χN Values Research Articles

Determining Structure and Thermodynamics of A-b-(B-r-C) Copolymers.

The self-assembly of block copolymers (BCPs) is dictated by their segregation strength, χN, and while there are well-developed methods for determining χ in the weak and strong segregation regimes, it is challenging to accurately measure χ of copolymers with intermediate segregation strengths, especially when copolymers have inaccessible order-disorder transition temperatures. χeff is often approximated by using strong segregation theory (SST), but utilizing these values to estimate the interface width (wm) of BCPs in the intermediate segregation regime often results in predictions that deviate significantly from measured values. Therefore, we propose using the extent of mixing, quantified as the normalized interface width wm/L0, where L0 is the block copolymer pitch, as a thermodynamic parameter. We experimentally measure wm and L0 for a series of lamellar A-b-(B-r-C) copolymers via resonant soft X-ray reflectivity and extract values of χeffN based on previous data collected for A-b-B copolymers. The composition profiles measured via reflectivity match the extracted χeffN values, while those calculated with SST predict much more mixed composition profiles. The extracted χeff values agreed quantitatively between copolymers of different molecular weights. We believe that this methodology will be well-suited for block copolymers used in lithographic applications due to their inaccessible order-disorder transition temperatures, intermediate values of χN, and the importance of wm for line edge roughness metrics.

Order-Disorder Transitions and Free Energies in Asymmetric Diblock Copolymers

Simulations of simple bead-spring models of asymmetric diblock copolymers are used to study the dependence of order-disorder transitions and free energies upon the invariant degree of polymerization N̅ and the fraction f of beads in the minority block. Well-tempered metadynamics is used to determine values of (χN)ODT along the lamellar-disorder and hexagonal-disorder transitions over the range 0.1875 ≤ f ≤ 0.5 for two models with different values of N̅ = 480 and 1920, where χ is an effective Flory–Huggins interaction parameter, N is the degree of polymerization, and (χN)ODT is a value of χN at the order-disorder transition (ODT). More extensive studies are performed for systems with f = 1/4, which undergo a hexagonal-disorder transition. Equivalent results for both phase boundaries and free energies are obtained for one pair of systems with different numbers of beads per chain but matched values of f = 1/4 and N̅, in agreement with the corresponding state hypothesis. Comparison of results for (χN)ODT for systems with f = 1/4 and several values for N̅ show a systematic decrease in (χN)ODT with an increase N̅, consistent with the expected approach to the self-consistent field (SCFT) prediction as N̅ → ∞. Results for the free energy per chain in the disordered and hexagonal phases of systems with f = 1/4 show that SCFT gives rather accurate predictions for the free energy in the ordered hexagonal phase but that the random-mixing approximation underlying SCFT significantly overestimates the free energy of the disordered phase.

Tilted Lamellar Phase of the Rod–Coil Diblock Copolymer: Dissipative Particle Dynamics Simulation

The tilted lamellar phase of a symmetric rod–coil diblock copolymer is studied by dissipative particle dynamics simulation over a wide range of degrees of block segregation. The microphase separation in the copolymer and the orientational ordering of rigid blocks in the tilted phase are investigated. For the first time, spatial profiles of the orientational order parameter and the density of monomer units of rigid and flexible blocks at various values of χN are calculated and the influence of this parameter on the value of the tilt angle is studied.

Dynamics and Viscoelasticity of Disordered Melts of Symmetric Diblock Copolymers

Simulations of coarse-grained models are used to study relationships among chain motion, composition fluctuations, and stress relaxation in unentangled melts of symmetric diblock copolymers. Measurements of the dynamic structure factor S(q,t) are reported as a function of wavenumber q, time t, and χN, where χ is the Flory–Huggins interaction parameter and N is degree of polymerization. The function S(q,t) is found to be a nearly exponential function of time, S(q,t) ∝ e–t/τ(q), for wavenumbers similar to or less than the wavenumber q* at which the static structure factor S(q) ≡ S(q,t=0) is maximum. The relationship between the decay time τ(q) and S(q) is used to define an effective wavenumber-dependent diffusivity D(q) for fluctuations of wavenumber q. The function D(q) is shown to change very little with changes in χN and to be a monotonically decreasing function of the nondimensional wavenumber qRg, where Rg is polymer radius of gyration. The linear shear stress relaxation modulus G(t) is inferred from measurements of the shear stress autocorrelation function. At low values of χN, far from the order–disorder transition (ODT), the modulus G(t) agrees with predictions of the Rouse model. Near the ODT, G(t) develops an additional slowly decaying feature arising from slow decay of composition fluctuations with q ∼ q*. The behavior of G(t) near the ODT is predicted nearly quantitatively by a modified version of the model of Fredrickson and Larson (FL), in which the prediction of the FL theory for the slowly decaying component is added to the prediction of the Rouse theory for contributions arising from single-chain relaxation, using the independently measured behavior of S(q,t) as an input to the theory.

