Statistical mechanics of soft matter: Derivation, analysis and implementation of dynamic density functional theories

Lead Research Organisation: Imperial College London
Department Name: Department of Chemical Engineering

Abstract

The term "soft matter" is used to describe materials which at room temperature can easily deform under external forces like gravity or pressure while their properties are governed by slow internal dynamics. Soft matter often plays a central role in engineering and biomedical science and has numerous practical applications. The proposed research focuses on a large class of soft matter systems, that of classical fluids, i.e. systems of particles which retain a definite volume and are at sufficiently high temperatures that quantum effects can be neglected. Of particular interest are colloidal fluids whose particles are of micrometer size, suspended in a bath of many more, much smaller and lighter particles, which cannot be described by continuum macroscopic formalisms such as the Navier-Stokes equation.

Modelling the dynamics of the full colloidal fluids is prohibitively expensive and in fact computationally intractable due to both the large number of particles and the wide range of length- and time-scales which must be considered. As a consequence, one must employ statistical mechanics approaches, with the most widely used method being dynamic density-functional theory (DDFT). However, previous DDFTs often involve approximations whose accuracy and validity cannot be ascertained a priori. As a consequence, the results obtained from such formulations are questionable and often inaccurate.

This proposal seeks funding for a comprehensive three-year research programme into a two-pronged novel theoretical and numerical investigation aimed at rationally understanding and systematically predicting the complex physical behaviour and properties of colloidal systems. The primary aim is the development of a generic DDFT formalism that would allow for the accurate, systematic and predictive modelling of physically relevant systems where all the neglected effects in previous idealised studies now come to the fore. This in turn will allow for step improvements to the performance and efficiency of a host of technologies and applications that rely crucially on particulate systems. The analytical work will be complemented by detailed numerical simulations that will act so as to verify the efficacy of the developed models, as well as aiding the development of a toolkit for practical applications. The research will be undertaken by a team from the School of Mathematics of the University of Edinburgh and the Chemical Engineering and Mathematics Departments at Imperial College London with complementary skills and strengths: Goddard (Complex Multiscale Systems, Statistical Mechanics, Analysis and Computations), Kalliadasis (Multiscale Fluid Dynamics, Theory and Computations) and Pavliotis (Stochastic Processes, Multiscale Analysis, Statistical Mechanics).

Planned Impact

The economic and societal impact of the proposed research will be realised through the development of modelling and predictive tools for soft matter systems. Such systems are key to several rapidly developing technologies in the UK's smart materials, specialty manufacturing sectors and biomedical industries. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can contribute to many aspects, from improvements in product and design all the way to production.

The research will also lead to the development of state-of-the-art numerical methodologies for the accurate and reliable multiscale simulations of soft matter flows. These codes will be of benefit to the control and optimisation of industrial processes and devices that exploit soft matter flows.

Finally, the team of Investigators has a strong track record of both science and technology transfer through short courses and workshops and training of high-calibre researchers. The impact plan we have devised is based on a range of routes to maximise the likelihood of success and to reach as wide a community as possible: training of the researcher including visits to leading researchers who could potentially be collaborators, publication in leading journals, conference presentations, provision of a website and research colloquium.

Publications

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Abdulle A (2017) Spectral Methods for Multiscale Stochastic Differential Equations in SIAM/ASA Journal on Uncertainty Quantification

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Carrillo J (2019) Long-Time Behaviour and Phase Transitions for the Mckean-Vlasov Equation on the Torus in Archive for Rational Mechanics and Analysis

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Craster R (2018) Cloaking via Mapping for the Heat Equation in Multiscale Modeling & Simulation

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Duncan A (2015) A Multiscale Analysis of Diffusions on Rapidly Varying Surfaces in Journal of Nonlinear Science

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Duncan AB (2016) Variance Reduction Using Nonreversible Langevin Samplers. in Journal of statistical physics

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Duncan AB (2016) Noise-induced transitions in rugged energy landscapes. in Physical review. E

