This dissertation has been microfilmed exactly as received 68-15,372
RHODES, James Franklin, 1938- RHEOLOGY OF VITREOUS COATINGS.
The Ohio State University, Ph.D., 1968 Engineering, chemical
University Microfilms, Inc., Ann Arbor, Michigan RHEOLOGY OP VITREOUS COATINGS
DISSERTATION Presented In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By ■ i James Franklin Rhodes, B. Ch.E., M.S.
******
The Ohio State University 1968
Approved by
Department of Ceram^ Engineering ACKNOWLEDGMENTS
The author wishes to express his thanks to all the individuals who have assisted him in the course of this investigation.
Special gratitude is due to Dr. B.W. King, Jr, and
Dr. R. Russell, Jr, for their encouragement, guidance and patience. The suggestions and advice of Mr. Lawrence E.
Muttart and Dr. W.B. Shook are gratefully appreciated. The assistance of Walter Grudzinski of the General
Motors Research Center in preparing the illustrations is gratefully acknowledged.
Special thanks are due to the author*s wife, Paula, for her encouragement and assistance in the preparation of this disseration.
The author is indebted to the National Science Foundation whose financial support allowed him to pursue this investigation.
The author expresses a final note of thanks to the three women -- his mother, wife and daughter — whose sacrifices made this work possible.
ii VITA
January 20, 1938 Born - Cleveland, Ohio
1963 B.Ch.E., University of Detroit Detroit, Michigan 1963-1967 Research Fellow - Department of Ceramic Engineering, The Ohio State University, Columbus, Ohio National Science Foundation Fellowship 19 66 M,S., The Ohio State University Columbus, Ohio
PUBLICATIONS Masterfs Thesis - "Leveling of Porcelain Enamels", March, 1 9 66, The Ohio State University Library TABLE OF CONTENTS
Page ACKNOWLEDGMENTS ii
VITA iii LIST OF TABLES vi
LIST OF ILLUSTRATIONS viii
INTRODUCTION 1
LITERATURE REVIEW 5 DISCUSSION OF LITERATURE 11
METHOD OF ANALYSIS 15
EXPERIMENTAL PROCEDURE 21
Preliminary Investigation 21
Final Experimental Procedure 27 Development of A wave Forming Method 27 Preparation of Measuring Apparatus 28 Procedure for Glass EF 9 52 Investigation of Glass-Powdered Metal Mixtures 35 Preparation of Glass EF 9 Powdered Nickel Wave Specimen 35 Measurement of Surface Tension of Glasses 607 and EF 9 56 EXPERIMENTAL DATA 41 ANALYSIS OF DATA 69
RESULTS AND DISCUSSION 103
Comparison of Leveling Models 122 Sources of Possible Error 124 CONCLUSIONS 125 iv Page SUMMARY 126
APPENDIX A - MATHEMATICAL MODEL 127
Description of Model and Assumptions 128 Theory 129
APPENDIX B - RAW MATERIALS AND EQUIPMENT DATA 164
APPENDIX C - CHEMICAL AND PHYSICAL DATA 168 BIBLIOGRAPHY 178
v LIST OP TABLES
No# Title Page
1 Particle Sizing Data 22
2 Powdered Metal Investigation Data 34 3 Average Amplitudes and Cumulative Firing Data, Specimen Series D, Glass 607 42 4 Average Amplitude and Cumulative Firing Data, Specimen Series E, Owens Illinois Glass EF 9 59 5 Wave Amplitude and Cumulative Firing Time Specimen Series F, 90$ Glass EF 9 - 10# Inco 123 Nickel 64
6 Surface Tension Data, Glass EF 9 66
7 Surface Tension Data, Glass 607 66 8 Time Corrected Data, Specimen Series D, Glass 607 72 9 Corrected Time Data, Specimen Series E, Owens Illinois Glass EF 9 84 10 Corrected Time Data, Specimen Series F, 90# Owens Illinois Glass EF 9 - 10# Inco 123 Nickel Powder 88
11 Data for the Graph of C^ versus -kh 105 12 Viscosity Calculation Data, Glass 607 109
13 Viscosity Calculation Data, Glass EF 9 111 14 Viscosity Calculation Data, 90# Glass EF 9 - 10# Powdered Nickel 112
15 Log Viscosity Temperature Data, Glass 607 113 16 Log Viscosity Temperature Data, Glass EF ; 114 vi No. Title Page
17 Log Viscosity Temperature Data, 90$ Glass EF 9 - 10$ Inco Nickel 115 18 Effect of Water on Glass Viscosity 119 19 Comparison of the Viscosity of Glassy Coatings and Glass-Metal Dispersion Coatings 121 3 20 Data for Determining C^/(kh) Ratio 123 21 Data for the Determination of the Effect of aA on the Function 1/1 + a2k2 /A 132
22 Derivatives and Integrals for Solution of Stress Functions 140
23 Calculation Data for kC^C^ Ratio 152 24 Calculation Data for kC^/C^ Ratio 156
25 Elastic Theory Solutions: Stresses, Displacements, and Constants 158
26 Viscosity-JTemperature Data, Glass 607 (After English (9) ) 169
27 Owens Illinois Corporation, Glass EF 9, Physical Data 170 28 Chemical and Physical Data, International Nickel Company, Type 123 Carbonyl Nickel Powder 171 29 Chemical Composition, Glass 607 172
30 Chemical Composition, Glass EF 9 173
31 Chemical Composition, Glass 604, English 174 32 Chemical Composition, Glass 4, Parmalee et al. (18) 175 33 Viscosity Data, Glass 604, (After English (9) ) 176
34 Surface Tension Data, Glass 4 Parmalee et al. (18) 177 vii LIST OF ILLUSTRATIONS
No. Title Page
1 Surface Tension Measuring Furnace 37 2 Surface Tension Measuring Furnace 38
3 Surface Tension vs Temperature, Glass 607 67
4 Surface Tension vs Temperature, Glass EF 9 68
5 Log l/a vs Corrected Time, Specimen El 89 6 Log l/a vs Corrected Time, Specimen E2 90
7 Log l/a vs Corrected Time, Specimen E3 91 8 Log l/a vs Corrected Time, Specimen E4 92
9 Log l/a vs Corrected Time, Specimen E5 93 10 Log l/a vs Corrected Time, Specimen E6 94
11 Log l/a vs Corrected Time, Specimen E7 95 12 Log l/a vs Corrected Time, Specimen E8 96
13 Log l/a vs Corrected Time, Specimen E9 97 14 Log l/a vs Corrected Time, Specimen E10 98
15 Log l/a vs Corrected Time, Specimen Ell 99 16 Log l/a vs Corrected Time, Specimen FI 100
17 Log l/a vs Corrected Time, Specimen F2 101
18 Log l/a vs Corrected Time, Specimen F3 102
19 C jj versus ■-kh 106
20 C5 versus -■kh 107 21 versus ~kh 108
Vili No. Title Page
22 Log Viscosity vs Temperature, Glass 607 116
23 Log Viscosity vs Temperature, Glass EF 9 117 24 Reference Diagram, Mathematical Model, Rheology of Vitreous Coatings 130 25 a/X versus 1/( 1 + a2k2 ) 1 3 3 26 kC2/C4 versus q 3.53 27 k C 2/ C 4 versus q 154 28 q vs -kC-j/C/j. 1 5 8
ix INTRODUCTION
The motto of the Society of Rheology is "Everything flows". Rheology is that branch of physics which Is concerned with the flow, and deformation of materials.
While it is true that to some extent everything does flow, the rheologlcal behavior of different types of real materials Is varied and exceedingly complex. The rheology of coatings, and especially vitreous coatings, Is no exception. Vitreous coatings are essentially glasses that are used as coatings for various types of substrates.
Those used on metal substrates are called porcelain enamels those used on ceramic substrates are called glazes.
Some porcelain enamels are used In a dry process whereby a finely ground glass is sprinkled on hot ware.
The enamel adheres to the piece which Is then reheated so that the glass can flow into a smooth coating. All glazes and most porcelain enamels utilize a "wet process" whereby finely ground materials are dispersed In a water suspension. This water suspension (called a slip) is applied to the substrate, dried, and fired. In the firing process, the coating fuses, becomes vitreous, flows out and bonds to the substrate. 1 2
Up to the ^>oint where the wet coating has dried, its rheological behavior is similar to a paint. At the beginning of thei fusion process, the coating acts as a sintering-reacting solid. Upon completion of reactions and fusion into a monolithic layer, it behaves as a glass.
The total rheological picture of vitreous coatings is thus a complex one. Since it is beyond the scope of any single investigation to cover all the rheological phenomena of a vitreous coating, this work will be confined to the study of the final rheological phase, the flow of the fused vitreous coating.
The glass industry has developed methods for exact measurements of viscosity for highly viscous systems.
These viscosity-temperature relations are used to determine control points for glass manufacturing and fabrication. The high viscosities (7.65 log poise and greater) are conven- tionally determiLned by measuring the rate of elongation of uniform glass fibers. Low viscosities (less than 5 log poise) are generally determined with rotational viscometers.
Paint technologists have also applied exact rheological measurements to the evaluation, development, and quality control of organic coatings. Plow cups and rotational viscometers are generally used to determine the rheological properties of paints.
Neither the methods of the glass technologist nor those of the paint technologist are readily applicable to studying the rheology of fused vitreous coatings. Vitreous 3
coatings generally involve blends of various glassy
components with crystalline oxide raw materials; in
addition, they quite often contain a precipitated
crystalline phase. Therefore, the methods developed by the glass technologist for homogeneous fibers cannot be used. Conventional rotational viscometers are convenient
for melts having viscosities less than 5 log poise. Many various methods of viscometry have been developed in the past and Rheological Abstracts (35) lists many newly developed methods each year. However, most methods require intricate apparatus. The porcelain enamel industry uses the flow button test (11) as a relative viscosity measure ment in the development and quality control of porcelain enamel glasses, but exact control points have not been established. The flow button test was developed by Kinzie
(12) and modified by Marbaker (13). Dekker (8 ) has developed a method for calculating viscosity-temperature curves from flow button data. Dekker*s method gives good agreement with his known data, but it is rather complicated to us?. There are presently no simple methods for studying the rheology of a fused vitreous coating.
Previous work on the leveling of porcelain enamels (22) has indicated that the rate of leveling of an irregularity of known dimensions on the surface of a vitreous coating could be used as a method for studying the rheological behavior of vitreous coatings. The rate of leveling can be measured in a fairly simple manner, and a rheometrlc 4
method based on the rate of leveling could be generally applied to all types of vitreous coatings.
This investigation was undertaken to develop a mathematical model for studying the leveling behavior of an idealized configuration of irregularities on the surface of a vitreous coating and to verify the model experimentally using a simple porcelain enamel system. The model can then be used as an analytical tool to study the rheology of coatings. LITERATURE REVIEW
A complete review of the viscosity measurements on glasses and vitreous coatings will not be attempted. Cox et al. (7), Yee (37) > and Dekker (8 ) have discussed the measurement of the viscosity of porcelain enamels. Taken together they have compiled a comprehensive bibliography on available methods and data. Morey (l4) has compiled an excellent bibliography on the measurement of the viscosity of glasses prior to 195^. In addition, many references on glass viscosity measurements can be found in Ceramic
Abstracts (38). The prime interest in this Investigation
Is the leveling of coatings and, as such, the literature on this subject will be reviewed in detail.
Historically, the first analysis of the leveling of coatings was done by Waring (29) In the study of paint.
An analysis was carried out for the leveling of brushmarks
In a pigment-vehicle mixture which has a yield value. It was assumed that the leveling was a function of surface tension, yield value, and the dimension of the brushmarks.
A relationship was derived of a form:
h = 5 l £ 8J
5 6
value h = depth of brushraark
d = width of brushmark
f = yield value
’■ surface tension
Some reasonable values were substituted into the derived relationship for d and . An arbitrary height was assumed as a criterion of levelability. These were substituted into the derived relationship and it was calculated that a pigment-vehicle mixture with a yield value less than 2.8 dynes per square centimeter should level satisfactorily.
The Los Angeles Paint and Varnish Club (4) conducted an analytical study of the brushability, sagging, and flow out of paints. An expression was developed for the rate of sagging of a vertical paint film. The rate of sagging was given by: v = a g t2 -_2T where V = rate of sagging
d = density of coating
T = thickness
= viscosity
A mathematical analysis of the forces causing the flow out of ridges in paints due to brushmarks was developed. The analysis was carried out to determine whether two conflicting requirements could meet simultaneously: high viscosity for little sagging and a zero elastic yield 7 point for perfect flow out. In the development, the effect of gravity was neglected and the driving force for flow out of the coating was considered to be only surface tension.
The fbllowing relationship was derived:
c * . w -k ) t = t 3 4 S
where 7} = viscosity
C s half width of a brushmark
S = surface tension
T s mean coating thickness
k = constant which is equal tounity
t = time required for a ridge of a radius of curvature, r0 , to flow out to a radius of curvature r
Smith, Orchard and Rhind-Tutt (27) also examined the factors influencing the extent of flow out of brushmarks in paints. The work of Waring (29) was discussed and shown to be only an approximation. The authors carried out a formal stress analysis based on the elastic theory of plane strain for brushmarks having a sinusoidal profile. They have outlined their method of approach and given their results for the stress distribution in a coating having regular sinusoidal brushmarks. They have shown that, for their model, the maximum shear stresses exist at the coating substrate interface. This maximum shearing stress is glvenby: 47T3 7 d h f * A 8
where f ■ maximum shear stress
* surface tension of the coating material
D = depth of the brushmarks
h = mean coating thickness
A. s wavelength of the brushmarks
Savins (24) and Asbeck (2) have discussed the fund amentals of rheology as applied to the leveling of paints and the application of rheological measurements to the evaluation of coatings.
Patton (19) has thoroughly reviewed the rheology of paints. An entire chapter of his book is devoted essentially to leveling and sagging phenomena. Relations were developed for leveling of parallel brushmarks assuming either Newtonian or Bingham flow. Patton used a method similar to that of Blackington et al. (4) to develop a leveling equation of Newtonian flow of brushmarks:
X k Tl zo t = 0.036 - JL - Log- C f x3 z
where wavelength of brushmarks
viscosity of coating
a surface tension of coating
X a mean coating thickness
Z0« half depth of the brushmark at time t = $
Z a half depth of the brushmark at time, t 9
t = time required for the half height of the crest to go from Zo to Z.
log denotes common logarithm
The interplay of sagging, slumping and leveling were
also discussed,
Rhodes (22) has studied the leveling of a porcelain
enamel. A model was developed for the leveling of
irregularities on the surface of a vitreous coating which
has the shape of sectors of spheres. The model was developed to simulate observed roughness in porcelain
enamel coatings and to derive a fundamental relationship between the surface tension of the coating, viscosity of
the coating, the geometry of the surface roughness, and
the rate of leveling of the coating. The derived
relationship is:
in”? . 22 / x3 „
h " 3 W where hQ -.-height of the peak at the time . chosen as t0
h = height of the peak at the time t
x = average thickness of the enamel coating
4a = distance from the peak of one spherical sector to the peak of the next
I f = surface tension of the coating 7\ = viscosity of the coating 10
t = time required for the height of the peak to go from hQ to h
In = natural logarithm
The validity of the above relation was checked experimentally and the constant term was found to be 3 .
It Is shown that the predominant variable In the
leveling of a vitreous coating Is the viscosity. DISCUSSION OP LITERATURE
While the other works mentioned give insight into the problems involved in the leveling of a coating, the main works of interest are those of Smith (27), Patton (19),
Blackington (4) and Rhodes (22). Blackington (4) and
Patton (19) have derived relationships for the leveling of paint brushmarks from fundamental principles. The brushmark models deal with a relatively simple geometrical configuration which can be formulated in a two dimensional analysis. No experimental verification of either work has been carried out. The application of any method of measurement but an optical one would apply a stress to the readily mobile paint systems and negate the results.
In addition, specimen preparation for any controlled study would be very difficult. The low viscosity paints do not readily lend themselves to the fabrication of specimens.
It is therefore easily understandable why no experimental proof has been carried out.
Rhodes' (22) development on porcelain enamels requires neglecting of interactions between the individual surface irregularity and surrounding ones. It is essentially a three dimensional problem which is solved by a two
11 12 dimensional method; mathematically it is not as simple a3 the brushmark model. A vitreous system such as was used by Rhodes offers a distinct advantage over a paint system, however. The highly viscous porcelain enamel aystem allows the necessary flexibility in the fabrication of specimens.
