Jingzhe Pan

A phenomenological model for the degradation of biodegradable polymers

This paper presents a phenomenological diffusion-reaction model for the biodegradation of biodegradable polymers. The biodegradation process is modelled using a set of simplified reaction-diffusion equations. These partial differential equations are non-dimensionalised giving two normalised parameters which control the interplay between the hydrolysis reaction and the monomer diffusion. The equations are firstly solved for simple cases of plates and pins. The numerical results are presented in the form of biodegradation maps which show the conditions where the biodegradation is controlled by auto-catalysed hydrolysis, non-catalysed hydrolysis, a combination of auto-catalysed and non-catalysed hydrolyses, or a combination of hydrolysis and monomer diffusion, respectively. The degradation maps provide a clear guide for the design of biodegradable fixation devices used in orthopaedic surgeries. Finally the diffusion-reaction equations are solved using the finite element method for strip and square meshes, showing how the model can be used to assist the design of sophisticated fixation devices.

A model for the sintering of spherical particles of different sizes by solid state diffusion

Abstract In this paper the numerical scheme developed by Pan and Cocks (Acta metall. 43, 1395–1406, 1995) is used to simulate the co-sintering process of two spherical particles of different sizes by coupled grain-boundary and surface diffusion. The numerical analysis reveals many interesting features of the co-sintering process. For example, it is found that the shrinkage between the two particles is not affected significantly by the size difference of the two particles as long as the difference is less than 50%. Based on the numerical results, empirical formulae for the characteristic time of the co-sintering process and for the shrinkage rate between the two particles are established. The empirical formulae can be used to develop constitutive laws for early-stage sintering of powder compacts which take into account the effect of particle size distribution. To demonstrate this, a densification rate equation for compacts with bimodal particle size distributions is derived.

A numerical technique for the analysis of coupled surface and grain-boundary diffusion

Abstract The continuity conditions across a junction between a grain-boundary and a free surface are discussed for the coupled surface and grain-boundary diffusion problem. An appropriate treatment of the continuity is proposed, which forms the basis of a numerical procedure for an arbitrary network of grains containing pores of arbitrary shape and size. The analysis technique, when combined with a suitable time integration algorithm, is able to simulate physical processes such as powder sintering, diffusional void growth and creep crack propagation etc. which are dominated by the coupled diffusion mechanism under many practical circumstances. Currently only two-dimensional problems are discussed. The basic principles proposed in this paper, however, are also valid for three-dimensional situations. Simple numerical examples of powder sintering are given while the full potential of the numerical technique for the above physical problems will be exploited in a forthcoming paper.

A model for simultaneous crystallisation and biodegradation of biodegradable polymers

This paper completes the model of biodegradation for biodegradable polymers that was previously developed by Wang et al. (Wang Y, Pan J, Han X, Sinka, Ding L. A phenomenological model for the degradation of biodegradable polymers. Biomaterials 2008;29:3393-401). Crystallisation during biodegradation was not considered in the previous work which is the topic of the current paper. For many commonly used biodegradable polymers, there is a strong interplay between crystallisation and hydrolysis reaction during biodegradation - the chain cleavage caused by the hydrolysis reaction provides an extra mobility for the polymer chains to crystallise and the resulting crystalline phase becomes more resistant to further hydrolysis reaction. This paper presents a complete theory to describe this interplay. The fundamental equations in the Avramis theory for crystallisation are modified and coupled to the diffusion-reaction equations that were developed in our previous work. The mathematical equations are then applied to three biodegradable polymers for which long term degradation data are available in the literature. It is shown that the model can capture the behavior of the major biodegradable polymers very well.

Modelling sintering at different length scales

Abstract Recent developments in modelling the sintering process of powder compacts are critically reviewed. Both free sintering and pressure assisted sintering are considered. The review approaches the sintering models from the point of view of an end user of the models and sets out to: assess how useful and accurate the existing models are, discuss what is missing from the existing models, and suggest the way forward. This naturally leads to the challenges and opportunities in industrial practice and scientific research of the sintering process. The structure of the review is arranged around the length scales of the various models, i.e. the atomic scale, the particle/grain scale and the component/continuum scale. At each scale, a black box description of the models is given and then the details and the issues of the models are discussed. As a secondary aim, the various numerical techniques for computer simulation of sintering are also reviewed. The review is not intended to explain the numerical techniques in detail, but to guide the reader in the right direction and point to appropriate literature.

Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration

Finite element formulations are developed to model microstructure evolution by a combination of grain–boundary diffusion, grain–boundary migration and free surface diffusion. The formulations are based on a unified variational principle which allows fully coupled processes to be analysed. For example, a process can be analysed which involves grain–boundary diffusion along a curved and migrating grain–boundary network, coupled with surface diffusion along internal and/or external free surfaces which intersect with the grain–boundary network. The numerical solution provides the velocities of each individual grain and the velocities of grain–boundaries and migrating surfaces. The finite element formulations, when combined with a time integration algorithm, form a numerical technique which can be used to simulate microstructural evolution in polycrystalline materials. The technique can be applied to a wide range of physical problems including: sintering of powder compacts; grain–growth; diffusive void growth and crack propagation; superplastic deformation and the morphological evolution of electronic thin films. Various numerical examples are presented to demonstrate the effectiveness of the numerical technique.

Computer simulation of superplastic deformation

Abstract In this paper, a numerical procedure is described to simulate the grain structure evolution of fine grain materials during superplastic deformation. The basic assumption is that the dominant mechanism for the deformation of the material under mechanical loading is grain-boundary sliding accomodated by grain-boundary diffusion. At each time step, the full set of equations which control the grain-boundary diffusion process are solved by the numerical technique suggested by Cocks [1–2] with a further extension to the situation of grains with arbitrary sizes and shapes. The grain boundary network is updated according to grain velocities obtained from the numerical analysis. Any four-rayed grain boundary junction is supposed to be unstable. Therefore a vanishing grain-boundary will lead to either a “neighbour-switching” [3] or “grain vanishing” event. It is demonstrated that the computer simulation based on these few assumptions is able to capture the main characteristics of microstructural evolution in fine grain materials during superplastic deformation. Most importantly, the grain size increases slightly and the grain shape is essentially preserved, remaining equiaxed after a hundred percent elongation.

Degradation mechanisms of bioresorbable polyesters. Part 1. Effects of random scission, end scission and autocatalysis.

A mathematical model was developed to relate the degradation trend of bioresorbable polymers to different underlying hydrolysis mechanisms, including noncatalytic random scission, autocatalytic random scission, noncatalytic end scission or autocatalytic end scission. The effect of each mechanism on molecular weight degradation and potential mass loss was analysed. A simple scheme was developed to identify the most likely hydrolysis mechanism based on experimental data. The scheme was first demonstrated using case studies, then used to evaluate data collected from 31 publications in the literature to identify the dominant hydrolysis mechanisms for typical biodegradable polymers. The analysis showed that most of the experimental data indicates autocatalytic hydrolysis, as expected. However, the study shows that the existing understanding on whether random or end scission controls degradation is inappropriate. It was revealed that pure end scission cannot explain the observed trend in molecular weight reduction because end scission would be too slow to reduce the average molecular weight. On the other hand, pure random scission cannot explain the observed trend in mass loss because too few oligomers would be available to diffuse out of a device. It is concluded that the chain ends are more susceptible to cleavage, which produces most of the oligomers leading to mass loss. However, it is random scission that dominates the reduction in molecular weight.

A constitutive model for stage 2 sintering of fine grained materials—I. Grain-boundaries act as perfect sources and sinks for vacancies

Abstract In this paper, a constitutive model for stage 2 sintering of powder compacts by grain-boundary diffusion is established. A three-dimensional material model composed of a uniform array of tetrakaidecahedra shaped grains with spherical pores at each grain-boundary vertex is adopted. A bounding theorem described by Cocks for grain-boundary diffusion problems is used to obtain a lower bound to the scalar strain-rate potential, through which the macroscopic strain rates and macroscopic stresses are related. Use of the theorem requires a set of kinematically compatible fields of grain-boundary separation rates, matter flux along the grain-boundaries and macroscopic strain-rates to be proposed. In this paper, we focus our attention on the classical grain-boundary diffusion problem where grain-boundaries are assumed to act as perfect sources and sinks for vacancies. The effects of an interface-reaction are discussed in an accompanying paper.

Analysis of degradation data of poly(l-lactide-co-l,d-lactide) and poly(l-lactide) obtained at elevated and physiological temperatures using mathematical models.

The degradation of resorbable polymeric devices often takes months to years. Accelerated testing at elevated temperatures is an attractive but controversial technique. The purposes of this paper include: (a) to provide a summary of the mathematical models required to analyse accelerated degradation data and to indicate the pitfalls of using these models; (b) to improve the model previously developed by Han and Pan; (c) to provide a simple version of the model of Han and Pan with an analytical solution that is convenient to use; (d) to demonstrate the application of the improved model in two different poly(lactic acid) systems. It is shown that the simple analytical relations between molecular weight and degradation time widely used in the literature can lead to inadequate conclusions. In more general situations the rate equations are only part of a complete degradation model. Together with previous works in the literature, our study calls for care in using the accelerated testing technique.