Microscopic Volume Element Research Articles

Multiscale analysis of mixed-mode fracture and effective traction-separation relations for composite materials

A multiscale framework for the analysis of fracture is developed in order to determine the effective (homogenized) strength and fracture energy of a composite material based on the constituent’s material properties and microstructural arrangement. The method is able to deal with general (mixed-mode) applied strains without a priori knowledge of the orientation of the cracks. Cracks occurring in a microscopic volume element are modeled as sharp interfaces governed by microscale traction-separation relations, including interfaces between material phases to account for possible microscale debonding. Periodic boundary conditions are used in the microscopic volume element, including periodicity that allows cracks to transverse the boundaries of the volume element at arbitrary orientations. A kinematical analysis is presented for the proper interpretation of a periodic microscopic crack as an equivalent macroscopic periodic crack in a single effective orientation. It is shown that the equivalent crack is unaffected by the presence of parallel periodic replicas, hence providing the required information of a single localized macroscopic crack. A strain decomposition in the microscopic volume element is used to separate the contributions from the crack and the surrounding bulk material. Similarly, the (global) Hill–Mandel condition for the volume element is separated into a bulk-averaged condition and a crack-averaged condition. Further, it is shown that, though the global Hill–Mandel condition can be satisfied a priori using periodic boundary conditions, the crack-based condition can be used to actually determine the effective traction of an equivalent macroscopic crack.

Magnetic field control of the potential distribution and current at microdisk electrodes

Au and Pt microdisk electrodes (6.4, 12.5, and 25 μm radii) have been used to investigate the influence of a uniform external magnetic field on faradaic reactions. Magnetohydrodynamic flow within a microscopic volume element adjacent to the microdisk surface results in an increase in steady-state limiting voltammetric currents for reduction and oxidation of both neutral and ionic electroactive molecules. The dependence of limiting currents on external field strength suggests the existence of a threshold value of the magnetic force necessary to induce convective flow. Above this threshold value, the voltammetric current is found to increase in proportion both to the external magnetic field strength and to the steady-state current measured in the absence of the field. The results are used to establish an empirical correlation between faradaic currents and the magnetic force acting on the solution within the depletion layer. In low ionic strength solutions, the external magnetic field also reduces the electrostatic driving force for electron-transfer. This magnetic field-induced reaction overpotential is shown to result from a steady-state displacement of charge-balancing counterions at the electrode-solution interface, analogous to that recently observed for a rotating microdisk electrode.

Magnetic Field Effects in Electrochemistry. Voltammetric Reduction of Acetophenone at Microdisk Electrodes

The influence of an external magnetic field on the electrochemical reduction of acetophenone (AP) at Pt and Au microdisk electrodes (radii = 0.1, 6.4, 12.5, and 25 μm) is described. Voltammetric measurements in CH3CN/[n-Bu4N]PF6 solutions containing between 3 mM and 8 M AP demonstrate that the mass-transfer-limited reduction of AP at a microdisk electrode may be significantly enhanced or diminished by an external magnetic field, depending on the redox concentration, the electrode radius, and the angular orientation of the microdisk relative to the field. A mechanism for the magnetic field effect is presented that considers the force arising from the divergent radial flux of electrogenerated radical anion (AP•-) through a uniform magnetic field. Viscous drag on the field-accelerated ions results in convective fluid flow that alters the rate at which electroactive AP is transported to the surface. Both lateral and cyclotron fluid motion can be established within a microscopic volume element (∼30 nL) near the electrode surface depending on the orientation of the magnetic field with respect to the microdisk. The earth's gravitational field is demonstrated to enhance or diminish the magnetic field-induced convective flow, depending on the relative directions of the magnetic and buoyancy forces at the electrode surface.

H NMR Characterization of Swelling in Cross-Linked Polymer Systems

A 1H NMR method capable of determining the level of swelling of microscopic volume elements (about 20 μm in diameter) within cross-linked materials is described. The fact that it is a microscopic swell measurement makes it extremely useful for the characterization of the swelling heterogeneities which may exist within common network systems, such as core/shell or other morphologies. The method utilizes the differences in chemical shift between solvent absorbed into the cross-linked polymer and that of solvent outside the polymer. This chemical shift difference is then correlated to macroscopic swelling (rather than cross-linking) through a simple model which encompasses both the effective chemical cross-links and the entanglement cross-links in the manner of classical swelling experiments. The analysis is demonstrated for styrene, divinylbenzene copolymer beads, cross-linked polycarbonates, ion-exchange cation resins and cross-linked poly(acrylic acid). A calibration is, in each case, developed with a ser...

Micro-Volume Time-Resolved Fluorescence Spectroscopy Using a Confocal Synchrotron Radiation Microscope

The confocal microscope SYCLOPS (SYCLOPS is not an acronym although the first syllable is derived from its main light source, a synchrotron radiation source) at the Daresbury Science and Engineering Research Council (SERC) Laboratory, U.K., has been designed to facilitate fluorescence decay measurements on microscopic volume elements. SYCLOPS utilizes the Daresbury Synchrotron Radiation Source (SRS) as a pulsed light source. Visible or UV excitation light is selected from the white synchrotron radiation spectrum with a bandpass filter matching the absorption band of the fluorophores. Decay curves of fluorescence intensity emitted by a pre-selected micro-volume in the sample can be acquired with standard time-correlated single-photon counting techniques. The fluorescence intensity was collected from a confocal spot with a volume of 3 μm3. The first lifetime measurements done with the instrument were carried out on a coumestrol l]benzopyran-6-one) solution and on different cellular compartments of living cells incubated with coumestrol.

Continuum mechanics for higher-stage micropolar materials. 4th Report. Discussions of thermodynamics for constitutive equations.

Constitutive equations for higher-stage micropolar materials must be discussed ther-modynamically as a preliminary to the construction of concrete constitutive equations. In the present paper, averaging the lst and 2nd laws of thermodynamics is the microscopic volume elements up to stage-2, the energy balance equation and the entropy inequality for the micropolar materials of stage-2 are formulated on the basis of the results of the previous reports. Thermodynamic relation-ships between the generalized forces (stress, couple stress, bicouple stress, heat flux density) and thermodynamic functions (free energy and dissipation function) which govern the constitutive equations are derived using the Clausius-Duhem inequality obtained by coupling the energy balance equation and entropy inequality. In this way, the thermodynamic limitations which the constitutive equations must satisfy are clarified, and the response functions and their arguments which characterize the constitutive equations are determined.