Simulation study of the coil-globule transition of a polymer in solvent

Molecular dynamics simulations are used to study the coil-globule transition for a system composed of a bead-spring polymer immersed in an explicitly modeled solvent. Two different versions of the model are used, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambdatheta separating the coil from the globule states. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. The results are analyzed in terms of a simple Flory-type theory of the collapse transition. The ratio of lambdatheta for the two models converges in the high density limit exactly to the value predicted by the theory in the random mixing approximation. Generally, the predicted values of lambdatheta are in reasonable agreement with the measured values at high rho, though the accuracy improves if the average chain size is calculated using the full probability distribution associated with the polymer-solvent free energy, rather than merely using the value obtained from the minimum of the free energy.     

Diagram of State of Stiff Amphiphilic Macromolecules

Summary: We studied coil-globule transitions in stiff-chain amphiphilic macromolecules via computer modeling and constructed phase diagrams for such molecules in terms of solvent quality and persistence length. We showed that the shape of the phase diagram essentially depends on the macromolecule degree of polymerization. Relatively short amphiphilic molecules always form a spherical globule in a poor solvent, and the coil-globule transition includes one or two intermediate conformations, depending on the chain's stiffness. These are a disk-like globule in case of high enough Kuhn segment length, and a pearl necklace-like structure of spherical micelles and a disk-like globule in case of relatively flexible chains. The phase diagram of a long stiff amphiphilic chain was found to be more complex still. Thus three specific regions can be distinguished in the poor solvent region, depending on the chain stiffness. These correspond to a cylindrical globule without any specific backbone ordering, a cylindrical globule containing blobs with collagen-like ordering of the chain, and co-existence of collagen-like and toroidal globules. In the intermediate transition region in this case, apart from the pearl necklace-like conformations with spherical micelles, necklace conformations can be also observed where the polymeric chain has collagen-like ordering within each bead.     

The size of a polymer in a symmetric solvent

Using a simple thermodynamic model, we derive an expression for the excluded volume parameter v of a polymer chain in a symmetric solvent (solvated by its own monomers). For a chain with a given segment length and number of monomers, this parameter determines whether the chain is collapsed or expanded. For the latter it determines the degree of expansion. Using a simple off-lattice version of Flory's model [P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, 1953)] and relaxing the assumption of incompressibility, we obtain the result v=(1-kappa)rho(0), where kappa is the dimensionless compressibility and rho(0) the number density of solvent. In the incompressible limit (in the sense that kappa-->0) the chain is expanded and the inverse of the solvent number density determines the degree of expansion of the chain. Using the van der Waals equation of state to estimate kappa (allowing for nonzero compressibility in a system that can undergo a gas-liquid phase transition), the model predicts that upon raising the temperature at constant pressure there is both a lower (coil to globule) and upper (globule to coil) Flory temperature. This is in quantitative agreement with experiment and computer simulations.     

Knots in globule and coil phases of a model polyethylene

We examine the statistics of knots with numerical simulations of a simplified model of polyethylene. We can simulate polymers of up to 1000 monomers (each representing roughly three CH(2) groups), at a range of temperatures spanning coil (good solvent) and globule (bad solvent) phases. We quantify the abundance of knots in the globule phase and in confined polymers, and their rarity in the swollen phase. Since our polymers are open, we consider (and test) various operational definitions for knots, which are rigorously defined only for closed chains. We also associate a typical size with individual knots, which are found to be small (tight and localized) in the swollen phase but large (loose and spread out) in the dense phases.     

