1

Single Polymer Chain Stretching by Atomic Force Microscopy

Jason E. Bemis

Submitted to the Graduate Faculty of Art and Science in

partial fulfillment of the requirements for the

degree of Doctor of Philosophy.

Department of Chemistry

University of Pittsburgh

1998

2

Table of contents

Specific Aims

Summary

Polymer Theory

< Freely jointed chain

< Worm like chain

< Tube model

Previous work

< Overview

< Polymer studied< 15F-6F-HB< PS-P2VP

< Conclusions< RTV11®

< Conclusions

Future work: stretching of end grafted polyelectrolyte chains< Summary

< Theory

< Synthesis

Conclusions

Acknowledgments

References

Page

4

5

6

9

11

13

1616282834

36

38

41

45

45

46

3

Glossary of terms and abbreviations

Appendix A< Solvent effects.

< Perturbed characteristic ratio< Perturbed persistence length

Appendix B< Detailed experimental of PS-P2VP< Detailed experimental of RTV11®

Appendix C< Additional Figures

49

5354

5555

58

4

Specific Aims

< Directly measure the force of rupture for a single covalent bond.

< Test molecular polymer theories pertaining to the force of uniaxial extension through

various media.

< Develop a comprehensive methodology and quantitative analysis for the manipulation of

single polymer chains.

6

Summary

Structural material is increasingly being constructed from polymer and composites as

alternatives to metal or other traditional sources. This provides a new challenge for engineers as

polymeric material often possesses novel mechanical properties with respect to previously used

material. Polymers often have a non-linear response to mechanical strain, exhibited in the yield and

fracture characteristics uncommon to metals. Computer aided design has become an advantageous

tool in the design of new polymer systems, which requires a detailed understanding of the material.

The theory of such systems often stems from molecular dimensions, but is tested on bulk material.

The recent development of AFM to manipulate single molecules1-5 offers the opportunity to test such

theories on the most fundamental level. We have shown that the ability to manipulate polymer chains

is common to many systems, such as coblock, random block, and homo polymers, as well as industrial

polymer blends such as RTV11® , the top coat of naval vessels.

Polymer physics is a rapidly growing field in response to the increasing demand for alternative

structural material. This work details the further study of the elastic properties of such polymeric

materials.

Polymer Theory

The understanding of adhesive properties of two surfaces in contact requires the detailed

comprehension of the molecular interactions between the surfaces. This is necessary in numerous

applications from the agglutination of two surfaces to the preparation of antifouling surfaces.

7

A B

Schematic representation of polymer attachment to anAFM tip. A, with a single chain attached; B, with multipleattachments.

Previous AFM studies of adhesion have focused on the intersurface properties from both

experimental and theoretical standpoints.6-11 Recent developments in AFM have allowed for the

manipulation of single polymer chains,1-5 providing an unprecedented opportunity to test

molecular theories of adhesion. We have further developed the study of molecular adhesion, by

investigating the interaction of a single commercial polymer chain between the tip of an AFM

and the surface.

Figure 1 shows a

schematic representation of

polymer attachment to an AFM

tip. If one or more chains are

attached to the tip after the tip

jumps free of the surface, the tip

will experience an attractive force

away from the surface until it

breaks free from the polymer chains. AFM has been shown to be capable of elastically

stretching single proteins when specific ligand-receptor pairs are used to bind the proteins to the

tip.5 Proteins have also been unfolded by reversibly attaching them to the tip. 4

8

Bond angle

Dihedral angle

Theta(Θ)

Bond length(Li) r

Freely jointed chain with random bond and dihedralangles. The bond length is equal tot he persistencelength and ½ the Kuhn segment length.

r'jni'1 Li(1)

+r 2,'nL 2(2)

Cn'+r 2,

nL 2(3)

The basis of many single polymer theories stem from Flory’s Freely Jointed Chain (FJC)

model.12,13 This model assumes that chemical bonds are free to rotate and posses a uniform

distribution of bond angles. The end-to-end distance, or chain vector (r) is described by Equation

1. This is the summation over all bond vectors (Li), the vector of the ith bond, over all monomers,

where n+1 is the degree of polymerization, Figure

2. The uniform distribution of bond angles

necessitates that the average chain vector over all

conformations is zero. The square of the chain vector averaged over all conformations however,

has a finite value. With the assumption that L is the average bond length the average square chain

vector simplifies to Equation 2, where the brackets denote the statistical average over all

conformations. Flory uses this relation to define the characteristic ratio (Cn) of a polymer,

Equation 3.12

By definition Cn is unity for freely jointed chains,

for other models, which do not assume that the

bond angle is free to rotate, Cn exceeds unity, and

will be discussed later.

9

+s 2,'+r 2,

6(4)

f' kTAk

ý((R)(5)

Ak'LCn(6)

Cn'2Cn(7)

Based on viscosity studies conducted by Debye, it is well known that +r2, is related to the

radius of gyration by Equation 4.14 In the absence of solvent effects the elastic properties

of such an ideal chain are given by the Kuhn statistical segment length (Ak), Equation 5. The

entropic force of uniaxial extension (f) is described by the inverse Langevin function (ý*), where

ý is defined as coth(R)-1/R. R is the extension ratio, the fraction of the contour length that the

polymer is extended. The freely jointed chain is defined as having a Kuhn length equal to the

bond length, Equation 6.

When solvent effects are taken into account Ak is given by the perturbed characteristic ratio, CNn,

multiplied by L. CNn is described by α, the intramolecular expansion factor, by Equation 7

(appendix A).

10

Cn'1%1&

&

( 2n

)(1& n)

(1& )2where: 'cos( )(8)

Angstroms

Angstroms

0 2 4 6 8 10 12 14-5050

2

4

6

8

Angstroms

Worm Like Chain-2 -1 0 1 2 3-50

5-4

-2

0

2

Angstroms

Freely Jointed Chain

-2 0 2-2

0

2

4

Freely Jointed Chain

Ang

stro

ms

Angstroms

Worm Like Chain

-5 0 5

0

2

4

6

Angstroms

Ang

stro

ms

A

B

C

D

A: Freely jointed chain in three dimensionsB: WLC with a fixed bond angle of 109.5 in three dimensionsC: FJC when confined to two dimensionsD: WLC with a fixed bond angle of 109.5 when confined totwo dimensions.

A more realistic model

for many systems is that of the

Freely Rotating Chain, or

Worm Like Chain (WLC). This

model assumes that the bond

angles are fixed at 180−θ, but

are free to rotate, giving rise to

a uniform distribution of

dihedral angles. Random

conformations were calculated

for both models and are shown

below (Figure 3). It is difficult

to see the three dimensional

effect of the rigid bond angle in

the two dimensional figure, so

the hypothetical case of

restricting the dihedral angle to

0 or 180, to force the molecule

into a plane was used to illustrate

the two models (Figure 3c and

11

Figure 4Cn as a function of degree of polymerizationand bond angle.

d). The restriction on the bond angle gives

rise to Equation 8. In the limit where n64, Cn

reduces to C4

, Equation 9. If n>200 or so Cn

is approximately equal to C4

. The effect of the

bond angle and degree of polymerization on Cn

is shown in Figure 4. The force of extension

for a WLC is given by Equation 10.15 The

characteristic length describing the force of

extension in the WLC model in is the

persistence length (q). This length is defined by Equation 11.16 The persistence length is the sum

12

C4

'

(1% )(1& )

(9)

Figure 5Schematic representation of the tube model. The red circles are fixed cross links constrainingthe available positions of the chain. Thepolymer is free to move within the green shadedarea.

q'+rCuo,'limn64L

1&(11)

f' kTq

( 1

4(1&R)2&

14%R)(10)

fc'NckT( &1/ 2)[1& 2

(1& 2 )2&

2

1& 2](12)

'

Ak

D'

2%

2

of the projections of the bonds onto the unit

vector (uo) of the polymer chain. The unit

vector of

the polymer chain is the vector of the first

bond divided by the bond length.

Essentially q

described how far the polymer extends in a

given direction before becoming random.

There are many other polymer

stretching theories. One of the more

successful has been the tube model developed

by Edwards and Doi.17-19 This model

describes the polymer as a tube which is free

to move within a network of fixed constraints,

Figure 5. On the fast time scale the polymer is

moving within the cross link confined tube,

13

fs'NskT( &

12)[

(1& 2)(1% )

(1& 2 )%

(1& 2 ) 2(1& 2 )

(1% 2)2( % )2%

2[2

1% 2%

2%

]%(1% 2)( % )

&

2

1& 2](13)

Figure 7Two chains entangled can slip along theslip link until another entanglement isencountered

1

λ

Figure 6At rest (red) the extension ratio (λ) is 1. Upon extension (blue) the extension ratioincreases by the fraction of the length atrest.

effectively filling the volume of the tube. On a

slower time scale the tube is allowed to move within

the network, giving it a net displacement, termed

reptation. This model gives rise to a similar force-

extension law tot he WLC and FJC models. The

total force is given by the sum of the cross link

constraints (fc) and the slip link constraints (fs). The

slip links are simply cross links that are not fixed

in space. The cross link contribution to the force

of uniaxial extension is described by Equation 12.

Nc is the number of cross links per length of the

tube, λ is the extension ratio, but is very different

than the previous extension ratio used with the

FJC and WLC models. Here the extension ratio is

defined as the distance the tube is extended over

the length of the tube at rest, Figure 6. λ ranges from unity to λmax, where λmax is the tube

diameter (D) over Ak. There is also a slip link contribution to the force of extension, Equation 13.

