CAPILLARY VISCOMETRY 355 I tions by substituting for the actual capillary length the sum of this length plus an increment of length equivalent to the additional pressure drop. The value of the equivalent increment of length to be used in any case can be evaluated by making measurements in capillaries of a given radius but several different lengths, with identically shaped ends. If the observed pressure drop for these capillaries for a given rate of flow is plotted versus their actual lengths, there will be a negative intercept indicating the pressure drop due to the end contributions. From this can be calculated the increment to be added to the actual length in calculations. Theory indicates that the equivalent increment of length for this end effect should be proportional to the capillary diameter and that the con- stant of proportionality, n, should be about 1.14 when the flow is laminar just outside both the entrance and exit ends of the capillary. This is the case at very low rates of flow. At higher rates the flow becomes turbulent outside the exit end and the theoretical value of n is reduced by half to 0.57. Experimental values of n generally fall in the broad neighborhood of unity. Another, often much larger, type of end effect sometimes appears in the viscometry ofviscoelastic materials (11). This is due to elastic adjustment which are required in the region of the capillary entrance and exit. Such effects have not been theoretically analyzed to any appreciable extent, but it does not appear that they are always proportional to the capillary diameter. If their magnitudes are expressed empirically in terms of an equivalent increment of capillary length, values of n as high as 100 are not uncommon for this contribution. However, since it has not been analyzed systematically it is difficult to correct for the effect. 4.6. Surface Tension Contributions The entrance and exit ends of a capillary viscometer are usua]ly con- nected with reservoirs having free surfaces of the experimental material. If the forces exerted by these two surfaces because of surface tension are not equal and opposite, there will be a corresponding contribution to the experimentally observed pressure drop. The forces exerted by the two surfaces will be equal only if their geometries are maintained identical during the entire course of a measurement. Such an arrangement is possible, but is complicated and is not often used. If the two surfaces have different geometries at any time during a measurement, the surface tension contribution to the pressure drop can be evaluated and corrected for only by calculating and if necessary integrating the appropriate components of surface tension for the surfaces concerned. Since a wide variety of surface geometries are used in capillary viscometry, no systematic correction scheme has been devised. Swindellg (12) gives

356 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS a comprehensive discussion of the calculation of surface tension corrections . for several surface geometries based on Bashforth and Adams' tables. (13) for the components of surface tension of curved surfaces. Fortunately conditions can usually be found where surface tension corrections are negligible. ¾.7. Drainage from Reservoir IFalls The rate of flow of experimental fluid through a capillary is most often determined by observing the fall of the meniscus in the reservoir from which the fluid enters the capillary. This observation must be combined with the geometry of the reservoir to give the rate of flow. In this procedure an error can arise if the fluid drains down off the capillary wall slowly enough in comparison with the rate of fall of the meniscus so that an appreciable film of fluid temporarily adheres to the wall during the time the fall of the meniscus is under observation. This film decreases the effective cross-sectional dimension of the reservoir by twice the film thick- ness. The result is that less fluid flows through the capillary in an observed interval of time than is calculated from the cross section of the reservoir. Correspondingly the uncorrected derived rate of flow is higher than the true rate, and too low a viscosity is obtained. This effect is usually taken into account by a procedure using the results of measurements including both directions of flow through the capillary thus measuring rates of both falling and rising meniscii in the reservoir. The procedure is sometimes complicated by the concomitant interchange of entrance and exit ends of the capillary. The end effects discussed in subsections 4.4 through 4.6 of this paper are altered by this interchange. Drainage corrections are discussed more thoroughly by Swindells, et al. (12). ¾.8. Dissipation of Heat During the course of any viscous process, by its very nature, mechanical energy is necessarily converted to heat. The rate of heat dissipation per unit volume is given by the product of the shearing stress and the rate of shear. Under gentle shearing conditions, which are often used in viscometry, this heating is not sufficient to cause any detectable tempera- ture rise in the experimental material. However, if measurements are made at high rates of shear, especially on materials of high viscosity, it is easy to reach shearing conditions where the heat dissipation in the flow process does cause a noticeable temperature rise in the flowing material. This temperature rise must then of course be taken into account in analyzing the results of the measurement. Taking this temperature rise into account is not as simple as measuring a single temperature for the flowing material. Since according to equation 8 the shearing stress and accordingly the rate of shear vary from zero at