An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

Critical phenomena and renormalization-group theory

We propose a model giving the conformation of a star shaped polymer by taking into account the radial variation of the monomer concentration ϕ( r). For an isolated star when increasing r (at the centre of the star r = 0), the variation of ϕ (r) is first given by a constant value (r f1/2 ν-1 l); wh ere f is the number of branches, N the number of monomers in a branch and ν and l are the excluded volume and the length associated to a monomer. For all these cases, it is shown that the size of a branch is always larger than that of a linear polymer made of N monomers. Beyond the overlapping concentration the star conformation is obtained from two characteristic lengths essentially : χ(c ) a radius inside which the branches of the other stars do not penetrate, this radius defines a domain where the conformation of a star is similar to that of an isolated one. Beyond χ(c) the interpenetration of branches is characterized by a screening length ξ(c) very similar to that found for semi-dilute solutions of linear polymers. For all these regimes the variation of the size of a star is predicted as a function of N, f, v and c.

https://hal.archives-ouvertes.fr/jpa-00209423/document

Star shaped polymers : a model for the conformation and its concentration dependence

http://static.sif.it/SIF/resources/public/files/va2012/likos_0710.pdf

Effective interactions in soft condensed matter physics

Tethered polymer chains refers to macromolecular chains that are attached into microstructures by their ends. Highly branched polymers, polymer micelles and end-grafted chains on surfaces are a few examples. This review brings out the common features of these seemingly widely disparate microstructures. Tethering can be reversible or irreversible and is frequently sufficiently dense that the chains are crowded. Densely tethered chains stretch to alleviate the interactions caused by crowding. They thus exhibit deformed configurations at equilibrium. These effects of tethering on the structure of the polymer chains are reflected in distinctive behavior and properties of microstructures containing tethered chains.

Tethered chains in polymer microstructures

Solutions of flexible polymers. Neutron experiments and interpretation.

Neutron coherent scattering techniques have been used for the determination of the conformation of polymer in bulk and experimental details are given about the application of this method to the study of polymeric systems. Measurements have been made for small and intermediate momentum ranges on a series of eight monodisperse deuterated polystyrenes of molecular weight ranging from 21,000 to 1,100,000. The results lead to the concluson that in amorphous state the conformation of the polymer molecule is indistinguishable from that in solvent and that the Debye scattering function which is valid for unperturbed chains applies for q ~1 as low as 10 Á. In order to explain such properties as rubber elasticity it has long been assumed that polymer chains in the liquid state obey Gaussian statistics.2,3 Below its glass transition temperature a polymer becomes hard and loses its elastic properties. It is believed,4 however, that this transition affects only the motion of the drawn segments and not their configuration which is expected to remain Gaussian. There is much indirect evidence in favor of this hypothesis but until recently it has been impossible to prove it directly. Moreover, many authors5,6 believe from theoretical considerations, electron microscopy, and diffraction results that there is more order in the amorphous state, taking the form of supramolecular structures. In this paper, we present the results of measurements of the conformation of polystyrene molecules in the bulk amorphous state. These measurements were made using the neutron small angle scattering apparatus at the Institut Laue-Langevin in Grenoble and at the CEA Saclay. Similar experiments in this field have already been published by Kirste7 and Ballard.8 The neutron SAS technique is particularly well suited to the problem for two reasons. (1) Neutron wavelengths available from the cold sources at each of the installations are of the order of 10 Á. It is thus possible to match the experimental wave vectors to the molecular dimensions. In light scattering experiments only radii of gyration of very large molecules can be measured, while X-ray experiments need extremely small scattering angles. (2) The difference between the neutron scattering length of deuterium and hydrogen is particularly useful since it will be shown that deuteration does not affect the thermodynamic properties of the chains and that there is perfect compatibility between deuterated and hydrogenous molecules. Thus, if deuterated molecules are embedded in a matrix of hydrogenous molecules, the scattering pattern of the former may be measured and the molecular dimensions and the statistical distribution of the chain segments deduced. The contrast is large enough for very low concentration of polymer to be used. In the first part of the paper we give details of the neutron technique as applied to the problem of polymer conformation, then the sample preparation is described and finally the results and their interpretation are discussed. Neutron Small Angle Scattering Technique For experiments on polymeric systems two principal properties differentiate neutron scattering from other techniques: the interaction with the sample depends only on the neutron-nucleus interaction leading to results described in part A of this section; the wavelengths available range from 1 to 12 Á giving values of the scattering vector which are inaccessible to light and X-ray scattering. The apparatus is described in part B. (A) Scattering Intensity. The intensity scattered by systems with long-range correlations (greater than 10 Á) takes the form of a central peak about the forward direction. The amplitude of this peak varies strongly from nucleus to nucleus. Isotopic substitution used in neutron scattering gives a powerful method of labeling molecules because it leaves the chemical properties of the molecules unchanged. We now evaluate the scattering cross section of a mixture of labeled and unlabeled polymer molecules with a view to its application to deuterated and nondeuterated polystyrene. Using the relation for the scattered intensity which we derived, we consider the choice of sample. (1) Scattering Cross Section. The neutron-nuclear interaction is characterized by a scattering length o¿ which takes into account the spin of the isotope of the nucleus considered. The scattering cross section a(q )per unit solid angle of a neutron of wavelength X scattered at an angle by a monoatomic system consisting of N atoms at positions r¡ is written9 v(q) = Z^(ei9irrri)) (1) il q is the scattering vector |q| = (4ir/X) sin 6/2, the bar over the scattering amplitudes denotes a spin and an isotopic averaging while the broken brackets indicate a thermal average of the function inside. a(q) may be written in the form a{q) = a2J^(ei“iri-rJ>) + Na2 (2) ij This relationship introduces the coherent scattering amplitude a and the incoherent scattering cross section a\\ defined by a = a¡ and by = 4 2 = 4 ( ^ a2) (3) and for which tables of experimental values exist.10,11 Thus the scattered intensity decomposes into a coherent intensity depending on the scattering vector and an incoherent intensity 2. Values of a and are given in Table I. It is evident from the table that the values of the scattering amplitudes of hydrogen and deuterium are very different. The transmission of the system T = I(x)/I(0) is given by the equation T = exp[-d(4na2 + ) ] (4) D ow nl oa de d vi a U N IV O F C IN C IN N A T I on J an ua ry 2 2, 2 02 1 at 0 0: 26 :5 4 (U T C ). Se e ht tp s: //p ub s. ac s. or g/ sh ar in gg ui de lin es f or o pt io ns o n ho w to le gi tim at el y sh ar e pu bl is he d ar tic le s. 864 Benoit, et al. Macromolecules where d is the density of the system and x the thickness of the sample. Expression 2 takes no account of multiple scattering and can only be used when T is of the order of unity. Let us consider a system of N deuterated molecules each composed of m atoms of which the scattering amplitudes are aQ and ai„ (a varying from 1 to m). If the values of the scattering vector q are such that q~x is large relative to the molecular dimensions, the structure factor of the molecule may be neglected (this is in fact always true for small scattering angles). In this case, it is possible to generalize to molecules the idea of coherent and incoherent scattering amplitudes A and A \\ such that