The slip link is depicted schematically in Figure 7. The polymer chains are allowed to move

13

Figure 8Force vs. Distance profile for PS60800-P2VP46900,the green line is the tip approaching the surface, the redline is the unloading curve. The solid blue line is theWLC fit to the data, the dotted magenta line is the FJCfit.

20

40

60

80

100

120

140

0 50 100 150 200 250 300length of chain [nm]

me

an

len

gth

of e

xte

nsi

on

[nm

]

Figure 9Four molecular weights of PS-P2VP (Table 1). Thex axis is the polymer chain length, where the valuesare estimated from the molecular weights asdescribed in the text. The lower blue solid line is themean tip-sample separation at the rupture of theelastic response. The upper red dashed line is thecontour length of the chain between the tip andsurface estimated from the maximum extensionratios obtained from the WLC model fits.

through the slip link, but are still

confined by the entanglement. The

measure of the slip link is η, when

η=0, denotes the absence of the

constraint, and when η=1, the slip link

acts as a cross link. η is typically

0.23.19 Ns is the number of slip links

per length of the tube.

There are many more polymer

stretching theories, such as Rotational

Isomeric State (RIS), which places

restrictions on the available dihedral

angles in the WLC model.20 There are

also many theories pertaining to volume exclusion

and solvent quality.21,22 The discussion of these

theories is beyond the scope of the current work.

Previous Work

Overview

A wide variety of polymers have been studied

including block copolymer, random block

copolymers and homopolymers. All of which

14

exhibit the characteristic force distance profile of polymer stretching, Figure 8. These different

polymer systems all exhibit polymer stretching, but with very different properties. Polymer

stretching was first observed in our research group by Boris Akhremitchev, while studying a

fluorinated amide-urethane random block copolymer.23 This prompted us to study the molecular

weight dependance of the length of the polymer stretching, which was accomplished through

studying polystyrene-b-poly-2-vinyl-pyridine (PS-P2VP). 24 There was a linear increase in the

mean length of stretching with increasing molecular weight, Figure 9. Through the statistical

analysis of the data it was demonstrated that the majority of the extension events constitutes the

stretching of a single polymer chain tethering the polymer surface to the tip. This was concluded

based on the persistence length obtained from fitting the data to the WLC model. The median

value was 2.9 D, in close agreement with the theoretical persistence length for a carbon-carbon

polymer backbone, 2.31 D, Figure 10. The polymer diameter was also calculated from fitted

values to the data, giving a single molecule dimension of 2.5 D, Figure 10. Lastly the length of

extension was compared to the estimated contour length of the polymer, Table 1. This shows that

very few (<5%) of the polymer stretching exceeds the contour length, which can easily be

accounted for by the polydispersity of the polymer.

Based on the success of PS-P2VP system, other polymers were investigated. RTV11®, a

15

Figure 10The red histogram represents persistence lengths obtained frommulti-chain WLC fits to elastic responses. This histogram is thesum of all polymers studied as there is no noticeable difference inpersistence lengths between molecular weights of blockcopolymers or homopolymers (see Table 2). The blue histogramis the corresponding chain diameters to the persistence lengths.

polymer with technological

applications as a fouling release

coating, was investigated to

determine the effect of polymer

stretching on biofouling.25 RTV11® is

a duplex polymer coating consisting of

polydimethyl siloxane (PDMS) and a

CaCO3 filler developed as a top coat

for marine vessels.26 It was shown that

the soft PDMS sample exhibited

polymer stretching which consisted of multiple chains. This was determined from the unrealistic

values obtained from fitted values. It was also shown that the due to the softness of the sample,

as well as the number of chains contributing to the stretching, that the total adhesion of the

organic particle to the surface was five times greater on the soft PDMS surface than on areas

supported by the rigid CaCO3 filler. These results were compared to those obtained from an end

grafted monolayer, which showed that PDMS does exhibit molecular level properties when the

number of chains stretched are few, or one.

These results show that polymer stretching is common to many systems. This

phenomenon has not been extensively been studied. It is clear that the capabilities of AFM to

investigate these properties needs to be established, so that a systematic investigation can be

conducted.

16

17

Figure 11Force plot taken at the edge of a micelle on thesurface of a fluorinated amide urethanepolymer.

15F-6F-HB

The fluorinated amide-urethane random block copolymer (15F-6F-HB) was synthesized

and characterized in the laboratories of Dr.

Toby Chapman.27-29 The major finding in our

study was that the stretching events were

laterally localized surrounding the edges of

micelles.23 Most of the polymer systems

studied so far have not exhibited this lateral

inhomogeneity, and as such have not shown

this lateral behavior in the stretching. It was also apparent that the majority of the stretching were

of a constant force with displacement, Figure 11. This type of behavior was also observed in the

PS-P2VP and RTV11® polymers studied, and is believed to be a slightly different response of the

polymer, which will be discussed in more detail later.

PS-P2VP

Experimental

PS7800-P2VP10000, PS13800-P2VP47000, PS52400-P2VP28100, PS60100-

P2VP46900, and PVP50000 were purchased from Polymer Source (Dorval, Canada) all with a

polydispersity index less than 1.11 and were used without further purification. PS29100 was

18

purchased from Aldrich with a polydispersity index of 1.04. Blocks are designated by PS for

polystyrene and P2VP or PVP for poly-2-vinyl-pyridine. Molecular weights of the individual

blocks are given after the designation for the block in units of Daltons .

Table 1

Summary of strand elongation length and force

polymer length of

blocks in nm

[total length]

mean

length of

elastic

response

[nm]

maximum

length of

elastic

response

[nm]

median

rupture

force of

elastic

response

[pN]

elastic

event

probability

relative

trigger

[nm]

z scan

size

[nm]

z scan

rate [Hz]

PS7800-

PVP10000

19/24 [43] 24.5 59.3 170 ±

40

0.30 50 130 4.88

PS13800-

PVP47000

33/112

[145]

42.0 92.6 210 ±

30

0.03** 30 300 5.58

PS52400-

PVP28100

126/67

[193]

44.5 94.1* 150 ±

25

0.52 100 300 1.95

PS60100-

PVP46900

145/112

[257]

68.1 268.8 90 ± 14 0.24 100 500 1.95

PS29100 70 53.5 111.1 280 ±

30

0.22 100 300 4.88

PVP50000 119 65.2 125.6 120 ±

20

0.44 100 500 9.77

All force measurements were conducted with a Digital Instruments (Santa Barbara, CA)

Nanoscope IIIa Multimode scanning force microscope, equipped with a fluid cell, in 10mM

19

sodium acetate. Force volumes of 16x16 force distance plots, each with 512 points, were

collected with Nanoscope software and analyzed using custom software written for Matlab (Math

Works Inc. Natick, MA). Persistence lengths and maximum extension ratios were obtained from

fitting the WLC model to the elastic response curves and minimizing χ2. Kuhn length and

maximum extension ratios were obtained from fitting an approximation30 of the FJC model to

the elastic response curves and minimizing χ2. Relevant experimental scanning parameters are

given in Table 1. Full experimental details are given in appendix B.

Results

The surface topography of the block copolymers is irregular with sporadic height features

on the order of 100 nm. There are ample areas (~1µm2) that are devoid of height features greater

than 15 nm. All force volume measurements were conducted in such areas so as to remove height

aberrations from the force plots. There are small (~50 nm wide, 5 nm high) ordered structures,

which may possibly be micellular in nature.31 These features are not observed in air, although this

may be due to the higher scanning forces induced by capillary forces. The topography of the

homopolymer appears similar to the block copolymer, with the absence of the aforementioned

structures. All samples were scanned in sodium acetate to decrease the force of adhesion by

means of the double layer formed around the silicon nitride tip. This decrease in force of adhesion

(1.7 nN to 1.1 nN) decreases the distance the tip travels when it jumps free of the surface, in turn

increasing the distance over which data is obtainable.

20

Figure 8 represents a typical force plot in which an elastic response is observed. At point

A in the retracting plot, the tip jumps free from the surface. It slowly goes towards zero

deflection until point B, 45 nm above the surface, where the tip begins to experience significant

elastic tension from the attached polymer chain. At point C, 58 nm from the surface, the chain

breaks free from the tip. This point is taken as the point of rupture, giving the length and force of

the elastic response, 58 nm and 180 pN in this case. After this point, the tip returns to zero

deflection (point D) for the remainder of the z travel (360 nm).

Table 1 summarizes the statistical analysis of data collected. The lengths of the blocks are

estimated from the covalent radii of carbon (0.77 Å), the C-C bond angle of 109.5o, and the

number of monomers for the chain. The data represented in Table 1 is comprised of different

types of elastic responses. The single elastic response shown in Figure 8 is the most frequent

response observed. Occasionally, multiple elastic events are observed in a single force plot, as

shown in Figure 12, which represent a smaller, but appreciable, content of the data. In rare cases,

force plots are observed where the same chain is repeatedly strained as seen in Figure 13.

Both the FJC and WLC models can be fit to the elastic response of the polymer samples,

as shown by Figure 8, the FJC being the dotted line. The statistical analysis of fitted values

obtained from both models is summarized in Table 2. When the maximum extension ratio is a free

parameter, the major difference in the two models is in the low force, low extension ratio regime

(from 25 to 45 nm tip-sample separation in Figure 8). The noise in the data (10-20 pN standard

deviation) at this low force, and the quality of the resulting fits to the two models precludes the

choice of one model over the other, as demonstrated by χ2 values. This is not entirely surprising

21

D'4 32qkTE

(14)

as the WLC model requires only a small persistence length to fit the data well. The noise also

results in shallow error functions for the persistence and Kuhn lengths.

