Stress-optical Law Research Articles

Analytical and experimental study on wave propagation problems in orthotropic media

Wave propagation problems in orthotropic media are studied jointly by analytical and experimental methods in this paper. Dynamic orthotropic photoelasticity, which studies experimentally the dynamic behavior of orthotropic materials on a macroscopic scale by employing orthotropic birefringent materials, is established. A dynamic stress-optic law for orthotropic birefringent materials is postulated and practical methods of calibrating dynamic mechanical constants and dynamic stress-fringe values are proposed. Meanwhile, time domain boundary element method (BEM) for wave propagation in orthotropic media, is also presented based on the theory of elastodynamics. A scheme of stress calculations that is necessary for strength analysis is established. The paper stresses on the applications in wave propagation problems in orthotropic media by demonstrating three examples. The semi-infinite orthotropic plates with and without a circular hole modeled by a unidirectional fiberreinforced composite under impact loading are analyzed. Time histories of birefringent fringe orders or stresses for specific points of the plates are obtained respectively from the two methods and compared with each other. Based on the above comparative study, the dynamic response of an underground workshop under seismic waves is studied by time domain BEM. The responses of displacements and stresses are solved. The effects of angle and frequency of incident waves and the degree of media anisotropy on dynamic response of the underground workshop are investigated.

Atomistic Monte Carlo simulation of steady-state uniaxial elongational flow of long-chain polyethylene melts: dependence of the melt degree of orientation on stress, molecular length and elongational strain rate

Following our recent work on the simulation of elasticity and birefringence of linear polyethylene (PE) melts [Mavrantzas and Theodorou, 1998; 2000], new results are presented for PE melts of chain length up to C1000, modelled in atomistic detail. The simulations have been performed with the End-Bridging Monte Carlo (EBMC) method, allowing large numbers of strained polymer melt configurations, thoroughly equilibrated at all length scales, to be sampled. These strained configurations are representative of the structure of PE melt systems under conditions of low-Deborah number steady-state uniaxial elongational flow in the x direction. The degree of orientation at the level of individual bonds and of entire chains and the melt anisotropic refractive index (birefringence) are presented and analyzed for a collection of PE melt systems of mean chain length C24 up to C1000 and polydispersity index 1.05 to 1.09 over a wide range of imposed stresses. The simulations demonstrate: a) a linear dependence of the conformation tensor component cxx, quantifying the overall chain orientation and deformation, and of the bond order parameter Sx on the imposed stress difference τxx-τyy, for all systems and strain-rates investigated, except for the shortest C24 system at stress levels exceeding about 350 atm, and b) conformity to the stress optical law (linearity between the anisotropy of the refractive index and the anisotropy of stress) in all cases except for the C24 system at the highest stress levels. The stress optical coefficient C is seen to go through a maximum at around C100 and to reach an asymptotic value beyond C500 , which is predicted within 30% of the experimental value for high-molecular weight linear PE melts. The linear relationship between Sx and cxx borne out of the melt simulations is in excellent agreement with Flory's theoretical treatment based on single freely-jointed chains. Results are also presented for the elongational viscosity ηE of the (unentangled) C24 and C78 PE melts and for its dependence on elongational strain rate. Values of ηE are calculated from the dependence of stress on the orienting field employed in the EBMC simulations. This field is converted to a strain rate by mapping the atomistic model onto the Hookean dumbbell, FENE dumbbell and Rouse models of viscoelasticity so that the unperturbed chain dimensions and chain self-diffusivity (obtained through equilibrium molecular dynamics studies of the same systems, Harmandaris et al., 1998) are preserved. The results obtained with all models: a) respect Trouton's law and b) display a continuous increase of ηE for these short PE melt systems with the applied elongational strain rate.