Conformational properties of rigid-chain amphiphilic macromolecules: The phase diagram

The coil-globule transition in rigid-chain amphiphilic macromolecules was studied by means of computer simulation, and the phase diagrams for such molecules in the solvent quality-persistence length coordinates were constructed. It was shown that the type of phase diagram depends to a substantial extent on the degree of polymerization of a macromolecule. Relatively short amphiphilic macromolecules in the poor-solvent region always form a spherical globule, with the transition to this globule involving one or two intermediate conformations. These are the disk globule if the Kuhn segment is relatively large and the string of spherical micelles or the disk globule in the case of relative flexible chains. The phase diagram of a long rodlike amphiphilic chain turned out to be even more complex. Namely, three characteristic regions were distinguished in the region of a poor solvent, depending on the chain rigidity: the region of a cylindrical globule without certain order in the main chain, the region of the cylindrical globule with blobs having the collagen ordering of the chain, and the region of coexistence of collagen-like and toroidal globules. In the intermediate transitional region, not only conformations of strings of spherical micelle beads but also the necklace conformations in which the polymer chain in each bead has collagen ordering can occur in this case.     

Amphiphilic comb macromolecules with different distribution statistics of side-chain grafting sites: Mathematical modeling

The influence of the distribution statistics of side-chain grafting sites on the conformational properties of amphiphilic comblike macromolecules immersed in a solvent that is poor for the main chain and good for the side chains was studied. It was shown that the coil-globule transition for macromolecules with the protein-like distribution of side-chain grafting sites occurs at higher temperatures, wherein the size of the proteinlike macromolecules is generally smaller than that of the corresponding regular macromolecules. Regardless of distribution statistics of side-chain grafting sites, the coil-globule transition of comb macromolecules passes through the step of the formation of the beads-on-a-string conformation composed of micelle-like beads. The temperature dependence curves of the heat capacity exhibit at least two maximums associated with the coil-globule transition per se and the coalescence of the beads into a single globule. The coil-globule transition temperature is slightly dependent upon the degree of polymerization of the main chain and drops with a decrease in the degree of polymerization of the side chains. It was found that comb macromolecules can form spherical, disklike, or cylindrical globules, depending on the structural parameters.     

An off-lattice Wang-Landau study of the coil-globule and melting transitions of a flexible homopolymer

The Wang-Landau Monte Carlo approach is applied to the coil-globule and melting transitions of off-lattice flexible homopolymers. The solid-liquid melting point and coil-globule transition temperatures are identified by their respective peaks in the heat capacity as a function of temperature. The melting and theta points are well separated, indicating that the coil-globule transition occurs separately from melting even in the thermodynamic limit. We also observe a feature in the heat capacity between the coil-globule and melting transitions which we attribute to a transformation from a low-density liquid globule to a high-density liquid globule.     

Kinetics of collapse and decollapse of a single double-stranded DNA chain

Transition of a single duplex DNA between elongated coil and condensed globule has been studied by the technique of single molecular observation with fluorescence microscopy. It has become clear that individual DNA chains undergo first-order phase transition. We have observed the time-dependent change of the DNA structure accompanied with the phase transition from coil to globule. The speed of the compaction was found to be almost constant along the DNA chain. It is indicated that the coil-globule transition exhibits the phenomenon of “nucleation and growth”. The process of the decollapse of a single DNA has also been observed, the time dependence of the long axis length being described as l ∼ t^1.8.     

Counterion mixing effects on the conformational transitions of polyelectrolytes. II. Counterion binding as measured by NMR spectroscopy of alkali metal poly(acrylate)s

Bound states of counterions during the coil‐globule transition of poly(acrylic acid) in water/organic solvent mixtures were investigated by NMR spectroscopy of alkali metal cations (Li⁺, Na⁺, Cs⁺). Accompanying the transition, the line widths of the respective NMR peaks significantly increased with increasing the organic solvent composition in the medium. Although this line width broadening suggests that some specific counterion binding with desolvation is involved with the coil‐globule transition, the most marked broadening was observed in higher organic solvent compositions than those of the coil‐globule transition region detected by the viscometry. Namely, the specific counterion binding with desolvation proceeds even after the polymer chain collapsed. This means in turn that such a strong counterion binding is not a prerequisite for the coil‐globule transition, at least at the stage of the onset. For the Li⁺/Cs⁺ mixed counterion system in 60 vol % DMSO, where our previous conductivity data suggested that the specific counterion binding occurred only for Cs⁺ during the coil‐globule transition induced on mixing with Li⁺, a significant increase in the line width was also observed only for Cs⁺. The coincidence between the conductivity and the NMR results for the Li⁺/Cs⁺ mixed counterion system strongly supports a working hypothesis, “size‐fitting effect,” that has been proposed to determine the counterion specificity observed for the conformational transitions of polyelectrolytes. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2132–2139, 2009     

