Langmuir 1991,7, 1815-1821
1815
Analysis of the Sensitivity of Surface Shear Instruments S. S. Feng,t R. C. MacDonald,t J. B. Ketterson,g and B. M. Abraham’*$ Department of Biochemistry, Molecular Biology, and Cell Biology and Department of Physics and Astronomy, Northwestern University, 2153 Sheridan Road, Evamton, Illinois 60208 Received January 30, 1991. I n Final Form: March 15, 1991 A cylindrically symmetric surface shear trough, utilizing a torsion pendulum, is widely used to probe the material properties of surface films. A systematic analysis of the sensitivity of such an instrument is presented. The performance is shown to be governed by four dimensionless factors: two geometric parameters involving the trough (the ratio of the wall to the rotor radius and the ratio of the fluid depth to the rotor radius) and two physical parameters (involving the bulk and surface viscosities and the Reynolds number of the substrate flow). To examine the effects of each of these four factors, a theoretical analysis is made of the effects of the depth and radius of the trough on both continuous and oscillatory rotor rotations. In each case the exact solution of the fluid flow problem is developed, from which the torque exerted on the rotor by the surface film and the surface velocity distribution are obtained. The effects of each of the four dimensionlessfactorson the performance of the surface shear trough are presented graphically. The sensitivity of the shear trough to the material properties of the surface film and to the driving frequency is discussed. Some ways to improve the design of surface shear trough instruments are suggested.
Introduction The surface shear trough is one of the oldest and most useful instruments for measuring the material properties of surface films in either steady or oscillatory motion.’” Generally, the surface shear trough employs a rotor with a “knife”edge (idealized to have zero thickness) that makes a line contact with the surface film. Using a theoretical hydrodynamic model, one may determine the surface viscosity of a film subject to a steady rotation or the complex (frequency dependent) surface viscosity of a film undergoing oscillatory rotory motion. The experimentally determined quantity is generally the torque exerted on the rotor by the motion of the surface film. The theoretical analysis is based on the solution of the substrate fluidflow problem; the effect of the adsorbed film is determined by the boundary conditions applied to the film-covered surface. The solution for steady motion in the idealized case of a ring of radius fi with zero thickness in line-contact with a plane semiinfinite surface film (i.e., in the absence of the effects of the wall radius and the depth of the trough) was given by several worker^.^*^-'^ The effects of trough dimensions on surface viscosity measurements were estimated for a trough with a finite radius of the trough wall” and a second knife-edgering coaxialwith the original ~ n e . ~ J * -The l ~ early studies usually involved only the t Departmentof Biochemistry,Molecular Biology, and Cell Biology.
Department of Physics and Astronomy. (1)Langmuir, I. Science 1936, 84,378. (2)Langmuir, I.; Schaefer, V. J. J. Am. Chem. SOC. 1937,59,2400. (3)Wazer, J. R. J. Colloid Sei. 1947,2, 223. (4)Brown, A. G.;“human, W. C.; McBain, J. W. J.Colloid Sci. 1983, 8,491. (6)Abraham, B. M.; Miyano, K.; Xu, S. Q.;Kettereon, J. B. Reu. Sci. Instrum. 1983,64 (2), 213. (6) Myers, R.J.; Harkins, W. D. J. Chem. Phys. 1937,5,601. (7)Goodrich, F.C.;Allen, L. H. J. Colloid Interface Sei. 1971,37(l), 68. (8)Goodrich, F.C.;Allen, L. H. J. Colloid Interface Sci. 1972,40 (3), 329. (9) Goodrich, F. C. In hogre88 in Surface and Membrane Science; Danielli, J. F., et al.,Ede.; Academic Press: New York and London, 1973; VOl. 7,p 161. (LO)Goodrich, F. C. Roc. R. SOC. London, A l%S, A310,359. (11)Lifshutz, N.;Hegde, M. G.; Slattery,J. C. J.ColloidInterface Sci. 1971, 37 (l),73. (12)de Bemard, L.Mem. Sero. Chim. Etat. 1956,41,287. (13)de Bernard, L. Roc. Int. Cong. Surf. Act. Subst,2nd, 1967,Z, 360. t
