Surface instability of viscoelastic thin films

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Submitted on 1 Jan 1993

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Surface instability of viscoelastic thin films

S. Safran, J. Klein

To cite this version:

S. Safran, J. Klein. Surface instability of viscoelastic thin films. Journal de Physique II, EDP Sciences,

1993, 3 (5), pp.749-757. �10.1051/jp2:1993164�. �jpa-00247868�

Classification

Physics

Abstracts

46.608 47.50 68.15

Surface instability ^of viscoelastic thin films

S. A. Safran and J. Klein

Department of Materials and Interfaces> Weizmann Institute of Science> Rehovot> Israel 76100

(Received 24

September

1992,

accepted

in

final

form 15

January

1993)

Abstract. The surfaces of thin>

liquid

films _can be unstable due to thinning _van der Waals interactions>

leading

to the formation of holes in the

initially

uniform film. These instabilities can

be

greatly

retarded in viscoelastic materials (and

completely

inhibited in elastic materials) even

when the finite

frequency

shear modulus, E> ^{is small}

compared

to the infinite

frequency

modulus,

G. This _occurs when E/G _»

(a/ho)~

_« l where _a is _a molecular size and ho is the film thickness.

We relate the growth rate of the

instability

to the

dynamic viscosity,

_a~ (w), with examples for the

cases of _a

polymer

brush> _an elastic fluid (gel)> ^and a transient

polymer

network, described

by

reptation dynamics.

1. Introduction.

While the

production

of thin films is

important

in many

applications,

_a crucial consideration is whether the films remain flat and uniform at ambient temperatures ^{in thermal}

equilibrium.

The ultimate

stability

of such films

depends

^of course on the

equilibrium

contact

angle

of the film material. Non-zero _contact

angles imply

that the

initially

uniform

film,

formed

perhaps by

a

non-equilibrium

_process such _as

spin-coating

_or molecular beam

epitaxy,

will

eventually

break up into disconnected islands. The kinetics of this process

usually

involves the formation of

holes

[I]

in the otherwise uniform film _; the holes grow and an island structure results

[2, 3].

How

quickly

these holes nucleate is

directly

related to the

metastability

of the

initially

uniform film.

In

polycrystalline

solid

films,

_a natural _source of surface instabilities

leading

to eventual hole formation and

dewetting

_are the intersections between

grain

boundaries

[1, 4]

while in

liquid films,

there exists _an intrinsic mechanism for hole formation i,ia the flow induced

by

_a

thinning

van der Waals interaction. The spontaneous rupture ^{of thin}

liquid

films under this

force has

already

been discussed for the _case of

simple,

viscous

liquids [5-7].

Recent

experiments

have focused _on thin films

[8]

of

polymers,

where both hole formation and

growth

have been observed. At

long times,

the holes arrange themselves into characteristic pattems which may be related to a «

spinodal

_{» type} of

instability

of the hole rim

[9].

Thus

far,

the

polymeric

films have been discussed

using

a

simple hydrodynamic

model where the

viscosity

of the film is _{a constant.} In this paper, we reexamine the spontaneous rupture ^of

initially~

uniform,

thin

liquid

films

using

_a linear viscoelastic model for the

dynamics.

When the interactions with the substrate _are such that the

equilibrium

state is _a disconnected _set of islands

or

droplets I,e., partial wetting

hole formation is _a necessary intermediate state ; we

focus _on the

dynamics

of the initial

instability

at the surface. For viscoelastic materials with _a

750 JOURNAL DE

PHYSIQUE

II N° 5

single

time which characterizes the _crossover from elastic to viscoelastic _{response we} show that _{even a} small shear elastic

restoring

force _can

greatly

retarde the surface

instability leading

to hole formation _; in elastic

materials,

where this _crossover time becomes

infinitely long

the

instability

can be

completely

inhibited

by

the

equilibrium

shear modulus, _even if it is small.

We illustrate this with _an estimate of the timescale for surface induced

thinning

of _a

polymer

« brush _»

[10] (I.e.,

_a

monolayer

of

polymer grafted

onto _a solid

surface).

