Resonantly enhanced neutron intensity in a surface segregated polymer blend

(1)

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Resonantly enhanced neutron intensity in a surface segregated polymer blend

L. Norton, E. Kramer, R. Jones, F. Bates, H. Brown, G. Felcher, R. Kleb

To cite this version:

L. Norton, E. Kramer, R. Jones, F. Bates, H. Brown, et al.. Resonantly enhanced neutron intensity in a surface segregated polymer blend. Journal de Physique II, EDP Sciences, 1994, 4 (2), pp.367-376.

�10.1051/jp2:1994134�. �jpa-00247967�

(2)

J. Phys. II IFance 4

(1994)

367-376 FEBRUARY 1994~ PAGE 367

Classification

Physics Abstracts

61.40 42.10 61,12E

Resonantly enhanced neutron intensity in

_a

surface segregated polymer blend

L.J. Norton

(~),

E.J. Kramer

(~),

^R.A.L. Jones

(~),

F-S- Bates

(~),

H-R- Brown

(~),

G-P- Felcher

(5)

and R. Kleb

(5)

(~) Department of Materials Science and Engineering and the Materials Science Center, Cornell University~ Ithacai NY14853, U-S-A-

(~) Cavendish Laboratory, Madingley Rd., Cambridgei CB3 OHE, U-K-

(~) Department of Chemical Engineering and Materials Sciencei University of Minnesota, Minneapolis, MN 55455, U-S-A-

(~) IBM-ARC, 650 Harry Roadi San Jose, CA 951201 U-S-A-

(~) Materials Science Divisioni Argonne National Laboratory, Argonne, IL 60439, U-S-A-

(Received

14 October1993, accepted 10

November1993)

Abstract. We have formed _a resonant cavity for _a massive particle, the neutron, in _an attractive potential between _a surface segregated polymer film and its substrate. Standing

waves occur when incident _waves interfere with the _waves reflected from the substrate. Multiple

internal reflections lead to _an enhanced intensity at resonant _wave vectors below the _wave vector for total reflection. At these resonant _wave vectors the enhanced neutron intensity incoherently

scatters from the hydrogen containing polymer of the potential well causing _a marked decrease in the reflectivity. The position of the resonant dips provides

an aid to fitting the reflectivity data by restricting the possible scattering length density profiles. This phenomena _may find application in the study of in-plane correlations by enhancing off-specular scattering.

Introduction.

In the last six years ^neutron

reflectometry

has advanced the

understanding

of

polymer

science

by providing

the

depth sensitivity

_necessary for _a

rigorous

test of theories _on bulk and interface

polymer thermodynamics

_[1-4]. The excellent contrast between blend components

provided by

selective deuteration

gives

_a

depth

resolution of10

1

_for

sharp gradients

in the composition

profile

normal to the

sample

surface. Since

phase

information is lost in

reflectivity experiments,

educated _guesses about the nature of the

sample

_are ^essential to successful and reliable data

analysis

_[5]. We have observed _an

interesting phenomena

that _can aid in the isolation of the

composition profile

that best fits the

reflectivity: resonantly

enhanced _neutron

intensity (REN)

that _occurs inside _a

potent1al

well

during

total external reflection. Jannink [6] has

predicted

(3)

368 JOURNAL DE PHYSIQUE II N°2

z

~x

Fig.

I. Illustration of the reflection geometry.

REN for neutron reflection from the

potent1al

well formed

by

_a

polymer

^adsorbate in deuterated solution. REN _can be

exploited

to

produce polarized,

monochromatic neutron beams

using

_a

Fabry-P4rot

neutron interferometer constructed with

magnetic

^films as

suggested by

Maaza et al. [7]. ^Flux enhancement has been demonstrated with

electro-magnetic

X-radiation in _a

resonant bearn

coupler by Feng

et al. [8] and with _a

stepped potent1al Wang

et al. [9] ^have

observed enhanced

X-ray intensity,

in _a thin

lipid

film _on _a

highly

reflective substrate. We have been able to prepare

polymer

film

samples

that self-assemble to form structures

showing

strong resonances; we will discuss _our

experimenta1findings

and suggest

applications

for the

phenomenon.

