Entropic Attraction Research Articles

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.

Controlling the chirality and number of strands of helices self-assembled from achiral block copolymers confined inside a nanopore: a simulation study.

Achiral block copolymers can self-assemble into helical structures when confined inside a cylindrical nanopore. However, controlling the chirality and the number of strands of helices is challenging. We present our simulation results of the influence of a chiral patch added to the confining nanopore on the structures and chirality of helices self-assembled from achiral cylinder-forming diblock copolymers under the confinement. Our results indicate that, when the designed patch is of proper geometry, it can induce the formation of helical structures and exhibit good control over their chirality. The bottom surface of the patch can induce the formation of a characteristic local structure near and parallel to it. It is the characteristic local structure that directs the formation of helices and of their chirality consistent with that of the patch. A large patch angle or the top/bottom surface of a weakly selective pore promotes the formation of double-helices compared to single-helices by enlarging the pitch of the helices near the patch or through the entropic attraction of the top surface of the pore to the minority blocks.

Effects of Geometric Confinement on Caging and Dynamics of Polymer-Tethered Nanoparticle Suspensions

Well-dispersed polymer-tethered silica nanoparticles exhibit soft glassy rheology and caging behaviors due to chain interpenetration driven by entropic attraction and geometric confinement. In this...

Combinatorial-Entropy-Driven Aggregation in DNA-Grafted Nanoparticles.

We use computer simulations and experiments to study the interactions between nanoparticles (NPs) grafted with self-complementary DNA strands. Each strand ends with a sticky palindromic single-stranded sequence, allowing it to associate equally favorably with strands grafted on the same particle or on different NPs. Surprisingly we find an attractive interaction between a pair of NPs, and we demonstrate that at low temperature it arises purely from a combinatorial-entropy contribution. We evaluate theoretically and verify numerically this entropic contribution originating from the number of distinct bonding patterns associated with intra- and interparticle binding. This entropic attraction becomes more favorable with decreasing inter-NP distance because more sticky ends can participate in making this choice.

Universal Gelation of Metal Oxide Nanocrystals via Depletion Attractions

Nanocrystal gelation provides a powerful framework to translate nanoscale properties into bulk materials and to engineer emergent properties through the assembled microstructure. However, many established gelation strategies rely on chemical reactions and specific interactions, e.g., stabilizing ligands or ions on the nanocrystals' surfaces, and are therefore not easily transferable. Here, we report a general gelation strategy via nonspecific and purely entropic depletion attractions applied to three types of metal oxide nanocrystals. The gelation thresholds of two compositionally distinct spherical nanocrystals agree quantitatively, demonstrating the adaptability of the approach for different chemistries. Consistent with theoretical phase behavior predictions, nanocrystal cubes form gels at a lower polymer concentration than nanocrystal spheres, allowing shape to serve as a handle to control gelation. These results suggest that the fundamental underpinnings of depletion-driven assembly, traditionally associated with larger colloidal particles, are also applicable at the nanoscale.

Depletion force between disordered linear macromolecules

When two macromolecules come very near in a fluid, the surrounding molecules, having finite volume, are less likely to get in between. This leads to a pressure difference manifesting as an entropic attraction, called depletion force. Here we calculate the density profile of liquid molecules surrounding a disordered rigid macromolecules modeled as a random arrangement of hard spheres on a linear backbone. We analytically determine the position dependence of the depletion force between two such disordered molecules by calculating the free energy of the system. We then use molecular dynamics simulations to obtain the depletion force between stiff disordered polymers as well as flexible ones and compare the two against each other. We also show how the disorder averaging can be handled starting from the inhomogenous reference interaction site model equations.

Entropic Effects in Solvent-Free Bidisperse Polymer Brushes Investigated Using Density Functional Theories.

