Workshop on Synergies Between Mathematics, Data Science, and Molecular Simulations in Materials Science

University of Birmingham, 3-5 July 2024

Scope:

This workshop is part of a series of workshops organised by the Network on Mathematical Data Science for Materials Science to bring together both leading experts and early career researchers in order to explore the state-of-the-art development in mathematical data science for materials science.

Organisers:

Dr Xiaocheng Shang (School of Mathematics, University of Birmingham)
Professor Benedict Leimkuhler FRSE (School of Mathematics, University of Edinburgh)
Professor Paola Carbone (Department of Chemistry, University of Manchester)

Participation & Registration

Participation is open to everyone and is free of charge. However, we would like to ask everyone who intends to participate to fill in this registration form so that we have a good estimate of the number of participants for catering purposes. Moreover, we are planning to run an early career researchers' session where you can give short talks, as well as a poster session where you can present your posters; please indicate your intention in the registration form by 3 June 2024.
All participants of the workshop are expected to adhere to the University of Birmingham School of Mathematics' Code of Conduct (see below).
The School of Mathematics has established a programme to offer funded child-care services to visiting researchers. To take advantage of this opportunity, it is recommended that you make contact with Dr Xiaocheng Shang (x.shang.1@bham.ac.uk) at the earliest convenience to ensure proper arrangements are made. In addition, we would like to draw your attention to the LMS Caring Supplementary Grants.

Venue:

The workshop will be hosted by the School of Mathematics at the University of Birmingham. All talks and presentations will be held in the Lecture Theater B (101) on the first floor of the Watson Building (R15 on the campus map). The Edgbaston campus has its own train station, called "University". Further information on how to get to the campus can be found here.

Confirmed Speakers:

Livia Bartók-Pártay (University of Warwick)
Paola Carbone (University of Manchester)
Michele Ceriotti (EPFL)
Dwaipayan Chakrabarti (University of Birmingham)
Christoph Dellago (University of Vienna)
Benedict Leimkuhler (University of Edinburgh)
Andrew Morris (University of Birmingham)
Matthias Sachs (University of Birmingham)
Friederike Schmid (University of Mainz)
Jia Shao (University of Birmingham)
Sandeep Shirgill (University of Birmingham)
Gabriel Stoltz (École des Ponts ParisTech)
Alessandro Troisi (University of Liverpool)
Mark Wilson (Durham University)
Zhenyu Jason Zhang (University of Birmingham)

Schedule:

Day 1 (3 July): 11:00-11:45 Christoph Dellago (University of Vienna)11:45-12:30 Friederike Schmid (University of Mainz)12:30-13:30 Lunch Break 13:30-14:15 Paola Carbone (University of Manchester)14:15-15:00 Sandeep Shirgill (University of Birmingham)15:00-15:30 Coffee Break 15:30-17:10 Early Career Short Talks17:10- Poster Session
Day 2 (4 July): 9:30-10:15 Benedict Leimkuhler (University of Edinburgh)10:15-11:00 Dwaipayan Chakrabarti (University of Birmingham)11:00-11:30 Coffee Break 11:30-12:15 Gabriel Stoltz (École des Ponts ParisTech)12:15-13:00 Michele Ceriotti (EPFL)13:00-13:45 Lunch Break 13:45-14:30 Alessandro Troisi (University of Liverpool)14:30-15:15 Livia Bartók-Pártay (University of Warwick)15:15-15:45 Coffee Break 15:45-16:30 Matthias Sachs (University of Birmingham)16:30-17:15 Zhenyu Jason Zhang (University of Birmingham)
Day 3 (5 July):9:30-10:15 Mark Wilson (Durham University)10:15-11:00 Andrew Morris (University of Birmingham) 11:00-11:30 Coffee Break 11:30-12:15 Jia Shao (University of Birmingham) 13:00- Lunch Break/End of the Workshop

Talk Titles and Abstracts:

