Bhushan B. Nanotribology and Nanomechanics: An Introduction

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1112 B. Bhushan

The condition for the prevention of sticking is F

> F

. By combining (20.39)

and (20.40), a requirement for the minimum distance S between structures which

will prevent sticking of the structures is given as [57]

S > 2δ = 2





= 2





(20.41)

The constant 2 takes into account two nearest structures. Using distance S,theﬁber

density, ρ, is calculated as

ρ =

(S + 2R)

(20.42)

Equation 20.42 was then used to calculate the allowed minimum density of ﬁbers

without sticking or bunching. In (20.41), it is shown that the minimum distance, S,

depends on both the aspect ratio λ and the elastic modulus E. A smaller aspect ratio

and higher elastic modulus allow for greater packing density. However, ﬁbers with

a low aspect ratio and high modulus are not desirable for adhering to rough surfaces

due to lack of compliance.

20.7.4 Numerical simulation

The simulation of adhesion of an attachment system in contact with random rough

surfaces was carried out numerically. In order to conduct 2D simulations it is ne-

cessary to calculate applied load F

as a function of applied pressure P

as an input

condition.Using ρ calculated by non-stickingcondition, Kim and Bhushan [57] cal-

culated F

(20.43)

where p is the number of springs in scan length L, which equals L/(S + 2R).

Fibers of the attachment system are modeled as one-level hierarchy elastic

springs (Fig. 20.13) [57]. The deﬂection of each spring and the elastic force arisen

in the springs are calculated according to (20.21) and (20.22), respectively. The ad-

hesion force is the lowest value of elastic force F

when the ﬁber hasdetached from

the contacting surface. Kim and Bhushan [57] used an iterative process to obtain op-

timalﬁber geometry– ﬁber radius and aspectratio. Ifthe appliedload, theroughness

of contacting surface and the ﬁber material are given, the procedure for calculating

the adhesion force is repeated iteratively until the desired adhesion force is satis-

ﬁed. In order to simplify the design problem, ﬁber material is regarded as a known

variable. The next step is constructing the design database. Figure 20.22a shows the

ﬂow chart for the construction of adhesion design database and Fig. 20.22b shows

the calculation of the adhesion force that is a part of the procedure to construct an

adhesion design database.

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1113

Calculate F

i ad

≤

Number of contacts = 0

Calculate elastic force, F

Yes

Pull of loop

Calculate number of contacts

Yes

Calculate Δl

New spring position, h

Calculate elastic force, F

Yes

Calculate Δl

New spring position, h

Guess initial spring position, h

Rough surface height data, z

Input parameters

FRRl

,, , ,,N

tσ

Δl ≤ 0

el n

–

< 0.001

Press down loop

Calculation of adhesion force

Yes

Guess initial fiber size, R and λ

Input parameters

,,, ,E'R

μ σ

μμ

new

–

< 0.001

Non-buckling condition

Non-sticking condition

Non-fiber fracture condition

Construction of adhesion

design database

Fig. 20.22. Flow chart for (a) the construction of adhesion design database and (b) the calcu-

lation of the adhesion force. In this ﬁgure, P

is the applied pressure, E is the elastic modulus,



is the adhesion coeﬃcient, R

is the tip radius, σ is root mean square (RMS) amplitude,R

is the ﬁber radius, λ is the aspect ratio of ﬁber, F

is the applied load, N is the number of

springs, k and l are stiﬀness and length of structures, Δl is the spring deformation, f

is the

elastic force of a single spring and f

is the adhesion force of a single contact [57]

1114 B. Bhushan

20.7.5 Results and discussion

Figure 20.23 shows an example of the adhesion design database for biomimetic at-

tachment systems consisting of single-level cylindrical ﬁbers with orientation angle

of 30° and spherical tips of R

= 100nm constructed by Kim and Bhushan [57].

The minimum ﬁber radius calculated by non-ﬁber fracture condition, which plays

a role of lower limit of optimized ﬁber radius, is also added on the plot. The plots in

Fig. 20.22 cover all applicable ﬁber materials from soft elastomer material such as

poly(dimethylsiloxane)(PDMS) tostiﬀer polymerssuch aspolyimideand β-keratin.

