Tesis sobre el tema "Active Brownian Particles"

Das Konzept von aktiven Brownschen Teilchen kann benutzt werden, um das Verhalten einfacher biologischer Organismen oder künstlicher Objekte, welche die Möglichkeit besitzen sich von selbst fortzubewegen zu beschreiben. Als Bewegungsgleichungen für aktive Brownsche Teilchen kommen Langevin Gleichungen zum Einsatz. In dieser Arbeit werden aktive Teilchen mit konstanter Geschwindigkeit diskutiert. Im ersten Teil der Arbeit wirkt auf die Bewegungsrichtung des Teilchen weißes alpha-stabiles Rauschen. Es werden die mittlere quadratische Verschiebung und der effektive Diffusionskoeffizient bestimmt. Eine überdampfte Beschreibung, gültig für Zeiten groß gegenüber der Relaxationszeit wird hergleitet. Als experimentell zugängliche Meßgröße, welche als Unterscheidungsmerkmal für die unterschiedlichen Rauscharten herangezogen werden kann, wird die Kurtose berechnet. Neben weißem Rauschen wird noch der Fall eines Ornstein-Uhlenbeck Prozesses angetrieben von Cauchy verteiltem Rauschen diskutiert. Während eine normale Diffusion mit zu weißem Rauschen identischem Diffusionskoeffizienten bestimmt wird, kann die beobachtete Verteilung der Verschiebungen Nicht-Gaußförmig sein. Die Zeit für den Übergang zur Gaußverteilung kann deutlich größer als die Zeitskale Relaxationszeit und die Zeitskale des Ornstein-Uhlenbeck Prozesses sein. Eine Grenze der benötigten Zeit wird durch eine Näherung der Kurtosis ermittelt. Weiterhin werden die Grundlagen eines stochastischen Modells für lokale Suche gelegt. Lokale Suche ist die Suche in der näheren Umgebung eines bestimmten Punktes, welcher Haus genannt wird. Abermals diskutieren wir ein aktives Teilchen mit unveränderlichem Absolutbetrag der Geschwindigkeit und weißen alpha-stabilem Rauschen in der Bewegungsrichtungsdynamik. Die deterministische Bewegung des Teilchens wird analysiert bevor die Situation mit Rauschen betrachtet wird. Die stationäre Aufenthaltswahrscheinlichkeitsdichtefunktion wird bestimmt. Es wird eine optimale Rauschstärke für die lokale Suche, das heißt für das Auffinden eines neuen Ortes in kleinstmöglicher Zeit festgestellt. Die kleinstmögliche Zeit wird kaum von der Rauschart abhängen. Wir werden jedoch feststellen, dass die Rauschart deutlichen Einfluß auf die Rückkehrwahrscheinlichkeit zum Haus hat, wenn die Richtung des zu Hauses fehlerbehaftet ist. Weiterhin wird das Model durch eine an das Haus abstandsabhängige Kopplung erweitert werden. Zum Abschluß betrachten wir eine Gruppe von Suchern.
Active Brownian particles described by Langevin equations are used to model the behavior of simple biological organisms or artificial objects that are able to perform self propulsion. In this thesis we discuss active particles with constant speed. In the first part, we consider angular driving by white Levy-stable noise and we discuss the mean squared displacement and diffusion coefficients. We derive an overdamped description for those particles that is valid at time scales larger the relaxation time. In order to provide an experimentally accessible property that distinguishes between the considered noise types, we derive an analytical expression for the kurtosis. Afterwards, we consider an Ornstein-Uhlenbeck process driven by Cauchy noise in the angular dynamics of the particle. While, we find normal diffusion with the diffusion coefficient identical to the white noise case we observe a Non-Gaussian displacement at time scales that can be considerable larger than the relaxation time and the time scale provided by the Ornstein-Uhlenbeck process. In order to provide a limit for the time needed for the transition to a Gaussian displacement, we approximate the kurtosis. Afterwards, we lay the foundation for a stochastic model for local search. Local search is concerned with the neighborhood of a given spot called home. We consider an active particle with constant speed and alpha-stable noise in the dynamics of the direction of motion. The deterministic motion will be discussed before considering the noise to be present. An analytical result for the steady state spatial density will be given. We will find an optimal noise strength for the local search and only a weak dependence on the considered noise types. Several extensions to the introduced model will then be considered. One extension includes a distance dependent coupling towards the home and thus the model becomes more general. Another extension concerned with an erroneous understanding by the particle of the direction of the home leads to the result that the return probability to the home depends on the noise type. Finally we consider a group of searchers.

