Mean-square Radius Of Gyration Research Articles

A molecular-dynamics simulation study of diffusion of a single model carbonic chain on a graphite (001) surface

Molecular-dynamics simulations have been used to study the diffusion of a short single model carbonic chain on the graphite (001) surface. The calculated diffusion coefficient (D) first increases, then decreases with increasing chain length (N). This abnormal behavior is similar to polymer lateral diffusion at the solid-liquid interface. Furthermore, we have studied the relation between the mean-square gyration radius and N. [Figure: see text].

Perimeter generating functions for the mean-squared radius of gyration of convex polygons

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent $\nu=1$.

Scattering functions of knotted ring polymers

We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We evaluate the form factor PK(q) of a Gaussian polygon of N = 200 having a fixed knot K for some different knots such as the trivial, trefoil, and figure-eight knots. Here the Gaussian polygons with different knots K have distinct values of the mean-square radius of gyration, R2(G,K). We obtain the Kratky plots of the form factors--i.e., the plots of (qR(G,K))2PK(q) versus qR(G,K)--for the different topological constraints and discuss nontrivial large-q behavior as well as small-q behavior for the scattering functions. We also find that the distinct values of R2(G,K) play an important role in the large-q and small-q properties of the Kratky plots.

Study on Phase Morphology of PP/PS Blends by Phase Contrast Micrographs

The relation of the phase morphology of polypropylene/polystyrene (PP/PS) blends and their composition and mixing temperature was investigated using digital image analysis. The characteristic length L and the mean square radius of gyration were defined to discuss the phase morphology of the immiscible system. A fractal dimension, D f , was introduced to describe the distribution of sizes of domains of the dispersed phase. To study the distribution uniformity of these domains, another fractal dimension, D c , was defined. The results showed that these fractal dimensions were effective parameters to describe the evolution of the phase morphology.

Correction-to-Scaling Exponents for Two-Dimensional Self-Avoiding Walks

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ≈ 0.01% accuracy up to N=4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Δ1=3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.

Monte Carlo simulations of a polyelectrolyte chain with added salt: Effect of temperature and salt valence

Using the cooperative motion algorithm, the effect of salt valence z(s) and of the reduced temperature T* on a single polyelectrolyte chain as well as on counterions and salt ions themselves is studied. The calculations show that both parameters strongly influence the polymer, causing it to undergo conformational changes. For a given number of the added salt cations (anions) n(s) and temperature T*, the chain takes more and more compact forms as z(s) increases (z(s) > 0). For fixed z(s), in turn, the polymer size reduces sharply as T* drops down from intermediate to low. For high T* configurational the entropy dominates the chain statistics and the mean-square radius of gyration (s2)1/2(T*,n(s),z(s)) approaches its athermal value. The low-temperature polymer collapse is also accompanied by a drop in the effective mean charge per monomer q*(T*,n(s),z(s)) (condensation of ions onto the chain) and the total inner energy e*(T*,n(s),z(s)). Furthermore, the local structure of the system is analyzed by means of pair-correlation functions g(ab)(r,T*,n(s),z(s)). At lower T* they possess sharp local maxima at small interparticle distances r that disappear as T* grows. The former observation indicates that at lower T* the ions tend to group themselves close to each other. In particular, it is concluded that the condensation is dominated by the multivalent salt ions carrying charges of opposite sign to that of monomers.

Elastic behavior of adsorbed single compact chains

Elastic behaviors of short single two-dimensional compact chains adsorbed on the attractive surface are investigated in this paper by using the enumeration calculation method. In our model a single compact chain is fixed with one of its end at a position above the impenetrable surface, and then it is pulled away from the attractive surface slowly through elastic force acting. We investigate the chain size and shape of adsorbed compact chains, such as mean-square end-to-end distance per bond 〈 R 2〉/ N mean-square radii of gyration per bond 〈 S 2〉 x / N and 〈 S 2〉 y / N, shape factors 〈 δ〉, and fraction of adsorbed monomers f a in order to illuminate how the size and shape of adsorbed compact chains change during the process of tensile elongation. Especially for strong attraction interaction there are some special behaviors in the chain size and shape during this process. If there exits adsorption interaction, single compact chain is first almost pulled down to the adsorption surface and then moves in the direction of force until to leave the adsorption surface. These changes become more obvious with strong adsorption interaction. Our calculation can show this elastic process of adsorbed compact chains visually and simply. On the other hand, some thermodynamics properties are also studied here. We use average energy per bond, average Helmholtz free energy per bond, elastic force f and energy contribution to elastic force f u to study the elastic behavior of adsorbed single compact chains in the process of tensile elongation. Elastic force f has a long plateau during the tensile elongation for strong adsorption interaction, which agrees well with experimental and theoretical ones. These investigations can provide some insights into the elastic behaviors of adsorbed protein chains.