Catalyst: Size Distribution in Self-Assembly Matters

Dafni Moatsou completed her PhD in chemistry at the University of Warwick in 2015, was a Marie Sklodowska-Curie fellow at the Adolphe Merkle Institute until 2017, and is a junior research group leader at the Karlsruhe Institute of Technology, where she focuses on sequence control and self-assembly of polymers. Rachel O’Reilly completed her PhD in 2003 at Imperial College London. She currently holds a chair in chemistry within the College of Engineering and Physical Sciences at the University of Birmingham. In August 2018 she became the head of the School of Chemistry. Her research targets the design, synthesis, and application of polymeric materials, where control over architecture, functionality, and reactivity are central to their application.

Solubility and interfacial segregation of salts in ternary polyelectrolyte blends.

Solid Polymer Electrolytes (SPEs) consisting of ternary blends of charged polymer, neutral polymer, and plasticizer or salt have received much interest for their low volatility and high flexibility of polymers with ion-selective conductivity of the charge-carrying backbone. It has been shown that in these polyelectrolyte blends, where the dielectric constant is relatively low, ionic correlations can significantly influence the miscibility, inducing phase separation even at negative values of χN. Here we present a comprehensive study of phase behavior and interfacial segregation upon the addition of a tertiary component in blends of charged and neutral homopolymers. Using a hybrid of self-consistent field and liquid state theories (SCFT-LS), we investigate the bulk miscibility and the distribution of ions across the interface, looking at interfacial adsorption and selectivity of the minority component. We demonstrate that the competition between ionic correlations and ion entropy induces complex charge-dependent selectivity that can be tuned by the value of Γ, the ionic correlation strength. We show that charge interactions can have a pronounced effect on the interfacial width and tension, especially at low χN.

Computer simulation study of polydispersity effect on the phase behavior of short diblock copolymers

Recent progress in producing short chain length diblock copolymers facilitates the fabrication of block copolymer nanostructures with extremely small feature sizes. In this work, phase diagrams for monodisperse, one-sided and two-sided polydisperse short diblock copolymer melts are constructed using dissipative particle dynamics (DPD) simulations. Detailed comparisons are carried out between these systems to elucidate the influence of chain length polydispersity on the phase behavior of short diblock copolymers. In particular, we find an unexpected stability of a bicontinuous structure over a wide composition range between 0.7 < f ≤ 0.8 in one-sided polydisperse system, where f is the volume fraction of the polydisperse component. In the case of the two-sided polydisperse system, due to the presence of highly asymmetrical block copolymers and the pulling-away of short blocks of these molecules into opposite domain, an up-shift is found in the order–disorder transition boundary to larger χN values over the composition range we investigated.

Highly Ordered Structure Formation in RAFT-Synthesized PtBOS-b-P4VP Diblock Copolymers.

Linear poly(4-tert-butoxystyrene)-b-poly(4-vinylpyridine) (PtBOS-b-P4VP) diblock copolymers are synthesized using reversible addition-fragmentation chain transfer polymerization. The self-assembly of four different PtBOS-b-P4VP diblock copolymers is studied using small-angle X-ray scattering and transmission electron microscopy and a number of interesting observations are made. A tBOS62 -b-4VP28 diblock copolymer with a weight fraction P4VP of 0.21 shows a disordered morphology of P4VP spheres with liquid-like short-range order despite an estimated value of χN of the order of 50. Increasing the length of the 4VP block to tBOS62 -b-4VP199 results in a diblock copolymer with a weight fraction P4VP of 0.66. It forms a remarkably well-ordered lamellar structure. Likewise, a tBOS146 -b-4VP120 diblock copolymer with a weight fraction P4VP of 0.33 forms an extremely well-ordered hexagonal structure of P4VP cylinders. Increasing the P4VP block of this block copolymer to tBOS146 -b-4VP190 with a weight fraction P4VP of 0.44 results in a bicontinuous gyroid morphology despite the estimated strong segregation of χN≅150. These results are discussed in terms of the architectural dissimilarity of the two monomers, characterized by the presence of the large side group of PtBOS, and the previously reported value of the interaction parameter, χ≅0.39, for this polymer pair.

