Translational Entropy Research Articles

Effective Repulsion Between Oppositely Charged Particles in Symmetrical Multivalent Salt Solutions: Effect of Salt Valence

Salt ions play critical roles in the assembly of polyelectrolytes such as nucleic acids and colloids since ions can regulate the effective interactions between them. In this work, we investigated the effective interactions between oppositely charged particles in symmetrical (z:z) salt solutions by Monte Carlo simulations with salt valence z ranging from 1 to 4. We found that the effective interactions between oppositely charged particles are attractive for 1:1 and low multivalent salts, while they become apparently repulsive for high multivalent salts. Moreover, such effective repulsion becomes stronger as z increases from 2 to 3, while it becomes weaker when z increases from 3 to 4. Our analyses reveal that the overall effective interactions are attributed to the interplay between ion translational entropy and electrostatic energy, and the non-monotonic salt-valence dependence of the effective repulsions is caused by the rapid decrease of attractive electrostatic energy between two oppositely charged particles with their over-condensed counterions of opposite charges when z exceeds 3. Our further MC simulations show that the involvement of local-ranged counterion–co-ion repulsions can enhance the effective repulsions through weakening the attractive electrostatic energy, especially for higher salt valence.

Active coacervate droplets are protocells that grow and resist Ostwald ripening

Active coacervate droplets are liquid condensates coupled to a chemical reaction that turns over their components, keeping the droplets out of equilibrium. This turnover can be used to drive active processes such as growth, and provide an insight into the chemical requirements underlying (proto)cellular behaviour. Moreover, controlled growth is a key requirement to achieve population fitness and survival. Here we present a minimal, nucleotide-based coacervate model for active droplets, and report three key findings that make these droplets into evolvable protocells. First, we show that coacervate droplets form and grow by the fuel-driven synthesis of new coacervate material. Second, we find that these droplets do not undergo Ostwald ripening, which we attribute to the attractive electrostatic interactions and translational entropy within complex coacervates, active or passive. Finally, we show that the droplet growth rate reflects experimental conditions such as substrate, enzyme and protein concentration, and that a different droplet composition (addition of RNA) leads to altered growth rates and droplet fitness. These findings together make active coacervate droplets a powerful platform to mimic cellular growth at a single-droplet level, and to study fitness at a population level.

Radical Mechanism of Ir III /Ni II -Metallaphotoredox-Catalyzed C(sp 3 )–H Functionalization Triggered by Proton-Coupled Electron Transfer: Theoretical Insight

Photoredox catalysis can be induced to activate organic substrates or to modulate the oxidation state of transition-metal catalysts via unique single-electron transfer processes, so as to achieve c...

Adsorption of Block-Polyelectrolytes on an Oppositely Charged Surface

Electrostatic interaction is widely considered to be the main driving force of the adsorption of polyelectrolytes (PEs) on an oppositely charged surface. Therefore, the alteration of charge distribution as a result of varying PE chain structures will directly affect the performance of the adsorption behaviors in the system at a certain monomer concentration. In this work, we studied the effects of chain structure (chain length, Nₘ) as well as sequence variables including charge fraction and blockiness on the adsorption behaviors of block-PEs on an oppositely charged planar surface in the salt-free system using PE-Poisson–Boltzmann equations. Due to the complicated competitions among the translational entropies of PEs and counterions, the conformational entropy of PEs, and the surface attractive interactions, we find that the adsorption amounts show rich behaviors. First, the adsorption amounts of charged ζc and neutral ζₙ monomers both show two-stage behaviors, that is, ζc and ζₙ drastically increase with Nₘ for short chains but moderately increase to an asymptotic value for relatively long chains. Second, the adsorption amounts of charged monomers show non-monotonic behaviors (increase first then decrease) with charge fraction. However, the adsorption amounts of neutral monomers monotonically decrease with charge fraction. Finally, when chain length and charge fraction are unchanged, the adsorption amounts of both neutral and charged monomers are independent of blockiness. Moreover, we find that overcharging can only appear when the bare surface charge density is low enough, while charge reversal and inversion can be observed in the entire range of the bare surface charge density with appropriate conditions. Additionally, our results indicate that the strength of charge reversal and inversion increases monotonically with chain length and charge fraction, while it changes non-monotonically with blockiness.