We now examine the supposition that the majority of the elastic responses are the probing

of one polymer chain. It is possible to estimate the number of chains between the tip and the

surface based on the length and diameter. The diameter of the chain can be estimated from the

Young modulus of the bulk and the persistence length, Equation 14.

Here D is the chain diameter, and E is the Young modulus (3

pN nm-1 for polystyrene). Figure 10 shows the persistence

lengths and corresponding diameters for the multi-chain WLC

fits. Kuhn lengths and corresponding diameters can be found in appendix C. The fourth root of

the persistence length in Equation 14 compensates for the large uncertainty of this parameter, so

that the diameter of the chains have been accurately determined to be 2.5 Å ± 0.5 Å. This

molecular scale diameter strongly suggests that a single chain is being extended, or that the chains

are being stretched in series.

An analysis of the length of the elastic response shows that it is unlikely that the chains

being extended are in series. Figure 9 shows the length of the elastic response as the lower line,

which is a linear increase with chain length with a slope of 0.2. The slope and its magnitude are

presumably a consequence of chain entanglement. Longer chains are more likely to be entangled

and the length to which they may be extended is proportionally smaller than the full length.32 The

upper line in Figure 9 is the estimated contour length of the chain between the tip and the surface

based on the extension ratio obtained from the WLC fits. The estimated contour length is

22

f. V

Z(15)

calculated by dividing the length of the elastic response by the maximum extension ratio. The

extension ratios obtained with the FJC are higher, and so the estimated contour length of the

chains based on that model will lie between the two lines shown. This estimated contour length of

the chains is significantly less than the molecular weight-based estimate of the length of chains

(Table 1). This discredits the interpretation that chains are being extended in series, in turn

reinforcing our conclusion that most of the time a single chain is being stretched. The maximum

length of the elastic response given in Table 1 shows that occasionally (~3%) there is an elastic

response that exceeds the estimated length of the chains. This can be accounted for by the small

amount of polydispersity.

We have also considered volume exclusion effects. In

poor solvent conditions and low tension, a polymer chain will

collapse upon itself, forming a series of Pincus blobs.21 This is

particularly important at low extensions and has a predicted force-distance dependence as

described by Equation 15.22 Where γ is the interfacial tension, V is the volume of the polymer

globule that remains on the surface, and Z is the distance the chain has been pulled out of the

globule. This behavior is apparent in Figure 8 from 15 to 25 nm tip sample separation. Our

experiments were not optimized to collect data in this region of the force curve, but fits to

Equation 15 are reasonable. As this is the region in which volume exclusion effects are prevalent,

we have ignored volume exclusion effects at higher extensions; regions of low tip sample

separation and decreasing force with distance were not included in the fits to the elastic models.

Figure 12 shows multiple elastic responses in a single force plot. We consider two

23

Figure 12PS60800-PVP46900 collected with a 100nm trigger, 500nm z scan size and 1.95Hz z scan rate.

A: The multi-chain, multi persistence length WLC fit is shown as the solid line with the individual responsesshown as the thin dashed lines. From left to right the persistence lengths and maximum extension ratios are0.19nm and 0.75, 1.2nm and 0.86, 2.2nm and 0.88, 6.3nm and 0.94, 2.1nm and 0.92, 0.47nm and 0.88. The fitgave a χ2 of 0.321.

B: Single chain fit with a single persistence length of 0.44nm and χ2 of 0.59. From left to right the maximumextension ratios are 0.86, 0.78, 0.73, 0.77, 0.84, 0.87. The solid line is the sum with the dotted thin lines as theindividual responses.

fundamentally different ways of explaining the origin of multiple elastic responses: from multiple

chains or from multiple attachments of a single chain, schematically represented in Figure 1. For

the case depicted in Figure 1b, there are two chains attached to the tip. Despite the low

polydispersity of the samples, few chains will have identical contour lengths. Neither the length of

the chain on the surface that is free from entanglement, nor the position of the tip attachment to

the chain are required to be the same for each chain. The observed elastic response will contain

24

contribution from the elastic deformation of both the long and short chains.

In the case of a single chain with multiple attachments to the tip, the rupture of the first

attachment exposes a greater contour length of the chain to stretching. In contrast to the

multichain model, the rupture of this first attachment is independent of the elastic deformation of

the second attachment.

Figure 12a was fit using the multichain model, and the individual elastic responses are

shown as dotted black lines above the sum. The multichain model yields higher persistence

lengths and extension ratios than the single chain model for intermediate elastic responses.33 As

detailed in the caption of Figure 12, the intermediate responses have higher persistence lengths

than those typically found.

Table 2

Fitted parameters for the wormlike chain and freely joined chain models.

polymer WLC FJC

median

persistence

length Å

mean

maximum

extension

ratio

mean χ2 median

Kuhn

length Å

mean

maximum

extension

ratio

mean χ2

PS7800-

PVP10000

3.0 ± 2.6 0.85 ± 0.04 0.70 ± 0.15 3.6 ± 2.7 0.94 ± 0.02 0.74 ± 0.17

PS13800-

PVP47000

2.4 ± 2.1 0.85 ± 0.11 0.41 ± 0.26 3.0 ± 4.0 0.91 ± 0.08 0.49 ± 0.35

PS52400-

PVP28100

2.9 ± 11.3 0.84 ± 0.08 0.40 ± 0.18 4.0 ± 11.6 0.94 ± 0.03 0.42 ± 0.18

25

PS60100-

PVP46900

4.5 ± 14.0 0.83 ± 0.10 0.72 ± 0.44 5.8 ± 10.5 0.88 ± 0.18 0.70 ± 0.44

PS29100 3.7 ± 21.1 0.90 ± 0.05 0.57 ± 0.27 3.7 ± 6.4 0.96 ± 0.02 0.63 ± 0.27

PVP50000 4.0 ± 3.2 0.81 ± 0.07 0.25 ± 0.14 5.8 ± 2.8 0.91 ± 0.05 0.31 ± 0.18

All

polymers

2.9 ± 10.5 0.82 ± 0.21 0.51 ± 0.30 3.8 ± 7.1 0.92 ± 0.09 0.56 ± 0.32

The maximum extension ratio is the ratio of the entire length of the chain between the tip and the

surface divided by the distance at which the chain ruptures. ± Values are standard deviation. All

polymers represents all molecular weights of block copolymer and homopolymer of PS and PVP

studied.

The single chain and the multi-chain models are distinguishable by how well they fit

multiple elastic responses to a single persistence or Kuhn length. For ease of discussion, we will

consider only the persistence length. When multiple elastic responses are fit by a single chain

model, the persistence lengths become very similar and can be fit quite well to a single persistence

length for all of the elastic responses. In the example given in Figure 12b, a single persistence

length of 4.4 Å gives a χ2 of 0.59. The multi-chain, single persistence length fit (not shown) gives

a 5.7 Å persistence length and a 0.76 χ2 as the characteristic parameters, but fails to fit the

intermediate responses. The persistence length of 4.4 Å for the single chain, single persistence

length, corresponds to a diameter of 2.8 Å, further suggesting that even for multiple elastic

responses in a single force plot a single chain is bridging the tip and the surface.

26

Based on the single chain model’s success over the multi-chain, the small persistence

lengths obtained with both the FJC and WLC corresponding to small diameters of chains, as well

as the statistical analysis of the extension lengths, we conclude that the majority of our data

represents the elastic stretching of a single polymer chain. It then follows that the polymer chain

is left on the surface, or else multiple chain dynamics would be observed for the majority of the

force plots. However, it seems reasonable that if the chain is extended to lengths comparable to

the length of the entire chain, one might suspect that the chain remains on the tip. Indeed, we

have observed in rare cases (5 of 700+) that entire force volumes (16x16 force plots) give rise to

experimentally identical force plots over a lateral distance of a micron (data not shown). Each of

these force plots exhibit the same elastic response, suggesting that there is indeed a single chain

on the tip that is repeatedly stretched. This is not the case in most of the force plots, and suggests

that the removal of a chain from the surface is a rare event.

In studies on titin, Gaub and coworkers showed that the multiple elastic responses were

due to separate domains unfolding in a single chain.4 Each domain unfolds under greater strain

than the previous, showing that multiple attachments of the chain to the surface or tip were not

significant in the elastic response. This is obviously not the case in Figure 12, as the third elastic

response ruptures under the least strain. In our study, there are no rigid polymer domains along

the length of the chain; consequently, we identify elastic responses which detach additional

27

Figure 13PS52400-PVP28100 collected with a 100nm trigger, 300nm scan size, and at ascan rate of 1.95 Hz. All plots are sequential within the same force volume. There was 62.5nm in lateral displacement between force plots.

A: Force-distance plot showing the tip approaching the surface (green) andsubsequently pulling out an entangled mess of chains (red).

B: Here the advancing tip reversibly relaxes the chains before they have detachedfrom the tip. The blue line is a fit to the elastic responses using the WLC model. The fit gave a 0.91 maximum extension ratio and 1.2Å persistence length

C: Subsequent force plot showing the same process as B. WLC fit gave a 0.93maximum extension ratio and 1.6Å persistence length.

polymer from the

surface/tip, allowing

weaker attachments to

be exposed and rupture

at different forces.

