V O L U M E 20, NO. 2, F E B R U A R Y 1 9 4 8 viscosity. The fully corrected viscosity will then be accurate to a0.01unit. The infinite dilution value, the intrinsic viscosity, is obtained by graphical methods from either the approximate or fully corrected inherent viscosities, depending upon the accuracy required. The intrinsic viscosity will be of the same order of accuracy as the inherent viscosities from which it was obtained. After a considerable amount of data has been obtained for any solute-solvent system over a wide range of polymer molecular weight, it is possible to derive a simple quantitative relation between the intrinsic and the inherent viscosity, the latter being evaluated at a constant, fixed concentration-e.g., 0.25 gram per 100 cc. ( 7 ) . When this has been done, a single determination of
155
efflux time of the solution will suffice to determine the intrinsic viscosity of the polymer. LITERATURE CITED
(1) Barr, G., “Monograph of Viscometry,” p 113, London, Oxford Press, 1931. (2) Cragg, L. H., J . Colloid Sci., 1. 2G1 (1946). (3) Jones, G., and Talky, S. K , Physics. 4, 215 (1933). (4) Riley, J. L., and Seymour, G. W.. IND. ENG.CHEM.,ANAL.ED., 18, 387 (1946). ( 5 ) Scarpa, O., Gam. chim. ital., 40, 271 (1910). (6) Wagner, R. H.. ANAL.CHEW,20, 155 (1948). (7) Wagner, R. H.. J . Polumer Sci., 2,21(1947). RECEIVED December 11, 1946. search Laboratories.
Communication 1127 from Kodak Re-
Application of ,Corrections in Viscometry of High-Polymer Sohtions R . H. WAGNER, Kodak Research Laboratories, Rochester, IY. I’
The importance of the relative densities and of the so-called kinetic-energy correction in the measurement of the relative viscosities of polymer solutions is discussed. Equations are derived relating the separate and combined contribution of these factors to the relative viscosity and to the inherent viscosity, an important polymer-characterizing function. A nomograph is presented by means of which the kinetic-energy correction contribution may be easily and quickly evaluated.
I
N T H E measurement of the relative viscosities of dilute highpolymer solutions by the capillary-tube method, it is often assumed that the simple ratio of the efflux times of the solution and the solvent is an adequate quantitative expression of the relative viscosity and of quantities derived therefrom- g., the specific viscosity, inherent viscosity, stc. This procedure may produce significant inaccuracies. This was recognized by Schulz (3)in his treatment of some of the errors incident to the application of the capillary viscometer in the measurement or evaluation of the specific viscosities of polymer solutions. In his analysis of the error produced by neglecting the so-called kinetic-energy term it is justifiably assumed that the densities of the solution and the solvent may be considered equal. The direct consequence of neglecting the density factor was not dealt with, probably because it is not dependent upon the dimensions of the viscometer. His treatment and results of the kinetic-energy error which are presented in a form not easily or immediately useful to the technologist, are less general both in form and in application than are given in this paper. The following argument, developed by a different approach and, unfortunately without prior knowledge of Schulz’s contribution, yields a result which is in quantitative agreement with his findings. I t is well knoIvn that the difference between the relative viscosity, ne, and the simple ratio of the effhx times, tito, depends upon (1) the ratio of the corresponding densities, d/da; and (2) the kinetic-energy (capillary-end) correction, which is given by the last term of Equation 1. The complete expression for the relative viscosity may be written as:
where V is the volume of the “measuring bulb” in a conventional capillary viscometer, I is the length of the capillary, and m i s it dimensionless coefficient, the kinetic-energy coefficient, which
we shall assume to.be unity. The coefficient of absolute viscosity of the solvent is represented by 70,and d and t are the density and the efflux time, respectively, of the solution. The corresponding quantities for the solvent are denoted by the subscript. I n the following discussion it is convenient to set d/do = p, t/to = a , and Vdo/8&& = K , so that Equation 1 becomes: (2)
The use of the term “inherent viscosity” to represent the value of the quantity, In n J C , a t finite values of C follows the suggestion of Cragg ( I ) . I t will be denoted by the symbol (71. This function should be carefully distinguished from the intrinsic viscosity, [VI, which is the infinite dilution value of the inherent viscosity:
