Mean-square radius of gyration of polymer chains

The calculations of the mean-square radius of gyration for more than thirty sorts of polymer chains are reviewed on the basis of a unified approach. A general expression of the mean-square radius of gyration was developed for polymer chains with side groups and/or heteroatoms. It consists of two parts. The first part is the mean-square radius of gyration of a model chain, in which every side group, R, was considered to be located in the centroid of the substituent flanking the related skeletal atom, and the second one is the total contribution of the square radius of gyration of every substituent around its centroid. Numerical calculations showed that the logarithmic relationship between the mean-square radius of gyration and the degree of polymerization becomes linear when x is greater than 100, and the dependence of the mean-square radius of gyration on the molecular weight can be expressed by the general formula 〈S²〉 = aM^b, which was supported by a number of experimental measurements. A comparison of our expression for the mean-square radius of gyration with that reported by Flory was made. The difference is obvious in the range of lower molecular weight, and gradually declines with increasing degree of polymerization.     

The radius of gyration for a ternary self-condensing vinyl polymerization system

A ternary self-condensing vinyl polymerization(SCVP) system consisting of monomers, inimers and core initiators is studied by the principle of statistical mechanics. Starting from partition functions for polymerization that constructed from two different viewpoints, explicit expressions of the size distribution function(SDF), equilibrium free energy, and the law of mass action are obtained. For two types of hyperbranched polymers(HBPs) formed in the system, the first, the second, and the third mean-square radii of gyration(MSRG) are presented under various conditions. Furthermore, the solvent effect and excluded volume effect are taken into account in the calculation of the MSRG. With these results, the effects of monomers, inimers and core initiators on the spatial dimension of HBPs are discussed. It is shown that the competition between polymers with and without a core in the growing process results in a significant influence on the average properties of the system.     

Mean square radius of gyration and hydrodynamic radius of jointed star (dumbbell) and H-comb polymers

Equations for the mean square radius of gyration and the hydrodynamic radius for jointed stars (dumbbells) and H-combs are derived, based on random flight statistics for each subchain. Comparision with literature data on computer simulations and experimental data for H-combs show good agreement for the g-value of the mean square radius of gyration even in good solvents. This suggests that for the mean square radius of gyration the relative dimension of a H-comb relative to the linear molecule of the same degree of polymerization is not altered significantly by long range interactions, as in the case of star polymers. For the hydrodynamic radius the situation is different. Fair agreement is found when comparing our results with viscosity measurements under θ conditions, while in good solvents the viscosity of the branched molecule is more reduced than predicted by our results.     

关于聚乙烯分子链均方回转半径的进一步讨论

本文考虑到氢原子的影响, 用矩阵代数的方法重新推导了聚乙烯分子链的均方回转半径。在我们得到的公式中若忽略氢原子的质量时, 就简化到Flory的公式。具体数值计算得到聚乙烯分子链均方回转半径对分子量的依赖关系是:〈S^2〉^1^/^2=0.44M^1^/^2比较θ-溶剂的实验数据: ⅹ〈S^2〉^1^/^2~w=0.45±0.08M^1^/^2~w是符合得很好的。     

含引发机制的Ａf—Ａg型缩聚反应Z—均回转半径研究

讨论了含引发机制的Ａf-Ａg型缩聚反应体系Z－均回转半径,利用枝状高分子成键的统计分析,给出组合因子分解公式,并得到均方回转半径的表达式;进一步结合数量分布函数,定义k次高分子半径,并给出计算k次高分子回转半径的循环公式,最后得到凝胶点前Z－均回转半径的解析表达式。     

Improved expression of the mean-square radius of gyration, 3. 1,4-polydienes

The mean-square radii of gyration of cis- and trans-1,4-polybutadiene and corresponding 1,4-polyisoprene were derived using Abe's and Flory's rotational isomeric scheme. Calculations performed using available experimental data showed that the dependence of the mean-square radius of gyration on the molecular weight for these stereoregular polydienes can be expressed as where a and b are constants characteristic of the polymer. This behaviour is analogous to that of vinyl polymers and poly-1,1-disubstituted ethylenes discussed in the preceding papers of the present series. The mean-square radius of gyration of 1,4-polyisoprene with predominantly trans units is more sensitive to stereoirregularity than with predominantly cis units. However, 1,4-polybutadiene chains are comparatively insensitive to the stereochemical composition.     