Figure 13

depicts three

consecutively collected

force plots. The top

force plot shows the tip

picking up a number of

polymer chains.

Subsequent force plots

demonstrate the

relaxation and

stretching of the same

group of chains. These particular force plots most likely involve multiple chains in series, as the

tip-sample separation is approximately 282 nm.34 The individual chains are estimated to be 192

nm, well below this length. The persistence length of 1.2-1.6 Å gives a diameter of 2.02-2.17 Å.

This suggests that if indeed these force plots involves multiple chains then they are in series. Of

the approximately 200,000 force plots examined, these three are the only ones that demonstrate

28

repeated stretching of polymer chains attached to the surface. If a z scan size is chosen that

allows the tip to travel above the surface a distance comparable to the mean length of the elastic

response, then the probability of repeatedly elongating the same polymer chains should increase.

We are currently investigating further the reversibility of polymer chain stretching. It is also

apparent that the force plots shown in Figure 13b and c exhibit elastic responses at the same tip-

sample separations (100-150 nm).

The rupture force of the elastic response is determined by how well the polymer is

attached to the tip, which in turn depends on the interfacial tension. There is no evidence of

specific interactions between the tip and the chain, and so in principle the rupture force will allow

for the distinction between blocks of a block copolymer. In an attempt to distinguish the blocks

that were studied here, we conducted contact angle measurements on P2VP films. Linear

regression of advancing contact angles of sodium chloride solutions gave a critical surface tension

of 65.8 ± 0.7 pN nm-1. The Zisman plot can be found in the appendix C. Pure water gave a

contact angle of 55 degrees, which through Young’s Equation gives an interfacial tension of 24.1

pN nm-1. Polystyrene has a surface energy of 33 pN nm-1 35 and a contact angle of 88 degrees,36

giving an interfacial tension of 30.5 pN nm-1. Assuming this similarity in interfacial tensions

propagates to the interfacial tension of the polymer and the tip, this difference is beyond the

accuracy of our force measurement.36 In the case of PS-P2VP the difference in interfacial

tensions of the two blocks with water are too close to distinguish. The use of block copolymers

with noticeably different interfacial tensions could help distinguish between polymer blocks, and

29

0 20 40 60 80 100contact time [msec]

mea

n le

ngth

of

exte

nsio

n [n

m]

30

40

50

60

70

80

90

100

110

120

Maximum force applied tosample [nN]

2.5 5 7.5 10 12.5

Figure 14PS13800-PVP47000 z scan speed 3300nm/sec. The lowerblue line is the mean tip sample separation at the rupture ofthe extraction event. The upper red line is the estimatedcontour length of the chain based on the maximum extensionratios obtained from the WLC fits.

possibly the mapping of surface

segregation, as well as monitoring

surface rearrangement as a function of

solvent quality.

The dependence of the elastic

response on contact time with the tip was

studied on PS13800-P2VP47000 (Figure

14). Contact time was increased by

increasing the maximum applied pressure

while maintaining a constant z scan speed

of 3300 nm/sec, and thus it is

undetermined if the dependence is a

function of contact time alone or if there

is also an applied pressure component. Neither the rupture force of the elastic response nor the

fitted values for the WLC and FJC model show a contact time dependence (Table 3, appendix C).

However, there is a dramatic increase in the probability of an elastic response on increasing

contact time (Table 3). It is also apparent that the relative probability of attaching a longer chain

increases with contact time, as shown by the increase in average extension length of the elastic

response (Figure 14). The red line is the estimated contour length of the chain based on the

maximum extension ratio from the WLC model. The increase in the average length of extension

presumably results from an improved attachment of the chain to the tip. This is no apparent

30

dependence in the rupture force of the elastic event on contact time, as shown in Table 3

(appendix C).

Conclusions

We have demonstrated the ability of AFM to elastically extend a single polymer chain

from a surface into the overlying solvent. We have shown that the WLC and FJC models fit the

nonlinear elastic response with equal success. Persistence lengths so obtained indicate the chains

have ca. 2.5 Å diameters, comparable to the estimated monomer size. The extension ratio

obtained from the fits, the length of the elastic response, and the estimated contour length of the

chains demonstrate that on average, we are extending the polymer chains approximately 35% of

their entire chain length.

RTV11®

The experimental details are given in appendix B. We have determined that the softness of

RTV11® is laterally inhomogeneous. Figure 15b is the Young modulus determined from force

volumes taken with a glutathione coated tip in water. There is a direct correlation of the high

Young modulus with the observed height features, Figure 15a. The height features are the result

of the CaCO3 filler. When the surface is imaged in air with tapping mode, where the interaction

forces can be minimized, it is apparent that surface is deformed in a non uniform manner when the

31

Figure 15A: Topography of RTV11®, collected in force volume mode witha glutathione coated tip under water.B: Young modulus calculated fromthe force plots using the springconstant measured to be 0.033 N/M. The Sneddon model incorporating aparabolodal tip shape was used. C: Simultaneously collectedadhesion map. The softer valleyshave 5-10 times higher adhesion.

tip comes into contact, Figure 16. The image on the left is

the topography of RTV11® when higher forces (the

cantilever amplitude is 0.48 times the free cantilever

oscillation amplitude) are applied to the surface. The

image to the right shows the same area when low forces

(the cantilever amplitude is 0.95 times the free cantilever

oscillation amplitude) are applied to the surface. There

are three small (~100 nm wide, 60 nm high) height

features that are observed at high forces but are not

apparent in the low force image. The wave pattern

observed at low force was reproducible on a number of

size scales and locations. It should also be noted that the

objects studied at low force appear to be 130 nm shorter

than when they are studied at high force. When the

surface is immersed in water this trend is more difficult to

detect, which is mostly a consequence of the low quality

of the images at low force, but a similar force dependance

of the topography is observed. This implies that there is a

soft “spongy” over layer of RTV11® that is easily

compressed when typical forces are applied. This describes the different observed adhesion by the

changes in the contact area, schematically represented by Figure 17. The contact area is greatest

32

Figure 16A: Topography of RTV11® collected in tapping mode in air with 0.48 times the free cantileveroscillation amplitude, i.e., when typical AFM forces are applied to the surface. The three smallheight features, as indicated by the arrows, are 60nm high.B: Topography of the same area collected in tapping mode with 0.95 times the free cantileveroscillation amplitude, i.e., a minimum of force is applied to the surface. The height featuresobserved in both images appear 130nm shorter when the lower force is applied.

where the tip presses into the surface near a rigid particle (Figure 17c). When it pushes on top of

the particle the tip is not able to push into the surface as far and the contact area is minimized

(Figure 17a). The contact area is directly related to the adhesion, by the interfacial tension. This

is consistent with the adhesion map Figure 15c, which is approximately the inverse image of the

height image, Figure 15a.

33

Figure 17Schematic representation of the contactarea on RTV11®. A: In the soft surface away from theCaCO3, there is an intermediate contactarea, and thus an intermediate adhesion.B: Next to the CaCO3 there is a slightincrease in the contact area, giving riseto the higher adhesion.C: On top of the CaCO3, there is littleindentation and so there is little contactarea. This results in the small adhesionobserved on the CaCO3.

When the surface is imaged underwater

polymer stretching is apparent, as shown in Figure 18,

which depicts a typical extension event for RTV11®.

The probability of a extension event depends directly

on the amount of force applied on the sample by the

descending tip, as it does for PS-P2VP. With a 10 nN

force applied to the surface, approximately 10-20% of

the force plot exhibit polymer extension. The length

which the polymer is stretched ranges from 100 to

600 nm but is typically near 250 nm. The force of the

rupture

ranges from 0.1 to 2.5 nN and is typically 1 nN.

Histograms of the length of extension are given in

appendix C.

The elastic response of RTV11® has been fit

to WLC, FJC and tube models. Persistence and

maximum extension ratios (Rmax) were obtained from

fitting elastic response curves to the WLC model by

minimizing χ2. Kuhn length and Rmax were obtained

from fitting elastic response curves to an approximation to the FJC model by minimizing χ2.29 Nc,

and Rmax were obtained by fitting elastic response curves to the cross link contribution of the tube

34

A

B

C 1/N

c

Rm

ax

λmax0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0 200 400 600 800 1000

tip sample separation [nm]

forc

e [n

N]

0 50 100 150

-3.5

-2.5

-1.5

-0.5

0.5

tip sample separation [nm]

forc

e [n

N]

0 50 100 150

-3.5

-2.5

-1.5

-0.5

0.5

Figure 18Force plot showing a typical polymerstretching event.A: The blue line is the fit to the FJC modelgiving a Kuhn length of 0.005 nm, amaximum extension ratio of 0.785, and aχ2 of 0.134.B: The blue line is a force plot with a fitof the tube model to the data with a lmax of100, Nc of 9.1 nm-1, a maximum extensionratio of 0.553, and a χ2 of 0.119.C: Parameters obtained from multiple fitsof the tube model to the data. The value ofλmax was not a fitting parameter. The blueline is 1/Nc, the distance [nm] betweencross links. The red line is the maximumextension ratio.

model. λmax was not a fitting parameter, a range of

values were entered and the best fit values for Nc

and Rmax obtained for each value (Figure 18c),

which provided similar errors, 0.119 ± 0.001.

