adhesion mechanics of ivy nanoparticles

Journal of Colloid and Interface Science 344 (2010) 533–540

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Journal of Colloid and Interface Science

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Adhesion mechanics of ivy nanoparticles

Yu Wu, Xiaopeng Zhao *, Mingjun ZhangMechanical, Aerospace, and Biomedical Engineering Department, University of Tennessee, Knoxville, TN 37996, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 June 2009Accepted 23 December 2009Available online 4 January 2010

Keywords:AdhesionContact mechanicsBiological materialsNanostructured materials

0021-9797/$ - see front matter � 2009 Elsevier Inc. Adoi:10.1016/j.jcis.2009.12.041

* Corresponding author. Fax:+1 865 974 6372.E-mail addresses: [email protected], [email protected] (

Adhesion mechanism of ivy has been of major research interest for its potential applications in high-strength materials. Recent experimental studies demonstrated that nanoparticles secreted from ivy ten-drils play an important role in adhesion. In this work, we investigate how various factors such as van derWaals interaction, capillarity, and molecular cross-linking influence the adhesion mechanics of ivy nano-particles. This paper provides guidelines in choosing different adhesive contact models. Understandingthe mechanics of ivy adhesion could potentially inspire the design and fabrication of novel nano-bio-materials.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Ivy is a fascinating climber. Using its tendrils, ivy can climb onwalls, trees, and many other surfaces. Though very small in size,the tendrils can provide surprisingly large forces. For example,Darwin [1] showed that an ivy disc of weight 0.5 mg can produceabout 2 lb pull-off force, which is over 1.8 million larger than theweight of adhering disc. Thus, understanding ivy adhesionmechanics may have important significance in material science.Using atomic force microscopy (AFM), Zhang et al. [2,3] observedivy secretes nanoparticles through adhering discs of the ivy root-lets that allow the plant to affix to a surface (see Fig. 1). This obser-vation suggests that the nanoparticles play a direct and importantrole for ivy surface climbing although the mechanics interpretationfor the adhesion is still not completely known. This paper attemptsto explore the adhesion mechanics of ivy nanoparticles using con-tact and fracture mechanics models. In particular, the computa-tional work here is greatly inspired by the seminal work of Gaoet al. in their research about the adhesion structures of gecko [4–8].

From mechanics point of view, adhesion is expressed in termsof the work of adhesion [9], the physical origins of which may in-clude: van der Waals interaction, electrostatic forces, capillarityand chemical bond. The adhesion also undergoes cross-linkingwith a corresponding increase in adhesion strength, and roughnesswith a decrease in strength. Continuum mechanics models of theadhesion between spherical surfaces which deform within theelastic limit are well developed. The inter-atomic forces at inter-faces were first explained by London [10], and soon after they were

ll rights reserved.

X. Zhao).

applied by Bradley [11], Derjaguin [12] and Hamaker [13] to theproblem of the forces between small particles. In an attempt tocharacterize the adhesive contact between elastic spheres, threemajor theories have been developed. These theories include: John-son–Kendall–Roberts (JKR) [14], Derjaguin–Muller–Toporov (DMT)[15], and Maugis-Dugdale (M-D) [16]. The JKR theory is applicableto large, soft, compliant materials with high surface energy. Theadhesion forces outside the area of contact are neglected and elas-tic stresses at the edge of the contact are infinite. Contrary to theJKR theory, the DMT theory applies to smaller, stiffer, less compli-ant materials with a low surface energy. The interaction forces out-side the contact area are taken into account, but these interactionforces are assumed not to deform the profile. The M-D theory is formaterials with property between JKR and DMT regimes, and can begoverned by a non-dimensional elasticity parameter defined asfollows

k � r09R

2pwE�2

� �1=3

ð1Þ

where E� ¼ 1� m21

� �=E1 þ 1� m2

2

� �=E2

� ��1 is the combined effectiveelastic modulus of the two contacting objects, Ei and mi are Young’smoduli and Poisson’s ratios of the two materials, respectively. R isthe equivalent radius of the two spheres given by 1=R ¼ 1=R1þ1=R2, Ri are the radii of the two contacting spheres, w is work ofadhesion, r0 the maximum attractive stress. This parameter maybe interpreted as the ratio of the elastic deformation of the surfacesat the point of separation (pull off) to the effective range of action ofthe adhesive forces. The JKR theory and DMT theory are each appro-priate to opposite extremes of the parameter k. When k increasesfrom zero to infinity there is a continuous transition from theDMT approximation to the JKR approximation [16,17].

