Fox-Flory Equation Research Articles

Synergistic Physical Properties of Cocured Networks Formed from Di- and Tricyanate Esters

The co-cyclotrimerization of two tricyanate ester monomers, Primaset PT-30 and 1,2,3-tris(4-cyanato)propane (FlexCy) in equal parts by weight with Primaset LECy, a liquid dicyanate ester, was investigated for the purpose of exploring synergistic performance benefits. The monomer mixtures formed stable, homogeneous blends that remained in the supercooled liquid state for long periods at room temperature, thereby providing many of the processing advantages of LECy in combination with significantly higher glass transition temperatures (315-360 °C at full cure) due to the presence of the tricyanate-derived segments in the conetwork. Interestingly, the glass transition temperatures of the conetworks after cure at 210 °C, at full cure, and after immersion in 85 °C water for 96 h were all higher than predicted by the Flory-Fox equation, most significantly for the samples immersed in hot water. Conetworks comprising equal parts by weight of PT-30 and LECy retained a "wet" glass transition temperature near 270 °C. The onset of thermochemical degradation for conetworks was dominated by that of the thermally less stable component, while char yields after the initial degradation step were close to values predicted by a linear rule of mixtures. Values for moisture uptake and density in the conetworks also showed behavior that was not clearly different from a linear rule of mixtures. An analysis of the flexural properties of catalyzed versions of these conetworks revealed that, when cured under the same conditions, conetworks containing 50 wt % PT-30 and 50 wt % LECy exhibited higher modulus than networks containing only LECy while conetworks containing 50 wt % FlexCy and 50 wt % LECy exhibited a lower modulus but significantly higher flexural strength and strain to failure. Thus, in the case of "FlexCy", LECy was copolymerized with a tricyanate that provided both improved toughness and a higher glass transition temperature.

Is the Universal Law Valid for Branched Polymers?

A matter of phis: The size of macromolecules is determined by the "universal calibration" based on the Flory-Fox equation (see scheme). The effect of the segmental density of real branched systems on this equation is investigated. To study this effect, accurately prepared polymers with discrete molecular properties were characterized in detail and precisely evaluated.

Synthesis and thermal studies of aliphatic polyurethane dendrimers: a geometric approach to the Flory–Fox equation for dendrimer glass transition temperature

A new convergent synthesis for polyurethane dendrons to generation 4, and dendrimers to generation 3, is presented with control of surface functionality. The systematic synthesis of twenty-six new dendritic materials has led to a study of the factors affecting Tg using widely accepted approaches. The established understanding of dendritic polymerTg behaviour is that the Flory–Fox models, used for linear polymers, are unsuitable as the high number of chain ends and globular nature of dendrimers requires special consideration. In our review of the accepted understanding we have shown that the conventional Flory–Fox models predict Tg∞ accurately and generate identical values to the established modifications of the Flory–Fox equation that consider ‘dendrimer-relevant’ aspects such as the non-zero values of ne/M at infinite molecular weight. We also present a new approach using the geometric parameters of dendrimer mass evolution suggesting that the Flory–Fox equation is indeed appropriate for determination of dendrimerTg∞.

Hydrophilic silica additives for disulfonated poly(arylene ether sulfone) random copolymer membranes

Hydrophilic silica (SiO 2) nanoparticles (average size = 7 nm), which act as inorganic acids at low pH (<2), were added together with a PEO–PPO–PEO triblock copolymer dispersant to a random disulfonated poly(arylene ether sulfone) copolymer in the potassium salt ( SO 3 −K +) form in order to control permeation and rejection characteristics of the homopolymer. The dispersants (shell) absorbed on the surface of SiO 2 nanoparticles (core) formed a distinctive core–shell structure. The PEO units located at the outside of the dispersant formed complexes with SO 3 −K + groups of BPS-20 via ion–dipole interactions. These interactions induced a compatible binary system following the Flory–Fox equation associated with glass transition temperature ( T g) depression and prevented extraction of the water-soluble dispersant even under aqueous conditions. The ion–dipole interaction, combined with hydrogen bonding between SiO 2 and the dispersant, caused SiO 2 nanoparticles to be well distributed within the BPS-20 matrix up to a limit of 1 wt.% of SiO 2 and minimized the formation of non-selective cavities within the matrix's hydrophilic water channels. The resulting BPS-20_SiO 2 nanocomposites showed improved salt rejection and reduced ionic conductivity. These trends are analogous to those of disulfonated copolymer systems, with polar groups creating hydrogen bonding or acid–base complexation with SO 3 −K + groups in BPS copolymers. Well dispersed SiO 2 nanoparticles in highly water-permeable desalination membranes are expected to result in an increase in salt rejection but very little change in water permeability. The addition of nanoparticles to desalination membranes may offset the permeability-selectivity tradeoff observed in polymer membranes. Above 1 wt.%, SiO 2 nanoparticles increased both the interchain distance between polymer chains and the water uptake. However, the increased hydrophilicity due to high SiO 2 content did not yield improved water permeation of the nanocomposite membranes. The SiO 2 nanoparticles acted as barriers, hindering water passage (restrictive diffusion) and lowering water permeability. Meanwhile, acidic hydroxyl groups (OH 2 +) on the SiO 2 surface in the sulfonate matrix led to improved ionic conductivity, but NaCl rejection capability decreased because the concentration of SO 3 −K + was diluted by highly absorbed water molecules, resulting in weakened Donnan exclusion. The mechanical properties and chlorine resistance of all BPS-20_SiO 2 nanocomposites were comparable to those of BPS-20.