Particles motion inside complex, irregular or crowded environments is a common phenomenon ranging from microscopic to macroscopic scales. It can be involved in everyday practical problems, like traffic, in fundamental biological mechanisms, like growth and reproduction of cells, and in important industrial or chemical applications, like oil catalysis. In many cases, transport in crowded environments is guided by 'active' elements, i.e. units that consume energy in order to produce motion. Among systems belonging in this class, the diffusion of hard-core particles in a channel so narrow they cannot pass each other, known as Single File Diffusion, has assumed a particular role. Single File Diffusion is responsible for the transport of ions in membrane channels, the diffusion in nano- and micro-porous materials and has been observed in many other natural and artificial systems. Aim of this thesis is to investigate Single File system of passive (purely diffusive) or active (self propelled) particles, focusing on the effects of the activity on the Single File motion and on the Single File properties in the presence of absorbing boundaries. Most of the work has been carried out developing analytical and numerical tools within the framework of the Stochastic Processes. By using single particle techniques in a microfluidic approach, we obtained an excellent comparison between experimental data and numerical model of particles emptying a Single File channel with open ends. In this thesis, after a brief introduction in the framework of confined diffusion processes, we will review the most relevant works in the theoretical and experimental literature of Single File Diffusion, with particular attention to an analytical technique, the Reflection Principle Method, which will be extensively used in this thesis. We will investigate the properties of Single File systems of diffusing particles in presence of two absorbing boundaries, with particular interest to the survival probability, i.e. the probability to find a particle between the boundaries at time t. We will provide an analytical solution of the emptying process, i.e. we calculate the probability characterizing the progressive decrease of the number of particles in the presence of absorbing boundaries, and for the survival probability of a Tagged Particle within the file, either in the presence or in the absence of a constant external force. We also characterize the trend of the characteristic survival times (also called Mean First Passage Times) as function of the system size and of the initial number of particles. We also investigate numerically the case when only the central particle is affected by the absorbing boundaries. We find an exponential decay of the survival probability, as it happens for normal diffusive processes, even in the presence of overcrowding. We will then introduce activity in a Single File system, through a Self-Propelled Particle model, for which we will provide a detailed characterization. In particular, within this model, particles can be either runners or tumblers, if their motion is dominated by straight runs or by changes of direction, respectively. Under Single File conditions, runners tend to form dynamical aggregates: these clusters are continuously formed and disassembled due to random fluctuations of the activity. For tumblers, the survival probabilities are still well described by the analytical theory developed for passive diffusing particles. Conversely, the formation of dynamical clusters enhances anomalous behaviours in the characteristic survival times of runners and induces a remarkable capacity to overcome the action of an external force.
Il moto di particelle in mezzi irregolari, complessi o affollati è un fenomeno comune, dalla scala microscopica a quella macroscopica. Lo si può incontrare tanto in situazioni comuni, come il traffico, quanto in meccanismi biologici, come la riproduzione e la crescita delle cellule, e in importanti processi chimici e tecnologici, come la catalisi di idrocarburi. In molti casi, il trasporto in mezzi confinati o affollati è guidato da elementi 'attivi', cioè unità che consumano energia per sostenere il loro stato di moto. Fra i diversi sistemi soggetti a confinamento, particolare rilevanza è rivestita dalla diffusione di sfere impenetrabili in un canale così stretto da non permettere il passaggio di più di una particella alla volta, conosciuto come diffusione in Single File. La diffusione in Single File è il meccanismo responsabile del trasporto di ioni attraverso la membrana cellulare, della diffusione in materiali micro e nanoporosi ed è stata osservata in molti altri sistemi naturali ed artificiali. Scopo di questa tesi è lo studio su scala mesoscopica di particelle passive (diffusive) o attive (auto-propellenti) in condizioni di Single File, con particolare attenzione all'effetto dell'attività sulla dinamica e sulle proprietà delle particelle nel caso siano presenti condizioni al contorno assorbenti. Gran parte del lavoro è stato svolto nello sviluppo di risultati analitici e numerici nel contesto dei Processi Stocastici. Inoltre, mediante tecniche di manipolazione ottica di singola particella in canali microfluidici, abbiamo ottenuto una eccellente confronto fra dati sperimentali e numerici per il processo di svuotamento di un sistema di particelle in condizioni di Single File. In questa tesi, dopo una breve introduzione ai processi diffusivi fortemente confinati, passeremo in rassegna i lavori più rilevanti della letteratura teorica e sperimentale sulla Single File Diffusion, con particolare attenzione ad un formalismo matematico, il Reflection Principle Method, che sarà applicato in maniera estensiva nel corso della tesi. Studieremo poi le proprietà di un sistema di particelle diffusive in Single File in presenza di condizioni al contorno assorbenti, concentrandoci sulla survival probability, cioè la probabilità di trovare una particella fra gli estremi del sistema al tempo t. Mostreremo come, in condizioni di Single File, abbiamo ottenuto una soluzione