# Creating macroscale effective interfaces encapsulating microstructural physics

Lead Research Organisation:
Imperial College London

Department Name: Mathematics

### Abstract

This proposal seeks funding for a comprehensive three year research

program into

methodologies for modeling microscale interfacial phenomena on the macroscale

level. A fundamental question stretching across many disciplines is:

Given a microstructured interface

can it be replaced by an effective ``averaged'' boundary condition

entirely posed upon a macroscale. If so, can it accurately reproduce the physical

effects created by the microstructure? Can this effective boundary condition be

derived rigorously, rather than in some ad-hoc fashion, and what are

the limitations in so doing?

The proposal aims to answer these questions, with the goal of being

able to accurately and efficiently predict complex physical behaviour

in three apparently unconnected

fields: in wave propagation for surface Rayleigh-Bloch

waves and for the reflection of waves from designer structured

surfaces, in the statistical mechanics of phase transitions on

micropatterned surfaces, and in modeling diffusions through

structured domains. These fields all share a complex structured

interface and the generic overarching Mathematical approach we propose will lead to effective

boundary conditions

encapsulating the dominant microscale Physics; this will represent a

considerable advance in each of these areas.

The primary Mathematical approach will be based around

Homogenization theory utilizing the discrepancy in lengthscales to

create asymptotics from multiple scales analysis. Homogenization is conventionally

used when the bulk material has short-scale fluctuations and the

solution varies on a long-scale, its use for

interfaces is much less well explored.

Importantly we also aim to enhance the

range of validity of homogenization theory away from long-wave,

quasi-static, regimes to ones that can vary on the same scale as

the microstructure. This analytical work will be complemented by

detailed

numerical simulations that will act to verify the efficacy of the

developed interfacial models. The work will be undertaken by a team

from the Mathematics Department at Imperial College London

with complementary skills and strengths: Pavliotis (Homogenization

theory, stochastic processes), Parry (Statistical mechanics, phase

transitions) and Craster (Wave propagation, homogenization, fluid mechanics).

program into

methodologies for modeling microscale interfacial phenomena on the macroscale

level. A fundamental question stretching across many disciplines is:

Given a microstructured interface

can it be replaced by an effective ``averaged'' boundary condition

entirely posed upon a macroscale. If so, can it accurately reproduce the physical

effects created by the microstructure? Can this effective boundary condition be

derived rigorously, rather than in some ad-hoc fashion, and what are

the limitations in so doing?

The proposal aims to answer these questions, with the goal of being

able to accurately and efficiently predict complex physical behaviour

in three apparently unconnected

fields: in wave propagation for surface Rayleigh-Bloch

waves and for the reflection of waves from designer structured

surfaces, in the statistical mechanics of phase transitions on

micropatterned surfaces, and in modeling diffusions through

structured domains. These fields all share a complex structured

interface and the generic overarching Mathematical approach we propose will lead to effective

boundary conditions

encapsulating the dominant microscale Physics; this will represent a

considerable advance in each of these areas.

The primary Mathematical approach will be based around

Homogenization theory utilizing the discrepancy in lengthscales to

create asymptotics from multiple scales analysis. Homogenization is conventionally

used when the bulk material has short-scale fluctuations and the

solution varies on a long-scale, its use for

interfaces is much less well explored.

Importantly we also aim to enhance the

range of validity of homogenization theory away from long-wave,

quasi-static, regimes to ones that can vary on the same scale as

the microstructure. This analytical work will be complemented by

detailed

numerical simulations that will act to verify the efficacy of the

developed interfacial models. The work will be undertaken by a team

from the Mathematics Department at Imperial College London

with complementary skills and strengths: Pavliotis (Homogenization

theory, stochastic processes), Parry (Statistical mechanics, phase

transitions) and Craster (Wave propagation, homogenization, fluid mechanics).

### Planned Impact

This proposal is unique as it bridges between the research programs of three well-established Investigators with complementary expertise with the aim to tackle a fundamental

scientific issue regarding structured surfaces. The immediate impact will be academic and felt across a spectrum of disciplines, such as applied mathematics, applied physics, statistical physics, optics, fluid dynamics and mathematical biology. The applications are truly vast as structured surfaces are all around us. The results and advances in modelling obtained through this proposal will impact these applications, for instance, the potential of the effective bulk modelling of waves upon biomedical sensors and the accurate modelling of thin fluid films, spreading and wetting over structured surfaces. Wetting phenomena have an enormously wide variety of industrial applications ranging from the macroscopic scale (oil recovery, drainage of water, cooling of reactors, deposition of pesticides) to the microscopic (microfluidics, ink jet printing). Furthermore, the development of efficient numerical and analytical techniques for the study of the narrow escape problem will be useful for researchers in cellular biology. Indeed, the techniques that methodologies that we will develop will lead to efficient analytical and computational techniques for calculating quantities such as the time needed for a reactive particle released from a specific organelle to activate a given protein on the cell membrane.

