Small-angle neutron scattering studies of molten and crystalline polyethylene

Small-angle neutron scattering studies have been made of molten and crystalline polyethylene using samples containing small amounts of deuterated polyethylene (PED) in a protonated polyethylene (PEH) matrix. Careful studies were made of PED aggregation effects, and by a combination of solution blending techniques and rapid quenching from the melt, it was possible to prepare samples with a statistical distribution of PED molecules in the PEH matrix. Measurements of radius of gyration $\text{(S}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{w}}$ at low κ [$\text{κ = (}\text{4π}\text{λ}$) sin $\text{? ≤ 2 × 10}{}^{\mathrm{?2}}\text{A}\text{?}{}^{\mathrm{?1}}$] in the melt and in the solid state gave very similar values which may be summarized as $\text{〈S}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{w}}\text{= (0.46 ± 0.05)M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{w}}$ for both phases. This correspondence of values indicates that on a rapid quench, diffusion is sufficiently slow that the molecule crystallizes with a similar spatial distribution of mass elements to that possessed in the melt. Measurements of scattering data over a wide κ range ($\text{6 × 10}{}^{\mathrm{?3}}\text{≤ κ ≤ 0.12}\text{A}\text{?}{}^{\mathrm{?1}}$) have also been made from samples showing no aggregation effects. Calculations indicate that it is difficult to fit this data in terms of models which postulate adjacent chain re-entry in one crystallographic plane for this type of sample.     

Investigation of the methyl torsion in isotactic polypropylene — comparison between neutron inelastic scattering spectra and normal coordinate calculations

Incoherent neutron scattering spectra for polypropylene and its partially deuterated analogue allow identification of the methyl torsion at 230 cm^?1. Comparison of the experimental spectra with the calculated densities of states shows discrepancies with the expected intensity of the methyl torsional band. Results from a stretch-oriented sample arranged so that the wave vector $\text{Q}$ is first parallel ($\text{Q}{}^{\mathrm{\parallel }}$) and then perpendicular ($\text{Q}{}^{\mathrm{\perp }}$) to the helical chain axis indicate that the torsion is more intense and probably has a small frequency dispersion around 240 cm^?1 for $\text{Q}{}^{\mathrm{\parallel }}$ and is weaker with a broad dispersion centred at 220 cm^?1 for $\text{Q}{}^{\mathrm{\perp }}$.     

Chain trajectory in solution grown polyethylene crystals: correlation between infra-red spectroscopy and small-angle neutron scattering

Infra-red (i.r.) measurements of the CD₂ bending vibration have been made for polyethylene single crystals of mixed isotopic composition under a wider range of conditions than hitherto. A comparison with data for mixed paraffin crystals shows that the polymer molecules exist in a compact conformation, consistent with the ‘superfolded’ structure which has emerged from small angle neutron scattering (SANS) measurements. However, the correlation splitting of the CD₂ bending vibration, a prominent feature of previous studies, is not generally observed. At ‘guest’ concentrations of 1%, SANS data indicate minimal isotopic fractionation, and it is only at such low concentrations that the band-shape of the CD₂ bending vibration primarily reflects intra-chain effects due to chain conformation. Previous SANS data at wider angles ($\text{10}{}^{\mathrm{?1}}\text{> s > 10}{}^{\mathrm{?2}}\text{A}\text{?}{}^{\mathrm{?1}}$) lead to an estimate of about 0.5 for the fraction of stem sites along a folded ribbon occupied by a single molecule. A study of the i.r. peak broadening leads to a similar value for the local concentration.     

