H
I L L I N 0 I
S
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
PRODUCTION NOTE
University of Illinois at
Urbana-Champaign Library
Large-scale Digitization Project, 2007.
UNIVERSITY QF ILLINOIS BULLETIN
ISSUED WEEKLTY
VoL XXV January 31, 1928 No. 22
[Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, under
the Act of August 24, 1912. Acceptance for mailing at the special rate of postage provided
for in section 1103, Act of October 3, 1917, authorized July 31, 1918.]
THE SURFACE TENSION OF
MOLTEN METALS
WITH
A DETERMINATION OF THE CAPILLARY
CONSTANT OF COPPER
BY
EARL E. LIBMAN
SBULLETIN No. 173
ENGINEERING EXPERIMENT STATION
uMaSH ax tas Uxxaanar Or ILnMetsO", UsaMa
Pmes. Thlkzx Oaum
N-
T- HE Engineering Experiment Station was established by
act of the Board of Trustees of the University of Illinois
on December 8, 1903. It is the purpose of the Station to
conduct investigations and make studies of importance to the
engineering, manufacturing, railway, mining, and other industrial
interests of the State.
The management of the Engineering Experiment Station is
vested in an Executive Staff composed of the Director and his
Assistant, the Heads of the several Departments in the College
of Engineering, and the Professor of Industrial Chemistry. This
Staff is responsible for the establishment of general policies gov-
erning the work of the Station, including the approval of material
for publication. All members of the teaching staff of the College
are encouraged to engage in scientific research, either directly or
in co6peration with the Research Corps composed of full-time
research assistants, research graduate assistants, and special
investigators.
To render the results of its scientific investigations available"
to the public, the Engineering Experiment Station publishes and
distributes a series of bulletins. Occasionally it publishes circu-
lars of timely interest, presenting information of importance,
compiled from various sources which may not readily be acces-
sible to the clientele of the Station.
The volume and number at the top of the-front cover page
are merely arbitrary numbers and refer to the general publica-
tions of the University. Either above the title or below the seal
is given the number of the Engineering Experiment Station bul-
letin or circular which should be used in referring to these pub-
lications.
For copies of bulletins or circulars'or for other information
address
THE ENGINxRING EXPERIMENT STATION,
..UNivsT O iIorrrs, - ' -,
SUaaA, ILLINOIS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULI ETIN No. 173
JANUARY, 1928
THE SURFACE TENSION OF MOLTEN METALS
WITH
A DETERMINATION OF
CONSTANT OF
THE CAPILLARY
COPPER
EARL E. LIBMAN
NATIONAL RESEARCH FELLOW IN PHYSICS
ENGINEERING EXPERIMENT STATION
PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA
CONTENTS
PAGE
I. INTRODUCTION . . . . . . . . . . . .. . 5
1. Historical Sketch . . . . . . . . . . . 5
2. Previous Determinations of Surface Tension of Molten
* M etals . . . . .. . . . . . . . . . . . . 6
3. Importance of Surface Tension Data for Metals . . 7
4. Acknowledgments . . . . . . . . . . . 8
II. THEORETICAL ASPECT OF PROBLEM . . . . . . . . 8
5. General Theory . . . . . . . . . . . 8
6. Theory of Method Employed in this Investigation . 13
7. Discussion of Errors . . . . . . . . . . 20
8. Test with Mercury . . . . . . . . . . . 28
III. APPARATUS USED IN INVESTIGATION-THE CRUCIBLES . . 29
9. Form of Crucibles Used . . . . . . . . . 29
IV. APPARATUS USED IN INVESTIGATION-THE VACUUM FURNACE 31
10. Housing . . . . . . . . . . . . 31
11. Heating Element and Container . . . . . . . 33
12. Vacuum System . . . . . . . . .. . 33
V. TEMPERATURE MEASUREMENT . . . . . . . . . 33
13. Method of Measuring Temperature . . . . . . 33
VI. THE X-RAY EQUIPMENT . . . . . ... . . . . 36
14. High Tension Apparatus . . . . . . . . . 36
15. Tube, Stand, and Filament Supply . . . . . . 36
16. Lead Room ............ . 36
VII. DETERMINATION OF CAPILLARY CONSTANT OF COPPER . . 37
17. Method of Procedure . . . . . . . . . . 37
18. Development, Intensification, and Measurement of
