Langmuir 1991, 7, 1281-1286
1281
Deformation of a Viscoelastic Droplet in an Electric Field. 1. Aqueous Cetyltrimethylammonium Bromide-Sodium Salicylate Solution in Poly(dimethylsiloxane) Keiichiro Adachi, Miyuki Tanaka, Toshiyuki Shikata, and Tadao Kotaka' Department of Macromolecular Science, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan Received September 11,1990. I n Final Form: January 14, 1991 Deformation of viscoelastic droplets in an alternating current (ac) electric field of 60 Hz was studied on aqueous solutions of cetyltrimethylammonium bromide-sodium salicylate (CTAB-NaSal) complex suspended in poly(dimethylsi1oxane)(PDMS). Under an ac field of ca. 106 V m-1, a droplet with the initial radius b deformed into an ellipsoid with the major and minor semiaxes X and Y, respectively. The degree of deformation D = (X- Y)/(X+ Y) vs the logarithm of time curve was a double-step sigmoidal shape characterized with, at least, two strengths and two retardation times: The fast process was attributed mainly to viscoelastic deformation due to the balance of the electric stress against the elasticity of the droplet, while the slow process was governed by the balance against interfacial free energy ylz. The time dependence of D was simulated with a phenomenological model composed of parallel combination of a Maxwell element representing the droplet viscoelasticity with the plateau (shear) modulus GN' and the relaxation time T~ and a Voigt element involving y l z / b and ~2 of the medium. With this model, the fast process of the deformation (or recovery) behavior may be interpreted as being governed essentially by the viscoelasticity of the droplet. With increasing time after application (switching off) of the field, viscoelastic relaxation took place within the droplet, and the following slow deformation (or recovery) process was thus governed by the same mechanism prevailing in the behavior of a viscous droplet in a viscous medium reported previously.
I. Introduction A droplet suspended in a dielectric medium deforms into an ellipsoid under a relatively low electric field but bursts under a high electric field.1-'3 The phenomena have been known for a long time, since as early as 1600.' However, we first reported that the deformation process of viscous droplets (suspended in viscous media) conforms to a single retardation time equation with the retardation time T = (TI + ~2)b/y12,where 7 1 2 is the interfacial free energy and and 772 are the viscosities of the droplet and the medium, re~pectively.'~J~ We also found that the recovery process is represented by a single relaxation time equation with the relaxation time equal to the retardation time T . An obvious extension of these studies is to examine the behavior of viscoelastic droplets suspended in viscous or viscoelastic media. The problem, however, has been left untouched. To study this problem, we need to have droplets with a suitable elastic modulus. We anticipated that since the recovery force of a deformed viscous droplet with the original radius b is proportional to 712/b, that is the order of 5 to 100 Pa at most, droplets with the modulus of this order of magnitude would exhibit deformation behavior different from purely viscous droplets. Recently Shikata (1) Gilbert, W. de Magnete; 1600; Book 2, Chapter 2. (2) Zeleny, J. Phys. Reu. 1917, IO, 1. (3) Adameon, A. W.Physical Chemistry of Surfaces, 4th ed.; John Wiley: New York, 1982. (4) Taylor, G. I. Proc. R. SOC. London, A 1934, 146, 501. (5) O'Konski, C. T.; Thatcher, H. C. J. Phys. Chem. 1953,57,995. (6) O'Konski, C. T.; Harris, F. E. J. Phys. Chem. 1957,61, 1172. (7) Garton, C. G.;Krauski, Z.Proc. R. Roc. London, A 1964,280,211. (8) Allan, R. S.; Mason, S. G.Proc. R. SOC. London, A 1964,267,45. (9) Tona, S.; Cox, R. G.; Mason, S. G. Philos. Trans. R. SOC. London, A 1971,269, 295. (10) O'Konski, C. T.; Gunter, P. L. J. Colloid Sci. 1964, 10, 563. (11) Moriya, S.; Adachi, K.; Kotaka, T. Polym. Commun. 1985, 26, 235. (12) Moriya, S.; Adachi, K.; Kotaka, T. Langmuir 1986, 2, 155, 161. (13) Nishiwaki, T.; Adachi, K.; Kotaka, T. Langmuir 1988,4, 170.