steady motion (continuous rotation). Recent studies involved forced oscillations,and only purely viscous surface films were considered.l5 It was concluded, qualitatively, that the wall and the bottom have little effect, so long as they are separated from the rotor by a distance of order of the rotor radius or larger, and that this kind of device is useful for determining the material properties of surface films over a wide range of viscosities. However, in the very small (surface) viscosity limit, the performance is significantly compromised by the finite thickness of a real knife edge. To date, a systematic analysis of the sensitivity of a surface shear trough with a knife-edged rotor has not been carried out for free oscillatory rotation, with or without boundaries. This is partly due to difficulties in solving the Navier-Stokes equations of the system for a transient-free oscillation with decay, especially in the presence of the trough boundaries. In the present study we systematically investigate the sensitivity of surface shear instruments. It is found that there are four dimensionless factors that affect the behavior of the system: two geometrical parameters (the ratio of the wall radius to the rotor radius and the ratio of the trough depth to the rotor radius) and two physical parameters (the ratio of the surface to the bulk viscosities and the substrate fluid flow Reynolds number). To separately study the effectsof these parameters, an analysis is made of four cases, namely steady and free oscillatory motions, each with and without trough wall and bottom effects. For the case of a steady rotation of the rotor (or the trough) or when the rotor is driven into a forced oscillation, the hydrodynamic equations reduce to a set of homogeneous Stokes equations. For the transient case (free oscillations with decay), where the rotor is part of a mechanical oscillator that is excited and subsequently decays due to the viscous drag of the trough, the equations are inhomogeneous. However, this transient case may be formally treated as homogeneous by introducing a complex frequency;the real part corresponds to the actual frequency ~~
114)Mannheimer.R.J.: Burton. R. A. J.Colloid Interface Sci. 1970. 32 (1);73. I
.
(15)Ray,Y.-C.;Lee,H.O.;Jiang,T.L.;Jiang,T.S.J.ColloidZnterface Sci. 1987,119,81.
0 1991 American Chemical Society
Feng et al.
1816 Langmuir, Vol. 7,No. 8, 1991
and the imaginary part corresponds to the reciprocal of the characteristic decay time. The axial symmetry of the problem makes it possible to apply the Hankel transformation to the equations of motion and to the stress-equilibrium boundary condition at the surface. In the limiting case when we neglect wall effects, the usual (infinite) Hankel transformation is applicable; when wall effects are considered, the finite Hankel transformation is employed. In applying the Hankel transformation to the stress-balance-boundary condition, it should be borne in mind that, although the velocity is continous everywhere on the surface film, there is a discontinuity in the velocity gradient at the line of contact between the surface film and the knife edge of the rotor. The jump in the velocity gradient across the contact line turns out to be proportional to the torque exerted by the surface film on the rotor. Using the above approach, we obtain the earlier solutions for the steady motion cases and solve the fluid flow problem for the cases of decaying free oscillations. The results for the torque exerted on a knife-edge rotor by the surface film and the associated surface velocity distribution (both deduced from the solution of the substrate fluid flow problem) are presented graphically, and we discuss the separate effects of the wall radius, trough depth, and the (complex) driving frequency. The sensitivity of the shear trough device to the material properties of the surface film for a given (complex)frequency is then discussed for both the steady rotation and oscillatory cases. Some suggestions for improving the design of surface shear instruments are given.
