Bhushan B. Nanotribology and Nanomechanics: An Introduction

Подождите немного. Документ загружается.

1092 B. Bhushan

20.5.4 Eﬀects of Hydrophobicity

To further test the hypothesis capillary forces, plays a role in gecko adhesion, the

spatular pull-oﬀ force was determined for contact with both hydrophilic and hy-

drophobic surfaces. As seen in Fig. 20.12a, the capillary adhesion theory predicts

that agecko spatulawill generatea greater adhesionforce when in contactwith a hy-

drophilic surface as compared to a hydrophobicsurface while the van der Waals ad-

hesion theory predicts that the adhesion force between a gecko spatula and a surface

will be the same regardless of the hydrophobicity of the surface [7]. Figure 20.12b

shows the shear stress of a whole gecko and adhesive force of a single seta on hy-

drophilic and hydrophobic surfaces. The data shows that the adhesion values are the

same on both surfaces. This supports the van der Waals prediction of Fig. 20.12a.

Huber et al. [47] found that the hydrophobicity of the attachment surface had an ef-

fect on the adhesion force of a single gecko spatula as shown in Fig. 20.12c. These

results show that adhesion force has a ﬁnite value for superhydrophobicsurface and

increases as the surface becomes hydrophilic. It is concluded that van der Waals

forces are the primary mechanism and capillary forces further increase the adhesion

force generated.

20.6 Adhesion Modeling

With regard to the natural living conditions of the animals, the mechanics of gecko

attachment can be separated into two parts: the mechanics of adhesion of a single

contact with a ﬂat surface, and an adaptation of a large number of spatulae to a nat-

ural, rough surface. Modeling of the mechanics of adhesion of spatulae to a smooth

surface, in the absence of meniscus formation, was developed by Autumn et al. [7],

Jagota and Bennison[52] andArzt et al. [3].As discussed previouslyin Sect. 20.3.2,

the adhesion force of multiple contacts F



can be increased by dividing the con-

tact into a large number (n) of small contacts, while the nominal area of the contact

remains the same, F



∼

√

. However, this model only considers contact with

a ﬂat surface. On natural, rough surfaces, the compliance and adaptability of setae

are the primary sources of high adhesion. As stated earlier, the hierarchical struc-

ture of gecko setae allows for a greater contact with a natural, rough surface than

a non-branched attachment system [82].

20.6.1 Spring Model

Bhushan et al. [20] and Kim and Bhushan [55–58] have recently approximated

a gecko seta in contact with random rough surfaces using a hierarchical spring

model. Each level of springs in their model corresponds to a level of seta hierar-

chy. The upper level of springs corresponds to the thicker part of gecko seta, the

middle spring level corresponds to the branches, and the lower level of springs cor-

responds to the spatulae. The upper level is the thickest branch of the seta. It is

75µm in length and 5µm in diameter. The middle level, referred to as a branch,

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1093

Contact angle (deg)

Adhesive force (nN)

Capillary adhesion prediction

c) Results supporting capillary forces

40 60 80 100 120

N-phil

Hydrophilic

SiO

Hydrophilic

van der Waals adhesion prediction

Shear stress (N/mm )

b) Results supporting van der Waals forces

Hydrophobic Hydrophilic Hydrophobic

Hydrophobic

GaS

Hydrophilic

SiO

Hydrophobic

T-phil

Glass

N-phob

T-phob

Adhesive force ( N)μ

Whole

gecko

data

Single

seta

data

0.05

0.10

0.15

0.20

0.25

Fig. 20.12. (a) Capillary and van der Waals adhesion predictions for the relative magnitude

of the adhesive force of gecko setae with hydrophilic and hydrophobic surfaces [7]. (b) Re-

sults of adhesion testing for a whole gecko (left) and single seta (right) with hydrophilic and

hydrophobic surfaces [7] and (c) results of adhesive force testing of a single gecko spatula

with surfaces with diﬀerent contact angles [47]

1094 B. Bhushan

has a length of 25µm and diameter of 1 µm. The lower level, called a spatula, is the

thinnest branchwith a length of 2.5µm and a diameter of about 0.1µm (Table 20.2).

Autumn et al. [6] showed that the optimal attachment angle between the substrate

and a gecko seta is 30° in the single seta pull-oﬀ experiment. This ﬁnding is sup-

ported by the adhesion models of setae as cantilever beams [31,79] (see Sect. 20.3.3

for more details). Therefore, θ was ﬁxed at 30° in the studies [20,55–58].

