Volume 25, Issue 9, 01 September 1954
 RHEOLOGY SYMPOSIUM


Composition of Apparent Shearing Forces During Shear Degradation of Polymers
View Description Hide DescriptionAn investigation has been made of the degradation of Vistanex polyisobutenes having viscosity‐average molecular weights from about 500 000 to 2 500 000 by shearing in solution in cetane at polymer concentrations from 5 to 20 weight percent, at nominal rates of shear from 9000 to 66 000 sec^{−1}, and at temperatures from 20° to 80°C. The initial apparent shearing loads, observed at the beginning of the degradation process, vary in unexpected ways with variations in nominal rate of shear, temperature, concentration, and initial molecular weight of the polymer. In some cases the observed variations are opposite to those which would be expected on the basis of viscoelastic resistance to shearing. These results can be explained on the basis of the hypothesis that the observed apparent shearing forces include some contribution associated with the energy used up in breaking chemical bonds in the polymer molecules during the degradation process. If this hypothesis is correct, valid viscosity values cannot be obtained directly from observed apparent shearing forces in viscometry measurements under conditions where shear degradation occurs.

Relaxation Time Spectrum of Dough and the Influence of Temperature, Rest, and Water Content
View Description Hide DescriptionThe effect of temperature, rest period, and water content on stress relaxation in doughs held at constant extension has been studied with a relaxometer previously described. Temperature was varied from 13 to 35°C, rest period from 2 to 120 min, and water content from 74.4 percent to 86.0 percent dry basis. It is found that the shape of the relaxation curves on a log time plot does not change with temperature in the above range but that the curve is shifted laterally along the log time axis. This permits the construction of master curves valid in the temperature range and extends the time scale of observation. The long relaxation time end of the distribution of relaxation times is affected most by rest period, decreasing as rest period is increased, and the short time end increases as water content is decreased. Activation energies from 11 to 24 kcal per mole are calculated for relaxation from the shift of the curve on the log time axis. There is an indication that the activation energy depends on temperature and increases with rest period.
A mathematical framework, based on the general Maxwell model, is discussed and the distribution function on the log time axis is interpreted in terms of viscosity and modulus of the relaxing mechanisms. Special reference is made to the ``box distribution'' and a viscosity is calculated.

Constant Stress Elongation of Soft Polymers: Plasticizer Studies
View Description Hide DescriptionConstant stress measurements have been made on lightly cross linked GR‐S and polyisobutylene of 7 million molecular weight. The modulus values as a function of temperature, time, and concentration of mineral oil plasticizer have been treated according to the ``reduced variables'' principles.
Since both plasticized and unplasticized samples showed the expected equivalence of temperature and time, the modulus values could be shifted to produce master curves for each composition. In the case of GR‐S the concentration of plasticizer could be similarly reduced to yield a single curve for all the data.
The polyisobutylene results, however, showed quite clearly that the plasticizer had a unique effect not paralleled by temperature or time. This effect has been attributed to the pseudoequilibrium modulus which apparently results from chain entanglements. Since the pseudoequilibrium modulus should be proportional to the number of chain entanglements per gram of polymer, it should vary as the square of the volume fraction of polymer. It is shown that a correction of this type improves the shift of the curves for polyisobutylene.
Changes in activation energy with the addition of plasticizer are apparently within experimental error (±2 kcal) in all cases.

Some Rheological Properties of Concentrated Polyisobutylene Solutions
View Description Hide DescriptionThree types of measurements have been made on concentrated polyisobutylene solutions. Dynamic measurements have been extended to low frequencies and the dynamic viscosities and rigidities determined in the region close to that in which pure viscous flow predominates. A modified Stormer viscometer has been used to determine the apparent (non‐Newtonian) viscosities of the same solutions. Pressures developed (normal stress) at the inner cylinder have also been measured in a concentric cylinder apparatus with the inner cylinder rotating. It is noted that superposition procedures currently used for dynamic data are apparently applicable to the viscosity and pressure data with the rate of shear taking the place of frequency. Relations between these phenomena are discussed in the light of a theory proposed by one of the authors.

Elastoviscous Properties of Amorphous Polymers in the Transition Region. II. Stress‐Relaxation of Butadiene‐Styrene Copolymers
View Description Hide DescriptionExperimental data have been obtained on the stress‐relaxation modulus E_{r} (t) of several copolymers in their transition regions. The new data are expressed as master curves which can be fitted to the Gauss Error Integral form of reduced equation, as previously proposed. It appears possible to characterize the time and temperature‐dependence of E_{r} of amorphous polymers in general by means of the parameters of this equation:E _{1} (glassy‐state modulus), E _{2} (rubbery‐state modulus), T_{d} (distinctive temperature, related to second‐order transition temperature), K_{d} (characteristic relaxation time at T_{d} ), and h (parameter related to steepness of master curve). In the butadiene‐styrene series, the effect of composition on these parameters has been determined.

