534
Vol. 63
GRAHAM WILLIAMS TABLE I11 AQREEMENTBETWEEN PARTICLE SIZEDISTRIBUTIONS FROM SEDIMENTATION AND ELECTRONIC SIZING Powder
Sand Glass beads Anhydrous monocalcium phosphate Monocalcium phosphate monohydrate Anhydrous dicalcium phosphate Dicalcium phosphate dihydrate Tricalcium phosphate Calcium pyrophosphate Insoluble sodium metaphosphate I n microna. Calculated from the
Dispersion fluid
Water Butyl alcohol Isobutyl alcohol 2-Ethylhexanol Ethanol 0 . 1 % Na hexametaphosphate in water Ethanol Ethanol Ethanol measured M , and u,, and equation 10.
Sieve Analysis Evaluation.-For those powders where ca. 80% or more of the particles have diameters greater than 44 p (325 mesh U. S. Screen), the particle size distributions as determined by sieving and sedimentation are identical, provided there is no significant amount of large particles to cause turbulent flow. Figure 4 demonstrates the agreement for a sodium isethionate sample. However, Fig. 5 shows that when ca. 35% or more of the particles of a powder have diameters smaller than 44 p , sieve analysis is inaccurate and fails to show significant differences for two powders having different particle size distributions. The results plotted in Fig. 6 offer a plausible explanation for the above described behaviors. Thus, for the case of sodium isethionate the geo-
Measured Mn"
5.3 23.6 4.6 0.61 2.5 2.05 1.16 2.08 3.0
m
1.96 1.38 2.38 3.54 2.22 1.96 1.95 1.95 2.07
Measured Mg'
ug
Ca1cd.b
Mg"
ug
1.96 1.38 2.38 3.54 2.22 1.96 .. .. 1.95 8.4 2.15 7.9 1.95 11.2 2.24 10.1 2.07
21.2 31.0 45.0 70.0 18.8 9 .O
2.00 1.34 2.18 3.50 2.22 2.09
20.7 32.5 43.5 71.5 16.8 8.0 4.4
metric mean diameter of a sieve cut as determined by sedimentation approximately equalled the average diameter of the two screen openings. However, for the anhydrous monocalcium phosphate sample investigated, the actual geometric mean diameter of a sieve cut is significantly different from the average diameter of the two screen openings. Figure 6 further shows that a significant portion of the powder lies outside the screen opening 1imits.l' I n addition, this figure demonstrates that exactly the same screens give different size classifications for different powders. Consequently, accurate sieve analysis is obtained only when the sieves are pre-calibrated against the more precise methods for the individual powder under investigation.
THE MEASUREMENT OF DIELECTRIC CONSTANT AND LOSS FACTOR OF LIQUIDS AND SOLUTIONS BETWEEN 850 AND 920 MC./SEC. BY MEANS OF A COAXIAL TRANSMISSION LINE BY GRAHAM WILLIAMS The Edward Davies Chemical Laboratories, University College of Wales, Aberystwyth, Wales Received July 3,I068
A coaxial line method of measuring the dielectric properties of liquid systems in the frequency range 250-1000 Mc./sec. (or higher) is described. The method is essentially that of Roberts, Westphal and von Hippel. Data are presented for typical systems of high and low E' and e" values showing results which compare favorably with other methods in this region. Evaporation losses, access of moisture and temperature are readily controlled for the liquids.
(1) C. N. Works, T. W. Dakin and F. W. Boggs. Proc. Inst. Radio Eng., 88, No. 4, 245 (1946). (2) C. N. Works, J . A p p l . Phya., 18, 605 (1947). (8) 8. J. Reynolds, Uenerol Elect&c Rev., 80, No. 9, 34 (1947). (4) J. V. L. Parry,Proc. Inst. Blectr. Eno., 98, pt. 111, 303 (1951). (5) D. L. Hollowry and G. J. A. Cassidy, ibid., 99, pt. 111, 364
fication of an i,mpedancemethod of Cole12by Marcy and Wyman. lS However, elaborate equipment is required, and in some cases only moderate reproducibility of results could be achieved. The present apparatus offers an excellent alternative, since the effects of stray inductance and capacitance are eliminated by confining the electric field between the coaxial conductors and treating the system in terms of distributed circuits. The apparatus could be used for lower frequency work, the lower limit being fixed 'by the length of coaxial transmission line available, and the measurements could be extended
(1952). (6) J. H. Beardsley, Reu. Sci. Inatr., 84, No. 2, 180 (1953). (7) R. A. Chipman, J . A p p l . Phys., 10, 27 (1939). (8) W. L. G. Gent, Trans. Faraday Soc., 60, 383 (1954). (9) H. Linhart, 2. physik. Cham.. BSS, 2 1 (1937). (IO) J. B. Bateman and G. Potapenko, Phgs. Rev., 67, 1185 (1940).