20.6.2 Single Spring Contact Analysis

In their analysis, Bhushan et al. [20] and Kim and Bhushan [55–58] assumed the

tip of the spatula in a single contact to be spherical. The springs on every level

of hierarchy have the same stiﬀness as the bending stiﬀness of the corresponding

branches of seta. If the beam is orientedat an angle θ to the substrate and the contact

load F is aligned normal to the substrate, its components along and tangential to

the direction of the beam, Fcosθ and Fsinθ, give rise to bending and compressive

deformations, δ

and δ

, respectively, as [95]

Fcosθl

3EI

,δ

Fsinθl

(20.16)

where, I = πR

/4andA

= πR

are the moments of inertia of the beam and the

cross-sectional area, respectively, and l

and R

are the length and the radius of

seta branches, respectively, and m is the level number. The net displacement, δ

⊥

normal to the substrate is given by

⊥

= δ

sinθ + δ

cosθ. (20.17)

Using (20.16) and (20.17), the stiﬀness of seta branches, k

, is calculated as [34].

πR

sin



cot

 (20.18)

For an assumed elastic modulus, E, of seta material of 10GPa with a load applied at

an angle of 60° to spatulae long axis, Kim and Bhushan [55] calculated the stiﬀness

of every level of seta as given in Table 20.3.

In the model, both tips of a spatula and asperity summits of the rough surface

are assumed to be spherical with a constant radius [20]. As a result, a single spatula

adhering to a rough surface was modeled as the interaction between two spherical

tips. Because β-keratin has a high elastic modulus [12, 75], the adhesion force be-

tween two round tips was calculated according to the Derjaguin–Muller–Toporov

(DMT) theory [26] as

= 2πR

(20.19)

where R

is thereduced radiusof contact,which is calculatedas R

(

1/R

+ 1/R

)

−1

;

– radii of contacting surfaces; R

= R

, R

= R/2. The work of adhesion, W

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1095

Table 20.3. Geometrical size, calculated stiﬀness and typical densities of branches of seta for

Tokay gecko (Kim and Bhushan, 2007a)

Level of seta Length

(µm)

Diameter

(µm)

Bending

stiﬀness

(N/m)

Typical density

(#/mm

)

III upper 75 5 2.908 14×10

II middle 25 1 0.126 –

I lower 2.5 0.1 0.0126 1.4–14 × 10

for elastic modulus of 10 GPa with load applied at 60° to spatula long axis

is then calculated using the following equation for two ﬂat surfaces separated by

a distance D [50].

= −

12πD

(20.20)

where H is the Hamakar constant which depends on the medium between the two

surfaces. Typical values of the Hamakar constant for polymers are H

air

= 10

−19

in the air and H

water

= 3.7×10

−20

J in the water [50]. For a gecko seta, which is

composed of β-keratin, the value of H isassumedtobe10

−19

J. The works of

adhesion of two surfaces in contact separated by an atomic distance D ≈ 0.2nmis

approximately equal to 66mJ/m

[50]. By assuming that the tip radius, R,is50nm,

using (20.19), the adhesion force of a single contact is calculated as 10 nN [55].

This value is identical to the adhesion force of a single spatula measured by Huber

et al. [46]. This adhesion force is used as a critical force in the model for judging

whether the contact between the tip and the surface is broken or not during pull-oﬀ

cycle [20]. If the elastic force of a single spring is less than the adhesion force, the

spring is regarded as having been detached.

20.6.3 The Multi-Level Hierarchical Spring Analysis

hierarchical levels in the attachment system on attachment ability, models with one-

[20, 55, 56], two- [20, 55, 56] and three- [55, 56] levels of hierarchy were simu-

lated (Fig. 20.13). The one-level model has springs with length l

= 2.5µmand

stiﬀness k

= 0.0126N/m. The length and stiﬀness of the springs in the two-level

model are l

= 2.5µm,k

= 0.0126N/m and l

= 25µm, k

= 0.126N/m for lev-

els I and II, respectively. The three-level model has additional upper level springs

with l

III

= 75µm, k

III

= 2.908N/m on the springs of the two-level model, which is

identical to gecko setae. The base of the springs and the connecting plate between

the levels are assumed to be rigid. The distance S

between neighboring structures

of level I is 0.35µm, obtained from the average value of measured spatula density,

8×10

−2

, calculated by multiplying 14,000setae/mm

by anaverageof 550 spat-

ula/seta [77] (Table 20.3). A 1 : 10 proportion of the number of springs in the upper

level to that in the level below was assumed [20]. This corresoponds to one spring

1096 B. Bhushan

One-level hierarchy

Two-level hierarchy

Three-level hierarchy

III

Fig. 20.13. One-, two- and

three-level hierarchical spring

models for simulating the ef-

fect of hierarchical morphol-

ogy on interaction of a seta

with a rough surface. In this

ﬁgure, l

I,II,III

are lengths of

structures, s

is space be-

tween spatulae, k

I,II,III

are

stiﬀnesses of structures, I, II

and III are level indexes, R is

radius of tip, and h is distance

between upper spring base of

each model and mean line of

the rough proﬁle [55]

at level III is connected to ten springs on level II and each spring on level II also

has ten springs on level I. The number of springs on level I considered in the model

is calculated by dividing the scan length (2000µm) with the distance S

(0.35µm),

which corresponds to 5700.