Scaling law for the radius of gyration of proteins and its dependence on hydrophobicity

The scaling law between the radius of gyration and the length of a polymer chain has long been an interesting topic since the Flory theory. In this article, we seek to derive a unified formula for the scaling exponent of proteins under different solvent conditions. The formula is obtained by considering the balance between the excluded volume effect and elastic interactions among monomers. Our results show that the scaling exponent is closely related to the fractal dimension of a protein's structure at the equilibrium state. Applying this formula to natural proteins yields a 2/5 law with fractal dimension 2 at the native state, which is in good agreement with other studies based on Protein Data Bank analysis. We also study the dependence of the scaling exponent on the hydrophobicity of a protein chain through a simple two‐letters HP model. The results provides a way to estimate the globular structure of a protein, and could be helpful for the investigation of the mechanisms of protein folding. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 207–214, 2009     

The kth radius of gyration for a polycondensation reaction with identical structural units AaBb

For a polycondensation reaction with identical structural units A_aB_b (i.e. each unit has a functional groups of type A and b of type B) the kth radius of gyration $ \begin{array}{*{20}c} {\left\langle {R^2 } \right\rangle _k = \sum\limits_n {n^k \cdot \left\langle {R_n^2 } \right\rangle P_n } } & {({\rm sum over all }n{\rm - mers; see Eqs}{\rm . (11) and (17))}{\rm }} \\ \end{array} $ is investigated. By means of a differential method, a recursion formula that holds true for both the pre-gel and the post-gel state is obtained for the evaluation of the kth radius.     

Temperature dependence of limiting viscosity number and radius of gyration of cellulose diacetate in acetone

Acetone solutions of a cellulose diacetate fraction were studied by viscosity and light scattering methods over the range 12.6–50.3². The temperature dependences of the limiting viscosity number [η], the mean-square radius of gyration 〈s²〉, and the second virial coefficient A₂ were determined. The unperturbed mean-square radius of gyration 〈s²〉_o and the expansion factor α, were estimated by using theoretical relations to the interpenetration function. It was found that dln 〈s²〉_o/dT is ?6.4 × 10^?3 deg^?1, while α, is close to unity over the whole temperature range studied. The viscosity results are interpreted to show that the draining effect is not negligible and the Flory viscosity parameter Φ slightly increases with increasing temperature. It is finally concluded that the value of ?6.9 × 10^?3 deg^?1 for dln [η]/dT can be ascribable to the rapid decrease in 〈s²〉_o.     

Measurements of persistence length and temperature dependence of the radius of gyration in bulk atactic polystyrene

Measurements of radius of gyration, $\text{〈S}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, and persistence length of polymer molecules in bulk atactic polystyrene have been made by means of low angle neutron scattering. The measured values of $\text{〈S}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ lie close to those measured in ?-solvents and have a molecular weight dependence of $\text{M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>+?</mtext>}}$ where ? = 0·04 ± 0·04. The molecular configuration agrees closely with that of a random coil with a persistence length $\text{～ 10}\text{A}\text{?}$. These measurements support the model of an unperturbed Gaussian coil for this polymer. Measurements of $\text{〈S}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ throughout the glass transition region indicate that the overall conformation of the polymer chain does not change on heating through this region.     