(11) W. P. Conner and C. P. Smyth. J . Am. Chem. SOC.,64, 1870 (1942). (12) R. H. Cole, Rev. Sei. Instr., l a ( 6 ) . 298 (1941). (13) H. 0. Marcy and J. Wyman, J . Am. Chem. Soc., 63, 3388 (1941).
The frequency range 100 to 1000 Mc./sec. has proved in the past to be a difficult region for the measurement of dielectric constant and loss factor. The difficultiesarise from residual lead and electrode inductance, resistance and capacitance, and the correction of these factors. Measurements have been achieved using re-entrant cavities1-6 and parallel transmission lines7-" and also by a modi-
MEASUREMENT OF DIELECTRIC CONSTANT BY
April, 1959
A
535
COAXIAL TRANSMISSION LINE
up to 3000 Mc./sec. by means of commercially available oscillators, with a more accurate vernier scale on the coaxial line. The components used are commercially available and although we have used the most sensitive of the immediately procurable detecting systems, alternative simpler versions can be used when reduced accuracy would suffice. Experimental The arrangement of the apparatus is shown in Fig. 1. The signal oscillator is a General Radio (G.R.) 1209 B unit oscillator operating in the range 250 to 920 Mc./sec. The line used is a G.R. 874 LBA slotted line, which is a 50 ohm, air dielectric coaxial transmission line, whose electric field is sampled by a probe that projects through a longitudinal slot in the fine. The voltage induced in the probe is measured by means of a G.R. DNT3 detector. This comprises a G.R. 874 MR mixer rectifier, a G.R. 1209 B unit oscillator operating between 250 and 920 Mc./sec., and a G.R. 1216 A unit i.f. amplifier. The latter is a four-stage high gain intermediate frequency am lifier operating a t 30 Mc./sec., with a band width of 0.7 dc./sec. The meter is calibrated in a decibel and linear scale, and has a precision step attenuator for measuring relative signal levels. The cell for liquid dielectrics is a 30 cm. length of 50 ohm coaxial transmission line which fits on the slotted line by means of flanges. The lower end of the cell is shorted by means of a brass plate, and a small slot at the shorted end (0.37 cm. X 0.15 cm.) allows liquid to enter the coaxial line. Liquid depths are varied continuously by firmly fixing the glass cylindrical container for the liquid (A, Fig. l ) , whose diameter is twice that of the air line ( L e . , 2 X 1.5 cm.), on to a small table having a vertical rack-and-pinion movement. Depths are found by measuring the external liquid height relative to a reference mark on the cell, the distance between the reference and the physical end of the line being known: a long-focus travelling microscope or cathetometer readable to dz0.02 mm. is used for this purpose. Variation in the liquid temperature can be achieved by a concentric jacket fused on to the glass cell (A of Fig. 1). Circulation of a thermostatically controlled transparent liquid (water or Nujol) allows ready control of the temperature. A is some 30 cm. in length and the depth of liquid is usually of the order of 5 cm. A Teflon ring acts a8 a stopper at the top of A, and serves to center the coaxial line in the cylindrical container A and to reduce evaporation losses: guard tubes or a dry air stream serve to prevent access of moisture when the liquid is hygroscopic. Frequencies beyond 920 Mc./sec. can be measured by choice of suitable harmonics of the signal oscillator: for instance, fairly accurate measurements have been achieved a t 1700 Mc./sec. using the second harmonic of 850 Mc./sec. The very small band width of the detector amplifier relative to the beat-frequency (30 Mc./sec.) allows this to be done for chosen harmonics. The purity of the standing-wave for the harmonic frequency can be checked from the symmetry of its pattern in the line and from the measured widths of the succession of nodal minima.