A rapid air quench of the specimen to a rigid state permits simple measurement of the leveling phenomenon. Rhodes' work shows excellent agreement between the theoretical and experimental relationship of the variables. There is, however, a difference between the derived and experimental constant terms. This can partially be attributed to the assumptions involved in the solution. The one basic assumption common in all three leveling models is that the stress under a curved surface caused by surface tension is transmitted hydrostatically through the cylindrical or prismatic volume which is under the crest of a wave or irregularity. The end result of this assumption is a discontinuous stress distribution whereby the volumes under a crest of a wave or irregularity are under a compressive stress while the volumes under a trough region undergo capillary suction. Since hydrostatic transmission is assumed, the stress distribution at the coating substrate interface is the same, i.e. alternate regions of suction, compression, suction etc. Intuitively, a diver several feet under the surface of a stormy sea feels the effects of the waves to a great extent; when he reaches the bottom, however, he can be oblivious of the storm which is occurring on the 13 surface. By analogy, the stress distribution at the coatlng- substrate interface should depend upon the thickness of the coating. In the work of Smith et al. (26) it is shown that for a sinusoidal brushmark, the stress, y, is Indeed a function of the coating depths
O y = - ^ a k 2 sin (kx) [l— | k2^2]
normal stress in y direction r = surface tension a = amplitude of sine wave
k - wave number
X = horizontal distance
y Z vertical distance measured from the mean coating surface into the coating
& y, in this case, is essentially the stress due to surface tension used by Blackington, Patton and Rhodes. It is readily seen that the stress would only be essentially independent of depth for a very thin coating with irregularities of a large wavelength. The work of Smith et al. is based on a study of a sinusoidally loaded beam which is discussed by Timoshenko (28). The work done is merely outlined and the assumptions involved in the solution are not clearly pointed out. As is the case with
Patton*s work, no experimental verification is given; the application of a continuum approach to the leveling problem is pointed out, however. 14
Mathematical models have been derived (19) (22) for the leveling of two configurations of irregularities on the surface of coatings. These expressions were derived for discontinuous stress distributions which were independent of the thickness of the coating. It has been shown (27), however, that the stress distributions in similar configurations are not independent of thickness as was assumed.
In order to use the leveling of a vitreous coating to study Its rheological properties, a valid model had to be derived. The actual stress distribution In the coating must be taken Into account In the development of the model. None of the previous models had sufficient
flexibility to allow their modification to accommodate a realistic stress distribution and evolve a realistic model.
In order to formulate a realistic model, a method of
analysis had to be developed which would be as rigorous as possible, physically accurate, and still solvable. METHOD OP ANALYSIS
Patton (19) ana Blackington (4) have generated mathematical models for the leveling of regular brushmarks in a paint coating. Rhodes (22) has studied the leveling of a vitreous coating. The work of Rhodes (22) was designed to explain observed phenomena in thin vitreous coatings; the flow out of irregularities in the surface of a porcelain enamel. As such, the method involves approximations which do not readily lend themselves to a more precise mathematical treatment. In addition, the model involves a three dimensional problem. Since vitreous coatings are not applied by brushing, parallel ridges, the configuration described by Patton and
Blackington, are never generated in vitreous coatings.
The brushmark problem is a two dimensional one, and mathematically much simpler, however, and similar patterns can be artifically generated in a vitreous coating.
Preliminary investigations showed that there was some apparent deviation from the Patton model for thick coatings.
This could be understood by an extension of Saint Venant's
Principle in mechanics. Saint Venant's Principle is given by Timoshenko (28) (p. 35) as:
15 16
"This principle states that if the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally but has negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed."
Patton's model deals with a series of circular arcs with alternating positive and negative radii of curvature.
The corresponding pressures under those arcs are also considered as alternating in sign and as being transmitted hydrostatically. By Saint Venant's Principle, the effects of the alternating surface stresses which are equal in magnitude should tend to neutralize one another in coating medium as the distance from the surface increases. This means that the stress distribution assumed by Patton would not be sufficiently valid for coatings whose depth is of the order of a wavelength of the surface waves or thicker.
Furthermore, the coating is a continuous medium and the discontinuous form of stress distribution assumed by Patton does not appear tenable.
Therefore, a more sophisticated analysis was carried out in a manner suggested by the work of Nadai (16) on estimating the pressures under parallel ranges of mountains.
Nadai (p. 195-199) solved the idealized case of uniform parallel sinusoidal mountain ranges as a plane strain problem. A stress function was formulated which would generate the applied loading pattern. The stress function 17 was then solved for the components of stress, strain, and displacement at any point under the mountain range using the methods of the theory of elasticity. In discussing the general expression for the displacement components for the bottom point of the valley, he states:
"The corresponding relation for the small components of velocity, (u,v) with which a point (x,y) underground moves steadily owing to, the viscosity of the rocks are given by replacing the modulus of rigidity, G, in the preceeding ' expressions by the coefficient of viscosity, u. Therefore an expression for the displacement component of a point from the theory of elasticity can be converted to an equation for the velocity component of a point in slow viscous flow by substituting the coefficient of viscosity for the shear modulus in the elastic expression,"
N.D.P. Smith (27) et a l . have investigated the stress distribution in a paint film having sinusoidal brushmarks on the surface. The approach used was to treat the coating as a two-dimensional elastic solid and utilize a plane strain solution. Their method was based on work given by
Timoshenko for the stress analysis of a beam which is subjected to a sinusoidally varying load on both sides.
N.D.P, Smith et al. state that the local stresses in the paint film are due essentially to the effects of the curvature of the surface of the brushmarks. They have merely outlined their method of approach, but they state that the normal stresses in the y direction on the plane, y = 0 are given by:
R 18
0 y = normal stress In y direction ~jf z surface tension
i = curvature
Their derived expression for the stress at any point in the body is:
Oy = sin kx [l - \ k2y2J where a - amplitude of the wave
k = wave number
y = vertical distance from the mean coating thickness
Therefore considering the plane y = 0:
0 y s - ^ a k 2 sin kx
The curvature, , is thus given by:
1 2 ^ - ak ain kx
In order to further consider the work of N.D.P. Smith et al., it is necessary to justify this key point. The approximation and its limitations are discussed with geometrical arguments by Poynting and Thomson (20)
(p. 6 and 158) and C.J. Smith (26) (p. 473). Their application was an explanation of Lord Rayleigh*s method for the determination of surface tension by means of ripples. A purely mathematical argument is given in
Appendix A. The method of Timoshenko (28) and N.D.P.
Smith (27) can thus be applied to determine the expressions for displacements and stresses in a coating subjected to a sinusoidally varying surface stress. 19
Jaeger (11) has discussed the rate of subsidence of
the surface of a viscous liquid of Infinite depth which has been distorted Into the shape of a conlne wave and has
shown that rate of subsidence of the surface can be expressed as the time rate of change of the amplitude of the wave.
The equations for the stresses at a point and the displacement of a point In an elastic solid can thus be determined from the elastic theory of plane strain. Using the analogy between the stream function of slow viscous
flow and the Airy Stress function of elastic theory, the equations for the elastic displacement of a point can be converted to expressions for the velocity of a point undergoing slow viscous flow. The genera: expression for the velocity of a point can then be evaluated toyield the velocity of the peak of the wave. The expression for velocity of the peak of the wave can then be integrated to yield an expression for the amplitude qf the wave at
any time, t.
The entire analysis has been carried out in Appendix
A. The expression for the amplitude of the wave at any
time, t, resulting from this analysis is:
lr) ^ * ,1 k T Cc t+cons tant
where a - amplitude of sir e wave
7\ - viscosity
k * wave number 20
7 * = surface tension
Cr = derived experimental variable 0 (function of wave number and coating thickness only)
t = time
This expression is thus a mathematical model for the leveling of a coating having a surface in the shape of parallel sinusoidal ridges. For a given specimen, having striations with a fixed uniform wavelength, a uniform coating thickness, and held at a constant temperature,
(assuming viscosity and surface tension are functions of temperature only) a graph of log i versus time should yield Cl a straight line if the model is valid. If all the other variables are known, the viscosity of the coating can be determined from the slope of the graph.
In order to check the validity of the derived model, a procedure had to be developed whereby the geometrical configuration described by the model could be generated experimentally and the experimental variables could be controlled. EXPERIMENTAL PROCEDURE
In order to develop as simple a method as possible, a porcelain enamel type of system was selected to check the validity of the derived relationship. A normal glaze system wherein the raw materials are mixed in a slip form, applied to a body and matured Into a glass during firing would pose added complexities due to reaction during firing. A fritted glaze on a wall tile specimen would also have been used, but a porcelain enamel system allowed more flexibility in quenching the specimens. For the Initial investigation,
Glass 604 whose leveling properties were previously studied
(22) was selected. The viscosity of Glass 604 was studied by English (9). The surface tension of a glass of similar composition was determined by Parmalee, Lyon and Harmon (1 8 ).
Preliminary Investigation 1,2 A one thousand gram batch of Glass 604 was prepared.
The glass was fused in a fire-clay crucible in a gas-fired pot furnace. The crucible was heated to 2340°F while empty; the raw batch was poured into it and heated to 2340°F. 1 All raw materials and equipment used in the course of this Investigation are given in Appendix B. 2 All chemical compositions are given in Appendix C. 21 22
After the entire batch reached temperature, a soak of 40 minutes at temperature was required before a fiber pulled from the melt was free from bubbles and unmelted batch.
The molten glass was then quenched in water and dried at 220°P• The glass was ground dry in a ball mill, and then separated into particle size fractions by a dry sieve method. The particle size fractions will be denoted as follows for the remainder of this investigation.
TABLE 1
PARTICLE SIZING DATA
Sieve Fraction Description Number
1 Through 100 mesh - Retained on 140 mesh 2 Through 140 mesh - Retained on 200 mesh
3 Through 200 mesh - Retained on 270 mesh
4 Through 270 mesh - Retained on 325 mesh
5 Through 325 mesh
It was shown that Inconel is a suitable substrate
for Glass 604. It was found initially in this work that
Inconel Alloy 600^ could also be used. Two inch square
specimens of Inconel 600 were prepared by scrubbing with
cleanser, rubbing with 325 mesh silicia, rinsing in tap
water, and wiping dry.
Sieve fractions 2,3,4, and 5 were tried as enamels.
Since the system had to be simple, no clays or soluble
^international Nickel Company 23 salts could be used to form a stable water suspension of the glass. Therefore, a dry process was selected whereby the dry glass powders could be sifted onto the substrate.
The powdered glass was poured into a 100 mesh sieve having a diameter of three inches and sifted onto the horizontal substrate. The sieve was held about 6 inches from the specimen and tapped gently to obtain a uniform coat ing. The 100 mesh sieve was used because it allowed a con venient rate of application. It was determined that the maximum temperature at which a specimen could be fired with out excessive reaction between the substrate and the glass was l620°F. In order to develop a smooth coating with minimum bubbles a firing time of six minutes was necessary.
Sieve fractions 2,3,4, and 5 were then applied as dry enamels to four square inch specimens of Inconel. Application weights of 0 .5 , 1.0 and 1.5 grams of each sieve fraction were dusted on each specimen and the specimens were fired at l620°F for six minutes. Fractions 3 and 4 gave the smoothest coatings. The specimen prepared from fraction 4 was smoother than the one prepared from fraction
3 but formed more bubbles. Specimens for use as base coated samples were then prepared using the one gram application weight and sieve fraction 3. The specimens were fired at l620°F for 12 minutes.
In order to generate a regular wave form with the powdered glass, a wave pattern was fabricated. A sheet 24
h. of dentist*s wax was placed on a half-inch thick flat piece of glass and heated in an electric dryer until it was sufficiently soft to allow easy working. A threaded rod having 1/8 inch thread was then pressed into the wax and rolled across it. The wax was allowed to cool and the surface was then replicated using a cold-setting metallur gical mounting plastic.^ The replica was thoroughly cleaned and dried. A small amount of sieve fraction 3 was poured onto the replica and levelled flush to the surface of replica. The initially coated specimens were placed flush against the replica surface, inverted and tapped gently to release the powdered glass from the replica.
The replica was removed and the specimen fired at l620°p for three minutes. The specimen had an excessive amount of bubbles in the waves. Various particle size fractions, firing times and temperatures were tried, but bubble free specimens could not be prepared. Glass 604 was not sufficiently fluid at the maximum temperature to which the metal could be heated without excessive glass-metal reaction to allow adequate bubble removal from the base coat. Proper firing of the wave form could not be achieved. A chemical analysis of a sample of Glass 604 that was used (see Appendix C) showed a much higher alumina content than was anticipated from the batch
^ruewax Dents - Ply Base Plate Wax, Dentists Supply Company k ^Quickmount Self Curing Acrylic Plastic, Fulton Metallurgical Products Corporation, Pittsburgh, Pennsylvania 25 composition. The exeess alumina was attributed to the solution of the fire-clay crucible in the melt.
Another glass whose viscosity was studied by English
(9), Glass 607, was found to be appreciably softer than
Glass 6o4. This glass was selected for use as an enamel.
In order to reduce contamination, Glass 607 was melted in platinum. A 500 gram batch was prepared and mixed thoroughly. The size C platinum dish to be used as a crucible was placed on an insulating brick base. To minimize the loss of volatile constituents, a section of alumina tube was placed around the crucible and an
Insulating fire brick lid was placed over the top of the crucible. The entire apparatus was placed in an electric kiln and heated to 2100°P. The vessel was then removed from the kiln and a portion of the glass batch was charged into the platinum crucible. The lid was replaced, and the appar&tus was again placed In the furnace. After 30 minutes, all traces of the raw batch had disappeared.
The glass was then soaked at 2100°P for two hours. The crucible was then removed, and the glass was quenched by pouring it onto a one Inch thick steel plate- which was resting In a pan of water. The melting process was repeated until the entire raw batch had been melted. A portion of the glass was then crushed to pass through a 40 mesh sieve by hand with an alumina mortar and pestle. It was then ground in an automatic mortar and pestle. In order to prevent formation of an excessive amount of fine 26 material, the grinding stopped every three minutes and the glass was shaken by hand through the series of sieves used previously. Any material not passing through a 100 mesh sieve was returned for further grinding. Sieve fractions
3,4, and 5 were dusted onto two inch square Inconel Plates using a one gram application weight in the same manner as was done with Glass 604. From the work of Bremond (5) on the study of glazes, it was determined that a coating viscosity of approximately 3.3 log poises was the optimum value for formation of a smooth vitreous coating. From the problems experienced during the investigation using
Glass 604, a viscosity In the range of 3 log poises appeared to be necessary. Using the viscosity curve drawn from the data of English (Appendix C), it was decided to fire specimens at l450°F, l480°F and 1500°F. These temperatures corresponded to viscosities of 3.3, 2.9, and
2.5 log poises respectively. Specimens of the sieve fraction used were fired at each of these three temperatures for 3£ minutes. The specimens of sieve fractions 3 and 4 fired at l480°F and 1500°F were essentially bubble free and very smooth. Specimens prepared from sieve fraction 5 were rough and contained excessive bubbles at all three temperatures. Base coated specimens were then prepared using sieve fraction 4 with a 36 gram per square foot application weight. These specimens were fired at l480°F for 3? minutes. Wave patterns were prepared by the replica method using all particle size fractions In 27
Table 1. The wave coats were fired at l400°F for minutes. All waves produced had appreciably lower bubble content than any ones produced using Glass 604. The coarse particle fractions (1 and..2) formed waves which were very low In bubble content, but they were very irregular. The coarse particles would not pack in the replica; when the replica was removed the particles would flow into Irregular waves • The fine particles packed well in the replica and formed regular waves when the replica was removed. In firing, however, the outer surface fused too rapidly to allow gas to escape from the inner portion of the wave.
The waves produced by the replica method also created very thick coatings upon flowing out. The replica method did not generate homogeneous bubble-free waves, and the heavy coatings that resulted did not allow sufficient flexibility in the control of coating thickness. A new method for forming the waves was then developed.
Pinal Experimental Procedure
Development of a Wave Forming Method
A grid was fabricated using 0.1094 inch diamter rod.
Two small pieces of sheet aluminum were drilled with a line of 0.1094 inch diameter holes on a 0.2188 inch centers.
Pieces of 0.1094 inch rod were then forced through the drilled holes to form a regular grid. The grid was placed on a coated specimen and sieve fraction 4 was dusted over it.
The grid was then removed and the wave coat was fired at 1400% for 1-| minutes. The wave that resulted was practically free from bubbles, was uniform, and approximately sinusoidal. Sieve fractions 1,2, and 3 were also dusted on to form waves using the grid method and fired In the same manner. Fractions 1 and 2 formed bubble-free waves, but particles of the glass scattered excessively while being dusted onto the plate. The waves thus formed were less uniform and rougher In contour than those prepared from fraction 3- While the waves formed from fraction 3 were not absolutely bubble free, they were judged to be the best due to their uniformity and smooth surface. Additional grids were fabricated from 0.0625,
0.0781 and 0.0937 inch diameter rods to allow the formation of different wavelengths.
Preparation of Measuring Apparatus
To measure the peaks and valleys of the waves, a g Federal C21 dial indicator was mounted In a Mitutoyo 7 7003 Comparator Stand. Since the thermal expansion coefficient of Glass 607 was less than that of the metal, the coated specimens all warped on cooling. The glass always formed the convex surface. The specimens could not be measured on the flat anvil of the comparator stand.
