数理解析研究所講究録
第 891 巻 1995 年 117-126

117

Transition of viscous fingering patterns in polymer solutions
Masami Kawaguchi$(j1|\square \not\subset\not\equiv)$

Department of Chemistry for Materials, Faculty of Engineering, MieUniversity

1515 Kamihama, Tsu, Mie 514 Japan

Abstract:

Viscous fingering patterns of aqueous hydroxypropyl

methyl oellulose (HPMC) solutions pushed by air in the Hele-Shaw oellwere observed as a function

pressure of 15 cm$H_{2}O$.

$\backslash of$isopropyl alcohol ,under a constant

We found a morphological transition fromside branching patterns to tip splitting ones with increasing isopropyl
alcohol content, accompanied with a decrease in surface tension.

Theobserved morphology transition was classified as a second-order one by

a-comparison of the average tip velocity as a function of isopropyl
alcohol content.

Introduction
Small anisotropies such as roughening the plates,1-3 introductionof a wire,4 and entrainment of a bubble,5 or using a nematic liquid

crystal6 in the flow have led to drastic changes in the morphology of theobserved fingering patterns in the Hele-Shaw cell.

Similar results

were obtained by computer simulation experiments by taking intoaccount anisotropy.

$7_{2}8$

In particular, Ben-Jacob and Garik concluded

that anisotropy was a neoessary factor for dendrite growth.9

paper,In thiswe describe the results of experiments on radial viscous

118

fingering in the Hele-Shaw oell of aqueous solutions of hydroxypropy
methyl oellulose (HPMC) involving isopropyl alcohol.

Experimental Section
An HPMC kindly supplied by Shin-Etsu Chemical

Materials.

Thecompany was purified by the method described previously.
molecular weight of HPMC was determined to be 271 $x10^{3}$ by an
intrinsic viscosity measurement in a 0.1 aqueous $NaC1$ solution at 25
group and the molar
The degree of substitution (DS) of
$1$

$M$

$\circ c^{10}$$OCH_{3}$substitution (MS) of OC${}_{3}H_{6}OH$

group had values of 1.8 and 0.18,

respectively, as obtained from the manufacturer.
Water purified by a Millipore Q-TM system was used.

Spectroscopic-grade quality isopropyl alcohol was used without furtherpurification.
HPMC dissolved in water and the resulting aqueous HPMC

solution and isopropyl alcohol was$mi^{\backslash }xed$

in 100 : $0,99$ : 1, 97: 3, 95: 5, 93 : 7, 90 : 10, and 85 : 15 volume ratios of water and isopropylalcohol.

The final HPMC concentration in all solutions was 1.5 $g/$.

In order to obtain high-contrast patterns in the fingering$100mL$experiments, the HPMC solutions were colored by adding methylene
blue.

Viscosities of the water and isopropyl alcohol mixtures weredetermined from their effluent times using a Ubbelohde viscometer at$25\circ c$.

Rheometer.

The shear flow measurements of HPMC solutions werecarried out in the shear rate range 0.1-400$s^{-1}$, and their dynamic

119

measurements were performed in the frequency range 0.031-12.4 rad.
$s^{-1}$

using an MR-300 Soliquid Meter produoed by Rheology Co. Ltd.

(Kyoto, Japan), with a coaxial cylinder geometry (od. 4.003 cm; id.

3.898 cm; immersion length, 0.897 cm).

The temperature of thesample chamber was maintained at 25 oC.

Surface Tension Measurements.

Surfaoe tensions of the HPMC

solutions and mixtures of water and isopropyl alcohol were determinedby using the same instruments as reported previously.11 A glass troughwith a diameter of 10 cm was filled with HPMC solutions and its

temperature was controlled within 25 oC.

The trough was placed on astage that can be moved up and down to adjust the position of thetrough.

The height of the trough was adjusted to touch the edge of asandblasted platinum plate $(24 x10x0.1mm^{3})$ connected to a Cahn

2000 electrobalanoe. The surfaoe tensions of the HPMC solutions were
determined from the electrobalanoe connected to a digital voltmeter.

We regarded it as important for attaining a constant surfaoe tension,
namely, an equilibrium value unless it does not remain constant over 10min.
The fingering experiments were prepared inHele-Shaw Cell.

radial geometry, in a rectangle Hele-Shaw oell made by using two plane-parallel glass plate $(0.8 x50x35 cm3)$ with eight silicone wafer spacersof 0.05 cm thickness clamped in between.

