Shape Of Micelles Research Articles

Green Synthesis of Polyesters Using Surfactant Catalysts in Microemulsions: The Role of Micelle Shape and Aggregation on Polymer Formation

Green Synthesis of Polyesters Using Surfactant Catalysts in Microemulsions: The Role of Micelle Shape and Aggregation on Polymer Formation

Effects of dilution and age on perfume release from two mixed‐surfactant systems containing hydrotropes

AbstractThis study investigates the encapsulation and controlled release of perfume within micellar systems. We explored how perfume‐surfactant interactions and micelle structure influence fragrance delivery. Two mixed‐micelle systems, one with branched‐tail and the other with linear‐tail surfactants, were subjected to varying perfume concentrations, dilution, and time. Using small‐angle neutron scattering (SANS), gas chromatography‐mass spectrometry (GC‐MS), and statistical analysis, we determined that micelle size and shape are significantly affected by perfume content and dilution. While both systems exhibited similar trends, the branched‐tail system uniquely formed a lamellar phase at higher perfume levels. Our findings reveal a strong correlation between micelle geometry and headspace perfume concentration, highlighting the role of surfactant tail structure. This research provides fundamental insights into perfume‐micelle interactions, with potential applications in materials science and active ingredient delivery.

Investigating micelle formation in systems with benzethonium chloride and n-lauroyl sarcosine sodium salt: The impact of thermal effects and ethyl lactate concentration using volumetric, statistical, acoustic, and molecular dynamic methods

Surface active agents (SAAs) are versatile molecules that possess the ability to serve as multifunctional ingredients in a wide range of consumer products across various industrial sectors. These molecules can act as wetting and dispersion agents, emulsifiers, foaming and anti-foaming agents, lubricants, and more. The objective of this study was to evaluate the effect of temperature, and ethyl lactate concentration on micelle formation in systems containing benzethonium chloride (BC) and N-lauroyl sarcosine sodium salt (NLSS). The experimental measurements were conducted at three equidistant temperatures i.e., 298.15 K to 318.15 K and pressure, P=0.1 MPa. Several volumetric as well as compressibility parameters including apparent molar properties (Vϕ,Kϕ,S), partial molar properties (Vϕ°,Kϕ,S°), transport properties (ΔtrVϕ°,ΔtrKϕ,S°), hydration number, etc. have been evaluated. A Bayesian two-factor design with uninformative priors was used to analyze the values of Vϕ°,and Kϕ,S°. It was possible to determine that the temperature factors, and the ethyl lactate (EL) addition significantly influence the results for Vϕ° while for Kϕ,S°it is not appreciable at 5 and 10 % in EL concentration when BC and NLSS systems is compared. The simulations of the micellization process show the impact of temperature, particularly in the presence of ethyl lactate, is evident in the structural and dynamic changes observed in the micelles. Specifically, as the temperature rises from 298.15 K to 318.15 K, the self-assembly process becomes more spontaneous and accelerated, leading to larger and more spherical micelles. These alterations in micelle size, shape, and orientation enhance their capacity to effectively solubilize hydrophobic substances. The computational simulations demonstrated that BC molecules exhibit spontaneous aggregation at the interface of polar-nonpolar systems, resulting in the formation of a monolayer. In this monolayer, the heads of the molecules are oriented towards the polar phase, while the hydrophobic tails are in the nonpolar phase. Similarly, the NLSS system exhibits a continuous and gradual process of self-organization over a period of 120 ns, where NLSS molecules naturally come together to create a single layer arrangement at the zigzag-shaped boundary.

Influence of chirality and sequence in lysine-rich lipopeptide biosurfactants and micellar model colloid systems

Lipopeptides can self-assemble into diverse nanostructures which can be programmed to incorporate peptide sequences to achieve a remarkable range of bioactivities. Here, the influence of peptide sequence and chirality on micelle structure and interactions is investigated in a series of lipopeptides bearing two lysine or D-lysine residues and tyrosine or tryptophan residues, attached to a hexadecyl lipid chain. All molecules self-assemble into micelles above a critical micelle concentration (CMC). Small-angle x-ray scattering (SAXS) is used to probe micelle shape and structure from the form factor and to probe inter-micellar interactions via analysis of structure factor. The CMC is obtained consistently from surface tension and electrical conductivity measurements. We introduce a method to obtain the zeta potential from the SAXS structure factor which is in good agreement with directly measured values. Atomistic molecular dynamics simulations provide insights into molecular packing and conformation within the lipopeptide micelles which constitute model self-assembling colloidal systems and biomaterials.