et al.14-16reported that aqueous solutions of cetyltrimethylammonium bromide (CTAB) complexed with sodium salicylate (Nasal) can be modeled by a Maxwell model with a single relaxation time T~ and a plateau (shear) modulus GN' of this order of magnitude. Interestingly, the 7, and GN' of the CTAB-NaSal solutions are easily controlled by changing the CTAB and Nasal concentrations. We thus employed CTAB-Nasal solutions suspended in poly(dimethylsi1oxane) (PDMS) as a model system and studied their deformation and recovery behavior under the electric field. 11. Theory Deformation of Viscous Droplet. First, we briefly summarize our previous work on viscous droplets. A spherical droplet of the initial radius b deformed into an ellipsoid of the major and minor semiaxes X and Y, respectively, is characterized by the degree of deformation D defined as4
D = (X-Y)/(X+ Y)
(1) where X is taken in the direction of the field. From the balance between the electric stress, the hydrodynamic stress, and the interfacial free energy 712, the time dependence of D is described with a single retardation time equation12
D(t) = DV.J - exp(-t/rV)l (2) where Dvmis the equilibrium degree of deformation and TV is the retardation time. According to Torza et al.? the equilibrium degree Dv.. of deformation was given by D,, = 9t,,K2bE2/16ylz (3) where €0 is the absolute dielectric constant of vacuum, K2 (14) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987,3, 1081. (15) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988,4, 354. (16) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1989,5, 398.
0 1991 American Chemical Society
1282 Langmuir, Vol. 7, No. 6, 1991
Adachi et al. For an elastic droplet, the dashpot A should be replaced by a spring A, as shown in Figure 1B. From eq 8, we expect the spring constant GAcould be 5G/4. However, we omit the factor 5/4 for the sake of simplicity. Then the retardation time TG for this model is given by
A
B
C
Figure 1. Models for a viscous droplet suspended in a viscous medium (A), an elastic droplet in a viscous medium (B), and a viscoelastic droplet in a viscous medium (C).
is the relative dielectric constant of the medium (and K1, that of the droplet was assumed to be K1 >> Kz), and E is the electric field strength. We found that the retardation time TV, on the other hand, is given by12
(4) 7v = (q+ B712)b/712 where CY and fl are constants. In our previous study,2 we found theoretically CY = = 1,but the observed retardation time appeared to conform better to eq 4 with CY = 2 and 0 = l.13 However, since the scattering of data points in the previous experiments12J3 was fairly large, here we simply express TV as (5) 7v = (711 + 712)b/712 We also found the the recovery process Dhck(t)from an ellipsoid of the degree of deformation D(t0) to the original sphere of the radius b conforms to a single relaxation time equation as
Db&(t) = D(t0) eXP(-t/Tback) (2') where t is the time elapsed after to and the relaxation time TW for the recovery process is practically equal to TV of the deformation process.11J2 Elastic Deformation. Next we examine deformation of an incompressible elastic droplet with shear modulus G under an electric field. If the droplet is uniformly (affinely)elongated into an ellipsoid, the elastic strain energy WG will be given by
(9) = q2/(G + 712/b) Since we have assumed a perfectly elastic droplet with zero internal friction, the TG reflects only the medium viscosity 712. In the case of a viscoelasticdroplet suspended in a viscous medium, the droplet undergoes stress relaxation during the deformation process. Then the spring GAin the model B should be replaced by a generalized Maxwell model. However, for CTAB-Nasal droplets that have virtually no distribution of relaxation times, a Maxwell element with GAand VA, as shown in Figure lC, is probably good enough. The deformation profile D(t) for this model C is given by solvinga second-orderdifferential equation shown in the Appendix. The solution D(t) after applying a constant stress a t t = 0 is conveniently cast into the following form that involves fast and slow processes 7G
D(t) = DF[1 -eXp(-t/T~)] + Ds[1 -eXp(-t/Ts)] (10) where TJ (J = F for the fast mode; and J = S for the slow mode) is given by 75
27m72
= (Tm
+ T v ) f [(Tm + 7 ~ ) ~ - 4 7 , 7 2 ] ~ ' ~ (11)
+ sign for TF and - sign for and the Tm equals (=V1/GNo), 71 equals VA/GB (=Vlb/712), 72 equals ~ B / G (=712b/712), B and (71 + 72) equals the relaxation time
with
TS,
VA/GA
TV of a viscous droplet defined by eq 5. On the other hand, DF and Ds are
DF = [(Tm
- T F ) / ( ~ s - 7~)lDv..
