# Titelaufnahme

## EnglishWhat makes a liquid a liquid? This question can be answered by the van der Waals theory, which states that attraction between molecular particles is necessary for a gas-liquid phase transition [1]. So let us take an example, the well-known Lennard-Jones potential [2], due to the r−6 dispersion attraction, the Lennard-Jones model exhibits a gas-liquid phase transition (e.g. Ref. [3, 4]) and can in the meantime be considered as a kind of warm-up exercise for anyone getting started with computer simulations in the field of soft matter physics. But what about polar models, where the attractive interaction is not as apparent as in the case of the Lennard-Jones model? Or, in the words of van Leeuwen and Smit "What makes a polar liquid a liquid" [5]? The interaction of two point dipoles can be attractive or repulsive, depending on their orientation to each other. For a gas of dipole moments μ it can be shown that the thermal average at temperature T results in an overall attractive dipole-dipole [6]. This indicates the existence of gas-liquid coexistence. But the aforementioned conclusion is not as clear as it looks. The existence of gas-liquid coexistence in dipolar fluids with no other contribution to attractive interaction than dipole-dipole interaction is a basic and open question in the theory of fluids and has created significant interest. It is perhaps comparable to the freezing transition in the hard sphere fluid discovered in early computer simulation work by Alder and Wainwright in the late fifties [7]. In the studies of dipolar fluid systems, two simple and often used representations are the dipolar soft sphere model, where the dispersion attraction in the Lennard-Jones potential is replaced by the interaction energy of a point dipole, and the dipolar hard sphere model, where the soft sphere repulsion in the dipolar soft sphere model is replaced by a spherical hard core potential [8–11]. About twenty years ago,a number of theoretical papers concluded the absence of a gas-liquid critical point for those systems due to the formation of reversible chains, which essentially possess no mutual interaction [12–15]. Those theoretical results are supported by different computer simulations done for dipolar systems. Dipolar hard spheres were analyzed for a gas-liquid phase separation by Calliol [16] using Monte Carlo simulations in the isothermal-isobaric ensemble (NPT ensemble, constant particle number N , pressure P , and temperature T) and in the Gibbs ensemble. He came to the conclusion that in the analyzed region of the phase diagram no gas-liquid transition exists. This is in contradiction to an earlier work of Ng et al. [17], who observed a gas-liquid transition for a system of 32 dipolar hard spheres using Monte Carlo simulations. Van Leeuwen and Smit [5] introduced a factor λ to allow an independent control of the dispersion attraction from the dipolar interaction in the Stockmayer fluid (Lennard-Jones plus dipole-dipole potential). They studied a range of λ with the focus on the limit λ → 0, where the system is equivalent to dipolar soft spheres. For λ < 0.3 they could not detect gas-liquid criticality in their simulations due to chain formation, which is not observed for higher λ. The pure dipolar soft sphere fluid in an applied field was studied by Stevens and Grest [18] using Gibbs ensemble Monte Carlo. They determined the critical parameters as a function of the applied field and conclude in the zero field case that coexistence most likely does not occur. Thermodynamics and structural properties of the dipolar Yukawa hard sphere fluid were studied by Szalai et al. [19]. They found that at high dipole moments the gas-liquid phase transition disappears, while chainlike structures appear in the low-density fluid phase. More recent molecular dynamics simulations of the Stockmayer fluid performed by Bartke and Hentschke [20, 21], which can be mapped onto the system studied by van Leeuwen and Smit, show that the threshold found in Ref. [5] does not exist. The conclusion that chain formation indeed leads to the disappearance of gas-liquid criticality in the dipolar soft sphere limit still appears to be valid. Sindt and Camp used an effective many-body isotropic interaction able to mimic the dipolar soft sphere model and clearly linked the transition disappearance to chain formation [22]. Rovigatti et al. [23, 24] observes self-assembly of dipolar hard spheres into closed rings and concludes that this excludes the possibility of critical phenomena. In a further study on a model of particles with dissimilar patches they support the conclusion that the absence of gas-liquid separation in DHS could be a consequence of extensive ring formation [25]. To shed more light on the gas-liquid transition debate they suggest further studies on the structural transition from chains to branched network for the dipolar hard sphere model [26]. Again, ring formation suggests a similar