Origin of Anomalously Large Depletion Zones in Like-Charged Colloid-Polyelectrolyte Mixtures.

Depletion zones in polyelectrolyte solutions in contact with like-charged flat surfaces are investigated. Using a coupled self-consistent field and Debye-Hückel approach, an explicit expression for the thickness δ of the depletion layer is derived. It is found that δ∼δ_{n}+cκ^{-1}, where δ_{n} is the depletion thickness at a neutral surface, c is a function of the electrostatic characteristics of the system, and κ^{-1} is the Debye length. It is argued that the theory still holds beyond the mean-field approximation, which is confirmed by quantitative agreement between our theoretical results and experiments.

Multiphase Coexistence in Binary Hard Colloidal Mixtures: Predictions from a Simple Algebraic Theory.

A general theoretical framework is proposed to quantify the thermodynamic properties of multicomponent hard colloidal mixtures. This framework is used to predict the phase behavior of mixtures of rods with spheres and rods with plates taking into account (liquid) crystal phases of both components. We demonstrate a rich and complex range of phase behaviors featuring a large variety of different multiphase coexistence regions, including two five-phase coexistence regions for hard rod/sphere mixtures, and even a six-phase equilibrium for hard rod/plate dispersions. The various multiphase coexistences featured in a particular mixture are in line with a recently proposed generalized phase rule and can be tuned through subtle variations of the particle shape and size ratio. Our approach qualitatively accounts for certain multiphase equilibria observed in rod/plate mixtures of clay colloids and will be a useful guide in tuning the phase behavior of shape-disperse mixtures in general.

Depletion interaction mediated by semiflexible polymers.

We present a simple mean-field theory to describe the polymer-mediated depletion attraction between colloidal particles that accounts for the polymer's chain stiffness. We find that for fixed polymer radius of gyration and volume fraction, the strength of this attraction increases with increasing chain stiffness in both dilute and semidilute concentration regimes. In contrast, the range of attraction monotonically decreases with chain stiffness in the dilute regime, while it attains a maximum in the semidilute regime. The obtained analytical expressions for the depletion interaction were compared with numerical self-consistent field lattice computations and shown to be in quantitative agreement. From the interaction potential between two spheres, we calculated the second osmotic virial coefficient B2, which appears to be a convex function of chain stiffness. A minimum of B2 as a function of chain stiffness was observed both in the numerical self-consistent field computations and the analytical theory. These findings help explain the general observation that semiflexible polymers are more effective depletants than flexible polymers and give insight into the phase behavior of mixtures containing spherical colloids and semiflexible polymers.

The depletion thickness in solutions of semi-flexible polymers near colloidal surfaces: analytical approximations.

We derive a simple, yet accurate approximate mean-field expression for the depletion thickness δsf of a solution of dilute semi-flexible polymers next to a hard surface. In the case of a hard wall this equation has the simple form δsf = δ0[1 - tanh(psf/δ0)], where psf accounts for the degree of flexibility and δ0 is the depletion thickness in the case of fully flexible polymers. For fixed polymer coil size, increasing the chain stiffness leads to a decrease in the depletion thickness. The approach is also extended to include higher polymer concentrations in the semidilute regime. The analytical expressions are in quantitative agreement with numerical self-consistent field computations. A remarkable finding is that there is a maximum in the depletion thickness as a function of the chain stiffness in the semidilute concentration regime. This also means that depletion attractions between colloidal particles reach a maximum for a certain chain stiffness, which may have important implications for the phase stability of colloid-polymer mixtures. The derived equations could be useful for the description of interactions in- and phase stability of mixtures of colloids and semi-flexible polymers.

Phase stability of colloidal mixtures of spheres and rods.

