strong elongational flows from the experimental side, Luap et al. (2005) measured birefringence together with the stress. For their two NMMD polystyrene melts, they found the stress optical rule fulfilled up to a critical stress of order 2.7 MPa or about ten times the plateau modulus. The fact that the \(\dot{\epsilon}^{-0.5}\) scaling sets in when the steady stress is of order the plateau modulus therefore suggests that the physics behind the \(\dot{\epsilon}^{-0.5}\) scaling is not related to finite extensibility of the chains. However, their measurements also show that the \(\dot{\epsilon}^{-0.5}\) scaling extends unchanged well beyond the onset of chain stretch in the non-Gaussian regime (Luap et al. 2005) evidenced by failure of the stress-optical rule. In other words, finite extensibility does come into play, but it is not the prime reason for the observed scaling nor does it lead to a drastic transition to an asymptotic behavior.

The purpose of this study is to gain further insight into the physics of strong elongational flows by application of small-angle neutron scattering (SANS). In pioneering work (Boué et al. 1982, 1983) on the application of SANS to understand dynamics in stretched polystyrene melts, the melts were rapidly extended by a stretch ratio of 3 at temperatures not far above the glass transition. The subsequent relaxation of the radius of gyration in the direction transverse to the stretching direction was subsequently followed as function of time from a sequence of quenched samples. The early reduction in the transverse direction predicted by the basic tube theory was not observed possibly due to Rouse relaxation during the quenching process. The minimum was, however, clearly observed later by Blanchard et al. (2005) for PI utilizing an in situ SANS device.

SANS measurements have also been performed on NMMD polystyrene samples quenched from steady elongational flow (Muller et al. 1990). While the measurements were performed in the linear viscoelastic regime and therefore do not cast light on the \(\dot{\epsilon}^{-0.5}\) scaling, the study does give credence to a straightforward analysis in terms of radii of gyration in the parallel and transverse direction, respectively. More recently, neutron scattering has been reported on the steady flow in a 4:1 planar contraction/expansion of NMMD polystyrene samples (Bent et al. 2003). However, while the inhomogeneous flow is steady in a Eulerian sense, it is not steady in a Lagrangian sense so it is not straightforward to compare with steady viscosity data. In fact while scattering from deformed polymer melts has been analyzed in considerable detail both experimentally and theoretically (Read 2004), we do not know of SANS measurements that may cast light on the molecular behavior behind the \(\dot{\epsilon}^{-0.5}\) scaling.

Experimental

Synthesis

To eliminate ambiguities due to polydispersity, we used anionic polymerization (Ndoni et al. 1995) to prepare polymer samples of narrow molar mass distribution. The key polymer is a symmetric triblock copolymer PS(H)–PS(D)–PS(H). The middle block (PS(D)) is polystyrene with hydrogen substituted by deuterium, while the two identical end-blocks are ordinary polystyrene (PS(H)). This polymer is mixed into a matrix of ordinary polystyrene of the same molar mass. In this way, we obtain a solution of deuterium labeled middle segments in a sea of ordinary polystyrene. We have labeled only the middle block with deuterium for two reasons. Firstly, it is anticipated that the molecular relaxation of stretch that inevitably will occur during quenching primarily will take place near the ends of the chain. Secondly, if we labeled the entire chain, the molecular length scale in the direction parallel to the macroscopic elongation would be too large to be measured accurately in the SANS facility used.

Chromatography

The molar masses of the polystyrene samples were determined by size exclusion chromatography (SEC) with toluene as the eluent and employing a column set consisting of a 5pm guard column and two 300 ×8 mm2 columns (PLgel Mixed C and Mixed D). The system is equipped with a triple detector system (a combined Viscotek model 200 differential refractive index (DRI) and differential viscosity detector plus a Viscotek model LD 600 right angle laser light scattering detector (RALLS)). On the basis of calibration with narrow molar mass polystyrene standards, the values of Mw and Mw/Mn were determined to be 145 kg/mol and 1.13 for the triblock-copolymer and 150 kg/mol and 1.07 for the homopolymer. Note that due to the calibration technique, these numbers refer to molar masses for non-deuterium substituted molecules. Thus, the molar volume for the triblock and homopolymer are the same within experimental error. The molar mass numbers given in the main text are to be interpreted in the same manner. Molar masses based on direct use of the DRI and RALLS signals were within error identical to the values obtained by calibration. The presence in the SEC setup of the RALLS detector enhances sensitivity toward high molar mass impurities. No such impurities were detected. The final molecular characteristics are summarized in Table 1.
Table 1