Detecting Shape of Hybrid Polymer/Surfactant Micelles: Cryo-Transmission Electron Microscopy, Small-Angle Neutron Scattering and Dynamic Light Scattering Study

In-depth study of shape of hybrid micelles in the micellar solutions of anionic surfactant potassium oleate, containing hydrophobic polymer poly(4-vinylpyridine) (P4VP) was conducted via cryo-transmission electron microscopy (cryo-TEM), small-angle neutron scattering (SANS) and dynamic-light scattering of visible light (DLS). Direct visualization of the solutions with cryo-TEM evidenced the coexistence of polymer-free spherical micelles and branched rodlike hybrid micelles with mean length of 200 nm and radius of 2 nm, governed by contour length of solubilized P4VP and length of hydrophobic “tail” of potassium oleate, respectively. The formation of branches in the hybrid micelles was explained by attaching the thermodynamically unfavorable end-caps of micelles to their polymer-loaded cylindrical fragments. By SANS it was shown that the cylindrical local shape and the radius of the micelles are independent of the concentration of embedded P4VP. Relaxation processes in the solutions were investigated with DLS. Three relaxation modes were observed for hybrid micelles, similar to polymer-free wormlike micelles. Fast and medium relaxation modes were attributed to diffusion of entangled micellar chains and their segments, respectively. The slow mode was related to electrostatic repulsion between similarly charged hybrid micelles.

Shape of AOT Reverse Micelles: The Mesoscopic Assembly Is More Than the Sum of the Parts.

AOT reverse micelles are a common and convenient model system for studying the effects of nanoconfinement on aqueous solutions. The reverse micelle shape is important to understanding how the constituent components come together to form the coherent whole and the unique properties observed there. The shape of reverse micelles impacts the amount of interface present and the distance of the solute from the interface and is therefore vital to understanding interfacial properties and the behavior of solutes in the polar core. In this work, we use previously introduced measures of shape, the coordinate-pair eccentricity (CPE) and convexity, and apply them to a series of simulations of AOT reverse micelles. We simulate the most commonly used force field for AOT reverse micelles, the CHARMM force field, but we also adapt the OPLS force field for use with AOT, the first work to do so, in addition to using both 3- and 4-site water models. Altogether, these simulations are designed to examine the impact of the force field on the shape of the reverse micelles in detail. We also study the time autocorrelation of shape, the water rotational anisotropy decay, and how the CPE changes between the water pool and AOT tail groups. We find that although the force field changes the shape noticeably, AOT reverse micelles are always amorphous particles. The shape of the micelles changes on the order of 10 ns. The water rotational dynamics observed match the experiment and demonstrate slower dynamics relative to bulk water, suggesting a two-population model that fits a core/shell hypothesis. Taken together, our results indicate that it is likely not possible to create a perfect force field that can reproduce every aspect of the AOT reverse micelle accurately. However, the magnitude of the differences between simulations appears relatively small, suggesting that any reasonably derived force field should provide an acceptable model for most work on AOT reverse micelles.

The Effect of Different Substances Embedded in Fullerene Cavity on Surfactant Self-Assembly Behavior through Molecular Dynamics Simulation.

Fullerene-based amphiphiles are new types of monomers that form self-assemblies with profound applications. The conical fullerene amphiphiles (CFAs) have attracted attention for their uniquely self-assembled structures and have opened up a new field for amphiphile research. The CFAs and CFAs with different substances embedded in cavities are designed and their self-assembly behaviors are investigated using molecular dynamics (MD) simulations. The surface and internal structures of the micelles are analyzed from various perspectives, including micelle size, shape, and solvent-accessible surface area (SASA). The systems studied are all oblate micelles. In comparison, embedding Cl- or embedding Na+ in the cavities results in larger micelles and a larger deviation from the spherical shape. Two typical configurations of fullerene surfactant micelles, quadrilateral plane and tetrahedral structure, are presented. The dipole moments of the fullerene molecules are also calculated, and the results show that the embedded negatively charged Cl- leads to a decrease in the polarity of the pure fullerene molecules, while the embedded positively charged Na+ leads to an increase.

