Modified Maxwell Model Research Articles

Effects of novel silane modification of zeolite surface on polymer chain rigidification and partial pore blockage in polyethersulfone (PES)–zeolite A mixed matrix membranes

A novel silane coupling agent, (3-aminopropyl)-diethoxymethyl silane (APDEMS) was used in this work to modify zeolite surface for mixed matrix membranes (MMMs). Both elementary analysis and XPS spectra confirm the chemical modification, while BET measurements show no changes in zeolite surface area and total pore volume after the modification. Polyethersulfone (PES)–zeolite 3A, 4A and 5A MMMs were fabricated at high processing temperatures using unmodified and chemical modified zeolite. SEM images of these MMMs indicate the interface between polymer and zeolite phases becomes better if modified zeolite is used. The effects of chemical modification of zeolite surface and zeolite loadings on the gas separation performance of these MMMs were investigated. Both permeability and selectivity of MMMs made from APDEMS modified zeolite are higher than those of MMMs made from unmodified zeolite at 20 wt% zeolite loading because of a decrease in the degree of partial pore blockage of zeolites. The permeability of all studied gases decreases with increasing zeolite content for PES–zeolite 4A-NH 2 MMMs, while for PES–zeolite 5A-NH 2 MMMs, the gas permeability decreases and then increases with an increase in zeolite loadings. This unique phenomenon implies that using large pore-size zeolite for MMMs would potentially offset the negative effects of partial pore blockage and polymer chain rigidification on permeability. A modified Maxwell model with adjusted parameters was applied to study the PES–zeolite 4A-NH 2 MMM system. The predicted permeability and selectivity show very good agreement with experimental data, indicating the modified Maxwell model is fully capable of predicting the gas separation performance of MMMs made from both unmodified and modified zeolite.

Predicting the effective thermal conductivity of unfrozen, porous foods

In this study, it was shown that effective thermal conductivity models that are functions only of the components’ thermal conductivities and volume fractions could not be accurate for both granular-type porous foods (“external porosity”) and foam-type porous foods (“internal porosity”). An extra parameter is needed to make the model sufficiently flexible to allow it to be applied to porous foods with a range of different structures. A number of effective thermal conductivity models contain the required extra parameter, and of these, Krischer’s model appears to have received the greatest use in the food engineering literature; however, for isotropic materials it is recommended that a modified Maxwell model be used instead, because it assumes an isotropic physical structure, unlike Krischer’s model, and because the numerical value of the extra parameter may be estimated based on whether the food has internal or external porosity. A new procedure for predicting the effective thermal conductivity of non-frozen porous foods is presented as a flowchart.

The effects of polymer chain rigidification, zeolite pore size and pore blockage on polyethersulfone (PES)-zeolite A mixed matrix membranes

The polyethersulfone (PES)-zeolite 3A, 4A and 5A mixed matrix membranes (MMMs) were fabricated with a modified solution-casting procedure at high temperatures close to the glass transition temperatures ( T g) of polymer materials. The effects of membrane preparation methodology, zeolite loading and pore size of zeolite on the gas separation performance of these mixed matrix membranes were studied. SEM results show the interface between polymer and zeolite in MMMs experiencing natural cooling is better (i.e., less defective) than that in MMMs experiencing immediate quenching. The increment of glass transition temperature ( T g) of MMMs with zeolite loading confirms the polymer chain rigidification induced by zeolite. The experimental results indicate that a higher zeolite loading results in a decrease in gas permeability and an increase in gas pair selectivity. The unmodified Maxwell model fails to correctly predict the permeability decrease induced by polymer chain rigidification near the zeolite surface and the partial pore blockage of zeolites by the polymer chains. A new modified Maxwell model is therefore proposed. It takes the combined effects of chain rigidification and partial pore blockage of zeolites into calculation. The new model shows much consistent permeability and selectivity predication with experimental data. Surprisingly, an increase in zeolite pore size from 3 to 5 Å generally not only increase gas permeability, but also gas pair selectivity. The O 2/N 2 selectivity of PES-zeolite 3A and PES-zeolite 4A membranes is very similar, while the O 2/N 2 selectivity of PES-zeolite 5A membranes is much higher. This implies the blockage may narrow a part of zeolite 5A pores to approximately 4 Å, which can discriminate the gas pair of O 2 and N 2, and narrow a part of zeolites 3A and 4A pores to smaller sizes. It is concluded that the partial pore blockage of zeolites by the polymer chains has equivalent or more influence on the separation properties of mixed matrix membranes compared with that of the polymer chain rigidification.

