Anomalous diffraction model of linear optical dichroism in shear-thickening polymer solutions

The anomalous diffraction approximation (ADA) has bees recently applied to interpret measurements of the linear optical dichroism induced by shear in shear-thickening polymer solutions. A conceptual problem in this application is discussed, and a minor modification to the interpretation is proposed which is concordant with earlier magneto-optic results, but retains the correct use of the ADA.     

Brownian dynamics simulations of shear-thickening in dilute polymer solutions

A versatile model describing the shear thickening behaviour of dilute polymer solutions in high shear flows is presented. The polymer macromolecules are modelled as Hookean elastic dumbbells which deform affinely during flow. In addition, the dumbbells feel a retractive anisotropic hydrodynamic drag and an isotropic Brownian force. Furthermore, it is assumed that high shear rate increases the probability of molecules forming associations and this is described through expressions for the frequencies of association and dissociation, without explicitly accounting for finite extensibility, hydrodynamic interaction or excluded volume effects. Thus, a reversible kinetic process is incorporated into the model, which results in two diffusion equations for the associated and dissociated dumbbells. Numerical simulations predict shear thickening for specific range of parameters, which are physically meaningful and related to molecular characteristics of the polymer. A comparison against experimental data reported in the literature revealed very promising results, thus confirming the ability of this model to predict shear thickening under a wide range of conditions, for various polymer models.Nomenclature A A factor in the frequency of association - B Frequency of dissociation - B ₀ Reference frequency of dissociation - c Concentration of polymer solution - c _i Concentration of singlets (i = 1) and doublets (i = 2) in the solution - c ^* The overlap concentration - D _t Translation coefficient of molecule - F _i (Q) Spring force for a singlet (i = 1) and for a doublet (i = 2) - F Frequency of association - F ₀ Reference frequency of association - H _i Dumbbell spring constant for a singlet (i = 1) and for a doublet (i = 2) - k Boltzman's constant - k _H Huggins constant - MW Molecular weight - MW _c Critical molecular weight for formation of entanglements - n Number density of molecules in the polymer solution - n ₀ Number density of dumbbells at equilibrium - n _i Number density of singlets (i = 1) and doublets (i = 2) - Q Vector defining the size and orientation of a dumbbell - t Time - T Absolute temperature - x Degree of multimerization - W Interaction energy between the two components of a doublet Greek letters a Dimensionless anisotropy parameter - Shear rate - _i Friction coefficient of singlets (i = 1) and doublets (i = 2) - _i Intrinsic viscosity of singlets (i = 1) and doublets (i = 2) - _red Reduced viscosity of solution - _sp Specific viscosity - Viscosity of the polymer solution of concentration c - _s Viscosity of the solvent - (t) White noise - K ^T Velocity gradient tensor - _Hi Time constant of a singlet (i = 1) and a doublet (i = 2) - ₁ Length scale of singlets (standard deviation of singlet lengths at equilibrium) - ₂ Length scale of doublets - T _p Stress tensor - T _xy Shear Stress (xy element of T _p ) - T _pi Contributions to the stress tensor of singlets (i = 1) and doublets (i = 2) - ₀ Equilibrium configuration distribution function of Q - _i Configuration distribution function of singlets (i = 1) and doublets (i = 2)     

剪切增稠液体液舱侵彻实验

为研究剪切增稠液体(shear-thickening fluid, STF)液舱对弹体的防护性能,制备特定规格剪切增稠液体,并开展弹体侵彻剪切增稠液舱实验研究。实验中采用高速相机记录液舱侵彻过程中空泡的演化情况,并测试得到了弹体的剩余弹速以及前后靶板变形数据。实验结果显示,剪切增稠液体可有效抑制液舱侵彻过程中空泡的增长,从而降低液舱结构的损伤程度。结合空泡扩展理论模型,并考虑液体密度以及黏度变化对空泡增长的影响,验证了剪切增稠液体在高速冲击下产生的局部密度增大以及固化现象是抑制空泡扩展的主要原因。此外,剪切增稠液体对弹体速度的衰减作用明显,且相同初始弹速下,剪切增稠液体液舱前后靶板变形明显小于水体液舱。将剪切增稠液体填充于舰船液舱防护结构,可显著提高液舱结构的防护性能。     

