Depletion Interaction Research Articles

Self-stratification and phase separation in drying binary colloidal films

HypothesisFilms that develop compositional heterogeneity during drying offer a promising approach for achieving tailored functionalities. These functionalities can be realized by strategically directing different components during the drying process. One approach to achieve this is through spontaneous size segregation of colloidal particles. Two variants thereof have previously been observed in binary suspensions: layer formation (self-stratification) due to kinetically driven concentration gradients, and micro-domain formation (phase separation) due to thermodynamic depletion interactions between the small and large species. Surprisingly, in the context of binary colloidal films, these phenomena have never been investigated concurrently during evaporation. ExperimentsWe show how we can achieve both self-stratification and domain formation in a single step. Using real-time 3D confocal fluorescence microscopy, we quantitatively unravel the effects of various parameters on the emergence of compositional heterogeneity. FindingsWe reveal that beyond a certain size ratio, micro-phase separation becomes a prominent mechanism dictating the final morphology. The initial volume fraction minimally affects the final domain size but significantly impacts self-stratification. Reducing the evaporation rate increases the domain size while minimizing stratification. Finally, reducing the colloidal electrostatic interaction by a small increase in salt concentration enhances phase separation yet reverses stratification. These findings unveil a strategy for harnessing two distinct size segregation mechanisms in a single film, forming a foundation for customizable self-partitioning coatings.

Depletion forces beyond the diluted limit

The determination of the effective interactions in many-body systems still possesses a tremendous challenge in Condensed Matter Physics. The problem mainly resides in the fact that, on the one hand, there is not a unique route to obtain the effective forces between the ”observable” constituents and, on the other hand, one typically considers that the diluted limit of those components always corresponds to the effective forces at all particle concentrations. In the particular case of colloidal dispersions, it is very important to have experimental, theoretical, and computational tools that allow us to determine, accurately and without using significant approximations, the effective interactions between colloids. In this contribution, we report on a detailed study of depletion potentials in colloidal mixtures at finite concentration, namely, binary and ternary mixtures of disks (in two dimensions or 2D) and spheres (in three dimensions or 3D). To this end, the depletion forces between the large colloidal species are determined through a recently developed scheme based on the so-called contraction of the description, which has been extended and built at the level of the bare forces, i.e., we basically consider that all particles interact through the bare potentials and the net force acting on a given particle is determined by the second’s Newton law. This scheme can be easily adapted to Molecular Dynamics simulations. To verify the physical consistency of the formalism, we explicitly show that in the diluted limit of large particles, the resulting depletion potential reproduces correctly the AO-Vrij limit. As further proof of the accuracy of the results, a comparison of the structure of the large colloids in the whole mixture with the one that results using exclusively the depletion potential is carried out. In 3D, the results are explicitly compared with the potential of mean force and those obtained with the integral equations formalism. We also report a new colloidal stability mechanism based on the use of two different species of depletant agents at low and equal concentrations. Although we highlight the fact that the depletion potential depends on the concentration of the large species, we show that this dependence can be eliminated when the colloidal dispersion is open to a reservoir of small particles, i.e., when the chemical potential of small particles is fixed. Finally, we report the effects of polydispersity on the depletion interaction between large colloids.

Effect of carboxymethyl cellulose binder on physical, thermal, and rheological properties of milk protein isolate/guar gum mixture powder

This study examined the effect of fluidized-bed agglomeration with a carboxymethyl cellulose (CMC) binder on the physical, thermal, and rheological properties of agglomerated milk protein isolate (MPI)-guar gum (GG) mixtures (AMGs). The agglomeration process produced larger and more porous particles, improving solubility and powder flowability. An exothermic event followed by an endothermic event was observed in DSC thermograms. These thermal events tended to occur at a higher temperature with lower enthalpy values as the binder concentration increased. All samples exhibited shear-thinning behavior, with apparent viscosity at 100 s−1 values (0.27–0.29 Pa s) for AMGs being lower than for non-agglomerated MPI-GG mixture (NAMG; 0.34 Pa s). Viscoelastic moduli values showed similar trends. Additionally, compared to NAMG, larger and more guar gum lumps occurred in the AMGs. These findings suggest that the agglomeration process with CMC promoted depletion interactions between the protein and polysaccharide components in the mixture system.

