January, 1927
INDUSTRIAL A N D ENGINEERING CHEMISTRY
of the curve, it is more transparent to red than to green light and consequently will give a higher reading. The relationship drawn between particle size and opacity (Curve 1) indicates that there is one particle size a t which the opacity of a pigment attains a maximum. It is seen that this size corresponds to that of French process zinc oxide. Any oxide larger than this decreases in opacity, and any oxide smaller than this also decreases in opacity. On the right, or large-size, arm of the curve the particles cause opacity by reflecting light from their surfaces. Therefore the smaller the particle size the greater the surface available and hence the greater the opacity. However, below the particle size giving maximum opacity the particles are so small that they n o longer can reflect the light, but instead only scatter or diffuse the light, with the result that the smaller the particles
s3
become the less will be their opacity until, for very small sizes, the suspension will be optically clear. Curves 2 and 3 clearly show that the pigment having a particle size of 0 . 2 4 will ~ have a maximum opacity, no matter what the relative refractive indices of pigment and vehicle. Therefore, zinc oxide, having this particle size, will give a maximum opacity when the pigment is dispersed in linseed oil (refractive index 1.475). Again, any white (transparent) pigment having this particle size will show the maximum opacity obtainable for that pigment. This may not be true for an absorbing material such as a black. Evidently, a very definite relation exists between the size of the particles and the wave length of the light for which the pigment will show a maximum opacity. This is being made the subject for further investigation.
The Relation of Yield Value to Particle Size By Henry Green and George S. Haslam THENEWJERSEY ZINC Co., PALMERTON, PA.
HE success of an investigation of yield value in its rela-
T
tion to particle size depends largely on the correctness with which the investigator visualizes the nature of the pigment vehicle structure. For this purpose, the use of the microscope is a sine qua non; attempts at visualization without the aid of microscopy usually lead to misconceptions such as the possibility of cubical and trigonal packing. Unfortunately, the nonuniformity of particle size, the irregularity of particle shape, and the force of flocculation make symmetry of structure (of the cubical and trigonal kind) impossible, and so the investigator is compelled to work with a model that is dficult to treat theoretically.
Figure 1-Diagram of Structure of Paint The pigment particles are grouped into flocculates and the flocculates, being in contact with one another, give a "continuous" structure to the mass.
Assume that there exists a mass of spheres or particles of equal diameter. These spheres can be packed cubically or trigonally. If we now imagine that each particle changes in size so that ultimately no two of them are exactly of equal ,diameter, then symmetrical arrangement becomes impossible; but if we next assume that each sphere changes into some irregular shape or form, and that these forms are suspended in a liquid medium where flocculation must occur, then the utter impossibility of symmetry becomes quite apparent. The act of incorporating pigments into a vehicle necessarily produces motion in the form of currents, so it is evident that if the pigment-vehicle ratio is sufficiently great the particles will collide, and when this happens the flocculating force will cause them to adhere. This random adherence of the particles into groups or flocculates produces a heterogeneous
structure-that is, one where the particle density (number of particles per unit volume of mixture) varies throughout the mass-the density being greatest in the flocculate and perhaps zero in the spaces between the flocculates. In a plastic mixture the flocculates are so close that they touch or interlock, thus making the structure continuous, though still heterogeneous. Nofe-A flocculate must not be thought of as a hard aggregate, but rather as a loose cluster of particles easily dispersed by stirring, and easily reformed again as motion ceases.
Figure 1 is a diagram of the structure of paint in cross section. This much the microscope can reveal, but it can tell us nothing about the nature of the flocculating force, or even indicate where it resides. About this force we can only speculate. Sulmanl has given a clear-cut description in which he shows that interfacial tension is sufficient to cause flocculation. This theory will be adopted here as a working hypothesis. Flocculation produces structure; the resistance to shear offered by this structure causes yield value;* that yield value is finite-that is, yield does not take place under infinitely small pressure as in liquids-is due, presumably, to the fact that the interfacial tension is finite. In the ultimate analysis there can be very little doubt that yield value arises from interfacial tension, but there is an intermediate step where it becomes necessary to think of this resistance as a quasifrictional phenomenon.
Sketch in Cross Section of T w o Layers of Particles When these layers are in contact (adjacent) a quasi-frictional resistance to shear arises.
Figure 2-Diagrammatic
The particles in each layer in Figure 2 are held together by the force of flocculation. Assume that the layers come together so that the projecting particles of a fit into the depressions of b. If a shearing force is applied so that a moves relative to b, in the direction of the arrow, then there will be a collision of particles which will stop flow unless the applied force is great enough to cause a rearrangement of the particles. 1 Bull.
I n s f . Mining M e f . , 18a (1919).
