Correction: Phase behavior of patchy colloids confined in patchy porous media.

Correction for 'Phase behavior of patchy colloids confined in patchy porous media' by Yurij V. Kalyuzhnyi et al., Nanoscale, 2024, 16, 4668-4677, https://doi.org/10.1039/D3NR02866F.

Phase behavior of patchy colloids confined in patchy porous media.

A simple model for functionalized disordered porous media is proposed and the effects of confinement on self-association, percolation and phase behavior of a fluid of patchy particles are studied. The media are formed by randomly distributed hard-sphere obstacles fixed in space and decorated by a certain number of off-center square-well sites. The properties of the fluid of patchy particles, represented by the fluid of hard spheres each bearing a set of the off-center square-well sites, are studied using an appropriate combination of the scaled particle theory for the porous media, Wertheim's thermodynamic perturbation theory, and Flory-Stockmayer theory. To assess the accuracy of the theory a set of computer simulations have been performed. In general, predictions of the theory appeared to be in good agreement with the computer simulation results. Confinement and competition between the formation of bonds connecting the fluid particles, and connecting fluid particles and obstacles of the matrix, gave rise to a re-entrant phase behavior with three critical points and two separate regions of the liquid-gas phase coexistence.

Protein Association in Solution: Statistical Mechanical Modeling.

Protein molecules associate in solution, often in clusters beyond pairwise, leading to liquid phase separations and high viscosities. It is often impractical to study these multi-protein systems by atomistic computer simulations, particularly in multi-component solvents. Instead, their forces and states can be studied by liquid state statistical mechanics. However, past such approaches, such as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, were limited to modeling proteins as spheres, and contained no microscopic structure-property relations. Recently, this limitation has been partly overcome by bringing the powerful Wertheim theory of associating molecules to bear on protein association equilibria. Here, we review these developments.

Behaviour of the model antibody fluid constrained by rigid spherical obstacles: Effects of the obstacle-antibody attraction.

This study investigates the behaviour of a fluid of monoclonal antibodies (mAbs) when trapped in a confinement represented by rigid spherical obstacles that attract antibodies. The antibody molecule is depicted as an assembly of seven hard spheres (7-bead model), organized to resemble a Y-shaped object. The model antibody has two Fab and one Fc domains located in the corners of letter Y. In this calculation, only the Fab-Fab and Fab-Fc attractive pairs of interactions are effective. The confinement is formed by the randomly distributed hard-spheres fixed in space. The spherical obstacles, besides the size exclusion, interact with beads of the antibody molecules via the Yukawa attractive potential. We applied the combination of the scaled particle theory, replica Ornstein-Zernike equations, Wertheim's thermodynamic perturbation approach and the Flory-Stockmayer theory to calculate: (i) the phase diagram of the liquid-liquid phase separation and the percolation threshold, (ii) the cluster size distributions, and (iii) the second virial coefficient of the protein fluid distributed among the obstacles. All these quantities were calculated as functions of the strength of the attraction between the monoclonal antibodies, and the monoclonal antibodies and obstacles. The conclusion is that while the hard-sphere obstacles decrease the critical density and the critical temperature of the mAbs fluid, the effect of the protein-obstacle attraction is more complex. Adding an attractive potential to the obstacle-mAbs interaction first increases the wideness of the T*-ρ envelope. However, with the further increase of the obstacle-mAbs attraction intensity, we observe reversal of the effect, the T*-ρ curves become narrower. At some point, depending on the obstacle-mAbs interaction, the situation is observed where two different temperatures have the same fluid density (re-entry point). In all the cases shown here the critical point decreases below the value for the neat fluid, but the behaviour with respect to an increase of the strength of the obstacle-mAbs attraction is not monotonic. Yet another interesting phenomenon, known in the literature as an approach toward the "empty liquid" state, is observed. The stability of the "protein droplets", formed by the liquid-liquid phase separation, depends on their local environment and temperature.

Integral equation theory for mixtures of spherical and patchy colloids. 2. Numerical results.