Solvent-free polymer-functionalized nanoparticles form a special type of colloid composed of inorganic cores self-suspended by their grafted coronas. In the absence of intervening solvent molecules, the fluidity of the system is provided by these tethered polymers as they fill the space. Here, we study the structure and interaction of neighboring polymer-grafted surfaces in the solvent-free condition using mean-field density functional theories. For opposing flat surfaces, the brush configuration and the associated energy landscape are semianalytically investigated given the incompressibility of the tethered entropic chains. The effect of brush polydispersity (including variations in both chain length and surface grafting density) is considered by two bidisperse models corresponding to different physical scenarios: one for opposing brushes uniformly mixed with two species at a fixed grafting density, and the other for opposing brushes with distinct chain lengths and grafting densities. The space-filling capabilities of the neighboring coronas differ not only by their ratio of radii of gyration for the composing polymers but also by their ratio of grafting densities. We show that the system energy depicts a steric repulsion as the brushes are compressed, which is typical for hairy particles in a solvent. However, as the interwall separation increases, the cooperative stretching of the chains leads to an entropic attraction between them, a unique characteristic of solventless systems. The corresponding brush profiles change from a bell-like shape to a more step-function-like feature as the interwall spacing increases significantly. The interwall separation associated with the overall free energy minimum therefore characterizes the favorable interparticle spacing for solvent-free polymer-functionalized particles. The limiting accessible parameter space of polymer sizes and grafting densities subjected to the space-filling constraint is comprehensively explored for representative interparticle spacing characterizing the compressed, relaxed, and stretched regimes for a given polymer species, respectively. Such information would be useful for guiding the design of experimental solvent-free polymer-functionalized nanoparticles.

Effect of polymer-coated gold nanoparticle stabilizers on drop coalescence

Polymer-coated gold nanoparticles (PGNPs) can be used as stabilizers in immiscible polymer blends, similar to block-copolymers (BCs). However, the PGNP gold cores increase the magnitude of the disjoining pressure (Π), i.e., the van der Waals interaction for unit area, in the film between the drops, favoring coalescence. This might explain the counterintuitive 70% drainage time (td) reduction for polymeric drops stabilized by PGNPs compared to those stabilized by BCs, as reported in recent flow-induced head-on collision experiments in extensional flow, despite PGNPs being more surface active. Knowledge of the mechanisms determining td is fundamental for designing effective PGNP compatibilizers. Here, we performed a parametric study of those experiments via boundary integral simulations, treating PGNPs as surfactants and utilizing for the first time a disjoining pressure expression which includes the effect of interfacial PGNPs (ΠPGNP). In particular, we varied the PGNP concentration and core size in ΠPGNP, the surface diffusivity (Ds) via the surface Peclet number, and the surface elasticity via the Marangoni number. Flow-induced coalescence was very sensitive to all three parameters. td was reduced up to 60% for touching 3 nm core diameter PGNPs, increasing significantly the coalescence probability for drop sizes <5 µm, but the soft coronas diminished this effect considerably. Thus, other causes, besides the enhanced Π, had to be simultaneously present to explain the dramatic experimental td reduction; the most likely is a Ds higher than its Stokes-Einstein relation estimate and the PGNP ligands being in a dry-brush regime, leading to entropic attraction between the drop interfaces.

Entropic Interactions between Two Knots on a Semiflexible Polymer.

Two knots on a string can either be separated or intertwined, and may even pass through each other. At the microscopic scale, such transitions may occur spontaneously, driven by thermal fluctuations, and can be associated with a topological free energy barrier. In this manuscript, we study the respective location of a trefoil () and a figure-eight () knot on a semiflexible polymer, which is parameterized to model dsDNA in physiological conditions. Two cases are considered: first, end monomers are grafted to two confining walls of varying distance. Free energy profiles and transition barriers are then compared to a subset of free chains, which contain exactly one and one knot. For the latter, we observe a small preference to form an intertwined state, which can be associated with an effective entropic attraction. However, the respective free energy barrier is so small that we expect transition events to occur spontaneously and frequently in polymers and DNA, which are highly knotted for sufficient strain lengths.

Structural arrest and dynamic localization in biocolloidal gels.

Casein micelles interacting via an entropic intermediate-ranged depletion attraction exhibit a fluid-to-gel transition due to arrested spinodal decomposition. The bicontinuous networked structure of the gel freezes shortly after formation. We determine the timescales of structural arrest from the build-up of network rigidity after pre-shear rejuvenation, and find that the arrest time as well as the plateau elastic modulus of the gel diverge as a function of the volume fraction and interaction potential. Moreover, we show using scaling from naïve mode coupling theory that their mechanical properties are dictated by their microscopic dynamics rather than their heterogeneous large scale structure.