Flow-induced mixing,demixing, and phase transitions in polymeric fluids

In polymer solutions or blends, flow can strongly influence the degree of mixing of the components. In a shearing flow, droplets in a dispersion can be broken down to sizes comparable to the dimensions of the polymer molecules themselves, thereby inducing molecular-scale mixing. Demixing can also occur when the two components of the mixture differ greatly in viscoelastic properties. Shear or extensional flow can induce polymer migration in nonhomogeneous flows or in flows with curved streamlines, and can render turbid solutions or blends that are otherwise transparent. Flow can also induce polymer gelation, and can induce ordering transitions in liquid crystals or block copolymers. Here, we review these phenomena, discuss proposed mechanisms, and assess the degree to which recent theories can account for the observations. Because the phenomena are complex, multiple experimental probes and theoretical methods are required to study them. Successful theories must incorporate polymer/polymer or polymer/solvent thermodynamics, critical phenomena, and phase transitions, as well as polymer theology and the kinetics of diffusion or crystallization. The experimental techniques used to study these phenomena are equally wide ranging, and include turbidity measurements, light, x-ray, and neutron scattering, fluorescence quenching, microscopy, and theology.     

A rheo-optical study of shear-thickening and structure formation in polymer solutions. Part I: Experimental

Simultaneous measurements of the optical and theological response of solutions of highly fractionated polystyrenes have been made, in-situ, to ascertain the connection between flow-induced structure formation and the phenomenon of shear-thickening. Transient and steady state viscosity, dichroism, birefringence and the associated orientation angles were measured in decalin and bromobenzene in the semi-dilute region using a couette device capable of shear rates up to 8,000 s^–1. A one-to-one correlation has been found between the occurrence of maxima in the dichroism and minima in the viscosity. While the size and shape of the shear-thickening structures could not be directly determined, results suggest they are intermediate in size between a cluster of entangled chains and a completely phase-separated liquid. For solutions exhibiting shear-thinning alone, no maximum in dichroism was observed, the signal instead showed a saturation behavior at high shear rates. Birefringence was found to be insensitive to the structure formation and attributable to that of the dissolved chains or entanglement regions. The kinetics of the structuring process leading to shear-thickening are instantaneous and completely reversible and there is a concentration window, above and below which only shear-thinning occurs.     

A rheo-optical study of shear-thickening and structure formation in polymer solutions. Part II: Light scattering analysis

Light scattering calculations based on Anomalous Diffraction Theory (AD), Rayleigh spheroids, and flexible macromolecules are used to propose a phenomenological explanation for the relationship between shear-thickening and structure formation in polymer solutions. Quantitative comparisons are made to experimental data for the rheo-optical behavior of fractionated polystyrene solutions presented in part I of this paper. Results from the ADA calculations suggest that the viscosity and dichroism behavior can be attributed to the production and growth of micron-size, optically isotropic structures during flow. The saturation dichroism behavior exhibited by the solutions which shear thin can be attributed to the formation of entanglement regions which achieve a fixed size and act as Rayleigh spheroids in their scattering behavior. The magnitude and shear rate dependence of the observed birefringence can be accounted for on the basis of the non-linear, flexible macromolecule model, implying that birefringence is governed by the polymer chains remaining in solution which do not take part in the structure formation. The latter result is consistent with the experimental observation that the birefringence dependence on shear rate is the same whether the solution exhibits shear thickening or shear thinning in its viscosity behavior.     

Formation of the shear-induced state in dilute cationic surfactant solutions

In cationic surfactant solutions a change of state occurs due to mechanical stresses. In the dilute regime of rodlike micelles the formation of a so-called Shear-Induced State (SIS) occurs above a critical shear rate. In this context dilute means that there is no sterical interaction between rodlike micelles, the solution is below the overlap concentration. Employing a mathematical model, it is shown that aggregation forces are weak compared to hydrodynamic forces. The mathematical formulation is based on a model of Israelachvili which describes the chemical potential of micelles. Hydrodynamic forces are calculated with a rigid-dumbbell model. SIS formation can be explained by the destruction of rodlike micelles.     

