Unperturbed Dimensions Research Articles

Some simulations on elastomers and rubberlike elasticity

A number of examples are given to illustrate the use of computer modeling to elucidate the structures of elastomeric polymer networks, and to provide guidance on controlling network structure to maximize mechanical properties. The first example involves simulations of the gelation process leading to the development of the network structure necessary for a material to exhibit reversible elastomeric properties. The goals here are to characterize the amounts and constitutents of the sol phase, and the structure of the gel phase sufficient for prediction of its mechanical properties. Attempts are also being made to simulate the structure and properties of networks having multimodal distributions of chain lengths, for example to suggest how trimodal distributions might give additional improvements beyond those shown by bimodal networks. Network thermoelasticity can also be elucidated by simulations of the temperature dependence of the unperturbed dimensions of the network chains. Strain-induced crystallization is also being simulated to generate chains having sequence distributions that are representative of various polymerization conditions. The chains are then placed alongside one another to determine matched-sequence runs that could lead to the formation of crystallites. Finally, the long-standing problem of clarifying how fillers reinforce elastomeric materials is being approached using simulations of chains in the vicinity of impenetrable filler particles. In this way it is possible to determine how reinforcement depends on the perturbation of distributions caused by this excluded-volume effect.

A novel method for estimating unperturbed dimension [ η] θ of polymer from the measurement of its [ η] in a non-theta solvent

A novel method based on the Einstein’s viscosity equation modified by Guth–Simha–Gold and the model on concentration dependence of polymer chain dimension proposed in our previous study has been developed to estimate the unperturbed dimension, [ η] θ , of a polymer (i.e., the intrinsic viscosity at theta conditions) from the measurement of the intrinsic viscosity, [ η], at non-theta conditions only. The [ η] θ values obtained in present method for polystyrene, poly(oxyethylene), poly(methyl methacrylate), poly(vinyl chloride) and poly(isobutene) are good consistent with their experimental values at theta conditions or with their calculated values according to Mark–Houwink–Sakurada equation at theta conditions by the aid of their K θ and molecular weight.

The effect of autoclaving on the dispersibility and stability of three neutral polysaccharides in dilute aqueous solutions

The dispersibility of three neutral polysaccharides, oat β-glucan, detarium xyloglucan and dextran in a dilute water–cadoxen mixture was studied by viscosity measurement. It was found that intrinsic viscosity measurement, with water–cadoxen mixtures as solvents, is a useful tool to distinguish polymer degradation from disruption of supramolecular aggregates. This approach, in conjunction with size exclusion chromatography, was used to study the effects of heat and pressure treatment on the dispersibility and stability of three polysaccharides in aqueous solutions. Autoclaving treatment at 121°C for 15 min may reduce the degree of aggregation. Following autoclaving in aqueous solution, the Huggins constants decreased from 0.66 to 0.42 for oat β-glucan and from 0.63 to 0.56 for detarium xyloglucan. It remains the same for dextran, indicating good solubility of this polymer in water. The current treatment did not cause evident changes in molecular weight and structures to detarium xyloglucan and dextran. However, degradation occurred with oat β-glucan. The Burchard–Stockmayer–Fixman approach was applied to estimate the unperturbed dimension of oat β-glucan and detarium xyloglucan on samples after autoclaving. The characteristic ratio C ∞ was found to be 7.3 for detarium xyloglucan and 4.7 for oat β-glucan, corresponding to the Kratky–Porod persistence lengths of 2.0 and 1.2 nm, respectively.

