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A growing two-dimensional cell model is defined as follows. In an area there are Poisson-distributed nuclei. Arising from these nuclei, grains start to grow simultaneously. All grains grow circularly with the same constant radial growth rate $$\dot R$$ . During the process of growth no new nuclei are formed. If two grains touch each other, growth is stopped there by formation of a straight grain boundary. We arbitrarily put a straight line, called Rosiwal's line, into the area. While grains are growing many straight grain boundaries and circular growth fronts cross Rosiwal's line. At a fixed fraction transformed, F(=crystallized area/total area), we consider the different extension rates of growth fronts (growing borders) along Rosiwal's line, v( $$\dot R$$ ⩽ v

The topological properties of a set of nuclei undergoing a phase transformation were investigated. The nuclei were spread out in a plane according to a Poisson distribution. All centres started to grow at the same moment and with the same constant rate. They grew circularly and free of shrinking. The mean numbers per nucleus of grain boundaries, triple points and growth fronts were calculated as functions of the degree of transformation, F (0⩽F⩽1). For these relationships we deduce plain and exact expressions.

A foil of polypropylene is heat-treated in such a manner that a «growing 2D cell-model» is formed. A Rosiwal's line is placed into the totally primarily crystallized foil. The nucleus-coordinates of grains intercepting Rosiwal's line are measured. From these «effective» nuclei we determine at fraction transformed F=1/4, 1/2 and 3/4 the distributions of grain-lengths and melt-lengths. Further we determine properties which are derived from the chord intercepts at a given F. Experimentally we find that the values (Poisson distribution of nuclei, chord intercepts, etc.) are in good agreement with the growing 2D cell-model

We show theexperimental verification of a didactic model which is named “one-dimensional (1D) process of solidification (crystallization) from nuclei at random to linear polycrystal”.

A filament of isotactic polypropylene, 15 μm diameter, is used to prove the theoretical predictions for a one-dimensional (1-D) process of solidification from nuclei at random to a linear polycrystal. The theoretical predictions were made in 1983. The agreement between theory and experiment is very good.

We propose to take the calculus of variations in order to compute the shape of a growing 2D spherulite in an uniaxial field of growth rate. We are concerned with the growth line (a path that is traveled in the shortest possible time from nucleus to a point (x1, y1), where a molecule just crystallizes) and the growth front (the times between spherulite and supercooled material). The Euler differential equation—a result of the calculus of variations—is derived for all uniaxial growth ratesv (x). Here we especially investigatev(x)=px+q.

A 2D spherulite grows in a linear-temperature field from a nucleus of polypropylene at (0,0). The growth lines and the growth fronts are computed by the calculus of variations. The agreement between theory and experiment is satisfactory

Each spherulite growing undisturbed by other spherulites around a spherical inclusion possesses directly a channel. The channel damages some properties of the polymer, e.g. the bond strength, corrosion rate and breakdown voltage

We investigate experimentally and theoretically the isothermic growth of two spherulites of different modification into a supercooled isotactic polypropylene film. The faster growingβ-spherulite grows around theα-spherulite, and finally theα-spherulite is symmetrically and completely included inβ. In contrast to literature but in agreement with experimental evidence we find that the grain boundary between the teardropshapedα-spherulite and the surroundingβ-spherulite consists of two parts, where one is always an arc of a logarithmic spiral. Thisα-spherulite ends always in a vertex. Its angle depends on the ratio of the two growth rates only. Behind the vertex an intrinsicβ-β-grain boundary exists, degenerating to a channel in bulk material. The growth fron of theβ-spherulite, which ends on the logarithmic spiral or on the intrinsic grain boundary during growth, consists of an arc of a circle continued by an arc of a logarithmic spirial, too.

A phase transformation in a one component polycrystal starts to grow instantaneously and with a constant rate from the grain boundaries. After the time t there exist regions of the old and of the new phase. The derivation of the statistical quantities characterizing this state at t is unknown yet. Therefore we reduce this complicate three-dimensional problem to a one-dimensional model analogously to the linear chain in lattice dynamics or to the one-dimensional energy band model. A closed analytical treatment of this model is practicable. It is derived in detail for those one-dimensional systems, which transform according to a rate law of n-th order (n ≧ 0). In einem einkomponentigen polykristallinen Gefuge beginne die Phasenumwandlung gleichzeitig und mit konstanter Wachstumsgeschwindigkeit an den Korngrenzen. Nach der Teit t liegen Bereiche aus alter und aus neuer Phase vor. Die Ableitung der statistischen Charakteristika dieses Zustandes zum Zeitpunkt t ist bisher nicht gelungen. Deshalb wird das schwierige dreidimensionale Problem auf eine Dimension reduziert, analog zur linearen Kette in der Gitterdynamik oder zum eindimensionalen Energiebandermodell. Eine analytisch geschlossene Zustandsbeschreibung mit der Zeit als Parameter ist moglich. Sie wird fur diejenigen eindimensionalen Systeme abgeleitet, deren Umwandlung als Prozes n-ter Ordnung ablauft (n ≧ 0).