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Dugdale

Lennard Jones

h0

1 1.5 2 2.5 3

z

z00

0.5

1

1.50

σσ

Fig. 3. Force-separation laws: Lennard–Jones potential; the Dugdale approxima-tion, the work of adhesion w ¼ r0h0, the cohesive zone is thus h0 ¼ 0:97z0.

Fig. 1. An AFM image of nanoparticles secreted from the adhering discs of ivy on asilicon wafer surface.

534 Y. Wu et al. / Journal of Colloid and Interface Science 344 (2010) 533–540

2. Problem formulation

2.1. Geometry of a nanoparticle

Based on the observation of AFM topography images (see Fig. 1),we postulate that the ivy nanoparticles may not be perfect nano-spheres, and assume that there are some irregular planes on thesurface. To understand the nano scale adhesion mechanism, weconsider a rigid/elastic spherical cap with a flat punch in contactwith a smooth rigid substrate, as shown in Fig. 2. The radius ofthe actual contact area is a ¼ R sin a; 0� 6 a 6 90�. Three models,generalized from DMT, JKR, and M-D theory to specified geometry,are presented to interpret the pull-off force.

2.2. DMT type model

In general, the surface force r, defined as the force per unit areabetween two half-spaces separated by a distance z, can be obtainedfrom the Lennard–Jones potential and is given by

rðzÞ ¼ 8w3z0

z0

z

� 3� z0

z

� 9 �

ð2Þ

α

Fig. 2. Schematic diagram of the contact between a nanoparticle and a rigidsubstrate.

where z0 is the equilibrium separation. To avoid self-consistentnumerical calculations based on the Lennard–Jones interactionmodel, it is easier to representing the surface force by the Dugdaleapproximation [18], in which the attractive stress is assumed to bea constant r0 with a critical separation h0 ¼ w=r0 (called the cohe-sive zone) and zero beyond this distance, as shown in Fig. 3.

Using the Dugdale type interaction law, we consider a simpleDMT type model in which the deformation of contact surfaces isneglected (rigid assumption) and assume a constant attractivestress r0 inside the contact area. The maximum adhesive force,i.e., the pull-off force to separate two rigid objects, can be calcu-lated as follows:

Fp ¼ pa2r0 þZ r0

a2pr0rdr ¼

pa2r0 þR h0

0 2pr0ðR cos a� hÞdh;

if h0 < R cos apa2r0 þ

R R cos a0 2pr0ðR cos a� hÞdh;

if h0 P R cos a

8>>>><>>>>:

¼ F0½1� cos2 að1� fÞ2�; f < 1F0; f P 1

(ð3Þ

where r0 is the radius of the cohesive zone, and

f ¼ h0

R cos a; F0 ¼ pR2r0 ð4Þ

There is a saturation of adhesion strength below a critical sizeR0 ¼ h0= cos að0� 6 a < 90�Þ or a critical angle a0 ¼ arccosðh0=RÞ.The contact achieves its theoretical strength F0 when R < R0 ora > a0. The perfect shape is simply a hemisphere with a ¼ 90�, un-der which condition the nanoparticle adhering to a flat rigid sub-strate would achieve the theoretical adhesion strength F0

regardless of the particle size R.

2.3. JKR type model

Since biological contacts usually consist of compliant materials[19], elastic deformation should be considered in the analysis ofadhesive contact.

Gao et al. proposed a JKR type model, which is consistent withlinear elastic fracture mechanics (LEFM) approximation, to deter-mine the pull-off forces of a cylindrical spatula [5]. Here we modelthe contact as an elastic spherical cap in contact with a rigid sub-strate, resembling a soft ivy nanoparticle in contact with a hardmaterial. We assume that adhesion forces outside the area of con-tact are negligible and elastic stresses at the edge of the contact areinfinite. The adhesive strength of such an adhesive joint can be cal-culated by treating the contact problem as a circumferential crack(see Fig. 4), in which case the stress field near the edge of the

Fig. 4. Schematic diagram of the contact between an elastic nanoparticle and a rigidsubstrate. The contact problem is treated as a circumferential crack with length 2a.

