Extrapolated viscosity

Thenceforth it became necessary to double extrapolate viscosity data to zero time and to infinite dilution. The time scale involved allows a schedule of data collection sufficient for double extrapolation to be conducted simultaneously, and a program has been worked out to accomplish this, which minimizes the amount of data required (8,20). 

As just stated, the L3 phase resembles structurally a bicontinuous microemulsion, which makes it interesting to compare their rheological properties. Viscosity measurements on an L3 phase [in the system cetylpyridinium chloride-hexanol-brine (0.2 M NaCl)] showed Newtonian flow behavior for the range of shear rates of 0.1-100 s" [113]. The viscosity of this highly interconnected, spongelike system is always very low and close to the solvent viscosity, even for a volume fraction of 0.2 it is less than 10 mPa s. (Similar viscosity values have been observed in the L3 phase of the system tetradecyldimethylamine oxide-hexanol-water [114].) It increases linearly with the volume fraction of the amphiphilic material, where it is interesting to note that extrapolation to zero concentration does not yield the solvent viscosity but a value about three times as high. A similar value for the extrapolated viscosity was also reported more recently for another L3 phase (in the system SDS pentanol-dodecane-water [115]), and it seems that this enhanced viscosity is a universal property connected to the structure of the L3 phase. 

The extrapolated viscosity follows the order D > E > B in both series of mesogens. Similar analysis of pyridinium zwitterions 71 and 72 in ClEster host showed that they both increase viscosity y, of the phase, and the increase is larger than for the sulfonium The effect is host dependent and in 6-CHBT the sulfonium zwitterions 5 and 73 have significantly lower impact on viscosity. 

Revised material in Section 5 includes an extensive tabulation of binary and ternary azeotropes comprising approximately 850 entries. Over 975 compounds have values listed for viscosity, dielectric constant, dipole moment, and surface tension. Whenever possible, data for viscosity and dielectric constant are provided at two temperatures to permit interpolation for intermediate temperatures and also to permit limited extrapolation of the data. The dipole moments are often listed for different physical states. Values for surface tension can be calculated over a range of temperatures from two constants that can be fitted into a linear equation. Also extensively revised and expanded are the properties of combustible mixtures in air. A table of triple points has been added. 

Strauss and Williamst have studied coil dimensions of derivatives of poly(4-vinylpyridine) by light-scattering and viscosity measurements. The derivatives studied were poly(pyridinium) ions quaternized y% with n-dodecyl groups and (1 - y)% with ethyl groups. Experimental coil dimensions extrapolated to 0 conditions and expressed relative to the length of a freely rotating repeat unit are presented here for the molecules in two different environments ... 

Which range should be considered The answer is the region near the origin of a plot like Fig. 2.2 for pseudoplastic materials. The slope of the tangent to a pseudoplastic curve at the origin is called the viscosity at zero rate of shear. Note that this is an extrapolation to a limit rather than an observation at zero shear (which corresponds to no flow). We shall use the symbol to indicate the viscosity of a polymer in the limit of zero shear, since the behavior is Newtonian (subscript N)in this region. 

When m = 1.0, as in Fig. 2.5, the exponent becomes zero and the viscosity is independent of 7 when m = 0.7, a factor of 10 change in 7 results in a decrease of viscosity by a factor of 2. This is approximately the case for the data in Fig. 2.5 for 7 values between 10" and 10" sec". Equation (2.14) and its variations are called power laws. Relationships of this sort are valuable empirical tools for extrapolating either F/A or t over modest ranges of 7. In such an application, the exponent m - 1 and the proportionality constant are... 

The inherent viscosity (I/C2) In (77/770). A plot of inherent viscosity versus concentration also extrapolates to [77] in the limit of C2 0. That this is the case is readily seen by combining Eq. (9.12) with the definition of the inherent viscosity and then expanding the logarithm ... 

Some quite viscous oils in the 450 650 mm /s are employed for high temperatures. Less viscous oils, down to 25 mm /s and lower at 40°C, are used in special greases for low temperatures. The maximum oil viscosity in a grease for starting medium torque equipment is about 100, 000 mm /s(= cSt) (4). Extrapolations for various oils can be made on viscosity—temperature charts, as shown in Figure 8, to estimate this approximate low temperature limit. 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Dilute Polymer Solutions. The measurement of dilute solution viscosities of polymers is widely used for polymer characterization. Very low concentrations reduce intermolecular interactions and allow measurement of polymer—solvent interactions. These measurements ate usually made in capillary viscometers, some of which have provisions for direct dilution of the polymer solution. The key viscosity parameter for polymer characterization is the limiting viscosity number or intrinsic viscosity, [Tj]. It is calculated by extrapolation of the viscosity number (reduced viscosity) or the logarithmic viscosity number (inherent viscosity) to zero concentration. 

