high molecular weight polymers takes place at temperatures below the equilibrium melting temperature, the resulting crystals will melt at a lower temperature than if they had formed nearer to the equilibrium temperature.

In a recent study, an equation was developed that gives a mathematical relationship between the melting temperature of a crystallite formed at nonequilibrium temperatures, the equilibrium melting temperature, and the temperature of crystallization. This relationship was found to agree very well with experimental values obtained from studies of the crystallization and subsequent melting of rubber under nonideal conditions.

A theoretical study of polymer crystallization mechanisms revealed that crystallization rates and isotherm shapes are sensitive to small amounts of structural irregularities in a polymer chain. This behavior results from a compositional change in the melt, which causes a marked decrease in the nucleation rate during crystallization. The theoretical conclusions, which were confirmed by experimental observations, indicate that chain irregularities as little as one mole percent should be detectable.

In

High Pressure Polymerization. Many molecules that have double or triple bonds in their chemical formulas-for example, carbon disulfide and nitrogen-do not form polymers under usual conditions. In general, when attempts are made to polymerize such molecules, the chemical equilibrium favors monomer instead of polymer production. However, if very high pressures are applied, the equilibrium will be displaced to favor polymerization. Reaction can be initiated by heat, catalysts, or ionizing radiation. work for the Office of Ordnance Research, materials were placed under pressure in a bomb and exposed to gamma radiation. When irradiated, perfluoroheptene polymerized to a moderate degree, whereas when unirradiated, no polymer formed. At 10,000 atm, 50 °C, and under irradiation, carbon disulfide formed a polymer which was previously obtained only at 50,000 atm and high temperature.

Thermal Stability of Polymers. The relationship between thermal stability and molecular structure of polymers was investigated at pyrolysis temperatures up to 1200 °C. In this program, sponsored by the Air Force, polymers which are originally highly crosslinked-such as polytrivinylbenzene and phenolic and epoxy resins and polymers which develop crosslinks at the pyrolysis temperatures such as polyvinylidene fluoride and polyacrylonitrile-were observed to yield carbonaceous residues and low molecular weight volatile fragments. When polymers which do not become crosslinked--such as polystyrene, polymethylene, and polytetrafluoroethylene--were heated to pyrolysis temperatures, only volatile products of low molecular weight were noted. In either case, the higher the temperature, the greater is the fragmentation of the degradation products. This research has yielded additional proof that the energy-absorption capacity of a polymer during thermal degradation is inversely proportional to the molecular size of the volatile fragments. Furthermore, it has demonstrated that polymers which leave a carbonaceous residue and liberate gases at the elevated temperatures generally give superior ablation resistance in missile

nose cones.

Free Radicals in Polymers. Free radicals can be observed, estimated quantitatively, and sometimes identified in very small amounts by electron spin resonance spectroscopy. As part of a program sponsored by the Air Force to investigate the reactions of free radicals in polymeric systems, a number of polymers, including polystyrene, cellulose, and fluorocarbon polymers, were irradiated and examined. Considerable new information was obtained on the mechanism of polymerization and on the chemical changes induced in these polymers by ionizing radiation. In addition, fluorocarbon monomers were observed to yield appreciable amounts of free radicals when irradiated at liquid-nitrogen temperatures.

Fungicidal Analysis. A study was undertaken to find a means for determining the quantity of 5,6-dichlorobenzoxazolinone in leathers. This compound is used as a fungicide to prevent mildew in leather, and an analytical technique was required by the Office of the Quartermaster General for treatment control. A colorimetric method was devised which consists of the conversion of 5,6-dichlorobenzoxazolinone to 2-amino-4,5-dichlorophenol by alkaline hydrolysis, diazotization of the aminophenol, coupling with resorcinol to give a colored azo compound, and photometric measurement of the color. A chloroform-water mixture is used to extract the fungicide. Vegetable tannins which would interfere with color measurements are precipitated with lead acetate and removed in the aqueous phase of the extract which is discarded. The interfering effect of chromium is also overcome, either because the water in the mixture prevents formation of a chrome-fungicide complex, or because the chrome itself is removed in the aqueous phase.

Meteorite Erosion of Materials. The possible hazard of meteors to the space-craft of the future has recently been made more significant by the discovery of a charged zone around the earth which attracts meteoric particles. When small masses, traveling at the velocities at which meteors travel, collide with solids, the transformed energy is so large that both the projectile and the target in the immediate vicinity liquefy after collision. The extent of this meteorite-erosion hazard is being investigated, in research sponsored by the Air Force, by examining the craters produced in collisions of liquid drops with liquids, and in collisions of metal spheres with metal plates, at velocities up to 20,000 ft/sec. An equation was derived relating the maximum depth of the cavity produced in a target liquid as a result of collision with a liquid drop, to the kinetic energy of the impinging drops.

Calcified Tissues Investigated. Hydrogen bonding in calcium-deficient apatites was investigated by infrared spectrophotometry in work supported in part by the U.S. Public Health Service. A direct correlation was found between the number of calcium ions missing per unit cell volume, obtained by chemical analysis and refractive index measurements, and the hydrogen bond content.