Solutions for Various Boundary Conditions We systematically investigate the surface shear trough for four cases: steady rotation, with and without trough wall and bottom effects (cases A and B, respectively), and the free oscillatory rotation, with and without trough wall and bottom effects (casesC and D, respectively). For cases A and B the earlier derived solutions are obtained; for cases C and D the boundary value problems for the substrate fluid flow in free oscillation (with decay) are solved. The viscoelasticity of the surface film is also considered. A. Steady Rotation for a Semiinfinite Fluid. Consider the case where the rotor continuously rotates with its knife edge contacting the surface film; this results in a steady rotation of the surface film and the substrate fluid caused by the combined effects of the fluid and surface film viscosities. The angular velocity of the rotor, o,is assumed to be small enough that the flow is laminar. Since the motion is steady, the viscosity of the surface film, p B ,is a real number. If a slow motion is assumed and the convective inertial forces are neglected (low Reynolds number flow), then the problem reduces to solving the continuity equation and the homogeneous Navier-Stokes equations div i, = 0
(la)
+ $725
(1b)
vp
=0
subject to the appropriate boundary conditions. Here 17 and p are the velocity and pressure fields and q is the fluid viscosity; p may be taken to be constant for the case of steady (or quasi-steady) motion. A cylindrical coordinate system (r,O,z) is taken. The origin is located at the center of the surface film and the z axis is positive downward. From the axial symmetry of the problem, it is apparent that i, has only azimuthal
components, i.e. V, = V, = 0 and V p V. Furthermore, V = V(r,z) only. Therefore, the equation of motion in the substrate phase becomes16
-
- -
In the absence of walls, it is required that V 0 as either r m or z m. We may therefore apply the standard (infinite) Hankel transform," namely
V(t,z)= rV(r,z)rJl(€r) dr
(3)
to eq 2 and make use of
toreduce that partial differential equation (inthe variables r and z ) to an ordinary differential equation (in z only)
The solution of this equation is straightforward. The boundary condition at the rotor knife edge is V(ri,O) = wri (6) and the stress equilibrium equation at the film-covered surface is
The standard Hankel transformation can also be formally applied to eq 7 by noting that the surface velocity field V(r,O) is continuous in the entire domain (0,m) but there is a discontinuity in the velocity gradient at r = ri. That is, although the requirement of Hankel transformation is not satisfied in the transformation domain (O,m), we can still multiply both sides of eq 7 by rJl(5r) and perform integration with respect to r from r = 0 to r = ri- and then from r = ri+to r = m, respectively. Thus, it can be found that
Further, it can be shown that the jump in velocity gradient at r = ri is proportional to the torque exerted by the surface film on the rotor
In this way, the dimensionle& expressions for the torque exerted by the surface film on the rotor, T* = P(X), and the velocity distribution of the surface film, V* = V* (r*,O), are given by
and (16) Landau, L. D., Lifshitz, E. M. Fluid Mechanics; Addison-Wesley Publishing Company, Inc.: Reading, MA, 1968, Chapter 11, p 6!. (17) Tranter,C. J. Integral Transforms in Mathematical Physics, 3rd ed.; John Wiley & Sone, Inc.: New York, 1966; Chapter IV.
