Application of scaling concepts to sedimentation phenomena in semidilute macromolecular solutions

Sedimentation velocity data on polystyrene in a good solvent (toluene) and in a theta solvent (cyclopentane), over a large concentration range are reported. Under good-solvent conditions the exponent β in the apparent scaling law describing the concentration dependence of the sedimentation coefficient (s ∝ c^β) in the semidiiute region is found to be concentration dependent. However, a power law fit to data for the highest molecular weight (M = 20.6 × 10⁶) in the concentration region (c < 2 kg m^?3) yields a value β = ?0.59, somewhat smaller than that (?0.54) predicted theoretically. This discrepancy and the observed curvature in logs vs. logc at higher concentrations are discussed. Under theta-solvent conditions, on the other hand, the concentration dependence of s in the semidilute regime can be represented by a simple power law, with β = ?1.0, in excellent agreement with the theoretical prediction. The crossover concentration c^*, separating the dilute and semidilute concentration regimes, was found to be well defined and located at c = 1/[η]. c^* varies with molecular weight as M^?0.73 and M^?0.50 under good-solvent and theta-solvent conditions, respectively.     

Small-angle x-ray scattering measurements on dilute solutions of polyisobutene by use of kratky cone collimation

SAXS measurements on dilute solutions of polyisobutene in n-hexane (M _ν = 25,600?3.55 × 10⁶, c < 53 g/liter) were carried out by combining conventional slit collimation with the new cone collimation technique. With a fractionated sample (M _ν = 25,600; U < 0.2) the radius of gyration (R = 48 Å), the cross-sectional radius of gyration (R_q = 3.9 Å), the molecular weight per unit length (M/L = 22.8 Å^?1), and the Porod persistence length (a^* = 8.1 Å) are found. The persistence length appears to be dependent on the molecular weight. The partial specific volume ν of polyisobutene in n-hexane also depends on the molecular weight according to ν = 1.025 + 105/M _ν (cm³/g).     

Dynamic light scattering of polystyrene single chains in a semidilute solution of poly(methyl methacrylate) and benzene

Translational diffusion and internal motion have been observed by dynamic light scattering of optically labeled single chains of polystyrene (PS) in a semidilute solution of poly(methyl methacrylate) and benzene for the case in which the dimension R_g of the PS chain is comparable to the correlation length of the matrix solution. The molecular weight M_w dependence of the hydrodynamic radius R_h is expressed as R_h ～ M, while R_h ～ M in pure benzene. The average linewidth Γ for internal motions (KR_g > 1) appears to depend on the magnitude K of the scattering vector approximately as Γ ∝ K⁴ at higher KR_g ( > 1), in contrast with the fact that Γ ∝ K³ approximately for KR_g > 1 in pure benzene. The scaling law for the K dependence of Γ does not hold in low-molecular-weight PS owing to the K dependence of Γ /K² for KR_g < 1.     

γ-Radiation-Initiated Polymerization of Acrylonitrile in Aqueous Solution and in Emulsion

The γ-radiation-initiated polymerization of acrylonitrile (AN) at 25[ddot] C has been studied, both in aqueous solution and in emulsion, at dose rates between 70 and 175 krad/hr. The effect of added emulsifier, sodium lauryl sulfate (SLS), on reaction rates, R_p, and on polymer molecular weight, M_n, has been investigated. G (monomer polymerized) values ranged from 7,500 in aqueous solution to 20,000 in bulk to 45,000 in emulsion, all based on the total energy absorbed. In the aqueous solution polymerization, where R_p is approximately first order in initial monomer concentration over the range 0.15 ± [AN]_o ± 1.06 moles/liter, addition of SLS increases R_p but does not influence the order of the reaction with respect to [AN]_o. In the emulsion system at 70 krad/hr and at a phase volume ratio AN/H₂O of 1/2, (PR = 1/2), R_p varies as [SLS]^0.1 over the concentration range 0.01 ± [SLS] ± 2.5± wt/vol of aqueous phase. At the same PR value, and at 80 krad/hr, M_n of the polymer (measured by viscometry in dimethylformamide solution) is effectively independent of [SLS] in the range of 0.01 ± [SLS] ± 10± wt/vol of aqueous phase. Initial R_p values are either independent of PR in the range 1/3 ± PR ± 1/1 or exhibit an insensitive and unsystematic dependence thereon. Based on measurements at 70 and at 175 krad/hr, the intensity exponent of R_p at PR = 1/2 is approximately 0.4.     