Components in test material

Components

Block copolymer

Homopolymer

PS(H)–PS(D)–PS(H)

PS(H)

Mw

145 kg/mol

150 kg/mol

Mw/Mn

1.13

1.07

The mass fraction of the deuterated middle block is 0.30 of the block copolymer, while the mass fraction of the copolymer in the mixture is 0.33 so the mixture contains 10% deuterium

Mechanical spectroscopy and filament stretching

We performed small amplitude oscillatory shear to determine the linear viscoelastic properties of the polystyrene melt. We used a plate–plate geometry on an AR2000 rheometer from TA Instruments. The measurements were performed at 150°C with the results as shown in Fig. 1.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-012-0622-1/MediaObjects/397_2012_622_Fig1_HTML.gif
Fig. 1

G and G for 150 kg/mol narrow molar mass distribution polystyrene at 150°C. Points represent data. Full red and green lines represent BSW fit with parameters G0 = 220 kPa, ne = 0.23, ng = 0.67, λm = 19 s, λc = 0.013 s. Blue lines represent the contributions to G(ω) from the Likhtman model for the Rouse dynamics (dashed-dotted line longitudinal mode relaxation, dashed line fast Rouse motion inside tube, full line sum of longitudinal modes and fast Rouse motion)

To establish a well-defined steady elongational flow, we use a filament stretching rheometer (McKinley and Sridhar 2002) equipped with a thermostat (Bach et al. 2003b). Prior to flow, the sample is allowed to equilibrate for a time longer than the estimated longest relaxation time. Then at times t > 0, the plates are separated from one another in such a fashion that the elongation rate at the plane of observation (the plane of symmetry) is kept constant at a given value \(\dot{\epsilon}\). The force at the end-plate is measured online, as is the filament diameter in the plane of observation. The plate motion is controlled to obtain a constant value of \(\dot{\epsilon}\). At time \(t = 3/\dot{\epsilon}\), the fluid in the plane of observation has undergone locally a Hencky strain of \(\epsilon = \dot{\epsilon}t = 3\). This means that the fluid in the plane of observation has been stretched by a factor e3 ≈ 20. The fluid in the plane of observation is the only part of the sample that experiences ideal elongational flow.

In Fig. 2, we show the development of the stress in the plane of observation as function of the imposed Hencky strain for elongation rates 0.001, 0.01, and 0.1 s − 1. It is seen that the stress increases monotonically reaching a constant value at a Hencky strain of about 2.5 for \(\dot{\epsilon} = 0.1~\mbox{s}^{-1}\) and somewhat earlier for \(\dot{\epsilon} = 0.001~\mbox{s}^{-1}\). We assume therefore that a steady flow situation has been established at Hencky strain 3. Also shown in Table 3 are the corresponding steady stresses divided by the modulus \(G_0 = 220~\mbox{kPa}\). It is seen that the steady stresses at \(\dot{\epsilon} = 0.001~\mbox{s}^{-1}\) are of order G0 while the steady stress at \(\dot{\epsilon} = 0.1~\mbox{s}^{-1}\) are about 30G0 indicating that the experimental window is well into the non-linear regime including stress levels for which Luap et al. found deviations from the stress optical rule.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-012-0622-1/MediaObjects/397_2012_622_Fig2_HTML.gif
Fig. 2

Axial stress as function of continuum stretch \(\epsilon = \dot{\epsilon}t\) in the plane of observation for elongational flow of polystyrene melt undergoing startup of elongational flow with constant elongation rates \(\dot{\epsilon}\). The stress appears to reach a steady value at ε ≈ 2.5

