1534
J. J. BIKERMAN
value. However, it has been reported by Smith14 that the width of such crystallites are from 360 to 390 A. The 400 A. length, therefore, does not seem to be unreasonable. (2) Again, if the present values of y and B are not far from reality the chains in the amorphous phase of polyethylene are highly oriented. Their configurations are much more extended than the random coil configuration. Thus the specific volume of the amorphous phase niay be quite different from the value extrapolated from the liquid state. This fact also supports the view on the growth of spherulites that the formation of a nucleus is induced by the growth of the first crystallite. (3) Equation 3 indicates that for polydispersed polymers the degree of crystallinity depends on the number average molecular weight. The number average molecular weights of commercial polyethylenes usually vary within a very narrow range.16 As a result, the crystallinity is not greatly affected by the molecular weight based on viscosity measurements. Acknowledgment.-The authors are indebted t o Dr. H. W. McCormick of the Physical Research Laboratory for his assistance in the centrifugation of polymer samples.
Vol. 62
counted in the term dXu,etc. I n order to count each type of undesirable configurations only once we arrived a t the present form of equation 7. To evaluate c $ ~ , one has to count the configurations containing groupings of different numbers of x, -x steps, such as “x, -x”, “x, -x, x”, etc., separately. Let qhXdenotes the number of configurations containing the largest grouping of x and -x steps. Then
,
The first part of 41%is the permutation within the x step grouping and the second part is the permutation outside this grouping. For the number of configurations & containing the next largest grouping of x and -x steps, the expression is 9% =
+
n, n-. - l)! [ ( ( n , - l)!n-.!
+
(n,
+ n-. - l)!
n,! (n-z
-
l)!l
I
The first bracket again is the permutation within the x step grouping. The first term in the second bracket is the permutation outside the x grouping and the second term which represents the configAppendix urations with another x step adjacent to the x The Evaluation of Q’.-In the derivation of grouping is subtracted from the first term. Thus equation 7, c $ ~4,~, , and c $ have ~ ~ ~ already been c $ is ~ the sum of all these @qx, dzX. . . . terms correcounted once in the term qjX; similarly 4xyzhas been sponding to all possible numbers of x step groupings. All the other 4 terms have to be evaluated similarly. (14) D. C. Smith, “Molecular Structure of Marlex Polymers,” It is quite obvious then when y becomes large 8’ is presented a t ACS meeting, Dallas, 1956. impossible to evaluate numerically. (15) H. Smith, J . Poly. Sei., 21, 563 (1956).
USE OF HYSTERESIS OF WETTING FOR MEASURING SURFACE TENSION1 BY J. J. BIKERMAN Department of Civil and Sanitary Engineering, Massachusetts Institute of Technology, Cambridge 39, Mass. Recewed June 16, 1968
The hysteresis of wetting observed when a artially wetted vertical slide is moved in and out of a liquid can be used to calculate the surface tension of the liquid. !&e vertical displacement of the liquid surface h2 - h1 = (2y/gp)’/e(M2:/a M l ’ / g ) , if A f 2 = 1-( 1 - F22/L2y2)‘/zand Ilfl = 1 - (I - F12/L2y2)’/2; F , and F2 are the capillary forces on the slide a t the two positions of the surface, L is the length of the 3-phase boundary, -1 is surface tension, g is acceleration due to gravity, and p the density of the liquid. The method was tested successfully on water and mercury and is suitable for the latter liquid.
I. Introduction I n the course of a study of the hysteresis of wetting it was noticed that the surface tension y of the liquid can be calculated from the hysteresis data. As a new method of measuring y , the procedure seems t o be advantageous for mercury only. Solids which give a zero contact angle with mercury are soluble in the latter and thus are apt to contaminate it (alt,hough the contamination may be insignificant in favorable instances2); thus the easier methods of measuring y cannot be used with safety, and those that can are not more convenient than the present one. At any rate it has a theoretical significance and thus deserves a description. (1) Presented a t the 134th National Meeting, American Chemical Society, Chicago, September, 1958. (2) E. A. Owen and A . P. Dufton, I’roc. Pliys. Soc. ( T m d o h ) , 38, 204 (1920).
11. Theory A vertical plate suspended in a liquid surface (see the continuous curve in Fig. 1) is subject to capillary force3 F, =
L ~ C O S ~ ~
~
(1)
L is the perimeter of the plate (that is the length of the 3-phase boundary line) and is the contact angle. If now the level of the liquid be raised t o the position indicated by the dashes while, because of the hysteresis of wetting, the 3-phase line retains its position, the contact angle increases to e2 and the capillary force diminishes t o F~ = L~
COS
e2
(2)
From F1 and Fz, y can be calculated without measuring el and e2. (3) E.Q.,J. J. Bikerman, “Suiface Chemistry.” 2nd edition, Aca. deniic Press, Inc., New York, N. Y . , 1958, p. 9.