Viscometric,light scattering,and size‐exclusion chromatography studies on the structural changes of aqueous poly(vinyl alcohol) induced by γ‐ray irradiation

The crosslinking processes of aqueous poly(vinyl alcohol) (PVA) by γ‐ray irradiation were studied by viscometry, dynamic and static light scattering (DLS and SLS), as well as size exclusion chromatography (SEC). Increases in the intrinsic viscosity ([η]), molecular weight (M_w), hydrodynamic radius (R_h), and radius of gyration (R_g), and a decrease in second virial coefficient (A₂) were observed after γ‐ray irradiation. However, both the values of [η] and A₂ for irradiated PVA fell below the data of unirradiated PVA solutions, suggesting a conformational change of PVA chains after γ‐ray irradiation. This structural change of PVA as a result of γ‐ray irradiation was also indicated by the decreases in R_g/R_h from 1.5 to 1.39 by SLS and DLS, and in Mark–Houwink exponent α_η from 0.54 to 0.26 by SEC‐Viscometry. The broadening of the M_w distribution (MWD) as indicated by the polydispersity index increased from 2.2 to 6.5 because of γ‐ray irradiation. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 214–221, 2000     

Diffuse interface between polymers: Structure and kinetics

The interfacial structure and diffusion kinetics of two compatible polymers, poly(methyl methacrylate) and poly(vinylidene fluoride) are studied in the melt. The interdiffusion rates of the two components are found to be unequal, giving unequal diffusion coefficients, a net mass flow across the interface, and an asymmetric interfacial composition profile. The structure and kinetics confirm the predictions of the reptation theory. The interfacial thickness d grows with t^1/2, and the interdiffusion coefficient is proportional to M^?2, where t is the time and M is the molecular weight. The scaling law for the interfacial thickness is therefore d ∝ M^?1t^1/2. The number of chains per unit area crossing the original interface reaches a constant value independent of diffusion time after a short induction time on the order of the tube disengagement time (about 0.1–10 s in the present cases depending on the molecular weights). The adhesive bond strength σ is scaled by σ ∝ t^1/4M^?1/2 and σ/σ∞ ∝ t^1/4M^?1/2 [1- (M_c/M)]^?1, where σ _∞is the σ at infinite molecular weight and M_c is the entanglement molecular weight.     

Crosslinked polymer chains with excluded volume: A new class of branched polymers?

In this note microgels with and without excluded volume interactions are considered. Based on earlier exact computations on Gaussian mircogels, which are formed by self-crosslinking (with M crosslinks) of polymer chains with chainlength N, Flory type approximations are used to get first insight to their behavior in solution. It is shown that two different types of microgels exist: A special type of branched polymer whose size scales as R ∝ N^2/5/M^−1/5, instead of R ∝ N^1/2. The second type are c^*-microgels whose average mesh sizes r are swollen and form self avoiding walks with a scaling law of the form r = a(N/M)^3/5.     

Chain dynamics in entangled polymers: Diffusion versus rheology and their comparison

This review article scrutinizes and reanalyzes the extensively available literature data on the tracer and self chain diffusion coefficients D_tr and D_s along with the corresponding zero‐shear viscosity η₀ to show that D_s ∝ M^?ν starts with ν > 2.0 and converges to the asymptotic scaling exhibited by D_tr ∝ M^?2.0 as the molecular weight M increases beyond M/M_e = 10–20, in contrast to the onset of the asymptotic scaling M³ for η₀ taking place typically for M/M_e ? 10–20. A coherent analysis of these observations leads to the suggestion that the observed crossover in D_s is due to the constraint release effect, which diminishes around M/M_e = 10–20 and is negligible in measurements of D_tr when the matrix molecular weight P is much greater than M. The contour length fluctuation (CLF) effect, which is believed to cause the molecular weight scaling of η₀ to deviate significantly from its limiting behavior of M³, has little direct influence on the chain diffusion. The absence of the CLF effect on D_s leads to a much stronger than linear dependence of the product η₀D_s on M, which has been observed previously. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1589–1604, 2003     

Data evaluation in light scattering from solutions of branched polyethylene

The molecular dimensions of branched polyethylene published in the literature were re‐evaluated to obtain the radii of gyration R_G and other relevant topological parameters. By means of the exponent a_s in the relationships R_G = b · M_s^a the reliability of the data and reasons for deviations were discussed. Linear and branched polyethylenes are molecularly dissolved only in the good solvent, tetralin, whereas in 1‐chloronaphthalene and 1,2,4‐trichlorobenzene the exponent a₂ is lower than expected for good solvents. Microgels and aggregates affect the exponents. After correction of the data by a two component separation method in combination with model calculations and the analysis of scattering curves, a consistent picture is obtained. This procedure is demontrated with fractions of branched polyethylene in the theta solvent diphenylmethane, which are indicating incomplete molecular dissolution.     

