Open Access Article

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Airidas
Korolkovas
*^{ab},
Sylvain
Prévost
^{a},
Maciej
Kawecki
^{b},
Anton
Devishvili
^{ac},
Franz A.
Adlmann
^{b},
Philipp
Gutfreund
^{a} and
Max
Wolff
^{b}
^{a}Institut Laue-Langevin, 71 rue des Martyrs, 38000 Grenoble, France. E-mail: korolkovas@ill.fr
^{b}Department for Physics and Astronomy, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden
^{c}Department for Physical Chemistry, Naturvetarvägen 14, 223 62 Lund, Sweden

Received
4th November 2018
, Accepted 19th November 2018

First published on 20th November 2018

Entangled polymers are deformed by a strong shear flow. The shape of the polymer, called the form factor, is measured by small angle neutron scattering. However, the real-space molecular structure is not directly available from the reciprocal-space data, due to the phase problem. Instead, the data has to be fitted with a theoretical model of the molecule. We approximate the unknown structure using piecewise straight segments, from which we derive an analytical form factor. We fit it to our data on a semi-dilute entangled polystyrene solution under in situ shear flow. The character of the deformation is shown to lie between that of a single ideal chain (viscous) and a cross-linked network (elastic rubber). Furthermore, we use the fitted structure to estimate the mechanical stress, and find a fairly good agreement with rheology literature.

Thanks to deuteration, small angle neutron scattering (SANS) can measure the structure of an individual polymer chain, called the form factor. Many previous studies have used extensional flow to characterize the relaxation of polymers (creep) over time.^{2–4} In this work we focus on shear flow, which poses more practical challenges, but has an advantage of eschewing the complicated time response, once steady state has been reached. The effect of a shear rate κ on the material structure is quantified with a dimensionless Weissenberg number Wi = κτ, where τ is the relaxation time specific to each fluid, typically a millisecond or more. For polymer melts,^{5} shear can be applied on a heated sample, which is then quenched below the glass transition temperature, and the molecular structure is later examined ex situ. Polystyrene (PS) has been measured with SANS using this technique, at an estimated shear rate of Wi = 4. An asymmetry of 1.7 was detected between the chain radii of gyration along the flow and the vorticity directions.^{6} More difficult, but also more industrially relevant experiments measure the fluid structure under in situ shear. Molten polymers like polydimethylsiloxane (PDMS) and polybutadien (PBD) are popular examples, thanks to their low glass transition temperature and comparatively low viscosity. In situ steady flow SANS experiments have not detected any anisotropy of the form factor for either of these samples. The highest shear rate for the PBD experiment^{7} in Couette geometry was Wi = 5.4 and for the PDMS experiment^{8} in cone-plate geometry it was Wi = 0.8. The only in situ shear experiment that has shown anisotropy of entangled polymers was performed in a Couette cell with PS at Wi ≈ 1, but since the relaxation time has not been reported, the Weissenberg number is uncertain.^{9} Anisotropy of 1.5 has also been detected in a dilute solution of long but unentangled PS chains.^{10}

Up to now, the form factor of entangled semi-dilute polymer solutions has not been characterized by SANS under in situ shear. The advantage of the semi-dilute condition is that it has a lower viscosity and glass transition temperature than a melt, facilitating its handling and enabling higher Wi. However, shear induces massive concentration fluctuations, leading up to complete demixing in the extreme case. The resulting SANS signal contains a strong contribution from the structure factor,^{11,12} hindering the single chain analysis.^{13} Fortunately, it is possible to use deuteration to match the contrast between the solvent and the polymer, fully canceling the inter-chain contribution to scattering, even at high density.^{14}