Analytical and Numerical Investigation of Star Polymers in Confined Geometries.

The analysis of the impact of the star polymer topology on depletion interaction potentials, depletion forces, and monomer density profiles is carried out analytically using field theory methods and techniques as well as molecular dynamic simulations. The dimensionless depletion interaction potentials and the dimensionless depletion forces for a dilute solution of ideal star polymers with three and five legs (arms) in a Θ-solvent confined in a slit between two parallel walls with repulsive surfaces and for the case where one of the surfaces is repulsive and the other inert are obtained. Furthermore, the dimensionless layer monomer density profiles for ideal star polymers with an odd number (f˜ = 3, 5) of arms immersed in a dilute solution of big colloidal particles with different adsorbing or repelling properties in respect of polymers are calculated, bearing in mind the Derjaguin approximation. Molecular dynamic simulations of a dilute solution of star-shaped polymers in a good solvent with N = 901 (3 × 300 + 1 -star polymer with three arms) and 1501 (5 × 300 + 1 -star polymer with five arms) beads accordingly confined in a slit with different boundary conditions are performed, and the results of the monomer density profiles for the above-mentioned cases are obtained. The numerical calculation of the radius of gyration for star polymers with f˜ = 3, 5 arms and the ratio of the perpendicular to parallel components of the radius of gyration with respect to the wall orientation for the above-mentioned cases is performed. The obtained analytical and numerical results for star polymers with an odd number (f˜ = 3, 5) of arms are compared with our previous results for linear polymers in confined geometries. The acquired results show that a dilute solution of star polymer chains can be applied in the production of new functional materials, because the behavior of these solutions is strictly correlated with the topology of polymers and also with the nature and geometry of confined surfaces. The above-mentioned properties can find extensive practical application in materials engineering, as well as in biotechnology and medicine for drug and gene transmission.

Morphological control of bundled actin networks subject to fixed-mass depletion.

Depletion interactions are thought to significantly contribute to the organization of intracellular structures in the crowded cytosol. The strength of depletion interactions depends on physical parameters such as the depletant number density and the depletant size ratio. Cells are known to dynamically regulate these two parameters by varying the copy number of proteins of a wide distribution of sizes. However, mammalian cells are also known to keep the total protein mass density remarkably constant, to within 0.5% throughout the cell cycle. We thus ask how the strength of depletion interactions varies when the total depletant mass is held fixed, a.k.a. fixed-mass depletion. We answer this question via scaling arguments, as well as by studying depletion effects on networks of reconstituted semiflexible actin in silico and in vitro. We examine the maximum strength of the depletion interaction potential U∗ as a function of q, the size ratio between the depletant and the matter being depleted. We uncover a scaling relation U∗ ∼ qζ for two cases: fixed volume fraction φ and fixed mass density ρ. For fixed volume fraction, we report ζ < 0. For the fixed mass density case, we report ζ > 0, which suggests that the depletion interaction strength increases as the depletant size ratio is increased. To test this prediction, we prepared our filament networks at fixed mass concentrations with varying sizes of the depletant molecule poly(ethylene glycol) (PEG). We characterize the depletion interaction strength in our simulations via the mesh size. In experiments, we observe two distinct actin network morphologies, which we call weakly bundled and strongly bundled. We identify a mass concentration where different PEG depletant sizes lead to weakly bundled or strongly bundled morphologies. For these conditions, we find that the mesh size and intra-bundle spacing between filaments across the different morphologies do not show significant differences, while the dynamic light scattering relaxation time and storage modulus between the two states do show significant differences. Our results demonstrate the ability to tune actin network morphology and mechanics by controlling depletant size and give insights into depletion interaction mechanisms under the fixed-depletant-mass constraint relevant to living cells.