analitica per il processo di svuotamento, cioè calcoleremo la probabilità che caratterizza la progressiva diminuzione del numero di particelle in presenza di condizioni al contorno assorbenti, e per la survival probability di una particella 'marcata' all'interno della Single File sia in presenza che in assenza di una forza esterna costante. Caratterizzeremo gli andamenti dei tempi caratteristici di sopravvivenza, chiamati Tempi Medi di Primo Passaggio, in funzione della taglia del canale e del numero iniziale di particelle. Indagheremo inoltre numericamente il caso in cui solo la particella centrale del sistema in Single File subisce l'effetto delle condizioni al contorno assorbenti. Osserviamo un decadimento esponenziale della survival probability, come accade nell'usuale moto Browniano, anche in presenza di estremo confinamento. Introdurremo l'attività nella Single File attraverso un modello di particelle Self-Propelled, di cui descriveremo le proprietà in dettaglio. In particolare in questo modello le particelle possono essere o runners o tumblers, a seconda che la loro traiettoria sia dominata da lunghi tratti rettilinei o da cambi di direzione. In condizioni di Single File, i runners tendono a formare aggregati dinamici: questi cluster vengono continuamente formati e distrutti dalle fluttuazioni casuali della forza propulsiva. Per i tumblers, le probabilità di sopravvivenza sono ben descritte dalla teoria analitica sviluppata per le particelle passive. Per contro, la formazione di cluster dinamici accresce i comportamenti anomali nei tempi caratteristici di sopravvivenza dei runners e ne induce una notevole capacità di opporsi all'azione di un campo esterno.

Les marches aléatoires avec recherche de cible peuvent modéliser des réactions nucléaires ou la quête de nourriture par des animaux. Dans cette thèse, nous identifions des stratégies qui minimisent le temps moyen de première rencontre d’une cible (MFPT) pour plusieurs types de marches aléatoires. Premièrement, pour des marches symétriques ou avec biais, nous déterminons la distribution des temps de première sortie par une ouverture dans une paroi en forme de secteur angulaire, d’anneau ou de rectangle. Nous concluons sur la minimisation du MFPT en termes de la géométrie du confinement. Deuxièmement, pour des marches alternant entre diffusions volumique et surfacique, nous déterminons le temps moyen de première sortie par une ouverture dans la surface de confine- ment. Nous montrons qu’il existe un taux de désorption optimal qui minimise le MFPT. Nous justifions la généralité de l’optimalité par l’étude des rôles de la géométrie, de l’adsorption sur la surface et d’un biais en phase volumique. Troisièmement, pour des marches actives composées de phases balistiques entrecoupées par des réorientations aléatoires, nous obtenons l’expression du taux de réorientation qui minimise le MFPT en géométries sphériques de dimension deux ou trois. Dans un dernier chapitre, nous modélisons le mouvement de cellules eucaryotes par des marches browniennes actives. Nous expliquons pourquoi le temps de persistance évolue expo- nentiellement avec la vitesse de la cellule. Nous obtenons un diagramme des phases des types de trajectoires. Ce modèle minimal permet de quantifier l’efficacité des processus de recherche d’antigènes par des cellules immunitaires
Random search processes can model nuclear reactions or animal foraging. In this thesis, we identify optimal search strategies which minimize the mean first passage time (MFPT) to a target for various processes. First, for symmetric and biased Brownian particles, we compute the distribution of exit times through an opening within the boundary of angular sectors, annuli and rectangles. We conclude on the optimizability of the MFPT in terms of geometric parameters. Second, for walks that switch between volume and surface diffusions, we determine the mean exit time through an opening inside the bounding surface. Under analytical criteria, an optimal desorption rate minimizes the MFPT. We justify that this optimality is a general property through a study of the roles of the geometry, of the adsorption properties and of a bias in the bulk random walk. Third, for active walks composed of straight runs interrupted by reorientations in a random direction, we obtain the expression of the optimal reorientation rate which minimizes the MFPT to a centered spherical target within a spherical confinement, in two and three dimensions. In a last chapter, we model the motion of eukaryotic cells by active Brownian walks. We explain an experimental observation: the persistence time is exponentially coupled with the speed of the cell. We also obtain a phase diagram for each type of trajectories. This model is a first step to quantify the search efficiency of immune cells in terms of a minimal number of biological parameters

Plusieurs mécanismes biologiques utilisent la polymérisation des filaments d'actine comme moteur mécanique. L'énergie chimique libérée à l'addition d'un monomère dans le filament est convertie en travail mécanique et une force est générée. Les filaments ainsi formés s'organisent grâce à des protéines liant l'actine et forment des structures qui diffèrent par leurs propriétés mécaniques et élastiques mais aussi de leurs fonctions dans les différents processus biologiques. Notre système expérimental permet d'étudier le lien entre les propriétés mécaniques et les mécanismes à l'origine de la production de la force. La polymérisation des filaments est directement initiée sur la surface de particules magnétiques. En présence d'un champ magnétique, ces dernières s'organisent en chaîne par des interactions dipôle-dipôle, et une force magnétique compressive est induite sur les filaments qui polymérisent. La polymérisation écartent les particules au cours du temps et en fonction de la force appliquée, la vitesse d'écartement des particules est ralentie. En suivant l'évolution de la distance entre particules, nous détaillons la relation force-vitesse et les propriétés mécaniques des filaments.