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Durán-Olivencia M (2018) General framework for nonclassical nucleation in New Journal of Physics

 
Description Many-body systems are ubiquitous in nature, ranging in scale from stellar clusters to soft matter and down to the quantum scale of electrons in atoms and molecules. Soft matter is material at sufficiently high temperatures that quantum effects can be neglected, and which easily deforms under external forces like gravity or pressure. An example are classical fluids, some of which e.g. atomic-molecular fluids (simple fluids), consisting of sufficiently small (10^{-10} m) particles, that their dynamics may be well-described by continuum
hydrodynamic equations down to the nanometer scale. In contrast, colloidal systems, with particles of micrometer size
suspended in a simple fluid bath, exhibit non-trivial effects on lengthscales comparable to the colloidal particle size and are thus out of reach of continuum models. This project is concerned with such systems.

Since the experimental observation of the Brownian motion of pollen particles in water in the 1820s, colloidal systems have motivated the development of statistical mechanics and helped confirm the molecular nature of matter. The physical size of colloidal particles makes them accessible experimentally,
their interactions may be easily tuned, and many of the forces governing their behaviour also govern simple fluids. However, despite the considerable attention they have received, a large number of key problems remain unresolved and many challenging aspects of their dynamics still elude us. In
particular, there is no mathematically reliable, systematic and physically consistent derivation of multiscale models or low-dimensional representations that would allow for the accurate and predictive modelling of physically relevant systems. Such modelling of colloidal fluids is a highly non-trivial task while recent advances in microscopy, bio-/microfluidics and nanoparticle deposition have increased the importance and range of applications of such fluids from cloud formation to bacteria dynamics and aerosol production. The proposed research aims to address these questions through a synergistic approach at the cross-road between applied mathematics, fluid dynamics, statistical mechanics and mathematical physics, allowing the prediction and elucidation of a wide range
of phenomena of great current interest in engineering, physics and biology. A major strength of this project is a balanced combination of analytical and computational techniques.
Exploitation Route The impact plan we have developed is based on the identification of several roots to societal-economic impact and impact on peers.

The following activities offer a means of dissemination of the results obtained in this proposal to fellow academics as well as industrial researchers: (i) Our national and international network of academic contacts; (ii) National and international conferences. These are leading international forums for the exchange of information on all aspects of applied mathematics, fluid flow and thermodynamics{statistical mechanics of fluids; (iii) We have strong records in publication of articles in the leading international journals of high impact factor (hence read by a wide audience) of our respective fields. It is anticipated that the results from this project will be submitted to high-profile international journals devoted to the publication of authoritative articles at the forefront of the topics of the proposed research; (iv) Research seminars in the UK and overseas; (v) Sabbatical and occasional visits by overseas scientists to the UK and visits to overseas institutions. Each Investigator has active international collaborations and regularly visits overseas institutions and receives visitors from abroad. This will aid to the dissemination of the research and increase its impact; (vi) Web-based dissemination, i.e. via the creation of an interactive website with a Wiki architecture that will provide the latest news, access to results, codes and relevant publications. In particular, the resulting numerical codes will be made available on the web as open source. This would also allow for contributions from our peers to the codes' further development. The project will thus contribute to the wider dissemination of our computational tools; (vii) As noted below there are linkages with industry, through other research.

This is fundamental research, however the applications are far reaching and have potential for commercial exploitation. IC has a dedicated commercial exploitation group, Imperial Innovations Ltd (IIL), and consultancy arm while the engineering faculty has a Knowledge Transfer Fellow and Intellectual Property is protected through IIL. There are also links with industry through other research and, therefore, we are ideally placed to disseminate the relevant parts of this research to those who could ultimately use it in applications. These industrial collaborations are vital resources that will be used to facilitate the wider impact of the research and to ensure that results are channelled towards appropriate applications.
Sectors Chemicals,Digital/Communication/Information Technologies (including Software),Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology

