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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 “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.

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.