A base plate was fabricated which would conform to the
^Federal Instrument Company
^Mitutoyo Instrument Company 29 bottom of the enameled specimen and would rest on the top of the comparator stand anvil. A small ring of one Inch conduit was placed against the bottom of an enameled specimen. A roll of modeling clay was placed around the bottom of the conduit to seal It to the specimen. Quick- 8 mount cold-setting acrylic plastic was poured into the conduit and allowed to harden. The resulting plastic cylinder thus had a surface which matched the curvature of the plate. The bottom of the plastic cylinder was then sanded smooth. The cylinder was positioned so that the maximum point of the curved surface of the top of the block was under the stylus of the dial gauge. The block was then glued to the base of the comparator stand.
Two inch squares of Inconel 600 were scrubbed with g Comet Cleanser, rubbed with 325 mesh silica, rinsed with tap water, and wiped dry. This method was shown (22) to be an adequate preparation of Inconel for enameling. In order to determine the coating thickness the metal plates were measured before enameling. The specimens could then be measured after the waves had leveled and the average coating thickness could be determined. A straight line was scratched across the middle of the specimen prior to the application of the base coat. Since the glass was clear the line could be seen easily. This was done so the measurements
^Quickmount Self Curing Acrylic Plastic, Pulton Metallurgical Products Corporation, Pittsburgh, Pennsylvania
9proctor and Gamble, Cincinnati, Ohio 30 would be made at the same point after each refire. An indention was cut at the end of the line with a small file to designate the side of the plate from which measurement was started.
The specimens were then weighed and a dust coat was sprinkled on them. The specimens were again weighed and the dusting repeated until a 0.5 gram coating weight was obtained. The coating was then fired horizontally at l480°F for 3% minutes. If a thicker base coat was necessary, second or third gram coats were applied and fired in the same manner. The coating was applied in this manner to allow the escape of bubbles formed during firing. The grid was placed on the fired base-coated specimen and the entire apparatus was weighed. It was found that approximately 1,3 grams of glass applied to the whole apparatus yielded ridges of the desired amplitude.
The grid was then removed and the.specimen placed gently
Into the furnace. The specimen was fired at l400°F for
1.15 to 2.15 minutes depending on the base coat thickness and the wavelength of the wave coat. The shortest wave length was fired for the shortest time to keep It from leveling totally. The back of the plate had some oxidation on It due to firing. Prior to measurement the back of the plate was rubbed with a fine piece of emery paper (Grit
4/0) so that the specimen would slide smoothly over the plastic anvil. The specimen was then pushed under the stylus of the dial indicator along the line which was 31 scratched on the metal surface of the plate, and the coating thickness at the bottom of the trough and the peak of the crest of the wave was recorded. The specimen was then placed in the furnace at the desired temperature for a time which was sufficient to allow measurable deformation to occur. The specimen was removed from the furnace and allowed to cool. The measurement of the waves was then repeated. This process was repeated until the coating was too smooth to allow accurate measurement of the wave. The specimen was given an additional firing, and a final measurement was taken to determine the average coating thickness.
It was found that upon prolonged refires, bubbles would develop in the Glass 607 coating. Long firing times also produced bubble formation. In order to study viscosities higher than 6 log poise, firing times greater than ten hours were necessary. Investigations carried out on the effect of water on the viscosity-temperature relationship of borate glasses (2 5 ) have shown that the presence of small amounts of water In a borate glass can
markedly reduce the viscosity of the glass at a given
temperature. Since Glass 607 had an exceptionally high
B20g content and the melting conditions could not be made
the same as those used by English (9), the viscosity data
could not be determined with sufficient accuracy to Judge the
developed mathematical model. Another glass, EF 9* which was 32 suitable to be used as a porcelain enamel on Inconel was obtained from the Owens-Illinois Glass Corporation.
Glass EP 9 was selected for three reasons: (l) An accurate viscosity-temperature curve was avaiable from the manufacturer. (2) The glass had a sufficiently low working point (Viscosity - 10^ poise) to allow it to be fired as an enamel without excessive oxidation of the substrate.
(3) The thermal expansion coefficient of the glass was sufficiently high to minimize warping of coated specimens.
Procedure for Glass EP 9
The properties of Glass EF 9, as1 listed by the Kimble
Glass Handbook (36), are given in Appendix C. The glass was received already ground. It was run through a series of sieves and sieve fractions 3*^, and 5 were retained.
Base coated specimens were prepared in the same manner as was done with Glass 6 0 7 . It was found that a 10 minute fire at l400°P was required to remove the bubbles from the base coat. Base coats of various thicknesses were formed by applying second and third coats as was necessary.
These coats were fired in the same manner. The wave coats were applied using sieve fraction 3 in the same manner as was done with Glass 6 0 7 . For the longer wave lengths a two minute 15 second fire was used when firing the wave coat. For the shorter wave lengths; the firing time used was one minute and 15 seconds; for the longer wavelengths, the firing time was two minutes and thirty seconds. The 33
waves were measured in the same manner as was done with
Glass 607. While a small amount of bubbles could not be removed from thick coatings, no bubble formation even during long refires was noticed. No other defects in the
coatings were noticed; thus the data obtained using Glass
EP 9 was judged to be satisfactory for evaluating the derived model.
Investigation of Gla33-Powdered Metal Mixtures
An investigation was then made to determine an inert
material which could be used as a dispersed phase in the
glass. In this manner the simple system could be made to
approach the condition of real porcelain enamel glasses which generally contain a dispersed phase that acts as an
opacifier. Oxide materials could not be used since the
low viscosity of the glass coating required for bonding
to the metal substrate could cause the solution of a
dispersed oxide phase. The viscosity of the glass matrix
and the volume fraction of the dispersed phase would both
be changed and evaluation of the rheological behavior of
the resulting system would be impossible. Powdered metals
appeared to offer the best possibility. The powdered metal
alloys evaluated are given in Table 2. 34
TABLE 2
POWDERED- METAL INVESTIGATION DATA
Nominal Composition Particle Size Supplier (Weight %) Number
Chromium Nickel
100 0 -325 mesh 1
40 60 -325 mesh 2
20 80 -325 mesh 2
0 100 -325 mesh 2
0 100 5-9 microns 3 The names and addresses of the suppliers are given in
Appendix B. Five gram samples with ten percent and twenty percent of the powders were prepared using Glass 607. The resulting mixtures were then dusted onto clean Inconel
specimens and fired at l480°F for 3% minutes. From a
visual observation, the 5-9 micron pure nickel powder
appeared to produce the best coating. All the other
coatings had a large number of pits and blisters which were
due apparently to the outgasslng of the metal powders.
The 5-9 pure nickel powder was chosen as the dispersed
phase to be used In the Investigation. Ten gram samples
containing ten, twenty and thirty percent by weight of the
5-9 micron nickel powder were prepared using sieve fraction
5 of Glass 6 0 7 . These samples were sintered for 30 minutes
at 1300°F. The mixture was sintered in order to produce
a homogeneous glass-metal composite which would not
separate while being dusted onto the substrate. The nickel 35
powder also outgassed during the sintering prior to applica
tion as a coating. The resulting composite was ground
by hand with a mortar and pestle; the material passing
through a 200 mesh sieve and remained on a 325 mesh
sieve was retained for use as an enamel. The resulting
enamels were dusted onto Inconel specimens and fired
at l480°F for 3% minutes. The enamel containing ten percent
nickel had a smooth surface and resembled a conventional
porcelain enamel. The enamels containing twenty and thirty
percent of the nickel powder had a rough surface and did
not appear to wet the substrate well. The specimens were
then refired at l400°F for ten minutes to determine if the
coatings were stable under refiring at a lower temperature.
Both the twenty and thrity percent nickel specimens
pulled together in globules thereby exposing the metal.
The ten percent nickel specimen remained a smooth vitreous
coating?. The 5-9 micron nickel powder could be thus used
as a dispersed phase if the amount used was approximately
ten percent.
Preparation of Glass EF 9-Powdered Nickel Wave Specimens
Due to the possibility of reaction between the water in
Glass 607 and the nickel powder it was decided to use Glass
EP 9. Five gram specimens of glass and 5-9 micron nickel
powders were prepared using ten, twenty and thirty percent
weight of the metal phase in sieve fraction 5 of Glass EF 9 .
These mixtures were sintered on nickel foil at 1200°F for 36 thirty minutes. The sintered composites did not bond to the foil at that -temperature and could be easily removed. The samples were crushed, ground and applied as was done with the composites made using Glass 607. Again it was found that the composite containing 10# nickel formed a smooth unified coating. The composites containing 20 and 30# of the dispersed phase also dewetted upon a prolonged refire at l400°P. A fifty gram batch of the mixture of the composite containing 10# of the dispersed nickel phase was then prepared. The specimens were prepared, fired, and measured in the same manner as was done previously for
Glass EP 9 without any dispersed phase.
Measurement of Surface Tension of Glasses 607 tend EF 9
A simplified drop weight method similar to that used by Cox and Parr (7) was selected to determine the surface tension of Glasses 607 and EP 9. The method has the advantages of being simple to carry out and requiring no extensive apparatus. The values determined by this method must be multiplied by a factor of 1.8 to yield the actual surface tension. The furnace which was constructed is shown in Figure 1 and Figure 2. The transite plate was drilled with three holes, two to accommodate the thermo couple wires and one to allow passage of the glass fiber.
A slot was also cut so the formation of the drop could be watched. A chromel-alumel thermocouple was passed through the transite lid and connected to a potentiometer with TRANSITE
0.5 0.5"l.D. FUSED SILICA TUBE 0.06 25" WALL
0 .75 —3 000 °F INSULATING BRICK
TRANSITE
SURFACE TENSION MEASURING FURNACE
FIGURE I u> -'J THERMOCOUPLE
FI BER
FURNACE PROPANE TORCH
POTENTIOMETER
C —INCONEL PAN
RING STAND
SURFACE TENSION MEASURING FURNACE FIGURE 2 39 temperature-compensated lead wires. The source of heat for the furnace was a Turner propane torch10.
Fibers were prepared by placing a specimen of the glass to be measured on an Inconel plate. The glass was heated with a propane torch until-It was soft enough to allow drawing of fibers. The flame was removed and a fused silica rod was dipped Into the softened glass and pulled to form a fiber. Sections between eight and sixteen
Inches long were broken from the fiber and checked for roundness and uniformity with a dial indicator comparator stand. Specimens uniform to 0.0005 inches were retained.
The furnace was supported on the edges of two
Insulating bricks In such a manner thajb the drop that would fall from the rod could be easily removed. The propane torch was aligned with the port in the side of the furnace and the temperature of the hot zone was controlled by adjusting the gas flow valve of the torch. The furnace was allowed to equilibrate at a temperature In the desired range. Initially a fiber of Glass 60J was lowered by hand
Into the furnace at 1550°F at the rate of five millimeters per minute. This rate was determined by Cox and Farr (7).
The location in the furnace where the drop fell was noted and the thermocouple bead was positioned at this point.
Another fiber was measured and the procedure was repeated.
The bead fell from the fiber at the same position. The
10Turner Corporation, Sycamore, Illinois 40
temperature registered by the thermocouple was taken as the
temperature of the glass bead. Two measurements were
made at each temperature. Whenever possible, both
measurements were made with the same fiber. The beads
which fell from the fibers were removed and weighed on a
Grammatic1'*’ balance. The average weight of the beads and
the average diameter of the fiber were then calculated.
The surface tension of the glass was calculated from the
following equation:
x 1.8
where m = mass of the drop (grams)
D = diameter of rod (centimeters)
I f - surface tension
g = acceleration of gravity cm/sec2
1.8 = correction factor of Cox and Parr
The temperature of the furnace was changed, allowed to
equilibrate, and the process repeated until sufficient
data points were obtained to plot surface tension-
temperature curves for Glasses 607 and EP 9 .
All experimental data is collected in Tables 3
through 7 .
■^E. Mettler, Zurich, Switzerland: Distributed by Fisher Scientific, New York, New York EXPERIMENTAL DATA
Table 3 contains the average amplitude, cumulative firing time, average coating thickness, wavelength, and firing temperature for specimen series D. This series comprises all specimens run with Glass 6 0 7 .
Table 4 gives the same data for series E which contains all specimens run using Glass EF 9 with no dispersed phase.
Table 5 gives the data for series F, Glass EF 9 with ten weight percent of powdered nickel as a dispersed phase.
Tables 6 and 7 contain surface tension data obtained for Glasses EF 9 and 607 respectively.
