INDUSTRIAL AND ENGINEERING CHEMISTRY
1472
NOMENCLATURE = total number of moles of oxygen not chemically combined 3 (02) = concentration of oxygen in liquid phase, moles per liter
p P
= = = = = = =
R t T O(
’
G
ACKNOWLEDGMENT
This paper presents results of a part of the research carried out for the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. DA-04-495 ORD-18 sponsored by the U. S. Army Ordnance Corps.
partial pressure, atmospheres total pressure, atmospheres universal gas constant, atmosphere liters per mole O K. temperature, C. temperature, OK. total volume, liters coefficient in Henry’s law, gram-mole per liter atmosphere O
Subscripts A = vapor above acid solution for physical equilibrium at bubble point in absence of decomposition products B = oxygen d = contribution of decomposition of acid between states 1 and 2 t o oxygen not chemically combined
1 2
= = = =
Vol. 41, No. 7
gas phase liquid phase state 1 of system state 2 of system LITERATURE CITED
(1) Egan, E. P., Jr., IND. ENG.CHEM,,37, 303-4 (1945). (2) Klemenc, A., and Rupp, J., 2. unorg. Chem., 194, 51-72 (1930). (3) Lynn, S., Mason, D. M., and Sage, B. H., IND. ENQ.CHEM.,46, 1953-5 (1954).
(4) Perry, J. H., and Davis, D. S., Chern. and Met. Eng., 41, 188-9
(1934). (5) Priy, H. A., Schweikert, C . E., and Minnich, B. H., IND.ENG. CHEM.,44, 1146-51 (1952). (6) Reamer, H. H., Corcoran, W. H.. and Sage, B. H., Ibid.,45,2699704 (19531. (7) Robertson, G. D., Jr., “Some Physicochemical Properties of the System Nitric Acid-Nitrogen Dioxide-Water. Kinetics of the Thermal Decomposition of Nitric Acid in the Liquid Phase,” thesis in chemical engineering, Part I, California Institute of
Technology, 1953.
(8) Taylor, G. B., IND.ENQ.CHEM.,17, 633-4 (1925). RECEIVED for review July 23, 1954.
ACCEPTED November 4, 1954.
Paint Viscosity and Ultimate Pigment Volume Concentration U
d
W. IC. ASBECIQ, G. -4. SCHERER2, AND MAURICE VAN LOO The Sherwin- Williams Co., Chicago, Ill.
A
PREVIOUS paper (2)demonstrated the usefulness of the hypothetical term “viscosity a t infinite shear velocity” applied to practical problems of viscometry involving paint systems. This theoretical point is obtained by extrapolating the plot of the logarithm of the apparent viscosity of the system at finite shear velocity, log )I, against the reciprocal of the square as ,measured on a high shear root of the shear velocity l/a velocity viscometer (3, 6, 11) to the point where the shear velocity is infinitely great. It represents a specific rheological state of the paint system, It is that point at wrhich all rheological structure, be it shear volocity or time sensitive, is removed. At this point only the viscosity of the supernatantliquid and the pigment siee distribution, shape factor, and concentration play major roles in determining the viscosity of the system. A t infinite shear velocity the viscosity of the pigment/binder svstem is directlv DroDortional to the viscosity of the binder, ” The Vand (10) and Brailey ( 5 ) equations, relating the concentration of dispersed pigment to the viscosity of the system, are valid. This is true because both derivations were carried out on the assumption that no rheological structure resided in the dispersion. This condition can be realized only at infinite shear velocities for any system containing rheological structure. Most paint systems fall into this category. The Vand equation states
constants of the system. K is the pigment shape factor and q is the immobilization constant which corresponds to the amount of vehicle immobilized during the collision of pigment particles. The data presented in this paper deal only with rheological systems in which the vehicle is essentially Newtonian in nature. This is true for most oil paint systems but not all, The validity of actual data applied to the Vand equation can be tested by 1 plotting log ?I- against log If the value of (I has (1 -c - (IC?. ?lo been chosen properly, the data will lie on a straight line with slope K . The choice of the Proper values of P and K requires considerable manipulation. The values can be established either roughly by solving a series of Vand equations simultaneously or by trial and error methods. Both are cumbersome and time consuming. Nomography does not seem to reduce the difficulty materially because of the complexity of the diagrams involved.