Secondary structure of globules of copolymers consisting of amphiphilic and hydrophilic units: Effect of potential range

The relation of the coil-globule transition in macromolecules consisting of amphiphilic and hydrophilic monomer units to the radius of action of the interaction potential is investigated by the method of computer-assisted experiments. The internal structure of globules formed by such macromolecules is significantly dependent on the radius of action of the potential. In the case of the long-range potential, the globule is characterized by the blob structure, while in the case of the short-range potential, a quasi-helical structure forms. In this structure, the skeleton of a macromolecule forms a helical turn, and the direction of twisting may vary from one turn to another. The coil-globule transition in such macromolecules proceeds through formation of the necklace conformation from quasi-helical micelle beads. For sufficiently long macromolecules, the dimensions of such globules are linearly dependent on the degree of polymerization.     

Simulation analysis of the temperature dependence of lignin structure and dynamics

Lignins are hydrophobic, branched polymers that regulate water conduction and provide protection against chemical and biological degradation in plant cell walls. Lignins also form a residual barrier to effective hydrolysis of plant biomass pretreated at elevated temperatures in cellulosic ethanol production. Here, the temperature-dependent structure and dynamics of individual softwood lignin polymers in aqueous solution are examined using extensive (17 μs) molecular dynamics simulations. With decreasing temperature the lignins are found to transition from mobile, extended to glassy, compact states. The polymers are composed of blobs, inside which the radius of gyration of a polymer segment is a power-law function of the number of monomers comprising it. In the low temperature states the blobs are interpermeable, the polymer does not conform to Zimm/Stockmayer theory, and branching does not lead to reduction of the polymer size, the radius of gyration being instead determined by shape anisotropy. At high temperatures the blobs become spatially separated leading to a fractal crumpled globule form. The low-temperature collapse is thermodynamically driven by the increase of the translational entropy and density fluctuations of water molecules removed from the hydration shell, thus distinguishing lignin collapse from enthalpically driven coil-globule polymer transitions and providing a thermodynamic role of hydration water density fluctuations in driving hydrophobic polymer collapse. Although hydrophobic, lignin is wetted, leading to locally enhanced chain dynamics of solvent-exposed monomers. The detailed characterization obtained here provides insight at atomic detail into processes relevant to biomass pretreatment for cellulosic ethanol production and general polymer coil-globule transition phenomena.     

Phase transitions of a polymer threading a membrane coupled to coil-globule transitions

We theoretically study phase transitions of a polymer threading through a pore imbedded in a membrane. We focus on the coupling between a partition of the polymer segments through the membrane and a coil-globule transition of the single polymer chain. Based on the Flory model for collapse transitions of a polymer chain, we calculate the fraction of polymer segments and the expansion factor of a polymer coil on each side of the membrane. We predict a first-order phase transition of a polymer threading a membrane; polymer segments in one side are discontinuously translocated into the other side, depending on solvent conditions and molecular weight of the polymer. We also discuss the equilibrium conformation of the polymer chain on each side of the membrane.     

