H. Henning Winter

Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point

We suggest a very simple memory integral constitutive equation for the stress in crosslinking polymers at their transition from liquid to solid state (gel point). The equation allows for only a single (!) material parameter, the strength S[Pas1/2], and it is able to describe every known viscoelastic phenomenon at the gel point. Measurements were performed on polydimethylsiloxane model networks with balanced stoichiometry for which the crosslinking reaction has been stopped at different degrees of conversion. At the gel point, the loss and storage moduli were found to be congruent and proportional to ω1/2 over a wide range of temperature (−50°C to +180°C) and five decades of frequency ω. The hypothesis is made that this behavior is valid in the entire range 0<ω<∞. This congruence hypothesis is consistent with the Kramers‐Kronig relation and leads to a constitutive equation which shows that, for our polymer, congruent functions G′(ω)=G″(ω) are as much a rheological property at the gel point as are infinite ...

Linear Viscoelasticity at the Gel Point of a Crosslinking PDMS with Imbalanced Stoichiometry

The evolution of linear viscoelasticity during cross‐linking of a stoichiometrically imbalanced polydimethylsiloxane (PDMS) was measured by small amplitude oscillatory shear. At the gel point (GP), stress relaxation was found to follow a power law, St−n, as described by the previously suggested gel equation. However, while stoichiometrically balanced gels (PDMS, polyurethanes) gave the specific exponent value of n=1/2, a higher exponent value, 1/2 Gv′(ω), and a higher rate of stress relaxation. GP was found to occur before the crossover point of the loss and storage moduli, G″(ω0,t), and G′(ω0,t), as measured during the cross‐linking reaction (reaction time, t) at constant frequency, ω0. This suggests new met...

Rheology of Polymers Near Liquid-Solid Transitions

Polymeric materials near the liquid-solid transition (LST) exhibit a very distinct relaxation pattern. The reference point for analyzing these patterns is the instant of LST at which relaxation becomes self-similar over wide ranges of the relaxation time. The universality of this transition and its consequences have been explored extensively during the past decade. This study will present an overview of rheological implications inherent in liquid-solid transitions of polymers. The LST can be most reliably detected in a dynamic mechanical experiment in which the frequency independence of the loss tangent marks the LST. A wide variety of rheological observations of materials in the vicinity of an LST are discussed with respect to their universality. It is shown that polymer chemistry, molecular weight, stoichiometry, temperature, inhomogeneities, etc. greatly influence the material behavior near the LST. However, the characteristic self-similar relaxation is shown by all investigated materials, independent of the nature of the LST (e.g., both, physically and chemically crosslinking polymers). Several theories predict chemical and rheological properties in the vicinity of an LST. They are briefly discussed and compared with experimental results. A variety of applications for polymers near LST are presented that either already exist or can be envisioned. The self-similar relaxation behavior which results in a power law relaxation spectrum and modulus is not restricted to materials near LST. Different classes of polymers are described that also show power law relaxation behavior. What makes the self-similar relaxation specific for materials at LST is its occurrence at long times with the longest relaxation time diverging to infinity.

Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G′(ω) andG″(ω), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersgi,λi,i = 1,2, ⋯N, of the discrete spectrum to obtain a best fit ofG′, G″, and it was found to be essential that bothgi andλi are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesgi,0,λi,0,N0. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ0,η, Jd,i andΛi of the creep complianceJ(t) can then readily be calculated using the Laplace transform.

Stopping of crosslinking reaction in a PDMS polymer at the gel point

SummaryA method has been developed to stop the crosslinking reaction of a polydimethylsiloxane system without disturbing the state of the sample. Oscillatory shear experiments on samples just before and just beyond the gelation point demonstrated the transition of the material from a viscoelastic liquid to a viscoelastic solid. At the gel point the loss modulus and the storage modulus were found to be identical over several decades of frequency and for temperatures ranging between -50°C and +180°C. Both moduli were proportional to the square root of the frequency.

The relaxation of polymers with linear flexible chains of uniform length

The analysis of dynamic mechanical data indicates that linear flexible polymer chains of uniform length follow a scaling relation during their relaxation, having a linear viscoelastic relaxation spectrum of the formH(λ) = n1GN0 × (λ/λmax)n1 forλ≤λmax. Data are well represented with a scaling exponent of about 0.22 for polystyrene and 0.42 for polybutadiene. The plateau modulusGN0 is a material-specific constant and the longest relaxation time depends on the molecular weight in the expected way. At high frequencies, the scaling behavior is masked by the transition to the glassy response. Surprisingly, this transition seems to follow a Chambon-Winter spectrumH(λ) = Cλ−n2, which was previously adopted for describing other liquid/solid transitions. The analysis shows that the Rouse spectrum is most suitable for low molecular-weight polymersM ≈ Mc, and that the de Gennes-Doi-Edwards spectrum clearly predicts terminal relaxation, but deviates from the observed behavior in the plateau region.

