Growth theories

Growth theories can be divided into equilibrium theories and kinetic theories. Equilibrium theories explain some features of the crystal thickness. They predict the existence of two minima in free energy, one at a finite crystal thickness and the other at infinite thickness. The crystal thickness associated with the 

The Lauritzen-Hoffman (LH) theory has been the dominant growth theory for polymer crystallization for the last 20 years. New experimental data have led to the revision and development of the theory during recent years. The original theory of Lauritzen and Hoffman (1960) is presented here in detail. Later modifications of the theory are outlined in the subsequent text. Finally, criticisms of the LH theory including the later developed versions are mentioned and discussed. 

The LH theory provides expressions for the linear growth rate (G), i.e. the rate at which spherulites or axialites grow radially, as a function of degree of supercooling (AT = — T, where is the 

This presentation of the LH theory considers a very simple case, namely a homopolymer of intermediate or high molar mass. Fold length fluctuation is not considered in this simple treatment. A secondary nucleus is first formed and it spreads out laterally at the rate g. The thickness of the stem along the growth (G) direction is b. It is not necessary for the steady-state structure of the growing crystals to be smooth. In fact two of the three regimes of crystallization that are defined by the LH theory are characterized by a surface which contains several patches on to which stems are deposited. 

This expression (for P) is derived from the WLF equation and h is Planck s constant, 7i is a constant, U is a constant (dimension J mol ), and is the temperature at which diffusion is stopped. Later versions of the LH theory use another expression derived from de Gennes s theory for self-diffusion (see Chapter 6) for the P parameter  

As in the case of small molecules, attempts have been made to develop theories that help to probe the mechanism of crystal growth. As with small molecules 

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Stoyanov S and Kashchiev D 1981 Thin film nucleation and growth theories a confrontation with experiment Current... 

The seeond-order dependenee of the growth rate on the supersaturation ean be explained by a number of growth theories. The most eonvineing, however, is that of Burton etal. (1951). In their BCF theory about the serew disloeation eentred surfaee spiral step, it is assumed that growth units enter at kinks with a rate proportional to cr and that the kink density is also proportional to cr whieh gives the faetor cr in the rate expression. 

The layout of this article is as follows Section 2 considers the equilibrium aspects of the crystals whilst Sects. 3 and 4 explain the growth theories, divided into nucleation and non-nucleation theories, respectively. Finally, Sect. 5 provides an overview and suggests future lines of investigation. 

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

Having discussed some equilibrium properties of a crystal, we now outline and contrast the bases of the growth theories which will be dealt with in more detail later. The theories may be broadly split into two categories equilibrium and kinetic. The former [36-42] explain some features of the lamellar thickness, however the intrinsic folding habit is not accounted for. Therefore, at best, the theory must be considered to be incomplete, and today is usually completely ignored. We give a brief summary of the approach and refer the interested reader to the original articles. The kinetic theories will be the topic of the remainder of the review. 

Growth theories of surfaces have received considerable attention over the last sixty years as summarized by Laudise et al. [53] and Jackson [54]. The well-known model of the crystal surface incorporating adatoms, ledges and kinks was first introduced by Kossel [55] and Stranski [56]. Becker and Doring [57] calculated the rates of nucleation of new layers of atoms, and Papapetrou [58] investigated dendritic crystallization. 

Using this thermodynamic picture, classic nucleation and growth theory was used to describe the phase transformation that occurs in these materials, despite the relatively unique synthesis method that is employed. The governing equation for homogeneous nucleation that describes the change in free energy associated with the formation of a spherical crystalline nucleus in an amorphous host is as follows ... 

Wu, Ruff and Faeth12491 studied the breakup of liquid jets with holography. Their measurements showed that the liquid volume fraction on the spray centerline starts to decrease from unit atZ/<70=150 for non-turbulent flows, whereas the decrease starts at aboutZ/<70=10 for fully developed turbulent flows. Their measurements of the primary breakup also showed that the classical linear wave growth theories were not effective, plausibly due to the non-linear nature of liquid breakup processes. 