The Rheological Unit in Raw Elastomers
View Description Hide DescriptionReasons are adduced for suspecting that in a raw elastomer subjected to continuous shear there exist groups of molecules which are held together by entanglement or by thixotropic attachments and move and rotate essentially as elastic solid bodies. Such coherent molecular groups are called rheological units.
If an elastomer in continuous shear consists essentially of closely‐packed rheological units, such units, rolling on each other, will transfer a rubbersoluble dye across the shear planes as the ink rollers transfer ink in a printing press. This predicted phenomena has been observed with a Mooney viscometer fitted with a smooth rotor, the face of which is covered with a thin film of dried cement containing a dye in high concentration. From the velocity of the color front a rough value can be computed for the mean diameter of the rheological units. The diameters for a series of different elastomers lie in the range from 1 to 35 microns.

Further Dynamic Investigations on Polymers
View Description Hide DescriptionUsing the vibration tester of The Franklin Institute, a check of the proportionality between stress and strain in dynamic testing disclosed that a proportionality exists when heating is eliminated. A comparison of values of the dynamic test and steady‐state experiment in the same range of rates of shear and shearing stress showed that the behavior is basically different; whereas in the dynamic test the viscosity is independent of both amplitude and shearing stress, in the steady‐state test experiment a very high dependence exists. The curve of viscosityvs rate of shear (D) is very similar to the one of dynamic viscosityvs frequency (f). A frequency shift of the order of two is involved in comparing D and f. The application of Ferry's method of reduced variables to a polyisobutylene solution,liquid polyesters, and solid plastics showed its validity. A precision test on the polyester showed that the temperature shift, Tρ/T _{0ρ0}, is necessary to make experimental values at different temperatures fit. The reduced curve of G′ vs frequency for different plastics has an S shape with different slopes in a very wide range of frequency around 10^{24}.

Stress and Strain Discussion of Dr. Philippoff's Paper
View Description Hide Description 
Effect of Rheological Behavior on Thermal Stresses
View Description Hide DescriptionSince the conventional elasticanalysis of thermal stress problems coupled with limiting creep rates and time‐dependent fracture stresses as (inelastic) design criteria, results in design procedures for thermal stresses (in heat exchangers, nuclear reactors, flight structures at supersonic speeds, etc.) of considerable unreality, the effect of various types of rheological behavior (viscoelastic, plastic, work hardening) on the level of thermal stresses is analyzed under simplified assumptions, such as uniaxial stress and polar or cylindrical symmetry. The effect on the thermal stress intensity of the rheological behavior of the material is shown to be very significant, particularly with respect to stress relaxation and the development of residual stresses.

Flow of Melts in ``Crosshead''‐Slit Dies; Criteria for Die Design
View Description Hide Description``Crosshead''‐slit dies are widely used for the extrusion of polymer sheeting, flat film, and blown film. In this paper we analyze the laminar flow of liquids in such dies, deriving an equation relating pressure to position along the slit for materials that obey the ``power law.'' The equation yields an index of uniformity (UI), which is the ratio of the rate of extrusion at the far end to that at the feed end, given by ,where n=the exponent of the power law, L=the length of the die from feed end to far end, and α=a parameter depending on the other die dimensions and the exponent n, having the unit (in)^{−1}. An illustrated procedure is given for designing dies that will make film whose caliper is satisfactorily uniform over its whole width. We show that the UI for center‐fed dies is always greater than for end‐fed dies of the same length.

Rotational Plastometry Applied to Molten Polyethylene
View Description Hide DescriptionA concentric cylinder rotational plastometer which automatically plots the shear strain vs time curve for each of a number of shear stresses is described. The flow behavior of molten polyethylene resins, measured with this instrument from 130°C to 210°C, is discussed. It is shown that, at each shear stress, the shear strain vs time curve for these resins is describable in terms of a viscosity, an elastic shear strain and a retardation time. As the shear stress is raised both the viscosity and retardation time decrease, while the elastic shear strain increases. The shear stress dependency of the viscosity and elastic shear strain are shown to be related and in a manner which suggests that the viscosity change is the result of an entropy effect.

Internal Friction and Dynamic Modulus of Cold‐Worked Metals
View Description Hide DescriptionThe complexities of the internal friction and dynamic modulus of metals in the cold‐worked condition may be understood reasonably well by separation of the phenomena into the following three distinct effects: (1) the ``nonlinear'' effect, characterized by strong amplitude dependence, frequency independence, and its disappearance after severe deformation; (2) the Köster effect, which shows rapid recovery after deformation, at temperatures well below the recrystallization temperature; (3) the ``viscosity'' effect, characterized by strong frequency and temperature dependence. The origin of each of these effects is briefly discussed from the point of view of dislocation theory.