The spring deﬂection Δl was calculated as

Δl = h−l

−z , (20.21)

where h is the positionof the spring base relative to the mean line of surface; l

is the

total length of a spring structure which is l

= l

for the one-level model, l

= l

+ l

for thetwo-levelmodel, andl

= l

III

for the three-levelmodel; and z is proﬁle

height of the rough surface. The elastic force F

arisen in the springs at a distance

h from the surface was calculated for the one-level model as [20]

= −k



i=1

Δl



1 if contact

0 if no contact

(20.22)

where p is the number of springs in the level I of the model. For the two-level model

the elastic force was calculated as [20]

= −



j=1



i=1

(Δl

−Δl



1 if contact

0 if no contact

(20.23)

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1097

where q is the number of springs in the level II of the model. For the three-level

model the elastic force was calculated as [55]

= −



k=1



j=1



i=1

kji

(Δl

kji

−Δl

kji



1 if contact

0 if no contact

(20.24)

where r is the number of springs in the level III of the model. The spring force when

the springs approach the rough surface is calculated using either (20.22), (20.23),

or (20.24) for one-, two- and three-level models, respectively. During pull-oﬀ,the

same equations are used to calculate the spring force. However, when the applied

load is equal to zero, the springs do not detach due to adhesion attraction given

by (20.19). The springs are pulled apart until the net force (pull-oﬀ force minus

attractive adhesion force) at the interface is equal to zero.

The adhesion force is the lowest value of elastic force F

when the seta has

detached from the contacting surface. The adhesion energy is calculated as

∞



(

)

dD (20.25)

where D is the distancethat the springbase movesawayfrom the contactingsurface.

The lower limit of the distance

D is the value of D where F

is ﬁrst zero when the

model is pulled away from the contacting surface. Also although the upper limit of

the distance is inﬁnity, in practice, the F

(D) curve is integrated to and upper limit

where F

increases from a negative valueto zero. Figure 20.14shows the ﬂow chart

for the calculation of the adhesion force and the adhesion energy employed by Kim

and Bhushan [55].

The random rough surfaces used in the simulations were generated by a com-

puter program [14,15]. Two-dimensional proﬁles of surfaces that a gecko may en-

counter were obtained using a stylus proﬁler [20]. These proﬁles along with the

surface selection methods and surface roughness parameters (root mean square

(RMS) amplitude σ and correlation length β

∗

) for scan lengths of 80, 400, and

2000µm are presented in Appendix 20.A. The roughness parameters are scale

dependent, and, therefore, adhesion values also are expected to be scale depen-

dent. As the scan length was increased, the measured values of RMS amplitude

and correlation length both increased. The range of values of σ from 0.01µmto

30µm and a ﬁxed value of β

∗

= 200µm were used for modeling the contact of

a seta with random rough surfaces. The chosen range covers values of roughnesses

for relatively smooth, artiﬁcial surfaces to natural, rough surfaces. A typical scan

length of 2000µm was also chosen which is comparable to a lamella length of

gecko.

1098 B. Bhushan

Calculate F, E

ad ad

iad

≤

Number of contacts = 0

Calculate elastic force, F

Yes

Pull off loop

Calculate number of contacts

Yes

Calculate , ,ΔΔΔlll

kji kj k

New spring position, h

Calculate elastic force, F

Yes

Calculate Δl

kji

New spring position, h

Guess initial spring position, h

Rough surface height data, z

Input parameters

Fk k k NN N l l l

n I II III I II III I II III

,, , , , , ,, ,

Δl

kji

≤ 0

el n

–

< 0.001

Press down loop

Fig. 20.14. Flow chart for the

calculation of the adhesion

force (F

) and the adhesion

energy (E

) for three-level

hierarchical spring model. In

this ﬁgure, F

is an applied

load, k

I, II, III

and l

I, II, III

are

stiﬀnesses and lengths of

structures, Δl

kji

, Δl

and Δl

are the spring deformations

on level I, II and III, respec-

tively, i, j and k are spring

indexes on each level, f

the elastic force of a single

spring and f

is the adhesion

force of a single contact [55]

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1099

20.6.4 Adhesion Results of the Multi-level Hierarchical Spring Model

by Kim and Bhushan [55]. The obtained varioususeful results and will be presented

next. Figure 20.15a shows the calculated spring force–distance curves for the one-,

two- and three-level hierarchical models in contact with rough surfaces of diﬀerent

values of root mean square (RMS) amplitude σ anging from σ0.01µmtoσ10µm

at an applied load of 1.6µN which was derived from the gecko’s weight. When the

springmodel is pressedagainstthe roughsurface,contact betweenthe springand the

rough surface occurs at point A; as the spring tip presses into the contactingsurface,

the force increases up to point B, B



or B



. During pull oﬀ, the spring relaxes, and

the spring force passes an equilibrium state (0 N); tips break free of adhesion forces

at point C, C



or C



as the spring moves away from the surface. The perpendicular

distance from C, C



or C



to zero is the adhesion force. Adhesion energy stored

during contact can be obtained by calculating the area of the triangle during the

unloading part of the curves (20.25).