For materials with _no

equilibrium

elastic shear

modulus,

but with _a time

dependent

_response to shear stress, the surface

instability

is similar to that of the Newtonian

fluid,

if the

viscosity

is

replaced by

the

viscosity

at the _«

frequency

_» scale of the

instability

these results in _a self-consistent

equation

for the

growth-rate

of the process. ^This

equation

is examined for _a

polymeric

_system where

reptation dynamics [I

I _are assumed and it is shown that the characteristic time scales for hole formation _can be

large.

The

theory

^is motivated

by

recent

experiments [12]

which indicated that the addition of

long-chain

and surfactant-like

polymers

can inhibit hole formation in

polymeric

thin

films, although

the

theory

does _not

directly

address these

experiments

in

multicomponent

systems.

2.

Hydrodynamic instability

of viscoelastic fluid films.

The standard discussion of surface instabilities in

thin, liquid

films

[6, 7]

focuses on the effect of the _van der Waals energy of _a

solid,

with _a normal in the I

direction,

covered

by

a thin film of local

height

h

(x,

y t

)

where t is the time. Films that tend to thin _are described

by

_{a van} der

Waals free energy per unit _area,

U(h) given by [5, 6]

U(h)=-

^~

~

(l)

12 aTh

where A is related to the difference between the Hamaker constants of the

liquid

and _a

molecule of the

liquid interacting

with the solid A

~ 0 for

thinning

films which show surface instabilities which

give

rise to holes. This energy can be related _{to a}

(negative) disjoining

pressure, 4

=

dF/dD where F

= D U/h is the total free energy and D is the volume.

Noting

that dF/dD

=

(aflaD

_{)~ +}

(aflah)(ahlaD )

we find _:

~

=

~

~

(2)

The

tendency

of the _van der Waals interaction to thin the film and thus enhance any surface fluctuations competes ^{with the surface} tension, _y, which tends _to flatten _a

nearly-flat

surface.

Large

wavevector fluctuations _are stable while small wavevector

fluctuations,

where the surface tension force is

negligible,

are unstable and holes _are nucleated _{at a} characteristic

lengthscale

determined

by

this

competition.

The

dynamics

of this process are described

by

the Navier-Stokes

equation

and _a linear

stability analysis

^of the

flat, incompressible film,

with initial

thickness, ho.

^{The linear}

stability analysis

is

appropriate

for surface disturbances

(h(x, y;t)- hoi who,

but a non-linear

analysis [7]

indicates that the linear

approximation gives

a reasonable

description

^of the process, ^at least for small initial

perturbations.

Since the flat film has zero

velocity,

^the

linearized,

Navier-Stokes

equation

_can be written _:

p

~~=-Vp-V4+V.v ₍₃₎

at

where _p is the

density,

V

=

(u,

_v,

w)

is the

velocity,

_p is the pressure, 4 is the

negative

disjoining

_pressure, ^and _v ^{is the} stress tensor. For _an

incompressible, isotropic,

viscoelastic

h(x,y)~ h

~ z

Fig.

I. Surface geometry ^{used in these} calculations.

material,

the _{stress tensor} is related

[13, 14]

to the strain tensor _e and to the rate of strain

(velocity gradient)

tensor k

(e,

_{g., d~~}

=

awlaz)

where the dot

signifies

the time

derivative, by

_:

?ij "

E~ij

+ '~

o~

dij

⁺

£ lcn

~~^{~~~~ ~~}

@ij

(T)

^dT

(4)

n o~

Here,

E is the

equilibrium

shear

modulus,

_~

~

is the

high frequency viscosity, corresponding

to

typical

molecular

motions,

and m~ are the _rates associated with the discrete

spectrum

^of relaxation times. For Newtonian

fluids,

E

=

C~

₌ 0 and the _term V. _v becomes

V~V.