Since the

scattering

vector is

always

normal to the

sample

surface in _a reflection experiment the

reflectivity provides

one-dimensiona1structural information. A neutron beam of

wavelength

is incident _on _a

sample

at _a

glancing angle

0 _m 1°

(see Fig. I)

and the

reflectivity

R

(the intensity

of the reflected beam divided

by

the incident

intensity)

is measured _as _a function of the

perpendicular

component ^of the neutron wave vector k ₌ (27r

/~) sin(0)

_[10]. To model the

reflectivity

R

= [r[~, ^where r is the

amplitude

of the reflected _wave ifi, the

time-independent Schr6dinger equation

fi2 _~2~fi

~2m

_dz2

^~~ ^

is solved for the component ^{of the} neutron wave function

perpendicular

to the

surface,

where

m is the neutron mass and the

effective

incident neutron energy E

=

h~k~/2m.

Since

27r/k

is

large compared

to the

spacing

^of

nuclei,

the

potential

can be considered continuous with 27rh~ b

~~

m V

where

b/V

is the

scattering length density,

the _sum of all nuclear

scattering lengths

b in _a volume V.

Inside _a uniform

layer

the

perpendicular

_component of the _wave vector ki is

given by

k; = 47r

For _an

infinitely

thick material of uniform

b/V,

total reflection of neutrons _occurs for k~ _<

k)

₌

47rb/V.

^In ^this case ki is

imaginary

^and ifi is evanescent in the material. For k~ > 47rb

IV,

above the critical _wave vector for total reflection kc, ki is real and the _wave propagates ⁱⁿ ^the material.

Consider _a

sample comprised

of three

layers

that form _a potential well with _a

scattering length density profile j(z)

_such

as shown

by

_a thin line in

figure

2. The

probability density

(4)

N°2 RESONANTLY ENHANCED NEUTRON INTENSITY IN POLYMERS 369

8 8 8 8

«

6 6/< 6

6/~

~

~ ~ ~O i' ~~

~w~ ^~

« ~w~ ^~ j

2 2

~

2 2

I

~

-

~

O o O

-2000 o 3000 -2000 O 3000

depth

I

_depth

I

a> b>

~~ ²

o

30 ₆ 7

[<

_j~ o 8

~

~° ' 4 ~°

j

o 6 ~i

~

i ^I ^' ^" Z

lo 2

~

o

I 2 _-

m

o

' _~

o ° ~

-2000 o 3000 Do

depth Ao oooo ooo3 ooo6 ooo9 oo12 oo15

C)

~

Fig.

2. The probability density, ~*~

(thick

line) _versus depth for _a given value of the per-

pendicular component ^of ^the neutron _wave vector, k, is superimposed _over the scattering length density profile

(thin line).

The potential is based _on numbers for deuterated

poly(ethylene

_propy-

lene) [dPEP]/poly(ethylene propylene) [PEP]

mixture. ~*~ is calculated assuming

b/V

has _an imagi-

nary component in the well of 0.00007

l~~,

a value based

on a well material of 90 i~ PEP and lo i~

dPEP. The surface barrier is 100 1 _thick _and _the _well _is ₁₄₀₀ 1 _thick. _The _effective _incident

energy E is shown with _an _arrow in units of b/V. In

(a)

k

= 0.0079 i~~ _and _is _above _the _critical

wave vector for total reflection kc

= 0.0051 l~~. _In

(b)

k

= 0.0035 l~~ _The _two _resonant _conditions

are shown in

(c).

The solid line is ~*~ calculated at ki _" 0.0030 l~~ _and _the _dashed _line _at _k2

= 0.0043

l~~,

the _wave vectors where the reflectivity R, plotted _versus k in

(d),

is

a minimum. An instrumental resolution of

6k/k

= 0.05 is assumed.

ifi*ifi(z)

is

plotted

in

figure

2a _as _a thick line for the condition when the incident neutron has E

(shown

with _an _arrow in units of

b/V)

less than the surface barrier

potent1al

_yet greater than the total reflection condition of the substrate Ec ₌

h~k)/2m.

The neutron tunnels

through

the surface barrier and propagates ⁱⁿ ^the ^low

b/V

mater1al _on the other side. In the well

region

ifi*ifi is _a

standing

_wave because of interference between the _wave transmitted

through

the barrier and the _wave reflected from the well-substrate interface. Since there is _no reflected

wave in the

infinitely

thick

substrate,

ifi*ifi ^is constant in that

region.