Mechanical and Optical Behavior of Double Network Rubbers

Stress, strain, and optical birefringence were measured for a series of peroxide-cured natural rubbers, having both isotropic and double network structures. The residual stretch (permanent set) for the latter ranged from 2.0 to 4.5, with elastic moduli that were an increasing function of this residual strain, as found in previous works. A small birefringence, ca. 10-5, was observed for the unstressed double networks, and its magnitude increased with increasing residual strain. The sign of this birefringence corresponded to extension of the (undeformed) double networks. Under stress, the birefringence followed the stress optical law. The double network properties were interpreted using the constrained-chain model of rubber elasticity, with the assumption of independent, additive contributions from the two component networks. The calculated results differed from the experimental findings, in particular underestimating the residual strain. This failure is a consequence of the overprediction of the stresses during compression, a limitation common to molecular theories of rubber elasticity. The modeling of the double networks does account qualitatively for the sign of their unstressed birefringence, which is due to the stress- optical coefficient being larger in tension than in compression. This particular deviation from the stress- optical law is known from both theory and experiment.

Atomistic simulation of the birefringence of uniaxially stretched polyethylene melts

The birefringence of uniaxially stretched, long-chain polyethylene (PE) melts is predicted through detailed atomistic simulations by employing the end-bridging Monte Carlo method. The method involves two steps: First, a large number of well-equilibrated, uniaxially stretched polymer configurations are sampled by invoking the methodology developed in our recent work on the simulation of polymer melt elasticity. A key feature in this step is the tensorial field axx which orients and, under certain conditions, deforms the polymer chains in the x direction, inducing anisotropy in the melt. Second, the structural characteristics of the oriented polymer configurations are analyzed and a description of their anisotropy at the monomer level is obtained. By transforming the polarizability tensor of each individual skeletal bond (or united atom group) from the coordinate frame of its principal axes to the laboratory frame, the ensemble average polarizability tensor per methylene group 〈α〉 of the uniaxially stretched polymer melt is calculated as a function of the segment order parameter Sx. The anisotropic melt refractive index Δn(≡nxx−nyy) is obtained from 〈α〉 by using the Clausius–Mossoti and Lorentz–Lorenz relationships. Results obtained for two linear PE melts (average chain length C78 and C200) verify the validity of the stress optical law for small enough imposed elongational flow rates axx. The calculated stress optical law coefficient C is found to be equal to (3.15±0.20)×10−9Pa−1 for the C78 melt and equal to (2.35±0.10)×10−9Pa−1 for the C200 melt. The experimentally measured value for high-molecular weight, linear, high-density PE melts is 2.20×10−9Pa−1.

ポリプロピレンの高速溶融紡糸過程における分子配向形成のオンライン計測

In order to clarify the mechanisms of molecular orientation in high-speed melt spinning of polypropylene (PP), the change of birefringence along the spinline was analyzed through the simultaneous on-line measurement of the diameter and optical retardation profiles. The optical retardation was evaluated using an optical system with a rotating polarizer, polarizing filters, quarter-wave plates and a He-Ne laser as the light source. The birefringence increased only slightly in the upstream portion of the spinline. On the other hand, there was a steep increase of birefringence, which became more significant with an increase in the take-up velocity, near the solidification point where crystallization appears to start. An additional increase of birefringence was notably observed in the downstream portion of the spinline where the thinning of the spinline was already completed. The as-spun PP fibers also showed a gradual increase of birefringence even after the spinning, indicating that the increase of birefringence after the solidification of the spinline was caused by the change in the super-structure of the fibers, not by the increase in tensile stress along the spinline because of the accumulation of the air-friction force. To analyze the applicability of the stress-optical law, tension and temperature profiles of the spinline were calculated based on the experimental results of the diameter profiles. The small increase of birefringence in the molten state was found to be due to the low stress-optical coefficient of PP. It was also found that at take-up velocities above 1km/min the steep increase of birefringence started at the position where the tensile stress of the spinline exceeded about 1MPa. These results suggest that the orientation-induced crystallization of the PP spinline starts at this stress level and also that the development of birefringence of PP fibers in the high-speed spinning process is significantly affected by the crystallization.