The scenarios where polymers may deform range from dilute, to semi-dilute, to melts. Moreover, a strong anisotropy can also be found in cross-linked polymer networks of gels and rubbers,^{15,16} as well as nanocomposites like polymer–clay.^{17,18} Mechanically, these materials are probed in either shear or extensional flow, applied in a steady, oscillatory, or stepwise mode, or even a superposition of multiple stimuli. As a rule of thumb, a strong deformation will stretch the polymer along flow and shrink it perpendicular to flow. However, the detailed shape of the form factor can have considerable differences in the various cases listed above. While many isotropic theories exist for equilibrium,^{19–21} anisotropic scattering patterns up to now have been analyzed in mostly ad hoc fashion: fitting 1D radial cuts,^{6} comparing angular sector averages,^{9} fitting ellipses to isointensity curves,^{6} and fingerprinting with spherical harmonics.^{4} In the present work, we develop a new approach to extract the underlying real-space structure directly from the data, not requiring any knowledge of the molecular motion. The observed form factor originates from the MSD between the monomers, which is a function of their index separation along the chain, see Fig. 1. At equilibrium, this function is a straight line (ideal random walk), while under a strong deformation it becomes some other, unknown curve. Our main novelty is to approximate this curve with a set of straight segments, or layers. This discrete model converges to the exact mathematical result when the number of segments is brought to infinity (a textbook definition of the Riemann integral). Luckily, in real-world experiments the MSD deviation from a perfect straight line is quite small, almost never exceeding ×2, so there is no need for an infinity of parameters for a good description, and only a few layers are sufficient. In this case, the model is convenient to integrate analytically, and the resulting formula is fitted to the 2D data, to determine the width and the slope of each layer. This structure is then fitted to reveal a power-law of ξ = 1.2, and that is our novel measure of structural non-affinity.

(1) |

(2) |

(3) |

(4a) |

=ρ_{1}ρ_{2}|F|^{2} + ρ_{1}(ρ_{1} − ρ_{2})|F_{1}|^{2} + ρ_{2}(ρ_{2} − ρ_{1})|F_{2}|^{2} | (4b) |

For convenience, the scattering length density (SLD) contrast has been defined as ρ_{1} = b_{D}/v − b_{S}/v_{S} and ρ_{2} = b_{H}/v − b_{S}/v_{S} for the two labels. The Fourier transform squared of each phase can be further decomposed into the diagonal (intra-chain) and the off-diagonal (inter-chain) terms:

|F_{1}|^{2} = C_{1}N^{2}P(q) + C_{1}(C_{1} − 1)Q(q), | (5) |

(6a) |

(6b) |

(7) |

Our sample was an entangled semi-dilute polymer solution, with a volume fraction ϕ_{1} = 0.172 of deuterated PS (575 kg mol^{−1}, N = 5127, PDI = 1.09) and ϕ_{2} = 0.0998 of hydrogenated PS (510 kg mol^{−1}, N = 5000, PDI = 1.1), purchased from Polymer Source. It was prepared by first dissolving the powdered PS mix in a glass beaker with a large amount of deuterated toluene, using a magnetic stirrer. After removing the stirrer, the solution was left in a ventilated fume hood for several days until the toluene has evaporated to the volume fraction quoted above, which was determined by weighing the dry and the dissolved polymer, minus the container. The detailed rheological characterization of a similar sample has been reported in ref. 24.

In our region of interest, (qR) = 1, the structure and the form factors as defined in eqn (1) and (2) both have a similar magnitude of (S(q) ≈ P(q)) = 1. Using the SLD values ρ_{1} = 0.47 × 10^{10} cm^{−2} and ρ_{2} = −4.5 × 10^{10} cm^{−2} we can estimate the ratio of the two intensities as

(8) |

Fig. 2 The scattering cross-section from a quiescent solution, multiplied by q^{2} to reveal the ideal random walk character (flat line) of the chain form factor. The solid black line is the Debye function, eqn (16), fitted to R = 27.12 nm. |

The form factor of an ideal random walk has a power law behaviour of P ∝ (qR)^{0} for qR ≪ 1 and P ∝ (qR)^{−2} for qR ≫ 1, as evidenced in Fig. 2. Eventually at high q the scattering starts to probe correlations inside the blob of size (λ) = 1 nm, which is a typical distance between the semi-dilute chains, called the mesh size. Within the blob (qλ ≫ 1) the excluded volume interactions are not screened, so the polymer form factor changes towards the scaling of P ∝ (qλ)^{−5/3}, which is a signature of a self-avoiding random walk (see textbook ref. 27 and 28). In addition, the scattering from density fluctuations at the chemical monomer level may become visible for the highest q-values (not measured here). On the opposite side of the spectrum, the ultra low q data also deviates from Debye, this time due to scattering from very slowly relaxing density inhomogeneities spanning large distances, likely hundreds of chains or more.^{12} Extreme viscoelastic samples like ours are difficult to fully equilibrate, as some residual flow persists for many hours if not days (one experiment has been running for almost 100 years^{29}). Even when left perfectly still, the sample may keep flowing due to an interplay of gravity and the capillary forces between the narrow gap of the rheometer plates. This can induce concentration fluctuations (see ref. 30–33), and while their amplitude may be tiny, when integrated over a long distance, a strong SANS signal can result at ultra low q.