Langmuir, Vol. 7, No. 8, 1991 1817
Serrsitiuity of Surface Shear Instruments
where Jl(s)is the first-order Bessel function of the first kind. The parameter X is defined by A =Vi/pa
(12)
where q (dyn s/cmz) and ps (dyn s/cm) are, respectively, the bulk and surface viscosities and ri is the rotor radius. The dimensionless torque and velocity, T* and V*, are related to the actual quantities, T and V , by
The solution for this case was given by Goodrich et B. Steady Motion with Finite Trough Dimensions. We define the radius and the depth of the cylindricaltrough as ro and h, respectively and apply the nonslip boundary conditions at the wall and the bottom of the trough V(ro,z)= 0 V(r,h)= 0 (14) In this case a finite Hankel tran~formation'~ is applied to the Navier-Stokes equation, eq 2, and then (formally) to the stress balance boundary condition at the surface film, eq 7 (there is a discontinuity in velocity gradient at r = ri and the jump across the ring is proportional to the torque exerted on the rotor). The finite Hankel transformation is based on the fact that any radial function can be expanded as a series of Bessel functions with arguments containing all consecutive zeros of the Bessel function.18 Following the same procedure as in case A and neglecting the mathematical details, and making use of the inverse theorem of finite Hankel transformation, we find the dimensionless torque, T*(A,ro*,h*),and surface velocity distribution, V*(r*,O),as 1 " tanh (€n*h*) [Jl(€n*/rO*) (15) - = 2 x T* n-i€,*IA + En* tanh (€,*h*)) Ji(€,,*) and
I'
the velocity in their work was nondimensionalized by T / 2 ~ r i (in q our notation). Consequently, the solution for the velocity still contains the torque as a scaling factor, and for its determination, the velocity boundary condition at r = ri, eq 6, had to be included. Our treatment, on the other hand, provides an explicit physical interpretation of the jump in the velocity gradient at the position of the knife-edged ring. Nevertheless, it is clear that the two treatments are mathematically equivalent. C. Oscillatory Rotation for a Semiinfinite Fluid. In the case of oscillatory motion the rotor oscillates while contacting the surface film. The decaying oscillations may be described by introducing a complex frequency, 52 = w + i / ~ . Here w is the oscillator frequency and T is a characteristic decay time. When studying oscillatory motion, we must regard the viscosity of the surface film as a complex quantity, pl = p1 - ipz, where pz and p1 represent the elastic modulus and the surface viscosity (loss modulus) of the surface film, respectively. For this case, the equation of continuity and the NavierStokes equations are given by div B = 0 vp
(19a)
+ qV2B = p-ai, at
where p is the mass density of the substrate fluid. As in the case of steady motion, due to the axial symmetry of the problem, the fluid velocity field, 0, and the surface displacement field, ii, have only azimuthal components in the cylindrical coordinates (r,O,z),i.e., B = B(O,u,O) and 0 = ii(O,u,O). In addition, u is a function of r, z, and t and u is a function of r and t . We write the time dependence of u and u in the form u = V(r,z)ei", u = U(r)ein* The Navier-Stokes equation then becomesls
(20)
We continue to have mixed boundary conditions on the surface film: the velocity at the rotor knife-edge is V(ri,O)= i52agi (224 and we require stress equilibrium on the surface film
here Jl'(s) denotes dJl(s)/ds and En* (n = 1, 2,3,...I are the consecutive zero8 of the Bessel function
J'(C,*)
=0 The dimensionless lengths are defined by
(17)
in eq 22a, a0 is the initial amplitude of the angular displacement of the rotor, i.e.
= a0ei*t (23) For the case where wall and bottom effects can be neglected, we require V 0 as r or z w; therefore, the normal Hankel transformation can be applied to both the equation of motion and the stress balance boundary condition (as in case A above). The dimensionless amplitudes of the torque, TO*= To*(A,Re), and surface velocity distribution, V*(r*,O),are obtained as +
This case was treated by Mannheimer and Burton.14A delta function strategem was utilized in their work to combine the mixed boundary conditions, eqs 6 and 7, into one. That single equation was then integrated from r = 0 to r = ro. This is actually equivalent to a piecewise application of the finite Hankel transformation from 0 to ri- and from ri+to ro (the finite Hankel transformation can only be applied if the function and its derivative are continuous in the transformation domain). Furthermore, (18)-, J.; Mullineux, N.Mathematica in Phyaica ondEngineering; Academic Press: New York, 1969; Chapter 111, p 127.
- -
and V*(r*,O)= T*$,
Jl(s)Jl(sr*)s ds [s2 + A(s2 + iRe)'/']
(25)
1818 Langmuir, Vol. No.8, 1991
Feng et al.
et- Im (pr#nh) a 10.
a
IO.
8. 6. 1. 4.
2. 0. 0.
c
20.
6.
7.
1.