Molecular dynamics of dissolution of a 36-chain cellulose Iβ microfibril at different temperatures above the critical pressure of water

The cellulose dissolution is an essential pretreatment process for the chemical conversion of lignocellulosic biomass into biofuels. Here, the dissolution of a 36-chain Iβ cellulose model with a hexagonal cross-section (M36HCS) in water is analyzed by Molecular Dynamics (MD) with CHARMM36/TIP3P (C36/TIP3P) force field using gradual heating at 25 MPa. Our simulations showed that the dissolution of M36HCS starts to occur between 560 K and 600 K, which agrees with experimental observations. In our system, conditions near the critical point of water reveal that translational and rotational entropies decrease, while the low hydration level increases vibrational entropy. This investigation theoretically shows that C36/TIP3P adequately reproduces the dissolution of M36HCS even in high-pressure water as corroborated in reactors.

Permut - Spatially Resolved Hydration Entropies from Atomistic Simulations

The thermodynamics of the first few surface water layers profoundly affects the overall free energy change in protein folding and unfolding, bilayer self-assembly, and ligand binding, and many other biomolecular processes. Whereas the enthalpy contribution can, in principle, be directly calculated from atomistic molecular dynamics simulations, the solvation shell entropy calculation suffers particularly from the poor sampling of the high dimensional configuration space. To furthermore understand, e.g., the effects of water molecules inside a binding pocket or the contribution of individual side chains to the hydration entropy, spatial resolution is required. To address these problems, we developed Per|Mut, a new method to calculate spatially resolved hydration entropies. In Per|Mut, we first employ a permutation reduction, which exploits the permutation symmetry of the identical water molecules to alleviate the sampling problem by the Gibbs factor N! without changing the physics. Next, we use a third-order mutual information expansion to obtain the hydration shell entropy. The spatial resolution and the decomposition into one- two- and three-body correlations of the translational and rotational entropies, as well as of the translation-rotation correlation, allows for an easy physical interpretation and characterization of the resolved entropy changes. In test-simulations, Per|Mut yielded accurate entropies for an argon gas test system and solvated alkanes. Applied to the solvation statistical mechanics of hydrated octanol and the protein Crambin, Per|Mut revealed the local effects of individual chemical groups or side chains on the solvation entropy.

Non-affinity adsorption of nanorods onto smooth walls via an entropy driven mechanism.

Preferential adsorption of nanorods onto smooth walls is investigated using dissipative particle dynamics in the absence of specific attraction and a depletant. Although the translational and rotational entropy of nanorods is significantly reduced after adsorption, the effective attraction between the nanorod and wall is clearly identified based on the distribution profile of rods. As the rod length increases, the attractive interaction grows stronger and clusters of aligned nanorods can emerge on the smooth wall. However, the presence of a depletion zone of nanorods adjacent to the adsorbed layer gives zero surface excess. These two regions correspond to the primary minimum and maximum mean force potentials observed. Since adsorbed nanorods lose their rotational and translational entropy, the strong adsorption of long nanorods has to be attributed to the entropy gain associated with the increase in free volume for the solvent in this athermal system. Nonetheless, as the surface roughness is present, entropy-driven attraction is lessened, similar to the depletion force between colloids.

Entropy-driven segregation in epoxy-amine systems at a copper interface.