Coarse-grained molecular dynamics modeling of the kinetics of lamellar block copolymer defect annealing

State-of-the-art block copolymer (BCP)—directed self-assembly (DSA) methods still yield defect densities orders of magnitude higher than is necessary in semiconductor fabrication despite free-energy calculations that suggest equilibrium defect densities are much lower than is necessary for economic fabrication. This disparity suggests that the main problem may lie in the kinetics of defect removal. This work uses a coarse-grained model to study the rates, pathways, and dependencies of healing a common defect to give insight into the fundamental processes that control defect healing and give guidance on optimal process conditions for BCP-DSA. It is found that bulk simulations yield an exponential drop in defect heal rate above χN∼30. Thin films show no change in rate associated with the energy barrier below χN∼50, significantly higher than the χN values found previously for self-consistent field theory studies that neglect fluctuations. Above χN∼50, the simulations show an increase in energy barrier scaling with 1/2 to 1/3 of the bulk systems. This is because thin films always begin healing at the free interface or the BCP-underlayer interface, where the increased A−B contact area associated with the transition state is minimized, while the infinitely thick films cannot begin healing at an interface.

Free energy difference of pitch variation and calculation of the order–disorder transition in block copolymer systems using thermodynamic integration

Thermodynamic integration and the calculations involved in using thermodynamic integration in a coarse-grained molecular dynamics model are explained and are used to measure the energy penalty of altering the pitch of polymers at various χN values. The minimum free energy occurred at a pitch that corresponded to the calculated natural pitch from scattering calculations. The penalty for pitches lower than the natural pitch was found to scale with χ and the proportional decrease in pitch, indicating a purely enthalpic driving force and resulting in a universal energy penalty curve. The penalty for pitches larger than the natural pitch is likely entropically driven. The order–disorder transition (ODT) was also measured in a manner similar to work done previously by calculating the free energy in the mixed state and of the demixed state. The intersection of these curves is the ODT at the pitch used. The measured ODT depends on the pitch used in the demixed calculation. If the pitch is too large, then the χN at the ODT will be overestimated. The ODT measurement was taken as a function of pitch and found to approach the expected ODT of 10.5 as the correct pitch was approached, thus improving the work done previously.

Analysis of glass transition and relaxation processes of low molecular weight polystyrene-b-polyisoprene diblock copolymers

The objective of this study is to analyze the glass transition temperature and relaxation processes of low molecular weight polystyrene-block-polyisoprene diblock copolymers with different compositions, synthesized via anionic polymerization. Thermal properties were investigated by differential scanning calorimetry and dynamic-mechanical thermal analysis, while the morphologies at room temperature were investigated by transmission electron microscopy and small-angle X-ray scattering. The χN values indicate that the diblock copolymers lie near the weak segregation regime. Three different experimental techniques were applied to determine the dynamic properties, i.e., linear viscoelastic shear oscillations, creep recovery experiments, and dielectric spectroscopy. The rheological experiments were performed above the order–disorder transition temperature where the diblock copolymers behave like a Maxwell fluid. Our results indicate that the presence of the polyisoprene segments strongly influences the monomeric friction coefficient and the tendency to form entanglements above the order–disorder temperature. Consequently, the zero-shear rate viscosity of a diblock copolymer is much lower than the zero-shear rate viscosity of the neat polystyrene block (the polystyrene precursor of the polymerization procedure). Dielectric spectroscopy enables the analysis of relaxation processes below the glass transition of the polystyrene microphase. Frequency sweeps indicate the dynamic glass transition of the polyisoprene blocks, which are partly mixed with the polystyrene blocks, which are always the majority component in the block copolymers of this study.

Test of a scaling hypothesis for the structure factor of disordered diblock copolymer melts

Coarse-grained theories of dense polymer liquids such as block copolymer melts predict a universal dependence of equilibrium properties on a few dimensionless parameters. For symmetric diblock copolymer melts, such theories predict a universal dependence on only χN and , where χ is an effective interaction parameter, N is the degree of polymerization, and is a measure of overlap. We test whether simulation results for the structure factor S(q) obtained from several different simulation models are consistent with this two-parameter scaling hypothesis. We compare results from three models: (1) a lattice Monte Carlo model, the bond-fluctuation model, (2) a bead–spring model with harsh repulsive interactions, similar to that of Kremer and Grest, and (3) a bead–spring model with very soft repulsion between beads, and strongly overlapping beads. We compare results from pairs of simulations of different models that have been designed to have matched values of , over a range of values of χN and N, and devise methods to test the scaling hypothesis without relying on any prediction for how the phenomenological interaction parameter χ depends on more microscopic parameters. The results strongly support the scaling hypothesis, even for rather short chains, confirming that it is indeed possible to give an accurate universal description of simulation models that differ in many details.