The new modelling to be undertaken will lead to effective boundary conditions that truly incorporate the microstructure. On one level this will lead to more rapid and simple computations and on another will allow the rapid design of smart structures or designer surfaces for real applications. This is in addition to the academic impact upon each subject and topic.

scientific issue regarding structured surfaces. The immediate impact will be academic and felt across a spectrum of disciplines, such as applied mathematics, applied physics, statistical physics, optics, fluid dynamics and mathematical biology. The applications are truly vast as structured surfaces are all around us. The results and advances in modelling obtained through this proposal will impact these applications, for instance, the potential of the effective bulk modelling of waves upon biomedical sensors and the accurate modelling of thin fluid films, spreading and wetting over structured surfaces. Wetting phenomena have an enormously wide variety of industrial applications ranging from the macroscopic scale (oil recovery, drainage of water, cooling of reactors, deposition of pesticides) to the microscopic (microfluidics, ink jet printing). Furthermore, the development of efficient numerical and analytical techniques for the study of the narrow escape problem will be useful for researchers in cellular biology. Indeed, the techniques that methodologies that we will develop will lead to efficient analytical and computational techniques for calculating quantities such as the time needed for a reactive particle released from a specific organelle to activate a given protein on the cell membrane.

The new modelling to be undertaken will lead to effective boundary conditions that truly incorporate the microstructure. On one level this will lead to more rapid and simple computations and on another will allow the rapid design of smart structures or designer surfaces for real applications. This is in addition to the academic impact upon each subject and topic.

### Publications

Achaoui Y
(2017)

*Clamped seismic metamaterials: ultra-low frequency stop bands*in New Journal of Physics
Achaoui Y
(2017)

*Clamped seismic metamaterials: Ultra-low broad frequency stop-bands*
Antonakakis T
(2014)

*An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces.*in Proceedings. Mathematical, physical, and engineering sciences
Antonakakis T
(2014)

*Homogenisation for elastic photonic crystals and dynamic anisotropy*in Journal of the Mechanics and Physics of Solids
Antonakakis T
(2013)

*Asymptotics for metamaterials and photonic crystals.*in Proceedings. Mathematical, physical, and engineering sciences
Antonakakis T
(2014)

*Moulding and shielding flexural waves in elastic plates*in EPL (Europhysics Letters)
Bierkens J
(2017)

*Limit theorems for the zig-zag process*in Advances in Applied Probability
Bierkens J
(2016)

*Limit theorems for the Zig-Zag process*
Bonnaillie-Noël V
(2016)

*Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations*in IMA Journal of Numerical AnalysisDescription | Analysis of stochastic systems with a multiscale structure. Understanding of noise induced transitions for multiscale stochastic systems. Use of these techniques to study problems in nonequilibrium statistical mechanics. Development of improved Markov Chain Monte Carlo techniques and application to problems in statistical mechanics. Development of new homogenization techniques for the study of complex media. Study of diffusion problems in complex media and composite materials. Data-driven approaches to the derivation of coarse-grained models. Control of thin film flows, falling liquid films. Control of spatiotemporal chaos. Rigorous and systematic study of wetting transitions. Rigorous study of time dependent density functional theory. |

Exploitation Route | Noise induced transitions for stochastic dynamical systems with multiple scales. Applications of homogenization theory/multiscale techiques to metamaterials. Time dependent density functional density theory. |

Sectors | Aerospace Defence and Marine Other |

Description | The analytical and computational techniques that have been developed are being used in order to develop algorithms for the calculation of transport coefficients and in order to optimize the performance of statistical algorithms such as Markov Chain Monte Carlo. The research supported by this grant also led to advances in the study of metamaterials and also in the physics of wetting phenomena. |

First Year Of Impact | 2014 |

Sector | Other |

Description | (MetaVEH) - Metamaterial Enabled Vibration Energy Harvesting |

Amount | € 4,018,875 (EUR) |

Funding ID | 952039 |

Organisation | European Commission |

Sector | Public |

Country | European Union (EU) |

Start | 01/2021 |

End | 12/2024 |

Description | EPSRC Platform Grant |

Amount | £1,616,110 (GBP) |

Funding ID | EP/L020564/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 05/2014 |

End | 06/2019 |

Description | EPSRC Programme Grant |

Amount | £2,551,402 (GBP) |

Funding ID | EP/L024926/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 06/2014 |

End | 07/2019 |

Description | EPSRC responsive mode |

Amount | £379,072 (GBP) |

Funding ID | EP/L025159/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 08/2014 |

