Mean-square Radius Of Gyration Research Articles

Interaction of a spherical particle with linear chains. II. Chains end-grafted at the particle surface

Linear flexible polymers end-grafted onto the spherical surface (“hairy sphere”) are simulated using the cooperative motion algorithm. The simulations are performed in two extremes of the surrounding matrix: (1) The hairy sphere immersed in a pure solvent of single beads and (2) the hairy sphere in a melt of linear chains. The static properties of the grafted chains are analyzed by calculating the polymer concentration profiles, the distributions of polymer centers-of-mass and of the free chain ends for various values of the sphere radius, various chain lengths, and variable coverage of the sphere surface by the grafting-chain ends (surface coverage). Ordering phenomena of the polymers and their intramolecular structure are taken into account by considering the orientation correlation parameter, the mean-squared radius of gyration and the mean-squared end-to-end distance as functions of the position of the polymer centers-of-mass with respect to the sphere surface. In case (2) some properties of the melt chains have also been analyzed. The simulations indicate that (1) the concentration profiles of the end-grafted chains under good solvent conditions are noticeably different from those in a melt, (2) in both cases, they are strongly affected by the values of the surface coverage, (3) as the sphere radius increases, the monomer concentration profile changes from concave to convex, (4) both the free ends and the centers of mass of the grafted chains reveal a tendency to concentrate at some distance from the surface, i.e., the profiles have noticeable maxima, (5) for small surface coverage the chain centers of mass penetrate into the sphere, especially when anchored to the smaller sphere in the melt, (6) depending on the position of the center-of-mass of the end-grafted chains, both tangential and radial ordering of the polymers relative to the sphere exists, (7) the end-to-end vectors and radii of gyration show a paraboliclike shape with minima at a finite distance from the surface.

Distribution of transverse distances in directed animals

We relate $\phi(\bf{x},s)$, the average number of sites at a transverse distance $\bf{x}$ in the directed animals with $s$ sites in $d$ transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbor exclusion in $d$ dimensions. For large $s$, $\phi(\bf{x},s)$ has the scaling form $\frac{s}{R_s^d} f(|\bf{x}|/R_s)$, where $R_s$ is the root mean square radius of gyration of animals of $s$ sites. We determine the exact scaling function for $d =1$ to be $f(r) = \frac{\sqrt{\pi}}{2 \sqrt{3}}erfc(r/\sqrt{3})$. We also show that $\phi(\bf{x}=0,s)$ can be determined in terms of the animals number generating function of the directed animals.

Solvent effect on single-chain collapse of poly(methyl methacrylate) in tert-butyl alcohol

Kinetic and static properties of the coil–globule transition of poly(methyl methacrylate) (PMMA) in pure tert-butyl alcohol were determined by static light scattering and compared with those of PMMA in the mixed solvent tert-butyl alcohol+water(2.5 vol %) in order to examine the effect of water on the chain collapse. The measurements were carried out for the molecular weight M×10−6=4.1 and 12.2 in the concentration range of 0.6×10−4–2.6×10−4 g/cm3, and the mean-square radius of gyration 〈s2〉 was determined as a function of the time after an abrupt decrease of temperature. PMMA chains collapsed to equilibrium globules within 90 min after quenching for M=1.22×107 and within 30 min for M=4.1×106. Chain aggregation due to phase separation became noticeable after the collapse of the chain because of an increase of observed molecular weight. For PMMA in the mixed solvent tert-butyl alcohol+water(2.5 vol %), the chain collapse process has been observed for periods from hours to weeks depending on the molecular weight and temperature, and the chain aggregation was negligibly small in the chain collapse process. The expansion factor α2=〈s2〉/〈s2〉0 obtained for fully collapsed chains in pure tert-butyl alcohol was represented by the theoretical prediction α3−α−C(α−3−1)=B(1−θ/T)M1/2 with the coefficients of B=0.0179 and C=0.054. For PMMA in the mixed solvent, the coil–globule transition curve has been expressed by the same equation with B=0.0164 and C=0.049, close to the above values. The small amount of water in the mixed solvent caused a drastic slowdown in the chain-collapse rate but had little effect on the coil–globule transition curve.