Improved expression of the mean-square radius of gyration, 2. Poly(1,1-disubstituted ethylene)s

The mean-square radius of gyration of poly(1,1-disubstituted ethylene)s is calculated according to a method already developed for poly(methyl methacrylate), poly(α-methylstyrene) and polyitaconate. During the derivation both the effect of side groups and the masses of skeletal atoms were taken into account. A hypothetical polymer chain was introduced, in which the mass of the substituents on every C^α was considered to be concentrated in their center of mass, and the virtual side bond vector runs from C^α to this center. The mean-square radius of gyration of poly(1,1-disubstituted ethylene)s consists of two parts, one of which is the mean-square radius of gyration of the hypothetical molecule described before and the other is related to the geometrical characteristics of the side groups. Numerical calculations indicated that the dependence of the mean-square radius of gyration of poly(1,1-disubstituted ethylene)s on the molecular weight is analogous to that of vinyl polymers, 〈S²〉 = aM^b, where a and b are constants characteristic of the polymer.     

Distribution function for the radius of gyration for theta chains attached at one end to an impenetrable surface

Chains that are unperturbed by the intramolecular excluded volume effect are examined in the free state and under conditions where one of the ends is attached to an impenetrable surface. The density of the attachment of the chains to the surface is so low that interchain interactions can be ignored. The shapes of the distribution functions for the squared end-to-end distance, r², and squared radius of gyrations, s², are evaluated in both situations. As is well-known from previous work, attachment of the chains to the impenetrable surface produces an increase in 〈r²〉 and (to a lesser extent) 〈s²〉. It also narrows the distribution function for r². There is, however, no detectable effect on the shape of the distribution function for s². The narrowing of the distribution function for r², coupled with the absence of any effect on the shape of the distribution function for s², is rationalized with a very simple model.     

Corrections to calculations of the radius of gyration of polydisperse gaussian coils when scattering measurements at low angle are missing

The initial slopes and intercepts of Zimm plots of scattering experiments yield the centroidal radius of gyration of the scattering particle. In many cases, it is not possible to measure at sufficiently low angle to obtain accurate initial slopes and intercepts. Errors for missing data can be corrected by using calculated scattering functions as models. Such corrections are given here for polydisperse Gaussian coils.     

Cyclic polysiloxanes

The preparation, fractionation and characterisation of synthetic cyclic polysiloxanes (R(CH₃)SiO)_x are reviewed. Particular emphasis is given to the cyclic poly(dimethylsiloxanes) (PDMS) (R is CH₃) cyclic poly(phenylmethylsiloxanes)(PPMS) (R is C₆H₅) and cyclic poly(vinylmethylsiloxanes) (PVMS) (R is CH₂=CH). The application of high performance liquid chromatography (HPLC) and analytical and preparative gel permeation chromatography (GPC) to the preparation and characterisation of cyclic polysiloxanes is discussed. The methods include preparation of sharp fractions of relatively high molar mass cyclic polymers (with heterogeneity indices typically ca. 1.05) and comparison of the GPC and retention volumes of cyclics with the corresponding linear polymers. Investigations of the cyclic polymers prepared have been made using a number of experimental methods and techniques. These include dilute solution viscometry and low angle neutron scattering. Comparisons are drawn between cyclic PDMS and other cyclic polysiloxanes and also between cyclic polysiloxanes and the corresponding commercially-important linear polymers.     

On-line measurement of the RMS radius of gyration and molecular weight of polyimide precursor fractions eluting from a size-exclusion chromatograph

The recent introduction of multiangle light-scattering detectors for size-exclusion chromatography has made possible the measurement of the root mean square radius of gyration (R_g) and molecular weight (M) of polymer fractions eluting from a size-exclusion chromatography column. The characterization of the dimensions of a polymer may be accomplished with only a few milligrams. The dimensions of a polyimide precursor prepared by the condensation of the meta-diethyl ester of pyromellitic dianhydride with para-phenylene diamine have been measured with this technique. The dependence of R_g on M across the distribution is compared with that predicted for a freely rotating chain, and with other similar polymers measured with hydrodynamic techniques.