Persistence and Kuhn lengths obtained from WLC

and FJC models respectively are unrealistically low,

well below a fraction of an angstrom. The fitted

responses of PDMS monolayers end grafted to the

silicon (100) face however yield parameters with

reasonable molecular dimensions. The median

persistence length obtained from fitting the polymer

stretching to the WLC model was 0.295 ± 0.058

nm, with a median Rmax of 0.80 ± 0.02. Median

Kuhn length was 0.444 ± 0.067 nm with a median

Rmax of 0.91 ± 0.025. Errors for the fits were χ2

and were typically less than 0.2, for both models.

Histograms of the persistence and Kuhn lengths are

given in appendix C. There are a number of

polymer extensions that are greater than the

estimated length of the average molecular weight,

35

tip sample separation [nm]

forc

e [n

N]

0 100 200 300 400 500

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Jump from thesurface

Figure 19Force plot showing an atypicalpolymer stretching event. Afterthe tip jumps from the surface aconstant force is repeatedlyobserved until 430 nm, where thesilicone tip returns to zero force.

which is most likely due to the high polydispersity. The

histograms of the lengths of extensions are given in

appendix C.

The majority of the elastic responses are non linear

increases in force, with one rupture following the initial

AFM tip jump from the surface. A few (<2%) of the force

plots demonstrating polymer stretching contain multiple

elastic responses, too few for a meaningful statistical

analysis. There is also a much different elastic response,

where after the tip jump from the surface the force remains

constant, but at negative deflection, as the tip moves away

from the surface. This type of elastic response usually appears in batches where subsequent force plots

show the same force-distance plot, as it was observed on PS-P2VP. Figure 19 shows one of the

140 sequential force plots collected, demonstrating this behavior. The forces of the plateau range

from 0.10 to 0.16 nN averaging 0.12 nN. The length at which the tip returned to zero force

ranged from 385 to 475 nm, and averaged 430 nm. The force plot in Figure 19 was reproduced,

within experimental error, in over 140 force plots collected sequentially over a lateral

displacement 1.8 µm, and thus it is concluded this is the result of polymer chains attaching to the

tip. This polymer chain is brought into contact with the surface and peeled off again, giving rise

to the flat force distance profile. The interfacial tension of the polymer being exposed to water

and the circumference of the polymeric cylinder determine the force of this type of stretching.23

36

As the polymer is pealed away from the surface the amount of the polymer exposed is

approximately the circumference. One half of the chain is in contact with the surface so the total

polymer exposed on the surface and on the chain is approximately the circumference. The force

of the extension is given by the relation F= γ D π, where γ is the interfacial tension (pN/nm), D is

the tube diameter (nm) and F is the force of the wetting of the polymer chain (pN). The

interfacial tension of PDMS, as determined from Young’s equation, is 47 pN/nm. This would

require the polymer stretching observed in Figure 19 to be the deformation of a polymer cylinder

or chain with a circumference of 2.5 nm and consequently a diameter of 8 Å.

Conclusions

The peptide Glutathione was used to emulate biofouling. Biofouling of marine vessels

drastically increases the use of non-renewable fuels and costs billions of dollars each year.38 A

current strategy to inhibit biofouling is to minimize the adhesion and maximize the release of

biofoulers using a polymer coating on exposed surfaces. One of the prerequisites for a minimally

adhesive coating is that it possess a low surface energy, which PDMS satisfies well with surface

energies from 20 to 25 pN/nm.39 A second quality thought to be desirable is a flexible coating,

which in principle will prevent mechanical locking with the fouler.40 The results presented here

show a conflicting view. The soft areas of the polymer deform to a greater degree, increasing the

contact area, and ultimately increasing the adhesion(Figures 15 and 17). There is also the added

issue of the polymer stretching, which is present only in the soft areas of the RTV11® surface.

These tethers to the surface serve to increase the adhesion, in some instances by more than the

36

adhesion to the surface. The softness of the anti fouling/fouling releases coating must be soft to a

certain extent, so as to prevent the mechanical locking of foulers, but conversely, must not be too

soft so as to allow large sample deformation.25

It is also clear from this work that a multi chain model needs to be developed, or at least a

way to estimate the effect of multiple chains have on q and Ak. The results of the fitting to the

theoretical models suggests that the stretching of multiple chains deceases the persistence length,

as well as the Kuhn statistical segment length. This effect is not clearly understood from a

theoretical stand point, and requires further work to access the exact nature.

Future Work: Stretching of end grafted polyelectrolyte chains

Summary

Physisorption of polymer chains provides a simple experimental method to probe the

capabilities of AFM to stretch polymer chains. The difficulty with this method is that it requires

statistical analysis of the length of the polymer stretching, as there is a distribution of chain lengths

free of entanglements, a well as the location of attachment to the tip. Physisorption also requires

statistical treatment of the force of rupture, as the area of attachment is not constant. This hinders

the assessment of the number of chains being stretched. A model system of monodisperse

polymers that are end grafted to the surface and the AFM tip is desirable to further the study of

polymer elasticity. The grafting of the polymer will provide a large force, increasing the signal,

and quantify the force to the point where the number of chains being stretched will be easily

37

assessed. It will also allow a study of bond strengths as a function of solvent quality. Another

problem with the polymer system studied previously is that it is difficult to determine the

theoretical persistence length. The effect of solvent on a chain attached to the surface and tip is

not straightforward. It is desirable to have a polymer where the persistence length can be

controlled. Polyelectrolyte chains have a strong dependance of persistence length with charge

density and ionic strength of the solution.41 This electrostatic contribution to the persistence

length, as described below, can increase the persistence length by as much as three orders of

magnitude, Figure 20. This will allow for a method to test the polymer properties determined by

stretching. However it is not entirely understood how the persistence length depends on the

electrostatic environment of the polymer. The geometry of the polymer chain has prevented exact

treatment of the electrostatic environment. Previous experimental methods have not satisfactorily

determined the persistence length as a function of ionic strength for DNA.42 The recent advent of

AFM to determine the persistence length may provide insight into the dependance of the

electrostatic environment. Below a polymer system is outlined which will provide a narrow

polydispersity (<1.2) of end grafted polyelectrolyte chains to a self assembled monolayer (SAM).

Grafting the opposite end to the polymer tip can easily be accomplished by functionalizing the tip

with the same SAM and exposure to UV light when in contact with the surface. The charge

density of the polymer can also be easily be controlled, as well as the ionic strength of the

environment, giving experimental control of the persistence length.

This polymer system will further the achievement of all three of the specific aims. The

covalent attachment of the polymer chain to the surface and the tip will clearly allow the

38

qt'qo%qe(16)

'

ji 4ie2zi

2

gogkT(17)

LB'e 2z 2

4 gogkT(18)

qe'LB

4(Leff )&2(19)

experimental determination of a single covalent bond strength. The control of the electrostatic

environment will allow the study of the effect that the medium plays on the polymer elasticity.

Finally, the ability of AFM to stretch single polymer chains will be further developed by this

technique. The covalent attachment of a polymer to the surface and the tip has not been

previously achieved. Polyelectrolytic polymers have been stretched by AFM, but there has been

no systematic study of charge density and ionic strength, which will be conducted in the

experiment outlined below.

Polyelectrolyte theory

The persistence length of a polyelectrolyte chain (qt) is the sum of the unperturbed persistence

length (qo) and the persistence length from electrostatic contributions(qe), Equation 16.41

The electrostatic contribution is a function of charge density on

the polymer chain, and the ionic concentration in solution.42 The

electrical double layer of ions that form around the ions of the

polymer can overlap, giving rise to an osmotic repulsion. This

will force the chain to have become more rigid, and thus persist

in a given direction to a greater degree, increasing the persistence

length, Figure 20. The characteristic distance that double layer

forms away from a charged surface is the Debye length (1/κ),

Equation 17. εo is the permittivity of free space, ε is the

relative permittivity, e is the elementary charge, zi is the valence

39

qt'0.033

C%qo(20)

Cf'L 2S

0.0162(22)

qt'0.0162

Leff2C

%qo(21)

of ion i, and ρ4i is the number density of ion i at infinite separation. This geometry does not apply

to cylindrical polymer chains. There has been considerable difficulty

with the strongly non-linear decay of the electric field near the

cylindrical axis, which has prevented an exact solution. There has

however been a number of approximations. The characteristic

electrostatic length scale on the polymer is the Bjerrum length (LB),

Equation 18. This is the distance where two charges have an

Coulombic interaction energy of kT. For low charge densities on

the polymer chain, i.e. when the distance separating the charges (Le)

on the polymer chain is greater than LB, the screening is termed to

be in the Debye-Huckel region. This means that Leff=Le. Leff is the

effective distance between charges and is used in Equation 19. When the distance between the

charges is less than LB, then it is considered strong screening, and Leff=LB. These approximations

lead to an instantaneous transition when Le=LB, which does not seem to be

realistic. There has been some theoretical work that suggests this is the case, especially for cases

when D<<κ-1.43,44

40

Per

sist

ence

leng

th [

nm]

concentration [mol/L] Charge fraction

Figure 20Effect of a monovalent ionic aqueous solution on the persistencelength of a variably charges polyelectrolyte.

These two regions leads to two relevant cases for the experiment proposed below which

will be conducted in monovalent electrolyte solutions. When the polymer chains is highly charged

(>30%) and low charge fraction (<30%). For the case of high screening Equation 16 reduces to

Equation 20. This equation clearly outlines a method to determine the unperturbed persistence

length, by plotting the persistence length obtained from the fitted data against the inverse of the

ionic concentration (C, mol L-1). This will provide a line with a slope of 0.033 and a y-intercept of

qo. The other case is when the charge fraction is below 0.3, and thus in the Debye-Huckel region,

for this case Equation 16 reduces to Equation 21. By again plotting the fitted persistence lengths

against the inverse concentration and fitting for a straight line qo can be determined from the y-

intercept. More importantly, the charge fraction can be determined by the slope of the line

through Equation 22,

where Cf is the charge

fraction, S is the slope of

the line, L is the bond

length and Le=L Cf-1.