The composition of an epoxy resin at the interface with the adherend is usually different from that in the bulk due to the enrichment of a specific constituent, a characteristic called interfacial segregation. For better adhesion, it should be precisely understood how epoxy and amine molecules exist on the adherend surface and react with each other to form a three-dimensional network. In this study, the entropic factor of the segregation in a mixture of epoxy and amine at the copper interface before and after the curing reaction is discussed on the basis of a full-atomistic molecular dynamics (MD) simulation. Smaller molecules were preferentially segregated at the interface regardless of the epoxy and amine, and this segregation remained after the curing process. No segregation occurred at the interface for a combination composed of epoxy and amine molecules with a similar size. These findings make it clear that the size disparity between constituents affects the interfacial segregation via the packing and/or translational entropy. The curing reaction was slower near the interface than in the bulk, and a large amount of unreacted molecules remained there. Finally, the effect of molecular shape was also examined. Linear molecules were more likely to segregate than round-shaped ones even though they were similar in volume. We believe that these findings, which are difficult to obtain experimentally, contribute to the understanding of the interfacial adhesion phenomena on a molecular scale.

Self-assembly of rod-coil diblock copolymer-nanoparticle composites in thin films: dissipative particle dynamics.

In this study, the assembled structures of rod-coil diblock copolymer and nanoparticle blends were studied via dissipative particle dynamics (DPD). Thin films were composed of soft confinement DPD fluid beads and the fluctuating film structure was maintained during the simulation process. Analysis of the position of nanoparticles was done in the smectic lamellar phase of the rod-coil polymer matrix, and density distributions of rods, coils, and nanoparticles were obtained as functions of the size of the nanoparticle and the DPD repulsion constant between the rod and the nanoparticle. The distribution of nanoparticles was explained by using the concept of translational entropy of nanoparticles, stretching energy of the polymer chain, relative repulsion enthalpy of nanoparticles to rods or coils, and the effect of the liquid crystalline rod.

On-Off of Co-non-solvency for Poly(N-vinylcaprolactam) in Alcohol-Water Mixtures: A Molecular Dynamics Study.

Poly(N-vinylcaprolactam) (PVCL) exhibits co-non-solvency in aqueous solutions of 2-propanol but not in methanol. What distinguishes the impact of these two cosolvents on the polymer conformational stability? We report a molecular dynamics simulation study on PVCL 50-mer and monomers dissolved in methanol-water and 2-propanol-water mixtures. We show that the alcohol-concentration dependence of the effective attraction between a pair of PVCL monomers closely resembles the conformational changes in a single PVCL 50-mer as well as the experimentally observed behavior for PVCL chains. We also found that, at the co-non-solvency maximum, the monomer-monomer attraction works over a long-range beyond the solvent-separated distance. Then, we correlate the long-range attraction to the appearance of a dense alcohol concentration accumulated between the monomers. Furthermore, we distinctly demonstrate that the co-non-solvency of PVCL monomers can be switched on/off by artificially tuning the alcohol size while keeping the energetic parameters. Thus, we conclude that the magnitude of the excluded volume effect in alcohol accompanying the gain in translational entropy of the solvent is crucial to the occurrence of PVCL polymer co-non-solvency.

Entropy-controlled cross-linking in linker-mediated vitrimers

Recently developed linker-mediated vitrimers based on metathesis of dioxaborolanes with various commercially available polymers have shown both good processability and outstanding performance, such as mechanical, thermal, and chemical resistance, suggesting new ways of processing cross-linked polymers in industry, of which the design principle remains unknown [M. Röttger et al., Science 356, 62-65 (2017)]. Here we formulate a theoretical framework to elucidate the phase behavior of the linker-mediated vitrimers, in which entropy plays a governing role. We find that, with increasing the linker concentration, vitrimers undergo a reentrant gel-sol transition, which explains a recent experiment [S. Wu, H. Yang, S. Huang, Q. Chen, Macromolecules 53, 1180-1190 (2020)]. More intriguingly, at the low temperature limit, the linker concentration still determines the cross-linking degree of the vitrimers, which originates from the competition between the conformational entropy of polymers and the translational entropy of linkers. Our theoretical predictions agree quantitatively with computer simulations, and offer guidelines in understanding and controlling the properties of this newly developed vitrimer system.