Unperturbed dimensions of poly(phenyl acrylate)

Thirteen fractions of poly(phenyl acrylate) have been prepared with weight-average molecular weight ranging from 0.158 × 10⁶ to 2.57 × 10⁶ g mol^?1. The temperature coefficient of the unperturbed dimensions and the glass transition temperature were found to be ?1.8 × 10^?3 deg^?1 and 55.6°C respectively. Good accord was obtained among different methods for establishing θ-conditions of 11.5°C in ethyl lactate. From viscometry, osmometry and light scattering under θ-conditions, as well as in a good solvent, the unperturbed dimensions were determined via several procedures yielding a value of $\text{[〈r}{}^2\text{〉}{}_{\mathrm{0w}}\text{M}\text{?}{}_{\mathrm{w}}\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 6.0 (±0.2) × 10}{}^{\mathrm{?9}}\text{cm g}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{mol}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. This corresponds to a steric factor υ = 2.37 (±0.08) and a characteristic ratio C_∞ = 11.3 (±0.8). The polymer chain is thus more rigid than poly(methyl acrylate), but less rigid than poly(phenyl methacrylate). With respect to its T_g and flexibility, poly(phenyl acrylate) bears a strong similarity to poly(benzyl methacrylate).     

Neutron scattering studies on solution-grown crystals of polyethylene: a statistical preference for adjacent re-entry

Neutron scattering studies of solution-grown polyethylene crystals demonstrate a strong statistical preference for adjacent re-entry, with 75% of stems occupying adjacent lattice sites. The number of sheets of stems for one molecule increases with molecular weight, with an average molecular weight per sheet of 21 000. These are the main features of computer models which give the best agreement with scattering data obtained over an angular range ($\text{q</1}\text{A}\text{?}{}^{\mathrm{?1}}$). Scattering in this q range is directly sensitive to typical stem separations. Any preference for alternate re-entry would result in a peak at $\text{q=0.7}\text{A}\text{?}{}^{\mathrm{?1}}$, which was not observed. The results are discussed in terms of possible crystallization mechanisms.     

Solution properties and chain flexibility of polythiolmethacrylates: 1. Poly(phenyl thiolmethacrylate) and poly(O-methyl phenyl thiolmethacrylate)

Dilute solution properties of poly(phenyl thiolmethacrylate) (PTPh) and poly(o-methylphenyl-thiolmethacrylate) (PTMPh) were studied by gel permeation chromatography, light scattering, osmotic pressure and viscosity measurements in different solvents. Relations between intrinsic viscosity [η], z-average root-mean-square end-to-end distance $\text{(}\text{r}\text{?}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and molecular weight $\text{M}\text{?}{}_{\mathrm{w}}$ were established. The unperturbed dimensions were calculated by different methods using $\text{M}\text{?}{}_{\mathrm{w}}$ and [η]. PTPh (σ = 2.26) and PTMPh (σ = 2.27) have practically the same flexibility, but both are more flexible than poly(phenyl methacrylate) (σ = 2.50). In our opinion, this is due to the change in the local intramolecular interaction and the higher flexibility of the sulphur-containing side group.     

Static and dynamic structure factors calculated for flexible ring macromolecules

The influence of hydrodynamic pre-averaging on the first cumulant Γ of the dynamic structure factor of Gaussian ring molecules has been calculated. Similar to star-branched macromolecules, the relative error $\text{ΔΓ}\text{Γ}$ introduced by the pre-average approximation exhibits a maximum at intermediate $\text{q = (}\text{4π}\text{λ}\text{)}$ sin $\text{θ}\text{2}$ which does not ocour for the corresponding linear chain. This maximum is related to the increase of segment density by ring formation. The static structure factor also shows a maximum in the Kratky plot (q²〈S²〉P(q) against q²) and has an asymptote at large q which is half that of linear chains. Furthermore, the S:R ratio ($\text{〈S}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{〈R}{}^{\mathrm{?1}}{}_{\mathrm{h}}\text{〉}$), which can be determined from combined integrated and quasi-elastic scattering, proves to be significantly lower than for the linear chain. Thus, ring molecules and open chains can easily be distinguished from combined integrated and quasi-elastic light scattering, without comparison with linear chains as has been necessary hitherto. The accuracy will be particularly high with instruments which allow the recording of both types of scattering measurements simultaneously.     