Light-scattering characterization of poly(ethylene terephthalate)

By using a closed-circuit filtration system, we have succeeded in clarifying poly(ethylene terephthalate) (PET) dissolved in hexafluoroisopropanol (HFIP). Such static properties as the radius of gyration R_g, the weight-average molecular weight M_w, and the second virial coefficient A₂ and such dynamic properties as the translational diffusion coefficient D, or its equivalent hydrodynamic radius R_h, and the second (diffusion) virial coefficient k_d were determined for several PET samples of different molecular weights by using light-scattering intensity and linewidth measurements. An empirical relation between D_o (or R_h) and M_w was established: R_h = (1.77±0.15)X10^?2 M with R_h and M_w expressed in units of nanometers and grams per mole, respectively. The empirical exponent α_D(ca. 0.58±0.01) is in good agreement with the less precisely determined intrinsic viscosity/molecular weight exponent α_η (ca. 0.71±0.02). Several intensity correlation functions were measured very precisely using long accumulation times. A Laplace inversion was performed using the singular-value decomposition technique. The approximate molecular weight distribution (MWD) determined by light-scattering spectroscopy was in reasonable agreement with a completely independent determination of MWD using gel permeation chromatography (GPC). It was interesting to note, though not surprising, that GPC showed emphasis on lower-molecular-weight fractions, while light-scattering emphasized higher-molecular-weight fractions. The agreement further strengthens some complementary aspects of the two techniques.     

The chain length dependence of self-diffusion in melts of polyethylene and polystyrene

The self-diffusion coefficients in melts of polyethylene fractions and polystyrene standards were measured by the NMR pulsed field gradient technique and compared with those measured by other techniques. The data agree very well if one takes into account the molar mass distribution of the samples and the free volume of the matrix. For molar masses much higher than the critical molar massM _c, reptation is confirmed,D M ^–2 holds. BelowM _e=M_c/2 the self-diffusion coefficients corrected for constant free volume show approximately the dependenceD M ^–1 confirming Rouse-like diffusion. This result was also obtained by investigating the self-diffusion of the molecules with different molar masses of a polyethylene fraction with a rather broad molar mass distribution aroundM _e andM _c, i. e. diffusion in a constant matrix. In the molar mass region betweenM _c and about 3 ·M _c the observed molar mass dependence of self-diffusion can be explained by tube formation. The constraint release model of Graessley seems to slightly overestimate the self-diffusion coefficients.     

Molecular dimensions and melt rheology of an all‐aromatic liquid crystalline polymer

The molecular dimensions and melt rheology of a thermotropic all‐aromatic liquid crystalline polyester (TLCP) composed of p‐hydroxy benzoic acid, hydroquinone, terephthalic acid, and 2,4‐naphthalenedicarboxylic acid is examined. The Mark–Houwink exponent (α) of 0.95 is estimated for the TLCP. The persistence length estimated from molecular weight (M) and intrinsic viscosity ([η]) data using the Bohdanecky–Bushin equation is about 95 Å, whereas that estimated from light scattering data is 117 Å. These persistence lengths and the observed α value, both higher than those for flexible polymers, suggest that the present TLCP is a semirigid polymer. The zero shear melt viscosity (η₀) varies with approximately M⁶ for molecular weight M > 3 × 10⁴ g/mol; below this molecular weight, η₀ varies almost linearly with M. Widely different entanglement molecular weights (M_e) are predicted, depending on the method used; the plateau modulus estimates M_e of about 8 × 10⁵ g/mol, whereas the ratio of mean square end‐to‐end distance and molecular weight (〈R²〉₀/M) predicts M_e's either too small (0.33 g/mol) or too large (2.5 × 10⁶ g/mol), depending on the theory used. Although the change in the molecular weight dependency of melt viscosity appears to be associated with the onset of entanglement coupling of the semirigid molecules, its origin needs further investigation. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2378–2389, 2001     

Amino Acid and Poly(Ethylene Glycol) Based Self-Organizing Polymeric Systems: Chemo-Enzymatic Synthesis and Characterization