Christoph Dellago (University of Vienna)Title: Machine Learning for Rare Event Simulations: From Transition Pathways to Reaction CoordinatesAbstract: The microscopic dynamics of many condensed matter systems occurring in nature and technology is dominated by rare but important barrier crossing events. Examples of such processes include nucleation at first order phase transitions, chemical reactions and the folding of biopolymers. The resulting wide ranges of time scales are a challenge for molecular simulation and numerous simulation methods have been developed to address this problem. Recently, machine learning methods have been proposed as a powerful way to further enhance such simulations. In my talk, I will discuss various machine learning approaches based on deep neural networks to sample rare reactive trajectories and identify the collective variable needed for the construction of low-dimensional models capturing the microscopic mechanism.
Friederike Schmid (University of Mainz)Title: Effective Dynamics of Passive Probe Particles in Steady-State Non-Equilibrium BathsAbstract: A popular approach to designing coarse-grained models for colloids dispersed in a solvent bath is to map their equations of motion onto effective Langevin equations (LE) or generalized Langevin equations (GLE) that include memory. Here we discuss the question what happens if the system is an inherently nonequilibrium system. We consider two cases: Driven colloids and colloids in a bath of active particles, and also the combination (driven colloids in an active bath). We study such systems by molecular simulations and use the simulation data to derive effective GLE descriptions.
First, we address the question whether it is possible to identify signatures of nonequilibrium just from looking at the trajectory of a single colloid, without having any information on the bath. It has been argued that non-equilibrium could lead to a breaking of the so-called second fluctuation dissipation theorem (2nd FDT). We present a general argument showing that typical memory reconstruction methods will result in GLEs that satisfy the 2nd FDT by construction, even for non-stationary, non-Hamiltonian, non-equilibrium systems.
It is however possible to detect non-equilibrium signatures when comparing the dynamics of colloids with that of bath particles. One striking example is lack of thermalization. Colloids in active baths are found to have a higher effective (kinetic) temperature than the bath particles. We investigate the microscopic mechanisms behind this behavior. We find that the active motion of the colloid cannot be simply attributed to the convective motion in the bath. Instead, the boundary of the probe contributes significantly to these adopted dynamics by causing active bath particles to spontaneously accumulate at the probe. This lead to a nontrivial nonmonotonic relation between the effective (kinetic) temperature and the size of the particle.
We then examine the influence of external forces on the effective dynamics described by a GLE equation. Specifically, we consider two types of forces that are highly relevant for microrheological studies: A harmonic, trapping force and a constant, “drag” force. We find that, in an active bath, the external force in the GLE is not equal to the physical external force, but rather a renormalized external force, which can be significantly smaller. The effect cannot be attributed to a mere temperature renormalization, which is also observed in addition.
[1] J. Shea, G. Jung, F. Schmid, Soft Matter, 18, 6965 (2022). Passive probe particle in an active bath: Can we tell it is out of equilibrium?[2] J. Shea, G. Jung, F. Schmid, Soft Matter 20, 1767 (2024). Force renormalization for probes immersed in an active bath.[3] J. Shea, G. Jung, F. Schmid, under review (2024). How boundary interactions dominate emergent driving of passive probes in active matter, https://arxiv.org/abs/2401.09227
Paola Carbone (University of Manchester)Title: Framework for a High-Throughput Screening Method to Assess Polymer/Plasticizer Miscibility: The Case of Hydrocarbons in PolyolefinsAbstract: Polymer composite materials require softening to reduce their glass transition temperature and improve processability. To this end, plasticizers (PLs), which are small organic molecules, are added to the polymer matrix. The miscibility of these PLs has a large impact on their effectiveness and, therefore, their interactions with the polymer matrix must be carefully considered. Many PL characteristics, including their size, topology, and flexibility, can impact their miscibility and, because of the exponentially large number of PLs, the current trial-and-error approach is very ineffective. In this work, we show that using coarse-grained molecular simulations of a small dataset of 48 PLs, it is possible to identify topological and thermodynamic descriptors that are proxy for their miscibility. Using ad-hoc molecular dynamics simulation setups that are relatively computationally inexpensive, we establish correlations between the PLs’ topology, internal flexibility, thermodynamics of aggregation, and degree of miscibility, and use these descriptors to classify the molecules as miscible or immiscible. With all available data, we also construct a decision tree model, which achieves a F1 score of 0.86 ± 0.01 with repeated, stratified 5-fold cross-validation, indicating that this machine learning method can be a promising route to fully automate the screening. By evaluating the individual performance of the descriptors, we show this procedure enables a 10-fold reduction of the test space and provides the basis for the development of workflows that can efficiently screen PLs with a variety of topological features. The approach is used here to screen for apolar PLs in polyisoprene melts, but similar proxies would be valid for other polyolefins, while, in cases where polar interactions drive the miscibility, other descriptors are likely to be needed.