* Green, THISJOURNAL, 16, 122 (1923).
INDUSTRIAL A N D ENGINEERING CHEMISTRY
54 a
b
C
d
First Series
VOl. 19, No. 1
at. the expense of its small ones, with a net result of an increase in average particle size. By controlling the temperature and time of heating, a zinc oxide can be produced of any desired particle size, within certain limits, and so a series can be formed from a single material. The oxide chosen was a well-known American process brand. It was heated at two temperatures only, 750" and 1025' C. This gave two materials, one fine- and the other coarsegrained, and both free from colloidal particles that would have been difficult to measure. By mixing these two in various proportions, a graded series was obtained ranging in particle size from that of the fine up to that of the coarse material. The oxides at the extremities of the series were measured; the average particle sizes of the intermediate ones calculated. After heating, the oxides were ground in a ball mill for 24 hours in order to destroy aggregation. N o t e I t was found that aggregation could be reduced, if the oxide was brought to temperature very slowly. A specially constructed rotating tube furnace was employed.
Any aggregates remaining were subsequently treated as single particles. A second series was made and run as a qualitative check on the first series. The results obtained from both series are given throughout the remainder of the paper. Figure 3-Small
Portions of Photomicrographs of Zinc Oxides Used in Experiments a-first series fine; b-first series coarse; c-second series fine; d-second series coarse. Magnification 1285 diameters.
However, as the particles are held in place by interfacial tension, there will be a resistance offered to their rearrangement. This resistance manifests itself as yield value. It is now possible to understand that particle size in its relation to yield value must be studied from two anglesthe amount of interfacial area and the number of points of contact (of collision) per unit area of cross section. In order to carry out this investigation, it was necessary to prepare a series of particulate substances that covered a range of particle sizes and at the same time possessed a constant wetting factor. I Preparation of Materials
There is no practical method for determining the interfacial tension between solid a n d liquid. Consequently, in order to feel reasonably sure that the series possessed a c o n s t a n t wetting factor, it was ne cess a r y to produce each member of the series from one original substance. The substance chosen was zinc oxide. It is possible to o b t a i n zinc oxides of various average particle Figure &Distribution Curve of Fine sizes; but as each would Oxide in First Series be made, .perhaps, by a different method, and particularly from dflerent ores, it could not be assumed that all would possess the same wetting factor. In order to insure, as much as possible, the constancy of the wetting factor, advantage was taken of the fact that if a zinc oxide is heated its large particles will grow
Figure 5-Distribution Curve of Coarse Oxide in First Series
Measurement of Particle Size
All measurements of particle size were made by the photomicrographic methode3 Since zinc oxide that has been sufficiently heated contains practically no colloidal particles and, furthermore, since heating tends to make the particles roundish, the photomicrographic method was particularly suitable for this work. Series 2 was composed of coarser grained oxides than series 1 (compare photomicrographs Figure 3). Consesequently, the results from each series are to be considered not as quantitative but simply as qualitative checks. The forms of the functions are the same, but the position of the curves differs for each series (Figures 8, 9, 12, and 13). The oxides were dispersed and mounted in damar4 and
* J . Franklin Ins;., 192, 637 (1921). 4
For details of method of mounting see Green, THISJOURNAL,
677 (1924).
16.
55
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
January, 1927
dl = End/%;
dl = Z n d 2 / 2 n d ; d3 = ZndJ/Znd2; d , = End'@nd*
d = size of individual particle; taken here as diameter of a sphere n = particle frequency
Besides these average diameters there are two others, the diameter (D)of the particle of average volume and the diameter ( A ) of the particle of average surface.
D
and
A =
4-n
1
I
ZOO
= qZnd3/Zn
I
~
I bo
1
7 1
I
c
I20
Figure 6-Distribution Curve of Fine Oxide i n Second Series
Figure 7-Distribution Curve of Coarse Oxide in Second Series
I $0
10 0
x 0
Y e l d Value D3"EJ/~~' Figure &Curves of Both Series Showing Relation of Yield Value to I. F. A. when P. of C.
Are Constant
In considering the meaning of "average particle size" of nonuniform particulate matter, it should be noted thqt there are various "averages" to be taken into account. Perrott and Kinney6 have given four of them; they will be written dl, da, da, and d4. 6
J . Am. Ceram. SOC.,6 , 417 (1923),
Figure 9-Curves of Both Series Showing t h e Relation of Yield Value t o P. of C. when I. F. A. Is Constant
P o i n t s of C o n t a c t a n d Interfacial Area
Let w be the percentage weight of the pigment in the paint, p the density of the paint, and N the number of particles per gram of pigment. Then wpN is the number of particles per cubic centimeter of paint. Consider the particles to be dispersed evenly throughout a unit cube of paint; then (wpN)'/a
vJ1. 19, No. 1
INDUSTRIAL A N D ENGINEERING CHEMISTRY
56
o/
0
50
Yield
Value
d?~os/~'
IS0
200
Figure 11-Yield Value Plotted against V These curves show effect of different degrees of wetting, keeping particle size roughly constant.