Synthesis, molecular, and morphological characterization of initial and modified diblock copolymers with organic acid chloride derivatives

AbstractSix well defined PS‐b‐PB1,2 diblock copolymers (PS: polystyrene and PB: polybutadiene) with almost 100% of 1,2 microstructure for the PB segment were synthesized. Size exclusion chromatography (SEC), membrane osmometry (MO) and proton nuclear magnetic resonance spectroscopy (1H NMR) were used for verification of the molecular characteristics and the 100% ‐1,2 addition for the PB blocks. Modification with heptanoyl or pentadecafluorooctanoyl chloride was accomplished via hydroboration and subsequent oxidation, leading to hydroxylated PB blocks and was verified with 1H NMR and Fourier transform infrared (FTIR) spectroscopy. Only two samples were modified with both organic acid chloride derivatives. Structural characterization was accomplished via transmission electron microscopy (TEM) and small‐angle X‐ray scattering (SAXS) in all cases. The self‐assembly was more evident in the modified copolymers with the corresponding halides due to the increase of the molecular weight of the modified PB block. Taking into consideration the χN values in each case and comparing the results with those of PS‐b‐PI copolymers already reported in the literature the discrepancies with the theoretical predictions are very small or minimal. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011

Effects of interaction range and compressibility on the microphase separation of diblock copolymers: Mean-field analysis

Using the random-phase approximation and self-consistent field calculations, we have investigated the effects of finite interaction range and compressibility on the order-disorder transition (ODT) and the lamellar structure of symmetric diblock copolymers. While the compressibility does not affect the ODT, both the values of chiN and bulk lamellar period at the ODT increase with increasing interaction range. On the other hand, both the free-energy density and bulk period of the lamellae increase with either increasing interaction range or decreasing compressibility. Even with a finite compressibility, the mean-field ODT is still a second-order phase transition. The scaling exponent of bulk lamellar period with chiN, however, decreases with increasing compressibility. Our mean-field analysis provides a well understood reference for the study of fluctuation effects in diblock copolymers with finite interaction range and compressibility.

An Approximate Linear Solvation Energy Relationships Model Based on Snyder’s Selectivity Parameters. Chromatographic Behavior of Some 1-Aralkyl-4-Arylpiperazines

In order to elucidate the influence of the mobile phase composition on the retention of some 1-aralkyl-4-arylpiperazines, Snyder’s selectivity and polarity concept for building a simple linear solvation energy relationships model of chromatographic behavior was used. Maximum variability of mobile phase selectivity was achieved by using ternary mixtures of solvents from three different corners of the Snyder selectivity triangle: methanol as a strong proton donor, acetone as a strong proton acceptor and dimethyl formamide as a dipole interactor. Water was added to keep elution strength constant. Mobile phase composition was varied using Simplex–lattice mixture design. Selectivity parameters were calculated on the basis of linear relationships between volume fractions of pure solvents and their χe, χd, χn values. For seven-substituted 1-aralkyl-4-arylpiperazines RM values were measured and correlated with previously calculated selectivity parameters. Mathematical models obtained are discussed according to the structural properties of the studied compounds.

Copper(II) Complexes of the Tetraazamacrocyclic Tertiary Amide Ligand Alanyl‐Cyclam (Eur. J. Inorg. Chem. 12/2006)

AbstractThe cover picture shows the structural changes in tertiary carboxamide groups induced by coordination of the electrically neutral nitrogen atom to a metal ion. Changing the hybridization from sp2 to sp3 results in pyramidalization at the nitrogen atom. This is expressed by the parameter χN which ranges from 0° (sp2) to 60° (sp3). Werner complexes with relatively weak M–Namide bonds are known with χN values between 30° and 47°. Full sp3 hybridization with complete lifting of the resonance and a value of 58° for χN is achieved in a complex in which the copper(II) ion is in the centre of the tetraazamacrocyclic amide ligand tert‐butoxycarbonyl‐alanyl‐cyclam. Details of its synthesis and structure are discussed in the article by R. Alsfasser et al. on p. 2357 ff.