Association of Lennard-Jones particles in nanoconfined aqueous solution: Theory and molecular dynamics simulations

We develop a new Gibbs free energy-based association model and perform series of molecular dynamics (MD) simulations to study the association behavior of Lennard-Jones (LJ) particles in nanoconfined aqueous solution. In the association model, the total Gibbs free energy is the sum of free energy cost in the nucleation of a densely packed cluster with an isotropic shape and a radius dependent interfacial tension coefficient, and the translation entropy of dispersed particles outside the cluster. Two key theoretical expressions – total Gibbs free energy formula and the thermodynamic equilibrium equation – are obtained to estimate the cluster aggregation number for a given LJ particle number and the corresponding total Gibbs free energy. Using MD simulations, we observe that the association state transforms from the stable dispersion state, through the reversible state, finally to the stable aggregation state as the LJ particle number increases. In the reversible state, the system reversibly switch in between the dispersion and aggregation states. The existence of three types of association states is also found for LJ particles with the varied well depth. According to our model, the Gibbs free energy curve shows from a single minimum, through two minima, finally to a single minimum, leading to the transformation of association states. The occurrence of reversible state transition is attributed to the free energy barrier of the order of thermal fluctuation in water between the dispersion and aggregation states. These findings deepen the understanding of nucleation/aggregation of hydrophobic gas molecules or particles in solution under nanoconfinement.

Role of local order in anomalous ion diffusion: Interrogation through tetrahedral entropy of aqueous solvation shells

Small rigid ions perturb the water structure around them significantly. At constant viscosity, alkali cations (Li+, Na+, and so on) exhibit an anomalous non-monotonic dependence of diffusivity on ion-size, in stark violation of the Stokes-Einstein expression. Although this is a well-known problem, we find that an entropic view of the problem can be developed, which provides valuable insight. The local entropy experienced by the solute ion is relevant here, which leads to the connection with local viscosity, discussed earlier by many. Due to the strong interactions with ions, the translational and rotational entropy of solvation water decreases sharply; however, an opposite effect comes from the disruption of the tetrahedral network structure of water near the charges. We compute the tetrahedral order of water molecules (qtet) around the ion and suitably defined tetrahedral entropy [S(qtet)] that is a contribution to the excess entropy of the system. Our results reveal that although the structural properties of the second shell become nearly identical to the bulk, S(qtet) of the second shell is found to play an important role in giving rise to the non-monotonic ion-size dependence. The detailed study of the static and dynamic fluctuations in qtet and the number of hydration water molecules provides interesting insights into correlation between the structure and dynamics; the smallest static fluctuation of qtet for the first hydration shell water molecules of Li+ is indicative of the iceberg picture. The study of fluctuation properties of qtet and the coordination number also reveals the role of the second hydration layer and could explain the anomalous behavior of the Rb+ ion.

Assembly of Polymer-Grafted Nanoparticles in Polymer Matrices.

Leibler pioneered the idea that long enough matrix polymers of length P will spontaneously dewet a chemically identical polymer layer, comprising chains of length N, densely end-grafted to a flat surface ("brush"). This entropically driven idea is routinely used to explain experiments in which 10-20 nm diameter nanoparticles (NPs) densely grafted with polymer chains are found to phase separate from chemically identical melts for P/N ≳4. At lower grafting densities, these effects are also thought to underpin the self-assembly of grafted NPs into a variety of structures. To explore the validity of this picture, we conducted large-scale molecular dynamics simulations of grafted NPs in a chemically identical polymer melt. For the NPs we consider, in the ≈5 nm diameter range, we find no phase separation even for P/N = 10 in the absence of attractions. Instead, we find behavior that more closely parallels experiments when all of the chain monomers are equally attractive to each other but repel the NPs. Our results thus imply that experimental situations investigated to date are dominated by the surfactancy of the NPs, which is driven by the chemical mismatch between the inorganic core and the organic ligands (the graft and free chains are chemically identical). Entropic effects, that is, the translational entropy of the NPs and the matrix, the entropy of mixing of the grafts and the matrix, and the conformational entropy of the chains appear to thus have a second-order effect even in the context of these model systems.