i
Figure 12-Yield
i
I
I
Value Plotted against Various Average Diameters 10 Constant
of First Series, Keeping
is the number of particles per unit area of cross section and will be taken as the number of points of contact (P. of C,). The number of particles per unit area of paint multiplied by the average particle area is the interfacial area of one layer of particles (I. F. A.). This area can be thought of as composed of the upper half of the I. F. A. of one layer plus the lower half of the I. F. A. of an adjacent layer; the plane of shear and the points of contact existing between these two layers (Figure 2). The I. F. A. for spheres is equal to rA2(wpN)Va. The formulas for P. of C. and I. F. A. are independent of the uniformity of the material. M e a s u r e m e n t of Yield Value
'
Yield value is the minimum tangential force per unit area which, when applied to a plastic material, is able to cause a lateral displacement of adjacent parts-that is, flow. A number of methods have been proposed for measuring yield value, the original and customary one being to use the intercept obtained by extrapolating the "straight" part of the plastic flow curve to the pressure axis.6 Unfortunately, the work of Buckingham shows that this intercept is not necessarily proportional to yield value. Buckingham's equation of flow,' however, apparently fails a t the critical point from which yield value must be obtained, and so the possibility of determining yield value from plastic flow curves
* Bingham and Green, Proc. A m . SOC.Testing Malerbls, 19, 640 (1919). I b i d . , 21, 1164 (1921).
Value Plotted a a i n s t Various Average Diameters of Second Series, g e e p i n g w Constant
Figure 13-Yield
is at present a somewhat doubtful one. In order to avoid the use of flow curves, the yield values given here were measured by means of the microplastometer.* By this method yield value is obtained from direct measurement of the amount of force necessary to just cause visible yield under the microscope. The method is not highly accurate perhaps, but it has the great advantagd of being free from uncertain theoretical deductions. Experimental R e s u l t s
CONSTANT POINTS OF CONTACT-The first experiment was to determine the variation of yield value with I. F. A., when the P. of C. were maintained constant. In order t o accomplish this, the following formula was used:
where K = P. of C. = constant for each series; a = density of the pigment; b = density of the vehicle. Various values of N were substituted in the formuIa and the corresponding values of w determined. These values of w were then used t o determine. I. F. A. from the formula given in the text.
The results are given in Figure 8 and Table I, and may be visualized as follows: Suppose a definite number of particles to be evenly dispersed throughout a cube of paint and that 8
Green and Haslam, THISJOURNAL, 17, 726 (1925).
INDUSTRIAL AND ENGINEERING CHEMISTRY
January, 1927
57
When I. F. A. is maintained constant, these particles are able to increase their volume without changing their number. As their volume increases, the I. F. A. will increase, and consequently the amount of interfacial tension for a given number of P. of C. This should where k = I. F. A. = constant for each series. After a suitable k is decided upon, the values for d s / D are substituted in the equaincrease the yield value as shown by the curve. The curve tion and the corresponding w calculated. The ratio ds/D would cuts the I. F. A. axis because the particles in that direction be equal to unity if the materials were of perfect uniformity in have become so small that with their particular interfacial regard to particle size. tension they have insufficient I. F. A. to produce a continuous Since the I. I?. A. was maintained constant, and as it was structure; consequently, the material ceases to be plastic and the yield value becomes zero. In carrying out this work, chosen large enough in the beginning to produce a continuous the coarse oxide and the fine oxide were each ground separately structure with the given interfacial tension, then, theoretin raw linseed oil a t pigment content of about 75 per cent ically, the structure must be continuous when the P. of C. by weight. These materials were allowed to age for a week are zero. Hence, the curve must cross the yield value axis, (so that the wetting would become approximately stabilized) , if produced beyond the last experimentally determined point, mixed in various proportions (to produce different average for a continuous structure always produces a yield value. particle sizes), and then diluted to the predetermined ZL' At this point the particles would be infinitely large. EFFECT OF WEmING-In order to investigate completely that would maintain a constant P. of C. the relationship of yield value to particle size, it is