41 42
TABLE 3
AVERAGE AMPLITUDES AND CUMULATIVE FIRING TIMES
SPECIMEN SERIES D
GLASS 607 ______
Specimen No. D19
AMPLITUDE (INCHES) CUMULATIVE FIRING'TIME (MINUTES)
0.0413 0
0.00160 4
0.00068 8
0.00040 12
Average Coating Thickness QQ053 Inches
Wavelength 0.1250 Inches
Firing Temperature 1388°F
Specimen No. D20
AMPLITUDE (INCHES) ______CUMULATIVE FIRING TIME (MINUTES)
0 .00374 0
0.00195 5 0.00098 10 0.00055 15 Average Coating Thickness 0.0070 Inches
Wavelength 0.1250 Inches
Firing Temperature 1340°F 43
TABLE 3 (Continued)
Specimen No. D21
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00485 0
0.00185 3 0.00060 6
Average Coating Thickness 0.0070 Inches
Wavelength 0.1250 Inches
Firing Temperature 1388°F
Specimen No. D22
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00362 0
0.00080 5
Average Coating Thickness 0.0100 Inches
Wavelength 0.1250 Inches
Firing Temperature 134o°f
Specimen No. D23
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES )
0.00457 0
0.00323 10
0.00153 20
0.00096 30
0.00057 40 44
TABLE 3 (Continued)
Specimen No. D23 (Continued)
Average Coating Thickness 0.0060 Inches
Wavelength 0.1250 Inches
Firing Temperature 1352°F
Specimen No. D24
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00497 0
0.00433 10
0.00387 30
0.00242 50
O.OOI69 70
0.00132 90
Average Coating Thickness 0.0092 Inches
Wavelength 0 .15625 Inches
Firing Temperature 1255°F
Specimen No. D25
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00412 0
0.00206 10 0.00088 20
Average Coating Thickness 0.0062 Inches
Wavelength 0.125 Inches
Firing Temperature 1352°F TABLE 3 (Continued)
Specimen No. D26
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES) 0.00380 0
0.00210 10
0.00130 20
0.00090 30
Average Coating Thickness 0.0049 Inches
Wavelength 0.125 Inches
Firing Temperature 1352°F
Specimen No. D28
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES) 0.0044 0
0.00158 10
0.00053 20
0.00027 30 Average Coating Thickness 0.0084 Inches
Wavelength 0.0125 Inches
Firing Temperature 1310°F
Specimen No. D30
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00412 0
0.00357 10
0.00297 25 46
TABLE 3 (Continued)
Specimen No. D3Q (Continued)
0.00242 40
0.00179 55
0.00125 70
0.00102 85
0.00077 100
0.00054 115
Average Coating Thickness 0.0090 Inches
Wavelength 0.125 Inches
Firing Temperature 1255°F
Specimen No. D31
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00265 0 0.00180 15
0.00127 30
0.00086 45
0.00054 60
0.00039 75 Average Coating Thickness 0.0097 Inches
Wavelength 0.125 Inches
Firing Temperature 1255°F 47
TABLE 3 (Continued)
Sfrecltnen No. D32
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00304 0
0.00273 15
0.00249 30 0.00210 53
0.00156 100
0.00115 145
Average Coating Thickness 0.0061 Inches
Wavelength 0.1250 Inches
Firing Temperature 1255°F
Specimen No. D33
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00431 0 0.00212 15
0.00086 30
0.00054 40
0.00037 50
Average Coating Thickness O.OO76 Inches
Wavelength O.I875 Inches
Firing Temperature 1358°F 48
TABLE 3 (Continued)
Specimen NO. D34
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00475 0
0.00329 8
0.00216 16
0.00134 24
0.00090 32
0.00057 40 Average Coating Thickness O.OO79 Inches
Wavelength O.I875 Inches
Firing Temperature 1358°F
Specimen No. D35
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00423 0 0.00178 20
0.00079 30
0.00062 35 0.00038 45 0.00022 55
Average Coating Thickness 0.0097 Inches
Wavelength O .1875 Inches
Firing Temperature 1335°F 49
TABLE 3 (Continued)
Specimen No. D36
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00485 0
0.00367 4
0.00251 8
0.00175 12 0.00112 16
0.00072 20
0.00041 24
0.00029 28
Average Coating Thickness 0.0100 Inches
Wavelength 0.1875 Inches
Firing Temperature 1350°F
Specimen No. D37
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00450 0
0.00313 10 0.00187 20
0.00117 30
0.0078 40
Average Coating Thickness 0.0097 Inches
Wavelength O .1875 Inches
Firing Temperature 1335°F 50
TABLE 3 (Continued)
Specimen No. D38
AMPLITUDE (INCHES) ______CUMULATIVE FIRING TIME (MINUTES)
0.00519 0 0.00419 5
0.00265 13 0.00163 21
0.00095 29 0.00065 37
Average Coating Thickness 0.0077 Inches
Wavelength 0.1875 Inches
Firing Temperature 1350°F
Specimen No. D3 8 -A
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00465 0
0.00313 5
0.00155 13
0.00055 21
0.00032 29
Average Coating Thickness O.OO87 Inches
Wavelength 0.1875 Inches
Firing Temperature 1350°F 51
TABLE 3 (Continued)
Specimen No, D39
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00451 0
0.00373 22
0.00296 44
0.00182 79 0.00125 100
0.00073 130
0.00052 155 Average Coating Thickness 0.0081 Inches
Wavelength 0.1875 Inches Firing Temperature 1310°F
Specimen No. D4l
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00561 0
0.00394 8
0.00212 18
0.00129 28
0.00082 38
0.00060 48 0.00050 58
Average Coating Thickness 0.0079 Inches
Wavelength 0,1875 Inches
Firing Temperature 1350°F 52
TABLE 3 (Continued)
Speclmten1 No. D43
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00610 0
0.00315 5 0.00130 10
0.00068 15 0.00032 20
0.00021 25 Average Coating Thickness 0.0129 Inches
Wavelength O.I875 Inches
Firing Temperature 1350°f
Specimen No. D45
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00610 0
0.00040 9 Average Coating Thickness 0.0106 Inches
Wavelength 0.2188 Inches
Firing Temperature l4lO°F
Specimen N o . D46
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00580 0
0.00323 6 0.00148 12 53
TABLE 3 (Continued)
Specimen No. P46 (Continued)
0.00080 18
0.00037 24
Average Coating Thickness 0.00980 Inches
Wavelength 0.2188 Inches
Firing Temperature 1375°F
Specimen No'. P47
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00551 0 0.00342 12
0.00195 24
0.00112 3 6
0.00068 48
Average Coating Thickness 0.00970 Inches
Wavelength 0.2188 Inches
Firing Temperature 1335°F
Specimen No. P48
AMPLITUDE (INCHES) ______CUMULATIVE FIRING TIME (MINUTES)
0.00572 0 0.00440 20
0.00350 35 0.00236 60
0.00163 80 54
TABLE 3 (Continued)
Specimen No. P48 (Continued)
Average Coating Thickness 0.0109 Inches
Wavelength 0.2188 Inches
Firing Temperature 1290°F
Specimen No. D49
AMPLITUDE (INCHES) 1 ' CUMULATIVE FIRING TIME (MINUTES)
0.00470 0
0.00309 62
0.00247 92 Average Coating Thickness 0.0140 Inches
Wavelength 0.2188 Inches
Firing Temperature 1250°F
Specimen No. D51
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00243 0
0.00165 30
0.00117 60
Average Coating Thickness 0.01367 Inches
Wavelength 0.1250 Inches
Firing Temperature 1215°F 55
TABLE 3 (Continued)
Specimen No. D52
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00641 0
0.00423 8
0.00212 18
0.00111 28
0.00054 38
Average Coating Thickness 0.01151 Inches
Wavelength 0.1250 Inches
Firing Temperature 1264°F
Specimen No. D53
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00386 0
0.00258 5 0.00161 10
0.00108 15
0.00067 20
Average Coating Thickness 0.01640 Inches
Wavelength 0.1250 Inches
Firing Temperature 1264°F 56
TABLE 3 (Continued)
Specimen No. D54
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
O .60345 0 0.00276 10
0.00218 20
0.00157 35
0.00129 45 Average Coating Thickness 0.0080 Inches
Wavelength O .1563 Inches
Firing Temperature 1288°F
Specimen No. D55
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00511 0 0.00362 12
0.00236 24
0.00161 34 0.00102 44
0.00069 5^
Average Coating Thickness 0.0098 Inches
Wavelength O .1563 Inches
Firing Temperature 1288°F 57
TABLE 3 (Continued)
Specimen'No . 'D56
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME ( MINUTES)
0.00603 0
0.00306 10
0.00143 20
Average Coating Thickness 0.0137 Inches
Wavelength 0.1563 Inches
Firing Temperature 1288°F
Specimen No. D57
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00610 0
0.00565 69
0.00520 144
Average Coating Thickness 0.0115 Inches
Wavelength 0.125 Inches
Firing Temperature 1175°P
Specimen No. D 58
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME(MINUTES)
0.00375 0
0.00217 5
0.00145 10
0.00117 15 58
TABLE 3 (Continued)
Specimen NO. D58 (Continued)
Average Coating Thickness 0.0066 Inches
Wavelength 0.2188 Inches
Firing Temperature l450°F
Specimen No. D59
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.00515 0 0.00504 30
0.00477 140
Average Coating Thickness O.OI67 Inches
Wavelength 0.125 Inches Firing Temperature______ll45°F _____■ ■ ■ - . ■ ■ 59
TABLE 4
AVERAGE AMPLITUDES AND CUMULATIVE FIRING TIMES
SPECIMEN SERIES E
OWENS ILLINOIS GLASS. EF' 9
Specimen El
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00227 0
0.00214 .1490
0.00191 3910
Average Coating Thickness O.OO65 Inches
Wavelength 0.2188 Inches
Firing Temperature 1165°F
Specimen E2
AMPLITUDE (INCHES) CUMULATIVE FIRING' TIME (MINUTES )
0.00509 0 0.00468 8
0.00401 23 0.00391 38
0.00274 58
0.00213 78
Average Coating Thickness 0.0091 Inches
Wavelength 0.2188 Inches
Firing Temperature 1260°F 60
TABLE 4 (Continued)
Specimen E3
Am p l i t u d e (i n c h e s ) c u m u l a t i v e f i r i n g t i m e (m i n u t e s ) 0.00426 0
0.00326 10
0.00259 20
0.00178 35
Average Coating Thickness 0.0070 Inches
Wavelength 0.2188 Inches
Firing Temperature 1320°F
Specimen E4 a m p l i t u d e (i n c h e s ) c u m u l a t i v e f i r i n g t i m e (m i n u t e s )
0.00357 0 0.00256 6
0.00169 12
0.00110 18
0.00070 24
Average Coating Thickness 0.0070 Inches
Wavelength 0.2188 Inches
Firing Temperature 1370°F 61
TABLE 4 (Continued)
Specimen E5
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00178 ■o'
0.00088 5 0.00021 10
Average Coating Thickness 0.0070 Inches
Wavelength 0.2188 Inches
Firing Temperature l440°F
Specimen E6
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES) 0.00500 0
0.00433 260
0.00325 940
0.00263 1495
0.00153 2755 Average Coating Thickness 0.0133 Inches
Wavelength 0.1250 Inches
Firing Temperature 1100°F
Specimen E7
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00281 0
0.00201 5 0.00118 14
0.00090 19 62
TABLE 4 (Continued),
Specimen E7 (Continued)
Average Coating Thickness 0,0060 Inches
Wavelength 0.2188 Inches
Firing Temperature l400°F
Specimen E8
AMPLITUDE (INCHES)______CUMULATIVE FIRING TIME (MINUTES)
0.0024-3 0 0.00226 1170
Average Coating Thickness 0.0204 Inches
Wavelength 0.1250 Inches
Firing Temperature 1032°f
Specimen E9
AMPLITUDE (INCHES) ______CUMULATIVE FIRING TIME (MINUTES)
0.00223 0 0.00109 5 0.00072 10
0.00049 15 0.00026 20
Average Coating Thickness 0.00451 Inches
Wavelength 0.2188 Inches
Firing Temperature l480°F 63
TABLE 4 (Continued)
Specimen E10
0.00308 0
0.00265 8160
0.00198 29500
Average Coating Thickness 0.0195 Inches
Wavelength 0.1250 Inches
Firing Temperature 995°F
Specimen Ell
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00226 0
0.00192 900
0.00168 1830
0.00134 3190 Average Coating Thickness 0.0204 Inches
Wavelength 0.1250 Inches
Firing Temperature 1060°F 64
TABLE 5
WAVE AMPLITUDE AND CUMULATIVE FIRING TIME
SPECIMEN SERIES F
90# GLASS EF 9 10# INCO 123 NICKEL .
Specimen No. FI
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES) 0.00241 0
0.00073 1180 0.0048 1900
0.00044 2560
Mean Coating Thickness O.OO78 Inches
Wavelength 0.1250 Inches
Firing Temperature ll80°F
Specimen No, F2
AMPLITUDE (INCHES) CUMULATIVE FIRING TIME (MINUTES)
0.00169 0 0.00150 12
0.00107 46
0.00088 76
0.00074 106
0.00051 221 • -
0.00029 361
Mean Coating Thickness 0.0072 Inches
Wavelength 0.1250 Inches
Firing Temperature 1260°F 65
TABLE 5 (Continued)
Specimen No. F3
AMPLITUDE (INCHES) _____ CUMULATIVE FIRING TIME (MINUTES)
0.00332 0
0.00317 6 0.00278 28
0.00234 92
Mean Coating Thickness 0.0071 Inches
Wavelength O.IO98 Inches
Firing Temperature_____1380°F______• 66
TABLE 6
SURFACE TENSION DATA
GLASS EF. 9
Spec. Temperature Drop Weight Fiber Diam T No. °F (ras) in x 103 dynes/cm
EF9-1 1637 9.4 11.5 182
EF9-2. 1744 11.5 15.5 166
EF9-3 1400 10.4 11.5 200
EF9-4 1503 11.8 14.0 186
EF9-5 1380 10.0 11.2 198
EF9-6 1235 9.5 9.8 214
EF9-7 1512 8.7 10.7 181
TABLE 7
SURFACE TENSION DATA
___ GLASS 607
Spec. Tempgrature Drop Weight Fiber Diam T No. (mg) in x 103 dynes/cm
607-1 1550 19.4 19.0 228
6©7-2 1749 17.2 17.3 222
607-3 1329 20.8 19.0 245 607-4 1840 6.6 7.0 214 Surface Tension (Dynes/cm) 250 250 230 230 290 270 270 210 00 20 40 60 80 2000 1800 1600 1400 1200 1000 ufc Tnin s Temperature Vs. Tension Surface eprtr (°F)Temperature ls 607 Glass iue 3 Figure 67 Surface Tension (Dynes/cm) 200 220 240 160 180 800 ufc Tnin Vs. Temperature Tension Surface 00 20 40 60 1800 1600 1400 1200 1000 eprtr (°F) Temperature ls EF9 Glass iue 4 Figure 68 ANALYSIS OP DATA
The expression derived for the degree of leveling
(Appendix A, Eq. 131 ) expressed in terms of the common logarithm is:
l0S * r °5 t +
constant
Therefore, for a given specimen fired at a given temperature, a graph of log i versus the time that the coating is a flowing will be a straight line if the model is valid. In the experimental procedure, the waves on a specimen were measured at room temperature. The specimen was placed in the furnace and fired for a given time. The specimen was removed from the furnace, placed on a block of metal and allowed to cool to room temperature. The amplitude of the waves was then measured. This process was repeated a number of times so the specimen was subjected to a number of cycles whereby it was raised abruptly from room temperature to furnace temperature, held at furnace temperature for a time and then quenched rapidly to room temperature. The cumulative firing time which was actually measured Included the time required for the specimen to reach furnace temperature (heat-up time) plus the time that 69 70 the specimen was maintained at the furnace temperature.
Previous work (22) has shown that for the same type of furnace and a similar substrate the heat-up time is one minute. This heat-up time must be multiplied by the number of firings that the specimen has undergone and subtracted from the measured cumulative firing time to yield the effective firing time. The effective firing time is thus the actual time of flow in the derived model.
It was assumed, that upon removal from the furnace, the surface of the coating is immediately quenched to a rigid state so that no further deformation of the surface occurred. Since the temperature difference between the furnace and the ambient air was generally greater than
1000°F, and the specimen reached room temperature in approximately three minutes, the assumption appeared reasonable.
In order to avoid the inclusion of conversion factors; it was decided to use the centimeter-gram-second system throughout the analysis. The actual experimental data was measured in inches; it is converted to centimeters in
Tables 8 through 10.
The values for log ^ and effective firing time for experimental series D, E, and P are given in Tables 8 through 10. Figures 5 through 21 are graphs of log — 81 versus effective firing time for all specimens prepared from
Glass EF 9, series E and F. For brevity, the log — versus 81 71 effective firing time data will not be presented graphically in this work. Graphs were prepared from this data, however, and the graphs were found to be essentially straight lines.
Some deviations from linearity in the initial stages of firing noted on several specimens was observed to be caused by the incomplete formation of a uniform wave surface in the initial preparation of the specimen.