Centrifugation-Mediated Crystal Growth of Attractive Colloids for Band Edge Lasing.

Attractive depletion interactions are utilized to organize colloidal particles into crystalline arrays with high crystallinity through spontaneous phase separation. However, uncontrolled nucleation frequently leads to the formation of crystalline grains with varied crystal orientations, which hampers the optical performance of photonic crystals. Here, colloidal crystals have been engineered with uniform orientation and high surface coverage by applying centrifugal force during the depletion-induced assembly of polystyrene particles. The centrifugal force encourages the particles to move toward the bottom surface, which fosters heterogeneous nucleation and supports rapid crystal growth, yielding densely-packed and uniformly-arranged crystal grains with high reflectivity. This study has observed that the nucleation and crystal growth behavior is significantly influenced by the salt concentration. Based on the pair potentials, the transition boundary has been quantitatively analyzed between fluid and crystal phases and identified the threshold for homogeneous nucleation. Utilizing the high-reflectivity colloidal crystals, band-edge lasing is achieved by dissolving the water-soluble dye into the aqueous suspensions. Upon optical excitation, a lasing emission characterized is observed by a narrow spectral width at the short-wavelength band edge. Notably, the laser wavelength can be adjusted by altering the salt concentration or particle diameter, offering a versatile approach to tuning the optical properties.

Depletion Strategies for Crystallized Layers of Two-Dimensional Nanosheets to Enhance Lithium-Ion Conductivity in Polymer Nanocomposites.

The assembly of long-range aligned structures of two-dimensional nanosheets (2DNSs) in polymer nanocomposites (PNCs) is in urgent need for the design of nanoelectronics and lightweight energy-storage materials of high conductivity for electricity or heat. These 2DNS are thin and exhibit thermal fluctuations, leading to an intricate interplay with polymers in which entropic effects can be exploited to facilitate a range of different assemblies. In molecular dynamics simulations of experimentally studied 2DNSs, we show that the layer-forming crystallization of 2DNSs is programmable by regulating the strengths and ranges of polymer-induced entropic depletion attractions between pairs of 2DNSs, as well as between single 2DNSs and a substrate surface, by exclusively tuning the temperature and size of the 2DNS. Enhancing the temperature supports the 2DNS-substrate depletion rather than crystallization of 2DNSs in the bulk, leading to crystallized layers of 2DNSs on the substrate surfaces. On the other hand, the interaction range of the 2DNS-2DNS depletion attraction extends further than the 2DNS-substrate attraction whenever the 2DNS size is well above the correlation length of the polymers, which results in a nonmonotonic dependence of the crystallization layer on the 2DNS size. It is demonstrated that the depletion-tuned crystallization layers of 2DNSs contribute to a conductive channel in which individual lithium ions (Li ions) migrate efficiently through the PNCs. This work provides statistical and dynamical insights into the balance between the 2DNS-2DNS and 2DNS-substrate depletion interactions in polymer-2DNS composites and highlights the possibilities to exploit depletion strategies in order to engineer crystallization processes of 2DNSs and thus to control electrical conductivity.

Kinetic stability of Fe-based nanoparticles with rheological modification by xanthan gum: A critical stabilization concentration and the underlying mechanism

Enhanced kinetic stability of Fe-NPs in groundwater is a focus in application of Fe-NPs for groundwater remediation. The effect of surfactants (Triton X-100 and SDBS) and polymers (XG, SA, CCS, PSS and PVP) on the kinetic stability of Fe-NPs were studied with sedimentation experiments. Polymers improved stability of nFe3O4 and XG had the best effect, while surfactants had minimal effect. There was a critical concentration (CSC) for XG to stabilize nFe3O4, which was 2.0 g/L. At such a concentration nFe3O4, nFe2O3, and nCuO did not settled in 10 h, while the settlement occurred below the concentration and increased with decreasing XG concentration. At CSC XG could stabilize 20 g/L of nFe3O4 for >30 days and 8.0 g/L of nZVI for 13 days. Rheology studies indicated that the enhanced stability was due to the entanglement of XG molecules in the concentration range of 0.5–2.8 g/L and the formation of a uniform entangled network at CSC concentration was responsible for non-sedimentation of Fe-NPs. At hyper-CSC concentrations under the regime of concentrated network (>2.8 g/L), the stability of nFe3O4 and nFe2O3 decreased due to depletion interaction. The rules for XG to stabilize particles and information about the critical concentration will improve XG application in groundwater remediation using Fe-NPs.

