Theta Conditions Research Articles

Morphologies of microphase-separated conformationally asymmetric diblock copolymers

The equilibrium morphological behavior of a series of conformationally asymmetric linear diblock copolymers is characterized via small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The linear diblock molecules of polyisoprene and poly(t-butylmethacrylate) (PtBMA) are prepared anionically over a range of PtBMA volume fractions 0.17 to 0.85. Solution light-scattering experiments are performed on PtBMA homopolymer at theta conditions, and the results were compared with PI data in the literature in order to characterize the degree of conformational asymmetry between the respective blocks. This conformational asymmetry is quantified by an e of 0.75. The experimental results are compared with morphological behavior calculated utilizing self-consistent mean field theory for a diblock system with e = 0.75. At middle to high PtBMA volume fractions, Φ PtBMA > 0.30, the experimental morphological behavior agrees well with the calculated behavior; the microphase boundaries are slightly shifted to higher volume fractions of the PtBMA block due to its larger Kuhn length. At Φ PtBMA < 0.30, however, discrepancies are found in the volume fraction dependence of experimentally determined morphological behavior and that calculated theoretically. Interestingly, extremely well-ordered cylindrical microstructures were observed for PI cylinder domains embedded in PtBMA matrices; these samples were prepared by solvent casting with no treatment, such as shearing, to enhance long-range order. These well-ordered cylinder structures contrast with PtBMA cylinders in a PI matrix on the opposite side of the phase diagram that have very poor long-range order.

Shape distribution and correlation between size and shape of tetrahedral lattice chains in athermal and theta systems

Chains embedded in the tetrahedral lattice have been produced by means of Monte Carlo simulation for athermal and for theta conditions. Nonreversal random walks (random walks without backfolding bonds) have been generated as a reference. Probability distributions of an asphericity factor δ*, of a prolatness factor S*, and of shape factors sfi* have been evaluated, the quantities being based on the orthogonal components of the squared radius of gyration taken along the principal axes of inertia. In addition, the correlation between δ* and other shape descriptors as well as between δ* and quantities characteristic of the size of configurations have been evaluated. In accordance with existing literature, the distributions H(δ*) and H(S*) are found to be very broad. The distributions H(sf1*) of the small and H(sf3*) of the large shape factor (sf1*⩽sf2*⩽sf3*, sf1*+sf2*+sf3*=1) are clearly distinct from each other, while the distribution of sf2* overlaps with that of sf1* and (slightly) with H(sf3*). Distributions of theta chains (“unperturbed” polymer) coincide fairly well with respective distributions of nonreversal random walks (which in turn are nearly identical to those of (off-lattice) random walks. As a matter of course, other shape descriptors are directly correlated with the asphericity factor δ*. Actually, for all systems evaluated, the global size of configurations is strongly correlated with δ* as well: The larger the asymmetry the larger are the dimensions of the configuration under consideration.

Dynamic light scattering and rheology of associating sulfonated polystyrene ionomers in non-polar solvents

Dynamic light scattering and reduced viscosity measurements on dilute solutions of sodium sulfonated polystyrene (NaSPS) ionomers of molecular weight 1 × 10 5 g mol −1 and sulfonation levels of 0, 0.62, 0.95 and 1.39 mol% in xylene are described and compared with earlier static light and small-angle neutron scattering studies. The unsulfonated polystyrene and 0.62 mol% ionomer are present in dilute xylene solutions as single expanded chains. The 0.95 mol% ionomer, however, forms an equilibrium between single chains and aggregates of three chains in the concentration range studied (i.e. closed association). The dimensions of both the aggregates and single chains with the latter ionomer are closer to those observed for polystyrene under theta conditions rather than in xylene. With the 1.39 mol% ionomer single chains are found to be in equilibrium with aggregates of all sizes (i.e. open association). The single chains are more compact than expected for polystyrene even in a theta solvent. The aggregates, however, are lower density structures. The viscosities of the ionomer solutions were found to be determined by the total volumes occupied by the single chains and aggregates only in very dilute solutions ( < 0.4 g dl −1). Comparison of the behaviour of these ionomers dissolved in xylene with previous rheological and light scattering studies on different ionomers dissolved in a range of other solvents suggest that the variations in the extent of single-chain collapse and aggregation behaviour with charge level observed in this work could be a common feature of randomly charged ionomers in non-polar solvents. The charge levels at which transitions between different types of behaviour occur are determined primarily by the strengths of the ionic interactions and the dielectric constant of the solvent.

