Mean-square Radius Of Gyration Research Articles

Linking probabilities of off-lattice self-avoiding polygons and the effects of excluded volume

We evaluate numerically the probability of linking, i.e. the probability of a given pair of self-avoiding polygons (SAPs) being entangled and forming a nontrivial link type L. In the simulation we generate pairs of SAPs of N spherical segments of radius rd such that they have no overlaps among the segments and each of the SAPs has the trivial knot type. We evaluate the probability of a self-avoiding pair of SAPs forming a given link type L for various link types with fixed distance R between the centers of mass of the two SAPs. We define normalized distance r by where denotes the square root of the mean square radius of gyration of SAP of the trivial knot 01. We introduce formulae expressing the linking probability as a function of normalized distance r, which gives good fitting curves with respect to χ2 values. We also investigate the dependence of linking probabilities on the excluded-volume parameter rd and the number of segments, N. Quite interestingly, the graph of linking probability versus normalized distance r shows no N-dependence at a particular value of the excluded volume parameter, rd = 0.2.

Transport Coefficients of Poly(diisopropyl fumarate) in Dilute Solution

Top of pageAbstract The intrinsic viscosity [η] was determined for 13 samples of poly(diisopropyl fumarate), each with the fraction of racemo diads fr = 0.22, in the range of weight-average molecular weight Mw from 4.02 × 104 to 8.59 × 105 in tetrahydrofuran at 30.0 °C. The translational diffusion coefficient D was also determined from dynamic light scattering measurements for 12 of them in the range of Mw from 4.95 × 104 to 8.59 × 105 under the same solvent condition. The data so obtained were analyzed on the basis of the corresponding transport theories for the unperturbed Kratky–Porod (KP) wormlike cylinder model combined with the quasi-two-parameter theory for the intramolecular excluded–volume effect. It is shown that the experimental values are in quantitative agreement with the perturbed KP theory ones by the use of the model parameter values consistent with those determined previously from analyses of the mean-square radius of gyration (S2) and second virial coefficient A2. An examination was also made of behavior of the Flory–Fox factor Φ and reduced hydrodynamic radius ρ−l as functions of Mw. Keywords: Poly(diisopropyl fumarate), Intrinsic Viscosity, Translational Diffusion Coefficient, Kratky–Porod Wormlike Chain Model, Excluded-Volume Effect, Quasi-Two-Parameter Theory

Some comments on the second virial coefficient of semiflexible polymers

A Monte Carlo study is made of the mean-square radius of gyration S(2) and second virial coefficient A(2) for the two freely rotating chains with the Lennard-Jones (LJ) 6-12 potential and the hard-sphere (HS) one in the range of the bond angle theta from 109 degrees (typical flexible chain) to 175 degrees (typical semiflexible or stiff chain) and in the range of the number n of bonds from 6 to 1000. It is shown that a value may be properly assigned to the collision diameter of the HS potential so that S(2) of the chain with the HS potential agrees well with that of the chain with the LJ one whose parameter values correspond to a good-solvent condition irrespective of the chain stiffness. It is then found that A(2) of the latter chain becomes remarkably smaller than that of the former as the chain stiffness is increased. The result implies that the binary-cluster approximation does not seem to work well for typical semiflexible and stiff polymers.

Molecular Architecture of Non‐Linear Polymers: Kinetic Modeling and Experimental Characterization of the System Methyl Methacrylate + Ethylene Glycol Dimethacrylate