The Rheology of Suspensions
View Description Hide DescriptionThe rheological properties of concentrated suspensions of sized glass spheres in an equal‐density medium has been investigated with the rotational viscosimeter. It has been found that the variation of fluidity with rate of shear can be expressed satisfactorily by a form of an expression suggested by Reiner relating the fluidity to the shearing stress, the fluidity at rest, and the fluidity at infinite shearing stress. It is also shown that values of the apparent fluidity are influenced by the time of shearing, the aging of the suspensions, and temperature effect attributed to the work of shearing.
The viscosity of suspensions of sized spheres at concentrations as great as 55 volume percent solids may be satisfactorily represented by either the relation of Mooney or of Maron, Madow, and Krieger; the identity of their relations is also shown. The constants for the equation relating viscosity and concentration are shown to be in satisfactory agreement with those predicted by Mooney. Constants for the viscosity‐concentration equation are also given for several examples of experimental work selected from the literature; the form of equation proposed by Mooney relating viscosity and concentration is generally useful though the constants may not agree with the predicted values.
It has been found that the apparent fluidity of the glass bead suspensions decreases with an increase in particle diameter when the suspending fluid is zinc bromide in aqueous glycerol though the viscosity is independent of particle diameter when the suspending fluid is the nonaqueous mixture of ethylene tetrabromide and diethylene glycol.
The viscosity of a bimodal system of glass beads was shown to decrease very significantly as the bulk density was increased by packing progressively smaller spheres into the interstices of the larger spheres of the bimodal system, in qualitative agreement with a relation proposed by Mooney.
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 REGULAR CONTRIBUTED ORIGINAL RESEARCH


Analysis of Systems Involving Difference‐Differential Equations
View Description Hide DescriptionIt is known that many physical systems, both time and space dependent, are described by difference‐differential equations. Although the exact solutions to such equations are known to be unique, their determination usually involves a segmented type of solution that is very laborious to obtain. An approximate method of determining any number of roots is described. This approximate procedure allows forward gain and delay to be freely chosen and gives immediately the maximum stable value of forward gain. In many cases, especially where an optimum value of damping constant or forward gain is to be found, it appears that this approximate procedure is more easily applied than any previously described method.

Transient Response of a p‐n Junction
View Description Hide DescriptionThe authors investigate the transient response of a p‐n junction in a circuit where the applied voltage is suddenly switched from the forward to the reversed direction. The behavior is studied as a function of the circuit resistance. The solution of the appropriate diffusionequation is carried out for a time‐dependent boundary condition, resulting in an integrodifferential equation. Analytical and numerical results for the current and the junction voltage as a function of time are then obtained by suitable approximations for the ``recovery'' and reverse periods. The ``recovery'' or reversal time t_{f} is the time required for the junction voltage to become negative and is given by,where τ is the lifetime of holes, I_{f} is the forward current, and I_{r} the initial reverse current, which is limited by the external resistance.

Lower Modes of a Concentric Line Having a Helical Inner Conductor
View Description Hide DescriptionThe natural, undamped modes of propagation in a concentric line having a circular outer conductor and a tape‐helix inner conductor are investigated. The investigation is primarily concerned with a study of the propagation characteristics of the modes and, by making suitable approximations, closed expressions for the propagation factors of a few of the lower modes are obtained. Numerical results are then presented for a system having typical geometrical parameters, and generalizations of a qualitative nature are indicated. In the course of the work, the natural‐mode field expressions are derived.

Minimum Noise Figure of Traveling‐Wave Tubes with Uniform Helices
View Description Hide DescriptionBy using a new theorem concerning noise in electron streams together with work by D. A. Watkins, an expression for the minimum noise figure of a traveling‐wave tube with a uniform helix is deduced. For reasonable values of tube parameters this minimum lies around 6 db.

Dielectric Constant and Loss Measurements on Barium Titanate Single Crystals While Traversing the Hysteresis Loop
View Description Hide DescriptionA small amplitude high‐frequency voltage has been superposed on a large amplitude low‐frequency sweeping voltage to study the dielectric behavior of bariumtitanatesingle crystals while traversing the hysteresis loop. The output of a capacitance bridge (filtered to pass only the high‐frequency component) is displayed on an oscilloscope as a function of either the applied sweeping voltage, the polarization, or the current to the sample. As a result, the bridge can be balanced for any portion of the sweeping voltage cycle desired. The dielectric constant and loss obtained in this manner have been studied as a function of measuring frequency, of sweeping frequency, of sweeping voltage and of rate of switching polarization (current). The measuring frequency and the switching current are shown to be the significant variables. The frequency dependence of the dielectric constant and loss can be described by a relaxation type of spectrum with a relaxation time of 5.5 microseconds.

Solute Redistribution by Recrystallization
View Description Hide DescriptionThe time‐dependent diffusion equation is solved using transformation calculus to obtain the solute concentration in a solvent after a segregating liquid‐solid interface is passed through the material. The results are presented in graphical and tabular form for typical experimental conditions and compared with previously published findings.

Matrix Analysis of Linear Time‐Varying Circuits
View Description Hide DescriptionThis paper presents a general mathematical technique for studying the response and stability of linear time‐varying circuits that contain parameters whose magnitudes vary in a periodic manner with the time. The method presented reduces the solution of the circuit differential equation to that of computing powers of matrices. For purposes of illustration, the method is used to determine the response of an electric circuit consisting of a constant resistance and inductance in series with a periodically varying capacitance. The cases of square‐wave and saw‐tooth‐wave variations of the variable parameter are considered in detail.
