The instability of a sessile drop of magnetic fluid

Journal of Magnetism and Magnetic Materials 39 (1983) 165-168 North-Holland Publishing Company

165

THE INSTABIUTY OF A SESSILE DROP OF MAGNETIC FLUID C. ZELLER and W.W. CHEN *

Pitney Bowes R&D, Norwalk, CT, USA The suitability of attracting drops of magnetic fluids from low energy surfaces for magnetic printing was investigated. As part of the fluid was pulled from the sessile drop, the shape was observed up to the rupture point as a function of field strength. Relations governing the break-off and predicting the shape of the deformation were developed.

1. Introduction

• printing, we established the conditions for removing ferrofluids from low energy surfaces.

Early liquid magnetic printing used an unstable suspension of magnetic toner particles which were deposited on the surface of a magnetized drum or tape passing through the fluid; whereas dry printing involves mechanically dusting the magnetized surface with toner. These processes unfortunately require transferring and fusing the toner to the paper. The inking process that we studied begins with a gravure roller applying a matrix of discrete drops of stable ferrofluids (FF) on a teflon coated roller. A magnetic head attracts selected drops across a gap to a moving paper surface (see fig. 1). Besides eliminating contact transfer and fusing steps, the advantages of this wet process over dry toning include reduction in background and simplified cleaning. To evaluate the feasibility of such * Current address: General Electric, Binghamton, NY, USA.

2. Theory A sessile drop of ferrofluid in a magnetic field gradient can rupture after deforming if the attractive force is sufficiently large. Not only does the magnetic pressure change the contact angle at the solid-liquid interface but also the curvature of the drop and the interfacial contact area. A complete calculation of the instability leading to a rupture accounts for the critical energy of the surface and the effects of advancing and receding angles and agrees well with measurements using ultrastable ferrofluids. The free surface of a FF drop is assumed unibound and axisymmetric (r = f(z)). Also, if the function f ( z ) is assumed unique, the potential energy U is ,/S + y'S' + W~+ Wm, where S is the drop's free surface, S' the area of the base, y the liquid's surface tension, ~,' the interracial tension difference liquid solid-solid gas, Wg the gravitational work and Wm the magnetic energy. S=

j0

2~rf(z)

J

1

[dz]

s' = ,¢2(0),

= pgCrfo°(Z - Zo)f2(z)dz, A) Doctor Blade B) Ink Reservoir C) Gravure Roller

D) Transfer Roller E) Paper F) M a g n e t i c Head

Fig. 1. ~hemaficrep~sentationof ~einktransferprocess.

Wm=foZ°dzfo/(Z)2~rrdrfoH(r'z)M(H)dH,

(2) (3) (4)

zo is the height of the drop's apex above the contact surface. In a general case the equilibrium

0304-8853/83/0000-0000/$03.00 © 1983 North-Holland

166

C Zeller, W. IV. Chen/ Instability o/a sessile drop of magnetic fluid

shape of a FF drop can be constructed and its stability analysed only with numerical methods. In our case an analytical parametric approximation for the unknown function f(z) is adequate and accurate. The functional used for the calculation has the form a ( 1 - A z 2 - B z ) 1/2, where a is the radius of the drop's base. With f(Zo)= 0 and the drop volume V0 = f~°~rfE(z)dz, the constants A and B are fully determined. Thus the total energy U depends on three independent parameters only, namely V0, a and z o. The equilibrium shape is found by minimizing U with respect to a and z o at constant volume. Since the surface wetted by the ferrofluid seems to have a critical energy higher than the external unwetted surface, a term AU= 2 ( y , - y*)AS',

AS'

<

0

(5)

must be added to U. AS' is the change in the contact area due to the magnetic field and y* is the critical energy of the wetted surface. The resuiting force returns the drop closer to the original area and increases appreciably the field necessary to rupture the drop.