0.4
Rei
-
ro’-ro/r,
PA IprPW)
Figure 1. Contour plots of the complex dimensionless torque
T* in the complexplane of the complex Reynolde number Re = pr?Q/s = Re1 + i R e 2 : (a) the amplitude constant and (b) the phase angle difference LYT = constant, where T* = T/2ria@&& = lT*l exp(iaT). These lots are from eq 24 for oscillatory motion with X = (1+ i ) X lo! For those values of Re = Re1 + iRe2 where the contours of I!PI = constant become dense, the instrument will be most sensitive to changes in the complex Reynolds number. By definition, Re is proportional to the complex frequence,Q = w + i / r ,where w is the oscillatorfrequency and T is a characteristic decay time. Parta a and b, therefore, also represent the (complex)frequency response of the (complex) torque for a given A.
where Re is a complex Reynolds number defined by
Figure 2. Contour plots of the dimensionless torque exerted on the rotor, T* = T/2rr&w = constant, in the ro*-h* plane for X = 101 from eq 15 for steady motion: (a) wall and bottom are near to the rotor and (b) wall and bottom are far from the rotor. The contours are almost vertical when h* > 0.5; hence the trough depth has a negligible effect in the experiment for h > rJ2, Note from part b that it is possible to find a series of combinations of h and ro which give the same dimensionless torque. Also, these two figures show that relative to the depth, the trough radius has a larger effect. The contours become denser when the trough
wall approaches the rotor.
nonslip boundary conditions a t the trough wall and bottom (as in case B), i.e. V(ro,z) = 0, V(r,h) = 0
Y = v / p being the kinematic viscosity of the fluid. The dimensionless complex amplitudes of the torque and the surface velocity, T* and V*, are defied slightly differently than that in eq 13 (the same notations T* and V* as used for steady motion are also used to refer to the same physical quantities for oscillatory motion). These different definitions imply that the complex torque and the surface velocity are both proportional to the initial value of the angular displacement of the rotor
Here TOis the complex amplitude of T , which we can write as
T = Toeint= ( T , + iT2)eint
The solution for this case has been obtained previous1y.lg The method involved applying the finite Hankel transformation to both the equation of motion and the stress equilibrium boundary condition on the surface film (including the discontinuity in the velocity gradient a t the rotor/film contact line, which is related to the torque exerted on the rotor). The expressions for the amplitude of the dimensionless complex torque, TO*= To*(X,ro*,h*,Re), and surface velocity distribution, V*(r*,O), are given by 1
tanh (8,*h*/r0*) X
Xr0*@,*
- &,*2 tanh (8,*h*/ro*)
(28)
with TIand TZreal. D. Oscillatory Motion for Finite Trough Dimensions. The hydrodynamic problem to be solved here is the same as for case C, but with the imposition of the
(19) Feng, S.5.;MacDonald, R.C.; Abraham, B. M. Longmuir 1991,
7,572.
Langmuir, Vol. 7, No. 8,1991 1819
Sensitivity of Surface Shear Instruments h'-h/ri
T*.T12xr+p.a
a
a
-3.
-2.
-1.
1.
2.
3.
4. IOPioA
h'-h/ri
>YtI
b 3.
Oa8
O+k
2.
0.2
1.
\
-3. 1.5
2.
2.5
ro'-fdfl
and tanh (b,*h*/ro*)
V*(r*,O)= 2 T 0 * C n-1
-1.
1.
2.
3.
4.
IOPioA
3.
Figure 3. Contour plota of the complex dimensionless torque !P in the ro*-h+ plane: (a) the amplitude I!Pl = constant and (b) the phase angle difference a~ = constant, where T+ = T/2ria,9-&1,0= I!Pl exp(iaT). These plota are from eq 29 for oscillatory motion with X = (1+ i) X 101 and Re = 0.6 + 2.0i. The conclusions are similar to those for parts a and b of Figure 2 except that the phase angle oscillates with increasing h at large r*. (D
-2.