result from Rovigatti and Dussi et al. [27] where they investigated the charged soft dumbbell model we also use in our work. The charged soft dumbbell model consists of two oppositely charged soft sphere sites, displaced through an axis of length d, and allows to approach the dipolar soft sphere limit d → 0. In their Monte Carlo simulations they were able to locate the gas-liquid critical point for dumbbells, but they were not able to track it down to the smallest dumbbell length due to the formation of rings. Despite the abundant evidence for the absence of a gas-liquid phase transition for purely dipolar systems, there are also researchers who come to the opposite conclusion. First of all, the already mentioned study by Ng et al. [17], who found gas-liquid coexistence for the dipolar hard sphere system. More than twenty years later, McGrother and Jackson [28] induced gas-liquid coexistence in a system of nonspherical hard-core dipolar molecules, i.e. they consider hard spherocylinders with central longitudinal point dipole moments. By using Monte Carlo simulations to calculate the equation of state and the free energy of dipolar hard sphere fluids at low temperatures and densities, Camp et al. [29] obtained evidence in favor of an isotropic fluid-to-isotropic fluid phase transition. Pshenichnikov and Mekhonoshin [30] utilized the Monte Carlo method to simulate dipolar hard sphere with open boundaries. They applied an extra field which confines the particles to a spherical region and observed a gas-like distribution within this region, or a pronounced clustering, depending on the strength of dipolar interaction. They interpret this as an indication for phase separation in the dipolar hard sphere bulk system. Ganzenmüller and Camp [31] used a fluid of charged hard dumbbells, each made up of two oppositely charged hard spheres, separated by the distance d, to track the gas-liquid coexistence towards the dipolar hard sphere limit d → 0. Via extrapolation of their grand-canonical Monte Carlo results obtained for finite dumbbell length, they found a gas-liquid critical point in the dipolar hard sphere limit. Almarza et al. [32] confirm the results of Ganzenmüller and Camp [31] using Monte Carlo to analyze a mixture of hard spheres and dipolar hard sphere. The critical parameters for the gas-liquid equilibrium, extrapolated based on their mixture results to the limit of vanishing neutral hard sphere concentration, are in accord with the extrapolation for the dumbbells approaching the dipolar hard sphere limit. Kalyuzhnyi et al. [33] examine the phase behaviour of the dipolar Yukawa hard sphere fluid using Monte Carlo simulations. Again the critical point may be tracked as the dipolar hard sphere limit is approached by decreasing the strength of the attractive Yukawa potential. They find a critical point for values of the Yukawa potential well depth which is used as control parameter and representing the ”distance” from the dipolar hard sphere limit, which are far lower than the limit set by the earlier study by Szalai et al. [19]. Continuation of this work in Ref. [34], however, results in the conclusion that phase separation is not observable beyond a critical value of the Yukawa energy parameter. Similar to the aforementioned work by Stevens and Grest [18], our colleague Jia [35] tracked the critical point for the dipolar soft sphere model as a parameter of the field strength and found critical values in the limit of vanishing field. The results were also confirmed as part of this work at zero field strength. A graphical representation to summarize the results on the localization of the gas-liquid critical point on dipolar systems is shown in Figure 1.1. Based on the above-mentioned results, the assembly of single dipolar particles into reversible dipole chains or other aggregates like rings and branched structures, has a significant effect on the gas-liquid coexistence of dipolar systems (an interesting study of gas-liquid phase separation in systems composed of rings and chains was presented by Pam et al. [36]). Therefore, different approaches, such as an external field or modifications to the dipolar potential, were introduced in several aforementioned studies with the main target to prevent or somehow control the assembly reversible aggregates. Also, those long chains are difficult to equilibrate in computer simulations and also hard to handle analytically. The original notion of gas-liquid phase separation of monomers is generalized to include phase separation in systems of polydisperse reversible aggregates, or even other types of transitions like the demixing of topological defects [37]. In this work, we present another attempt to track the gas-liquid critical point in a system approaching the dipolar soft sphere limit. We study a system of charged soft dumbbells using the molecular dynamics technique. The charge-to-charge separation on the dumbbell d can be controlled and we start