A network theory of constrained elastic recovery in concentrated polymer solutions

Summary The stress-strain-history relations derived byGreen andTobolsky for a relaxingGaussian molecular network with temporary junctions, generalized to allow for a distribution of junction mean lifetimes, have previously been used to calculate the elastic recovery which occurs in a polymer solution in a state of steady shear flow when all the stress components are instantaneously made zero. The equations predict that the instantaneous part of the recovery involves, in addition to the expected shear recovery, an expansion in directions normal to the previous lines of flow.The present paper contains the corresponding calculations for the case in which this expansion is not allowed to take place, the liquid being constrained so that planesx ₂=const. move rigidly in directions parallel to thex ₁-axis during recovery as well as during the steady shear flow;x ₁,x ₂,x ₃ denote coordinates relative to a rectangularCartesian coordinate system fixed in space. It is shown that the relation between shear stress and the time integral of shear strain is of the linear type much studied in the literature; the calculation of the behaviour ofp ₁₁–p ₂₂, the difference of two normal stress components, is new. It is shown that ifp ₂₁, the shear stress component, be instantaneously made zero (following steady shear flow),p ₁₁–p ₂₂ also decreases instantaneously but in general takes time to reach the value zero; this is true even for the case in which all junctions have the same mean lifetime when the instantaneous shear recovery is not followed by any delayed recovery. The magnitudes of both instantaneous and ultimate shear recovery are calculated; it is found that the latter differs by a factor of two from that calculated on the hypothesis that the stress tensor in steady shear flow is an isotropic function of the strain tensor describing the ultimate shear recovery. The transient behaviour associated with the start of steady shear flow is also considered. Inertial forces are neglected throughout.

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Zusammenfassung Die Beziehungen zwischen der Spannung und dem Zeitintegral der Verformung, die vonGreen undTobolsky für ein sich entspannendes Gaußsches Netzwerk mit zeitweiligen Vernetzungen abgeleitet wurden, werden mit Berücksichtigung der Verteilung der mittleren Lebensdauer der Vernetzungen verallgemeinert. Diese Beziehungen sind schon früher benutzt worden, um die elastische Erholung zu berechnen, die erfolgt, wenn in einer Polymerlösung, die sich in einem Zustand stationärer Scherströmung befindet, alle Kräfte momentan gleich Null gesetzt werden. Die Gleichungen sagen aus, daß der Momentanteil der Erholung nicht nur eine voraussichtliche Schererholung, sondern auch eine Ausbreitung in den Richtungen normal zu den Strömungslinien einschließt.Die vorliegende Abhandlung enthält die diesbezügliche Rechnung für den Fall, wo diese Ausbreitung nicht stattfinden kann, da die Flüssigkeit so umschlossen ist, daß die Ebenen normal zurx ₂-Achse sich unnachgiebigerweise in den Richtungen parallel mit derx ₁-Achse während der Erholung als auch während der stationären Scherströmung bewegen.x ₁,x ₂, undx ₂ bezeichnen die Koordinaten bezüglich eines räumlich festen rechtwinkeligen, cartesischen Achsenkreuzes. Man erhält eine lineare Beziehung zwischen der Schubspannung und dem Zeitintegral der Schiebung, die im Schrifttum schon viel behandelt worden ist. Die Berechnung des Verhaltens der Differenz der zwei Normalspannungskomponentenp ₁₁ undp ₂₂ ist neu. Es wird gezeigt, daß wenn, bei stationärer Scherströmung, die Schubspannungskomponentep ₂₁ momentan Null gesetzt wird, dannp ₁₁ bisp ₂₂ auch momentan abnimmt, aber im allgemeinen ist eine gewisse Zeit erforderlich, den Nullwert zu erreichen. Das trifft sogar dann zu, wenn alle Vernetzungen dieselbe mittlere Lebensdauer haben und wo der momentanen Schuberholung nicht eine verzögerte Erholung folgt. Die Beträge der momentanen und endgültigen Schererholung wurden berechnet. Die so ermittelte endgültige Schererholung beträgt die Hälfte derer, die man, von der Annahme ausgehend, daß der Spannungstensor der stationären Scherströmung eine isotrope Funktion des Verzerrungstensors, der die endgültige Schererholung beschreibt, sei, berechnet. Das vorübergehende Verhalten am Anfang der stationären Scherströmung ist auch in Betracht gezogen. Dagegen sind Trägheitskräfte durchweg vernachlässigt.

