Polystyrene-Polylactide Bottlebrush Block Copolymer at the Air/Water Interface
pubs.acs.org/Macromolecules Published on Web 09/28/2009
r2009 American Chemical Society
Macromolecules 2009, 42, 9027–90339027
Polystyrene-Polylactide Bottlebrush Block Copolymer at
the Air/Water Interface
Lei Zhao,†Myunghwan Byun,†Javid Rzayev,‡and Zhiqun Lin*,†
†Department of Materials Science and Engineering, Iowa State University, Ames, Iowa 50011, and
‡Department of Chemistry, University at Buffalo, The State University of New York, Buffalo,
New York 14260-3000
Received July 23, 2009; Revised Manuscript Received September 17, 2009
ABSTRACT: Hydrophobic ultrahigh molecular weight bottlebrush block copolymer and linear block
and exhibiting unique assembly behaviors at the air/water interface, which cannot be addressed by the classic
theory of Langmuir monolayer of amphiphilic copolymers. New models were proposed to illustrate these
intriguing surface behaviors. The self-assembled structure of Langmuir monolayer of bottlebrush block
copolymer was determined by a combination of AFM measurement, thermal annealing, and enzymatic
degradation experiment. To the best of our knowledge, this is among few studies on hydrophobic block
technique to hydrophobic block copolymers.
As one of the most typical film preparation methods, the
Langmuir-Blodgett (LB) technology has been widely utilized to
produce copolymer films with mono- or multimolecule layers.1
These copolymer films with well-controlled thickness have been
attracting considerable attention due to their broad range of
potential applications in microlithography,2devices,3and bio-
mimetic thin films.4The overall property of these copolymer
Langmuir films is closely related to their surface morphology,
which is dictated by a number of parameters, including the
solution concentration, surface pressure, and temperature.2,5-7
The interfacial behavior of amphiphilic copolymers at the air/
water interface has been extensively studied since the pioneering
work of Eisenberg and Lennox.5,8-10The so-called “pancake”
and “brush” models have been established and proved quite
effective in understanding the air/water interfacial behavior of a
variety of amphiphilic copolymers, such as linear block copoly-
mers,2,6,7,11,12star copolymers,13-17comb block copolymers,18
dendritic polymers,19-21etc. The “pancake” model refers to the
morphology of copolymer LB film formed at low surface
pressure. In this model, the hydrophilic blocks spread over the
water surface, forming the pancake-like morphology, while the
hydrophobic blocks aggregate and sit on the hydrophilic “pan-
cake” to reduce the surface energy. Under higher compression
pressure, the “pancake” is transformed into the “brush”; i.e., the
spread hydrophilic blocks are expelled into the water subphase,
yielding brush-like morphology (“brush” model). To date, only
PI-PMMA,25have been reported to form LB monolayer in
which neither block is water-soluble; these copolymers have
surface active blocks with hydrophilic groups (e.g., ester group
in PMMA26), thereby possessing a relatively strong affinity on
the presence of hydrophilic or water surface active blocks is
essential for the formation of Langmuir monolayer because they
facilitate the tethering of whole copolymer chain on the water
surface. So far, self-assembly of LB films has been primarily
focused on amphiphilic copolymers. By contrast, few studies on
hydrophobic copolymers at the air/water interface have been
In this paper, both bottlebrush and linear block copolymers of
hydrophobic polystyrene-polylactide were found, for the first
time, to be capable of forming the Langmuir monolayers on the
water surface. The systematic studies showed that they displayed
unique assembly behaviors at the air/water interface, which can
no longer be understood by the classic models for amphiphilic
The polystyrene-polylactide (PS-PLA) block copolymers
were selected in the present study because of the biodegrable
nature of PLA blocks.27Thin films of these copolymers are
promising in biomedical and pharmaceutical application.28No-
tably, all previous research on PS-PLA block copolymers are
limited in their self-assembly in bulk and thin films,29-32and no
self-assembly of Langmuir monolayer has ever been reported.