Coarse grained MD simulation of bulk and interfacial behavior of mixture of CTAB/SDS surfactants.

This study involves simulating the process of inhibiting corrosion through the formation of micelles by surfactants and their deposition on iron (Fe) surfaces. The primary focus is on examining CTAB/SDS mixtures in aqueous solutions with different concentrations. Micelle properties, including size, shape, aggregation number, cluster size, and surfactant diffusion, were calculated and validated with experimental data. The coarse-grained Fe surface was modeled and validated against experimental water contact-angle data. Subsequently, the deposition of CTAB/SDS mixtures on the Fe surface and air-water interface was studied systematically. We found that the relative ratio of CTAB/SDS in the solution directly influences surfactant deposition behavior, which might impact the corrosion inhibition efficiency. All the MD simulations were performed using the GROMACS software with MARTINI2 force field and Martini polar water. The molecules are packed using PACKMOL software. Both NVT and NPT simulations are caried out at temperature and pressure of 303 K and 1 bar respectively, with a nonbonded interaction cut-off (rcut) of 1.1 nm. The LJ potential was shifted from 0.9 nm to rcut, while the electrostatic potential was shifted from 0.0 nm to rcut. For electrostatics, reaction-field coulomb type is used, relative dielectric constant (epsilon-r) and the reaction field dielectric constant (epsilon-rf) are equal to 2.5 and infinity respectively. The dielectric constant below rcut is epsilon-r, and beyond the cut-off is epsilon-rf. Coulomb-modifier used as potential-shift which leads to shift in the coulomb potential by a constant such that it is zero at the rcut. This makes the potential of the integral of the force . The neighbor list was updated every 10 steps, employing a neighbor list cut-off equal to rcut. Using a polar water model, we used a constant time step of 0.02 ps throughout the simulation. The used epsilon-r = 2.5, is recommended for polar water.

Sodium dodecyl sulfate micelles: Accurate analysis of small-angle X-ray scattering data through form factor and atomistic molecular dynamics modelling

The structure of micelles of sodium dodecyl sulfate (SDS) is probed via analysis of small-angle x-ray scattering (SAXS) data with the aim to fit the data over an extended wavenumber q range. This provides detailed information on the micelle shape, which can be described as a polydisperse triaxial ellipsoidal core-shell structure according to model form factor fitting. This model was necessary to fit the data over a wide q range, which is not accurately represented by simpler models such as biaxial ellipsoidal core-shell structures. Data for SDS (at fixed concentration) in a NaCl concentration series revealed increasing structure factor effects with decreasing salt concentration. This reflects decreased charge screening on the headgroups. The structure factor could be modelled using a simple hard sphere structure factor. The analysis of form factor was complemented by atomistic molecular dynamics (MD) simulations, starting from an unbiased initial configuration of a defined number of molecules in a box. The MD configurations were used to calculate the form factor using the software CRYSOL (for small-angle scattering analysis of solution scattering, traditionally for proteins, here for micelles) and accounting for the boundary layer hydration effects. Good agreement with experimental data was found for systems with association numbers close to p = 60. This association number is consistent with that obtained from analysis of the form factor (in the case where structure factor effects could be neglected) and from model-free analysis of the forward scattering intensity. It is also in agreement with prior literature and our findings in regard to form factor parameters are also compared to previous reports.

Harnessing the potential of fatty Acid-Surfactant-Based micellar gel for enhanced topical delivery of Apremilast in psoriasis treatment

Apremilast (APR) is a potent anti-psoriatic agent that inhibits the phosphodiesterase 4 enzyme. Due to the poor oral bioavailability and associated systemic side effects the clinical applicability of APR has been constrained. Nanotechnology-based carrier system presents a novel option to increase the efficacy of the topical treatment of APR. The current investigation deals with the development of fatty acid-surfactant conjugate-based hybrid mixed micellar gel (HMMG) for the topical delivery of APR. The developed micelles exhibited an average size of 83.59 ± 4.46 nm, PDI of 0.239 ± 0.047, % entrapment efficiency of ∼ 94.78 ± 3.98 %, with % practical drug loading of ∼11.37 ± 3.14 %. TEM analysis revealed the spherical shape of micelles. The hybrid micelles were further loaded in a carbopol®934P gel base for ease of application. Ex vivo permeation study revealed enhanced permeation and ∼ 38-fold higher retention in deeper layers of skin from a hybrid micellar gel. In vivo, assessment demonstrated augmented efficacy of APR-HMMG as compared to 0.1 % betamethasone valerate. Also, APR-HMMG showed no sign of irritation, suggesting superior safety as a topical application. Thus, the proposed formulation strategy represents a viable avenue for enhancing the therapeutic efficacy of various anti-psoriatic moieties.

