Theta Solutions Research Articles

Molecular-Weight-Dependent Infiltration and Adsorption of Polymers into Nanochannels.

The mesoporous silica shell coating hydrogenolysis nano-catalysts alters the molecular weight distributions of cleaved polymer chains compared to catalysts without a shell. The shell, composed of radially aligned narrow cylindrical nanopores, reduces the formation of low-valued gaseous products and increases the median molecular weight of the product, thus enhancing the value of the products for polymer upcycling. To understand the role of the mesoporous shell, we have studied the spatial distribution of polystyrene chains, used as a model polymer, in the nanochannels in both the melt phase and solution phase. In the melt, we observed from small-angle X-ray scattering experiments that the infiltration rate of the polymer into the nanochannels is inversely proportional to the molecular weight, which is consistent with theory. In theta solution experiments using UV-vis spectroscopy, we found that the shell significantly enhances polymer adsorption compared to nanoparticles without pores. In addition, the degree of polymer adsorption is not a monotonic function of molecular weight but initially increases with the molecular weight before eventually decreasing. The molecular weight for the peak adsorption increases with the pore diameter. This adsorption behavior is rationalized as resulting from a balance between the mixing entropy gain by surface adsorption and the conformational entropy penalty incurred by chains confined in the nanochannels. The spatial distribution of polymer chains in the nanochannels is visualized by energy-dispersive X-ray spectroscopy (EDX), and inverse Abel-transformed data reveals a less uniform polymer distribution along the primary pore axis for longer chains.

An Asymptotic Solution for Call Options on Zero-Coupon Bonds

We present an asymptotic solution for call options on zero-coupon bonds, assuming a stochastic process for the price of the bond, rather than for interest rates in general. The stochastic process for the bond price incorporates dampening of the price return volatility based on the maturity of the bond. We derive the PDE in a similar way to Black and Scholes. Using a perturbation approach, we derive an asymptotic solution for the value of a call option. The result is interesting, as the leading order terms are equivalent to the Black–Scholes model and the additional next order terms provide an adjustment to Black–Scholes that results from the stochastic process for the price of the bond. In addition, based on the asymptotic solution, we derive delta, gamma, vega and theta solutions. We present some comparison values for the solution and the Greeks.

Effect of sodium hydroxide on the interactions of polyethylene glycol macromolecules with water molecules

The rheological properties of dilute water solutions of polyethylene glycols with different molecular mass (1000, 3000, 6000, 8000, 20,000) were investigated in the temperature range 293-323°K in the presence of sodium hydroxide (0.1 mol/l). The intrinsic viscosities, Huggins constants, mean-square distance between the ends of the macromolecular chain, crossover concentrations and activation energies of viscous flow have been determined vs. temperature and molecular mass of polymers. It has been established that the macromolecular coils in the presence of sodium hydroxide (0.1 mol/l) in aqueous solutions of polyethylene glycols accept the unperturbed state as in theta solutions and the intrinsic viscosity becomes very sensitive to the molecular mass of polymers. The new approach for estimating the interaction energy per volume fraction between the macromolecule and solvent molecule has been proposed.

Riemann theta solutions and their asymptotic property for a (3 $$+$$ + 1)-dimensional water wave equation

In this paper, we consider the (3\(+\)1)-dimensional water wave equation \(u_{yzt}+u_{xxxyz}-6u_{x}u_{xyz}-6u_{xy}u_{xz}=0.\) Based on Bell polynomials, we obtain its Hirota bilinear equation which enable us to acquire Riemann theta solutions including arbitrary differentiable functions. Furthermore, we analyze the asymptotic property of these solutions and reveal the relationship between these solutions and the soliton solutions.

Monte Carlo Simulation Studies of the Size and Shape of Regular Three Generation Dendrimers

By the use of a dynamic Monte Carlo method, three‐generation dendrimers of type F1F2F2 with functionalities F1 = 3 to 6 and F2 = 2 to F1–1 are simulated and investigated as functions of spacer lengths, with a total chain length up to 200 000 segments. Highly diluted athermal and theta solutions are studied as well as properties of non‐reversal random walk dendrimers for comparison. For athermal conditions mean square dimensions obey the same scaling law regarding the total number of bonds as linear chains, irrespective of functionality and complexity. Likewise the theta parameter is the same for all systems investigated. Dimensions of theta systems are slightly expanded as compared to non‐reversal random walk data, the effect increasing with increasing F1 and F2 while shape data near to theta conditions fairly well coincide with those of non‐reversal random walks.