When the total reflection condition is met, ^E <

Ec,

and E is greater than the

potential

energy for the well

mater1al, Ewejj, standing

_waves with

resonantly

enhanced

intensity

_can

occur. In

figure 2b,

there is

tunneling through

the barrier with the

boundary

conditions _on

the _wave function at the well sides

resulting

in little constructive interference between the transmitted _wave and the _wave reflected

by

the substrate. The

intensity

inside the well is less than

just

outside the

sample

where ifi*ifi has a maximum of

4,

the condition _at total reflection

(5)

3m JOURNAL DE PHYSIQUE II N°2

fl120

(

~ _loo

~~' ⁸⁰ i'

E 60

)

w

40 E

I

₂₀

§

P o

o oo2 o oo3 o oo4 o oos

k

(I")

Fig.

3. The maximum value of the probability density, ~*~, in the well is plotted _versus ^the

incident _wave vector, k, for the scattering length density profile shown in figure 2. The attenuation of

~*~ because of incoherent scattering is not included in this calculation.

in the absence of loss mechanisms

(incoherent scattering

_or

absorption). However,

in

figure 2c,

a resonant condition is achieved. The well thickness and _wave vector of the _neutrons inside

the film _are such that nodes of the _wave function _occur at the well boundaries. The neutron

wave is

internally

^reflected many times,

constructively interfering

to

produce

_a

standing

wave

with _a maximum in the

probability density

ifi*ifi m 33 for the lowest order _resonance, shown _as

a solid thick

line,

_an enhancement of

nearly

ten times the maximum value outside the

sample.

The

higher

_energy resonance with two antinodes inside the well is shown _as _a dashed line in

figure

2c. The existence of _even

higher

order states is

prevented by

the shallowness of the well.

The resonant _wave vectors _are

analogous

to the

eigenstates

of _a _square

potential

well. How- ever, since the

potential

has _a finite thickness wall _on one side, the states are not bound but have _a finite lifetime. In

figure

3 the maximum value of ifi*ifi ^is

plotted

_versus the incident

neutron wave vector k for

)(z)

of

figure

2. At each resonant k there is _a

sharp peak

that _corre-

sponds

to _an increased lifetime in the well. The lifetime _can be estimated from the

Heisenberg

uncertainty

principle

~

m

~

" hkAk

For this

)(z),

the lifetime at the lowest order _resonance,

ki,

_Ti _* 1.30 _x 10~~ _s

and,

for

k2,

T2 * 0.27 x 10~~ _s. _We _take Ak _to be the FWHM of the

peak

at kn

(k;nc;dent).

If _no loss mechanism _were present in the

film,

R ₌ I at k < kc and the _resonances would be unobservable.

However,

if the well material is _a _neutron absorber _or

incoherently

scatters

neutrons, resonant

dips

_are found in R below kc. When the lifetime in the well increases _on

resonance the

probability

of

being

absorbed _or

incoherently

scattered increases,

decreasing

R at

the resonant _wave vectors. The

scattering length

^of a nucleus is modeled _as _a

complex quantity

b = a +

ill

where the

imaginary

component ^describes

absorption,

which is

significant

for _a few

naturally occurring

nuclei such _as

~Li, 1°B,

^ll~cd and

15~Gd,

_or incoherent

scattering

from nuclei with

spin

such _as

hydrogen (protons). Figure

2d shows _a linear

plot

^of R _versus

k,

for the

potential ) _(z)

_shown _in

_figures

_2a-c, _where

_b/V

_{of the} _well _material _is

_complex.

_{R is} _decreased

by

about 30 ~o at the first and 20 ~o at the second order resonant wave vectors,

assuming

an

instrumental resolution

Ak/k

₌ 0.05. We have observed

dips

in the

reflectivity,

indicative of

resonantly

^enhanced neutron

intensity,

in _a system that

naturally

creates _a

potent1al

well.

(6)

Experiment.

With neutron

reflectometry

_we

probed

the

j~(z)

^of a thin film blend of

hydrogenated

and

deuterated

polymer.

The small entropy ^of

mixing

^of

long

chain

polymer

molecules is

easily outweighed by

the

microscopic

difference between the two components ^{of the} system

producing phase separation

below _a critical temperature. ^In

addition,

at

high

^molecular

weights

the lower

polarizability

of the C-D bond

compared

with C-H bond encourages the deuterated

polymer

to segregate to the air surface. For _a

sample

with _one bulk

equilibrium phase

and

sufficiently long molecules,

surface

segregation [I, iii

of the

highly

reflective deuterated

polymer depletes

the thin film reservoir of

deuterium, reducing

the bulk

b/V.