Determination of the main structural parameters of lysozyme that affect its enzymatic activity

The main characteristics of lysozyme globules in solutions, including the sizes of cores and fringelike surface layers, are determined from the data of calorimetry (lysozyme-denaturation temperatures in solution) and light scattering, via the method of atomic contributions, and from the Flory-Fox equation. It is shown that the enzymatic activity of lysozyme is governed by fringelike surface-layer loops that contain amino acid residues forming the active sites of lysozyme. Methylresorcinol affects the enzymatic activity of lysozyme owing to a change in the properties of fringelike surface layers.

Hydroplasticization of Polymers: Model Predictions and Application to Emulsion Polymers

The plasticization of a polymer by solvent has a dramatic impact on both its thermal and mechanical behavior. With increasing demand for zero volatile organic compound materials and coatings, water is often the sole solvent used both in the polymer synthesis and in formulation and application; latex colloids derived from emulsion polymerization are a good example. The impact of water on the glass transition temperature of a polymer thus becomes a critical physical property to predict. It has been shown here that in order to do so, one simply needs the dry state glass transition temperature (T(g)) of the (co)polymer, the T(g) of water, and the saturated weight fraction of water for the sample in question. Facile calculation of the later can be achieved using water sorption data and the group additivity method. With these readily available data, we show that a form of the Flory-Fox equation can be used to predict the hydroplasticized state of copolymers in exceptional agreement with direct experimental measurement. Furthermore, extending the prediction to include the impact of the degree of ionization for pH responsive components, only with extra knowledge of the pK(a), was also validated by experiment.

The structure and glass transition behavior of PLLA under the influence of gamma radiation

The influence of gamma radiation on the structure and glass transition behavior of poly-L-lactide (PLLA) has been studied. Since PLLA exposed to high-energy radiation in the presence of air is prone to chain scission reactions and large degradation, changes in molecular weight were obtained by the gel permeation chromatography (GPC). Alterations in the glass transition behaviour were investigated by differential scanning calorimetry (DSC). The apparent activation energy, ?H*, for the glass transition was determined on the basis of the heating rate dependence of the glass transition temperature, Tg. Our findings support the fact that chain scission is the main reason for the decrease of Tg and ?H* with the absorbed dose. At low doses, despite large changes in molecular weight, only small alterations are observed in the glass transition temperature and apparent activation energy. Further increase in the absorbed dose introduces not only significant changes in the molecular weight but also in the glass transition temperature and the activation energy. Such glass transition behaviour is a manifestation of a well-known effect of molecular weight on Tg, postulated by the Fox-Flory equation. However, all the observed alterations are small and tolerable at absorbed doses required for sterilization (up to 25 kGy in most circumstances). Furthermore, this study reveals that the radiation-induced changes in the molecular weight and the glass transition temperature occur in a predictable and fairly accurate manner. Therefore, gamma radiation can be used not only for sterilization but also for tailoring desirable end-use properties of PLLA based implants.

DSC Characterization of Polylactic Acid Synthesized by Non-Catalyzed Direct Polycondensation under Vacuum

Polylactic acid (PLA) was synthesized from l-lactic acid by direct polycondensation under vacuum without the use of catalysts (NC-DP). It was studied in order to adapt it to on-site cell production where safety treatment, compact capacity, and simple operations are required to reduce production costs. Experiments were conducted at polymerization temperatures (Tp) between 150 and 250°C and under reduced pressure. The resulting PLA was characterized using gel permeation chromatography (GPC) to determine the molecular weight as well as by differential scanning calorimetry (DSC) to examine the thermal properties. The maximum weight average molecular weight of PLA obtained was 90 kDa after 89 h at 200°C. Above 200°C, PLA was thermally degraded, and racemization occurred as PLA became atactic. Indeed, PLA only exhibited a glass transition temperature (Tg) and exhibited neither a crystallization temperature nor a melting point. The relations between Tg and the molecular weight were in agreement with the Flory–Fox equation.