Generalized Theoretical Study of the Effect of Molecular Bond Optical Tensor Components on the Mean-Squared Optical Anisotropy ⟨γ2⟩ of Model Bead-Spring Linear Homopolymer Chains

The conformationally averaged mean-squared optical anisotropy ⟨γ2⟩ of homopolymer chains under the theta solution condition is simulated using a rotational isomeric state (RIS) scheme with a generalized optical tensor description based on model bead-spring macromolecules. The effect of torsion angle values, number of minimum energy torsion states, probability (statistical weights) of the conformational states, and values of the components of the polarizability tensor for the bead are studied in this manner for the first time. Chains having either one or two interdependent rotatable bonds in the repeat unit are simulated. We find that under certain conditions of the relative orientations and values of the optical polarizability components with regards to the plane of the backbone bond angle, a higher probability for the gauche (less extended) conformation actually leads to a higher value of the optical anisotropy. In other cases, in conjunction with expectations from known experimental results, the optical anisotropy decreases with an increase in the probability of the gauche conformational state consistently over the range of values. We observe that, in most cases, higher values of the difference in the polarizability components lead to higher values of the chain optical anisotropy.

Computer simulation of linkage of two ring chains

We performed off-lattice Monte Carlo simulations of links of two model ring chains with chain length N up to 32,768 in the theta solution or amorphous bulk state by using a random walk model (Model I), and molecular dynamics simulations of two model ring chains in solution with excluded volume interaction (Model II) to investigate topological effects on the geometry of link and ring conformation. In the case of Model I, the mean squared linking number, its distribution, and the size of two chains with fixed linking number are investigated. Our simulation results confirm the previous theoretical prediction that the mean squared linking number decays as pe(-qs(2)) with the distance of centers of chain mass s, where p and q are found to be chain length dependent and q asymptotically approaches to 0.75 as chain length increases. The linking number distribution of two chains has a universal form for long chains, but our simulation results clearly show that the distribution function deviates from the Gaussian distribution, a fact not predicted by any previous theoretical work. A scaling prediction is proposed to predict the link size, and is checked for our simulations for the Model II. The simulation results confirmed the scaling prediction of the blob picture that the link with linking number m occupies a compact volume of m blobs, and the size of the link is asymptotic to R(L) ≈ bN(ν)m(1/3-ν), where N is the chain length, and v is the Flory exponent of polymer in solutions.

Comparison of Calculated and Measured Critical Flow Rates for Dragging Linear Polymer Chains through a Small Cylindrical Tube

Using recently developed analytical Green’s function/numerical inverse Laplace transform methods, we calculate the hydrodynamic drag and confinement forces on a linear Gaussian chain inside an interacting and impenetrable cylindrical tube in the free draining limit. Equating the two forces leads to an estimation of the critical (minimum) flow rate (qc) to drag the chain through the tube. The estimated qc is compared with our measured qc in theta solutions as well as with our previous scaling argument for a variety of experimental conditions (solvent quality and tube radius, R). Satisfactory agreement between the calculated and observed critical flow rates reveals that the previous description by de Gennes of a linear chain confined in a tube as a series of hard spheres (blobs) significantly underestimates the hydrodynamic drag force, such that the predicted qc for the hard sphere model is 102–103 times higher than observed. The calculations also confirm our previous scaling argument that qc decreases with increasing R, thus departing from the hard-sphere blob prediction that qc is independent of R. More importantly, the calculation describes how interactions with the tube walls affect chain confinement.

Viscoelasticity and primitive path analysis of entangled polymer liquids: From F-actin to polyethylene

We combine computer simulations and scaling arguments to develop a unified view of polymer entanglement based on the primitive path analysis of the microscopic topological state. Our results agree with experimentally measured plateau moduli for three different polymer classes over a wide range of reduced polymer densities: (i) semidilute theta solutions of synthetic polymers, (ii) the corresponding dense melts above the glass transition or crystallization temperature, and (iii) solutions of semiflexible (bio)polymers such as F-actin or suspensions of rodlike viruses. Together, these systems cover the entire range from loosely to tightly entangled polymers. In particular, we argue that the primitive path analysis renormalizes a loosely to a tightly entangled system and provide a new explanation of the successful Lin-Noolandi packing conjecture for polymer melts.