41

C H2CHC H2 C H2 CH

N

nCH2 C H C H2

Figure 21Styrene terminated poly-4-vinyl pyridine

Polyelectrolyte experimental

Overview

The preparation of the end grafted polyelectrolyte surface has been previously done by

Granick.45,46 A mixed SAM of octadecyltriethoxysilane (OTS) and icosenyltriethoxysilane (OLE)

system was used on Si wafers. The polymer that is grafted to the SAM is styrene-terminated

poly-4-vinyl pyridine, Figure 21. The OLE chains have a carbon-carbon double bond at the end,

which upon UV (256 nm) exposure forms a carbon-carbon bond with the olefin group of the

terminal styrene of the polymer.

In order to graft the polymer chains to the tip, a silicon tip will be functionalized with the

OLE/OTS SAM. This will allow the free polymer chain end to form a covalent bond with the

OLE chains on the tip. The tip will be brought into contact with the surface, where it will be

exposed to UV light, forming the carbon carbon bond.

42

Li2CuCl4

THF

(1) Mg

Br

4

Br

9

Br Br

4

(2)

(1) Mg

THF

4

SiO

(2)

SiO

9 3

Scheme 1

The alkylation of P4VP is accomplished post grafting by immersion into a solution of

methyl iodine in DMF. The degree of quanternization is controlled by immersion time with 10%

at 10 minutes, and 96% at 5 hours.

OLE synthesis

OLE is synthesized by a two step Grignard reaction, outlined in Scheme 1.

The undecenyl bromide is converted to the Grignard reagent upon addition of magnesium, in a

solution of LiCl and CuCL2 in THF. This solution is added to 1,9 dibromononane over the period

of 1.5 hours. The presence of the copper catalyst reduces the homo-coupling of the Grignard

reagent, effectively giving only the cross-coupled product, 19-eicosenyl bromide.48 After 4 hours,

below 10oC, the product is extracted into hexane, which is subsequently washed with an aqueous

solution of salt and sodium bicarbonate. The extract is then purified by fractional distillation,

collecting from 145-160oC. The product is converted into the Grignard reagent by the addition of

magnesium. This is subsequently added to tetraethoxysilane over 1 hour. After a 5 hours the

product is extracted into hexane, and purified by vacuum distillation, collecting from 175-180oC

and 0.05 mmHg.

43

Monolayer formation

The substrate first was cleaned by ultrasonication in ethyl acetate for 10 minutes. OTE

and OLE are then hydrolyzed separately by the following procedure. 210 mg of silane is added to

a solution of 25ml of THF and 125 mg of 1.31 N HCl. This is allow to sit for at least 6 hours,

where 278 mg of the OTS solution and 833 mg of the OLE solution are added to cyclohexane.

The silicon wafer is immersed in this solution for 30 minutes. A 2 g/L solution of the styrene

terminated P4VP in 1-methyl-2-pyrrolidinone is prepared. This solution is spread between a

quartz plate and the functionalized silicon. Exposure to UV light (256 nm) for 5 minutes end

grafts the polymer olefin groups to the OLE olefin group.

Quanternization

A 2.5% (w:w) solution of CH3I in DMF is used to alkylate the P4VP. By controlling the

time the polymer surface is immersed the degree of alkylation is controlled. At 10 minutes 10%

of the polymer is ionized, at 100 minutes 60%, 5 hours 96%. This degree of control will allow for

the Debye-Huckel region (<30% ionized) of ion screening as well the strong screening to be

experimentally tested.

Polymer grafting to the tip

The polymer may be grafted to the tip as well as the silicon surface. Silicon cantilevers are

commercially available in a wide range of spring constants, from 0.01 nN/nm to well over 100

nN/nm. The silicon cantilevers can be functionalized in the same manner as the silicon wafer.

44

This will provide the tip with a coating of olefin groups. When the tip is brought into contact

with the surface inside of the AFM cell the free end of the P4VP, also containing a styrene

terminal group, can be grafted to the OLE of the tip by UV excitation. This can be done in a

number of ways. The easiest would be to constantly irradiate the sample through the bottom of

the sample cell. The AFM in our research group is uniquely adapted with a central bore opening

through the piezo stack. This allows for fiber optics to be fed through the scanner into the sample

cell. The monolayer substrate is thus required to transmit enough UV light for the process to

occur. The OLE/OTS monolayer has been grafted to a variety of substrates, including mica,

glass, and silicon wafers. It may be possible to use the SAM method to functionalize quartz,

which would allow for the UV excitation to pass into the sample cell. If it is not possible to excite

the sample from below the grafting can be accomplished from the side with the removal of the

fluid cell o-ring. This will allow for the fluid to evaporate, but this is not a problem on the minute

time scale, allowing for the experiment to be conducted.

45

Conclusions

The ability of AFM to stretch single physisorbed polymer chains has been demonstrated and a

statistical analysis of the data has been developed. The ability to stretch a single polymer chain

that is covalently bound to both the substrate and the AFM tip will be developed. Theories

pertaining to polyelectrolyte chains will also be tested in a novel manner through the stretching of

a single chain of various charge densities through solutions of various ionic strengths.

Acknowledgments

I earnestly thank Boris Akhremitchev for countless hours of stimulated conversation, and advice,

without which none of this would be possible. I give special thanks to Dr. Gilbert Walker, whose

guidance has been invaluable throughout. I also give thanks to Dr. Toby Chapman, for supplying

the fluorinated polyamide urethane polymer.

46

References

(1) Chatellier, X; Senden, T. J.; Joanny, J. F.; DiMeglio, J. M. Europhysics letters, 1998, 41, 303-

308.

(2) Florin, E. L.; Moy, V. T.; Gaub, H. E. Science, 1994, 264, 415-417.

(3) Noy, A.; Vezenov, D. V.; Kayyem, J. F.; Meade, T. J.; Lieber, C. M. Chem. Biol. 1997, 4,

519-527.

(4) Rief, M.; Gautel, M.; Oesterhelt, F.; Fernadez, J. M.; Gaub, H. E. Science, 1997, 276, 1109-

1112.

(5) Rief, M.; Oesterhelt, F.; Heymann, B.; Gaub, H. E. Science, 1997, 275, 1295-1297.

(6) Aime, J. P.; Elkaakour, Z.; Odin, C.; Bouhacina, T.; Michel D.; Curely J.; Dautant A. J. Appl.

Phys. 1994, 76, 754-762.

(7) Burnham, N. A.; Colton, R. J.; Pollock, H. M. Nanotechnology, 1993, 4, 64-80.

47

(8) Hoh, J. H.; Cleveland, J. P.; Prater, C. B.; Revel, J. P.; Hansma, P. K. J. Am. Chem. Soc.

1992, 114, 4917-4918.

(9) Magonov, S. N.; Reneker D. H. Annu. Rev. Mater. Sci. 1997, 27, 175-222.

(10) Sasaki, M.; Hane, K.; Okuma, S.; Bessho, Y. Rev. Sci. Instrum. 1994, 65, 1930-1934.

(11) Zhulina, E. B.; Walker, G. C.; Balazs, A.C. Langmuir, 1998, 14, 4615-4622.

(12) Flory, P.J. In Statistical Mechanics of Chain Molecules, 1969, Interscience Publishers, New

York.

(13) Also referred to as the random flight, the random coil, or in the limit n6 4, the Gaussian

chain.

(14) Debye, P. J. Chem. Phys. 1946, 14, 636.

(15) Bustamante, C.; Marko, J.; Siggia, E.; Smith, S. Science, 1994, 265, 1599-1600.

(16) Yamakawa, H. In Helical Wormlike Chains in Polymer Solutions, 1997, Springer, New

York.

48

(17) Doi, M.; Edwards, S. F. In The Theory of Polymer Dynamics, 1986, Oxford, Oxford.

(18) Ball, R. C.; Doi, M.; Edwards, S. F.; Warner, M. Polymer, 1981, 22, 1010-1018.

(19) Edwards, S. F.; Vilgis, T. A. Rep. Prog. Phys. 1988, 51, 243-297.

(20) Volkenstein, M. V. In Configurational Statistics of Polymer Chains, 1963, Interscience,

New York.

(21) Pincus, P. Macromolecules, 1976, 3, 386-388.

(22) Halperin, A.; Zhulina, E. B. Europhysics letters, 1991, 15, 417-421. Halperin, A.; Zhulina,

E. B. Macromolecules, 1991, 24, 5393-5397.

(23) Akhremitchev B. B.; Mohney B.; Marra K.; Chapman T.; Walker G. C. Langmuir, 1998, 14,

3976-3982.

(24) Bemis, J, E.; Akhremitchev, B. B.; Walker G. C. Langmuir, 1998, submitted.

(25) Bemis, J. E.; Akhremitchev, B. B.; Al-Maawali, S.; Walker G. C. to be submitted.

49

(26) Griffith, J. U. S. Patent 5,449,553, 1995.

(27) Chapman, T. M.; Marra, K. G. Macromolecules, 1995, 28, 2081-2085.