Quenching and scattering

Assuming that steady flow conditions are established around Hencky strain 2.5, a sequence of samples obtained by quenching the polystyrene bridge at Hencky strain 3 should contain molecular configurations representative of steady elongational flow. Thus, when the polymers in the plane of observation have reached Hencky strain ε = 3, the liquid bridges were quenched by a series of cooled nitrogen jets thereby providing samples for the SANS experiments. Due to the atactic nature of the anionically synthesized polystyrene, this polymer does not crystallize but undergoes a glass transition around 100°C whereby the molecular configurations are frozen. All cooling equipment was made of stainless steel. Consequently, the cooling equipment acted as a heat sink in the oven even when the gas flow was turned off. The influence on filament temperature was investigated by placing a thermocouple in a filament and measuring the temperature. When the oven temperature was set to 130.0°C, the measured filament temperature was 126.7±0.5°C. We use this temperature as the filament temperature for all elongational measurements, but the uncertainty in the temperature is a concern especially since the polymer time constants are highly dependent on the temperature.

The interpretation of the scattering results in terms of steady flow configurations hinges of course on the assumption that the polymer chains are frozen before they have time to relax. Hence, we need to know the time needed to quench the entire filaments below the glass transition temperature. Because of the high flow rate of the cooling gas, we estimate that the cooling rate of the sample is limited primarily by temperature gradients within the sample, whereas the surface of the filament obtains the temperature of the cooling gas immediately. The internal gradient is governed by a time constant \(\tau_k = \rho C_{\rm p} R^2/k\) where ρ, Cp, and k are the density, heat capacity, and thermal conductivity, respectively, and R is the radius of the cylindrical shaped sample during quenching. Using R = 1 mm and the physical constants in Table 2, we obtain τk = 3 s. With a gas temperature of 30°C, we find that a cooling time of 1 s is sufficient to cool the sample below the glass transition. In reality, the molecular motion is slowed considerably well before the glass transition is reached. Since this is more than two orders of magnitude smaller than the Rouse time, we feel confident that the quenched samples do indeed represent steady flow conditions.
Table 2

Physical properties of probe melt at experimental conditions (126.7°C)

ρCp

k

τw

η0

τR

2 106 J/m2 K

0.17 W/mK

970 s

70 MPa s

160 s

Density ρ, heat capacity Cp, longest relaxation time τw, zero shear-rate viscosity η0, and Likhtman Rouse time τR

The small-angle neutron scattering experiments were performed at the SANS-2 instrument, the Paul Scherrer Institute in Switzerland, using standard data treatment (Strunz et al. 2004). The solid PS samples were fixed to standard sample holders with tape and mounted in the beam so only the central part of the sample is exposed to neutrons (see Fig. 3). The SANS data shown in this report are all obtained with a sample-to-detector distance equal 5.7- and 6-m pin-hole collimation with 16-mm-diameter entrance hole and 5-mm exit hole next to the sample. The neutron wavelength was 4.6 Å with 10% resolution. Instrumental setting with 11.6-Å neutron wavelength was also applied, but did not give reliable data due to very low signal-to-noise ratio, and is accordingly not included in this report. Correspondingly, we do not include results obtained at smaller sample-to-detector distances, since these data do not contribute to the relevant length scales studied. The results are shown in Fig. 4 left column and summarized in Table 3. The scattering statistics reflects the sample volume scattered from: Samples stretched at the highest rate were pre-stretched more than the other samples and therefore ended up having smaller final diameter, lower volume and thus worse statistics.
https://static-content.springer.com/image/art%3A10.1007%2Fs00397-012-0622-1/MediaObjects/397_2012_622_Fig3_HTML.gif
Fig. 3

Schematic illustration of the SANS setup, showing the central part of the elongated PS sample exposed to the neutron beam

https://static-content.springer.com/image/art%3A10.1007%2Fs00397-012-0622-1/MediaObjects/397_2012_622_Fig4_HTML.gif
Fig. 4