,
MEASUREMENT OF SURFACE TENSION BY HYSTERESIS OF WETTING
Dec., 1958
Figure 2 represents the vertical cross-section of a meniscus at a vertical plane wall (shaded) for el > 90". The equation of the curve is simply r l R = hgp
cos e dB = g p h dh
I
(3)
if R is the radius of curvature a t a point h cm. below the Rat portion of the meniscus, g is acceleration due t o gravity, and p is the density of the liquid minus the density of air. As Fig. 2 shows, R d8=ds; ds is the length of the surface curve between 8 and 8+d8, and 8 is the variable angle between this curve and the vertical. On the other hand, ds = -dh/cos 8, see insert in Fig. 2. Hence, R = -dh/cos 8 de. Setting this into 3, we obtain -y
1535
Fig. 1.-Vertical cross-section of a meniscus a t a vertical plate (shaded). Because of the hysteresis of wetting, the 3-phase line does not move when the liquid surface is raised from the continuous to the discontinuous curve; el is the initial and e2 is the resulting contact angle.
(4)
and after integration between 8 = O1 and 8 = go", that is also between h = hl and h = 0 y
(1
- sin e,)
= .3gphI2
(5 )
Thus, hl = ( 2 y / g p ) ' / 2 ( 1 - sin O1)'/2. But sin el = (1 - cos2 8 , ) ' / 2 and cos 8, = F1/Lr from equation l. Hence
Analogously
and
Equation 8 rather t,haii equation 6 or 7 must be used to calculate y because neither 121 nor hz is measured in the method presented here, but only the difference h2 - hl. Since equation 8 does not make clear how y varies with the measurable quantities, the approximation
Fig. 2.-Sketch for calculating the relation between the contact angle (e,) a t a vertical wall (shaded) and the vertical distance of the 3-phase line from the flat portion of the meniscus.
of the top plate when screw 5 was turned. The pitch of the screw was 0.0635 cm. When assembled, screw 5 was in sleeve 3, and the top plate was resting on cylinder 7. The value of hz - hl was not exactly equal to the displacement shown by scale 4 because of the change in the volume of the meniscus, caused by this displacement. The correc(9) tion term evidently was Ah = ( F s - F 1 ) / g p A cm., A being is helpful4; the coefficient k , a pure number, varies the area of the vessel; it had to be added to tne difference between the readings on scale 4. In our experiments, A only from 1.00 t o 1.25 when F / L y increases from was 60 cm.2 or 19.0 cm.,2 and Ah ranged between 0.001 and zero t o 0.80 or 8 varies between 37 and 143". 0.003 cm. The absence of any measurable shift of the 3-phase line 111. Experiments is best proved by measuring F Iand FZat different displaceThe vertical plate was suspended from the beam of an ments of plate 8. Thus, if displacements of 0.06 and 0.09 analytical balance, whose pans previously were removed, and cm. give identical values for y, there certainly was no moveequilibrated with a weight at the other end of the beam. ment of the air-liquid-solid boundary during the experiment. Then a dish filled with liquid was raised until the liquid When the displacements were greater (e.g., over 0.18 cm.), touched the lower edge of the plate. An additional weight the calculated value of y usually was too small, undoubtedly W1 now was needed to maintain the horizontal position of because the 3-phase boundary did not remain stationary. A considerable error can be introduced by suspending the the beam; this weight was reckoned positive if added to the equilibrating load, and negative if hanged a t the side of the slide on a thin thread or a fine hook. The extension of the plate. Next the liquid level was raised by a definite length, thread or the hook depends on the load; if the latter varies equal to h2 - hl of equation 8, and again the weight (W2)by the value Fz - F1, the extension also changes, thus affectingthe true value of h2 - hl. As, in the present series needed for balance was determined. A convenient device for raising the dish is illustrated in of experiments, h2 - hl was less than 0.1 cm., the change of Fig. 3. It consists of three separate parts shown one above 0.001 cm. in the thresd or hook length would cause an error the other in the sketch. The bottom plate (1) carries two of more than 2% in y, see equation 9. The values of F1 and Fz to be put into (8) or (9) also were slender columns (2, 2) and a threaded sleeve (3) with a scale (4). The middle part is a screw ( 5 ) provided with a a little different from the weights Wl and W Zneeded for pointer (6). The top part (8) is a thick plate in which 3 equilibrium. This was caused by buoyancy. The buovcylindrical indentations were drilled, one for the screw and ancy correction of Fl was always small and estimated from two for the guiding columns ( 2 , 2 ) which prevented rotation the downward displacement of platform 8, Fig. 3, required to withdraw the slide from the liquid. The additional (4) NOTEADDED IN THE PRooF.-According t o J. Guastslla, J . correction for F2 was gpzua(h2 - hl), if w and 6 are the width Chim. p h y s . , 51. 688 (1051), B diffrretitisl forni of eqiintiort (9). \\,it11 nnd the thicliness of the slide; for instance, 8'2 - F, was k 1 , was derived by It. Mrttaloii, "TIiBsr de Doctorst," Lpon, lYd8. snittller than W z - W1by 980.4 X 2.5 X 0.005 X 0.067