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2017
Atualmente é possível observar um crescente desenvolvimento da tecnologia e, nomeadamente, de robots. Cientistas e engenheiros estão a trabalhar com escalas cada vez mais pequenas, como a micro e a nano-escala. A sociedade pode beneficiar destes desenvolvimentos de diversos modos, como por exemplo, na área da Saúde. Pensemos então na possibilidade de construir micro-robots que são capazes de levar substâncias específicas, como medicamentos, a partes do corpo humano em necessidade. Esta tarefa parece f´acil se pensarmos nos mais recentes desenvolvimentos de robots à escala humana, mas o desafio está no facto de que os micro-robots sofrem muitas colisões provenientes do meio em que estão, devido a flutuações térmicas. Tendo este facto em conta, a descrição matemática destas partículas usando teorias da Mecânica torna-se um desafio. Para estudar o movimento dessas partículas, baseamo-nos no movimento de partículas criadas pela Natureza, como por exemplo, um espermatozóide. O seu movimento não é apenas aleatório pois é claro que existe uma direção preferencial de velocidade, à qual chamamos velocidade de deriva. Este termo de velocidade de deriva diz-nos se a partícula é ativa ou não, sendo então uma partícula Browniana passiva. Para descrever as equaçõees de movimento desta partícula, utiliza-se o conceito de processo de Wiener que também é denominado por movimento Browniano. Reduz-se então um problema de vários corpos a um problema de um corpo e para isso é necessário incluir conceitos de Cálculo Estocástico, cujo fator principal é o processo de Wiener, e o resultado será uma equação diferencial estocástica como equação de movimento da partícula em estudo. Apesar disso, note-se que não é possível aplicar esta redução a todos os sistemas, pois é necessário preencher os requisitos apropriados, por exemplo esta redução é aplicada quando há interesse em estudar as propriedades globais do sistema e/ou o comportamento mecânico em vez da estrutura molecular e/ou interações químicas ou quando se pretende fazer um estudo simples dos resultados das simulações e do comportamento do sistema. Com esta redução obtêm-se cálculos computacionalmente mais económicos e, além disso, é possível simular sistemas maiores, assim como maiores escalas de tempo, o que permite usar um incremento de tempo maior. Contudo, é importante garantir que os modelos simplificados são capazes de reproduzir as propriedades físicas relevantes. Utilizando esta equação diferencial estocástica, é possível realizar simulações e explorar várias possibilidades tais como diferentes valores da velocidade de deriva, diferentes tempos totais de simulação, diferentes viscosidades do fluído, introduzir obstáculos (como por exemplo uma parede) e diferentes potenciais de interação entre as partículas. Para integrar numericamente as equações de movimento, implementou-se o método de Euler-Maruyama. Este método baseia-se no tão conhecido método de Euler que é utilizado para resolver equações diferenciais ordinárias numericamente. Considerou-se um sistema de duas dimensões e partículas de forma circular. Foram feitas simulações com partículas Brownianas passivas a partículas ativas considerando condições periódicas de fronteira de onde se concluiu que as partículas Brownianas passivas possuem um comportamento puramente difusivo ao longo do tempo, ou seja o deslocamento quadrático médio varia linearmente com o tempo, enquanto que as partículas ativas possuem um comportamento balístico, ou seja o deslocamento quadrático médio varia quadraticamente com o tempo, para tempos menores do que o inverso do coeficiente de difusão rotacional e comportamento difusivo para tempos maiores do que este. Nas simulações seguintes considerou-se a presença de uma caixa quadrada no sistema e partículas com a mesma posição inicial. Estas simulações foram realizadas para proceder ao estudo da função densidade de probabilidade. Aqui, para além de termos estudado o impacto de se considerar diferentes valores da velocidade de deriva e diferentes tempos totais de simulação, também se considerou o impacto de um fluido diferente. Observou-se que, no caso das partículas Brownianas passivas, a função densidade de probabilidade convergiu de uma distribuição normal para uma distribuição uniforme ao longo do tempo. No caso das partículas ativas, observou-se que a função densidade de probabilidade aumentou nas regiões próximas das paredes e que diminui nas restantes regiões. Para um tempo fixo e variando a velocidade de deriva das partículas ativas, observou-se o mesmo. Foi possível concluir que, com a presença de paredes, as partículas ativas acumulam-se perto destas e a velocidade deste processo depende