Role of reptation in the rate of crystallization of polyethylene fractions from the melt

The theory of polymer crystallization with chain folding is extended to include the effect of reptation in the melt on the rates of crystallization G_I and G_II in régimes I and II. The result is that the pre-exponential factors for G_I and G_II contain a factor $\text{1}\text{n}$, Where n is the number of monomer units in the pendant chain being reeled onto the substrate by the force of crystallization; n is proportional to the molecular weight. The predicted fall in growth rate with increasing molecular weight is found experimentally in nine polyethylene fractions M_z=2.65 × 10⁴ to M_z=2.04 × 10⁵, corresponding to n_z=1.90 × 10³ to 1.45 × 10⁴. The data on these fractions are analysed to find the reptation or ‘reeling’ rate r and the substrate completion rate g. The values g_nuc～0.5/n_z cm s^?1 and r_nuc～21/n_z cm s^?1 at 400K are obtained from the data in conjunction with nucleation theory adapted to account for reptation assuming a substantial degree of regular folding. These results are consistent with a melting point in the range of ～142° to ～145°C. (The analysis using T°_m(∞)=145°C gives values of such quantities as σ σ_e and α that are quite similar to those deduced in earlier studies.) An estimate of g (denoted g_expt) that is independent of the molecular details of nucleation theory gives g_expt～0.4/n_z cm s^?1 and r～17/n_z cm s^?1 at 400K. Calculations of the reptation rate from r_1,2 = (force of crystallization ÷ friction coefficient for reptation in melt), where the friction coefficient is determined from diffusion data on polyethylene melts, leads to r_1,2～17/n_z to 34/n_z cm s^?1 at at 400K, or g_1,2～0.4/n_z to 0.8/n_z cm s^?1. The conclusion is that the reptation rate characteristic of the melt is fast enough to allow a significant degree of adjacent re-entry or ‘regular’ folding during substrate completion at the temperature cited, and that the substrate completion process is governed jointly by the activation energy for reptation $\text{Q1}{}_{\mathrm{D}}$ and the work of chain folding q. The nucleation theory and the friction coefficient theory approaches are compared, and the formulations found to be essentially equivalent; the ‘reeling’ rate r is found to be proportional to $\text{(}\text{1}\text{n}\text{)A}{}_0\text{(Δf)v}{}_0\text{exp}\text{[?}\text{(Q1}{}_{\mathrm{D}}\text{+q)}\text{RT}\text{]}$, where v₀ is a frequency factor, and A₀(Δf) is the force of crystallization on the pendant chain. The data analysis on the fractions confirms the detailed applicability of régime theory. The growth rate theory presented allows the possibility that the growth front may be microfaceted in régime I.     

Intramolecular mobility and its role in the glass transition temperature of ionic polymers

The glass transition temperature of ionic polymers is found to depend on the molar ionic cohesive energy, the universal gas constant and a factor (n) that is related to intramolecular interactions. The polyphosphates, polyacrylates and X, Y-ethylene ionenes polymer families are shown to follow the equation interrelating the above-mentioned variables. The significance of n can be understood by deriving various rules of constant specific heat at T_g (ΔC_p). The results of this analysis show that approximately $\text{2}\text{3}$ of ΔC_p is attributable to intersegmental interactions, while $\text{1}\text{3}$ of ΔC_p is related to intramolecular forces. Similar relationships were established between the degrees of freedom of the polymer repeat unit and ΔC_p. These rules are shown to be related to Wunderlich's ‘bead’ concept.     