The relaxation of concentrated polymer solutions

The focus of this paper is on the viscoelastic properties of concentrated polymer solutions and polymer melts. Dynamic mechanical measurements were performed on various polystyrene/ethylbenzene solutions with polymer concentrations ranging from 40% up to 100% and temperatures from Tg+30°C up to 70°C (230°C for polymer melts). The basis polymers are two commerical grade polystyrenes (BASF) with M _W = 247 kg/mol and 374 kg/mol, respectively. To avoid solvent loss due to evaporating during the measurements, a special sealing technique was used.A phenomenological model which describes quantitatively the relaxation spectrum of concentrated polymer solutions from the flow regime up to the glass transition regime is developed. The relaxation data of the respective polymer melt and the glass transition temperature of the solution are the only input parameters needed. The temperature dependence is described by a universal, concentration invariant WLF-equation. The relaxation spectra are divided into two parts accounting for the entanglement and the segmental relaxation modes, respectively. The relaxation strength related to the flow and entanglement regime scale with c ^2.3, whereas the segmental relaxation strength does not alter with concentration. All relaxation times change with concentration proportional to c ^3.5. Flow curves can be calculated from these relaxation spectra and thus, our results are useful for engineering applications.Roman Symbols a _T Time temperature superposition shift - factor - a _c Time concentration superposition - shift factor in the flow regime - a _c Time concentration superposition - shift factor in the glassy regime - b _T Modulus temperature superposition - shift factor - b _c Modulus concentration shift factor - in the flow regime - b _c Modulus concentration shift factor - in the glassy regime - B Virial coefficients - c Polymer mass fraction kg/kg - c ₁ WLF-parameter - c2 WLF-parameter K - g Relaxation strength of a relaxation Pa mode - G(t) Relaxation modulus Pa - G Storage modulus Pa - G Loss modulus Pa - GN Plateau modulus of linear flexible Pa polymers - (x) Delta function: (0) = 1, - (x<>0)=0 - h() Damping function - H() Relaxation spectrum Pa - J ₀ ^N Recoverable compliance Pa^–1 - m Mass kg - M _c Critical molecular weight kg/mol - M _e Entanglement molecular weight kg/mol - M _w Weight average molecular weight kg/mol - M Number of datapoints - n Scaling exponent - N Number of discrete relaxation modes - T Temperature °C - T _g Glass transition temperature °C - V Volume 1 Greek Symbols Scaling exponent - _f Thermal expansion coefficient K^–1 - Scaling exponent - Shear deformation - Shear rate s^t–1 - Relaxation time s - _c Characteristic relaxation time of thes Cross model - _e Entanglement relaxation time s - Viscosity Pa s - ₀ Zero shear viscosity Pa s - ₀ First normal stress coefficientPa s2 - Segmental friction coefficient - Frequency rad/s Indices f Flow and entanglement regime - g Glass transition regime - i Count parameter - p Polymer - ref Reference state - s Solvent Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

Molecular orientation in non-Newtonian flow of dilute polymer solutions around spheres

A novel approach is presented to study the benchmark problem of flow around spheres in model dilute solutions of monodisperse samples of atactic polystyrene in di-octyl phthalate. Spheres are held stationary on flexible cantilevers of known spring-constant, k, while the polymer solutions are pumped past at controlled flow rates, allowing access to a wide range of Deborah number. In this way the non-Newtonian forces experienced by the spheres can be measured as a function of Deborah number, while detailed observations and measurements of birefringence are made, enabling assessment of macromolecular strain and orientation. In addition the flow field around a sphere has been measured in an a-PS solution. Experiments have been performed on a single sphere and on two spheres axially aligned in the direction of flow. The extensional flow around the downstream stagnation point of the single sphere is found to play a pivotal role in the development of molecular strain and stress, resulting in flow modification and subsequent non-Newtonian behaviour. The flow birefringence in the wake is found to modify severely the flow around a second, downstream, sphere, affecting the non-Newtonian forces encountered by the second sphere. This provides an explanation for the time interval dependent terminal velocity often observed when two spheres follow the same path through viscoelastic liquids.     

An AFVS model for polymer solutions: New scaling relationship

The altered free volume state (AFVS) approach developed by Kulkarni and Mashelkar has been extended to develop a model for the viscosity of polymer solutions. The effect of polymer concentration, molecular weight and electrolyte concentration is shown to be satisfactorily explained in the framework of AFVS model. The regimes of viscosity behaviour have been defined for the systems investigated and the transitions involved identified. The approach leads to the identification of a surprisingly simple and unique scaling parameterΦ _P, which enables excellent concentration-molecular weight superposition.     

