The size distribution for theA g RB f−g Model of polymerization

This paper gives the equilibrium distribution of polymer sizes for Flory'sA _g RB _f–g model of polymerization. In this model, the polymers are composed of structural units withg functional groups of the typeA and (f-g) functional groups of the typeB. Reaction is subject to three conditions: (1) Functional groups of the typeA react only with those of typeB, and vice versa. (2) Intramolecular reactions do not occur [and therefore only branched-chain (noncyclic) polymers and formed]. (3) Subject to conditions (1) and (2), all functional groups are equally reactive. The derivation employs Stockmayer's statistical mechanical method (first used on Flory'sRA _f model), coupled with a recursion giving the number of distinct polymers which may be assembled fromk units of theA _g RB _f–g type. We also give distributions for a limiting case of theA _g RB _f–g model, the so-calledA _g RB model. This paper completes the solution of the Smoluchowski coagulation equation (monodisperse case) for the kernelsa _ij =A + B(i +j)+ Cij. The proof will be given in another publication.     

The Development of Gas-flow Modulation for High-frequency MTDSC measurements

The purpose of this study was to investigate the feasibility of modulating the temperature programme of a conventional DSC by use of an alternating gas-flow system. Modulated temperature differential scanning calorimetry (MTDSC) is an important thermal analysis technique but suffers from a limited applicable frequency range due to the mass of the sample and DSC cell leading to the impingement of thermal conductivity effects. We suggest that the frequency limit can be increased by replacing the cell as the source of temperature modulation with an external gaseous source, directed towards the sample and reference pans. In this evaluation, an alternating gas-flow was passed through a line to a forced gas-flow accessory (FGFA). The FGFA consisted of two matched cylinders containing chambers that allowed pre-temperature-equilibration of the stream of gas before it was passed over the sample and reference pans. The development of this device revealed the essential practical requirements of gas-flow modulation for high-frequency temperature modulation. These include the following: an appropriately sealed tunable gas supply to both sample and reference pans, an effective method for high-frequency cycling of the gas-flow rate, a small aperture to deliver the flowing gas directly over the pan and a temperature equilibration chamber. The results from samples of quenched PET and amorphous Saquinavir indicate that gas-flow modulation is indeed feasible, with the FGFA able to raise the attainable temperature modulation frequency by an order of magnitude compared to conventional MTDSC. This revised version was published online in August 2006 with corrections to the Cover Date.     

Selective hydrogenation of butadiene-styrene copolymers using a Ziegler-Natta type catalyst 2. Thermal properties

Three types of butadiene-styrene copolymers, diblock, triblock and random, were partially hydrogenated in their elastomeric units in order to determine the influence of hydrogenation extent on their thermal properties. The hydrogenation reactions were carried out using a Ziegler-Natta type catalyst and the extent of hydrogenation was evaluated by FTIR spectroscopic technique. The percentage of crystallinity was determined by differential scanning calorimetry (DSC), considering the low density polyethylene (LDPE) as reference since the saturation of elastomeric units with low content of 1,2-vinyl bonds gives a structure which resembles to LDPE, thus semi-crystalline polymers were obtained. On the other hand, the glass transition temperature (T_g) for the hydrogenated and non-hydrogenated copolymers as well as their heat of fusion, were also determined by DSC. An equation to evaluate the T_g of partially hydrogenated random copolymers is proposed considering both the saturated and unsaturated fractions. The thermo-oxidative behaviour of certain hydrogenated and non-hydrogenated copolymers was evaluated by thermogravimetric analyses (TGA). The results obtained by TGA suggest that a minimum saturation level is necessary in order to improve the thermo-oxidative resistance of the polymers.     

A characterization of probability distributions by mean values

Let (X) be a measurable complex function onR;X, Y, Z be i.i.d. random variables; and (t, u, v)=E(tX+uY+nZ), wheret, u, vR. In this paper we describe a class of function (x) such that the distribution ofX, Y, Z is determined by the funetion (t, u, v). The main result is a generalization of the author's characterization of normal and stable distributions.     

High-density properties of hard spheres within a modified Percus-Yevick theory: The role of thermodynamic consistency

Using an integral-equation approach based upon an approximation for the tail function, the equilibrium properties of a system of hard spheres are studied with special concern for the behavior in the region of close packing. The closure adopted is such that full, internal consistency is ensured in the thermodynamics of the model with respect to both the two zero-separation theorems as well as to the more standard virial and fluctuation routes to the equation of state. The scheme also makes use of the continuity properties of the tail function and of the cavity distribution function at contact. These properties are explictly tested in the low-density limit up to the fourth derivative. The theory generates an equilibrium branch bounded on the high-density side by a point corresponding to a packing fraction0.78, a value which closely matches Rogers' least upper bound for the densest packing of spheres. The pair structure of the fluid in the state of random close packing is also compared to the type of local order predicted by the theory at similar densities.     

Statistics of transfer matrices for disordered quantum thin metallic slabs

In the quantum transport problem of a tight-binding Anderson model, the statistics of eigenvalues for the transfer matrices of thin disordered slabs is studied. Numerical simulations indicate that the probability distribution of nearest neighbor eigenvalue spacing and the ₃ statistics have already become close to that of the Gaussian orthogonal ensemble for sample lengths of the order of the mean free path, provided that transverse localization effects are not important. An intuitive argument is given why this should occur independently of the size of the matrix. Therefore, good mixing of the channels is not essential for obtaining Gaussian orthogonal ensemble type statistics and universal conductance fluctuations.     

A long-time tail for random walk in random scenery

Consider a simple random walk on ^d whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn _k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let _k =E(n _k )–q ^–1. Our main result is a proof that (*) lim _k k ^d/2 _k = (1 –q)q ^{d/2 – 2}(d/2) ^d/2. Since it is known thatq ^{– 1} _k =E(n ₁ n _{k + 1} B) –E(n ₁ B)E(n _{k + 1} B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of _k (1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of _k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of _k cannot be faster than (*).     

Anomalous diffusion in a random velocity field

We construct diffusions in random velocity fields which present anomalous superdiffusive behavior. The mean square displacement can be made to have any power lawt for 1<2. Higher moments and characteristic functions are also investigated.     

Multiple scattering in random media I. Restricted two-body additive approximation

We present a microscopic theory of the problem of finding the properties of a particle interacting with potentials located at random sites. The sites are governed by a general probability distribution. The starting point is the multiple scattering equations for the amplitude k ₁|T |k ₂ in terms of the individual scattering amplitudes k ₁|T |k ₂. We work with quantitiesA defined by k ₁|T |k ₂=k ₁|T |k ₂exp[i(k ₁–k ₂)R ]. The theory is based on a splitting of the fundamental equation forA into equations for the mean A and the fluctuationsAA . Neglect of the fluctuations yields the quasicrystalline approximation. We rearrange the equation forAA to isolate the collective part of the fluctuations. We then make the simplest microscopic truncation which is thatAA is a restricted two-body additive function of the site positions. With the contribution of the collective fluctuations, this yields results forA that are accurate to ordert ⁴.Work supported in part by the National Science Foundation under Contract No. NSF DMRWork supported in part by the National Science Foundation under Contract No. NSF DMR