End | 08/2017 |

Description | Control theory for multiscale stochastic systems |

Organisation | Free University of Berlin |

Department | Institute of Mathematics |

Country | Germany |

Sector | Academic/University |

PI Contribution | I brought my expertise on multiscale analysis for diffusion processes such as homogenization theory, averaging, and also statistical inference for multiscale diffusions |

Collaborator Contribution | Stochastic Optimal Control |

Impact | Optimal control of multiscale systems using reduced-order models (with W. Zhang, J.C. Latorre and C. Hartmann). J. Comp. Dyn. 1(2), pp 279-308, (2014). Numerical Methods for Computing Effective Transport Properties of Flashing Brownian Motors (with J.C. Latorre and P.R. Kramer). J. Comp. Phys., 257 Part A, pp. 57-82, (2014). Corrections to Einstein's relation for Brownian motion in a tilted periodic potential (with J.C. Latorre and P.R. Kramer). J. Stat. Phys. 150(4), 776-803 (2013). |

Start Year | 2012 |

Description | Optimizing the performance of Markov Chain Monte Carlo algorithms |

Organisation | University of Edinburgh |

Department | School of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This collaborative research project is concerned with the optimization of the performance of Markov Chain Monte Carlo algorithms that used for sampling from probability distributions in high dimensional spaces. My contribution to this project was the necessary mathematical techniques for studying this problem, in particular the development of techniques for studying spectral problems for hypoelliptic operators |

Collaborator Contribution | They brought their expertise on molecular dynamics and on spectral theory for differential operators. They also made contributions on the numerical analysis of algorithms used in Molecular dynamics simulations. |

Impact | Variance Reduction using Nonreversible Langevin Samplers (with A. Duncan and T. Lelievre). J. Stat. Phys. (2016) doi:10.1007/s10955-016-1491-2 Langevin dynamics with space-time periodic nonequilibrium forcing (with R. Joubaud and G. Stoltz). J. Stat. Phys., to appear J. Stat. Phys., 158(1) pp 1-36 (2015). Optimal nonreversible linear drift for the convergence to equilibrium of a diffusion (with T. Lelievre and F. Nier). J. Stat. Phys. 152(2) 237-274 (2013). |

Start Year | 2012 |

Description | Optimizing the performance of Markov Chain Monte Carlo algorithms |

Organisation | École des ponts ParisTech |

Country | France |

Sector | Academic/University |

PI Contribution | This collaborative research project is concerned with the optimization of the performance of Markov Chain Monte Carlo algorithms that used for sampling from probability distributions in high dimensional spaces. My contribution to this project was the necessary mathematical techniques for studying this problem, in particular the development of techniques for studying spectral problems for hypoelliptic operators |

Collaborator Contribution | They brought their expertise on molecular dynamics and on spectral theory for differential operators. They also made contributions on the numerical analysis of algorithms used in Molecular dynamics simulations. |

Impact | Variance Reduction using Nonreversible Langevin Samplers (with A. Duncan and T. Lelievre). J. Stat. Phys. (2016) doi:10.1007/s10955-016-1491-2 Langevin dynamics with space-time periodic nonequilibrium forcing (with R. Joubaud and G. Stoltz). J. Stat. Phys., to appear J. Stat. Phys., 158(1) pp 1-36 (2015). Optimal nonreversible linear drift for the convergence to equilibrium of a diffusion (with T. Lelievre and F. Nier). J. Stat. Phys. 152(2) 237-274 (2013). |

Start Year | 2012 |

Title | IN SITU CONTROL OF FLUID MENISCI |

Description | A system includes a non-vertical channel containing a fluid forming a fluid meniscus having a capillary length and a contact angle ?. The channel in cross-section has a perimeter length |S |and an area |O|. The cross-section of the non-vertical channel is selected so as to define a constant Lagrange multiplier ?, where ?= |S |cos ? / |O|. A functional F [G*] ? |G*| - cos ? |S* | + (1/a2)G* +? |O*| is minimised to define a minimum value F0 =MinF. At a critical transition where F=0, the fluid defines a smooth arc of length [G*] that divides the cross-section of the channel into two parts. |O*| is the cross-sectional area of the fluid, which has a curve of length |S* | in contact with the channel, and G* represents a vertical position of the centre of mass of the fluid multiplied by the cross-sectional area |O*|. How far the fluid meniscus extends along the channel is controlled by one or more parameters of the functional F [G*]. |

IP Reference | WO2015121629 |

Protection | Patent granted |

Year Protection Granted | 2015 |

Licensed | No |

Impact | ISIS Project 9544 involves a method for controlling fluid menisci in micro- and nano-fluidic devices from mesoscopic to a nanoscopic scale. The method can be used to control how far the fluid meniscus extends along the channel without emptying, and can be used to force the draining of much smaller channels than previously possible. This meniscus deformation has several advantages that could find applications in a number of science and technology sectors: Increasing the liquid-gas or fluid- |