3. Experiments We observed the shapes of isolated FF drops of different masses resting on a horizontal surface in

Magnetic Head

t z o (~m)

Fig. 2. Deformation of a sessile drop of magnetic fluid in an axially symmetric field.

a variable vertical magnetic field H. A perpendicular magnetic head, whose axis of symmetry coincided with that of the drop (fig. 2), provided the field. Careful experiments showed that the field could be derived from the following magnetic scalar potential: 305 I l o g [ ( d + 0.3 - z)2 + x2 + y 2 ] , 0 ~
(6)

where I was given in amperes and d, x, y, z in millimeters. The origin of coordinates was at the center of the drop's base and d was the distance between substrate and magnetic head. Photographs of stationary drops showed contact angles between the liquid and solid surfaces and related parameters such as the drop's height and radius at its base. For every FF there existed at a certain volume of the sessile drop a field value, at which rupture occurred. When the field slightly exceeds the critical value, the drop disintegrates along the magnetic field direction, implying a threshhold effect, which may be explained as a specific instability of the drop. A detailed study has related the drop volume with the critical parameters giving the break-off conditions. Various non-volatile ferrofluids (e.g. diester or glycol based) were used. Their saturation inductions ranged between 100 and 200 G and surface tensions were about 30 d y n / c m . Each drop rested on a glass plate plasma coated with teflon. According to the material used, the contact angle varied between 53 ° and 84 ° corresponding to a ratio Y/Yc, ranging between 1.25 and 1.82. For a drop on uncoated glass the contact angle was vanishingly small, corresponding to almost complete wetting. In the above range of y/yc the experiments show three general geometrical features: (a) The radius of the base does not change appreciably during the drop's deformation: (b) The ratio of the critical height of the drop (just before rupture) to the height of a cone of the same volume and base is a constant (0.8); (c) The volume of the remooed drop is approximately equal to the volume of a sphere having a diameter equal to the critical height of the drop (see fig. 3). Assertions a and b have been theoreti-

C Zeller, W. W, Chen / Instability of a sessile drop of magnetic fluid

contact angle 0 > 90. For 0 = 90, ~'/Yc = 2 and the limiting value of V ' / V o should be equal to unity leading to k = 0.79. This agreement is impressive and supports the validity of eq. (8). The magnetic force attracting the drop was calculated from the following measurements: (1) shape and size of the drop, (2) magnetization curve of the fluid, (3) field distribution and calibration of the magnetic head. The magnetic force is the negative gradient of the magnetic energy Wm. In the range of our data the critical magnetic force F m is 2 ~ra'/. This semi-empirical critical magnetic force can be compared to the force of adhesion of the sessile drop F~ given by

zo* +

5-

4-

o

3-

j

+

% : 1.2 . 1.37•

21.61+ 1.73•

i-

1.820 i

1

2

i

3

i

i

4

5

i

D'

Fig. 3. A plot of the apex before rupture versus the diameter of the removed drop (dimensions are in arbitrary units).

cally demonstrated. Assertion c leads to the semiempirical equation V'

3 y

--

-1

Y

2),

(7)

where V' is the volume of the removed drop and Vo the volume of the initial sessile drop (V0 = 4~rR3o/3 if spherical). "//'y¢ is calculated from the contact angle at zero field. The consistency of the equations can be verified in two ways: (1) measured values of V ' / V o versus 7/7c give k = 0.78 when eq. (7) is fitted to the experimental values (fig. 4); (2) a non-wetting surface is defined by a

8~rR07)

Fa =

Thus, the ratio F m / F~ is F m / F. = ~ -~

-

•

clean surface

•

stain surface

.8 .7

.5 .4 .3 .2 .i

1.4

116

+

(Tc

~1/6

1)

4. Conclusion

e .6"

1.2

(9)

Experiments show that rupture of the drop occurs when a >_.3.

.9

1

1.

Consistent with eq. (7) for Y/Yc < 2, the ratio F m / Fa < 1. The dimensionless parameter a for the present problem is given by the ratio of the magnetic force to the critical magnetic force. Approximating the magnetic force by M s O H J a z ) V o, where "bHJOz is the magnetic field gradient at the center of the basis of the drop, we found s

vo

(8)

3%)2/3

M [ ~Hz)R20

.~¢-!

1.0-

167

1.8

~

~,

%

Fig. 4. A plot of the volume of the removed droi~ versus the surface tension of the fluid.

Minimizing the potential energy of a sessile FF drop accurately predieis the shape and the rupture conditions for the attracted fluid. The amount removed, although not predicted, was correlated with the starting volume and the fluid parameters. Impractically large fields were required to use this technique for magnetic printing. The causes include the difficulty in haaintaining surface energies below 20 erg/cm after repeated wetting and insuf-

168

C Zeller, W. W. Chen / Instability of a sessile drop of magnetic fluid

ficient saturation magnetization for fluids with high surface tensions and low viscosities.

Acknowledgements The authors are grateful to T. Cruz-Uribe for his advice, encouragement and editing of this

paper. They wish to thank R. Pryor and J. Popplewell for timely assistance. Pitney Bowes kindly supported this research.