Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

The condensation and ordering of models of empty liquids

We consider a simple model consisting of particles with four bonding sites ("patches"), two of type A and two of type B, on the square lattice, and investigate its global phase behavior by simulations and theory. We set the interaction between B patches to zero and calculate the phase diagram as the ratio between the AB and the AA interactions, epsilon(AB)*, varies. In line with previous work, on three-dimensional off-lattice models, we show that the liquid-vapor phase diagram exhibits a re-entrant or "pinched" shape for the same range of epsilon(AB)*, suggesting that the ratio of the energy scales - and the corresponding empty fluid regime - is independent of the dimensionality of the system and of the lattice structure. In addition, the model exhibits an order-disorder transition that is ferromagnetic in the re-entrant regime. The use of low-dimensional lattice models allows the simulation of sufficiently large systems to establish the nature of the liquid-vapor critical points and to describe the structure of the liquid phase in the empty fluid regime, where the size of the "voids" increases as the temperature decreases. We have found that the liquid-vapor critical point is in the 2D Ising universality class, with a scaling region that decreases rapidly as the temperature decreases. The results of simulations and theoretical analysis suggest that the line of order-disorder transitions intersects the condensation line at a multi-critical point at zero temperature and density, for patchy particle models with a re-entrant, empty fluid, regime. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3657406]

Three-dimensional patchy lattice model for empty fluids

The phase diagram of a simple model with two patches of type A and ten patches of type B (2A10B) on the face centred cubic lattice has been calculated by simulations and theory. Assuming that there is no interaction between the B patches the behavior of the system can be described in terms of the ratio of the AB and AA interactions, r. Our results show that, similarly to what happens for related off-lattice and two-dimensional lattice models, the liquid-vapor phase equilibria exhibit reentrant behavior for some values of the interaction parameters. However, for the model studied here the liquid-vapor phase equilibria occur for values of r lower than 1/3, a threshold value which was previously thought to be universal for 2AnB models. In addition, the theory predicts that below r = 1/3 (and above a new condensation threshold which is < 1/3) the reentrant liquid-vapor equilibria are so extreme that it exhibits a closed loop with a lower critical point, a very unusual behavior in single-component systems. An order-disorder transition is also observed at higher densities than the liquid-vapor equilibria, which shows that the liquid-vapor reentrancy occurs in an equilibrium region of the phase diagram. These findings may have implications in the understanding of the condensation of dipolar hard spheres given the analogy between that system and the 2AnB models considered here. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4771591]

Harvesting dynamic aspects from a real rational map : the bulb of period 5

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.

Mapping stability : real rational maps of degree zero

In memory of our beloved Professor José Rodrigues Santos de Sousa Ramos (1948-2007), who João Cabral, one of the authors of this paper, had the honor of being his student between 2000 and 2006, we wrote this paper following the research by experimentation, using the new technologies to capture a new insight about a problem, as him so much love to do it. His passion was to create new relations between different fields of mathematics. He was a builder of bridges of knowledge, encouraging the birth of new ways to understand this science. One of the areas that Sousa Ramos researched was the iteration of maps and the description of its behavior, using the symbolic dynamics. So, in this issue of this journal, honoring his memory, we use experimental results to find some stable regions of a specific family of real rational maps, the ones that he worked with João Cabral. In this paper we describe a parameter space (a,b) to the real rational maps fa,b(x) = (x2 −a)/(x2 −b), using some tools of dynamical systems, as the study of the critical point orbit and Lyapunov exponents. We give some results regarding the stability of these family of maps when we iterate it, specially the ones connected to the order 3 of iteration. We hope that our results would help to understand better the behavior of these maps, preparing the ground to a more efficient use of the Kneading Theory on these family of maps, using symbolic dynamics.

