Interchain Pressure Research Articles

A new perspective on monomeric friction reduction in fast elongational flows of polystyrene melts and solutions

Primitive chain network (PCN) and non-equilibrium molecular dynamics (NEMD) simulations indicate that the nonuniversality seen in the elongational rheology of linear polymer melts and concentrated solutions is caused by monomeric friction reduction. We present, in the context of tube models with varying tube diameter, a new constitutive equation based on enhanced relaxation of stretch (ERS) and find good agreement with experimental evidence and with predictions of the extended interchain pressure (EIP) model. The stretch evolution equations of the ERS and the EIP model can be converted in a form, which allows expressing them in terms of monomeric friction reduction. Monomeric friction coefficients obtained from the ERS model and those used in PCN and NEMD simulations show at least qualitative agreement. While the reduction of monomeric friction is incorporated in PCN and NEMD simulations by empirical correlations between friction coefficient and segmental orientation or elongational stress fitted to elongational viscosity data, the EIP and ERS models provide analytical and parameter-free relations of monomeric friction reduction as a function of chain stretch.

Elongational viscosity and brittle fracture of bidisperse blends of a high and several low molar mass polystyrenes

Elongational viscosity data of four well-characterized blends consisting of 10% mass fraction of monodisperse polystyrene PS-820k (molar mass of 820 kg/mol) and 90% matrix polystyrenes with a molar mass of 8.8, 23, 34, and 73 kg/mol, respectively, as reported by Shahid et al. Macromolecules 52: 2521–2530, 2019 are analyzed by the extended interchain pressure (EIP) model including the effects of finite chain extensibility and filament rupture. Except for the linear-viscoelastic contribution of the matrix, the elongational viscosity of the blends is mainly determined by the high molar mass component PS-820k at elongation rates when no stretching of the lower molar mass matrix chains is expected. The stretching of the long chains is shown to be widely independent of the molar mass of the matrix reaching from non-entangled oligomeric styrene (8.8 kg/mol) to well-entangled polystyrene (73kg/mol). Quantitative agreement between data and model can be obtained when taking the interaction of the long chains of PS-820k with the shorter matrix chains of PS-23k, PS-34k, and PS-73k into account. The interaction of long and short chains leads to additional entanglements along the long chains of PS-820k, which slow down relaxation of the long chains, as clearly seen in the linear-viscoelastic behavior. According to the EIP model, an increased number of entanglements also lead to enhanced interchain pressure, which limits maximal stretch. The reduced maximal stretch of the long chains due to entanglements of long chains with shorter matrix chains is quantified by introducing an effective polymer fraction of the long chains, which increases with the increasing length of the matrix chains resulting in the excellent agreement of experimental data and model predictions.

Modeling elongational viscosity and brittle fracture of polystyrene solutions

Elongational viscosity data of well-characterized solutions of 3–50% weight fraction of monodisperse polystyrene PS-820k (molar mass of 820,000 g/mol) dissolved in oligomeric styrene OS8.8 (molar mass of 8800 g/mol) as reported by André et al. (Macromolecules 54:2797–2810, 2021) are analyzed by the Extended Interchain Pressure (EIP) model including the effects of finite chain extensibility. Excellent agreement between experimental data and model predictions is obtained, based exclusively on the linear-viscoelastic characterization of the polymer solutions. The data were obtained by a filament stretching rheometer, and at high strain rates and lower polymer concentrations, the stretched filaments fail by rupture before reaching the steady-state elongational viscosity. Filament rupture is predicted by a criterion for brittle fracture of entangled polymer liquids, which assumes that fracture is caused by scission of primary C-C bonds of polymer chains when the strain energy reaches the bond-dissociation energy of the covalent bond (Wagner et al., J. Rheology 65:311–324, 2021).