Counter-ion adsorption and electrostatic potential in sodium and choline dodecyl sulfate micelles - a molecular dynamics simulation study.

Choline-based surfactants are interesting both from the practical point of view to obtaining environmental-friendly surfactants as well as from the theoretical side since the interactions between the choline and surfactants can help to understand self-assembly phenomena in deep eutectic solvents. Although no significant change was noticed in the micelle size and shape due to the exchange of the sodium counter-ion by choline in our simulations, the adsorption of the choline cation over the micelle surface is stronger than the adsorption of the sodium, which leads to a reduction of the exposed surface area of the micelle and remarkable effects over the electrostatic potential. The choline neutralizes the surface charge of the surfactant better than sodium; however, this is partially compensated by a stronger water orientation around the SDS micelle. The balance between the contributions from the surfactant, the counter-ion, and water to the electrostatic potential leads to a complex pattern with alternate regions of positive and negative potential at the micelle/water interface which can be important to the incorporation of other charged species at the micelle surface as well as for the interaction between micelles in solution. To evaluate the effects of the counter-ion substitution, micelles of sodium dodecyl sulfate (SDS) and choline dodecyl sulfate (ChDS) were studied and compared by means of molecular dynamics simulations in aqueous solution. In both cases, the simulations started from pre-assembled micelles with 60 dodecyl sulfate ions and 240-ns simulations were performed at NPT ensemble at T = 323.15K and P = 1bar using the Gromacs software with the OPLS-AA force field to describe dodecyl sulfate and choline, Åqvist parameters for sodium, and SPC model for water molecules.

Investigation of the shapes and sizes of soft molecular aggregates in solution via partial molar volume and dynamic light scattering measurements

Soft molecular aggregates are naturally dynamic and deformable nanoscale objects of various shapes and sizes. However, soft nanoparticle shape can be difficult to measure and often requires advanced instrumentation or methods. Yet, foreknowledge of soft aggregate overall shape is often critically important in many experiments that measure the aggregate’s internal structure and properties. Small-angle neutron and x-ray scattering are often used to measure soft nanoparticle shape. Molecular dynamics simulations can also give an indication of soft molecular shape but, like small-angle scattering methods, requires considerable expense and training. To complement these approaches a more straightforward experimental method of determining soft molecular aggregate shape using a combination of partial molar volume and dynamic light scattering measurements (PMV-DLS) is presented. PMV-DLS is inexpensive, requires modest instrumentation, and can be performed in any laboratory. In this article, the PMV-DLS technique is demonstrated on two separate soft molecular aggregates: AOT reverse micelles and Triton X-100 micelles. The shapes of these aggregates were investigated over a range of solution properties. In both cases, the penetration of continuous phase solvent molecules into the aggregate was found to play an important role in shape determination. AOT reverse micelles were observed to transition from ellipsoidal to spherical as the solution water content increased. Triton X-100 micelle shape was found to be critically dependent upon the amount of water that partitions into the surfactant polyethylene oxide head group region.

Many-body dissipative particle dynamics simulations of micellization of sodium alkyl sulfates.

We present a study of micelle formation in alkyl sulfate surfactants using the simulation method of many-body dissipative particle dynamics (MDPD). We parametrise our model by tuning the intermolecular interactions in order to reproduce experimental values for the chemical potential and density at room temperature. Using this approach, we find that our model shows good agreement with experimental values for the critical micelle concentration (CMC). Furthermore, we show that our model can accurately predict CMC trends, which result from varying properties such as surfactant tail length and the salt concentration. We apply our model to investigate the effect of aggregation number on various micellar properties, such as the shape of individual micelles and the fraction of bound counterions. We show that micelles become aspherical at large aggregation numbers, in line with experimental predictions, and that longer tail surfactants are generally more spherical at all aggregation numbers compared to those which are shorter. We find excellent agreement between our simulations and experimental values for the degree of counterion binding, a factor that is crucial to accurately studying micellar shape, but one that is typically overlooked in the existing literature.