The Hele-Shaw oell was

plaoed on a white board on which there are concentric circles with theradii of 1, 2, 3, 4, 5, 7, and 9 cm.

Air was injected through a small

hole (0.2 cm in diameter) at the center of the top plate under a constantpressure of 15 cm$H_{2}O$, which is applied by the hydrostatic pressure

generated using a water filled cylinder with the air in a cup with avolume of 3000 $mL$ .

The cup volume was enough large to maintain a

120drop in the pressure less than 1% during the experiments. In order toavoid lifting of the plates in the experiments eight clamps were applied

symmetrically around the boundary. Records of the developed patterns

were taken by a CCD video camera/recorder method. The images ofthe recorded patterns were printed out by a video printer.
Results and Discussion
An aqueous solution of an HPMC displays weak shear-thinningbehavior: the shear viscosity is almost constant in the lower shear rates
than 100$s^{-1}$, regarding as a fast Newtonian region, which is called zero-

shear viscosity, and above the shear rate the viscosity gradually
decreases with an increase in the shear rate and finally approaches a less
10% reduction of the zero-shear viscosity at the highest shear rate of

400$s^{-1}$measured in the present study. The aqueous HPMC solution was

surfaoe active and its surfaoe tension showed a little time-dependence
and finally attained an equilibrium value of 47.0 $mN/m$ after 30 min.

Isopropyl alcohol shows a lower surfaoe tension than that of theaqueous HPMC solution and addition of isopropyl alcohol to the HPMC
solution should lead to a decrease in its surfaoe tension. As expected,
the surfaoe tensions of the HPMC solutions were decreased with anincrease in isopropyl alcohol, and above at 10 vol. % isopropyl alcohol
the surfaoe tension showed a shorter time-dependenoe than that of thepure aqueous HPMC solution. The equilibrium surfaoe tensions aresummarized in Table 1 as a function of isopropyl alcohol content.
Shear viscosities of the HPMC solutions in the presenoe of

isopropyl alcohol also showed shear-thinning and the zero-shear

viscosity data are listed in Table 1.

An increase in the shear viscosity

121with isopropyl alcohol content would be mainly attributed to the factthat the viscosity of isopropyl alcohol is more twioe as large as that ofwater.

Furthermore, from the zero-shear viscosity and solvent

viscosity data, we can calculate the relative viscosities of HPMC
solutions and the magnitudes decrease above the isopropyl alcohol
contents lager than 5 vol. % as shown in Table 1.

This means that

HPMC chains have more compact forms in aqueous isopropyl alcohol

solutions with an increase in isopropyl alcohol. The solution properties

can be interpreted reasonably by taking account of isopropyl alcohol asa poor solvent for HPMC.
of isopropyl alcohol content on theFigure 1 displays an$effe_{\backslash }ct$morphological transition of the viscous fingering patterns.

Without

isopropyl alcohol (Figure la) the side branching fingers are stable andsome tips in the finger grow with a dendrite pattern.

On increasing

isopropyl alcohol content (Figures lb and lc) the side branching growthis suppressed and the interfacial pattern becomes less dense.

At 7 vol.

% isopropyl alcohol the tip splitting growth in {he pattern is dominatedand a less side branching is observed.

Further increasing isopropyl

alcohol content (Figures le and lf) leads to a tip splitting pattern, which

is similar to that obtained using a Newtonian isotropic liquid.1-5
Recent studies based on diffusion limited aggregation $mode1^{12,13}$suggest that the finger in the pattem first becomes slightly asymmetric,
then the tip of the finger splits, and a highly branched and apparently
dendrite structure is produced with a decrease in surfaoe tension.

The

surfaoe tension dependenoe on the pattern transition is opposite to thepresent finding.

The pattern growth continuously produoes a newinterface between the HPMC solution and air and its speed is much
shorter than the time to attain the equilibrium surfaoe tension of the122

HPMC solutions, resulting that a real surface tension in the fingering
front should be higher than the equilibrium one due to a slower
diffusion rate of HPMC chain than the solvent.

Such dynamics will

influence the transition of the viscous fingering patterns, but at themoment there are not any reasonable explanations.

As mentioned above, isopropyl alcohol should play a role inshrinking HPMC chains and it is expected to lead to the less elastic

repulsive energy with an increase in isopropyl alcohol.