Effect of a controlled pre-deformation history on extensional viscosity of dilute polymer solutions

We use a modified filament stretching rheometer to quantify the influence of a known controlled pre-shear history on the transient extensional viscosity of a dilute polymer solution. Two different types of pre-deformation are explored; both influence the subsequent stretching significantly, albeit in opposite ways. Small-amplitude oscillatory straining parallel to the direction of stretching enhances strain hardening and accelerates the tensile stress growth toward the steady-state value. Conversely, steady torsional shearing orthogonal to the direction of stretching retards strain hardening and results in a delayed approach to steady-state elongational flow. In both cases, the final steady-state extensional viscosity is the same as that observed with no pre-shearing. Calculations using a finitely extensible nonlinear elastic Peterlin dumbbell model qualitatively capture the trends observed in experiments, enabling interpretation of these observations in terms of the degree of polymer chain stretching imposed by the flow before extensional stretching.     

The shear viscosity dependence on concentration,molecular weight,and shear rate of polystyrene solutions

The solution viscosity of narrow molecular weight distribution polystyrene samples dissolved in toluene and trans-decalin was investigated. The effect of polymer concentration, molecular weight and shear rate on viscosity was determined. The molecular weights lay between 5 10⁴ and 24 10⁶ and the concentrations covered a range of values below and above the critical valuec ^*, at which the macromolecular coils begin to overlap. Flow curves were generated for the solutions studied by plotting log versus log . Different molecular weights were found to have the same viscosity in the non-Newtonian region of the flow curves and follow a straight line with a slope of – 0.83. A plot of log ₀ versus logM _w for 3 wt-% polystyrene in toluene showed a slope of approximately 3.4 in the high molecular weight regime. Increasing the shear rate resulted in a viscosity that was independent of molecular weight. The sloped (log)/d (logM _w ) was found to be zero for molecular weights at which the corresponding viscosities lay on the straight line in the power-law region.On the basis of a relation between _sp and the dimensionless productc · [], simple three-term equations were developed for polystyrene in toluene andt-decalin to correlate the zero-shear viscosity with the concentration and molecular weight. These are valid over a wide concentration range, but they are restricted to molar masses greater than approximately 20000. In the limit of high molecular weights the exponent ofM _w in the dominant term in the equations for both solvents is close to the value 3.4. That is, the correlation between _sp andc · [] results in a sloped(log _sp)/d(logc · []) of approximately 3.4/a at high values ofc · [] wherea is the Mark-Houwink constant. This slope of 3.4/a is also the power ofc in the plot of ₀ versusc at high concentrations. a Mark-Houwink constant - B ₁,B ₂,B _n constants - c concentration (g · cm^–3) - c ^* critical concentration (g · cm^–3) - K, K constants - K _H Huggins constant - M molecular weight - M _c critical molecular weight - M _n number-average molecular weight - M _w weight-average molecular weight - n sloped(log _sp)/d (logc · []) at highc · [] - PS polystyrene - T temperature (K) - shear rate (s^–1) - critical shear rate (s^–1) - viscosity (Pa · s) - ₀ zero-shear viscosity (Pa · s) - _s solvent viscosity (Pa · s) - _sp specific viscosity - [] intrinsic viscosity (cm³ · g^–1) - dynamic viscosity (Pa · s) - | ^*| complex dynamic viscosity (Pa · s) - angular frequency (rad/s) - density of polymer solution (g · cm^–3) - ₁₂ shear stress (Pa) Dedicated to Prof. Dr. J. Schurz on the occasion of his 60^th birthday.Excerpt from the dissertation of Reinhard Kniewske: Bedeutung der molekularen Parameter von Polymeren auf die viskoelastischen Eigenschaften in wäßrigen und nichtwäßrigen Medien, Technische Universität Braunschweig 1983.     

Velocity measurement of start-up flow of polymeric solutions in a planar contraction channel

The start-up flow of polymeric solutions in an abrupt contraction channel with a rectangular cross-section was experimentally studied. Aqueous solutions with 0.2 wt% and 1.0 wt% of polyacrylamide were used as test fluids. Temporal changes in velocity were measured with a laser Doppler velocimeter. The velocity overshot just after the onset of the flow. The changes in velocity caused by the rearrangement of the velocity distribution were observed. A three-dimensional flow structure and the development of a vortex region were found near the entrance to the contraction. A decrease in the axial velocity just upstream from the contraction was observed. The velocity profile is related to the three-dimensional nature of the flow and the elongational rheological properties of the test fluid. The experimental results indicate that three-dimensional analysis of viscoelastic flows is required especially for the flow in rectangular channels. Received: 6 July 1998 Accepted: 1 December 1998     

