Research Article An Analytical Model for Adsorption and

Research ArticleAn Analytical Model for Adsorption and Diffusion ofAtomsIons on Graphene Surface

Yan-Zi Yu Jian-Gang Guo and Yi-Lan Kang

Tianjin Key Laboratory of Modern Engineering Mechanics School of Mechanical Engineering Tianjin UniversityTianjin 300072 China

Correspondence should be addressed to Jian-Gang Guo guojgtjueducn

Received 24 July 2015 Accepted 16 September 2015

Academic Editor Ungyu Paik

Copyright copy 2015 Yan-Zi Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Theoretical investigations are made on adsorption and diffusion of atomsions on graphene surface based on an analyticalcontinuous model An atomion interacts with every carbon atom of graphene through a pairwise potential which can beapproximated by the Lennard-Jones (L-J) potential Using the Fourier expansion of the interaction potential the total interactionenergy between the adsorption atomion and a monolayer graphene is derived The energy-distance relationships in the normaland lateral directions for varied atomsions including gold atom (Au) platinum atom (Pt) manganese ion (Mn2+) sodium ion(Na1+) and lithium-ion (Li1+) on monolayer graphene surface are analyzed The equilibrium position and binding energy of theatomsions at three particular adsorption sites (hollow bridge and top) are calculated and the adsorption stability is discussedThe results show that H-site is the most stable adsorption site which is in agreement with the results of other literatures Whatis more the periodic interaction energy and interaction forces of lithium-ion diffusing along specific paths on graphene surfaceare also obtained and analyzed The minimum energy barrier for diffusion is calculated The possible applications of present studyinclude drug delivery system (DDS) atomic scale friction rechargeable lithium-ion graphene battery and energy storage in carbonmaterials

1 Introduction

In recent decade graphene has been paid much attention asan ideal candidate for the development of high-sensitivitysensors [1ndash3] drug delivery system (DDS) [4ndash7] recharge-able lithium-ion graphene battery [8ndash10] lithium storagein carbon materials [11ndash15] and so on due to its uniquecharacteristics such as the thinnest known 2-dimensionalnanomaterial the lowest resistivity and extreme mechanicalstrength The characteristics of graphene have been widelyinvestigated in the field of biology chemistry and physicsHowever there is still a common unsolved major scientificquestion for its applications It is to develop an analyticalmodel to quantitatively describe the adsorption and diffusionof atomsions on graphene surface which can be furtherapplied for the interaction between nanoparticles or atomicforce microscopy (AFM) tip and graphene surface

For this problem lots of theoretical and experimentalresearches have been made and some interesting research

results are achieved In atomic and nanoscale general the-oretical research methods include density functional the-ory (DFT) first principle and molecular dynamics (MD)simulation A systematic density functional study was per-formed on the adsorption of Cu Ag and Au adatomson pristine graphene especially accounting for van derWaals (vdW) interactions by the vdW-DFT and PerdewBurke and Ernzerhof plus long-range dispersion correction(PBE + D2) methods [16] According to the differences ofthe total energies at the three adsorption sites they foundthat the diffusion of Cu and Au took place along carbon-carbon bonds while the Ag adatoms could diffuse almostunrestrictedly on graphene sheet Using density functionaltheory theoretical calculations were also carried out forthe diffusion of Si and C atoms on a monolayer graphenesurface and between the multilayer graphene layers [17] Theresults showed that the diffusion barrier for a Si atom onmonolayer graphene surface was extremely small and thediffusion of Si atombetween bilayer graphenewas also almost

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 382474 10 pageshttpdxdoiorg1011552015382474

2 Journal of Nanomaterials

barrier-freeHowever the diffusion of aC atomonmonolayergraphene was far more difficult than that of the Si atom Bythe same methods the adsorption of Li atom on graphenewas investigated [9] Three different adsorption sites weretaken into consideration According to their researches thehollow site was found to be the most stable followed by thebridge and top sites Moreover for Li atom the minimumdiffusion energy path where Li atom migrates from a hollowsite to a neighboring hollow site should be a path throughthe bridge site between them What is more the adsorptiondiffusion and desorption of hydrogen on graphene werestudied by DFTmethod [18]They concluded that the atomichydrogen adsorbate produced by bombarding the surfacewith an atomic beam was not possible under equilibriumconditions and below room temperature diffusionwas frozenout and a beam-deposited atomic hydrogen adsorbate couldbe maintained though completely out of equilibrium