Photographs . . . . . . . . 40
19. Capillary Constant of Copper . . . . . . . 40
LIST OF FIGURES
No. Page
1. Surface of a Liquid Showing Spheres of Cohesional Activity . . . . . 10
2. Sections of a Curved Surface Cut by a Perpendicular Pair of
Normal Planes . . . . . . . . . . . . . . . . . 11
3. Free Surface of a Liquid Connected with a Reservoir . . . . . . . 12
4. Plane and Capillary Depressions . . . . . . . . . . . . . 13
5. Details of Plane Depression . . . . . . . . . . . . .. 14
6. Capillary Surface Approximated by Sphere and by Ellipsoid . . . '. . 15
7. Surface Tension Forces Acting on Capillary Surface . . . . . . . 18
8. Special Crucible . . . . . . . . . . . . . . . . 19
9. Surface Depression d Due to Two Parallel Planes at Distance I . . . . 21
10. Variation of Surface Depression d with I for Values of a and h as Given 25
11. Reproduction from X-ray Photographs of Plane and Capillary
Depressions for Mercury . . . . . . . . . . . . . . 27
12. Dimensions of Special Crucible . . . . . . . . . . . . . 28
13. View of Furnace Water Jacket with Levelling Table and Rotation
M echanism . . . . . . . . . . . . . . . . . . 29
14. Sectional Details of Vacuum Furnace . . . . . . . . . . . 30
15. Furnace Housing . . . . . . . . . . . . . . . . . 31
16. Alundum Furnace Core ..... . . . . . . . . . 32
17. Furnace Core with Heating Element in Place . . . . . . . . 33
18. Alundum Reflecting Boxes . . . . . . . . . . . . .. . 34
19. Furnace Arrangement, Pumping System, and Temperature
Measuring Instruments . . . . . . . . . . . . . . 35
20. Temperature-Resistance Calibration Curve . . . . . . . . . 36
21. 100 000-Volt Kenetron Rectifying Set . . . . . . . . . . . 37
22. Diagram of Connections for 100 000-Volt Rectifier . . . . . . . . 38
23. Insulated Tube and Battery Stands . . . . . . . . . . . . 39
24. Solid and Liquid Copper in Equilibrium at the Melting Point . . . . . 40
25. Reproduction of Typical X-ray Photograph from the Negative
of Which Measurements Were Made . . . . . . . . . . . 40
26. Variation of Capillary.Constant of Copper with Temperature . . . . . 41
LIST OF TABLES
1. Capillary Constant of Metals Melting at Temperatures above 1000 deg. C. 6
2. Surface Tension of Metals Melting at Temperatures below 1000 deg. C. 7
3. Reservoir Corrections .. . . . . . . . . . . . . 22
4. Results of Preliminary Determination of Capillary Constant of Copper 43
5. Temperature Calibration of Molybdenum Furnace Coil . . . . . . 43
6. Film Measurements for Case of Pure Copper . . . . . . . . . 44
7. Distortion Correction on Film Measurements for Case of Pure Copper 47
8. Shrinkage Correction on Film Measurements for Case of Pure Copper . 47
9. Film Measurements Corrected for Distortion and Shrinkage for Case of Pure
Copper . . . . . . . . . . . . . . . . . . . . 48
10. Calculation of Approximate Capillary Constant for Pure Copper . . . 49
11. Calculation of Corrected Capillary Constant for Pure Copper . . . . 50
THE SURFACE TENSION OF MOLTEN METALS WITH A
DETERMINATION OF THE CAPILLARY CONSTANT
OF COPPER
I. INTRODUCTION
1. Historical Sketch.-When a tube of small bore is placed verti-
cally with its lower end submerged in a liquid, the latter rises or drops
within the tube and stands at a higher or lower level than the liquid
without. This phenomenon is due to the mutual attractions of the par-
ticles of the liquid for each other and for those of the tube. It is most
marked in the case of very fine bore tubes, bores of hair size called cap-
illaries (Latin "capilla," meaning hair). This and all phenomena due
to the same cause are grouped under the head of "capillary action."
The first to note capillary phenomena was Leonardo da Vinci,*
but accurate observations were not made until 1709 when Francis Hawks-
beet studied the action of liquids with glass tubes and plates. His work
was augmented by Dr. James Jurin, I and Sir Isaac Newton devoted the
31st query in the last edition of his Optics to molecular forces in which
he mentions capillarity.
Alexis Claude Clairaut§ appears to have been the first to attempt
a mathematical investigation of the phenomena, but without success.
