Mathematical Theory of Exchange-driven Growth

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3<\mu+\nu\leq4,$ we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.Comment: Mistakes in the uniqueness proofs are fixed. Some typos are corrected. Some references are adde

The role of zero-clusters in exchange-driven growth with and without input

The exchange-driven growth model describes the mean field kinetics of a population of composite particles (clusters) subject to pairwise exchange interactions. Exchange in this context means that upon interaction of two clusters, one loses a constituent unit (monomer) and the other gains this unit. Two variants of the exchange-driven growth model appear in applications. They differ in whether clusters of zero size are considered active or passive. In the active case, clusters of size zero can acquire a monomer from clusters of positive size. In the passive case they cannot, meaning that clusters reaching size zero are effectively removed from the system. The large time behaviour is very different for the two variants of the model. We first consider an isolated system. In the passive case, the cluster size distribution tends towards a self-similar evolution and the typical cluster size grows as a power of time. In the active case, we identify a broad class of kernels for which the the cluster size distribution tends to a non-trivial time-independent equilibrium in which the typical cluster size is finite. We next consider a non-isolated system in which monomers are input at a constant rate. In the passive case, the cluster size distribution again attains a self-similar profile in which the typical cluster size grows as a power of time. In the active case, a surprising new behavior is found: the cluster size distribution asymptotes to the same equilibrium profile found in the isolated case but with an amplitude that grows linearly in time

Teachers predicting self-regulation skills of children:The relationships among teacher beliefs, teaching intentions and preschoolers’ self-regulation skills

We designed this study to find out if teachers' beliefs and intentions would have an impact on children's self-regulation skills and to see if teachers would estimate self-regulation skills of children properly. There were 43 teachers and 190 preschool children as participants. We asked teachers to report their beliefs about teaching and we also observed their teaching intentions during classroom activities. For measuring self-regulation skills of children, we created story stems which were presented to participant children individually and scored with a four-point coding system. We presented the same stories for teachers. Findings showed that there was a positive correlation between the reports of children and teachers on two dimensions: waiting for own turn and having control on negative behaviour. Teacher beliefs and self-regulation skills of children were also found to be significantly correlated with each other

Generalized Phase Field Models with Microscopic Potentials

In this thesis we study the solidification process of systems with intrinsicanisotropy. We aim at finding a bridge between the microscopic mechanismsand macroscopic description. This is achieved by generalizing the currentphase field models in a way to incorporate microscopic physics and usingasymptotic techniques to obtain macroscopic results. Upon analysis,expressions for physically relevant quantities are obtained. Also it isfound that classical interface relations for both stationary and movinginterfaces hold. These conditions are presented in various representations.Exemplary numerical calculations are carried out to illustrate the potentialof the method as an additional tool in the study of interfaces. Furthermore,a concrete physical system with realistic parameters is considered to showhow one can use the ideas developed here in order to get results that are ofinterest to other scientific communities, e.g. materials scientists andphysicists

Molecular Structure and Surface Organization: A Study of Liquid/Vapor Interfaces Using Newly Developed Sum Frequency Methods

Vibrational sum frequency spectroscopy (VSFS) is a surface specific nonlinear technique that provides vibrational spectra of molecules at interfaces. Studies presented in this thesis use VSFS to examine surface structure of common liquids at liquid/vapor interfaces. The goal of this work is to correlate molecular structure and molecules functionality with the structure adopted by molecules at liquid surfaces. The first part of this work describes the instrumental design and components of the newly developed VSFS. New methods are developed to overcome experimental difficulties associated with data collection for an entire spectral region of interest. The methods for data analysis and spectral post processing are also presented. Surface vibrational studies of linear alkanes ranging in length from 9 to 17 carbon atoms investigated dependence of surface order on chain length. The linear alkane liquid surfaces evince a surprising degree of conformational order and this surface structure becomes more disordered as chain length increases. Halogenated alkane studies showed that replacing one terminal methyl group of linear alkanes with CH2X (X=Cl or Br) for chain lengths 10, 14 and 16 leads to a mixed surface layer primarily occupied by CH2X. These results are attributed to dipole-dipole interactions between the CH2X and are indicative of a higher surface activity of the halogenated ends compared to the unsubstituted ends. Additional experiments have investigated the effect of stronger interactions on the surface structure of 1-, 3-, and 5-nonanols and their ketone analogs namely 1-, 3-, and 5-nonanones. Results showed an increasing order as the OH or =O groups advances toward the center of the molecule. A surprising result is a slightly lower surface order for nonanones compared to nonanols hints at the relative importance of simple dipolar interactions and highly directional hydrogen bonding in determining the surface structure. In addition, vibrational studies are carried out for 1- and 3-octanol at liquid/vapor interfaces. The results confirm observations from the nonanols by having a higher order for the 3-octanol. However, monolayer of 1-octanol on aqueous surface have a much higher surface order compared to 3-octanol. The increased order is attributed to the strong hydrogen bonding with water molecules