Impact of a poly(ethylene glycol) corona block on drug encapsulation during polymerization induced self-assembly.

Polymerization induced self-assembly (PISA) provides a facile platform for encapsulating therapeutics within block copolymer nanoparticles. Performing PISA in the presence of a hydrophobic drug alters both the nanoparticle shape and encapsulation efficiency. While previous studies primarily examined the interactions between the drug and hydrophobic core block, this work explores the impact of the hydrophilic corona block on encapsulation. Poly(ethylene glycol) (PEG) and poly(2-hydroxypropyl methacrylate) (PHPMA) are used as the model corona and core blocks, respectively, and phenylacetic acid (PA) is employed as the model drug. Attachment of a dithiobenzoate end group to the PEG homopolymer - transforming it into a macroscopic reversible addition-fragmentation chain transfer agent - causes the polymer to form a small number of nanoscopic aggregates in solution. Adding PA to the PEG solution encourages further aggregation and macroscopic phase separation. During the PISA of PEG-PHPMA block copolymers, inclusion of PA in the reaction mixture promotes faster nucleation of spherical micelles. Although increasing the targeted PA loading from 0 to 20 mg mL-1 does not affect the micelle size or shape, it alters the drug spatial distribution within the PISA microenvironment. PA partitions into either PEG-PHPMA micelles, deuterium oxide, or other polymeric species - including PEG aggregates and unimer chains. Increasing the targeted PA loading changes the fraction of drug within each encapsulation site. This work indicates that the corona block plays a critical role in dictating drug encapsulation during PISA.

Temperature dependence of micelle shape transitions in copolymer solutions: the role of inter-block incompatibility.

The nature of the transition between worm-like and spherical micelles in block copolymer dispersions varies between systems. In some formulations, heating drives a transition from worms to spheres, while in other systems the same transition is induced by cooling. In addition, a sphere-worm interconversion can be accompanied either by an increase or a decrease in the core solvation, even if the direction of the temperature dependence is the same. Here, self-consistent field theory is used to provide a potential explanation of this range of behaviour. Specifically, we show that, within this model, the dependence of the transition on the incompatibility χBS of the solvophobic block B and the solvent S (the parameter most closely related to the temperature) is strongly influenced by the incompatibility χAB between B and the solvophilic block A. When χAB is small (χAB < 0.1), it is found that increasing χBS produces a transition from worm-like micelles to spheres (or, more generally, from less curved to more curved structures). When χAB is above 0.1, increasing χBS drives the system from spheres to worm-like micelles. Whether a transition is observed within a realistic range of χBS is also found to depend on the fraction of solvophilic material in the copolymer. The relevance of our calculations to experiments is discussed, and we suggest that the direction of the temperature dependence may be controlled not only by the solution behaviour of the solvophobic block (upper critical solution temperature-like versus lower critical solution temperature-like) but also by χAB.

Hybrid Polymer-Surfactant Wormlike Micelles for Concurrent Use for Oil Recovery and Drag Reduction.

We report on the effect of a hydrocarbon (n-dodecane) on the rheological properties and shapes of the hybrid wormlike micelles (WLMs) of a surfactant potassium oleate with an embedded polymer poly(4-vinylpyridine). With and without hydrocarbon solutions, the hybrid micelles exhibit the same values of viscosity at shear rates typical for hydraulic fracturing (HF) tests, as solutions of polymer-free WLMs. Therefore, similar to WLMs of surfactants, they could be applied as thickeners in HF fluids without breakers. At the same time, in the presence of n-dodecane, the hybrid micelles have much higher drag-reducing efficiency compared to microemulsions formed in polymer-free systems since they form "beads-on-string" structures according to results obtained using cryo-transmission electron microscopy (cryo-TEM), dynamic-light scattering (DLS), and small-angle X-ray scattering (SAXS). Consequently, they could also act as drag-reducing agents in the pipeline transport of recovered oil. Such a unique multi-functional additive to a fracturing fluid, which permits its concurrent use in oil production and oil transportation, has not been proposed before.