On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow

The hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow is studied in the framework of three distinct theoretical models. For ideal kinematics of uniaxial elongation, we show that the kinetic theory of FENE dumbbells and its FENE-L approximation present a hysteresis when plotting polymer stress versus average molecular extension. Similar behaviour is obtained for ideal extensional kinematics using a FENE-P constitutive equation with a spectrum of finite extensibility parameters. Finally, a numerical simulation of the filament-stretching device shows that spatial inhomogeneities of the stress and average conformation fields also lead to hysteretic behaviour with a single-mode FENE-CR constitutive equation. In all three cases, hysteretic behaviour results from the combined effect of dispersity and non-linearity. We also address the validity of the stress-optic law for FENE dumbbells in relaxation following start-up of uniaxial extension. The simulation results show that the stress-optic coefficient remains constant at low strains only. Plots of stress-optic coefficient versus birefringence show hysteresis as well. This rules out a modified stress-optic law for FENE dumbbells wherein the stress-optic coefficient would be a function of the second moment of the configuration distribution function alone. Finally, it is shown in the Appendix that a proper selection of the spectrum of finite extensibilities can be made so that the multi-mode FENE-P model gives essentially the same stress response as the kinetic theory of FENE dumbbells in transient uniaxial extension.

Photoelastic studies for composite dynamics

The orthotropic birefringent composites suitable for the study of dynamic photoelasticity are investigated and the determination of residual birefringence in the materials is briefly described. A stress-optic law for dynamic photoelastic analysis of orthotropic birefringent composites is postulated based on the static stress-optic law of Hyer and Liu. Subsequently, practical methods of calibrating dynamic mechanical constants and dynamic stress-fringe values are proposed. With dynamic strain measurements and time domain BEM for anisotropic media, three calibration specimens (0°, 90° and 45°) are used to verify the proposed stress-optic law in uniaxial-stress fields and a plate of unidirectional fiber-reinforced birefringent composite under impact loading, with the loading direction parallel, perpendicular and at 45° to the fiber direction, is analyzed to verify the proposed stress-optic law in biaxial-stress fields. Results show that the dynamic stress-optic law for orthotropic birefringent composites is valid in the two cases.

Flow birefringence and stress optical law of viscoelastic solutions of cationic surfactants and sodium salicylate

The optical and rheological properties of different viscoelastic solutions of surfactant are studied in order to gather experimental data used to calculate the value of the stress optical coefficient C. Three surfactants of the same family (CTAB) have been chosen; they differ by the length of the hydrocarbon chain; it concerns the dodecyltrimethylammonium bromide (C15H34BrN or DoTAB), the myristyltrimethylammonium bromide (C17H38BrN or MyTAB), and the hexadecyltrimethylammonium bromide (C19H42BrN or CTAB). Different parameters like the temperature of the solution and the salinity of the solvent have been made to vary. Flow birefringence experiments and rheological measurements are performed on these solutions in order to study the dependence of the extinction angle , of the birefringence intensity and of the shear stress with the shear rate . These data are used to check the stress optical law which turns out to be valid in a wide range of shear rates. The stress optical coefficient C is then computed: it is found to vary with the salinity of the solvent and the temperature of the solution for a given surfactant. Then, for all solutions of this work the variations of C are related to the variations of the polarizability anisotropy and the persistence length.

Stress tensor and stress-optical law in entangled polymers

The molecular origin of the stress in entangled polymers is reconsidered here, in order to ascertain whether or not the pressure that the confined chain exerts on its `tube of constraints' should be accounted for. Indeed, the classical expression for the stress includes only traction along the chain. The analysis shows that the pressure terms significantly contribute to the stress provided flow or deformation creates a long-lasting local anisotropy of the chain confinement. The effects of such anisotropy is described by means of elliptical cross-sections of the constraining tubes. The new expression for the stress tensor thus obtained properly degenerates into the classical one for circular cross sections. It is also shown that the stress-optical law remains obeyed.