necessary Table I-Points of Contact = Constant to consider the effect that wetting has on the problem. This is difficult to accomplish because wetting cannot be measured. YIELDVALUE I. F.A. N A2 W Dynes/sq. cm. Sq. cm./sq. cm. X 10-12 Sq. microns Per cent The curves in Figures 10 to 13, however, are of some value 2.45 X 108 in this respect. Figure 10 is obtained from a series of zinc 0.142 56.8 oxides, by plotting yield value against u'. These oxides were 59.8 0.148 0.156 63.2 grown by heating, and all four were made from a single orig0.167 66.7 inal material; therefore, it can be assumed that their wetting 0.180 70.8 75.4 0.198 factor is constant throughout. If these materials were per81.1 0.223 87.2 0.263 fectly uniform9 then for any given w each would have the 2nd Series: P,of C. 1.295 X 108 same I. F. A. ... 1.065 1.614 0.262 53.8 10.1 1,128 1.427 0.277 57.5 I. F. A. = 7rA2 ( ~ p N ) ~ / sA.2 N 2 / 3= (D'/da) (l/~aDs)~ia 26.0 1.200 1.239 0.295 61.7 P
36.5 65.6 101.5 152.5
1.310 1.457 1.694 2.101
1.052 0.864 0.676 0.488
0.322 0.358 0.416 0.516
66.6 72.2 78.9 87.5
( D l d d (l/7ra)2/a When the uniformity is perfect, D = da, and I. F. A. = ~ 1 1 (wp/a)2/a 3
~
b = 0.932
a = 5.7
CONSTANT INTERFACIAL AREA-The next experiment was the reverse of the first; here the variation of yield value with change of P. of C. was determined, keeping I. F. A. constant. The results are given in Figure 9 and Table 11. When the I. F. A. is maintained constant for a series, it will be found that the pigment-vehicle ratio is also nearly constant. This ratio would be constant if it were possible to deal with uniform instead of nonuniform materials. Consequently, in visualizing the results of this experiment the easiest procedure is to disregard the lack of uniformity and simply imagine a cube of paint in which the particles are undergoing subdivision while their total weight remains the same. As subdivision increases the I. F. A. does not change (I. F. A. is the surface per square centimeter of cross section and not the total surface throughout the cube) but the P. of C. do; consequently, yield value should increase. Materials that had been aged for a week were used here, in a manner similar to the method employed in the first experiment. Table 11-Interfacial YIELD VALUE
Dynes/sq. cm.
c.
0.41 1.09 1.63
...
... 2.40 ...
30.0
...
44.0 59.0 81.3 25.5 44.0 52.0 64.2 81.0 108.1 127.1 149.1 169.5
2.97 3.17 3.34 2nd Series: 0.318
I
D
Microns
W
Per cent
Summary
I . P . A . = 1.475
1 s t Series:
4.1 17.3 16.8
Area = Constant
do Microns
P. OF X 10-8
0.738 1.062 1.318 1.531 1.703 1.858 1.979 2.095
a
sg. cm./sq. cm. 1.339 1.157 1.172 0.791 1.041 0.666 0.938 0.597 0.858 0.553 0.789 0.515 0.730 0.486 0.677 0.465 0.630 0.447 I . F. A . 1.722 sq. cm./sp. cm. 1.719 1.438 1.500 1.037 1.330 0.882 1.193 0.792 1.083 0.729 0.992 0.683 0.915 0.646 0.848 0.617 0.791 0.592
-
-
5.7
b
-
0.932
As their wetting factor is constant, then where the I. F. A. became insufficient to cause a continuous structure for one, it would also be insufficient for the other three, and the four curves would intersect a t a common point on the w axis. If the influence of nonuniformity is disregarded, the effect of increasing particle size is to swing the curve upward about a pivotal point on the w axis. In actual practice there are no perfectly uniform materials and so the pivotal point is not stationary. In Figure 11 is a series of curves obtained from materials of roughly the same average particle size but with widely different wetting factors. The parallelism of these curves is no doubt due to the similarity in particle size, but their position is determined by the degree of wetting; the more completely wet the higher the position of the curve. Antimony oxide is the better wet and therefore has the weaker flocculating tendency, while with gas black the opposite is true. If the formula of these curves were known, it would probably be possible to determine interfacial tension between solids and liquids, and the surface tension of solids. Figures 12 and 13 show the various average diameters plotted against yie!d value, keeping w constant.
62.3 72.5 75.0 75.0 74.6 74.0 73.3 71.8 70.8 70.2 78.2 80.1 79.8 79.3 78.3 77.2 76.0 75.0
A n attempt has been made to visualize the structure of a pigment-vehicle mixture such as occurs in a slightly plastic paint. The relation of particle size to this structure has been analyzed and has been shown to involve interfacial area per unit area of cross section (I. F. A.) and the number of points of contact per unit area of cross section (P. of C.). Curves have been plotted showing the relationship between yield value, I. F. A,, P. of C., and the percentage weight and volume of pigment; also between yield value and the various average diameters of a nonuniform material.
'For definition of uniformity ree footnote 4.