For substrates with _a

b/V

greater than the bulk

polymer,

a

potential

well is formed.

The

polymer

system we studied _was _a mixture of

poly(ethylene propylene), (PEP),

with its deuterated

analogue (dPEP),

in which

nearly

all the

hydrogen

atoms _were

replaced by

deuterium. The components ^had

nearly

identical

degrees

of

polymerization, NpEP

= 2140 and

NdPEP _" 2360 [12], ^and were cc-dissolved in toluene _at _an init1al dPEP volume fraction of 0.137. The solution _was then _spun cast onto a silicon substrate that had been

stripped

of its native oxide and then annealed at 70 °C for 44 h to allow the surface

layer

to reach

equilib-

rium with the bulk

polymer.

The

samples

were cooled

quickly (m

4

min)

to a temperature ^of -55 °

C,

the

glass

transition temperature ^of PEP. Neutron reflection

experiments

_were carried out at

this,

temperature to prevent _any

change

in the

composition profile during

the _measure-

ments.

Figure

4a shows _a

plot

of R _versus k for this

sample.

The inset

enlarges

the

region

of k _< kc. The solid line

through

the data is _a fit based _on

j~(z)

^shown ^with a thin line in

figure

4b. The fit _was determined

by

_an iterative process [2] and _a detailed discussion of the

profile

will be

presented

in _a future

publication

[13].

Immediately,

_one notices in

figure

4a two

dips

in R below kc

corresponding

to the resonant conditions of the

potent1al

well shown in

figure

4b. The loss of reflected

intensity

in this

sample

is _a result of incoherent

scattering

from the

highly protonated

bulk

polymer

film _as the _wave

undergoes multiple

internal reflections. In

figure 4b,

_ifi*ifi is

plotted

with thick lines _versus

depth

for the two resonant _wave vectors. The maximum neutron

intensity

at the first order

resonance is about 64, sixteen times the

intensity just

above the

sample

^surface. For this

instrumental resolution of

Ak/k

m 0.05, R is decreased from i

by

about 17 l~. For the second

order,

the maximum ifi*ifi QS 27 with R decreased

by

8 l~.

Discussion.

In reflection

experiments

there is

always

_a

question

of

uniqueness

^of

fit,

_a consequence of the loss of

phase

information.

Although complementary experiments

with direct space

profiling techniques

_are

best,

certain features of the data _can

guide

^and limit the

possible profiles.

For

example,

_a noticeable feature of the data at k > kc _are

oscillations,

_or thickness

fringes,

_a result of interference between _waves reflected from the

air-polymer

and

polymer-silicon

interface. The

period

of these

fringes provides

_a

precise

determination of the total

polymer

film thickness

and,

with the initial film PEP concentration, _mass conservation _can be maintained for each

attempted

fit.

The resonant features

provide

_an additional constraint to the

fitting procedure

since the

position,

^width ^and

depth

of the

dips

_are

strongly

tied to the well

thickness,

the barrier thickness and the

hydrogen

density in the well mater1al.

Borrowing

from the _case of the infinite _square

well,

the resonant wave vectors or

eigenstates

_are located at kn

= 7rn

IL

where _n is the quantum ^number and L is the well thickness. In this

experiment

^the

positions

are shifted

(7)

jo

~~-l ^W

X'o~'

(

~o5

j~-2 ^f

~ ~

i

xli~

) ^li~

oj

G 2 3 4 5

~

~ -30-j

lo~ k(xlo A

j~-5

oooo coos cola o.ols oo20 oo25 oo30

k (Aa-j

a)

80 8

60 ⁶

«

(~

£H ~

~~

w ~~

7

>

1

I j

20 / ²

2 _-

I /

-

o ^o

-2000 o 3000

depth

I b)

Fig.

4. Reflectivity, R, is plotted _versus the perpendicular _component of neutron _wave vector, k, in figure

(a)

for the sample described in the text. The inset is _a log-linear plot in the region of

total reflection. Neutron reflectometry measurements _were performed using the POSY II reflectometer at the Intense Pulsed Neutron Source of Argonne National Laboratory _[26]. The spectrum of _wave

lengths in each pulse _ranges from 3 to 12 I. _Three _incident angles _were used to extend the range of k

probed. Error bars _are not included for clarity. The best fit _was determined by minimizing the x~ _per point. Values less than 6 for data taken at each angle _were achieved for the fit corresponding to the solid line

through

the data calculated from the

scattering length

density _versus depth profile shown in

(b)

with _a thin line. The neutron _wave function squared, ~*~, _versus depth for _a given value of

the perpendicular component ^of ^the neutron _wave vector is superimposed _over the scattering length density profile in figure

(b)

at ki

= 0.0028 l~~

(solid

heavy

line)

and at k2

# 0.0042 l~~

(dashed

line).