Method.-The method of measurement is due to Roberts and von Hip~e1.l~They obtain the relation tan h w d =
wd
s
- j tan &XO
jBd {1
(1)
- j s tan Blxol
where yz is the propagation constant of the liquid, d is the liquid depth, s = Emin/Emax is the standing wave ratio, $0 is the distance of the first node from the liquid surface, XI, is the wave length in the air = 27r/X1. Writing the portion of the line and right-hand side of equation 1 in the form x j!j x =
(sa
Bld ( 1
-
1) tan &ro
(14) S.
(1946).
.
+ sz tanz j%zo]' y e -
+
s(l Bld { 1
1I
Fig. 1. I
I
' ' * '
(2)
Roberta and A. R. van Hippel, J . A p p l . Phys., 11, 610
I
I
I
1.0
2.0 3.0 4.0 Liquid depth (cm.). Fig. 2.-Water at 900 Mc. and 18.5'. Observed SWR = Emi./Emax; xo = distance of first node from liquid surface.
x and 3 are determined from experimental data. Expressing x j.y = CejI, we have
+
C
=
v'FTj7;
~=tan-1(y/x)
Writing yzd = Tejr, then by means of correlation chartsls T and r are found from C and f . Since yz = a 2 jP2 where a2 is the attenuation const,ant and p2 is the phase constant for the liquid, then
+
a2
= *-T
COS
7
T sin .82=---- d
7
The real and imaginary factors of the dielectric constant are found by the relations €'
= n2(1
- (raZ/P22):
E"
= 2n*(az/Pz)
(3)
where n is the generalized index of refraction for the liquid and is equal to XIB~!~~. For large s values, the direct decibel difference between maxima and minima was used, since (db response),al
+ tana 61x0) + sa tan2 plxo)
A
- (db response)min= -20
logs
For small values of s, the width of minimum method was used. l5 A 3 db displacement from the voltage minimum as indicated by the amplifier, (15) W. B. Westphal, in "Dieleotrio Materials and Applications," Chapman and Hall, 1854, p. 67.
GRAHAM WILLIAMS
536
Vol. 63
a 1% error in the liquid depth introducing an error of 1.3 to 3.5 % in e' and of 0.6 to 1.6% in e". The latter is more significant at small 8 values. Using these data to calculate 7 via
0.15
0.10 b'
gives with e m = 4.5l5 a value 7 = (8.910.2) X 10-l2see. If we take ea = 80.8,lSd m = 4.5 and 0.05 e" = (3.9 f 0.1) the value becomes 7 = (9.0 f 0.2) X 10-l2sec. At the same temperature, the best value in the literature is (9.6 i 0.4)16:other values are 9.5 a t 20°17 and 8.5 a t 21°.'* Thus it is evident that the experimental procedure and 6.5 7.5 8.5 9.5 method are suitable for the measurement of large log f (CIS). dielectric constants to an accuracy of about 2% Fig. 3.-Bu;NPic in dioxane: 1.03 X 10-3 M at 17'. and the loss factor to about 4%. The curve is a calculated Debye function for e" = 0.145: Water was also investigated a t 800 Mc. using the 7 = 620 x lo-'* sec. The experimental uncertainty in the procedure described by Little.le The dielectric observations is given by the diameter of the circles. constant was accurate to f 1.5%) the loss factor is required for the points of twice minimum power. to AS%. The relaxation time calculated from Then the loss factor was (9.8 f 0.5) X 10-l2sec., which is in good agreement with the results quoted above. 8 u A / ~ However, the experimental procedure is extremely where A is the distance between points of twice tedious compared with the Roberts-von Hippel minimum power. If A is very small, then the method, and cannot be recommended for general points of four times minimum power may be em- application. ployed. This corresponds to a 6 db displacement (2) Tetra+butylammonium Picrate in Dioxfrom the voltage minimum and s = rA'/t/3~1, me.