Conformations and dynamics of adsorbed protein-like chains

Conformations and dynamics of adsorbed protein-like chains are investigated by using Monte Carlo simulation based on the modified orientation-dependent monomer–monomer interaction (ODI) model. The chain size and shape of adsorbed protein-like chains, such as mean-square end-to-end distance 〈 R 2〉, mean-square radius of gyration 〈 S 2〉 xy (or 〈 S 2〉 z ), shape factors 〈 s f i * 〉 ( i = 1 , 2 , 3 ) , and 〈 δ *〉 are discussed here. At the same time, fraction of adsorbed segment f a and average orientation of bond 〈 P 2(cos θ)〉 are also investigated. The adsorbed protein-like chains trend to be more flat when adsorption interaction energy becomes strong. Different kinds of interactions (such as contact interaction, sheet interaction, spin–spin interaction, helical interaction, and adsorbed interaction) are considered in detail. Dynamics of adsorbed protein-like chains are investigated by calculating their diffusion coefficients, and we find that there exist the relationships of D x y ∼ N − γ x y and D z ∼ N − γ z , and the values of γ xy and γ z are 4–5 times larger than that of general self-avoiding walk (SAW) chains. These investigations may provide some insights into adsorption of proteins.

The scaling behavior of the molecular parameters of the hyperbranched polymers made by self-condensing vinyl polymerization

The hyperbranched polymers can be made by self-condensing vinyl polymerization without gelation transition. The average molecular weights, as well as the average sizes, can reach infinite values as the reaction is quantitatively completed, and the scaling forms of the molecular parameters should exist. In the paper, based on a recursion formula, the scaling form of the number fraction distribution and the number of the n-mers are given analytically as the conversion of double bonds is near 1. The mean square radius of gyration for very large hyperbranched polymers is calculated explicitly to give a scaling exponent. Finally, a scaling relation associated with the fractal dimension and the polydispersity exponent is given clearly.

Preparation and Crystallization Kinetics of New Structurally Well-Defined Star-Shaped Biodegradable Poly(l-lactide)s Initiated with Diverse Natural Sugar Alcohols

This study presents syntheses, structural characterization, and crystallization kinetic investigation of new structurally well-defined star-shaped poly(l-lactide)s (PLLAs). First, a series of new 3- to 6-arm star-shaped PLLAs were synthesized through SnOct(2) catalyzed ring-opening polymerization of (l)-lactide with natural sugar alcohols of glycerol, erythritol, xylitol, and sorbitol as the favorable initiators. Subsequently, their chemical structures were characterized by means of GPC, NMR, and viscometer with respect to the star-shaped structures, demonstrating the well-defined arm structures as evidenced on the g(1/2)/g' values, where g and g' denote the ratios of mean-square radius of gyration and intrinsic viscosity of a star-shaped polymer to those of a linear structural reference with similar absolute molecular weight. Furthermore, spherulite morphologies and growth rates were studied by a polarized microscopy (POM) for the synthesized star-shaped PLLAs with different molecular weights, and it was found that the more arms of a star-shaped PLLA finally resulted in a lower spherulite growth rate. With regard to the crystallization kinetics of these star-shaped PLLAs, isothermal and nonisothermal crystallization were examined by differential scanning calorimeter (DSC). It was found that Avrami exponent n values of isothermal crystallization were almost independent of the isothermal crystallization temperature T(c) for different series of star-shaped PLLAs. In contrast, the values of Avrami exponent n were observed to strongly depend on the star-shaped structures with different arms, implying their distinct nucleation mechanisms, and the more arms of a star-shaped PLLA led to a slower isothermal crystallization rate. On the basis of a modified Avrami equation, new light was shed on the nonisothermal crystallization kinetics for the star-shaped PLLAs, and the activation energies were found to vary from 146.86 kJ/mol for the linear PLLA EG-3 to 221.23 kJ/mol of the star-shaped S-3, demonstrating much decreased crystallizabilities of star-shaped PLLAs with more arms.