The dashedlines in each plotrepresentthe limits of ﬁber fracturedue to the adhesion

force. For a soft material with E = 1 MPa in Fig. 20.23a, the range of the desirable

ﬁber radius is more than 0.3µm and that of the aspect ratio is approximately less

than 1. As elastic modulus increases, the feasible range of both ﬁber radius and as-

pect ratio also increase as shown in Fig. 20.23b and 20.23c. In Fig. 20.23, the ﬁber

radius has a linear relation with the surface roughness on a logarithm scale.

If the applied load, the roughness of contacting surface and the elastic modu-

lus of a ﬁber material are speciﬁed, the optimal ﬁber radius and aspect ratio for the

desired adhesion coeﬃcient can be selected from this design database. The adhe-

sion databases are useful for understanding biological systems and for guiding the

fabrication of biomimetic attachment systems. Two case studies [57] are calculated

below.

Case study I: Select the optimal size of ﬁbrillar adhesive for a wall climbing

robot with the following requirements:

• Material: polymer with E ≈100MPa

• Applied pressure by weight <10kPa

• Adhesion coeﬃcient < 5

• Surface roughness σ<1 µm.

The subplot of adhesion database that satisﬁes the requirement is at second col-

umn and second row in Fig. 20.23b. From this subplot, any values on the marked

line can be selected to meet the requirements. For example, ﬁber radius of 0.4 µm

with aspect ratio of 1 or ﬁber radius of 10 µm with aspect ratio of 0.8 satisﬁes the

speciﬁed requirements.

Case study II: Compare with adhesion test for a single gecko seta [6,7]:

• Material: β-keratin with E ≈10GPa

• Applied pressure = 57kPa (2.5µNonanareaof43.6µm

)

• Adhesion coeﬃcient = 8to16

• Surface roughness σ<0.01µm.

Autumn et al. [6,7] showed that in isolated gecko setae contacting with surface

of a single crystalline silicon wafer, a 2.5 µN preload yielded adhesion of 20 to

40µN and thus a value of adhesion coeﬃcient of 8 to 16. The region that satisﬁes

the above requirements is marked in Fig. 20.23c. The spatulae of gecko setae have

an approximate radius of 0.05µm with an aspect ratio of 25. However, the radius

corresponding to λ = 25 for the marked line is about 0.015µm. This discrepancy

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1115

Fig. 20.23. Adhesion design database for biomimetic attachment system consisting of single-

level cylindrical ﬁbers with orientation angle of 30° and spherical tips of 100 nm for elastic

modulus of (a) 1MPa,(b) 100 MPa and (c) 10 GPa [57]. The solid lines shown in Figs. (b)

and (c) correspond to the cases I and II, respectively, which satisfy the speciﬁed requirements

1116 B. Bhushan

Fig. 20.23. (continued)

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1117

Fig. 20.23. (continued)

1118 B. Bhushan

is due to the diﬀerence between simulated ﬁber model and real gecko setae model.

Gecko setae are composed of three-level hierarchicalstructure in practice, so higher

adhesion can be generated than a single-level model [20,55,56]. Given the simpli-

ﬁcation in the ﬁber model, this simulation result is very close to the experimental

result.

20.8 Fabrication of Biomimetric Gecko Skin

On the basis of studies found in the literature, the dominant adhesion mechanism

utilized by geckos and other spider attachment systems appears to be van der Waals

forces. The complex divisions of the gecko skin (lamellae-setae-branches-spatulae)

enable a large real area of contact between the gecko skin and mating surface.

Hence, a hierarchical ﬁbrillar micro/nanostructure is desirable for dry, superadhe-

sive tapes. The development of nanofabricated surfaces capable of replicating this

adhesion force developed in nature is limited by current fabrication methods. Many

diﬀerent techniqueshavebeen used in an attempt to create [32,64,65,82,96,97]and

characterize [19,37,68] bio-inspired adhesive tapes.

Gorb et al. [37] and Bhushan and Sayer [19] characterized two polyvinylsilox-

ane (PVS) samples from Gottlieb Binder Inc., Holzgerlingen, Germany: one con-

sisting of mushroom-shaped pillars (Fig. 20.24a) and the other sample was an un-

structured control surface (Fig. 20.24b). The structured sample is inspired by the

micropatterns found in the attachment systems of male beetles from the family

chrysomelidae,and are easier to fabricate.Both sexes possess adhesivehairs on their

tarsi, however, males bear hair extremely specialized for adhesion on the smooth

surface of female’s covering wings during mating. The hairs have broad ﬂattened

tip with grooves under the tip to provide ﬂexibility. The structured samples were