For non-Newtonian

fluids,

_a

simple

relation exists for the

Laplace

transforms

[15]

of the stress and the

velocity gradient

tensor :

To (W

)

~ 'l (W

)

@ij(W

(5)

where

~(w)=j+~~+z~[~ ₍₆₎

~ n

In these

equations,

the

Laplace

variable _{w can} be

complex,

in

general. Equation (3)

_can be

Laplace

transformed _to read _:

p wv

(w )

₌

vn(w )

+ ~

(w ) v2v _{(w ) (7)}

where n

= p +

4. Similarly,

the

incompressibility

condition _can be written _:

w~(w )

=

(u~(w

+

v~(w )) (8)

where the

subscripts signify

derivatives. The

Laplace

transforms of the normal and

tangential

stress

boundary

conditions at the

liquid

surface characterized

by

a surface tension

y, can be

respectively

written _{as :}

P

(w )

+ 2

1~

(w )(w~)~

h ^~ Y

V~h(x,

_{y ;}

w

) (9)

u~(w

+

w~(w )

=

v~(w )

₊

w~(w ⁾

= 0.

(10)

The

stability analysis

is

performed by solving

these

equations

in the lubrication

approxi-

mation

(for

a

rigorous

treatment see Ref.

[7])

where it is assumed that the

length

scale for fluctuations in the

velocities,

_pressures and film

height

in the _x and y directions is much

larger

than the film

thickness, ho

^the wavevector q associated with the

instability

is small

compared

with

I/ho.

We also _assume that the deviations from _a flat

film, H(x,y;w)=

[h(x,

_{y ; w}

) hoi

are small. The

growth

rate of the

instability

is shown to be related _to the

value of _w that satisfies these

equations

and the results derived below indicate that

752 JOURNAL DE PHYSIQUE II N° 5

w scales _as

q~

so that to a first

approximation,

_terms in _w in

equation (7)

_can be

neglected.

Consistent with this

approximation

is the

neglect

of

spatial gradients

of the

velocity

in the

x and y directions in this

equation

and _one finds

~

(w ) u~~(w )

=

n~(w (i1)

~

(w vzz(w

₌

ny(w ) (12)

Solving

for _u and v with the

boundary

conditions of _no

slip

at _{z =} 0 and

equation (10)

with the

x and _y derivatives

again neglected (I,e., u~(w

=

0 and

v~(w )

₌ 0 at _z

=

ho,

^where we use

[16]

_a linearization about

ho),

we find _:

u

(w )

= ~

~(~ ^{JI~(w )(z~}

²

^{zho) (13a)}

v w

)

= ~

lI~ ^{(w ) (z~}

²

^{zho ) (13b)}

To this order, the

tangential

stress

boundary

^condition

implies

p = y

V~h. (14)

Thus,

_p and

by equation (2)

Hare

independent [17]

of _z.

Finally,

the

incompressibility relation, equation (8)

and the

no-slip boundary

condition determine w as

w

(w )

₌

~

i

(n~

+

n~~) z2

ho (

(15)

where _we have

again [16]

linearized about

ho,

^{and used n=}

~ (h)

_y

^V~h

where all the

variables _are the

Laplace

transforms

(I.e., lI(w )).

The

equation

of motion for the surface is

given

from the kinematic

boundary

condition which relates the _z

component

^{of the}

velocity

at the surface _z

=

h

(x,

_{y ;} t

),

V~, to the total time derivative of h _:

fl+ (V~.V)h=V~.1. (16)

Keeping

terms linear in

h(x,

_y

t)

and

Laplace transforming

this

equation yields

wh(x,

_{y ; w t}

ws(w ) (17)

where the

subscripted w~(w

^is the _z direction

velocity

w

(w )

evaluated at z ₌

ho. Linearizing

the

expression

for

~

in

equation (2)

about

ho

and

using equation (15)

we thus find _:

wH(x,

_{y ; w}

=

~~ yv2(v2H(x,

y ; w

))

+

~

~

v2H(x,

_{y ;}

w

j (18)

3

71(w)

2

who

Consider a

single

Fourier mode of _a fluctuation in the

height

with _wavevector q with

H(x,

_{y ; w}

=

H~(w) exp(iq r)