(12)
Ds = [(Ts - T m ) / ( 7 s - ~F)IDV, (13) For the recovery after deformation up to the time to, the D(t) is given by
WG(32*b3D2/9)[G + (4/5)7/b121 (6) where the second term in the right-hand side of eq 6 represents contribution of the interfacial free energy as calculated by O'Konski and T h a t ~ h e r .They ~ also calculated the electric energy Wel for an ellipsoidal droplet with small D (an an exceedingly high K1 over K2 of the medium: K1 >> K2) and derived the equation
D(t) = DF[l- eXp(-to/7~)]exp(-t/TF) + Ds[l - exp(-t0/rs)l exp(-t/Ts) (14) where t is the time after switching off the electric field. We note that when VA (-a) >> BB, the TF for the fast mode reduces to eq 9, but the TQfor the slow mode reduces to
we,= - 2 ~ t , , ~ ~ b+~8D/5)E2 (1
(15) 7S = Tm + 91b/Y12 that is virtually infinity (7s m) and model C reduces to model B. On the other hand, if TA (-4) T, where stress relaxation takes place, it behaves as a viscous droplet. Obviously when Tm is short, the fast mode is unobservable and the deformation curve becomes single step. Figure 3 shows log ( D / b P ) plotted against time t for recovery processes observed for the same systems shown in Figure 2 after the deformation process has reached an equilibrium. Some of the curves have a break and two straight line portions, indicating existence of fast and slow recovery processes. Magnitude of Deformation. The values of D, (=DF Ds)and DFare determined from the deformation curves shown in Figure 2 and compared with the theoretical values of Dv,and DF by using eqs 3 and 12,respectively, with the necessary quantities listed in Tables I and 11. The results are listed in Table I11in which the reduced degrees of deformation d defined by d = D / b P are listed. We also compare the observed D, with the calculated values of Dv, in Figure 4. We see that the observed D, values are by a factor of 1.3to 1.5 larger than the calculated Dvm values. The solid line in the figure represents Dm =
+
1.42Dvm.
To explain this discrepancy we estimated possible experimental errors involved in the field strength E, the droplet radius b, and the interfacial free energy 7 1 2 to be
Adachi et al.
1284 Langmuir, Vol. 7, No. 6, 1991
-
I
CTAB NaSai(X,Y)/PDMS-lO CTAB NaSaI(005.0D5) 0 CTAB NaSal(0075,008) A CTABNaSal(O05.006) A CTABNaSal (0075,009) 3 - B CTAB NaSaI (0 05.03) n CTAB Nasal (0075.03) e CTAB Nasal (0 05.04) 0 CTAB Nasal (0075,O 5 )
-10
i
0
50
100
us
Figure 3. Recovery process of the same droplets shown in Figure 2. Table 111. Reduced Degree of Deformation Equilibrium. sample code CTAB (0.05, 0.05) CTAB (0.05, 0.06) CTAB (0.05, 0.3) CTAB (0.05, 0.4) CTAB (0.075, 0.08) CTAB (0.075, 0.09) CTAB (0.075, 0.3) CTAB (0.075, 0.5)
observed dF d, 4.65 38.1 6.95 31.3 7.51 37.6 26.9 4.52 43.9 3.30 41.2 32.3 28.5
dF 5.99 5.82 3.50 0.14 3.48 2.77 2.36 0.22
aP. and d, at
theoretical* dF" dv- dv-" 6.34 28.9 41.2 6.15 22.8 32.6 3.48 24.2 34.5 0.14 21.5 30.8 3.60 29.9 42.7 2.84 26.8 38.3 2.39 24.5 35.0 0.22 16.0 22.9
a The reduced degress d = Dl(bE2) are given by unig of 10-lo V-* m. The theoretical values with superscript "corr" are calculated with corrected interfacial free energy yE.