do tempo total de simulação e/ou da velocidade de deriva. Este resultado era esperado pois é sabido da teoria que quando uma partícula ativa interage com uma parede, existe uma assimetria entre o movimento de chegada e o de partida da parede. Quando a partícula se aproxima da parede irá ficar próxima da parede até que a orientação da sua velocidade se altere para uma orientação oposta à parede e aí a partícula irá nadar para longe da parede. Esta assimetria faz com que exista uma tendência de acumulação de partículas ativas próximo das paredes. O estudo da acumulação de partículas ativas nas paredes é importante em aplicações, por exemplo no caso em que se supõe que um micro-robot leva uma dada substância a partes específicas do corpo humano, estes robots podem ter tendência a acumular próximos de superfícies no corpo o que pode resultar em elevadas concentrações da substância em questão em locais desejáveis ou não podendo haver, portanto, efeitos secundários indesejáveis. Ao variar o coeficiente de difusão translacional, observaram-se várias dependências entre o coeficiente de difusão translacional e a velocidade de deriva e também entre o coeficiente de difusão translacional e o tempo total de simulação. Nas simulações com partículas Brownianas passivas observou-se que, fixando o tempo de simulação, ao aumentar o coeficiente de difusão translacional a curva da função densidade de probabilidade era mais larga. Ao longo do tempo, para qualquer coeficiente de difusão translacional estudado, observou-se que a função densidade de probabilidade tornou-se mais larga. No caso com partículas ativas e com variação do coeficiente de difusão translacional, foram observadas as mesmas diferenças de comportamento em relação às partículas Brownianas passivas aquando a introdução das paredes: ao aumentar a velocidade de deriva nos quatro casos de diferente coeficiente de difusão translacional observou-se que o processo de acumulação de partículas próxima da parede era mais rápido. Para um valor de velocidade de deriva fixado, observou-se que a função densidade de probabilidade aumenta próximo das paredes ao longo do tempo nos quatro casos de diferente coeficiente de difusão translacional considerados. Por fim, consideraram-se interações entre partículas de modo a melhor entender o movimento colectivo de partículas, para isso estudou-se o caso com um potencial puramente repulsivo e com um potencial repulsivo e atrativo (potencial de Lennard-Jones). O primeiro caso estudado foi com duas partículas tangentes uma à outra no início das simulações. No caso com partículas Brownianas passivas, observou-se que, com qualquer dos potenciais considerados, as partículas repelem-se e afastam-se uma da outra. O mesmo acontece com partículas ativas apesar de que após um certo tempo a distância entre elas fica aproximadamente estável. De seguida, ainda considerando os potenciais, estudou-se um sistema com muitas partículas para se estudar a função densidade de probabilidade para diferentes valores da velocidade de deriva e do tempo total de simulação. Aqui observou-se o mesmo comportamento do que no caso em que as simulações realizadas sem se considerar potencial referido anteriormente: a função densidade de probabilidade no caso de partículas Brownianas passivas converge para uma distribuição uniforme e no caso com partículas ativas, a função densidade de probabilidade atinge valores mais elevados próxima das paredes ao longo do tempo e este processo, mais uma vez, ocorre de modo mais rápido quando se consideram valores da velocidade de deriva mais elevados. No entanto, foram observadas algumas diferenças entre ambos os potenciais: as partículas Brownianas passivas, ao considerar o potencial puramente repulsivo, necessitam de mais tempo até que se distribuam uniformemente pelo espaço e, portanto, exige-se mais tempo até que a função densidade de probabilidade convirja para uma distribuição uniforme; as partículas ativas acumulam-se mais depressa junto das paredes quando se considera o potencial puramente repulsivo do que o potencial de Lennard-Jones. Existem questões interessantes e importantes em termos de aplicações a serem estudadas no futuro. Por exemplo, se se considerar a existência de obstáculos no fluido será um caso interessante de se estudar o comportamento das partículas ativas e a sua acumulação, por exemplo, para diferentes geometrias do obstáculo. Outra questão interessante será o movimento ativo quiral no qual as partículas ativas nadam em trajetórias circulares. A existência de mais conhecimento científico nestes tópicos irá permitir um maior controle sobre partículas ativas, em particular, partículas ativas criadas pelo Homem tais como os micro- e nano-robots mencionados atrás.
