Air as a Dispersion Medium F. B. Stieg N L Industries, Inc., Titanium Pigment Division. South Amboy. New Jersey 08879
Past experience in the paint industry has suggested that the white hiding power of porous Ti02-containing coatings may be predicted by considering air voids as part of the dispersion medium after solvent evaporation. The relative numbers of pigment-air and pigment-binder interfaces present in the dry paint film are calculated from a previously published PVC-CPVC relationship. A weighted-average refractive index for the composite air-binder “medium” is then used to obtain an effective Fresnel reflectivity, from which white hiding power is calculated using a previously developed empirical equation. Predicted values appear as accurate as those determined experimentally. A nomograph is offered from which hiding power per pound of Ti02 may be obtained from a knowledge of formula PVC, pigmentation CPVC, and the effective PVC of the Ti02 content (plus binding power index for latex paints).
The use of porous paint films is older than the paint industry itself, for prior to the commercial availability of opaque synthetic white pigments only natural glue and casein-bound “white washes” were used for both interior and exterior painting by the common man. These coatings, which possibly may deserve a place in the general category of microvoid organic coatings, although their content of organic binder was minimal, owed their opacity to the presence of an excess of pigment. Natural calcium carbonate, lime, talc, and clay were used to produce the desired “high dry hiding”; we would call them extenders in the paint industry today, but they served quite effectively as white pigments in such porous films. As the paint industry developed, it was not forgotten that porosity due to a high level of pigmentation was the cheapest way to develop white hiding power. Unfortunately, however, it was a device which could be utilided only in flat or lusterless finishes-and, equally unfortunately, porosity was not considered to be a desirable characteristic, even for flat wall paint films. The term porous immediately implied permeability to water vapor and/or liquid, which in turn implied poor protection of the substrate, difficult stain removal, and poor recoat characteristics. The paint formulator’s problem thus became one of achieving a balance between the three parameters of cost, hiding power, and porosity. Today, the relatively new concepts of using discrete microvoids as a replacement for pigment make it possible to formulate high-gloss coatings and to use the low refractive index of air t o develop hiding power without producing permeable films. There may, however, be some clue in the story of how the paint industry learned to predict the performance of porous pigmented coatings that will aid in doing the same for “pigmentless paint.” Two parallel paths of investigation led to the ability to predict dry hiding power for porous organic coatings pigmented with titanium dioxide. One of them was concerned with the hiding power of titanium dioxide in the presence of extender (Stieg and Ensminger, 1961), and the other, with film porosity (Stieg, 1959). Hiding Power of Nonporous Films
It had long been known that the opacifying power of titanium dioxide in organic coatings did not follow Beer’s law of concentration (Stieg, 1957). As pigment volume concentration, PVC, was increased, it was observed that the hiding power contributed per unit weight of pigment became smaller and smaller until it reached a minimum
a t the point a t which high dry hiding made its appearance. Figure 1 is a hiding power curve for a typical enamel-grade titanium dioxide. It was theorized that this loss of efficiency was the result of interference effects produced by light scattered, out of phase, by adjacent pigment particles-and therefore probably a function of particle separation. The spacing of particles in a paint film could be calculated, of course, if it were assumed that they were uniformly distributed and all of the same size. A comparison of average hiding power data over a wide range of PVC’s for both rutile and anatase titanium dioxide with calculated pigment spacings showed that this assumption was apparently justified (Stieg, 1967). The following empirical equations were found to correctly relate hiding power and PVC data for these two pigments: for rutile HP = 370[1
- (PVc/0.74)”3]
(1)
and for anatase HP = 298[1 - (PVC/0.74)”3] (2) The term (PVC/0.74)1/3is the reciprocal of the center-tocenter spacing between particles expressed in pigment diameters. Table I is a comparison of determined and calculated hiding power values, in square feet of coverage a t 0.98 contrast ratio (CR) per pound of pigment, for rutile titanium dioxide (Stieg, 1967). Further study of eq 1 and 2 disclosed that the constants 370 (for rutile) and 298 (for anatase) were directly proportional t o the Fresnel reflectivity, as calculated from the refractive index of the pigment and that of the alkyd resin in which the determinations had been made (Stieg and Ensminger, 1961)
F
=
(nl
-
n2I2/(nl
+
n2)*
(3)
where F is the Fresnel reflectivity, nl is the refractive index of the pigment, and n2 is the refractive index of the medium. The two equations, (1) and (2), could therefore be rewritten as one
HP = 4430F[1 - (PVC/0.74)1’3]
(4)
The application of this formula to many practical paint systems over a period of years led to the discovery of a number of conventions which had to be observed. (1) The term within the brackets could never have a numerical value of less than 0.5-corresponding to a PVC of 9.25%. Below this PVC, which was that a t which pigInd. Eng. Chem., Prod. Res. Develop., Vol. 13, No. 1, 1974
41
I80
7
I
.1
N
160
I ,lo
I
.15
80
.20
.10
.40
.30
Figure 1. Hiding power curve TABLE I. Calculated us. Determined Hiding PowerQ
Calcd
Determined
PVC
Calcd
Determined
181 153 131 113
181 155 131 112
30 35 40 45
96 82 69 57
96 84 70 58
~
a
Square feet per pound of pigment (0.98CR).