Two-dimensional contour plots of neutron scattering data and fits as obtained from sample of PS–PSd–PS/PS mixture of polystyrene triblock copolymer mixed into a similar size polystyrene homopolymer. The data represent experiments under a equilibrium conditions and bd steady-state elongational flow at strain rates: b 0.001 s − 1, c 0.01 s − 1, and d 0.1 s − 1. Left column is the measured scattering intensity, middlecolumn is a 2D Debye fit, and the rightcolumn is the difference between the two first columns. The coordinates in each box are the scattering vectors \(q_x \in [-0.075~{\AA}^{-1},+0.075~{\AA}^{-1}]\), and \(q_z \in [-0.075~{\AA}^{-1},+0.075~{\AA}^{-1}]\)

Table 3

Elongational stress and SANS data for hot-stretched and quenched samples

Sample

\(\dot{\epsilon}\) (s − 1)

Wi

WiR

Rg ∥  (Å)

Rg ⊥ (Å)

Rm (Å)

λ ∥ 

λ ⊥ 

σ (kPa)

σ/G0

epsd 0001al3.0

0.001

1.0

0.16

66

41

48

1.4

0.89

210

0.95

epsd0001al5.7

0.001

1.0

0.16

77

43

52

1.6

0.93

210

0.95

epsd001bl3.0

0.01

10

1.6

147

32

53

3.2

0.69

1,380

6.3

epsd001bl5.7

0.01

10

1.6

155

31

53

3.4

0.67

1,380

6.3

epsd001cl3.0

0.01

10

1.6

149

33

54

3.3

0.72

1,370

6.2

epsd001cl5.7

0.01

10

1.6

157

32

54

3.4

0.69

1,370

6.2

epsd001d10l3.0

0.01

10

1.6

152

34

56

3.3

0.74

1,330

6.0

epsd001d10l5.7

0.01

10

1.6

147

35

56

3.2

0.76

1,330

6.0

epsd01cl3.0

0.1

100

16

228

23

49

4.9

0.50

5,700

26

epsd01cl5.7

0.1

100

16

259

24

53

5.6

0.52

5,700

26

epsd01a+b+cl5.7a

0.1

100

16

269

22

51

5.8

0.48

6,100

28

Here \(R_{\rm m}^3 = R_{{\rm g}\parallel}R_{{\rm g}\perp}^2\) while λ ∥  = Rg ∥ /Rg0 and λ ⊥  = Rg ⊥ /Rg0 with the effective \(R_{{\rm g}0}=46~\mbox{\AA}\). Also σ = σzz − σrr and G0 = 220 kPa. All samples were stretched to Hencky strain 3