Deviations in the final stages of firing were observed in specimens where excessive bubble formation occurred during the firing process. Prom the linearity of Figures
5 through 15, it is shown that the model is valid to the extent of the logarithmic dependence of the amplitude of the wave upon the actual time of flow. 72
TABLE 8
TIME CORRECTED DATA
SPECIMEN SERIES D
GLASS 607;
Specimen No. D19
1/a (CENTIMETERS) ”1 Log l/a CORRECTED TIME (MINUTES)
242 2.384 0
625 2.796 3
1470 3.167 6
2500 3.398 9
Specimen No. D20 l a (CENTIMETERS) -1 Log l/a CORRECTED TIME (MINUTES)
267 2.427 0
513 2.710 4 1020 3.009 8
1818 3.260 12
Specimen No. D21
1/a (CENTIMETERS)“1 Log l/a CORRECTED TIME (MINUTES)
206 2.314 0
540 2.732 2 1666 3.222 4 73
TABLE 8 (Continued)
Specimen No. D22 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
276 2.441 0
1250 3.097 4
Specimen No. D23 l/a (CENTIMETERS r 1 Log l/a CORRECTED TIME (MINUTES)
219 2.340 0
310 2.491 9 654 2.816 18
1042 3.018 27 1754 3.244 36
Specimen No. D24
l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
201 2.303 0
231 2.364 9 258 2.412 28
413 2.6l6 47
592 2.772 66
758 2.880 85
Specimen No. D25
l/a (CENTIMETERS)-1 Log l/a CORRECTED TIME(MINUTES) 243 2.386 0 485 2.686 9 1136 3.054 18 74
TABLE 8 (Continued)
Specimen No. D26 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
263 2.420 0
476 2.678 9 769 2.886 18
1111 3.046 27
Specimen No. D28 l/a (CENTIMETERS)“1 Log l/a CORRECTED TIME (MINUTES)
227 2.344 0
633 2.801 9
1887 3.276 18
3704 3.569 27
Specimen No. D30 i—I to (CENTIMETERS)"1 Log l/a CORRECTEDTIME (MINUTES)
243 2.386 0
280 2.447 9
337 2.528 23
413 2.616 37
559 2.747 51
800 2.903 61
980 2.991 79
1299 3.114 93
1852 3.268 107 75
TABLE 8 (Continued)
Specimen No. D31 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
377 2.576 0
556 2.741 14
787 2.892 28
1163 3.066 42
1852 3.268 56
2564 3.409 70
Specimen No. D32 l/a (CENTIMETERS)"1 Log l/a ' CORRECTED TIME (MINUTES)
329 2.517 0
366 2.563 14
402 2.6o4 28
476 2.678 52
641 2.807 96
870 2.939 140
Specimen No. D33 . l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
232 2.365 0
472 2.674 14
1163 3.066 28
1852 3.268 37
2703 3.432 46 7 6
TABLE 8 (Continued)
Specimen No. P34 l/a (CENTIMETERS)rl Log l/a ' CORRECTED' TIME (MINUTES) 210 2.322 0
304 2.482 7 463 2.666 l4
746 2.873 21 llll 3.046 28
1754 3.244 35
Specimen No. 035 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
236 2.373 0
562 2.750 19 1266 3.102 28
1613 3.208 32
2632 3.420 41
4543 3.658 50
Specimen No. D36 l/a (CENTIMETERS)-1 Log l/a CORRECTED TIME (MINUTES)
206 2.314 0
272 2.435 3
398 2.600 6
571 2.751 9
893 2.951 12 TABLE 8 (Continued)
Specimen NO. D36 (Continued) l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
1389 3.143 15
2439 3.387 18 3448 3.538 21
Specimen No. D37 l/a (CENTIMETERS)"1 .Log l/a CORRECTED TIME (MINUTES)
222 2.346 0
319 2.504 9
535 2.728 18
934 2.970 27
1282 3.108 36
Specimen No. D38 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
193 2.286 0
239 2.378 4
377 2.576 11
613 2.787 18
1053 3.022 25
1538 3.187 32 78
TABLE 8 (Continued)
Specimen No. D38-A l/a (CENTIMETERS)"1 Leg l/a CORRECTED TIME (MINUTES)
215 2.332 0
319 2.504 4
645 2.809 11 1818 3.260 18
3125 3.495 25
Specimen No. D39 l/a (CENTIMETERS) “1 Log l/a CORRECTED TIME (MINUTES)
222 2.346 0
268 2.428 21
338 2.529 42
549 2.740 76
800 2.903 96
1370 3.137 125
1923 2.384 149
Specimen No. D4l l/a (CENTIMETERS ) "1 Log l/a CORRECTED TIME (MINUTES )
178 2.250 0
254 2.405 7
472 2.674 16
775 2.889 25 1220 3.086 34 79
TABLE 8 (Continued)
Specimen NO. P4l (Continued): l/a (CENTIMETERS)"1 Log l/a CORRECTED'TIME (MINUTES)-
1667 3.222 43
2000 3,301 52
Specimen No. P43 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
164 2.215 0
317 2.501 4 769 2.886 8
1471 3.168 12
3125 3.495 16 4762 3.678 20
Specimen No. D45 l/a (CENTIMETERS)-1 Log l/a CORRECTED TIME (MINUTES)
164 2.215 0
2500 3.398 8
Specimen No. P46 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
172 2.236 0
310 2.491 5
676 2.830 10
1250 3.097 15 2703 3.^32 20 80
TABLE 8 (Continued)
Specimen No. D47 l/a (CENTIMETERS)-1 Lo r l/a CORRECTED TIME (MINUTES)
181 2.258 0
292 2.465 11
515 2.710 22
895 2.951 55
1470 5.167 44
Specimen No. D48 l/a (CENTIMETERS)-1 Lo r l/a CORRECTED TIME (MINUTES)
175 2.243 0
227 2.356 19
286 2.456 55
424 2.627 57 613 2.787 76
Specimen No. D49 l/a (CENTIMETERS)-1 Lo r l/a CORRECTED TIME (MINUTES)
215 2.328 0
324 2.510 61
405 2.607 90
Specimen No. D51 l/a (CENTIMETERS)"1 Log l/a______CORRECTED TIME (MINUTES)
412 2.615 0
606 2.782 29
855 2.952 58 81
TABLE 8 (Continued)
Specimen No.' D52 l/a (CENTIMETERS) ‘"1 Log l/a CORRECTED TIME (MINUTES)
156 2.193 0
236 2.373 7 472 2.674 16
901 2.955 25 1852 3.268 34
Specimen No. D53 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
259 2.413 0
388 2.589 4
621 2.793 8
926 2.967 12
1493 3.174 16
Specimen No. D54 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
290 2.462 0
362 2.559 9
459 2.662 18
637 2.804 32
775 2.889 4l 82
TABLE 8 (Continued)
Specimen No. D55 .1 l/a (CENTIMETERS) ' LOg l/a CORRECTED TIME (MINUTES):
196 2.292 0
27 6 2.440 11
424 2.627 22
621 2.793 31
980 2.991 40
1449 3.161 49
Specimen No. D 56 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
166 2.220 0
327 2.515 9
699 2.844 18
Specimen No. D57 l/a (CENTIMETERS)-1 Log l/a CORRECTED TIME (MINUTES)
164 2.215 0
177 2.248 68
192 2.283 142
Specimen No. D58
l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
267 2.427 0
46l 2.664 4 83
TABLE 8 (Continued)
-1 l/a (CENTIMETERS)' Log l/a CORRECTED TIME (MINUTES)
690 2.839 8
855 2.932 12
Specimen Nb . D59 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
194 2.289 0
198 2.297 29 210 2.322 . 13.8 84
TABLE 9
CORRECTED TIME DATA
SPECIMEN SERIES E
OWENS ILLINOIS GLASS EF 9
Specimen No. El l/a (CENTIMETERS) " 1 Log l/a CORRECTED TIME (MINUTES)
440 2.643 0 467 2.669 1489
524 2.719 3908
Specimen No. E2 l/a (CENTIMETERS)”1 Log l/a CORRECTED TIME (MINUTES)
196 2.292 0
214 2.330 7
249 2.396 21
254 2.405 35
365 2.562 54
469 2.671 73
Specimen No. E3
l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
234 2.309 0
306 2.486 9
386 2.587 18
562 2.750 32 85
TABLE 9 (Continued)
Specimen No. E4 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME' (MINUTES)
280 2.447 0
391 2.592 5
592 2.772 10
909 2.958 15
1428 3.154 20
Specimen N o . E5 l/a (CENTIMETERS)'1 Log l/a CORRECTED TIME (MINUTES)
562 2.750 0
1136 3.055 4
4762 3.678 8
Specimen NO. E6 l/a (CENTIMETERS)"1 . Log l/a.. CORRECTED TIME (MINUTES)
100 2.301 0
225 2.352 259 308 2.489 948
380 2.580 1492
654 2.816 2751
Specimen N o . E7 l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
356 2.551 0
498 2.697 4 86
TABLE 9 (Continued)
l/a (CENTIMETERS)**1 Log l/a CORRECTED TIME (MINUTES):
847 2.928 12
1111 3.046 16
Specimen No. E8
!/a (CENTIMETERS)_1 Log l/a CORRECTED TIME (MINUTES)
4ll 2.614 0
442 2.645 1169
Specimen No. E9 l/a (CENTIMETERS)~1 Log l/a CORRECTED TIME (MINUTES)
448 2.651 0
917 2.962 4
1389 3.142 8 2041 3.310 12
3846 3.585 16
Specimen No. E10 l/a (CENTIMETERS ) ~1 Log l/a CORRECTED TIME (MINUTES')
325 2.512 0
377 2.576 8159
505 2 .703 29^99 87
TABLE 9 (Continued)
Specimen No. Ell l/a (CENTIMETERS)"1 Log l/a CORRECTED TIME (MINUTES)
442 2.645 0
521 2.717 899
595 2.775 1828
______746 2.873 318? 88
TABLE 10
CORRECTED TIME DATA
SPECIMEN SERIES F
90# OWENS ILLINOIS GLASS EF 9 It# INCO 123 NICKEL POWDER
Specimen No. FI l/a (CENTIMETERS)-1 Log; l/a CORRECTED TIME (MINUTES)
164 2.215 0
523 2.719 1179 820 2.914 1898
894 2.951 2557
Specimen No. F2
l/a (CENTIMETERS)_1 Log l/a CORRECTED TIME (MINUTES)'
232 2.365 0
262 2.418 11
368 2.566 44
446 2.649 73 532 2.726 102
782 2.893 216
1350 3.130 355
Specimen No. F3
l/a (CENTIMETERS)“1 Log l/a CORRECTED TIME (MINUTES)
119 2.076 0
124 2.093 5
142 2.152 26
168 2.225 89 89
Log Vs. Corrected Time Specimen El
3.4
3.2
3.0
2.8
2.6
2.4
2.2 1000 2000 3000 4000 Corrected Time (minutes)
Figure 5. 90 Log Vs. Corrected Time Specimen E2
3.4
3.2
3.0
2.6
2.4
2.2 20 4 0 60 80 Corrected Time (minutes)
Figure 6 91
Log j[ Vs. Corrected Time Specimen E3
3.4
3.0
-l« o 2.8
2.6
2.4
2.2 D 20 30 40 Corrected Time (minutes)
Figure 7 92
Log *5- Vs. Corrected Time Specimen E4
3.4
3.2
3.0
2.6
2.4
2.2 20 Corrected Time (minutes)
Figure 8 93 Log— Vs. Corrected Time Specimen E5
3.6
3.4
3.2
3.0
2.8
2.6
2.4 ) 20 30 40 Corrected Time (minutes)
Figure 9 Log a 2.2 2.6 2.4 3.2 2.8 3.4 3.0 o g s Cretd Time Log -g- Corrected Vs. orce Tm (minutes) Time Corrected pcmn E Specimen iue 10 Figure 6 94 95
Log jj[ Vs. Corrected Time Specimen E 7
3.4
3.2
3.0
2.6
2.4
2.2 ) 20 30 40 Corrected Time (minutes)
Figure II 96
Log Vs. Corrected Time Specimen E8
3.4
3.2
3.0
5 0 0 1000 Corrected Time (minutes)
Figure 12 97
Log - Vs. Corrected Time Specimen E9
3 . 6
3.4
3.2
-|o o» 3.0
2.8
2.6
2.4 20 30 40 Corrected Time (minutes)
Figure 13 98
Log -5- Vs. Corrected Time Specimen EIO
3.4
3.2
3.0
-lo o> 2 .8
2.6
2.4
2.2 3 20 30 Corrected Time (minutes)
Figure 14 99
Log 5 - Vs. Corrected Time Specimen E 11
3.4
3.2
3.0
2.6
2.4
2.2 1000 2000 3000 4000 Corrected Time (minutes)
Figure 15 Log a 2.2 2.4 2.0 3.2 3.0 2.8 2.6 o^ V. orce Time Corrected Vs. Log^- 00 0 0 0 4 0 0 0 3 0 0 0 2 1000 pcmn FI Specimen orce Tm (minutes) Time Corrected iue 16 Figure 100 101
Log£ Vs. Corrected Time,
Specimen F2
3.2
3.0
2.8
2.4
2.0 100 200 300 400 Corrected Time (minutes) Figure 17 102
Log^j Vs. Corrected Time Specimen F3
3.2
3.0
2.8
-lo o> 3 2.6
2.4
2.2
2.0 20 40 60 80 Corrected Time (minutes)(minutes)
Figure 18 RESULTS AND DISCUSSION
In order to evaluate the derived model as a rheological t o o l , equation (131)t Appendix A, is modified to a form which is more compatible with graphical data from a graph of log 1 versus time. Since the graph is linear, the a constant could be eliminated and a simple expression for viscosity could be formulated utilizing the slope of the graph. The modified equation is:
4.608 fclog T" 3
where 7/- viscosity (poises)
- surface tension dynes/cm
k - wave number (cm-1)
C5 - dimensionless function of wave number and thickness
= slope of the graph of log — versus time (time in minutes) a
60 - conversion from minutes to seconds
The value of viscosity determined by this method can be
used to plot a viscosity-temperature curve for glasses
607 and EF 9 which can be compared with the curves from
literature. The dimensionless constant, C,_, is defined by: 103 o5 . - k C2/C»
* C 3 / C 4
where k C2 and k Ci a functions of wave number and C4 04 mean coating thickness Very accurate values of the constant (See Appendix A)
can be calculated individually using the data from
Appendix A. However, for simplicity in use, a graph of
versus kh was constructed. The required data for determining a graph of versus kh is given in Table 11.
The graphs of versus kh are given by Figures 19, 20, and 21. Three figures are necessary due to the wide range of values of for the range of kh of interest in this investigation.
The required data for the solution of equation (l)
for all the specimens that were run in this investigation
are listed in Tables 14 and 15.
In order to check the validity of the mathematical
model, the viscosity-data calculated from the model
(Tables 14 and 1 5 ) must be compared to the temperature-
viscosity data from literature (Tables 26 and 2 7 , Appendix
C) for Glasses 607 and EF 9» All viscosity data for Glass
607 is plotted fl>r Comparison in Figure 22; the data for
Glass EF 9 is plotted in Figure 23.
The data obtained from the leveling model for Glass EF 9
shows very good agreement with the data obtained from the
Kimble Glass Technical Data Book (36). 105
TABLE 11
DATA FOR THE GRAPH OF VERSUS -kh
-kh -kC2/C ^ -kCi/04 °5
0.01 0.000001 1.00010 0.000001
0.03 0.000017 1.00090 0.000017
0.05 0.00008 1.00249 0.00008
0.07 0.00023 1.00488 0.00023
0.10 0.00066 1.00990 0.00065
0.15 0.00221 1.02200 0.00216
0.20 0.00517 1.03844 0.00498
0.25 0.00992 1.05875 0.00936
0.30 0.01677 1.08236 0.01529
0.40 0.03769 1.13690 0.03315
0.50 0.06897 1.19657 0.05764
0.70 0.16005 1.31102 0.12208
0.90 0.27807 1.39441 0.19941
1.00 0.34l6l 1.41998 0.24057
1.20 0.46761 1.43925 0.32490
2.00 0.82273 1.28260 0.64l45 5.00 0.99901 1.00450 0.99453
10.00 1.00000 1.00000 1.00000 C» (dimensionless) io*l 1.0 4 2 3 4 8 3 2 4 6 2 3 0 ' 3 6 81.0 6 4 3 2 10“' ------J1 J1L1 1 .J LJL1J — C® Vs.-kh k (dimensionless) -kh iue 19 Figure 3 6 810 6 4 3 2 106 Ce (dimensionless) rt 0.10 4 .5 .0 0.25 0.20 0.15 C k (dimensionless) -kh b V s iue 20 Figure . -kh 0.30 0.35 107 C» (dimensionless) 5*1 .0 0 00 00 0.08 0.06 0.04 002 0.00 ------_ L k (dimensionless) -kh » Vs. -kh C» Fgr 21 Figure . 0.10 108 TABLE 12 VISCOSITY CALCULATION DATA ______GLASS 607______1 t kh Specimen Firing log TBT k c5 7 , No. Temp. (Minutes) log (cm)"-1 cm”l Dimensionless Dimensionless dynes/cm Op
D19 1383 6 0.783 19.81 0.2662 0.0106 239 D20 1340 12 0.833 19.81 0.3521 0.0235 243 D21 1388 4 0.908 19.81 0.3521 0.0235 239 D22 1340 4 0.644 19.81 0.5020 0.0571 243 D23 1352 20 O.56O 15.82 0.2413 0.0080 244 D24 1255 40 0.385 19.81 0.4360 0.0462 248 D25 1352 15 0.552 19.81 0.3116 0.0170 244 D26 1352 20 0.480 19.81 0.2463 0.0082 244 D28 1310 10 0.410 19.81 0.4222 O .0361 245 D30 1255 75 O .665 19.81 0.4523 0.0430 248 D31 1255 60 0.735 19.81 0.4878 0.0529 “ 248 D32 1233 150 0.455 19.81 0.3066 0.0163 248 D33 1358 30 0.730 13.19 0.2547 0.0093 242 D34 1358 35 O .922 13.19 0.2647 0.0105 24§ D35 1335 30 0.773 13.19 0.3250 0.0190 243 D36 1350 18 1.103 13.19 0.3351 0.0210 242 D37 1335 20 0.455 13.19 O .3250 0.0190 243 D38 1350 20 0.575 13.19 0.2580 0.0097 242 D38a 1350 20 O .981 13.19 0.2915 0.0140 242 TABLE 12 (Continued)
1 Specimen Firing t loga k kh c5 No. Tgmp. (Minutes) log (cm)~l cm“l Dimensionless Dimensionless dyn^/cm
D39 1310 100 0.721 13.19 0.2714 0.0112 245 D4l 1350 30 0.760 13.19 0.2647 0.0105 242 d 43 1330 16 1.280 13.19 0.4323 0.0370 242 d 45 l4l0 8 1.183 11.32 0.3060 0.0162 239 d 46 1375 15 0.880 11.32 0.2826 0.0135 24l d 47 1335 30 0.565 11.32 0.2795 0.0125 243 d 48 1292 60 0.443 11.32 0.3141 0.0173 246 D49 1250 60 0.180 11.32 0.4035 0.0402 248 D51 1215 58 0.317 19.81 0.6871 0.1130 250 D52 1264 30 0.986 19.81 0.5786 0.0801 243 D53 1264 16 0.761 19.81 0.8244 0.1650 248 D54 1288 32 0.331 15.82 0.3217 0.0185 246 D55 1288 20 0.330 15-82 0.3941 0.0300 246 D 56 1288 10 0.340 15.82 0.5509 0.0706 246 D57 1175 142 0.068 19.81 0.5780 0.0804 252 D58 1450 5 0.253 11.32 0.1900 0.0041 237 D59 1145 138 0.033 19.81 0.8294 0.1680 254 110 TABLE 13 VISCOSITY CALCULATION DATA GLASS EF 9 ____
—■ c Firing t log a k kh 5 r Temp. (Minutes) log (cm)” cm Dimensionless Dimensionless ies/ °F
1145 3908 0.076 11.32 0.1863 0.0041 218
1260 73 0.479 11.32 0.2626 0.0102 210
1320 20 0.232 11.32 0.2019 0.0048 204
1370 15 0.562 11.32 0.2019 0.0043 200
1440 8 0.928 11.32 0.2019 0.0048 193
1100 2751 0.515 19.81 0.6685 0.1100 224
1400 16 0.495 11,*32 0.1731 0.0032 197 1032 1169 0.031 19.81 1.0254 0.2410 230
1480 16 0.895 11.32 0.1302 0.0015 190
995 29^99 0.191 19.81 0.9802 0.2201 233 1060 1387 0.228 19.81 1.0254 0.2405 227 TABLE 14 VISCOSITY CALCULATION DATA 90# GLASS EF 9 10# POWDERED NICKEL 'I " Specimen Firing t log a k kh C5 T; No. Temp. (Minutes) log (cm) cm-1 Dimensionless Dimensionless dynes/cm °F
F - 1 1180 1179 0.564 19.81 0.392 0.036 215 F - 2 1260 44 0.201 19.81 0.362 0.024 210
F - 3 1380 26 0.076 11.32 0.204 0.005 201 TABLE 15 LOG VISCOSITY TEMPERATURE DATA ______GLASS 607
Specimen Temperature (°P) log V
D19 1388 3.698 D20 1340 4.330 D21 1388 3.808 D22 1340 4.348 D23 1352 4.156 D24 1255 5.487 D25 1352 4.464 D26 1352 4.323 D28 1310 4.745 D30 1255 5.493 D31 1255 5.443 D32 1255 5.536 D33 1358 4.202 D34 1358 4.199 D35 1335 4.489 D36 1350 4.158 D37 1335 4.453 D38 1350 4.118 D38A 1350 4.078 D39 1310 4.823 D4l 1350 4.239 D43 1350 4.284 D45 1410 3.539 d 46 1375 3.8o4 d 47 1335 4.377 D&8 1292 4.928 d 49 1250 5.692 D 51 1215 6.128 D52 1264 5.194 D53 1264 5-346 D54 1288 4.959 D55 1288 4.964 D 56 1283 5.018 D57 1175 7.037 D58 1450 3.360 D59 1145 7.664 114
TABLE 16 LOG VISCOSITY-TEMPERATURE DATA ______GLASS EF 9______
Specimen______Temperature (°F)______log
E - 1 1145 6.821
E - 2 1260 4.672
E - 3 1320 4.012 E - 4 1370 3.578
E - 5 1440 3.072 E- 6 1100 7.532
E - 7 1400 3.480 E - 8 1032 8.730
E - 9 1480 2.880
E - 10 995 9.341
P — : 11 1060 8.282 115
TABLE 17 LOG VISCOSITY TEMPERATURE DATA 90# GLASS EP 9 10# INCO NICKEL
Specimen Temperature (°p) log
P - 1 1180 6.592 P - 2 1260 5.124
P - 3 1380 4.738
0 Log Viscosity (log poise) 12 10 6 01 8 4 2 1 0 20 30 40 50 1600 1500 1400 1300 1200 1100 ------Log Viscosity Vs. Temperature Vs. Temperature Viscosity Log L- Temperature (°F) (°F) Temperature ls 607 Glass iue 22 Figure - eeig Model Leveling •- x~ English (9) Englishx~ 116 Log Viscosity (log poise) 14 12 16 10 6 4 8 o Vsoiy s Temperature Vs. Viscosity Log 00 20 40 1600 1400 1200 1000 x-Owens-Illinois (36) (36) x-Owens-Illinois -eeig Model •-Leveling Temperature (°F) (°F) Temperature ls EF9 Glass iue 23 Figure 118
The viscosity data obtained from the model for Glass
607 does not show the same agreement, however. At 1300°F,
the discrepancy is a factor of 10^. For a given viscosity of 7 log poise, the temperature from the leveling model is
80°F lower than the data from literature. The literature viscosity for Glass EF 9 was obtained from a specimen of glass which had the same chemical composition and was smelted in the same manner. Glass 607, however, was prepared in the laboratory in an attempt to duplicate a given
composition. It smelted at a lower temperature than the glass prepared by English (9 ) in order to minimize the loss of volatile constituents. The chemical analysis of Glass
607, which was duplicated on two analyses show 0.22 percent of material which is unaccounted in the analysis.