Interfacial Properties of a PEG2000−PLA50 Diblock Copolymer at the Air/Water Interface

In this paper, we compared the interfacial behaviors of spread films of poly(ethylene glycol) (PEG2000), poly(d,l-lactide) (PLA50), and a mixture of PEG2000 and PLA50, to understand the properties of a PEG2000−PLA50 diblock copolymer at the air/water interface. This was achieved (i) by analyzing the surface pressure−surface area curves obtained on a Langmuir trough and (ii) by modeling the dilatational properties of the films according to a modified Maxwell model. The properties of the films composed of a mixture of PEG2000 and PLA50 showed the influence of PEG2000 before the nucleation transition of PLA50. Then, increasing the lateral compression of the mixed monolayer led to the expulsion of PEG2000 segments with presumably their irreversible desorption into the bulk. In the case of the copolymer, the covalently bound PEG2000 segments were segregated at low surface coverage forming a stable film with the PEG corona oriented toward the water phase. This situation was conserved until high surface coverage (10−15 mN/m). Afterward, compression led to the penetration of PEG2000 into the tridimensional layer oriented toward the air/phase. These results allow better understanding and modulation of the composition and the hydrophilic character of interfaces formed during emulsion processes and thus improvement of the control of the surface properties of drug delivery systems.

Viscoelastic modeling of natural and synthetic textile yarns

Viscoelastic models were employed to analyze the stress–strain behavior of cotton, acrylic, and polyester yarns. The potential model and the modified Maxwell model gave the best fits as regards the stress–strain behavior of these yarns. The results of the potential model resembled those of the modified Maxwell model with an infinite relaxation time. The potential model is suitable for explaining the rearrangement undergone by the yarns when no slippage between fibers under strain occurs. When slippage occurs, Maxwell models are able to explain the stress–strain behavior of the yarns. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 2062–2067, 2000

Structural developments in supercooled alkali tellurite melts

Brillouin light scattering has been used to determine the high-frequency complex mechanical modulus of alkali tellurite liquids and glasses, as a function of the temperature. In order to adequately describe the temperature dependence of this modulus, a modified Maxwell model for linear viscoelastic systems was developed. Accordingly, the modulus comprises relaxational components and a temperature-dependent static modulus, the magnitudes of which are determined by the equilibrium volume fraction of kinetically arrested domains. In binary alkali tellurites the structure degrades with increasing temperature by means of the dissolution of network rings, which involves the conversion of ${\mathrm{TeO}}_{4}$ trigonal bipyramids into ${\mathrm{TeO}}_{3}$ trigonal pyramids. This decrease in network connectivity is believed to be responsible for the decrease in the elastic modulus and to cause the release of structural components into a viscoelastic state. At high temperatures structural dynamics can be described with a single relaxation time.

Effect of Hydrophobic Modification of a Nonionic Cellulose Derivative on the Interaction with Surfactants. Rheology

The interactions between various surfactants (anionic, cationic, and nonionic) and a nonionic ethyl(hydroxyethyl)cellulose (EHEC) polymer or a hydrophobically modified analogue (HM-EHEC) have been examined. The study has been performed as a comparative investigation between the hydrophobically modified polymer and the unmodified parent polymer using rheology. The rheological results have been analyzed with the aid of a simple modified Maxwell model. Information about the low-frequency behavior was extracted from this model, and the rheological features were presented in terms of the complex viscosity and a characteristic relaxation time. In the presence of an ionic surfactant, the rhelogical measurements revealed significant polymer/surfactant interaction for both EHEC and HM-EHEC, but the interaction peak, observed in the different rheological quantities, was more pronounced and located at a lower surfactant concentration for the hydrophobically modified polymer. In the presence of a nonionic surfactant,...