aSANS data obtained by measuring on three samples simultaneously

Analysis and results

Polymer relaxation times and stresses

The linear viscoelastic moduli in Fig. 1 are modeled by a continuous spectrum of relaxation times 1990.
$$ H(\lambda) = n_{\rm e} G_0 \left[ \left( \frac{\lambda}{\lambda_{\rm m}} \right)^{n_{\rm e}}+\left( \frac{\lambda}{\lambda_{\rm c}} \right)^{-n_{\rm g}}\right] h\left(1 - \frac{\lambda}{\lambda_{\rm m}}\right) $$
(1)
$$ G(t) = \int_0^{\infty} H(\lambda) \frac{\exp (-t/\lambda)}{\lambda} \, {\rm d} \lambda \label{eq:G} $$
(2)
with parameters ne = 0.23, \(G_0=220~\mbox{kPa}\), ne = 0.23, ng = 0.67, \(\lambda_{\rm m}=19~\mbox{s}\), and \(\lambda_{\rm c}=0.013~\mbox{s}\) (Jackson et al. 1995) at 150°C. The part associated with ne is associated with the viscoelastic behavior, while the part with ng is associated with the glassy behavior. G0 is the elastic modulus in fast deformation of the hypothetical material in which the glassy behavior is absent. Basically G0 = G(0) when the glassy part of the relaxation spectrum is omitted. The parameter G0 as defined here is denoted the plateau modulus by Jackson et al. (1995) although the physical interpretation is closer to the entanglement modulus proposed by Likhtman and McLeish (2002).
The longest relaxation time and the zero shear-rate viscosity are extracted from the viscoelastic part of the relaxation modulus,
$$\begin{array}{rll} \eta_0 & = & \int_0^{\infty} G(s) {\rm d}s \\&=& {n_{\rm e} G \lambda_{\rm m}}/{(1+n_{\rm e})}\approx 780~\mbox{kPa\,s},\;(150^{\circ}\mbox{C}) \end{array} $$
(3)
$$\begin{array}{rll} \tau_{\rm w} &=& \int_0^{\infty} G(s)s\, {\rm d}s/ \int_0^{\infty} G(s){\rm d}s \\ &=& \lambda_{\rm m} (1+n_{\rm e})/(2+n_{\rm e}) \approx 10~s ,\;(150^{\circ}\mbox{C}) \end{array} $$
(4)
The temperature dependence for polystyrene is described by a shift factor given by the WLF equation (Bach et al. 2003a)
$$ \log a_T =\frac{-c_1^0 (T - T_0)}{c_2^0 + (T - T_0)} $$
(5)
where \(c_0^1 = 8.86\), \(c_0^2 = 101.6\) K, \(T_0 = 136.5^{\circ}\)C, and log is the logarithm to base 10. We measured the temperature in the filament stretching apparatus to be 126.7°C. Consequently, the time constants at experimental conditions are about 97 longer than at 150°C. Our estimates of the longest relaxation time and zero shear-rate viscosity under the experimental conditions are shown in Table 2.
In addition to the longest relaxation time, we would like to estimate a time scale for stretch relaxation. Such a time scale is often taken to be represented by the Rouse time τR. Unfortunately, the BSW spectrum is not useful for extraction of a Rouse time from the present data. Basically Rouse scaling would require that G(ω) and G(ω) both scale as ω0.5 for large ω while the data have a higher slope represented by the BSW parameter ng = 0.67. Therefore, a Rouse time cannot be extracted from the BSW spectrum in a meaningful fashion. To estimate a Rouse time, we use the (Likhtman and McLeish 2002) relaxation modulus for the Rouse dynamics of a linear chain in a tube
$$ G_{\rm LR}(t)= \frac{G_{\rm e}}{Z}\left(\frac{1}{5}\sum\limits_{p=1}^{Z-1}\exp\left(- \frac{p^2 t}{\tau_{\rm R}}\right) + \sum\limits_{p=Z}^{N}\exp\left(-\frac{2 p^2 t}{\tau_{\rm R}}\right)\right) $$
(6)
where Ge is the modulus, τR is the Rouse time, and N is the total number of Kuhn links. The first term describes longitudinal mode relaxation in the tube while the second term describes Rouse relaxation not constrained by the tube. The high frequency part of the loss modulus may be fitted with the parameters Ge = 220 kPa, Z = 9, N = 200, and τR = 1.16 s as shown by the blue lines in Fig. 1. The lowest curve (dashed-dotted) represents the longitudinal mode relaxation. The next curve from below represents the unconstrained Rouse dynamics, and the top curve is complete Rouse dynamics given as the sum of the two individual contributions. It appears from Fig. 1 that longitudinal mode relaxation is insignificant compared to unconstrained Rouse motion for this low value of Z. The complete Likhtman fit to the data including also escape from the tube and constraint release is indistinguishable from the BSW fit and therefore not presented. The Likhtman parameter \(\tau_{\rm e}=\tau_{\rm R}/Z^2=0.020\) s could be compared to the BSW parameter λc = 0.013 s. Given the differences in approach, this difference does not seem alarming. We base therefore the analysis on the Likhtman Rouse time which when shifted to experimental conditions (126.7°C) becomes 160 s.