Kinetic analysis of the crystallization of poly(p-phenylene sulphide)

Growth rates of spherulites were measured in poly(p-phenylene sulphide) crystallized from the melt and the quenched glass over the temperature range 100°C–280°C, possibly the most extensive overall range yet reported for any polymer and, as such, most propitious for study of régime III crystallization. For a medium M.wt. polymer, a régime II → III transition was obtained at 208°C using values of transport parameters common to many polymers ($\text{U1 = 1400}\text{cal mol}{}^{\mathrm{?1}}$, T_∞ ? T_g = 30°C) together with experimentally determined values of T⁰_m(315°C) and T_g(92°C). Under these conditions, the régime III/II slope ratio was found to be 2.07 (i.e. only 3.5% higher than predicted by régime theory), and reasonable estimates of surface free energies and of the work of chain folding were obtained. Other choices of the transport terms, including WLF and zero values, did not allow successful kinetic analyses. Although a régime I → II transition is predicted to occur at the high-temperature end of our growth-rate data, we found no experimental evidence for it. For a low M.wt. polymer, our analysis showed that régime III kinetics is obeyed at low temperatures, while at higher ones there is a continuous departure from that behaviour without, however, full attainment of régime II kinetics.     

Rotational energetics in vinyl polymer liquids: 1. Poly(vinyl acetate)

Mobilities of liquid poly(vinyl acetate) as expressed by the Vogel equation, $\text{?}\text{In}\text{μ =}\text{B}\text{T}\text{? T}{}_0$), are extrapolated to negative pressure using previously determined pressure coefficients. The extrapolations are extended to $\text{P1 = ?b (}\text{Tait}\text{)}$ where the liquid volume becomes ‘infinite’ and the polymer chains are in the hypothetical ‘isolated’ state. By an analysis of the rotational kinetics and relationships developed previously, this procedure leads to $\text{U1}$ and $\text{V1}$ for the ‘isolated’ chain, where $\text{U1}$ is the energy difference between the rotational states and $\text{V1}$ is an energy barrier between these states. It is found that the greatest increase occurs between $\text{U1}$ for the ‘isolated’ chain and U for the liquid at atmospheric pressure, with relatively little further increase up to 2000 bars. The energy barrier V, on the other hand, increases more uniformly over the entire pressure range from $\text{P1}$ to 2000 bars. On the basis of the analyses of the rotational energetics and kinetics, an approach to a unified molecular interpretation of shear-induced lowering of viscosity (non-Newtonian viscosity) and shear-induced crystallization in certain flowing polymer liquids is suggested.     

Study of thermal regeneration of spent activated carbons: Thermogravimetric measurement of various single component organics loaded on activated carbons

Thermogravity analysis of the activated carbons loaded with 32 different single component organics showed that TGA curves could be classified into three distinct groups with regard to their shapes. The organics that belong to Group (I) are rather volatile and TGA curves can be explained by equdibrium desorption model. Group (II) organics are relatively easy to decompose and TGA curves were interpreted in terms of first-order cracking kinetics. The parameters included in these models were obtained from the measured TGA curves by utilizing half desorbed temperature T_{$\text{1}\text{2}$} and reciprocal slope of TGA at T_{$\text{1}\text{2}$}, ΔT. Group (III) consists of phenol, β-naphtol, lignin etc. and gave high residuals on activated carbons after heating up to 800°C. This suggests that these organics are the ones that are critical to the ordinary thermal regeneration method.A rough classification of organics into these groups was done by using the boiling point and the aromatic carbon content.     

Rotational energetics in vinyl polymer liquids: 2. Other polymers

The procedures described for poly(vinyl acetate) in the preceding paper are applied to compute the effects of pressure on the Vogel T₀ and B between $\text{P1 = ?b (}\text{Tait}\text{)}$ and P = 2000 bars for polystyrene, linear polyethylene, polyisobutylene, and polydimethylsiloxane. From these are derived the pressure dependences of the rotational energies U and V, where U is the energy difference between the rotational states and V is a rotational barrier. The $\text{U1}$ for the ‘isolated’ chain is corrected by the factor $\text{Z1}$, which appears to be an intramolecular cooperative unit, to yield $\text{U7}$, the energy difference between the rotational states for a single main-chain bond in the ‘isolated’ chain. A major conclusion of this work is that statistical mechanical calculations of the conformational (i.e. rotational) properties using ‘isolated’ chain values for U will give erroneous results when applied to the real polymer liquid. The same conclusion based on a different approach has been reported previously.     

