DETERM~XATION O F LOSDOS-VAN DEI< I V A A L S
,Jiirie, 1962
CONSTASTS
1077
heat capacity can be calculated accurately from
TABLE V
COMPARISOX OF OBSERVED AND CALCULATED VAPORHEAT spectroscopic data. Although water does meet the requirements for a suitable reference substance for CAPACITY OF CARBON DISULFIDE vapor-flow calorimetry, and has been studied sucC,”, cal. deg.-l mole-’
cessfully,16 it has two disadvantages. First, the heat capacity of water vapor varies rapidly and 11.175 11.172 +0.003 325.65 non-linearly with pressure in the temperature range 11.565 + ,003 367.65 11.568 of interest because of association through hydrogen 11.883 - ,003 407.10 11.8PO bonding. Second, water has somewhat unfavorable 12.218 12.209 + ,009 453.35 boiling and condensation behavior related to the - ,025 12,484 32.509 502.25 high surface tension, which makes it difficult Lo a Five point Lagrangian interpolation of the values at even handle in a flow calorimeter. These two disadvantemperatures in Table IV. tages make water less suitable as a reference subsomewhat atypical of the usual organic vapors stance than carbon disulfide, which is a normal unstudied by vapor-flow calorimetry, but that prop- associated compound. erty may well be an advantage instead of a disBenzene is another useful substance for testing advantage for a reference substance. The low the accuracy of vapor-flow ca10rimeters.l~ It is speciJc heat magnifies the I C ~ / term F ~ in eq. 3 that is more convenient to work with than either carbon necessary because of the temperature profile wit’hin disulfide or water, but the heat capacity of benzene the calorimeter. If accurate results are obtained cannot be calculated from spectroscopic data with with carbon disulfide, for which the term is magni- as high accuracy as for the two simpler molecules. fied, more confidencc can be placed in result’s ob1. P. McCullough, R. E. Pennington, and G. Waddington, tained with typical organic vapors for which thc J . (16) A m . Chem. Soe., 7 4 , 4439 (1952). term is small or negligible. (17) D. W. Scott, G. Waddington, J. C. Smith, and H. M. €Tuffinan, Water is the only other liquid for which the vapor J . Chern. Phys., 16, 565 (1947). T , OK.
Obsd.
The0r.Q
Diff.
ON THE DETERRTINATIOS OF LONDON-VAN DER WAALS COSSTANTS FROM SUSPENSION AND EMULSION VISCOSITY AKD SURFACE ESERGY DBTA BY THOMAS GILLESPIE AND RALPHM. WILEY Physical Research Laboratory, T h e Dow Chemical Company, Midland, Michigan Receaued December 7 , 1981
A method has been developed for determining the London-van der Waals constant and the minimum distance of separation between emulsion drops and suspension particles frcm rheological and surface enerpy data. Data of Albers and Overbeelr for W/O emulsions and new data for nearly spherical and irregular solid particles in oil have been treated by this method with results which are directly opposite to the conclusions of Albers and Overbeek. The alternative theory suggests that the London-van der Waals constants are of the same order as commonly used in aqueous media, emulsifier layers enter in, and the minimum distance of separation is small and that even with “spherical” particles there can be a finite area of contact.
I. Introduction The forces between suspension particles and between emulsion drops usually are a complex function of electric double layer repulsion, Born repulsion, London-van der Waals attraction, and short range attraction such as hydrogen bonding. When oil is the continuous phase there is reason to believe that in some cases the force is due largely to London-\-an der Waals attraction and Born repulsion.1j2 In such simplr cases one might hope to measure these forces by appropriate rheological measurements and in this way obtain a check on various theories for the non-Xewtonian flow of suspensions and emulsions as well as data of intrinsic fundamental interest. In what follows, it is shown that there are two cases. In one of these the particles or drops may be considered to be in “point contact” and we have developed a method of determining the London-van (1) W. Albers a n d J. T h . G. Overbeek, J. CoZZozd Sei., 1 4 , 510 (19,59) (2) tV. .klk!is &nd J. Th. C. Overberk. zbzd., 15, 589 (1960).
der Waals constant and the minimum distance of separation for this type of system. In the other case, there is a finite area of contact and this complication prevents the separate determination of these quantities and thry can only be estimated approximately.