A Monte Carlo study of effects of chain stiffness and chain ends on dilute solution behavior of polymers. I. Gyration-radius expansion factor

A Monte Carlo (MC) study is made of the mean-square radius of gyration 〈S2〉 and the gyration-radius expansion factor αS for the freely rotating chain of bond angle 109° and with the Lennard-Jones (LJ) 6-12 intramolecular potentials between beads in a cutoff version for the number n of bonds in the chain ranging from 10 to 1500 at the reduced temperature ranging from 3.6 to 8.0, which is defined as the absolute temperature multiplied by the Boltzmann constant and divided by the depth of the well of the LJ potential. It is shown that the ratio 〈S2〉/n approaches asymptotically a constant independent of n for very large n at the value 3.72±0.05 of the reduced temperature, which value is equal to the reduced Θ temperature Θ* of the MC model system, and that possible effects of chain ends on 〈S2〉 and therefore on αS are negligibly small. Taking the values of 〈S2〉 at Θ* as the unperturbed ones, αS2 is evaluated from those at various reduced temperatures higher than Θ*. It is then found that the behavior of αS2 may be well explained in the quasi-two-parameter scheme or is in good agreement with that of real experimental data. Further, the binary cluster integral for a bead in the chain is found to be much smaller in magnitude than that for a single isolated bead at reduced temperatures higher than Θ*, the result being consistent with a previous finding.

Solution Characterization of Monodisperse Atactic Polystyrenes by Static and Dynamic Light Scattering

Anionically polymerized polystyrene standards have been characterized by static and dynamic light scattering in toluene at 25°C. The molecular weights ranged from 3000 to two million g/mol. An appropriate standard for light scattering intensity was developed and the absolute Rayleigh ratio for toluene was determined for the experimental conditions. The molecular weight dependence of the second virial coefficient A 2 could be represented as a power law. The present results are in good agreement with previous measurements of A 2 . For the higher molecular weight polymers, the mean-square radius of gyration was determined. The present results could also be represented as a power law, which was in good agreement with previous results. Dynamic light scattering was analyzed to obtain the mutual-diffusion coefficient as a function of concentration and molecular weight. These results were then used to obtain the self-diffusion coefficient, hydrodynamic radius, and virial coefficient for the mutual-diffusion coefficient. Hydrodynamic radii could be represented as a power law, which was also in good agreement with previous reports. The virial coefficient for the mutual friction was calculated as well. The relationship between molecular weight distribution and the polydispersity index determined from dynamic light scattering was calculated and compared to actual data. The samples used here conformed to the manufacturers' reported weight to number-average molecular weight ratios.

Simulation of size exclusion chromatography for branched polymers formed by simultaneous long-chain branching and random scission

AbstractIn free-radical olefin polymerization, polymer transfer reactions can lead to chain scission as well as to the formation of long-chain branches. The dimensions of branched polymers so formed are investigated theoretically, by focusing our attention on the effects of scission frequency that is often difficult to determine. Scission reactions decrease the weight-average molecular weight significantly. When the polymer molecules are fractionated by the degree of polymerization (DP), however, the averages of branching density P ρ ̅and the mean-square radius of gyration of polymers with DP = P, <s2>P, are essentially unchanged, as long as the scission frequency is not very large. In a simple relationship between the ratios g and g’ of radii of gyration and hydrodynamic radii, g’ = gb, the value of b may change as a function of DP, and a simplified model function b = A log P + B was used to investigate the elution curve of size exclusion chromatography (SEC). It is shown that, although the molecular weight distribution (MWD) calibrated relative to standard linear polymers in SEC is much narrower than the true MWD, the use of a light scattering photometer to determine the absolute weight-average molecular weight within the elution volume makes the obtained MWD very close to the true one. On the basis of the <s2>P vs. P relationship, the calibrated MWD can be converted reasonably well to the true MWD. This simplified transformation method can be applied without knowing the scission frequency or the DP dependence of b. Once the true MWD is successfully estimated, a reliable parameter for the scission reaction can be determined.

Influence of excluded volume on the conformational property of tail-like chain on the simple cubic lattice

The influence of excluded volume on the conformational property of linear tail-like chain with one end attached to a flat surface is investigated by means of dynamic Monte Carlo method. Conformational properties such as mean-square end-to-end distance 〈 R 2〉, mean-square radius of gyration 〈 S 2〉 and mean asphericity parameter 〈 A〉 are calculated for random walking (RW) and self-avoiding walking (SAW) tail-like chains on the simple cubic lattice. We find that the EV has nearly the same effect on 〈 R 2〉 as on 〈 S 2〉: (1) 〈 R 2〉 SAW/〈 R 2〉 RW≈〈 S 2〉 SAW/〈 S 2〉 RW∝ n 0.204±0.05, where n is the chain length, and (2) the limiting value of 〈 R 2〉/〈 S 2〉≈7.7 for both chains. The distribution P( R) of the SAW tail-like chain can be expressed as a R 4 correction of that of the RW one. We find that the value 〈 A〉 of the SAW tail-like chain is bigger than that of the RW tail-like chain for all chain lengths, and the limiting values are 0.446±0.006 and 0.403±0.005 respectively.