A “linear polycrystal” represents the one-dimensional model of a polycrystalline wire with bamboo-structure. It is formed by crystallization of a thread of a one-component melt within a capillary tube. In the paper is assumed, that the nuclei of crystallization are independently and randomly distributed along the thread, and that their crystallographic orientations are equidistributed in all directions. Furthermore it is assumed, that all nuclei are preformed and that they (and therefore their crystals) have a cubic crystalstructure. Four differently defined processes of crystallization, forming four different linear polycrystals, are considered. TheE- andG-moduls of these four linear polycrystals are calculated only from the knowledge about their conditions of crystallization and the elastic constants of cubic single crystal. We found: 1) The moduls of the linear polycrystals have the same value, independently whether their growth has been started at the same instant or successively. 2) If the growth-rate for all crystals has a distribution (anisotropic cubic growth) or has a constant value (isotropic growth), theE-moduls (and also theG-moduls) of their linear polycrystals vary only by about 1%.

An isothermal phase transformation from a single crystal A to another single crystal B is theoretically investigated along preferred lines (Rosiwal's lines). It is supposed that the nuclei of the B-phase are Poisson distributed within the single crystal A. From these nuclei the B-grains grow instantaneously, equioriented, and in the form of cuboids with three different growth ratesv x ,v y andv z . If the B-grains touch, growth stops and they form 8 larger B-grain. We derive for this microstructure, at timet along the three Rosiwal's lines (X-line,Y-line,Z-line), the distribution densities of the lengths of the A-phases an Well as of the B-phases using the theory of probability. The two-dimensional model (z = 0) is considered in detail, idealizing the transformation within a thin layer.

The microstructure of a two-dimensional cell growth model at each fraction transformed along Rosiwal's line is characterized. Rosiwal's line cut grains and yields chord intercepts. By the use of probability theory we derive the mean number of chord intercepts per unit length as well as the dependence of the distribution density of the length of these chord intercepts on the fraction transformed. Furthermore, other quantities are derived which appear along and around Rosiwal's line.

The two-dimensional cell model is defined by the following conditions of its formation: athermal nuclei may be randomly (Poisson) distributed in a thin amorphous film. From these nuclei spherulites may start to grow instantaneously, circularly, and at the same rate. Where two spherulites touch growth stops, and all points of contact form a straight intersection between neighboring spherulites. Finally, if the film contains spherulites only, each spherulite (cell) is completely limited by straight intersections, forming an irregular polygon. All irregular polygons of the film together form a planar irregular net of straight intersections. In quantitative microscopy this net is characterized by a “lineal analysis.” A straight line (Rosiwal's line) is arbitrarily put into the plane of the net. The net divides Rosiwal's line into chord intercepts of different lengths. The distribution of these lengths is analytically derived in the present paper by use of the theory of probability.

In der vorliegenden theoretischen Arbeit wird unter bestimmten Voraussetzungen die Verteilungsfunktion fur die Abstande benachbarter Kristallkeime abgeleitet. Ausgegangen wird dabei von einer unterkuhlten Schmelze in einer Kapillare. Unter Anwendung der Theorie zufalliger Prozesse last sich die Keimabstands-Verteilungsfunktion deduzieren, wobei der mittlere Keimabstand Parameter ist. Davon ausgehend werden fur drei spezielle Erstarrungsprozesse die im naherungsweise „eindimensionalen” Gefuge auftretenden Kornlangenverteilungen abgeleitet. At first in this theoretical article a melt in a capillary tube is considered, when appearing a number of nuclei by undercooling. The distribution function of the distances between neighbouring nuclei is deduced by use of the theory of stochastic processes. The average of these distances is the parameter in the distribution function. Under this supposition for the nuclei-distances the distribution functions of the length of crystals, forming by three different ways of crystallization in the capillary tube, is deduced.