(28) Chapman, T. M.; Benrashid, R.; Marra, K. G.; Keener, J. P. Macromolecules, 1995, 28, 331-

335.

(29) Zhuang, H.; Marra, K. G.; Ho, T.; Chapman, T. M.; Gardella, J. A. Macromolecules, 1996,

29, 1660-1665.

(30) For extension ratios below 0.8 the inverse Langevin function was approximated by a 20th

order polynomial. For higher extension ratios tan(R) was used.

(31) Stamouli, A.; Pelletier, E.; Koutsos, V.; Vegte, E. V. D.; Hadziioannou, G. Langmuir, 1996,

12, 3221-3224.

(32) Wang, Y.; Winnik, M. A. J. Phys. Chem., 1993, 97, 2507-2515, and references therein.

(33) In both the single chain and multichain models the last elastic response is from one chain. This

demands the same fitted values for the last response for both models. In Figure 12b (single chain

50

model fit), the six elastic responses were fit to a single persistence length, resulting in a slightly

different result for the last response, than that for the multichain model used in Figure 12a.

(34) The vertical tip sample separation is 230 nm. With 125 nm of lateral offset between the first and

third plot the total tip sample separation is 262 nm. The maximum extension ratio is 0.93 giving an

estimated contour length of 282 nm.

(35) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic press: San Diego, 1992;

pg. 204.

(36) Parsonage, E.; Tirrell, M.; Watanabe, H.; Nuzzo, R. G. Macromolecules, 1991, 24, 1987-1995.

(37) The small amount of uncertainty in the sensitivity determination propagates to approximately

10% when using calibration standards of known force constants. As a result, there does not appear

to be a trend in the rupture force of the elastic responses, as shown in Table 1.

(38) Wynne, K. Naval Research Reviews, 1997, 49, 2-3.

(39) Meyer, A. E.; Baier, R. E.; King, R. W. J. Chem. Eng. 1988, 66, 55-62.

51

(40) Mera, A. E.; Fox, R. B.; Bullock, S.; Swain, G.; Schultz, M. P.; Gatenholm, P.; Wynne, K. J.

Navel Research Reviews, 1997, 49, 4-8.

(41) Fleer, G. J.; Stuart, M. A.; Scheutjens, J. M.; Cosgrove, T.; Vincent, B. In Polymers at

Interfaces, 1993, Chapman and Hall, London.

(42) Odijk, T., J. Pol. Sci. Pol. Phys. Ed, 1977, 15, 477-491.

(43) Manning, G. S. J. Chem. Phys., 1969, 51, 924-929.

(44) Fixman, M. J. Chem. Phys., 1979, 70, 4995-5002.

(45) Sukhishvili, S. A.; Granick, S. Langmuir, 1997, 13, 4935-4938.

(46) Peanasky, J.; Schneider, H. M.; Granick, S. Langmuir, 1995, 11, 953-962.

(47) Tamura, M.; Kochi, J. Synthesis, 1971, 303-305.

49

Principal Symbols and Conventions

rij Vector connecting monomers i and j (nm).

r Vector connecting the ends of a polymer chain. Chain vector. End-to-

end distance (nm).

r2 Statistical average of the square end-to-end distance over all

conformations (nm2).

Li Length of the ith bond (nm)

L Contour length of the polymer chain, i.e. the chain vector when the

polymer is fully extended (nm).

θ The supplement to the bond angle. In Equation B1 θ is the angle of

approach of the cantilever.

Cn Characteristic ratio.

s2 The square of the radius of gyration averaged over all conformations

50

(nm2).

si Vector from the center of gravity to monomer i (nm).

Rmax The maximum extension ratio obtain prior to polymer chain

detachment.

R Extension ratio, the fraction of the contour length that the polymer is

extended.

Ak The Kuhn statistical segment length (nm).

qo unperturbed persistence length (nm), also denoted by q.

uo Unit vector for the first segment of the polymer chain.

α Intramolecular expansion factor when in conjunction with the radius of

gyration. When in conjunction with the tube model α is the inverse of

λmax.

51

k Boltzmann constant 1.381 x 10-5 aJ K-1

T Temperature (K)

qt Total persistence length (nm).

qe The electrostatic contribution to the persistence length (nm).

LB Bjerrum length. The distance at which to charges have an Coulombic

interaction energy of kT.

Le The distance between charges on a polyelectrolyte chain (nm).

Leff The effective distance between charges on a polyelectrolyte chain (nm).

e Elementary charge (1.602 x 10-19 C).

ε Relative Permittivity.

εo Permittivity of free space (8.854 x 10-39 C2 aJ-1 nm-1).

52

ρi Ion number density of ion i (#/nm3).

κ Inverse Debye length (nm-1).

zi Valency of ion i.

ρ4

The ion density at infinite separation (#/nm3).

C Bulk ion concentration (mol L-1).

Cf The fraction of the monomers of a polyelectrolyte that are charged.

S Slope of the linear fit.

Kref The spring constant of the reference cantilever, when used to calibrate

the spring constant of an unknown cantilever (nN nm-1).

δtest The sensitivity determined on the reference cantilever (V/nm).

K Spring constant of the cantilever (nN nm-1).

53

Rf Flory radius (nm).

E Young Modulus (nN nm-2).

D Chain diameter (nm).

γ Interfacial tension (nN nm-1).

V Volume of the polymer chain in a globule in contact with the surface

(nm3).

λ The extension ratio used with the tube model, ranging from 1 (rest) to

λmax.

λmax The maximum extension obtainable in the tube model, the tube

diameter divided by Ak.

η The strength of the slip link, ranges from 0 (no slip link) to 1 (cross

link), typically 0.23.

Ns Number of slip links per distance of the tube (nm-1).

54

Nc Number of cross-links per distance of the tube (nm-1).

fc The cross-link contribution to the uniaxial extension within the

framework of the tube model (nN).

fs The slip link contribution to the uniaxial extension with in the

framework of the tube model (nN).

φ Term in the tube model.

Z Distance that the polymer has been pulled out of the globule (nm).

χ2 Error in the fit to the data.

f Force of uniaxial extension for a polymer chain (nN).

ý Langevin function, coth(X)-1/X.

ý* Inverse Langevin function.

CNn Perturbed characteristic ratio

55

β Cos(θ)

56

Glossary of Abbreviations

FJC Freely joined chain

WLC Worm like chain

PSxxxx polystyrene, where xxxx is the molecular weight of the block in

Daltons.

PVPxxxx or

P2VPxxxx

poly-2-vinyl-pyridine, where xxxx is the molecular weight of the block

in Daltons.

P4VPxxxx poly-4-vinyl-pyridine, where xxxx is the molecular weight of the block

in Daltons.

AFM Atomic Force Microscope.

CMC Critical Micelle Concentration.

PDMS poly-dimethyl-siloxane

57

RTV Room Temperature Vulcanized

RTV11® Duplex coating of PDMS and calcium carbonate filler.

THF Tetrahydrofuran

RPM Revolutions per minute

CCD Charge coupled device

15F-6F-HB Fluorinated poly amide-urethane random block copolymer

SAM Self assembled mono layer

OTS octadecyltriethoxysilane

OLE icosenyltriethoxysilane

DMF Dimethylformamide

RIS Rotational Isomeric State

53

+r 2,' 2+r 2,oA(24)

Cn'+r 2,o

nL 2A(2)

Cn'+r 2,

nL 2A(3)

+r 2,

nL 2'

2+r 2,o

nL 2

Cn'2CnA(4)

Ak'L CnA(5)

Appendix ASolvent perturbations

Characteristic ratioIt is well known that the intramolecular expansion factor describes the solvent effects on +r2,o.

1

When α is less than unity the solvent is considered a poor solvent and the polymer brushcollapses upon itself. Conversely, when α is greater than unity the polymer brush expands dueto the good solvent-polymer interactions. When α is unity the solvent is termed a θ solvent,an idealized solvent that has no interaction with the polymer. Here it is necessary to comparethe unperturbed, and perturbed states, which will be denoted by a subscript o, for unperturbed,and it’s absence for the perturbed. While in terms of the characteristic ratio the perturbed statewill be denoted by a superscript N.

From Equation A2 (Equation 3) the unperturbed Cn, we introduce the perturbed characteristicratio (CNn) as the ratio of the perturbed square end-to-end distance to nL2.

From A1 it follows that:

Which directly leads to Equation A4 (Equation 7).

which now describes the solvent effect on Ak.

55

+r 2,o

nL 2'

2q

L&1A(6)

Cn'2qL&1

Cn'2qL&1A(7)

2Cn'2qL&1A(8)

Persistence length

Flory has shown that q can be described as follows.1

From Equation A2 it is clear that

Taking into account the perturbation of the solvent on the persistence length it follows that:

From A4 it can be written that:

References

(1) Flory, P.J. In Statistical Mechanics of Chain Molecules, 1969, Interscience Publishers,New York, chapters 1,2,4.

56

K'Kref(sensitivity& test

testcos( ))B(1)

Appendix B

PS-P2VP experimental detailsPS7800-P2VP10000, PS13800-P2VP 47000, PS52400-P2VP28100, PS60100-

P2VP46900, and PVP50000 were purchased from Polymer Source (Dorval, Canada) allwith a polydispersity index less than 1.11 and were used without further purification. PS29100 was purchased from Aldrich with a polydispersity index of 1.04. Blocks aredesignated by PS for polystyrene and P2VP or PVP for poly-2-vinyl-pyridine. Molecularweights of the individual blocks are given after the designation for the block in units ofDaltons. Block copolymers and polystyrene were dissolved in toluene (J.T. Baker) thatwas previously filtered with a 0.45µm nylon membrane. PVP was dissolved in THF (EMscience); all solutions were approximately 2g/L.1 Samples were prepared by spin castingonto 12mm diameter glass cover slips (Fischer Scientific) at 1000 RPM for 2 minutesusing a Headway Research ED101D photo resist spinner.