A chemo/regio selective enzymatic methodology has been designed to synthesize amphiphilic copolymers based on amino acid diesters and poly(ethylene glycol) [PEG]. The condensation polymerization was catalyzed by immobilized Candida antarctica lipase B (Novozyme 435) under solvent-less conditions. The synthesized polymers 3a–c were derivatized with long chain acid chlorides by chemical acylation to get the amphiphilic polymers 4a–c. The physical properties of the synthesized amphiphilic polymers viz: aggregation number, critical micelle concentration (CMC), radius of gyration (R_g), hydrodynamic radius (R_h) and particle size distribution were studied by static and dynamic light scattering (SLS and DLS) techniques. The polymers were found to be promising in drug delivery applications.     

Strain criterion in nonlinear creep and recovery in concentrated polymer solutions

Transient and steady-state rheological data are reported for several anionic polystyrene solutions in tritolylphosphate (1. 6 < cM/ρM_c < 7). Here c is the concentration of the solution, M is the molecular weight, ρ the density of the undiluted polymer, and M_c the molecular weight between entanglements as determined from zero-shear viscosity. The polystyrene used had M_w = 410,000 and M_w/M_n < 1.06. Data are also given for solutions of polyisobutylene and poly(vinyl acetate) with larger M_w/M_n. The results give a critical strain γ′ ∝ c⁻¹ such that linear viscoelastic behavior was obtained in a simple shear deformation with shear less than γ′. A simplified version of the constitutive equation of Bernstein, Kearsley, and Zapas is used with an empirical strain function F (γ) which contains γ′ as a parameter to discuss transient and steady-state behavior in terms of the distribution of relaxation (or retardation) times determined for linear viscoelastic responce. Features of the dependence of the steady-state viscosity η_k, recoverable compliance R_k, the first-normal stress function N_k⁽¹⁾ on shear rate k are discussed in terms of F (γ) and the distribution of relaxation times to conclude that the latter plays a dominant role in the behavior observed in the range of k usually studied. The results predict that the reduced functions η_k/η₀, R_k/R₀, and N_k⁽¹⁾/N₀⁽¹⁾ should depend on η₀R_0k, and that the functional form depends markedly on the distribution of relaxation times, at least in the range η₀R₀k < 10². Comparison with the mechanistic model of Doi and Edwards shows a similar F (γ) but substantial differences in the reduced functions caused by a very narrow distribution of relaxation times in the model.     

Diffusion of Nanoparticles in Semidilute Polymer Solutions: A Multiparticle Collision Dynamics Study

The diffusion of nanoparticles immersed in semidilute polymer solutions is investigated by a hybrid mesoscopic multiparticle collision dynamics method. Effects of polymer concentration and hydrodynamic interactions among polymer monomers are focused. Extensive simulations show that the dependence of diffusion coefficient D on the polymer concentration c agrees with Phillies equation D-exp (-αc^δ) with a scaling exponent δ≈0.97 which coincides with the experimental one in literature. For increasing nanoparticle size, the scaling prefactor α increases monotonically while the scaling exponent always keeps fixed. Moreover, we also study the diffusion of nanoparticle without hydrodynamic interactions and find that mobility of the nanoparticle slows down, and the scaling exponent is obviously different from the one in experiments, implying that hydrodynamic interactions play a crucial role in the diffusion of a nanoparticle in semidilute polymer solutions.     

Kinetics of fluid spreading on viscoelastic substrates

The spontaneous spreading of non‐film‐forming fluids on the surfaces of aqueous solutions of poly(2‐acrylamido‐2‐methyl‐propanesulfonic acid) and its chemically crosslinked gels was studied. The experiments were performed in the same concentration range for the solutions and gels, far above the overlap concentration of the polymer solutions. The leading edge (R) of the spreading liquid showed a power‐law behavior with time t: R = K(t + c)^α, where α is the spreading exponent and K is the spreading prefactor. α and K were significantly different for the polymer solutions and gels. Here c was a constant that depended on the initial conditions of the spreading liquids. Depending on the polymer concentration, α of the polymer solutions varied between the upper (3/4) and lower (1/10) theoretical limits for viscose liquids and solids, respectively. This indicates that no universal scaling law exists for the spreading process on viscoelastic surfaces. On the polymer gels, which were elastic substrates, universal values of α could be observed and could be expressed as R ∝ (t + c)^0.45 and R ∝ (t + c)^0.3 for miscible and nonmiscible spreading liquids, respectively; they showed no dependence on the polymer concentration or network mesh size. This shows that on an elastic gel surface, spreading is more or less similar to that on a solid surface. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 562–572, 2005     

