Spin-coating on nanoscale topography and phase separation of diblock copolymers

CRANN researchers are interested in mathematical modelling of all aspects of the process of spin-coating of diblock copolymers, with the aim of removing expensive trial and error design cycles. Of particular interest is the flow of the polymer during spin-coating, and also during the subsequent annealing process. Also of considerable interest is the chemical process of phase-separation and self-assembly of the diblock copolymer. Existing models in the literature rely heavily on computationally expensive Monte-Carlo simulation methods. The modelling work performed during the study group in summarized in this report. The report is split into four main sections, with discussion and suggestions for experiments in the concluding section. The content of the sections is as follows: Section 0.2: Mathematical modelling of spin-coating onto a flat substrate; no annealing considered. Section 0.3: Modelling of spin-coating onto a substrate with topography (i.e. trenches); no annealing considered. Section 0.4: Flow of polymer during annealing. Section 0.5: Models for self-assembly of polymers into nanostructures. Sections 0.2 to 0.4 are focussed on the fluid flow problems for the polymer, and go some way to providing useful answers to Problem 1. On the other hand, Problem 2 was found to be extremely challenging, and the efforts described in section 0.5 represent only a relatively modest impact on this problem

Nanomatemàtiques: modelització matemàtica a la nanoescala

En aquest article exposem tres problemes estudiats recentment al grup de Matemàtica Industrial del Centre de Recerca Matemàtica, en els quals l'aproximació del continu resulta vàlida per a descriure fenòmens a la nanoescala: 1. Transferència de calor en nanofluids: els resultats experimentals que confirmen l'increment de la capacitat de transferir calor dels nanofluids respecte dels fluids estàndard són sovint contradictoris. Mitjançant una anàlisi de capa límit mostrarem com el model matemàtic utilitzat en nombroses ocasions per justificar l'increment en la transferència de calor dels nanofluids preveu, de fet, una disminució d'aquesta propietat. 2. Fusió de nanopartícules: les nanopartícules mostren un increment abrupte de la velocitat de transició de fase a mesura que el seu radi decreix. Presentarem un model matemàtic que descriu aquest fenomen. El model preveu temps totals de transició sòlid-líquid que concorden amb les observacions experimentals. 3. Increment del flux d'un fluid en nanotubs de carboni (CNT): mostrarem que els resultats experimentals sobre l'increment de flux en nanotubs de carboni es poden explicar mitjançant les equacions estàndard de la dinàmica de fluids amb la incorporació d'una capa d'extinció (depletion layer) a la interfície entre el fluid i el sòlid.In this paper we discuss three problems recently studied within the Industrial Maths Research Group at the Centre de Recerca Matemàtica, where continuum theory may be applied to describe nanoscale phenomena: 1. Heat transfer with nanofluids: Experimental results concerning the remarkable heat transfer characteristics of nanofluids are at times contradictory. We apply a boundary layer analysis to show that a standard model which has been used by many authors to predict an improvement in heat transfer with increasing nanoparticle concentration in fact shows a decrease. 2. Nanoparticle melting: Nanoparticles often exhibit a sharp increase in melting rate as the size decreases. A mathematical model will be presented which predicts this phenomena and explains the experimentally observed abrupt melting of the smallest nanoparticles. 3. Enhanced flow in carbon nanotubes (CNTs): This model shows that the experimentally observed enhancement can be explained using standard flow equations but with a depletion layer between the liquid and solid interfaces. The results also provide one physical explanation for the Navierslip condition

Stress effects of silica particles in a semiconductor package molding compound

The stresses induced on a silicon chip encased in an epoxy compound are considered due to absorption of moisture and the presence of silica particles in the coating. A range of different approaches are considered including a one-dimensional model for the curvature due to the absorption of water in a bi-lateral sheet, numerical simulations for the stress at the molding compound-silicon die interface and a two-dimensional model in the complex plane

Slow and fast diffusion in a lead sulphate gravity separation process

A model for the growth of lead sulphate particles in a gravity separation system from the crystal glassware industry is presented. The lead sulphate particles are an undesirable byproduct, and thus the model is used to ascertain the optimal system temperature configuration such that particle extraction is maximised. The model describes the evolution of a single, spherical particle due to the mass flux of lead particles from a surrounding acid solution. We divide the concentration field into two separate regions. Specifically, a relatively small boundary layer region around the particle is characterised by fast diffusion, and is thus considered quasistatic. In contrast, diffusion in the far-field is slower, and hence assumed to be time-dependent. The final system consisting of two nonlinear, coupled ordinary differential equations for the particle radius and lead concentration, is integrated numerically

A soft sensor for the Bayer process

A soft sensor for measuring product quality in the Bayer process has been developed. The soft sensor uses a combination of historical process data recorded from online sensors and laboratory measurements to predict a key quality indicator, namely particle strength. Stepwise linear regression is used to select the relevant variables from a large dataset composed of monitored properties and laboratory data. The developed sensor is employed successfully by RUSAL Aughinish Alumina Ltd to predict product strength five days into the future with R-squared equal to 0.75 and to capture deviations from standard operating condition

Nanomatemàtiques: modelització matemàtica a la nanoescala

En aquest article exposem tres problemes estudiats recentment al grup de Matemàtica Industrial del Centre de Recerca Matemàtica, en els quals l'aproximació del continu resulta vàlida per a descriure fenòmens a la nanoescala: 1. Transferència de calor en nanofluids: els resultats experimentals que confirmen l'increment de la capacitat de transferir calor dels nanofluids respecte dels fluids estàndard són sovint contradictoris. Mitjançant una anàlisi de capa límit mostrarem com el model matemàtic utilitzat en nombroses ocasions per justificar l'increment en la transferència de calor dels nanofluids preveu, de fet, una disminució d'aquesta propietat. 2. Fusió de nanopartícules: les nanopartícules mostren un increment abrupte de la velocitat de transició de fase a mesura que el seu radi decreix. Presentarem un model matemàtic que descriu aquest fenomen. El model preveu temps totals de transició sòlid-líquid que concorden amb les observacions experimentals. 3. Increment del flux d'un fluid en nanotubs de carboni (CNT): mostrarem que els resultats experimentals sobre l'increment de flux en nanotubs de carboni es poden explicar mitjançant les equacions estàndard de la dinàmica de fluids amb la incorporació d'una capa d'extinció (depletion layer) a la interfície entre el fluid i el sòlid.In this paper we discuss three problems recently studied within the Industrial Maths Research Group at the Centre de Recerca Matemàtica, where continuum theory may be applied to describe nanoscale phenomena: 1. Heat transfer with nanofluids: Experimental results concerning the remarkable heat transfer characteristics of nanofluids are at times contradictory. We apply a boundary layer analysis to show that a standard model which has been used by many authors to predict an improvement in heat transfer with increasing nanoparticle concentration in fact shows a decrease. 2. Nanoparticle melting: Nanoparticles often exhibit a sharp increase in melting rate as the size decreases. A mathematical model will be presented which predicts this phenomena and explains the experimentally observed abrupt melting of the smallest nanoparticles. 3. Enhanced flow in carbon nanotubes (CNTs): This model shows that the experimentally observed enhancement can be explained using standard flow equations but with a depletion layer between the liquid and solid interfaces. The results also provide one physical explanation for the Navierslip condition

Functions used in Continuous Detection of Ore Moisture using Microwaves

Two functions that were created for use in the 2019 paper by Mark McGuinness, Sean Bohun, Vincent Cregan, William T. Lee, Gary O'Keeffe, and Jeff Dewynn