V O L U M E 19, NO. 2, F E B R U A R Y 1 9 4 7 Ferguson, Bassett, IND. ENG.CHEM.,ANAL.ED., 1 4 , 4 9 3 (1942). Ganshin, A. 8., K h i m . hi'ashinostroenie, 8, 8-10 (1939). Gibson, G. P., J . SOC.Chim. Ind., 58, 317-19 (1939). Hepp, H. J., and Smith, D. E., IND.ENG.CHEM., - 4 ~ aED., ~. 17, 579 (1945).
Hickman, K., and Weyerts, W., J . Am. Chem. SOC.,52, 4714 (1930).
Hill, J. B., and Ferris, S.W., I n d . Eng. Chem., 19, 379 (1927). Kester, E. B., and Andrews, R., ISD.ENQ.CHEhi., AN.AL.ED., 3, 373 (1931).
Lecky, H. S., and Ewell, R. H., Ibid., 12, 546 (1940). Leslie, E. H., and Geniesse, J. C., Ind. Eng. Chem., 18, 591 (1926).
Leslie, R. T., and Schicktana, S. T., Bur. Standards J . Research, 6, 379 (1931).
Loveless, A. TV. T., I n d . Eng. Chem., 18, 826 (1926). Marshall, M .J.,Ibid., 20, 1379 (1928). Marshall, M.J., and Sutherland, B. P., Ibid., 19, 735 (1927). Means, E. A , , and Xewman, E. L., IND.ESG. CHEM.,ANAL.ED., 8 , 231 (1936).
Palkin, S.,and Hall, S. -4., Ibid., 14, 901 (1942),
123 Peters, W. A., and Baker, T., I n d . Eng. Chem., 18, 69 (1926). Podbielniak, W. J., Technical Advisory Committee of the Petroleum Industry War Council, Report HcAC-6, Fig. 13 (1944).
Richards, A. R., IND.ENG.CHEM.,ANAL.ED., 14, 177 (1942). Rossini, F. D., and Glasgow, -4.R., Jr., B u r . Standards J . Research, 23, 509 (1939).
Rothrnan, S.C., IND. ESG.CHEY.,4 . y ~ED., ~ . 5, 338 (1933). Schwarta, A. M., and Bush, M. T., Ibid., 3, 138 (1931). Selker, R.1. L., Buck, R. E., and Lankelma, H. P., Ibid., 12, 353 (1940).
Simons, J. H., Ibid., 10, 29 (1938). Suen, Taeng-Jiueq, Ibid., 13, 519 (1941). Tongberg, C. O., Quiggle, D., and Fenske, 11.1. R., I n d . Eng. Chem., 26, 1213 (1934).
Tooke, J. W., IND.ENG.CHEM.,ha^. ED.,10, 214 (1938). Towne, R. S., Ibid., 16, 584 (1944) Turk, -A., and Matusaak, A , , Ibid., 14, 72 (1942). Vickery, H. B., and Pucher, G. W., Ibid., 6, 372 (1934). Wagner, E. C., and Simons, J. K., Ibid., 5, 183 (1933).
Instrument for' Measurine Rheological Properties of Elastic Fluids HERBERT GOLDBERG AND OTTO SANDVIK, Kodak Research Laboratories, Rochester, N. Y .
A n instrument is described with which the elastic constants of gels may be measured easily. The gel is placed in the annular space formed by two concentric cylinders, one of which is oscillated mechanically at frequencies ranging from 10 to 6000 cycles per minute. After the resonance frequency and maximum amplitude of the system have been observed the elastic constants of the system may be calculated. A modified apparatus allows measurements to be taken while the sample is being sheared at various rates, and rapid changes of shear modulus may be recorded photographically.