Molecular shape of poly(2-ethyl-2-oxazoline) chains in THF

The molecular shape of well-defined poly(2-ethyl-2-oxazoline) (PEOz) chains in tetrahydrofuran (THF) at room temperature has been studied by a combination of static and dynamic light scattering and viscometry in the framework of the hard-sphere model. Data of the z-average radius of gyration as well as equivalent sphere radii such as the hydrodynamic radius (RH), the viscometric radius (RV), and the thermodynamic radius (RT) give some evidences of sphere-like structure. The unperturbed molecular dimension ((〈RG2〉0/Mw)∞1/2=624×10−4nm) determined by an indirect method and the persistence length q=0.70nm determined by the Bushin–Bohdanecky method appear to be quite small compared to those of common flexible polymers. In particular, the R value (24.0), defined by 〈RG2〉1/2/q, being of a considerably higher order than unity allows an estimation that the chain possibly has a type of “random Gaussian globule”. Based upon the experimental results so far obtained, it can be concluded that the chains of PEOz in THF at 25°C assume basically a shape that is very similar to that of a sphere in the least draining limit. Both the branch-like structure of the PEOz molecule and the intramolecular polar interaction between the nitrogen of the main chain and the carbonyl carbon in the side group are considered to be responsible for the chain contraction.

Determination of molecular dimensions of linear polyethylene by light scattering

The molecular dimensions of linear polyethylene published in the literature were reevaluated in order to discuss the radii of gyration R G and their dependence on molar mass, M. The relationships R G= bM a and the exponents a of this equation in comparison with the unperturbed dimensions of linear polyethylene obtained from viscometric measurements allowed the discussion of the reliability of the data and the reasons for deviations. The exponent a determined by light scattering for linear polyethylene in good solvent, tetraline, showed molecular dissolution, whereas in 1-chloronaphthalene and trichlorobenzene, the exponent a is lower than expected for good solvents. Microgels and aggregates of molecules affect the exponents. Radii of gyration measured directly by light scattering at fractions of linear polyethylene in the theta solvent, diphenylmethane, indicate incomplete molecular dissolution and the presence of branched structures.

Ultracentrifugal Characterization of Poly(ethylene terephthalate) in Dilute Solutions

The well-fractionated poly(ethylene terephthalate) (PET) samples ranging in weight average molecular weight Mw from 0.9 to 16×104 have been studied by sedimentation velocity and equilibrium, and viscometry in hexafluoroisopropanol (HFIP) and m-cresol at 30°C. The second virial coefficient A2 determined in HFIP by the sedimentation equilibrium method is found to have high dependence on Mw expected for the helical wormlike chains in very good solvents. The constant Φ1/3P−1 in the Mandelkern-Flory equation is 2.54×106 for PET in HFIP and 2.57×106 in m-cresol, being in excellent agreement with that reported for several polymer-solvent systems. The dependences of sedimentation coefficient s0 on Mw agree with also those expected for good solvents. The Mark-Houwink-Sakurada relations established with HFIP and m-cresol as the solvents are [η]=2.95×10−4Mw0.72 in HFIP at 30°C (3.1×104<Mw) [η]=4.97×10−3Mw0.48 in HFIP at 30°C (3.1×104>Mw) [η]=4.04×10−4Mw0.68 in m-cresol at 30°C (2.4×104<Mw)The values of steric factor σ estimated from the unperturbed chain dimensions range from 1.3 to 1.5, irrespective of the solvent nature, and such small σ values suggest that the extent of steric hindrance to the internal rotation of PET chain is significantly low.

Study of interactions between a novel elastomer and solvents

The flexibility parameters of chlorosulfonated polyethylene indicate chain stiffness comparable to other synthetic elastomers. Similar conclusion is also reached from a study of the hydrodynamic behavior. The Mark-llouwink constants, obtained from the molecular weight dependence of [η], also indicate a fairly coiled up configuration for the molecular chain. The parameters characterizing the stiffness of the elastomer chain, such as the effective bond length (b), the steric factor (σ) also reveal that the chlorosulfonated polyethylene chain resembles the other elastomers so far as the chain flexibility is concerned. Unperturbed dimension values obtained from θ-solvent data and other theories of dilute polymer solutions show good agreement with each other.