TABLE 11. Effects of Film Porosity
PI 0 0.10 0.20 0.30 0.40 0.50 0.60
F
Re1 H P
0.0836 0.0928 0.1026 0.1133 0.1248 0,1371 0,1504
1 .oo 1.11 1.23 1.36 1.49 1.64 1.80
E2
1.50 1.45 1.40 1.35 1.30 1.25 1.20
ment particles were separated by a full pigment diameter, there was no further increase in pigment efficiency, and hiding power was directly proportional to pigment concentration (Stieg, 1959). (2) When extenders were present, the term PVC was no longer that of the paint formula but became equivalent to the effective PVC of the titanium dioxide in the dry paint film, PVCerr. This effective PVC was calculated using only those film components serving as diluents for the titanium dioxide-vehicle solids (NV) and fine-particle-size extenders. (See eq 5 . ) The latter were defined as those possessing an average particle diameter of the same order of magnitude as that of the titanium dioxide itself (about 0.2-0.3 ~ c ) . . Figure 2 shows the relationship of effective PVC to the hiding power of titanium dioxide expressed in square feet per pound. vol TiOz PVC& = (5) vol fine extender vol TiOz vol NV
+
+
(3) The term PVC could never become larger than the critical PVC, CPVC, of the titanium pigment used (Stieg, 1962). The CPVC represents the PVC existing a t the end point of the spatula rub-out oil absorption test (Stieg, 1956, 1958) and therefore the maximum condition of packing that will be produced in a practical paint film. Hiding power a t the CPVC, or a t an effective PVC equivalent to the CPVC, was found to be the minimum value obtained. Above the CPVC, air was introduced into the 42
a
.35
*
.40
&
. 4 5 .50
Figure 2. Relationship of effective PVC to hiding power
.50
PVC
10 1.5 20 25
L
.30
EFFECTIVE PVC
0
PVC
.25
.20
Ind. Eng. Chem., Prod. Res. Develop., Vol. 13,No. 1, 1974
films, as indicated by density measurements, and eq 4 no longer appeared to apply. It was recognized that the increase in hiding power above the CPVC was due to the replacement of pigmentvehicle interfaces with pigment-air interfaces-the lower refractive index of air resulting in a higher value for the Fresnel reflectivity (eq 3)-but no method of calculating the degree of replacement in a specific formulation was a t first apparent.
Film Porosity While the CPVC has been mentioned as the pigment: binder ratio above which film porosity resulted in increased dry hiding power, its primary significance to the paint formulator was the fact that it also represented a transition point a t which film properties such as gloss, vapor Permeability, washability, enamel holdout, and color uniformity in deep-tone flat wall paints passed from an acceptable level to somewhat less than acceptable (Asbeck and Van Loo, 1949; Stieg, 1956; Stieg and Burns, 1954). It was therefore helpful to have some numerical system for indicating the relative porosity of a paint film so it might be used as a guide in formulation. Such a system was devised, making use of the relationship existing between the amount of binder required to produce an impermeable film, as indicated by the CPVC, and the amount of binder supplied by the paint formula, as indicated by the PVC (Stieg and Ensminger, 1961). Since the CPVC represents the closest possible packing of the pigmentation, any deficiency of binder is replaced by an equal volume of air upon the evaporation of solvent. The result of the CPVC-PVC relationship was expressed as a “porosity index” (PI) CPVC(1
PI=1PVC(1
- PVC)
-
(6)
CPVC)
Equation 6 was found to correlate well with water vapor permeability, enamel holdout, and dry hiding power. Numerically, the porosity index, PI, is equivalent to the volume of air in the dry paint film, expressed as a percentage of the total binder demand. It is therefore different from the term porosity as usually defined, since it is independent of the total volume of the film. For this reason, the ratio of PI to (1 - PI) becomes equal to the ratio of air to binder in the dry paint film. If the assumption is made that the relative numbers of pigment-air and pigment-binder interfaces are directly proportional to their respective volumes, this ratio may be used to calculate the average refractive index, fiz, for the mixture of air and binder which serves as the m‘edium. Air is thus considered as a m e d i u m , rather than as a pigmentary microvoid (Stieg, 1970a).