The large quantity of BgO^ (41.25#) in Glass 607 and the
relatively low smelting temperature (2100°F) lead to the possibility of the missing 0.22 percent being water. This
theory is strengthened by the formation of bubbles in the
Glass 607 coating. A similar effect was noted in glazes (34)
due to the release of dissolved water which formed bubbles
and eventually pinholes. Under microscopic examination, these
bubbles were found to be distributed through the glass rather
than just at the glass metal interface. Fine bubbles were
found distributed throughout a coating which was refired at
a temperature which corresponded to a viscosity of 5*5 log
poise according to the model. The coating was initially
clear and bubble-free upon preparation. The bubbles developed 119 when the specimen was fired for approximately two hours. The viscosity of the glass was sufficiently high that the bubbles did not rise in the glass, but appeared only to grow in size.
The bubbles did not come from the substrate but rather from the glass which further strengthens the theory on the presence of water,
Scholze (25) discusses the effect of small amounts of water on the viscosity of a Borosilicate glass. For the composition discussed by Scholze the following effects of water vapor in the glass upon the temperatures corresponding to several viscosity points were noted. Some of the results are summarized in the following table:
TABLE 18
EFFECT OF WATER ON GLASS VISCOSITY
Log Viscosity (Log Poise)
% h2o 13 8 4
0.00 287°C 354°C 553°C
0.248 267°C 267°0 537°C
0.490 239°C 320°C 524°C
Scholze data shows that small amounts of water can markedly
lower the viscosity of a glass which contains BgO^. The
viscosity curve obtained from the leveling model indicates
that Glass 607 prepared in the course of this investigation
is apparently much softer than the Glass 607 studied by
English. This variation can probably be attributed to the
presence of water in the glass. 120
Prom Figure 23, the agreement between the experimental ^ Q and known viscosity data for Glass EP 9 from 10 to 10^ poise 3 discounts an error of 10 in the mathematical development.
The validity of the leveling method for the study of the viscosity of coatings is shown by this agreement.
No conclusive results were obtained from the coatings which contained the dispersed metal phase. In specimens prepared from Glass EP 9, the apparent viscosity of the glass-powdered metal composites increased with time. Micro scopic examination showed that bubbles were present in the: metal containing specimens and were not present in a simple glass coated specimen which had been refired in a similar manner. Prom the data supplied by International Nickel
Company (Table 28, Appendix C), the apparent density of the nickel powder ranges from 2.0 to 2.7 grams/cc. The powder is open and spongy. The density of Glass EP 9 is 2.57 grams/cm^
(Table 27, Appendix C). Since the apparent density of the nickel and the density of the glass phase are approximately equal, the ten weight percent of powdered nickel should also yield ten volume percent of dispersed phase. This fact was confirmed by microscopic examination. From a microscopic examination of Specimens PI and 2, it was determined that the total volume fraction of bubbles plus powdered metal was approximately twenty percent. The expression derived for * relative viscosity of dispersed systems (10) (30) (3 2 ) is:
** ■ n-oj '2 -5 das)
♦NOTE: This equation is misprinted in reference (32) 121
where ^|r » relative viscosity
C s volume fraction of dispersed phase
Relative viscosity is defined by the following relationships
relative viscosity
7|0 * viscosity of the pure liquid
« viscosity _of the mixture
For ten volume percent of a dispersed phase, the visoosity
of the mixture should be 1.30 times the viscosity of the pure
liquid, i.e.7/r = 1.3. If the bubbles and the dispersed nickel phase (approximately twenty volume percent) interacted
as if twenty volume percent of a solid phase were present,
the resulting value for7^ r from equation (332) would be 1 .7 .
The comparison of the viscosities of the glass-metal coating
and the simple glass coating as determined from the model
is given in Table 19.
TABLE 19
COMPARISON OF THE VISCOSITY OF
GLASSY COATINGS AND GLASS-METAL DISPERSED COATINGS
10# NICKEL
Specimen logflo Hr
FI 6.592 6.10 3.1
F2 5.124 4.70 2.68 122
The Increase in viscosity due to the presence of a dispersed phase is much greater than predicted by theory. The viscosities determined for specimens in series P were calculated from the initial slopes of the plots of i versus firing time for specimens FI, F2 and F3. These were the miniiTrtjm viscosities since all viscosities apparently increased with time. The possibilities which can explain the observed phenomena are:
1. A continuous change in composition of the coating
during firing.
2. A yield point phenomenon in the coating due to the
presence of bubbles and dispersed metal phase.
The likely explanation is a combination of both effects * but no definite conclusions can be drawn from the experimental data.
Comparison of Leveling Models
For thin coatings, the derived model should simplify essentially to the model derived by Patton since the stress distributions will then be similar. The function denoted
can be resolved into the following form:
= Constant X (kh)^
From Table 11, the values of C5 for various values of kh which correspond to thin coatings can be determined. It is
judged that a coating having kh2, 0.20 is thin. 123
TABLE 20
DATA FOR DETERMINING, Cy/Jkh)3 RATIO,
kh c5 C y ^ k h ) 3 0.02 5.1 x 10 0 .64
0.05 8 x 10"5 0.64 0.10 6.4 x 10 "4 0.64
0.20 5.0 x 10 “3 O .63
Therefore, as seen from Table 20, for thin coatings:
C5 « 0.64 k3h3
Substituting this relationship into equation (122),
substituting — - for k, converting to common logarithms A and multiplying out yields:
log i - 4.6 x 10"3 T h 3^ a ** constant
Denoting the amplitude at t = 0 as a0 > the above equation
becomes:
log — - 4.6 x 10“3 H A Patton's equation translated into the above notation is:
log fl£ c 36 x 10"3 h3 If ^ 7j Pattonfs result is therefore different only in the constant
term which multiplies the variables. Patton’s constant is
greater by a factor of 7*8. Some variation can be anticipated
due to the difference in basic assumed geometry. The variables
have essentially the same basic relationship in both models;
the more general model thus simplified mathematically to the
specific case. 124
Sources of Possible Error
Above the softening point of a glass (log n m 7 .6 5 ) the effect of the previous thermal history can have some bearing on the viscosity of the glass. The leveling method Is
limited approximately to viscosities of less than 10 log poise by the time required for measurable deformation of the waves to occur. Since the annealing point viscosity of a glass is 14.5 log poise, it is thought that all effects of previous thermal history are annealed out of the specimen in a period of time which is very short as compared to the total firing time. The primary source of error in the measurements in this investigation was temperature. Some drift in the potentiometer-controller occurred on the long firing times.
The temperature override during the heating cycle of the furnace could contribute to the variation between literature and experimental data.
The specimens for measuring viscosities above 8 log poise required very thick coatings in order to allow measurable flow to occur in a reasonable time. These thick
coatings did not allow the formation of completely bubble-
free base coats. The presence of these bubbles could also
contribute some inaccuracies. CONCLUSIONS
The results obtained from this investigation have shown that the derived mathematical model as expressed in the
following form is a valid expression for describing the
leveling of a vitreous coating.
log - = - 3=- k T ’Cj-t + constant (131) a 2 '
Through the derived model, the leveling behavior of a vitreous coating can be used as a tool to study the
rheological behavior of a vitreous coating. The leveling
analysis has been used to study the rheology of coatings in
the viscosity range of 10^ to 10^ poise and has been shown
to be valid. The powdered metals tried as dispersed phases
for simple porcelain enamel systems in this investigation
were not satisfactory. The metal appears to act as a
catalyst for removing dissolved gases from the vitreous
phase.
125 SUMMARY
This investigation was undertaken to develop and verify a realistic mathematical model for the leveling of vitreous coatings. The leveling phenomenon could then be used as a tool to study the rheologlcal behavior of vitreous coatings•
A rigorous mathematical analysis of the problem was carried out and a leveling model was formulated. An experimental technique was developed whereby the validity of the derived model could be checked. Experiments were then performed which verified the derived model and showed its validity as a rheometric method. In addition to being used as a measuring device, the derived model can also be used to predict the behavior of vitreous coatings at elevated temperatures from known Theological data.
126 APPENDIX A
MATHEMATICAL MODEL
127 128
Description of Model and Assumptions
In order to explain the Theological behavior of a
vitreous coating during firing, a model was assumed which
would lend Itself to a mathematical analysis. The model
is a homogeneous vitreous layer resting on a substrate.
The free surface of the homogeneous layer has parallel
waves with a sinusoidal profile.
The model is first analyzed as an elastic solid.
Then, using the analogy between the theory of slow viscous
flow and the theory of elasticity, an analytical expression
is derived for the rate of leveling of the surface. The
rate of leveling is expressed as a function of the geometry
of the model, the surface tension of the coating, and the
viscosity of the coating.
The assumptions made regarding the model can be summar
ized as follows:
1. The coating is a homogeneous, Newtonian liquid.
2. The coating wets the substrate and bonds to It.
3. Chemical reaction between the coating and the
substrate is negligible.
4. The effects of gravity are negligible.
5. The surface tension and viscosity of the coating
are uniform across the specimen.
6. The mean thickness Is uniform across the specimen.
7. The amplitude of the wave Is, In all cases, much
smaller than Its wavelength.
8. The wavelength remains constant throughout the firing process. 9. The substrate has a smooth homogeneous surfact
10. There Is no change In the physical properties
or chemical composition during firing.
11. There is no movement at the coating substrate
interface.
i 12. Edge effects can be neglected.
13- The contributions of surface diffusion and
evaporation-condensation may be neglected.
Theory
The free surface of the coating was assumed to ha% profile in the shape of a sine wave. If the waves are plane waves, the surface of which are para lei to the » direction, the profile in the x-y plane can be expresse as follows:
y = a sin kx
where: y - the vertical height from the ave
plane of the surface
a = amplitude
k = wave number =
A. = wavelength
x - horizontal distance
This model is illustrated in Figure 2**. The x and directions are shown and the z direction Is perpendleu1 to the plane of the paper.
The coordinates are taken such that y equals zero the average plane of the surface and x equals zero at a where the wave crosses the average plane in a positive 129
9. The substrate has a smooth homogeneous surface.
10. There is no change in the physical properties or chemical composition during firing.
11. There is no movement at the coating substrate interface. 12. Edge effects can be neglected. 13. The contributions of surface diffusion and
evaporation-condensation may be neglected.
Theory
The free surface of the coating was assumed to have a profile in the shape of a sine wave. If the waves are plane waves, the surface of which are parallel to the z direction, the profile in the x-y plane can be expressed as follows: y - a sin kx (2) where: y = the vertical height from the average
plane of the surface a = amplitude 271 k = wave number = A = wavelength x - horizontal distance
This model is illustrated in Figure 24. The x and y directions are shown and the z direction is perpendicular to the plane of the paper. The coordinates are taken such that y equals zero at the average plane of the surface and x equals zero at a line where the wave crosses the average plane in a positive slope Reference Diagram Mathematical Model Rheology of Vitreous Coatings
+y -Free Surface y = a sin kx
Coating
Substrate
Figure 24 131
The distance from the average plane of the surface to the substrate is designated as h.
The force which causes the coating to level is due to the surface tension since it was assumed that gravity can be neglected. The pressure difference under a curved surface as compared to a flat surface is given by the La Place equation.