where 7 is the viscosity of a pigment dispersion at infinite shear velocity, 70 is the viscosity of the binder, c is the volumetric percentage of pigment present in the dispersion, and K and q are
where 70, and c conform to the notations in the Vand equation, and k is very nearly proportional t o K , the form factor, while U is a function of q, the immobilization constant. The ratio of K to k is constant to within less than 10% for all useful values of p or U . This equation can be thrown into a number of forms that can be solved graphically quite readily.
1 Present address, Carbide and Carbon Chemicals Co., South Charleston, W. Va. * Present address, Earlham College, Richmond, Ind.
HYPERBOLIC EQUATION
Fortunately the Vand equation and the improved form suggested by Brailey can be converted semiempirically into a simple hyperbolic form which conforms very closely to the results obtained by these equations. The theoretical grounds for this hyperbolicequation are described by Mooney (9)and Maron ( 7 , 8 ) . It is of the type
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1955
I
20
14 l6
I
2 0
0
2
4
I
I
1
1
1
1
1
I
6
8
IO
12
14
16
18
e0
I/=
Figure 1. Viscosities of suspensions from Vand data
1.95 and.45.5 for Figure 2 and 1.95 and 55.5 for Figure 3, respectively. An interesting concept is inherent in the constant U ; it is the concentration of pigment in a pigment/binder system a t which the regular, tightly paoked system of pigment particles have their interstices just exactly filled with the suspending medium and the particles are stacked in their most favorable rheological configuration. Such a system would have an infinitely great viscosity. However, it need not necessarily be solid. It is the critical point above which, if the system is sheared, a volume increase occurs and dilatency results.
c 4t t / 3
Table I lists the viscosities of suspensions as calculated by Brailey (6),employing a form factor K = 2.5 and an immobilization constant q = 1.16, and as obtained from the hyperbolic equation. These data are plotted in Figure 1 in a hyperbolic form as I/c, the reciprocal of the pigment concentration, versus
t-
l/log 5,the reciprocal of the logarithm of the specific viscosity
1x1
VI
of the system a t infinite shear velocity. The plot is a straight line with the data showing negligible deviation from linearity. The constants k and U are obtained from this plot, k being the cotangent of the line, and U the intercept on the abscissa expressed in reciprocal units of pigment concentration. For these data k = 1.10 and U = 71y0 pigment. The values of K and k seem to check well when it is considered that the Vand and Brailey derivations were carried out on the basis of the natural logarithm, while the graphical computations were accomplished on the basis of the logarithm of 10. The ratio of K to k is very nearly equivalent t o 2.3, the factor for converting the logarithm from the base e to the base 10. Consequently K is very nearly equal to 2.3 k.
Table I.
Relative Viscosities of Suspensions
Brdey
C
0
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.46 0.50 0.55
‘la
1 1.15 1.356 1.640 2.050 2.662 3.680 5.53 9.35 20.0 70.9 437
L/C
20.0 10.0 6.6 5.0 4.0 3.3 2.86 2.50 2.22 2.00 1.82
Hyperbolic Form log 7 8 0 0.06070 0.13226 0.21484 0.31175 0,42521 0,66585 0.74272 0,97081 1.30103 1 ,85065 2.64048
l/log
‘18
m
16.45 7.55 4.65 3.21 2.35 1.77 1.35 1.03 0 77 0.54 0.38
1473
2
0
2.
Figure
I
I
I
4
’
6
I
Viscosities of titanium linseed oil systems
I
I
8
I IO
dioxide-bodied
ULTIMATE PIGMENT VOLUME CONCENTRATION
It is interesting to speculate whether other physical methods besides rheological measurements may lead to a similar point. This would require that a previously dispersed pigment/binder system have its particles brought to closest packing and the volume percentage of pigment and vehicle determined. No current technique in general use accomplishes this directly.