11. Theoretical Considerations We have been estimating for some time the energy of interaction between colloid particles in shear-thinning systems using a rheological theory based 011 the arguments of Goodevr.3-5 In this theory the work done in shearing a suspension is considered to be due to the sum of hydrodynamic effects (such as that due to the disturbance of flow caused by the presence of susprnded material) and the work necessary to rupture links which are continually re-forming under shear. When the shear rate is large this theory leads to the relation3-5 (3) C. F. Goodeve, Trans. Faraday Soc., 86, 342 (1939) ( 4 ) T. Gillespie, J. Collozd Sca., 16, 219 (1960). (5) T. Gillespie, J . PoZyme7 Sci., 46, 383 (1960).
S = p*G + CI mation, we may use the corresponding expression (1) where S and G are the shear stress and shear rate, for flat parallel plates respectively, p* is the “Yewtonian viscosity,” (10) and C1 is the ultimate dynamic yield value. The suspensions examined in the present work where u is the area of contact. were Newtonian up to a critical concentration at We cannot solve for A and h,, as we did in the which they became non-Newtonian. We assumed simpler case by making use of eq. 5 . However, we all links were effective and from previous ~ o r k ~ can - ~ solve for u. The result is this leads to g = -
&here E-4 is the energy of interaction and Ki is the rate constant for collisions at unit shear rate in the absence of Brownian movement. N is the number of particles per cc. and L T o is the number of particles per cc. a t the critical concentration. I n the interests of clarity, memay take K1equal to its value for a suspension of uniform particles, i.e.
where a is the particle radius6 By measuring the ultimate dynamic yield value, C1, at various concentrations the energy of intercan be determined using eq. 2 and 3. action EA& From Bradley’s early work7 and subsequent work by HamakerYwe may write as an approximation for identical spheres in point contact (4)
A is the effective London-van der Waals constant and h, is the distance corresponding to the maximum energy which commonly is called “the distance of minimum separation.” From the work of Bradley7 and Hamaker* it can be shown quite easily that
+ Y Z Z- 2~12)
A = 96~h,,,~(r11
(5)
where yll and ya2are the surface ciiergy of the dispersed phase and the continuous phase, respectively. y12is the surface energy descriptive of the interface between the dispersed and continuous phases. These surface energies may be measured directly or with the help of the rclatioii Y l l = TIL
+
cos 0
(6)
- 2712
(9)
Ye2
where e is the contact angle. Combining eq. 4 and 5
J+ herr
6 = YI1
+
Y22
Using eq. 7 and 8, the effective London-van der Waals constant and h, can be determined from the value of EA determined from flow measurements and the value of 6 from surface measurements. If there is a finite area of contact between particles eq. 4 would have to be modified. As an approxi(6) M. yon Smoluehowski, Z . physzk. Chem. (Leipaig), 92, 129 (1917).
(7) R. S. Bradley, Phzl. MuQ., 13, 853 (1932). (8) € I . C.Hamaker, Physzca, 4,1058 (1937).
EA 26
The value of u determined from eq. 11 using flonand surface measurements can he compared with direct microscopic observations. 111. Experimental TKOtypes of solid particles were chosen for suspension vkcosity measurements. One of these was Neo-Novacite silica obtained from RIalvin Minerals Company, Hot Springs, Arkansas. The particles were typical of silica, being like small pieces of ground glass, i.e., roughly spherical with flat portions. By microscopic examination we found the number average particle radius was 3 p , the average axial ratio was 1.27, and the average length of the apparent line of contact of particles a t rest was 3.2 p . The second suspension contained crosslinked polystyrene beads (X .P. S.). They were made by polymerizing an emulsion of styrcne with 8% divinylbenzene in water using a very small particle size silica as emulsifier. The number average radius was 1.5 p . The beads had a considerably narrower size distribution than the Xeo-Xovacite silica and, although they were not perfect spheres, the axial ratio was small for most of them. To make the suspensions, the particles were dispersed in dioctyl phthalate using a high speed laboratory mill. No emulsifiers were used. The flox properties were assessed with a Ferranti-Shirley cone arid plate viscosimeter at 2 5 ” . In each test the instrument wab run at a shear rate of 670 set.-' for half an hour before taking measurements. Typical shear rate-shear stress c u v e s are shown in Fig. 1 and 2. The surface properties of emulsion I V of Albers and Overbeek2 were assessed with a du Nouy ring balance. The emulsion contained 1% sorbitan monostearate and 207, water dispersed in a mixture of carbo: tetrachloride in benzcne of the same density as water at 25 .