Figure 4. (a)The dimensionless torque exerted on the rotor, !P = T 2rrtp,u, vs the relative viscosity X = qri/c(.plotted from eq
I
10 or steady rotation. It is evident from the figure that the surfaceshear trough is relatively insensitive for very smallsurface viscosities (very large A), since for X > 108 both the torque and the derivative of the torque with respect to X become quite small. (b)The sensitivity of the instrument, S(X) = dT+/(!Pdh), vs the relative viscosity X plotted from eq 33 for steady motion. For X < 10-3 (i.e., for very viscous surface films) the sensitivity of the surface shear trough (percent change in !P per unit change in A) is constant at roughly 80%. For X > 109 (Le., for very fluid surface films) the sensitivity of the instrument is very small,say, less than 1%.
X
Aro*&+ - tn*'tanh (&*h*/ro*) J 1 (€n* / ro*>J1(tn*r*/ ro* 1 [J[(€,*)I
'
(30)
where tn*(n = 1,2,3, ...) are again the zeros of the Bessel function, Jl(s),and the fin* (n = 1 , 2 , 3 , ...) are related to €n* by (&*)' = ((,,*)' + i(r0*)'Re (31) in which Re is the complex Reynolds number defined by eq 26.
Results and Discussion From the above analyses, it is clear that the behavior of a surface shear trough for steady or oscillatory rotation is completely specified by, at most, four dimensionless parameters. Two are geometrical parameters, the ratio of the wall radius to the rotor radius, ro* = ro/ri, and the ratio of the trough depth to the rotor radius, h* = h/ri* Two are physical parameters, the relative viscosity, A = qri/p, (where p, is real for steady rotation and complex for oscillatory motion), and a complex Reynolds number in the substrate fluid flow under oscillatory rotation, Re = pQri'/q. Thus, P = TY(ro*,h*,A) in steady motion and To* To*(ro*,h+,A,Re)in oscillatory motion, respectively. When ro* and h* go to infinity, the effects of the trough dimensions are neglected and cases B and D become
equivalent to cases A and C, respectively. To evaluate the effects of finite trough dimensions, the solutions for cases B (for steady motion) and D (for oscillatory motion) are appropriate; however, to analyze the roles of the relative viscosity, A, and the complex Reynolds number, Re, the much simpler solutions for the ideal cases A (for steady motion) and C (for oscillatory motion) can be employed. Complex Reynolds Number Effects. It is clear from the above analyses that the Reynolds number is irrelevant in the case of steady rotation. The dimensionless torque, the surface velocity distributions, etc., are independent of the rotation speed. The measured driving torque and the velocity distributions are simply porportional to the rotation rate, which may have any value (so long as there is no turbulence generated in the substrate fluid flow). In contrast to the situation for steady rotation, in oscillatory motion the complex Reynolds number is a dominant parameter and affects all results (although it must, of course, be small enough that the substrate fluid flow remains laminar). Assuming the angular displacement of the rotor is described by eq 23, the complex torque on the rotor can be written as !P = I!PI exp(i(Qt+ aT)) (32) where I P I is the amplitude and (YT is a phase angle difference; Le., the torque lags the displacement by aT.
Feng et al.
1820 Langmuir, Vol. 7,No. 8, 1991 lOPl0 h
a
1.
-
D,8 '
0.6
"
0.4 .'
0. -1.
0.2
"
-2.
0.5 -3.
-3.
-2.
-1.
0.
2.
1.
1.
1.5
2.
2.5
3.
3.
-2.5 -3.
1
av (in ndh*)
-3.4 -3.
I III I
\ -2.
-1.
I 0.
1.
2.