S
EVERAL experimental procedures are available for studying the elastic and viscous properties of liquids. Some are based on observation of the behavior of the liquid as stress is suddenly applied or removed; others depend on the response of the liquid t o alternating stresses of constant, frequency and amplitude. The first method was used by Kendall ( 4 ) whose instrument consisted of two concentric cylinders between which the liquid was introduced, means to apply a torque of known magnitude t o the inner cylinder for a short time, and a photographic arrangement to record its motion as a function of time. Van Wazer (6) used a similar apparatus but measured the torque set up in the liquid when the outer cylinder is suddenly set in motion. Various relationships between chemical structure and rheological behavior of certain resilient liquids have been investigated with these apparatus and yield points have been measured. I n order t o go further and derive quantitative data from response curves thus obtained, it would he necessary to treat analytically the behavior of elastic bodies under transient conditions, a complex mathematical problem. The other procedure consists in subjecting a system, of which the liquid is a part, to sinusoidal forces of known frequency and amplitude and measuring its response. The equations applying to such vibrations are simple and well known, and quantitative data may he derived comparatively easily from the shape of steady-state resonance curves. With suitable apparatus, the fundamental constants of the liquid can he measured a t Lvidely different frequencies ranging from a few cycles to several hundred thousand cycles per minute. Such measurements have been made by Philipoff ( 5 ) and Ferry (3). Holyever, their instrunientation is complex and comprises a number of parts that must be finished by skilled instrument makers. I n some cases elaborate
calibration procedures are required before quantitative data can he obtained. There seems to be a need for an instrument of simple design, which can he built and serviced easily and will furnish quantitative results. It should he suitable for research work as \yell as for routine measurements by semiskilled operators. Since practically all elastic liquids show a change of viscosity and shear modulus if stress is applied, there should be provision for making measurements while continuous shear is taking place. Two instruments which fulfill these conditions are described in this paper. The first is more easily constructed and is useful in measuring the modulus of rigidity n-hile the liquid under study is essentially free of stress. The second instrument may also be used for measurements of rigidity under stress as well as for ordinary measurements of viscosity and some other tests reported in the literature. SIMP LE E L 4 STOBlETER
.In apparatus which may be used for measuring the modulus of rigidity while the liquid is essentially undisturbed is shown in Figure 1.
It consists of a turntable, 1, mounted on hall bearings and rigidly connected t o a rocker arm, 3. The rocker arm bears against an eccentric cam, 4,through ball bearings, 5 and 6. Cam 4 is driven through a pulley, 2, by a resistor-controlled shunt direct-current motor or by an induction motor and a variablespeed transmission. It may be rotated a t speeds ranging from 60 to 6000 r.p.m. The eccentricity of the cam should be such as to give the turntable an angular amplitude of approximately one degree, hut a smaller amplitude may be used if the liquid is easily disturbed. As the length of the connecting rod formed by ball bearings 5 and 6 is more than fifty times the eccentricity of
124
A N A L Y T I C A L CHEMISTRY
Table I. Dimensions of Cylinders for Simple Elastometer Symbol
1 2
Inside Diameter, Cm. Length, Cm. Outer Cylinders 1.1 6 6 1.25
It
R ,,
v.,
Remarks
“1
Standard cylinder hfade of glass tubing for very thin liquids
Inner Cylinders Moment of Inertia, G r a m Sq. Cm.
Figure 3. Models of Gel for Computing Shear Modulus over a pin mounted a t the bottom of the outer cylinder. It serves as a lateral guide and keeps the two cylinders concentric.
