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Emergence of Large Particles in Cluster-Cluster Aggregation

Periodic Reporting for period 1 - AggregationKinetics (Emergence of Large Particles in Cluster-Cluster Aggregation)

Reporting period: 2017-03-01 to 2019-02-28

Growth processes are ubiquitous in nature and social sciences. Despite the differences in appearances, the underlying mechanisms are few such as coagulation and exchange. Consequently, the collective properties of these systems share many commonalities. The kinetic theory of growth processes is a vibrant branch of non-equilibrium statistical mechanics that has enjoyed a resurgence of interest in recent years. As far as the mathematically interesting and practically important issues are concerned it is crucial to know the dominant behaviour of the solutions of the equations that describes these growth systems. The dominant behaviour can be manifested either as the special mathematical form (such as self-similar form) or the formation of extremely large clusters. This Action made significant advances in the analysis of such dominant behavior by analysing long time behaviour of the most commonly used growth models (aggregation and exchange). The research program also helped develop an improved understanding of formation of extremely large clusters.

In simple terms, the proposal achieved the following high level objectives
A. Established, for the first time, fundamental mathematical properties of solutions of exchange-driven growth model (such as existence, uniqueness, non-existence of solutions etc.)
B. Analyzed the large time behaviour of exchange-driven growth models and differential aggregation models
C. Developed an understanding, based on average dynamics, the mechanism behind the generation of extremely large clusters by identifying the regimes that such abrupt behavior does not take place
D. Employed the mathematical advances reached in Parts A,B and C in order to get genuine results in applications in inkjet printing applications.
Research Output from the Work Packages (explained below)
-Esenturk E., Mathematical theory of exchange-driven growth, Nonlinearity, (31), 3460-3483, (2018)
-Esenturk E., Connaughton C., Exchange-driven growth with source and sink, preprint
-Esenturk E., Velazquez J., Large time behavior of exchange-driven growth, submitted
-Esenturk E., Caginalp G., Asymptotic behavior of non-linear differential aggregation equations, in prep.
-Esenturk E., Connaughton C, Tse C., Stringer J., Smith P., Ta D., Coffee stain effect revisited, in prep.

Revised Work Package 2
In this work package (published), I obtained the first rigorous results for the fundamental mathematical properties of the mean field rate equations of this process. I showed that two different types of behavior arise depending on whether interaction kernel is symmetric or not. For the non-symmetric case, global existence of solutions and uniqueness of solutions were shown for kernels not growing faster than linear. On the other hand, for symmetric kernels it was proven that solutions exist locally for kernels growing faster than the 3/2th power (but slower than the 2nd power) of the cluster size. Hence, I identified the gelling regime for faster growing kernels while existence is completely lost for kernels growing faster than quadratic.

Revised Work Package 3
In this Work Package (submitted to journal) we studied the large time behavior of EDG equations. We rigorously proved, for a large class of kernels, that the EDG system approaches to equilibrium. For a class of kernels satisfying certain monotonicity conditions or for systems with small mass we were able to provide explicit convergence rates.

New Work Package 5
In this completed study (preprint), we presented results on exchange-driven process with the possibility of breakages non-zero clusters to form one-clusters (a cluster of single unit element). The effect is contingent and proportional to the available “active zero clusters”, that is the available volume to be occupied. We showed that for systems with active zero clusters in isolated systems, the classical indefinite growth trend is broken and the system approaches to an equilibrium even without the inclusion of a sink.

Work Package 4
This project is jointly done with an experimental group from the mechanical engineering department of Sheffield University (Patrick Smith’s group). I have been developing models for the clumping of ink particles during the process of evaporation of sessile droplet on a substrate. This is still a work in progress. The goal is to test the developed model against experimental measurements

Revised Work Package 1
This is the same work package proposed in the original program. I have had preliminary findings, but this is still a work in progress.
The detailed progress and expected results in the near future is explained above.


The beneficiaries of this MC Action are
• Academic Beneficiaries
• Research-driven Sectors (industrial or socio-economical)
• Wider Public Beneficiaries

1 Academic Beneficiaries
The primary academic beneficiaries of this project are mathematicians, analysts in particular. However, many components of the proposal borrow strategies, techniques from cross disciplines. The means of achieving this large scale productivity was ensured in the following ways:

Academic engagement: I achieved this at different target levels.
Internally, during the duration of fellowship, I co-supervised a MSc student on the exchange driven growth model.
Also within Warwick, I expanded my collaborations and interacted more closely with the faculty (e.g. Prof Stefan Gorsskinsky).
Externally, I visited as part of my secondment, the Hausdorff Centre of Mathematics twice in the springs of 2017 and 2018. There I met junior academics working in kinetic theory which helped foster future collaborations (such as the joint junior group application in the upcoming kinetic theory trimester program). Also, my interaction and collaboration with Prof J Velazquez grew into a research article.

Workshop organisation: I organized a a workshop in Sept 2018 which gave me the chance to advertise and communicate my work.

Dissemination of research results: As of now I completed 3 articles (1 published, 1 under review, 1 preprint) and preparing 2 more to be completed in the near future.
I attended 2 international conferences (Singapore, May 2017 and Sao Paulo, August 2018). I also made multiple research visits to communicate with scientist from other disciplines such as Prof Nikolai Brilliantov (Leicester Physics), Dr Micahel Herzog (Cambrdige Geography)

2 Research-driven Sectors
The proposal, in addition to theoretical advancements, gives great opportunities for exploitation of results through research driven sectors such as additive manufacturing industry, weather forecast agencies and economics.
Links to industry. Work Package 4 has the goal of quantifying the role of clumping of printer inks which is generally an undesired effect in the industry.

3 Wider Public Beneficiaries
To increase students’ awareness on the importance of mathematics in real life I will give hourly lessons/demonstrations on the subject of 3D printing and its future in our lives. This topic is chosen due to its direct relevance to Work Package 4. These lessons will be conducted in partnership with London Innovation Society (which I already contacted).