3. loo10 I
,
Figure 6. Contour plots of the complex dimensionless torque T+ in the complex plane of the relative viscosity, X = qri/p, = X I + iXa: (a)the amplitude lPl=constant and (b)the phase angle difference aT = constant, where P = T / 2 r i a & t J l = IPI exp(iaT). These plots are from eq 24 for oscillatory motion with Re = 0.5 + 2.0i. From these two figures it may be concluded that the surface shear trough becomes insensitive when both the real and imaginary components of the complex relative viscosity become large. In the region of XI < 101and A2 < 101,the contours for either the constant amplitude or the constant phase angle difference of the complex torque have a significant variation in contour density. Beyond this region the contours become widely spread, which implies a low sensitivity of the instrument. Figure 1shows contour plots of I P land CUTas functions of the complex Reynolds number (Re = Re1 + iRe2), generated from eq 24 for A = (1 + i) X lo2. For those values of Re where the contours of constant IPI become dense, the instrument will be most sensitive to changes in the complex Reynolds number. From its definition (eq 26) we can see that Re is proportional to the complex frequency, 52; therefore, Figure 1 also represents the (complex) frequency response of the (complex) torque for a given A. The Effects of Finite Trough Dimensions. Parts a and b of Figure 2 show contour plots of the dimensionless torque, P, in the ro*-h* plane when the wall and the bottom are far from and near to the rotor, respectively. They are obtained from eq 15 for case B (steady rotation) for X = 102. The contours are almost vertical when h* > 0.5; hence the trough depth has a negligible effect in the experiment for h > ri/2, Note from Figure 2b that it is possible to find a series of combinations of h and ro which give the same dimensionless torque. Relative to the depth, the trough radius has a larger effect. The contours become denser when the trough wall approaches the rotor. Parts a and b of Figure 3 show the contours of the constant I P I and CUTin the ro*-h* plane, respectively, obtained from eq 29 (case D for oscillatory motion) for A
Figure 6. Distribution of the dimensionless surface velocity V* = Viagin = IV*( exp (iav) along the dimensionless radial coordinate r* = r/ri. These figures are plotted from eq 25 for oscillatory motion with X = qrj/p, = (1 + i ) x 101 and Re = 1 + i: (a) IV*l vs r*. The amplitude of the dimensionless velocity increases from zero at the center of the surface film to 1 at the knife-edge of the rotor and then decreases to almost zero (3% of the maximum) at r = 3.0ri. There L a sharp peak at r = ri, and correspondinglya discontinuity in the velocity gradient (the slope of the curve). It is this jump in the velocity gradient which produces the torque on the rotor by the surface film. (b) a y vs r*. The phase angle difference is zero at the position of the knife-edgeof the rotor. The actual phase angle difference between the angular displacement and the velocity at the knife-edgering is (7r/2) + arctan (l/w). The phase angle difference increases from r = ri to r = 0 and beyond r = rj it increases nearly linearly with increasing r. The velocity always lags the angular displacement in phase (everywhereon the surface);the greater the distance from the knife-edge of the rotor, the larger the phase angle difference. = (1+ i) X lo2. The conclusions are similar to those for Figure 2 except that the phase angle oscillates with increasing h a t large r*. Qualitatively, the effect of adecrease in the trough depth is equivalent to an increase in the fluid viscosity and the effect of a reduction in the radius of the trough wall or an increase in the radius of the rotor are equivalent to incrementa in both the surface viscosity and the bulk viscosity. The effect of the trough radius is larger than that of ita depth; however, the rotor radius cannot be increased without limit, since ultimately turbulence must set in. Sensitivity of the Instrument. Figure 4a shows the behavior of the dimensionleas torque as a function of the relative viscosity A = vri/pfi obtained from eq 10 (case A for steady motion). It is clear from the figure that the surface shear trough is not sensitive for the measurement of very small surface viscosities (very large A). From the figure it is evident that, roughly for A > lo9,both the torque and the derivative of the torque with respect to A are small. A more accurate assessment of surface shear trough behavior is possible if we define the sensitivity of a surface