It is to A. von Segner¶ that the idea of "surface tension" is due. Follow-
ing this idea Gaspard Monge** asserted that supposing the adherence
of the particles of the liquid to be sensible only at the surface and in a
direction tangent to it, the entire phenomena may be explained by math-
ematical analysis, and this was done in 1804 by Thomas Young.tt It
remained for Laplace, however, to give a mathematical analysis of the
subject based on only two assumptions, the constancy of the angle of
contact and the existence of attractive forces between the particles of the
liquid. This appeared in the "Mecanique Cl1est," Supplement to Tenth
Book, 1806. The assumption of constant angle of contact was shown to
be unnecessary by Gauss. It
In 1831 Simeon Denis Poisson published his "Nouvelle Theorie de
l'Action Capillarie." He maintained that there is a rapid variation of
*Pogg. Ann. cI, p. 551.
tPhil. Trans. 1711, 1712.
PPhil. Trans. 1718, 1719.
§"Theorie de la Figure de la Terre," Paris, 1808.
TComment. Soc. Reg., G6tting. 1, 1751.
***'Mmoires de I'Acad. des Sciences," 1787.
t t"Essay on the Cohesion of Fluids," Phil. Trans. 1805, p. 65.
t:"Principia Generalia Theorise Figurae Fluidorium in Statu Aequilibrii," 1830.
5
ILLINOIS ENGINEERING EXPERIMENT STATION
density of the liquid at its surface and gave numerous reasons for this be-
lief. His results are, however, identical with those of Laplace and Gauss
and there is as yet no way to distinguish between the two theories.
W. Thomson* first applied thermodynamics to capillarity. This
phase of the subject was further treated by Gibbst and still more en-
larged by Van der Waals.t
The theory as it stands today is that of Laplace and Gauss with the
thermodynamical aspect due to Thomson, Gibbs, and Van der Waals.
2. Previous Determinations of Surface Tension of Molten Metals.-
The accurate determination of the surface tensions of molten metals has
been limited to those metals whose melting points are so low that they
can be handled in glass or transparent quartz. These are bismuth, lead,
tin, cadmium, and zinc Concerning all other metals, we know (except
for copper considered in this work) hardly more than the general order
of magnitude of this important property. The reason for this state of
affairs is the great difficulty involved in handling accurate measurements
at high temperatures upon materials in opaque containing vessels. All
measurements (except on the metals just noted) have been carried out
in air which oxidizes the surface and thus contaminates the very portion
whose cleanliness is most essential. Attempts made to avoid oxidation
by covering the surface with powdered charcoal but substitute one con-
taminating source for another. It is, therefore, not surprising that the
data obtained are conflicting and unreliable. The following tables in-
dicate the nature of the results up to date. Except for the low melting
TABLE 1
CAPILLARY CONSTANT OF METALS MELTING AT TEMPERA-
TURES ABOVE 1000 DEG. C.
a2 = 2t/gp in cm.2 p = density
t = surface tension in dynes/cm. g = acceleration due to gravity
Observer
Metal
Smith 1914 Gradenwitz 1899 Heydweiller 1897 Quinke 1868
Silver.......... 0.1852 0.145 0.1594
Gold........... 0.1129 0.069
Copper......... 0.2885 0.1444
Iron............ 0.2648
*Proc. Roy. Soc. 9, 1858.
t"Equilibrium of Heterogeneous Substances," Connecticut Acad. Trans. 3, 1876.
'"Thermodynamische Theorie der Kapillaritat unter voraussetzung kontinuerlicher
Dicteanderung," Verhandl. Akad. von Wetensch., Amsterdam, 1893.
THE SURFACE TENSION OF MOLTEN METALS
TABLE 2
SURFACE TENSION OF METALS MELTING AT TEMPERATURES
BELOW 1000 DEG. C.
Surface Tension in dynes/cm.
Metal
300°C 350°C 400oC 450°C 500°C
Bismuth......... 376 373 370 367 363
Lead ........... 442 438 438 431
Tin............. 526 522 518 514 510
Cadmium........ 628 625 622 618
Zinc ............. 755 751
metals noted no attempt has been made to determine the surface tension
at more than one temperature, which is usually in the vicinity of the
melting point.
Some results of previous work on capillary constant of metals are
presented in Table 1.
All previous work on surface tensions of molten metals melting at
temperatures below 1000 deg. C. has been rendered obsolete by the care-
ful work of Hogness* giving the results presented in Table 2.
3. Importance of Surface Tension Data for Metals.-
(a) Practical
When a substance is molten the only forces acting upon it are grav-
ity and the intermolecular attractions that manifest themselves in
the phenomenon of surface tension. It follows that the behavior of the
molten materials will be closely related to their surface tensions and that
an extension of our knowledge of this property should lead to a greater
insight into the peculiarities exhibited by molten metals.