Exploring relationship of the state of N-dodecyl betaine in the solution monomer, at the interface and in the micelle via configurational entropy

Interfacial adsorption and bulk micellization of surfactant are common in natural world, and have prominent applications both in production and life. Surfactant molecules circularly flux among solution bulk, interface, and micelle in surfactant solution. Molecular dynamics (MD) simulations are carried out to compare the state of surfactant among solution monomer, interfacial monolayer, and micelle. Surface coverages from extremely dilute adsorption to saturated adsorption are all considered. Using the quasi harmonic approximation method, configurational entropies of N-dodecyl betaine (NDB) in different environments are obtained. The NDB monomers possess the largest freedom owing to spatial and non-aggregated superiorities. Entropy decomposition, including translational, rotational and vibrational entropies, is calculated to give further comparison and analysis. A new Born-Haber cycle is proposed to depict the state of NDB in different environments and to present entropy change of NDB itself in different mass transport processes. In particular, entropy changes of NDB in physical adsorption process below and above critical micelle concentration are obtained. The NDB molecule exhibits entropy decrease in interfacial adsorption, bulk micellization, and the transport of surfactant from micelle to surface layer.

The Phase Behavior of Nanoparticle Superlattices in the Presence of a Solvent.

Superlattices of nanoparticles coated by alkyl-chain ligands are usually prepared from a stable solution by evaporation, therefore the pathway of superlattice self-assembly critically depends on the amount of solvent present within it. This work addresses the role of the solvent on the structure and the relative stability of the different supercrystalline phases of single-component superlattices (simple cubic, body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed). The study is performed with a molecular theory for nanoparticle superlattices introduced in this work, which predicts the structure and thermodynamics of the supercrystals explicitly treating the presence and molecular details of the solvent and the ligands. The theory predicts a FCC-BCC transition with decreasing solvent content due to the competition between the translational entropy of the solvent and the entropy and internal energy of the ligands. This result provides an explanation for recent experimental observations by in situ X-ray scattering, which reported a FCC-BCC transition during solvent evaporation. The theory also predicts the effects of the length and surface coverage of the ligands and the radius of the core on the phase behavior in agreement with experimental evidence and previous molecular dynamics simulations. These results validate the use of the dimensionless softness parameter λ (ratio of ligand length to core radius) to predict the phase behavior of wet superlattices. Our results stress the importance of explicitly considering the presence of the solvent in order to reach a complete picture of the mechanisms that mediate the self-assembly of nanoparticle superlattices.

Structural Determinants of Lipid Membrane Thickening at Close Distances

Membrane fusion and fission are key processes underlying, e.g., synaptic transmission. For a quantitative characterization of these processes at the molecular level, understanding the physics of lipid bilayer interactions at close distances is essential. We have observed increased membrane thickness for early membrane fusion intermediates in SNARE-mediated membrane fusion, resembling similar effects observed previously for multilamellar-membrane stacks. To reveal the underlying structural changes and molecular causes, we have carried out all-atom molecular dynamics simulations of double-membrane systems at decreasing mutual distances. Indeed, for distances below 1.5nm, thickening is seen also in the simulations, although to a smaller extent (11%). We also observed a decreased area per lipid and increased lipid chain order parameters, strongly reduced translational and rotational entropy for water molecules between the membranes, as well as markedly slowed and tightly correlated lateral diffusion of the inwards-facing lipid molecules. Particularly for close distances, the dipolar lipid head groups become increasingly tilted, which we identified as the main determinant for the observed thickening.