Unperturbed Chain Dimensions of Poly(di-n-hexylsilane), Poly(methyl-n-propylsilane), and Poly(di-n-butylsilane)

Unperturbed chain dimensions of poly(di-n-hexylsilane) (PDHS), poly(methyl-n-propylsilane) (PMPrS), and poly(di-n-butylsilane) (PDBS) have been evaluated by a combined use of static light scattering, size exclusion chromatography (SEC), and SEC−multiangle laser light scattering. The θ solvents and characteristic ratios (C∞'s) of the individual polysilanes were determined as follows: PDHS, a mixed solvent of n-hexane (58.2 wt %) and 2-propanol (41.8 wt %) at 25 °C and C∞ = 42.5; PMPrS, a mixed solvent of n-hexane (62.6 wt %) and 2-propanol (37.4 wt %) at 25 °C and C∞ = 19.9; PDBS, n-hexane at 19.1 °C and C∞ = 42.3. These C∞ values show that the polysilanes in the θ state are much more extended and rigid than ordinary carbon backbone polymers. On the basis of the results, the conformational characteristics of the three polysilanes are discussed by reference to their UV spectra.

Atomistic Monte Carlo simulation of steady-state uniaxial elongational flow of long-chain polyethylene melts: dependence of the melt degree of orientation on stress, molecular length and elongational strain rate

Following our recent work on the simulation of elasticity and birefringence of linear polyethylene (PE) melts [Mavrantzas and Theodorou, 1998; 2000], new results are presented for PE melts of chain length up to C1000, modelled in atomistic detail. The simulations have been performed with the End-Bridging Monte Carlo (EBMC) method, allowing large numbers of strained polymer melt configurations, thoroughly equilibrated at all length scales, to be sampled. These strained configurations are representative of the structure of PE melt systems under conditions of low-Deborah number steady-state uniaxial elongational flow in the x direction. The degree of orientation at the level of individual bonds and of entire chains and the melt anisotropic refractive index (birefringence) are presented and analyzed for a collection of PE melt systems of mean chain length C24 up to C1000 and polydispersity index 1.05 to 1.09 over a wide range of imposed stresses. The simulations demonstrate: a) a linear dependence of the conformation tensor component cxx, quantifying the overall chain orientation and deformation, and of the bond order parameter Sx on the imposed stress difference τxx-τyy, for all systems and strain-rates investigated, except for the shortest C24 system at stress levels exceeding about 350 atm, and b) conformity to the stress optical law (linearity between the anisotropy of the refractive index and the anisotropy of stress) in all cases except for the C24 system at the highest stress levels. The stress optical coefficient C is seen to go through a maximum at around C100 and to reach an asymptotic value beyond C500 , which is predicted within 30% of the experimental value for high-molecular weight linear PE melts. The linear relationship between Sx and cxx borne out of the melt simulations is in excellent agreement with Flory's theoretical treatment based on single freely-jointed chains. Results are also presented for the elongational viscosity ηE of the (unentangled) C24 and C78 PE melts and for its dependence on elongational strain rate. Values of ηE are calculated from the dependence of stress on the orienting field employed in the EBMC simulations. This field is converted to a strain rate by mapping the atomistic model onto the Hookean dumbbell, FENE dumbbell and Rouse models of viscoelasticity so that the unperturbed chain dimensions and chain self-diffusivity (obtained through equilibrium molecular dynamics studies of the same systems, Harmandaris et al., 1998) are preserved. The results obtained with all models: a) respect Trouton's law and b) display a continuous increase of ηE for these short PE melt systems with the applied elongational strain rate.

A new relation between the intrinsic viscosity and the molecular mass of polymers derived from the blob model: determination of the statistical segment length of flexible polymers

Using the relation between the unperturbed dimensions parameter on one hand and the intrinsic viscosity and the statistical segment length of the polymers on the other, as predicted by the two-parameters theory, we have modified Han's equation [Polymer, 20 (1979) 1083], which is derived from the blob theory. According to the proposed equation, plotting [η]/M1/2 versus M3ν−1.5, where ν is the exclude volume index, we obtain straight lines from the slopes of which we calculate the statistical segment length of flexible polymers. The value of ν is obtained from the value of the exponent of the Mark–Houwink–Sakurada equation. The proposed method is useful in the high molecular mass region in which the Stockmayer–Fixman–Burchard equation is not valid.