Scaling relations for brittle fracture of entangled polystyrene melts and solutions in elongational flow

The criterion for brittle fracture of entangled polymer liquids [Wagner et al., J. Rheol. 62, 221–223 (2018)] is extended by including the effects of finite chain extensibility and polymer concentration. Crack initiation follows from rupture of primary C–C bonds, when the strain energy of entanglement segments reaches the energy of the covalent bond. Thermal fluctuations will concentrate the strain energy on one C–C bond of entanglement segments, leading to bond scission and rupture of polymer chains followed by crack initiation and fast crack growth. In start-up flows, entanglement segments characterized by long relaxation times, i.e., predominantly those in the middle of the polymer chain, will be the first to reach the critical strain energy and will fracture. Recent experimental data of Huang [Phys. Fluids 31, 083105 (2019)] of fracture of a monodisperse polystyrene melt and of several solutions of monodisperse polystyrenes dissolved in oligomeric styrene are in agreement with the scaling relations for critical Weissenberg number as well as Hencky strain and stress at fracture derived from this fracture criterion and the extended interchain pressure model [Narimissa, Huang, and Wagner, J. Rheol. 64, 95–110 (2020)].

Elongational viscosity scaling of polymer melts with different chemical constituents

Morelly et al. (Macromolecules 52:915-922, 2019) reported transient and steady-state elongational viscosity data of monodisperse linear polymer melts obtained by filament-stretching rheometry with locally controlled strain and strain rate and found different power law scaling of the elongational viscosities of polystyrene, poly(tert-butylstyrene) and poly(methyl-methacrylate). Very good agreement is achieved between data and predictions of the extended interchain pressure (EIP) model (Narimissa et al. J. Rheol. 64, 95-110 (2020)), based solely on linear viscoelastic characterization and the Rouse time τR of the melts. The analysis reveals that both the normalized elongational viscosity and the normalized elongational stress are dependent on the number of entanglements (Z) and the ratio of entanglement molar mass Mem to critical molar mass Mcm of the melts in the linear viscoelastic regime through {eta}_E^0/left({G}_N{tau}_Rright)propto {left({M}_{mathrm{em}}/{M}_{mathrm{cm}}right)}^{2.4}{Z}^{1.4} and {sigma}_E^0/{G}_Npropto {left({M}_{mathrm{em}}/{M}_{mathrm{cm}}right)}^{2.4}{Z}^{1.4} Wi , while in the limit of fast elongational flow with high Weissenberg number Wi={tau}_Rdot{varepsilon} , both viscosity and stress become independent of Z and Mem/Mcm, and approach a scaling which depends only on Wi, i.e. ηE/(GNτR) ∝ Wi−1/2 and σE/GN ∝ Wi1/2. When expressed by an effective power law, the broad transition from the linear viscoelastic to the high Wi regime leads to chemistry-dependent scaling at intermediate Wi depending on the number of entanglements and the ratio between entanglement molar mass and critical molar mass.

Investigating the Transition between Polymer Melts and Solutions in Nonlinear Elongational Flow

The Doi–Edwards tube model, coupled with relaxation mechanisms, such as reptation, contour length fluctuation, and constraint release, allows us to quantitatively predict the linear viscoelastic properties of entangled polymers. However, for nonlinear elongational flows, large discrepancies between theoretical predictions based on the tube model and experimental results still persist today. This is in particular obvious for the experimentally observed strong qualitative differences in extensional flow of entangled polystyrene (PS) melts and solutions despite having the same number of entanglements and exhibiting the same linear viscoelastic behavior. The cause of this non-universality is often attributed either to a monomeric friction reduction or to an interchain pressure effect. In this work, we investigate the changes in extensional flow behavior going from polymer solutions to the melt state. For this purpose, we measure with a filament stretching rheometer the nonlinear extensional responses of differently long PS chains, both in the melt state and diluted in short-chain matrices of the same polymer at varying concentrations. These concentrations have been chosen sufficiently high, such that the chains stay entangled. This allows us to discuss the influence of concentration and molar mass on the steady-state elongational viscosity to highlight scaling relations. The purpose of the present work is to conduct well-defined experiments to further investigate how the steady extensional viscosity of polymer solutions and blends varies with the concentration and with the molecular weight of the chains.