Theory of the substrate length in polymer crystallization: Surface roughening as an inhibitor for substrate completion

A model is proposed for the physical origin of the substrate length L that appears in the customary treatment of the regime I→II growth rate transitions which occur in certain polymers during crystallization from the melt. (A previous analysis of growth rate measurements showed that L ≈ 0.77 μm at the I→II transition in polyethylene). L is treated as a ‘persistence length’ between defects that have the capacity to inhibit substrate completion. The defects are pictured as resembling the Greek letter Ω (omega) in their most extended state; in their normal state they are represented as hemispherical or disc-like amorphous patches that are pinned onto the substrate. The omega defect can form on the substrate by drawing in a portion of one of the cilia, loose loops, or interlamellar links that are characteristic of the ‘variable cluster’ representation of the molecular morphology of lamellar semicrystalline polymers. The formulation relates L to the equilibrium free energy of formation of the omega defect, which is viewed as being principally entropic. Thus we derive $\text{L∝(}\text{stem width}\text{) ×}\text{exp}\text{(?}\text{ΔS}\text{R}\text{)}$. From the known value of L for polyethylene, it is determined that the experimental entropy of formation of the defect is ΔS_expt. = ? 12.6±1.5 cal mole^?1 deg^?1. This is justified on basic grounds by first applying nucleation theory to estimate the number of chain units $\text{n}{}_{\mathrm{\Omega }}$ in the defect of critical size. Then from partition functions for once- and twice-pinned polymer chains on a surface, which gives $\text{ΔS = ?fR}\text{ln}\text{n}{}_{\mathrm{\Omega }}$ with f～ 2.0 to 2.5 depending on defect shape, one arrives at a theoretical estimate of ΔS for the omega defect in polyethylene that is in good agreement with the experimental value. This indicates that the omega defect model for L is reasonable on energetic grounds. It is shown further that the model is consistent in a number of respects with what is known about the I→II transition and L. Criteria for the occurrence of I→II transitions are presented, and the range of validity of the theory is discussed. It is noted that the I→II transition may be diffuse or absent in many cases, either because the equilibrium distribution is not attained or because the lifetime of the defects is too short in comparison with the residence time. Thus in many polymers, regime I may be missing so that regime II (with its locally rough growth front) will persist up to quite high temperatures, i.e., up to the practical limit of slow growth.     

Isothermal crystallization of poly(ethylene-terephthalate) of low molecular weight by differential scanning calorimetry: 1. Crystallization kinetics

The isothermal crystallization of poly(ethylene-terephthalate) (PETP) fractions, from the melt, was investigated using differential scanning calorimetry (d.s.c.). The molecular weight range of the fractions was from 5300–11750. Crystallization temperatures were from 498–513 K. The dependence of molecular weight and undercooling on several crystallization parameters has been observed. Either maxima or minima appear at a molecular weight of about 9000, depending on the crystallization temperature. The activation energy values point to the possibility of different mechanisms of crystallization according to the chain length. A folded chain process for the higher $\text{M}\text{?}{}_{\mathrm{<mtext>n</mtext>}}$ chains and an extended chain mechanism for the lower $\text{M}\text{?}{}_{\mathrm{<mtext>n</mtext>}}$ chains. The values of the Avrami equation exponent n vary from 2–4 depending on the crystallization temperature; non-integer values are indicative of heterogeneous nucleation. The rate constant K depends on T_c and $\text{M}\text{?}{}_{\mathrm{<mtext>n</mtext>}}$, showing maxima related to the T_c used. The plot of log K either vs. (ΔT)^?1 and (ΔT)^?2 or $\text{T}{}_{\mathrm{<mtext>m</mtext>}}\text{T(ΔT)}$ and $\text{T}{}^2{}_{\mathrm{<mtext>m</mtext>}}\text{T(ΔT)}{}^2$ is linear in every case.     