Conformational properties of the bisphenol-A polycarbonate using the RMMC method

The conformational properties of the bisphenol-A polycarbonate (PC) were studied using the rotational-isomeric-state metropolis Monte Carlo (RMMC) simulations adopting a polymer consistent forcefield (pcff). The conformations of a single PC chain were generated at theta condition in a wide range of molar mass. When the maximum bond number for non-bonded interactions is set to 5, the conformation corresponds to the unperturbed state. The probability that the dihedral bond pairs of diphenyl propane (DPP) groups exist at two minima of energy where the bond angles are (50.5, 50.5) and (−49.5, 129.5) is 3.4×10−4 at 300K. The probability of the trans–trans form of diphenyl carbonate (DPC) groups is 6.3×10−4 at 300K, and is reduced to 2.1×10−4 as temperature increases to 600K. The limiting unperturbed dimension in terms of mean-square end-to-end distance over molar mass (〈r2〉/M) is 1.01Å2mol/g. From intrinsic viscosity calculated with the radius of gyration (S), it was found that the gyration or the conformation of the PC chain at 300K is closer to that of dilute solutions in chloroform at 293K or in p-dioxane at 303K than in tetrahydrofuran (THF) solution at 297K. In future studies, the conformation obtained from the RMMC method here will be used as an initial structure to perform the molecular dynamics simulation in order to investigate interfacial properties of the PC bulk.

Unperturbed dimensions and the theta temperature of dextran in ethylene glycol solutions

The unperturbed molecular dimensions of dextran samples (of different molecular weights) have been evaluated in ethylene glycol solutions from viscosity measurements at 25°C, 30°C, 35°C, 40°C and 45°C. The unperturbed dimension, K 0, has been determined from extrapolation methods, i.e, Kurata–Stockmayer–Fixman (KSF), Berry and Inagaki–Suzuki–Kurata (ISK) equations. The hydrodynamic expansion factor, α η , as well as the unperturbed root-mean-square end-to-end distance, 〈 r 2〉 0 1/2, found for dextran samples in ethylene glycol solutions, indicated that the polymer coils are contracted as the temperature is raised from 25°C to 45°C. The long-range interaction parameter, B, was also evaluated and a significant decrease is found for the dextran/ethylene glycol system between 25°C and 45°C. The theta temperatures, Θ, were determined as 328.53, 328.02 and 325.93 K from the temperature dependence of the interaction parameter, respectively.

Solution properties of poly(diphenoxyphosphazene) below the θ temperature obtained by SEC/MALLS

Dilute solution properties of poly(diphenoxyphosphazene), PDPP, in THF at 25°C have been studied by size exclusion chromatography, using simultaneously multiangle light scattering and differential refractive index detectors. The anomalous elution behavior and the dependence of the dimensions, i.e. radius of gyration, of the polymer on the molecular weight are discussed. Coefficients for the scaling law and unperturbed dimensions have also been calculated. Molecular Dynamics simulations of some model compounds were performed in order to analyze the conformational characteristics of the P–N–P and N–P–N pairs of skeletal bonds. The conformational model thus obtained indicates that the PDPP chain is mainly formed by sequences of about four skeletal bonds in the trans conformation separated by one bond in cis, with negligible incidence of cis– cis orientations. The chain thus generated is very rigid so that the characteristic ratio of the unperturbed dimensions C N increases with the number of repeat units of the chain x, up to values as high as x≈800, reaching an asymptotic value of ca 30 when x→∞, in very good agreement with the experimental result.