The longest relaxation time is used to define a non-dimensional elongation rate in terms of the Weissenberg number \(\mbox{Wi} = \dot{\epsilon}\tau_{\rm w}\). It appears from Table 3 that the experiments cover the range from Wi = 1.0 to Wi = 100. For low values of Wi, one would estimate the steady stress to be \(\sigma_{\rm zz}-\sigma_{\rm rr} \approx 3\, \eta_0 \dot{\epsilon}\). Indeed for \({\rm Wi} = 1.0 (\dot{\epsilon} = 0.001\, \mbox{s}^{-1}\)), this gives a prediction of 210 kPa in agreement with the observation in Table 3. However, for Wi = 10 and 100, the dynamics in steady flow will be outside of linear dynamics. We here make the stresses non-dimensional with the modulus, G0 = 220 kPa.

Scattering

For a-PS, it is known (Fetters et al. 2009) that the mean-square end-to-end distance is given by
$$ \langle R^2\rangle_0 = 0.437 \, M $$
(7)
where M is the molar mass. The fully extended length is Rmax = nl cos(θ/2) where n is the number of backbone bonds, l is the bond length, and θ the bond angle. The Kuhn length \(b = \langle R^2\rangle_0/R_{\rm max} = 18~\AA\). The deuterated middle block has M = 45 kg/mol whereby the corresponding number of Kuhn links N = 60 and the equilibrium gyration of gyration Rg0 = 57 Å.
Principally we will limit our discussion to characteristics parallel and perpendicular to the stretching direction, respectively. Analyzing the data in terms of narrow intensity sectors of the two-dimensional data in the directions perpendicular and parallel to the stretch directions alone have, however, a very low signal to noise ratio, as due to the small sample volume constituted by the thin (1 mm diameter) filaments obtained at ε = 3, and broadening the sectors will introduce errors when averaging in circular sectors, as is the common procedure. Hence, we opted for the simple assumption of analyzing the full 2D data array using an elliptical Gaussian distribution whereby we could include all data in the fitting procedure. Thus, we assume that the distances between segment pairs n and m is Gaussian with variance \(|n - m|b^2 \lambda^2_j \) in the j-direction so the pair distribution function becomes:
$$ \psi = \prod\limits_{j=1}^3 \biggl( \frac{3}{2\pi | n-m | b^2 \lambda_j^2}\biggr)^{1/2} \exp \left(-\frac{3(x_j/\lambda_j)^2}{2 |n-m| b^2} \right) $$
(8)
Here the λj (=1 at equilibrium) are denoted the stretch factor in the j-direction. The corresponding expression for the scattering function is a modified Debye function of the form,
$$ S(\boldmath{q}) = \frac{2}{x^4}(\exp(-x^2) + x^2 - 1)\label{eq:Debye} $$
(9)
where in our situation,
$$ x^2 = (q_\parallel R_{{\rm g} \parallel})^2 + (q_\perp R_{{\rm g} \perp})^2\label{eq:x} $$
(10)
Here we have defined Rg ∥  = λRg 0 and R g ⊥  = λ ⊥ Rg 0. The iso-intensity contour lines are thus assumed elliptical with constant eccentricity, reflecting that the labeled middle part of the polymer chain is assumed to be stretched uniformly at the length scales approached. The experimental data did not, within statistical error bars, signify deviation from such behavior.
We extract the independent parameters Rg ∥  and Rg ⊥  in Eqs. 9 and 10 from the 2D scattering data, I(q) by minimizing the function f(q) = I(q) − cS(q) in a least squares sense. Here c is an additional free parameter determined also in the minimization procedure, but of no further significance. The assumption of simple elliptical Gaussian intensity distribution is a fair assumption and fits very well the experimental data. Generally, the block copolymer structure factor is characterized by a peak given by the correlation between the covalently bonded PS(H)- and PS(D)-polymer blocks. This peak vanishes, however, due to the dilution with PS(H) polymers. The random phase approximation (RPA) result for such binary systems of dilute block copolymers in homo and the structure factor of the free middle block by itself can be distinguished (see Fig. 5), but the error induced using Gaussian coil statistics without chain interaction is limited to less than 20%, as discussed below. Strongly deformed polymer systems, like mixtures of high molar-mass homopolymers or crosslinked polymer networks, lead typically to considerable lozengic type scattering patterns (Boué 1991; Straube et al. 1995; Hayes et al. 1996; Read and McLeish 1997; Westermann et al. 1998). This is, however, not observed in our case. The main origin for their lozengic shape scattering pattern is polydispersity effects, but also dangling ends of the chains may contribute. The dangling ends do not carry stress and have accordingly relaxed contributing isotropically to the 2D patterns and thus leading to the lemonlike lozengic structure. With the present triblock copolymer architecture, however, we have successfully avoided this effect and are left with a scattering pattern that is well described by an elliptical shape within a q-range which is easily accessible by the SANS technique. In Fig. 4, the middle column represents the resulting model fits, while the right column represents the residuals. The three sets of data—experimental, simulation, and residuals—are all shown on the same intensity scale, making comparison easy. We see that the modified Debye function with the parameters Rg ∥  and Rg ⊥  summarized in Table 3 describes the scattering data to an approximation better than experimental statistics. The largest discrepancy is located near the forward scattering (\(q \lesssim 0.02~\mbox{\AA}^{-1}\)) where the analytical calculation does not take the shadow of the beam-stop into account. The top row in Fig. 4 represents measurements for an un-stretched sample. The corresponding parameters determined by the modified Debye expression are Rg ∥  = Rg ⊥  = 46 Å which is about 20% lower than the literature Rg ∘  = 57 Å. This discrepancy may be explained in terms of the RPA to the blend of triblock (ABA) and homopolymer (A) system at equilibrium (Maschke et al. 1993). For the present system, the triblock has a total of N = 200 Kuhn links, with middle block 60 Kuhn links. The copolymer has a volume fraction 0.33 in a homopolymer with 200 Kuhn links. The corresponding normalized scattering function described by the RPA and the simple Debye function are shown in Fig. 5. It appears that the application of the Debye expression does indeed lead to the radius of gyration at equilibrium being underestimated by about 20%. Since we are only interested in the relative deformation ratios, no extra correction is required. In Table 3, we also show values of a mean experimental radius of gyration Rm defined from the experimentally determined parameters in various anisotropic samples as \(R_{\rm m}^3 = R_{{\rm g}\parallel} R_{{\rm g} \perp}^2\). In this way, the mistake in the determination of Rg,0 cancels and with Rg,0 ~Rm = (53 ±2) Å a good agreement with the estimated value is retrieved.
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Fig. 5