4l 3
“e I
Data for the viscosities of two model pigment/binder systems, consisting of 35, 30, 25, 20, 15, and 10% titanium dioxide and magnesium silicate, respectively, dispersed in a bodied linseed oil are shown in Figures 2 and 3 plotted on the same coordinates as Figure 1. The viscosities as measured a t finite high shear velocities on the high shear rotational viscometer were extrapolated to infinite shear velocity by the technique described in the previous papers (2,3, 11). The points as measured again seem to fall on a satisfactory straight line. This indicates very good correlation between the theoretical derivations and the practical results obtained by the use of the concept of the viscosity of a system a t infinite shear velocity. The apparent similarity between the derived forms of the equations and the more practical, empirical hyperbolic forms employed in the graphs is also well illustrated. The constants k and U again are obtained readily. They are
0
0
2
4
6
8
10
I /C
Figure 3.
Viscosities of magnesium silicatebodied linseed oil systems
The critical pigment volume concentration (CPVC) (1) and pigment packing factor (PPF) ( 4 ) techniques can be modified to approximate these conditions. These methods both depend on the principle of measuring the volume, including the interstitial voids, occupied by a given quantity of pigment under the specific conditions of dispersion or agglomeration resulting from the interaction of the particular combination of pigment and binder
INDUSTRIAL AND ENGINEERING CHEMISTRY
1474
employed. The critical pigment volume concentration is a useful constant for paint systems, above and below which the physical attributes of paint films change rather abruptly. The CPVC's may differ materially for the same pigment system when used with various vehicles, depending on the dispersive capacities of the vehicles for the pigment. Very briefly, the CPVC test method consists of filtering the vehicle from a given amount of paint whose volume percentage of pigment is known, using a special cell. The volume of the remaining pigment cake, including the interstitial voids, is then measured by use of a nonwetting liquid. The ratio of the packed pigment volume to the true pigment volume is equivalent t o the CPVC. I n the case where the particles of the suspension are monodispersed, this measurement theoretically should yield a result similar to the value of U as extrapolated by the use of rheological measurements outlined, if the packing systems are the same in both cases. The particles would be substantially in their tightest packing configuration, fulfilling the prerequisites outlined for the theoretical value of U . However, most paintlike pigment/ binder systems have their particles agglomerated to some extent, the degree depending upon the dispersive capacity of the vehicle for the particular pigments involved, As a result the filter cake volume in the CPVC test is larger than the ideal, close packed condition. This makes the CPVC lower than that for the same pigment system under conditions of complete dispersion. Figure 4. The problem, then, becomes one of deUltimate pigsigning a method for obtaining this ultiment volume mate pigment volume concentration of concentration cell practical pigment/binder systems. The simplest modification of the CPVC and PPF techniques would consist of tamping the CPVC cake to a tight mass before the volume determination is made. The tamper may consist conveniently of a glass rod, one end of which is flattened out to a diameter of about one half inch and ground flat on the bottom. It is advantageous t o grind a slight bevel around the edge, with the greater diameter at the bottom, so that the packing may be accomplished to the very edge of the cell. The packing is carried out while suction is applied to the bottom of the cell until a firmly packed cake is formed. The volume of this cake is then determined by the usual CPVC technique, and the results calculated as the quotient of the true volume to the packed volume of the pigment. The results are expressed in percentage as the UPVC, the ultimate pigment volume concentration. Relatively good check results are obtained with the same pigment dispersed in different vehicles. Thus the UPVC of 20% dispersions of titanium dioxide, magnesium silicate and a 1:1 volume combination of the two dispersed in bodied linseed oil and raw linseed oil, respectively, give the results shown in Table 11. Both raw oil and bodied oil, which have dispersive capacities which generally differ decidedly for the same pigment, show about the same values. This seems to be well within experimental error.