Y. Wu et al. / Journal of Colloid and Interface Science 344 (2010) 533–540 535

contact area (crack tip) has a square-root singularity [5,16]. Thestress intensity factor for mode I [20]

KI ¼ rffiffiffiffiffiffipap

K�1ðaÞ ¼Fp

affiffiffiffiffiffipap K�1ðaÞ ð5Þ

where the geometry factor K�1ðaÞ varies in a narrow range between0.4 and 0.6 for 0� 6 a 6 90� [21]. Here we simply set K�1 ¼ 0:5.

In the LEFM approximation, the strain energy release rate(when the crack varied by unit area) is

G ¼ 12

K2I

E�ð6Þ

The coefficient 12 arises because the substrate is undeformable.

We are faced with half a Griffith crack.As a bridge over surface physics and fracture mechanics, the

Griffith criterion shows the equilibrium between the strain energyrelease rate and the work of adhesion: to extend the crack by unitarea, the work w is needed; it is taken from the elastic field and/orthe potential energy of the system [22]

G ¼ w ð7Þ

From Eqs. (5)–(7), we obtain the pull-off force

Fp ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pE�wR3 sin3 a

pð8Þ

Since the theoretical adhesion strength Fth is

Fth ¼ p sin2 aR2r0 ð9ÞFp 6 Fth ð10Þ

E�

sina6

pRr20

8wð11Þ

The model will achieve the maximum strength and the satura-tion occurs at a critical effective elastic modulus

E�cr ¼pR sinar2

0

8wð12Þ

2.4. M-D model

Combining the results of fracture mechanics with contact the-ory, Maugis presents a transition solution in closed form in whichthe adhesion forces outside the area of contact are taken into ac-count [16]. Intimate contact is maintained over a central regionof radius a; adhesion forces extend to a radius r0. In the annulusa 6 r 6 r0 the surfaces separate slightly by a distance increasingfrom zero to h0. By using the Dugdale approximation, the adhesionforces at the crack lips (the air gap) are assumed to have a constantvalue r0, the theoretical stress, up to a maximum separation h0 be-yond which it falls to zero. This internal loading acting in the airgap leads to a stress intensity factor Km, which compensates forthe stress intensity factor KI due to the external applied loading.This cancellation suppresses the stress singularities, ensures thecontinuity of stresses, and determines the distance r0 � a of thecohesive zone.

We consider the contact model geometry shown in Fig. 2. Anelastic spherical cap is shown to be in contact with a rigid smoothplane. The profile of the bottom half of the spherical cap is given by

f ðrÞ ¼0; r < affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 � a2p

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � r2

p; a 6 r 6 R

(ð13Þ

We assume that the elastic displacement of the surface of thebottom half of the spherical cap under applied load P (compressionpositive) can be approximated by the deduction of crack opening inthe Boussinesq theory [16,23,24]

DuPðrÞ ¼ �P

2aE�þ P

paE�arcsin

ar¼ 2

ffiffiffiffiap

rKI

E�arccos

ar; r P a ð14Þ

where

KI ¼ �Pffiffiffiffiffiffiffiffiffiffiffi

4pa3p ð15Þ

is the stress intensity factor due to the external loading.Internal loading (adhesion forces) acting in the crack lips will

reduce the crack opening. Assuming that a constant adhesion force�r0 is applied on a length r0 � a, the elastic displacement of thecrack lips due to r0 is [16]

DurðrÞ ¼ 2ffiffiffiffiap

rKm

E�arccos

ar

� 4r0

paE�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

0 � a2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2 � a2p

� a arccosar

�

� ar20

Z minðr;r0Þ

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 � s2p

s2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

0 � s2q ds

375; r P a ð16Þ

where r0 is the radius of the end of the cohesive zone, and

Km ¼r0ffiffiffiffiffiffipap

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � a2p

þ c2

aarccos

ac

�ð17Þ

is the stress intensity factor due to the inner loading r0. The stresssingularities at the edge of the contact disappears if

KI þ Km ¼ 0 ð18Þ

The crack opening in case of both external applied loading P andinternal attraction r0 is thus

hðrÞ ¼ f ðrÞ þ DuPðrÞ þ DurðrÞ ð19Þ

The energy release rate G is computed by the J-integral [25], andby Dugdale approximation is simply