Shape Recognition of Nanoparticle-Imprinting Materials Enhanced by Depletants

The recognition of nanoparticles by imprinted materials via entropic depletion attraction is investigated by dissipative particle dynamics simulations. It is found that the depletion attraction exits between nanoparticles and imprinted materials with complementary shapes based on the interaction energy U(H) and association energy Ea. The strength of the attractive depletion grows with increasing size of a perfectly matched target/cavity (TC) pair owing to the increment in their overlapping excluded volumes. The uptake of targets can be significantly enhanced by increasing the concentration of depletant ϕD. The selective recognition between perfectly matched and mismatched TC pairs is also studied, and a very high selectivity can be achieved at an optimal ϕD. The kinetics of the recognition process reveals that small nanoparticles migrate fast and access the cavity easily but move out of the cavities eventually due to their weak association energy. Finally, the synergetic effect of entropic depletion and e...

Interactions, Structure, and Dynamics of Polymer-Tethered Nanoparticle Blends.

We report on the structure, jamming, and dynamics of blends of self-suspended hairy silica nanoparticles grafted with poly(ethylene glycol) (PEG) and poly(methyl methacrylate) (PMMA). We find that favorable enthalpic attraction between tethered PEG and PMMA chains augment previously reported entropic attractions between tethered polymer chains in self-suspended suspensions to enhance particle-particle correlations, increase jamming, and slow down chain dynamics. As with their single-component counterparts, the hairy SiO2-PEG/SiO2-PMMA nanoparticle blends exhibit soft glassy rheological behavior and both the energy dissipated at yielding and the plateau elastic modulus display strong maxima in the symmetric case. A comparison of the small angle X-ray scattering (SAXS) measurements with theoretical analysis from density functional theory (DFT) reveals that the addition of SiO2-PMMA to a self-suspended SiO2-PEG suspension initially leads to a higher degree of stretching of the corona chains, which produces stronger interdigitation of the tethered chains, enhanced jamming, and slower polymer relaxation than observed in the single-component materials. By means of an analysis of the heat of mixing released upon blending tethered and untethered PEG and PMMA chains, we find that the strong enthalpic attraction between the grafted polymer chains enhances entropic attractive forces produced by the space-filling constraint on tethered ligands in self-suspended suspensions to produce entangled-polymer-like physical properties in polymers with molecular weights below the thresholds normally associated with the transition to an entangled state.

Phase separation of comb polymer nanocomposite melts.

In this work, the spinodal phase demixing of branched comb polymer nanocomposite (PNC) melts is systematically investigated using the polymer reference interaction site model (PRISM) theory. To verify the reliability of the present method in characterizing the phase behavior of comb PNCs, the intermolecular correlation functions of the system for nonzero particle volume fractions are compared with our molecular dynamics simulation data. After verifying the model and discussing the structure of the comb PNCs in the dilute nanoparticle limit, the interference among the side chain number, side chain length, nanoparticle-monomer size ratio and attractive interactions between the comb polymer and nanoparticles in spinodal demixing curves is analyzed and discussed in detail. The results predict two kinds of distinct phase separation behaviors. One is called classic fluid phase boundary, which is mediated by the entropic depletion attraction and contact aggregation of nanoparticles at relatively low nanoparticle-monomer attraction strength. The second demixing transition occurs at relatively high attraction strength and involves the formation of an equilibrium physical network phase with local bridging of nanoparticles. The phase boundaries are found to be sensitive to the side chain number, side chain length, nanoparticle-monomer size ratio and attractive interactions. As the side chain length is fixed, the side chain number has a large effect on the phase behavior of comb PNCs; with increasing side chain number, the miscibility window first widens and then shrinks. When the side chain number is lower than a threshold value, the phase boundaries undergo a process from enlarging the miscibility window to narrowing as side chain length increases. Once the side chain number overtakes this threshold value, the phase boundary shifts towards less miscibility. With increasing nanoparticle-monomer size ratio, a crossover of particle size occurs, above which the phase separation is consistent with that of chain PNCs. The miscibility window for this condition gradually narrows while the other parameters of the PNCs system are held constant. These results indicate that the present PRISM theory can give molecular-level details of the underlying mechanisms of the comb PNCs. It is hoped that the results can be used to provide useful guidance for the future design control of novel, thermodynamically stable comb PNCs.