Dynamic simulation of freely-draining, flexible bead-rod chains: Start-up of extensional and shear flow

We present a study of the rheology and optical properties during the start-up of uniaxial extensional and shear flow for freely-draining, Kramers bead-rod chains using Brownian dynamics simulations. The viscous and elastic contributions to the polymer stress are unambiguously determined via methods developed in our previous publication [1]. The elastic contribution to the polymer stress is much larger than the viscous contribution beyond a time of 5.3λ1/N2 where N is the number of beads in the chain and λ1 is the longest relaxation time of the chain. For small Wi (at arbitrary strains) and for small strains (at arbitrary Wi) the stress-optic law is found to be valid. The stress-optic coefficients based on the shear stress and first normal stress difference are equal for all Wi (even when the stress-optic law is not valid) suggesting the stress-optic coefficient is in general a scalar quantity rather than its most general form as a fourth order tensor.We show that a multimode FENE-PM or Rouse model describes the rheology of the bead-rod chains at small strains, while the FENE dumbbell is an accurate model at larger strains. We compare the FENE-PM and FENE model to experimental extensional stress data of dilute polystyrene solutions and find that a multimode FENE-PM with a Zimm relaxation spectrum describes the data well at small strains while a FENE dumbbell with a conformation dependent drag is in quantitative agreement at larger strains.

Shear birefringence measurements on polymer thin films deposited on quartz resonators

We have measured the second-order stress birefringence in a 2 μm film of poly-(methyl-methacrylate) sandwiched between the top electrode of a quartz resonator and a second, semitransparent gold layer. The gold–film–gold system forms an optical Fabry–Perot resonator with characteristic dips of reflectivity at certain angles of incidence, which are the coupling angles of the different optically resonant modes. When modulating the driving voltage of the quartz resonator with a frequency in the kHz range and referencing the detection electronics to the modulation, shifts in the coupling angle in the range of 10−3° are easily detected. They result from a pseudo-dc component in the stress-optic signal, which is proportional to the square of the shear strain. This second-order stress birefringence is related to normal stresses in the film via the stress-optic law. We compare our findings to predictions from the theory of finite elasticity.

Dynamic simulation of freely draining flexible polymers in steady linear flows

We present a study of the rheological and optical behaviour of Kramers bead–rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coefficient. We have developed a computer algorithm to differentiate the Brownian from the viscous stress contributions which also avoids the order (δt)−1/2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the effective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we find an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead–rod chain scales as N3. Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi, Wi ≈ 0.06N2, where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi−1/2 and the first normal stress as Wi−4/3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead–rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi.

Flow birefringence of poly(n-hexyl isocyanate) solutions

A correction for optical quantities measured by a slit rheometer is proposed in order to obtain the corresponding optical properties at wall. This correction together with the Rabinowitch correction enables studies of the relation between the optical and the mechanical properties. A simple optical flow cell of a slit geometry was designed for the simultaneous measurements of the mechanical and the optical properties of flowing birefringent materials. The setup and the correction were applied to investigate the flow birefringence of the isotropic solutions of a liquid crystalline polymer, poly(n-hexyl isocyanate). The apparent birefringence defined as the refractive index difference between the ordinary light and the extraordinary light was found to increase with the increasing shear rates and was higher for solutions of high concentrations at a given shear rate. The stress optical law was checked. The ratio of the apparent birefringence to the shear stress was found to be independent of the shear rate at concentration of 10 wt%. For solutions of 15 and 20 wt%, the ratios increase with the shear rate when the shear rate is small, and they reach a maximum then decrease with increasing shear rate. The optical-stress behavior of PHIC solutions qualitatively follows the theoretical prediction of the Doi theory.