Sandeep Shirgill (University of Birmingham)Title: Modelling Antimicrobial Effects of Bioactive Glass Fibres on Chronic Wound Biofilms Using PDEsAbstract: Chronic wounds, such as diabetic foot ulcers, are a drain on global health services and remain a major area of unmet clinical need. Chronic wounds are characterised by a bacterial biofilm (densely aggregated colonies of bacteria encased by a matrix of extracellular polymeric substances (EPS)), which hinders innate immune response and can prevent wound healing. Bioactive glass (BG) fibres doped with antimicrobial metal ions can offer a promising treatment for chronic wound infections. From in vitro experimental studies, it has been found that silver and copper-doped BG fibres work synergistically to eradicate Pseudomonas aeruginosa biofilms, where P. aeruginosa is one of the most common bacterial species found in chronic wound infections. Specifically, the copper released by the BG fibres during dissolution reduces the EPS within the biofilm making the bacteria more susceptible to treatment by the released silver.
Benedict Leimkuhler (University of Edinburgh)Title: Thermostat Schemes for Neural Network Training Abstract: Our recent work has explored the efficient design and parameterisation of Langevin algorithms for molecular and statistical sampling tasks, as well as Adaptive Langevin which can adjust friction automatically to offset implicit and explicit stochastic perturbation, e.g. additive noise perturbation or gradient noise due to subsampling in big-data applications. In this talk I will focus on the use of thermostat methods for a broad range of sampling and optimisation challenges in machine learning.
Dwaipayan Chakrabarti (University of Birmingham)Title: Programming Self-Assembly of Colloidal Photonic CrystalsAbstract: Colloidal particles in the size range of hundreds of nanometres appeal as building blocks for photonic crystals with a complete photonic band gap in the visible [1]. The self-assembly of colloidal photonic crystals, which offers a low-cost, scalable fabrication technique, has, however, proved elusive [2,3]. Colloidal diamond, which has been the most sought-after colloidal photonic crystal over nearly three decades, continues to prove a challenging target for programmed self-assembly, despite recent breakthroughs [4,5]. In this talk, I will demonstrate, using a variety of computer simulation techniques, how we can encode facile self-assembly pathways in designer colloidal patchy particles to address certain long-standing challenges, concerning the self-assembly of diamond-structured photonic crystals in particular [6-9]. Moreover, I will present an enantiomorphic pair of single-gyroid crystals comprising colloidal spheres as targets for chiral photonic crystals and establish two distinct routes for programmed self-assembly of each single colloidal gyroid enantiomorph from rationally designed patchy spheres [10]. In the context of contemporary experimental research, I will discuss how our design rules push the frontiers of colloidal self-assembly in the quest for colloidal photonic crystals at optical frequencies, with a variety of applications in lasing, sensing, and energy harvesting to name a few.
[1] J. D. Joannopoulos, P. R. Villeneuve and S. Fan, Nature 386, 143 (1997).[2] G. von Freymann, V. Kitaev, B. V. Lotsch and G. A. Ozin, Chem. Soc. Rev. 42, 252 (2013).[3] Z. Cai et al., Chem. Soc. Rev. 50, 5898 (2021).[4] W. Liu et al., Science 2016, 351, 582 (2016).[5] M. He et al., Nature 585, 524 (2020).[6] D. Morphew, J. Shaw, C. Avins and D. Chakrabarti, ACS Nano 12, 2355 (2018).[7] A. B. Rao, J. Shaw, A. Neophytou, D. Morphew, F. Sciortino, R. L. Johnston and D. Chakrabarti, ACS Nano 14, 5348 (2020).[8] A. Neophytou, V. N. Manoharan and D. Chakrabarti, ACS Nano 15, 2668 (2021).[9] A. Neophytou, D. Chakrabarti and F. Sciortino, Proc. Natl. Acad. Sci. USA 118, e2109776118 (2021). [10] W. Flavell, A. Neophytou, A. Demetriadou, T. Albrecht and D. Chakrabarti, Adv. Mater. 35, 2211197 (2023).
Gabriel Stoltz (École des Ponts ParisTech)Title: A Mathematical Analysis of Autoencoders in the Context of Importance SamplingAbstract: Sampling high dimensional probability measures is often made difficult by the multimodality of the target probability distribution. The MCMC scheme under consideration needs to pass through low probability regions to switch from one mode to another, which is a rare event. An approach to making these transitions less rare is to identify a few selected (nonlinear) degrees of freedom of the system, which are at the origin of the slow mixing behavior, compute the associated free energies, and perform some importance sampling based on the latter function. Various tools have now recently been developed in molecular simulation to automatically find the most relevant nonlinear degrees of freedom hindering sampling, based on machine learning tools such as autoencoders [1]. I will present a methodology to leverage these models for better sampling, and will also provide a mathematical analysis of the approach, relating it to principal manifolds and providing an interpretation based on conditional expectations [2].
[1] Z. Belkacemi, P. Gkeka, T. Lelièvre, G. Stoltz, Chasing collective variables using autoencoders and biased trajectories, J. Chem. Theory Comput. 18(1), 59-78 (2022)[2] T. Lelièvre, T. Pigeon, G. Stoltz and W. Zhang, Analyzing multimodal probability measures with autoencoders, J. Phys. Chem. B 128(11), 2607-2631 (2024)