Residual stress build-up in thermoset films cured below their ultimate glass transition temperature

The stress build-up during isothermal cure below the ultimate glass transition temperature of epoxy and acrylate films is investigated in detail. Four systems are studied; two acrylates and two epoxies, with different crosslink densities. Relaxation modulus and film shrinkage are measured simultaneously during cure. The stress build-up is measured independently using a bi-layer beam bending technique. A model for the build-up of cure stresses is proposed, in which stresses are generated by the cure shrinkage and decay by viscoelastic relaxation. The relaxation is described by a simple, modified Maxwell model. Owing to the absence of memory in the Maxwell model, the resulting equation is simple and numerical stress computation straightforward. The stress build-up over time is thus simulated for the four model systems based on the relaxation and shrinkage data, and the simulations compared with the experimentally observed stress build-up. The model successfully predicts the cure stresses where more standard elastic methods fail. It is found that the amount of stress build-up during cure varies greatly between the different systems. In general, a higher crosslink density results in higher stress build-up. The stress on cure ranged from less than 1% of the total stress on cure and cool-down in a lightly crosslinked epoxy to more than 30% of the total stress in densely crosslinked epoxies and acrylates. Finally simple approximations for estimating the stress levels after cure and cool-down from basic material properties, e.g. modulus and cure shrinkage, are proposed.

Physiologic aspects of drug therapy and large artery elastic properties.

Vasoactive drugs alter smooth muscle tone not only in arterial resistance vessels, but also in large conduit arteries. The resultant changes in smooth muscle tone alter both conduit vessel size and stiffness and hence influence pulsatile components of left ventricular afterload. The effects of smooth muscle relaxation and contraction on arterial elastic properties are complex and have not been fully characterized. Several recent studies have utilized a new intravascular ultrasound technique to study the effects of changes in smooth muscle tone on brachial artery elastic mechanics in normal human subjects in vivo. Smooth muscle relaxation with nitroglycerin improves isobaric brachial artery compliance without significantly altering arterial wall stiffness as measured by incremental elastic modulus (Einc). The improvement in compliance with smooth muscle relaxation is the net result of factors that: (1) increase wall stiffness (increased tension in parallel elastin and collagen fibers); (2) decrease wall stiffness (decreased tension in the smooth muscle and its associated series elastic component); and (3) increase vessel lumen size. Using a modified Maxwell model for the arterial wall, smooth muscle relaxation is also shown to shift the predominant elements contributing to wall stress and Einc from smooth muscle and the collagen fibers in series with the smooth muscle to collagen fibers in parallel with the smooth muscle. A better understanding of the mechanisms contributing to changes in arterial elastic mechanics following alterations in smooth muscle tone will help in developing pharmacologic therapies aimed at reducing pulsatile components of left ventricular afterload.

Contribution of collagen, elastin, and smooth muscle to in vivo human brachial artery wall stress and elastic modulus.

The contributions of collagen, elastin, and smooth muscle to arterial mechanical properties in the in vivo human artery are not known. We used a recently developed intravascular ultrasound technique to measure total brachial artery wall stress and incremental elastic modulus (Einc) in seven normal human subjects at baseline and after intra-arterial norepinephrine (1.2 micrograms) and nitroglycerin (100 micrograms). Then we applied a modified Maxwell model to estimate the elastic modulus of elastin (EE); the recruitment of collagen fibers supporting wall stress; and the differential contributions of collagen, elastin, and smooth muscle to wall stress and Einc over a wide range of pressure and smooth muscle tone. With this model, EE was 3 x 10(6) dynes/cm2. Collagen fibers were recruited increasingly as transmural arterial pressure increased and reached a value of approximately 5% to 6% at 100 mm Hg under each of the conditions studied. Isobaric smooth muscle contraction resulted in a small decrease in total wall stress and no significant change in total Einc while shifting the predominant element contributing to these mechanical parameters from collagen in parallel with the smooth muscle to collagen in series with the smooth muscle. In contrast, isometric smooth muscle contraction produced large increases in total wall stress (from 0.11 x 10(6) dynes/cm2 after nitroglycerin administration to 1.35 x 10(6) dynes/cm2 after norepinephrine administration) and Einc (from 3.84 x 10(6) dynes/cm2 after nitroglycerin administration to 57.8 x 10(6) dynes/cm2 after norepinephrine administration) entirely as a result of the additional contribution of the smooth muscle and its associated series collagen. This study describes a technique for determining arterial elastic properties and a model that can be used to estimate a number of mechanical parameters of the human brachial artery in vivo. This technique may be useful in studies of the arterial elastic properties of arteries in patients with vascular pathology.

Assessment of smooth muscle contribution to descending thoracic aortic elastic mechanics in conscious dogs.