Determination of the glass transition temperature of latex films: Comparison of various methods

The glass transition temperature ( T g) for two latices with different styrene/butadiene compositions was determined by the thermal SPM probe resonance frequency method. The results were compared with the T g values obtained by differential scanning calorimetry (DSC), dynamical mechanical analysis (DMA), process rheometer (PR) and thermo-mechanical analyzer (TMA) measurements. The T g values detected by the thermal SPM method agreed well with the T g values obtained by DSC and calculated by the Fox–Flory equation. DMA, on the other hand, showed a significantly higher T g value for both latices than those obtained from theoretical calculations, the thermal SPM method and DSC. The T g obtained from the PR curve was slightly higher for the latex with a low styrene content, whereas good agreement was obtained with the thermal SPM data for the latex with a high styrene content. The glass transition temperature determined by TMA agreed fairly well with the thermal SPM data for the latex with the low styrene content, whilst the value of T g for the second latex was much less than those obtained by the other methods. The thermal SPM method detects changes in thermal behavior (thermal diffusivity, heat capacity) during heating of the latex films rather than changes in the mechanical properties. Information about the sample history could be seen in the thermal SPM curves, which was further associated with the degree of latex film formation, especially when the roughness of the films was taken into consideration.

Glass transition temperature of allyl methacrylate‐n‐butyl acrylate gradient copolymers in dependence on chemical composition and molecular weight

AbstractGlass transition temperatures (Tgs) of P(AMA‐co‐BA) copolymers and the corresponding homopolymers, where AMA is allyl methacrylate and BA isn‐butyl acrylate, obtained by means of atom transfer radical polymerization were measured using differential scanning calorimetry. Because of the (pseudoliving) nature of this polymerization technique an increase in molecular weight (MW) is produced as the reaction progresses, which gives rise to an increase inTgs. This increment can be adequately described by the Fox–Flory's equation in both homopolymers. However, in the spontaneous gradient copolymers of P(AMA‐co‐BA), the expected increase inTgwith the augment of the monomer conversion is compensated by the enrichment of BA as the polymerization reaction progresses. These opposite effects with respect to theTgvalues almost balance each other, and therefore no significant influence on the MW or on conversion is found. This fact establishes thatTgs can be used to describe the profile of these gradient copolymers, and can be theoretically determined because of its dependence on the molar fraction in the copolymer. From this dependence on chemical composition along with the experimental behavior, a prediction of theTgvariation with the MW was performed. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1845–1855, 2007

Effect of the molecular weight and architecture on the size and glass transition of arborescent polyisobutylenes

AbstractThis article discusses the characterization of arborescent (hyperbranched) polyisobutylenes (arb‐PIBs) by size exclusion chromatography and differential scanning calorimetry, in comparison with linear PIB standards. The radius of gyration (〈r〉1/2 = Rz), measured from the angle dependence of light scattering of high‐molecular‐weight arb‐PIBs, was significantly larger than the hydrodynamic radius (Rh) from size exclusion chromatography/viscometry, and the Rh values were significantly smaller than Rh of linear PIBs. The glass‐transition temperature of arb‐PIBs having a branch molecular weight higher than the critical entanglement molecular weight was dependent on both the total number‐average molecular weight and BR up to BR ∼ 15. A modified Fox–Flory equation is proposed to describe the effect of architecture on the thermal transition. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1770–1776, 2006

Molecular dynamics of oligofluorenes: a dielectric spectroscopy investigation.

The molecular dynamics were investigated in a series of "defect-free" oligofluorenes up to the polymer by dielectric spectroscopy (DS). The method is very sensitive to the presence of keto "defects" that when incorporated on the backbone give rise to poor optical and electronic properties. Two dielectrically active processes were found (beta and alpha process). The latter process (alpha) displays strongly temperature dependent relaxation times and temperature- and molecular weight-dependent spectral broadening associated with intramolecular correlations. The glass temperature (Tg) obeys the Fox-Flory equation and the polymer Tg is obtained by DS at 332 K. The effective dipole moment associated with the alpha process is 0.27 +/- 0.03 D.