W
Table 11. Ultimate Pigment Volume Concentrations Titanium Dioxide
Raw linseed oil Bodied linseed oil
50.4
51.4
Magnesium Silicate 58.7 59.5
1:1 Mixture Titanium Dioxide : Magnesium Silicate 55.7
55.1
Vol. 47, No. 2
The test process as outlined is carried out easily for systems which are not too highly agglomerated. However, such pigment/ binder systems as raw linseed oil dispersions of titanium dioxide are usually agglomerated to a considerable degree. With this type of dispersion, the pigment mass has a decided tendency to slip from under the tamper, and considerable patience is required to compress the cake to its densest volume. The greater the degree of agglomeration of the pigment system, the greater this tendency seems t o be. A new cell has been employed to circumvent this difficulty. This consists of a straight walled glass cylinder with an accurately ground and closely fitted plunger. The bottom of the cylinder is sealed with a fritted-glass plate. The cell is illustrated in Figure 4. A known volume of the material to be tested is placed in the cell, and the liquid portion is filtered as in a CPVC test. The plunger is then inserted and the cake compressed, with continued suction on the glass frit, until the smallest volume of the pigment mass is obtained. Slight rotation of the plunger is beneficial during this operation. The volume of the compressed pigment cake is then read on a suitable calibration scale on the plunger or cylinder. Calculations that are similar lead to the value of UPVC. Elimination of the filtration step hastens the process of measuring the UPVC materially. This can be accomplished by bringing the nonvolatile portion of the vehicle employed in the original dispersion of the pigment system below that required to fill the voids of the packed pigment mass a t the UPVC by use of the vehicle substitution method (1, 4). A given volume of the dispersion is then placed on a glass plate and the volatile portion of the vehicle allowed to evaporate. The film must not be allowed to dry completely before the pigment paste is scraped from the plate and placed in the cell, where the cake is compressed to its smallest volume, with the vacuum applied as before. Either cell can be employed for this purpose.
Figure 5 .
Z-Nomograph to obtain U and k
The UPVC method might be used as a substitute for the present oil absorption methods in characterizing the packing behavior of pigments independently of the dispersing vehicle. Z-NOMOGRAPH
A number of other dispersions were studied to determine whether the physical packing condition as obtained in the ultimate pigment volume concentration method correlates with the rheological packing conditions obtained by the viscosity measurements. These consisted of the same titanium dioxide and magnesium silicate pigments described in Figures 2 and 3, this time dispersed in raw linseed oil instead of the bodied linseed oil. A 1:l by volume system of both of these pigments dispersed in
*
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1955
bodied and raw linseed oils was also included. As before, 35% dispersions of these systems were ground on a three-roll laboratory mill and then diluted to 30, 25, 20, 15, 10, and 5% pigment by volume with the same vehicle used in the original dispersions. The 20% dispersions of these series were employed in the UPVC studies. The viscosities of each of these dispersion series and the viscosities of the vehicles were then determined on the rotational high shear viscometer and their viscosities extrapolated t o infinite shear velocity. 100
1475
.^
F
-
0 0
I
I
0
IO
0
I
I
l
l
I
1
I
I
I
I
I
2
3
4
5
6
7
8
9
IO
l / W X
F
Figure 7.
m
0
I
I
0
I
2
3
4 I / T
Figure 6.
5
6
7
8
9
IO
x 100
Z-Nomograph for titanium dioxide in raw linseed oil
The method of plotting the results by the method illustrated in Figures 1, 2, and 3 requires extensive calculation and has a tendency to make the extrapolation of U somewhat inexact because l / c is usually small a t this point. Therefore, another system was resorted to in order to obtain U and k for these data. The equation
100
Z-Nomograph for commercial one-coat exterior house paint
pigment concentration on an arbitrary linear scale on a line parallel to the abscissa, starting from the viscosity a t infinite shear velocity of the vehicle. This automatically reduces the g, value of the dispersions to gs, the specific viscosity a t infinite shear velocity, since g,/go = gs. By connecting the gm values with the corresponding c values, and projecting the lines, they are found to intersect reasonably well at a point. For the dispersions of Figure 6, U is 38y0 pigment and V is 0.54. Consequently, k = V / U = 1.4. Knowing this point for any dispersion of pigment in a Newtonian vehicle, the viscosity at infinite shear velocity of any given concentration of that pigment in the vehicle can be determined. C0,MPARISON OF UPVC AND U
U and k were determined for the various dispersions of titanium dioxide, magnesium silicate, and the 1:l mixture in raw and bodied linseed oils, respectively, by employing the above procedure. The results are listed in Table 111.