Elastic behavior of short compact chains confined in double parallel boundaries

Adsorption of short two-dimensional compact chains confined in the double attractive parallel planar boundaries is investigated by using enumeration calculation method in this paper. First, we calculate the chain size and shape of adsorbed compact chains, such as mean-square end-to-end distance per bond 〈 R 2〉/ N, mean-square radii of gyration per bond 〈 S 2〉 x / N and 〈 S 2〉 y / N, shape factor 〈 δ〉 and fraction of adsorbed segments f a to illuminate that how the size and shape of adsorbed compact chains changes during the process of tensile elongation. There are some special behaviors in the chain size and shape for strong attraction interaction. In the meantime, compact chains can reach to the stable state with large distance between two parallel boundaries D. On the other hand, some thermodynamic properties, such as average energy per bond, Helmholtz free energy per bond, elastic force f and energy contribution to elastic f U are also investigated in order to study the elastic behavior of compact chains adsorbed on the double attractive parallel planar boundaries. These investigations may provide some insights into the thermodynamic behaviors of adsorbed compact chains.

Rheological and structural studies of liquid decane, hexadecane, and tetracosane under planar elongational flow using nonequilibrium molecular-dynamics simulations

We report for the first time rheological and structural properties of liquid decane, hexadecane, and tetracosane using nonequilibrium molecular-dynamics (NEMD) simulations under planar elongational flow (PEF). The underlying NEMD algorithm employed is the so-called p-SLLOD algorithm [C. Baig, B. J. Edwards, D. J. Keffer, and H. D. Cochran, J. Chem. Phys. 122, 114103 (2005)]. Two elongational viscosities are measured, and they are shown not to be equal to each other, indicating two independent viscometric functions in PEF. With an appropriate definition, it is observed that the two elongational viscosities converge to each other at very low elongation rates, i.e., in the Newtonian regime. For all three alkanes, tension-thinning behavior is observed. At high elongation rates, chains appear to be fully stretched. This is supported by the result of the mean-square end-to-end distance of chains (R(ete2)) and the mean-square radius of gyration of chains (R(g2)), and further supported by the result of the intramolecular Lennard-Jones (LJ) potential energy. It is also observed that (R(ete2)) and (R(g2)) show a different trend as a function of strain rate from those in shear flow: after reaching a plateau value, (R(ete2)) and (R(g2)) are found to increase further as elongation rate increases. A minimum in the hydrostatic pressure is observed for hexadecane and tetracosane at about epsilon(msigma2/epsilon)1/2=0.02. This phenomenon is shown to be associated with the intermolecular LJ potential energy. The bond-bending and torsional energies display similar trends, but a different behavior is observed for the bond-stretching energy. An important observation common in these three bonded-intramolecular interactions is that all three modes are suppressed to a small value at high elongation rates. We conjecture that a liquid-crystal-like, nematic structure is present in these systems at high elongation rates, which is characterized by a strong chain alignment with a fully stretched conformation.

Computer Simulation of Complexes of Dendrimers with Linear Polyelectrolytes

Brownian dynamics computer simulations have been carried out of complexes formed by charged dendrimers and oppositely charged linear polymer chains of different degree of polymerization Nch. Bead−rod freely jointed models in the Debye−Huckel approximation without hydrodynamic interactions have been considered. Mean-square radii of gyration together with the radial density distribution functions have been calculated separately for a complex, a dendrimer, and a linear chain in a complex. The mean-square radius of gyration, the different monomer radial distribution functions, and the static structure factor for a dendrimer in a complex with long enough chains are very close to those for a single neutral dendrimer. The monomers of the linear chains with Nch equal to the number of the dendrimer's terminal charged groups are located very close to these terminal groups. For longer chains the total number of the chain monomers adsorbed onto a dendrimer exceeds the number that is necessary for a dendrimer neutrali...

