MEASURING VISCOSITY OF SEMISOLIDS 245 y dU x '•yx X Figure 3. Diagram illustrating shear rate based on plug deformation, producing a change of angle 0. in a two-dimensional system this angle would have doubled with the same magnitude of applied stress. Thus, in our case, the rate of change of angle 0 is: or dO 1. dt - 2 d0 •-• = 3e,• (3) EQUATION FOR THE SYSTEM If three identical cylindrical sample plugs of radius r, height H, and volume V, are pressed between two lubricated plates by a normal force F, then the shear stress for each sample becomes 're.= •Syy = cry,,, = • = • V (4) where (V/H) refers to the cross-sectional area of the sample. For shear rate, the angle 0 in the coordinate form (Figure 1) gives (substituting r with X/•/XH for the cylindrical geometry) tan = - = ß r •

246 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS which after differentiating provides From simple coordinates (Figure 1) =•] • 'dH (•) r2 + H2/4 4V + XH 3 sec 2 = = r 2 4V substituting in this Equation 5 and rearrangement provides HX/2 d0 = 6•' ß dH 4V + XH 3 (5) dO H •/2 dH or 2'--= 12X/-•-V' '-- (6) dr 4V + XH 3 dr which is equivalent to/• by Equation 3. Substituting Equations 4 and 6 into Equation 1 and using (-)ve sign for compressive stress yields 2 FH H 1/2 dH = •1' 12X/•' '-- 9 V 4V + XH 3 dt Designating the quantity 4V/X = a (constant for a given sample size), and rearrange- ment gives fo r F fi•,rl •_( 1 ) dH -- ß ß . 54• dt = HX/2 (7) 1 + H3/a The above expression may be integrated between the limits of H o (sample height at time 0) and H (sample height at time t) with the use of the following series expansion: 1 - 1 - x + x 2 - x 3 -I- ........ l+x such that H 3 --lx= a or H3X H3X 4V 4Xr2H or H 2r which provides where K 1 = (Ho 27v F 7a ] 7a/1 -- • (8)