     

On-line measurement of elasticity and viscosity in flowing polymeric liquids

A new slit-die rheometer (the Stressmeter) for on-line and sample measurement of the viscosity, , and the first normal stress difference, N ₁, in steady shear flow for molten polymers and other high-viscosity liquids is described. Two liquid-filled transverse slots, located in one die wall near the center station, give pressures P ₂ and P ₃ from whose difference the wall shear stress is calculated. In the other die wall at a location opposite the center of the P ₂ slot is a flush-mounted transducer, giving a pressure P ₁. N ₁ is calculated from the hole pressure P ^* = P ₁–P ₂. A metering pump, used to measure the flow rate Q, is supplied with melt either from an extruder (online mode) or from a pressurized sample cylinder (sample mode). The wall shear rate is calculated from Q and ; the Weissenberg-Rabinowitsch correction and a new small-viscous-heating-correction algorithm (affecting ) are used. Viscous heating corrections are small; entrance and exit errors are negligible. The instrument is tested by comparing its results with those obtained from cone-plate and capillary rheometers. Measurement ranges extend to = 200 kPa, = 3000 s^–1, and temperature = 250°C.Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Two calorimetric glass transitions do not necessarily indicate immiscibility: The case of PEO/PMMA

Differential scanning calorimetry has been used to examine blends of a poly(ethylene oxide) (PEO), M_n = 300 g/mol, and a poly(methylmethacrylate) (PMMA), M_n = 10,000 g/mol, across the complete composition range. The relatively low molar mass of the PEO minimizes interference from crystallization. In the midrange of composition, ～25–70% PEO, two broad, but distinct, glass transitions are resolved. These are interpreted as distinct glass transitions of the two components, as anticipated by the self‐concentration model of Lodge and McLeish. The composition dependence of the observed transitions is well described by the self‐concentration approach, using lengthscales of approximately two‐thirds of the Kuhn length. The results are compared with previous measurements on PEO/PMMA blends and other miscible systems. The principal, general conclusion is that one should actually expect two glass transitions in a miscible polymer blend or polymer solution; the rule of thumb that two transitions indicate immiscibility is incorrect. Furthermore, attempts to rationalize two transitions on the basis of incomplete segmental mixing, or other unspecified “nanoheterogeneity,” may not be justified in many cases. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 756–763, 2006     

Twinning and growth kinetics of lamellar grains in a diblock copolymer solution

The initial stages of growth of the lamellar phase in a block copolymer solution were observed with polarizing optical microscopy (POM). Measurements were made on a poly(styrene‐b‐isoprene) diblock copolymer with block molecular weights of 15 and 13 kg/mol, respectively, dissolved in dioctyl phthalate with 70% polymer by volume. Upon cooling from above the order–disorder transition temperature, 89.5 °C, to temperatures from 87.5 to 88.5 °C, four distinct types of grain were observed: ellipsoidal single grains, twinned ellipsoidal grains, 2‐fold twinned grains, and spherulites. The relative populations were distributed as 50% single ellipsoids, 25% twinned ellipsoids, 10% 2‐fold twinned grains, and 15% spherulites. These grain types cover a range of lamellae orientation. For example, the surface of a 2‐fold twinned grain is composed of lamellar edges, whereas the spherulite surface is composed of lamellar planes. The specific grain types that arise give insight into the thermodynamic and kinetic forces governing lamellae ordering. Furthermore, growth front velocities of individual grains were measured after rapid quenches from above T_ODT. These results were compared to the predictions of Goveas and Milner. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 405–412, 2005     

