Abstract:
Manipulating fluids at the nanoscale within networks of channels or chemical lanes is a crucial challenge in developing small scale devices to be used in microreactors or chemical sensors. In this context, ultra-thin (i.e., monolayer) films, experimentally observed in spreading of nano-droplets or upon extraction from reservoirs in capillary rise geometries, represent an extreme limit which is of physical and technological relevance since the dynamics is governed solely by capillary forces. In this work we use kinetic Monte Carlo (KMC) simulations to analyze in detail a simple, but realistic model proposed by Burlatsky \textit{et al.} \cite{Burlatsky_prl96,Oshanin_jml} for the two-dimensional spreading on homogeneous substrates of a fluid monolayer which is extracted from a reservoir. Our simulations confirm the previously predicted time-dependence of the spreading, $X(t \to \infty) = A \sqrt t$, with $X(t)$ as the average position of the advancing edge at time $t$, and they reveal a non-trivial dependence of the prefactor $A$ on the strength $U_0$ of inter-particle attraction and on the fluid density $C_0$ at the reservoir as well as an $U_0$-dependent spatial structure of the density profile of the monolayer. The asymptotic density profile at long time and large spatial scale is carefully analyzed within the continuum limit. We show that including the effect of correlations in an effective manner into the standard mean-field description leads to predictions both for the value of the threshold interaction above which phase segregation occurs and for the density profiles in excellent agreement with KMC simulations results.

Abstract:
We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear systems. We are interested in the minimization of time, energy and final manifold in transfer problems. The state variables space X and, respectively, the control variables space U, are considered to be Hilbert spaces. The linear operator T(t) which defines the solution of the linear control system is a strong semigroup. Our analysis is based on some results from the theory of linear operators and functional analysis. The results obtained in this paper are based on the properties of linear operators and on some theorems from functional analysis.

Abstract:
Catalytically active particles suspended in a liquid can move due to self-phoresis by generating solute gradients via chemical reactions of the solvent occurring at parts of their surface. Such particles can be used as carriers at the micro-scale. As a simple model for a carrier-cargo system we consider a catalytically active particle connected by a thin rigid rod to a catalytically inert cargo particle. We show that the velocity of the composite strongly depends on the relative orientation of the carrier-cargo link. Accordingly, there is an optimal configuration for the linkage. The subtlety of such carriers is underscored by the observation that a spherical particle completely covered by catalyst, which is motionless when isolated, acts as a carrier once attached to a cargo.

Abstract:
We study theoretically the effects of spatial confinement on the phoretic motion of a dissolved particle driven by composition gradients generated by chemical reactions of its solvent, which are active only on certain parts of the particle surface. We show that the presence of confining walls increases in a similar way both the composition gradients and the viscous friction, and the overall result of these competing effects is an increase in the phoretic velocity of the particle. For the case of steric repulsion only between the particle and the product molecules of the chemical reactions, the absolute value of the velocity remains nonetheless rather small.

Abstract:
We study the equilibrium properties of a lattice-gas model of an $A + B \to 0$ catalytic reaction on a one-dimensional chain in contact with a reservoir for the particles. The particles of species $A$ and $B$ are in thermal contact with their vapor phases acting as reservoirs, i.e., they may adsorb onto empty lattice sites and may desorb from the lattice. If adsorbed $A$ and $B$ particles appear at neighboring lattice sites they instantaneously react and both desorb. For this model of a catalytic reaction in the adsorption-controlled limit, we derive analytically the expression of the pressure and present exact results for the mean densities of particles and for the compressibilities of the adsorbate as function of the chemical potentials of the two species.