Apart from density function theory method MD simu-lation methods have also been widely used in the study ofgraphene interacting with atomion The diffusion processesof the lithium-ion on a fluorinated graphene (F-graphene)surface were investigated by means of a direct molecularorbital-molecular dynamics (MO-MD) method [19] It wasfound that the diffusion coefficient of lithium-ion on F-graphene surface was close to that of normal graphene (H-graphene) but the thermal behavior on the F-graphene sur-face wasmuch different from that on theH-graphene surfaceThe direction of lithium-ion diffusion could be controlledby fluorinated substitution (F-substitution) on the graphenesheet around room temperature But at higher temperaturesthe lithium-ion diffused freely on surface and edge region ofF-graphene The interaction of Mg atom with graphene wasalso researched by using MD simulations [20] It was foundthat the Mg atom vibrated strongly on the graphene surfaceeven at high temperature (1000K) the diffusion of Mg isdifficult

There are relatively few experimental researches onadsorption and diffusion of atomsions on graphene surfaceThe viability of utilizing functionalized graphene oxide withgood solubility and stability biocompatibility and target-specificity was demonstrated as a drug nanocarrier for con-trolled loading and folate receptor-targeted drug delivery ofanticancer drugs [21] It revealed that mastering the stabilityof equilibrium for graphene was vital to investigate a newdrug delivery system for cancer therapy The possibilityof higher lithium storage capacity was explored by con-trolling layered structures of graphene nanosheet materials[12] It enlightened that the spacing equilibrium distanceof atomion on and between graphene sheets is critical forthe future large-scale high-capacity lithium-ion batteriesThe diffusion of a cobalt bis-terpyridine Co(tpy)

2-containing

tripodal compound (1∙2PF6) noncovalently adsorbed onthe surface of a single-layer graphene (SLG) was stud-ied by scanning electrochemical microscopy [22] It wasproved that surface diffusion is the main mechanism forthe observed decrease in the electrochemical response dueto radial diffusion of the adsorbed molecules outward fromthe microspots onto the unfunctionalized areas of the SLGsurface

In order to quantitatively understand atomic adsorptionand diffusion on graphene surface from the perspective ofatomic scale an analytical continuousmodel is put forward inthis work based on the Fourier expansion and Lennard-Jonespotential With the theoretical model equilibrium positionbinding energy and adsorption stability of the atomsions(Au Pt Mn2+ Na1+ and Li1+) on monolayer graphenesurface are calculated and analyzed The different metalatomsions selected in this work possess many applicationprospects For example Mn2+ is an ideal candidate for drugdelivery and Li1+ is a common element for high-energydensity rechargeable alkali batteries The dependence ofdiffusion energy and diffusion forces on the path is illustrated

2 Theoretical Model

A monolayer graphene is assumed to be perfect flat andinfinite compared to atomic scale (as shown in Figure 1(a))Due to its periodic honeycomb lattice structure a two-dimensional lattice vector parallel to graphene can be defined[23ndash25]

l = 1198971a1 + 1198972a2 (1)

where 1198971and 1198972are integers and a1 and a2 are the unit lattice

vectors in the plane (as shown in Figure 1(a)) It is evidentto know that the translation of a single adsorption atom byl will take it from one position above a surface lattice cellinto an equivalent position above another different latticecell (as shown in Figure 1(a)) Therefore the total interactionpotential 119906(r) takes the form of

119906 (r + l) = 119906 (r) (2)

where r is the position of the adsorption atom with respect tothe graphene surface

Thus the periodic nature of the total interaction potential119906(r) is a Fourier series

119906 (r) = sum119892

119908119892(119911) exp (119894g sdot 120591) (3)

where 119911 is perpendicular distance between the adsorptionatom and graphene surface 120591 is the two-dimensional trans-lation vector and g is the multiples of the reciprocal latticevectors b1 and b2

g = 2120587 (1198921b1 + 1198922b2) (4)

where 1198921and 119892

2are integers and b1 and b2 are defined by

a1 sdot b1 = a2 sdot b2 = 1

a1 sdot b2 = a2 sdot b1 = 0(5)

Equation (5) indicates that b1 and b2 are perpendicular to a2a1 and of length csc120593119886

1 csc120593119886

2 respectively where 120593 is

Journal of Nanomaterials 3

x

y

zAdsorptionatomion

a1

a2

(a)

1

(1) H-site(2) B-site(3) T-site

3

2

(b)