Surface tension is the governing factor in all processes "of welding,
soldering, and joining. It has recently been shown (A. W. Coffman, doc-
tor's dissertation, 1927, Chemistry Department, University of Illinois)
that those fluxes that are most effective in soldering are just those whose
presence causes a lowering of the surface tensions of the metals that be-
come molten in the soldering operation.
The mutual solubilities of the molten metals, and the behavior of
such solutions upon cooling and before solidification, should likewise be
so related to the surface tensions that a knowledge of the latter will allow
a prediction of the former.
*Jour. Am. Chem. Soc. 42, No. 12, 1920.
ILLINOIS ENGINEERING EXPERIMENT STATION
In casting operations where sharp outline is necessary the surface
tension of the molten metal is the governing factor. Metals with high
surface tension will not flow into sharp corners or crevices and if the lat-
ter be sufficiently fine an enormous pressure would be required to force
the liquid into them. The previously-held idea that the metal must
expand, or at least not shrink, on cooling, in order to get sharp castings,
is giving way before the suspicion that those materials whose addition
increase detail in casting are just those that cause a lowering in the sur-
face tension of the melt.
It appears, therefore, that in the progress of metallurgy surface ten-
sion data are a growing necessity, and the fact that most of its applica-
tions are still in a formative state is due more to the lack of such data
than to any doubt concerning their importance.
(b) Theoretical
In its search into the properties of the universe, science has con-
cerned itself greatly with increasing our knowledge of the characteristics
of the few elements that make up the world upon which we live. A funda-
mental property is surface tension, and it is of great scientific interest
that this be added to those properties already known for the elements.
At present the greater part of the scientific world is investigating
the structure of the atom and molecule. The property of the cohesion
of fluids (and of solids as well) is one intimately connected with atomic
and molecular structure and any knowledge gained concerning the phe-
nomenon, surface tension, which is a direct result of those cohesive
forces, is a step in the solution of this great problem.
4. Acknowledgments.-This investigation has been a part of the reg-
ular work of the Engineering Experiment Station of the University of
Illinois, of which Dean M. S. Ketchum is the director, and of the De-
partment of Physics, of which Prof. A. P. Carman is the head.
The work was carried out under a National Research Fellowship.
The author wishes to acknowledge the cooperation of Professor
Chas. T. Knipp, of the Department of Physics, in the early stages of
the work, and the assistance of Mr. John Thews, Assistant in Physics,
who worked with him throughout the course of the investigation.
II. THEORETICAL ASPECT
5. General Theory.-An exhaustive treatment of the general theory
of capillarity would begin with the assumption that each particle of the
liquid is attracted by all the other particles. From this assumption the
THE SURFACE TENSION OF MOLTEN METALS
theory would progress through involved mathematical analysis to the
final formulas required. Such a treatment of the subject is to be found
in many excellent texts (see particularly Mathieu "Theorie de la Cap-
illaritV" and Minchen "Hydrodynamics").
For the present purpose, however, a more direct route will be
taken. Beginning with the same assumption it will be shown that a
liquid is subjected at every point of its surface to a pressure normal to its
surface at that point. Attention will then be drawn to the fact that if
there were no cohesive forces and the liquid were, instead, covered on
its surface with a stretched elastic skin (like a piece of sheet rubber) the
same condition of normal pressure would result. A calculation will next
be made of the tension in this elastic skin necessary to cause the exist-
ing pressure, and this tension is the force called "surface tension."
Note that in actuality no "skin" exists. Due to the cohesive forces
the liquid acts in a peculiar manner. This action is the same as would
occur if, instead of the cohesive forces from within, there were an elas-
tic skin without. That is, the liquid acts as if it were possessed at its
surface of a skin stretched so that it sustains a tension t, called "the sur-
face tension."
Every particle of a liquid is attracted by every other particle not
only by gravitational forces but also with cohesional forces directly pro-
portional to the masses of the attracting particles and to some unknown
function of the distance between them. These cohesional forces are sen-
sible only through a very small distance e, which distance is called
the radius of cohesional activity. The fact that liquids, and indeed crys-
tals as well, cohere with forces that cannot be due to gravitation alone
is well known. Experiments by Quinke and others have shown that if
such forces as the cohesional forces postulated above do exist, they can-
not be sensible over distances exceeding 5 X 10- 5 mm. The mutual grav-
itational forces have been shown to be small in comparison with the
cohesional forces, and in what follows they will be neglected.
