An analysis is made of the distribution function and its moments for a linear combination of three randomly chosen orthogonal components of the so‐called radius of gyration of an unrestricted random‐flight chain, and certain averages of moments are obtained of the three‐dimensional distribution W (L12L22L32), where L1 ≤ L2 ≤ L3 are the orthogonal components of the radius of gyration along the principal axes of inertia of the chain The strong departures of the chain shape from spherical symmetry indicated by these results are confirmed and complemented by Monte Carlo studies of unrestricted random walks on a simple cubic lattice A surprisingly high ratio of principal components is found for chains with 50 and 100 bonds, 〈L32〉: 〈L22〉: 〈L12〉 ≃ 117:27:1

Shape of a Random‐Flight Chain

Random Walks: Theory and Selected Applications

The relation, KIcY2 = (Σ*/2.35)3, where KIc is plane strain fracture toughness, Y is the local value of yield stress in the crack-tip plastic zone, and Σ* is the cleavage stress of an unnotched bulk specimen, is shown to be a reasonable approximation to the data from a large number of investigations. The relation is shown to be consistent with the results of metallographic studies of subcritical crack formation. Methods of extrapolating KIc to high strain rates and low temperatures are suggested.

The variation of K Ic with temperature and loading rate

L’industrie de traitement de surfaces (TS) utilise de nombreux produits chimiques, en particulier des metaux toxiques et des substances organiques, qui sont connus pour etre nocifs vis-a-vis des humains et de l’environnement. En raison d’une reglementation de plus en plus stricte, les effluents contenant des metaux lourds (Zn, Cu, Ni, Fe, Al, Cr, etc.) et de la charge organique caracterisee par le parametre DCO (demande chimique en oxygene) doivent etre traites. Il existe une grande variete de procedes de traitement des eaux usees provenant de la filiere traitement de surface. En general, les systemes conventionnels dedetoxification se composent principalement d’un ajustement de pH, d’une oxydation des cyanures et d’une reduction des chromes, suivi d’une precipitation sous forme d’hydroxydes et d’une clarification, et parfois d’une etape d’adsorption sur charbonpour enlever la DCO residuelle. Cependant, la technologie sur charbon actif presente plusieurs problemes tels que la saturation rapide et des problemes de colmatage et de regeneration des reacteurs. Comme les resines d’echange d’ions, leur utilisation est restreinte due au cout eleve. Pour ces raisons, la plupart des petites et moyennes entreprises ne peuvent pas utiliser de tels traitements. Ainsi, afin de repondre a ces problematiques, de nombreuses etudes sont menees pour trouver des alternatives peu couteuses, qui soient efficaces et acceptables pour un usage industriel. Dans cette etude, la bio-adsorption sur un bio-adsorbant d’amidon reticule a ete utilisee pour retenir les metaux et la DCO contenus dans differents effluents industriels. L’adsorption a ete etudiee en fonction du temps de contact, de la masse d’adsorbant et de la charge polluante (concentration, nature). L’influence de ces parametres sur l’efficacite d’adsorption a ete evaluee en utilisant une methode conventionnelle en mode cuveedite methode en batch. Les resultats ont montre que le materiau presente des capacites d’adsorption elevees vis-a-vis des ions metalliques, ce qui permet de diminuer les concentrations metalliques en dessous des valeurs reglementaires. Des resultats similaires ont ete obtenus vis-a-vis de la charge organique, et ceci quelque soit le type d’effluent. Afin d’evaluer l’impact des rejets industriels provenant de quatre entreprises differentes de TS, nous avons effectue des tests ecotoxicologiques normalises bases sur l’utilisation de graines de laitue Lactuca sativa. Nous avons mesure le taux de germination, la longueur et la masse deslaitues. Les resultats ont permis de comparer la toxicite des quatre rejets: les rejets contenant du cuivre et du nickel ont un impactbeaucoup plus eleve que ceux contenant du zinc ou de l’aluminium. De plus, les tests de germination effectues a l’aide de solutions synthetiques ont confirme que les melanges de metaux presentent une toxicite plus eleveeque celle de leurs constituants consideres isolement. Les testsde phytotoxiciteont ete egalement realises sur les rejets industriels avant et apres adsorption. Ces tests biologiques ont confirme l’efficacite du procedepour diminuer fortement la toxicite du rejet. L’abattement chimique et la reduction de la toxicite ont demontre que la bio-adsorption sur un materiau non-conventionnel peut etre une etape de finition interessante pour la detoxification des rejets industriels.Enfin, les tests biologiques normalises sont rapides, faciles a mettre en œuvre et parfaitement reproductibles. Ils pourraient etre utilises en routine pour evaluer les impacts d’un rejet industriel, meme lorsque celui-ci satisfait aux exigences reglementaires

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Développement de la bio-adsorption pour décontaminer des effluents de rejets industriels : abattement chimique et gain environnemental