Thermodynamic properties and structure of binary mixtures of patchy and spherical colloids are studied using a recently developed theory [Y. V. Kalyuzhnyi, et al., Soft Matter, 2020, 16, 3456]. The theory is based on a solution of the multidensity Ornstein-Zernike equation and provides completely analytical expressions for the structure factors of these systems and for all their major thermodynamical quantities. The considered mixtures are made up of particles of different size and with a different number of patches. A set of molecular simulation data has been generated to enable a systematic comparison and to access thus accuracy of the theoretical predictions. In general, the predictions of the theory appear to be in good agreement with computer simulation data. For the models with a lower number of patches (np = 1, 2) the theoretical results show very good accuracy. Less accurate are the predictions for the four-patch versions of the model. While theoretical results for the radial distribution functions are, generally, relatively accurate for all the models, results for thermodynamics deteriorate with increasing concentration of the spherical colloids. Possible ways to improve the theory are briefly outlined.

Aggregation, liquid-liquid phase separation, and percolation behaviour of a model antibody fluid constrained by hard-sphere obstacles.

This study is concerned with the behaviour of proteins within confinement created by hard-sphere obstacles. An individual antibody molecule is depicted as an assembly of seven hard spheres, organized to resemble a Y-shaped (on average) antibody (7-bead model) protein. For comparison with other studies we, in one case, model the protein as a hard sphere decorated by three short-range attractive sites. The antibody has two Fab and one Fc domains located in the corners of the letter Y. In this calculation, only the Fab-Fab and Fab-Fc attractive pair interactions are possible. The confinement is formed by the randomly distributed hard-sphere obstacles fixed in space. Aside from size exclusion, the obstacles do not interact with antibodies, but they affect the protein-protein correlation. We used a combination of the scaled-particle theory, Wertheim's thermodynamic perturbation theory and the Flory-Stockmayer theory to calculate: (i) the second virial coefficient of the protein fluid, (ii) the percolation threshold, (iii) cluster size distributions, and (iv) the liquid-liquid phase separation as a function of the strength of the various pair interactions of the protein and the model parameters, such as protein concentration and the packing fraction of obstacles. The conclusion is that hard-sphere obstacles strongly decrease the critical density and also, but to a much lesser extent, the critical temperature. Also, the confinement enhances clustering, making the percolating region broader. The effect depends on the model parameters, such as the packing fraction of obstacles η0, the inter-site interaction strength εIJ, and the ratio between the size of the obstacle σ0 and the size of one bead of the model antibody σhs; the value of this ratio is varied here from 2 to 5. Interestingly, at low to moderate packing fractions of obstacles, the second virial coefficient first slightly decreases (destabilization), and the slope depends on the observation temperature, but then at higher values of η0 it increases. The calculated values of the second virial coefficient also depend on the size of the obstacles.

Integral equation theory for a mixture of spherical and patchy colloids: analytical description.

An analytic theory for the structure and thermodynamics of two-component mixtures of patchy and spherical colloids is developed. The theory is based on an analytical solution of the multidensity Ornstein-Zernike equation supplemented by the associative Percus-Yevick closure relations. We derive closed-form analytic expressions for the partial structure factors and thermodynamic properties using the energy route for the model with arbitrary number of patches and any hard-sphere size ratio of the particles. To assess the accuracy of the theoretical predictions we compare them against existing and newly generated set of computer simulation data. In our numerical calculations we consider the model with equal hard-sphere sizes and one patch. Very good agreement between results of the theory and simulation for the pair correlation functions, excess internal energy and pressure is observed for almost all values of the system density, temperature and composition studied. Only in the region of low concentrations of spherical colloids the theoretical results become less accurate.

Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in Random Porous Media: TPT of Wertheim and Scaled Particle Theory.