AbstractA general kinetic approach allowing the prediction of the molecular architecture of non‐linear polymers is applied to the study of the copolymerization of methyl methacrylate (MMA) with ethylene glycol dimethacrylate (EGDMA). Dynamic predictions of molecular weight distributions, sequence length distributions and mean square radius of gyration are possible before and after gelation. A set of experiments concerning the copolymerization of MMA and EGDMA was carried out in toluene solution at 60 °C for which classic radical kinetics is a good approximation. The time evolution of key polymer properties was followed using a SEC system with a refractive index detector coupled with MALLS allowing the determination of absolute weight average molecular weight and apparent molecular size distributions as well as z‐average radius of gyration. Special focus was given to assess the influence of the initial amount of cross‐linker on the dynamics of the non‐linear structure build‐up of these products. A kinetic scheme comprising 23 different chemical species and 76 chemical reactions was used in the modeling studies of this chemical system. Most of the kinetic parameters used in the simulations have been collected from previous studies. For experiments at low monomer conversion (up to about 0.5) a good agreement between predictions and experimental measurements is observed for molecular weights and z‐average radius of gyration by fitting a small number of parameters describing gel effect (with a conversion dependent but chain length independent termination rate parameter) and the relative propagation on pendant double bonds. However, predicted values of weight‐average molecular weights and z‐average radius of gyration before gelation are too low at higher monomer conversions with non‐linear systems. The likely cause is the presence of intramolecular reactions which should not be neglected in these circumstances.

Effect of salt solutions on chain structure of partially hydrolyzed polyacrylamide

The effect of salt solutions (NaCl, Na2SO4 and CaCl2) on the conformational properties of partially hydrolyzed polyacrylamide (HPAM) was investigated by using static laser light scattering (SLLS). The special interaction between CaCl2 solution and HPAM was also researched. Experimental results show that the chain structure of HPAM is interrelated with the charge density, the kind and the concentration of salt solutions. The mean-square radius of gyration (R z) and the second virial coefficient (A 2) of HPAM decrease with increasing concentration of salt solutions, and the salt effect tends towards the maximum when the concentration of salt solution is increased to some amount.

Self‐Assembly of Cylinder‐Forming ABA Triblock Copolymers under Cylindrical Confinement

AbstractThe self‐assembly of two types of linear ABA triblock copolymers confined in cylindrical nanopores is studied using simulated annealing. The effects of pore size and block copolymer chain architecture on morphology, chain conformations and bridging fraction are investigated. For the bulk cylinder‐forming copolymers, novel structures such as helices and stacked toroids form, which depend sensitively on the pore size. Several significant differences between the two types of copolymers are predicted and explained based on the differences in their chain conformations and chain architectures. A simple model is proposed to explain the mean square radius of gyration for the bridge and loop chains.magnified image

A Monte Carlo Study of the Second Virial Coefficient of Semiflexible Regular Three-Arm Star Polymers

A Monte Carlo (MC) study is made of the second virial coefficient A2 along with the mean-square radius of gyration for regular three-arm star and linear freely rotating chains of bond angles θ ranging from 109° (typical flexible chain) to 175° (typical semiflexible or stiff chain) with the Lennard–Jones 6-12 potentials between beads corresponding to a good solvent system, in the range of the total number n of bonds from 30 to 900. On the basis of the MC values of A2 so obtained, an examination is made of effects of chain stiffness on the ratio of A2 of the star chain to that of the linear one, both chains having the same n and θ. It is then found that is rather insensitive to change in θ (chain stiffness) in contrast to the cases of the ratios gS and gη related to and the intrinsic viscosity [η], respectively, which remarkably decrease with increasing θ.

Mean-Square Radius of Gyration and Second Virial Coefficient of Poly(diisopropyl fumarate) in Dilute Solution

The mean-square radius of gyration <S2> and second virial coefficient A2 were determined from light scattering measurements for 13 samples of poly(diisopropyl fumarate) (PDiPF), each with the fraction of racemo diads fr=0.22, in a range of weight-average molecular weight from 4.02×104 to 8.59×105 in tetrahydrofuran at 30.0 °C. From a simultaneous analysis of the present data for <S2> and A2 on the basis of the Kratky–Porod (KP) wormlike chain with excluded volume, the KP model parameters for PDiPF, i.e., the stiffness parameter λ−1 and the shift factor ML are determined to be 113 Å and 89 Å−1, respectively, the ML value being consistent with that estimated on the basis of chain conformations of PDiPF. The λ−1 value indicates that PDiPF is stiffer than typical flexible polymers but not so stiff as typical semiflexible polymers such as poly(n-hexyl isocyanate).