Small-angle neutron scattering by semicrystalline polyethylene

The angular dependence of the intensity of neutron scattering by deuterated polyethylene chains (PED) dispersed in the protonated host (PEH) in the semicrystalline state has been computed over the scattering range $\text{0 < μ < 0.2}\text{A}\text{?}{}^{\mathrm{?1}}$, where μ = (4π/λ)sin(θ/2), for various morphologies that differ according to the mode of chain re-entry into the crystalline lamellae. The scattering functions vary strongly with the tendency for adjacent re-entry. The intensity in the range $\text{0.04 < μ < 0.10}\text{A}\text{?}{}^{\mathrm{?1}}$ is 30–40% greater for the regularly folded morphology than for the ‘switchboard’ model. Calculations for the latter approximate the experimental results of Schelten et al. Optimum agreement with their scattering intensities, and the chain dimensions and degree of crystallinity as well, is achieved by a statistical model in which the chain preferentially re-enters the crystalline layer with a probability of about 0.7. Re-entry, when it occurs, is not required to be adjacent to the preceding sequence.     

Model calculations of the phase behaviour of rod-like particles in solution: Polydisperse systems having asymmetric distributions of rod-lengths

Model calculations have been made of the phase behaviour, and compositions of the phases in the biphasic range, for polydisperse rod-like particles in solution. The special cases of a most-probable distribution (m.p.d.), an empirically modified most-probable distribution and asymmetric Gaussian distributions are considered. It is shown that the presence of species of very low molecular length in the m.p.d. result in $\text{v}{}^{\mathrm{<mtext>o</mtext>}}{}_{\mathrm{?}}\text{7}$, the volume fraction of polymer for which the system becomes wholly-anisotropic, being close to unity and thus the biphasic range for the case of this distribution is far larger than that normally found experimentally. The empirically-modified m.p.d. gives a narrower biphasic range than that for the m.p.d. Calculations for the asymmetric Gaussian distribution make it clear that $\text{v}{}^{\mathrm{<mtext>o</mtext>}}{}_{\mathrm{?}}\text{7}$ is determined by the low molecular length species in the distribution whilst $\text{v}{}^{\mathrm{<mtext>o</mtext>}}{}_{\mathrm{?}}\text{1}$, the volume fraction of polymer at which the system goes from isotropic to biphasic, is determined by the high molecular length species of the distribution. It is suggested that these distributions and those considered earlier are unable to adequately explain the observed dielectric and viscosity behaviour of rod-like molecules in solution. It is necessary to introduce a small high molecular weight tail to a chosen distribution in order to obtain agreement betwee between the calculate and the observed behaviour.     

Solution properties of poly(propylene sulphide)

Poly(propylene sulphide) (PPS) initiated by cadmium bis(phenyl allyl thiolate) was fractionated from benzene-methanol, and the fractions were examined by viscosity, osmotic pressure and light scattering determinations. Gel permeation chromatography with Styragel columns was found unsatisfactory in tetrahydrofuran (THF) and toluene at room temperature. Polymer stored in the dark at room temperature underwent a very slow degradation which broadened the distribution in the fractions. The following quantitative relationships were found at 25°C:

Extrapolations to derive the unperturbed coil dimensions indicate a possible, but doubtful, solvent effect. The value derived from the θ-mixture gives $\text{A = 696 × 10}{}^{\mathrm{?11}}\text{cm mol}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{g}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ and a steric factor σ = 1.49, consistent with a high ‘chain flexibility’ expected from the relatively long C-S-C linkages.     

Small-angle neutron scattering studies of polyethylene crystallized at high pressure

The technique of small-angle neutron scattering (SANS) has been used to study the chain configuration in pressure crystallized polyethylene. Two narrow molecular weight fractions of deuterated molecules (PED) of M_w 23 000 and 54 000 were solution blended with a protonated matrix polymer of M_w 81 500. Although pressure crystallization was shown to have produced a clustering of the PED molecules, the radii of gyration $\text{〈}\text{S}{}^2\text{〉}{}_{\mathrm{z}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ were, nevertheless, shown to be consistent with a model in which the PED molecules possessed rod-like configurations. The predicted rod lengths were in close agreement with the molecular stem lengths of the PEH matrix polymer, which were independently determined by nitric acid etching. Furthermore, a doubling of the PED molecular weight produced no change in the value of $\text{〈S}{}^2\text{〉}{}_{\mathrm{z}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. This was interpreted in terms of a chain folding mechanism in which a molecule is bounded by the surfaces of a lamellar block and is therefore unable to increase its' rod length.     