Nuclei are Poisson-distributed within a plane. Out of the nuclei, grains begin to grow instantaneously, circularly and at a constant rate. The forming microstructure at each fraction transformed, consisting of grains and untransformed regions, is characterized in this work by use of a straight line (Rosiwal's line), which passes arbitrarily through the plane. Along this line we obtain chord intercepts of grains and of untransformed regions of different lengths, independent of the position of the straight line. The distributions of these lengths are essentially determined by the distribution of the nuclei and from the conditions of growth. From these assumptions we derive, by use of probability theory, the distributions of the chord intercepts. Jn Part 1, only the distribution density of the chord intercept of the untransformed regions is studied. From these results several other statisticai quantities of the microstructure are also deduced.

Die Metalldispersitat von 0,5% PdAl2O3-, 0,5% PtAl2O3- und 1,5% PtAl2O3-Katalysatoren wurde mit Hilfe de Rontgenkleinwinkelstreuung untersucht. Unter Verwendung einer nach dem Kratky-Prinzip arbeitenden Kamera wurden die Differenzstreukurven Katalysator-Tonerdetrager ermittelt und aus diesen unter Voraussetzung globularer Metallpartikeln und Abwesenheit interpartikularer Streuung mit Hilfe der Guinierschen Naherung die Guinier-Radien RG errechnet. Wahrend bei der Pd-Katalysatorprobe stets positive Differenz-streukurven gemessen wurden, nahmen diese bei den Pt-Katalysatoren ab Streuwinkeln >0,022 rad negative Werte an. Deshalb konnte nur fur die 0,5% PdAl2O3-Probe die Porodsche Invariante und der Porod-Radius RP bestimmt werden. Aus RG und RP wurden die Parameter fur eine zwei- und eine einparametrige Verteilung abgeleitet. Ein Vergleich der erhaltenen Ergebnisse mit Werten aus Rontgenweitwinkelmessungen ergab befriedigende Ubereinstimmung. On the Dispersion of Metals in Supported Catalysts. II. Small Angle X-ray Investigation. on Platinum/Alumina and Palladium/Alumina Catalysts The metal dispersion in 0.5% Pd/A12O3. 0.5°% Pt/Al2O3 abd 1.5% Pt/Al2O3 catalysts was investigated by small angle X-ray scattering, Using a Kratky camera the difference scattering curves catalyst-carrier were evaluated. The assumption was made that the scattering centres are globular and that there is no interparticular scattering. From the Guinier approximation the Guinier radii RG of the metal particles were determined. With the Pd catalyst always positive difference scattering curves were found, whereas negative values were found with the Pt catalysts at a scattering angle greater than 0.022 rad. Therefore only for the sample with 0.5% Pd the Porod invariant and the Porod radius RP could be determined. From RG and RP the parameters of a two- and of a one-parametric distribution were calculated. The results are in good agreement with data from wide angle X-ray diffraction measurements.

About 1000 nuclei were produced and developed within a filament of a glass ceramics with a diameter of 43 μm and a length of 14 mm. All crystallized centres are spheres to a good approximation with a diameter of about 7 μm. The x coordinates of all nuclei, where the x axis equals the axis of the filament, were microscopically measured. From these results we prove that the nuclei are statistically independently distributed in the x direction. Furthermore, the y coordinates of all nuclei were measured. From these results we prove that the nuclei are distributed within the volume and not on the surface of the filament.

Das nadelformige Kristallisat, das sich in einer Kapillare nach einem definierten Ausscheidungsprozes im Loslichkeitsgleichgewicht bei der Temperatur T0 einstellt, wird stochastisch modelliert. Ausgegangen wird von Kristallisationskeimen kubischer Kristallstruktur, die langs der eindimensional erstreckten Losung poissonverteilt sind, deren kristallografische Orientierungen uber alle Richtungen gleich-wahrscheinlich verteilt sind und die nach einer plotzlichen Temperaturerniedrigung auf T0 alle gleichzeitig zu wachsen beginnen. Zwei extreme Wachstumsprozesse werden betrachtet: 1. Diffusionsbestimmtes (isotropes) Wachstum; 2. Einbaubestimmtes (anisotropes) Wachstum mit {100}-Flachen als Wachstums- und Gleichgewichtsflachen. Prozesbedingte Unterschiede zwischen den beiden Kristallisaten werden quantitativ abgeleitet. (Beispielsweise sind Kristallnadeln mit -Achsen beim einbaubestimmten Wachstum mit groserem Masseanteil vorhanden als beim diffusionsbestimmten Wachstum.) Die Kornlagenverteilungen beider Kristallisate werden deduziert. Sie sind nahezu identisch.