TIC log g* = -
u-c
Table 111. Constants U and k for Linseed Oil Dispersions
where V = kU of the previous hyperbolic equation can be plotted as a nomograph employing the familiar Z-form. This has a number of advantages. The system of plotting is illustrated in Figure 5. The logarithm of the specific viscosity of the paint system a t infinite shear velocity is the ordinate, while the concentration of pigment in the system is the abscissa. If straight lines are drawn connecting qm/qO on the ordinate with the corresponding pigment concentration on the abscissa, the continuation of these lines for two or more sets of data will intersect a t a point which characterizes U and V . U is equivalent to the pigment concentration obtained when a line is drawn parallel to the log q,/qo axis through the point, and intersecting the c axis a t U . V is the linear distance the point lies below the abscissa expressed as a proportion of the logarithmic decade used for plotting qm/qO taken equivalent to 1. k is equal to V / U . When this type of plot is used, practically no calculations are required in order to obtain U and k from the data employed to obtain the viscosity of the system a t infinite shear velocity. In fact it is possible to employ the same graph paper on which the q m extrapolation is made. Figure 6 shows the data obtained for titanium dioxide dispersed in raw linseed oil. The basic graph is that of the logarithm of apparent viscosity of the dispersion plotted against the reciprocal square root of the shear velocity. The values for infinite shear velocity can be extrapolated by laying a straight line through the points obtained a t finite shear velocity and continuing the line to intersect the ordinate. The Z-nomograph can be superimposed on this graph by laying off the
Raw linseed oil
Bodied linseed oil
Titanium Dioxide U k 38 1.4 44 1.9
Magnesium Silicate U k 65 1.5
67
1.7
1 : 1 Mixture Titanium Dioxide: Magnesium Silicate
U 46 55
k 1.4 l.Q
Comparing Tables I1 and 111, the physical packing system obtained by the UPVC procedure and the rheological packing system arrived a t through viscosity measurements are not the same. Evidently the rheological packing is looser than the physical packing. I n fact, the rheological packing systems seem t o differ for the same pigment dispersed in various vehicles. This substantiates the preliminary report made in the previous paper (8). The theoretical significance of this fact is not yet apparent. The validity of the Vand equation and its hyperbolic counterpart for complex multipigment/multivehicle systems was substantiated on commercial paints. Figure 7 shows the results obtained for a commercial one-coat exterior house paint plotted in the same manner as the data for Figure 6. The various percentages of pigment were obtained by centrifuging the pigment out of a portion of the paint and diluting the original material with the proper proportions of the supernatant vehicle. The correlation between the theoretical and practical values is well within experimental error. Consequently, the viscosity a t infinite shear velocity of any pigment/binder system containing a Newtonian vehicle can be
INDUSTRIAL AND ENGINEERING CHEMISTRY
1476
predicted at any concentration of the pigment if two constants and the viscosity of the binder are known. These constants can be obtained readily by determining from finite shear velocity measurements on the high shear viscometer a t two or more levels of concentration of that pigment in the given vehicle and following the procedures as outlined.
1.
log
2.
log??,
70
Vol. 47, No. 7
+ d sD kc u = log to+ -
= log q,
~
u-c
CONCLUSIONS
.I 8 .I 6 .I 4 .I 2 .I0
tFigure 8.