*
.
I
A CTAB kaSol(005 CSjIPDMS-10 0 CTABNaSa1(0075 CsjlPDMS-10
8
4t
,(talc) F i g u r e 4. Plot of D, against Dv. calculated with the physical quantities listed in Table I.
as much as 3 ?6 , 3'36, and 10'36 , respectively. They are far too small to account for the discrepancy. Two sources of errors are considered to account for this discrepancy. One is a possible change in the interfacial free energy ylz induced by the electric field. The other is the dielectric heating by the ac field. The former phenomenon was
reported on the interfacial free energy between mercury and electrolyte solutions as due to a structural change of the electric double layer at the mercury-solution interface.31~' We expect that a similar effect of the electric field might change the structure of the present CTABNasal solution/PDMS interface so that the 7 1 2 might also change depending on the field strength. However, it is necessary to test this possibility by experiments since the electricfield strength for the interface of the present system is expected to be much lower than that in the double layer of the mercury-solution interface. As to the second possibility, we previously estimated the rate of the temperature change by an electric field to be ca. 0.01 K/s.13 Thus we estimate the change in temperature is less than 5 K. Such a temperature rise will not cause the change of 7 1 2 as much as 40% . We thus estimated the interfacial free energy 7 1 2 ~ corrected for the effects of the electric field to be 0.70712 assuming the ratio DmlDVm is totally due to the change in 7 1 2 by the electric field. We then recalculated the magnitude of deformation dFCon(=DFCon/bE2)and dvmCOn using eqs 12 and 3 with the reevaluated values 7 1 2 ~as mentioned above. The results are also compared in Table I11 with the corresponding observed values of dF and d m . We see that the agreement between the observed and recalculated values is reasonably good. Retardation Times. Parts A-C of Figure 5 show the dependence Of T F and 7s on the viscoelasticrelaxation time T~ of the droplet for CTAB-NaSal(O.05, Cs)/PDMS-lO, CTAB-Nasal (0.075, Cs)/PDMS-lO, and CTAB-Nasal (0.75, Cs)/PDMS-3systems, respectively. The TF(D)and Q(D) are the retardation times determined from the deformation curves by assuming the deformation process conforming to eq 10, and reading the time t at which D(t) reaches 0 . 6 3 0 ~and 0.63Ds + DF, respectively. The TF(R) and TS(R)are, on the other hand, the relaxation times for the recovery process determined from the slope(s) of the recovery curves, log (Db-lG2) versus t shown in Figure 3. For the PDMS-3 system, however, we could not obtain reliable deformation curves, so that Figure 5C shows only the relaxation times determined from the recovery curves. As seen in Figure 5, both 7s(D) for the deformation process and q ( R ) for the recovery process increase with increasing 7111 and hence with r]l. The behavior was common for TV of viscous droplets.12J3 We see that the Q(D) appears to be slightly longer than the 7s(R), especially when 7, is long. The ratio of 7s(D) to rs(R)for CTAB-NaSal(O.O5,0.05)/PDMS-10 is roughly 2 and that for CTAB-Nasal (0.075, O.OS)/PDMS-lO, 1.6. On the other hand, the TF were observable only for the solutions with relatively long 7131, as anticipated from eq 11. Both are nearly independent of T,, and thus of TF(D)and TF(R) 71 in the observable range of T ~again , as anticipated from eq 11. However, we see a surprisingly large discrepancy between the two values: The TF(R)are approximately 10 times longer than the TF(D),which in turn agree rather well with the values calculated by eq 11with the corrected values of 7 2 . The observed characteristic times are compared with the model values calculated with eq 11. To be consistent with the speculation on the field dependence of 712 to explain the discrepancy between the observed and calculated Dvmvalues, we calculated the model values of 7 s and TF in two ways: ~J"(D) (with J = S or F) for the deformation process was calculated with the corrected interfacial free energy 71zE and SJ(R) for the recovery (17) J. 0'.M. Bockris, M. A. Sac. London, A 1963,274,55.