A general simulation method for computing conformational properties of single polymer chains

Rotational isomeric state (RIS) theory is the standard method for computing the conformational statistics of polymer chains. It applies to chains under theta conditions, either in the melt or in theta solution. RIS statistical weights can also be used in a Monte Carlo scheme for generating independent chain conformations with the correct statistics. One practical drawback of RIS methods is that statistical weights must be derived before any chain properties can be calculated. This can be tedious for all but relatively simple chains. Here, an efficient method is presented for computing the properties of theta chains without the intermediate step of deriving statistical weights. The method—‘RIS’ Metropolis Monte Carlo (RMMC) simulation-allows computation of the same type of properties as does RIS theory. It shares certain approximations with RIS theory but is not, strictly speaking, a rotational isomeric state method, as it allows bond torsion angles to vary continuously.

Concentration dependence of static chain properties 2, off-lattice Monte Carlo simulations

Monte Carlo simulations of polymer chains in off-lattice systems under theta and athermal conditions have been performed in order to support results obtained for cubic lattice chains in assemblies of varying concentration. Global as well as local quantities are investigated in detail in order to point out the differences as well as the similarities of the results and the reliability of the two models, especially with respect to local packing effects. Furthermore, the results are discussed with reference to theoretical predictions of the scaling theory. Probably the most important finding is the fact that an isolated θ chain and a chain in bulk exhibit totally different microscopic structures (which is in full concordance with our previous lattice simulations), although the global dimensions of bulk polymers tend to converge to the values of an isolated θ chain.

Morphologies of microphase‐separated conformationally asymmetric diblock copolymers

The equilibrium morphological behavior of a series of conformationally asymmetric linear diblock copolymers is characterized via small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The linear diblock molecules of polyisoprene and poly(t-butylmethacrylate) (PtBMA) are prepared anionically over a range of PtBMA volume fractions 0.17 to 0.85. Solution light-scattering experiments are performed on PtBMA homopolymer at theta conditions, and the results were compared with PI data in the literature in order to characterize the degree of conformational asymmetry between the respective blocks. This conformational asymmetry is quantified by an e of 0.75. The experimental results are compared with morphological behavior calculated utilizing self-consistent mean field theory for a diblock system with e = 0.75. At middle to high PtBMA volume fractions, Φ PtBMA > 0.30, the experimental morphological behavior agrees well with the calculated behavior; the microphase boundaries are slightly shifted to higher volume fractions of the PtBMA block due to its larger Kuhn length. At Φ PtBMA < 0.30, however, discrepancies are found in the volume fraction dependence of experimentally determined morphological behavior and that calculated theoretically. Interestingly, extremely well-ordered cylindrical microstructures were observed for PI cylinder domains embedded in PtBMA matrices; these samples were prepared by solvent casting with no treatment, such as shearing, to enhance long-range order. These well-ordered cylinder structures contrast with PtBMA cylinders in a PI matrix on the opposite side of the phase diagram that have very poor long-range order.

Fracture of flexible polymer chains in dilute solution under transient extensional flow

Applying the technique of Brownian dynamics simulation, we have studied the fracture process of flexible polymer chains when they encounter an extensional flow field of transient character. For this purpose, a mathematical model was made of an experimental device used earlier by other authors to study fracture of polystyrene in dilute solution. The polymer/solvent system studied was a very dilute solution in theta conditions. The polymer molecule was modeled as a FENE bead-spring chain, including a modification to allow for chain fracture. The simulation results showed that the fracture yield depended strongly on flow rate and on molecular weight. We have characterized the molecularweight dependence of the critical flow rate which is necessary for fracture to occur. The distribution of the resulting fragments is interpreted in terms of the conformation of the chains prior to fracture.

Electrostatic persistence length of NaPSS polyelectrolytes determined by a zero average contrast SANS technique

The electrostatic persistence lengths b c of NaPSS polyions have been measured as a function of ionic strength I in the solutions using a small-angle neuron scattering (SANS) technique combined with a zero average contrast (ZAC) method. This ZAC method provides the optical theta condition distinguishing the intermolecular scattering functions S 2( q), and thereby the scattering functions of a single chain S 1( q) free from the intermolecular interferences were obtained. The resulting b c values which were derived by the analysis of S 1( q) are proportional to I −1 2 . This work confirms and extends an earlier SANS study where the contrast-match was used.

