The hydrogel of Wyoming bentonite shows characteristic slip patterns under critical stress σ₀, i.e. the shear strength for slip fracture. In the previous paper, the present authors reported that the shear strength of the bentonite gels are related to the shear modulus, G, as follows:
σ₀=γ₀G, (1)
where γ₀ is a proportional constant bearing nearly equal value to the critical strain γ_c obtained experimentally i.e. 1/10∼1/30, and G varies depending upon the concentration. Those experiments were carried out at the rate of strain ranging from 10^-3 to 10^-4sec.^-1
In this report, the dependence of σ₀, γ_c and G on the strain rate was examined. Fig. 1 gives the stress-time curves obtained at various rate of compression (Rsec^-1). The critical shear stress or the shear strength σ₀ increases with the increasing strain rate, especially above 10^-3sec^-1 (Fig. 2). At the lower rates than 5×10^-5sec^-1, no slip fracture was observed but the specimens showed viscous flow. Fig. 3 shows the dependence of G on the strain rate R, where G takes almost constant value in the range of strain rate, 10^-3∼10^-1sec^-1, but decreases apparently at lower rates of strain than 10^-3sec.^-1 The critical strain γ_c obtained experimentally remains almost constant at the strain rate between 10^-3∼10^-1sec^-1 as shown in Fig. 4, when the linear relation (1) holds aproximately between σ₀ and G. Dependence of σ₀/G on the rate of strain is discussed further.

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Flow curves of disperse systems were determined in the wide range of shearing rates as illustrated in Figs. 2, and 3. Silica gel-water, clay-water, and zinc oxide-water systems containing its own deflocculating agents respectively, were shown as examples for the suspension, and benzene-water and nitrobenzene systems having stabilizing additives for the emulsion.
All the flow curves were recognized as the so-called Ostwald's flow pattern. The analysis was performed using Eyring's equation (1), Cross's equation (4) and Casson's equation (5), and the following conclusions were derived;
(1) Eyring's equation is valid in the whole range only in the case of silica gel-water system having 2% solid content.
(2) The n-value in the Cross's theory is recognized to show the width of the range of shear rate within which structual break-down is completed and α-value indicates the initiation of break-down of agglomelating structure.
(3) Casson's plots are employed to decide the existence of yield value, but no yielding characters are proved in all cases as illustrated in Figs. 6∼9.

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The analytical method for obtaining the relation between the peeling rate and the peeling strength has recently been reported by using the numerical solution of peeling stress distribution. It has been experimentally known that the relation changes with temperature at which it was peeled. This paper presents the data for temperature dependence of peeling strength of polyisobutylene (PIB) at temperature far above its glass transition.
The analysis of the relation between the peeling rate and the peeling strength showed that our theoretical estimation predicts the data fairy well at temperatures lower than 20°C, but the coincidence is not so good at temperatures higher than 50°C. From the analysis of the relation, three essential parameters are derived. They are P₀, the constant peeling strength at lower peeling rate, P_∞, the constant peeling strength at higher peeling rate and τ, the relaxation time of the three element viscoelastic model which is assumed to show the mechanical property of adhesive interlayer.
The temperature dependence of τ are similar to that of a_T for PIB and they agree with WLF equation except for PIB-PTFE (polytetrafluoroethylene) system. But the value of the constants differs from the value of their equation. The temperature dependence of τ for PIB-PVC (polyvinyl chloride) system shows an abrupt change at a temperature between 70°C and 90°C. The temperature dependence of P₀ and P_∞ also changes abruptly at the corresponding temperature. It may be considered that the property of adhesive interlayer changes at the temperature and that the change is accounted for by the glass transition of PVC. The temperature dependence of τ for PIB-PTFE system also seems to reflect the rheological properties of PTFE.
The mechanical property of adhesive interlayer may be determined by the mechanical properties of both the materials forming the interlayer.

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The coefficient of rolling friction for a hard sphere and a cylinder over a polymeric material are derived as a function of mechanical loss. A few assumptions were made to clarify the quantitative relation between rolling friction and dynamic modulus of polymeric material. Viscoelastic properties of polymeric material are represented by the generalized Voigt model. Contact deformation of base material owing to the rolling object is assumed to be a periodic phenomenon. From this assumption, the strain and stress for surface deformation is expressed by Fourier series and the stress is also obtained as a function of the dynamic modulus. Spectrum of the relaxation time of polymeric materials is assumed to be moderate and this leads to the correspondence between the rolling velocity and the angular frequency of the dynamic modulus.
The coefficient of rolling friction for a rigid sphere is obtained as follows, λ_s=β_s(W/E₁)^1/4r^-3/4, where W, r, and E₁ are the load, the radius of the sphere, and the real component of complex the dynamic modulus, respectively. β_s is a function of mechanical loss. For a small range of mechanical loss, β_s is linearly proportional to mechanical loss. λ_s is entirely proportional to mechanical loss. Temperature dependence of λ_s is explained by that of mechanical loss and modulus E₁. The rolling velocity corresponds to the angular frequency of dynamic modulus and correspondence between them is obtained as follows v=0.7(Wr/E₁)^1/4ω. From these relations, the coefficient of rolling friction for any temperature or any rolling velocity can be calculated from the dynamic data, or the dynamic data can be presumed from the rolling friction measurement. Consistency of the rolling frictional measurement for several polymers and theoretical results is quantitatively very good.
The coefficient of sliding friction for a hard sphere over a polymeric materials were measured. Temperature dependence of sliding friction of NBR resembles that of mechanical loss. Above glass temperature, however, it deviates from the tendency of mechanical loss curve. This may be due to the ploughing effect.