Vitamin C hinders radiation cross-linking in aqueous poly(vinyl alcohol) solutions

Poly(vinyl alcohol) (PVA) is a promising semi-crystalline material for biomedical applications. It is soluble in water and can be formed into hydrogels by freezing and thawing or crystallizing from an aqueous theta solution such as that of polyethylene glycol (PEG). Radiation cross-linking caused by sterilization or high dose irradiation of concentrated PVA solutions could compromise some properties of these hydrogels. Therefore, we hypothesized that radiation cross-linking of PVA solutions and PVA–PEG theta gels could be prevented by using the antioxidant vitamin C as an anticross-linking agent. Our hypothesis tested positive. Vitamin C concentrations of 0.75 and 4.5 mol/mol of PVA repeating unit could prevent cross-linking in 17.5 wt/v% PVA solutions made with PVA molecular weight of 115,000 g/mol irradiated to 25 and 100 kGy, respectively. Vitamin C also prevented cross-linking in 25 kGy irradiated PVA–PEG theta gels containing up to 5 wt% PEG and decreased the viscosity of those up to 39 wt%.

Polystyrene/Poly(methyl methacrylate) Blends in the Presence of Cyclohexane: Selective Solvent Washing or Equilibrium Adsorption?

Cyclohexane has been frequently used as a selective solvent to remove PS layers or domains from polystyrene:poly(methyl methacrylate) (PS:PMMA) blends and for reorganization or self-assembly of polymer brushes and block copolymers. We have found that cyclohexane is not efficient at PS removal, observing significant residual PS at PMMA surfaces. In contrast, 1-chloropentane was found to be a far greater selective solvent (i.e., residual PS was essentially nonexistent). These results were compared to PMMA surfaces after PS was allowed to adsorb to the surface from a dilute theta solution in cyclohexane. Using near-edge X-ray absorption fine structure spectroscopy and inverse gas chromatography, coupled with self-consistent mean-field theory calculations, we have demonstrated that selectively washing a polymer from a polymer blend is nearly identical to adsorption of a polymer to a “soft” surface from a dilute solution. Improved knowledge about the effects of selective solvents will improve experimental analysis of washed systems as well as manipulation of block copolymers and polymer brushes for reorganization or self-assembly.

Nanorods assembly of polystyrene under theta condition

Different polystyrene (PS) nanostructured materials have been constructed through the template technique in its theta solution with different concentrations. Polystyrene nanorods can be obtained at theta temperature in cyclohexane with a concentrated solution of C = 140.0 g/L. Below this concentration, mixed PS nanotubes and rods or pure nanotubes are obtained. The measurements from scanning electron microscope (SEM) and transmission electron microscope (TEM) confirmed all the rod or tubular structure.

The adsorption and partitioning of self-avoiding walk polymer chains into pores from a bulk theta solution

Monte Carlo simulations of self-avoiding walk (SAW) polymer chains on a simple cubic lattice were used to study the adsorption and partitioning of the chains into pores from a bulk Θ solution. The behavior of the SAW chains confined in pores at the bulk Θ temperature was found to differ from that of the Gaussian chains. When confined in a slit or in a square channel, the radius of gyration of a Θ chain perpendicular to the direction of the confinement was shown to increase in comparison with the radius of gyration in bulk solution. The solvent quality in the pore was shown to be better than the Θ solvent, causing the expansion of the chain in the pore. When terminally anchored onto a solid surface, the reduced polymer-surface interaction at the critical adsorption point, εwca, was found to be −0.252±0.001, which differs from the value of εwca for chains without excluded volume effects, a category including random walks and non-reversing random walks on the lattice. When partitioning into a pore modeled as a slit, the polymer-surface interaction at the critical point was shown to be the same as the critical adsorption point and does not shift with the slit width. But when partitioning into a pore modeled as a square channel, the polymer–surface interaction at the critical point was shown to shift with the width of the square channel. At the critical condition point, the partition coefficient of the SAW Θ chains was greater than one in both types of pores. These results differ from the results of the Gaussian chains; instead, they are similar to that of SAW chains in athermal solvent.