Comparison of the behaviour of polymers in supercritical fluids and organic solvents via small-angle neutron scattering

Small-angle neutron scattering has been used to study the effect of temperature and pressure on the phase behaviour of semidilute solutions of polymers dissolved in organic and supercritical solvents. Above the theta temperature (TΘ), these systems exhibit a "good solvent" domain, where the molecules expand beyond the unperturbed dimensions in both organic solvents and in CO2. However, this transition can be made to occur at a critical "theta pressure" (PΘ) in CO2 and this represents a new concept in the physics of polymer-solvent systems. For T &lt; TΘ, and P &lt; PΘ, the system enters the "poor solvent" domain, where diverging concentration fluctuations prevent the chains from collapsing and allow them to maintain their unperturbed dimensions.

Synthesis, Dilute Solution Properties, and Chain Flexibility of Poly(Difluorobenzyl Methacrylates)

The synthesis and dilute solution behavior of poly(2,5-difluorobenzyl methacrylate) (P2,5-DFBM) and poly(2,6-difluorobenzyl methacrylate) (P2,6-DFBM) are described. The polymers were characterized by viscosity and classical light-scattering and size exclusion chromatography (SEC) measurements. The Kuhn-Mark-Houwink-Sakurada (KMHS) relationships were established. Unperturbed dimensions <r 0 2>1/2, the rigidity factor σ, the characteristic ratio C ∞, and the thermodynamic parameters were determined using the Stockmayer-Fixman treatment. The double substitution in the 2,5 and 2,6 positions by fluorine atoms increases the conformational parameters. The presence of substituents such as ─CH3─ or fluorine atoms close to the main chain (2,6 position) affects the internal rotation potential in the same way.

The evaluation of polyethylene chain dimensions as a function of concentration in nonadecane

This study reports the findings from a polyethylene chain dimension study as a function of concentration in concentrated solutions. The concentration range, in terms of volume fraction (O), ranged from 0,25 to 1,0. The solvent was nonadecane (d 40 ) while the technique of assay was small angle neutron scattering at 423 k. Over the range of concentration covered <R 2 ) 0 ∼ O 0.0 an the second virial coefficient (A 2 ) is 0 within the limits of experimental error. Thus the unperturbed chain dimension is retained over a wide concentration range. This demonstrates that the complete screening of excluded volume effects occurs long before the melt-state is reached ; this is in accord with the prediction of the Flory-Edwards meanfield theory approach.

Evaluations of forcefields for aromatic polysiloxanes, and some applications to poly(diphenylsiloxane)

Two forcefields were evaluated for possible use in polysiloxanes containing aromatic phenyl or phenylene groups, either as side chains or as part of the chain backbone. The criterion was reproducing results on the crystal structure of the cyclic diphenylsiloxane trimer, and both forcefields were satisfactory. The better of the two was used to obtain conformational energies for estimating some of the statistical properties of the corresponding polymer poly(diphenylsiloxane) (PDPS) [Si(C6H5)2–O–]. The calculations were based almost entirely on the rotational isomeric state (RIS) theory, but the possibility of using a Metropolis Monte Carlo method was also considered. Since the stiffness of the chains was of primary interest, the quantities calculated were the unperturbed dimensions and its temperature coefficient and the radii of gyration. Degrees of polymerization ranged from 40 to 400, and temperatures from 300 to 1000K. The “characteristic ratio” (of the mean-square unperturbed dimensions to those of the corresponding freely-jointed chain) was found to be approximately 65 in the limit of very long chains. This is an order of magnitude larger than that of the much studied and very flexible poly(dimethylsiloxane), and the associated chain stiffness this indicates for PDPS seems to be consistent with its high transition temperatures and other properties.

A conformational model for poly(dichlorophosphazene) derived from molecular dynamics simulations