Abstract:
Using a set of evolution equations [J.G. Amar {\it et al}, Phys. Rev. Lett. {\bf 86}, 3092 (2001)] for the average gap-size between islands, we calculate analytically the asymptotic scaled capture-number distribution (CND) for one-dimensional irreversible submonolayer growth of point islands. The predicted asymptotic CND is in reasonably good agreement with kinetic Monte-Carlo (KMC) results and leads to a \textit{non-divergent asymptotic} scaled island-size distribution (ISD). We then show that a slight modification of our analytical form leads to an analytic expression for the asymptotic CND and a resulting asymptotic ISD which are in excellent agreement with KMC simulations. We also show that in the asymptotic limit the self-averaging property of the capture zones holds exactly while the asymptotic scaled gap distribution is equal to the scaled CND.

Abstract:
We present an \textit{exactly solvable} model of a monomer-monomer $A + B \to \emptyset$ reaction on a 2D inhomogeneous, catalytic substrate and study the equilibrium properties of the two-species adsorbate. The substrate contains randomly placed catalytic bonds of mean density $q$ which connect neighboring adsorption sites. The interacting $A$ and $B$ (monomer) species undergo continuous exchanges with corresponding adjacent gaseous reservoirs. A reaction $A + B \to \emptyset$ takes place instantaneously if $A$ and $B$ particles occupy adsorption sites connected by a catalytic bond. We find that for the case of \textit{annealed} disorder in the placement of the catalytic bonds the reaction model under study can be mapped onto the general spin $S = 1$ (GS1) model. Here we concentrate on a particular case in which the model reduces to an exactly solvable Blume-Emery-Griffiths (BEG) model (T. Horiguchi, Phys. Lett. A {\bf 113}, 425 (1986); F.Y. Wu, Phys. Lett. A, {\bf 116}, 245 (1986)) and derive an exact expression for the disorder-averaged equilibrium pressure of the two-species adsorbate. We show that at equal partial vapor pressures of the $A$ and $B$ species this system exhibits a second-order phase transition which reflects a spontaneous symmetry breaking with large fluctuations and progressive coverage of the entire substrate by either one of the species.

Abstract:
We study the equilibrium properties of a model for a binary mixture of catalytically-reactive monomers adsorbed on a two-dimensional substrate decorated by randomly placed catalytic bonds. The interacting $A$ and $B$ monomer species undergo continuous exchanges with particle reservoirs and react ($A + B \to \emptyset$) as soon as a pair of unlike particles appears on sites connected by a catalytic bond. For the case of annealed disorder in the placement of the catalytic bonds this model can be mapped onto a classical spin model with spin values $S = -1,0,+1$, with effective couplings dependent on the temperature and on the mean density $q$ of catalytic bonds. This allows us to exploit the mean-field theory developed for the latter to determine the phase diagram as a function of $q$ in the (symmetric) case in which the chemical potentials of the particle reservoirs, as well as the $A-A$ and $B-B$ interactions are equal.

Abstract:
We study the effect of spatial confinement on the strength of propulsive diffusiophoretic forces acting on a particle that generates density gradients by exploiting the chemical free energy of its environment. Using a recently proposed simple model of a self-propelling device driven by chemical reactions taking place on some parts of its surface, we demonstrate that the force significantly increases in the presence of confining walls. We also show that such effects become even more pronounced in two-dimensional systems.

Abstract:
The use of ultra-thin, i.e., monolayer films plays an important role for the emerging field of nano-fluidics. Since the dynamics of such films is governed by the interplay between substrate-fluid and fluid-fluid interactions, the transport of matter in nanoscale devices may be eventually efficiently controlled by substrate engineering. For such films, the dynamics is expected to be captured by two-dimensional lattice-gas models with interacting particles. Using a lattice gas model and the non-linear diffusion equation derived from the microscopic dynamics in the continuum limit, we study two problems of relevance in the context of nano-fluidics. The first one is the case in which along the spreading direction of a monolayer a mesoscopic-sized obstacle is present, with a particular focus on the relaxation of the fluid density profile upon encountering and passing the obstacle. The second one is the mixing of two monolayers of different particle species which spread side by side following the merger of two chemical lanes, here defined as domains of high affinity for fluid adsorption surrounded by domains of low affinity for fluid adsorption.