Entropic attraction: Polymer compaction and expansion induced by nano-particles in confinement.

We investigated nanoparticle (NP)-induced coil-to-globule transition of a semi-flexible polymer in a confined suspension of ideal NP using Langevin dynamics. DNA molecules are often found to be highly compact, bound with oppositely charged proteins in a crowded environment within cells and viruses. Recent studies found that high concentration of electrostatically neutral NP also condenses DNA due to entropically induced depletion attraction between DNA segments. Langevin dynamics simulations with a semi-flexible chain under strong confinement were performed to investigate the competition between NP-induced monomer-monomer and monomer-wall attraction under different confinement heights and NP volume fractions. We found that whether NP induce polymer segments to adsorb to the walls and swell or to attract one another and compact strongly depends on the relative strength of the monomer-wall and the NP-wall interactions.

Re-entrant solidification in polymer–colloid mixtures as a consequence of competing entropic and enthalpic attractions

In polymer-colloid mixtures, non-adsorbing polymers dispersed with much larger colloids provide a universal yet specific entropic attraction between the colloids. Such so-called depletion interaction arises from an osmotic-pressure imbalance caused by the polymers and is considered to be independent of temperature. Here we show that, for the most commonly used polymer-colloid depletion systems, the polymer undergoes a crossover from non-adsorbing to adsorbing and that, consequently, the effective colloidal interactions depend on temperature. We also find that a combination of the enthalpic (polymer bridging) and entropic (polymer exclusion) interactions, both attractive, leads to a re-entrant regime where the colloids are dispersed and form solids both on heating and on cooling. We provide a simple model to explain the observed transitions and to fill the theoretical gap at the polymer-adsorption crossover. Our findings open possibilities for colloidal self-assembly, the formation of colloidal crystals and glasses, and the behaviour of temperature-controlled viscoelastic materials.

Structure factor of blends of solvent-free nanoparticle-organic hybrid materials: density-functional theory and small angle X-ray scattering.

We investigate the static structure factor S(q) of solvent-free nanoparticle-organic hybrid materials consisting of silica nanocores and space-filling polyethylene glycol coronas using a density-functional theory and small angle X-ray scattering measurements. The theory considers a bidisperse suspension of hard spheres with different radii and tethered bead-spring oligomers with different grafting densities to approximate the polydispersity effects in experiments. The experimental systems studied include pure samples with different silica core volume fractions and the associated mean corona grafting densities, and blends with different mixing ratios of the pure samples, in order to introduce varying polydispersity of corona grafting density. Our scattering experiments and theory show that, compared to the hard-sphere suspension with the same core volume fraction, S(q) for pure samples exhibit both substantially smaller values at small q and stronger particle correlations corresponding to a larger effective hard core at large q, indicating that the tethered incompressible oligomers enforce a more uniform particle distribution, and the densely grafted brush gives rise to an additional exclusionary effect between the nanoparticles. According to the theory, polydispersity in the oligomer grafting density controls the deviation of S(q) from the monodisperse system at smaller q, and the interplay of the enhanced effective core size and the entropic attraction among the particles is responsible for complex variations in the particle correlations at larger q. The successful comparison between the predictions and the measurements for the blends further suggests that S(q) can be used to assess the uniformity of grafting density in polymer-grafted nanoparticle materials.

Monte Carlo simulation studies on the effect of entropic attraction on the electric conductivity in polymer nano-composites.