IV. Discussion Table I summarizes the data obtained with the suspensions and Emulsion IT7 of Albers and Overbeek. For their other emulsions there were not enough data to determine the ultimate dynamic yield value with confidence. I n Table I, a refers to the most frequent particle or drop radius. E A was determined from yield value measurements and eq. 2 and 3. For the emulsion data N o was taken to be For the silica the surface energy value was obtained from Bradley’ while for the polymer beads it mas obtained from measurements of the surface tension of the monomer mixture. The London-van dcr Waals constant and h,, in the case of the emulsion were determined by using ey. 7 and 8 which assume “point contact.” When these equations were applied to the suspension data the calculated values of h,, were of the order of 1200 k , which is very large, hence we calculated the area of contact from eq. 11. Using eq. 10, we calculated the value of hm for values of A in the range 1 0 - 1 3 to 10-1’ erg as suggested by quantum theory. The figures in the last row in Table I indicate that the area of contact is relatively small. Assuming that the contact area was circular, we calculated the apparent area of contact from the value of 3.2
June, 1962
DETERMINATION OF LONDO~X-VSN DER WAALS CONSTAXTS
1079
for the average apparent length of the line of contact. On this basis the contact area was estimated to be about twice the figure given in Table I. TABLE 1 SUMMARY OF THE EXPERIMENTAL AND DERIVED DATA Emulsion I V (ref. 2)
3.5 10
31.40 27.1 4.3 50 0 3.9
x
10-12
2.3
Silica D.O.P.
3 3160 33.8 30.4 3.4" 57.4 2.76 10-'2-10-" 0.8-2.4
Area of contact Area of av. 0 0.024 particle a Obtained from eq. 6 using cos 0 equal to 1.
X.P.S. D.O.P.
1.5 400 32 30.4 1.6" 59.2 0.34 10-1"10-11 0.8-2.4 0.012
The results we have obtained are realistic and suggest that the technique outlined above should be of considerable value, particularly for systems in which there is no complication due to an area of contact. For more precise work, one mould have to work with systems with a much narrower size distribution or modify the equations to take into account known effects of size distribution on particle adhesion7 and possible effects on the rheological kinetics.6 It should be emphasized that the arguments presented above ca,nnot be used without serious modification when electric double layer repulsion, etc., are not negligible. Finally, we believe that the present analysis is preferable to the treatment of emulsion viscosity data presented by Albers and Overbeek.2 They outlined a method of determining A/hm2by equating the force to pull a doublet apart to the Stokes drag of the continuous phase at the point when the S ws. G plot becomes linear. Their values of A/hm2are much smaller than those calculated from eq. 5 . When applied to our data their technique leads to the same conclusions as for W/O emulsions, namely, that the London-van der Waals constants are very small or h, is larger than we realize. As an explanation for their results they took A ta be 4 X
2
I
0
3
5
4
6
S x I O 3 (Dyn8s/Crn21.
Fig. 1.-The relation between shear stress and shear rate for silica dispersions in dioctyl phthalate. The numbers refer to the fractional solids volume. 500
400
3 00
G ISec-? 200
I00
0
0
I
3
2 Sx
lo3
4
5
6
(Dynes/Cmzl
Fig. 2.-The relation between shear stress and shear rate for crosslinked polystyrene beads in dioctyl phthalate. The numbers refer to the fractional solids volume. l O - l 5 erg, regardless of the emulsifier, and h, of the Le., about twice the length of an order of 40 8., emulsifier molecule. The present work would suggest that the emulsion drop plus its emulsifier layer acts as an entity and h, is the distance between the outer portions of the emulsifier layer.