Homoleptic Naked Clusters of Endohedral Zintl Ions

Nanoclusters are of interest because of their remarkable catalytic properties and their application in nanotechnology fields. There is a strong preference toward bimetallic systems due to their superior catalytic properties compare to monometallic systems. Synthetic protocols are developed by using polyatomic main group clusters (Zintl ions) for making new bimetallic nano-catalysts. In this study, new, free standing, transition metal stabilized Zintl ions have been isolated. They are promising candidates to be used both in nanotechnology and heterogeneous catalysis. [M@Pb&lt;sub&gt;12&lt;/sub&gt;]&lt;sup&gt;2-&lt;/sup&gt; (M = Ni, Pd, Pt) are the &lt;em&gt;first isolated&lt;/em&gt; free-standing centered icosahedra without any attendant of ligands. The anions contain naked Pb&lt;sub&gt;12&lt;/sub&gt; icosahedra clusters encapsulating Group 10 transition metals and posses near perfect I&lt;sub&gt;h&lt;/sub&gt; point symmetry. These endohedral naked metal clusters have nontraditional spherical aromatic electronic structures. They are the smallest members of the nanoparticle growth sequence, M&lt;sub&gt;13&lt;/sub&gt; (centered icosahedra), with hcp lattices. Their positive chemical shifts in &lt;sup&gt;207&lt;/sup&gt;Pb NMR spectra contradict theoretical chemical shift calculations of isostructural aromatic clusters. A new type of Zintl ion, [Ni@Pb&lt;sub&gt;10&lt;/sub&gt;]&lt;sup&gt;2-&lt;/sup&gt;, has also been synthesized. The anion contains Ni atom centered in a closo-Pb&lt;sub&gt;10&lt;/sub&gt;&lt;sup&gt;2-&lt;/sup&gt; bicapped square antiprism, and possesses virtual D&lt;sub&gt;4d&lt;/sub&gt; point symmetry. It is the first isolated ten atom, homoatomic naked Zintl ion cluster. The Pt and Pd analogs, and non-metallated Pb&lt;sub&gt;10&lt;/sub&gt; and Pb&lt;sub&gt;12&lt;/sub&gt; clusters have also been observed in the gas phase by LDI-TOF-MS. NMR spectroscopy was used to investigate unusual dynamic behaviors of the one focus [Ni@Pb&lt;sub&gt;10&lt;/sub&gt;]&lt;sup&gt;2-&lt;/sup&gt; and two focus [Ni&lt;sub&gt;2&lt;/sub&gt;Sn&lt;sub&gt;17&lt;/sub&gt;]&lt;sup&gt;4-&lt;/sup&gt; anions and both demonstrated surprising global atomic mobility. The latter has a capsule like structure having central Sn atom with coordination number of eight. This extraordinary high coordination of the central Sn is more akin to solid state compounds. The [Ge&lt;sub&gt;9&lt;/sub&gt;Ni&lt;sub&gt;2&lt;/sub&gt;(PPh&lt;sub&gt;3&lt;/sub&gt;)]&lt;sup&gt;2-&lt;/sup&gt; anion is the new example of Zintl ion, which do not adopt Wadian type structure. Another Ni-Ge system with exciting structure, [Ni&lt;sub&gt;6&lt;/sub&gt;Ge&lt;sub&gt;13&lt;/sub&gt;(CO)&lt;sub&gt;5&lt;/sub&gt;]&lt;sup&gt;4-&lt;/sup&gt; is also isolated. The anion has interpenetrating biicosahedral structure which is unique to Zintl ion chemistry. The potential interconversion of [Ge&lt;sub&gt;9&lt;/sub&gt;Ni&lt;sub&gt;2&lt;/sub&gt;(PPh&lt;sub&gt;3&lt;/sub&gt;)]&lt;sup&gt;2-&lt;/sup&gt; and [Ni&lt;sub&gt;6&lt;/sub&gt;Ge&lt;sub&gt;13&lt;/sub&gt;(CO)&lt;sub&gt;5&lt;/sub&gt;]&lt;sup&gt;4-&lt;/sup&gt; have been investigated through ESI-MS studies

Large time behaviour of exchange-driven growth

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel K(j,k). Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel (i) K(j,k)=bjak and (ii) K(j,k)=jak+bj+εβjαk. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass ρs above which there is no equilibrium. We prove that if the system has an initial mass below ρs then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above ρs then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm

A Comparison of Playfulness Levels of the Secondary Students in Kayseri and Trabzon Cities

The aim of the study is to draw a comparison of 11-14 age group secondary school students playfulness level to play games which consist of physical activity. It also aims to evaluate playfulness level in terms of various parameters. For this purpose, 11-14 age group students’ playfulness level were examined according to cities, gender, and ages. A total of 523 11- 14 year-old students, including 303 from Kayseri and 220 from Trabzon, and some various schools formed a study group for the research conducted in the relational model. “Playfulness scale for playing games that consist of physical activity of children of 10-14 age group” that was developed by Hazar (2014) was used in the data collection. Based on the research, the level of playfulness of the participants was found to be "good" at 3.40-4.19. Students living in Kayseri and Trabzon were found to have "good" in terms of social adaptation, wish to play game, wish to win, and in general. Students in Kayseri "good" level had a playfulness at "medium" level in Trabzon. Due to the comparison of male and female student scores, it was found that there was a significant difference in favor of male students in game play overall, game passion, and risk taking dimensions. Girls are more likely to have social adaptation, playfulness, and a meaningful difference generally. Also, girls who live in Kayseri are more likely to be students. They wish to play game, social adaptation, wish to win, and risk taking. Meaningful differences can however be seen based on the scores of the 11 and 12 year students

Broadband THz modulators based on multilayer graphene on PVC

In this study we present the direct terahertz time-domain spectroscopic measurement of CVD-grown multilayer graphene (MLG) on PVC substrate with an electrically tunable Fermi level. In a configuration consisting MLG and injected organic dopant, the transmitted intensity loss of terahertz radiation was observed with an applied voltage between 0 and 3.5 V. We showed that MLG on PVC devices provided approximately 100 % modulation between 0.2 and 1.5 THz at preferentially low operation voltage of ca. 3V. The observed modulation bandwidth in terahertz frequencies appears to be instrument limited. © 2016 IEEE