Elongational flow opto-rheometry for polymer melts

Polymer melt elongation is one of the most important procedures in polymer processing. To understand its molecular mechanisms, we constructed an elongational flow opto-rheometer (EFOR) in which a high precision birefringence apparatus of reflection-double path type was installed into a Meissner's new elongational rheometer of a gas cushion type (commercialized as RME from Rheometric Scientific) just by mounting a small reflecting mirror at the center of the RME's sample supporting table. The EFOR enabled us to achieve simultaneous measurements of tensile stress σ(t) and birefringence Δn(t) as a function of time t under a given constant strain rate $$\dot \varepsilon _0$$ within the range of 0.001 to 1.0s−1. σ(t) can be monitored upto the maximum Hencky strain ɛ(t) of 7 as attained, in principle, with RME, while the measurable range of the phase difference in the birefringence was 0 to 250 π (0 to 79 100 nm for He-Ne laser light) within the accuracy of ±0.1 π (±31.6 nm) up to ɛ(t) ∼ 4. The performance was tested on an anionically polymerized polystyrene (PS) and a low density polyethylene (LDPE). For both polymers σ(t) first followed the linear viscoelasticity rule in that the elongational viscosity, $$\eta _E (t) \equiv \sigma (t)/\dot \varepsilon _0$$ , is three times the steady shear viscosity, 3η o(t), at low shear rate $$\dot \gamma$$ , but the η E (t) tended to deviate upward after a certain Hencky strain $$\varepsilon (t) = \dot \varepsilon _0 t$$ was attained. The birefringence Δn(t) was a function of both Hencky strain $$\varepsilon (t) = \dot \varepsilon _0 t$$ and strain rate $$\dot \varepsilon _0$$ in such a way that the stress-optical law holds with the stress-optical coefficient C(t) = Δn(t)/δ(t) being equal to the ones reported from shear flow experiments. Interestingly, however, for PS elongated at low strain rates the C(t) vs σ(t) relation exhibited a strong nonlinearity as soon as σ(t) reached steady state. This implies that the tensile stress reaches the steady state but the birefringence continues to increase in the low strain-rate elongation. For the PS melt elongated at high strain rates, on the other hand, C(t) was nearly a constant in the entire range observed. For LDPE with long-chain branchings, σ(t) exhibited tendency of strain-induced hardening after certain critical strain, but C(t) was nearly a constant in the entire range of σ(t) observed.

The confined flow of polyethylene melts past a cylinder in a planar channel

This paper is concerned with the comparison of the results of numerical simulation of confined flow past a cylinder to birefringence data for two polymer melts. The Phan-Thien and Tanner (PTT) constitutive equation and the Rivlin-Sawyers (RS) constitutive equation with the Papanastasiou, Scriven, and Macosko (PSM) damping function were each fit to the shear viscosity and extensional viscosity data of both linear low-density polyethylene (LLDPE) and low-density polyethylene (LDPE) melts to determine the values of the model parameters. Finite element calculations were carried out using the 4 × 4SUPG and 4 × 4SU methods for the PTT model and the method developed by Dupont, Marchal, and Crochet for the RS model. Isochromatic birefringence patterns calculated from the predicted stress field and the stress-optic law were compared to birefringence data. Good agreement was found between the birefringence data and the numerical predictions, except in the immediate vicinity of the cylinder surface. Large extensional stresses were observed and predicted along the centerline downstream of the cylinder for LDPE. This behavior was not observed or predicted for LLDPE. Stress fields obtained from birefringence measurements for LDPE flowing past three cylinders in a channel indicate an effect of deformation history on the flow behavior of LDPE. It is shown that the PTT model does not correctly predict the rheological behavior of LDPE as a function of shear history because the time scale of structural recovery is much longer than the relaxation time associated with viscoelasticity.

Rheology, self-diffusion, and microstructure of charged colloids under simple shear by massively parallel nonequilibrium Brownian dynamics

The simple shearing of a suspension of charge-stabilized, colloidal particles close to the melting line is investigated by massively parallel, nonequilibrium Brownian dynamics (NEBD) simulation. The suspension undergoes a discontinuous transition from a distorted fluid structure to an ordered ‘‘string’’ phase. Comparisons between simulations of 43 000, 4725 particles, and previous NEBD work on ≤500 particles proves that shear-induced ordering is not an artifact of the small system sizes. We also show that the shear-rate dependence of the rheological properties obtained from NEBD is different than those obtained from nonequilibrium molecular dynamics (NEMD), a consequence of the solvent damping not being present in NEMD. The validity of the Ree–Eyring model for viscosity and the stress-optic law for colloids are tested. Further, a type of generalized Stokes–Einstein relationship is discovered for systems under shear.