Freezing-mediated formation of supraproteins using depletion forces

HypothesisLong-acting formulations such as microparticles, injectable depots and implantable devices can realize spatiotemporally controlled delivery of protein drugs to extend their therapeutic in vivo half-lives. To efficiently encapsulate the protein drugs into such drug delivery systems, (sub)micron-sized protein particles are needed. The formation of micronized supraproteins can be induced through the synergistic combination of attractive depletion forces and freezing. The size of the supraproteins can be fine-tuned from submicron to several microns by adjusting the ice crystallization rate through the freeze-quench depth, which is set by the target temperature.MethodsSupraprotein micron structures were prepared from protein solutions under various conditions in the presence and absence of nonadsorbing polyethylene glycol. Scanning electron microscopy and dynamic light scattering were employed to determine the sizes of the supraproteins and real-time total internal reflection fluorescent microscopy was used to follow the supraprotein formation during freezing. The protein secondary structure was measured before and after micronization by circular dichroism. A phase diagram of a protein–polyethylene glycol mixture was theoretically predicted to investigate whether the depletion interaction can elucidate the phase behavior.FindingsMicronized protein supraparticles could be prepared in a controlled manner by rapid freeze-drying of aqueous mixtures of bovine serum albumin, horseradish peroxidase and lysozyme mixed with polyethylene glycol. Upon freezing, the temperature quench initiates a phase separation process which is reminiscent of spinodal decomposition. This demixing is subsequently arrested during droplet phase separation to form protein-rich microstructures. The final size of the generated protein microparticles is determined by a competition between phase separation and cooling rate, which can be controlled by target temperature. The experimental phase diagram of the aqueous protein–polyethylene glycol dispersion aligns with predictions from depletion theory for charged colloids and nonadsorbing polymers.

Depletion Interactions at Interfaces Induced by Ferromagnetic Colloidal Polymers.

The pair-interaction force profiles for two non-magnetic colloids immersed in a suspension of ferromagnetic colloidal polymers are investigated via Langevin simulations. A quasi-two-dimensional approach is taken to study the interface case and a range of colloidal size ratios (non-magnetic:magnetic) from 6:1 up to 20:1 have been considered in this work. Simulations show that when compared with non-magnetic suspensions, the magnetic polymers strongly modify the depletion force profiles leading to strongly oscillatory behavior. Larger polymer densities and size ratios increase the range of the depletion forces, and in general, also their strength; the force barrier peaks at short distances show more complex behavior. As the length of the ferromagnetic polymers increases, the force profiles become more regular, and stable points with their corresponding attraction basins develop. The number of stable points and the distance at which they occur can be tuned through the modification of the field strength H and the angle θ formed by the field and the imaginary axis joining the centers of the two non-magnetic colloids. When not constrained, the net forces acting on the two colloids tend to align them with the field till θ=0∘. At this angle, the force profiles turn out to be purely attractive, and therefore, these systems could be used as a funneling tool to form long linear arrays of non-magnetic particles. Torsional forces peak at θ=45∘ and have minimums at θ=0∘ as well as θ=90∘ which is an unstable orientation as slight deviations will evolve towards θ→0∘. Nonetheless, results suggest that the θ=90∘ orientation could be easily stabilized in several ways. In such a case, the stable points that the radial force profiles exhibit for this orthogonal orientation to the field could be used to control the distance between the two large colloids: their position and number can be controlled via H. Therefore, suspensions made of ferromagnetic colloidal polymers can be also useful in the creation of magnetic colloidal tweezers or ratchets. A qualitative explanation of all the observed phenomena can be provided in terms of how the geometrical constraints and the external field modify the conformations of the ferromagnetic polymers near the two large particles, and in turn, how both factors combine to create unbalanced Kelvin forces that oscillate in strength with the distance between the two non-magnetic colloids.