The effect of non-conductive nano-particles on the electrical percolating network formation and the electrical conductivity of conductive nano-particles in polymer matrices is investigated using Monte Carlo simulations and a percolation theory. Both conductive and non-conductive nano-particles are modeled as spheres but with different diameters. Non-conductive nano-particles are up to four times bigger than conductive nano-particles. Equilibrated configurations for mixtures of nano-particles are obtained via Monte Carlo simulations and are used to estimate the probability (P) of forming an electrical percolating network and the percolation threshold conductive nano-particle volume fraction (phi(c)). As the volume fraction (phi(nc)) of non-conductive nano-particles increases, phi(c) decreases significantly, thus increasing the electrical conductivity. When non-conductive nano-particles mix with conductive nano-particles, they make the effective interaction energy W(r) between conductive nano-particles attractive, which should facilitate the formation of the electrical percolating network. For a given phi(nc), phi(c) increases slightly with an increase in the non-conductive nano-particle diameter (sigma(nc)). We also carry out simulations with non-conductive nano-particles of different structures and find that phi(c) is relatively insensitive to the non-conductive nano-particle structure.

Entropy-Driven Pattern Formation of Hybrid Vesicular Assemblies Made from Molecular and Nanoparticle Amphiphiles

Although an analogy has been drawn between them, organic molecular amphiphiles (MAMs) and inorganic nanoparticle (NP) amphiphiles (NPAMs) are significantly different in dimension, geometry, and composition as well as their assembly behavior. Their concurrent assembly can synergetically combine the inherent properties of both building blocks, thus leading to new hybrid materials with increasing complexity and functionality. Here we present a new strategy to fabricate hybrid vesicles with well-defined shape, morphology, and surface pattern by coassembling MAMs of block copolymers (BCPs) and NPAMs comprising inorganic NPs tethered with amphiphilic BCPs. The assembly of binary mixtures generated unique hybrid Janus-like vesicles with different shapes, patchy vesicles, and heterogeneous vesicles. Our experimental and computational studies indicate that the different nanostructures arise from the delicate interplay between the dimension mismatch of the two types of amphiphiles, the entanglement of polymer chains, and the mobility of NPAMs. In addition, the entropic attraction between NPAMs plays a dominant role in controlling the lateral phase separation of the two types of amphiphiles in the membranes. The ability to utilize multiple distinct amphiphiles to construct discrete assemblies represents a promising step in the self-assembly of structurally complex functional materials.

Nonspecific bridging-induced attraction drives clustering of DNA-binding proteins and genome organization

Molecular dynamics simulations are used to model proteins that diffuse to DNA, bind, and dissociate; in the absence of any explicit interaction between proteins, or between templates, binding spontaneously induces local DNA compaction and protein aggregation. Small bivalent proteins form into rows [as on binding of the bacterial histone-like nucleoid-structuring protein (H-NS)], large proteins into quasi-spherical aggregates (as on nanoparticle binding), and cylinders with eight binding sites (representing octameric nucleosomal cores) into irregularly folded clusters (like those seen in nucleosomal strings). Binding of RNA polymerase II and a transcription factor (NFκB) to the appropriate sites on four human chromosomes generates protein clusters analogous to transcription factories, multiscale loops, and intrachromosomal contacts that mimic those found in vivo. We suggest that this emergent behavior of clustering is driven by an entropic bridging-induced attraction that minimizes bending and looping penalties in the template.

Wall-induced orientational order in athermal semidilute solutions of semiflexible polymers: Monte Carlo simulations of a lattice model

An athermal solution of semiflexible macromolecules with excluded volume interactions has been studied at various concentrations (dilute, semidilute, and concentrated solutions) in a film of thickness D between two hard walls by grand canonical Monte Carlo simulations of the bond fluctuation lattice model. Analyzing profiles of orientational order parameters across the film, we find that for thick films two phase transitions occur at chemical potentials of the polymers (or polymer densities, respectively) where the bulk polymer solution still is in the disordered isotropic phase. At rather small polymer densities, polymers accumulate at the walls due to an entropic attraction and undergo a transition to two-dimensional nematic order. Due to the properties of the lattice model, this order has Ising character, and the simulation results seem to be compatible with a second-order transition. Increasing the polymer density, nematically ordered "wetting" layers form at both walls; the increase of thickness of these layers is compatible with a logarithmic divergence when the chemical potential of the isotropic-nematic transition in the bulk is approached. In a system of finite width, D, between the walls, this leads to capillary nematization, exhibiting a reduction of the transition chemical potential inversely proportional to D. This transition exists only if D exceeds some critical value Dc, while the transition from the isotropic phase to the two-dimensional nematic state is suggested to persist down to ultrathin films.