_{; we} find _an

equation

for the allowed value of _{w ;}

~~3

~ 3 _'l

(l)

~~~~~ ~~~ ~~~~

where

q(

=

~

(20)

2

aTh(

y

This result has the _same form _as the

growth

rate derived

[6, 7]

for hole formation in _non- viscoelastic

fluids,

with the

proviso

that the

viscosity

_~, which is constant in those systems, ^is

replaced by

the

Laplace transform, ~(w) (Eq.(6)),

evaluated

self-consistently

at the

frequency,

_w, ^{of the}

instability.

The

present

^{derivation shows that this} substitution is consistent with the lubrication

approximation

and indicates how corrections may be included.

Positive values of

w indicate

[15]

that the surface fluctuation is unstable and will grow,

leading

to hole

formation,

while _{w ~} 0 indicates

stability

of the

flat,

uniform film _to surface

height

fluctuations.

Instability

is obtained

only

for wavevectors q ~ q~ where the _van der Waals interaction dominates the surface tension term.

For _an elastic materials with _an

equilibrium

^shear

modulus, (I.e.,

E # 0 and

C~

₌ 0 in

Eq. (6)),

^the

growth

rate is

given by

_:

~° ~

"q~ q~ _q(

+

)/~

₂

⁽²¹⁾

Y _o q

where

~~3

a =

fi (22)

3 ~~

We _see that _at small

enough

values of q, w ~ 0 and the

system

^{is stable. For}

larger

values of q the

system

may be

unstable,

while _as q - oJ, the surface tension guarantees w ~ 0 and

stability.

For _a viscoeslastic materials with _a

single,

characteristic relaxation time _v~

=

I/w~

which separates ^{the short-time elastic} response

(with

_a modulus,

E)

from the

long-time

viscous

behavior,

_{one can} write

equation (6)

in the form _:

~(w)= ~~+~ )~ ₍₂₃₎

~

From

equation (19)

one sees that there will be nvo surface

modes,

_a stable _one with

w ~ 0 and unstable _one. These solutions are

given by

:

~ ~

j _~~q)

=

~la _(q)~

+ 4 b~~~ ~~~~

where

a(q)

= w~ +

aq~jq~ qji ₍₂₅₎

and

b(q)

=

E/~

~

+

aq~

_[q~

q(]

_w~

₍₂₆₎

The

physical meaning

of these solutions is discussed below.

3.

Stability

of

polymer

brushes,

gels

and solutions.

We first

recapitulate

the known results for _a Newtonian fluid

(E

₌

0, C~

= 0 in

Eq. (6))

and

write

~~ =

Gv

(27)

where G is _a modulus

(typically

_{an energy} of order kT per unit molecular

volume)

and

754 JOURNAL DE PHYSIQUE II N° 5

v a characteristic molecular time scale.

Equation (19)

indicates that the

growth

rate is maximal at

q$

=

q(/2

and the fastest

growing mode,

_w~ is

given by

w~ v =

p

~

(28)

where

p~

=

yh( q(/12

_G

₍₂₉₎

and the

subscript

A denotes that

p

is related to the Hamaker constant

by equation (20).

We note that for most

systems,

one can set all the energy scales

(in

_y, G and

A)

to order kT.

Assuming

a

single microscopic length,

_{a, we} find that

p~

~

(a/ho)~

to within _a constant of order

unity.

For films

greater

than several molecular

thickness,

_we thus expect

p

« I.

Thus,

even the

fastest growing

mode

usually

satisfies w~ v =

p~ (a/ho)~

« l.

We next consider elastic materials.

Equation (21)

indicates that the maximal

growth

rate

occurs as above at

q$

=

q(/2

and the value of the _w~ is

given by

:

w~ v = p

~

p

~

(30)

where

p~

₌ E/G is the ratio of the shear elastic modulus to the

viscosity

modulus. We

immbdiately

_see that if

p~

_~

p~,

_w~ is

always negative

and the thin film is stable to surface undulations

[18].

Since

p~ (a/ho)~

can be very small for _a film whose thickness is much greater ^than a molecular

size,

_{even a very} small

equilibrium

shear modulus _a small

degree

of network formation _can reduce the

growth

rate of the

instability

and _even stabilize the film

against

holes. We note that thin

polymeric

films _can show surface effects which _can lead _to vitrification and

possibly

network-like behavior

[19].

A _more

realistic,

but still

phenomenological description