Quantitative evaluation of the Gibbs—DiMarzio theory of the glass transition

The applicability of the Gibbs—DiMarzio (G—DM) theory of the glass transition (T_g) is quantitatively evaluated for PS, PVC, PαMS and PMMA. The analysis was conducted under the assumption that both the inter-/intramolecular energy ratio (r) and the effective chain segment density (n) remain constant while the fractional free volume at T_g(V₀) varies as a function of the reciprocal degree of polymerization ($\text{10}{}^3\text{P}\text{?}$). Based upon reduced parametric plots of $\text{T}{}_{\mathrm{<mtext>g</mtext>}}\text{T}{}_{\mathrm{<mtext>g\infty </mtext>}}$versus$\text{10}{}^3\text{P}\text{?}$, the results showed that the G-DM equations were satisfactory for PS and PVC but unsuccessful in the cases of PαMS and PMMA. For the former cases the analysis indicated that when 0.015 ? V₀ ? 0.045 optimum agreement occurred at n=1.80, r=10.5 and n=1.36, r=0.95, respectively. Although potential n, r values were obtained for PαMS when the allowable V₀ range was expanded to 0.010–0.050, none of these combinations satisfied all of the analytical requirements. No agreement for the PMMA data sets could be obtained even when this less stringent V₀ criterion was adopted. Attempts to improve this situation by incorporating ‘beads’ and ‘flexes’ into the statistical mechanical equations are also considered.     

Calculation of SANS intensity for polyethylene: effect of varying fold planes and fold plane roughening

The intensity of the small angle neutron scattering (SANS) for polyethylene crystallized in the lamellar habit from the melt at large supercoolings is calculated for μ = 0.01 to μ = 0.14 [$\text{μ = (}\text{4π}\text{λ}\text{)}\text{sin}\text{(}\text{θ}\text{2}\text{)}$]. Computations are made on models which allow various amounts and types of chain folding and varying degrees of ‘tight’ or ‘regular’ folds. The models that fit the SANS data best have folding along lattice planes in which the stem separation is larger than 0.5 nm (5 Å) or which allow for fold plane roughening on a variety of fold planes. the ‘leapfrog’ type folds mentioned by Sadler were also considered, and a possible cause for their existence suggested. As an example, the variable cluster model gives a good account of the SANS data with the surface roughening suggested by nucleation theory with fold planes [110], [200], and [310], or a mixture. Even though the conditions of crystallization used in preparing the SANS specimens (large supercoolings) were conducive to the maximum surface disorder, the probability of ‘tight’ or ‘regular’ folding, p_tf, was found to be ～0.7 for the best models. This corresponds closely to the theoretical lower bound $\text{p}{}_{\mathrm{tf}}\text{=}\text{2}\text{3}$ which is rigorous for the case of non-tiled stems. The probability of strictly adjacent re-entry in a single specified fold plane, p_ar, was ～0.4 to ～0.7 depending on the particular model chosen. The best models fit not only the SANS data, but also the liquid and crystal density, degree of crystallinity, and characteristic ratio (or radius of gyration). None of the models show the density anomaly inherent in the switchboard or random re-entry models of Yoon and Flory.     

Dielectric loss of an oligomeric poly(propylene oxide) in the liquid region above Tg

Dynamic dielectric studies of oligomeric poly(propylene oxide) (PPO) of $\text{M}\text{?}{}_{\mathrm{n}}\text{=3034}$, between ?10° and 40°C at 0.1, 1, and 10 KHz, reveal a glass transition and a T >T_g liquid-liquid transition. Analysis of $\text{d}\text{?′}\text{d}\text{T}$ in the liquid region of PPO also indicates the presence of T₁₁. The activation enthalpies for the T_g and T₁₁ transitions have been calculated to be 39 and 18 kCal mol^?1, respectively. The T₁₁ transition in poly(propylene oxide) has been assigned to the motion of the entire polymer molecule.     

Pulsed n.m.r. of cis-polyisoprene: 1

Proton spin-spin (T₂), and spin-lattice (T₁) relaxation time measurements are reported for six monodisperse cis-polyisoprenes ($\text{M}\text{?}{}_{\mathrm{n}}$ from 2000 to 200 000) over the temperature range from ?50° to 170°C. At low temperatures (?30° to 10°C) T₁ and T₂ are determined by the short range segmental motions but above 10°C T₂ is sensitive to the long range motions. When $\text{M}\text{?}{}_{\mathrm{n}}\text{? 30 000 T}{}_2$ becomes influenced by the presence of entanglements which produce a transient network structure and this confers on the spin-spin relaxation a pseudo-solid-like response. Similar behaviour is observed in crosslinked networks produced by irradiation. The results are discussed in terms of the types of motion occurring in amorphous polymers above T_g and the analogy with dynamic mechanical measurements is discussed.     