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To investigate the tensile fracture mechanism of rubber vulcanizates, in particular to determine the relationship between the tensile properties and polymer structure, styrene-butadiene copolymers (SBR) with various styrene contents were studied in this paper. The following conclusions were obtained from the results of tensile measurements and physical properties.
(1) Failure for the SBR vulcanizates (carbon black filled), at the temperature region over the necking phenomena, occurred mainly at the site of the main chain, not between rubber molecule and carbon black.
(2) If the number average molecular weight, network chain density and the type of combined sulfur are considered nearly the same, the same molecular motional region (which was discussed from the view of viscoelastic study and nuclear magnetic resonance study in this paper, but it can be mentioned as the glass transition temperature in general) was shifted for the higher temperature side with the increase of styrene content in SBR, and the characteristic temperature for the failure properties (the inflection point for the tensile strength-temperature curve and the maximum point for the ultimate elongation-temperature dependence) was also shifted in the same quantity.
(3) Shifting the tensile data in the same quantity as the transition temperature, only one master curve was obtained for the tensile strength, even if the styrene contents were changed. For the ultimate elongation, however, the master curves were not the same for the various contents of styrene, but the value of the ultimate elongation after shifting decreased with the incresae of styrene parts in polymer. This result was successfully interpreted by the hypothesis that the elongation of polymer molecule depends on the carbon atom in butadiene monomer unit, not on the carbon atom in styrene unit.

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Following the previous paper¹⁾ on the tensile properties of the noncrystallized amorphous rubber, the properties of natural rubber (NR) crystallized in a high elongated state are reported in the present studies. The gum vulcanizates of various crosslink densities and the filled vulcanizates of various carbon black contents were prepared. The tensile properties of these samples were studied, with particular reference to their ultimate elongation and their comparative studies were made with the molecular chain mobility and the polymer structure. The following conclusions were obtained.
(1) The crystallized rubber showed “the large” ultimate elongation up to about 100°C, while noncrystallized rubber showed it no more than about -20°C. The abrupt increase of ultimate elongation in the transition region is consistent with the previous finding.
(2) The large ultimate elongation at high temperature is interpreted as the result of increase in strength, based on the crystallization on stretching, and of increased mobility of molecular chains. The ultimate elongation and the temperature of maximum ultimate elongation were correlated with the network chain length. That is, while the former is reciprocally proportional to the square-root of the crosslink density, the latter is related to the crosslink density inverse.
(3) The influence of the apparent network chain density on the tensile properties of the filled vulcanizates, is apparently opposite to the case of the gum vulcanizates. This tendency is considered to be due to the fact that the rubber molecules constitute the heterogeneous dense structure on the filler surface. In the region far from the filler particles where the molecules take part in the crystallization on stretching, the molecular chain length between the crosslinks is larger as the filler content increases. Thus the temperature of maximum ultimate elongation increases. On the other hand, the ultimate elongation is assumed to be determined by the apparent network chain density.

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Time-concentration dependence of ultimate properties for polyvinyl chloride-dioctyl phthalate (PVC-DOP) sheets in the rubbery and glassy regions were examined simultaneously with time-temperature dependence at various volume fractions of PVC (v₂) ranging from 0.191 to 0.699.
By the time-concentration superposition, the data of tensile strength σ_b and ultimate strain α_b composed respective master curves on a plot of log σ_bρ_s/ρ vs. log t_b and log (α_b-1) vs. logt_b, where ρ is weight of PVC per unit volume of the sheet, ρ_s is a ρ arbitrarily selected as ref6rence and t_b is the breaking time observed by the test at a constant strain rate. The shift distance seemed to be equal to those obtained from time-concentration superposition of the tensile creep compliance ρD/ρ_s. The compliance D₂ was defined by the relation, D₂=(α-1)/ασ(1-λ₀ρ), based on a molecular theory of rubberlike elasticity by Sato, where σ, α denote the tension and the relative extension and λ₀ is average degree of contraction of network chains at the natural state. The observed compliance for the specimens of PVC coincided with the above compliance D₂ up to very high extension. For smaller extensions α, the relation, D₁=(α-1)/ασ, was satisfactory.
The experimental shift distances were larger than those given by the WLF equation, and excessive parts of the distances seemed to be represented by the Arrhenius equation.
A failure envelope was also constructed from these ultimate properties data. The data obtained for the specimens at high v₂, however, did not form a single master curve but gave a domain similar to that for natural rubber in time-temperature regions where chrystallization proceeded.

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Time and temperature dependence of ultimate properties in fluid fracture is reported for PVC-DOP solution (volume fraction of PVC, v₂=0.04) and liquid polybutadiene (PBD). The method used was such that small quantity of the sample was injected into air from a narrow nozzle and let it fall in drops, namely the sample overloaded by the drop weight. A cylindrical part of the pendant from the nozzle was elongated and took time to break. The time was defined as breaking time t_b. The drop weight W corresponding to load at break and associated t_b were measured with various drop sizes at different temperatures. Six temperatures were selected in the range from 30°C to 80°C for PVC-DOP solution and from 11°C to 35°C for liquid PBD. In elongation processes, a change of drop sizes was very small and almost negligible. Weight vs. time curves at different temperatures can be superposed into an almost unified master curve by the horizontal shift. It seems that the shift factor for the ultimate properties follows the prediction of the Arrhenius equation.
Furthermore, a relation between the shift factors a_T for both fluid fracture of PVC solution and creep deformation of PVC-DOP sheets was examined. The creep data were obtained in the temperature range from 2.5°C to 29.5°C for PVC-DOP sheets having different volume fractions of PVC ranging from 0.191 to 0.699.
It is found that the shift factor a_T for the fluid fracture is almost equal to that predicted from the values of a_T for creep deformation. Thus the values of log a_T are given by the Arrhenius equation above 30°C, and close to the WLF equation below 30°C.