Recently, a novel ultrahigh molecular weight PS-PLA bottle-
brush block copolymer (BBCP) was synthesized by a combina-
tion of living radical and ring-opening polymerizations.29
PS-PLA BBCP is a comb-like macromolecule with highly
backbone. The backbone is much longer than the branches. The
steric effect of adjacent branches causes the backbone to stretch
structure and large cross-sectional area of cylindrical shape of
BBCP result in little entanglement between the BBCP melts.34,35
The unique properties described above, together with their
large domain size and domain spacing originating from the
ultrahigh molecular weight of BBCP, make BBCP a perfect
candidate toexplore theirself-assembly behavior atthe air/water
*To whom correspondence should be addressed. E-mail: zqlin@
9028Macromolecules, Vol. 42, No. 22, 2009Zhao et al.
In this paper, the Langmuir isotherms and surface morphol-
ogies of BBCP were presented. The structure of deposited LB
films of BBCP was then determined by the AFM measurements,
thermal annealing, and enzymatic degradation experiments. A
linear block copolymer (LBCP) of PS-PLA with relatively
similar molecular length and composition ratio to BBCP was
chemical architecture on the air/water interfacial behavior of
PS-PLALangmuirmonolayers (Figure1b). Combined withthe
copolymersystemsatthe air/waterinterfacewasfinally revealed.
Previously synthesized PS-PLA BBCP with an ultrahigh
molecular weight (MW) of 1200000 g/mol was utilized in the
study.29PS-PLA LBCP with MW of 34000 g/mol and PS
volume fraction of 71.4% was synthesized according to a litera-
ture procedure.36The dimension of PS-PLA BBCP was esti-
using Material Studio 4.1. The length of PS-PLA LBCP in fully
form (99.9%, Fisher Chemicals) and Proteinase K (Sigma,
lyophilized powder) were purchased and used without further
purification. The BBCP and LBCP chloroform solutions at a
concentration c = 2 ? 10-3g/mL were prepared, and a 12 μL
solution was placed on the LB trough for each experiment.
Surface pressure-area (π-A) isotherms of block copolymer
monolayers were obtained with the R&K Langmuir-Blodgett
Before LB deposition, the LB trough was carefully cleaned with
1:1 H2O2:NH3OH solution overnight and subsequently rinsed
with DI water (NanoPure, >18 MΩ cm) five times. After
chloroform evaporated for 30 min, the monolayer film was
compressed at a rate of 150 μm/s.
Si substrate used for depositing LB films was cleaned with a
mixture of sulfuric acid and Nonchromix, followed by rinsing
substrate was withdrawn at a rate of 35 μm/s while keeping the
pressure constant. The scratch experiment was carried out by
the desired temperature for a certain amount of time. The
enzymatic degradation was performed by vertically immersing
the deposited LB film into the degradation solution, which was
buffer (pH = 8.6) at 37 ?C for 1 h in an oil bath.
Morphologies of LB films were examined by atomic force
microscopy (AFM; Dimension 3000) in the tapping mode. The
scanning rate was 2 Hz. Each sample was imaged at more than
five locations to ensure the reproducibility of features observed.
The interdomain distance was obtained by performing 2D fast
Fourier transform (FFT) of the AFM height images. The aggre-
gation number of the domains was estimated by the surface area
of domains and arms obtained from the AFM images divided by
the molecular area obtained from the isotherm.
Results and Discussion
The chemical structure and molecular morphology of
PS-PLA BBCP are depicted in Figure 2a,c. A detailed informa-
tion onBBCPcan befound inourprevious work.29The PS(red)
and PLA (white) branches are densely grafted along the poly-
methacrylate backbone through the ester group. The backbone
can be divided into two parts with the same length, namely, one
PS arms (i.e., PS part). The volume fraction of PS is calculated
to be 60% using the known densities of FPS= 1.04 and FPLA=
1.25 g/mL. The length of BBCP backbone, LBBCP, was about
90 nm (Figure 1a), and the lengths of PLA and PS arms are 6.3
and 6.5 nm (Figure S1), respectively, obtained by performing
energy minimization at room temperature using Material Studio
4.1. Because of steric crowding of PLA and PS arms, the BBCP
adopts a relatively rigid cylindrical morphology in the solid
state, with the aspect ratio of the length to diameter, L/D = 9
(Figure 1a). The chemical structure of PS-PLA LBCP is
shown in Figure 2b; the MW of PS and PLA blocks are
23000 and 11000 g/mol, respectively. The LBCP assumes a
random coil conformation in both the chloroform solution and
the melt state due to the absence of the steric effect of arms as in
the case of BBCP. The length of a fully stretched LBCP is
estimated to be 79 nm with PS block of 45 nm and PLA block
of 34 nm (Figure 1b).