A constitutive analysis of nonlinear shear flow

We analyse shear stress and normal stress data obtained by cone-partitioned-plate (CPP) shear rheometry in recent years. The data sets of Schweizer et al. (Rheol. Acta 47, 943–957, 2008) and Costanzo et al. (Macromolecules 49, 3925–3935, 2016; & Fluids 4, 28, 2019) on nearly monodisperse polystyrene melts and solutions are considered to be among the most reliable shear data available. The Doi-Edwards independent alignment (DEIA) model (J. Chem. Soc., Faraday Transactions 2: Molecular and Chemical Physics 74, 1802–1832, 1978a,b) allows for quantitative description of the steady-state values of shear viscosity eta left(dot{gamma}right) and first normal stress coefficient {psi}_1left(dot{gamma}right) , while it underpredicts the stress overshoot of the stress growth coefficient of the shear stress, η+(t), and fails in predicting a stress overshoot of the stress growth coefficient of first normal stress difference, {psi}_1^{+}(t) . On the other hand, the extended interchain pressure (EIP) model (J. Rheol. 64, 95–110, 2020) provides an excellent prediction of the stress overshoots of both shear stress and first normal stress difference, while overpredicting the steady-state shear viscosity and the first normal stress coefficient. We demonstrate that the shear stress overshoot is the result of a combination of orientational stress overshoot and stretch overshoot, while the normal stress overshoot depends solely on the overshoot of the stretch. Based on these considerations, we propose a novel constitutive approach consisting of a combination of the DEIA and the EIP model, and predictions of this approach are found to be in quantitative agreement with the data sets of Schweizer et al. and Costanzo et al. within experimental accuracy.

Elongational rheology of polystyrene melts and solutions: Concentration dependence of the interchain tube pressure effect

The elongational viscosity data of Bach et al. [Macromolecules 36, 5174–5179 (2003)] and Huang et al. [ACS Macro Lett. 2, 741–744 (2013); Macromolecules 48, 4158–4163 (2015); Macromolecules 46, 5026–5035 (2013)] on monodisperse polystyrene melts and concentrated polystyrene solutions in oligomeric styrene represent a unique benchmark for improving the tube model of Doi and Edwards with respect to its predictive capabilities of nonlinear viscoelasticity and especially chain stretch. By relaxing one of the basic assumptions of the original tube model of Doi and Edwards [J. Chem. Soc., Faraday Trans. 2 Mol. Chem. Phys. 74, 1802–1817 (1978); J. Chem. Soc., Faraday Trans. 2 Mol. Chem. Phys. 74, 1818–1832 (1978)], i.e., the assumption of a constant tube diameter, and assuming that chain stretch is inversely proportional to a deformation-dependent tube diameter, the extended interchain pressure (EIP) theory [Wagner and Rolon-Garrido, AIP Conf. Proc. 1152, 16–31 (2009); Wagner and Rolon-Garrido, Korea Aust. Rheol. J. 21, 203–211 (2009)] allowed a parameter-free modeling of the elongational viscosity of monodisperse polystyrene melts. Here, we demonstrate that when the dependence of the interchain pressure effect on polymer concentration and molar mass of the oligomeric solvent is considered, the EIP model agrees with experimental evidence that at low Weissenberg numbers W i R = e ˙ τ R < 1 melts and solutions show extension thinning behaviors, while at W i R ≅ 1 solutions switch to extensional thickening or show a more or less constant steady-state elongational viscosity and melts continue with extension thinning behavior with a scaling of η E ∝ W i R − 0.5. We explain quantitatively the effects of molar mass of solvent and of polymer concentration on the elongational viscosity in the investigated concentration range from 10% to 100% (melt), based solely on the linear-viscoelastic characterization. For polystyrene dissolved in oligomeric styrene with molar mass larger than a quarter of the entanglement molar mass of the melt, M o s ≥ M e m / 4 = 3.3 kg / mol, we predict a universal relation for the steady-state elongational stress at W i R ≫ 1, which is independent of polymer concentration and molar mass of the solvent, while the stretch potential of polystyrene dissolved in oligomeric styrene with M o s < 3.3 kg / mol increases with decreasing molar mass of solvent and decreasing polymer concentration in agreement with available experimental evidence.The elongational viscosity data of Bach et al. [Macromolecules 36, 5174–5179 (2003)] and Huang et al. [ACS Macro Lett. 2, 741–744 (2013); Macromolecules 48, 4158–4163 (2015); Macromolecules 46, 5026–5035 (2013)] on monodisperse polystyrene melts and concentrated polystyrene solutions in oligomeric styrene represent a unique benchmark for improving the tube model of Doi and Edwards with respect to its predictive capabilities of nonlinear viscoelasticity and especially chain stretch. By relaxing one of the basic assumptions of the original tube model of Doi and Edwards [J. Chem. Soc., Faraday Trans. 2 Mol. Chem. Phys. 74, 1802–1817 (1978); J. Chem. Soc., Faraday Trans. 2 Mol. Chem. Phys. 74, 1818–1832 (1978)], i.e., the assumption of a constant tube diameter, and assuming that chain stretch is inversely proportional to a deformation-dependent tube diameter, the extended interchain pressure (EIP) theory [Wagner and Rolon-Garrido, AIP Conf. Proc. 1152, 16–31 (2009); Wagner and Rolon-Garrido, Korea Aust. Rheol...