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In connection with the evaluation of polymer stability and chemical antioxidants it is a very urgent work to make clear the degradation mechanism of crosslinked polymers.
There have long been two extremely different opinions about the mechanism of scission of network chain polymers, especially about natural rubber vulcanizates. One is that only random chain scission of this vulcanizates occurs in the oxidation reaction from the experiments and theory by A. V. Tobolsky. Another is that only their crosslink scission occurs in the oxidation reaction from the experiments and theory by Berry and Watson.
The results of recent studies of crosslinked polymers including this vulcanizates show that both the scissions may occur at the some time in many cases.
So the authors have made an attempt to derive the general equations for both the scissions for the purpose of finding out the quantitative amounts of scission chains using the experimental results and judging the relative estimation of the scissions.
We have generalized the case where one cross-linkage is cut in the minimum model of perfect chain network, and the following equation has been derived for the vulcanizates in which both the scissions occur.
N_t/N₀=f_t/f₀={1-4kx/k+2x·Q(t)/M₀}·e^{-2x2/k+2x·Q(t)/M₀}+2kx/k+2x·Q(t)/M₀·e^{-4x2/k+2x·Q(t)/M₀}
(1)
The Eq. (1) in both the cases of k→0 and k→∞, conforms to that proposed by Tobolsky. Assuming the more practically probable mechanism in the scission of cross-linked polymer, the following Eq. (2) obtained by modification of Eq. (1) is derived.
N_t/N₀=f_t/f₀={1-8kx/k+2x·Q(t)/M₀}·e^{-2x2/k+2x·Q(t)/M₀}+6kx/k+2x·Q(t)/M₀·e^{-4x2/k+2x·Q(t)/M₀}-2kx/k+2x·Q(t)/M₀·e^{-8x2/k+2x·Q(t)/M₀}
(2)
The theoretical relation between Q(t)/M₀ and N_t/N₀ is shown and discussed.

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Four kinds of natural rubber were used for specimens, the kinds A, B and C were cured with sulphur and the kind D was cured with peroxide. The kinds A, B and C had different initial densities of crosslinking.
The actual density, N₀ for the four samples, was obtained by using the equation of f(0)=N₀ RT (α-α^-2) and by the swelling method using the equation of Flory-Rhener.
According to the results obtained already, both the scissions of main chains and crosslink sites occurred at the same time for A, B and C, but the only main-chain scission occurred for the sample D.
The Eq. (1) was obtained for the sample D because k=0 in the Eq. (3) of
[f_t/f₀={1-2k/M₀·q_m(t)}·e^{-x/M₀·q_m(t)}+k/M₀·q_m(t)·e^{-2x/M₀·q_m(t)}]
(f_t/f₀)D=e^{-xD/M₀·q_m(t)} (1)
or
q_m(t)=-N_{0, D}ln(f_t/f₀)_D (2)
As the Eq. (3) of [f_t/f₀={1-2k/M₀·q_m(t)}·e^{-x/M₀·q_m(t)}+k/M₀·q_m(t)·e^{-2x/M₀·q_m(t)}] is applicable to the samples of A, B and C, and q_m (t) is equal to the four samples under the same condition, the Eq. (4) is established by substituting the Eq. (2) into the Eq. (3).
-N_{0, D}ln(f_t/f₀)_D=(f_t/f₀)_A-(f_t/f₀)_D^{N_{0, D}/N_{0, A}}/(f_t/f₀)_D^{N_{0, D}/N_{0, A}}-2·M₀/k_A·1/(f_t/f₀)^{N_{0, D}/N_{0, A}} (4)
Using the experimental data of (f_t/f₀)_A, (f_t/f₀)_D and the known value of N_{0, A}, N_{0, D} and M₀, k_A is calculated from the Eq. (4).
Similarly k_B, k_C and k_D are obtained.

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In many of the typical thermosetting acrylic polymers, functional groups are incorporated as comonomers so that crosslinking can occur. In this study N-butoxymethyl amide was used as comonomer to introduce functional groups and melamine resin was added so that crosslinking can occur through self-condensation of the base acrylic polymer as well as through reaction with melamine.
Chemical stress relaxation of swollen films in boiling water was determined. In continuous relaxation, a rapid decay of stress occurred at the early stage which probably is due to the loss of unreacted melamine. Subsequent decay of stress was found to follow the following function consisting of three exponential terms:
f(t)/f(0)=Aexp(-k₁t)+Bexp(-k₂t)+Cexp(-k₃t)
With intermittent relaxation, when the degree of crosslinking is low, relative stress f(t)/f(0) was found to decrease at first and then to increase sharply at around 60∼100min. This increase indicates the crosslinking caused either by the unreacted melamine resin or the recombination of the scissions that had occurred earlier.

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The decay of crosslinked epoxy resin acrylic polymer films under stress was studied. Stress relaxation in hot water was found to fit well with a function consisting of three exponential decay terms. Polymer systems with higher amount of epoxy resin gave lower rate of decay, or a higher proportion of the term of longer relaxation time, indicating the difference in the type of crosslinking. It is considered that different types of crosslinking give relaxations of different decay rates (k₁, k₂ and k₃), and the fractions (A, B and C) of the three decay terms are dependent on the ratio at which those different crosslinks exist.
Stress decay in boiling water proceeds about twice as fast as in air at the same temperature. This is considered to be due to some scissions of bonds caused by water in addition to thermal degradation that occur both in air and in water. The rate of stress decay increased when epoxy resin of higher molecular weight was used.
The activation energies of stress decay term were estimated. The values are 30.9 and 6kcal/mol for polymer systems of zero (k₁) and 10% (k₂) epoxy resin, respectively.