Early investigators found contradictory evidence that vascular smooth muscle activation reduces the elastic modulus of the arterial wall under isotonic conditions but increases it under isometric conditions, concomitant with increased pulse-wave velocity. We examined the individual contributions of aortic constituents to the elastic modulus of the aortic wall to determine if isobaric analysis produces an accurate assessment of vascular smooth muscle activation. We used a modified Maxwell model assuming an incremental elastic modulus (Einc) composed of the elastic modulus of elastin fibers (EE), the elastic modulus of collagen fibers (EC) affected by the fraction of collagen fibers (fC) recruited to support wall stress, and the elastic modulus of the vascular smooth muscle (ESM) according to the following formula: Einc = EE+EC x fC+ESM.Einc was assessed in eight conscious dogs using descending thoracic aortic pressure (microtransducer) and diameter (sonomicrometry) measurements. Stress-strain relations in the control state and during activation of smooth muscle by continuous administration of phenylephrine (5 micrograms.kg-1 x min-1) were obtained by transient occlusions of the descending aorta and inferior vena cava. Results were as follows: EE was 4.99 +/- 1.58 x 10(6) dynes/cm2 (mean +/- SD), and EC was 965.8 +/- 399.8 x 10(6) dynes/cm2, assessed during the control state. Phenylephrine administration increased the theoretical pulse-wave velocity (Moens-Korteweg equation) from 5.25 +/- 1.03 m/s during the control state to 7.57 +/- 2.53 m/s (P < .005). Active muscle exhibited a unimodal stress-strain curve with a maximum stress of 0.949 +/- 0.57 x 10(6) dynes/cm2 at a corresponding strain value of 1.299 +/- 0.083. The maximum value observed corresponded, on the pressure-diameter curve of the active artery, to a pressure of 234.28 +/- 46.6 mm Hg and a diameter of 17.94 +/- 1.6 mm. The maximum ESM derived from the stress-strain relation of the active muscle was 8.345 +/- 7.56 x 10(6) dynes/cm2 at a strain value of 1.283 +/- 0.079. This point was located at 208.01 +/- 40.8 mm Hg and 17.73 +/- 1.41 mm on the active pressure-diameter curve. During activation of vascular smooth muscle, Einc decreased (P < .05) when plotted against internal pressure but increased (P < .05) when plotted against strain, over the operative range.(ABSTRACT TRUNCATED AT 400 WORDS)

Measurement of elasticities of very dilute polymer solutions in shear and elongational flows.

Elasticities were measured under the conditions of shear and elongational flows of PEO and Separan solutions below the concentration level of several tens of ppm. The first normal stress difference in a shear flow was measured by utilizing reactions exerted by a jet issuing from a capillary, and the elongational stresses were also obtained separately with reactions or pressure drops produced in the flow through small orifices. The following points were clarified: the first normal stress difference τ11-τ22 had a relationship of the power law type with the shear rate γ, that is, τ11-τ22&prop;γ. For the PEO 5 ppm solution, the power law index s was 2.0, as predicted with the Maxwell model, and in this case, the relaxation time was determined as 0.0016 second. For the other solutions used, we had 1.2< s < 2.0 and the measured first normal stress differences could be expressed with a modified Maxwell model. The elongational stresses measured with the two methods of jet reaction and pressure drop generally agree with each other. The elongational stress can be expressed, albeit approximately, with the Maxwell model or the modified Maxwell model depending on the concentration of the solution used. In some cases, the elongational stresses of PEO solutions, at least qualitatively, agree with predicted values of the yo-yo model proposed by Ryskin.

A Modified Maxwell and a Nonexponential Model for Characterization of the Stress Relaxation of Agar and Alginate Gels

ABSTRACTCompressive stress relaxation curves of agar and alginate gels of different gum concentration (1–3% and 0.5–2%, respectively) were fitted by a two parameter nonexponential empirical model and a three term modified Maxwell model with two fixed relaxation times (10 and 100s). The asymptotic portion of the residual (unrelaxed) stress calculated by the two models for each gel had a similar magnitude, and therefore could serve as an objective measure of the gels degree of solidity. The coefficients of the modified Maxwell model provided a simple but meaningful means to compare, in quantitative terms, the differences in the relaxation time spectra that were associated with the gels' stiffness and strength.

Experimental estimation of elongational stresses of dilute polymer solutions and a related examination of some constitutive equations

Elongational stresses of dilute polymer solutions have been estimated by utilizing the flow through small orifices under the condition of no vortex upstream of the orifice plane. The flow was approximated with a linearly converging flow towards an apex of a cone, its validity being partially confirmed by the measured center velocities, and the elongational stresses are determined from the measured thrusts of dilute polymer solutions. On the other hand, elongational stresses were theoretically obtained with the modified Maxwell model and the second order fluid. A comparison was made between the experimental and the theoretical results and the following points were clarified; below an elongational rate of 2 × 10 4 s −1 the modified Maxwell model gives elongational stresses close to the experimentally determined ones, but above that elongational rate it deviates from the experimental results. The second order fluid is not sufficient to describe the stresses in this kind of elongational flow and an acceleration term such as δ 2 e ij /δ t 2 may be necessary in this case.