Primary Structure of Poly(N-isopropylacrylamide) Synthesized by Radical Polymerization. Effects of Polymerization Solvents

The mean-square radius of gyration 〈S2〉, second virial coefficient A2, and intrinsic viscosity [η] were determined in methanol at 25.0 °C for two kinds of poly(N-isopropylacrylamide) (PNIPA) synthesized by radical polymerization in tert-butanol and benzene by the use of azobis(isobutyronitrile) as an initiator. In all cases of the three quantities, it is found that the value is smaller for the PNIPA synthesized in benzene than for that synthesized in tert-butanol and the difference between the two kinds of PNIPA increases with increasing the weight-average molecular weight Mw. The average chain dimension of the former PNIPA should then be smaller than that of the latter. Since the two kinds of PNIPA were shown to have the same stereochemical composition (and also the same chain end group), the difference in the average chain dimension may be regarded as arising from the difference in the primary structure, i.e., the number of branch points. Considering the fact that the average dimension of a given polymer chain in general decreases with increasing the number of branch points, it may be concluded that the number is larger in the former PNIPA than in the latter. The behavior of the interpenetration function Ψ and the Flory–Fox factor Φ as functions of Mw is also examined, confirming the conclusion.

MEAN-SQUARE RADIUS OF GYRATION AND DIPOLE MOMENT OF POLY(METHYLPHENYLSILOXANE) CHAINS

The mean-square radius of gyration 〈S2〉, the mean-square dipole moment 〈D2〉, the mean-square end-to-end distance 〈R2〉 and their temperature coefficients of unsymmetrical disubstituted poly(methylphenylsiloxane) (PMPS) chains, as a function of stereochemical structure, confomational energies and length of polymers, were studied by using an improved configurational-confomational statistical method based on the rotational-isomeric-state theory. It is found that the increase in isotacticity of PMPS chains causes a significant decrease in the unperturbed dimensions and increase in their temperature coefficients. This correlation of the properties of PMPS and its stereochemical structure is different from that of vinyl monosubstituted polystyrene and disubstituted poly(α-methylstyrene) chains. Dependence of the characteristic ratio of 〈S2〉 on conformational energy Eω′ of C6H5…C6H5 four-bond interactions increases markedly with the degree of isotacticity increasing, the dependence on the conformational energy Eδ...

Time Programmed Feed of Semi‐Batch Reactors with Non‐Linear Radical Copolymerizations: An Experimental Study of the System Styrene + Divinylbenzene Using SEC/MALLS

AbstractSummary: The radical crosslinking copolymerization of mono and divinyl monomers was experimentally studied with a 2.5 dm3 semi‐batch reactor using styrene + divinylbenzene as a model system. The analysis of products was carried out by SEC with a MALLS detector. The influence of the feed policy of divinylbenzene on the time evolution of the copolymer molecular weights and z‐average mean square radius of gyration was assessed. A detailed kinetic model, in the absence of intramolecular reactions but taking into account the presence of the two isomers m‐ and p‐ in the commercial divinylbenzene and the different reactivities of the various radicals and double bonds was developed; most parameters have been collected from previous kinetic studies, and only two have been regressed using our measured molecular weights. These results can be used to improve the production of branched/crosslinked polymers with controlled molecular architecture.

A Monte Carlo Study of the Intrinsic Viscosity of Semiflexible Regular Three-Arm Star Polymers