Conformation of Polymer Chain in the Bulk

Using the neutron scattering technique, we have measured the temperature dependences of the characteristic correlation lengths in semidilute polymer solutions in the vicinity of the theta (Θ) point. From the analysis of the data in the Guinier range, with a sample containing a few labeled chains, we find that the radius of gyration Rg has the dependence Rg∼τ0.125±0.005, where τ is the reduced temperature (T−Θ)/Θ. The scattering experiment from unlabeled samples, in the intermediate range of the momentum transfer, shows a decrease of the screening length ξ, ξ∼τ−0.25±0.01. These two dependences are in agreement with a recent calculation [M. Daoud and G. Jannink, J. Phys. (Paris) (to be published)] based on de Gennes’s theory that the theta temperature is in fact a tricritical point. A scaling law in molecular weight and concentration is tested below Θ. A temperature concentration diagram is drawn with the above mentioned experimental results.

Experimental determination of the temperature–concentration diagram of flexible polymer solutions by neutron scattering

POLYMER molecules dispersed in a good solvent expand1 significantly beyond the size expected from purely random orientation of the segments. In the bulk, however, there are theoretical reasons2 which indicate that the molecule should recover its “Gaussian” size. For a long time experimental evidence for this assertion was missing. Several people (G. Allen and S. F. Edwards, personal communication and refs. 3 and 4) have recently recognized that such evidence may be obtained from a small angle neutron scattering experiment. One of them4 has already published results on polymethyl methacrylate. We discuss here some results for atactic polystyrene.

Dimensions of a Flexible Polymer Chain in the Bulk and in Solution

Polystyrene networks prepared by anionic polymerization have been characterized by small-angle neutron scattering. Two kinds of systems have been examined:

(A) networks with labelled branch points allowing characterization of the spatial distribution of crosslinking points; (B) networks containing a low proportion of chains labelled with perdeuterated polystyrene in order to characterize the conformation of individual elastic chains of the polymeric network. The dependence of the results on swelling and uniaxial extension is discussed.

J. P. Cotton

Papers

Solutions of flexible polymers. Neutron experiments and interpretation.

Conformation of Polymer Chain in the Bulk

Experimental determination of the temperature–concentration diagram of flexible polymer solutions by neutron scattering

Dimensions of a Flexible Polymer Chain in the Bulk and in Solution

Characterization of polystyrene networks by small‐angle neutron scattering