of

polymeric

materials considers their viscoelastic response, with _a characteristic

frequency

_w~

(which

_may be molecular

weight dependent

_as shown

below)

which

separates

^{the short-time elastic} response

(w

_{~ w~) from} the

long-time

viscous behavior.

Using equation (24)

_we find that the

growth

rate of the fastest

growing

unstable surface mode is _:

w~v=)(p~-p~-w~v)+)~/(p~-p~-w~v)~+4p~w~v. (31)

In the limit that

p~

»

p~

which _can be satisfied _even for a very small shear modulus since

p~ (a/ho)~

« l, we find that _:

w~ m w~

) (32)

E

For

long

characteristic times

(w~

_v « I

) compared

to molecular relaxation times and for small ratios of

p~/p~,

the

growth

rate can be very small indeed because

(I)

it is scaled

by

^the

(small)

characteristic

frequency

^of the viscoelastic material and

(it)

because _{even a} shear

modulus,

E which is small

compared

to that of _a solid _or the infinite

frequency

modulus of the

polymer,

results in _a very small value of

p~/p~

for films much thicker than _a molecular dimension. If the inverse

growth

rate is much slower than the

experimental

observation

time,

the system ^is

effectively

stable

against

surface undulations. Hole

growth

will not be observed and is

inhibited for all

practical

purposes.

As _a

specific example,

we consider _a set of

polymer

chains end-attached _to the surface of _a solid substrate. For the case where there is _no solvent, this is known as a

polymer

melt brush

[10].

Hole formation is

potentially possible

when these ends _are

reversibly

attached _{; we}

focus _on the surface

instability

of the brush. The shear modulus _can be estimated

by applying

a

lateral force to the top of the brush which shears the brushes

by displacing

each brush end _at the surface

by

a

distance,

_x in the direction of the force per unit _area, F. This results in the

stretching

of the brushes _{to a new}

length, L,

with _a

consequent

^{elastic tension} per

chain, f, along

each chain. The value of _x is such that the

shearing force,

F is

exactly

balanced

by

^the

component ^{in the} direction x of the elastic tension in the brush due _to the stretched chains _:

F

=

«fx/L (33)

where the tension per chain is

given by

^the derivative of the

stretching

free energy with

respect

to the chain

length

_:

f

=

3

kTL/Nb~

_for

an ideal chain in _a

melt,

where N is the

degree

of

polymerization.

This results in _a modulus

given by

3

kTv«~

(34)

E

=

Emeit

_~

~2

where _v is the

segmental volume,

b is the _monomer

size,

and _« is the number of chains per unit _area of surface. This result has

previously

been derived

by

Fredrickson _et al.

[20] using

a

slightly

different

approach.

For _a brush in _a

good solvent,

_a similar argument

[21]

indicates that _at low shear

strains,

E

= E~~j = 3 kT« ^~~~ The ratio E/G which _enters in

equation (30)

is small, since the surface

density (for

_a

given

melt brush

height)

is

proportional

_to

N~~

_{while G is}

independent

of molecular

weight

and

depends only

_on the

entanglements

between chains

[20]. However,

this small modulus is sufficient to stabilize the brush

against

surface undulations in the limit of

large N,

since the film

height, ho,

^is

proportional

to N and in

equation (30) p~ hi

^~ Thus, ⁱⁿ

equation (30), p~

^{N~ ~} »

p~

^{N~ ~ and the} system ^{is stable for}

asymptotically large

N.

(For

_a brush in _a

good

solvent the exponents are

slightly different,

but the conclusion is

unchanged.)

Of _course a melt brush with

irreversibly grafted

chains will _never dewet and form holes.

However,

_one can

imagine

that _a melt with

reversibly

attached chains _can be modeled _{as a} viscoelastic material with no

equilibrium

shear modulus

(I.e.,

at infinite

time)