Mechanism for thermal shrinkage of oriented polystyrene monofilaments

Thermal shrinkage behaviour of oriented atactic polystyrene monofilaments, which show a brittle-to-ductile transition in the vicinity of Δn = ?2 × 10^?3 (at a temperature of 20°C and a stretching rate of 100%/min), is investigated by thermomechanical analysis (t.m.a.). The maximum contraction ratio $\text{ΔL}{}_1\text{L}{}_0$ at a constant rate of heating decreases linearly with increasing external tensile stress, and the limiting contraction ratio (at zero external load) has a broad maximum near the transition point. The time dependence of the isothermal contraction ratio $\text{[ΔL(∞) ? ΔL(t)]}\text{ΔL(∞)}$ is represented by an exponential function and an Arrhenius type temperature dependence is realized with an activation energy of several tens of kcal/mol. The degree of birefringence after isothermal shrinkage (Δn(∞)) increases linearly with an increase in external tensile load. Degrees of birefringence before isothermal shrinkage (Δn(0)) and Δn(∞) are related to each other almost linearly. The two-state model is effective in understanding these thermal shrinkage behaviours.     

Universal relation for polymer crystallization rates from the melt

Analysis of spherulitic growth rate data for a number of linear polymers has shown that the temperature at maximum growth rate, $\text{T}{}^1$, is related to the glass transition temperature, T_g, through the empirical equation, $\text{T}{}^1\text{= 1.26 T}{}_{\mathrm{g}}$. The universal master curve for the temperature dependence of the growth rate of crystals from the melt in reduced Gandica—Magill coordinates, $\text{ln}\text{(}\text{G}\text{G}{}^1\text{) =}\text{f(T ? T}{}_{\mathrm{\infty }}\text{)}\text{(T}{}_{\mathrm{m}}\text{? T}{}_{\mathrm{\infty }}\text{)}$, is possible only on the condition that the following empirical equation holds: $\text{0.26 =}\text{T}{}_{\mathrm{\infty }}\text{T}{}_{\mathrm{g}}\text{?}\text{T}{}_{\mathrm{\infty }}\text{T}{}_{\mathrm{m}}$. Finally, limits of variation of the ‘conformational’ contribution to the excess entropy, and of the free volume fraction at $\text{T}{}^1$ were evaluated for some polymers.     

Macromolecular conformational volume and thermodynamics of crystal melting

Conformational parts of the thermodynamical function change on melting, in particular the conformational volume ΔV_conf, the conformational entropy ΔS_conf and the pressure variation of the latter are analysed for an ideal polymer crystal. These quantities can be calculated from the data for low molecular compounds combined with the three-state rotational isomer model (with adjacent gauche states of opposite sign forbidden). The assumption of intramolecular dominance in the Gibbs energy difference between conformers in a polymethylene chain results in a negative value for ΔV_conf and a positive value for the quotient $\text{(}\text{?ΔS}{}_{\mathrm{<mtext>conf</mtext>}}\text{?P}\text{)}{}_{\mathrm{T}}$ which contradicts the available experimental data for polyethylene crystal fusion. The disagreement might be reconciled by the postulation of intermolecular contribution to the Gibbs energy difference between conformers favouring the existence of trans sequences of bonds in the chain and the correlation of molecular orientations in neighbouring chains.     