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In order to clarify the mechanism underlying the deformation of crystalline polymers, the change in their morphology with uniaxial elongation has been studied by electronmicroscopy for the polyethylene films having the so-called“a-axis orientation”and the b-axis orientation, both of which can be assumed to have more simplified textures than those of isotropic spherulitic films.
A commercial blown film was used as the sample of the so-called“a-axis orientation”and films giving the b-axis orientation were prepared by moving them through the gap between the two metal plates having a thermal gradient. In the former case, the stress-strain curve depends considerably on the drawn direction as against the machine direction of the blowing, whereas such an anisotropy has not been observed in the latter case. This anisotropy may be attributable to the firm interconnection between lamellae, which probably resulted from the screw dislocation during the crystallization, in the case of the blown films.
The yield stress was not various among the samples studied, suggesting that the intra-lamellar (inter-ribbon) slip governs the large deformation. The electronmicroscopy revealed that only small unfolding occurred, and that locally, in all the cases studied.

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A series of studies were made concerning the rheological behavior of the two polymer components with respect to their mechanical mixture and to the degree of mixing. And an investigation was made, as one of the series, of well annealed polypropylene and polyethylene, mechanically mixed, with respect to their behavior of relaxation from tensile stress, as a function of the fraction of polypropylene component.
From a dilatometoric experiment made by using an automatic recording apparatus, it was shown that the mixture had two melting regions corresponding to the original melting temperature of each component; this suggests that the mechanical mixture retains two characteristic phases of each component in its crystallized texture.
The parameter n representing the degree of mixing was evaluated from the stress relaxation data. Referring to the n value, it was concluded that the polypropylene-polyethylene systems are in a state of extremely poor mixture.

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This report covers the study of the rheological properties of polymers by stress relaxation method, especially on the effect of molecular weight and molecular weight distribution on relaxation spectra by Procedure X. Monodisperse polystyrenes and blends of them were used for the experiment.
The following results were obtained:
(1) The box portion of relaxation spectrum gives information on the molecular weight distribution of the polymer. Stress relaxation technique has been found to be convenient to the determination of molecular weight distribution.
(2) In case of blends of monodisperse polystyrenes of molecular weights being larger than the critical molecular weight, M_C, the maximum relaxation time, τ_m, is proportional to 3.4 powers of the mean values of the weight average molecular weight of these blends, M_w, as previously reported for monodisperse polystyrene. At the same time, however, τ_m has been found proportional to 2.0 powers of the average values of number average molecular weight, M_n, even when their molecular weights are lower than M_C. This result is one of the important conclusion of this work.
(3) The polymer with a molecular weight lower than M_C acts like a plasticizer, so the wedge portion as well as the box portion in the relaxation spectrum shifts to the shorter τ region.

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High polymers have generally larger internal loss than metals. Therefore we can damp undesirable vibrations of some machines or tools by using them as structural materials in place of metals. But unfortunately high material damping often seems incompatible with high stiffness or small temperature coefficients of stiffness or damping itself.
It is the aim of this paper to study experimentally such interrelations between internal loss and stiffness for many practical polymers.
The temperature dependence of complex dynamic Young's moduli of twelve thermoplastic resins and three thermosetting resins, of which the latter the dynamic viscoelastic properties have hitherto been but scantily reported, has been measured of dozens of cps and in the temperature range of -30∼+100°C.
It has been shown that the loss tangents of these polymers are approximately proportional to the temperature coefficients of storage moduli.
The proportion factor K(T) for amorphous polymers in the glass transition region depends roughly on activation energy of relaxation time and absolute temperature.
A method of calculating approximate values of activation energy from measurement of K(T) at a fixed frequency has been studied about an example of NBS polyisobuthylene measured by Ferry et al.
Experimental results on the explicit relations between the storage modulus and the loss tangent are given for many polymers and commented on briefly from the standpoint of structural damping.

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Thermodynamic properties of amorphous polymers are hereunder discussed on the basis of some refined cell model. It is assumed that each polymer chain is surrounded by six nearest neighbours which form a hexagonal lattice (Fig. 1). We shall further assume the Lennard-Jones potential between the chain units along the axes of polymer chains.
The configurational energy and cell partition function are thus obtained. Then the reduced equation of state can be written in the form
pV/T=[1-0.831(V)^-1/2]^-1+T^-1[4.263(V)^-11/2-6.635(V)^-5/2] (1)
where the reduced variables are defined by
T=sλkT/3(1-f)ε^*r^*, V=v/v^*, p=λv^*p/3(1-f)ε^*r^*
λ: bond length
s: the parameter as a measure of chain flexibility
r^*, ε^*: the parameters of distance and energy in the Lennard-Jones potential
v^*: the characteristic value for the van der Waals volume v per chain unit
f: free volume fraction
Eq. (1) shows the principle of corresponding state. From this equation, we can calculate the thermal expansion coefficient and isothermal compressibility for the van der Waals volume. We can treat them for the rubbery state as well as glassy state, considering the growth of holes' or free volume owing to the micro-Brownian motions of the chain segments.
In Table I, we show the predicted values of molecular force constants, lattice energies and cohesion energy densities for a few amorphous polymers. They give fairly good agreement with available experimental data.

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Applications of condenser pick-up for rheometers in measuring displacement were studied.
(1) By a condenser pick-up very large angular displacements (±70°) were easily measured and recorded. It was found that the amplitude dependence of logarithmic decrement of non-Newtonian polyelectrolyte solutions in free damping oscillation apparatus was very small. However, the samples which seemed to have very weak thixotropic structures evidently revealed the amplitude dependence of viscoelasticity, and the dependence, in general, became more and more obvious with decrease in frequency of oscillation.
(2) The vertical oscillation of the rod in a so-called electromagnetic transducer were easily detected by a condenser pick-up, and the viscoelasticity of liquid materials in the audiofrequency range were measured electrically with no experimental difficulty as was often experienced in the transducer method. The viscoelasticity in this modified rheometer was calculated in the method as was described above.
(3) A short comment on the creep and its recovery behaviors in concentrated polyelectrolyte solutions was described, where large shear deformations were also recorded by a condenser pick-up.