Comparison of normalized scattering intensities as modeled by the RPA (dotted line) and the Debye expression (full line), respectively. The RPA is computed from Eqs. 1520 with parameters N = 200, b = 18.4 Å, f = 0.35, and ϕ = 0.33

Discussion

Three distinct elongation rates have been considered each with multiple measurements:
  • \(\dot{\epsilon}=0.001~\mbox{s}^{-1}\), Wi = 1, and WiR = 0.16: The stress is equal to \(3 \eta_0 \dot{\epsilon}\) which is an indication that the dynamics is within the linear range or at least not far into the non-linear range. This also agrees with expectations at a Weissenberg number equal to 1, although \(\dot{\epsilon}= 0.001~\mbox{s}^{-1}\) may represent the upper limit for the application of linear viscoelasticity. The stretch factors indicate that significant deviation from isotropy occurs in the linear viscoelastic range.

  • \(\dot{\epsilon}=0.01~\mbox{s}^{-1}\), Wi = 10, and WiR = 1.6. The stress is significantly smaller than \(3 \eta_0 \dot{\epsilon}\) which is an indication that the dynamics is outside of the linear range as would be expected at Wi = 10. Specifically the stress is larger than five times the modulus, which is the upper limit that can be predicted by the Doi–Edwards model without stretching. Consequently, according to the Doi–Edwards picture, the chains must be stretched relative to their equilibrium length. This agrees also with the Rouse time obtained from the Likhtman method. According to Rouse theory, the coil stretch transition occurs at WiR = 0.5. Hence, some degree of chain stretching at WiR = 1.6 is not unreasonable.

  • \(\dot{\epsilon}=0.1~\mbox{s}^{-1}\), Wi = 100, and WiR = 16. These data are in the fully non-linear region, with significant chain stretching taking place indicated by σ/G0 ≈ 27 as well as the stretch ratios. The value of λ ∥  may be questioned as discussed below.

