Lattice models in materials science diffusion, trabecular bone remodelling and linear elastic networks

by

This thesis presents the results of investigations on three independent research topics of modern biophysical and materials science research: substitutional diffusion in binary alloys, the remodelling process in trabecular bone and the prediction of mechanical properties of self assembling, amphiphilic bilayers. The basic description of all three projects is based on lattice models, a highly successful class of models that are used in several fields of modern physics to describe physical processes. For the diffusional process in alloys, which on a microscopic scale manifests itself in a discrete site exchange between one atom and a neighbouring vacancy, it was investigated how this microscopic description can be reconciled with a macroscopic continuum model. In a computer simulation exact microscopic averages were used to determine macroscopic properties, like Onsager’s coefficients. These were then compared to theoretical predictions of different accuracy. Following the same strategy – comparing averaged results from microscopical simulations with purely continuum mechanical calculations – interdiffusion problems were investigated. It was shown that for obtaining an appropriate macroscopic description it is essential to fully include the behaviour of the vacancy in the description, which is – due its complexity – often omitted. For the investigations on remodelling of trabecular bone, bone’s architecture was mapped onto a lattice and the local mechanical state of each element was determined by a simplified mechanical model. A local remodelling law was then used to translate this mechanical information into a signal that determined the rate of change of the architecture at that special point. This rate of change was given by a stochastic description, i.e. the remodelling law gave the probabilities for bone formation and resorption, respectively. The development of the model was guided by the aim to give a good balance in the accuracy of the description of the mechanical and biological part. The simple, but fast, algorithm to assess the mechanical properties of the structure gave the possibility to test a variety of biological hypotheses, concerning the special form of the remodelling law. It was shown that a stochastic description of the remodelling process demands the formulation of both, a formation and a resorption probability, since – in contrast to conventional simulations with deterministic rate equations – a pure net effect does not suffice to describe the process. Furthermore it was shown that a non-linear remodelling law is a better candidate to describe the remodelling process in real bone than a linear one. Finally the model was used to describe osteoporosis, a wide spread disease affecting trabecular architecture. It was concluded that in the features attributed to osteoporosis one has to distinguish between normal ageing of bone’s architecture and additional changes that stem from pathological alterations in the regulatory system. A simple concept was introduced to model the mechanical properties of self-assembled membranes. The (amphiphilic) molecules forming the membrane are assumed to occupy a regular lattice, nearest neighbours are connected by linear, elastic springs. Different spring constants are assumed for different atomic pairs. The full elastic matrix of a given structure was solved and the elastic modulus, the Poisson ratio and the bending rigidity of the system determined. It was shown that the bending rigidity exhibits a pronounced concentration dependence, varying over orders of magnitude in a small concentration regime, giving very flexible membranes at one end (bending rigidities of the order of kT ), very stiff ones at the other (bending rigidities up to three orders of magnitude larger than kT ).