Small-Angle X-Ray Scattering from Polystyrene Polymacromonomers in Cyclohexane

Small-angle X-ray scattering measurements have been made on seven polymacromonomer samples consisting of polystyrene with a fixed side chain length of 15 styrene residues in cyclohexane at 34.5 °C to obtain the z-average mean-square radius of gyration ‹S2›z and the particle scattering function P(θ). The dependence of ‹S2›z on the weight-average molecular weight Mw, which ranges from 5.1×103 to 1.8×105, is described by the wormlike chain with the model parameters explaining previous light scattering ‹S2›z data for high molecular weights (6×105–7×106), when the effect of the chain diameter d and that of side chains near the main-chain ends on the polymacromonomer contour length L are incorporated into analysis. Scattering profiles in the form of the Kratky plot are also explained by the recent theory for the wormlike chain with a circular cross section unless the axial ratio L/d is less than 2, i.e., unless Mw is lower than 2.6×104. The conventional cross-section plot, i.e., the plot of ln[kP(θ)] vs. k2 with slope −d2/16, is found to be applicable to samples with L/d>2, where k denotes the magnitude of the scattering vector. It is concluded from further analysis of the ‹S2›z data combined with previous transport coefficient data that the wormlike cylinder model, a smeared model for regular comb polymers, consistently explains the molecular weight dependence of ‹S2›z, intrinsic viscosity, and translational diffusion coefficient for the polystyrene polymacromonomer in cyclohexane over a broad range of molecular weight. The applicability of a semiflexible comb model consisting of wormlike main and side chains to ‹S2›z and P(θ) is also examined.

Conformational end effects in unperturbed chains and star-branched polymers

End effects in unperturbed chains with a specified number of bonds, n, have previously been characterized using the conformations at individual bonds and the mean square unperturbed dimensions of subchains situated at various positions along the chain. Both criteria have found that the end effects are extremely weak in typical flexible chains, such as unperturbed polyethylene. Of course, these criteria detect no end effect at all in a freely jointed chain. This conclusion requires modification if the end effects are defined differently, using the mean square radius of gyration of the n+1 chain atoms about chain atom i, denoted 〈 s i 2 〉 0 . The values of 〈 s i 2 〉 0 depend on i in all chains, including the freely jointed chain. The range for the volume specified by 〈 s i 2 〉 0 3 / 2 suggests that the average density of chain segments about a terminal atom is roughly 1/3 of the average density of chain segments about the atom midway along the contour of any long unperturbed chain. The severity of these end effects in finite chains depends on the detailed local structure, becoming more severe for more extended chains. The end effect detected by 〈 s i 2 〉 0 in regular freely jointed star-branched polymers becomes more severe as the number of branches increases.

Models for Excluded Volume Interaction between an Unfolded Protein and Rigid Macromolecular Cosolutes: Macromolecular Crowding and Protein Stability Revisited

Statistical-thermodynamic models for the excluded volume interaction between an unfolded polypeptide chain and a hard sphere or hard rod cosolute are presented, permitting estimation of the free energy of transfer of a polypeptide chain with fixed radius of gyration from a dilute (ideal) solution to a solution containing volume fraction ϕ of either cosolute. Also presented is a general thermodynamic description of the equilibrium between a unique native state and a manifold of unfolded or partially unfolded states of a protein distinguished by their respective radii of gyration. Together with results of a Monte Carlo calculation of the distribution of radii of gyration of four different unfolded proteins published by Goldenberg in 2003, these models are used to estimate the effect of intermolecular excluded volume upon an experimentally measurable apparent two-state constant for equilibrium between native and nonnative conformations of each of the four proteins, and upon the experimentally measurable root mean-square radius of gyration of the unfolded protein. Model calculations predict that addition of inert cosolutes at volume fractions exceeding 0.1 stabilizes the native state relative to unfolded states by an amount that increases strongly with ϕ and with the size of the native protein relative to the size of inert cosolute, and results in significant compaction of the manifold of unfolded states. Predicted effects are in qualitative and/or semiquantitative accord with the results of several published experimental studies.

Amyloid-β Protofibrils Differ from Amyloid-β Aggregates Induced in Dilute Hexafluoroisopropanol in Stability and Morphology

The brains of Alzheimer's disease (AD) patients contain large numbers of amyloid plaques that are rich in fibrils composed of 40- and 42-residue amyloid-beta (Abeta) peptides. Several lines of evidence indicate that fibrillar Abeta and especially soluble Abeta aggregates are important in the etiology of AD. Recent reports also stress that amyloid aggregates are polymorphic and that a single polypeptide can fold into multiple amyloid conformations. Here we demonstrate that Abeta-(1-40) can form soluble aggregates with predominant beta-structures that differ in stability and morphology. One class of aggregates involved soluble Abeta protofibrils, prepared by vigorous overnight agitation of monomeric Abeta-(1-40) at low ionic strength. Dilution of these aggregation reactions induced disaggregation to monomers as measured by size exclusion chromatography. Protofibril concentrations monitored by thioflavin T fluorescence decreased in at least two kinetic phases, with initial disaggregation (rate constant approximately 1 h(-1)) followed by a much slower secondary phase. Incubation of the reactions without agitation resulted in less disaggregation at slower rates, indicating that the protofibrils became progressively more stable over time. In fact, protofibrils isolated by size exclusion chromatography were completely stable and gave no disaggregation. A second class of soluble Abeta aggregates was generated rapidly (<10 min) in buffered 2% hexafluoroisopropanol (HFIP). These aggregates showed increased thioflavin T fluorescence and were rich in beta-structure by circular dichroism. Electron microscopy and atomic force microscopy revealed initial globular clusters that progressed over several days to soluble fibrous aggregates. When diluted out of HFIP, these aggregates initially were very unstable and disaggregated completely within 2 min. However, their stability increased as they progressed to fibers. Relative to Abeta protofibrils, the HFIP-induced aggregates seeded elongation by Abeta monomer deposition very poorly. The techniques used to distinguish these two classes of soluble Abeta aggregates may be useful in characterizing Abeta aggregates formed in vivo.

