There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Enabling nanotechnology with self assembled block copolymer patterns

In this review we intend to provide a relatively comprehensive summary of the work of supramolecular hydrogelators after 2004 and to put emphasis particularly on the applications of supramolecular hydrogels/hydrogelators as molecular biomaterials. After a brief introduction of methods for generating supramolecular hydrogels, we discuss supramolecular hydrogelators on the basis of their categories, such as small organic molecules, coordination complexes, peptides, nucleobases, and saccharides. Following molecular design, we focus on various potential applications of supramolecular hydrogels as molecular biomaterials, classified by their applications in cell cultures, tissue engineering, cell behavior, imaging, and unique applications of hydrogelators. Particularly, we discuss the applications of supramolecular hydrogelators after they form supramolecular assemblies but prior to reaching the critical gelation concentration because this subject is less explored but may hold equally great promise for helping ...

Supramolecular Hydrogelators and Hydrogels: From Soft Matter to Molecular Biomaterials

Selectivity in propene polymerization with metallocene catalysts.

Novel injectable neutral solutions of chitosan form biodegradable gels in situ

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 ...

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

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...

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

Dynamic mechanical measurements allow direct determination of the instant at which a network polymer gels. In such an experiment, the evolution of G′(t,ω0) and G″ (t,ω0) is measured in small amplitude oscillatory shear as a function of cross-linking time t. The frequency ω0 is kept constant throughout. At the beginning of the experiment, G″ is orders of magnitude larger than G′, and at completion of reaction, this order is reversed. It recently has been suggested by Tung and Dynes that the gel point (GP) might occur at the time at which G′ and G″ cross each other. However, there is much dispute whether GP occurs exactly at the crossover or just somewhere in its vicinity. This study resolves the dispute by modeling the rheological behavior at GP: There is only one class of network polymers for which GP coincides with the crossover. This class of polymers exhibits, when reaching GP, power law relaxation G(t) ∼ t−n with a specific exponent value n = 1/2. Examples are stoichiometrically balanced network polymers and networks with excess cross-linker, however, only at temperatures much above the glass transition. Otherwise, the power law behavior would be masked by vitrification. Power law relaxation seems to be property of polymers at GP in general. However, some polymers have a different exponent value, n ≠ 1/2, in which case the crossover occurs before GP (for n 1/2); i.e. the crossover cannot be used for detecting GP. While there are no networks known to us with n 1/2. A new method is suggested for measuring GP of these imbalanced networks.

Can the gel point of a cross-linking polymer be detected by the G′ – G″ crossover?

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.

/pdf/rheology-of-polymers-near-liquid-solid-transitions-51fps6zhlz.pdf

Rheology of Polymers Near Liquid-Solid Transitions

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 parametersg

 i
 ,λ

 i
 ,i = 1,2, ⋯N, of the discrete spectrum to obtain a best fit ofG′, G″, and it was found to be essential that bothg

 i
 andλ

 i
 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 valuesg

i,0,λ

i,0,N
0. 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 parametersJ
0,η, J

 d,i
 andΛ

 i
 of the creep complianceJ(t) can then readily be calculated using the Laplace transform.

H. Henning Winter

Papers

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

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

Can the gel point of a cross-linking polymer be detected by the G′ – G″ crossover?

Rheology of Polymers Near Liquid-Solid Transitions

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