Polyacrylamide in water: molecular weight dependence of 〈R2〉 and [η] and the problem of the excluded volume exponent

A commercial sample of polyacrylamide (CALGON) has been carefully purified, analysed and fractionated. With suitable care, the molecular characterization can be carried out by light scattering and viscosity measurements, without disturbances arising from microgels or aggregates. The molecular weight dependence of the radius of gyration and intrinsic viscosity are given by:

$\text{〈R}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 7.49 × 10}{}^{\mathrm{?}}\text{M}{}^{0.64}\text{(}\text{A}\text{?}\text{)}$

$\text{[η] = 9.33 × 10}{}^{\mathrm{?3}}\text{M}{}^{0.75}\text{(cm}{}^3\text{/g)}$

The value of the exponent ν = 0.64 in the $\text{〈R}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ equation has been cross-checked by a study of the asymptotic behaviour of the scattering envelope. Such a value is higher than that expected in a good solvent (ν = 0.60) and should lead to a higher exponent in the intrinsic viscosity dependence. However, it agrees more closely with the published dependence of sedimentation and diffusion coefficients.     

A calculation of the elastic constant c33 of a graphite crystal as a function of temperature

A calculation is presented of the elastic constant C₃₃ of a graphite crystal as a function of temperature up to 2500 K, taking into account the anharmonic contribution and the changes in interlayer interactions due to the large lattice thermal expansion. Parametric variations in the theory show that the anharmonic contribution to C₃₃ depends principally on the parameter ($\text{?}{}^2\text{C}{}_{33}\text{?e}{}^2{}_{\mathrm{zz}}$) Comparison of theoretical results with the experimental data, which is mainly from neutron scattering experiments, shows that the data can be accounted for if ($\text{?}{}^2\text{C}{}_{33}\text{?e}{}^2{}_{\mathrm{zz}}$) lies in the range 7–10 × 10¹³ dynes/cm². A theoretical estimate of ($\text{?}{}^2\text{C}{}_{33}\text{?e}{}^2{}_{\mathrm{zz}}$) based on Lennard-Jones potentials between atoms in adjacent basal planes gives a value of 9·07 × 10¹³ dynes/cm².     

Properties of extremely high molecular weight polystyrene in solution

Extremely high molecular weight polystyrenes with a $\text{M}\text{?}{}_{\mathrm{w}}$ in the range 10.8 × 10⁶ to 2.2 × 10⁷ were prepared by emulsion polymerization initiated with a heterogeneous initiator at 30°C, which has a ‘living character’. Samples of polystyrene were characterized by light scattering and viscometry in toluene and benzene at 25°C, and in θ-solvent cyclohexane at 34.8°C. Also determined were the relationships of mean-square radius of gyration 〈s²〉 (m²) and the second virial coefficient A₂ (m³ mol kg^?2) on the molecular weight, which for toluene and benzene are described in equations: Toluene (25°C) $\text{〈s}{}^2\text{〉=1.59 × 10}{}^{\mathrm{?23}}\text{M}\text{?}{}_{\mathrm{w}}{}^{1.23}$; $\text{A}{}_2\text{=4.79 × 10}{}^{\mathrm{?3}}\text{M}\text{?}{}_{\mathrm{w}}{}^{\mathrm{?0.63}}$; Benzene (25°C) $\text{〈s}{}^2\text{〉=1.23 × 10}{}^{\mathrm{?22}}\text{M}\text{?}{}_{\mathrm{w}}{}^{1.20}$; $\text{A}{}_2\text{=2.59 × 10}{}^{\mathrm{?3}}\text{M}\text{?}{}_{\mathrm{w}}{}^{\mathrm{?0.59}}$. The parameters in the Mark-Houwink-Sakurada equation were established, for extremely high molecular weight polystyrene in toluene and in benzene, at 25°C into the form giving for $\text{[η] (}\text{m}{}^3\text{kg}{}^{\mathrm{?1}}\text{): [η] = 8.52 × 10}{}^{\mathrm{?5}}\text{M}\text{?}{}_{\mathrm{w}}{}^{0.61}$; $\text{[η] = 1.47 × 10}{}^{\mathrm{?4}}\text{M}\text{?}{}_{\mathrm{w}}{}^{0.56}$. The mentioned relations, as well as the obtained values of Flory parameter ?₀ and of ratio $\text{[η]}\text{M}\text{?}{}_{\mathrm{w}}{}^{0.5}$ were compared with solution properties of high molecular weight polystyrene with narrow molecular weight distribution prepared by anionic polymerization by Fukuda et al.     