Universal viscometric functions for dilute polymer solutions

The Gaussian approximation has been shown previously to be excellent for the treatment of hydrodynamic effects in dilute polymer solutions. However, the computational time required to find the viscometric functions in simple shear flow is prohibitively long for chains with a large number of beads. Here we introduce a new approximation which retains the accuracy of the Gaussian approximation but is significantly less computationally intensive. Thus the rheological behavior of long chains may be explored. Extrapolation of results obtained numerically for long chains to the infinite chain length limit is shown to lead to predictions independent of model parameters. As a result, within the context of the approximation introduced here, universal viscometric functions for dilute polymer solutions in simple shear flow under theta conditions are obtained.

Conformation of thermally denatured RNase T1 with intact disulfide bonds: A study by small-angle X-ray scattering

Small-angle X-ray scattering of RNase T1 with intact disulfide bonds was measured at 20° and 60°C in order to get insight into the structural changes of the protein caused by thermal denaturation. The radius of gyration increases from R G=1.43 nm to R G=2.21 nm. The conformations of the molecules at 60°C are similar to those of ring-shaped random walk chains. However, the molecules are more compact than one would expect under theta conditions due to attractive interactions between the chain segments. The volume needed for free rotation of the thermally unfolded protein molecules about any axis in solution is five times greater than in the native state whereas the hydrodynamic effective volume is increasing only two times.

The ionic strength dependence of the structure and dynamics of polyelectrolyte solutions as seen by light scattering: The slow mode dilemma

The ionic strength dependence of the structure and dynamics of polyelectrolyte solutions was investigated by static and dynamic light scattering. Narrow molecular weight distribution sodium poly(styrenesulfonate) (NaPSS) standards in aqueous NaCl solutions were chosen as model systems. The study covers moderate polymer concentrations (∼0.5–50 g/L) where interparticle interactions rather than single-polyion properties are dominant. Two diffusive modes characterized as fast and slow were detected in most cases. Scattering amplitudes of these modes were evaluated from light scattering data, which was not done in previous investigations. Amplitudes were normalized by the scattering of a benzene standard. Attention was paid mainly to the occurrence of the slow diffusive mode, which is interpreted as the dynamics of large multichain domains. Experimental results show that the occurrence of the slow mode is not connected with any kind of a sharp transition at some critical conditions. The amplitude of the slow mode (scattering intensity associated with the slow mode) changes smoothly as a function of the ratio of the polymer concentration to the solution ionic strength. At sufficiently high polymer concentrations, the slow mode can even persist until theta conditions are reached (4.17 M NaCl at 25 °C). It was shown that the absolute scattering intensity associated with the slow mode increases upon decreasing ionic strength, which is a direct proof that the structures responsible for the slow mode (domains) really form in solution and cannot be associated with the presence of some impurities, stable aggregates due to the polymer incomplete solubility, etc. Multiangle dynamic light scattering data show that dimensions of the domains increase upon decreasing ionic strength, too. The fast and slow diffusive modes are always widely separated on the time scale and the ionic strength dependence of diffusion coefficient cannot be characterized as a ‘‘splitting of diffusion coefficient.’’

The theta condition for linear polymer chains in continuous space and three dimensions

Configurational properties of isolated flexible linear chains of lengths in the range 72⩽N⩽576 have been investigated close to the theta regime by continuous space simulations using a configurational bias Monte Carlo algorithm. The polymer model consisted of beads interacting through a nontruncated Lennard-Jones potential and connected by a Gaussian distribution of link vectors. We use two criteria in order to characterize departures from ideal behavior at finite N, first the ratio of the mean squared end-to-end distance and the mean squared radius of gyration, and second the end-to-end vector distribution. Both the criteria lead, within the statistical errors, to the same prediction for the theta temperature as N→∞; the ratio criterion gives kBTθrat/ε=4.167±0.035 and the distribution criterion gives kBTθdis/ε=4.184±0.035, in close agreement with previous estimates for the same model. Deviations from ideal behavior were found to be independent of whether the finite number of beads constitute a whole chain or merely an inner part of a much larger chain. The N-dependencies of several configurational properties including the moments of the end-to-end distribution and the asphericity have been examined at the N→∞ theta temperature. Nonideal effects are manifest in different ways dependent on the property being considered. For instance the radius of gyration shows a slight contraction effect which diminishes with increasing chain length and which shows remarkable quantitative agreement with the prediction of tricritical renormalization group theory, while the end-to-end distribution shows slight expansion effects. It is suggested that each property emphasizes nonideal effects in slightly different ways.