Our shear experiments were conducted on PAXY (Laboratoire Léon Brillouin, Saclay, France), with a narrower q range set at 0.05–0.5 nm^{−1}, where the scattering is fully described by the Debye function. We have used a custom-made vertical sealed cone-plate shear cell,^{34} designed for both SANS and NSE instruments and allowing a smaller liquid volume than typical Couette cells,^{9} which can be a considerable advantage for costly and rare deuterated samples. It is also well suited for shearing fluids which exhibit non-linear viscoelastic phenomena such as the rod-climbing effect. A vertical cone-plate geometry is a necessity for rheo-NSE and allows a direct measurement of both structure (SANS) and dynamics (NSE) in the same setup. In our experiment the shear rate was κ = 300 s^{−1}, corresponding to Wi = 30. Data collection has lasted 4.25 h per spectrum, at a temperature of 45 °C, which is the same as used at D11 for the quiescent measurement.

In this article we only report data from the SANS experiment carried over one day. After that, the experiment continued for three more days with NSE, which will be a separate subject. However, for full disclosure we note that after these four days of shearing, we have spotted some wear of the cell sealing, causing aluminum and teflon impurities to have leached into the sample. Solid particles are known to give rise to Porod scattering of P ∝ q^{−4}, and fortunately there was no trace of it in the range covered by PAXY, where the Debye law P ∝ q^{−2} dominates. As the SANS data was collected during the first day of shearing, the impurities at that stage must have been very dilute and hence invisible to the beam. On top of that, the particle size must have been much greater than the polymer radius of gyration, falling outside of the SANS range. Such big particles cannot interfere with the polymer dynamics, as that is only possible in polymer-nanocomposites where the two components are similar-sized.^{18} These specialty materials require advanced chemical synthesis and cannot be produced by just using mechanical friction to grind up some aluminum dust. Therefore, even if we would have had a considerable percentage of sample contamination, its effect could not have altered the entanglement physics, but only lowered the overall polymer density. This means that the actual Wi may have been 29 instead of 30 we claim. Either way, it is unlikely that these impurities could have altered the polymer form factor beyond the uncertainty of the fit (15%), as explained in the next section.

(9) |

(10) |

〈e^{iq·r}〉 = e^{−〈(q·r)2〉/2} | (11) |

〈(q·r)^{2}〉 = 〈x^{2}〉q^{2}_{x} + 2〈xy〉q_{x}q_{y} + 〈y^{2}〉q^{2}_{y} + 〈z^{2}〉q^{2}_{z} | (12) |

(13) |

(14) |

〈(q·r_{nm})^{2}〉/2 = a|n − m|/N | (15) |

(16) |

(17) |

b = R^{2}[(αq_{x})^{2} + (βq_{z})^{2}] | (18) |

c = R^{2}[(γq_{x})^{2} + (δq_{z})^{2}] | (19) |

(20) |

Fig. 3 (a–c) The scattered intensity under shear P(q), divided by the quiescent signal P_{iso}(q). This plot removes the Debye envelope 1/q^{2}, highlighting the structural changes induced by the shear. The data (a) is fitted with the analytical function (b), and their difference is plotted in (c), showing that the fit accounts for 85% of the signal or more. (d) The inferred mean square distance (MSD) between two monomers, for different directions probed by the scattering vector q. The dotted lines are a sketch of what the true function may look like. The straight black lines show our piecewise approximation, eqn (17). The dashed black line is the isotropic MSD found at equilibrium, eqn (15). |