Michele Ceriotti (EPFL)Title: (Why) Do We Need Symmetry in Atomic-Scale Machine Learning?Abstract: Symmetry, and in particular rotational equivariance, has become one of the central concepts that underlies machine-learning models for atomic-scale simulations. It is usually understood as beneficial for the accuracy and transferability of models, and it has influenced deeply the theory and practical implementation of the most widely used models in the field, at variance with other domains of geometric learning, in which the most performing models are not intrinsically consistent with rotational symmetry.
I will give a brief overview of the current understanding of the design space of equivariant models, highlighting their advantages, but also the constraints they introduce on the architecture of machine-learning models.
I will then discuss some recent results that take a different approach - building a model that is not intrinsically equivariant, and making its predictions rigorously equivariant a posteriori. I will show that this approach can be at least as accurate as an intrinsically-equivariant architecture, challenging some of the common wisdom in the field, and suggesting that, counterintuitively, relaxing symmetry requirements might be beneficial to the design of better atomistic models.
Alessandro Troisi (University of Liverpool)Title: Digital Materials Discovery in Organic ElectronicsAbstract: This talk will provide an overview of the three main approaches used within the group to aid the discovery of new materials (i) traditional bottom up construction of physical models (ii) high-throughput virtual screening (iii) machine learning (ML). The relation between the three approaches will be discussed including a methodology for selecting the best strategy given time and budget constraints. The challenge of dealing with experimental and highly biased datasets and the difficulty of measuring “novelty” of a given ML-based prediction will be discussed. The talk will explore how linking the prediction with the knowledge of the chemical supply chain can substantially shorten the gap between predictions and experimental verification.
Livia Bartók-Pártay (University of Warwick)Title: High-Throughput Computational Thermodynamics: A Unique Insight into the Potential Energy SurfaceAbstract: During the past decade, we have adapted the Bayesian statistical approach, nested sampling, for exploring the potential energy surface of atomistic systems. Nested sampling's primary advantage lies in its automatic sampling of thermodynamically relevant configurations across the entire configuration space, from gas phase to crystalline solid structures, in proportion to their phase space volume. This enables the calculation of the partition function at any temperature, facilitating the evaluation of key thermodynamic properties such as free energy and heat capacity. Nested sampling has proven particularly effective for sampling phase transitions, allowing us to calculate pressure-temperature phase diagrams for various materials and model systems. This unbiased and predictive sampling approach has revealed surprising properties and new structures in several systems, enhancing our understanding of interatomic potential models and guiding improvements, including those in machine-learned potentials. The talk describes the main features of the nested sampling method and highlights a few such example applications.
Matthias Sachs (University of Birmingham)Title: (Hyperactive) Learning of Coarse-Grained Dynamics: From Data Acquisition to Equivariant Representations of Stochastic Heat Bath ModelsAbstract: Rigorously derived dynamics of coarse-grained particle systems via the Mori-Zwanzig projection formalism take the form of a (generalized) Langevin equation with, in general, configuration-dependent friction and diffusion tensors. Learning these dynamics typically requires i) the learning of a (conservative) inter-particle forcefield and ii) the learning of the residual heat bath dynamics. In this talk, I will first present the Hyperactive Learning framework, an accelerated Bayesian active learning technique for the learning of conservative force fields. The framework improves upon previous approaches for active learning of atomic force fields by biasing exploration during the data acquisition stage towards configurations of high predictive uncertainty. In the second part of this talk, I will introduce a class of equivariant representations of matrix-valued functions of particle configurations that allows for efficient learning of configuration-dependent friction and diffusion tensors from data. Besides satisfying the correct equivariance properties with respect to the Euclidean group E(3), the resulting heat bath model satisfies a fluctuation-dissipation relation and can be extended to include additional symmetries, such as momentum conservation to ensure correct hydrodynamic properties of the particle system.
Zhenyu Jason Zhang (University of Birmingham)Title: Leveraging Molecular Modelling for Material Science – A Wish List by an ExperimentalistAbstract: Taking advantage of the molecular insight generated by molecular modelling is highly desirable by experimentalists who deal with complex mixtures. Two case studies in which we tried to synchronise the modelling and experimental works will be disussed in my talk: 1) aggregation of polyaromatic compounds where Molecular Dynamics (GROMACS) was used to help us understand the experimental findings such as Dynamic Light Scattering and Diffusion Nuclear Magnetic Resonant spectroscopy [1,2,3,4]; 2) adsorption kinetics and interfacial configuration of fibronectin fragment, which was investigated using both advanced experimental techniques including Single Molecule Force Spectroscopy and Quartz Crystal Microbalance [5], alongside by a fully Atomistic Molecular Dynamics simulations using the NAMD package [6].