Superlattice formation in binary mixtures of block copolymer micelles

Two distinct diblock copolymers, poly(styrene-b-isoprene) (SI) and poly(styrene-b-dimethylsiloxane) (SD), were codissolved at various concentrations in the polystyrene selective solvent diethyl phthalate. Two SI diblocks, with block molar masses of 12,000-33,000 and 30,000-33,000, and two SD diblocks, with block molar masses of 19,000-6000 and 16,000-9000, were employed. The size ratio of the smaller SD micelles (S) to the larger SI micelles (L) varied from approximately 0.5 to 0.6, based on hydrodynamic radii determined by dynamic light scattering on dilute solutions containing only one polymer component. Due to incompatibility between the polyisoprene and polydimethylsiloxane blocks, a binary mixture of distinct SI and SD micelles was formed in each mixed solution, as confirmed by cryogenic transmission electron microscopy. When the total concentration of polymer was increased to 20-30%, the micelles adopted a superlattice structure. Small angle X-ray scattering revealed the lattice to be the full LS13 superlattice (space group Fm3c) in all cases, with unit cell dimensions in excess of 145 nm. A coexistent face-centered cubic phase composed of SD micelles was also observed when the number ratio of S to L micelles was large.     

Temperature- and pH-triggered reversible transfer of doubly responsive hairy particles between water and a hydrophobic ionic liquid

This article reports on the synthesis of thermo- and pH-sensitive polymer-brush-grafted silica particles ("hairy" particles) and the study of their phase-transfer behavior between water and a hydrophobic ionic liquid, 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)amide ([EMIM][TFSA]), in response to temperature and pH changes. The hairy particles were made by the surface-initiated atom transfer radical polymerization of methoxytri(ethylene glycol) methacrylate (TEGMMA) and tert-butyl methacrylate with a molar ratio of 100:11 in the presence of a free initiator and the subsequent removal of tert-butyl groups. The cloud points (CPs) of poly(TEGMMA-co-methacrylic acid), obtained after the treatment of the free polymer with trifluoroacetic acid, in both water and [EMIM][TFSA]-saturated water increased with the increase in pH and can be tuned over a wide temperature range. The hairy particles moved spontaneously from the aqueous phase to the [EMIM][TFSA] phase upon heating at 80 °C and returned to the aqueous layer upon cooling at 10 °C. This process can be repeated many times regardless of whether the pH of the aqueous phase is 2.99, 5.00, or 7.02. UV-vis absorbance measurements showed that the transfer temperature (T(tr)) of hairy particles from water to [EMIM][TFSA] increased with the increase in the pH of the aqueous phase. A linear relationship was observed between the T(tr) of hairy particles and the CP of the corresponding free polymer. By taking advantage of the tunability of the T(tr) of hairy particles, we demonstrated the pH-driven reversible transfer of hairy particles at a fixed temperature by changing the pH of the aqueous phase and multiple phase-transfer processes by controlling both the temperature and pH.     

An expansion method for calculating molecular properties IV.

The energies of the 2pσ and 2sσ states of H₂ ⁺ and HeH²⁺ and those of the a ³, A ¹, b ³, B ¹Σ⁺ states of HeH⁺ are calculated up to second order in the heavy nuclear charge. For the one-electron systems the first-order equation is solved analytically employing a special separation technique for degeneracies. The consideration of screening enables a recently proposed interchange result for the degenerate case to be tested. The results for the 2sσ states are good while those for the 2pσ states are only good at small internuclear distances.

For the two-electron systems the method employed is a single perturbation approach for two perturbations in the presence of degeneracy. The results for HeH⁺ appear to be good only at small internuclear distances. The corresponding results for the excited states of H₂ are omitted since an interesting problem arises at first order.