Theoretical estimation of the effect of solvent on unperturbed dimensions: 1. Isotactic poly(vinyl alcohol)

The effect of the quality of solvent on the unperturbed chain dimensions $\text{r}{}_0{}^2$ of perfectly isotactic poly(vinyl alcohol) (iso-PVA) was studied by means of some model calculations according to the matrix method of Flory et al. These investigations were based on the assumption that the quality of solvent affected mainly the energy of the intramolecular hydrogen bond, while the magnitude of the interactions between other groups, hardly varied with the change of solvent. The individual conformation energies in a chain of iso-PVA were estimated from the analysis of structurally similar polymers, as well as the results obtained with 2,4-pentanediol. It was found that the characteristic ratio $\text{C =}\text{r}{}_0{}^2\text{nl}{}^2$ increased in non-polar solvents, with energy of the intramolecular hydrogen bond stabilizing the stretched zig-zag conformation of chain. Conversely, the characteristic ratio, monotonously decreased with increasing polarity of solvent to the value of about 6, typical of a flexible random coil. The existence of an analogous relationship, valid for other polymers, able to form an intramolecular hydrogen bond in the chain, was also discussed.     

A study of molecular orientation in poly(methyl methacrylate) by means of laser-Raman spectroscopy

Measurements have been made of the polarization dependence of the intensity of the Raman scattering from uniaxially oriented samples of poly(methyl methacrylate). Lines at wavenumber shifts 486, 562, 604 and 1732 cm^?1 were studied. With the right-angle scattering geometry used, four combinations of polarization directions of incident and observed scattered light could be chosen and the angle between the draw direction and the direction of the incident laser beam could be varied. The observed intensities were found to fit equations derived in an earlier publication. Five parameters which would determine the scattered intensity for any scattering geometry and combination of polarization vectors were derived. They were shown to contain information about the distribution of orientations of the structural units of the polymer despite the fact that their values also suggest that either the nature of the structural units or the interactions between them are modified by drawing. Values of $\text{cos}{}^2\text{θ}$ and $\text{cos}{}^4\text{θ}$ deduced for the structural units, where θ is the angle between a unique axis in a typical unit and the draw direction, are in good agreement with those obtained by other authors from broadline n.m.r. measurements. For the sample with the highest birefringence, ?14.8×10^?4, the values of $\text{cos}{}^2\text{θ}$ and $\text{cos}{}^4\text{θ}$ were 0.52±0.03 and 0.43±0.04 respectively.     