TABLE 111. Calculated us. Determined Hiding Power
HP2 ft3/gal
lb of TiOI/gal
PVC
PI
PVCerr
F
Calcd
0.1007 0 ,0889 0.1173
310 240 373 303 253 424 358 307 205 219 231 234 228
AI
Determd 331 252 344 283 260 400 355 310 207 211 221 234 240
~~
2.12 1.66 2.46 1.93 1.59 2.84 2.22 1.83 2.04 2.12 2.22 2.31 2.42
0.550 0.550 0.600 0.600
0.600 0.650 0.650 0.650 0.655 0.680 0.710 0.740 0.775
1.409 1.471 1.333 1.384 1.425 1.269 1.310 1.343 1.475 1.424 1.368 1.317 1.262
0.182 0.058 0.333 0.233 0.150 0.462 0.380 0.314
0.225 0.186 0.263 0.219 0.187 0.307 0.258 0.222 0.303 0.327 0.359 0.395 0.441
0.050
0.152 0.265 0.367 0.477
1.80 .50
0.1062
0.0976 0.1322 0,1224 0.1148 0 .0881 0.0978 0.1095 0.1209 0.1342
-
I
1.70 1.60
.55
w E
1.50
z Z
.BO
-
.65
-
300 250
>
1.40
3
.zo
2w
z
Y
c
\
1.30
.70 .75
-
.80
-
Y
1.20
150
4
/
I I , /
1.10 100
1.00
0
.IO
.20
.30
.40
.50
.60
POROSITY INDEX (P.1,)
Figure 4. Dry hiding power nomograph
Figure 3. Effect of porosity on relative hiding power
TABLE IV. Calculated us. Determined Hiding Powera
HP, ftz/gal For a binder with a refractive index of 1.50 LOOP1 + 1.50(1 - PI) n2 = PI + (1 - PI)
(7)
+
= 1.00 0.50(1 - PI) From the values of i22 calculated using eq 7, it is possible to calculate a new value for the Fresnel reflectivity, F, which increases as the average refractive index of the composite medium decreases, using the formula
+ 0.5PI 4.22 - 0.5PI
.-i 1 1.22
(8)
Table I1 shows the calculated effect of film porosity on average refractive index, Fresnel reflectivity, and hiding power. If relative hiding power from this table is plotted against porosity index, the curve produced closely approximates the straight line observed in practice (Stieg, 1969) (see Figure 3 ) . Practical Applications. T o demonstrate the relative accuracy of this procedure, data for a number of flat wall paints pigmented above their CPVC’s with various combinations of rutile titanium dioxide and normal extendersand for which hiding power in square feet per gallon at 0.98 contrast ratio had already been determined-were used to calculate expected dry hiding power, employing eq 4,6, and 8. The results are tabulated in Table 111. Statistical analysis of the calculated and determined
CPVC
PI
0.400 0.354 0.400 0.354
0.08
0.24
x
iiz
1.00 1.46
1.00 1 . 3 8 0.08 0.75 1.45 0.24 0.75 1 . 3 8
-
F
Calcd
Determd
0.0918 0.1087 0.1173 0,1315
289 343 385 432
290 333 376 420
a Conditions: PVC = 0.42; PVC,fr = 0.318; 2.91 lb of TiOI/gal (3.46 lb/gal of 84% Ti02 content pigment).