but with _a finite relaxation time which characterizes the _crossover from the intermediate time shear

modulus,

E, to the very short time

modulus, G,

_as discussed above. Hole formation for the _case where the

grafting

locations _are free to

equilibrate

_on the substrate

(but

with _a

long

relaxation

time)

may be

govemed by equation (32).

For

long chains,

the ratio

p~/p~ l/N~

_{and when this}

small factor is

multiplied by

_a small value of _w~, the system can be

effectively

stable with hole formation inhibited _over any reasonable

laboratory

time scale.

Another system with _a finite shear modulus is _a

physical gel (where

the crosslinks _are not

permanent

so hole formation _can

potentially occur). Again,

the system ^{will be elastic} at times shorter than ml ^{and viscous} at

longer

^time scales. The

growth

rate will

again

be

given by equation (32)

and _{even a} value of E/G « I may still result in _a very small value of the fastest

growing

surface

mode,

_w~, since

p~/p~

« I _; for small

enough

values of the characteristic

frequency

_w~, the system may appear to be stable _to hole

formation,

while in fact it is unstable, but with _an

extremely

slow

growth

rate.

We _now consider _a

simple,

non-crosslinked

polymeric

system ^with no

equilibrium

shear

modulus, but with very

long

relaxation times. For this _case, the

longest

relaxation times

[I I]

are associated with the

reptation

of the

polymer

chains. The

reptation

modes of the chains have characteristic times which scale with the

disengagement

time for chain relaxation

[I II,

v~ To

N~/N~ (35)

where _To is _a

(microscopic) segmental jump time, (of

order

r),

N is the

degree

of

756 JOURNAL DE PHYSIQUE II N° 5

polymerization

and

N~

is _{an «}

entanglement

_»

degree

of

polymerization

which

depends

_on the

particular polymer

and concentration but is

typically

of order 200 in _a melt. Because of the factor of

N~,

these times _are much

longer

than the

microscopic

time scales and _can lead to _a dramatic

slowing

down of the surface undulation

instability

and thus of hole formation and

growth.

In the

reptation

model, the

complex viscosity

_can be written

[I II

~1(W ₌

G°

Td

f ) /~) _(36a)

where

f(x)

=

(

(1 ~~@~ ^(36b)

x

and

G°

_is

a modulus that is

independent

of molecular

weight

but

depends

on concentration. In

the limit where

p~

ml and

(G° _v~)/(G~)»

_I, the small _w limit of

equation (36) yields

w~ v~ =

gp

~ where g =

G/G°. _Thus,

_{the maximum}

growth

rate, _w~, is small _not

only

because the film is

relatively

thick

(p~

« I

),

but because the characteristic time

scale,

_v~ is very

long, scaling [22]

with

N~.

_For

long enough

chains, this

again

_may stabilize the surface

instability

_on

laboratory

time scales.

In this context, it is

interesting

to note recent

experiments

in which thin films of _a

polymer

melt _were

investigated.

In _one

study [23],

which we call the thick _case, films of

polystyrene

with N

=

10~

_and of thickness _ca 500 _{nm were} heated up to 70 °C above their

glass

transition temperature ^for

periods

of two weeks and

longer,

^and remained

quite

stable

(I,e.,

uniform and

unbroken).

In _more recent work

[8],

which _we refer to _as the thin _case,

polystyrene

films with N

- 300 and of thickness 50-100 _nm, heated to the _same temperature, were observed _to

commence

dewetting (holes began

to form and

grow)

within _ten minutes. While _no

systematic

study

has been carried _out to date

(in

the work

quoted,

both the film

thickness, ho,

^{and the}

longest (N-dependent)

relaxation times _were

varied),

the

striking

differences in

film

stability

_on increase of

ho

^{and N} are

qualitatively

in accord with _our discussion. Our

predictions

for the fastest

growing mode,

_w~ indicate that the ratios of the times