Let AB, Fig. 1, represent the surface of a fluid and A'B' an imagi-
nary surface parallel to AB and a distance e below it equal to the radius
of cohesional activity. Consider a particle of liquid at P. About P
draw a sphere of radius e. Of this sphere only the hemisphere abc ex-
ists in the liquid so the cohesional forces acting upon P come only from
the particles in this hemisphere. From the symmetrical grouping of
these particles about P it is obvious that the resultant force is in the di-
rection Pb which is normal to the surface at P.
Consider next the particle Q. The action of the liquid upon the
particle at Q is that of the particles within a,, mbi, ncl, that portion of the
sphere with radius e about Q which exists within the liquid. The re-
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 1. SURFACE OF A LIQUID SHOWING SPHERES OF COHESIONAL ACTIVITY
sultant force is obviously in the direction Qbj and is, moreover, less than
at P for here we have some liquid above Q exerting an upward force.
Finally, consider a particle at R. The particle is completely surrounded
by liquid in all directions for a distance at least equal to e. The result-
ant force upon it is, therefore, zero, as it is equally attracted in all di-
rections, and the same is true of any other particle below A'B'.
From the foregoing analysis we see that every particle in the ex-
tremely thin layer of thickness e at the surface is being urged inward by
a force normal to the surface. If the surface be plane it is evident from
its symmetry that this force will be the same over the entire surface.
If, however, the surface be curved, the force will vary from point to point
and be a function of the curvature of the surface which becomes a con-
stant when the curvature vanishes. Thus, this inwardly directed force
distributed over the surfaces causes a pressure within the liquid (in ad-
dition to the hydrostatic pressure H) which will be indicated by K + p
(curvature), where p (curvature) represents some function of the cur-
vature which vanishes when the curvature becomes zero; that is, when
the surface becomes flat. The total pressure is then P = K + H + p.
This first term K, the pressure due to cohesion existing within the liquid
when its surface is flat, is exceedingly large (of the order of 10 000 at-
mospheres). Being constant and within the liquid it does not manifest
itself with change of shape of the liquid and is not the term which gov-
erns the capillary phenomena. It is the third term, p (curvature),
which plays the important r6le. This term, depending upon the cur-
vature, changes with a change in shape of the surface and it is to this
variation of pressure with surface that the phenomena of capillarity are
due.
THE SURFACE TENSION OF MOLTEN METALS
FIG. 2. SECTIONS OF A CURVED SURFACE CUT BY A PERPENDICULAR PAIR
OF NORMAL PLANES
Now suppose that the liquid possessed no cohesive attractive force
but was surrounded with an elastic skin (in tension) at its surface, and
also subjected to a constant pressure (as might be accomplished by plac-
ing the containing vessel in a room and raising the air pressure to the de-
sired amount). The elastic skin would exert a force upon the surface
everywhere normal to it, and this would result in a pressure within the
liquid. This pressure would vary with the curvature of the surface and
when the surface became flat the pressure would vanish. Hence, if the
elastic skin possessed such a tension that the pressure within the liquid
due to it were p (curvature), and the constant pressure (exerted by the
air as indicated above) were equal to K, then the whole pressure within
the liquid would be P = H + K + p (curvature) precisely as before.
Consider now any curved surface (Fig. 2). At any point E of the
surface let us erect a normal N. Then through N let us draw two mu-
tually perpendicular planes. These cut the surface in two plane curves
CD and C'D'. The curve CD has a certain radius of curvature R1 at E.
The curve C'D' has another radius of curvature R2 at E. If we rotate
the pair of planes about N the curves CD and C'D' will change, and so
will their radii of curvature Ri and R2. The sum I + - will, however,
R, R2
remain a constant, and is therefore a property, not of the curves, but of
the surface at the point E. It is called the mean curvature of the sur-
face at the point E.
Now if an elastic skin is stretched over a curved surface in such a
way that it possesses throughout a constant tension T, the pressure it
exerts upon the surface at any point E is known to be
/1 1\
p = T -+-
(R& IR2
I . 2
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 3. FREE SURFACE OF A LIQUID CONNECTED WITH A RESERVOIR
and this vanishes when the surface becomes a plane. This then is the
function p (curvature) and the tension T is called the "surface tension."
The actual pressure P existing within the liquid is P = K + p + H.