Birefringence of Rubber during Creep and Recovery

Measurements of creep and recovery of lightly cross-linked polybutadiene, carried out in both shear and compression geometries, reveal two deviations from the stress optical law: the birefringence increases during the constant stress creep, and the birefringence is not zero during the stress-free recovery. Most interestingly, the sign of the birefringence during recovery was found to be opposite to that during creep. This implies that the chain segments have a net orientation opposite to the macroscopic deformation.

Use of coupled birefringence and LDV studies of flow through a planar contraction to test constitutive equations for concentrated polymer solutions

Laser Doppler velocimetry and flow-induced birefringence are used to measure the rate of deformation and the principal components of the refractive index tensor in a 5% polyisobutylene (PIB) solution in tetradecane (C14) flowing along the centerplane of an abrupt 3.97:1 planar contraction. The stress optical law is used to interpret the birefringence data in terms of the normal stress difference, which is used to calculate a transient elongational viscosity defined along the centerplane. These measurements are compared directly to predictions of six multimode, differential constitutive models (Oldroyd-B, White–Metzner, Acierno et al., Giesekus, Bird–DeAguiar, and Phan-Thien–Tanner) that are fit to steady and small amplitude oscillatory shear flow data for the PIB/C14 solution. The fluid exhibits slight elongational thickening followed by apparent extensional thinning at higher elongation rates. We believe that this ‘‘thinning’’ behavior is due to the decreased residence time of the polymer molecules in the high-strain-rate region as the flow rate (and maximum elongation rate) is increased. The nonlinear constitutive equations, except for the White–Metzner model, are virtually indistinguishable in their description of the dynamical response of the fluid in this experiment; however, the Phan-Thien–Tanner model gives the best quantitative fit to the data. These results point to the need for experiments in which the fluid flowing along the centerline is subjected to a greater total elongational strain.

A Chain-Intrinsic Fluorescence Study of Orientation-Strain Behavior in Uniaxially Drawn Poly(ethylene terephthalate) Film

The chain-intrinsic fluorescence technique was used to investigate the development of orientation in the noncrystalline phase of PET film during drawing at 90 °C. It was confirmed that orientation develops faster at higher strain rate. In the low draw ratio (precrystallization) regime, the orientation-strain data at high strain rate can be fitted to the affine network model, but at low strain rate, where relaxation effects are substantial, this model is not applicable. We found considerable deviation from the linear stress-optical law : it appears that a stress threshold must be reached before significant orientation takes place and that the threshold value increases with strain rate. Crystallization onset seems to increase the rate of development of orientation at the lower strain rate, probably by providing additional junction points. At both high and low strain rates, however, there eventually occurs an abrupt decrease in the rate of development of orientation (and crystallinity) which coincides with a sharp increase in stress. By comparing our data to some results in the literature on uncrystallizable PEMT, we infer that the slowing of orientation development arises from chain slippage and that the increase in stress largely results from the reinforcing effect of crystallites and an increase in polymer viscosity arising from the interconnection of crystallites.

Measurements of equi-biaxial stress in adhered polyimide films by tilted beam polarized light microscopy

The equi-biaxial tension, developed during the drying process of the adhered polyimide films on a rigid substrate, was determined by measuring the out-of-plane birefringence of the dried films. Using a polarized light microscope and a tilting sample stage, the out-of-plane birefringence of a linear polyimide film that was dried during a given thermal process was determined to be (7.0 ± 0.5) × 10 −3. The equibiaxial tension was calculated by using the stress-optical law to be around 60 MPa, in agreement with that by the commonly used bending beam method. Due to the additional spatial resolution of stress distribution and the higher accuracy for measuring soft polymer coatings, the birefringence method described here is considered to be a useful alternative to the bending beam technique, which measures only the macroscopically averaged stress and its accuracy is generally limited by the huge modulus ratio between the rigid substrate and the polymer.