The effective incident energy E is shown with _an _arrow in units of

b/V.

(8)

f 200

~

w

~ lso

. .

i'

. .

~w . .

% loo

. . . .

~ . .

~ ~ . . .

§

. .

"

.

j

⁵⁰ _. .

"

~

.

E "

. ".

.

« .

~ . . .

a . .

~ o ^"

o loo 200 300 400 soo

Barrier Thickness

Fig.

5. Using the real and imaginary _components of the three layer potential of figure 2 the maximum achievable value of the probability density ~*~ at the antinode in the well is plotted for the resonant conditions ki

(squares)

and k2

(circles)

as the surface barrier thickness is increased while

holding the width of the the well constant at 1400 1.

slightly

to lower values of k

primarily

because _one side of the well

(that

formed

by

the dPEP

segregation)

is of

finite,

rather than _an infinite

potential height

and width.

Thus, subtracting

L from the total

polymer

film thickness

gives

_one _an estimate of the adsorbed surface

layer

thickness.

The amount of reflected beam lost to incoherent

scattering,

_can be correlated with the

concentration of the

hydrogen-containing

PEP and the enhanced

intensity

in the well. This information aids

fitting

since the bulk

polymer

in these

samples

has _a

b/V

less than that of the silicon and therefore does not

strongly

^affect the

shape

of the

reflectivity

above kc. For _a

qualitative understanding

_one _can examine the transmission

coefficient, T, through

_a barrier derived

using

the WKB

approximation

in the limit where the

"optical"

