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Use of the full-network model at the highlight of the logarithmic strain measure
Published in Per-Erik Austrell, Leif Kari, Constitutive Models for Rubber IV, 2017
The full-network model has been left apart for long due to numerical difficulties of computation. By using a Padé approximant of the inverse Langevin function, the model becomes relevant. Due to its accurate representation of a macromolecular network, this model is a good opportunity for a better understanding of the network deformation. By using the logarithmic invariants and the macromolecular full-network model, it has been shown that the theoretical response of rubber-like material is independent of the state of strain for small strains. It has also been demonstrated that for a fixed loading intensity, the molecular chains are significantly more extended in uniaxial tension than in equibiaxial tension. Consequently, the well known upturn observed on stress-strain curves, which indicates that chains are reaching their limit of extension, appears earlier in uni-axial tension than in equibiaxial tension. These observations are completely consistent with those observed when Treloar’s data are plotted by using the logarithmic invariants.
Magnetic Anisotropy of Nanocomposites Made of Magnetic Nanoparticles Dispersed in Solid Matrices
Published in Mahmood Aliofkhazraei, Advances in Nanostructured Composites, 2019
where VNP is the nanoparticle volume, the magnetization of the nanocomposite is equal to zero in the absence of an external magnetic field, and increases according to the Langevin function (Jacobs and Bean 1963) when magnetic field is applied, and in the absence of interactions (Figure 3b) (similar to a system of paramagnetic atoms). In this case the saturation magnetization of nanocomposite is () Msat,NC=nmNP,
An affine full network model for strain-induced crystallization in rubbers
Published in Alexander Lion, Michael Johlitz, Constitutive Models for Rubber X, 2017
A. Nateghi, M.A. Keip, C. Miehe
where k is the Boltzmann constant and L−1 is the inverse Langevin function. Thus, the total configurational entropy of the semi-crystalline chain with respect to the fully-crystalline chain is () s=s1+s2.
Ionic liquid templated novel porous anionic cobalt(II) coordination framework based on rod-shaped metal-carboxylate chains
Published in Inorganic and Nano-Metal Chemistry, 2019
Jian-Hua Qin, Ya-Dan Huang, Fei-Fei Li
Here S = 1 and u is the well-known Langevin function, u = coth[JS(S + 1)/kT] – kT/JS(S + 1). The least-squares analysis of magnetic susceptibilities data led to g = 2.09, J = –2.99 cm−1, and R = 8.98 × 10−5.
Manganese(II) and zinc(II) coordination polymers based on 2-(5-bromo-pyridin-3-yl)-1H-imidazole-4,5-dicarboxylic acid: synthesis, structure and properties
Published in Journal of Coordination Chemistry, 2019
Yafang Ge, Guoting Li, Dongxia Fu, Lina Liu, Benlai Wu
Compounds based on bis-chelated MnII ions usually display interesting magnetism [31, 44, 45], and thus the temperature-dependent magnetic susceptibilities (χM) of polycrystalline sample 1 were measured at a field of 1000 G from 2.0 to 300 K. As shown in Figure 5, the χMT value of 4.65 cm3 mol−1 K per MnII at room temperature is slightly more than the spin-only value of 4.38 cm3 mol–1 K per MnII (S= 5/2, g= 2). Upon cooling from 300 K, the χMT value decreases continuously down to 0.59 cm3 mol–1 K at 2 K, while the χM value reaches a maximum value around 4 K. The magnetic behaviors of 1 indicate a typical antiferromagnetic interaction between MnII ions. Its χM–1 vs. T plot from 85 to 300 K follows the Curie–Weiss law with C= 4.69 cm3 mol–1 K and θ=–7.28 K, and the negative θ value confirms antiferromagnetic interactions between neighboring MnII ions. Based on the above structural analysis, the system can be treated as a magnetically regular chain where the antiferromagnetic coupling is exchanged by imidazoledicarboxylate bridges. The interchain magnetic interactions in 1 should be negligible because the longer bridge between the imidazoledicarboxylate-chelating MnII and the Npyridyl-binding MnII excludes efficient magnetic couplings, owing to the obviously non-coplanar structure between the imidazoledicarboxylate and pyridyl in the ligand (HL)2– (vide supra). Consequently, the intrachain magnetic interaction (J) in 1 can be evaluated using the classical spin expression derived by Fisher for isotropic Heisenberg chains (H=–J∑SiSi+1) [46]: where u is the well-known Langevin function defined as u = coth[JS(S+ 1)/kT] – kT/[JS(S+ 1)] with S= 5/2. The best fit of the experimental magnetic susceptibility data of 1 in the whole temperature range to Equation (1) gives the following parameters: J=–0.71(4) cm–1, g= 2.05(2), and R= 2.7 × 10–6 (the agreement factor R=∑[(χMT)obsd – (χMT)calc]2/∑[(χMT)obsd]2). The small J value indicates weak antiferromagnetic coupling within the imidazoledicarboxylatebridged manganese(II) helical chains, which is comparable to the J value of –1.34 cm–1 found in the reported 4,5-imidazoledicarboxylate-bridged MnII 3D polymer [Mn3(IMDC)2(H2O)4] (IMDC = 4,5-imidazoledicarboxylate) [38], but is larger than the J value of –0.11 cm–1 found in the reported 4,5-imidazoledicarboxylate-bridged MnII zigzag chain polymer [Mn(phen)(Hdcbi)]n (phen = 1,10-phenanthroline, Hdcbi = 4,5-imidazoledicarboxylate) [39].