Molecular dynamics (MD) simulations have been performed for an oligomer of poly (dichlorophosphazene) (PDCPN) containing 56 repeating units i.e. (PCl2N)56, seeking for the orientations of the rotational angles over the P–N skeletal bonds. Two different molecular systems were simulated. In the first one, the chain was packed into a cubic lattice (i.e. a cubic box with periodic boundary conditions) having a side length of 22.1Å in order to reproduce a density of about 1g/cm3, while in the second one, the chain was assumed to be alone in vacuo. In both systems, the allowed orientations for the skeletal bonds are cis (φ=0) and trans (φ=180) with a slight preference for this second orientation. However, a more detailed analysis shows that the lattice produces more extended conformations, and therefore larger molecular dimensions, than the isolated molecule. Thus, the chain in the lattice is formed by sequences of two or three bonds in the trans conformations separated by trans–cis (the average number of bonds in the all-trans conformation is 〈nt〉≈2.7) with negligible incidence of cis–cis conformations. The trans sequences are shorter (i.e. 〈nt〉≈1.7) in the isolated molecule and the presence of cis–cis orientations is noticeable. The a priori probabilities for all the allowed orientations of each pair of bonds obtained by MD simulations can be reproduced by a very simple RIS model that was employed for the evaluation of the unperturbed dimensions of long chains. The values obtained for the characteristic ratio were C∞≈15.7 in the case of the lattice and 8.0 for the isolated molecule, the former value being in better agreement with experimental results than the later one. Long range interactions, mainly due to van der Waals forces, may be responsible for the differences observed in the two systems simulated here, and could also explain the widely different experimental values of molecular dimensions reported for this kind of polymers.

Solution properties of poly(dimethyl siloxane)

The solution properties of poly(dimethyl siloxane) (PDMS) were studied with light scattering (LS), gel permeation chromatography/light scattering (GPC/LS), and viscometry methods. PDMS samples were fractionated, and the weight-average molecular weights, second virial coefficient, and the z-average radius of gyration of each fraction were found according to the Zimm method with the LS technique. In this work, the molecular weight range studied was 7.5 × 104 to 8.0 × 105. Molecular weights and molecular weight distributions were determined by GPC/LS. The intrinsic viscosities of these fractions were studied in toluene at 30 °C, in methyl ethyl ketone (MEK) at 20 °C, and in bromocyclohexane (BCH) at 26 °C and 28 °C. The Mark–Houwink–Sakurada relationship showed that toluene was a good solvent, and MEK at 20 °C and BCH at 28 °C were θ solvents for PDMS. The unperturbed dimensions were calculated with LS and intrinsic viscosity data. The unperturbed dimensions, expressed in terms of the characteristic ratio, were found to be 6.66 with different extrapolation methods in toluene at 30 °C. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2678–2686, 2000

Predicted Influence of N-Acetyl Group Content on the Conformational Extension of Chitin and Chitosan Chains

ABSTRACT Conformational analysis of chitosan molecules has been performed using the MM3(92) force field to investigate the role played by the acetamido groups on the stiffness of these chains. A high dielectric constant value was needed to model an aqueous environment and to reproduce the distribution of the N-acetyl glucosamine group orientation that is observed by NMR. Disaccharidic fragments, differently substituted at C2, were selected as models for chitin and chitosan chains. Their conformational space has been explored by means of adiabatic mapping of the glycosidic Φ,Ψ torsion angles. Although the overall features of all the potential energy surfaces created appear similar, the accessible conformational space of a glycosidic bond is affected by the nature of the substituent at C2 on the non-reducing residue of the disaccharide unit. This is illustrated by the differences in the calculated partition functions together with the predicted average homonuclear and heteronuclear coupling constants. Computed maps were used to predict polymeric unperturbed dimensions, characteristic ratio and persistence length of idealized chitin and chitosan chains, by Monte Carlo methods. Pure chitosan is predicted to be more coiled than pure chitin chains. At low N-acetyl group contents, chain extension appears to be dependent on the degree of substitution. Average chain dimensions increase monotonically for increases in content up to 60% of N-acetyl groups, but show no significant variation at higher contents. For molecules consisting of 50% amino and 50% N-acetylated residues, random, alternate and block patterns of substitution have been investigated. It has also been shown that the spatial extension of the polymer chains is dependent on the primary structure. Comparison with the literature experimental data is difficult because of the extreme diversity of the reported conformationally dependent values. However, such study provides a unique insight into the dependence of these two factors (degree of acetylation and distribution of acetyl groups) on the stiffness and flexibility of different chitin and chitosan chains.