The unperturbed molecular dimensions of poly(ethylene oxide) in aqueous solutions from intrinsic viscosity measurements and the evaluation of the theta temperature

Intrinsic viscosity measurements were carried out on five well characterized fractions of poly(ethylene oxide) in aqueous solutions at 24.9°, 34.9°, and 45.5°C. The Stockmayer-Fixman extrapolation was applied to the data: it yields the unperturbed dimensions K₀ of the chain. The unperturbed root-mean-square end-to-end distance $\text{〈}\text{R}\text{?}{}^2\text{〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_0$ calculated for the polymer fractions in water indicate that the polymer molecules are expanded in this solvent as the temperature is raised. The temperature coefficient of unperturbed dimension, $\text{d In}\text{〈}\text{R}\text{?}{}^2\text{〉}{}_0\text{d}\text{t}\text{= 0.024}\text{K}{}^{\mathrm{?1}}$, calculated for poly(ethylene oxide) in water using the present data is about 100 times higher than the literature values of 0.23 (±0.02) × 10^?3 K^?1 and 0.2 (±0.2) × 10^?3 K^?1, respectively, obtained from force-temperature (‘thermoelastic’) measurements on elongated networks of the polymer in the amorphouse state and form viscosity measurements on this polymer in benzene. A value of θ=108.3°C was obtained from the temperature dependence of the interaction parameter B in the Stockmayer-Fixman equation.     

Quasi-elastic light scattering: separability of effects of polydispersity and internal modes of motion

The influence of a size distribution on the angular dependence of the quasielastically scattered light is studied for (i) large hard spherical particles and (ii) large flexible chain molecules. For the spherical particles the angular dependence is shown to depend solely on the size distribution and the particle scattering factor. Combination of conventional elastic light scattering with quasielastic light scattering allows the determination of the z-average radii moments $\text{r}{}^{\mathrm{<mtext>n</mtext><mtext>?</mtext>}}{}_{\mathrm{z}}\text{(n = ?1, 1, 2, …)}$ which define the size distribution. Flexible chains — linear and branched ones — show in any case a linear dependence of the apparent diffusion constant $\text{D}{}_{\mathrm{app}}\text{=}\text{Г}\text{q}{}^2$ on $\text{q = (}\text{4π}\text{λ}\text{)}$ sin $\text{θ}\text{2}$, when q becomes large. This behaviour represents the flexibility of the spring-bead model with strong hydrodynamic interaction. The initial part on the other hand form a straight line when D_app is plotted against q². The intercept of this straight line is the z-average diffusion constant while the slope is proportional to the z-average mean square radius of gyration. Thus, the polydispersity can be estimated from D_z and 〈S²〉_z while the asymptote at large q-values is determined by the internal flexibility of the molecule.     

Small-angle neutron scattering studies of isotropic polypropylene

Small-angle neutron scattering studies have been made of molten and crystalline polypropylene using samples containing small amounts of deuterated polypropylene in a protonated polypropylene matrix. The specimens were characterized by small- and wide-angle X-ray scattering to determine the d-spacing and the degree of crystallinity χ and by gel permeation chromatography to determine molecular weight, M_w, and molecular weight distribution. The degree of crystallinity was varied from 0.5 to 0.7, the d-spacing from 120 to 250 Å and the molecular weight from 34 000 to 1 540 000. Clustering was not observed. The radius of gyration $\text{〈s}{}^2\text{〉}{}_{\mathrm{w}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the tagged molecules was approximately proportional to $\text{M}{}_{\mathrm{w}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and almost independent of d and χ. In the melt similar values were obtained which are, within experimental uncertainties, the same as in a θ-solution. For 〈s²〉_wk²? 1 the scattering law approaches a k^?2 dependence. The results are discussed with reference to the chain-folded model but a fit cannot be obtained over all molecular weights. A simple random coil model fits the neutron scattering data partly but this does not explain the origin of the d-spacing.     

Determination of type and width of a size distribution from the z-averages of the radii moments rnz

Relationships are given between the z-average radii moments $\text{r}\text{?}{}^{\mathrm{n}}{}_{\mathrm{z}}$ and the common moments $\text{r}\text{?}{}^{\mathrm{n}}$ of a size distribution. Instructions are given for finding the type and width of a size distribution from measurements of the $\text{r}\text{?}{}^{\mathrm{n}}{}_{\mathrm{z}}$ moments.