Effects of Some Nonionic Polymeric Additives on the Crystallization of Calcium Carbonate

The effect of water-soluble, nonionic polymers on the crystallization of calcium carbonate is reported. The polymers used were poly(vinyl alcohol) (PVA), polyacrylamide, poly(N-isopropyl acrylamide), poly(N-vinyl pyrrolidone), and poly(ethylene oxide). In addition, several smaller molecules containing hydroxyl groups (methanol, ethanol, ethylene glycol, glycerol, dextran, and maltodextran) were also examined. Crystallization began by mixing together, with or without a polymer or other additive, calcium chloride and ammonium carbonate. The first crystals to form in all cases were vaterite, possibly with a very small amount of calcite. Subsequent behavior depended on the polymer or other additive present. With most additives, the vaterite transformed within ca. 1 h to calcite, probably by dissolution and recrystallization. However, with PVA, the formation of calcite was inhibited, and when enough PVA was added, the vaterite remained. With less PVA, aragonite formed. After ca. 9 days, aragonite was the exclusive crystalline phase, except for a very small amount of early-forming calcite. The distinguishing characteristics of PVA seem to be its ability to hydrogen bond and its forming nearly theta solutions in water over a range of temperatures including room temperature. From a theta solution, the polymer tends to adsorb nonspecifically on all solid surfaces. The common formation of aragonite in biological organisms may arise from the organism's proteins acting in a way similar to that of PVA.

Thermodynamics and partitioning of homopolymers into a slit-A grand canonical Monte Carlo simulation study.

Grand canonical ensemble Monte Carlo simulation (GCMC) combined with the histogram reweighting technique was used to study the thermodynamic equilibrium of a homopolymer solution between a bulk and a slit pore. GCMC gives the partition coefficients that agree with those from canonical ensemble Monte Carlo simulations in a twin box, and it also gives results that are not accessible through the regular canonical ensemble simulation such as the osmotic pressure of the solution. In a bulk polymer solution, the calculated osmotic pressure agrees very well with the scaling theory predictions both for the athermal polymer solution and the theta solution. However, one cannot obtain the osmotic pressure of the confined solution in the same way since the osmotic pressure of the confined solution is anisotropic. The chemical potentials in GCMC simulations were found to differ by a translational term from the chemical potentials obtained from canonical ensemble Monte Carlo simulations with the chain insertion method. This confirms the equilibrium condition of a polymer solution partition between the bulk and a slit pore: the chemical potentials of the polymer chain including the translational term are equal at equilibrium. The histogram reweighting method enables us to obtain the partition coefficients in the whole range of concentrations based on a limited set of simulations. Those predicted bulk-pore partition coefficient data enable us to perform further theoretical analysis. Scaling predictions of the partition coefficient at different regimes were given and were confirmed by the simulation data.

Free energy and confinement force of macromolecules in a slit at full equilibrium with a bulk solution

Nondilute athermal and theta solutions of nonadsorbing flexible macromolecules in equilibrium with repulsive slit-like pores were examined by the lattice Monte Carlo simulations. The free energy of confinement Δ A/ kT and the force f/ kT exerted by polymers on the slit were computed as a function of the slit width D in a wide range of bulk concentrations φ. The free energy and force profiles in nondilute solutions were found to deviate considerably from the ideal chain theory; the perturbation of chains by a presence of the slit walls were substantially reduced in nondilute solutions. The free energy and force functions appropriate for nondilute solutions were derived by fitting the simulation data. Further, the relative pressure p I/ p E exerted by the nonadsorbing confined molecules on the slit walls was calculated. The depletion effect relevant to colloid stabilization was found in dilute solutions to be slightly weaker for excluded-volume chains than for ideal chains. The relative pressure equation was modified to cover semidilute solutions, by using the mean-field and scaling expressions of the osmotic pressure. Both the relative pressure p I/ p E and the intra-slit concentration profiles φ I( x) in tandem display a suppression of the depletion effect with increasing φ in simidilute solutions.

Weak-to-strong penetration transition of macromolecules into a slit in theta solvent

Partitioning of polymer chains in the theta solution with a confined space of a slit was studied in a wide range of concentrations by using lattice Monte Carlo simulations. The slit width was equal to or greater than the radius of gyration of the theta chains. In the low concentration limit, the partitioning of the theta chains was indistinguishable from that of athermal chains compared at the same chain dimension. At higher concentrations but below the overlap concentration in the solution surrounding the slit, the partition coefficient was nearly independent of the concentration. With a further increase in the concentration, the partition coefficient increased toward unity, as the motional unit in the solution changed from a single chain to the concentration blob of the semidilute theta solution and the blob size decreased with an increasing concentration. The increase in the partition coefficient occurred, however, at concentrations much higher than those for the athermal solutions that showed a similar weak-to-strong penetration transition. We ascribe the delay to the absence of the positive second virial coefficient and the presence of chain clusters. The latter was verified in the density correlation functions of the bulk solutions as well as in the snapshots of the solutions.