Simulation of polymeric flows in the injection moulding process

Recent progress in the simulation of polymeric flows of two key problems in the injection moulding process, carried out by a team at Cornell University, is briefly described. For the filling of cooled thin cavities, the fluid is characterized by a power-law viscosity with exponential temperature dependence, and interaction between the transient thermal boundary-layer and the core flow in a domain with moving boundary is essential. The earlier procedure of Hieber and Shen is modified in two aspects: a boundary-integral formulation replaces the finite-element treatment of the pressure, and an ‘energy integral’ approach is used for the transient temperature. The second problem is the steady visco-elastic flow in the juncture region where sudden changes of the geometry and large strain rates occur. The constitutive equation is postulated according to the Leonov model. The main features in the numerical implementation are: integration along a streamline to determine the elastic deformation tensors for a given velocity field, and finite-element treatment (in time-dependent form) of the pressure and fields for given stresses. In an example where the contraction ratio is 7:1, results for nominal Deborah number exceeding 100 show no numerical instability. (However, for this problem, the true Weissenberg number, i.e. the ratio of local first-normal-stress difference to shear stress turns out to be generally O(10).) The predictions also correlate very well with experimental birefringence measurements.     

Gelation of polymeric nanoparticles

We review how, starting from polymeric nanoparticles, to generate clusters of fractal morphology and to expand the entire space and interconnect to form gels, through either Brownian motion or intense shear-induced aggregation. In the case of Brownian motion-induced gelation, specific techniques developed to obtain uniform structure of gels under both reaction-limited and diffusion-limited cluster aggregation conditions have been described. In the case of intense shear-induced gelation as a newly developed technique, our focus is on its principle, theoretical development and advantages with respect to Brownian motion-induced gelation in practical applications. We consider gelation of both rigid and soft particles. As a physical process, the bonding between the particles within gels is owed to van der Waals attractions, thus being easily broken. However, in the case of soft particles that can coalesce upon contact, the coalescence can allow the particles to stick together forming permanent gels. In this case, the gel structure can be controlled by controlling the degree of coalescence. Techniques used to control the degree of coalescence have also been described.     

Shear induced phase transitions in highly dilute aqueous detergent solutions

Aqueous solutions of some cationic detergents show rheopectic behaviour at very low concentrations. Rheological measurements and the electrical anisotropy of the streaming solutions indicate that the rheopectic behaviour of the solutions is due to a shear induced phase transition.     

Viscosity of suspensions in polymeric solutions

Viscosities of suspended particles in polymeric solutions depend upon dissolved polymer concentration, volume fraction of particles and shear rate. In this analysis of viscosity data, relative viscosity was defined as the ratio of suspension viscosity to solution viscosity at the same shear stress rather than shear rate. These relative viscosities reached asymptotic values at high shear stress for all concentrations of dissolved polymer and for all particle loadings. At a given particle loading, the asymptotic values of relative viscosity were nearly independent of the concentration of dissolved polymer. Realtive viscosities were correlated with volume fraction by the one-constant equation of Maron and Pierce.     

The rheology of suspensions of vinylon fibers in polymer liquids. II. Suspensions in polymer solutions

Summary The rheological properties of vinylon fiber suspensions in polymer solutions were studied in steady shear flow. Shear viscosity, first normal-stress difference, yield stress, relative viscosity, and other properties were discussed. Three kinds of flexible vinylon fibers of uniform length and three kinds of polymer solutions as mediums which exhibited remarkable non-Newtonian behaviors were employed. The shear viscosity and relative viscosity ( _r ) increased with the fiber content and the aspect ratio, and depended upon the shear rate. Shear rate dependence of _r was found only in the low shear rate region. This result was different from that of vinylon fiber suspensions in Newtonian fluids. The first normal-stress difference increased at first slightly with increasing fiber content but rather decreased and showed lower values for high content suspensions than that of the medium. A yield stress could be determined by using a modified equation of Casson type. The flow properties of the fiber suspensions depended on the viscosity of the medium in the suspensions under consideration.With 16 figures and 1 table     

Phenomena of high polymer solutions while flowing through capillaries at high rates of shear

Generally solutions of high polymers show a shear-rate dependent flow behaviour and so the properties of these fluids have to be measured under conditions of shear corresponding to the practical service. Capillary viscometry is suitable for achieving high rates of shear but relaxation phenomena can effect the results, which is proved experimentally. The flow behaviour of a lubricant blended with a high polymer additive is measured and a graphical representation of the dependence of flow behaviour on temperature and rate of shear is recommended.