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The flow properties of molten EVA copolymer in the region of compratively higher shear rate were measured in connection with investigation of the behaviors and forming mechanism of melt fracture by means of a capillary type rheometer. The relation between the actual shear rate and viscosity obtained by means of the differential method indicates that their flow pattern is non-Newtonian and viscosity decreases with increase of shear rate.
They are described by two straight lines of different slope. The shear rate at their intersecting point corresponds to the critical shear rate where melt fracture forms.
The EVA copolymer differs slightly in the flow curve from low density polyethylene; it seems, however, that their difference is due to the difference in molecular weight distribution rather than in the materials themselves.
And also, the fluidity of the molten EVA copolymer having about 3 in the ratio of M_w/M_n may be equivalent to that of low density polyethylene having from 4 to 7 and, generally, EVA copolymer has higher fluidity than low density polyethylene does. It is expected that the fluidity will increase with incroporation of vinylacetate in the same molecular weight distribution and the same average molecular weight.
The critical shear rate increases with the melt index and temperature. It can not be found that the materials themselves and the molecular weight distribution gives direct influence to the critical point of melt fracture forming as far as the melt index is defined as the parameter. The critical viscosity η_c where melt fracture forms decreases almost in a straight line with increase of melt index.
It is found from the studies of end correction and behavior of formation of melt fracture that melt fracture occurs at inlet of the die, and it supposed that formation of melt fracture has been caused by the result of the elastic turbulence of flow pattern due to failure of recoverable shear strain at inlet of the die.

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It is shown how to determine parameters of a material obeying Casson's equation by some rheometers.
Casson has proposed an equation for the flow of varnishes relating to the shear stress F to the rate of shear D. This relation has the form
F1/2=k0+k1D1/2,
k0 and k1 being constants depending on the properties of suspensions. This equation is generally called Casson's equation.
We have determined the parameters k0 and k1, that is, the viscosity η corresponding to k1 and the yield value τf corresponding to k0, by a rotating coaxial cylinder viscometer and a cone and plate viscometer.
In the case of a rotating coaxial cylinder viscometer, η and τf are determined from the relationship between the torque M and the angular velocity Ω as follows:
η=(1/a2-1/b2)/(4πhtan2θ)
τf=(1/a+1/b)2M0/8πh
In the case of a cone and plate viscometer, there is a following relationship between the torque M on the plate and the angular velocity Ω for a non-Newtonian liquid
Ω=1/2∫τaτbf(τ)/√τ(τ-c)dτ, c=3M/2πa3
Making use of this relation, we can get η and τf as follows:
η=3α/(2πa3tan2θ), τf=3M0/2πa3)
Further we have tried to find the flow curve f(τ)=γ from the known relationship between Ω and M. The flow curve is approximately given by
f(τ)=3/α3τ-(1/2+3/α2)∫τ0ξ(3/α2-1/2)Ω(ξ)dξ.

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Calendering is a continuous operation of driving a pair rolls to compress softened materials into a sheet of uniform thickness. Gaskell presented a theory of the steady isothermal flow of an incompressible Newtonian liquid in a symmetrical calender. In his treatment, it is assumed that the boundaries of the liquid are approximately in parallel. It is the purpose of this paper to take into consideration not only the velocity component v_x but also the velocity component v_y and we have determined v_x and v_y and the pressure p as a function of the position variables x and y.
With regard to the motion of the liquid, the following assumptions are made: (1) that the liquid is Newtonian; (2) that the liquid is incompressible; (3) the motion of the liquid is laminar; (4) that the motion is steady; (5) that no force acts on the liquid; (6) that the motion is symmetrical about the x axis; (7) that the separation at the nip 2H₀ is small in comparison with the roll radius R; (8) that the radii of the rolls are equal and the rolls rotate at the same speed; (9) and that there is no relative motion between the rolls' wall surfaces and the liquid in immediate contact with the wall. We shall consider only the immediate region of the nip.
Let us take a rectangular coordinate system whose origin is at the middle point between the centers of the rolls.
We get
φ=Σ²_i=1B_i(α_isinhk_iY/k_i²+β_iY)cosk_iX,
where B_i, α_i, β_i and k_i are constants. x/R=X, y/R=Y.
The velocity component v_x and v_y are determined by v_x=-∂φ/∂Y and v_y=∂φ/∂X. Also, p is obtained.

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A theoretical investigation of treating the dependence of viscosity on concentration and molecular weight is carried out by making use of the expression of viscosity obtained from the theory of viscoelasticity in temporarily crosslinked network structure. According to the theoretical equation (1), the viscosity depends not only on the“molecular”length but also on the“chain”length: here the term“molecule”represents the total chain of a molecule, while the term“chain”represents the submolecule between the adjacent crosslinks. Since the chain length depends on concentration, the concentration dependence of viscosity is obtained by investigating the concentration dependence of chain length. It is known from the network theory that the chain length is proportional to C^-1 in low concentration, (Eq. 7), and to C^-2 in high concentration, (Eq. 8). Accordingly, it is predicted that the viscosity is proportional to M^3.5 C^3.5 in low concentration range, (Eq. 10), and to M^3.5 C⁶ in high concentration range, (Eq. 12), when the effect of terminal chains is neglected and the friction constant of a segment is assumed to be independent of concentration.
When the effect of terminal chains and the concentration dependence of friction constant of a segment are taken into account, the viscosity is expressed as Eq. (15) in low concentration and as Eq. (16) in high concentration.
Judging from the above results, two kinds of polymer entanglement can be considered; one is the first order entanglement which may result from the overlap of two molecules in low concentration and the other is the second order entanglement which may result from the overlap of more than two molecules in high concentration.