Our goal is to study the singular and collective dynamics of active particles and compare them with passive Brownian particle dynamics. We introduce the Wiener process (also known as Brownian motion) and stochastic differential equations after which we present the numerical method used here: the Euler-Maruyama method. The concepts of both passive Brownian particles and active particles are explained and we introduce the concept of drift velocity. We then study the influence of a wall in these dynamics, calculating the probability density function when we consider different conditions such as different total simulation times, different values of drift velocity and different fluids. The existence of two distinct potentials is also considered: one that is purely repulsive and the Lennard-Jones potential, which is attractive and repulsive. Again, we calculate the probability density function in a system with several particles and study cases with different total simulation times and differing values of drift velocity. We consider a two-dimensional system for our conclusions and circular shaped particles. When comparing the motion of a singular particle, we can conclude that passive Brownian particles show a purely diffusive behavior, i.e. the mean square displacement is linear over time, and active particles show a diffusive behavior for longer times, i.e. times longer than the inverse of the rotational diffusion coefficient, whereas for shorter times, i.e. times shorter than the inverse of the rotational diffusion coefficient, they show a ballistic behavior, i.e. the mean square displacement shows a quadratic dependence of time. If we assume the existence of a wall in our system, we notice that the probability density function increases near the walls when fixing the value of drift velocity, while it converges to zero in the remaining regions. Physically, this means that there is an accumulation of particles near the wall since they stay there hitting the wall until their velocity direction changes. We also conclude that for passive Brownian particles, there is a convergence of the probability density function from a normal to a uniform distribution. However, when we consider active particles, the probability density function increases near the wall and it converges to zero in the remaining regions over time. We conclude that this occurs more rapidly as the value of drift velocity increases. When comparing different fluids, i.e. fluids with distinct values of translational diffusion coefficient (and so distinct values of rotational diffusion coefficient), we conclude that, for passive Brownian particles, the curve of distribution representing the probability density function becomes wider when the translational diffusion coefficient is larger when keeping the time fixed while, if we vary the time, this curve will be wider over time. For active particles, we concluded that the probability density function increases near the walls over time for a smaller translational diffusion coefficient. For a fixed time of simulation, the larger the value of drift velocity is, the higher the probability density function will be near the walls, which also increases when the translational diffusion coefficient is lowered. If we assume a repulsive potential between two passive Brownian particles tangent to each other, we conclude that they move away from each other. When we consider two active particles, they move away until they are outside the interaction range of the repulsive potential and then the distance between them stays approximately stable. If we consider several particles, again, we see that, for passive Brownian particles, there is a convergence of the probability density function to a uniform distribution. When considering active particles, the probability density function increases near the wall over time and it converges to zero in the remaining regions. When increasing the value of drift velocity this occurs faster than if we were considering a smaller value of drift velocity. When considering the Lennard-Jones potential between two particles tangent to each other, the conclusions are the same as for the repulsive potential. In our work we observed that when the depth of the potential well converges to zero, there is a weaker attraction between the particles, which leads to a non-aggregation state. In the opposite case, in which the depth of the potential well is much larger than one, the aggregation between the particles is very strong. This last result does not depend on the value of the drift velocity. When considering several particles, we conclude the same for the probability density function as in the repulsive potential case. For passive Brownian particles, the process of converging to a uniform distribution is slower when considering the purely repulsive potential than the Lennard-Jones potential. For active particles, the process of accumulation of particles near the wall is faster when the repulsive potential is considered than with the Lennard-Jones potential.