thickness of the

barrier, A,

is

large

_[14],

~n ~

~-2A

~~~~~

A "

~

lG(X)dX

and

~(x)

₌

) _jv(x) _Ej.

The barrier

shape

influences the transmission

through

the limits of

integration,

which define the effective barrier thickness at

E,

and the barrier

height, ~t(x).

Since

tunneling

occurs with

higher probability

for thinner barriers and

higher

neutron _en-

ergies

there will be _a barrier thickness where the neutron _wave function will have _a maximum enhancement determined

by

the balance of _two effects. Below this

optimum

thickness the barrier is

leaky:

the transmission

probability

of the neutron is

large

and the _wave enters and leaves the well

easily.

Above the thickness for maximum

enhancement,

the lifetime of the neutron in the well is

high

but the

probability

of it

entering

the well is small and the

resulting

(9)

enhancement is decreased. To illustrate this

point,

_we

plot

in

figure

5 the maximum value of the

probability density

in the well at the two resonant _wave vectors,

ki (squares)

and k2

(circles),

_as the surface barrier thickness is varied

()(z)

of

Fig. 2).

There is _a maximum at a barrier thickness of _m 150

1

_for

ki

For

k2,

the maximum enhancement _occurs for _a thicker

barrier than for

ki

at

(m

270

I)

_since

_higher

_energy _neutrons _are _more

effectively trapped by

_a

thicker barrier. In Jannink's paper [6]

analytic expressions

for the

dependence

of T _on _a _more

complicated profile

are derived and

presented

_as three-dimensional

plots

of T _versus _a reduced

wave vector and _a parameter

describing

the well

shape.

Thus

fitting

the resonant

dips

in the

reflectivity

below k

= kc

provides

added constraints _on the form of the

j~(z)

over and above those obtained

by fitting

the data _at k _> kc.

However,

the _resonance enhanced neutron

intensity

_can

potent1ally provide

_more than

just

information about the one-dimensional

scattering length density profile

when

coupled

with other

scattering techniques.

One

example

would be

experiments analogous

to those done

by

Wang

et al. with resonant enhanced

X-ray

induced fluorescence _[9] where _one could monitor

enhanced emissions of prompt _gamma or

alpha particles

from neutron

absorbers,

similar _to Neutron

Depth Profiling

NPD [15].

Thus,

the

depth

of _trace _amounts of _a

specific

type ^of nucleus could be

separated

from the overall

scattering length density profile.

Resonantly

enhanced neutron

intensity

also shows

promise

in

breaking

the

ambiguous

in-

terpretation

of

specular

data and

providing

information

complementary

to

j~(z) ^through

en-

hancement of the

off-specular scattering

that holds information about strucjure in the

plane

of the

sample parallel

to the incident

beam,

x-direction of

figure

I.

Length

scales _are estimated to be about _a micron for the

grazing

incidence geometry. Substant1al _progress is

already being

made to sort out the

origin

of

off-specular scattering

from these

in-plane

correlations. Wu

recently

derived

expressions

that relate

scattering

from interfaces with

in-plane density

fluc- tuations to the

perpendicular

structure

probed by specular

reflection _as _a _means _to limit the

possible potent1als

that describe R [16]. ^At

pulsed

neutron sources, where it is

possible

to

measure the

specular

and

off-specular scattering simultaneously, experimenters

_are

recording

rich patterns ^that

result,

for

example,

from strong correlations between _two

rough

^surfaces ^of

a

bilayer

_[17] _or from ordered but

imperfect

block

copolymer

films [18].

We also suggest ^the use of REN to assist in

elucidating in-plane

structure via enhanced off-

specular scattering.

Since the lifetime of the _neutron in the mater1al is

substantially

increased at resonance, the

plane

_wave

traveling parallel

to the surface will have _a

higher probability

of

sampling

the

in-plane

structure. With _a

complementary

mathematical

analysis,

_a

comparison

of the

off-specular scattering

on and off _resonance _may

provide

_a tool for

sorting

out the

origin

of the

off-specular scattering

and thus

help distinguish

between different features of the

in-plane

structure. To broaden the

utility

of this

technique artificially

constructed

potential

wells would optimize the lifetime _or

intensity

enhancement. Because the

degree

of confinement is affected

by

the surface

inhomogeneities phase separation

in

lipid layers,

where contrast is

achieved

through

selective

deuteration,

could be

investigated

with the membranes

deposited

on top ^of a low

scattering length density

well material.

In

addition, understanding in-plane

correlations in

polymer alloys

could benefit from the careful

study

of

off-specular scattering.

For

example,

the _crossover from one-dimensional to three-dimensional structures

during spinodal decomposition

in thin films in the _presence of

an attractive surface _can be

measured, building

_on the information from direct _space

depth profiling

_[19,

_20].

Additional

topics

^of interest include:

capillary

waves at immiscible

polymer

interfaces where the surface tension has been reduced

by copolymer

brushes [3,

21],

correlations within

grafted polymer

brushes

[22-24]

and deformation of

polymer

brushes [25]. ^These topics ^have ^been ^treated

recently by

theorists but have not been

experimentally investigated.

(10)

Enhancing

the

intensity

of incident neutron beams

using

resonant structures may well allow

in-plane

information _on these to be obtained.

Conclusions.

An attractive

potential

well for _neutrons _was formed in _a surface

segregated polymer

film between _a silicon substrate and _a

highly

reflective thin

layer

of deuterated

polymer.

At resonant

wave vectors, ^below ^the ^critical wave vector for total

reflection,

nodes of the _wave function _occur at the well boundaries.

Multiple

internal reflections increase the lifetime of the _neutron thus

enhancing

the

intensity

in the well. With

long

lifetimes the

probability

of incoherent

scattering

from the

mostly hydrogenated

well material increases

leading

to a decreased

intensity

at the resonant wave vectors. The

reflectivity

features

provide

sensitive additional constraints for

fitting

b

IV profiles.

In

addition,

the

resonantly

enhanced neutron

intensity

_may ^find

application

for

investigations

of

in-plane

correlations

through complementary analysis

of both the

specular

and

off-specular scattering.

Acknowledgments.

We

gratefully acknowledge primary funding

from the National Science Foundation

Polymer Program

^NSF-DMR grant ^number ^922-3099 ^with

additiona1support

for FSB _on NSF-DMR grant ^number 895-7386. LJN is funded

by

_a

Department

of Education

Polymer Fellowship.

We

appreciate

discussions with Sushil

Satija

and

Alamgir

Karim. The

reflectivity

measurements were done with the

help

of William Dozier at IPNS at

Argonne

National

Laboratory

that is

funded

by

the US

Department

of

Energy

under contract W-31-109-ENG-38. We benefited

from the _use of the National Nanofabrication

Facility

and the facilities of the Materials Science Center at Cornell

University

both of which _are

supported by

the NSF.

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