K is a constant called the "intrinsic pressure"; this constant K cancels
out in the consideration of capillary phenomena. p is the pressure with-
in the liquid due to the hypothetical elastic surface skin possessing sur-
face tension T. Hence the total pressure at any point within the liquid
is
P=K+H+T( + (2)
Consider the free surface of a liquid which is subjected only to
gravity and the reactions of the walls of its containing vessel. No mat-
ter what the form of the surface may be, the container may always be
considered joined by a canal to a reservoir sufficiently large so that the
liquid in it has a plane surface. Thus let AB, Fig. 3, represent the free
surface of a liquid connected by a canal, as shown, with a reservoir in
which the liquid has a portion CD of its surface plane. The pressure at
p' must be the same as that just within the surface at p and at the same
height. The pressure at p' is, by equation (2), K + H (since at p"
1 + = 0), and that at p is K + T (I + I. Equating these,
Ri R2 \Ai R2
H = T ( - + But H = gp (k + z) where p is the density of
2T
the liquid. Hence, writing - = a2 (the "capillary constant"),
gp
a - + - k (3)
where k is obviously a constant depending for its value upon the selec-
tion of the origin. In this analysis no mention has been made of the
fact that any particular arrangement was being considered. 0 is arbi-
trarily selected as the origin from which to measure z. Hence equation
THE SURFACE TENSION OF MOLTEN METALS
/
/
/
/
/
/
/
/
/
0.
7) (b)
FIG. 4. PLANE AND CAPILLARY DEPRESSIONS
(3) is perfectly general and holds for all liquids subject only to gravity.
This is the fundamental equation required for a discussion of the meth-
od used in this investigation. Note that + - is the curvature at
R, Rs
the point whose ordinate is z.
6. Theory of Method Employed in this Investigation.-If a vertical
plane be dipped into the level surface of a fluid which does not wet the
plane, the fluid is depressed at the plane, that is, the line of contact of
the fluid with the plane will be lower than the level of the fluid. The
magnitude h (Fig. 4) of the depression will depend upon the properties
of the liquid and the solid forming the plane, that is, upon the surface
tension T1 and the angle of contact 01.
h = F, (Ti, 01) (4)
If a capillary tube be placed with its lower end beneath the level
surface of a liquid which does not wet the tube, the liquid will stand
within the tube at a lower level than the surface without. The depres-
sion H (Fig. 4) is again a function of 02 and T2, or
H = F2 (T2, 2) (5)
Now if the material forming the plane is the same as that compos-
ing the tube, and if the liquid is the same in both cases, then 01 = 02,
T, = T2, and
h = F1 (T, 0) (6)
H = F2 (T, 0)
e//
r'9 I
0
ILLINOIS ENGINEERING EXPERIMENT STATION
IG. 5. DETAILS O LANE D7) ()PRESSION
FIG. 5. DETAILS OF PLANE DEPRESSION
two equations connecting the two unknowns T and 0 which can then be
determined when h and H are known.
To determine the form of Fi (T, 0), consider an infinite plane dipped
vertically into the plane, horizontal surface of an infinite liquid which
does not wet it. The liquid will then be depressed in the neighborhood
of the plane and its surface will be cylindrical with generators parallel
to the plane. Let p be any point on the surface and make two sections
through p, one perpendicular to the plane and the surface and repre-
sented by OB, Fig. 5, the other through p perpendicular to the first sec-
tion and the surface and represented by CD, Fig. 5, which obviously is a
1
generator of the surface and therefore a straight line. The curvature - of
this last section is zero and so the equation of the surface is, from equa-
tion (3),
a2 1
z - - k. (7)
2 R1
Take the origin at the point of greatest depression at the plane
1
(Fig. 5). Then from the figure it is plain that when z = h, - = 0, and
substituting in (7) gives h = - k, so that equation (7) may be rewritten
a2 1
(z - h) - . (8)
2 R1
Now
1 _ de _ ddz _ dosin
R1 ds dz ds dz
and as z varies from 0 to h, varies from 0 - - to 0.
2
THE SURFACE TENSION OF MOLTEN METALS
8
---
H
Al
*rV
A
8
(a)
FIG. 6. CAPILLARY SURFACE APPROXIMATED BY SPHERE AND BY ELLIPSOID
Therefore
(z-h) =- a sin 4 or 2 (z - h) dz = a' sin 4 de4, whence
2 dz Jo fJ-e
2
h2 = a2 (1 - sin 0) (9)
2T
and, since a2 = -,
gUp
FI (T, 0) = 2 (1 - sin 0) *
gp
To determine the function F2 (T, 0), consider a large vessel with a
capillary tube attached, and suppose the vessel of sufficient size so that
the surface of the liquid it contains has a plane portion. If the tube is
sufficiently small the surface in the capillary resembles a sphere and if
it be assumed truly spherical there results the arrangement indicated in
Fig. 6.