Langmuir isotherm (i.e., surface pressure-area, π-A, plot) of
the BBCP is shown in Figure 3a. The continuous pressure
riseindicated the formationof Langmuir monolayer.1The entire
isotherm can be divided into three typical regions based on the
slope of the isotherm (i.e., the pressure increasing rate with the
molecular area).1,5They are (i) gas state region at π = 0 mN/m,
(ii) liquid state region at π = 0-13 mN/m, and (iii) condensed
state region at π > 13 mN/m. Figure 4 shows the representative
Figure 1. Schematic representations of (a) a newly synthesized bottle-
brush copolymer (BBCP) with PLA (yellow) and PS (blue) chains
densely grafted on the backbone (black) and (b) a fully stretched linear
in yellow and blue, respectively. The length and diameter of block
copolymers are labeled in the schematic.
Figure 2. Chemical structures of (a) bottlebrush copolymer (BBCP)
and (b) PS-b-PLA linear block copolymer (LBCP). (c) Molecular
morphology of BBCP established by Material Studio 4.1, in which PS
and PLA chains are labeled in red and gray, respectively.
ArticleMacromolecules, Vol. 42, No. 22, 20099029
AFM heightimagesofLangmuirmonolayersobtainedatπ= 1,
5, 13, and 30 mN/m; dot-like domains with a broad size
distribution were observed. Table 1 summarizes the domain
height, size, and their surface and aggregation number. With
the increase in surface pressure, the domain shape and the
aggregation number of the domains did not show obvious
change; however, the number of domains dramatically increased
with the surface coverage increasing from 29.84% at π =
1 mN/m to 66.12% at π = 30 mN/m. Moreover, the initially
dispersed dot-like domains transformed into the island-like
morphology at the condensed state region (i.e., region iii). It is
noteworthy that the domain height, however, showed little
difference at different surface pressures (Table 1).
At the first glance, the LB isotherm and dot-like morphology
the latter assumes the “pancake” morphology on the water
surface (i.e., the hydrophobic blocks aggregate into dot-like
surrounding the hydrophobic blocks). However, the fact that no
domain height difference was observed at different surface
pressures cannot be explained by the models of amphiphilic
copolymers that are described previously. According to the
theory of amphiphilic copolymers at the air/water interface, the
domain height should be largely increased with the increase in
surface pressure due to the formation of brush-like structure.5,11
Thus, to fully understand the surface behavior of the hydro-
phobic BBCP, it is necessary to systematically explore the fine
study of AFM measurement through a mechanical scratching
test, thermal annealing, and enzymatic degradation.
AFM Study. The LB film deposited at π = 5 mN/m was
gently scratched with a razor blade to remove a small portion
of the film and subsequently examined with AFM (Figure 5).
the exposed Si substrate and the interdomain region in the LB
film was about 0.7 nm (Figure S2). The result suggested that a
spreading phase should exist between the dot-like domains;
the LB film if the space between the dot-like domains was
empty (i.e., not occupied by the spreading phase). This was
further revealed in the AFM phase images (Figure 5b,c), in
the dot-like domain (i.e., domain in Figure 5c) were clearly
evident. In AFM phase images taken in the tapping mode, the
magnitude of phase shift is directly related to the elastic
phases in the LB film can be readily discriminated; one phase,
composed of the dot-like domain, appeared dark, and the
other appeared light and was in the form of cylindrical arms
surrounding the dark domains, which constituted the spread-
ing phase. These arms were closely connected to the dot-like
domain radially (Figure 5c). Moreover, the aggregation num-
ber of the domains estimated from isotherm andAFM images
microscopy technique. The length and width of the arm were
40 and 20 nm, respectively. Considering the twisting of poly-
of BBCP, either the PS part or the PLA part described
previously. It has been reported that both PS38and PLA39
are able to form the spreading phase at the air/water interface.
Therefore, to identify which part of BBCP composed the dot-
like domain and the other formed the spreading phases
adjacent to it, thermal annealing study was performed.
Thermal Annealing. The molecular weights (MWs) of
PS and PLA branches in the PS-PLA BBCP are 3000 and
Figure 3. Pressure-area isotherms of the Langmuir monolayer of (a)
state, and (iii) condensed state.