Rigidity percolation model of polymer fracture

A theory of the fracture of polymers with network microstructure was developed that was based on the vector, or rigidity percolation (RP) model of Kantor and Webman, in which the modulus, E, is related to the lattice bond fraction p, via E ～ [p ? p_c]^τ. The Hamiltonian for the lattice was replaced by the strain energy density function of the bulk polymer, U = σ²/2E, where σ is the applied stress and p was expressed in terms of the lattice perfection via the bond density ν, with the entanglement molecular weight, ν = ρ/M_e and appropriate measures of crosslink density for rubber, thermosets, and carbon nanotubes. The stored mechanical energy, U, was released by the random fracture of νD_o[p ? p_c] over stressed hot bonds of energy D_o ≈ 330 kJ/mol. The polymer fractured critically when p approached the percolation threshold p_c, and the net solution was obtained as σ = (2EνD_o [p ? p_c])^1/2 with a fracture energy, G_1c ～ [p ? p_c]. The fracture strength of amorphous and semicrystalline polymers in the bulk was well described by, σ = [ED_oρ/16 M_e]^1/2, or σ ≈ 4.6 GPa/M_e^1/2. Fracture by disentanglement was found to occur in a finite molecular weight range, M_c < M < M*, where M*/M_c ≈ 8, such that the critical draw ratio, λ_c = (M/M_c)^1/2, gave the molecular weight dependence of the fracture as G_1c ～ [(M/M_c)^1/2 ? 1]². The critical entanglement molecular weight, M_c, is related to the percolation threshold, p_c, via M_c = M_e/(1 ? p_c). Fracture by bond rupture was in accord with Flory's suggestion, G/G* = [1 ? M_c/M], where G* is the maximum fracture energy. Fracture of an ideal rubber with p = 1 was determined not to occur without strain hardening at λ > 4, such that the maximum stress, σ = E (λ ? 1/λ) = 3.75E. The fracture properties of rubber were found to behave as σ ～ ν, σ ～ E, and G_1c ～ ν. For highly crosslinked thermosets, it was predicted that σ ～ (Eν)^1/2, σ ～ (X ? X_c)^1/2, and G_1c ～ ν^?1/2, where X is the degree of reaction of the crosslinking groups and X_c defines the gelation point. When applied to carbon nanotubes (SWNT and MWNT) of diameter d and hexagonal bond density ν = j/b², the nominal stress as a function of diameter is σ(d) = [16 ED_o(p ? p_c) j/b]^1/2/d ≈ 211/d (GPa.nm) and the critical force, F_c(d) ≈ 166 d (nN/nm), in which j = 1.15, b = 0.142 nm, E ≈ 1 Tpa, and D_o = 518 kJ/mol. For polymer interfaces with Σ chains per unit area of length L and width X ～ L^1/2, G_1c is then ～ [p ? p_c], where p ～ ΣL/X. The results predicted by the RP fracture model were in good agreement with a considerable body of fracture data for linear polymers, rubbers, thermosets, and carbon nanotubes. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 168–183, 2005     

Transverse Relaxation of Scalar‐Coupled Protons

In a preliminary communication (B. Baishya, T. F. Segawa, G. Bodenhausen, J. Am. Chem. Soc. 2009 , 131, 17538–17539), we recently demonstrated that it is possible to obtain clean echo decays of protons in biomolecules despite the presence of homonuclear scalar couplings. These unmodulated decays allow one to determine apparent transverse relaxation rates R₂^app of individual protons. Herein, we report the observation of R₂^app for three methyl protons, four amide H^N protons, and all 11 backbone H^α protons in cyclosporin A. If the proton resonances overlap, their R₂^app rates can be measured by transferring their magnetization to neighboring ¹³C nuclei, which are less prone to overlap. The R₂^app rates of protons attached to ¹³C are faster than those attached to ¹²C because of ¹³C–¹H dipolar interactions. The differences of these rates allow the determination of local correlation functions. Backbone H^N and H^α protons that have fast decay rates R₂^app also feature fast longitudinal relaxation rates R₁ and intense NOESY cross peaks that are typical of crowded environments. Variations of R₂^app rates of backbone H^α protons in similar amino acids reflect differences in local environments.