Impact of micelle characteristics on cholesterol absorption and ezetimibe inhibition: Insights from Niemann-Pick C1-like 1 binding and molecular structure

Yamanashi et al., conducted a study on the absorption of cholesterol and β-sitosterol, as well as the inhibitory effect of ezetimibe (EZE). They used CaCo-2 cells to simulate the intestines and investigated how different mixed micelles, acting as carriers, were absorbed into these cells through the Niemann-Pick C1-like 1 (NPC1L1) protein. The study focused on the impact of micelle shape, size, and zeta potential on absorption and the inhibitory effect of EZE. I utilized small-angle X-ray scattering and a zeta potential measuring device to measure these characteristics. The findings revealed a two-step mechanism: NPC1L1 selectively bound micelles based on their shape and size, and once bound, the absorption was regulated by the molecular structure of the micelle components. EZE's inhibitory effect changed with micelle composition, influencing micelle size and shape. EZE initially acted on the micelle’s shape and size, and then NPC1L1 selectively bound micelles based on their shape and size, allowing EZE to directly inhibit absorption by interacting with NPC1L1. This groundbreaking discovery challenges existing concepts and holds significant implications for researchers in drug development, as well as physicians and pharmacists.

Stimuli-Responsive Systems Based on Polymer-like Wormlike Micelles of Ionic Surfactants and Their Modern Applications

Solutions based on polymer-like micelles of ionic surfactants attract the attention of scientists as stimuli-responsive systems that can transform from a low-viscosity Newtonian fluid to a viscoelastic solution having a viscosity of up to 10 kPa s and an elastic response. The shape of micelles and their length are determined by the balance of hydrophobic and electrostatic interactions; therefore, the use of various low molecular weight substances or functional nanoparticles as additives can significantly change the rheological properties of such solutions and impart new stimuli-responsive properties to them. This review presents the most commonly used methods for controlling the properties of multicomponent solutions of wormlike surfactant micelles, including nanocomposite systems. Modern areas of practical application of such systems and prospects for their development are described.

Aqueous Binary Mixtures of Stearic Acid and Its Hydroxylated Counterpart 12-Hydroxystearic Acid: Fine Tuning of the Lamellar/Micelle Threshold Temperature Transition and of the Micelle Shape.

This study examines the structures of soft surfactant-based biomaterials which can be tuned by temperature. More precisely, investigated here is the behavior of stearic acid (SA) and 12-hydroxystearic acid (12-HSA) aqueous mixtures as a function of temperature and the 12-HSA/SA molar ratio (R). Whatever R is, the system exhibits a morphological transition at a given threshold temperature, from multilamellar self-assemblies at low temperature to small micelles at high temperature, as shown by a combination of transmittance measurements, Wide Angle X-ray diffraction (WAXS), small angle neutron scattering (SANS), and differential scanning calorimetry (DSC) experiments. The precise determination of the threshold temperature, which ranges between 20 °C and 50 °C depending on R, allows for the construction of the whole phase diagram of the system as a function of R. At high temperature, the micelles that are formed are oblate for pure SA solutions (R = 0) and prolate for pure 12-HSA solutions (R = 1). In the case of mixtures, there is a progressive continuous transition from oblate to prolate shapes when increasing R, with micelles that are almost purely spherical for R = 0.33.

Micelle conformation of sodium alkyl sulfate surfactants with different hydrophobic chain length: A molecular dynamics study

Based on all-atom molecular dynamics (MD) simulation, micelle properties and microstructures of sodium alkyl sulfate (SAS) with different hydrophobic chain length in explicit aqueous solution are investigated. With the growth of hydrophobic alkyl chain (C8 ∼ C16), the shape of micelle gradually transform from spherical micelle into ellipsoidal or cylindrical aggregate. Longer hydrophobic tail chains collapse inside the micelle and results the smaller exposed surface, which leads to the decreased solvent accessible surface area (SASA) and hydrogen bond. The conformation of C8 ∼ C14 micelles is solid sphere or ellipsoid and the effect of expanding space by the hydrophobic tail chains is dominant. And the hydrophobic tail chains of C16 micelle collapse inside the micelle and extend to the micelle shell but do not expose to the surface of the micelle. The motion of shorter alkyl chain in micelle is restrictive and longer alkyl chain is active. Longer hydrophobic alkyl chain prompts the SAS molecules to move form aqueous solution to form micelle spontaneously due to the increase of entropy.