Structures of stiff macromolecules of finite chain length near the coil‐globule transition: A Monte Carlo simulation

Using a coarse‐grained model of a semiflexible macromolecule, the equilibrium shapes of the chain have been studied varying both the temperature and the chain stiffness. We have applied Monte Carlo techniques using the bond fluctuation model for a chain length of N = 80 effective monomers, and two different types of interactions: a potential depending on the angle between successive bonds along the chain to control the chain stiffness, and an attractive interaction between non‐bonded effective monomers to model variable solvent quality. In a diagram of states where chain stiffness and inverse temperature are used as variables, we find regions where the chain exists as coil, as spherical globule, and as toroidal globule, respectively. Some of these regions are not limited by sharply defined boundaries, but rather wide two‐state coexistence regions occur in between them, where also intermediate metastable structures (such as rods and disks) occur. Recording histograms of energy, orientational order parameters, etc., which exhibit a two‐peak structure in the two‐state coexistence regions, we perform a subensemble analysis of the individual structures corresponding to these peaks.     

Macromolecules in a blend of poor and amphiphilic solvents

The globular state of the homopolymer macromolecule in a blend composed of a poor solvent and an amphiphilic solvent (substrate), whose molecules tend to be aligned with the solvent concentration gradient in the inhomogeneity region, was theoretically studied. The size of a homogeneous globule and the substrate concentration in its volume were calculated in terms of a bulk approximation. After the transition of the macromolecule from the coil to the globule state, its volume first decreases with a decrease in temperature and then begins to grow due to substrate molecules penetrating the globule. The substrate concentration in the globule insignificantly exceeds that outside the globule at identical second virial coefficients of interaction between monomer units and between substrate molecules. The expression for the free energy functional depending on the volume fractions of the components and on the orientation of substrate molecules was examined in the ground-state approximation. The orientation effect leads to narrowing of the surface layer and to a decrease in the surface tension of the homogeneous globule, thereby increasing its stability with respect to the transition to the unfolded-coil state.     

Transitions of tethered polymer chains: a simulation study with the bond fluctuation lattice model

A polymer chain tethered to a surface may be compact or extended, adsorbed or desorbed, depending on interactions with the surface and the surrounding solvent. This leads to a rich phase diagram with a variety of transitions. To investigate these transitions we have performed Monte Carlo simulations of a bond fluctuation model with Wang-Landau and umbrella sampling algorithms in a two-dimensional state space. The simulations' density-of-states results have been evaluated for interaction parameters spanning the range from good- to poor-solvent conditions and from repulsive to strongly attractive surfaces. In this work, we describe the simulation method and present results for the overall phase behavior and for some of the transitions. For adsorption in good solvent, we compare with Metropolis Monte Carlo data for the same model and find good agreement between the results. For the collapse transition, which occurs when the solvent quality changes from good to poor, we consider two situations corresponding to three-dimensional (hard surface) and two-dimensional (very attractive surface) chain conformations, respectively. For the hard surface, we compare tethered chains with free chains and find very similar behavior for both types of chains. For the very attractive surface, we find the two-dimensional chain collapse to be a two-step transition with the same sequence of transitions that is observed for three-dimensional chains: a coil-globule transition that changes the overall chain size is followed by a local rearrangement of chain segments.     

Microstructuring of a polymer globule in solution in the presence of an amphiphilic substance

Microstructuring in the bulk of a polymer globule in a solution that contains dimeric amphiphilic molecules, in particular, surfactants, is studied in terms of the weak-segregation theory. An inhomogeneous structure can result from a decrease in free energy with the orientation of amphiphilic molecules in the region of inhomogeneity owing to the interaction of hydrophobic and polar parts of the molecules with the solvent. For the sake of simplicity, we discuss the case of identical second virial coefficients of the interaction of monomer units and amphiphilic molecules with different energies of interaction of the hydrophobic and polar parts of the molecule with the solvent. By comparing the free energy for different types of microstructures, we predict that, with deterioration in the quality of the solvent, there is an initial formation of a homogeneous globule followed by formation of a body-centered cubic structure; a hexagonal cylindrical structure; and, finally, a lamellar structure. For a low degree of amphiphilicity, the transition from a homogeneous globule to only a lamellar structure occurs. An increase in the concentration of the amphiphilic substance in the surrounding solution hinders the formation of a globule but facilitates its microstructuring, which is also promoted by an increase in the volume of the amphiphilic molecule and the difference in the interaction energies of its hydrophobic and polar parts with the solvent. Phase diagrams of a globule??s state at different values of model parameters are plotted.     

