The Aquatic Biota and Groundwater Quality of Springs in the Lincoln Hills, Wisconsin Driftless, and Northern till Plains Sections of Illinois

ID: 8307INHS Technical Report prepared for Environmental Protection Trust Fund Commission and Illinois Department of Natural Resources Division of Energy and Environmental AssessmentU of I OnlyRestriction applied due to concern over geolocation information of springs on private property

Rheological implications of completely monotone fading memory

In the constitutive equation modeling of a (linear) viscoelastic material, the “fading memory” of the relaxation modulusG(t) is a fundamental concept that dates back to Boltzmann [Ann. Phys. Chem. 7, 624 (1876)]. There have been various proposals that range from the experimental and pragmatic to the theoretical about how fading memory should be defined. However, if, as is common in the rheological literature, one assumes that G(t) has the following relaxation spectrum representation: G(t)=∫₀∞ exp(−t/τ)[H(τ)/τ]dτ, t > 0, then it follows automatically that G(t) is a completely monotone function. Such functions have quite deep mathematical properties, that, in a rheological context, spawn interesting and novel implications. For example, because the set of completely monotone functions is closed under positive linear combinations and products, it follows that the dynamics of a linear viscoelastic material, under appropriate stress–strain stimuli, will involve a simultaneous mixture of different molecular interactions. In fact, it has been established experimentally, for both binary and polydisperse polymeric systems, that the dynamics can simultaneously involve a number of different molecular interactions such as the Rouse, double reptation and/or diffusion, [W. Thimm et al., J. Rheol., 44, 429 (2000); F. Léonardi et al., J. Rheol. 44, 675 (2000)]. The properties of completely monotone functions either yield new insight into modeling of the dynamics of real polymers, or they call into question some of the key assumptions on which the current modeling is based, such as the linearity of the Boltzmann model of viscoelasticity and/or the relaxation spectrum representation for the relaxation modulusG(t). If the validity of the relaxation spectrum representation is accepted, the resulting mathematical properties that follow from the complete monotonicity of G(t) allows one to place the classical relaxation model of Doi and Edwards [M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans. 2 74, 1789 (1978)], as a linear combination of exp(−t/τ*) relaxation processes, each with a characteristic relaxation time τ*, on a more general and rigorous footing

On the scaling of molecular weight distribution functionals

When formulating a constitutive equation model or a mixing rule for some synthetic or biological polymer, one is essentially solving an inverse problem. However, the data will not only include the results obtained from simple step strain, oscillatory shear, elongational, and other experiments, but also information about the molecular weight scaling of key rheological parameters (i.e., molecular weight distribution functionals) such as zero-shear viscosity, steady-state compliance, and the normal stress differences. In terms of incorporating such scaling information into the formulation of models, there is a need to understand the relationship between various models and their molecular weight scaling, since such information identifies the ways in which molecular weight scaling constrains the choice of possible models. In Anderssen and Mead (1998) it was established formally that the members of a quite general class of reptation mixing rules all had the same molecular weight scaling. The purpose of this paper is to first introduce the concept of a generalized reptation mixing rule, which greatly extends the class examined by Anderssen and Mead, and then show that all such rules have the same molecular weight scaling. The proof is similar to that given by Anderssen and Mead, but uses the implicit function theorem to establish the uniqueness of the mean values which arise when invoking various integral mean-value representations for the molecular weight distribution functionals considered. The rheological significance of the new generalized two-parameter mixing rule, proposed in this paper, is examined in some detail in the conclusions. In particular, it is used to established how one must construct a mixing rule for a general polydispersed polymer where the molecular dynamics involves some single, some double and some higher levels of multiple reptation. The work of Maier et al. (1998) and Thimm et al. (2000) is then utilized to illustrate and validate this proposal

Fraxinus nigra Pott

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Goodyera pubescens (Willd.) R. Br.

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Vascular Flora of Hooper Branch Savanna Nature Preserve, Iroquois County, Illinois

INHS Technical Report prepared for Illinois Department of Natural Resources, Division of Natural Heritag

Phemeranthus rugospermus (Holz.) Kiger

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