Leonidas N. Gergidis

Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone

The classical linear theory of elasticity has been largely used for the ultrasonic characterization of bone. However, linear elasticity cannot adequately describe the mechanical behavior of materials with microstructure in which the stress state has to be defined in a non-local manner. In this study, the simplest form of gradient theory (Mindlin Form-II) is used to theoretically determine the velocity dispersion curves of guided modes propagating in isotropic bone-mimicking plates. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g, introduced in the strain energy, and (b) the micro-inertia term h, in the kinetic energy. The plate was assumed free of stresses and of double stresses. Two cases were studied for the characteristic length: h=10(-4) m and h=10(-5) m. For each case, three subcases for g were assumed, namely, g>h, g<h, and g=h. The values of g and h were of the order of the osteons size. The velocity dispersion curves of guided waves were numerically obtained and compared with the Lamb modes. The results indicate that when g was not equal to h (i.e., g not equal h), microstructure affects mode dispersion by inducing both material and geometrical dispersion. In conclusion, gradient elasticity can provide supplementary information to better understand guided waves in bones.

The effect of boundary conditions on guided wave propagation in two-dimensional models of healing bone.

Guided wave propagation has recently drawn significant interest in the ultrasonic characterization of bone. In this work, we present a two-dimensional computational study of ultrasound propagation in healing bones aiming at monitoring the fracture healing process. In particular, we address the effect of fluid loading boundary conditions on the characteristics of guided wave propagation, using both time and time-frequency (t-f) signal analysis techniques, for three study cases. In the first case, the bone was assumed immersed in blood which occupied the semi-infinite spaces of the upper and lower surfaces of the plate. In the second case, the bone model was assumed to have the upper surface loaded by a 2mm thick layer of blood and the lower surface loaded by a semi-infinite fluid with properties close to those of bone marrow. The third case, involves a three-layer model in which the upper surface of the plate was again loaded by a layer of blood, whereas the lower surface was loaded by a 2mm layer of a fluid which simulated bone marrow. The callus tissue was modeled as an inhomogeneous material and fracture healing was simulated as a three-stage process. The results clearly indicate that the application of realistic boundary conditions has a significant effect on the dispersion of guided waves when compared to simplified models in which the bones surfaces are assumed free.

Brownian Dynamics Simulations on the Self-Assembly Behavior of AB Hybrid Dendritic—Star Copolymers

The micellization behavior of hybrid dendritic-star copolymers with solvophilic dendritic units is studied by means of Brownian dynamics simulations. The critical micelle concentration and the micelle size and shape are examined for different solvophobic/solvophilic ratios r as a function of the number of the dendritic and linear arms. Hybrid dendritic-star copolymers with one dendritic and up to three solvophobic linear branches form spherical micelles with preferential aggregation number. Those with two dendritic arms and three solvophobic branches form micelles with wide aggregation numbers only for small values of r. For hybrid dendritic-star copolymers with three dendritic arms and two or three solvophobic linear arms, micelles with wide aggregation numbers are also formed but for slightly higher values of r. Our results for the aggregation number are compared with existing results of other architectures obtained at the same temperature, and an inequality for the aggregation number is proposed.

Dendritic brushes under good solvent conditions: a simulation study.

The structural properties of polymer brushes, formed by dendron polymers up to the third generation, were studied by means of Brownian dynamics simulations for the macroscopic state of good solvent. The distributions of polymer units, of the free ends, of the dendrons centers of mass, and of the units of every dendritic generation and the radii of gyration necessary for the understanding of the internal stratification of brushes were calculated. Previous self-consistent field theory numerical simulations of first-generation dendritic brushes suggested that at high grafting densities two kinds of populations are evident, one of short dendrons having weakly extended spacers and another with tall dendrons having strongly stretched spacers. These Brownian dynamics calculations provided a more complicated picture of dendritic brushes, revealing different populations of short, tall, and in some cases intermediate height dendrons, depending on the dendron generation and spacer length. The scaling dependence of the height and the span of the dendritic brush on the grafting density and other parameters were found to be in good agreement with existing theoretical results for good solvents.

Dendritic brushes under theta and poor solvent conditions

The effects of solvent quality on the internal stratification of polymer brushes formed by dendron polymers up to third generation were studied by means of molecular dynamics simulations with Langevin thermostat. The distributions of polymer units, of the free ends, the radii of gyration, and the back folding probabilities of the dendritic spacers were studied at the macroscopic states of theta and poor solvent. For high grafting densities we observed a small decrease in the height of the brush as the solvent quality decreases. The internal stratification in theta solvent was similar to the one we found in good solvent, with two and in some cases three kinds of populations containing short dendrons with weakly extended spacers, intermediate-height dendrons, and tall dendrons with highly stretched spacers. The differences increase as the grafting density decreases and single dendron populations were evident in theta and poor solvent. In poor solvent at low grafting densities, solvent micelles, polymeric pinned lamellae, spherical and single chain collapsed micelles were observed. The scaling dependence of the height of the dendritic brush at high density brushes for both solvents was found to be in agreement with existing analytical results.