DNA condensation induced by cationic surfactant: a viscosimetry and dynamic light scattering study

The compaction of DNA induced by two simple amphiphiles, cetyltrimethylammonium bromide [CTAB] and dodecyldimethylamine oxide [DDAO], has been investigated by means of combined viscosity and dynamic light scattering measurements, to demonstrate the formation of soluble DNA/surfactant complexes, undergoing a coil-globule transition, upon the increase of the amphiphile concentration. In both of the two systems investigated, the complexation process reaches a maximum for a value of the surfactant to DNA phosphate groups molar ratio of about X = 1. Below this critical concentration, the coil and the globule state coexist in the solution, as clearly shown by the bimodal size distribution obtained from the light scattering intensity correlation functions. Some suggestions are given to support a molecular mechanism responsible for the complex formation, both in the case of a cationic surfactant (CTAB) and of a pH-dependent neutral or cationic amphiphile (DDAO), where the hydrophobic interactions play an important role.     

Semiflexible amphiphilic polymers: cylindrical-shaped, collagenlike, and toroidal structures

A coarse-grained model is used to study the conformational properties of semiflexible polymers with amphiphilic monomer units containing both hydrophilic and hydrophobic interaction sites. The hydrophobically driven conformational transitions are studied using molecular dynamics simulations for the chains of varying stiffness, as characterized by intrinsic Kuhn segment lengths that vary over a decade. It is shown that the energy of hydrophobic attraction required for the realization of the coil-to-globule transition increases with increasing chain stiffness. For rather stiff backbone, the coil-to-globule transition corresponds to a first order phase transition. We find that depending on the chain stiffness, a variety of thermodynamically stable anisometric chain morphologies are possible in a solvent selectively poor for hydrophobic sites of amphiphilic monomer units. For flexible chains, the amphiphilic polymer forms a cylindrical globule having blob structure with nearly spherical blobs. With increasing stiffness, the number of blobs composing the globule decreases and the shape of blobs transforms into elongated cylinder. Further increase in stiffness leads to compaction of macromolecules into a collagenlike structure when the chain folds itself several times and different strands wind round each other. In this state, the collagenlike structures coexist with toroidal globules, both conformations having approximately equal energies.     

Secondary globular structure of copolymers containing amphiphilic and hydrophilic units: Computer simulation analysis

The coil-globule transition in copolymers composed of amphiphilic and hydrophilic monomer units has been studied by the computer simulation technique. It has been shown that the structure of globules formed in such systems substantially depends on the rate at which the solvent quality worsens. The globule resulting from slow cooling is cylindrical, and its core contains a large amount of hydrophilic groups. The globule formed upon rapid cooling takes the helical conformation, in which all hydrophilic groups are displaced to the periphery. One helix turn of such globules contains 3–5 units. In both cases, the backbone of the polymer chain forms a typical zigzag-shaped structure with an average angle between neighboring bond vectors of about 60°. This fact implies that globules of copolymers consisting of amphiphilic and hydrophilic units comprise secondary structure components.     

Transition in swollen polymer networks induced by intramolecular condensation

It is predicted that the net repulsion between the segments of a polymer network and a poor solvent can cause a phase transition marked by a sudden change in the degree of swelling. This is analogous to the “coil–globule” transition recently predicted by Ptitsyn to occur for a macromolecule in solution. The critical conditions for the transition. as well as phase diagrams, are calculated for the gel in free swelling and under uniaxial tension, which facilitates the transition. The transition depends on the gel being formed of chains crosslinked while greatly swollen by a diluent and also having a high degree of crosslinking. It is concluded that it would be difficult to attain the conditions necessary for the transition in the free-swelling case, but that it should be possible for gel under tension.