On the universality in the extensional rheology of monodisperse polymer melts and oligomer dilutions thereof

The startup and steady extensional viscosities were measured on two narrow molar mass distributed (NMMD) poly(methyl methacrylates) (PMMA) diluted in 57% 2.1 kg/mole oligomer methyl methacrylates. The oligomer is short enough to be random configured and un-entangled though it is still a Kuhn chain. The weight-based average molecular weights of the PMMAs are 86 kg/mole and 270 kg/mole with polydispersites of 1.08 and 1.09 respectively. The extensional viscosities were in theoretical agreement with a constant ‘interchain pressure’ model, representing the maximal level of strain hardening in a Kuhn fluid. This has been observed for similar (styrene) oligomer diluted NMMD polystyrenes before, when the styrene oligomers were Kuhn chains. The original ‘interchain pressure’ model by Marrucci and Ianniruberto (Macromolecules 37(10):3934–3942, 2004), represents the minimal level of strain hardening in a Kuhn fluid. It has been shown previously to predict the extensional viscosities of NMMD polystyrene melts, and it is also in agreement with the extensional viscosities of the 86 kg/mole NMMD PMMA melt as well.

The transition between undiluted and oligomer-diluted states of nearly monodisperse polystyrenes in extensional flow

We have measured the startup and steady extensional viscosity of two narrow molar mass distributed (NMMD) polystyrenes, a 910 kg/mole and a 545 kg/mole, diluted in a NMMD 4.29 kg/mole styrene oligomer, with a wide concentration range from 90 down to 17%. The constant interchain pressure model, proposed by Rasmussen and Huang (Rheol Acta 53(3):199–208 (2014a)), predicts the extensional viscosity well for the dilutions with lower concentrations. However, for the 70 and 90% 545 kg/mole samples which represent the transition between the diluted and undiluted states, the model predictions are less satisfactory. Another concept based on interchain pressure, proposed by Wagner (Rheol Acta 53(10):765–777 (2014)), also shows agreement with the measured data.

Constant interchain pressure effect in extensional flows of oligomer diluted polystyrene and poly(methyl methacrylate) melts

The constant ‘interchain pressure’ idea has been addressed, to evaluate if it is an adequate quantitative assumption to describe the fluid mechanics of oligomer diluted entangled NMMD polymer systems. The molecular stress function constitutive framework has been used with the constant interchain pressure assumption. Furthermore, the maximal extensibility based on the number of Kuhn steps in an entanglement has been used based on the relative Pade inverse Langevin function. The model predictions agree with the extensional measurements on all previously published poly(methyl methacrylate)s and almost all published oligomer diluted NMMD polystyrenes. The only deviation is on the most diluted and largest molecular weight case of an 18% 1880 kg/mol polystyrene in oligomer diluent. In this case, the maximal extensibility is not needed.