Determination of interaction parameters for highly incompatible polymers

Experiments and calculations were performed for the ternary system cyclohexane/polystyrene/polyisobutylene (CH/PS/PIB) to study the possibilities of determining the Flory-Huggins interaction parameters χPS/PIB between these highly incompatible polymers. To that end χCH/PIB was determined (vapour pressure measurements and additional thermodynamic information) as a function of composition; χCH/PS and its concentration dependence could be taken from earlier experiments. Furthermore, the cloud point curve and some tie lines of the ternary system were measured. In the subsequent evaluation of these data, the phase diagram was calculated and χPS/PIB (as a function of concentration) adjusted until the theoretically calculated binodal line matches with the measured cloud points. The polymer/polymer interaction parameter thus obtained increases at 35°C from 0.416 in the limit of pure PS to 0.449 in the limit of pure PIB. This result agrees reasonably well with the prediction of the solubility parameter theory and is in good accord with information stemming from light scattering experiments in a ternary system under ‘optical theta conditions’.

Study of mixed solvent quality in a polystyrene-dioctyl phthalate-polystyrene system

Polymer solvent interactions in the ternary system high molecular weight polystyrene (HMPS), low molecular weight polystyrene (LMPS), and dioctyl phthalate (DOP) have been characterized by means of intrinsic viscometry (IV), dynamic light scattering (DLS), and static light scattering (SLS). Excluded volume exponents have been extracted from the scaling of intrinsic viscosity and translational diffusivity with polymer molecular weight for a mixed solvent of 13 wt % LMPS/87 wt % DOP. The value of the excluded volume exponent, v = 0.45, indicates that HMPS in the mixed solvent DOP/LMPS has apparently assumed a reduced conformation relative to the theta condition. However, SLS measures of the second virial coefficient (A 2 ) confirm that DOP is a theta solvent at our experimental temperature of 22°C and indicate that the addition of LMPS increases A 2 . SLS also suggests that neither solvent component is strongly preferentially adsorped into the HMPS coil. Our system then is a mixture of a theta solvent and a good solvent that exhibits poor solvent scaling behavior. We believe this to be the first demonstration of such behavior in a system that does not exhibit appreciable preferential adsorption. We conclude by examining our observations in the context of current descriptions of mixed solvent thermodynamics.

A field theory of random heteropolymers near solid surfaces: Analysis of interfacial organization and adsorption–desorption phase diagram

Understanding the interfacial organization of heteropolymers near solid surfaces is an issue of fundamental interest that is relevant for many technological and biological applications. In this paper we address several questions pertaining to the surface-induced ordering and the adsorption–desorption phase behavior of a dilute solution of two-letter random heteropolymers interacting with a solid surface. Our analysis is based on a statistical field theoretic formulation of the propagator for the problem of interest. We employ the replica trick to alleviate analytical difficulties which arise when considering averaging over the sequence distribution of the A and the B units. In order to highlight the effects of the surface, we consider the situation wherein the intersegment interactions are of the excluded volume type while the segment–surface interactions of the A and B segments are arbitrarily different. Within the replica symmetric solution, we show that proper coarse-graining of the interaction potentials leads to exact analytical expressions for the self-consistent propagator of the heteropolymer at theta conditions and for the case where excluded volume interactions prevail. One of our interesting findings is that heteropolymers undergo an adsorption–desorption transition in the vicinity of a surface that interacts with the different types of segments in arbitrarily different ways. This is consistent with our previous numerical findings for much more restricted circumstances. We explicitly analyze the influence of the fluctuations in the sequence distribution on the conformational organization of the adsorbed chains and obtain the scaling behavior of properties of interest in the vicinity of the adsorption–desorption threshold with respect to the disorder strength and other polymer interaction parameters. Specifically, invoking the Ehrenfest theorem we find that the adsorption–desorption transition at theta conditions is a second-order phase transition while in the case where excluded volume interaction prevails the transition becomes first order. We also obtain exact analytical expressions for the adsorption–desorption threshold. The threshold exhibits quite an unusual dependence on the strength of the disorder. Finally, we compute the point of onset of repulsive forces between plates that confine a random copolymer solution as a function of chain sequence distribution. We suggest specific experiments employing the surface force apparatus that could directly test our predictions.