Salt effect on thermodynamics and kinetics of a single RNA base pair.

Due to the polyanionic nature of RNAs, the structural folding of RNAs are sensitive to solution salt conditions, while there is still lack of a deep understanding of the salt effect on the thermodynamics and kinetics of RNAs at a single base-pair level. In this work, the thermodynamic and the kinetic parameters for the base-pair AU closing/opening at different salt concentrations were calculated by 3-µsec all-atom molecular dynamics (MD) simulations at different temperatures. It was found that for the base-pair formation, the enthalpy change is nearly independent of salt concentration, while the entropy change exhibits a linear dependence on the logarithm of salt concentration, verifying the empirical assumption based on thermodynamic experiments. Our analyses revealed that such salt concentration dependence of the entropy change mainly results from the dependence of ion translational entropy change for the base pair closing/opening on salt concentration. Furthermore, the closing rate increases with the increasing of salt concentration, while the opening rate is nearly independent of salt concentration. Additionally, our analyses revealed that the free energy surface for describing the base-pair opening and closing dynamics becomes more rugged with the decrease of salt concentration.

Experiments and Simulations Probing Local Domain Bulge and String Assembly of Aligned Nanoplates in a Lamellar Diblock Copolymer

Within ordered poly(styrene-b-methyl methacrylate) (PS-b-PMMA) block copolymer (BCP) lamellae, oriented nanoplates [gadolinium trifluoride doped with ytterbium and erbium, GdF3:Yb/Er (20/2 mol %)] grafted with polyethylene glycol assemble into aligned strings at small interparticle separations. Herein, we investigate the origin of these aligned assemblies using X-ray scattering, electron microscopy, and hybrid particle/self-consistent field theory simulations. From previous reports, the insertion of a nanoplate in a BCP microdomain is expected to perturb the polymer chains and produce a local domain bulge as the PS/PMMA interface distorts to optimize conformational entropy. While experimental techniques are unable to directly resolve this small distortion, the two-dimensional simulations of the equilibrium BCP nanocomposite structure clearly show bulge formation around the nanoplates. As a function of particle separation, the potential of mean force (PMF) calculation reveals a global minimum corresponding to an equilibrium interparticle spacing of 7.0 nm, which agrees well with a mean experimental value of 6.4 nm. Furthermore, the PMF calculation exhibits a small activation barrier due to the high curvature penalty between two nanoplates at a separation distance of 21.7 nm. Ultimately, nanoplate strings form to fulfill the criteria where the energy benefit of decreasing interfacial area and minimizing chain stretching outweighs the energy penalty associated with reducing the translational entropy affiliated with evenly distributing the nanoplates. The simulations also illustrate a narrow tolerance for orientation angles, supporting the high degree of nanoplate alignment observed experimentally. We anticipate that the ability to align and couple anisotropic nanoparticles in BCPs presents opportunities to create functional polymer nanocomposites with orientation-dependent properties.

Force driven transition of a globular polyelectrolyte.

We have systematically studied behavior of a flexible polyelectrolyte (PE) chain with explicit counterions, subjected to a constant force at the terminal ends. Our simulations reveal that in the hydrophobic regime, a PE globule abruptly opens to a coil state beyond a critical force Fc. At the transition point, the polymer shape shows large scale fluctuations that are quantified in terms of end-to-end distance Re. These fluctuations suggest that the system coexists in globule and coil states at the transition, which is also confirmed from the bimodal distribution of Re. Moreover, the critical force associated with the globule coil transition exhibits a nonmonotonic behavior, where surprisingly, Fc decreases with Bjerrum length lB in the limit of small lB, followed by an increase in the larger lB limit. Furthermore, this behavior is also validated from a theory adopted for the PE. From the free energy analysis, we have demonstrated that predominantly, the competition between the intrachain repulsive energy, counterion's translational entropy, and adsorption energy leads to the novel feature of nonmonotonic behavior of force.