Figure 1 Schematics of the graphene where (a) the rhombus connecting the centers of four adjacent hexagonal is selected as a two-dimensional primitive unit cell (b) hexagon with three particular adsorption sites hollow (on top of a hexagon H-) top (on top of a C-Cbond T-) and bridge (on top of a carbon atom B-)

the angle between a1 and b1119908119892(119911) in (3) can be expressed inthe following forms

119908119892(119911) =

1

119860

sum

11989711198972

sum

119896

int

119860

exp (minus119894g sdot 120591) 119906119904(119911 120591 + l +m

119896) 119889120591

=

1

119860

sum

119896

exp (119894g sdotm119896) int

infin

0

exp (minusig sdot t) 119906119904(119911 t) 119889t

(6)

where 119860 is the area of unit cell 119906119904(119911 120591 + l +m

119896) is the partial

interaction potential between the single atom and the carbonatom in graphene plane 119896 indicates the 119896th atom in a unit celland the position of the atom in the plane is given by l + m

119896

and t = 120591 + l +m119896

Commonly the average interaction potential peratomion on graphene is defined by the total energyof an atomion a sheet of graphene and the graphenewith atomion adsorption which consider the effects oftemperature chemical bond charge transfer van der Waalsinteraction and so forth [9 10 26 27] In this paper for theconvenience of discussion the interaction potential is onlydefined by van der Waals interaction Even so the model inthis paper is also suitable for taking into account the effectsof other factors mentioned above Two empirical potentialscommonly used are the L-J potential and theMorse potentialBy referring the reader for detail of the Morse potential andits applications [28 29] this paper adopts the form of 6-12L-J potential to determine the van der Waals interactionpotential between the adsorption atom and the 119894th carbonatom in the graphene

119906119904(r119894) = 4120576 [(

120590

119903119894

)

12

minus (

120590

119903119894

)

6

] (7)

where r119894 120576 and 120590 denote the distance between the adsorption

atom and the 119894th carbon atom in the graphene the potential

well depth and the distance when the interaction potential isequal to zero respectively Then

1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(8)

where 119902 = 2 is the number of the carbon atoms in the unit celland119870

2and119870

5are themodified Bessel function of the second

kind Substitute (8) into (3) and then the total interactionpotential 119906(r) can be derived as

119906 (r) asymp 1199080(119911) + sum

119892gt0

119902

sum

119896=1

119908119892(119911) exp [119894g sdot (m

119896+ 120591)]

= 119862 (119911) + sum

119892gt0

119902

sum

119896=1

119863(119892 119911) exp [119894g sdot (m119896+ 120591)]

(9)

where 119892 is the magnitude of g

119892 =

1003816100381610038161003816g1003816100381610038161003816=

4120587

radic3119886

(119892

2

1+ 119892

2

2minus 11989211198922)

12

119862 (119911) = 1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119863 (119892 119911) = 119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(10)

As shown in Figure 1(a) when 120591 is chosen to be zero at ahollow point and the unit cell which contains two carbonatoms is marked by a rhombus thenm

119896for the two atoms in


the unit cell indicated in Figure 1(a) are located at (13 13)119886(23 23)119886 where 119886 is the distance between the centers oftwo nearest hexagons Summation over m

119896gives a value of

cos[(21205873)(1198921+ 1198922)] so (9) can be further written as

119906 (119903)

= 119862 (119911)

+ 2sum

119892gt0

119863(119892 119911) exp (119894g sdot 120591) cos [21205873

(1198921+ 1198922)]

(11)

Because 119863(119892 119911) in (11) decreases rapidly with the increaseof 119892 and the 120591-dependent terms are important only over ashort range of distance 119911 [23 25] there are approximatelysimplified expressions of |g| = 4120587radic3119886 and cos[(21205873)(119892

1+

1198922)] = minus12 As shown in Figure 1(a) 119909 and 119910 coordinates are

marked 120591 = x+y can be expressed in terms of a1 and a2 with

x = 119909119886

(a1 minus a2)

y = 119910

radic3

(a1 + a2) (12)

Thus the total energy between an adsorption atomion andgraphene can be derived as

119906 (119903) = 119862 (119911) minus 119863(

4120587

radic3119886

119911)

sdot [2 cos(2120587119910

radic3119886

) cos(2120587119909119886

) + cos(4120587119910

radic3119886

)]

(13)

If we introduce the following dimensionless variables 119860lowast =119860119886

2 119911lowast = 119911119886 119892lowast = 119892119886 120590lowast = 120590119886 119909lowast = 119909119886 119910lowast = 119910119886 and119906

lowast= 119906120576 then the dimensionless form of total energy can be

written as

119906

lowast(119903)