On the dyadic scattering problem in three-dimensional gradient elasticity: an analytic approach

The investigation of the direct scattering problem of an elastic dyadic incident field from a spherical inclusion, is the main outcome of this work, in the case where the scatterer and the host environment dispose microstructure. The framework of the method is based on the implication of Mindlins gradient theory. The development of the method is fully analytic and gives successively several byproducts, which are indispensable for the solution of the scattering problem but constitute also independent results of their own theoretical and practical value. So the numerable set of Navier eigendyadics is constructed, which is proved to be a basis for every dyadic field obeying the dynamic gradient elasticity equation. This permits the construction of a useful spectral representation for every gradient elasticity field. Furthermore, the set of dyadic spherical harmonics is built, which stands for the extension of the well-known spherical vector harmonics to the dyadic realm. Every dyadic field restricted on the unit sphere can be expanded in terms of these spherical dyadic harmonics. The orthogonality relations of these functions are determined in close form and this is the prerequisite for the fully analytic treatment of the boundary conditions involving the scattering problem under consideration.

Complexation of Polyelectrolyte Micelles with Oppositely Charged Linear Chains

The formation of interpolyelectrolyte complexes (IPECs) from linear AB diblock copolymer precursor micelles and oppositely charged linear homopolymers is studied by means of molecular dynamics simulations. All beads of the linear polyelectrolyte (C) are charged with elementary quenched charge +1e, whereas in the diblock copolymer only the solvophilic (A) type beads have quenched charge -1e. For the same Bjerrum length, the ratio of positive to negative charges, Z+/-, of the mixture and the relative length of charged moieties r determine the size of IPECs. We found a nonmonotonic variation of the size of the IPECs with Z+/-. For small Z+/- values, the IPECs retain the size of the precursor micelle, whereas at larger Z+/- values the IPECs decrease in size due to the contraction of the corona and then increase as the aggregation number of the micelle increases. The minimum size of the IPECs is obtained at lower Z+/- values when the length of the hydrophilic block of the linear diblock copolymer decreases. The aforementioned findings are in agreement with experimental results. At a smaller Bjerrum length, we obtain the same trends but at even smaller Z+/- values. The linear homopolymer charged units are distributed throughout the corona.

A study on Rayleigh wave dispersion in bone according to Mindlin's Form II gradient elasticity

The classical elasticity cannot effectively describe bones mechanical behavior since only homogeneous media and local stresses are assumed. Additionally, it cannot predict the dispersive nature of the Rayleigh wave which has been reported in experimental studies and was also demonstrated in a previous computational study by adopting Mindlins Form II gradient elasticity. In this work Mindlins theory is employed to analytically determine the dispersion of Rayleigh waves in a strain gradient elastic half-space. An isotropic semi-infinite space is considered with properties equal to those of bone and dynamic behavior suffering from microstructural effects. Microstructural effects are considered by incorporating four intrinsic parameters in the stress analysis. The results are presented in the form of group and phase velocity dispersion curves and compared with existing computational results and semi-analytical curves calculated for a simpler case of Rayleigh waves in dipolar gradient elastic half-spaces. Comparisons are also performed with the velocity of the first-order antisymmetric mode propagating in a dipolar plate so as to observe the Rayleigh asymptotic behavior. It is shown that Mindlins Form II gradient elasticity can effectively describe the dispersive nature of Rayleigh waves. This study could be regarded as a step toward the ultrasonic characterization of bone.

Entropic effects, shape, and size of mixed micelles formed by copolymers with complex architectures.

The entropic effects in the comicellization behavior of amphiphilic AB copolymers differing in the chain size of solvophilic A parts were studied by means of molecular dynamics simulations. In particular, mixtures of miktoarm star copolymers differing in the molecular weight of solvophilic arms were investigated. We found that the critical micelle concentration values show a positive deviation from the analytical predictions of the molecular theory of comicellization for chemically identical copolymers. This can be attributed to the effective interactions between copolymers originated from the arm size asymmetry. The effective interactions induce a very small decrease in the aggregation number of preferential micelles triggering the nonrandom mixing between the solvophilic moieties in the corona. Additionally, in order to specify how the chain architecture affects the size distribution and the shape of mixed micelles we studied star-shaped, H-shaped, and homo-linked-rings-linear mixtures. In the first case the individual constituents form micelles with preferential and wide aggregation numbers and in the latter case the individual constituents form wormlike and spherical micelles.

A theoretical and experimental study of bone's microstructural effect on the dispersion of ultrasonic guided waves

Ultrasonic evaluation of bone has been based on the classical linear elastic theory. However, this theory cannot adequately describe bones mechanical behavior since the microstructure is neglected. In this study, we use the simplest form of gradient theory (Mindlin FormII) to theoretically derive the velocity dispersion curves for free isotropic bone-mimicking plates and we investigate whether that theory can characterize the modes propagating in real bones better than the Lamb wave theory. Two additional terms are included in the constitutive equations representing the characteristic length in bone: (a) the gradient coefficient g and (b) the micro-inertia term h whose values were at the order of the osteons size. The velocity dispersion curves of guided waves were numerically obtained for four combinations between g and h and were superimposed on the time-frequency representations of the signals obtained from ex-vivo measurements. For the first time it was made feasible to detect fast waves with velocity higher than the S0 mode. Overall the gradient theory seems to be more efficient in mode identification than the classical theory, providing thus better understanding of ex-vivo measurements.