Microfabrication techniques have opened up new ways to study the dynamics of microsystems expanding the range of applications in microengineering and cell biology. Among three-dimensional microfabrication techniques, two-photon polymerization enjoys a unique set of characteristics that make it appealing for designing complex structures of arbitrary form. During the last decades, two-photon polymerization has evolved from the first structure fabricated with this technique, a coil with a diameter of 7 μm and a total length of approximately 34 μm (by Maruo et al.), to generate sophisticated systems like remotely driven micromachines. In the present thesis, we address two main applications of microfabrication. On the first line of research, the design, and fabrication of efficient and self-powered micro-robots have been a very active research topic. Motile micro-organisms like E. coli may provide an optimal solution to generate propulsion in artificial microsystems. It has been demonstrated that microstructures can be transported when released on a layer of swarming bacteria, suspended in a bacterial bath, or covered by surface adhering bacteria. Although it is possible to obtain a net movement in the mentioned cases, the displacement is stochastic and self-propulsion characteristics are hard to reproduce. In this thesis, we investigate possible design strategies for bio-hybrid micro shuttles having a defined number of propelling units that self-assemble onto precisely defined locations. One of the biggest issues involved in the optimization design process of the microshuttles is an irreversible adhesion of structures in the substrate, which often is caused by Van der Waals attraction. To overcome this problem we use different stabilization methods with unsuccessful results. Looking for a less invasive and biocompatible strategy we investigate the possibility of changing the sign of Van der Walls forces turning them from attractive to repulsive. To this aim, we develop a method that demonstrates to reduce the adhesion observed before. So, the final design aims at minimizing friction and adhesion with the substrate while optimizing propulsion speed and self-assembly efficiency. Finally, using a mutated strain of E. coli the microshuttle can be remotely controlled by dynamic structured light patterns for reaching an optimal control of the motion of the structures. In a different direction of microfabrication applications, 3D microstructures can also offer new opportunities to address more fundamental problems in the soft matter dynamic. On this second line of research, we have designed and used complex 3D microstructures to investigate the Brownian dynamics and hydrodynamics of propeller shaped particles, as well as to probe effective interactions in colloidal systems, like critical Casimir forces. In the dynamics of microhelices we use optical tweezers to study the mechanic and hydrodynamic properties of micro-fabricated helices suspended in a fluid. For the case of rigid helices, we track Brownian fluctuations around mean values with a high precision and over a long observation time. Through the statistical analysis of fluctuations in translational and rotational coordinates, we recover the full mobility matrix of the micro-helix including the off diagonal terms related with roto-translational coupling. Exploiting the high degree of spatial control provided by optical trapping, we can systematically study the effect of a nearby wall on the roto-translational coupling, and conclude that a rotating helical propeller moves faster near a no-slip boundary. We also study the relaxation dynamics of deformable micro-helices stretched by optical traps. We find that hydrodynamic drag only weakly depends on elongation resulting in an exponential relaxation to equilibrium. In connection with the versatility of microfabrication by two-photon polymerization, we find the study of interaction in colloidal systems. At macroscopic scales, thermal fluctuations of a physical property on a system are typically negligible, but at the micrometer and nanometer scales instead, fluctuations become generally relevant and they give rise to novel and intriguing phenomena such as critical Casimir effect. Critical Casimir forces are induced between colloidal objects suspended in a critical binary mixture undergoing strong thermal fluctuations. So far, most of the experiments and proposed models consider the interaction between simple geometrical objects such as two spheres, or a single sphere and a plate. In the last part of this thesis, we propose a novel 3D printed microprobes consisting of the main body and two handles that can be optically trapped to directly measure effective forces and torques between colloidal objects with non spherical shapes. The organization of this thesis is as follows. Chapter 1 gives a general introduction to the physical phenomenon behind the 3D microfabrication technique employed in our experiments, two-photon polymerization. We describe the differences between two phenomena: single-photon absorption and two-photon absorption, and explain the effectiveness of using two-photon polymerization for reaching a resolution of 100 nanometers in microfabrication. Then we present an experimental characterization of the voxel size of our custom-built two-photon polymerization set-up. We explain the sample preparation steps for microfabrication as well as the development of an innovative low-refractive index layer for eliminating irreversible adhesion of SU-8 microstructures. Chapter 2 provides a general introduction to E. coli motility, the propulsion mechanism of these bacteria, and the circular trajectory developed by the microorganism when swimming near a rigid boundary. Besides, we briefly explain the possibility of using synthetic biology to obtain light-driven strains of E. coli by the expression of Proteorhodopsin on the bacteria membrane. In Chapter 3 we combine two-photon polymerization technique and genetically modified bacteria to create a biohybrid microshuttle. We start with a basic microshuttle design whose propulsion is obtained from four E. coli bacteria. After integrating ramps in another microshuttle model for minimizing the circular trajectory showed in the microstructure trajectory, we make major changes in the distribution of microchambers inside the last model named catamaran microshuttle. Exploiting the ability of a mutated strain of E. coli expressing proteorhodopsin, we successfully control the microrobot steering by illuminating our sample with green light patterns. In Chapter 4 we design and use 3D microhelices from two different materials to investigate, through optical tweezers, Brownian dynamics, and hydrodynamics of this kind of chiral particles. Through the statistical analysis of fluctuations in translational and rotational coordinates, we study the roto-translational coupling element from the mobility matrix of the micro-helix. Besides, we conclude that a rotating helical propeller moves faster near a no-slip boundary. For the case of a deformable micro=helix, we find that hydrodynamic drag only depends on its elongation. Finally, Chapter 5 presents the design for a microprobe to measure critical Casimir forces using holographic optical tweezers. We show a characterization experiment for a micro-cube with two handles, concluding that a third handle will improve the stability of a microprobe inside the sample.

