Dana Mackey

Two-way diffusion model for short-exposure holographic grating formation in acrylamide- based photopolymer

A theoretical model for formation of a short-exposure holographic grating is presented. The model accounts for both monomer and polymer diffusion and distinguishes between short polymer chains capable of diffusing and long polymer chains that are immobile. It is shown that the experimentally observed decrease of diffraction efficiency at higher spatial frequency can be predicted by assuming diffusion of short-chain polymers away from the bright fringes. The time evolution of the refractive-index modulation after a short exposure is calculated and compared with experimental results. The effects of diffusion coefficients, polymerization rates, intensity, and spatial frequency of recording on the properties of weak diffraction gratings are investigated by numerical simulations.

Study of the photoinduced surface relief modulation in photopolymers caused by illumination with a Gaussian beam of light

The formation of surface relief profile in photopolymerizable systems when illuminated with a focused beam of light is simulated numerically using a two-way diffusion model that takes account both for monomer and short polymer chain diffusion. The concentration and spatial distribution dynamics of monomer and short and long polymer chains are calculated. The surface profile is obtained from the calculated component concentrations considering different densities of monomer and polymer. The influence of the illumination time, intensity and spot diameter on the surface profile dynamics is discussed. A good agreement between the calculated and the experimentally measured profiles is observed, thus demonstrating the successful application of the two-way diffusion model to this system.

Modelling random antibody adsorption and immunoassay activity

One of the primary considerations in immunoassay design is optimizing the concentration of capture antibody in order to achieve maximal antigen binding and, subsequently, improved sensitivity and limit of detection. Many immunoassay technologies involve immobilization of the antibody to solid surfaces. Antibodies are large molecules in which the position and accessibility of the antigen-binding site depend on their orientation and packing density. In this paper we propose a simple mathematical model, based on the theory known as random sequential adsorption (RSA), in order to calculate how the concentration of correctly oriented antibodies (active site exposed for subsequent reactions) evolves during the deposition process. It has been suggested by experimental studies that high concentrations will decrease assay performance, due to molecule denaturation and obstruction of active binding sites. However, crowding of antibodies can also have the opposite effect by favouring upright orientations. A specific aim of our model is to predict which of these competing effects prevails under different experimental conditions and study the existence of an optimal coverage, which yields the maximum expected concentration of active particles (and hence the highest signal).

Mathematical modelling at secondary school: the MACSI-Clongowes Wood College experience

In Ireland, to encourage the study of STEM (science, technology, engineering and mathematics) subjects and particularly mathematics, the Mathematics Applications Consortium for Science and Industry (MACSI) and Clongowes Wood College (County Kildare, Ireland) organized a mathematical modelling workshop for senior cycle secondary school students. Participants developed simple mathematical models for everyday life problems with an open-ended answer. The format and content of the workshop are described and feedback from both students and participating teachers is provided. For nearly all participants, this workshop was an enjoyable experience which showed mathematics and other STEM components in a very positive way.

A Diffusion Model for Spatially Dependent Photopolymerization

Photopolymers represent an attractive class of optical recording materials due to properties such as high refractive index modulation, dry film processing, low cost, etc. Applications include holographic data storage disks, optical interconnections, memories and filters. This paper addresses the dynamics of short-exposure holographic grating formation; a new model is proposed to explain the experimental observations of low diffraction efficiency in high spatial frequency gratings.

Direct immunoassays and their performance – theoretical modelling of the effects of antibody orientation and associated kinetics

The orientation and activity of antibodies immobilized on solid surfaces are of direct relevance to many immunosensing applications. We therefore investigate a mathematical model which estimates the fraction of antibodies which are available for reaction in a randomly adsorbed sample. Numerical simulations are presented which highlight the separate effects of antibody orientation, accessibility and loss of binding ability on the amount of captured antigen. The assay response can then be expressed as a function of total antibody density and used for optimizing the surface coverage strategy under various conditions.

Theoretical modeling of the effect of polymer chain immobilization rates on holographic recording in photopolymers

This paper introduces an improved mathematical model for holographic grating formation in an acrylamide-based photopolymer, which consists of partial differential equations derived from physical laws. The model is based on the two-way diffusion theory of [Appl. Opt.43, 2900 (2004)10.1364/AO.43.002900APOPAI1559-128X], which assumes short polymer chains are free to diffuse, and generalizes a similar model presented in [J. Opt. Soc. Am. B27, 197 (2010)10.1364/JOSAB.27.000197JOBPDE0740-3224] by introducing an immobilization rate governed by chain growth and cross-linking. Numerical simulations were carried out in order to investigate the behavior of the photopolymer system for short and long exposures, with particular emphasis on the effect of recording parameters (such as illumination frequency and intensity), as well as material permeability, on refractive index modulation, refractive index profile, and grating distortion. The model reproduces many well-known experimental observations, such as the decrease of refractive index modulation at high spatial frequencies and appearance of higher harmonics in the refractive index profile when the diffusion rate is much slower than the polymerization rate. These properties are supported by a theoretical investigation which uses perturbation techniques to approximate the solution over various time scales.

A Competitive Random Sequential Adsorption Model for Immunoassay Activity

Immunoassays rely on highly specific reactions between antibodies and antigens and are used in biomedical diagnostics applications to detect biomarkers for a variety of diseases. Antibody immobilization to solid interfaces through random adsorption is a widely used technique but has the disadvantage of severely reducing the antigen binding activity and, consequently, the assay performance. This paper proposes a simple mathematical framework, based on the theory known as competitive random sequential adsorption (CRSA), for describing how the activity of immobilized antibodies depends on their orientation and packing density and generalizes a previous model by introducing the antibody aspect ratio as an additional parameter which could influence the assay behaviour.

Optimising Copying Accuracy in Holographic Patterning

We propose a partial differential equations model for the formation and evolution of a holographic grating in a photopolymer system and use perturbation methods and numerical simulations in order to investigate the dynamical mechanism by which distortions of the illumination pattern arise during recording. The parameters of interest are diffusion and photopolymerization rates as well as exposure time, for which we seek to determine regimes which allow for high fidelity copying.

Modelling Two-Dimensional Photopolymer Patterns Produced With Multiple-Beam Holography

Periodic structures referred to as photonic crystals attract considerable interest due to their potential applications in areas such as nanotechnology, photonics, plasmonics, etc. Among various techniques used for their fabrication, multiple-beam holography is a promising method enabling defect-free structures to be produced in a single step over large areas. In this paper we use a mathematical model describing photopolymerisation to simulate two-dimensional structures produced by the interference pattern of three noncoplanar beams. The holographic recording of different lattices is studied by variation of certain parameters such as beam wave vectors, time and intensity of illumination.