Extension of Wertheim's thermodynamic perturbation theory and its combination with scaled particle theory is proposed and applied to study the liquid-gas phase behavior of polydisperse hard-sphere square-well chain fluid confined in the random porous media. Thermodynamic properties of the reference system, represented by the hard-sphere square-well fluid in the matrix, are calculated using corresponding extension of the second-order Barker-Henderson perturbation theory. We study effects of polydispersity and confinement on the phase behavior of the system. While polydispersity causes increase of the region of phase coexistence due to the critical temperature increase, confinement decreases the values of both critical temperature and critical density making the region of phase coexistence smaller. This effect is enhanced with the increase of the size ratio of the fluid and matrix particles. The increase of the average chain length at fixed values of polydispersity and matrix density shifts the critical point to a higher temperature and a slightly lower density.

Melting upon cooling and freezing upon heating: fluid-solid phase diagram for Svejk-Hasek model of dimerizing hard spheres.

A simple model of dimerizing hard spheres with highly nontrivial fluid-solid phase behavior is proposed and studied using the recently proposed resummed thermodynamic perturbation theory for central force (RTPT-CF) associating potentials. The phase diagram has the fluid branch of the fluid-solid coexistence curve located at temperatures lower than those of the solid branch. This unusual behavior is related to the strong dependence of the system excluded volume on the temperature, which for the model at hand decreases with increasing temperature. This effect can be also seen for a wide family of fluid models with an effective interaction that combines short range attraction and repulsion at a larger distance. We expect that for sufficiently high repulsive barrier, such systems may show similar phase behavior.

Two- and three-phase equilibria of polydisperse Yukawa hard-sphere fluids confined in random porous media: high temperature approximation and scaled particle theory.

We have studied the phase behavior of polydisperse Yukawa hard-sphere fluid confined in random porous media using extension and combination of high temperature approximation and scaled particle theory. The porous media are represented by the matrix of randomly placed hard-sphere obstacles. Due to the confinement, polydispersity effects are substantially enhanced. At an intermediate degree of fluid polydispersity and low density of the matrix, we observe two-phase coexistence with two critical points, and cloud and shadow curves forming closed loops of ellipsoidal shape. With the increase of the matrix density and the constant degree of polydispersity, these two critical points merge and disappear, and at lower temperatures the system fractionates into three coexisting phases. A similar phase behavior was observed in the absence of the porous media caused, however, by the increase of the polydispersity.

Modeling phase transitions in mixtures of ß-γ lens crystallins.

We analyze the experimentally determined phase diagram of a γD-ßB1 crystallin mixture. Proteins are described as dumbbells decorated with attractive sites to allow inter-particle interaction. We use thermodynamic perturbation theory to calculate the free energy of such mixtures and, by applying equilibrium conditions, also the compositions and concentrations of the co-existing phases. Initially we fit the Tcloudversus packing fraction η measurements for a pure (x2 = 0) γD solution in 0.1 M phosphate buffer at pH = 7.0. Another piece of experimental data, used to fix the model parameters, is the isotherm x2vs. η at T = 268.5 K, at the same pH and salt content. We use the conventional Lorentz-Berthelot mixing rules to describe cross interactions. This enables us to determine: (i) model parameters for pure ßB1 crystallin protein and to calculate; (ii) complete equilibrium surface (Tcloud-x2-η) for the crystallin mixtures. (iii) We present the results for several isotherms, including the tie-lines, as also the temperature-packing fraction curves. Good agreement with the available experimental data is obtained. An interesting result of these calculations is evidence of the coexistence of three phases. This domain appears for the region of temperatures just out of the experimental range studied so far. The input parameters, leading good description of experimental data, revealed a large difference between the numbers of the attractive sites for γD and ßB1 proteins. This interesting result may be related to the fact that γD has a more than nine times smaller quadrupole moment than its partner in the mixture.

Protein aggregation in salt solutions.