Memory effect in the chain-collapse process in a dilute polymer solution

The effect of temperature perturbation on a single-chain-collapse process was studied for poly(methyl methacrylate) with the molecular weight M(w)=1.05 x 10(7) in the mixed solvent of tert-butyl alcohol+water (2.5 vol %). In the chain-collapse process after a quench from the theta; temperature to a temperature T(1), the temperature was changed from T(1) to T(2) at the time t(1) after the quench and returned to T(1) at the time t(1)+t(2). In the three stages at T(1), T(2), and T(1), measurements of the mean-square radius of gyration of polymer chains were carried out by static light scattering and the chain-collapse process was represented by the expansion factor as a function of time. An effect of chain aggregation on the measurements was negligibly small because of the very slow phase separation. For the negative temperature perturbation (T(1)>T(2)), the chain-collapse processes observed in the first and third stages were connected smoothly and agreed with the collapse process due to a single-stage quench to T(1). A memory of the chain collapse in the first stage at T(1) was found to persist into the third stage at the same temperature T(1) without being affected by the temperature perturbation of T(2) during t(2). The memory effect was observed irrespective of the time period of t(2). The positive temperature perturbation (T(1)<T(2)) showed an acceleration of the chain-collapse process.

Elastic behavior of adsorbed polymer chains

Elastic behaviors of single polymer chains adsorbed on the attractive surface are first investigated using Monte Carlo simulation method based on the bond fluctuation model. We investigate the chain size and shape of adsorbed chains, such as mean-square radius of gyration S2, mean-square bond length b2, shape factors sf(i) and delta*, and the orientation of chain segments P2<cos theta>, to illuminate how the shape of polymer chains changes during the process of tensile elongation. There are some special behaviors of the chain size and shape at the beginning of elongation, especially for strong attraction interaction. For example, mean fraction of adsorbed segments decreases abruptly in the region of small elongation ratio and then decreases slowly with increasing elongation ratio. In fact, the chain size and shape also changes abruptly for small elongation ratio with strong attraction interaction. Some thermodynamics properties are also investigated here. Average Helmholtz free energy increases fast for elongation ratio lambda<1.15, especially with strong attraction, and increases slowly for lambda>1.15. Similar behaviors are obtained for average energy per bond. Elastic force (f ) and energy contribution to force (f(U)) are also studied, and we find that elastic force decreases abruptly for lambda<1.15, and there is a minimum of elastic force for strong attraction interaction, then increases very slowly with increasing elongation ratio. However, there are different behaviors for weak attraction interaction. For energy contribution to force (f(U)), there is a maximum value for strong attraction interaction in the region of lambda<1.15. Some comparisons with the atomic force microscopy experiments are also made. These investigations may provide some insights into the elastic behaviors of adsorbed polymer chains.

Self-avoiding walks and polygons on the triangular lattice

We use new algorithms, based on the finite lattice method of series expansion, to extendthe enumeration of self-avoiding walks and polygons on the triangular lattice to length 40and 60, respectively. For self-avoiding walks to length 40 we also calculate series for themetric properties of mean-square end-to-end distance, mean-square radius ofgyration and the mean-square distance of a monomer from the end points. Forself-avoiding polygons to length 58 we calculate series for the mean-square radiusof gyration and the first 10 moments of the area. Analysis of the series yieldsaccurate estimates for the connective constant of triangular self-avoiding walks,μ = 4.150 797 226(26), and confirms to a high degree of accuracy several theoretical predictions for universalcritical exponents and amplitude combinations.