= ln q,’C = f(C) [VI =C--co lim 171 f f(c)
(171
13)
the concentration being expressed in units of grams per 100 cc. COYTRIBUTIOK OF THE DENSITY FACTOR
To the Relative Viscosity. The contribution of the density factor, p , to the relative viscosity is given by the equation:
where A p is the amount by which the density ratio, p, differs from unity. Since, in most cases, the quantity, K ( a * - l ) / a ,is small compared with a, Equation 4 reduces to: (ATr’dens
%
a13
(5)
This approximation is admissible, inasmuch as the error produced in making it is of second-order magnitude. A similar
ANALYTICAL CHEMISTRY
156
Figure 1. Nomograph for Calculating the Kinetic-Energy Contribution to the Inherent Viscosity
principle is involved in all subsequent approximations made in this paper. To the Inherent Viscosity. The variation or uIlcertainty produced in the inherent viscosity, A { ? ) , by a given uncertainty in the relative viscosity is: Atlll =
a(?) X -
Aqr =
9 7l.C
(6)
Hence we have:
,
Absolute values of d,
-
do exceeding 0.8 are not likely to be 1 .Ad . found in practice, from which it follows that *- X - and do AC * ap/C = 0.010. Thus it may be concluded that the relative
densities need not be taken into consideration in the viscometry of polymer solutions, provided the allowable uncertainty in the inherent viscosity is equal to or greater than 0.01 of a unit, CONTRIBUTIOK OF THE KINETIC-ENERGY FACTOR
TO the Relative Viscosity. This contribution is given by the second term of Equatiun 2 or: by combining Equations 4 and 6. General Considerations. It is apparent from the equations above that the magnitude of the errors will depend only upon Ab, C, and 01, and will be completely independent of the dimensions of the viscometer. The magnitude of Ap is, in general, sufficiently small so that the total error related to it is usually within the allowable uncertainty of the measurement. It can be shown that A@ is the product of the solute concentration, C, and 1 Ad the quantity, - X -, where A d ] AC is the slope of the densitydo AC concentration curve. This curve is essentially linear up to concentrations of a t least 2.00 grams per 100 cc.; its sign and magnitude are determined by the relative magnitudes of the densities of the solute and solvent, d, and do, respectively. Values of d, 1
do and - X do systems.
Ad are given in Table I for twelve solute-solvent AC
(Av,)~.,.
CY2 - 1 BK X CY
Since p is unity to within 1 or (A?,)k.,.
(8)
2y0,Equation 8 may be written:
-1 a CY2
X
(9)
without significant error. To the Inherent Viscosity. By combining Equations 2 and 8 with Equation 6 the exact expression for the kinetic-energy contribution to the inherent viscosity is obtained. This expression : Ai7Ik.s. =
K(CY2
- 1) - l)]C
[a2 f K(a2
(10)
may be simplified to: A(7)k.c.
K(a2 CY2
- 1)
c
since K(a2 - 1)is small compared with a’.
(11)
V O L U M E 20, NO. 2, F E B R U A R Y 1 9 4 8 Table I.
157
Density Factors of Solute-Solvent Systems at 25”
c.
Solute Solvent Cellulose ace- Acetone tate butyrate Cellulose ace- Acetone tate Polyvinyl ace- Acetone tate Polyvinyl ace- Benzene tate Polvvinvl Water
Densitya of Solute,
Density of Solvent,
1.50
0.784
0.72
0,009
1.35
0.784
0.57
0.006
1.19
0.784
0.42
0.007
1.19
0.873
0.33
0.006
1.30
0.997
0.30
0.002
0.997 0.873 1.032 0.860
0.27 0.19 0.12 0.05
0.003 0.002 0.001 0.001
a.
do
1 d. - d o
Ad
do
(natural) 0.08 0.000 Polyisobutylene Benzene 0.95 0.873 -0.52 -0,004 1.469 Polyisobutylene Chloroform 0.95 -0.56 -0.005 1.469 Rubber Chloroform 0.91 (natural) a , Solute density values were obtained from tables in Meyer’s book ( 8 ) ; and from data given by Wearmouth ( 4 ) .
Table 11. Inherent Viscosity 0.10
0.25 0.50 1 .oo
1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
K Values Required to Produce a Concentration, Grams
0.125 0.051 0.021 0.011 0.006 0.004 0.003 0.003 0.002 0,002 0.002 0,002 0.002
0.250 0.051 0.021 0.011 0.006 0.005 0,004 0.004 0.003 0.003 0.003 0.003 0.003
0,500
0.053 0.023 0.013 0,008 0.006 0.006 0.005 0.005 0.005 0.005 0.005 0.005
be obtained a t the afore-mentioned minimum concentration to ascertain the required value of K . Table 11, which gives values of K required to make A ( q ) k e . equal to 0.01 for various inherent viscosities a t various concentrations, should be helpful in estimating this maximum a. The various combinations of vptolda and VI1 necessary to produce this K can be readily determined. The use of the nomograph in this capacity is most useful in the designing of new viscometers. In determining the optimum dimensions of a viscometer to be used in polymer measurement, the dimensions should be based upon the characteristics of the solvent of minimum (kinematic) viscosity likely to be employed. A viscometer so designed when used with solvents of greater viscosity will give rise to increasingly smaller kinetic-energy errors. COMBINED EFFECTS