G ¼ r0hðr0Þ ð20Þ

Using the Griffith criterion (7) and the singularity cancelationcondition (18), we have the equilibrium relation

w ¼ r0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � a2

p� ar0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

a2 � g2

sþ 4ar2

0

pE�ffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 1

parccos

1g� gþ 1

�ð21Þ

from which g ¼ r0=a can be solved. The pull-off force can be written

Fp ¼ �Pmin ¼ffiffiffiffiffiffiffiffiffiffiffi4pa3p

KI ð22Þ

where

KI ¼ r0

ffiffiffiffiap

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffig2 � 1

pþ g2 arccos

1g

� �ð23Þ

2.5. Guidance for model selection

Three models for calculating the force of the adhesion of a nano-particle with a rigid substrate are considered. The appropriatemodel to use depends on the elasticity and geometry of thenanoparticles. To assist in model selection, a diagram has beenconstructed with coordinates E� and a, as shown in Fig. 5. While

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

Effective elastic modulus of ivy nanoparticles E* (GPa)

Angl

eα

(°)

I

II

III

IV

Fig. 5. A diagram to guide the model selection. I zone: M-D model; II zone: JKR typemodel; III zone: JKR Dugdale model; IV zone: DMT type model.


applied compression may increase the radius of the intimate con-tact area, it is worth noting that the angle a is an intrinsic param-eter related to the original shape of nanoparticle only, and thusshould be measured without any applied load. The boundaries inthe diagram are drawn according to the criterion that the pull-offforces calculated by the two models are the same on the boundary.In this illustration the parameters are chosen as R = 35 nm,w ¼ 0:025 J=m2; A ¼ 0:7� 10�19 J, r0 = 108.72 MPa, h0 = 0.23 nm.The reasoning for the parameter setting will be explained in Sec-tion 3.1.

2.5.1. The M-D zone (I zone)Since in the M-D theory the distribution of surface traction is

built up from (1) the Hertz pressure associated with a intimatecontact of radius a, and (2) the Dugdale cohesive stress, the M-Dmodel can only be used under conditions where the surface profileof the nanoparticle under the effect of a normal load and surfaceforces is very similar to that described by Hertz theory. The contactshould be non-conforming contact, in which the shapes of thebodies are dissimilar enough that, under no load, they only touchat a point. So the angle a must be small enough to achieve goodapproximation. In the M-D zone the pull-off force is given by Eq.(22).

2.5.2. The JKR type zone (II zone)The JKR type model is based on the LEFM approximation, which

is only valid if the length r0 � a of the cohesive zone is small incomparison to the crack length 2a. We can get the criterion fromEq. (5) by setting g ¼ 2

a� arcsinK2

I

11:16Rr20

ð24Þ

On the other hand, the error introduced by neglecting the adhe-sion force outside the contact area will be unacceptable if Eq. (24)is not satisfied.

In the JKR type zone the pull-off force is given by Eq. (8) withsaturation value Eq. (9).

2.5.3. The JKR Dugdale zone (III zone)The cohesive force in the annulus a 6 r 6 r0 should not be ne-

glected when a is small. We can replace the saturation force inEq. (9) with Eq. (3), thus the critical effective elastic modulus

E�cr ¼E0 1� cos2 að1� fÞ2h i2

; f < 1

E0; f P 1

8<: ð25Þ

where f is the same as that in Eq. (4), and

E0 ¼pRr2

0

8Dc sin3 að26Þ

We name this zone as ‘‘JKR Dugdale model” in our model dia-gram. The pull-off force is given by Eq. (8) with saturation valueEq. (3).

2.5.4. The DMT type zone (IV zone)The DMT type model is applicable when effective elastic modu-

lus E� of nanoparticle is large. The critical value of E� can be com-puted from Eq. (25), and the pull-off force is given by Eq. (3).

The model diagram (Fig. 5) provides an intuitive guidance to theselection of models. Once the effective elastic modulus E� andgeometry angle parameter a are in hand, we can find the appropri-ate model by locating the coordinates in the model diagram, andthe pull-off force can be obtained straightforward.

3. Pull-off force of individual ivy discs

Regardless of the physical origin, adhesive strength can be mod-eled using a ‘work of adhesion’ w; that is, the external work done toseparate unit area of the adhering surfaces. The physical processesinvolved may include van der Waals forces, electrostatic forces,capillary forces, and glue effects [9].