Sensitivity of Surface Shear Instruments shear trough instrument as the percent change in the torque P per unit change in the relative viscosity X
which can be deduced from eq 10. We plot the sensitivity, SIvs X in Figure 4b, from which we can conclude that for X < 10-3 (i.e., for very viscous surface films) the sensitivity of the surface shear trough is constant at roughly 80% (the increment in the measured torque caused by increasing the surface viscosity is 80% of the measured torque itself). For X > 109 (i.e., for very fluid surface films) the sensitivity of the instrument is less than 1%(the decrease in the measured torque caused by decreasing the surface Viscosity is less than 1%of the measured torque itself). It was previously concluded by Goodrich from a similar study of case A, that the surface shear trough could be remarkably sensitive for X up to lo2 (which, in the case of water as the substrate and with ri = 1cm, corresponds to p 8 = 1V dyn s/cm). There are, however, practical difficulties in making an ideal knife-edged ring. A real ring, with either a finite thickness or azimuthal departures from cylindrical symmetry, will have associated stresses that contribute to the measured torque. If the surface viscosity is very small in comparison to the bulk viscosity, the torque caused by the surface shear stresses becomes insignificant compared to that caused by the bulk shear stresses.9 However, it is clear from our study that this limitation of the surface shear trough is inherent since it becomes insensitive for X > lo3,no matter how perfect the knife-edge is made. From Figure 4 we see that both the dimensionless torque and its derivative with respect to X rapidly approach zero beyond X = l e , even if the ideal knife edge could be made perfect. For water and a rotor radius of 1cm, A > 103is equivalent to pa < 1od dyn s/cm. Usually, most synthetic and biological membranes have static surface viscosities of this magnitude. For example, for a red blood cell membrane, it ;s reportedme21th& ps = 0.6 X lo4 dyn s/cm. A similar study can also be made for the measurement (20) Chien, S.; Sung,K.-L. P.; Skalak, R.;Uaami, 5.Biophys. J. 1978, 24, 46.
Langmuir, Vol. 7, No. 8, 1991 1821 of the complex surface viscosity in oscillatory motion. Figure 5 shows contour plots of the amplitude and the phase angle difference of the complex torque, P,in the complex plane of A = XI + i X 2 obtained from eq 24 (case C for oscillatory motion). From these two figures similar conclusionsaa for the case of steady rotation can be drawn; the surface shear trough becomes insensitive when the real and/or imaginary componentsof the complex relative viscosity become large. We note that in the region XI < lo2and XZ < lo2the contours for either the constant amplutide or the constant phase angle difference of the complex torque have a significant variation in contour density. Beyond this region the contours become widely spread, which implies a low sensitivity of the instrument. Surface Velocity Distribution. Figure 6 shows the distributions of the amplitude and the phase angle difference of the complex dimensionless surface velocity given by eq 25 (case C for oscillatory rotation) for X = (1 + i) X 102 and Re = 1 + i. The amplitude of the dimensionless velocity (Figure 6a) increases from zero at the center of the surface film to 1at the knife-edge of the rotor and then decreases to almost zero (3% of the maximum) at i =3.0ri. There is a sharp peak at r = ri and, correspondingly, a discontinuity in the velocity gradient (the slope of the curve). It is this jump in the velocity gradient which produces the torque on the rotor by the surface film. The phase angle difference is zero at the position of the knife-edge of the rotor (Figure 6b). Since the dimensionless velocity is defined by V* = V/ia&, the actual phase angle difference between the angular displacement, a,and the velocity at the knife-edge ring is (?r/2) + arctan ( 1 / w ~ ) . The phase angle difference increases from r = ri to r = 0 and beyond r = ri it increases nearly linearly with increasing r. The velocity always lags the angular displacement in phase (everywhere on the surface) and the greater the distance from the knife-edge of the rotor, the larger the phase angle difference.
Acknowledgment. This study was supported by NIH Grants JX36634 and GM38244. (21)Feng, S. S. R8d Blood Cells Tank-Treading in a Shear Flow. Doctoral Thesis, Columbia Univereity, New York, 1988.