= 120582119902(

2120590

lowast12

5119911lowast10

minus

120590

lowast6

119911lowast4) minus 120582 (120572 minus 120573)

sdot [2 cos( 2120587radic3

119910

lowast) cos (2120587119909lowast) + cos( 4120587

radic3

119910

lowast)]

(14)

where dimensionless parameters 120582 120572 and 120573 have the follow-ing forms

120582 =

2120587

119860lowast

120572 =

120590

lowast12

30

(

119892

lowast

2119911lowast)

5

1198705(119892

lowast119911

lowast)

120573 = 2120590

lowast6

(

119892

lowast

2119911lowast)

2

1198702(119892

lowast119911

lowast)

(15)

Furthermore let Flowast = minusnabla119906lowast(rlowast) and then the normal andlateral interaction forces between the atomion and graphene

can be obtained in the Cartesian coordinate system as shownin Figure 1(a)

119865

lowast

119909= minus2120587120582 (120572 minus 120573) [2 cos( 2120587

radic3

119910

lowast) sin (2120587119909lowast)]

119865

lowast

119910= minus

4120587120582

radic3

(120572 minus 120573)

sdot [sin( 2120587radic3

119910

lowast) cos (2120587119909lowast) + sin( 4120587

radic3

119910

lowast)]

119865119911

lowast= 4120582119902(

120590

lowast12

119911lowast11

minus

120590

lowast6

119911lowast5)

minus 120582 [(

11

2119911lowast+ 119892

lowast)120572 minus (

5

2119911lowast+ 119892

lowast)120573]


119910


radic3

119910

lowast)]

(16)

In the same way the lateral interaction force between theatomion and graphene in the polar coordinate system canbe derived

119865

lowast

119905= minus2120587120582 (120572 minus 120573)


120588

lowast sin 120579) sin (2120587120588lowast cos 120579)] (17)

where 120588lowast and 120579 are dimensionless polar coordinates and 120588lowastcoincides with the 119909lowast-axis when the angle 120579 = 0

3 Results and Discussions

Equation (13) describes the interaction potential between asingle adsorption atomion and graphene via the L-J potentialand the first-order approximation of Fourier expansionObviously it is dependent on the position of the adsorptionatomion on graphene and the parameters of L-J potentialwhich is varied for different atomion In this work severalkinds of metal atoms and ions are selected including Au PtMn2+ Na1+ and Li1+ If the L-J potential parameters for theseatomsions are specified [30ndash32] the interaction potential ofthese atomsions can be obtained in terms of the positionvector above the graphene surface

For a lithium-ion the L-J potential parameters can bespecified as 120590lowast = 1005 and 119911lowast = 0983 The variations ofinteraction potential between the lithium-ion and grapheneare illustrated in Figure 2 with respect to the positions inthe 119909- and 119910-directions respectively It can be seen fromFigure 2 that the potential curves are periodic in bothdirections and there are three periodic equilibriumpositionscorresponding to three particular adsorption sites of lithium-ion on graphene surface They are hollow (on top of ahexagon H-) top (on top of a C-C bond T-) and bridge(on top of a carbon atom B-) sites (as shown in Figure 1(b))respectively At three adsorption sites the equilibrium heightand interaction potential at the equilibrium which can becalled binding energy are different Figure 3 shows thevariations of interaction potential of the lithium-ion at three


minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 20

H-site H-site H-site

B-site B-site

ulowast

xlowast

(a)

minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 35302520


B-site

T-site T-site T-site T-site

B-site

ulowast

ylowast

(b)

Figure 2The variations of interaction potential between the lithium-ion and graphene in the (a) 119909-direction and (b)119910-direction respectively

20

15

10

5

0

minus5

minus10

minus1510 15 20 25 30

minus7

minus8

minus9

minus10085 090 095 100 105 110 115

ulowast

zlowast

HBT

Figure 3 The variations of interaction potential of the lithium-ionat three adsorption sites along the 119911-direction

adsorption sites along the 119911-direction Although the variationtrends are almost identical for the three sites the equilibriumheight and binding energy are different The equilibriumheight and binding energy at the H-site are the lowestfollowed by those at the B-site and those at the T-site are thehighest

Furthermore via the equilibrium equations of lithium-ion in the 119911-direction 119865lowast

