Theoretical Study Of Dynamic Properties Of Material Nonlinear Growth

Theoretical investigation of stochastic growth dynamics has been one of the mostactive research topics in the field of dynamic system analysis. In the process ofmaterial growth, the growth surfaces can exhibit kinetic roughening phenomena, andusually possess self-affine fractal structure. From the viewpoint of stochasticnonlinear dynamic equations for describing material growth processes, the dynamicproperties of material growth are studied theoretically in this thesis. The main work issummarized as the follow:Firstly, in order to investigate the physical mechanisms which lead to theanomalous roughening, based on local slope theory, the anomalous scaling behaviorof the stochastic dynamic growth equations describing material growth processes isstudied using Flory-type scaling approach, and the general expressions of anomalousscaling exponents are derived for both local and nonlocal cases. The results show that:(i) Both the Kardar-Parisi-Zhang (KPZ) and Sun-Guo-Grant (SGG) equations withspatiotemporal correlated noises exhibit the normal self-affine scaling properties inboth the strong-and weak-coupling regions. However, the Villain-Lai-Das Sarma(VLDS) equation possesses the anomalous scaling.(ii) In the nonlocal cases, theanomalous scaling exponents are related not only to the dimension of the substrate,but also to the nonlocal decay exponents. In the limit that long-range interactiondecays to zero, the nonlocal growth equations can recover to the corresponding localones.Secondly, the conserved dynamic growth equations and the corresponding localslopes in1+1and2+1dimensions are studied using numerical simulations to furtherexplore the anomalous scaling behavior of the conserved dynamic growth processes.The obtained values of critical exponents numerically are consistent with theanalytical results. The main results show that:(i) In1+1dimensions, both the linearmolecular-beam epitaxy (LMBE) and Villain-Lai-Das Sarma (VLDS) equationspossess anomalous dynamic scaling behavior, while their local slopes don't exhibitthe anomalous scaling. Both the SGG and its local slope equations exhibit normalscaling behavior.(ii) In2+1dimensions, anomalous scaling exponents in both LMBEand VLDS equations tend to zero, which implies very weak anomalous scaling.Furthermore, the theoretical explanation is also given about universality class issuesof thenonlinear stochastic continuum equation for epitaxial growth processes, it is found that2+1dimensional Escudero and controlled VLDS (CVLDS) equationsbelong to the same universality class.Thirdly, the time-fractional stochastic equations are introduced to describememory effects in growth dynamic processes, and the dynamic scaling behavior isstudied using scaling analysis and numerical scheme. The main results show that:(i)All these time-fractional stochastic growth equations exhibit normal Family-Vicsekscaling.(ii) The time-fractional orders affect dynamic scaling exponents significantly,and these time-fractional growth equations belong to a family of continuouslychanging universality classes.(iii) The results obtained by scaling analysis andnumerical stimulations are consistent with each other for larger order of thetime-fractional derivative. The time-fractional growth system shows more evidentcharacteristic of growth instability in comparison with the local growth equations.Finally, the space-fractional stochastic dynamic equations are studiednumerically to investigate nonlocal effects in stochastic dynamic processes. Theresults show that:(i) The scaling exponents of the space-fractionalKardar-Parisi-Zhang (SFKPZ) equation exhibit very weak dependence on thefractional orders, while the critical exponents depend evidently on the fractionalorders for the space-fractional Edwards-Wilkinson (SFEW) equation.(ii) Neitherfinite-size nor finite-time effects are found distinctly in the SFEW or SFKPZ system.The values of the global and the local roughness exponents are same, which impliesthat anomalous behavior does not occur in the SFEW and SFKPZ equations.(iii) Thespace-fractional quenched (SFQ) model for driven interfaces in random media isproposed. The numerical results show that, near the critical point, both the criticalexponents and external driven forces are monotonously increasing functions of thespace-fractional orders. The SFQ model exhibits intrinsic anomalous behavior, whichdiffers from the space-fractional growth equations.The investigation of this thesis provides a deeper understanding about dynamicscaling, especially anomalous scaling behavior, in material growth processes. Theintroduction of the stochastic fractional differential equations also deepened ourunderstanding of memory and nonlocal effects in stochastic dynamic growthprocesses.