We determined the phase boundaries of aqueous mixtures containing colloidal rod-like fd-viruses and polystyrene spheres using diffusing-wave spectroscopy and compared the results with free volume theory predictions. Excluded volume interactions in mixtures of colloidal rods and spheres lead to mediated depletion interactions. The strength and range of this attractive interaction depend on the concentrations of the particles, the length L and diameter D of the rods, and the radius R of the spheres. At strong enough attraction, this depletion interaction leads to phase separation. We experimentally determined the rod and sphere concentrations where these phase transitions occur by systematically varying the size ratios L/R and D/R and the aspect ratio L/D. This was done by using spheres with different radii and modifying the effective diameter of the rods through either the ionic strength of the buffer or anchoring a polymeric brush to the surface of the rods. The observed phase transitions were from a binary fluid to a colloidal gas/liquid phase coexistence that occurred already at very low concentrations due to the depletion efficiency of highly anisotropic rods. The experimentally measured phase transitions were compared to phase boundaries obtained using free volume theory (FVT), a well established theory for calculating the phase behavior of colloidal particles mixed with depletants. We find good correspondence between the experimental phase transitions and the theoretical FVT model where the excluded volume of the rod-like depletants was explicitly accounted for in both the reservoir and the system.

Phase separation in mixed suspensions of bacteria and nonadsorbing polymers.

The shapes of bacteria can vary widely; they may, for instance, be spherical, rod-like, string-like, or curved. In general, bacilli are highly anisotropic. For research and (bio)technological purposes, it can be useful to concentrate bacteria, which is possible by adding nonadsorbing polymers. The induced phase separation originates from a polymer-mediated depletion interaction, first understood by Asakura and Oosawa. Here, it is shown that free volume theory (FVT) can semi-quantitatively describe the phase transitions observed when adding sodium polystyrene sulfonate polymers to E. coli bacteria [Schwarz-Linek et al., Soft Matter 6, 4540 (2010)] at high ionic strength. The E. coli bacteria are described as short, hard spherocylinders. FVT predicts that the phase transitions of the mixtures result from a fluid-ABC crystal solid phase coexistence of a hard spherocylinder-polymer mixture.

Phase behavior of binary hard-sphere mixtures: Free volume theory including reservoir hard-core interactions.

Comprehensive calculations were performed to predict the phase behavior of large spherical colloids mixed with small spherical colloids that act as a depletant. To this end, the free volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20, 559 (1992)] is used as a basis and is extended to explicitly include the hard-sphere character of colloidal depletants into the expression for the free volume fraction. Taking the excluded volume of the depletants into account in both the system and the reservoir provides a relation between the depletant concentration in the reservoir and that in the system that accurately matches with computer simulation results of Dijkstra et al. [Phys. Rev. E 59, 5744 (1999)]. Moreover, the phase diagrams for highly asymmetric mixtures with size ratios q ≲ 0.2 obtained by using this new approach corroborate simulation results significantly better than earlier FVT applications to binary hard-sphere mixtures. The phase diagram of a binary hard-sphere mixture with a size ratio of q = 0.4, where a binary interstitial solid solution is formed at high densities, is investigated using a numerical free volume approach. At this size ratio, the obtained phase diagram is qualitatively different from previous FVT approaches for hard-sphere and penetrable depletants but again compares well with simulation predictions.

Defying the Gibbs Phase Rule: Evidence for an Entropy-Driven Quintuple Point in Colloid-Polymer Mixtures.

Using a minimal algebraic model for the thermodynamics of binary rod-polymer mixtures, we provide evidence for a quintuple phase equilibrium; an observation that seems to be at odds with the Gibbs phase rule for two-component systems. Our model is based on equations of state for the relevant liquid crystal phases that are in quantitative agreement with computer simulations. We argue that the appearance of a quintuple equilibrium, involving an isotropic fluid, a nematic and smectic liquid crystal, and two solid phases, can be reconciled with a generalized Gibbs phase rule in which the two intrinsic length scales of the athermal colloid-polymer mixture act as additional field variables.

Algebraic equations of state for the liquid crystalline phase behavior of hard rods.