Dimensional Changes in Dentures. A study sponsored by the American Dental Association and the Federal dental services established that the dimensional changes of dentures in service are too small to be of clinical significance. In the investigation more than 200 dentures were

105

measured, some over a period of six years. They had been made of 12 different types of polymers processed by a variety of currently used methods. Acrylic resin dentures processed with simple conventional equipment, employing compression molding, were just as accurate and as dimensionally stable as dentures made with other types of resins using complex and costly equipment.

Dental Amalgams from Spherical Particles. The particle size of the alloy (Ag-Sn-Cu-Zn) used in making dental amalgams was investigated, independently of other variables, in a study sponsored by the American Dental Association and the Federal dental services. A standard alloy com

Data were obtained on the mechanism of polymerization and on the chemical changes induced in polymers by ionizing radiation. Here free radicals produced by ultraviolet radiation are studied by electron spin resonance (page 105).

position was used in the form of spherical particles produced by spray atomization. The particles were separated into eight different sizes, the smallest 1-4 microns and the largest 105-150 microns in diameter. Particles from 15-50 microns diameter exhibited the highest strength of all those studied. The study showed that an alloy in this spherical shape produces an amalgam equal or superior to those in current use, and offers a new approach to the control of critical amalgam properties, such as early strength, setting time, and flow, and promises major advances in the simplification of manufacture.

2.3. SPECIAL TECHNICAL SERVICE

PROGRAMS

2.3.1. APPLIED MATHEMATICS

The Bureau's applied mathematics facility conducts basic and applied research and renders advisory services in various mathematical fields. These services are available to other Government agencies as well as to the Bureau's staff. Modern computing equipment is used by the facility in support of its program.

During the past year the Bureau emphasized activities in statistical and numerical analysis, mathematical physics, and operations research. Extensive assistance was rendered in these areas and in digital computation. Special attention was given to problem formulation and analysis in order to select and develop numerical methods for the solution of problems in engineering and the physical sciences, utilizing both automatic and nonautomatic computing machines. An appreciable share of the mathematical program was devoted to government problems of business management and operation, sometimes called data-processing problems. Significant progress was achieved in the exploration of the utility of modern digital computers in the mechanical translation of scientific publications, for which there is a continuing urgent need.

As in previous years, the Bureau's applied mathematics program was strengthened by the active interest and support of other Government agencies. The Office of Naval Research and the USAF Office of Scientific Research supported fundamental and applied research in numerical analysis and mathematical physics, respectively. The National Science Foundation continued to support the compilation of a handbook of mathematical functions. and mathematical research related to information retrieval. The study of mechanical translation of scientific publications was supported by the U.S. Army Research Office and the Office of the Chief Signal Officer, U.S. Army.

Combinatorial Analysis. Work continued in combinatorial analysis, a branch of mathematics concerned with the arrangements of finite sets of objects. It was applied to the selection of the best pattern of linkages in transportation or communication networks, in the determination of the most efficient method for encoding messages to provide automatic correction of

possible transmission errors, and in the design of experiments to maximize the useful information obtained from a given number of measurements.

Eigenvalue Theory. The determination of eigenvalues for operators is a central problem in mathematical physics which frequently involves considerable numerical difficulty. A method of "generalized special choice" has been developed for determining lower bounds to the eigenvalues of selfadjoint linear operators. Numerical results carried out for the anharmonic oscillator are satisfactory even when perturbation theory fails. The method has also been used to estimate the eigenvalues of the spheroidal wave equation. The numerical results obtained by this method were found to be excellent in regions where difficulties were experienced with other procedures.

Matrix Theory. Various schemes were investigated for the solution of matrix eigenvalue problems. Extensive research continued in the development of techniques for obtaining the characteristic roots, the determinant, and the P-condition number of matrices. Several new results in doubly stochastic matrices have been obtained through the exhibition of the permanent as an inner product in a suitable space. Inequalities were derived which lead to new bounds for the determinants of nonnegative Hermitian matrices.

Approximation Theory. Research in approximation theory was emphasized because of its fundamental importance in numerical analysis. Extensive investigations of best approximation by nonlinear families were continued. A study was conducted concerning the approximation to convex functions by means of convex polynomials and trigonometric polynomials. The results contain some explicit constructions and estimates of accuracy. Tchebycheff approximation by rational functions was investigated, with emphasis on the computational aspect. Significant research on infrapolynomials and their generalizations was also performed.

Numerical Experimentation. In areas of numerical analysis where no theory exists or existing theory is merely suggestive, numerical experimentation may provide insight into a method of problem solution. Such experiments were performed in the numerical solution of nonlinear partial differential equations. The partial differential equation determining the pressure distribution in a gas-lubricated bearing was studied. The final aim, once a method of solution was found, was to compute a set of design curves for such bearings. Also, a system of three nonlinear, ordinary differential equations arising in a study of the internal structure of stars was investigated.

Machine Translation. Further progress was made on the automatic Russian language translation scheme being developed by the Bureau. In contrast with other machine translation projects, the Bureau project is characterized by emphasis on syntax in the conventional sense and by a system of predictions. A Russian word in a sentence "predicts" certain other grammatical forms; for example, a transitive verb predicts a direct object.

A significant innovation has been the development of a procedure called "profiling," by which clause and phrase boundaries are recognized mechani