Benoit, theory

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

Benoit Champagne (95), Laboratoire de Chimie Thdorique Appliqude, Facultds Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium Marshall G. Cory (357), Quantum Theory Project, University of Florida, Gainesville, Florida 32611... 

Smoluchowski (1908), Einstein (1910), Ornstein Zernike (1914, 1918). In a textbook on scattering HIGGINS Benoit ([136], Sect. 7.6) consider the fluctuation theory from a different point of view. 

The theory of LS from copolymer solutions was pioneered by Stockmayer et al. and later by Bushuk and Benoit, subsequent developments being due to Benoit, Inagaki et al Molecular weight determination constitutes only a part of the most recent review of the field by Benoit and Froelich149 ... 

A direct consequence of the development of hydrodynamic volume theory In SEC has been the universal calibration method as introduced by Benoit (17). Universal calibration methodology is based upon the fact that retention in SEC can be described as a function of the hydrodynamic volume of polymer molecules. 

This chapter summarizes important data for intrinsic viscosity and translational friction coefficient of polypeptides. The first half of the chapter discusses the data obtained in helicogenic solvents and in helix-breaking solvents. It is actually a supplement to the review article by Benoit et al. (61), in which such data published by 1967 were surveyed critically. The second half of the chapter is concerned with the helix-coil transition region. The context here is largely descriptive because of the lack of relevant theory. 

Ammonium, Metal(Metal Ammonium). Solutions of mercury arid the alkali metals in liq ammonia were first prepd and studied by Weyl (Ref 1). According to his views, the metals are joined to the nitrogen of ammonia, forming substituted ammonium radicals. Many investigators later studied the solns of metals in liq ammonia and while a few of the investigators were in favor of Weyl s theory(Joannis in 1892, Moissan in 1898 and Benoit in 1923), others were against it(Seeley in 1871, Ruff and Geisel in 1906, Kraus in 1908, Biltz in 1920 etc)... 

Table 6.3 provides a summary of the different microscopic techniques that have been applied to hydrate studies and the type of information that can be obtained from these tools. The following discussion provides a brief overview of the application of diffraction and spectroscopy to study hydrate structure and dynamics, and formation/decomposition kinetics. For information on the principles and theory of these techniques, the reader is referred to the following texts on x-ray diffraction (Hammond, 2001), neutron scattering (Higgins and Benoit, 1996), NMR spectroscopy (Abragam, 1961 Schmidt-Rohr and Spiess, 1994), and Raman spectroscopy (Lewis and Edwards, 2001). 

An introduction to the theory of neutron scattering can be found, for instance, in a books by Hansen and Donald (1986) and Higgins and Benoit (1994). The scattering function for a single macromolecule is known for some models of... 

These results are now in terms of correlation functions of P2Ct) and, in the field on cases, of P (t), as in linear dielectric and other response theories. The asymmetry given by the last term of the field on expression reveals the interesting possibility of obtaining both P correlations (from the field off response) and P correlations (from the asymmetry). One easily verifies that Benoit s result is obtained for diffusional reorientations, as then = (l/5)exp(-6D t), -(2/15)exp(-6Dt), and PoP2PlPl(t)> = (2/15)exp(-2D t)7 but this form of dynamics is not assumed in the development End effects of joint correlations Eire included in eqs.13 and 14. 

Effects due to macromolecular shape changes during single-contact interactions can, within the Benoit-Benmouna theory, be included in an ad-hoc fashion by renormalizing the single-chain structure factor to make it concentration dependent. This approach is often used to describe polymer solutions up to the concentrated region. 

Fig. 21. Dimensions of cellulose nitrates. Circles, data of Holtzer, Benoit and Doty (126)- Full curve, present theory for flexible chain with expansion factor dashed curve, warmlike chain without expansion factor...

The first worker to consider the detailed effect of a strong external force field on the molecular dynamics of an isotropic molecular liquid seems to have been Benoit. This treatment may be developed considerably and extended. In this review, however, the technique of computer simulation is used in combination with reduced model theory (see Chapter I) in an attempt to probe more deeply into the fundamental physical characteristics of the molecular liquid state. 

By employing a very strong external field, a gedankexperiment may be set up whereby the natural thermal motion of the molecules is put in competition with the aligning effect of the field. This method reveals some properties of the molecular liquid state which are otherwise hidden. In order to explain the observable effects of the applied fields, it is necessary to use equations of motion more generally valid than those of Benoit. These equations may be incorporated within the general structure of reduced model theory " (RMT) and illustrate the use of RMT in the context of liquid-state molecular dynamics. (Elsewhere in this volume RMT is applied to problems in other fields of physics where consideration of stochastic processes is necessary.) In this chapter modifications to the standard methods are described which enable the detailed study of field-on molecular dynamics. 

Benoit JM, Buisson JP, Chauvet O, Godon C et al (2002) Low-frequency Raman studies of multiwalled carbon nanotubes experiments and theory. Phys Rev B 66(7) 073417-1-073417 ... 

M. Profeta, M. Benoit, P. Mauri, and C. J. Pickard (2004) First-Principles Calculation of the NMR Parameters in Ca Oxide and Ca Aluminosilicates the Partially Covalent Nature of the Ca-O Bond, a Challenge for Density Functional Theory. J. Am,. Chem. Soc. 126, p. 12628... 

The fundamental principle of separation by SEC was described by Benoit and coworkers in 1967. They found an excellent correlation between the elution volume and a dynamically based molecular size, the hydrodynamic volume Vjj for a wide range of species and large-scale molecular architectures [9]. Their theory assumed a thermodynamic separation principle considering that the elution volume is independent of the flow rates. Recently, it has been proved that the radius of gyration is more appropriate than the hydrodynamic volume [10]. The radius of gyration is defined as the mean square distance away from the center of gravity [11]. Its mathematical definition is ... 

J.M. Benoit, J.P. Buisson, O. Chauvet, C. Godon, and S. Lefrant, Low-flequency Raman studies of multi walled carbon nanotuhes Experiments and theory, Phys. Rev. B, 66,073417 (2002). 

The previously discussed theories were developed for monodisperse diblock copolymers, which are not TPEs. However, Leibler s mean-field theory has been extended to include polydispersity (Leibler and Benoit, 1981) and to include triblock, star, and graft copolymers (Olvera de la Cruz and Sanchez, 1986 Mayes and Olvera de la Cruz, 1989). In the former case, polydispersity corrections tend to lower x N corresponding to the ODT. As would be expected from the analogy between blends and diblocks, triblocks will phase separate at higher xN values than the corresponding diblocks. This theory predicts a monotonic increase in the critical value of x A as the symmetry of the triblock increases, to a maximum of about 18 for the symmetric triblock. Surprisingly, the minimum xN value that separates the order and disordered regions in triblocks does not necessarily correspond to the critical point. 

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