[1] D. Simionesie, G. O’Callaghan, J.R.H. Manning, T. Düren, J.A. Preece, R. Evans, and Z.J.Zhang* “Combined experimental and computational study of polycyclic compound aggregation: the impact of solvent composition” Polycyclic Aromat. Compd. 43 (2023) 3790[2] D. Simionesie, G. O’Callaghan, J.L.L.F.S. Costa, L. Giusti, W.J. Kerr, J. Sefcik, P. Mulheran, and Z.J.Zhang* “Clustering behaviour of polyaromatic compounds mimicking natural asphaltenes” Colloids Surf., A 603 (2020) 125221 [3] D. Simionesie, G. O’Callaghan, R. Laurent, J.A. Preece, R. Evans, and Z.J. Zhang* “Combined experimental and computational study of polyaromatic hydrocarbon aggregation – isolating the effect of attached functional groups” Ind. Eng. Chem. Res. 58 (2019) 20505[4] J.L.L.F.S. Costa, D. Simionesie, Z.J. Zhang,* and P.A. Mulheran* “Aggregation of model asphaltenes: a molecular dynamics study” J. Phys.: Condens. Matter 28 (2016) 394002[5] E. Liamas, R.A. Black, P.A. Mulheran, R. Tampé, R. Wieneke, O.R.T. Thomas, and Z.J. Zhang* “Probing fibronectin adsorption on chemically defined surfaces by single molecule force microscopy and a quartz crystal microbalance” Sci. Rep. 10 (2020) 15662[6] E. Liamas, K. Kubiak-Ossowska, R.A. Black, O.R.T. Thomas, Z.J. Zhang,* and P.A. Mulheran* “Adsorption of fibronectin fragment on surfaces using fully atomistic molecular dynamics simulations” Int. J. Mol. Sci. 19 (2018) 3321
Mark Wilson (Durham University)Title: Coarse-Grained Models for Soft Matter: Exploring Top-Down and Bottom-Up Models for Studying Self-Assembly and Self-OrganisationAbstract: For many soft matter systems, phenomena of interest occur on length and time scales that cannot be addressed by atomistic simulation today. Here, coarse-grained (CG) models come into play, with typical simulations taking advantage of fewer interaction sites, longer time steps, and faster navigation through phase space. However, the problem of how to move successfully from an atomistic to a coarse-grained model remains a difficult one! The twin problems of representability and transferability are inherent to most coarse-grained models, and nowhere is this more relevant than in models designed to study self-assembly and/or self-organisation in solution.This talk addresses the process of making successful coarse-grained models to study self-assembly and self-organisation in the context of amphiphilic molecules; covering self-assembly to form micelles, chromonic aggregation to form structured molecular stacks, self-organisation to form lyotropic liquid crystalline phases, and adsorption of molecules at a polymer surface. These are all problems where changes in concentration and molecular environment place a severe test on the transferability of a model. I will cover top-down and bottom-up approaches to coarse-graining in the context of coarse-grained MD (CG-MD), dissipative particle dynamics (DPD), and many-body dissipative particle dynamics (MDPD) and discuss the best strategies for improving both representability and transferability.
Andrew Morris (University of Birmingham) Title: Strong Electron Correlations in Novel Battery MaterialsAbstract: The lithium nickel manganese cobalt oxides, LiNixMnyCozO2 (x+y+z=1) (NMC) are a promising family of materials for the cathodes of lithium-ion batteries (LIB). The Ni-rich NMCs especially, exhibit excellent performance as high-voltage cathode materials, enabling batteries with high energy densities and high capacities of around 200-275mAh/g. However Ni-rich NMCs are prone to structural instabilities and oxygen loss leading to electrode degradation, a hurdle that must be overcome before widespread commercialisation.Density-functional theory (DFT) is now the standard modelling technique for atomistic physics, chemistry and materials science. It allows us to solve a single-particle Schrödinger-like equation for the energies of electrons in molecules and solids, thereby allowing us to deduce the material’s crystal structure and properties. I introduce DFT and show how it can clarify the complex behaviour of a class of LIB cathodes, the tungsten niobates. Indeed, this behaviour may then be rationalised within the much simpler crystal-field theory. However, for describing the crystal and electronic structure of NMCs, DFT falls short due to a lack of ability to account for strong electron correlations. I introduce the more advanced dynamical-mean-field theory (DMFT) and show that it correctly describes the electronic properties of the NMC family. This, in turn allows us to uncover the mechanism of oxygen loss on delithiation of NMC.The talk should be accessible for the non-expert in DFT, and the emphasis is on the different levels of theory, and where they fall short.
Jia Shao (University of Birmingham)Title: Sampling Rare Event with Subset Simulation: Applications and BenefitsAbstract: Rare events encompass natural catastrophes and anthropogenic hazards that occur with low frequency and extreme magnitude. A major challenge in simulations is the accurate sampling of rare events, which are critical for understanding phenomena like phase transitions and chemical reactions at the atomic level. Sampling high-dimensional probability measures is often complicated by the multimodality of the target probability distribution.In this talk, I will demonstrate a Subset Simulation (SubSim) algorithm designed to enhance the efficiency of rare event sampling. Subset Simulation is an elegant and adaptive method that can simulate multi-variables problem by sampling from conditional distributions of rare failure probabilities. I will also discuss the benefits and the potential applications of this algorithm extend beyond civil engineering and materials science.