A Monte Carlo (MC) study is made of the intrinsic viscosity [η] and also of the mean-square radius of gyration <S2> for regular three-arm star freely rotating chains of bond angles θ=109°, 165°, and 175° and with the Lennard–Jones 6-12 potentials between beads having the parameter values corresponding to the Θ temperature, in the range of the total number n of bonds in the chain from 60 to 300. Three kinds of approximate values of [η] are calculated by the use of the Kirkwood–Riseman (KR) approximation, the Zimm rigid-body ensemble approximation which gives an upper bound [η](U) to [η], and by the Fixman method which gives a lower bound [η](L), the KR value of [η] being designated [η](KR). On the basis of the three kinds of MC values of [η] so obtained, the behavior of the ratio gη of [η] for the star chain to that for the linear one, both having the same n, is examined as a function of the reduced contour length λL as defined as the total contour length L of the corresponding Kratky–Porod (KP) wormlike chain divided by its stiffness parameter λ−1, the values of λL having been determined from an analysis of the present and previous MC data for <S2> on the basis of the KP chain. It is found that the KR value gη(KR) of gη as defined by [η](KR)(star)/[η](KR)(linear) lies between the values of an upper bound gη(U) and a lower one gη(L), which are defined by [η](U)(star)/[η](L)(linear) and [η](L)(star)/[η](U)(linear), respectively, irrespective of the values of λL. Further, the difference between the two bounds becomes very small for small λL, indicating that gη(KR) may give a good approximate value for semiflexible or stiff chains.

Transition from a chain-collapse process to a chain-aggregation process of poly(methyl methacrylate) in a mixed solvent

The transition from a chain-collapse process to a chain-aggregation process was studied by static light scattering on poly(methyl methacrylate) of the molecular weight mw=1.22x10(7) in the mixed solvent tert-butyl alcohol+water (2.5 vol%). The concentration c of the solutions ranged from 0.5 to 2.5x10(-4) g/cm3 and the measurement was carried out at appropriate time intervals up to the time t(h)=3250 after the quench to 35.0 degrees C which was a few degrees below the phase separation temperature. The molecular weight Mw and mean-square radius of gyration<s2>z were estimated from each scattering curve determined at the finite concentrations. At the initial stage of 100 h, <s2>z decreased rapidly with the time t indicating the chain collapse, while at the later stage of 3000 h, <s2>z increased very slowly indicating the chain aggregation. The chain-collapse process and chain-aggregation process could be analyzed separately, though the two processes overlapped appreciably at the higher concentrations. The former process depended slightly on the concentration, while the latter process showed the exponential growth of ln Mw approximately ct and ln<s2>z approximately ct. In the plot of ln<s2>z versus ln Mw, the chain-collapse process was depicted by different lines depending on the concentration, while the chain-aggregation process was described by a single straight line. The transition from the former to the latter process occurred distinctly near 200 h after the quench irrespective of the concentration.

Monte Carlo simulation of the molecular properties of poly(vinyl chloride) and poly(vinyl alcohol) melts

NPT Monte Carlo simulations were performed to calculate the molecular properties of syndiotactic poly(vinyl chloride) (PVC) and syndiotactic poly(vinyl alcohol) (PVA) melts using the configurational bias Monte Carlo move, concerted rotation, reptation, and volume fluctuation. The density, mean square backbone end-to-end distance, mean square radius of gyration, fractional free-volume distribution, distribution of torsional angles, small molecule solubility constant, and radial distribution function of PVC at 0.1 MPa and above the glass transition temperature were calculated/measured, and those of PVA were calculated. The calculated results were compared with the corresponding experimental data and discussed. The calculated densities of PVC and PVA were smaller than the experimental values, probably due to the very low molecular weight of the model polymer used in the simulation. The fractional free-volume distribution and radial distribution function for PVC and PVA were nearly independent of temperature.

Complex Formation of DNA with Oppositely Charged Polyelectrolytes of Different Chain Topology: Cylindrical Brushes and Dendrimers

The complex formation between DNA (pUC19-supercoiled DNA, 2686 base pairs) and some polycations of different chain topologies in aqueous solution was studied by light scattering, gel electrophoresis, and AFM. The investigated polycations comprised cylindrical brush polymers with quaternized poly(vinylpyridine) and polyethylene imine side chains as well as a fifth generation dendrimer thus covering a broad molar mass regime of 3 × 104 g mol-1 < Mw < 1 × 107 g mol-1 and very different chemical charges/molecule, Z+, of 127 < Z+ < 5500. Irrespective of the polycation, the complexes formed in dilute solution exhibited a similar size in terms of the mean square radius of gyration, 〈Rg2〉, i.e., 30 nm < Rg < 40 nm (excess of DNA) and 40 nm < Rg < 55 nm (excess of polycation). At a large excess of either DNA or polycation, the complexes were shown to coexist with the uncomplexed molecules of the excess component and did not vary in size with increasing weight fraction of the minority component. Only if the number ...