For Marx, the task of establishing how a capitalist economy can reproduce itself is not limited to a particular period of production. The reproduction examples that he carves out in the final part of Capital, volume 2, show how balanced reproduction can take place over an extended number of years. Despite the limitations he faced, with a lack of formal modelling tools, computing power and waning personal health - the reproduction schemes were one of his last contributions to political economy - Marx was able to devise complex numerical examples, in which somehow a 10 per cent rate of growth is sustained in each period of production. It is not for nothing that he has been described as the father of modem growth theory. 

Ott, A.E. (1967) Marx and modem growth theory , German Economic Review, 10 189-95. 

The classical crystal growth theory goes back to Burton, Cabrera and Frank (BCF) (1951). The BCF theory presents a physical picture of the interface (Fig. 6.9c) where at kinks on a surface step - at the outcrop of a screw dislocation-adsorbed crystal constituents are sequentially incorporated into the growing lattice. 

B. Lewis, "Nucleation and Growth Theory" in Crystal Growth, BJl. Pamplin, Ed., Pergamon Press, Oxford (1980). 

Interactions between solute and solvent molecules can have a significant effect on the shape of a crystal. This can be accounted for by specific adsorption of the solvent molecule on ciystal faces. Current oystal growth theories indicate that when interactions between solute and solvent are strong the solute molecules are solvated and a solvation layer dsts at the oystal-liquid interface which likely can vary as a function of ciystal face. Crystal growth requires desolvation of die solute molecule and desolvation of the surface site on the crystal. The molecule then surface diffuses imtil it reaches an incorporation (kink) site. 

Lower et al. (1998b) can shed more light on this phenomenon. They again used AFM, SEM, TEM, SEM-EDS, electron diffraction, and XRD to study the reactions between 0.5 and 500 mg/L of Pb with hydroxyapatite at pH 6 and a reaction temperature of 22 °C. A commercial hydroxyapatite was used at sorbent concentrations of 0.5 g/L. Reactions were observed over a 2 h period. At high initial Pb concentrations, Pb solution concentrations dropped from 500 mg/L to <100 mg/L. At concentrations of 0.5-100 mg Pb/L, after reaction, Pb levels dropped to less than 15 pg/L. In both cases, hydroxyapatite dissolved and hydroxypyromorphite formed. The authors applied some nucleation and crystal growth theory developed... 

It is also important to emphasize that conventional consciousness of colloid or fine particle technology, like better dispersion and control of rheological properties of dipping solution, are not to be overlooked. The growth of nanoparticles in liquid phase is almost exactly regulated by the nuclei-growth theory as well as stability of lyophobic colloids suggested half a century ago. 

It became necessary to understand how crystals grow at the atomic level so as to form a deeper understanding of why crystals can take a variety of forms. This was achieved through the layer growth theory put forward in the 1930s by Volmer, Kossel, and Stranski on the structure and implication of the solid-liquid interface, the spiral growth theory by Frank in 1949, and the theory of morphological... 

The concept of dislocations was theoretically introduced in the 1930s by E. Orowan and G. I. Taylor, and it immediately played an essential role in the understanding of the plastic properties of crystalline materials, but it took a further twenty years to understand fully the importance of dislocations in crystal growth. As will be described in Section 3.9, it was only in 1949 that the spiral growth theory, in which the growth of a smooth interface is assumed to proceed in a spiral step manner, with the step serving as a self-perpetuating step source, was put forward [7]. 

Why polyhedral forms bounded by smooth interfaces can grow, whilst maintaining their polyhedral forms, was not properly accounted for until the layer-by-layer growth theory (which considers atomic process of crystal growth) formulated by Kossel and Stranski appeared. 

Rayle, D.L. Cleland, R.E. (1992). The acid growth theory of auxin-induced cell elongation is alive and well. Plant Physiol. 99,1271-1274. 

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