Diffusing Wave Spectroscopy Measurements of Colloidal Suspension Dynamics.

Diffusing wave spectroscopy (DWS) is used to measure the dynamics of charged silica particles between the volume fractions 0.065 ≤ ϕ ≤ 0.352 (weight percentages from 12.7 to 55.8 wt %). The short-time diffusivity averaged over the scattering vectors sampled by DWS (ϕ) decreases with an increasing concentration. An effective hard-sphere model that accounts for hydrodynamic interactions and a double-layer repulsion fits the values up to an effective volume fraction , where is the excluded shell radius normalized by the particle radius = b/a = 1.3. While DWS measurements of diffusivity are sensitive to repulsive interactions, we show that they are relatively insensitive to attraction, such as those due to secondary minima in the interaction potential or weak depletion interaction.

Diffusion dynamics for an infinite system of two-type spheres and the associated depletion effect

We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in Rd, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive short-range dynamical interaction – known in the physics literature as a depletion interaction – between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximising the contact number of n identical spheres in Rd. As support material, we propose numerical simulations in the form of movies.

Polyethylene glycol regulates the pitch and liquid crystal behavior of cellulose nanocrystal-based photonic crystals

Inspired by iridescent color in natural creations, cellulose nanocrystal (CNC) photonic crystals artificially created by nanotechnology have great application prospects due to their potential to control light propagation in the linear and nonlinear regimes. One of the most important development directions of photonic crystals is the diversification of colors, usually by adjusting the pitch. However, few researchers notice the effect of polymer molecular weight and content on pitch regulation and the interaction between polymer and CNC liquid crystals. Polyethylene glycol (PEG) were used as polymers to regulate the pitch of CNC photonic crystals and investigate the changes in microstructure, crystal structure, thermal properties, and liquid crystal texture of the composites by changing the PEG content and molecular weight. Different photonic crystal construction systems show that when the molecular weight of PEG is 0.4 k, it can be filled between CNCs to regulate the pitch of photonic crystals, while when the molecular weight of PEG is 20 k, it cannot always be filled between CNCs in evaporation-induced self-assembly (EISA) process due to the depletion interaction, which cannot effectively regulate the pitch. This study reveals the relationship between PEG and CNC liquid crystals, which supports the development of photonic crystals and the pitch regulation.

Enzymatic cleaving of entangled DNA rings drives scale-dependent rheological trajectories.

DNA, which naturally occurs in linear, ring, and supercoiled topologies, frequently undergoes enzyme-driven topological conversion and fragmentation in vivo, enabling it to perform a variety of functions within the cell. In vitro, highly concentrated DNA polymers form entanglements that yield viscoelastic properties dependent on the topologies and lengths of the DNA. Enzyme-driven alterations of DNA size and shape therefore offer a means of designing active materials with programmable viscoelastic properties. Here, we incorporate multi-site restriction endonucleases into dense DNA solutions to linearize and fragment circular DNA molecules. We pair optical tweezers microrheology with differential dynamic microscopy and single-molecule tracking to measure the linear and nonlinear viscoelastic response and transport properties of entangled DNA solutions over a wide range of spatiotemporal scales throughout the course of enzymatic digestion. We show that, at short timescales, relative to the relaxation timescales of the polymers, digestion of these 'topologically-active' fluids initially causes an increase in elasticity and relaxation times followed by a gradual decrease. Conversely, for long timescales, linear viscoelastic moduli exhibit signatures of increasing elasticity. DNA diffusion, likewise, becomes increasingly slowed, in direct opposition to the short-time behavior. We hypothesize that this scale-dependent rheology arises from the population of small DNA fragments, which increases as digestion proceeds, driving self-association of larger fragments via depletion interactions, giving rise to slow relaxation modes of clusters of entangled chains, interspersed among shorter unentangled fragments. While these slow modes likely dominate at long times, they are presumably frozen out in the short-time limit, which instead probes the faster relaxation modes of the unentangled population.