Based on simplifications of previous numerical calculations [H. Graf and H. Löwen, Phys. Rev. E 59, 1932 (1999)1063-651X10.1103/PhysRevE.59.1932], we propose algebraic free energy expressions for the smectic-A liquid crystal phase and the crystal phases of hard spherocylinders. Quantitative agreement with simulations is found for the resulting equations of state. The free energy expressions can be used to straightforwardly compute the full phase behavior for all aspect ratios and to provide a suitable benchmark for exploring how attractive interrod interactions mediate the phase stability through perturbation approaches such as free-volume or van der Waals theory.

Phase stability of dispersions of hollow silica nanocubes mediated by non-adsorbing polymers.

Although there are theoretical predictions (Eur. Phys. J. E 41, 110 (2018)) for the rich-phase behaviour of colloidal cubes mixed with non-adsorbing polymers, a thorough verification of this phase behaviour is still underway; experimental studies on mixtures of cubes and non-adsorbing polymers in bulk are scarce. In this paper, mixtures of hollow silica nanocubes and linear polystyrene in N,-N-dimethylformamide are used to measure the structure factor of the colloidal cubes as a function of non-adsorbing polymer concentration. Together with visual observations these structure factors enabled us to assess the depletion-mediated phase stability of cube-polymer mixtures. The theoretical and experimental phase boundaries for cube-depletant mixtures are in remarkable agreement, despite the simplifications underlying the theory employed.

Scattering from colloidal cubic silica shells: Part II, static structure factors and osmotic equation of state.

HYPOTHESIS: The shape of colloidal particles affects the structure of colloidal dispersions. The effect of the cube shape on the thermodynamics of colloidal cube dispersions has not yet been studied experimentally. Static light scattering measurements on colloidal cubic silica shells at finite concentrations allows us to measure the structure factor of colloidal cube fluids and to test theoretical predictions for the equation of state of hard convex superballs. EXPERIMENTS: Hollow silica nanocubes of varying concentrations in N,N,-dimethylformamide were studied with static light scattering. The structure factor was extracted from the scattering curves using experimental form factors. From this experimental structure factor, the specific density of the particles, and the osmotic compressibility were obtained. This osmotic compressibility was then compared to a theoretical equation of state of hard superballs. FINDINGS: The first experimental structure factors of a stable cube fluid are presented. The osmotic compressibility of the cube fluid can be described by the equation of state of a hard superball fluid, showing that silica cubes in N,N,-dimethylformamide with LiCl effectively interact as hard particles.

Scattering from colloidal cubic silica shells: Part I, particle form factors and optical contrast variation.

HYPOTHESIS: Colloidal cubic silica shells, prepared from cuprous oxide cubes, with a typical size of 100 nm are promising model particles for scattering studies on dilute, as well as concentrated fluids, of non-spherical colloids. EXPERIMENTS: Small angle X-ray scattering, and static light scattering are employed to determine form factors of cubic silica shells and silica covered cuprous oxide cubes. Contrast variation experiments are performed to assess the refractive index and optical homogeneity of the cubic silica shells, which is important for the extension of the scattering study to concentrated dispersions of cubic shells in Part II (Dekker, submitted for publication). RESULTS: The experimental form factors, which compare well to theoretical form factors, manifest cubic silica shells that are dispersed as single stable colloids with a shape intermediate between a sphere and a perfect cube. Contrast variation demonstrates that the silica shells are optically homogeneous, with a refractive index that is independent of the shell thickness. The results presented here open up the possibility to extract structure factors from light scattering measurements on concentrated cube dispersions in Part II.

Entropic patchiness drives multi-phase coexistence in discotic colloid-depletant mixtures.