.THETA. Conditions in Binary and Multicomponent Polymer Solutions Using a Perturbed Hard-Sphere-Chain Equation of State

Theta (Θ) conditions in binary and multicomponent polymer solutions are calculated using a perturbed hard-sphere-chain (PHSC) equation of state. Θ conditions are identified by roots of the spinodal criterion in the limit of infinite polymer molecular weight and zero polymer-segment fraction. Because the thermodynamic properties of mixtures are described by an equation of state, both upper and lower Θ temperatures can be calculated. For binary systems (one polymer and one solvent), experimentally determined Θ conditions and the critical solution temperature for a single polymer molecular weight are used to obtain equation-of-state parameters. Parameters obtained in this manner yield nearly quantitative representation of upper and lower Θ temperatures as a function of pressure and upper and lower critical solution temperatures as a function of polymer molecular weight. For ternary systems (two solvents and one polymer), the dependence of the Θ temperature on solvent composition can be described using parameters obtained from binary data.

Moderately concentrated solutions of polystyrene, part 5. Static and dynamic light scattering in bis(2‐ethylhexyl) phthalate

AbstractStatic and dynamic light scattering data are reported on dilute and moderately concentrated solutions of a high molecular weight polystyrene (Mw = 3.61 × 106) in bis(2‐ethylhexyl) phthalate under Flory Theta conditions. The data cover a concentration range with 0.03 × [η]c × 3.5, with several concentrations large enough that the product cMw exceeds the value necessary for entanglement behavior. The results show that a certain intermolecular scattering function H(q, c) often approximated by unity should not be neglected in the evaluation of the correlation length in the static scattering from moderately concentrated solutions of flexible chain polymers. An approximate form for H(q, c) for moderately concentrated solutions is consistent with a soft spherically symmetric repulsive potential among the chains. The dynamic scattering show two distinct groups of relaxation rates at all concentrations, but the interpretation of the two modes changes as the concentration increases from low concentrations ([η]c &lt; 1) to higher concentrations. At low concentrations the slower mode corresponds to mutual diffusion, and the faster mode to intramolecular dynamics. For concentrations with [η]c &gt; 2.4 the slow mode is viscoelastic in origin, and the faster mode is diffusive. The behavior is compared with theoretical predictions in both regimes.

Neglected turbidity correction in light scattering experiments—a source of erroneous values of the second virial coefficient

Neglecting turbidity leads to an overestimation of the second virial coefficient in light scattering experiments. Although this effect is totally negligible for polymers in good solvents, it is dominant for systems consisting of large compact particles and for polymers at theta conditions. The turbidity correction is proportional to the characteristic constant of light scatterning K and thus inversely proportional to λ 4. The true virial coefficient can be obtained by extrapolation of Kc/ R to vanishing value of K. Light scattering of a well characterized sample of polystyrene in cyclohexane at three different wavelengths and several temperatures has been used to demonstrate the correction for turbidity. The true second virial coefficient for polystyrene in cyclohexane at various temperatures was obtained. Theta temperature and the entropic factor ψ were found to be 35.4°C and 0.21, respectively. Alternative procedures for evaluating the correction using light scattering data measured at a single wavelength were also offered.

Characterization of poly(3‐octylthiophene). I: Molecular characterization in dilute solution

AbstractThe poly(3‐alkylthiophenes) are a class of electrically conducting polymers which are of particular scientific and commercial interest due to their solubility and meltability. A series of poly(3‐octylthiophene) (P3OT) fractions having molecular weights in the range Mw=30,000 to 400,000 were generated by preparative gel permeation chromatography (GPC). Although the polymer remained weakly aggregated when dissolved in pure THF, the addition of an organic salt (1 wt.% tetrabutylammonium p‐toluenesulfonate) resulted in complete molecular dissolution. We were thus able to completely and consistently describe the polymer's single‐chain properties using three dilute solution characterization methods: viscometry, GPC, and integrated light scattering. The molecular weight dependence of the intrinsic viscosity and the radius‐of‐gyration were established, and were characteristic of solutions near the theta condition, indicating poor solvent quality. Calculation of the persistence length from our data revealed that the polymer backbone is only 2 to 3 times more rigid than common flexible polymers, despite its conjugated ring structure.

The Mark-Houwink-Kuhn-Sakurada exponent of polymers with long side groups: is [formula omitted] a reliable criterion of the theta state?

The intrinsic viscosity data for poly(decyl acrylate), poly(cetyl acrylate), poly(vinyl laurate), poly(vinyl stearate) and two polymethacrylates with long mesogenic side groups are reviewed and analysed by means of three sensitive plots. Unusual values of the Mark-Houwink-Kuhn-Sakurada exponent a 0 under theta conditions or in good solvents ( A 2 ≠ 0 but a 0 = 1 2 ) are assigned to a combination of the effects of long-chain branching, chain thickness and, in the last case, excluded volume.