Pinching dynamics and extensional rheology of dense colloidal suspensions with depletion attractions

We study the extensional flow properties by characterizing the capillarity-driven pinching dynamics of dense colloidal suspensions at a constant volume fraction ϕ=0.40 with polymer-induced depletion interactions using a dripping-onto-substrate (DoS) protocol. Methacrylate copolymer particles with dimethylacrylamide copolymer brushes are suspended in a refractive-index- and density-matched mixture of 80 (w/w)% glycerol in water with NaCl added to screen the electrostatic repulsions. Depletion attractions between the colloids are introduced by adding polyacrylamide polymers of weight and dispersity. The addition of polymer delays and modifies the pinch-off dynamics of the dense suspensions, depending on the size and dispersity of the polymer. The extensional relaxation time λE of suspensions collapses as a function of the normalized free volume polymer concentration c/c∗ with the corresponding polymer solutions, indicating that the elastic properties of the polymer solutions control the extensional time scale. Following the results of our previous study [Soetrisno et al., Macromolecules 56, 4919–4928 (2023)], the polymer size determines the scaling exponent of λE for colloid-polymer mixtures in the dilute regime and high dispersity shifts the concentration where the scaling of λE transitions from power-law to linear. The filament lifespans tf of colloid-polymer mixtures and of polymer solutions collapse onto a master curve as a function of c/c∗ when normalized by the filament lifespan of the corresponding fluid without polymer tf,0. These results provide insight into the role of the polymer size in dictating the pinching dynamics and extensional rheology of colloid-polymer mixtures and further suggest that the shear and extensional responses of these mixtures can be separately tuned through the concentrations of the two constituents.

Janus particles with tunable patch symmetry and their assembly into chiral colloidal clusters

Janus particles, which have an attractive patch on the otherwise repulsive surface, have been commonly employed for anisotropic colloidal assembly. While current methods of particle synthesis allow for control over the patch size, they are generally limited to producing dome-shaped patches with a high symmetry (C∞). Here, we report on the synthesis of Janus particles with patches of various tunable shapes, having reduced symmetries ranging from C2v to C3v and C4v. The Janus particles are synthesized by partial encapsulation of an octahedral metal-organic framework particle (UiO-66) in a polymer matrix. The extent of encapsulation is precisely regulated by a stepwise, asymmetric dewetting process that exposes selected facets of the UiO-66 particle. With depletion interaction, the Janus particles spontaneously assemble into colloidal clusters reflecting the particles’ shapes and patch symmetries. We observe the formation of chiral structures, whereby chirality emerges from achiral building blocks. With the ability to encode symmetry and directional bonding information, our strategy could give access to more complex colloidal superstructures through assembly.

Lubrication behavior of mixed colloidal systems: Effect of interactions between polysaccharides and oil droplets/ protein particles