hiding power data in Table I11 yields a correlation coefficient of 0.98-which is quite high-and a standard error of estimate of 12.66 ft2/gal. This degree of accuracy is of the same order of magnitude as the standard deviation of the ASTM D1738 method for determining the white hiding power of similar flat wall paints. Latex Paints A similar procedure may be followed in latex paints with two minor changes: the “binding power index” n must be used in the calculation of porosity, and a correction must be made to compensate for the titanium dioxide content of latex grade pigments. The binding power index is a term which compensates for the relatively lower binding ability of latex vehicles as compared to solution systems (Stieg, 1970b). It is determined experimentally for a given latex system by determining the PVC a t which dry hiding is first apparent for a series of paints pigmented only with a single extender-for which the CPVC either is already known or is determined, i.e. Ind. Eng. Chem., Prod. Res. Develop., Vol. 13, No. 1, 1974
43
PVC(1
-
CPVC) (9)
x =
CPVC(1 - PVC) where x is the binding power index, PVC is the PVC of zero porosity, and CPVC is the CPVC of extender used. The calculation of porosity is modified as CPVC(1 LP=1PVC(1
-
-
PVC) X
(10)
CPVC)
where L P is the latex porosity and x is the binding power index. The calculation for the average refractive index becomes (1.49 is the refractive index of a typical latex binder) (Stieg, 1970a)
+ 1.49(1 - LP) L P + (1 - LP) = 1.00 + 0.49(1 - LP) = 1.00 + 0.49(1 - PI)x 1.OOLP
n2 =
(11)
(12) The correction for titanium dioxide content is necessary because of the high treatment level used to develop high oil absorption in some latex-grade pigments. Typical values may range as low as 80%. Practical Applications. A good example of the hiding power differences produced by extender oil absorption and latex-compared-to-solution binder is provided by a group of four paints prepared for an earlier paper (Stieg, 1970a). These flat wall paints were all pigmented with the same latex-grade titanium pigment a t the same PVC. Two of them had an alkyd binder; the other two were a direct latex replacement. For each vehicle system, the extender was varied from a low oil absorption calcium carbonate to a high oil absorption clay. The data appear in Table IV. The last two formulas in Table IV were latex, as disclosed by the less-than-unity value of x , and the second and fourth formulas contained the high oil absorption extender, as disclosed by their higher PI values. The results show that the calculated values predict the higher dry hiding of the latex us. that of solution formulations, and the higher dry hiding of the high oil extender us. that of the low oil extender.
44
Ind. Eng. Chem., Prod. Res. Develop., Vol. 13, No. 1, 1974
In the process of preparing this paper the author devised a dry hiding power nomograph for rutile titanium dioxide which is reproduced here for the first time (Figure 4). The numerous calculations previously described may be reduced to the drawing of two straight lines if the formula PVC, the pigmentation CPVC, and the effective PVC of the titanium dioxide content are known. The broken lines on the nomograph illustrate the calculation for the first formulation in Table I11 (CPVC = 0.50). When a latex paint is involved, the apparent CPVC, CPVC'. is calculated using the binding power index, n CPVC(X) CPVC' = 1
- CPVC(1 -
(13) x)
Summary While it has been customary to think of the air bubble in microvoid coatings as performing a pigmentary function, past experience in the paint industry has led to the successful prediction of dry hiding power in porous paint films by calculating the effect of occluded air on the average refractive index of the medium in which titanium dioxide is dispersed after the evaporation of solvent. Literature Cited Asbeck, W. F., Van Loo, M., Ind. Eng. Chem.. 41 (7), 1470 (1949) Stieg, F. B., Off. Dig.. Fed. Paint Varn. Prod. Ciubs. 28, No, 379, 695 (1956). Stieg, F. B . , Off. Dig.. Fed. Paint Varn. Prod. Ciubs. 29, No. 388, 439 (1957). Stieg, F. B . , A m e r . PaintJ.. 43, 106 (1958) Stieg, F. B.. Off. Dig.. Fed. Paint Varn. Prod. Clubs, 31, No. 408, 52 (1959), Stieg, F. B.. Off. Dig.. Fed. SOC. Paint Techno/., 34, No. 453, 1065 (1962). Stieg, F. B., J. Paint Techno/.,39, No. 515, 703 (1967). Stieg, F. B.. J. Paint Techno/..41, No. 531, 243 (1969) Stieg, F. B., J. Oil Colour Chem. Ass.. 53, 469 (1970a) Stieg, F. B., J. Paint Techno/., 42, No. 545, 329 (1970b). Stiea. F. B.. Burns. D. F., Off. Dig.. Fed. Paint Varn. Prod. Clubs. 26, N i . 349,81 (1954). Stieg, F. E., Ensminger, R. I . , Off. Dig.. Fed. SOC. Paint Techno/., 33, NO. 438, 792 (1961).
Receiued for reuiew August 27, 1973 Accepted November 29, 1973 Presented at the Division of Organic Coatings and Plastics Chemistry, 166th National Meeting of the American Chemical Society, Chicago, Ill., Aug 1973.