URL http://www.imperial.ac.uk/complex-multiscale-systems
 
Description This project is a step change up from the current state-of-the-art in statistical mechanics of soft matter, in particular colloidal fluids. Such systems enable key, rapidly developing technologies in the UK's biomedical, smart materials, pharmaceutical and printing industries, to name a few. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can generate valuable insight into many aspects, from optimal product design all the way to production. These are increasingly competitive and knowledge-intensive sectors offering long term research and engineering opportunities. Characterising and understanding the interplay between fluid flow, surfaces and particles, one of the main aims of the project, are amongst the basic research questions that must be addressed. This should substantially contribute to the UK's capability to develop integrated multiscale models describing soft matter, with obvious economic benefits: enhanced competitiveness of the UK micro-/nanofluidics sector and underpinning of the development of fluid processes and formulated products.
First Year Of Impact 2016
Sector Chemicals,Energy,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology
Impact Types Economic

 
Title Development of new statistical mechanics methodologies for soft matter 
Description Our overarching objective is a general phase-space DDFT in two-three dimensions including all necessary effects for the study of various engineering, physical and biological applications. Existing DDFTs are focussed towards the validation of the formalism and numerical codes to tackle real-world problems are, as yet, not developed. We propose here a step change in the application of DDFT to such problems, which will require a synergistic approach combining modelling, analysis and validation of the (generally unconstrained) approximations currently used in all DDFTs, and accurate and efficient numerics. Such validation will also require accurate numerical codes for the underlying stochastic dynamics. Finally, a rigorous analysis of the approximations, where possible, can lead to insights which are not obtained from more heuristic arguments. 
Type Of Material Improvements to research infrastructure 
Provided To Others? No  
Impact This project is a step change up from the current state-of-the-art in statistical mechanics of soft matter, in particular colloidal fluids. Such systems enable key, rapidly developing technologies in the UK's biomedical, smart materials, pharmaceutical and printing industries, to name a few. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can generate valuable insight into many aspects, from optimal product design all the way to production. These are increasingly competitive and knowledge-intensive sectors offering long term research and engineering opportunities. Characterising and understanding the interplay between fluid flow, surfaces and particles, one of the main aims of the project, are amongst the basic research questions that must be addressed. This should substantially contribute to the UK's capability to develop integrated multiscale models describing soft matter, with obvious economic benefits: enhanced competitiveness of the UK micro-/nanofluidics sector and underpinning of the development of fluid processes and formulated products. 
URL http://www.imperial.ac.uk/complex-multiscale-systems
 
Title Development of theoretical and computational tools for soft matter 
Description One of our main aim sis the derivation, analysis and understanding of reduced, low-dimensional models that will allow the systematic predictive analysis of soft matter in a wide range of physically and biologically relevant settings. This will also guide and motivate numerical experiments to validate the numerical predictions, thus making the investigation of new phenomena involving classical fluids possible, as well as allowing the development of a numerical toolkit for general soft matter systems. 
Type Of Material Computer model/algorithm 
Provided To Others? No  
Impact The research aims to provide both physical insight in soft matter, e.g. colloidal systems behaviour, initially in open and ultimately in confined geometries and the appropriate modelling/predictive tools for such systems. These should be of great interest to researchers and engineers whose technological processes involve at some stage colloidal fluids as they will enable them to tackle classes of problems that have hitherto been inaccessible to them. Also to workers in the commercial/private sector with interests and/or stakes in the development of predictive models for systems utilising colloidal fluids. In terms of applications, these are quite extensive from small-scale separations to nanofluidics where a central problem is how to control and manipulate fluids at small scales, crucial for a variety of applications. Controlled nanofluidics could offer the possibility to stably confine molecules to very small spaces, or to subject them to desired forces. In this respect, nanofluidics can be used as an engineering tool to extract fundamental knowledge for e.g. biological applications, such as nuclear pore complexes and the transport of nanoparticles in the circulatory and respiratory systems. 
URL http://www.imperial.ac.uk/complex-multiscale-systems