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Blood flowing in capillary cannot be treated as a homogeneous fluid, since the diameter of capillary is of the same dimension as that of red cells. The hematocrit is about 45%. Therefore, when blood flows in capillary, the width of the“compartment”between the two adjacent red cells will be nearly equal to the thickness of the red cell. For the sake of simplicity, we shall make assumptions as follows: (1) the surfaces of red cells are perpendicular to the axis of capillary; (2) since the“compartments”are very narrow, there can be no relative motion of plasma to be contained between the red cells with respect to them; (3) the capillary is an infinite small cylindrical tube; (4) the flow of blood in capillary is steady; (5) blood is incompressible; (6) the effect of gravity is negligible; (7) there is no slip on the wall of the tube. From the above assumptions, we may treat the flow of blood in capillary as if an infinite rod flows steadily through an infinite cylinder filled with plasma.
Let us take a cylindrical coordinate system (r, θ, z), the z-axis being the axis of the capillary. Then the velocity component v_z will be a function of r alone. Solving the equations of continuity and of momentum, we get the discharge Q=πΔp (R⁴-r₀⁴)/8_η. If blood is regarded as a homogeneous fluid of apparent viscosity η_b, the discharge will be given by Q₀=πΔpR⁴/8η_b according to the Poiseuille equation. Consequently, the ratio of Q to Q₀ is given by Q/Q₀=η₀/η·{1-(r₀/R⁴}.
The apparent viscosity η_a of blood in our model is related with the discharge by the relation Q=πΔpR⁴/8η_a. By equating this expression with the obtained formula, we get η_a/η={1-(r₀/R)⁴}^-1. This relationship has been derived by Whitmore from a different stand point of view under certain conditions.

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Among various methods of viscosity measurement, the capillary methods are the most usual ones, by which the coefficient of viscosity is obtained on the assumptions that there is no slip at the capillary wall and that the fluid is homogeneous throughout the capillary, and so on.
So it is important to study first of all whether these assumptions hold or not in the actual flow. As regards the wall effect, measurements of the flow of various liquids using capillaries of different materials have hitherto been reported, but some of them are of doubtful accuracy and ambiguous in details of their experiments.
To obtain systematic and accurate data on this problem, measurements were made of efflux time of ion-exchanged water, human blood and plasma, using an Ostwald type viscometer with and without paraffine coating.
Correction in the thickness of the paraffine layer was made based on the Hagen-Poiseuille equation. Moreover, the change of the efflux time with the time elapsed was measured by using another capillary at the same time as shown in Figs. 1 (B) and 2 (B) to correct the effect due to denaturation of the samples such as blood and plasma.
The result shows that the efflux time is slightly increased by the paraffine coating and the order of the increase is as follows: water (-0.92∼+1.64%)<plasma(+0.15∼+2.54%)<blood(-0.56∼+4.2%).

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Monodisperse poly-α-methylstyrenes in benzene at 30.0°C and in cyclohexane at the theta temperature of 39.0°C have been studied. The two samples are numbered 312 for M_v=1.89×10⁵ and 317 for M_v=4.79×10⁵. The complex rigidity and the steady-shear viscosity have been measured by means of torsion crystals at the frequencies of 19.6kc, 39.2kc, 79.4kc and 117.7kc, and a Ubbelohde dilution type viscometer respectively. The intrinsic rigidity and the limiting relaxation time (divided by K_p) are given by the extrapolation to zero concentration.
The theory of Tschoegl, extended to partially-free-draining and non-Gaussian chains respectively through h and ε, may be summarized in the dimensionless functions of intrinsic rigidity and the relaxation time factors as follows,
[G']M/RT=Σ_pω_s²K_p²/(1+ω_s²K_p²), [G"]M/RT=Σ_pω_sK_p/(1+ω_s²K_p²)
ω_s=ω(τ_p)₀/K_p=ωη_s[η]M/RT, K_p=1/λ'_pΣ_pλ'_p^-1
where ω_s is the generalized angular frequency and λ'_p is the eigenvalue of Zimm. The numerical evaluations have been performed with a high-speed computer.
In the case of benzene solutions the ε'_s have been derived from the steady-shear intrinsic viscosities divided by those of theta solutions using Ptitsyn's equation, and the mean value is 0.15. The dimensionless plots of intrinsic rigidity shows that the most fitting curves for the data are the theoretical ones of Tschoegl for ε=0.15 and h=10. Consequently the degree of draining h is the order of 10, and the first relaxation time factor K₁ is 0.474. Then the first limiting relaxation time(τ₁)₀'s are estimated at 1.29×10^-6sec for 312 and 5.97×10^-6sec for 317.
In the theta solvent of cyclohexane, ε is zero and the dimensionless plots are explained by Zimm's dispersion curves i.e. h=∞. It is the non-draining case, so that K₁=0.422, and(τ₁)₀=7.17× 10^-7sec for 312 and (τ₁)₀=2.64×10^-6sec for 317.
The Tschoegl theory has been quantitatively confirmed with a research in infinite dilution, and the random coil polymer is considered as partially-free-draining molecule in good solvent.

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The compressibility of polymer solutions was measured by the ultrasonic method (in Mc range). The partial specific compressibility of polymer in the solution was determined, and they yielded information about the polymer in the solution.
The information on the polymer were divided into the following two parts: (1) the response of a spherical polymer chain in the solution as a pearl necklace model, (2) the response of a mixing unit of the polymer chain as a pearl in necklace.
The response of a spherical polymer chain was proportional to the intrinsic viscosity of each solution, and was related to the chain dimentions as the viscometric expantion parameter was. On the other hand, the compressibility of a mixing unit of the polymer was the specific value of each component.