Treatments of intrinsic viscosity and glass transition temperature data of poly(2,6-dimethylphenylmethacrylate)

A useful relationship, ln( T g)=ln( T g,∞)− m[ η] − ν , between intrinsic viscosity and glass transition temperature for a series of homologous polymers was obtained by combining the Mark–Houwink–Kuhn–Sakurada (MHKS) relation for intrinsic viscosity and molecular mass, and the Fox–Flory equation for glass transition temperature and number-average molecular mass. This relationship was applied to poly(2,6-dimethylphenylmethacrylate) (PDMPh) in a variety of solvents (ideal to good) such as toluene, tetrahydrofuran/water, tetrahydrofuran, and chlorobenzene systems. The parameter α estimated by this procedure in toluene, tetrahydrofuran/water, tetrahydrofuran, and chlorobenzene systems are 0.50 6, 0.51 1, 0.56 7, and 0.67 3, respectively which are in agreement with those of Mark–Houwink–Kuhn–Sakurada values by less than 5% differences. The T g,∞ quantity estimated from this equation also is within the standard deviation of that obtained from the Fox–Flory method. The m quantity is increasing as the thermodynamic quality of the solvent improves, therefore, m may be considered as an indicator of coil conformations in a given solvent.

Investigation into new materials for optical devices: the miscibility and solvent effects of azobenzene derivative blends

AbstractWith the increasing demand for novel devices with optical applications the search for new materials to data store and process becomes a priority. By introducing blends, tailor made properties and low cost give added advantage. Miscibility is an essential requirement for a new material, this research thus involves miscibility studies of poly(4‐(N‐(2‐methacryloyloxyethyl)‐N‐ethylamino)‐4′‐nitroazobenzene)90‐co‐(methyl methacrylate)10, (azobenzene derivative) with polymethyl‐methacrylate (PMMA), polyvinylacetate (PVAc) and polyvinylchloride (PVC) prepared in tetrahydrofuran (THF), and/or dimethylformamide (DMF) and/or dichloromethane (CH2Cl2). The glass transitions, solvent and varying molecular weight effect were investigated, since these all primarily influence the miscibility. THF was found to encourage miscibility at specific compositions of PVAc and PVC blends. However, with CH2Cl2 and DMF immiscibility is encouraged. The Fox–Flory equation was applied to the blends analyzing the PVC blends in DMF as deviations from ideality. Different molecular weights of PMMA were identified as immiscible regardless of solvent. PMMA's lower solubility in THF and CH2Cl2 compared to the azobenzene derivative, displayed the existence of PMMA islands. In all blends the favorable and unfavorable interactions between polymer–solvent–polymer systems are considered. Furthermore, the miscibility effect on increasing the MMA content of the azobenzene derivative was also investigated. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Physicochemical properties of commercial starch hydrolyzates in the frozen state

The physical properties of commercial starch hydrolyzate (varying in dextrose equivalent values, DE, from 0.5 to 42) solutions in the frozen state were related to their composition. At 20% (w/v) hydrolyzate concentration, an inverse linear relationship between DE and apparent glass transition temperature ( T g′) of the unfrozen solute matrix was observed. The T g′ temperatures remained relatively constant for solute concentrations below 40% (w/v); above this concentration the T g′ was depressed, possibly due to the plasticizing effect of additional water entrapped in the glassy state during non-equilibrium freezing. Simple predictive models (based on the Flory–Fox equation; 1/ T g=∑{ w i / T g i }) were found to predict reasonably well the T g′ value of ‘binary’ monodisperse and polydisperse hydrolyzate mixtures of varying proportions between the two components. Linear relationships were also found between T g′ measured (by calorimetry) and T g′ predicted (based on the Flory–Fox model, and using the oligosaccharide composition of the hydrolyzates and the respective T g′ values of the pure components). The rate of oxidation of ascorbic acid has been measured in the presence of starch hydrolyzates at temperatures between −4 and −16°C. Both the Arrhenius and WLF (Williams–Landel–Ferry) kinetic models were found to describe the temperature dependence of reaction rate constants reasonably well. However, knowing the T g′ value of the amorphous maltodextrin was not sufficient to predict its cryostabilization behavior.

Variation of intrinsic viscosity in the hydrolysis of hydroxyethylcellulose, and its relationship with resistance to enzymatic degradation

A mathematical model for the quantitative analysis of kinetic data obtained in the hydrolysis of cellulose ethers is described. The model is applied to solution viscosity measurements obtained during the enzymatic hydrolysis of O-(2-hydroxyethyl)cellulose (HEC). Data analysis involves the derivation of the intrinsic viscosity dependence on the enzymatic hydrolysis time, from the analytical expressions of the Martin equation. The scope of the method and its relationship to the resistance to enzymatic degradation of cellulose derivatives are discussed. The importance of the substitution pattern (represented by the degree of substitution) and molecular size (end-to-end distance derived from the Flory-Fox equation), on the biodegradability of HEC is demonstrated.