A constitutive analysis of the extensional flows of nearly monodisperse polyisoprene melts

Here two particular formulations [M.H. Wagner, S. Kheirandish, O. Hassager, Journal of Rheology 49 (6) (2005) 1317–1327; H.K. Rasmussen, Q. Huang, Rheologica Acta 53 (3) (2014) 199–208; G. Marrucci, G. Ianniruberto, Macromolecules 37 (10) (2004) 3934–3942] of the ‘interchain pressure’ [39], incorporated into the molecular stress function method [M.H. Wagner, S. Kheirandish, O. Hassager, Journal of Rheology 49 (6) (2005) 1317–1327], are used to assess the extensional [J.K. Nielsen, O. Hassager, H.K Rasmussen, G.H. McKinley, Journal of Rheology 53 (6) (2009) 1327–1346; G. Liu, H. Sun, S. Rangou, K. Ntetsikas, A. Avgeropoulos, S.-Q. Wang, Journal of Rheology 57 (1) (2013) 89–104] and shear viscosities [D. Auhl, J. Ramirez, A.E. Likhtman, P. Chambon, C. Fernyhough, Journal of Rheology 52 (3) (2008) 801–835] of narrow molecular weight distributed (NMMD) polyisoprene melts. These two formulations are expected to represent the highest [M.H. Wagner, S. Kheirandish, O. Hassager, Journal of Rheology 49 (6) (2005) 1317–1327, G. Marrucci, G. Ianniruberto, Macromolecules 37 (10) (2004) 3934–3942] and lowest level [H.K. Rasmussen, Q. Huang, Rheologica Acta 53 (3) (2014) 199–208] of the ‘interchain pressure’. The needed Rouse times are here defined as τR/τmax ∝ (M/Me)−1.4 with a proportional factor of 1.4, achieved based on the viscosity measurement. τmax is the maximal relaxation time, M the molecular weight and the entanglement molecular weight Me=(4/5)ρRT/GN0 [M. Doi M, S.F. Edwards, The Theory of Polymer Dynamics; Clarendon Press: Oxford (1986)]. ρ is the density, R the gas constant, T the temperature and GN0 the plateau modulus. The method by [M.H. Wagner, S. Kheirandish, O. Hassager, Journal of Rheology 49 (6) (2005) 1317–1327, G. Marrucci, G. Ianniruberto, Macromolecules 37 (10) (2004) 3934–3942] predicts start-up of extensional viscosities significantly below the measured value. The formulations by [H.K. Rasmussen, Q. Huang, Rheologica Acta 53 (3) (2014) 199–208] seem to be in agreement with both the start-up of extension as well as the shear flow of all NMMD polyisoprenes. Potential non-isothermal effects were addressed computationally using the pseudo time principle, assuming the most critical case of adiabatic heating.

A hierarchical multi-mode MSF model for long-chain branched polymer melts part III: shear flows

A novel hierarchical multi-mode molecular stress function (HMMSF) model for long-chain branched (LCB) polymer melts is proposed, which implements the basic ideas of (i) the pom-pom model, (ii) hierarchal relaxation, (iii) dynamic dilution, (iv) interchain pressure and (v) convective constraint release relaxation mechanism. Here, the capability of this approach is demonstrated in modelling the steady shear data of a broadly distributed long-chain branched polymer melt with only two non-linear parameters, a dilution modulus and a convective constraint release (CCR) parameter.

A hierarchical multi-mode MSF model for long-chain branched polymer melts part II: multiaxial extensional flows

In part I, a novel hierarchical multi-mode molecular stress function (HMMSF) model for long-chain branched (LCB) polymer melts has been proposed, which implements the basic ideas of (i) the pom-pom model, (ii) hierarchal relaxation, (iii) dynamic dilution, and (iv) interchain pressure. Here, the capability of this approach is demonstrated in modelling the extensional viscosity data of a broadly distributed long-chain branched polymer melt in uniaxial, equibiaxial, and planar extensional deformations with only a single non-linear parameter, the dilution modulus, which quantifies the fraction of dynamically diluted chain segments.