Figure 1. Simple Elastometer Measuring Rigidity and Dynamic Viscosity of Gels Essentially at Rest the cam, the motion of the turntable can be considered truly sinusoidal. Since i t is important to avoid the presence of spurious and misleading resonance points, an attempt was made to detect possible disturbances which may be oaused by loose bearings or uneven drive. A mirror was attached to the turntable and a light beam coming from a stationary source was reflected by the mirror and intercepted by a photoelectric cell placed behind a triangular aperture and connected to an oscillograph. Because of the shape of the aperture, the deflection on the oscillograph screen was proportional to the angular displacement of the turntable. Inspection of the trace showed that the motion was sinusoidal a t all frequencies and free from transient disturbances. In view of later experience, however, this test is not considered necessary if ordinary care is taken in the construction of the instrument. A cup, 10,which may serve as a thermostatic water jacket is fastened to the turntable. Several cylinders, 7, of various inside diameters are provided, and adapted to be mounted inside the thermostatic cup, concentric with the axis of rotation of the turntable. These cylinders are filled with the liquid to be tested. In order to facilitate the filling operation, it is advantageous to split the cylinders longitudinally and hold the two parts together by one or two rings. An inner cylinder, 8, carrying a mirror, 9, is suspended on a thread or silk fiber, affording a negligible restoring torque compared with that supplied by the liquid. A hole is provided at the lower end of cylinder 8, which fits loosely
Several different inner cylinders are necessary if substances of widely varying rigidities are to be measured. Their lengths and diameters can be identical, but their moments of inertia should cover a range of about 1 to 50. This may be done conveniently by loading the upper face of a standard inner cylinder with external weights. I n practice, two outer cylinders and five inner cylinders have been found sufficient for most purposes. Their dimensions are shown in Table I. The frequency of vibration of the turntable +nd outer cylinder is easily measured with a stroboscope or other commercially available speed indicator. The motion of the inner cylinder is magnified optically by a mirror, 9, which deflects a light beam coming from a stationary source. The measurements are carried out by noting the width of the trace of the light beam a s a function of the frequency, or simply the frequency and amplitude a t which the mavimum deflection occurs. Traces of typical resonance curves measured on gels of aluminum soaps in Varsol or gasoline are shown in Figure 2. All data are identified by the symbols of the two cylinders used as listed in Table I. Interpretation of Resonance Curves. Consider various resonant systems shown schematically in Figure 3. They all comprise two mass elements having moment of inertia, connected through springs n i t h negligible mass, and a viscous damping unit. . Well-known relationships of classical mechanics may be applied to any of these systems-e.g.,
a% 5
40-
y
=
M
dv -, v dt
=
dx
-,
dt
dT
=
dx -,C and T = Rv
where T is the torque applied, v is the annular velocitv. x is the annular displacement, C ‘is the complLnce (reciprocal rigidity), R is the viscous damping constant of the liquid layer, f is the frequency, M is the moment of inertia, j = and o =2?rj. From these relationships the differential equation determining the relative position of the masses as a function of time may be written. Upon integration, the simple relationships shown in
n,
~a L
T
W
F Q E O J E h C Y CYCLES D E R MINUTE
Figure 2. Resonance Curves of Gels Observed in Elastometer 0-0-0 6 %, X-X-X 9 %.
- 12%
V O L U M E 19, N O . 2, F E B R U A R Y 1 9 4 7
125
Figure 3 are obtained. They apply to steady-state conditions and give the magnitude and phase of the ratio of the maximum velocities of the two masses as a function of the frequency. If these amplitude ratios are plotted against frequency, bell-shaped curves similar to those shown in Figure 2 are obtained. If X I and x2 are the instantaneous displacements of the two masses in any of the systems shown in Figure 3, and XI and X 2 are their displacement amplitudes, then 21
VI
= =
XI sin wt XI wcos wt
In order to apply this formula to the experimental results, i t would be necessary to consider part of the liquid occupying the space between the two cylinders rigidly attached to the outer cylinder and the rest attached to the inner cylinder, with a spring without mass acting between them. Shear would occur a t the interface of the liquid layers only. Actually, however, mass elements and elastic elements are distributed uniformly throughout the liquid. The effects of the uniform distribution can generally be neglected unless a large outer cylinder is used in combination with an inner cylinder of low moment of inertia. I: such a case a better approximation may be obtained if in the computations one third of the moment of inertia of the liquid is added to that of the inner cylinder. .4 better way of accounting for the effects of distributed mass is outlined in the section entitled “;\Iechanical and Equivalent Electrical Model”.