Figure 4. AFM height images of the BBCP Langmuir monolayers
obtained from the chloroform solution at various transfer pressures:
(a) π = 1 mN/m, (b) π = 5 mN/m, (c) π = 13 mN/m, and (d) π =
30 mN/m. Chloroform was allowed to evaporate for 30 min before the
LB deposition. Scan size = 3 μm ? 3 μm and z scale = 50 nm for all
images. Scan size = 0.5 μm ? 0.5 μm for the inset in (d).
9030 Macromolecules, Vol. 42, No. 22, 2009Zhao et al.
entanglement MWs, that is, 14000 g/mol for PS40and 4000
mostly free of entanglements above its glass transition
temperature, Tg.29,34,35DSC measurement showed that the
BBCP had two well-defined Tgs at 54 and 104 ?C, corre-
sponding to the Tgof PLA and PS, respectively.29Two LB
films deposited at π = 5 mN/m were annealed at 95 ?C for
12 h and 170 ?C for 5 h. Figure 6 shows AFM images of
each height image. The interdomain distance, λC-C, ana-
lyzed by performing 2D fast Fourier transform of the height
respectively. The domain height, however, remained the
same, and the shape of the arms was well maintained as
evidenced in the phase image (Figure 6d). On the basis of
these observations, it is clear that no obvious changes
occurred after thermal annealing at 95 ?C, which was far
above the Tgof PLA. However, when annealed at 170 ?C, a
temperature well above the Tg of PS, the domain
height markedly decreased, and the phase image was
dramatically changed (Figure 6f). Taken together, these
results suggest that the originally formed dot-like domain
consist of the PS part of BBCP, while the arm-like structures
were formed by the PLA part. When annealed at 170 ?C, the
PS chains within the PS part of BBCP became mobile. This
led to the separation of PS chains from one another and
subsequent merging with separated PS chains from the
neighboring PS part to yield a continuous PS phase, as
shown in Figure 6f.
It is worth noting that, as schematically illustrated in
blocks can be arranged to yield the images observed in
Figures 5 and 6. In model A, the domains are composed of
the PS part (i.e., PS domain), which are separated by the
spreading PLA part (i.e., PLA arms). By contrast, in model
B, the PLA arms situate around the PS domain as well as
beneath it; in other words, the PS domain sits on the top of a
continuous PLA layer. In order to determine which model is
appropriate for the self-assembly of PS-PLA BBCP at the
Table 1. Height, Size, and Surface Coverage of BBCP Dot-like Domains Obtained from AFM Images
π = 1 mN/m
π = 5 mN/m
π = 13 mN/m
π = 30 mN/m
3.67 ( 0.69 3.93 ( 0.873.64 ( 0.673.82 ( 0.89
68.75 ( 8.20 66.72 ( 7.7966.07 ( 4.87
29.84 37.95 53.0066.12
Figure 5. (a) AFM height image of the BBCP Langmuir monolayer
obtained afterscratchingoffthetopportionofmonolayer.Scan size=
(a). Phase scale = 50?. (c) Close-up of the phase image in (b); the
domain is enclosedwith red dashlineand armswith white. Scan size=
0.3 μm ? 0.3 μm and phase scale = 50?. The original Langmuir
monolayer was obtained at π = 5 mN/m.
Figure 6. AFM height and phase images of (a, b) original Langmuir
monolayer, (c, d) monolayer annealed at 95 ?C for 12 h, and (e, f)
monolayer annealed at 170 ?C for 5 h. (a), (c), and (e) are height
z scale=50 nm, and phase scale=50?. The original Langmuir mono-
layer was obtained at π = 5 mN/m.
Figure 7. Schematic illustration of two models PS-b-PLA block co-
polymer at the air/water interface.
ArticleMacromolecules, Vol. 42, No. 22, 20099031
air/water interface seen in Figures 5 and 6, an enzymatic
degradation study was conducted.
