Abstract
Piezoelectric quartz (SiO2) crystals are widely used in industry as oscillators. As a natural mineral, quartz and its relevant silicates are also of interest in geoscience and mineralogy. However, the nucleation and growth of quartz crystals are difficult to control and not fully understood. Here we report successful solid-state crystallization of thin film of amorphous GeO2 into quartz on various substrates, including Al2O3, MgAl2O4, MgO, LaAlO3 and SrTiO3. At relatively low annealing temperatures, the crystallization process is spherulitic: with fibers growing radially from the nucleation centers and the crystal lattice rotating along the growth direction with a linear dependence between the rotation angle and the distance to the core. For increasingly higher annealing temperatures, quartz crystals begin to form. The edges of the sample play an important role in facilitating nucleation followed by growth sweeping inward until the whole film is crystallized. Control of the growth allows single crystalline quartz to be synthesized, with crystal sizes of hundreds of microns achieved on sapphire substrates, which is promising for further piezoelectric applications. Our study reveals the complexity of the nucleation and growth process of quartz and provides insight for further studies.
Highlights
T O STUDY complex biological and ecological systems, a range of mathematical models have been proposed, among which the Lotka–Volterra model [1], [2] and consumerresource model [3], [4] have been playing prominent roles
The Lotka–Volterra model essentially describes the evolution of the predator and prey populations
By extending the concept of resources dynamics, we study the eco-evolutionary dynamics under the different resource models using two-player two-strategy games
Summary
T O STUDY complex biological and ecological systems, a range of mathematical models have been proposed, among which the Lotka–Volterra model [1], [2] and consumerresource model [3], [4] have been playing prominent roles. GONG AND CAO: DIFFERENT ENVIRONMENT FEEDBACK IN FAST-SLOW ECO-EVOLUTIONARY DYNAMICS AND RESULTING LIMIT CYCLES further analyze the stability of interior equilibria, we fix some parameters and conduct bifurcation analysis for one payoff parameter and the time-scale parameter. Through both oneparameter and two-parameter analyses, it is shown that both systems can exhibit Hopf bifurcations in the specific parameter conditions.
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