produced at room temperature by pouring two-compound polymerizing PVS into

the holed template lying on a smooth glass support. The fabricated sample is com-

prised of pillars that are arranged in a hexagonal order to allow maximum packing

density. They are approximately 100µm in height, 60µm in base diameter, 35µm

in middle diameter and 25 µm in diameter at the narrowed region just below the

terminal contact plates. These plates were of about 40µm in diameter and 2µ min

thickness at the lip edges. The adhesion force of the two samples in contact with

a smooth ﬂat glass substrate was measured by Gorb et al. [37] using a home-made

microtribometer. Results revealed that the structured specimens featured an adhe-

sion force more than twice that of the unstructured specimens. The adhesion force

was also found to be independentof the preload. Moreover, it was found that the ad-

hesive force of the structured sample was more tolerant to contamination compared

to the control and it could be easily cleaned with a soap solution.

Bhushan and Sayer [19] characterized the surface roughness, friction force, and

contact angle of the structured sample and compared the results to an unstructured

control.As shown in Fig. 20.25a, the macroscalecoeﬃcient of kinetic friction of the

structured sample was found to be almost four times greater than the unstructured

sample. This increase was determined to be a result of the structured roughness

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1119

50 mμ

Fig. 20.24. SEM micrographs

of the (a) structured and (b)

unstructured PVS samples.

SH: shaft, NR: neck region,

LP: lip [19]

of the sample and not the random nanoroughness. It is also noteworthy that the

static and kinetic coeﬃcients of friction are approximately equal for the structured

sample. It is believed that the divided contacts allow the broken contacts of the

structured sample to constantly recreate contact. As seen in Fig. 20.25b, the pillars

also increased the hydrophobicity of the structured sample in comparison to the

unstructuredsample as expecteddue to increasedsurfaceroughness [22,92].A large

contact angle is important for self cleaning [10], which agrees with the ﬁndings of

Gorb et al. [37] that the structured sample is more tolerant of contamination than

the unstructured sample.

20.8.1 Single Level Hierarchical Structures

Oneof the simplestapproachesto create asingle levelhierarchicalsurfaceemployed

an AFM tip to create a set of dimples on a wax surface. These dimples served as

a mold for creating polymer nanopyramidsshown in Fig. 20.26a[82]. The adhesive

force to an individual pyramid was measured using another AFM cantilever. The

force was found to be about 200 µN. Although each pyramid of the material is

capable of producing large adhesive forces, the surface failed to replicate gecko

adhesion on a macroscale. This was due to the lack of ﬂexibility in the pyramids. In

1120 B. Bhushan

150

b) Contact angle (deg)

Structured

120

a) Coefficient of friction

Static friction

Kinetic friction

Unstructured

Structured Unstructured

Fig. 20.25. (a) Coeﬃcients of

static and kinetic friction for

the structured and unstruc-

tured samples slid against

magnetic tape with a normal

load of 130 mN.

(b) Water contact angle for

the structured and unstruc-

tured samples [19]

order to ensure that the largest possible area of contact occurs between the tape and

mating surface, a soft, compliant ﬁbrillar structure would be desired [52]. As shown

in previouscalculations, the van der Waals adhesiveforce for two parallel surfaces is

inversely proportional to the cube of the distance between two surfaces. Compliant

ﬁbrillar structures enable more ﬁbrils to be in close proximity of a mating surface,

thus increasing van der Waals forces.

Geim et al. [32] created arrays of nanohairs using electron-beam lithography

and dry etching in oxygen plasma (Fig. 20.26b (left)). The original arrays were

created on a rigid silicon wafer. This design was only capable of creating 0.01N

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1121

10 mμ

2 mμ 2 μm

Fig. 20.26. SEM micrographs of (a) three pillars created by nano-tip indentation [81],

(b) (left) an array of polyimide nanohairs and (right) bunching of the nanohairs, which leads

to a reduction in adhesive force [32], and (c) directed self-assembly based method of produc-

ing high aspect ratio micro/nanohairs [81]

of adhesive force for a 1 cm

patch. The nanohairs were then transferred from the

silicon wafer to a soft polymer substrate. A 1 cm

samplewasabletocreate3N

of adhesive force under the new arrangement. This is approximately one-third the

adhesive strength of a gecko. The fabricated a Spiderman toy (about 0.4N) with

a hand covered with molded polymer nanohairs, Fig. 20.27. The demonstrated that