Enzymatic Degradation. The motivation of this study was
to selectively degrade PLA blocks in the LB film; thus, the
precise location of the PLA phase can be identified. PLA is a
biodegradable polymer which can be degraded by either an
enzymatic approach or alkaline hydrolysis.27,42Given the
fact that the alkaline hydrolysis may cleave the ester groups
that connected the PS branches to the backbone, enzymatic
degradation using Proteinase K was performed instead,
thereby keeping the PS part intact. As shown in Figure 8a,
the dot-like domain morphology was retained after being
soaked in the Tris-HCl buffer solution for 1 h. The morpho-
of water or the shearing effect when the LB film was with-
with Proteinase K for 1 h, although the circular domain
existed, they gathered together to yield large aggregates,
leaving behind nothing in the region between the aggregates
(Figure 8b,c). The enzymatic degradation result, first of all,
confirmed the fact that the dot-like domain was made up of
the PS part; otherwise, it would have degraded and dissolved
in buffer. More importantly, this result suggests that for the
self-assembly of PS-PLA block copolymers at the air/water
interface model B is the mechanism that underpinned the
observed morphologies in Figures 5 and 6. According to
model A, when PLA phases are removed by enzymatic
treatment, the PS domains should be well retained on the
Si substrate. In marked contrast, if the LB film adopts the
arrangement depicted in model B, the PS domains can be
readily stripped off the substrate after degrading PLA be-
substrate,43a water layer may thus form on the Si substrate
during the PLA degradation. The decreased interaction
between the Si substrate and PS can lead to the aggregation
of PS domains which was composed of dot-like domains of
Thus, we believe that model B better describes the morpho-
Interfacial Behavior of PS-PLA Block Copolymer. Upon
the evaporation of chloroform in the gas state region (π =
like domains to reduce its surface energy (Figure S3), where
only extremely small PS domains exist in the LB film.
Although PLA is a hydrophobic polymer,39it contains ester
water and results in attractive interaction between PLA and
water subphase as compared to PS.39Therefore, the PLA
part spread over the water subphase, contacted with one
another, and prevented the PS parts from aggregating. A
monolayer was thus formed at low pressure (Figure 9a).
With the increase in surface pressure (i.e., in the liquid state
region ii in Figure 3a), there was a strong elastic repulsive
force between the PLA parts; this is analogous to the
behavior of spreading phase in amphiphilic block copoly-
mers.5,10As a result, a continuous pressure increase was
observed in this region. However, due to the intrinsic hydro-
phobic nature of PLA,39it can hardly be expelled into the
water subphase to form brush-like morphology as com-
monly seen in amphiphilic block copolymers. Thus, PLA
arms that surrounded the PS domain had to rearrange
themselves to release the elastic stress (Figure 9b). The
surface area of the spreading PLA phases was largely de-
creased via interdigitation, leading to the increased density
of dot-like domains (Figure 9b). Finally, the PLA arms
were strongly compressed in the condensed state region
(Figure 9c). The condensed LB film was incompressible,
and the surface pressure increased dramatically with little
decrease of the molecular area (i.e., region iii in Figure 3a).
Under this high surface pressure, the PLA arms were ex-
pelled apart (Figure 9c); the PS domains became connected,
yielding an island-like morphology (Figure 4d). Because
certain PLA arms may still exist between the PS domains,
the PS domains in the island was loosely packed as shown in
the inset of Figure 4d. By contrast, under high surface
pressure, amphiphilic block copolymers form more contin-
uous island morphology, as the spreading phase was com-
pletely expelled into water subphase.44
To explore the effect of bottlebrush structure on the
formation of LB films, a linear block copolymer (LBCP) of
PS-PLA with close molecular length and composition ratio
to BBCP was investigated for comparison. The LBCP had
indicating that LBCP has similar assembly behavior at the
per molecule; this is due to the size difference of two
copolymers. The AFM images of Langmuir monolayers of
As evidenced in Figure 10a, dot-like domains with relatively
well-defined shape originating from the flexible nature of
LBCP chains were observed; the domain size and height are
59 and 3.35 nm, respectively, similar to that of BBCP at the
same surface pressure (i.e., 68.75 and 3.67 nm, respectively;
in LBCP (i.e., 71.4%) than in BBCP (i.e., 60%), the PS
domains in LBCP were found to have a large surface cover-
age of 37.36% compared to 29.84% in BBCP. No arm-like
structures were seen in the phase image (Figure 10b); this is
because the linear PLA blocks are flexible and their MW of
entanglement.41Thus, the PLA blocks entangled and spread
on the water surface (Figure 11a). On the other hand,
bottlebrush PLA blocks in BBCP are rigid and have a large
Figure 8. (a) AFM height image of Langmuir monolayer after immer-
sion in the buffer solution without the addition of enzyme for 1 h. (b)
AFM height image of Langmuir monolayer undergoing enzymatic
degradation for 1 h. (c) Corresponding phase image of (b). Scan
size = 3 μm ? 3 μm, z scale = 50 nm, and phase scale = 50?. The
original Langmuir monolayer was obtained at π = 5 mN/m.