Investigation of the spatial segment density distribution (SSDD) of a Gaussian coil, generated using an off-lattice Monte Carlo method, revealed detailed information about the instantaneous shape. This information is used to compare shape models based on the principal components of the radius of gyration with an analysis in terms of block copolymer shape employing characteristic separation parameters. To speed up an accurate analysis, an analytical procedure for calculating both eigenvalues and -vectors of the radius of gyration tensor is developed. The observed averaged shapes are essentially bimodal (dumbbell-like) and do not depend on chain length. The numerical results confirm that an ideal homopolymer is to a large extent segregated into two equal parts. The influences of the left and right parts of a chain on the total shape are presented. These results, together with the evaluation of the density of special segments and the distribution of the separation angle between the parts, reveal the approxim...

Bimodality in the spatial segment density distribution of Gaussian chains

Probability density functions for the orthogonal components of the radius of gyration, Si, of a random‐flight chain were computed using an indirect numerical integration algorithm which was based upon a multiple convolution proposed by Forsman and Hughes. Computations were performed for relatively long chains of 100–1000 statistical segments t. By using a reduced radius of gyration ξi = (π / 23) Si〈Si2〉−1 / 2 the distributions converged to a limiting function at about t = 500, except for extremely small radii, ξi ≤ [10(2t)1/2]−1. Additional computations using the one‐dimensional distributions yielded the corresponding probability densities for both the two‐ and three‐dimensional (polar) radii. The computed functions were in excellent agreement with all their limiting properties previously determined by analytical methods.

Computed Probability Density Functions of the Radii of Gyration of Long Random-Flight Chains

Distribution functions for the one‐dimensional radii of gyration were calculated for random flight chains of 2, 4, and 10 statistical segments, and were compared with approximate distributions calculated with the “long‐chain” eigenvalues. The correct distributions are broader and flatter than their long‐chain counterparts and have larger moments. The true distributions are not well represented by the long‐chain approximation when the number of statical segments t is less than 10. But, considering the rate at which the two distributions approach each other with increasing t, we feel confident that the long‐chain eigenvalues would be adequate for t ≥ 100. Indeed, for many practical considerations, one might take the long‐chain distributions as reasonable approximations to the correct ones, even for t ≥ 10.

Probability Density Functions of the Radii of Gryation of Short Random Flight Chains

Average polymer segment densities and thermodynamic properties of polymer adsorbed at liquid–solid interfaces were computed by extension of the polymer adsorption theory of Forsman and Hughes. Expressions were derived for the total free energy of adsorbed polymer chains by using the Flory-Huggins theory to represent free energy of mixing. A square-well potential was used to represent segment–surface interaction, and configurational entropy was calculated from the probability density function for the radius of gyration of random-flight chains. For each specified amount of surface coverage the free energy of the adsorbed polymer was minimized by varying the density of segments normal to the adsorbing surface and using a modified gradient search algorithm on a digital computer. Two different segment densities were considered, and they both gave qualitatively the same results. The two densities were (1) the sum of two Gaussian distributions and (2) a two-step density distribution. Isotherms were then calculated by equating the partial molal free energy of polymer at the surface to that of polymer in bulk solution for each specified amount of surface coverage. The results showed that for the initial region of the isotherms the distribution of polymer segments normal to the surface consisted of a high-density layer adjacent to the surface and a low-density “tail” extending far out into the solution. At higher amounts of adsorbed polymer, i.e., in the general concentration range of the pseudo-plateau, the tail of the polymer density distribution was predicted to thicken, and a single Gaussian distribution best described the segment density. Predicted adsorptions were in good agreement with reported experimental values.

https://www.onlinelibrary.wiley.com/doi/pdf/10.1002/pol.1970.160081017

Use of numerical methods in the prediction of polymer adsorption

Poly(methyl methacrylate) tensile bars were prepared containing nearly spherical polytetrafluoroethylene particles in concentrations from one to a thousand particles per gauge length of the bars. Particle diameters varied from 0.0035 to 0.018 in. Exhaustive tensile tests were performed at sufficiently high strain rate to assure brittle fracture and the results analyzed statistically by the theory of extreme values as proposed by Epstein. The results suggested that the polytetrafluoroethylene particles themselves did not act as flaws, but that they intensified the stress field on natural flaws which acted as the origin of fracture. Assuming a Laplace distribution as the underlying distribution of tensile strength (not to be confused with observed distribution of tensile strengths) gave predicted fracture statistics in good agreement with experiment.

W. C. Forsman

Papers

Computed Probability Density Functions of the Radii of Gyration of Long Random-Flight Chains

Probability Density Functions of the Radii of Gryation of Short Random Flight Chains

Use of numerical methods in the prediction of polymer adsorption

Statistics of extreme values applied to the brittle fracture of poly(methyl methacrylate) containing polytetrafluoroethylene particles