How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particles

We investigate the influence of strong directional, or bonding, interactions on the phase diagram of complex fluids, and in particular on the liquid-vapour critical point. To this end we revisit a simple model and theory for associating fluids which consist of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the interactions between each pair of spots have strengths [image omitted], [image omitted] and [image omitted]. The theory is applied over the whole range of bonding strengths and results are interpreted in terms of the equilibrium cluster structures of the coexisting phases. In systems where unlike sites do not interact (i.e. where [image omitted]), the critical point exists all the way to [image omitted]. By contrast, when [image omitted], there is no critical point below a certain finite value of [image omitted]. These somewhat surprising results are rationalised in terms of the different network structures of the two systems: two long AA chains are linked by one BB bond (X-junction) in the former case, and by one AB bond (Y-junction) in the latter. The vapour-liquid transition may then be viewed as the condensation of these junctions and we find that X-junctions condense for any attractive [image omitted] (i.e. for any fraction of BB bonds), whereas condensation of the Y-junctions requires that [image omitted] be above a finite threshold (i.e. there must be a finite fraction of AB bonds).

Criticality of colloids with distinct interaction patches: The limits of linear chains, hyperbranched polymers, and dimers

We use a simple model of associating fluids which consists of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the bonding interactions between each pair of spots have strengths epsilon(AA), epsilon(BB), and epsilon(AB). The theory is applied over the whole range of bonding strengths and the results are interpreted in terms of the equilibrium cluster structures of the phases. In addition to our numerical results, we derive asymptotic expansions for the free energy in the limits for which there is no liquid-vapor critical point: linear chains (epsilon(AA)not equal 0, epsilon(AB)=epsilon(BB)=0), hyperbranched polymers (epsilon(AB)not equal 0, epsilon(AA)=epsilon(BB)=0), and dimers (epsilon(BB)not equal 0, epsilon(AA)=epsilon(AB)=0). These expansions also allow us to calculate the structure of the critical fluid by perturbing around the above limits, yielding three different types of condensation: of linear chains (AA clusters connected by a few AB or BB bonds); of hyperbranched polymers (AB clusters connected by AA bonds); or of dimers (BB clusters connected by AA bonds). Interestingly, there is no critical point when epsilon(AA) vanishes despite the fact that AA bonds alone cannot drive condensation.

Three-dimensial patchy lattice model: Ring formation and phase separation

We investigate the structural and thermodynamic properties of a model of particles with 2 patches of type A and 10 patches of type B. Particles are placed on the sites of a face centered cubic lattice with the patches oriented along the nearest neighbor directions. The competition between the self- assembly of chains, rings, and networks on the phase diagram is investigated by carrying out a systematic investigation of this class of models, using an extension ofWertheim's theory for associating fluids and Monte Carlo numerical simulations. We varied the ratio r epsilon(AB)/epsilon(AA) of the interaction between patches A and B, epsilon(AB), and between A patches, epsilon(AA) (epsilon(BB) is set to theta) as well as the relative position of the A patches, i.e., the angle. between the (lattice) directions of the A patches. We found that both r and theta (60 degrees, 90 degrees, or 120 degrees) have a profound effect on the phase diagram. In the empty fluid regime (r < 1/2) the phase diagram is reentrant with a closed miscibility loop. The region around the lower critical point exhibits unusual structural and thermodynamic behavior determined by the presence of relatively short rings. The agreement between the results of theory and simulation is excellent for theta = 120 degrees but deteriorates as. decreases, revealing the need for new theoretical approaches to describe the structure and thermodynamics of systems dominated by small rings. (C) 2014 AIP Publishing LLC.