Figure 9. Schematic stepwise representation of the packing of micro-
structures of BBCP: (a) at the low pressure (i.e., between regions i and
ii), where PLA arms (yellow) highly spread over the water surface and
PS (blue) form domains on the top of PLA; (b) at the intermediate
pressure (i.e., region ii), where the rearrangement of PLA arms occur;
and (c) at the high pressure (i.e., region iii), where the PLA arms are
highly compressed and PS domains become connected.
9032 Macromolecules, Vol. 42, No. 22, 2009 Zhao et al.
cross-sectional area so that they can be readily visualized by
AFM (Figures 1a, 5, and 6).
contrast sharply with amphiphilic copolymers11and the
BBCP in the present study; the latter showed little change
in the domain height as thepressure increased (Table 1). The
decrease of domain height can be attributed to the folding of
chains have been reported on the assembly of linear
PLA-PEO block copolymer at the air/water interface.39
Because of the folding of PLA chains as a result of applied
compressive surface pressure, the surface stress was released
and the space between PS domains was filled with PLA
chains, thereby leading to the reduction in the height differ-
ence between the PS domain and the spreading PLA phase
(Figure 11b). At high pressure, π = 13 mN/m, the folding of
PLA chains may be completed (Figure 11c), and the inter-
domain space was fully occupied by PLA chains. Conse-
quently, a smooth surface of LBCP Langmuir monolayer
was yielded (Figure 10e); however, the light PS domain and
the dark PLA phase can still be readily resolved in the
corresponding phase image (Figure 10f). In the case of
BBCP, the PLA bottlebrushes cannot be folded due to the
present of rigid backbone; thus, the PS domains were still
highly dispersed at π = 13 mN/m (Figure 4c), and no varia-
tion in the domain height was observed (Table 1). At very
high pressure, π = 30 mN/m, the LBCP LB film was highly
compacted (Figure 10g,h), and the surface pressure drama-
tically increased with a slight decrease of molecular area
(region iii in Figure 3b).
as linear PS-PLA block copolymers. These hydrophobic block
copolymers exhibited unique self-assembly as a function of sur-
face pressure at the air/water interface. The self-assembled
structure of a Langmuir monolayer of bottlebrush block co-
polymer was determined by combining the studies of AFM
measurement, thermal annealing, and enzymatic degradation.
The influence of chemical architecture on assembly behavior of
explored. Rather than undergoing “pancake” to “brush” transi-
tion as commonly seen in amphiphilic block copolymers, for the
pressure transformed into interdigitated and eventually island-
Langmuir monolayer was unchanged despite the increase in
surface pressure. On the other hand, for the PS-PLA LBCP,
the pancake-like morphology produced at the low pressure
transitioned into topologically featureless morphology at the
high pressure through the folding of flexible PLA chains. The
present study not only complements the well-known models of
self-assembly of amphiphilic block copolymers at the air/water
interface but also expands the use of the Langmuir-Blodgett
technique to hydrophobic block copolymers.
Figure 11. SchematicillustrationofthepackingofmicrostructuresofPS-
b-PLA LBCP at different surface pressures: (a) at the low pressure (i.e.,
water surface and PS (blue) form domains on the top of PLA); (b) at the
more PLA to occupy the space between PS domains; and (c) at the high
pressure (i.e., region iii), where the folding of PLA chains is complete,
thereby leading to the formation of a topologically continuous film.
Figure 10. AFMheightimages(a,c,e,andg)andcorrespondingphase
images (b, d, f, and h) of Langmuir monolayers of PS-b-PLA LBCP
obtained from the chloroform solution at various surface pressures: (a)
and (g) and (h) π = 30 mN/m. Chloroform was allowed to evaporate
30 nm, and phase scale = 50?.
ArticleMacromolecules, Vol. 42, No. 22, 2009 9033
Acknowledgment. We gratefully acknowledge the support
from the National Science Foundation (NSF CAREER Award,
of PLA and PS arms of BBCP, section analysis of AFM height
material is available free of charge via the Internet at http://
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