Large Step Shear Strain Experiments with Parallel‐Disk Rotational Rheometers

A new method is proposed and tested for step shear strain experiments with parallel‐disk rotational rheometers. The method is applicable to large strains, i.e., outside the linear viscoelastic region. The nonhomogeneity of the strain in the parallel‐disk rheometer is accounted for by a correction term which is similar to the well‐known Rabinowitsch correction in capillary rheometry. The transient shear relaxation moduli of a low‐density polyethylene (150°C) and of a polystyrene (180°C) from this method agree very well with equivalent data from a cone‐and‐plate rheometer. The two different geometries give an overlapping set of data; small‐strain data (γ=0.1−5) from cone‐and‐plate and large‐strain data (γ=0.4−25) from parallel disk. The step strain data support the separability of the relaxation modulus into time‐ and strain‐dependent functions. The strain dependence is well approximated by a sigmoidal function. The data were obtained with a Rheometrics dynamic spectrometer having a maximum angular displace...

Rheology and structure of isotactic polypropylene near the gel point: quiescent and shear-induced crystallization

Abstract Crystallizing isotactic polypropylene (iPP) develops large-scale spherulites and thick threads, large enough for observation by optical microscopy, and undergoes a liquid-to-solid transition as an expression of increased connectivity of the structure. In order to relate the time scales of structural and rheological changes, we measured time-resolved small-angle light scattering (SALS) and transmittance properties in a single experimental run, which then was repeated in an optical microscope for direct observation of growth of large-scale structures, and in a rheometer for mechanical spectroscopy. The results for quiescent and shear-enhanced melt crystallization of a high molar mass iPP are presented. In quiescent crystallization, iPP nuclei are only formed in the beginning (and not in later stages) and grow simultaneously at the same rate, which leads to spherulites of equal size. The critical gel point is reached close to the instant of the maximum of density fluctuations, but before spherulites impinge. Crystallinity estimates from Hv SALS (estimation method of Stein) are much higher than values from DSC. The discrepancy may be caused (in addition to the simplifying assumptions in the estimate) by the enhanced crystallization in the rheo-optical cell due to surface and sample loading effects. Shear flow induces anisotropic molecular conformations, preferably in the high molecular weight component of the iPP sample. The resulting orientation fluctuations (of highly oriented long chains and less oriented short chains) cause (1) an increase in nucleation rate, (2) possibly an increase in crystallization rate and (3) formation of highly elongated structures (threads) which are visible in the optical microscope and in anisotropic SALS patterns. The threads thicken until, at later stages, additional spherulites start to grow, presumably from the shorter chains and nucleated on by the threads.

Interrelation between continuous and discrete relaxation time spectra

Abstract A relation has been derived between the continuous and the discrete forms of linear viscoelastic relaxation-time spectra. Both forms can be interconverted, and they are equivalent in their ability to reproduce G′(ω), G″(ω), or G(t) data. The linear superposition of even only a few Maxwell-modes is able to mimic the continuous form. This rapid conversion leads to the ‘parsimonious’ spectrum which models the linear viscoelastic relaxation with the smallest possible number of modes. Typical experimental spectra of broadly distributed and of monodisperse polymers provide the means for testing the proposed relations. Effects of noise and data density were studied for G′, G″-data sets.

Fourier transform mechanical spectroscopy of viscoelastic materials with transient structure

Abstract A new technique is presented to measure the frequency dependent complex modulus simultaneously at several frequencies instead of consecutively as in a frequency sweep. For this purpose, the strain in a dynamical mechanical experiment is prescribed as a superposition of several different modes (three in our case). The resulting stress is decomposed into sinusoidal components, each of them characterized by their frequency, amplitude, and phase shift with respect to the corresponding strain component. Phase shift and amplitude are expressible in a frequency dependent complex modulus. A single experiment gives, therefore, values for the complex modulus at a set of prescribed frequencies. The method was demonstrated on three stable viscoelastic fluids and was applied to determine the instant of sol-gel transition (gel point) of a crosslinking polymer.