Protein aggregation is broadly important in diseases and in formulations of biological drugs. Here, we develop a theoretical model for reversible protein-protein aggregation in salt solutions. We treat proteins as hard spheres having square-well-energy binding sites, using Wertheim's thermodynamic perturbation theory. The necessary condition required for such modeling to be realistic is that proteins in solution during the experiment remain in their compact form. Within this limitation our model gives accurate liquid-liquid coexistence curves for lysozyme and γ IIIa-crystallin solutions in respective buffers. It provides good fits to the cloud-point curves of lysozyme in buffer-salt mixtures as a function of the type and concentration of salt. It than predicts full coexistence curves, osmotic compressibilities, and second virial coefficients under such conditions. This treatment may also be relevant to protein crystallization.

Theoretical and numerical investigations of inverse patchy colloids in the fluid phase.

We investigate the structural and thermodynamic properties of a new class of patchy colloids, referred to as inverse patchy colloids (IPCs) in their fluid phase via both theoretical methods and simulations. IPCs are nano- or micro- meter sized particles with differently charged surface regions. We extend conventional integral equation schemes to this particular class of systems: our approach is based on the so-called multi-density Ornstein-Zernike equation, supplemented with the associative Percus-Yevick approximation (APY). To validate the accuracy of our framework, we compare the obtained results with data extracted from NpT and NVT Monte Carlo simulations. In addition, other theoretical approaches are used to calculate the properties of the system: the reference hypernetted-chain (RHNC) method and the Barker-Henderson thermodynamic perturbation theory. Both APY and RHNC frameworks provide accurate predictions for the pair distribution functions: APY results are in slightly better agreement with MC data, in particular at lower temperatures where the RHNC solution does not converge.

Solution of the mean spherical approximation for polydisperse multi-Yukawa hard-sphere fluid mixture using orthogonal polynomial expansions.

The Blum-Hoye [J. Stat. Phys. 19 317 (1978)] solution of the mean spherical approximation for a multicomponent multi-Yukawa hard-sphere fluid is extended to a polydisperse multi-Yukawa hard-sphere fluid. Our extension is based on the application of the orthogonal polynomial expansion method of Lado [Phys. Rev. E 54, 4411 (1996)]. Closed form analytical expressions for the structural and thermodynamic properties of the model are presented. They are given in terms of the parameters that follow directly from the solution. By way of illustration the method of solution is applied to describe the thermodynamic properties of the one- and two-Yukawa versions of the model.

Phase coexistence in a polydisperse charged hard-sphere fluid: polymer mean spherical approximation.

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.

Equation of state and liquid-vapor equilibria of one- and two-Yukawa hard-sphere chain fluids: theory and simulation.

The accuracy of several theories for the thermodynamic properties of the Yukawa hard-sphere chain fluid are studied. In particular, we consider the polymer mean spherical approximation (PMSA), the dimer version of thermodynamic perturbation theory (TPTD), and the statistical associating fluid theory for potentials of variable attractive range (SAFT-VR). Since the original version of SAFT-VR for Yukawa fluids is restricted to the case of one-Yukawa tail, we have extended SAFT-VR to treat chain fluids with two-Yukawa tails. The predictions of these theories are compared with Monte Carlo (MC) simulation data for the pressure and phase behavior of the chain fluid of different length with one- and two-Yukawa tails. We find that overall the PMSA and TPTD give more accurate predictions than SAFT-VR, and that the PMSA is slightly more accurate than TPTD.

Phase coexistence in polydisperse charged hard-sphere fluids: mean spherical approximation.

Taking advantage of the availability of the analytic solution of the mean spherical approximation for a mixture of charged hard spheres with an arbitrary number of components we show that the polydisperse fluid mixture of charged hard spheres belongs to the class of truncatable free energy models, i.e., to those systems where the thermodynamic properties can be represented by a finite number of (generalized) moments of the distribution function that characterizes the mixture. Thus, the formally infinitely many equations that determine the parameters of the two coexisting phases can be mapped onto a system of coupled nonlinear equations in these moments. We present the formalism and demonstrate the power of this approach for two systems; we calculate the full phase diagram in terms of cloud and shadow curves as well as binodals and discuss the distribution functions of the coexisting daughter phases and their charge distributions.