3.1. Adhesion force due to van der Waals attraction

Since van der Waals force usually plays dominant role in adhe-sion of nanometer length scale [4,5], we compute the pull-off forcedue to van der Waals attraction. The work of adhesion w is Dupréenergy of adhesion

w ¼ c1 þ c2 � c12 ¼ Dc ð27Þ

where c1 and c2 represent the surface energy of the two surfaces,respectively, and c12 is the interface energy between the two mate-rials. In the case that the objects are made from the same material,w is twice the surface energy since the interface energy c12 is zero.According to Lennard–Jones potential

r0 ¼64

ffiffiffi3p

27

ffiffiffiffiffiffiffiffiffiffiffiffipDc3

A

rð28Þ

z0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA

16pDc

sð29Þ

h0 ¼9ffiffiffi3p

16z0 ð30Þ

where A ¼ p2q1q2C is the Hamaker constant, and C the London –van der Waals constant. The parameters for the van der Waals inter-action are selected to average values as follows [13,26–29]:

Dc ¼ 0:025 J=m2; A ¼ 0:7� 10�19 J ð31Þ

The nanoparticles secreted from ivy tendrils are measured byAFM to have an average diameter of nearly 70 nm [2,3], which cor-responds to radius R ¼ 35 nm.

The pull-off force with respect to single nanoparticle can nowbe calculated by using the aforementioned models. The resultsare plotted in Figs. 6–10. Clearly, the pull-off force increases witheither increasing effective elastic modulus or increasing contactangle, which means an increase in pull-off force can be achievedby applying a stiffer particle or a larger contact area.

10−1 100 101 1020

1

2

3

4

5

6


Pull−

off f

orce

Fp (n

N)

DMT type modelM−D modelJKR Dugdale modelJKR type model

Fig. 6. Variation of pull-off force corresponding to single ivy nanoparticle with theeffective elastic modulus E� . Adhesion is due to van der Waals attraction for a ¼ 1� .According to the model diagram, the pull-off force in this region should adopt theresult computed from M-D model. The results from other models will bring largeerror and thus unacceptable. The M-D model trends to the DMT type model whenE� > 250 GPa.

2 4 6 8 10 120

1

2

3

4

5

6

7


Pull−

off f

orce

Fp (n

N)


Fig. 7. Variation of pull-off force corresponding to single ivy nanoparticle with theeffective elastic modulus E� . Adhesion is due to van der Waals attraction for a ¼ 3� .The curves of M-D model and JKR Dugdale model intersect at E� ¼ 8:31 GPa.According to the model diagram, the pull-off force in the region E� < 8:31 GPashould adopt the result computed from M-D model, then it goes along the JKRDugdale curve with increasing E� , and finally achieves the saturation value atE� ¼ 11:34 GPa and stays in the DMT type zone.

0.5 1 1.5 2 2.5 3 3.50

2

4

6

8

10

12


Pull−

off f

orce

Fp (n

N)


Fig. 8. Variation of pull-off force corresponding to single ivy nanoparticle with theeffective elastic modulus E� . Adhesion is due to van der Waals attraction for a ¼ 6� .The curves of M-D model and JKR type model are nearly coincide with each other inthe region E� < 0:41 GPa. According to the model diagram, the pull-off force in theregion 0.68 GPa < E� < 3:26 GPa should adopt the result computed from JKRDugdale model, then it achieves the saturation value and enter the DMT type zone.

1 2 3 4 5 60

50

100

150

200

250


Pull−

off f

orce

Fp (n

N)


Fig. 9. Variation of pull-off force corresponding to single ivy nanoparticle with theeffective elastic modulus E� . Adhesion is due to van der Waals attraction for a ¼ 45� .According to the model diagram, the pull-off force in the region E� < 4:60 GPashould adopt the result computed from JKR type model, then it goes through JKRDugdale model and enters the DMT type zone at E� ¼ 4:77 GPa. The pull-off forcefrom M-D model is apparently underestimated.


There are clear saturations of adhesion strength over a criticaleffective elastic modulus or below a critical angle (as shown inFig. 10 as green solid curve) for JKR type model. We can understandthe saturations by recalling the ratio E�= sin a ða – 0Þ in Eq. (11),which could be considered as the criterion of saturation.