Take the center of the sphere as origin. Equation (3) then is
z a- + - k.
2 R, R2
ILLINOIS ENGINEERING EXPERIMENT STATION
1 1
At point B, + = 0 and z = R + H, so that R + H = - k,
R1 R2
and the equation becomes
z - (R + H) = a + (10)
the equation of surface. For point A, z = R, and if any pair of mutually
perpendicular normal sections through A be taken it is obvious, since
the surface is one of revolution, that the radii of curvature will be the
same for both sections, that is,
1I 1 1_
R1 R2 R
a2
at A. Hence equation (10) becomes for A, H = a- From the figure
R
it is seen that r = R sin 0 - or = - cos and
2 R r
a2 Hr
H = - - cos 0 or a2 - (approximate). (11)
r - cos 0
Equation (11) is approximate, for it is based upon the assumption
that the surface in the capillary tube is a sphere: this is true only for the
very smallest tubes. If the tube is of the size employed in this work
(about 3 mm. radius) then the surface flattens out at A and departs from
the spherical form, resembling a prolate ellipsoid of revolution. A more
accurate expression for a2 will next be obtained on the assumption that
the surface within the capillary is an ellipsoid, and it will be found that
the result is but to add a correction term to (11).*
Figure 6 represents an axial section through the surface in the tube
which, being a surface of revolution, has every axial section the same.
Taking the vertical axis of the ellipsoid as a, and the horizontal axis as
-, then the equation of the section is
m
x2 y
- )+ --= 1 whence y = /a2 - m2x2
Also,
tan dy - mx m2r
dx x= -r Va2 - m22 x= -r Va2 - mx2
*See Desains, Ann. Chem. Phys. (3), 51, 417 1857.
THE SURFACE TENSION OF MOLTEN METALS
But 0 = ± + ,cot = - tan co, cot = m2r (12)
2 a2 - m2x2
From (3) we have the equation of the surface as
2 1 1
z= - R- + -k
At pointB, z = a + Hand 1+ = 0. Hence a + H =- k
Ri R2
and
z - (a + H) = ( + (equation of surface) (13)
2 R, Rt/
At the point A, since the surface is one of revolution, 1 1 1
R1 R2 RA
the curvature of the ellipse at A, while z = a
1_ = d2Y + (di y -= dly
RA dx' \dx) dx2 x = 0
Since dy =0, 1 d2y - m2
dx X =0 RA dx2 x = 0 a
and substituting in (13)
H a2m2 (A)
a
Using this in conjunction with (12) gives
r2
m2 = a4 (B)
- r2 tan2
H2
Since m is a function of 0 as shown in (12) it follows that F2 (T, 0)
a
has the form given by (A), namely, (2T m2, and the attempt might
gp a /
be made to determine - as a function of 0 and substitute. Because of
a
the mathematical difficulties involved it is found more expedient to
pursue another course. Consider a section of the capillary at 0, Fig. 6b,
and determine the total force upon it from beneath; then determine the
total force upon it from above. Equating the two gives the desired
relation involving a and m which is required in order to complete the
analysis under consideration.
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 7. SURFACE TENSION FORCES ACTING ON CAPILLARY SURFACE
The pressure on the section at 0 from beneath is, by equation (2),
K + H + p (where p = T ( + 2) is the pressure transmitted to the
section by the liquid, and due to the elastic skin at B). Since the surface
at B is plane this last term is zero. The hydrostatic pressure is gp (H +
a) and so the total pressure from beneath is K + gp (H + a) and the
total force is irr2 [K + gp (H + a)].
The total force exerted on the section at 0 from above is that due
to the intrinsic pressure K, that due to the weight of the liquid above the
section at 0, and that due to the elastic skin. The total force due to the
pressure K is itr2K. That due to the weight of the liquid above the sec-
tion at 0 is gp (volume of liquid above section at 0) = gp r 27rxydx.
2rgp f xydx = 2rgp fx a2 - m2x2 dx
27gp[- (a2 _m r)2 a 3
=2g3m2 3m2
If, then, F represents the force exerted by the elastic skin, the force
from above is r2'K + F + 27gp [-(a' 2- mr2) a3
3mt2 3M2
Equating the forces from above and below on the section at 0
rr2 [K + gp (H + a)] = 7r2K + F + 2rgp[- (a m2r2)2+ 3
or
rr'gp (H + a) = F + 2rgp (a3m 3m (14)
Note that the intrinsic pressure K has cancelled out, as indeed it does in
all capillary phenomena.