Unperturbed dimensions of atactic poly(dibenzyl itaconate)s

The unperturbed dimension and temperature character of poly(dibenzyl itaconate)s (PDBzI) are studied by a revised rotational isomeric state (RIS) method. The improved formulas of the mean-square radius of gyration, deduced by the pseudo-stereochemical equilibrium approach, may be used to investigate the configurational-conformational properties of atactic polymers with large side groups [poly(itaconates) for instance]. The calculated results showed that poly(itaconates) have larger dimension of the molecule than other vinyl polymers. Comparison of the dimension between considering and without considering side groups showed that the effect of large side groups on the unperturbed dimension for short-chain polymers is more obvious than that of long-chain polymers and, if the dimension of side groups increases, the effect also increases. The dimension differences of PDBzI between short-chain and long-chain polymers are investigated by the relation of characteristic ratios and temperature coefficients with temperature. Moreover, the dependence between the temperature coefficients and the tacticity of chains shows that the temperature characters of the isotactic, syndiotactic and atactic PDBzI chains have remarkable difference.

Free energy barrier for compact chains escaping from a small sphere

In this paper, the process of compact polymer chains escaping from a small sphere to a large one in the view of thermodynamics is investigated in detail based on the pruned-enriched-Rosenbluth method (PERM), which is quite efficient for the three-dimensional polymers on the simple-cubic lattice. In our simulation, three representative states of a polymer chain during the escaping process are studied, and some statistical properties of the chain size and the chain shape, such as mean-square radius of gyration per bond 〈 S 2〉/ N and the shape factor 〈 δ ∗〉 are investigated. Our aim is to illuminate how the size and shape of the compact chains change during the escaping process. The changes of 〈 S 2〉/ N and 〈 δ ∗〉 are not monotone and it is due to the fact that the chain should stretch itself in the escaping process. In the meantime, some thermodynamic properties are also calculated here. The hole is designed to be small enough to allow only one monomer at a time and it thus reduces the number of allowed chain conformations and breaks contacts between monomers at the beginning of the process. Additionally, we discuss the free energy barrier per bond H 2 − H 1 = Δ H of a compact chain, and here H 2 is the maximum free energy per bond during the process and H 1 is the minimum one when the compact chain is within the small sphere. Averaging free energy barrier over chain length N is convenient for the comparison with different chain lengths. Δ H as a function of chain length N and radius r 1 of the small sphere is also studied and our result shows that Δ H for longer chains is lower means that it is relatively easier for each bond in longer chains to surmount the free energy barrier to escape. Some discussions about the self-avoiding walk (SAW) and swollen chains are also made for the comparison, and our results also show that the restriction of the small sphere on the SAW and the swollen chains is more effective because of their relatively looser intrinsic structure.

Coarse-Grained and Reverse-Mapped United-Atom Simulations of Long-Chain Atactic Polystyrene Melts: Structure, Thermodynamic Properties, Chain Conformation, and Entanglements