Entropy-driven equilibrium phase behaviour of hard particle dispersions can be understood from excluded volume arguments only. While monodisperse hard spheres only exhibit a fluid-solid phase transition, anisotropic hard particles such as rods, discs, cuboids or boards exhibit various multi-phase equilibria. Ordering of such anisotropic particles increases the free volume entropy by reducing the excluded volume between them. The addition of depletants gives rise to an entropic patchiness represented by orientation-dependent attractions resulting in non-trivial phase behaviour. We show that free volume theory is a simple, generic and tractable framework that enables to incorporate these effects and rationalise various experimental findings. Plate-shaped particles constitute the main building blocks of clays, asphaltenes and chromonic liquid crystals that find widespread use in the food, cosmetics and oil industry. We demonstrate that mixtures of platelets and ideal depletants exhibit a strikingly rich phase behaviour containing several types of three-phase coexistence areas and even a quadruple region with four coexisting phases.

Phase behaviour of colloids plus weakly adhesive polymers.

The phase behaviour of a colloidal dispersion mediated by weakly adhesive polymers is considered. The polymers are depleted but are weakly adhesive and hence comprise a non-zero polymer concentration at the colloid's surface, in contrast to the classical assumption in depletion theories involving a zero polymer concentration at the surface. The theory is composed of a generalized free-volume theory for colloid-polymer mixtures and a self-consistent mean-field theory for polymers at surfaces. It is found that the weak adhesion of the polymers shifts the phase stability of the colloid-polymer mixtures to higher polymer concentrations as compared to assuming a full depletion effect. The predicted phase diagrams employing the new theory are consistent with experiments on mixtures of silica spheres coated with stearyl alcohol and polydimethylsiloxane in cyclohexane and with Monte Carlo simulation results.

Controlled nanoparticle formation by diffusion limited coalescence.

Polymeric nanoparticles (NPs) have great application potential in science and technology. Their functionality strongly depends on their size. We present a theory for the size of NPs formed by precipitation of polymers into a bad solvent in the presence of a stabilizing surfactant. The analytical theory is based upon diffusion-limited coalescence kinetics of the polymers. Two relevant time scales, a mixing and a coalescence time, are identified and their ratio is shown to determine the final NP diameter. The size is found to scale in a universal manner and is predominantly sensitive to the mixing time and the polymer concentration if the surfactant concentration is sufficiently high. The model predictions are in good agreement with experimental data. Hence the theory provides a solid framework for tailoring NPs with a priori determined size.

How flow changes polymer depletion in a slit.

A theoretical model is developed for predicting dynamic polymer depletion under the influence of fluid flow. The results are established by combining the two-fluid model and the self-consistent field theory. We consider a uniform fluid flow across a slit containing a solution with polymer chains. The two parallel and infinitely long walls are permeable to solvent only and the polymers do not adsorb to these walls. For a weak flow and a narrow slit, an analytic expression is derived to describe the steady-state polymer concentration profiles in a Θ-solvent. In both Θ- and good-solvents, we compute the time evolution of the concentration profiles for various flow rates characterized by the Peclet number. The model reveals the interplay of depletion, solvent condition, slit width, and the relative strength of the fluid flow.

Interfacial tension between benzene and water in the presence of caprolactam.

We studied the physical properties and the concentration profile of benzene+water+caprolactam mixtures near the fluid-fluid interface using self-consistent field (SCF) theory. This yields the interfacial tension which plays an important role in describing the stability of transient liquid droplets of one phase in the other. The studies were performed at a fixed temperature of 313K. Flory-Huggins binary interaction parameters and the compound lattice segment numbers are input parameters for the applied SCF theory. These parameters were derived from activity coefficient relations, which are used to describe experimental liquid-liquid and vapor-liquid phase equilibrium measurements. Using first principles, the benzene-water interface was studied and the resulting interfacial tension was found to be in agreement with experimental values. This study illustrates that caprolactam accumulates at the benzene-water interface, acting as a weak surfactant. The interfacial tension is also demonstrated to be affected by the caprolactam concentration and the SCF results are in fair agreement with the experimental observations.