Structure nomograph titanium dioxide in raw linseed oil
Although these theoretical relationships hold only a t infinitely high shear velocities, the viscosity at brushing shear velocities usually do not differ very materially from this value for most conventional oil paints. As a result, the four factors, 70,c, U,or p and k or K , will influence the brushing characteristics of paints strongly. Changes in any of these factors would alter the brushing characteristics very nearly according to these relationships. STRUCTURE EQUATION
In order to make the prediction for brushability of oil paints exact, or t o predict the viscosity of pigment/binder systems a t any high shear velocity, it is necessary to know the relationship between the rheological structure of a paint system and the concentration of pigment present in the system. This has been found empirically to follow quite closely the relationship
where is the slope of the viscosity curve in the basic plots of Figures 6 and 7. This equation, too, can be solved with a Z nomograph by plotting q , S against c. A plot of this type is shown in Figure 8 for the data obtained from Figure 6. The point of intersection of the projection of the v W S versus c lines determine the constants 01 and 6. The factors required for solving this equation are obtainable from the basic viscosity/shear velocity measurements required for obtaining v m r U , and k . More work is required before the limits of the structure relationship can be ascertained. The exact theoretical significanceof the constants CY and 6 is not apparent a t this time. For practical purposes, however, it seems possible with the aid of four constants, U ,k , CY and 6, which can be obtained easily by empirical means, to predict the viscosity of most oil paint systems at any given concentration of pigment, viscosity of vehicle, and high shear velocity down t o about 300 sec.-l This is accomplished by the equation
For the brushing shear velocity, D”2is equal to 100. It is usually easier to solve this equation in several steps through the three basic equations that make it up.
The Vand and Brailey equations relating the viscosity of suspensions to the concentration of pigment present may be employed in practical viscosity measurements of paint systems. Because of the difficulty of determining the constants K , the pigment shape factor and q, the immobilization constant for these derived equations, a hyperbolic form giving results very similar to the theoretical equations is employed. The two constants k and U are obtained for this equation by graphical computation. k is very nearly proportional to K , while U , the densest rheological packing system for the particular dispersion is a function of p. This rhelogical packing system is not equivalent to the densest physical packing system obtained by determining the ultimate pigment volume concentration (UPVC) of that dispersion. The UPVC method may find application in the paint industry to characterize pigments as a substitute for present oil absorption tests. A prediction of the viscosity of any concentration of pigment in a given Newtonian vehicle may be made for any chosen high shear velocity by combining the equations with a second empirical hyperbolic equation, correlating the structure of pigmented systems with concentration of pigment. The two constants required for the structure equation, 01 and 6, can be obtained from the same data required t o determine k and U . These basic data are obtained from the high shear viscosity curves of the system a t two or more levels of pigmentation, LITERATURE CITED
(1) Asbeck, W. K., and Van Loo, M.,-IND.ENG.CHEM.,41, 1470 f 1 949). --I. \--
(2) Ibid., 46, 1291 (1954). (3) Asbeck, W. K., Laiderman, D. D., and Van Loo, M., J . Cotloid Sei., 7, 306 (1952). (4) Asbeck, W. K., Laiderman, D. D., and Van Loo, M., Ofic. Dig. Federation Paint & Varnish Production Clubs, No. 326, 156 , (1952). (5) Brailey, R. H., Division of Paint, Varnish and PlaBtics Chemistry preprint, p. 49, 120th Meeting ACS, New York, September 3-7, 1951. (6) Hull, H. H., J . Colloid Sci., 7 , 316 (1952). (7) Krieger, I. M., and Maron, S. H., Ibid., 6, 528 (1951). (8) Maron, S. H., Madow, €3. P., and Krieger, I. M., Ibid., 6, 584 (1951). (9) Mooney, M., Ibid., 6, 162 (1951). (10) Vand, V., Nature, 155, 364 (1945) ; J . Phys. & Colloid Chem.’ 52, 277 (1948). (11) Wachholtz, F., and Asbeck, W. K., Kolloid-Z., 93, 280 (1940); 94, 66 (1941).
RECFJ~ED for review November 12, 1954. ACCEPTEDDecember 15, 1954. Presented before the Division of Paint, Plastics and Printing Ink Chemistry at the 126th Meeting, ACS, New York, September 13, 1954.
Correction In the article, “Hydroforming Reactions, Effect of Certain Catalyst Properties and Poisons [W. P. Hettinger, Jr., C. D. Keith, J. L. Gring, and J. W. Teter, IND.ENG.CHEM.,47, 719 (1955)] “Hydroforming” should n o t have been capitalized. The authors used the name “hydroforming” uncapitalized as a generic term referring to all hydroreforming processes.