All AFM measurements were conducted using a Digital Instruments (SantaBarbara, CA) Nanoscope IIIa Multimode scanning force microscope, typically in forcevolume mode. The spring constants were calibrated using a Park Scientific Instrumentsforce constant calibration cantilever (Sunnyvale, CA), according to Equation B1. 5 δtest isthe sensitivity (V/nm) on the reference standard, Kref=0.157 nN/ nm, and θ is the angle of

the cantilever in respect to the surface, taken as 11degrees. Tip shapes were characterized by scanninga grating of 700 nm high, 20o cone angle, 10 nmradius tips (NT-MDT Moscow, Russia) to ensureminimal contamination of the tip. Tip radii wemeasure are typically on the order of 50 to 100 nm. All measurements were done in 0.45µm filtered

10mM sodium acetate buffer using a Digital Instruments fluid cell. Sodium acetate waspurchased from EM Science. Water was purified using a Barnstead NANO pure filter to 18MΩ resistivity.

Force volumes of 16x16 force verses distance plots, each with 512 points werecollected using Nanoscope IIIa software and analyzed using custom software written forMatlab (Math Works Inc. Natick MA). Persistence lengths and maximum extension ratioswere obtained from fitting elastic response curves to the WLC model and minimizing χ2. Kuhn length and maximum extension ratios were obtained from fitting elastic responsecurves to an approximation6 of the FJC model and minimizing χ2. Relevant experimentalscanning parameters are given in Table 1.

Contact angle measurements were conducted using a home built apparatus. Thesample was housed in an environmental chamber continually flushed with humidifiednitrogen. Images were collected using a CCD camera and video capture software. AZisman plot of PVP50000 was constructed from a series of aqueous NaCl solutions 0,5.43, 10.46, 14.92, 22.62, and 25.92% (w/w) (appendix C).

57

RTV11® experimental detailsGE RTV11® was purchased from Pittsburgh Electrical Inc. (Pittsburgh, PA) and was

used without further purification. Silanol terminated Polydimethylsiloxane (PDMS) 26000Daltons was purchased from Gelest inc. (Tullytown, PA) with a polydispersity index of 2-3 andwas used without further purification. Silicon (100) (Virginia semiconductor, Fredericksburgh,VA) wafers were used as the substrate for the PDMS; monolayers were prepared according topreviously reported methods.7 RTV11® films were prepared from dip-coating glass cover-slips(Fisher scientific), and the excess was blown off with pressurized air.

All AFM measurements were conducted using a Digital Instruments (Santa Barbara,CA) Nanoscope IIIa Multimode scanning force microscope. All aqueous environments usedfor imaging were previously filtered with a Barnstead NANO pure filter to 18 MΩ resistivity.Silicon nitride probes were purchased from Digital Instruments (Santa Barbara, CA). 200µmlong 22µm wide cantilevers were used with a announced spring constant of 0.06 nN/nm. Whenadditional force analysis was warranted spring constants were calibrated with a Park ScientificInstruments force constant calibration cantilever (Sunnyvale, CA), according to Equation B1.5

Sharpened Silicon Microlevers™ were purchased from Park Scientific Instruments (Sunnyvale,CA). Gold-coated tips were purchased from Bioforce Laboratories Inc. (Ames, IA).Glutathione-coated tips were prepared by placing gold-coated tips into a aqueous solution ofGlutathione (Sigma) that was made basic by the addition of a few drops of a 50% solution ofNaOH (EM Science). Glutathione is a common three amino acid peptide, Glu-Cys-Gly. Thecysteine sulfur is easily grafted to a gold-coated tip via a gold-thiol linkage. Tip radii wereobtained by scanning a silicon grating of 700nm high 20o cone angle 10-nm radius tips (NT-MDT, Moscow, Russia). A parabolodal tip model was then fit to the data to calculate the tipradius.

Force volumes were collected using Nanoscope IIIa software and analyzed using customsoftware written for Matlab (Math Works Inc. Natick MA). The elastic response curves werefit to theoretical models by minimizing χ2, as discussed in the text.

References

(1) Static light scattering reveals that the CMC of PS-P2VP solutions in toluene isapproximately 65µg/ml,2 and as such, much of the work with PS-P2VP has been above theCMC. Adsorption of polystyrene-polyethylene oxide block copolymers from selective solventsabove the CMC provide inhomogeneous film thickness and surface coverages.3 However, thesurface morphology of films made from toluene solutions of PS-P2VP block copolymers in theconcentration range of 0.1 to 6 µg/ml are surprisingly uniform.4 Part of the reason that the

58

films are so homogenous, even when the film is made from a solution above the CMC, is thatthere is an appreciable concentration of unimers above the CMC.2 Above 6 µg/ml there is aseries of “worm-like”ridges.4 By using solutions below this second critical concentration, thismorphology is avoided, simplifying the force-distance analysis.

(2) Tang, W. T. Ph.D. Thesis, Stanford University, 1987.

(3) Munch, M. R.; Gast, A. P. Macromolecules, 1990, 23, 2313-2320.

(4) Pelletier, E.; Stamouli A.; Belder, G. F.; Hadziioannou, G. Langmuir, 1997, 13, 1884-1886.

(5) Tortonese, M.; Kirk, M. Micromachining and Imaging, 1997, 3009, 53-60.

(6) For extension ratios below 0.8 the inverse Langevin function was approximated by a 20th

order polynomial. For higher extension ratios tan(R) was used.

(7) Bouhacina T.; Michel D.; Aime J.; Gauthier S. J. Appl. Phys., 1997, 82, 3652-3660.

59

Length of polymer extention [nm]

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140 160 180 200

Num

ber

of o

ccur

renc

es

Figure 3Histogram of length of extension for thePDMS monolayer. The solid vertical lineis the estimated length of the PDMS chainbased on the average molecular weight. The large number of polymer stretchinggreater than that length is most likely dueto the high polydispersity (2-3).

surface tension [dyne/cm]

cosi

ne a

dvan

cing

ang

le

65 70 75 80 85

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 2Zisman plot of poly-2-vinyl-pyridine50000. Data (blue line and Xs) wasobtained from 0, 5.43, 10.46, 14.92,22.62, and 25.92% (w/w) NaClsolutions. Linear regression (red dashedline) estimates the critical surface tensionas 65.82 pN nm-1.

The red histogram represents Kuhn lengths obtained from FJC fits to elastic responses. Thishistogram is the sum of all polymers studies; there is no noticeable difference in Kuhn lengthsbetween molecular weights of block copolymers or homopolymers. The blue histogram is thecorresponding diameters for those Kuhn lengths.

Appendix C

60

Table 1summary of contact time dependence on elastic responses

contact time [msec] 17.9 35.8 53.8 71.7 95.2 All times

maximum appliedforce [nN]

2.2 4.5 6.7 9.0 11.2

medianpersistencelength Å

2.2 ± 1.0 1.9 ± 0.8 2.0 ± 0.9 2.5 ± 3.0 2.4 ± 1.9 2.3 ± 2.1

Worm-like

Chain

meanmaximumextension

ratio

0.78 ±0.17

0.85 ±0.03

0.85 ±0.03

0.81 ±0.10

0.82 ±0.08

0.85 ±0.11

mean χ2 0.67 ±0.36

0.47 ±0.02

0.30 ±0.18

0.35 ±0.21

0.35 ±0.20

0.41 ±0.26

medianKuhn length

Å

2.7 ± 1.2 2.9 ± 1.1 2.5 ± 0.92 3.8 ± 6.3 3.0 ± 2.2 3.0 ± 4.0

FreelyJoinedChain

meanmaximumextension

ratio

0.88 ±0.13

0.94 ±0.01

0.94 ±0.02

0.91 ±0.08

0.92 ±0.05

0.91 ±0.08

mean χ2 0.77 ±0.48

0.56 ±0.10

0.38 ±0.28

0.36 ±0.23

0.46 ±0.32

0.49 ±0.35

Median rupture forceof elastic response

[nN]

0.19 ±0.02

0.34 ±0.03

0.28 ±0.02

0.18 ± 0.01

0.23 ±0.01

Probability of elasticresponse

0.03 0.04 0.05 0.08 0.17

PS13800-P2VP47000. The maximum extension ratio is the ratio of the entire length of thechain between the tip and the surface divided by the distance at which the chain ruptures. ±Values are standard deviation. All times represents the sum of all the contact times studied. Contact time was obtained by increasing the relative trigger while maintaining a constant zscan speed of 3300 nm/sec. The elastic event probability is taken as the number of forceplots exhibiting at least one elastic response divided by the total number of plots examined.

61

0

5

10

15

20

25

30

0 0.5 1 1.5 2 2.5 3Persistence length [nm]

Num

ber

of o

ccur

renc

es

Histogram of persistence lengths obtained byfitting the elastic response of end grafted PDMSmonolayers to the WLC model.

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Num

ber

of o

ccur

renc

es

Kuhn length [nm]

Histogram of Kuhn lengths obtained by fittingthe elastic response of end grafted PDMSmonolayers to the FJC model.