Although anelastic modulus determined from the dynamic measurements of HPMC

solutions was much less than a viscous one, its magnitude clearlydecreased with an increase in isopropyl alcohol content.

Thus, larger

elastic modulus would be related with the generation of the side
branching pattems for HPMC solutions with lower isopropyl alcohol
contents than 5 vol. %.

A variety of the side branching patterns wereobserved for viscoelastic materials such as clay slurries.14
In addition, the tip velocity is also a useful quantity to understandthe dynamics of the viscous fingering$\backslash$Figure 2 demonstrates theaverage tip velocity for the HPMC solutions, measured at the outermost
part when it is 7 cm from the center, as a function of isopropyl alcohol
content. The velocity has a maximum at around 3 vol. % isopropyl

alcohol and it gradually decreases with an increase in isopropyl alcoholcontent.

A comparison of Figure 1 and Figure 2 shows that theobserved transition is characterized as second-order sinoe the tipvelocity itself is continuous but the morphology changes.

A similar

plot in which there is the maximum in the velocity of pattem growth

was observed for the experiments by electrochemical deposition.15
Our main aim here has been to report the experimental discovery of themorphological transition of the viscous finger patterns in the123homogeneous polymer solutions shown in Figures la-lf without any

anisotropy.
References
1. E. Ben-Jacob, R. Godbey, N. D. Goldenfeld, J. Koplik, H. Levine,
T. Muller, and L. M. Sander, Phys. Rev. Lett., 55, 1315 (1985).

2.

V. Horvath, T. Vicsek, and J. Kertesz, Phys. Rev., A35, 2353

(1987).

3. J. D. Chen, Exp. Fluids, 5, 363 (1987).

4.

G. Zocchi, B. E. Shaw, A. Libchaber, and L. P. Kadanoff, Phys.

Rev., A36, 1894 (1987).

5.

Y. Couder, O. Cardoso, D. Dupuy, P. Tavernier, and W. Thom,Europhys. Lett., 2, 437 (1986).

6. A. Buka, J. Kertesz, and T. Vicsek, Nature, 323, 424 (1986).

7. S. K. Sarkar and D. Jasnow, Phys. Rev., A39, 5299 (1989).

8.

E. Ben-Jacob, P. Garik, T. Mueller, and P. Grier, Phys. Rev.,

A38, 1370 (1988).

9. E. Ben-Jacob and P. Garik, Nature 343, 523 (1990).
10.

T. Kato, T. Tokuya, and A. Takahashi, Kobunshi Ronbunshu (inJapaneses), 39, 293 (1982).

11. M. Kawaguchi and R. Nishida, Langmuir, 6, 482 (1990).
12. J. Nittmann, G. Daccord, and H. E. Stanley, Nature, 314, 141(1985).

13. S. Liang, Phys. Rev., A33, 2663 (1986).
14.

H. Van Damme, F. Obrecht, P. Levitz, L. Gatineau, and C.

Laroche, Nature, 320, 731 (1986).

15. Y. Sawada, A. Dougherty, and J. Gollub, Phys. Rev. Lett., 56,1260 (1986).

124

Table 1.

Surfaoe tension and viscosity data of the HPMC solutions

Isopropyl alcohol013571015

47.0

46.3

45.4

43.9

42.4

40.1

36.2

viscosity (Pa.s) 0.114

0.122

0.135

0.143

0.148

0.160

0.198

content(vol. %)Surface tension$(mN/m)$Zero Shear

Solvent
viscosity(mPa.s)

0.890

0.926

1.00

1.08

1.17

1.31

1.60

Relativeviscosity

128

132

135132

126

126

124

125

a$b$

$e$

$f$

$c$

Figure 1. Morphology transition in viscous fingering patterns of the$\phi=0$
HPMC solutions as a function of isopropyl alcohol content$\phi=10$$\phi=7$ vol. %;$\phi=5$ vol. %;$\phi=3$ vol. %;vol. %;
The concentric circles in the photographs
vol. %; $\phi=15$ vol. %.
have the radii of 1, 2, 3, 4, 5, 7, and 9 cm, respectively.
$\phi:a,$

$b,$
$f,$

$c,$

$d,$

$e,$

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$O\infty\tilde{\Xi}$$\sim$$\underline{\triangleright})$$\overline{OO}$$\overline{>O}$$\phi/$

vol. %

Figure 2. The average tip velocity of the viscous flngering
pattern as a function of isopropyl alcohol