UoB School of Mathematics Code of Conduct:

The School of Mathematics is committed to providing a welcoming, inclusive and safe community for all. We expect co-operation and support from all staff, students and visitors to help ensure a harassment-free environment where everyone is treated with courtesy, respect and dignity. Examples of harassment include, but are not limited to:

offensive or belittling comments related to age, body size, disability, ethnicity, gender, gender identity and expression, physical appearance, sexual orientation, socio-economic status and religion;
inappropriate language (this does not need to be aimed directly at an individual(s) for it to contravene the code);
deliberate intimidation;
harassing photography or recording;
inappropriate physical contact;
unwelcome sexual or other forms of attention.

We also reject complicity that knowingly promotes, encourages, or protects discrimination or unprofessional behaviour on the part of others. Should anyone experience or witness behaviour contravening the School of Mathematics’ Code of Conduct, please intervene where you feel comfortable doing so. We strongly urge you to report this behaviour using this online form.
Such behaviours are not welcome in the School of Mathematics and will not be tolerated. Alternatively, please speak to a member of staff to whom you feel comfortable disclosing this information. The Director of Equality, Diversity and Inclusivity in the School of Mathematics is Dr Sara Jabbari in Watson 113 (s.jabbari@bham.ac.uk).

Funding:

We acknowledge funding from the Isaac Newton Institute for Mathematical Sciences via its Network Support for the Mathematical Sciences initiative (EPSRC Grant Ref: EP/V521929/1), the London Mathematical Society via its Joint Research Groups in the UK scheme, as well as the CCP5 via its Workshops Funding scheme.

Page updated

Google Sites

Report abuse