Ion Binding Properties of Polycarboxylates Using Terbium(III) as a Fluorescent Probe: Viscosities and Coordinated Water Molecules in Polycarboxylate−Terbium(III) Complexes in Aqueous Solutions

The binding properties of trivalent ions to polyacrylate and its low molecular weight analogs (monomer, dimer, and trimer) were investigated in dilute aqueous solution (10 -2 -10 -3 M) using Tb 3+ ion as a fluorescent probe. The fluorescence intensity and lifetime of the Tb 3+ ion depend directly on the number of water molecules bound to their inner coordination sphere. The more efficiently ligands coordinate to Tb 3+ ion, the more water molecules are expelled and, consequently, the greater the fluorescence intensity and lifetime observed. Lifetime measurements in H 2 O and D 2 O showed that the number of water molecules coordinated to the Tb 3+ ion are 6.5, 6.1, 4.9, and 3.6 for monomer, dimer, trimer model compounds, and polyacrylate, respectively. The viscosities of Tb 3+ -polyacrylate complexes were measured in the presence of a large excess of sodium bromide. Viscosities (η sp /c) of the polyacrylate solutions follow Huggins' equation, and on addition of Tb 3+ ion, the intrinsic viscosities decrease abruptly from 700 to 40 mL/g. The mean end-to-end distances, 1/2 , for polyacrylate in Tb 3+ -polymer complexes were calculated using the Flory-Fox equation and were found to be reduced from 125 to 48 nm on addition of Tb 3+ . These results indicate that of the nine water molecules coordinated to Tb 3+ ion in aqueous solution, five to six are replaced with carboxylate groups attached to the polymer chain wrapped around the Tb 3+ .

Influence of the physical characteristics of the polymer on the glass transition of various chemically modified PVCs

AbstractVinylchloride–vinylthiocresol and vinylchloride–vinylthionaphthol copolymers of various compositions were prepared by the substitution reaction of PVC with sodium thiocresol and sodium thionaphthol, respectively, in different solvents and at different temperatures. The variation of the glass transition temperature of these copolymers with chemical com‐position does not follow the Flory–Fox equation, but is found to depend on the solvent and the temperature at which the copolymer is obtained. The comonomer sequence distribution and stereosequence content of the above copolymers, and those of vinylchloride–vinyl‐thiobenzene copolymer, were determined by 1H‐ and 13C‐NMR spectroscopy. The depen‐dence of comonomer distribution and stereoregularity content with chemical composition for the three series of copolymers, prepared in different solvents and temperatures, are similar. The Tg deviations for the three series of copolymers were attributed to the con‐tribution of physical characteristics of the PVC structure, originating from polymer–solvent interactions. © 1995 John Wiley &amp; Sons, Inc.

The use of GPC coupled with a multiangle laser light scattering photometer for the characterization of polymers. On the determination of molecular weight, size and branching

AbstractThe use of gel permeation chromatorgraphy (GPC) coupled with the multiangle laser light scattering (MALLS) detector for the determination of molecular weight and size distribution and for the identification of polymer branching is demonstrated on several examples of both organic and water souluble polymers. The influence of the second virial coefficient and the way of the light scattering data evaluation on the results obtained is shown. Measurements of the root mean square radius are presented for two series of polystyrene and polymethylmethacrylate standard, and the results are compared with the radii calculated from the Flory‐Fox equation. © 1994 John Wiley &amp; Sons, Inc.

Glass transition temperature and molecular parameters of polymer

A new relationship, which correlates the glass transition temperature ( T g) with other molecular parameters, is developed by using Flory's lattice statistics of polymer chain and taking the dynamic segment as the basic statistical unit. The dependences of T g on the chain stiffness factor ( σ 2), dynamic stiffness factor ( β = − d ln σ 2 dT ) and molecular weight of polymer are discussed in detail based on the theory. The theory is compared with experimental data for many linear polymers and good agreement is obtained. It is shown that T g is essentially governed by the chain stiffness factor at T g. Moreover, a simple correlation between the parameter K g of the Fox-Flory equation ( T g = T ∞ g − K g M n ) and other molecular parameters is deduced. The agreement between theoretical predictions and experimental measurements of K g has been found to be satisfactory for many polymers.