In Fig. 6, we show the steady stress and stretch ratios as function of the Weissenberg number \(Wi = \dot{\epsilon}\tau_{\rm a}\). Given that there are only three data points, it is not possible to rule out a coil-stretch transition completely. However, it does appear that the data are reasonably well approximated by expressions of the form
$$\Delta \sigma/G_0 \sim \mbox{Wi}^{\alpha} $$
(11)
$$\lambda_{\parallel} \sim \mbox{Wi}^{\beta} $$
(12)
$$\lambda_{\perp} \sim \mbox{Wi}^{-\gamma} $$
(13)
with α ≈ 0.6, β ≈ 0.25, and γ ≈ 0.125. Keep in mind that while the stresses refer to the entire molecules, the stretch factors refer just to the middle deuterated block. The power α ≈ 0.6 is in fact equal to the scaling observed by Luap et al. (2005). It is within experimental uncertainty equivalent to the \(\dot{\epsilon}^{-0.5}\) scaling reported by Bach et al. (2003a). The measurements cover two orders of magnitude in the steady elongational rate within the non-linear range.
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Fig. 6

Steady flow stress, Δσ/G0 (\(\Box\)), stretch ratios in flow direction, λ ∥  = Rg ∥ /Rg0 (∆), and perpendicular to the flow direction, λ ⊥  = Rg ⊥ /Rg0 (∇) as functions of non-dimensional elongation rate, \({\rm De} = \dot{\epsilon}\tau_d\) for polystyrene of molar mass 150 kg/mol. The dotted lines have slopes 0.6, 0.25, and −0.125, respectively. The stretch ratios refer to the deuterated middle segment of the chain only

Although both λ ∥  and λ ⊥  are given separately in Table 3 (using Rg,0 = 46 Å), in view of the limited q-range which was accessed in the experiment, the perpendicular data are to be trusted more than in the parallel direction. The max q-range considered in Fig. 4 is between qRg = 0.01×46 and 0.07×46, i.e. 0.5Rg to 4Rg. Hence, the parallel direction is fully out of range, shifted to un-measurable intensities in the beam stop area, and tabulated values are strongly co-determined by the perpendicular strain axis. For the latter axis, since the chain size is reduced due to incompressibility, the radius of gyration shifts to higher q and the simple Debye equation which is then Guinier-like fulfills still better the valid Guinier regime within the measured q-range. The microscopic deformation ratios λ ∥  are then ~1.5, 3.2, and 5.4. As an alternative to the direct determination of λ ∥ , we may also estimate the effective microscopic deformation ratios for the middle segment of the triblock chain as \(\lambda_\parallel^{e} = 1/\lambda_{\perp}^2\). This would give values ca. 1.3, 2, and 4 which are lower than those in Table 3. The power law for the parallel direction is virtually unchanged, however. The data and line in Fig. 6 are just shifted upward with a factor of about 1.2.

It remains to address the question whether the chain stretching approaches the fully stretched non-Gaussian limit. For Gaussian behavior, one would expect \(\Delta \sigma \sim \lambda_{\parallel}^2 - \lambda_{\perp}^2\) which for large λ ∥  would correspond to β = α/2 as observed within experimental accuracy. This may be taken as indirect evidence in support of the suggestion that finite extensibility is not the main mechanism behind the scaling observed in non-linear elongational flow (Luap et al. 2005). The maximum stretching one would anticipate can be estimated from the number of Kuhn segments in the deuterated middle segment as \(\sqrt{60}= 7.7\). Hence while the measurements are probably within the Gaussian range, the values at Wi = 90 may be probing the upper limit Gaussian statistics.

Acknowledgments

We are grateful to the Danish Technical Research Council for financial support to the Danish Polymer Centre, to the Danish Natural Science Council for financial support in the DANSCATT grant and to the Paul Scherrer Institut for use of their neutron beam facility, and the SoftComp EU-Networks of Excellence.

Copyright information

© Springer-Verlag 2012