Using the spring force-distance curves, Kim and Bhushan [55] calculated the

adhesion coeﬃcient, the number of contacts per unit length and the adhesion en-

ergy per unit length of the one-, two- and three-level models for an applied load of

1.6 µN and a wide range of RMS roughness as seen in the left graphs of Fig. 20.15b.

The adhesion coeﬃcient, deﬁned as the ratio of pull-oﬀ force to applied preload,

represents the strength of adhesion with respect to the preload. For the applied load

of 1.6µN, which corresponds to the weight of a gecko, the maximum adhesion co-

eﬃcient is about 36 when σ is smaller than 0.01µm. This means that a gecko can

generate enough adhesion force to support 36 times its bodyweight. However, if σ

is increased to 1 µm, the adhesion coeﬃcient for the three-level model is reduced

to 4.7. It is noteworthy that the adhesion coeﬃcient falls below 1 when the con-

tacting surface has an RMS roughness σ greater than 10µm. This implies that the

attachment system is no longer capable of supporting the gecko’s weight. Autumn

et al. [6,7] showedthat in isolated gecko setae contactingwith the surface of a single

crystalline silicon wafer, a 2.5µN preload yielded adhesion of 20 to 40µN and thus

a value of adhesion coeﬃcient of 8 to 16, which supports the simulation results of

Kim and Bhushan [55].

Figure 20.15b (top left) shows that the adhesion coeﬃcient for the two-level

model is lower than that for the three-level model, but there is only a small diﬀer-

ence between the adhesion forces between the two- and three-levelmodels, because

the stiﬀness of level III for the three-level model is calculated to be higher com-

pared to those of levels I and II. In order to show the eﬀect of stiﬀness, the results

for the three-level model with springs in level III of which the stiﬀness is 10 times

smaller than that of original level III springs are plotted. It can be seen that the

three-level model with a third level stiﬀness of 0.1 k

III

has a 20–30% higher ad-

hesion coeﬃcient than the three-level model. The results also show that the trends

in the number of contacts are similar to that of the adhesive force. The study also

investigated the eﬀect of σn adhesion energy. It was determined that the adhesion

energy decreased with an increase of σ. For the smooth surface with σ0.01µm, the

adhesion energies for the two- and three-level hierarchical models are 2 times and

1100 B. Bhushan

Distance ( )mμ

–60

–2 –1

0123

–40

–20

–60

–2 –1

0123

–40

–20

–60

–2 –1

0123

–40

–20

–60

Spring force ( N)μ

–2 –1

0123

–40

–20

= 0.01 mσ μ

= 0.1 mσ μ

=1 mσ μ

=10 mσ μ

The effect of multi-level hierachical structure

B’’

= 1.6 m * = 200 mμμ, β

B’ B

C’’ C’ C

One-level

Two-level

Three-level

Fig. 20.15. (a) Force-distance

curves of one-, two- and

three-level models in contact

with rough surfaces with

diﬀerent σ values for an

applied load of 1.6 µN

20 Gecko Feet: Natural Hairy Attachment Systems for Smart Adhesion 1101

Adhesion energy per unit length (×10 J/mm)

–12

10110

–1

–2

σ ( m)μ

Three-level

Two-level

One-level

The effect of multi-level hierachical structure

= 1.6 m * = 200 mμμ, β

2000

Relative increase of adhesion energy (%)

50.01

σ ( m)μ

500

1000

1500

0.05 0.1 0.5 1 3 10 30

3000

Number of contacts per unit length (mm )

–1

1000

2000

Relative increase of contact numbers (%)

100

150

Adhesion coefficient ( )FF

ad n

Three-level (0.1 )k

III

Three-level ( )k

III

Two-level

One-level

200

Relative increase of adhesion coefficient (%)

100

150

110

Fig. 20.15. (continued) (b) The adhesion coeﬃcient, the number of contacts and the adhesion

energy per unit length of proﬁle for one- and multi-level models with an increase of σ value

(left ﬁgures), and relative increases between multi- and one -level models (right side)foran

applied load of 1.6 µN. The value of k

III

in the analysis is 2.908 N/m [55]