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As a preparative step to treat the intrinsic viscosity of a stiff chain macromolecule in a solution, we have investigated the hydrodynamic behavior of a slightly curved rigid rod in a shear field.
Regarding the longitudinal axis of the rod as a differentiable space curve, we can represent the curve as follows using the Bouquet's formula: r(s)=r⁽⁰⁾(s)+κ₁r⁽¹⁾(s)+κ₁²r⁽²⁾(s)+κ₁'r⁽³⁾(s)+κ₁κ₂r⁽⁴⁾(s)+ …, where κ₁ and κ₂ are the curvature and the torsion of the curve at the middle point of it and the prime stands for the differentiation with respect to s, the contour length of the curve. The force distribution F on the longitudinal axis when the rod is in the shear field can be estimated by the Oseen's method: 1/8πμ∫^a_-aT(s, s₀)·F(s)ds=-V_r(s₀), where a is the half length of the rod, . μ the viscosity of the fluid, T the interaction tensor and V_r is the relative velocity to the rod. The Brownian motion is not taken into account here. Expanding T, F and V_r into power series of (κ₁, κ₁', …; κ₂, κ₂', …), we can obtain a set of equations instead of the above equation. The first equation of the set is identical with that for the straight rod derived by Burgers. The motion of the rod is to be determined upon the conditions of the balance of the force and of the moment of force.
Concrete calculation is made of a rod curved like a circular arc, and the intrinsic viscosity for the case is estimated as follows:
[μ]=2/3πρa/b1/σ-1.80√1+(a/b)²(κ₁a)²σ-2.13/10(σ-1.80),
where ρ and b are the density and the radius of the cross-section, of the rod and σ=log2a/b.

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Some rheological behaviors of paints containing magnetic powders (γ-Fe₂O₃) were investigated from the view point of interfacial phenomena, compared with the paints containing non-magnetic powder (α-Fe₂O₃), and the following results were obtained:
Both α-and γ-Fe₂O₃ paints without dispersing agent, show the structural viscosity, and its tendency is more remarkable in the former paint.
By the addition of the dispersing agents, the structure forming action of particles in α-Fe₂O₃ paints is weakened and its flow character is altered to a Newtonian pattern. On the other hand, in the γ-Fe₂O₃ paints, the structure still remains and Bingham-type flow is maintained.
These behaviors were considered to be caused by the formation of different structures in each paints. In the α-Fe₂O₃ paints, the particles form three-dimensional honey-comb structures as shown in Fig. 10 (b), only with the interparticular attractive forces. In the case of the γ-Fe₂O₃ paints, particles were arranged as chains or closed chains shown in Fig. 10 (a), by magnetic attractive force.
The applicability of Casson's equation, the appearence of 2nd Newtonian region and the thixotropic behaviors for the γ-Fe₂O₃ paints will mean the degradation of the chain to smaller units by shearing force, together with the rearrangement and orientation.

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The effect of polar pendant groups on flow properties of pigment dispersions were studied with xylene solution of acryl co-polymers.
The co-polymer having hydroxyl group as the pendant group (F-1), the co-polymer having carboxyl group (F-2) and the co-polymer without any pendant group (F-3) were polymerized in xylene and diluted to 33% non volatile content. The pigments chosen were titanium dioxide as polar pigment and phthalocyanin blue as non-polar pigment. These pigments were dispersed in the above mentioned co-polymer solution in 1.5-16.2vol% concentration, and the flow properties were determined.
TiO₂-F-1 systems showed Newtonian flow in this concentration range. With any other systems, however, Newtonian flow was observed only in the lower concentration range. The Cassons's equation agreed very well in all the systems studied. The yield value S₀ and residual viscosity η_∞ were calculated from these data.
The hydrodynamic volume of pigments in these systems has been discussed in terms of Mooney's equation. It appears that in the polar pigment--- polar vehicle system, the hydrodynamic volume ratio of pigment is unaffected by the concentration of dispersed phase, but in the other systems, the hydrodynamic volume ratio is dependent on the concentration of dispersed phase.
In comparing the calculated values of a parameter indicating the cohesive force of flocculate, it is concluded that the hydroxyl group has much more effect on the flow properties of titanium dioxide dispersion than the carboxyl group. Of phthalocyanin blue dispersions, only slight difference has been observed as made regarding whether with or without polar pendant groups, or in their, flow properties.