-
1536
J. J. BIKERMAN
Vol. 62
-0.015 g. ( L e . , FZ was -0.013 g.); water tended to expel the plate. The value of hz - hl was 0.635 (0.076 0.013)/1.00 X 60 = 0.0650 cm.; 1.00 is the density of water and 60 cm.2 was the area of the dish. With these values, equation 9 affords y = 73.5 g. sec.-Z. The considerable error presumably was caused by the low precision of the values of Fl and F z which, as mentioned above, could be measured only to the nearest 0.001 g. From the many measurements performed on mercury menisci it appears that the surface tension of a mercury pool kept in a closed balance box in air rapidly decreases. The value of y calculated from the two first measurements on a particular pool was greater than those derived from the subsequent readings; and when the greater (absolute) value of F was determined first, the calculated y was greater than when the measurement of the smaller force preceded that of the larger one. The following description of a t'ypical experiment will make these statements clearer. An aluminum foil, 0.0052 cm. thick and with a perimeter L of 6.10 em., was brought in contact with mercury in a dish of 60 cm.2 area a t 24.6-24.7' and a relative humidity of approximately 30%. The W1was -0.800 g. When the dish was raised 0.0635 cm., the needed weight increased to - 1.723 g. The dish was lowered to the initial position and IB1 was found to be -0 791 g. After another ascentdescent-ascent the weight was -1.677 + -0.774 + -1.652 g. Thus the difference W1 - W Zwas 0.923, 0.886 and 0.878 for the three successive pairs of values. When another sample of mercury was taken from the same bottle but the slide a t once immersed in it for approximately 0.1 em., the first weight (W,) was -1.721 g.; after lowering the dish by 0.0635 cm. the weight (W1) was -0.780 g.; thus the difference Wl - W zwas 0.941 g., that is eater than the greatest value of W1 W zobserved when %?was measured before W Z . The above description is summarized in the table in which arrows indicate the sequence of measurements.
+
7-
.4 -2
I
I
,
1
'
I
Fig. 3.-Instrument used to raise the liquid level by small measurable increments. See the text for the meaning of the numbers. g. cm./sec.a when the slide wa8 2.5 cm. wide and 0.005 cm. thick, and water was raised 0.067 cm. Although the balance was sensitive to 0.0001 g., readings could be made to the nearest milligram only because of the powerful dumping of the system, made clear by equations I and 2. To satisfy the requirement for a sturdy suspension and to have only a small buoyancy correction, the following form of the vertical plate was selected. A piece (for instance 2.5 X 1.3 X 0.005 cm.) of a thin metal foil was introduced between the bottom edges of two parallel beryllium-copper plates (of about 10 X 5 X 0.03 cm.) so that the major part of the foil (e.g., 2.5 cm. wide and 0.8 cm. in height) remained outside, and the 3 metal members were glued or soldered together. Then a hole was drilled near the top edge of the Be-Cu plates; it served to suspend the plates directly on a hook of the balance. A typical experiment with water may now be described. A strip of silver foil was coated with a polystyrene. Its L was 5.00 qm. and the buoyancy correction was, as calculated in a prevlous F g r a p h . 1.3 mg./mm. immersion. After the contact wit water at 21 O and a relative humidity of 50% the additional weight W 1was +0.075 g. (and Fl was +0,076 g.); the ositive sign means that the plate was pulled into wat,er. &hen the dish was raised 0.0635 cm., W Zwas
+
-
-Wl and -W2 values in g. wt. -Wi 0.800 0.791 L i" L ;.774L 1.721 1.677 1.652 1.723 -Wz 0.941 0.886 0.878 Vi - W2 0.923 Taking, in this experiment, W1 to be the average of the two earliest data, that is, equal to -0.790, and analogously Wzto be -l/2(1.723 1.721) = -1.722g.andcorrectingfor buoyancy, we obtain F1 = -0.780 g. wt. and Fz = -1.696 g. wt. The distance hz hl was 0.646 cm. Equation 6 then gives y = 464 g , sec. a. This is a reasonable value. As would be expected, the y of mercury depended on acci; dental surface contamination more than on the bulk purity of the sample. There was no definite difference between the yvalues of purest commercial mercury straight from the bottle and of mercury purified only by spraying through dilute nitric acid; however, a thorough drying of the latter sample proved necessary. They was independent of the material of the slide (aluminum, silver coated with silver sulfide, metal coated with polystyrene or with polyethylene).
+
-
Acknowledgment.-The financial assistance of the Lord Manufacturing Company of Erie, Pennsylvania, is gratefully acknowledged.
.