The lubrication behavior of food systems is critical for sensory perception and consumer acceptance. This behavior depends on the components present in food and their interactions. In the present study, binary systems were designed to systematically investigate the effect of the interactions (i.e. repulsive and attractive interactions) between components on lubrication properties. Whey protein-stabilized emulsion droplets or whey protein aggregates functioned as dispersed particles, while polysaccharides were present in the continuous phase. Polysaccharides with different molecular characteristics, i.e. xanthan (charged, stiff), pectin (charged, semi-flexible) and guar gum (non-charged, flexible), were selected. We observed that bridging due to attractive interactions between polysaccharides and particles (oil droplets or proteins aggregates) induced extensive particle aggregation, leading to increased friction coefficients. Such bridging was reduced as the concentration of polysaccharides increased, decreasing both cluster size and friction coefficient. In the case of systems with the same charge, weaker depletion interactions led to the formation of particle aggregates, resulting in increased friction coefficients. This was mostly observed for pectin. Conversely, addition of xanthan and guar gum did not cause extensive aggregation and a more homogeneous microstructure was observed. As a result, lower friction coefficients were obtained. The friction coefficients were reduced further when more xanthan or guar gum were added, the highly charged xanthan being more effective. Overall, for both types of interactions, systems with a more homogeneous microstructure (i.e. smaller clusters) resulted in lower friction, indicating that the microstructure of the systems was the primary determinant of lubrication behavior.

Two-dimensional ordering of bacteriorhodopsins in a lipid bilayer and effects caused by repulsive core between lipid molecules on lateral depletion interaction: A study based on a thermodynamic perturbation theory

Using binary hard disk mixture models, we studied the two-dimensional ordering of bacteriorhodopsins in a lipid bilayer. The phase diagrams were calculated using the thermodynamic perturbation theory. We examined two types of effective interactions to discuss the lateral depletion effects caused by repulsive core interaction between lipid molecules. The results indicate that the core repulsions drastically broaden the coexistence region for the fluid-ordered phase.

Generalized geometric criteria for the absence of effective many-body interactions in the Asakura–Oosawa model

We investigate variants of the Asakura–Oosawa (AO) model for colloid-polymer mixtures, represented by hard classical particles interacting via their excluded volume. The interaction between the polymers is neglected but the colloid-polymer and colloid-colloid interactions are present and can be condensed into an effective depletion interaction among the colloids alone. The original AO model involves hard spherical particles in three spatial dimensions with colloidal radii R and the so-called depletion radius δ of the polymers, such that the minimum possible center-to-center distance between polymers and colloids allowed by the excluded-volume constraints is R + δ. It is common knowledge among physicists that there are only pairwise effective depletion interactions between the colloids if the geometric condition δ/R&amp;lt;2/3−1 is fulfilled. In this case, triplet and higher-order many body interactions are vanishing and the equilibrium statistics of the binary mixture can exactly be mapped onto that of an effective one-component system with the effective depletion pair-potential. Here we rigorously prove that the criterion δ/R&amp;lt;2/3−1 is both sufficient and necessary to guarantee the absence of triplet and higher-order many body interactions among the colloids. For an external hard wall confining the system, we also include a criterion which guarantees that the system can be exactly mapped onto one with effective external one-body interactions. Our general formulation also accounts for polydisperse mixtures and anisotropic shapes of colloids in any spatial dimension. In those cases where the resulting condition is only sufficient, we further demonstrate how to specify improved bounds.

Polysaccharide-dependent depletion strategy to fabricate pickering emulsion gels

Understanding the associations between polysaccharide depletion and emulsion gelation is of great importance for achieving stable Pickering emulsion gels, but there is inconsistency in the relevant studies. In this work, we report a versatile strategy, by using depletion attractions of anionic polysaccharides - hyaluronic acid and sodium alginate, to realize the gelation of Pickering emulsions stabilized with various particles of different characteristics. The depletion interactions imposed by different ratios of a polysaccharide were demonstrated to be conducive to the formation of emulsion gels. With a higher addition of polysaccharides, emulsions underwent (1) enhanced interfacial adsorption at weak depletion, (2) bridging flocculation at moderate depletion, and (3) depletion flocculation at high depletion, which all led to the arrested dynamics and gelation of Pickering emulsions. The prepared emulsion gels exhibited good stabilities and favorable rheological properties including high gel strength, shear sensitivity, and thermostability. The universality of this study was further confirmed by using different types of Pickering stabilizers as well as oil phases. The polysaccharide-dependent depletion provides a simple strategy to design Pickering emulsion gels with tunable rheological behaviors for varied purposes.