required

for hole formation for the thick and thin _{cases can} be _{enormous ;} the time

required

increases

by

a

factor of

108

_{for an increase in} both the film thickness and the molecular

weight by

_one order of

magnitude.

For the 50 _nm film with N

m 300

(taking

a _m lo

h,

v m 10~ ^~~

s)

the

growth

rate is

estimated _to be 500 _s while the 500nm film with N

m10~,

the estimated

growth

rate is

loll _s and the

system

^is

effectively

stable _on

laboratory

time scales.

Acknowledgments.

The authors

acknowledge

useful discussions with N.

Dan,

G.

Fredrickson,

J. F.

Joanny,

S.

Milner,

G.

Reiter,

A.

Silberberg,

^{and R.} Yerushalmi-Rozen.

They

are

grateful

for the

support

^{of Minerva} foundation. J. K, and S.A.S,

acknowledge partial _support

from the

Ministry

^{of Science and}

Technology,

^{Israel under} grants ^{03M4040 and 3543-1-91}

respectively.

References

[Ii SRoLoviTz D. J. and SAFRAN S. A., J. Appl.

Phys.

60 (1986) ^247.

[2] SROLOVITz D. J. and SAFRAN S. A., J. Appt. Phys. 60 (1986) 255.

[3] SEKIMOTO K., OGUMA R, and KAWASAKI K., Ann. Phys. 176 (1987) 359.

[4] JIRAN C. and THOMPSON C. V., J. Elec. Mater. 19 (1990) 1153 and in Thin Solid Films, in press.

[5] VRIJ A., Discuss.

Faraday

Soc. 42 (1967) 23.

[6] RUCKENSTEIN E. and JAIN R. K.,

Faraday

Trans. ii 70 (1974) 132.

[7] WILLIAMS M. B. and DAVIS S. H., J. Colloid Int. Sci. 90 (1982) 220.

[8] REITER G., Phys. Rev. Lett. 68 (1992) 75.

[9] BROCHARD-WYART F., REDON C. and SYKES C., C. R. Acad. Sci. (Paris) 314 (1992) 19.

[10] ALEXANDER S., J. Phys. France 36 (1977) 983

MILNER S. T., WITTEN T. A. and CATES M., Macromolecules 21 (1988) 2610.

[I Ii _DE GENNES P. G.,

Scaling Concepts

in

Polymer Physics

(Cornell

University

Press, Ithaca,

1979)

DOI M. and EDWAROS S.,

Theory

of

Polymer Dynamics

(Oxford

University

Press, Oxford, 1986).

[12] YERUSHALMi-ROzEN R., KLEIN J, and FETTERS L. J., to be

published.

[13] MIDDLEMAN S., Chem. Engng. Sci. 20 (1965) 1037.

[14] GOLDIN M. _et al., J. Fluid Mech. 38 (1969) 689.

[15] Note that here _{we use} the

Laplace

and not the Fourier transform of the time

dependent

stress tensor the

quantity

_w which determines

stability

is thus real, and the time

dependence

is

~wt

[16] Since

lI~(w

^and

lI~(w already

have _terms linear in H(x, _{y w} )

= h(x, _{y w} ho, we evaluate the

boundary

conditions at _{z =} h~.

[17] This is consistent with the third Navier-Stokes equation, the analogue of equation (12) for

w.

Using

the

incompressibility

condition, equation (8), to solve for _w indicates that

w is

proportional

to q~. Thus to first order terms in q in the Navier-Stokes equations, we can set w~~ = 0

implying

that lI~ ₌ 0.

[18] This can also be _seen from the full

dispersion relationship

of

equation (19).

Similar restabilization has been considered in the context of jet breakup in references j13, 14]. While the present, dynamic argument

yields

the

growth

rate of unstable modes, _an estimate of the modulus needed to prevent

instability

_can also be obtained from

energetic

considerations.

[19] KREMER K., J. Phys. France 47 (1986) 1269.

[20] FREDRICKSON G. H., AJDARI A., LEIBLER L. and CARTON J., Macromolecules 25 (1992) 2882.

[2 Ii It is of interest that at this level, the modulus of _a

polymer

bnlsh in _a good solvent is

significantly larger

than that of _a melt brush, (E~~j/E~~j~) ^II (b

,$

=

s/b _» I, where _s

= « ~'~ is the _mean

interanchor spacing (s » b) and the segmental volume is taken _{as v}

=

b~.

[22]

Experimentally,

it is found that the

longest

relaxation times in entangled

polymer

systems ^scale as

N~_{~, but this does}

not

change

the present ^discussion (see ^FERRY J. D., Viscoelastic Properties of

Polymers,

3rd Edition,

(Wiley,

NY, 1980)).

[23j STEINER U., KRAUSCH G., SCHATZ G, and KLEIN J., P/~1ys. ^{REV. Lett. 64} (1990) Ii19.