Neutron scattering studies of semicrystalline polymers in the solid state: The structure of isotropic polyolefins

It has been shown that on solidification from the melt, the radius of gyration (R_w) of organic polymers does not change significantly. This shows that any conformational changes occurring involve only fragments of the polymer chain, and these are distributed throughout several lamellae in space. It is not always possible to differentiate between a coil or an array of sets of stems since at high molecular weight, for both structures, R_w shows an $\text{M}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{w}}$ dependence. At low molecular weight, the length of the stems which constitute the array can be recovered. In polypropylene and quenched polyethylene this is shown to be approximately equal to twice the X-ray lamellar thickness (d) showing that rods which constitute the array contain two stems d and 2d in length. The latter act as tie molecules between lamellae. The structure is supported by chemical etching experiments which can differentiate between the tie molecules and folds in the amorphous phase (Figure 6b) and show the presence of two stems d and 2d in length in polypropylene and polyethylene. Finally polypropylene, which is crystallized by ‘seeding’, folds fully in such a way as to preserve the tie molecules and the chain occupies two lamellae. Polyethylene crystallized under high pressure on the other hand, in which $\text{d ? 2000}\text{A}\text{?}$, is fully folded and each lamella contains complete polymer molecules. No evidence has been obtained for the nature of the re-entry in any of these structures.     

Effect of composition on freezing points of model hydrocarbon fuels

A study was made of the effect of composition on the freezing points (crystallization) of model hydrocarbon jet fuel type mixtures. Solutions of higher n-alkanes (C₁₂–C₁₇) in several solvents were emphasized. Freezing points (T_m) of solutions of single alkanes were found to conform with the Van't Hoff equation. From the slopes and intercepts of plots of concentration (In X) versus$\text{1}\text{T}{}_{\mathrm{m}}$, heats (ΔH_m) and entropies (ΔS_m) of fusion and extrapolated freezing points of pure alkanes (T_m,o) were derived. For isoparaffinic solvents the derived T_m,o values were in good agreement with the literature. For ΔH_m and ΔS_m, only the even carbon numbered alkanes exhibited values similar to literature data. The behaviour of the odd numbered alkanes can be explained on a basis of solid-phase transitions. For alkanes in other solvents (particularly aromatics), there is sometimes substantial disagreement between derived and literature data depending on the solvent. For mixtures of two alkanes in a single solvent, significant changes or reversals in slope were observed for $\text{1}\text{T}{}_{\mathrm{m}}$ plotted against In X for one of the two alkanes, suggesting interaction between the solutes. At C₁₆ concentration range of 0.2 to 2%, the addition of C₁₃ can depress the freezing point, but at C₁₆ concentrations ? ≈4% changes in X (C₁₃) had negligible effect on T₅.     

Paramagnetische elektronenresonanz (EPR) in polykristallinen graphiten bei hohen temperaturen

EPR measurements were carried out on polycrystalline graphites in a temperature range of 300–1300 K. A special temperature unit permitted stable and interference-free adjustment of the sample temperature up to 1400 K (Fig. 1). The powdered samples based on natural (RWA) and an artificial extruded graphite (RWS) respectively were separated into grains of small size (≤ 3 μm), dispersed and evacuated, to avoid troublesome oxygen and skin-depth effects. It was possible to obtain the crystallite tensor components of the g-factor (Figs. 2, 3) and the crystallite line-width parameters (Fig. 4) from the asymmetrical powder spectrum by computer analysis, assuming an axial symmetry of the crystallites within the grains. One can thereby determine the average orientation of the crystallites within the grains, expressed by $\text{sin}{}^2\text{φ}$, where φ is the angle between the symmetry axis of the grain and the perpendicular on the basal plane of a crystallite. The anisotropy $\text{Δg = g}{}_3\text{? g}{}_1\text{= ?(}\text{1}\text{T}\text{)}$ of the graphites examined is smaller than that of a monocrystalline graphite (Fig. 3). The thermal variation of Δg, ΔH_⊥ and ΔH_∥ can be expressed by the phenomenological equations (4) and (6). But the physical meaning of the degeneracy temperatures T′ and T″ is not clear.A simplified model of the density of states was used for computing the susceptibility χ₀ of the free charge carriers (eqn (8) and Fig. 5). This corresponds with our measurements (Fig. 6). The susceptibility of the polycrystalline graphites is smaller than that of a mono-crystalline graphite. This may result from a slightly modified band structure, i.e. a smaller band overlap.