Final Activity Report Summary - MANY-BODY COLLOIDS (Many-body interactions in charged colloidal suspensions)
The electrostatic interactions in colloidal suspensions have been investigated with the focus on the many-body effects, the effective interactions and the properties of colloidal crystals. The effective interactions and their applicability to predict structure and thermodynamic properties have been carefully analysed. We have shown that the microion-macroion correlations including the many-body effects are important and need to be accounted for when calculating the pressure of the colloidal suspension. On the other hand, at high enough densities, the macroion-macroion correlations can usually be neglected. The internal consistency of the Poisson-Boltzmann cell model and the renormalised Jellium model has been tested and compared. With added salt the Poisson-Boltzmann cell model has been found to be more consistent in describing the thermodynamics and the structural properties of the system.
The effects of the many-body interactions on the properties of colloidal crystals have been studied by means of numerically solving the Poisson-Boltzmann equation. By displacing one of the colloids inside an FCC unit cell we have determined the elastic constants in the colloidal crystals at different conditions. The Cauchy relation states that two of the three independent elastic constants in an fcc crystal have to be equal if the interactions are pairwise additive. Deviations from the Cauchy relation are a fingerprint of important many-body interactions in the system. We have shown that the Cauchy relation is satisfied at high salt concentration, where the interactions are heavily screened, but not at low salt, where the many-body effects are to be expected. Our numerical results agree well with the recent experiments by Reinke et al. in Konstanz.
Colloidal molecules and their ground states on patterned substrates have also been studied by means of Monte Carlo simulations. The ground states of dimers, trimers and tetramers on square and hexagonal lattices have been analysed and the phase diagram obtained. The results compare well with the experiments and predict new phases that have not zet been seen experimentally.
Finally, the chain formation in confined quasi-2D superparamagnetic colloids has been experimentally and numerically studied. The colloids interact via a purely repulsive interaction potential, but still under certain conditions couples and longer chains start to form. The experimentally observed phases have also been found in Monte Carlo simulations and these simulations have been further useful in determining the experimentally unknown parameters.
In scope of the project, we also devoted part of the time to study more biological subjects. The first project was simulating the chemotactic motion of bacteria E. coli. We have simulated the motion of large clusters of bacteria by modelling the individual cell response to the external food gradients. We have found external conditions in which the bacteria perform simple diffusion with exponential jump-length distribution and also conditions under which the Levy flights with power-law jump length distribution are observed.
The second biological project was developing a mathematical model of the fission yeast cell growth. The polar growth of fission yeast cells is governed by the localised delivery of the protein Tea1 onto the membrane, where it can react with the freely diffusing protein Mod5p and form a protein complex. We have written down the reaction-diffusion equations for the distribution of the proteins along the membrane, solved and analysed them and showed that the growth can be effective only when the protein complex is immobilised by the membrane.
1a) The ground state structure of 2D systems: the energy of different crystal lattices in 2D will be compared, obtained by summing up effective pair interactions and by solving the full nonlinear Poisson-Boltzmann equation. Generally accepted opinion is that the hexagonal lattice is the ground state for any 2D system with long-range pair interactions. However, there have been studies, where lattices other than hexagonal were found to be stable in systems with finite-range interaction potentials. It is therefore expected that by summing up effective pair interactions we will always end up in the hexagonal ground state, while after including the many-body effects (Poisson-Boltzmann), other ground states might be found.
1b)The macroscopic properties of crystals: elastic constants, compressibility, phonon spectrum and other macroscopic properties will be studied, again within full many-body Poisson-Boltzmann theory and within the pair approximation. Since there are pronounced many-body contributions to the total interaction energy, the macroscopic quantities are also expected to deviate from the pairwise predictions. It is important to show how the macroscopic properties depend on state variables, where the pair description is successful and where it breaks down.
1c)Building up clusters: the idea is to build up small crystallites of 2, 3, 4, ... N particles and to analyse the many-body contributions to the total interaction energy. The three-body interaction has been calculated and measured this way and the four-body was measured as well, but from these studies it was difficult to predict how the many-body series continues. Are there cancellations and does the series converge? By studying the energy of clusters of different sizes the total many-body contribution as a function of number of particles can be obtained and the previous fundamental question can be answered. By doing the same with crystallites of different lattice symmetry we can possibly find some universal characteristics of many-body effects.
2. Colloids on substrates: this part of the project will be carried out in strong collaboration with the experimental group of Clemens Bechinger (Stuttgart) and Hans Hennig von Grünberg (Graz). The idea is to continue our work on domain formation in 2D colloidal systems on light substrates. By performing Brownian dynamics and Monte Carlo simulations we will be able to predict the phase diagram of colloidal monolayers on the periodic light substrate. We will focus on:
2a) Crystal melting: how well can the renowned Kosterlitz-Zhouless-Halperin-Nelson-Young74,75 theory describe the melting of colloidal crystals in 2D with and without substrate potential? Is the floating solid phase predicted by Halperin / Nelson75 stable as they predict? The above mentioned theory is fundamental and in the past very successfully applied, but has not yet been rigorously tested. Colloidal systems manipulated by laser substrate are an ideal model system for this task.
2b) Defects and their effect on the macroscopic properties: it is easy to introduce defects into the system, experimentally, as well as in the numerical simulations. The effect of defects on the melting scenario will be studied in detail.
2c) Nucleation and crystal growth: finally nucleation and crystal growth are going to be addressed. Playing with the substrate potential a nucleation seed can be creates and the process of nucleation observed under controlled conditions.
The group in Stuttgart will perform experiments with 2D colloids on a periodic laser substrate which will directly correspond to our numerical simulations.