The adhesion force of individual ivy discs can be computed. Weassume that the ivy nanoparticles are tightly packed together, seeFig. 11. The average attached area of individual Boston ivy discs isS ¼ 1:22 mm2 [30]. The minimum force needed to separateindividual ivy disc from substrate is

F ¼ FpS

ð2RÞ2ð32Þ

For the convenience of comparison, the van der Waals parame-ters are set to be the same as those aforementioned. The pull-off

force is plotted against the effective elastic modulus of ivy nano-particles, as shown in Fig. 12, including two different values of a.Compared with the experimental measurements 8.8 N by Darwinin 1876 [1], the results show that the van der Waals attraction isapparently strong enough to afford the adhesion, provided thatthe nanoparticles have a reasonable stiffness (E� > 0:13 GPa fora ¼ 45�, or E� > 0:37 GPa for a ¼ 30�). The Young’s modulus valuesof various polymer can be quantitatively determined by indenta-tion experiments using atomic force microscope, and were foundto range from 1 kPa for a gel to 100 GPa for a fiber [31,32].

Most of the surfaces of ivy discs that have been in contact with asubstrate for a few hours were covered with a substance, whichseemed to have floated on the surface like a viscous layer and final-ly hardened [30]. It is claimed that adhesive curing is also reflectedin changes in effective elastic modulus [33]. The value of Young’s

0 10 20 30 40 50 60 70 80 900

50

100

150

200

250

300

350

400

450

Angle α (°)

Pull−

off f

orce

Fp (n

N)

DMT type modelJKR type modelSaturation zone for JKR type modelM−D model

Fig. 10. Variation of pull-off force corresponding to single ivy nanoparticle with theangle a. Adhesion is due to van der Waals attraction for E� ¼ 3 GPa. According to themodel diagram, the curve of pull-off force will access into the M-D zone, DMT typezone, and then JKR type zone in sequence with increasing a. The M-D model isapparently invalid when a > 10� .

Fig. 11. Geometry of tightly packed ivy nanoparticles. The radius of effectivecontact area a ¼ R sin a.

0 1 2 3 4 5 60

10

20

30

40

50

60


Pull−

off f

orce

F (N

)

α=45°, JKR type zoneα=45°, JKR Dugdale zoneα=45°, DMT type zoneα=30°, JKR type zoneα=30°, JKR Dugdale zoneα=30°, DMT type zone

Fig. 12. Variation of pull-off force corresponding to individual ivy disc with theeffective elastic modulus E� . Adhesion is due to van der Waals attraction fordifferent geometry a ¼ 30� and a ¼ 45� , respectively.


modulus increased 10-fold within the curing process, with whichto go along is the increase of adhesion force.

Temperature coefficient of surface energy varies in a narrowrange [29]. Taking the average value of �0:5� 10�4 Jm�2 �C�1, weobtain the reduction �0:003 Jm�2 when temperature rises from�20 �C to 40 �C, which is negligible compared to the supposedvan der Waals surface energy of ivy nanoparticles Dc ¼ 0:025Jm�2. Thus the ivy can achieve robust adhesion even in extremeweather.

3.2. Influence of cross-linking

The cross-linking play a significant role in the multilayer adhe-sion mechanism, as described by Berglin et al. [35]. The establish-ment of adhesion ivy discs starts with the secretion and theadhesion of the nanoparticles to the substrate. Subsequently, thisis followed by the formation of cohesive interactions (cross-link-ing) with the next layer of nanoparticles that are adsorbed ontop of the first layer. The cross-linking holds together the multi-layer stiff nanoparticles, just like nanocomposite materials to someextent [36]. It yields just before the nanoparticles would otherwisedetach, and dissipates large amounts of energy as it yields.

On the other hand, the cross-linking can also fix the nanoparti-cles in its original position which best match the surface asperities,and thus ensures the junctions are broken simultaneously to earnthe best interest from the van der Waals forces.

3.3. Pull-off force due to capillarity

For a smooth sphere contacting with a smooth substrate, an in-crease in adhesion force is generally observed when a thin film ofliquid is introduced at the contact interface either through adsorp-tion or by deposition. Bowden and Tabor concluded that the in-crease in adhesion was due to the meniscus formation aroundthe contact area [37,38], as shown in Fig. 13.

The radius of the meniscus is

q ¼ h0

cos h1 þ cosðh2 þ /Þ ð33Þ

where h1 and h2 are the contact angles of the liquid on substratesurface and particle surface, respectively, and h0 is the Dugdalecohesive distance, and

/ ¼ arccos cos a� h0

R

� �ð34Þ

and the geometric constraint condition must be satisfied

/ < p� h1 � h2 ð35Þ

Fig. 13. A schematic diagram of the meniscus with uniform radius formed at thecontact interface of a smooth sphere against a smooth substrate.