Three-Dimensional patchy lattice model for empty fluids

The phase diagram of a simple model with two patches of type A and ten patches of type B (2A10B) on the face centred cubic lattice has been calculated by simulations and theory. Assuming that there is no interaction between the B patches the behavior of the system can be described in terms of the ratio of the AB and AA interactions, r. Our results show that, similarly to what happens for related off-lattice and two-dimensional lattice models, the liquid-vapor phase equilibria exhibit reentrant behavior for some values of the interaction parameters. However, for the model studied here the liquid-vapor phase equilibria occur for values of r lower than 1/3, a threshold value which was previously thought to be universal for 2AnB models. In addition, the theory predicts that below r = 1/3 (and above a new condensation threshold which is < 1/3) the reentrant liquid-vapor equilibria are so extreme that it exhibits a closed loop with a lower critical point, a very unusual behavior in single-component systems. An order-disorder transition is also observed at higher densities than the liquid-vapor equilibria, which shows that the liquid-vapor reentrancy occurs in an equilibrium region of the phase diagram. These findings may have implications in the understanding of the condensation of dipolar hard spheres given the analogy between that system and the 2AnB models considered here.

Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawlucki's extension of the Puiseuxlemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

A quantum molecular similarity analysis of changes in molecular electron density caused by basis set flotation and electric field application

Quantum molecular similarity (QMS) techniques are used to assess the response of the electron density of various small molecules to application of a static, uniform electric field. Likewise, QMS is used to analyze the changes in electron density generated by the process of floating a basis set. The results obtained show an interrelation between the floating process, the optimum geometry, and the presence of an external field. Cases involving the Le Chatelier principle are discussed, and an insight on the changes of bond critical point properties, self-similarity values and density differences is performed

Liquid-gas phase transition in nuclear matter from realistic many-body approaches

The existence of a liquid-gas phase transition for hot nuclear systems at subsaturation densities is a well-established prediction of finite-temperature nuclear many-body theory. In this paper, we discuss for the first time the properties of such a phase transition for homogeneous nuclear matter within the self-consistent Green's function approach. We find a substantial decrease of the critical temperature with respect to the Brueckner-Hartree-Fock approximation. Even within the same approximation, the use of two different realistic nucleon-nucleon interactions gives rise to large differences in the properties of the critical point.

Intramolecular coupling as a mechanism for a liquid-liquid phase transition

We study a model for water with a tunable intramolecular interaction Js, using mean-field theory and off-lattice Monte Carlo simulations. For all Js>~0, the model displays a temperature of maximum density. For a finite intramolecular interaction Js>0, our calculations support the presence of a liquid-liquid phase transition with a possible liquid-liquid critical point for water, likely preempted by inevitable freezing. For J=0, the liquid-liquid critical point disappears at T=0.

Critical behavior of a water monolayer under hydrophobic confinement

The properties of water can have a strong dependence on the confinement. Here, we consider a water monolayer nanoconfined between hydrophobic parallel walls under conditions that prevent its crystallization. We investigate, by simulations of a many-body coarse-grained water model, how the properties of the liquid are affected by the confinement. We show, by studying the response functions and the correlation length and by performing finite-size scaling of the appropriate order parameter, that at low temperature the monolayer undergoes a liquid-liquid phase transition ending in a critical point in the universality class of the two-dimensional (2D) Ising model. Surprisingly, by reducing the linear size L of the walls, keeping the walls separation h constant, we find a 2D-3D crossover for the universality class of the liquid-liquid critical point for L/h=~50, i.e. for a monolayer thickness that is small compared to its extension. This result is drastically different from what is reported for simple liquids, where the crossover occurs for , and is consistent with experimental results and atomistic simulations. We shed light on these findings showing that they are a consequence of the strong cooperativity and the low coordination number of the hydrogen bond network that characterizes water.

Onko yrityksen taloudellisella tilalla vaikutusta Tekes-rahoitusta saaneen yrityksen menestymiseen