x2 = X , sin wt v, =
3 v2
=
x,wcos wt
x1/x2
This shows that the displacement amplitude ratio, as measured
in the apparatus and plotted in Figure 2 , is equal to the velocity amplitude ratios shown i n the formulas of Figure 3. It is possible, therefore, to calculate with these formulas the compliance of the elastic element of the vibrating system, its shear modulus, and damping. Computation of Shear Modulus. The frequency a t which the maximum occurs and the height of the resonance curve depend on the magnitude and position of the damping element in the system. However, if resonance curves for different positions a n J a series of values of the damping element including R = 0 and R = m are plotted, it is seen that the actual resonance frequency is identical with the resonance frequency of the same system without damping within 3Yc as long as the velocity amplitude ratio is greater than 3. This is demonstrated in Figure 4 , where the ratio of the resonance frequency of a typical system to that of the same system without damping is plotted as a function of the corresponding velocity ratio. As this velocity ratio is usually greater than-3, it is”justifiab1e in most cases to neglect damping in calcuIations of shear modulus. The systems shown in Figure 3,a, b, and c, may thus be reduced to that shown in Figure 3,d. The compliance of the elastic element is then given by
c=
In order to derive the relationship between the shear modulus of the liquid and the compliance measured in the apparatus, the liquid may be divided into three zones, according to Figure 5 . If 1 is the depth of immersion a is the thickness of the end layer r , , r1 are the radii of the cylinders F is the shearing force s is the surface of contact between adjacent layers G is the shear modulus of the liquid C is the compliance of the liquid d v / d z is the shear then d v / d x = F/sGanddv =
F
-
sG
dx
The compliance of part -4of the liquid is (Figure 6) :
Ca = f d C a
with d a = =
C* = (1)
1/X2W2
= Jda/Fx =
dv/x
f dv/Fx2 fd x / G s x 2
f&
For part B, one finds in a similar way:
\I
-
For part C, the engineer’s torsion formula may be used:
and the total compliance is given by I
I
Figure 5. Symbols Used in Computing Shear Modulus
09
,
f’
’ res ,/ fres
Figure 4. Relationship between Resonance Frequency of Damped Gel and -4mplitude at Resonance
I 1 I d L - 4
dl
I
Figure 6. Symbols’ Used in Computing Shear Modulus
In practice, it is always possible to make the outer cylinder long enough so that CB and CC are large compared with C A . I t is then permissible to put:
Shear moduli of some aluminum soaps gelled in Varsol were calculated according to Equation 1 and are shown in Figure 7. They are plotted against “effective rate of shear”_ that is, the rate of shear which, if it were maintained continuously, . would cause the same dissipation of energy in a Newtonian liquid as the actual rate of shear generated by oscillating motion. For sinusoidal motion, one has
126
$
ANALYTICAL CHEMISTRY
looot 500
L
2
0
4
6
8
10
EFFECTIVE
I2
I4
16
20
18
22
I
24
RBTE OF S H E b R S E C - ’
Figure 7. Shear Moduli of Aluminum Soap Gelled in Varsol as Function of Effective Rate of Shear
0-01) 12 %,
&if.
=
9
pmax..
X - X - X 9 %, 0-0-0 6 %
At low frequencies pmnx. =
Aw -
r2
component, separate units will slide appreciably over each other, causing permanent deformations. On assuming that the elastic units do not break to any large extent, it is seen that steady shearing forces cannot cause flow through RD. The viscosity measured in the MacMichael viscometer should be represented therefore by Rs alone and will be called “steady-flow viscosity”. Figure 8,b, which is essentially identical with Figure 3,c, admits an interpretation which is less mechanical in nature. The liquid is considered to be built up of separate units, M1, M z , attached to each other through elastic bonds, C . These bonds are of different strengths and as stress is applied some of them will break, allowing the liquid to flow through R until new bonds are formed. In this case an equilibrium between breaking bonds and healing bonds would establish itself, depending upon the rate of shear, and R would have to account for dissipation of energy during both elastic and plastic deformations. The difference between the two models lies in the way the steady-flow viscosity, Rs, is accounted for. However, if RS is large, it does not affect t o a great extent the values found for R Dand hence it is difficult to judge the models from the numerical results as such. If RD for the model of Figure 8,a, and R for Figure 8,b, are plotted against the effective rate of shear, it appears that smoother curves are obtained for the model shown in Figure 8,a.