This thesis consists of research work in the broad area of soft condensed matter theory with a focus on active matter. The study of long wavelength, low frequency collective behavior of active particles (bacterial suspensions, fish schools, motor-microtubule extracts, active gels) forms an interesting modification to liquid-crystal hydrodynamics, in which the constituent particles carry permanent stresses that stir the fluid. Activity introduces novel instabilities and many novel aspects emerge. Our works focus on the dynamics, order, fluctuations and instabilities in these systems. In particular, we investigated the dynamics, order and fluctuation properties emerging from effective hydrodynamic descriptions of translationally ordered active matter and also studied those in microwave-driven quantum Hall nematics. We also investigated the rheological properties of active suspensions subjected to an applied orienting field. A summary of the works carried out is as follows. Translationally ordered active phases – active smectics and active cholesterics: Active or self-propelled particles consume and dissipate energy generating permanent stresses that stir the fluid around them. The collective behavior of systems of active particles, in systems with translational order, pose interesting questions and possibilities of new physics that differ strikingly from those in systems at thermal equilibrium with the same spatial symmetry. We developed the hydrodynamic equations of motion for (a) an active system with spontaneously broken translational symmetry in one direction, i.e., smectic and (b) the simplest uniaxially ordered phase of active chiral objects, namely, an active cholesteric. We analyze the fluctuation properties as well as the nature of characteristic instabilities that these systems can display and make a number of predictions. For example, in the case of an active smectic, we show that active stresses generate an effective active layer tension which, if positive, sup-presses the Landau-Peierls effect, leading to long-range smectic order in dimension d =3 and quasi-long-range in d =2, in sharp contrast with thermal equilibrium systems. Negative active layer tension in bulk systems, however, lead to a spontaneous Helfrich-Hurault undulation instability of the layers, accompanied by spontaneous flow. Also, active smectics, unlike orientationally ordered active systems, normally have finite concentration fluctuations. Similarly, for the case of cholesterics we show that cholesteric elasticity intervenes to suppress some of the instabilities present in active nematics. xi Numerical simulation of active smectics: We present results from a Brownian Dynamics simulation, with no hydrodynamic interaction, of a system of apolar active particles form-ing translational liquid-crystalline order in a suspension. The particles interact through a prolate-ellipsoidal Gay-Berne potential. We model activity minimally through different noise temperatures for movement along and normal to the orientation axis of each particle. We present preliminary results on the disruptive effect of activity on smectic order for the parameter values investigated. Future work will test the predictions of our theory [1] on active smectics. Rheology of active suspensions near field-induced critical points : Shear induces orientation of active stresses in a suspension, through flow alignment. Depending on the sign, activity then either enhances or reduces the viscosity. The change in viscosity, in the zero frequency limit, is proportional to the product of the magnitude of active stress and the system relaxation time. A strong enough orienting field can make the system approach a critical point and the relaxation time diverges. We show that, this results in the divergence of viscosity at zero frequency making the system strongly viscoelastic. Depending on the sign, activity strengthens or reduces the effect of the field. We also investigate the rheological property of an active suspension with mixed polar and nematic oreder. Active quantum Hall systems: We construct the hydrodynamic theory for a 2d charged active nematic with 3d electrostatics. We have investigated the interplay of the Coulomb interaction and activity in these systems. We show that activity competes to enhance the charge density fluctuations normally suppressed by long-ranged Coulomb interactions. The charge structure factor Sq of the corresponding passive charged nematic goes to zero as q, whereas in charged active nematics, activity leads to a nonvanishing charge structure factor at small wavenumber. We also show how the effect of an applied magnetic field can be incorporated into the dynamics of the system and leave scope for further studies on these effects.