Concentration dependence of the global and anisotropic dimensions of confined macromolecules

The chain dimension of nondilute polymer solutions confined to a slit of the width D were studied using lattice simulations. It was found that the chain compression induced in good solvents by the concentration O in confined and unconfined theta solutions. At intermediate slit widths, a region was noted where coils are squeezed along all three axes. This region is manifested as a channel on a three-dimensional surface (D,O) in both good and theta solvents. The coil anisotropy, given by the ratio of the parallel and perpendicular components of the chain dimensions R / , reaches high values at strong confinements, where coils form quasi-two-dimensional pancakes. The concentration-induced reduction of the global chain dimensions in good solvents is almost fully transmitted the parallel component . The computed effects of concentration and confinement were compared with the predictions of mean-field and scaling theories, and implications of the results to ultrathin films and layered nanocomposites were discussed. In addition, the distribution functions of the components of the end-to-end distance R perpendicular and parallel to the plates, W(R x ) and W(R y ), were calculated. The function W(R x ) combined with the concentration profile O(x) along the pore provided details of the chain structure close to walls. A marked difference in the pace of the filling up of the depletion layer was noticed between chains in theta and good solvents. From the distribution function W(R x ) and W(r y ), the highly anisotropic force-elongation relations imply the deformation of chains in confined solutions and ultrathin bulk films.

Variance reduced Brownian simulation of a bead-spring chain under steady shear flow considering hydrodynamic interaction effects

To obtain numerical estimates for the properties of a model for polymers in dilute theta solutions in its long-chain limit we follow a stochastic approach to polymer kinetic theory. The model takes into account configuration-dependent hydrodynamic interaction (HI) and simplifies to the Zimm bead-spring chain model in the case of preaveraged HI, for which parameter-free “universal ratios” such as the ratio between radius of gyration and hydrodynamic radius are known. The Chebyshev polynomial method and a variance reduction simulation technique are used to implement an efficient Brownian dynamics simulation. We resolve the full dependence of several characteristic ratios versus both chain length and hydrodynamic interaction parameter, we extrapolate their values to determine universal behaviors, and compare with analytical and experimental results.

Extensional viscosity of dilute polystyrene solutions: Effect of concentration and molecular weight

This paper reports a detailed investigation of the steady shear and extensional properties of mono-disperse polystyrene solutions for a range of molecular weights from 1.95 to 20 million and a range of concentrations from 69 ppm to 777 ppm. The steady shear and dynamic properties are reasonably well described by the Zimm model. The relaxation time and polymer viscosity evaluated using the Zimm model exhibit the expected scaling with concentration and molecular weight for theta solutions. In an extensional flow, these solutions show strain hardening and need about 5.5 to 6 strain units to reach steady state. The stress growth depends both on strain rate and strain. However, at moderate values of strain and when the Weissenberg number exceeds 6, the extensional stress growth depends only on total strain. The steady state extensional viscosity for each fluid was observed to depend on the magnitude of Weissenberg number (Wi). For Wi<6, the steady state extensional viscosity was observed to be an increasing function of the strain rate. At a Wi∼10, the steady state extensional viscosity exhibits a maximum and surprisingly, for Wi>10, the steady state extensional viscosity is a decreasing function of the strain rate. The results indicate that even dilute solutions, with concentrations as low as 0.21 times the critical concentration, do show extension thinning under certain conditions. The transient and steady state extensional viscosities are found to be proportional to molecular weight (Mw) and concentration (c). This result is rather unexpected as the Zimm model would predict a scaling with c and Mw0.5 for the transient viscosity and scaling with c and Mw1.5 for the steady state viscosity. On the other hand, the Rouse model predicts a scaling with c and Mw (for transient extensional viscosity) and with c and Mw2 (for steady state extensional viscosity). Hence the data appear to follow a Rouse-like behavior for the transient viscosity. This implies that the advent of the stretching reduces the hydrodynamic interaction and a free draining behavior is obtained. As a result, predictions using the Zimm model parameters, estimated from shear data, are unable to predict the transient extensional viscosity. The data are analyzed using a constitutive equation that incorporates an anisotropic drag coefficient along with a Rouse spectrum of relaxation time. Such a model captures the extensional behavior of the solutions.