- r1
B is the displacement amplitude, w = 2rf
b
0
nhere pmax. is the maximum rate of shear occurring in the oscillating motion
RS
PI-
% !\
Since the phase angle of the ratio, m, of the velocities of the two cylinders is 90” a t resonance, the modulus of that ratio a t resonance is
Iml = v‘mz
+1
or approximately
b’u
I
Iml = m for m>3
Therefore, the effective rate of shear a t resonance is given by: Peff. =
mAw 4 2 - Ti 2
Tz
Damping. The finite height of the resonance curves shown in Figure 2 is due to the dissipation of energy in the liquid through Figure 8. Possible RIodels for Guiding Interpretation of Results shear, a phenomenon associated with viscosity. It was shown in Figure 3 that the numerical value of the damping constant (viscosity) necessary to account for the height of a measured 7 7000 resonance curve depends on the position occupied by the damping 6 6000 element in the vibrating system. I n these calculations i t was 5 5000 assumed that the ratio of the applied torque to the angular velocity is constant for a liquid and independent of the rate of shear 4 4000 applied. This is strictly correct for Sewtonian liquids only. However, shear experiments carried out i n the Clark-Hodsman ( 2 ) viscometer showed that a t least in the case of aluminum soap cc w 3 3000gels in Varsol no measurable breakdown occurred even for much Lo E iu 0 lower frequencies and larger amplitudes than those used in the z a 2. >D present experiments. It was considered permissible therefore to b2000 postulate .the Xewtonian law of viscosity in the folloving con2 0 siderations. 0 3 L“ 0 The viscosity of the liquids under study was measured with the > 0 I MacMichael ( I ) viscometer for several rates of shear. An at0 [L I tempt &-asthen made to find a position for the damping element a a W I in the vibrating system which would account for the steady shear f0 vr measurements as well as for the actual height of the resonance I 1000 curves obtained by the dynamic method outlined in this paper. This can be done in many ways, depending on the nature of the 0 8 800 liquid under consideration. 0 7 700 -
7
N
5
In the present case, aluminum soap compounds gelled in Varsol were used, and two possibilities which make use of mechanical models shown in Figure 8,a and b, were considered. I n
06
600
-
0 5
500
-
I
V O L U M E 19, NO. 2, F E B R U A R Y 1 9 4 7
127
e
tude is applied to the inner cylinder through a torsion wire and its motion is observed and plotted as a function of the frequency. From the resonance curves thus obtained, rigidity and damping of the liquid under test may be calculated as shown before, except that in the computations allowance must be made for the rigidity of the torsion wire as well as for the inertia of the inner cylinder. A schematic view of this apparatus is shown in Figure 11.
7
OB 1
0 8 01
I5
,
1
I
2 25 3
4
8
5 6
EFFECTIVE R b T E
r
IO
15
OF S H E E R
SEC,-'
10 4 0 50
20
Figure 10. Dynamic Viscosity of Aluminum Soap Gelled i n Varsol as Function of Rate of Shear 0 - 0 - 0 6 %, X - X - X 9 %, 0-0-0 12 %
With the representation of Figure 8,a,both the dynamic viscosity and the shear modulus would depend upon the presence of the same elastic units and it should be expected, therefore, that both quantities vary with the concentration of these units, according to the same general law. It is shown in Figure 9 that this is the case. The shear modulus and the RD viscosity are plotted as a function of the concentration for an aluminum soap gelled in Varsol. Both increase with concentration a t the same rate, and independently of the rate of shear a t which the viscosity is measured. The dynamic viscosity may be calculated by the following formulas, which are based on the concept shown in Figure 8,a:
8
res.