(3)
where: ^ p = pressure difference under a
curved surface as compared to a
flat surface 'f - surface tension
R1,R2 = PrlnciPle radii of curvature Since, in the model, one of the radii is infinite, the above relationship simplifies to:
(4>
Prom elementary calculus, the curvature, p, in a
Cartesian coordinate system is given by: (5)
i+(y'<)2 3/"2 where: p = curvature R = radius of curvature
y” = d2y/dx2 y ‘ = dy/dx 132
The profile of the free surface of the model is given by: y = a sin kx (2)
Therefore: y 1 = ak cos kx (6)
y" = ak2 sin kx (7) -ak2 sin kx and (8) 1+ a2 k2 cos2 kxV 3/2
To determine the effect of the ratio of the amplitude (a) to the wavelength (A), values for the function: 1 (9) 1 + a2 k2 cos2 kx
for various values of ^ (assuming cos kx = l) are given
in Table 21 and plotted in Figure 25 versus A TABLE 21 DATA FOR THE DETERMINATION OF THE EFFECT
OF a / \ ON THE FUNCTION 1/ 1+ a2k2
a2k2 1 + a2k2 1/1 + a2k2 a/A 1 2 7T a/A 1/4 0.^5 1.57 2.46 2.46 0.29 1/8 0.12 0.79 0.62 1.62 0.62 1/12 0.083 0.52 0.28 1.28 0.78 1/16 0.063 0.39 0.15 1.15 0.37 1/20 0.05 O .31 0.10 1.10 O .91 1/24 0.04 0.26 0.07 1.07 0.93 1/100 0.01 0.06 O.OO36 1.0036 0.99
i a i Fronj this data, It is seen that for ratios of
the a 2 k 2 i cos kx term can be considered negligible as compared to 1 and the curvature can be approximate by:
i = -ak2 sin kx (10) a/ \ dimensionless
o o CJ1
o ho £
T)
<5* c + ct> Q N ro yr O) M o N o oo M Therefore, the pressure due to surface tension under a curved surface with a sinusoidal profile can be approximated by: p - = -ak2 y sin kx (ll) The stress applied to the viscous layer thus varies sinus idally. According to Timoshenko and Goodier (28) and N.D.P. Smith et al.(27)i the stress distribution and deflecti ns of a body subjected to a sinusoidally varying load can be determined as a plane strain problem using the basic relations of the theory of elasticity. The relations used are as follows. The equilibrium equations for plane strain for an elastic solid (neglecting body forces) are given by: & Ox ■ . (5dxy * 0 (12) 6 x 6 y 6 O^y 5 (J'xy = 0 (13) 6 y 6 x V 2(0>cry) = o an) where ^x, Oy = the normal stresses at a chosen point in the x,y planes respectively • 0*xy = shear stress on either plane at a chosen point These equations are satisfied by the following relationships: O'* = A% - (15) 6 y2 135 b y ■ (16) <5 x ^ x y = 6 ■ $ (17) 6 x 6y where: (p - Airy stress function The above condition is true provided that v2[v20] ■ o The strain components at a point are given by: e ' x = - 1 + v [d-toov- by] (is) E L J e y 1+ V [ (l-y)cTy - b x] (19) E e Xy - A i±Aj^ (20) E where: \) - Poissons Ratio E = Young's Modulus ^ x ^ y = normal strains at a point in (x,y) plane € xy = shear sti-’ain The strains are related to displacements by the following expressions: 6 u = e x (21) "S’* V = e*y (22) 6 y 136 where: u = x component of displacement at a given point v = y component of displacement at a given point For a sinusoidal load (according to Timoshenko and Goodier (2) p. 48-50) the equation for the stress function can be satisfied by taking the Airy Stress Function of the following form: 0 (x,y) sin 2n7Tx\ F (y) (24) where: n = 1,2 3 ...n X P Tf In this derivation =-~—277 is equal to k. A Now if 0 is a valid stress function., it must satisfy the following partial differential equation: ° Substituting for0in the above equation yields a simple fourth order differential equation which can be integrated to yield F (y). The method is as follows: For 0 = |sin kxj F (y), the derivatives are as follows: <50 . k F (y) cox kx (26) 6 * -k2 F (y) sin kx (27) 6 x ~ ( 5 ^ 0 = -k^ F (y) cos kx (28) (5 - k^ F (y) sin kx (29) x <£t0 . a m kx (30) T (y) <5 137 The fourth and higher order derivatives will be denoted by a superscript in parentheses. Lower order derivatives will use the prime notation. = -k2 sin kx F" (y) (31) Substituting equations 29, 30, and 31 into equation 25 yields the following expression: k4F(y) 3in kx — 2 k 2 sin kx F " y + F ^ y ) sin kx = 0 (32) Dividing through by sin kx and rearranging terms: F ^ ( y ) - 2 k 2 F" (y) -k4F(y) = 0 (33) This is a linear differential equation with constant coefficients. The general integral of this equation is given by:|Timoshenko and Goodier (28) p. 46-47 and Wylie (31) p. 91 problem 19| P(y) = ci cosh k y + C 2 sinh ky+C^y cosh ky+C^y sinh ky(34) The complete general stress function is thus given by: sin kx| C-i cosh k y + C p sinh k y + C o y cosh ky+C,.y sinh ky 0= 2 ^ 4 (35) Using the method of Timoshenko (28), the general stress function can be substituted into the expressions for the stresses (equations 15-17) to yield the general expressions for stress at any point. Using the fundamental relationships between stress, strain and displacement (equations 18-23), general expressions for the displacement of any point in the model can be determined. The general solutions for stress and displacement can then be made particular solutions using the boundary conditions of the model. 138 In the following solution, the following nomenclature will be used in order to simplify manipulations: sin kx = A■(x) (3 6 ) F (y) - B (y) (37) The partial derivatives will be denoted as follows: = A ’(x) (38) 0 * = A"(x) (39) 0 * d L . b l z l z A"1 (x) (40) 6 x3 Similarly for B(y): ^ B (y) = B ’(y) (41) (5x = B" (y) (42) 6 y <33,BCy) = s'" (y) (43) 6 y The stress functior}0#can then be written using equations 36 and 3 7 - 0 = 0 (x,y) = A(x) B(y) (44) Substituting this relationship into equations 14, 15 and 16 yields: C fy * A"(x) B(y) (45) = B"(y) A(x) (46) (fxy j= -B'(y) A'(x) (47) 139 For an incompressible liquid, Poissons1 ration,^, is equal to 1/2. Substituting this value in equation 18 yields: = ~ ^y ) Defining L as D and substituting equations 45 and 46." 4E D £b "(y) A (x) - A"(x) B(y) (49) Similarly for equations 19 and 20: £y = -D [ y (y) A(x) - A" (x) B(y ) (50) £xy =" B> (y) A' (x) (51) E The expressions for displacements in terms of strain equations 21, 22, and 23 now become: V -r—6 = 6 y = D B"(y) A(x) - A"(x) B(y) (53) 5 y Equation 52 can be integrated to yield: u = D B"(y) / a (x ) dx - A»(x) B(y)j+0(y) (54) Similarly for equation 53 v = -D B'(y) A (x) - A"' (x) J B(y) dyj+ 0(x) (55) where: 0 ( y ) = arbitrary function of y 0(x) = arbitrary function of x The expression for shear strain is: e xy = (23) Introducing the equivalent functions in equation 23 and differentiating the expressions for u and v (equations 140 54 and 55) with respect to y and x respectively and substituting equation 51 for the following expression is obtained: -4D B' (y) A'(x) = D ^ B ”I (y) jA(x)dx-A'(x)B'(y)J + 0'(y) -D [Bf (y) A ' (x) -A"1 (x) x y*B(y) dyj + 0 ' (x) (56) For clarity and ease of substitution, the derivatives and integrals which will be used in the solution of the gen eral stress function will be arranged together in Table 22. TABLE 22 DERIVATIVES AND INTEGRALS FOR SOLUTION OF STRESS FUNCTIONS 0 (x,y) = sin kx F(y) (24) Introducing equation 34 for F(y) 0 (x,y) * sin kx C-^ cosh k y + C 2 sinh ky-f C3y cosh ky + C^y sinh kyj (57) and 0(x,y) = A(x) B(y) (58) A(x) - sin kx (59) A 1(x) = k cos kx (60) A"(x) - -k2 sin kx (6l) A"1 (x) = -k3 cos kx (62) B(y) = C cosh k y + C g sinh ky + C^y cosh ky +Ci|.y sinh ky (63 ) B'(y) = C^k sinh ky + Cgk cosh ky + yk sinh ky •+ cosh ky + C4 yk cosh ky+sinh icyl (64) l4l B" (y) = C-^k2 cosh ky + C2k2 sinh ky + £yk2 x cosh ky + 2k sinh ky + yk2 sinh ky + 2k cosh ky j (65) J a ( x ) dx = - i cos kx (66) / Q 1 B(y) dy = ■— sinh k y + ££ cosh ky k k c3 ky sinh ky - cosh kyj + — 2 k ky cosh ky - sinh ky k 2 L J B"' (y) = C-jk^ sinh ky + C2k^ cosh ky + C3 yk^ sinh ky + 3 k2 cosh ky (68) + C4 * yk3 cosh ky + 3k2 sinh ky' In order to evaluate the constants in the general solution of the stress function it is necessary to introduce the boundary conditions which exist in the model (See Figure 24. The boundary conditions are: 1) O^y r 0 at y = 0 for all values of sine kx (See Timoshenko and Goodier (28) p. 4 7). 2) The displacement, u, in the x direction is zero at the coating-substrate Interface, u = 0 at y = -h 3) The displacement, v, in the y direction is zero at the coating-substrate interface, v * 0 at y ■ »h 4) For a point at the free surface at the crest of the wave, (x,y) = ^ » a )t “ A p l4l B"(y) ® Cjk2 coah ky + C2k2 sinh ky -f Cg £yk2 x cosh ky + 2k sinh kyj + yk2 sinh ky + 2k cosh ky (65) ] A(x) dx = - i cos kx (66) /' k Q J B(y) dy = — - sinh ky+cosh ky /■ k k ky sinh ky - cosh ky (67) 3k [ c4 ky cosh ky - sinh ky [ ■] B'" (y) = sinh ky + Cgk^ cosh ky + C- yk^ sinh ky + 3k2 cosh ky (68) + C, *yk3 cjsh ky + 3k2 sinh ky In order to evaluate the constants in the general solution of the 3tress function it is necessary to introduce the boundary conditions which exist in the model (See Figure 2^. The boundary conditions are: 1) 0*Xy : 0 at y - 0 for all values of sine kx (See Timoshenko and Qoodier (28) p. 47). 2) The displacement, u, in the x direction is zero at the coating-substrate interface, u = 0 at y = -h 3) The displacement, v, in the y direction is zero at the coating-substrate interface, v « 0 at y ■ -h 4} For a point at the free surface at the crest of the wave, (x,y) = _ A _ , a), Cfy = ^ p 142 From boundary condition (l) CTxy * 0 at y = 0 From equation 4^: cfxy = - B'(y) A' (x) (47) Substituting in equation 47 for B’(y) and A'(x) from Table 22 ^xy = £-k cosh xj £c]k sinh ky+ C2k cosh ky + C3 |yk sinh ky + cosh ky (69 ) + Jyk cosh ky sinh ky j for y = 0 , O^xy = 0 sinh ky = 0 cosh ky = 1 Substituting these values in equation 69: 0 = C-£0+ C2k + C3 + C^* 0 Therefore: C2 = - 23. (70) k From boundary condition 4, at the free surface at the point (x,y) = ^ /\p = - y ak2 = Cfy (71) From equation 45 Cfy = A" (x) B(y) (45) Substituting in 45 from Table 22 yields: (5y - [-k2 sin kxj cosh ky + C2 sinh ky (72) + C3y cosh ky -f C^y sinh ky] 143 Equating 71 and 72, evaluating at (x,y) = ( A. , a) and factoring out cosh ka: cO S h t e [c1 + o2 j " ) + c„a + C.,a i slhh ka 3° ~r w4a | cosh ka ] ] 2 Dividing through by k , substituting for C3 from equation 70, regrouping, and substituting the identity: sinh ka _ o o 'b TTW - tanh ka = cosh ka + Cg (tanh ka-ka) (74) + C^a tanh ka J Expanding tanh ka in a power series for small values of ka tanh ka as ka - j. 2k^a^ 3 ^ ------15 Therefore: tanh ka - ka « ka - k^a^ 2k^a^ -ka 3 15 This investigation is limited to small values of ka, ka« ^ For ka = 7-: k^a^ 1 * - 3— Therefore (tanh ka-ka) will be neglected. Since kC^ = b -1 (by definition) equation 74 becomes: c4 r Cn Ta = cosh ka \61 + — -— ka tanh ka L 1 b -1 Solving explicitly for yields: Cl = y a ______(75) cosh ka 1 + — ( s r r r 144 Expanding cosh ka in a power series, multiplying out and p 2 neglecting powers of ka higher than k a : CL = L 2 ______(76) 1 + — — k2a2 + kga2 b-1 2 Rearranging the denominator yields: CX = ------O ------(77) 1 + .,.■?■ ^ (k.-l.)---- k2a2 2 (b-1) The maximum value for the term in brackets occurs when (b-l)=-l. This term then becomes: 2 + (b-1) = -b 2 (b-1/ For ka = 0.25,. the maximum value of ka in this investigation, the denominator in equation 77 then becomes: 1 - = 1 - 0.031 and 0.031 can be considered negligible compared to 1. Therefore equation 77 simplifies to: C1= 7a (78) In order to complete the evaluation of the constants, the equations for displacement 54 and 55- must be evaluated using boundary conditions (2) and (3) and the arbitrary functions 0 (x) and (j) (y) must be eliminated. Substituting values from Table 22 into equation 53 yields: u =-|jr jpi^2 °osh ky + C2k2 sinh ky + Cg (yk2 cosh ky + 2k sinh ky) (79) + C4 (yk2 sinh ky + 2k cosh ky)] Jji cos kxj 145 - [Cx coah ky + Cg sinh ky + C^y coah ky + C^y sinh kyj £k coa kxj + 0(y) On rearranging and combining terms equation 79 becomes u = D [-k cos kxj £ 2CX cosh ky + 2 0 ^ cosh ky +2Ci,y sinh ky + bosh ky (80) k + 2Cg sinh ky + §Inh kyj+ Similarly from equation 55: v =-D jsin kxj sinh-f 2Cgk cosh ky (81) + 2C^yk sinh ky + 2C^yk cosh kyj-f 0(x) In order to simplify manipulation, the displacement, u, will be expressed as follows: u = F(x) G(y) + 0(y) (82) where: F(x) = -D sin kx = function of x only G(y) = ^20.^ cosh ky + 2C^y cosh ky + 2Cuy sinh ky + 2 C ^. cosh ky •f 2Cg sinh ky + 2C3k sinh kyj 1 = k function of y only From the boundary condition (2): u - 0 at y = -h Therefore equation 82 becomes: 0 = G (-h) F(x)+ 0(-h) Therefore: G (-h) F(x) = - 0 ( - h ) (83) 146 G(-h) and 0(-h) are Independent of x so equation 83 can be true only if G(-h) = 0 A. v (84) 0 ( - h) = o Expressing the displacement, v, in a form similar to equation 82: v = -P(x) Q(y) + 0 (x) (85) where: P(x) - -D sin kx = function of x only Q(y) = [20-jk sinh ky + 2C2k cosh ky + 2Cgyk sinh ky + 2C^yk cosh kyj * function of y only Prom the boundary condition v = 0 at y =• -h 0 = -P(x) Q(-h) + 0(x) Therefore: Q(-h) P(x) - 0 (x) (8 6 ) or: 0(x) = jconstanlj P(x) Denoting this constant as H: 0(x) = H P (x) (87) In order to further evaluate the arbitrary functions it is necessary to compare the expression for shear strain obtained directly from the Airy stress function (equation 51) with that obtained from differentiating the expressions for displacements (equations 80 and 8l). Equation 56 relates these two expressions. Prom equation 87: 0 ( x ) = H A (x) therefore: 0 ' (x) = H A ’ (x) (88) Substituting equation 88 and values for A'(x), A”* (x) and J A(x)dx from Table 22 Into equation 56, factoring out cos kx and rearranging: D cos kx |-4k B'y] = D cos kx f-2k B^y + kH * f i i (89) " jB(y)dy B,n (y)] + 0 ' y Transposing and combining terms in equation 83 yields: - 0 ' (y) : D [cos kxj [ 2kB^yj+ kH -k3 J B(y)dy - 1 B’" (»)] <90) Now 0 (y) is a function of y, only, therefore 0 'y must be a function of y only. By equation 90jthis can only be true if 0 (y = 0 Since 0 * y Is zero, 0 (y) must be a constant. By equation 84, however, $ (,-fl) = 0 Therefore: 0(y) = o Also, from equation 90 since: cos kx ^ 0 for all x Then: [2kB'y+ kH -k3 J B(y)dy - i B'f»y] (91) must equal zero for all values of y. Solving for each of the terms in equation 91 separately 2KB'(y) = [2Cxk 2 + 2C3yk2 + 20 ^ ] sinh ky + + £ 2C2k2 + 2c3k + 2Cjjyk2] cosh ky -V?J&(y)dy = [”cik2 “C3yk2 -f c^kj sinh ky + [ -C2k2 + C3k “C^yk^ cosh Ky - ■£ B'1' (y) = [-C^k2 -C3yk2 -3C^k] sinh ky + [-Cgk^ ”^3^ -C4k2y ] cosh ky The sum of these three terms is zero, therefore kH = 0 k # 0 H Therefore: H = 0 i I Since, by equation 81: 0(x) = H P (x) 0 (x) z 0 Thus both arbitrary functions obtained from partial integration of the strain equations are zero. Therefore equation 80 becomes: u = D 1 cos kxj £ 2C1k2 cosh Ky + 2Cgk2 sinh ky + 2C3yk2 cosh ky + 2C^yk2 sinh ky + 20,5k sinh ky (92) + 2Cij.k cosh ky Similarly equation 81 becomes: v = -D["sin kxir2Cnk sinh