our simulations in an“easy regime”, where gas-liquid phase separation is readily observed and the assembly of chains is energetically unfavourable. The charge-to-charge separation, d, is systematically decreased, keeping the dumbbell dipole moment, respectively the dumbbell charges, constant. Using this procedure we can approach the dipolar soft sphere limit in such a way that difficulties like the reversible association of particles develop “slowly”, which allows the extrapolation to the desired limit. Even at the smallest d (= 0.0001), we do observe a transition terminating in a critical point, which suggests a gas-liquid critical point in the dipolar soft sphere limit. This result is in accordance with the previous simulation study by Ganzenmüller and Camp [31] for dipolar hard spheres. To explain our simulation results, we apply different simple models. However, neither the extension of Flory’s lattice theory to reversibly aggregating polymers [38, 39], nor the defect model put forward by Tlusty and Safran [37], yields a consistent description of the simulation results. Only the developed van der Waals mean field theory provides a close to quantitative description of the critical parameters obtained from the simulation. The theory combines the Onsager approach to dipolar liquids [40] with the idea that the basic unit is not the single dipole, but rather a small reversible aggregate. To support the results for charged soft dumbbell in the dipole limit, i.e. d → 0, we also conduct simulations for the dipolar soft sphere model in its pure form and with an additional parameter which controls the soft sphere repulsion. We see that the soft repulsive interaction can be used as an effective means for limiting the aggregation in the dipole limit. The resulting short reversible chains, which are easy to equilibrate, do exhibit gas-liquid phase separation. Subsequently to the investigation at the dipole limit we study the gas-liquid phase transition in the dumbbell system for d >> 1 and observe critical parameters in the whole simulated range, i.e. up to d = 7. On the basis of the aforementioned mean field theory, we construct a qualitative description of the critical behavior in the range of 1 < d < 7. In this range it seems likely that the charged soft dumbbell model behaves similar to an ionic system and therefore relates to the restricted primitive model of ionic liquids. The restricted primitive model is also a model whose gas-liquid transition has attracted interest both from the analytical (e.g., Refs. [41, 42]) as well as from the simulation side (e.g., Refs. [43, 44]). In order to connect to the restricted primitive model results, we simulate an ionic model, where we have eliminated the rigid dumbbell bond. Thus the dumbbell model becomes equal to the restricted primitive model, except that the hard core repulsion is replaced by soft sphere repulsion. We also observe a gas-liquid critical point for this model and discuss the relation to the charged soft dumbbell system, as well as to the numerical predictions of the Debye-Hückel theory. In addition to the investigation of the phase transition, we have a look at the cation-anion pairing in terms of Bjerrum association in our ionic model, which serves as an important basic model in the interpretation of experimental data obtained from colloidal suspensions, liquid salts, and salt solutions [45]. Finally, we make additional use of the expertise gained in the course of our work on the phase behavior of charged and dipolar systems in the context of osmotic equilibria driven by Coulomb interactions. Specifically, we extend previous Monte Carlo simulations of osmotic equilibria in Lennard-Jones systems by Schreiber in our work group [46, 47]. We employ the Monte Carlo algorithm developed by Schreiber and Hentschke and apply it to an ionic solute in a dipolar solvent. The algorithm allows us to perform Monte Carlo simulations in the T pπ ensemble, with the temperature T , the external pressure p, and the osmotic pressure π across the membrane. Thus we can study the dependency of the solute concentration on osmotic pressure and vice versa, which can then be compared with experimental data. The detailed study of osmosis is also of special interest due to the multitude of technical and biological problems connected with it. The present work is structured as follows. An introduction to the simulation techniques used here is given in chapter 2. Chapter 3 contains the description of the various model systems studied using the aforementioned techniques. In the subsequent three chapters the results of the charged soft sphere model (chapter 4), the dipolar soft sphere model (chapter 5), and the ionic soft sphere model (chapter 6) are presented and discussed. Chapter 7 is an introduction to the topic of osmotic pressure followed by our numerical results. The last chapter provides a summary of this work and is followed by the acknowledgement and appendix. |