Fully corrected values of the relative viscosity and of the inherent viscosity can be calculated from the quantities 01, C, 8, and K by means of the following equations:
I
A i ? k.s. = 0.010 100 cc. 1.000 2.000 0.060 0.055 0.025 0.032 0.016 0.023 0,012 0,020 0.011 0.020 0.020 0,010 0.020 0.010 0.020 0.010 0,020 0.010 0.020 0,010 0,020 0.010 0.020 0.010
General Considerations. In order to minimize the magnitude of the kinetic-energy correction, conditions should be provided which will produce the lowest practicable value of R. The factor V / 8 d of K (see Equation 2) is determined by the dimensions of the viscometer; a well-designed instrument should have as small a ratio of V/1 as possible consistent with size and operational requirements. The magnitude of the remaining factor of K will depend on the kinematic viscosity, qo/do, of the solvent, and its outflow time, to. The choice of solvent, and hence the kinematic viscosity, is usually fixed by solubility considerations. The magnitude of the outflow time can, for a given solvent, be adjusted between fairly wide limits by choosing the capillary bore and the average pressure head to be used in the viscometer (initial design). Very large values of t o should be avoided, however, as a method of minimizing K , since any advantage thus derived will be offset by the excessive time required in making measurements. To facilitate the evaluation of K from known or given values of T o , do, to, V , and 1; and also of A [ q ) k e , , from K , a, andC, a nomograph, shown as Figure 1,has been prepared. Its use should be readily understood by the following example, shown in Figure 1 as a broken line: The kinematic viscosity of acetone a t 25” C. of 0.0038 stoke, combined with the experimentally found value of 100 seconds for to, yields, on Scale 1, the indicated quantity. For an instrument whose V/I is 0.053 sq. em., we obtain K equal to 0.055 on Scale 3. If the ratio of efflux times, 01, is 2.500 for a solution containing 0.300 gram per 100 cc., a value of A { 7 ] h e . = 0.015 is indicated. This quantity, if significant, should be added to In a/C. The nomograph may be used in the reverse direction to determine the required conditions if the kinetic-energy factor is to be kept within a certain predetermined maximum. To this end the following procedure should be adopted: The maximum allowable value of A ( v } k . e . and the minimum concentration of solution planned to be used are aligned to produce a point on Scale 5. This point is then aligned with the maximum value of a likely t o
{VI
In
7+
=
CY
a2AP
(ZAP + I ) X K X - 1) + K ( a 2 - 1) ( A B + l ) C
+
[a2
(a2
1
In a -
c
K(a2
AB +
77
+
(13)
- 1)
a2C
Table I11 illustrates typical results obtained with viscometers of different dimensions. These serve as examples of the magnitude of the correction factors for polymers of four representative viscosities.
Table 111. Typical Data Obtained with Different Viscometers polyvinyl acetates in acetone, A B = 0.007 C . Viscometer 1, K = 0.0375; viscometer 2, K = 0.0067) C, L B g X Q s Polymer G/100 Cc. a C C C a: (v)
(System:
Viscometer 1
2.76
0.525 1.050 0.250 0,500 1.000 0.250 0.500
Very high viscosity [7] 4.75
-
0.240 0.490 1.000
Low viscosity
0.525 1.050 0,250
Low viscosity,
-
[q] =
0.16
Medium viscosity [q]
1.00
High viscosity [q] =
1,000
1.083 1.163 1.257 1.559 2.257 1.847 2.925 6.026 2.594 5.372 13.868
0.15 0.14 0.92 0.89 0.81 2.36 2.15 1.80 3.97 3.43 2.63
0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007
0,011 0.009 0.058 0.045 0,029 0,110 0.068 0.037 0,134 0.074 0.038
0.17 0.16 0.98 0.94 0.85 2.48 2.22 1.84 4.11 3.51 2.67
0,007 0,007 0,007 0,007 0.007 0.007 0.007 0.007 0,007 0,007 0,007
0.004 0.002 0,010 0,008
0.17 0.16 0.98 0.94 0.85 2.48 2.22 1.84 4.11 3.51 2.67
Viscometer 2
[VI
= 0.16
Medium viscosity [7]=
1.00
High viscosity [q] =
2.76
Very high viscosity [7] = 4.75
0.500
1,000 0.250 0.500 1.000 0,240 0,490 1,000
1.08 1.172 1.270 1.584 2.320 1.852 3.000 6.185 2.661 5.534 14.200
0.16 0.15 0.96 0.92 0.84 2.46 2.20 1.82 4.08 3.49 2.66
0.005
0.018 0.011 0.006 0.024 0.013 0.007
LITERATURE CITED
(1) Cragg, L. H., J . Colloid Sci., 1,261-9 (1946). (2) Meyer, K. H., “High Polymers,” Vol. I V , New York. Interscience
Publishers, 1942.
(3) Schulz, G. V., Z . Eleklrochem., 4 3 , 4 7 9 (1937). (4) Wearmouth, W. G., J . Sci. Instruments, 19, 132 (1942). RECEIVEDDecember 14, 1946. Presented at the High-Polymer Forum a t the 110th Meeting of the AMERICAN CHEMICAL SOCIETY, Chicago, Ill. Communication 1129 from the Kodak Research Laboratories.
e