Tämän tutkielman tavoite on yrityksen tulevan menestymisen tai epäonnistumisen ennustaminen. Tutkimuksen kohteena ovat tutkimus ja kehitys rahoitusta vuosina 2007- 2012 Tekesiltä saaneet mikro- ja pk yritykset. Ennustusta pyrittiin tekemään näille yrityksille niiden rahoitusprojektien alkuhetkeltä (hakemushetki). Tutkimus toteutettiin toimeksiantona Tekesille ja se jatkaa aikaisempaa tutkielmaa, ”Onko projektin henkilöriskitasolla tai talousriskitasolla yhteyttä t&k- projektin onnistumiseen?”. Tämän tutkimuksen johtopäätös oli, ettei varsinkaan talousriskillä ja projektin onnistumisella ole yhteyttä ja sen vuoksi niistä tarvittiin lisää tietoa. Teoreettisessa osuudessa käsitellään yrityksen kasvua ja onnistumisen tai epäonnistumisen ennakointia. Teoria pohjautuu aiheen aikaisempaan kirjallisuuteen ja sen tarkoitus on pohjustaa tutkielmassa tehtyjä valintoja yrityksien menestymisen ennustamiseksi. Tutkimus on toteutettu kvantitatiivisena tutkimuksena. Aineisto koostuu 430 Tekesin t&k- rahoitusta saaneista mikro- ja pk yrityksistä. Empiirisen osuuden tarkoitus oli selvittää, voidaanko koko yrityksen, menestystä ennustaa, jotta rahoitus voidaan kohdistaa paremmin, eli selvittää minkälaisia yrityksiä tulisi rahoittaa, jotta rahoitus kohdistuisi menestyjille. Tämän lisäksi tutkimuksessa haluttiin löytää oikeat muuttujat, joiden avulla yrityksen menestymisen ennustaminen on mahdollista. Menestymisen haluttiin kuvaavan koko yrityksen liiketoiminnan menestymistä. Menestymisen mittariksi valittiin liikevaihdon kasvu ja tarkasteltavaksi ajanjaksoksi aika yrityksen t&k- projektin hakemusvaiheesta, kolmen vuoden päähän projektin päättymisestä. Yritykset jaettiin menestyjiin liikevaihdon kasvun mukaan, ja niistä luotiin neljä koria: “Huiput”, “Kasvut”, “Hiipujat” ja “Konkurssit” ja koreja lähdettiin tutkimaan erilaisten mittarien avulla. Valitut mittarit olivat Tekesin riskiarvio (talousriski, henkilöriski, kehitysriski ja markkinariski), tilinpäätöksen tunnusluvuista kannattavuuden, maksuvalmiuden ja vakavaraisuuden tunnusluvut (liikevoittoprosentti ja sijoitetun pääoman tuottoprosentti, quick ratio ja omavaraisuusaste), projektin omarahoitus (yrityksen hakemusvaiheessa ilmoittama muun rahoituksen lähde, joka voi olla vieraanpääoman ehtoista, oman pääomanehtoista tai tulorahoitusta), sekä muut mittarit (uutuusarvo, jalostusarvo, tuottavuus ja lainojen perimättäjättöhakemukset). Riippumattomuutta testattiin ristiintaulukoinnilla, khiin neliötestillä ja kontingenssikerroin C:n avulla omavaraisuusasteen tunnusluvun osalta yhdistämällä ”Huiput”, ”Kasvut” ja ”Hiipujat” yhteen kotiin ja vertaamalla niitä konkurssiin menneisiin yrityksiin. Lopuksi konkurssiennusteista testattiin pienille yrityksille tarkoitettua Laitisen yhdistelmälukua. Tutkimuksessa havaittiin seuraavaa. Parhaiten yrityksen tulevan kasvun tai konkurssin ennakoivat havainnot seuraavissa muuttujissa: onko hanke rahoitettu lainalla vai avustuksella sekä mikä yrityksen hakemusvaiheen omavaraisuusaste, omarahoitusosuus tai tuottavuus on. Yrityksen hyvä rahatilanne projektin hakemusvaiheessa ei ennusta yrityksen liikevaihdon kasvua. Täydellistä ennustetta kasvulle ei voida tehdä. Laitisen yhdistelmäluku koettiin heikoksi tunnusluvuksi konkurssia ennustettaessa. Tämän takia, sitä muokattiin eri tavoin parhaan mahdollisemman ennusteen saamiseksi. Paras lopputulos saatiin muuttamalla tunnusluvun kriittistä arvoa pienemmäksi.