Fractal Dimension of Randomly Branched Polymers in a Good Solvent

We propose a concept of subchains for randomly branched polymers. Asa direct application of this concept, the asymptotic expression ofthe average mean square radius of gyration is determined to give thefractal dimensions, in which the excluded volume effect is taken intoconsideration. Furthermore, we investigate a scaling relation that isassociated with the Flory exponent ν, the fractal dimensiondf and the polydispersity exponent τ.

Finite-size and asymptotic behaviors of the gyration radius of knotted cylindrical self-avoiding polygons.

Several nontrivial properties are shown for the mean-square radius of gyration R2(K) of ring polymers with a fixed knot type K. Through computer simulation, we discuss both finite size and asymptotic behaviors of the gyration radius under the topological constraint for self-avoiding polygons consisting of N cylindrical segments with radius r. We find that the average size of ring polymers with the knot K can be much larger than that of no topological constraint. The effective expansion due to the topological constraint depends strongly on the parameter r that is related to the excluded volume. The topological expansion is particularly significant for the small r case, where the simulation result is associated with that of random polygons with the knot K.

Comment on the universality class of self-avoiding walks on a two-layer square lattice

We have calculated exactly the mean-square radius of gyration and the mean-square distance of a monomer from the origin for n-step self-avoiding walks on a two-layer square lattice up to 24 steps. We estimated two universal amplitude ratios and the results provide additional evidence that self-avoiding walks on a two-layer lattice belongs to the same universality class as the self-avoiding walk on a planar lattice.

Conversion Dependence of the z-Average Mean Square Radii of Gyration for Hyperbranched Polymers with Excluded Volume Effect

The concept of subchain for hyperbranched polymer systems is investigated. Taking ABf type polycondensation reaction as an example, the subchain number distribution is obtained explicitly. As a direct application of the subchain number distribution, the mean square radius of gyration is given, where the excluded volume effect is taken into consideration. Furthermore, the z-average mean square radius of gyration is evaluated as the function of the conversion of functional groups, in which the contribution of unreacted monomers in hyperbranched polymer systems is included and excluded, respectively.

Monte Carlo study of cycloamylose: Chain conformation, radius of gyration, and diffusion coefficient

Cyclic (1 --> 4)-alpha-D-glucan chains with or without excluded volume have been collected from a huge number (about 10(7)) of linear amylosic chains generated by the Monte Carlo method with a conformational energy map for maltose, and their mean-square radii of gyration <S(2)> and translational diffusion coefficients D (based on the Kirkwood formula) have been computed as functions of x (the number of glucose residues in a range from 7 to 300) and the excluded-volume strength represented by the effective hard-core radius. Both <S(2)>/x and D in the unperturbed state weakly oscillate for x < 30 and the helical nature of amylose appears more pronouncedly in cyclic chains than in linear chains. As x increases, these properties approach the values expected for Gaussian rings. Though excluded-volume effects on them are always larger in cycloamylose than in the corresponding linear amylose, the ratios of <S(2)> and the hydrodynamic radius of the former to the respective properties of the latter in good solvents can be slightly lower than or comparable to the (asymptotic) Gaussian-chain values when x is not sufficiently large. An interpolation expression is constructed for the relation between the gyration-radius expansion factors for linear and cyclic chains from the present Monte Carlo data and the early proposed asymptotic relation with the aid of the first-order perturbation theories.

SELF-AVOIDING WALKS ON THE 4–8 LATTICE

We have calculated exactly the number, the mean-square end-to-end distance, the mean-square radius of gyration, and the mean-square distance of a monomer from the origin for self-avoiding walks on the 4–8 lattice up to 42 steps by computer. We estimated the connective constant and the critical amplitudes. Our numerical results are consistent with the theoretical prediction by Cardy and Saleur on the universality for certain amplitude ratios.

On the number of benzenoid hydrocarbons.

We present a new algorithm which allows a radical increase in the computer enumeration of benzenoids b(h) with h hexagons. We obtain b(h) up to h = 35. We prove that b(h) approximately const.kappa(h), prove the rigorous bounds 4.789 < or = kappa < or = 5.905, and estimate that kappa = 5.16193016(8). Finally, we provide strong numerical evidence that the generating function summation operator b(h)z(h) approximately A(z) log(1 - kappa z), estimate A(1/kappa) and predict the subleading asymptotic behavior. We also provide compelling arguments that the mean-square radius of gyration <R(g)(2)>(h) of benzenoids of size h grows as h(2 nu), with nu = 0.64115(5).