A hierarchical multi-mode MSF model for long-chain branched polymer melts part I: elongational flow

A novel hierarchical multi-mode molecular stress function (HMMSF) model for long-chain branched (LCB) polymer melts is proposed, which implements the basic ideas of (i) the pom-pom model, (ii) hierarchal relaxation, (iii) dynamic dilution and (iv) interchain pressure. Here, the capability of this approach is demonstrated in modelling uniaxial extensional viscosity data of numerous broadly distributed long-chain branched polymer melts with only a single non-linear parameter, the dilution modulus.

From melt to solution: Scaling relations for concentrated polystyrene solutions

The scaling relations established for the relaxation modulus of concentrated solutions of polystyrene (PS) in oligomeric styrene [Wagner, Rheol. Acta 53, 765–777 (2014); Wagner, J Non-Newtonian Fluid Mech. (2015)] are applied to the solutions of PS in diethyl phthalate (DEP) investigated by Bhattacharjee et al. [Macromolecules 35, 10131–10148 (2002)] and Acharya et al. [AIP Conf. Proc. 1027, 391–393 (2008)]. The scaling relies on the difference ΔTg between the glass-transition temperatures of the melt and the glass-transition temperatures of the solutions. ΔTg can be inferred from the reported zero-shear viscosities, and the Baumgaertel, Schausberger, and Winter (BSW) spectra of the solutions are obtained from the BSW spectrum of the reference melt with good accuracy. Predictions of the extended interchain pressure (EIP) model, which is based on the assumption that the relative interchain pressure in the melt and in the solutions is identical, are compared to the steady-state elongational viscosity data of PS/DEP solutions. The Rouse stretch relaxation times as calculated from the respective scaling relation are a factor of 2–3 higher than the Rouse time defined by the experimentally observed upturn of the elongational viscosity at WiR=ε̇ τR→1. This may be caused by DEP being a good solvent at room temperature and/or polymer degradation. Using the experimentally determined Rouse times, quantitative agreement is obtained between experimental data and model. Except for a possible influence of solvent quality, linear and nonlinear viscoelasticity of entangled PS solutions can thus be obtained from the linear-viscoelastic characteristics of a reference polymer melt and the shift of the glass-transition temperature between melt and solution.

Interchain tube pressure effect in the flow dynamics of bi-disperse polymer melts

The constitutive equation as reported by Rasmussen and Huang (Rheologica Acta 53:199–208, 2014b), explaining the flow dynamics of oligomer (containing a least two Kuhn step)-diluted narrow molecular weight-distributed polymers were extended to general bi-disperse polymer melt system. It was assumed that the Rouse time of a particular polymer chain is dependent on the total number of Kuhn steps of the polymers in direct contact with the considered polymer chain. This number of Kuhn steps is proportional to the weight average molecular weight, M w , replacing the involved molecular weight in the equation as reported by Rasmussen and Huang (Rheologica Acta 53:199–208, 2014b). Two separate stretch evolution equations for the long and the short polymer respectively, were introduced to handle the two involved Rouse times. Experimentally, the bi-disperse polystyrene systems of Nielsen et al. (J Rheol 50:453–476, 2006) and polyisoprene systems as reported by Read et al. (J Rheol 56:823–873, 2012) were within quantitatively agreement with the derived model. This included both startup of extension as well as shear flow. One exception was observed. In the most diluted polyisoprene blend, the measured extensional viscosities were under predicted by the model.