Slope x | Slope z | Thickness | |
---|---|---|---|

Layer 1 (high q) | α = 1.005 | β = 0.976 | ν = 0.09 |

Layer 2 (low q) | γ = 1.556 | δ = 0.835 | 1 − ν = 0.91 |

Using the parameters from Table 1, the piecewise model of eqn (17) is plotted in Fig. 3d with solid black lines for the q_{x} and q_{z} directions. Quite obviously, a realistic polymer structure cannot have sharp kinks, so we have fitted two smooth curves (dotted red and blue) to our piecewise model. These fits are made with a semi-empirical function

(21) |

(22) |

N_{3} = σ_{xx} − σ_{zz} = σ_{0}(α^{2} − β^{2}) = 1.3 × 10^{5} Pa | (23) |

Since we could not measure this quantity with our shear apparatus, we compare it with the available literature data of a similar polymer. Ref. 36 reports oscillatory shear results for polyisoprene of M_{w} = 170 g mol^{−1}, which is 3 times shorter than our polystyrene, but also 3 times denser as they have used a melt instead of a semi-dilute solution. Judging from the dynamical moduli data in Fig. 3c of that study, the cross-over frequency, which corresponds to Wi = 1, is at ωa_{T} = 6 × 10^{−3}. To compare with our conditions of Wi = 30, we look at their Fig. 4c and frequency ωa_{T} = 0.18. Reading off the stress axis we find N_{3} = 6 × 10^{4} Pa, which is half of the magnitude that we could infer from our piecewise fit of SANS. It shows that our structural data analysis is reasonably consistent with an independent rheology perspective. We attribute the remaining discrepancy of (2) partly to the difference of sample chemistry, but mostly to the uncertainty of the SANS data in the high q region, which is the important bit for calculating the stress. A more precise comparison with rheology may become available in the future, by improving the resolution and the counting time of the SANS setup, and by collecting data in more directions than just the xz plane.

We have extracted the chain deformation parameters, Table 1, using a purely structural model, without any recourse to molecular theories. Nevertheless, to understand why the chain deforms in this particular way, a molecular explanation is needed. Currently no definitive theory exists, but the main contender in this arena is GLaMM,^{37} a tube theory^{38} with several modifications. In essence, the many-chain fluid is simplified with just a single chain trapped in a tube, which is the mean field of other chains, and the overall dynamics are described in a self-consistent way. This model can accurately reproduce the rheology of entangled polymer melts, although SANS studies have not reached a consensus yet, with some authors claiming a strong support of tube theory,^{39} others report no evidence of any tubes,^{40} and others still demonstrate kinetic trends opposite to theoretical predictions.^{4} The debate centers on how exactly does the tube relax, and how is it affected by a strong deformation.

There is considerable universality between entangled polymer melts and semi-dilute solutions, especially in the linear regime Wi < 1, where GLaMM could be applied. At higher shear the universality breaks down, as polymer solutions display enhanced concentration fluctuations, which can reach length scales considerably larger than the molecule radius of gyration.^{31,33,41–43} Therefore, a single average chain in a tube may not be enough to describe the whole fluid. Furthermore, the shape of an individual molecule is known to fluctuate between highly stretched and collapsed states, a phenomenon called tumbling dynamics.^{44} The mean field assumption, a core tenet of tube theory, becomes questionable given such inhomogeneities. Finally, we note that the form factor measured by SANS is a fundamentally static quantity (contains no units of time), and could be consistent with many different dynamical models. Given the above limitations, it would be premature to interpret our findings in terms of the current tube theories.

Instead, we offer an explanation based on the fact that an entangled polymer liquid is an intermediate case between a rubber and an ideal Rouse chain. A piece of rubber responds to stress with an affine deformation, meaning that the exponent in eqn (21) is ξ = 1. It is widely believed that entangled polymers, at a large scale, have this rubber-like affine response.^{16,45} However, on the short scale, a non-affine liquid-like response is expected. In this regime, unentangled polymers are well described by the Rouse model, which contains the following forces: spring, random, and shear (see Chapter 4 in ref. 28 for details):

(24) |

(25) |

The pairwise distance for polymers is obtained by summing all the Rouse modes (dumbbells):

(26) |

(27a) |

(27b) |

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