Confined polymer networks: The harmonic approach

Confinement of phantom networks of Gaussian chains between parallel surfaces is simulated through the application of a harmonic potential 12Hx2 to each chain bead at a distance x from the mean plane. We consider infinite, square (cubic) networks with a topological dimensionality of two (three), in addition to three-dimensional networks with a finite number of layers along one dimension. The partition function stays Gaussian and its integration reduces to the search of the eigenvalues of cyclic, or quasicyclic matrices. As applied to an isolated chain the confinement energy is 75% the exact value obtained by Casassa’s exact approach, and the accuracy is expected to improve in the network case. At strong compressions, the confinement energy per chain EH is about the same for all the networks, but smaller than for a single chain with the same contour length. Taking EH in kBT units, M=EH−1 gives the number of chains in the correlation domain. Therefore, the external potential breaks up the network into smaller and smaller correlated domains, whose statistics remain unperturbed (“blobs”). The unperturbed mean-square radius of gyration reproduces previous results on the two- and the three-dimensional networks. The enormous collapse in the three-dimensional case is evidenced once again.

Universal amplitude ratios for three-dimensional self-avoiding walks

We have calculated exactly the number, the mean-square end-to-end distance, the mean-square radius of gyration, and the mean-square distance of a monomer from the origin for n-step self-avoiding walks on the simple cubic, diamond, body-centred cubic and face-centred cubic lattices, respectively, up to 20, 30, 16 and 13 steps by a computer. Two universal amplitude ratios are estimated.

Ordered Arrangements of Semiflexible Polymers at the Interface with Solids

The molecular arrangements and conformations of dense systems of semiflexible polymer chains near solid surfaces have been investigated by Monte Carlo simulations. At variance with the results obtained for more flexible chains, the mean square end-to-end distance and the mean square radius of gyration of chains in contact with the surfaces are found to be much higher than in the bulk. This is related to the increased length of the surface trains and to the increased tendency of such trains to form rod-like strands. As a result, the first layer of polymer units in contact with the surfaces consists of two-dimensional domains of locally parallel chain segments. The width of these domains is several times the transverse diameter of the chains.

Comparison of the diffusion coefficients of linear and cyclic alkanes

Monte Carlo simulations of linear and cyclic alkanes were performed on a coarse-grained high coordination lattice. The simulations were performed at 473K for CNH2N+2 and CNH2N where N had the values of 60, 100, and 316. The results indicated: (i) at low molecular weights, cyclic alkanes have lower diffusion coefficients than linear alkanes, and (ii) at high molecular weights, they have higher diffusion coefficients than linear alkanes. The lower diffusion coefficient of the small cyclic alkanes was attributed to the high local density within the volume defined by the smaller mean square radius of gyration of the cyclic alkanes. The high local density of cyclic alkane segments resulted in a decrease in the mobility of the beads. The crossover in diffusion coefficients was observed around the entanglement molecular weight of linear alkanes, which suggests that the linear alkanes are more susceptible to the effects of entanglements than are the cyclic alkanes.

Long chain branch polymer chain dimensions: application of topology to the Zimm–Stockmayer model

An explicit topological approach to the dimensions of LCB polymers is presented. It is based on the Wiener number, a topological descriptor which is shown in this study to be related to the topological radius of the macromolecule, the mean-square radius of gyration, the g-ratio, and the intrinsic viscosity within the Rouse–Zimm range. The new theory enables the treatment of the highly complex hyperbranched polymers, which are difficult to handle by the classical theory of Zimm and Stockmayer. The agreement with the measured g-values of model polyethylenes, synthesized by Hadjichristidis et al., is fairly good for star-like polymers and satisfactory for pom–pom type of structures, whereas for crowded comb-type species the calculated g-values are underpredicted. Extension of the approach is shown to cyclic structures for which the Kirchhoff number replaces the Wiener number.

Gyration-Radius Expansion Factor of Oligo- and Poly(α-methylstyrene)s in Dilute Solution

The mean-square radius of gyration 〈S2〉 was determined from small-angle X-ray scattering and light scattering measurements for atactic oligo- and poly(α-methylstyrene)s (a-PαMS) with the fraction of racemic diads fr = 0.72 in toluene at 25.0 °C, in 4-tert-butyltoluene at 25.0 °C, and in n-butyl chloride at 25.0 °C in the range of weight-average molecular weight from 1.04 × 103 to 5.46 × 106. The gyration-radius expansion factor αS was then evaluated from the values of 〈S2〉 so determined along with those of 〈S2〉ϑ previously determined for the same samples in cyclohexane at 30.5 °C (ϑ). It is shown that the plots of αS against the scaled excluded-volume parameter z for a-PαMS along with those for atactic polystyrene (a-PS) and atactic and isotactic poly(methyl methacrylate)s previously studied form a single-composite curve, confirming the validity of the quasi-two-parameter scheme that all expansion factors are functions only of z irrespective of the differences in polymer species (chain stiffness and loc...