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The chemorheological study of rubbers was developed by A.V. Tobolsky by means of chemical stress relaxation measurement. We should also propose a different method by means of dynamic-mechanical measurement on chemorheological study for the same purpose as he has conceived, i.e., investigate both the cross linking and the scission reactions of thermosetting type of polymers.
For the investigation of the cross linking and the scission reactions two types of coating films were used here as shown in Table I, Chinese tung oil-modified phenolic resin and epoxy resin ester for comparison with it.
Dynamic modulus E₁ of the samples used here has been exhibited rubbery plateau at the higher temperature ranges, where the values of dynamic modulus E₁ are indepedent on the frequencies of measurement, as shown in Figs. 1 and 2. Time dependency of dynamic modulus E₁ at the higher temperature region is shown in Fig. 3 in straight lines within shorter time ranges.
Dynamic modulus E₁ obtained at the plateau region is expressed by Eq. (1) according to rubber elasticity.
E₁=3_ρRT/M_c (1)
Density of cross links ν (mol/g) in polymer is expressed by Eq. (2) from Eq. (1).
ν=E₁/(3_ρRT) (2)
The slopes of the straight lines of dynamic modulus E₁ at each temperature shown in Fig. 3, therefore, may correspond to the reaction-rate constant k of cross linking for the sample, respectively. As such dependence of the reaction-rate constant k on temperatures is known to be expressed by the Arrhenius Eq. (3).
k=Aexp(ΔH/RT) (3)
In this way the apparent activation energy ΔH of the cross linking reaction for the four samples can be calculated as shown in Table III, from replot in the form of log k vs. 1/T as shown in Fig. 4. From the break of slopes of the straight lines in Fig. 4 we can obtain two different values of ΔH for a sample. This phenomenon will be accounted for by the fact that there are two types of cross linking reactions for a sample over the temperature range used herein.
In order to ascertain what kinds of cross linking reactions had taken place during the baking process we continued the chemical experiments as shown in Figs. 5, 6, 7 and 8, respectively. It is clear that there are two heat generations due to chemical reactions from the results of Differential Thermal Analysis (DTA) of the samples as shown in Fig. 5, i.e., the heat generation located in lower temperature region takes place both in air and N₂ flow, whereas the higher temperature in air only. From the results shown in Figs. 6, 7 and 8, the heat generation located in lower temperature region seems to be caused by the cross linking reaction after degradation of hydroperoxide formed in the drying oil of the sample, the higher temperature may be due to another type of cross linking reaction accompanied with the degradation of drying oil.
Those results of chemical experiments approximately agree with the results obtained dynamic-mechanical measurements mentioned above.

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The solution grown crystals of polypropylene prepared from xylene dilute solution show a larger crystalline absorption peak in the temperature dependence of dynamic loss modulus than that of the melt crystallized bulk specimen. The solution grown crystals are effective in studying the viscoelastic properties of the polymer crystals. In order to make clear the relationship between the morphology and viscoelastic properties of the polymer, it is further necessary to clarify the characteristics of the crystals in the bulk crystallized specimen.
The tensile dynamic and loss moduli, E' and E" of the mats of crystals grown in dilute solution and those obtained by an acid etching of the bulk crystallized specimen, were measured at a frequency of 110c/s over a temperature range from -20 to 170°C by using a direct reading dynamic viscoelastometer, Vibron DDV-II.
The crystalline absorption peak of the solution grown crystal precipitated at 90°C, whose degree of crystallinity and crystal perfection was higher than that of the crystals precipitated at 70 and 80°C., appeared at higher temperature than the others without decreasing the peak intensity. The degree of crystal perfection of the crystal precipitated at 70°C could be increased by annealing. In this case, however, the intensity of crystalline absorption peak decreased, when the annealing temperature was taken above 135°C.
The crystal obtained from the acid etching of annealed bulk specimen below 135°C showed almost the same viscoelastic behavior as that of the solution grown crystal. Therefore, the less intensive crystalline absorption peak of the melt crystallized bulk specimen could be attributable to the existence of amorphous region between the crystal lamellae. On the other hand, the decrease of the crystalline absorption peak, as pointed out above, for the crystal annealed above 135°C could not be explained in terms of the two phase model composed of amorphous and crystalline phases, and would have to be attributed to the properties of the crystal itself.

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Many experiments on the flow of molten polymers have hitherto been performed by means of relatively short capillary tubes. The static pressure of the flow is measured indirectly by the load of plunger. In this paper the results of experiments made with the use of comparatively long tubes-not capillary-are described. As for the pressure, the elastic deformation of the tube according to the static pressure of the flow was measured directly by the wire strain gage. Polyethylen and polypropyrene were used as testing materials.
The shearing stress vs. shear rate curve can be approximated by two straight lines as shown in Fig. 3. The shear rate at this crooked point of the flow curve corresponds to that of the maximum swelling ratio, and over the region of this shear rate appears the vibration of pressure.
The swelling mechanism by Barus effect was investigated by introducing the Voigt model after McIntosh, on the other hand, and it was assured that the experimental tendency had good agreement with the theoretical ones as shown in Fig. 5. So it seems that Barus effect can be explained by a modified Voigt model of Fig. 6. The flow rate at which the smooth figure of ejection is obtained is calculated from Eq. (10). The facts thus obtained by this study will serve as the design for dies of injection machine.

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It is well known that normal stresses appear as well as shear stress when polymer melts are subjected to shear.
Various manifestations of this effect have been described in literature, but comparatively few experiments have been made to measure this effect quantitatively.
The measurement of the flow birefringence in the slit-like duct is considered to give facilities to get normal stress effect of polymer melts in shear flow.
The slit-like ducts which have side walls made of glass were made for this purpose. The photographs of isochromatic patterns were taken and the pressure gradient along with the flow were measured on the flow of polyethylene melts. From these experiments the normal stress difference S₁₁-S₂₂ were obtained through the following equation.
S₁₁-S₂₂=√(Δn₁₂/c)²-4S₁₂²=√(nλ/cw)²-4(yΔp/Δx)²
where c is stress optical coefficient, Δn₁₂ is flow birefringence in the“1-2”plane and S₁₂ is shear stress.
Using the values of S₁₁-S₂₂ the elastic energy in capillary flow that is imparted to the melt at the entrance of the capillary were calculated, and these results were compared with the values of total entrance pressure loss subtracted Couette loss which were obtained from the capillary experiment.
As the results of these experiments it has been ascertained that the former is far less than the latter.
This result is considered to show that the greater part of the total entrance pressure loss in capillary flow has been consumed as the viscous energy loss at the entrance region, and that the elastic energy imparted to the melt and therefore normal stress difference S₁₁-S₂₂ or recoverable strain S_R in shear flow cannot be obtained by Philippoff¹⁸⁾-Bagley's¹⁹⁾ method in a capillary experiment.

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