3. Anisotropic interactions; the existing Poisson-Boltzmann code will be upgraded to study the systems with charge and shape anisotropy. One of the aims is to study the angular dependence of DNA-DNA interactions with realistic charge distribution. Generally, the many-body interactions in systems of non-spherical objects are an unexplored and exciting field which can be and will be studied in scope of the proposal.
The effects of the many-body interactions on the properties of colloidal crystals have been studied by means of numerically solving the Poisson-Boltzmann equation. By displacing one of the colloids inside an FCC unit cell we have determined the elastic constants in the colloidal crystals at different conditions. The Cauchy relation states that two of the three independent elastic constants in an fcc crystal have to be equal if the interactions are pairwise additive. Deviations from the Cauchy relation are a fingerprint of important many-body interactions in the system. We have shown that the Cauchy relation is satisfied at high salt concentration, where the interactions are heavily screened, but not at low salt, where the many-body effects are to be expected. Our numerical results agree well with the recent experiments by Reinke et al. in Konstanz.
Colloidal molecules and their ground states on patterned substrates have also been studied by means of Monte Carlo simulations. The ground states of dimers, trimers and tetramers on square and hexagonal lattices have been analysed and the phase diagram obtained. The results compare well with the experiments and predict new phases that have not zet been seen experimentally.
Finally, the chain formation in confined quasi-2D superparamagnetic colloids has been experimentally and numerically studied. The colloids interact via a purely repulsive interaction potential, but still under certain conditions couples and longer chains start to form. The experimentally observed phases have also been found in Monte Carlo simulations and these simulations have been further useful in determining the experimentally unknown parameters.
In scope of the project, we also devoted part of the time to study more biological subjects. The first project was simulating the chemotactic motion of bacteria E. coli. We have simulated the motion of large clusters of bacteria by modelling the individual cell response to the external food gradients. We have found external conditions in which the bacteria perform simple diffusion with exponential jump-length distribution and also conditions under which the Levy flights with power-law jump length distribution are observed.
The second biological project was developing a mathematical model of the fission yeast cell growth. The polar growth of fission yeast cells is governed by the localised delivery of the protein Tea1 onto the membrane, where it can react with the freely diffusing protein Mod5p and form a protein complex. We have written down the reaction-diffusion equations for the distribution of the proteins along the membrane, solved and analysed them and showed that the growth can be effective only when the protein complex is immobilised by the membrane.
1a) The ground state structure of 2D systems: the energy of different crystal lattices in 2D will be compared, obtained by summing up effective pair interactions and by solving the full nonlinear Poisson-Boltzmann equation. Generally accepted opinion is that the hexagonal lattice is the ground state for any 2D system with long-range pair interactions. However, there have been studies, where lattices other than hexagonal were found to be stable in systems with finite-range interaction potentials. It is therefore expected that by summing up effective pair interactions we will always end up in the hexagonal ground state, while after including the many-body effects (Poisson-Boltzmann), other ground states might be found.
1b)The macroscopic properties of crystals: elastic constants, compressibility, phonon spectrum and other macroscopic properties will be studied, again within full many-body Poisson-Boltzmann theory and within the pair approximation. Since there are pronounced many-body contributions to the total interaction energy, the macroscopic quantities are also expected to deviate from the pairwise predictions. It is important to show how the macroscopic properties depend on state variables, where the pair description is successful and where it breaks down.
1c)Building up clusters: the idea is to build up small crystallites of 2, 3, 4, ... N particles and to analyse the many-body contributions to the total interaction energy. The three-body interaction has been calculated and measured this way and the four-body was measured as well, but from these studies it was difficult to predict how the many-body series continues. Are there cancellations and does the series converge? By studying the energy of clusters of different sizes the total many-body contribution as a function of number of particles can be obtained and the previous fundamental question can be answered. By doing the same with crystallites of different lattice symmetry we can possibly find some universal characteristics of many-body effects.
2. Colloids on substrates: this part of the project will be carried out in strong collaboration with the experimental group of Clemens Bechinger (Stuttgart) and Hans Hennig von Grünberg (Graz). The idea is to continue our work on domain formation in 2D colloidal systems on light substrates. By performing Brownian dynamics and Monte Carlo simulations we will be able to predict the phase diagram of colloidal monolayers on the periodic light substrate. We will focus on:
2a) Crystal melting: how well can the renowned Kosterlitz-Zhouless-Halperin-Nelson-Young74,75 theory describe the melting of colloidal crystals in 2D with and without substrate potential? Is the floating solid phase predicted by Halperin / Nelson75 stable as they predict? The above mentioned theory is fundamental and in the past very successfully applied, but has not yet been rigorously tested. Colloidal systems manipulated by laser substrate are an ideal model system for this task.
2b) Defects and their effect on the macroscopic properties: it is easy to introduce defects into the system, experimentally, as well as in the numerical simulations. The effect of defects on the melting scenario will be studied in detail.
2c) Nucleation and crystal growth: finally nucleation and crystal growth are going to be addressed. Playing with the substrate potential a nucleation seed can be creates and the process of nucleation observed under controlled conditions.
The group in Stuttgart will perform experiments with 2D colloids on a periodic laser substrate which will directly correspond to our numerical simulations.
3. Anisotropic interactions; the existing Poisson-Boltzmann code will be upgraded to study the systems with charge and shape anisotropy. One of the aims is to study the angular dependence of DNA-DNA interactions with realistic charge distribution. Generally, the many-body interactions in systems of non-spherical objects are an unexplored and exciting field which can be and will be studied in scope of the proposal.