-A 1.03 X 10-2 M solution of the salt was where A' is the distance between points of four examined as an example of a low dielectric constant, times minimum power. low loss medium. The dielectric constant values The observed widths must be corrected for the were accurate t o I 0.5%) and the loss factors to air line attenuation. l5 il%. The frequency range 300 to 900 Mc. was Some Typical Results. (1) Water.-Water pro- covered, and using a Hartshorn-Ward dielectric vided a check of the accuracy of the apparatus test set, the range 6 to 30 Mc. also was measured. and method for a high dielectric constant, medium From the results it was found possible to construct loss liquid. The variation of s and xo with varying a complete Debye curve corresponding to a single depth is shown in Fig. 2. A study of these curves relaxation time. The accuracy of the relaxation which can be calculated for given dielectric param- time is dependent upon the accuracy of (eo - e m ) . eters, serves to emphasize the variations in the The curve is shown in Fig. 3, where the measured accuracy of the results likely to bo .achieved with em) for varying conditions, e.g,, for different xo and d values. e"(max). is found to coincide with The evaluation of some of these results is shown in the solution and 7 is 620 X 10-l2sec. Table .I. (3) Other Systems. (i).-A 1.03 M solution of benzophenone in xylene was studied at 900 Mc. TABLE I in order to test the reproducibility of e' and e n as VALUESOF E' AND e" WITH VARYINGLIQUID DEPTHFOR the liquid depth is varied for a low dielectric conWATER AT 900 M c . AND 18.5' stant medium. The results me summarized in d(am.) e' en Table 11. The relaxation time was calculated 2,625 84.4 4 . 5 f 0.10 using the Debye dilute solution equation. The
*
I
-
2.669 2.803 2.849 2.891 2.965 4.463
83.9 82.2 82.6 82.7 82.6 80.7
4.1 & . I O
3.76 f 3.90 =k 3.91 f 3.96 f 3.70 f
.I6 .10 .10 .10 .10
The uncertainty quoted for the loss factor is that involved in reading the von Hippel charts for this system. The deviation of the e' values from 80.216 arises principally from two factors: the errror owing to the meniscus in the depth (d), and the air attenuation in the line which has not been taken into account in the calculation of these results. The former becomes less the greater the depth: (16) E. H.Grant, T. J. Buchanan and H. F. 26, 158 (1957).
Cook, J . Chsn. Phur..
TABLE I1 1.03
M BENZOPHENONE IN XYLENE AT 900 Mc. 10'9
x
7,
d(om.1
e'
e"
8ec. (calcd.)
2.99 4.75 5.32
3.65 3.65 3.65
0.230 f 0 . 0 1 2 . 2 4 2 f .OS .254 f .10
20.0 21.0 21.7
AND 18O 101s r.
x
#eo. (lit.)
22 (20) 18.1(21) 20.4(22)
(17) C. H. Collie, J. V. Hasted and R. M. Ritson, Proc. Phys. SOC., (London). 60, 145 (1948). (18) J. A. Saxton and J. A. Lane, Proc. Roy. Soc., (London), 8318,
(1952).
400 (19) V.
I. Little, Proc. P b s . Boc., (London), 66B,175 (1953). (20) D. H. Whiffen and H. W. Thompson, Trans. Faraday Soc., 49A. 114 (1946). (21) F. J. Cripwell and G. B. B. M. Sutberland, ibid., 149 (1946). (22) E. Fiecher, 2. Naturlorechung, Ca. 707 (1949).
I
EVALUATION OF DIELECTRIC DATAFOR LIQUIDSAND SOLUTIONS
April, 1959
literature values refer to benzene as solvent; the viscosity difference from xylene is negligible. (ii) .-Pure chlorobenzene was studied over the frequency range 300 to 900 Me. a t 12’. The results are shown in Table 111.