An extended interchain tube pressure model for elongational flow of polystyrene melts and concentrated solutions

Abstract An extended interchain tube pressure model for polymer melts and concentrated solutions is presented, based on the idea that the pressures exerted by a polymer chain on the walls of an anisotropic confinement are anisotropic (Doi and Edwards, 1986). In a tube model with variable tube diameter, chain stretch and tube diameter reduction are related, and at deformation rates larger than the inverse Rouse time τR, the chain is stretched and its confining tube becomes increasingly anisotropic. Tube diameter reduction leads to an interchain pressure in the lateral direction of the tube, which is proportional to the 3rd power of stretch (Marrucci and Ianniruberto, 2004). In the extended interchain tube pressure model, it is assumed that chain stretch is balanced by interchain tube pressure in the lateral direction, and by a spring force in the longitudinal direction of the tube, which is linear in stretch. The elongational viscosity data of Huang et al. (2013) are in agreement with the assumption that dilution of polystyrene by oligomeric styrene does not change the relative interchain tube pressure. Quantitative agreement between highly nonlinear viscoelastic experiments in elongation and predictions for polystyrene melts and concentrated solutions of polystyrene in oligomeric styrene is obtained based exclusively on the relaxation modulus of the polymer melt, the volume fraction of polymer in the solution and the time-temperature shift caused by the reduction of the glass transition temperature Tg of the polymer in solution relative to Tg of the melt.

Scaling relations for elongational flow of polystyrene melts and concentrated solutions of polystyrene in oligomeric styrene

A consistent model of the rheology of polymer melts and concentrated solutions is presented, based on the idea that the pressures exerted by a polymer chain on the walls of an anisotropic confinement are anisotropic (Doi and Edwards. The Theory of Polymer Dynamics, Oxford University Press, 1986). In a tube model with variable tube diameter, chain stretch and tube diameter reduction are related, and at deformation rates larger than the inverse Rouse time τ R, the chain is stretched and its confining tube becomes increasingly anisotropic. Tube diameter reduction leads to an interchain pressure in the lateral direction of the tube (Marrucci and Ianniruberto. Macromolecules 37:3934-3942, 2004). Chain stretch is balanced by interchain tube pressure in the lateral direction, which is proportional to the third power of stretch, and by a spring force in the longitudinal direction of the tube, which is linear in stretch. Analyzing elongational viscosity data of Huang et al. (Macromolecules 46:5026-5035, 2013a; ACS Macro Letters 2:741-744, 2013b) shows that dilution of polystyrene by oligomeric styrene does not change the relative interchain tube pressure. Based on this extended interchain pressure concept, scaling relations for linear viscoelasticity and elongational viscosity of polystyrene melts and concentrated solutions of polystyrene in oligomeric styrene are presented based exclusively on the relaxation modulus of a reference polymer melt, the volume fraction of polymer in the solution, and the time-molar-mass shift as well as the time-temperature shift caused by the reduction of the glass transition temperature T g of the polymer in a solution relative to T g of the melt.

Interchain tube pressure effect in extensional flows of oligomer diluted nearly monodisperse polystyrene melts

We have derived a constitutive equation to explain the extensional dynamics of oligomer-diluted monodisperse polymers, if the length of the diluent has at least two Kuhn steps. These polymer systems have a flow dynamics which distinguish from pure monodisperse melts and solutions thereof, if the solvent has less than two Kuhn steps, e.g. is not a chain. The constitutive equation is based on a phenomenological tube-based model within the methodology of the molecular stress function approach. The nonlinear dynamics have been explained as a consequence of a constant thermal interchain pressure originating from the short polymer chains (e.g. the oligomers) on the wall of the tube containing the long chains. The nonlinear dynamics are uniquely defined by the Rouse time and the maximal extensibility of the long polymer chains. Both are linked to the entanglement length. The relation between the Rouse times and entanglements have been established based on published extensional experiments on nearly monodisperse polystyrene melts. The constitutive equation has shown agreement with the experimental startup of and steady extension data from Huang et al. (Macromolecules 46:5026–5035, 2013a) based on 285 and 545 kg/mol polystyrenes diluted in styrene oligomers containing 3.3 (1.92 kg/mol) and 7.3 (4.29 kg/mol) Kuhn steps.