V.Devanathan, and Muller, K. R o c . R.
Langmuir, Vol. 7, No. 6, 1991 1285
Figure 5. Retardation times of the fast and slow processes TF(D)and TS(D)for the deformation (D)and those (relaxation times), TF(R)and TS(R),for the recovery (R) process plotted against the viscoelastic relaxation time T~ of the drop phase. Solid lines indicate and TF" calculated with YE and dash-dot lines indicate times (relaxation times) calculated with the corrected retardation times the observed YIZ. (A) CTAB-NaSal(O.05, C,)/PDMS-lO system; (B) CTAB-Nasal (0.075, Cs)/PDMS-lO system; and (C) CTABNasal (0.075, C,)/PDMS-3 system. In Figure 5C, T F and TF" are not distinguishable on the figure. process was calculated with the observed 7 1 2 , because the deformation proceeds under the imposed electric field but the recovery process proceeds under no field. The solid calculated ) with TI^^, lines indicate rgeon(D)and T F ~ ~ ' ( D and the dash-dot lines indicate TQ(R) and m(R) calculated with 7 1 2 . Comparing the results, we notice that, although the data points are scattered, the values of 7seorr and TS calculated ) TS(R), with eq 11appear to roughly agree with T S ( D and respectively. The slight difference between them might reflect that the effects of the electric field on 7 1 2 is qualitatively in agreement with our speculation that the 7 1 2 under the electric field is lower than that in the absence of the field. However, the observed TS(D)/TQ(R) ratios are still somewhat larger than the ylzE/ylz ratio. As is seen in parts A and B of Figure 5, the observed TF(D)for the deformation process agrees well with the calculated T F ~ O " . However, the fact that the observed 7F(R) are ca. 10 times longer than the observed ~ F ( Dis) rather unexpected. Moreover, the calculated 7F which, we expect, might be equal to TF(R)is smaller than the calculated 7FCorr(*~F(D)). This trend is opposite to the observation. The assumed field dependence of 7 1 2 is too small and just opposite in tendency to explain the 10times difference in the observed TF(R)/TF(D).We speculate that GN' might also gradually decrease under the electric field, and thus not only 7 1 2 but also GN' might be different between for the deformation and recovery processes. Of course, we must check this point through some other appropriate experiments. Reexamination of Deformation Behavior. Since we were able to reasonably well correlate the deformation profiles with the physical characteristics of CTAB-NaSal solutions incorporating their viscoelastic properties, we now reexamine the two-step deformation behavior by closely comparing the observed and the model C deformation curves for CTAB-NaSal (0.05, Cs) suspended in PDMS-10. Figure 6 shows the result. For simplicity, the data are plotted in the formD(t)/D, versus log (t/s). The solid curves show the model C curves calculated by eq 10 with the appropriate physical quantities listed in Tables I and I1 together with the corrected ylzEvalues: The dashdot lines represent model DF(t)/DVm curves, the dash-
I
I
!
CTAB: NaSal(0.05,Cs) o A
&=0.05
cS=0,06
-I
Qk-/' 0I og ( f /Is ) 4
I
-I
I
I
I
I
2
I
I
3
Figure 6. Comparison of the normalized deformation curves between the experimental data and D(t)/Dv- calculated with the quantities listed in Tables I and I1 on the basis of eqs 10-13. The dash-dot and dash-dot-dotlines indicate the contributions of the fast and slow processes.
dot-dot lines the Dg(t)/Dv, curves, and the solid curves the sum of two curves. We see that the simple model C reasonably well represents the overall feature of the deformation profiles. Examining the profiles, we notice the following. When 7, is short, the fast mode disappears for two reasons: TF becomes also short (cf. eq 11)and DFbecomes small relative to Ds (cf. eq 12).
Adachi et al.