The viscosity ratio or relative viscosity, Tj p is the ratio of the viscosity of the polymer solution to the viscosity of the pure solvent. In capillary viscometer measurements, the relative viscosity (dimensionless) is the ratio of the flow time for the solution t to the flow time for the solvent /q (Table 2). The specific (sp) viscosity (dimensionless) is also defined in Table 2, as is the viscosity number or reduced (red) viscosity, which has the units of cubic meters per kilogram (m /kg) or deciUters per gram (dL/g). The logarithmic viscosity number or inherent (inh) viscosity likewise has the units m /kg or dL/g. For Tj g and Tj p, the concentration of polymer, is expressed in convenient units, traditionally g/100 cm but kg/m in SI units. The viscosity number and logarithmic viscosity number vary with concentration, but each can be extrapolated (Fig. 9) to zero concentration to give the limiting viscosity number (intrinsic viscosity) (Table 2). 

Fig. 9. Plots of viscosity number (/c) and the logarithmic viscosity number (/c) vs concentration. Extrapolations to 2ero concentration...

Extrapolation to infinite dilution requites viscosity measurements at usually four or five concentrations. Eor relative (rel) measurements of rapid determination, a single-point equation may often be used. A useful expression is the following (eq. 9) (27) ... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

The number-average molecular weight of dimethylsiloxane can also be determined from the intrinsic viscosity [Tj, dL/g] (extrapolated to zero viscosity) ia toluene or methyl ethyl ketone according to the foUowiag equatioa (339,340) ... 

It shoiild be noted that the influence of substituting solvents of widely differing viscosities upon the interfacial area a can be very large. One therefore should be cautious about extrapolating data to account for viscosity effects between different solvent systems. 

Equations (22-86) and (22-89) are the turbulent- and laminar-flow flux equations for the pressure-independent portion of the ultrafiltra-tion operating curve. They assume complete retention of solute. Appropriate values of diffusivity and kinematic viscosity are rarely known, so an a priori solution of the equations isn t usually possible. Interpolation, extrapolation, even precuction of an operating cui ve may be done from limited data. For turbulent flow over an unfouled membrane of a solution containing no particulates, the exponent on Q is usually 0.8. Fouhng reduces the exponent and particulates can increase the exponent to a value as high as 2. These equations also apply to some cases of reverse osmosis and microfiltration. In the former, the constancy of may not be assumed, and in the latter, D is usually enhanced very significantly by the action of materials not in true solution. 

Experimental measurements of viscosity almost always are recommended when dealing with slurries and extrapolations should be made with caution. Most theoretically based expressions for liquid viscosity are not appropriate for practical calculations or require actual measurements to evaluate constants. For nonclustering particles, a reasonable correlation may be based on the ratio of the effective bulk viscosity, /ig, to the viscosity of the liquid. This ratio is expressed as a function of the volume fraction of liquid x in the slurry for a reasonable range of compositions ... 

For SEC employing light scattering or viscosity detection it is often necessary to extrapolate the log M versus retention time data derived from the integration to account for the lack of signal sensitivity at one or another extreme of the distrihution. If the characteristics of the column are known to he linear... 

Figure 3-56. Viscosity performance correction chart for centrifugal pumps. Note do not extrapolate. For centrifugal pumps only, not for axial or mixed flow. NPSH must be adequate. For Newtonian fluids only. For multistage pumps, use head per stage. (By permission. Hydraulic Institute Standards for Centrifugal, Rotary, and Reciprocating Pumps, 13th ed.. Hydraulic Institute, 1975.)...

Mobility of Ions in D20. The viscosity of liquid D20 at room temperature has a value 1,232 times the viscosity of H20. Since the D2O and HaO molecules are so similar in other respects, we should expect the mobility of ions dissolved in D20 to be smaller than in H20. The conductivity of potassium chloride and potassium acetate was measured in mixtures of D20 and H20 up to a composition containing 97 per cent of D20.1 The values for ions in D2O, given in Table 7, were obtained by extrapolation from values obtained in the mixed solvent containing a few per cent of H20. As was expected, the conductivity in D20 was found to be smaller than in H20. But the change was not quite so great as the change in the viscosity, as is shown by the ratios in the last column of Table 7. We must conclude that, for some or all of the ions, the... 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

A convenient form of 3-parameter equation which extrapolates to a constant limiting apparent viscosity (m0 or /jlqc) as the shear rate approaches both zero and infinity has been proposed by Cross(i3) ... 

Frequently occurs that extrapolations do not have a common value at their origin ordinates. These deviations may be caused by inadequate lineal extrapolations. The above mentioned is the routine method used for [q] determination. The procedure is laborious and consumes a considerable amount of time and reactive because of this, several equations were developed which estimate intrinsic viscosity at one single concentration and do not require a graphic. They are known as "single-point" methods. 

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