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- The 9-antroate chromophore as a fluorescence probe for waterPublication . Costa, Fernando Pestana da; Costa, Sílvia B.; Melo, Eurico C.; Prieto, Manuel; Maçanita, António L.
- On the positivity of solutions to the Smoluchowski equationsPublication . Costa, Fernando Pestana daThe dynamics of cluster growth can be modelled by the following infinite system of ordinary differential equations, first proposed by Smoluchowski, [8], where cj=cj(t) represents the physical concentration of j-clusters (aggregates of j identical particles), aj,k=aj,k≥0 are the time-independent coagulation coefficients, measuring the effectiveness of the coagulation process between a j-cluster and a k-cluster, and the first sum in the right-hand side of (1) is defined to be zero if j = 1.
- The 9-Anthroate chromophore as a fluorescent probe for waterPublication . Costa, Fernando Pestana da; Maçanita, António L.; Costa, Sílvia B.; Melo, Eurico C.; Santos, HelenaWater quenches the fluorescence of methyl 9-anthroate with a rate constant showing little dependence on solvent viscosity or polarity. In dioxane, at 20ºC the value of the rate constant is 9.6 X 10^6 M^(-1) s^(-1) , and the activation energy found for the process is 14.1 kJ mol^(-1). The quenching process is entropy-controlled and is likely to involve a hydrogen-bonded complex as an intermediate. Since the fluorescence lifetime of methyl 9-anthroate does not depend on the solvent properties other than its hydrogen-bonding ability, the concentration of nearby water can be estimated directly. Values of 3, 54, and 14 M were obtained for the solubilization site of methyl 9-anthroate in micelles of Triton X-100, sodium dodecyl sulfate (SDS), and dodecyltrimethylammonium chloride (DTAC), respectively. From the ring current effect of the anthroate group on the 'H NMR chemical shifts of the surfactant protons, it is concluded that the anthroate fluorescent probe is preferentially located in the surface region of the SDS and DTAC micelles; however, in Triton X-100, it resides in the micelle interior near the phenoxy groups of the surfactant molecule.
- Asymptotic behaviour of low density solutionsPublication . Costa, Fernando Pestana daThe asymptotic behaviour of solutions to the generalized Becker-Döring equations is studied. It is proved that solutions converge strongly to a unique equilibrium if the initial density is sufficiently small.
- A finite-dimensional dynamical model for gelation in coagulation processPublication . Costa, Fernando Pestana daWe study a finite-dimensional system of ordinary differential equations derived from Smoluchowski’s coagulation equations and whose solutions mimic the behaviour of the nondensity-conserving (geling) solutions in those equations. The analytic and numerical studies of the finite-dimensional system reveals an interesting dynamic behaviour in several respects: Firstly, it suggests that some special geling solutions to Smoluchowski’s equations discovered by Leyvraz can have an important dynamic role in gelation studies, and, secondly, the dynamics is interesting in its own right with an attractor possessing an unexpected structure of equilibria and connecting orbits.
- Existence and uniqueness of density conserving solutions to the coagulation-fragmentation equations with strong fragmentationPublication . Costa, Fernando Pestana da
- Convergence to equilibria of solutions to the coagulation-fragmentation equationsPublication . Costa, Fernando Pestana daThe Coagulation-Fragmentation equations are a model for the dynamics of cluster growth and consist of a countable number of non-locally coupled ordinary differential equations, modelling the concentration of the various clusters. The framework for the study of these equations is to see them as a nonlinear ODE in an appropriate infinite dimensional Banach space. A number of results about existence, uniqueness, density conservation, and asymptotic behaviour of solutions have been obtained in recent years. In the present paper we review some of the more recent results on the asymptotic behaviour of solutions as time tends to infinity. We pay special attention to the different behaviour of solutions in the weak-star and in the strong topologies of the phase space. This distinction can be interpreted in terms of a dynamic phase transition in the physical system modelled by the equations. It is shown that a balance between the relative strength of coagulation and fragmentation is crucial for that distinct behaviour to take place.
- Instantaneous gelation in coagulation dynamicsPublication . Costa, Fernando Pestana da; Carr, J.The coagulation equations are a model for the dynamics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters. For a certain class of rate coefficients we prove that the density is not conserved on any time interval.
- Testes e exames resolvidos de equações diferenciaisPublication . Costa, Fernando Pestana daEste texto consiste numa colectânia de testes e exames resolvidos da disciplina de Equações Diferenciais, leccionada e regida (ou co-regida) pelo autor aos cursos de licenciatura em Química, Eng.ª Aeroespacial, Eng.ª do Ambiente e Eng.ª Mecânica do Instituto Superior Técnico, entre 1994 e 1998.
- On the dynamic scaling behaviour of solutions to the discrete Smoluchowski equationsPublication . Costa, Fernando Pestana daIn this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations $$\dot{c}_{j} = \sum_{k=1}^{j-1}c_{j-k}c_{k} - 2c_{j}\sum_{k=1}^{\infty}c_{k}, j = 1, 2, \ldots$$ with general exponentially decreasing initial data, with density $\rho,$ have the following asymptotic behaviour $$\lim_{j, t \rightarrow\infty, \xi = j/t fixed, j \in {\cal J}} t^{2}c_{j}(t) = \frac{q}{\rho}\, e^{-\xi/\rho},$$ where ${\cal J} = \{j: c_{j}(t)>0, t>0\}$ and $q =\gcd \{j: c_{j}(0)>0\}.$