F has now to be found. Across every line of unit length in the sur-
face there acts the force T of surface tension in a direction normal to
the line and tangent to the surface. Along the line of contact of the
THE SURFACE TENSION OF MOLTEN METALS
FIG. 8. SPECIAL CRUCIBLE
liquid surface with the tube there acts, therefore, on every unit length a
force T normal to the line of contact and making an angle of (ir - 6)
with the vertical, Fig. 7. The vertical component of this force is
T cos (-r - 0) = - T cos 0 and the total vertical force exerted upon the
surface is F = -2rr T cos 0. Introducing this in equation (14)
ra 2 (a2 - m'r2) 2a"
rH ra 3m2r 3m2r
- cos 6 - cos 0
Note that this is the same as equation (11) with the quantity in brackets
as a correction term. Gathering the useful equations (A), (B), and (C)
we have
r2
m = -r__
a 4
H
a2- rL
- cc
- r7tan2 0
s 0 +
fr 2 (a2 - m2r24) 2a'
1ra 3m2r I 3mi rJ
- cos 0
ILLINOIS ENGINEERING EXPERIMENT STATION
Equations (9), (11), and (15) are the fundamental equations used to
calculate the capillary constants from observed data. They are used as
follows: From the values of h and H obtained experimentally we get
by means of equations (9) and (11) approximate values for a2 and 0.
Substituting these in equations (15) we get the correction term to our
approximate value of a2. (If desired this corrected value of a' can again
be used for a more accurate value of the correction term although in the
present work the first application was sufficient.) The detailed method
of use of these equations on the experimental data is to be found in
Tables 10 and 11.
7. Discussion of Errors.-The experimental details will be given
later, but before entering into a discussion of errors it is necessary to
give in barest outline a sketch of the method sufficient to make intel-
ligible the matter which is to follow. The plane depression h and cap-
illary depression H being arranged within a vacuum furnace, an X-ray
photograph is taken through the entire furnace. On the photograph h,
H, and r are measured. The vertical plane against which the depression
h occurs, and the capillary tube within which the depression H takes
place, are both part of a special crucible shown in Fig. 8 and completely
described further on.
In the theoretical treatment the case of an infinite liquid into which
is dipped an infinite vertical plane has been considered. The mathe-
matical analysis availed itself of this infinite liquid only to insure that
the surface was, at some point, sufficiently removed from the influence
of the plane to be horizontal. Hence, in the experimental work, it is
necessary only to insure that the containers used are sufficiently large to
allow the liquid to have a portion of its surface horizontal. If this is
done the theoretical formulas that we have deduced are applicable. If
this is not done, then an estimation of the error involved is necessary.
In order to be certain that the surface within the container is flat
it would be necessary to use a crucible of prohibitive dimensions. It
is, therefore, necessary to determine the error involved in the use of
apparatus of practical size.
Consider the case of two parallel vertical planes dipped into the
liquid, and assume that they are such a distance 1 apart that the surface
between them, at the midpoint, is depressed an amount d below the
original level.
From equation (3) for the free surface of the liquid
z - + - - k.
2 R, R2
THE SURFACE TENSION OF MOLTEN METALS
FIG. 9. SURFACE DEPRESSION d DUE TO TWO PARALLEL PLANES AT DISTANCE I
At B, Fig. 9, the surface is flat, and 1+ vanishes. Also, at B
R, R2
1
z = h + d and so k = - (h + d). Since the surface is cylindrical, - = 0
everywhere, and the equation becomes
a2
z - (h + d) =-2R
2R,
Now
dx2
1 d+ / J
and so
Integrating
d2z
- (h +d) = a dz2 3
S dx
2 [ ( +± )J ( dz\ 2l
Il +\)
[ ]d2 C - a2
z - (h + d) =1- dz 9
(dx I
dz
where C is a constant of integration. When z = h, dx
a2, hence C = d2 + a2 and
= 0 and d2 = C -
d2 + a2 - [h + d - Z]2 a2d
S+ (d -)21,
1+w
(18)
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 3
RESERVOIR CORRECTIONS
/2d/aa
0.76
0.59
0.46
0.37
0.30
0.24
0s00
'-C 0000000000000000OO
|| 00 Co Co Co Co 00 Co ^(O N O NO
Nm"CONloNO-cOżOOOO oNN
Co oo Cooo Co - m o Coo oo Coo
1 01 0 0 001 0 C 0!0 0c 0 C! 00
0 000 00 0000000
Q OOO00OO0CO'-C-O~O~O
0000' i^o