A coarse-grained model of atactic polystyrene, in which meso and racemo diads are represented as single “superatoms,” parametrized using Iterative Boltzmann Inversion, has been subjected to connectivity-altering Monte Carlo simulations in order to simulate monodisperse atactic polystyrene melts of molar mass up to 210000 g mol-1 at 500 or 413 K and 1 bar. Analysis of the Monte Carlo results reveals excellent equilibration of chain conformations at all length scales. Chain dimensions, as determined from the mean square end-to-end distance, the mean square radius of gyration, and simulated Kratky plots of the single-chain scattering function, are in excellent agreement with experiment. The equilibrated long-chain configurations are reduced to entanglement networks via topological analysis with the CReTA algorithm. The resulting Kuhn length of primitive paths provides an excellent estimate of the molar mass between entanglements and of the entanglement tube diameter extracted from plateau modulus measurements. The distribution of strand lengths between entanglements, when appropriately reduced, follows the same master curve as previously determined distributions of polyethylene, cis-1,4 polybutadiene, and poly(ethylene terephthalate). A new strategy is introduced for reverse mapping the long-chain coarse-grained configurations into detailed united-atom configurations in a manner that preserves the sequence of diad types along the chains. This strategy employs local “flip” Monte Carlo moves to relax the reverse-mapped configurations. Relaxation starts using bonded interactions only, and proceeds by gradually introducing nonbonded interactions. Final relaxation is achieved via short-time canonical molecular dynamics simulation. Predicted wide-angle X-ray diffraction patterns from reverse-mapped configurations are indistinguishable from those of short-chain melts equilibrated directly in the united atom representation using molecular dynamics, and in favorable agreement with experiment. Distributions of torsion angles and pairs of successive torsion angles in the reverse-mapped configurations exhibit some deviations from the corresponding distributions of directly equilibrated short-chain united atom melts and from experimental NMR measurements.

Characterization of Rodlike Poly(n-hexyl isocyanate) Macromonomers and Their Polymacromonomers by Light Scattering, SAXS, Intrinsic Viscosity, and Scanning Force Microscopy

SEC-MALS, small-angle X-ray scattering (SAXS), and viscosity measurements were made on tetrahydrofuran (THF) and n-hexane of a series of 4-vinylbenzyl and methacrylate ended poly(n-hexyl isocyanate) (PHIC) macromonomers (VB-HIC-n and MA-HIC-n, where n is a degree of polymerization of HIC and in a range from 21 to 192), together with higher molecular weight PHIC chains than the macromonomers. The molecular weight dependence of z-average mean-square radius of gyration z and intrinsic viscosity [η] of the macromonomers and PHIC chains in THF at 25 °C were quantitatively described by the wormlike chain model with the stiffness parameter (λ−1)=63nm, the molecular weight per unit contour length (ML)=725 nm−1, and the hydrodynamic diameter (dB)=1.6 nm. The SAXS scattering profile of VB-HIC-57 in n-hexane at 25 °C was also perfectly described in terms of the straight cylinder model. The results imply that the macromonomers may be regarded as a rigid rod molecule in THF and n-hexane. Cylindrical brushes consisting of polystyrene as a main chain and PHIC chains as a side chain were prepared by homopolymerizations of VB-HIC-46 and their dimensional properties were investigated in THF at 25 °C by SAXS and SEC-MALS measurements. The molecular weight dependence of z of the brushes was quantitatively explained by the wormlike cylinder model with the parameters of λ−1=48 nm, ML=2.40×104 nm−1, the cross-sectional radius of gyration of the cylinder o1/2=4.66 nm, and the end effect (δ=16.7 nm) arising from side chains near the main-chain ends. It was concluded, therefore, that the main chain stiffness of the brush remarkably increases by the presence of densely located rodlike side chains. Direct observation of single brush of the poly(VB-HIC-46) deposited on a mica was made by scanning force microscopy (SFM) to reveal the cylindrical brushes consisting of rodlike side chains with ca. 15 nm in a thickness.

Prediction of mean square radius of gyration of tree-like polymers by a general kinetic approach

This paper describes a kinetic method to predict the z-average molecular mean square radius of gyration of tree-like polymers formed by irreversible reactions, assuming Gaussian chains. It is based on the population balance equations for the two-sided molecular distributions of pendant chains associated with every chemically distinguishable kind of bonds. An automated method for the solution of those equations is valid both before as well as after gelation for complex kinetic schemes. Examples of its use are presented with polycondensation systems leading to hyperbranched polymers, the anionic polymerization of mono- and divinyl monomers and a radical polymerization with terminal branching and transfer to polymer.