1286 Langmuir, Vol. 7,No. 6,1991 I
CTAB:NaSol(O 075.0.3)/PDMS-10
o b=l.l7mm
* I
n
Dv&B/ (GA+ GB + Gc) and then the deformation can be represented by combination of two single retardation equations, of which the parameters can be calculated with equations similar to eqs 11-13. Anyway the equilibrium degree of deformation is obviously given by eq 3 and the overall profile by eq 2, with the retardation time given by eq 5.
Conclusions log( tls)
Figure 7. Comparison of the deformation processes of CTABNasal (0.075,0.3) dropleta with radii of 1.17 and 0.57 mm. The dashed line and dash-dot lines show the calculated D(t) curves for b = 0.57 and 1.17 mm, respectively.
Effect of Droplet Size. From eqs 5 and 11, we see that the retardation times are dependent on the droplet size. We tested this by comparing deformation curves of CTAB-NaSal(0.075, 0.3) droplet with b = 1.17 mm and b = 0.57 mm suspended in PDMS-10. The result is shown in Figure 7. As expected from eqs 5 and 11,the droplet with larger b exhibits a longer retardation time for the slow process resulting in enough separation of TF and TS. The dash-dot line and dashed line are the calculated deformation curves for b = 1.17 and 0.57 mm, respectively, The calculated curves appear to explain well the time dependence of the both curves. The droplet with b = 1.17 mm exhibits a double-step retardation curve with the fast process around log t = 0. However the theoretical curve does not show a clear fast process. The tendency that the observed D(t) curve exhibits clearer double step behavior than the calculated one is also seen in curves 2 and 3 of Figure 6. We consider that the effect of inertia may be one of origins of the discrepancy: In the early stage of the fast process, the drop deforms slower than the theoretical rate of deformation but in the later stage, the drop may exhibit an overshoot of D(t) due to the effect of inertia because the rate of elastic deformation in the fast process is high. The effect of inertia might have caused the clearer double step behavior. Effect of Medium Viscoelasticity. Finally we discuss the effect of medium viscoelasticity. It is obvious that when the drop phase is viscous but the medium is viscoelastic, the deformation of the drop is also modeled by model C in Figure 1: GA and V A now represent the viscoelasticity of the medium and VB the viscosity of the drop phase. Thus the deformation behavior can be also expressed theoretically by eqs 10-13. In the case that both the drop and medium phases are viscoelastic, the dashpot VB of model C in Figure 1also should be replaced by another Maxwell element with spring constant Gc and viscosity VC. In this case the droplet might exhibit threestep behavior, if the viscoelastic relaxation times are adequate: A t t = 0 the drop deforms instantaneously by
(1)Droplets of CTAB-Nasal solution suspended in PDMS media may exhibit double-step retardation curves when the viscoelastic relaxation time of the droplet is adequate. The fast process is due to the elastic deformation, and the slow process is due to the viscous deformation. (2) The deformation behavior of viscoelastic droplets is explained semiquantitatively by the phenomenological model consisting of a parallel arrangement of a Maxwell and a Voigt element: The former represents the viscoelasticity of the drop phase and the latter the interfacial free energy and the viscous drag due to the medium.
Acknowledgment. This work was supported in part by a Grant-in-Aid for ScientificResearch from the mini st^ of Education, Science and Culture (Mombusho), J a p h (63550667). Appendix From model C shown in Figure 1,we obtain a differential equation about the degree of deformation DAof the dashpot VA of the Maxwell element which should reflect the size of the droplet
The initial conditions are DA = 0 and dDA/dt = 0 at time t = 0. Then the solution of eq A1 on DA provides the expression of the total deformation D(t) of model C
where A1 and
A2
are the roots of the following equation:
(I*?s)A2+(?+n*+18)h+ GAG, GB
1= O
(A2)
The two roots, A1 and A2, take negative values. Thus, numbering the two roots so that lAll > IA4, defining the retardation time TF = -AI-' for the fast process and 7s = 4 2 - l for the slow process, and rearranging the terms, we finally obtain eqs 10-14 in the text.