Assessment of elongational stresses of dilute polymer solutions and a related examination of some constitutive equations.

Elongational stresses of dilute polymer solutions have been assessed by utilizing the flow through small orifices under the condition of no vortex upstream of the orifice plane. The flow was approximated with a linearly converging flow toward an apex of a cone, its propriety being confirmed by the measured velocities, and the elongational stresses were determined from the measured thrusts of dilute polymer solutions. On the other hand, elongational stresses were theoretically obtained with the modified Maxwell model and the second order fluid. A comparison was made between the experimental and the theoretical results and the following points were clarified ; below the elongational rate 2×104(s-1) the modified Maxwell model gives elongational stresses close to the experimentally determined ones, but above the elongational rate it deviates from the experimental results. the second order fluid is not sufficient to describe the stresses in this kind of elongational flow and an acceleration term such as δ2eij/δt2 may be necessary in this case.

On the role of penetrant structure in diffusion in structured polymers

Marked differences between the diffusional behaviour of linear and non-linear penetrant molecules in structured polymers exist, which remain hitherto unexplained. The existing theories of transport in polymers have been re-examined with a view to explain such differences. Direct and indirect experimental evidence has been analysed to support the view-point that linear molecules diffuse by a segmental motion. A diagnostic method in the context of a modified Maxwell model has been proposed to differentiate between different diffusive phenomena. The role of co-operative segmental motion for the polymer matrix has been quantitatively examined. Possible technological implications of the findings of the work have been elaborated on.

Diffusion in network polymers: model development and evaluation

In spite of the importance of diffusion in network polymers, there exist no theoretical models with predictive ability. A two-phase concept has been used to develop a modified Maxwell model to establish diffusivity-structure correlation in network polymers for the first time. The physical significance of the model parameters has been elaborated and existing data examined to substantiate the model. Pragmatic implications of the model are discussed.

Thermal diffusivity and conductivity of crystalline polymers

AbstractThe thermal diffusivity and conductivity of the isotropic and drawn samples of five semicrystalline polymers—nylon 6, poly(ethylene terephthalate), poly(butylene terephthalate), polybutene‐1, and poly(4‐methylpentene‐1)—were measured by the flash method over the temperature range 100–350 K. The temperature dependence of the thermal diffusivity was found to follow a simple phenomenological pattern, while a more detailed understanding of the temperature dependence and the effect of orientation on thermal conductivity was obtained by using the modified Maxwell model.

Thermal conductivity of oriented crystalline polymers

AbstractThermal conductivities of six oriented semicrystalline polymers which range from 0.37 to 0.63 in crystallinity and 1 to 5 in draw ratio λ (up to about 15 for two polymers) have been measured between 100 and 340 K. It was found that for increasing λ the conductivity K∥ (along the draw direction n̂) increases rapidly while K⊥ (normal to n̂) decreases slightly; K∥ also increases with temperature, but K⊥ shows no simple pattern in temperature dependence. These general features can be reproduced reasonably well at low draw ratio (λ &lt; 5) by the modified Maxwell model, and the discrepancy in details may be attributed to the fact that the model does not take into account the possible anisotropy of the amorphous phase of the oriented polymers. At high draw ratio the intercrystalline bridge effect becomes important, and one must resort to the Takayanagi model, but the lack of corroborating x‐ray data has rendered a detailed comparison impossible.

Thermal conductivity of highly oriented polyethylene

We have measured the thermal conductivity of oriented polyethylene both along and perpendicular to the draw direction for draw ratio λ between 1–25 and in the temperature range of 120 to 320K. The results for λ ⩽ 5 have been analysed in terms of the modified Maxwell model while the further increase of thermal conductivity along the draw direction at higher λ has been explained by the Takayanagi model. The Young's modulus and thermal conductivity along the draw direction for the sample with λ = 25 were found to be 64 GN/m 2 (at 220K) and 140 mW/cm K (at 300K), respectively, which are extremely high values for polymers. This material, which is a good electrical insulator and yet has a high thermal conductivity, may be useful in electrical applications requiring large dissipation of heat.