-
Strain
For most liquids, J
7
Therefore,
(3) \Vl
)rea.
With the experimental apparatus of Figure 1
I n Figure 10 the dynamic viscosities have been plotted against the effective rate of shear for different concentrations of aluminum soap in Varsol. There appears to be a linear relationship between the logarithms of dynamic viscosity and rate of shear, ELASTOMETER ADAPTED FOR MEASUREMENT UNDER STEADY SHEAR
The principle of measurement carried out with the more complete elastometer is similar to that described in the first part of this paper. The liquid under study is introduced into the annular space between two concentric cylinders. In the present case, however, an alternating torque of variable frequency and ampli-
A shaft, 1, runs in ball bearings mounted on the lower platform, 2. An outer cylinder, 3, may be clamped to the upper end of this shaft, and a heavy flywheel, 4, is fastened to its lower end. The assembly, 1, 3, 4, can be rotated uniformly at various speeds through a belt, 5 , driven by a governor-controlled motor. The upper platform, 6, carries the mechanism producing the alternating torque. It consists of a lever arm, 7, a connecting rod, 8, the double cam, 9, which forms a crank of adjustable eccentricity, and the flywheel, 10. The flywheel, 10, is driven from a continuously variable-speed transmission, not shown in the drawing. With this arrangement, oscillating motions of amplitudes as high as 20" may be applied to the shaft, 11, which carries the upper torsion wire, 12, and the inner cylinder, 13. A cap, 14, is provided to keep the inner cylinder centered and to prevent some liquids from being forced out when the outer cylinder is rotated. The lon-er torsion wire, 15, also serves to maintain the axis of both cylinders in alignment and a weight, 16, is used to keep the torsion wires under tension. For some applications it is useful to apply torque to the inner cylinder quickly and independently of the oscillating and shearing motions transmitted through the upper torsion wire and the liquid. This can be accomplished by rotating the xyeight, 16, through a reversible motor. The motion of the inner cylinder is magnified optically by a mirror, 17, which deflects a light beam from the light source, 18, to a scale on a drum not shown in the picture. This drum is designed to hold strips of photographic paper, and to rotate about a horizontal axis. It may be driven from the flywheel, 4,through a belt, and thus photographic records may be made, showing the amplitude of the oscillation of the inner cylinder as a function of the azimuth of the outer cylinder, which is proportional to the time. For certain types of measurements it is desirable to maintain the average position of the light beam on the recording drum independently of the motion of the inner cylinder. This can be accomplished by moving the frame, 19, carrying the light source around the circular track which is formed by the edge of the upper platform, 6. In order t o prevent evaporation of the solvent from the gel under test, an atmosphere of the solvent may be maintained in the apparatus by wrapping a sheet of transparent plastic, such as Kodaloid or Cellophane, around rings, 20, fastened t o the lower and upper platforms.
ANALYTICAL CHEMISTRY
128 Table 11. Possible Dimensions of Torsion Wires and Cy1inders Symbol a b
:
Diameter, inch
0.013
0.016 0,018 0.020
Steel Torsion Wires Length, Rigidity, inches dyne-om. per radian 2 1 6 . 9 X 103 41.2 2 2 60.0 111. 2
modulus is carried out in the same way as for the simple elastometer. I t is necessary, however, to allow for the rigidity of the upper torsion wire in the final computation. The shear modulus and the dynamic viscosity may be calculated from the observed resonance frequency and velocity ratio by the following formulas based on the model of Figure 12, which does not incorporate elements accounting for steady-shear viscosity. SHEAR h f O D U L E b
Inner CvlindPra ~~~~~.~ -