537
10-’2 sec. a t 12’ from the Cole-Cole circular arc. It is to be noted that their value for e’ a t this
temperature is 5.88 f 0.02, whilst the present work obtains 5.84 f 0.03. This indicates the accuracy to which low dielectric constants can be measured. As this work was being completed the thesis of TABLEI11 F. C. De VOS*~ was received. This describes simiPURECHLOROBENZENE AT 12’ FOP DIFFERENT AIR WAVE lar measurements using a coaxial line with a numLENGTHS (XI) ber of fixed filled cells having Teflon windows. Xl(cm.) a’ 6” 10’0 X e”/f This thesis also provides an excellent survey of other 0.275 3.06 5.90 33.32 high frequency dielectric methods. 5.86 .200 2.86 42.94 Grateful thanks are offered to Messrs. Courtauld’s 5.83 .172 49.84 2.86 Educational Trust Fund for a grant which has 5.81 .124 74.73 3.10 made this work possible. The author also wishes 5.81 .093 99.28 3.10 to thank Dr. Manse1 Davies for his advice and constant encouragement during this work. A Taking the mean e “ / f to be (3.0* 0.1) X 10-lo, maintenance grant from the Department of and e m to be na, ie., (1.522)2 = 2.32, then using Scientific and Industrial Research is gratefully equation 4, 7 = (13.6 f 0.5) X sec. acknowledged, and also helpful correspondence Hennelly, Heston and Smyth2aobtained 12.5 X from Professor Bottcher and Dr. F. C. De VOS. (23) E. J. Hennelly, W. M. Heaton and C. P. Smyth, J . Am. Chsm. Sac., TO, 4102 (1948).
(24) F. C. De Vos, Theah, Leiden, 1958.
THE EVALUATION OF DIELECTRIC DATA FOR LIQUIDS AND SOLUTIONS BYGRAHAM WILLIAMS The Edward Davies Chemical Laboratories, The University College of Wales, Aberystwyth, Wales Rscsivsd Bsptsmkr 86, I968
A number of useful modifications and correlations of current methods of evaluating the dielectric parameters for polar
fi%lute uids and solutions are described and illustrated by examples. These include the comparison and correlation of the solution” and “pure li uid” equations for the relaxation time, simple gra hical methods for determining the relaxa-
tion time and its distribution Tactor in the general cases, and the calculation of $pole moments via Guggenheim’s relations using Smyth’s dielectric parameters.
The Dilute Solution and Pure Liquid Equations for a Single Relaxation Time.-The general equations for the dielectric constant (E’) and loss factor (e”) of a polar medium possessing a single relaxation time ( 7 ) are e’ = e m
eo +1 +
em
w272
Here EO is the static dielectric constant of the medium and em is its value a t an angular frequency (0) so high that all orientational polarization has vanished. For a dilute solution of a solute having dipole moment p in a non-polar solvent, Debye,’ twentyfive years ago, made the approximation ’
after allowances for the small solvent absorption, is precisely defined by a single relaxation time. Of the various ways of evaluating the parameters of equations 1, three convenient procedures due to Cole may be quoted: (i) the circular arc plot of e” against e’; (ii) the linear plot of e’ against s” X w ; (iii) the linear plot of e‘ against e”/@. All gave 7 = 353 f 6 X 10-l2 sec. and the corresponding value of (eo - e,) = 0.99 f 0.01. Comparison of (lb) and (2) gives (EO
- em)
N?r/.LlC
=:
+
K 6750 ___ kT
where K = (€1 2)2. Using the Debye dilute solution equations we obtained p = 11.4 0.1 D: Maryott2 found for the same solute in benzene p = 11.7 D. Accordingly, using the above value of (EO - E m ), K =.(EO - e m ) / 5 . 8 9 X lo-’ = 16.8. Equation 2, in which K = (el 2)2, can be arrived a t in various ways.8 In one of them el replaces e‘ the real (and frequency dependent) dielectric constant of the solution; in another (e1 2)2 is substituted as an approximation for (eo 2)(e, 2). The values of these factors are (e1 2)2 = 18.8; (e’ 2)2 = 29.2 to 19.6 over the measured frequency range; (eo 2)(em 2) = 24.3. It is clear that even the best of these factors (18.8) would give a 7-value 20% below that
*
+
In this relation €1 is the static dielectric constant of the non-polar solvent and c is the molar concentration. We wish firstly to compare the relaxation times deduced by the use of equations 1 and 2 for the typical case of a 6.54 X M solution of tri-nbutylammonium picrate in m-xylene at 17”. Measurements in these laboratories from 196 kc./sec. to 1700 Mc./sec. show that this system, (1) P. Debye, “Polar Moleculea,” Chemical Catalog Co., New York. N. Y.,1929, Chapter V.
+ ++
+
+
+
+
(2) A. A. Maryott, J . Rawarch Natl. Bur. Standards, 41, 1 (1948). (8) C. J. F. B(lttcher, “Theory of Electrio Polariaation,” Elrevier PubCo., 1962, p. 374.