0 10 20 30 40 50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4

4.5

the water contact angle of substrate θ1 and ivy nanoparticle θ2 (°)

Pull−

off f

orce

F (N

)

with fixed θ1=58°

with fixed θ2=60°

Fig. 16. Variation of pull-off force of individual ivy disc with the water contactangle h1 and h2. Adhesion is due to capillary force for h0 ¼ 3 nm; a ¼ 30� . The reddashed curve: with a fixed water contact angle of substrate h1 ¼ 58� (for the glass


In addition to being an effective approximation to the Lennard–Jones potential, the Dugdale model exactly reproduces adhesiondue capillary forces exerted by a meniscus [9]. The Dugdale attrac-tive stress can be given by

r0 ¼cL

qð36Þ

where cL is the liquid/vapor surface tension.The effective work of adhesion is then given by

w ¼ r0h0 ð37Þ

The pull-off force to separate an ivy nanoparticle from smoothsubstrate can be calculated by using the Dugdale model, as de-scribed in Section 2.2

Fp ¼F0 cos2 að2f� f2Þ; f < 1F0 cos2 a; f P 1

(ð38Þ

where f and F0 are the same as those in Eq. (4).

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

the height of the meniscus h0 (nm)

Pull−

off f

orce

F (N

)

Fig. 14. Variation of pull-off force of individual ivy disc with the height of themeniscus (Dugdale cohesive distance) h0. Adhesion is due to capillary force fora ¼ 30�; h1 ¼ 58� (for the glass substrate [39]), h2 ¼ 60� .

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

the angle of the base of the cap α (°)

Pull−

off f

orce

F (N

)

Fig. 15. Variation of pull-off force of individual ivy disc with the angle of the base ofthe cap a. Adhesion is due to capillary force for h0 ¼ 3 nm; h1 ¼ 58� , h2 ¼ 60� .

substrate); the blue curve: with a fixed water contact angle of ivy nanoparticleh2 ¼ 60� .

Taking the surface tension cL ¼ 72:5 mJ=m2 for water/air, thepull-off forces to separate individual ivy discs from substrate arecalculated and the results are depicted in Figs. 14–16.

It is clear that the capillary force decreases with increasingheight of the meniscus, and also with increasing water contact an-gle of each surface. So the optimized capillary force can beachieved with thin water layer and hydrophilic material. However,even the optimized capillary force can not afford the force mea-sured by Darwin independently. The dominant adhesion forceshould be van der Waals attraction.

4. Conclusions

We have applied contact and fracture mechanics models tostudy the nano scale ivy adhesion mechanism, and compared thedifferences among DMT, JKR, and M-D models in calculating thepull-off force between individual ivy discs and a rigid substrate.The appropriate model used in a specific application depends onthe elasticity and geometry of the nanoparticles. To assist in themodel selection, a model diagram has been constructed. We sug-gest to select the appropriate model by locating the coordinatesin the diagram. The pull-off force can then be easily obtained.The contributions to adhesive force from van der Waals attractionand capillarity are computed, respectively. The influence of cross-linking to adhesion due to van der Waals attraction is also takeninto consideration. Our study demonstrates that van der Waalsattraction plays a dominant role in ivy nano scale adhesion. Theadhesion force due to van der Waals attraction increases with aconcomitant increase in effective elastic modulus and contact area.However, the adhesion force of stiff particles with large effectiveelastic modulus will drop significantly with increased surfaceroughness [9,34]. So a carefully selected effective elastic modulusis required for robust adhesion. Adhesion due to capillary force isstrongly influenced by water contact angle of each surface. In thecurrent configuration, the capillary force is much smaller thanvan der Waals attraction even with hydrophilic material. Theabove conclusions suggest that ivy nanoparticles with diameterof 70 nm can achieve best adhesion by possessing reasonable effec-tive elastic modulus, large contact area, and small water contactangle. The understanding of nano scale bio-adhesion mechanismcould be extended by taking more factors into consideration, such


as inelastic–viscoelastic or plastic-deformation, and chemicalbonding.

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adhesion mechanics of ivy nanoparticles

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