ky + 2Cok cosh ky L JL 1 * (93) + 2C3ky sinh ky + 2C^ky cosh kyj The expressions for the displacements, equations 92 and 93> can now be evaluated using boundary conditions (2 ) and (3 ) to yield expressions for the constants C2 and 149 in terms of C^. All the arbitrary constants of the general solution of the stress function will then have been evaluated. Equation 93, the expression for the y component of the displacement of any point (x,y) can be rewritten: v = -D £sin kxj £ 2kj £ C1 sinh ky + Cg cosh ky + C3y sinh ky + C4y cosh kyj. At the substrate, (y e -h), the expression in braces^ J,must be identically zero since there Is no displacement of the coating at the fixed surface and sin kx Is not always zero. Therefore: C, sinh (-kh)+ C2 cosh (-kh) + Co (-h) sinh (-kh) (95) + (-h) cosh (-kh) = 0 Denoting (-kh) as q, dividing by cosh q and using- the identity: 9, = tanh q cosh q Equation 95 becomes: tanh q + Cg -C^h tanh q -C^h = 0 (96) Equation 92, the expression for the x component of displace ment, u, can be rewritten: u = D i cos kxj [2kJ ^C^k cosh ky-f Cgk sinh ky +C^yk cosh ky + C^yk sinh ky + sinh ky (97) + C4 cosh kyj The x component of the displacement is also zero at the coating-substrate interface (y = -h) for all values of x. 150 Therefore, the term in braces^ ^is Identically zero since cos kx Is not always zero. Therefore: C-,k cosh (-kh) + Cpk sinh (-kh) + Co (-kh) cosh -kh 3 (98) C/j^ (-kh) sinh C-kh) + sinh (-kh) + cosh (-kh) * 0 Substituting equation JO into equation 98: C2 = - ^ 3 _ (70) k Simplifying, dividing through by k cosh (-kh) and again denoting (-kh) as q yields: °i - c3h -(cuh) ssrt+r = 0 (99) Again using the identity: sinh 9 = tanh q cosh' q Equation 89 becomes: Ci - C3h - Cjjh tanh q + = 0 (100) k Equations 96 and 100 can now be solved simultaneously to evaluate the remaining constants in terms of C^. Dividing equation 96 by tanh q yields: cl + f 2 - c h - C.h : 0 (101) tanh q 3 h s t t ; Subtracting equation 101 from equation 100 and simplifying: -Ci|h tanh q + ^ 4 c2 C^h (102) k tanh q tanh q Multiplying by k tanh q and rearranging: [— 55— 2 -kh tanh q + tanh q + kh = kCg (103) 151 Since (-kh) is defined as q, equation 103 can be written - q tanh2 q + tanh q - q (104) ^ 4 Equation 104 yields a ratio of C2 to C4 in terms of the experimental parameter (-kh). It is convenient to calculate kC2 for various values of (-kh) within the range of interest. -£4 kC2 for various values of (-kh) and the data required to -54 calculate kC2 are given in Table 23. Figure 26 is a plot ~G4 of the data from Table 23. A logarithmic plot was used to expand the scale. C4 must now be expressed in terms of C-^. To do this, we multiply equation 100 by k, yielding: C-jk-f C3 (-kh) + (-kh) tanh q + C4 = 0 (105) Substituting q for (-kh), Equation 70 for Cdividing by C4 and rearranging yields: _ n - q tanh q - 1 (106) C4 Ci* kC Using values for - —=■ from Table 23 and corresponding values c4 of q in the range of interest, a table can be constructed giving Cl^ for values of q. Clk for chosen values of q 04 C4 and the data required to solve equation 106 are given in Table 24. To determine the maximum value of equation 106 can be written as a function of q alone by substituting 152 TABLE 23 CALCULATION DATA FOR RATIO C 4 kC2 q tanh q tanh2q q tanh2q C 4 O.Ol 0.01000 0.00010 0.000001 0.000001 0.03 0.02999 0.00090 0.000027 0.000017 0.05 0.04996 0.00249 0.00012 0.00008 0.07 O .6989 0.00488 0.000342 0.00023 0.10 0.09967 0.00993 0.00099 0.00066 0.15 0.14889 0.02217 0.003326 0.002226 0.20 0.19738' 0.03896 0.007792 0.00517 0.25 0.24492 0.05999 0.01500 0.00992 0.30 0.29131 0.08486 0.02546 0.01677 0.40 0.37995 0.14436 0.05774 0.03769 0.50 0.46212 0.21355 0.10677 0.06897 0.70 0.60437 O .36526 0.25568 0.16005 0.90 0.71630 0.51308 0.46177 0.27807 1.00 0.76159 0.58002 0.58002 0.34l6l 1.20 0.83365 0.69497 0.83396 0.46761 2.00 0.96403 0.92935 1.85870 0.82273 5.00 0.99991 0.99982 4.99910 0.99901 10.00 1.00000 1.00000 10.00000 1.00000 kCa/C4 dimensionless 10 * 0 1 4 3 - -i 8 2 6 8 6 4 4 3 3 8 2 2 6 3 10 3 6 10 8 6 4 3 2 C C V. q Vs. /C4 kCt q- dimensionlessq- iue 26 Figure 3 6 8 6 4 3 2 153 kCt /C4 dimensionless -4 kC* Vs.q /C4 dimensionless - q 4 6 8 8 6 4 iue 27 Figure 10"' 2 154 155 the equivalent expression for as given by equation 104. Multiplying equation 104 q; kC2 2 q tanh^q Substituting equation 107. Into equation 106, combining and simplifying yields: ^l^ s ? r p i q [tanh q -x] -X (iQB) - -q sech q-i Differentiating with respect to q and setting the result equal to zero to obtain the maximum: 2 2 o 2q sech q tanh q - 2q sectn q = 0 (109) This expression 109 simplifies to: tanh q = — (110) q Solving equation 110 by trial and error from Table 33 yields: qmax Using the data from Table 24, a plot can be made of q versus -k^l to allow simple evaluation of the ^ 1 ratio. This Cij. C^. graph Is given in Figure 28. All the arbitrary constants in the solution of the stress function have now been evaluated. The expressions for the stresses, displacements and the constants are listed in Table 25. I 156 TABLE 24 CALCULATION DATA FOR ^£l RATIO c4 3 kC2 tanh q kC2-tanh q kC! "04 "C4 C4 0.01 0.000001 0.01000 0.01000 1.00010 0.03 0.000017 0.02999 0.02997 1.00090 0.05 0.00008 0.04996 0.04988 1.00249 0.07 1.000232 0.06966 0.06966 1.00488 0.10 0.00066 0.09967 0.09901 1.00990 0.15 0.002216 0.14889 0.14667 1.02200 0.20 0.005172 0.19738 0.19221 1.03844 0.25 0.009920 0.24492 0.23500 1.05875 0.30 0.01677 0.29131 0.27454 - 1.08236 0.40 0.03765 0.37995 0.34226 1.13690 0.50 0.06897 0.46212 0.39315 1.19657 0.70 0.16005 0.60437 0.44432 1.3H02 0.90 0.27807 0.71630 0.43823 1.39441 1.00 0.34161 0.76159 0.41998 1.41998 1.20 0.46761- 0.83365 0.36604 1.43925 5,00 0.99901 0.99991 0.00090 1.00450 10.00 1.00000 1.00000 0.00000 1.00000 ’ O ' (dimensionless) 1.0 V. / 4 |/C C k Vs. q kC, Figure 28 (dimensionless) 157 158 TABLE 25 ELASTIC THEORY SOLUTIONS: STRESSES, DISPLACEMENTS, AND CONSTANTS Cfx = sin kx [Cik2 cosh ky + C2k2 sinh ky + C3 |yk2 cosh ky + 2k sinh ky +C4 Jyk2 sinh ky + 2k cosh ky j = £-k2 sin kxj ^Cj cosh ky + Cgk sinh ky (72) + C3y cosh ky + C^y sinh ky] ^xy “ [ cos kx] [ clk sinh ky + C2k cosh ky + C 3 yk sinh ky + cosh ky (6 9) + C4 yk cosh k y + pinh ky j u = D i cos kxj j ^ ^ k 2 cosh ky + 2C2k2 sinh ky + 2C3yk2 cosh ky+ 2C^yk2 sinh ky (97) + 203k sinh ky <+ 2C^k cosh kyj v = -D [sin kx] [ 2k] [Ct sinh ky (92) + C 2 cosh ky + C^y sinh ky + C^y cosh kyj Cx = - / a (78) C2 = - £3 (7 0) k k*''! - -q2 sech2 q - 1 (108) C4 ^£2 - q tanh2 q + tanh q - q (10*0 159 The solution of the elasticity problem is now complete. An expression for the displacement of a point from an elastic solution of a problem can be converted to an expression for the velocity of a point in an analogous viscous flow problem. This transition can be accomplished by simply substituting the coefficient of viscosity 7^ for the shear modulus, G, in the elastic solution. The expression for v, the y component of the displacement, can be converted to an expression for the velocity of the point in viscous flow. Since the expression for the displace ment is general, the velocity for any given point can be determined. In this development the velocity of the point at peak of the wave, (x,y) * a) will be determined; the expression for velocity can then be integrated to yield an expression involving the amplitude of the wave and time of flow which can be measured experimentally. Prom the expression for v, equation 92, the equation for the displacement of the peak of the sine wave (x,y = (^t.,a) can be determined from equation 92 yields v = -D [2k sin kx] cosh ky (111) Lk lkCl , ky\ tanh ky+ I °2k + ky I c 4 c 4 j \ c 4 Evaluating equation 111 at the point (x,y) = where: n = 0 ,1,2 ...n• • • 160 v = -D |2k| [ £4 cosh ka] f ^ C3 ka\ tanh ka I j Lk J LI C4 + CJI I (112) ♦ R M 1 The constant D can be expressed in terms of the shear modulus by first relating it to the modulus of elasticity. By 3 definition: D = where: E = modulus of elasticity, E is given in terms of the shear modulus, G, by the basic relationship: E = 2 [l + 1/] G (113) where: V- Poisson's Ratio For an incompressible fluid, Poisson's ratio is 0.5» therefore equation 113 becomes E = 3G Therefore: D = (11*0 Equation 112 for displacement can now be expressed in terms of the shear modulus. By analogy with viscous flow, if G is replaced by 7^ » the coefficient of viscosity, the expression for the displacement of a point becomes the expression for the velocity of the point during viscous deformation. The y component of the velocity, £ v*(a)j , of the crest of the wave where (x,y) = a) is given by: v*(a) = -D* [2C4 cosh kaj [{“§4 + ^ ka| tanh ka H ] 161 where D* = ~ 47? The amplitude (a) of the wave changes with time and Is no longer a constant. In a similar manner, the y component, [v*(a)j , of the velocity of the trough of the wave, the point corresponding to: (x,y) = -a) Is given by: v*(*a) = -D* [-2C^ cosh (-ka)] (116) kCn C 1 + 1 2 (-ka) tanh (-ka)-f *-ka L T C4 C4 The net velocity of approach of the crest and trough of the wave is given by: v*( N ) ' v*(a> - v*(-a) (117) where: v* - net approach velocity Substituting equations 115 and 116 into 117: v*( n ) = ~D* (2C4 cosh ka) [ | ^ + 5 ka) tanhka+!?f+ta] (118) - |-D* [-2C4 cosh (-ka)j “ |l3 kaj tanh (-ka) + - ka ti 4 ' Upon substituting the Identities: cosh (-ka) = cosh (ka) tanh (-ka) = -tanh (ka) and simplifying, equation 109 becomes: v*( n ) = "D* [2C4 cosh ka] [“Cjj ka tanh ka (H9) 2kCg 1 162 Since by equation 70, C3 = -kC2 , equation 119 can be .expressed: v*(pg) = -D* [ 2Cijj ^jjcosh ka {1-ka tanh ka|i (120) The expression in braces j lean be expressed as: cosh ka - ka sinh ka (121) The hyperbolic sine and the hyperbolic cosine can be expanded in power series as follows (bibliography reference 33) cosh ka = 1 + ■+ -(KsJ— + ... 21 0 4' . sinh, U ka , -= ka1 + , (ka) . (lea)5 ■ ■ + 3! ' 5! For small values of ka, the series expansions can be approximated by the first two terms: 2 cosh ka « 1+ (122) sinh ka sss ka 4 (123) G Upon substituting equation 122 and 123 into 121, multiplying out, and simplifying: cosh ka - ka sinh ka 1 - 1,^) - (124) 2 8 For: ka S 0,25 (ka)2 < 0.031 2 ~~ / \ 4 (ka) < 0.0005 ~~B---- The last terms can thus be considered negligible as compared to 1. Equation 120 thus becomes: 2kC^ v*( N ) = -2D* C4 (125) G4 163 Using the identity: c4 = ^l/kC- and substituting In equation 125: ';2kC2 = -2D* kC- (126) v* ( N ) kC- 0lt . The approach velocity, v* ^ ^ y Is time rate of change of the y component of the distance between the crest and trough of the wave, 2a: da v* = 2 ( N ) dt (127) Equating 126 and 127 and dividing by 2: LkCi/ClJ (128) By equation 7 8 : G1 = "If ^ Substituting equation 78 in equation 128 yields: kC, ?/c4 (129) kC^Tc -dt Multiplying both sides of 129 by and integrating: In i - 2D* k J kC2/C4‘ t + constant (130) _kC1/C4_ ^ 2/04 Substituting for D* and denoting as C5* equation 130 becomes: 1 In - = k^ c5t + constant (131) APPENDIX B RAW MATERIALS AND EQUIPMENT DATA 164 165 The raw materials used in the preparation of Glass 607 were: Sodium Borate Decahydrate Reagent Grade J.T. Baker Chemical Company Phillipsburg, New Jersey Minusil 15 Micron Silica Pennsylvania Glass Sand Corporation Pittsburgh, Pennsylvania Sodium Carbonate Reagent Grade J.T. Baker Chemical Company Phillipsburg, New Jersey Calcium Hydroxide Reagent Grade J.T. Baker Chemical Company Phillipsburg, New Jersey T 61 Alumina -325 mesh Aluminum Company of America Pittsburgh, Pennsylvania The crucibles used in the preparation of Glass 607 were: Denver Fire Clay Crucible Type N Platinum Iridium Dish Size C The furnace used in the preparation of all the enameled specimens was: Hevi Duty Type 66 Hevi Duty Electric Company Milwaukee, Wisconsin The furnace temperature was controlled by: Foxbord Potentiometer Controller , Model 4041-40E Foxbord Instrument Company Foxbord, Massachusetts using a chromel-alumel thermocouple. 166 The dial indicator used in the measurement of the specimens was: Federal Model C21 Full Scale: 0.200 inch-0.0001 inch increments Federal Products Corporation Providence, Rhode Island The dial indicator was mounted in a comparator stand: Model 7003 Mitutoyo Instrument Company New York, New York The replicas used In the preliminary investigation and the curved anvils used to measure the amplitudes of the waves and the coating thicknesses were fabricated with: Quickmount Cold Setting Mounting Plastic Fulton Metallurgical Products Corporation Pittsburg, Pennsylvania The metal substrate used was: Inconel - Alloy 600 International Nickel Company New York, New York Supplied by: B.K. Williams and Company Columbus, Ohio The powdered metal used In the Investigation were supplied by: (Numbers correspond to those used in Table 2) 1. Metal Hydrides Incorporated Beverly, Massachusetts 2. Charles Hardy Division Chemetals Corporation New York, New York 3 . International Nickel Company New York, New York Type 123 Carbonyl Nickel Powder 167 The temperature of the surface tension apparatus was determined with a: Leeds and Northrup Potentiometer No. 723990 Leeds and Northrup Company Philadelphia, Pennsylvania APPENDIX C CHEMICAL AND PHYSICAL DATA 168 169 . TABLE 26 VISCOSITY-TEMPERATURE DATA GLASS 607 (AFTER ENGLISH (9) ) Temperature Log Viscosltv Op Xlog poises! 2021 1.28 1832 1.30 1584 1.763 1436 3.51 1292 6.43 1224 7.88 170 TABLE 27 OWENS ILLINOIS CORPORATION GLASS EF 9 PHYSICAL DATA Viscosity Data Working Point (log = 4) 700°C Softening Point(log « 7 .65) 580°C Annealing Point(log = 13.0) 479°C Strain Point (logj^ 14.5) 443°C Thermal Expansion Coefficient: 0-300°C 74 x lO”7 per °C Density 2.57 gm/cc Index of Refraction: 1.56 Data obtained from Kimble Glass Technical Data Book (36) TABLE 28 CHEMICAL AND PHYSICAL DATA INTERNATIONAL NICKEL COMPANY TYPE 123 CARBONYL NICKEL POWDER Average Particle Size — 4-7 Microns Apparent Density — 2.0-2.7 graras/cc Typical Chemical Analysis Carbon 0.05-0.10$ Oxygen 0.10 Sulfur less than 0 001$ Iron less than 0 01$ Nickel balance 172 TABLE 29 CHEMICAL COMPOSITION GLASS; 607 Oxi<3e Compositions Theoretical Analyzed sio 2 35.22 34.60 A1203 0.84 0.51 — Fe2°3 0.07 MgO ----- 0.01 CaO 0.16 0.27 Na20 23.71 23.14 b2°3 39.99 41.25 TOTAL 99.99 99.78 173 TABLE 30 CHEMICAL COMPOSITION GLASS EF 9 Nominal Oxide Composition Weight $______Mole % Si02 15.7 18.3 49.6 50.0 b 2°3 AlgO^ 5.7 3.9 V. 3.4 2.5 Ka20 3.8 4.3 CaO 4.6 5.8 BaO 4.9 2.2 Li20 . 3.0 6.9 ZnO 9.7 6.3 174 TABLE 31 CHEMICAL COMPOSITION GLASS 604 ENGLISH (9) Theoretical Analyzed Si02 64.72 63.0 AlgO^ 0.72 1.78 PSgO^ 0.06 — MgO 0.01 CaO 0.14 o.i4 Na20 19.95 18.80 14.45 b 2°3 13.73 175 TABLE 32 CHEMICAL COMPOSITION GLASS 4 PARMALEE et .al:. 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