Phase Segregation in Gradient Copolymer Melts

Local segregation in melts of copolymers with composition gradients along their backbones is analyzed. The transition to a lamellar periodic structure as the effective degree of incompatibility χN increases is studied for symmetric copolymers with various composition gradients. A numerical self-consistent mean-field (SCMF) technique is used to characterize the ordered lamellar state in the weak and strong segregation regimes, and the random phase approximation (RPA) is used to calculate the scattering function analytically and find the location of the critical order−disorder transition for each melt. The critical point increases from (χN)c = 10.495 for block copolymers to (χN)c = 29.25 for a fully tapered linear gradient copolymer. For broad composition gradients the equilibrium lamellar repeat length is shorter for a given value of χN, and the unit cell composition profile is more sinusoidal. The dependence of the equilibrium repeat distance on χN is nearly universal when renormalized by the critical point of each copolymer.

Structure-dependent magnetic susceptibility of sharp poly(ethylene oxide) fractions

Nine sharp fractions of poly(ethylene oxide) (PEO) glycol with number-average molar masses (M n) in the range from 0.6 × 103 to 20 × 103 (PEO-0.6 to PEO-20) were characterized by magnetic susceptibility χ measured in the temperature interval 293 K to 378 K. In contrast to the liquidlike PEO-0.6 with temperature-invariant χ, the values of χ for each of the remaining solid samples, after the initial increase, exhibited two plateaus separated by a relatively narrow temperature interval of their second increase. The jumps of χ at lower and higher temperatures were attributed to a solid-state transition of unspecific nature and to the melting of the crystal fraction, respectively. The temperature-invariant values of χn in the melt state above T m pass through a minimum for the sample PEO-2.0 and then increase again with (Mn) to a limiting value χ∞ = −0.622 × 10−6. It is concluded that a considerable contribution of the molar-mass-dependent “paramagnetism” χP = χ − χd (where χd is the diamagnetic contribution estimated by Kirkwood's equation) to the total magnetic susceptibility of PEO fractions reflects distortions of the spherical symmetry of the electron shells around chain atoms resulting from the discontinuous change of both inter- and intrachain interactions as the (Mn) increases through and above the critical crossover molar mass (Mcr ) = 2 × 103.

Elastic properties of homopolymer-homopolymer interfaces containing diblock copolymers

We study the elastic properties of homopolymer/homopolymer interfaces containing diblock copolymers by means of a theory of Gaussian fluctuations. The interfacial tension and the bending rigidity of the interface in the two-phase coexistence region are calculated from the power spectrum of capillary modes. Our theory shows that in the limiting case of a pure binary homopolymer mixture, while the interfacial tension increases monotonically with increasing χN (where χ is the Flory-Huggins parameter and N is the homopolymer molecular weight) the bending rigidity does not. The bending rigidity increases rapidly at first for small values of χN, but then decreases with further increase of χN. In the presence of diblock copolymers, the interfacial tension always decreases with increasing diblock copolymer volume fraction at a given χN. However, the bending rigidity can show either a decrease or an increase depending on χN and the ratio γ between the molecular weights of a diblock copolymer and that of a homopolymer. Our results for the surface pressure and the bending rigidity are further compared with results based on scaling arguments of wet polymer brushes.

Microphase separation in block copolymer/homopolymer blends: Theory and experiment

The process of microphase separation in diblock copolymer/homopolymer blends is studied both theoretically and experimentally for an asymmetric diblock copolymer and for homopolymer concentration less than 25%. The degree of polymerization of the added homopolymer (Nh) covered all possible cases; N≈Nh, N&amp;gt;Nh, and N&amp;lt;Nh. SAXS and rheology are employed and provide the order–disorder transition temperature through the discontinuous changes of the structure factor and the storage modulus. The minority phase can solubilize only a small amount of added homopolymer; addition of a higher amount results in the formation of nonequilibrium structures. Theoretical calculations performed in the strong segregation limit provide the period and the critical value of χN for the stability of the disordered phase. The theory predicts that (χN)c always increases with the addition of the majority phase. When the minority phase is added, (χN)c can increase (N≫Nh and N⩾Nh) or decrease (N≪Nh). The experimental results are in good agreement with the theoretical predictions for the two extreme cases; N≫Nh and N≪Nh, but differ when the minority component is added with N⩾Nh. These results demonstrate that the degree of compatibility between the two blocks of the diblock AB can be effectively controlled by adding a small amount of homopolymer A or B.