119911(119909

lowast 119910

lowast) = 0 the equilibrium

height 1199110at three adsorption sites can be calculated which

is 0241717 nm 0248800 nm and 0249502 nm respectivelyCompared with the results in the literature [30] there is agood agreement The equilibrium height is very important

for the design of lithium-ion battery based on grapheneelectrode materials which can be used in predicting theminimum interlayer spacing 2119911

0of two parallel graphene

sheets so that the embedded Li1+ undergoes no net forceAccording to our theoretical results the minimum interlayerspacing for Li1+ should be approximately 048ndash050 nmWith the equilibrium heights at three adsorption sites thebinding energy 119906

0of lithium-ion can also be calculated

The minimum value of binding energy is minus004059 eV atthe H-site subsequently minus003746 eV at the B-site and themaximum value is minus003717 eV at the T-site Thus H-site isthemost stable adsorption site andT-site is themost unstablesite

Similarly when the L-J potential parameters of otheratomsions are given the variations of interaction potentialof these atomsions can be illustrated with respect to thepositions in the 119909- 119910- and 119911-directions respectively (asshown in Figure 4) It can be found that the variation trends ofinteraction potential are almost the same in three directionsfor these atomsions Three adsorption sites (H- B- andT-site) exist for every atomion on graphene surface Theequilibrium height and binding energy of these atomsionscan be calculated and listed in Tables 1 and 2 Similarconclusions can be drawn that the order from low to high forboth equilibrium height and binding energy is H-site B-siteand T-site and H-site is the most stable adsorption site Inaddition the equilibrium height and binding energy of everyatomion are different Obviously Pt atom adsorbed at the H-site is the most stable among the atomsions selected in thework

As the discussions above the interaction potentialsbetween adsorption atomsions and graphene are not con-stant but dependent on their positions on graphene surfaceFor different adsorptions sites the energy for desorption ofthese atomsions is different When an atomion migratesfrom an adsorption site to another it must get over an energybarrierThe energy or force to drive the atomion is defined as


minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)

Figure 4 The variations of interaction potential of several kinds of metal atomsions along the (a) 119909-direction (b) 119910-direction and(c) 119911-direction respectively

Table 1 Equilibrium height 1199110of atomsions on the three particular adsorption sites

Equilibrium height (unit nm) Position AtomionMn2+ Au Pt Na1+ Li1+

1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502

Table 2 Binding energy 1199060of atomsions on the three particular adsorption sites

Binding energy (unit eV) Position AtomionMn2+ Au Pt Na1+ Li1+

1199060

H minus007915 minus003127 minus010358 minus002936 minus004059B minus007084 minus003051 minus010084 minus002826 minus003746T minus007013 minus003043 minus010053 minus002814 minus003717


0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)

Figure 5 The variations of (a) interaction potential and (b) interaction force between a lithium-ion and graphene for different diffusiondirections 0∘ 15∘ 30∘ and 60∘ angles from the 119909-axis

diffusion energy or diffusion force Obviously the diffusionenergy and diffusion force are dependent on the diffusionpath

For a lithium-ion the periodic interaction potentialsalong the 119909- and 119910-directions are illustrated in Figure 2 Theperiod is 119886 in the 119909-direction and radic3119886 in the 119910-direction Ifits initial adsorption site is assumed to be H-site in orderto migrate along the 119909-direction a lithium-ion must getover an energy barrier between H-site and B-site and theminimum diffusion energy is 000313 eV In the same way inorder to migrate along the 119910-direction a lithium-ion mustget over an energy barrier between H-site and T-site andthe minimum diffusion energy is 000342 eV Similar resultscan also be obtained for other atomsions according to theperiodic interaction potentials illustrated in Figure 4 and thebinding energies listed in Table 2

The initial adsorption site is still assumed to be H-siteand a lithium-ion is assumed to migrate along differentdirections such as 0∘ 15∘ 30∘ and 60∘ angles from the 119909-axisThe dimensionless interaction potential and interaction forcebetween the lithium-ion and graphene for different diffusiondirections are illustrated in Figure 5 It can be seen fromthese figures that the periods of interaction potential andinteraction force are varied for different diffusion directionsand the minimum diffusion energy and diffusion force arealso varied that is the diffusion of atomsions on graphenesurface is path-dependent Simultaneously it can be foundthat along the same lattice orientation such as 0∘ and 60∘angles the variations of interaction potential and interactionforce are nearly the same These phenomena have beenverified by the atomic scale friction experiments of AFM tipon graphene surface [33ndash35]

Furthermore the diffusion fromdifferent initial positionsalong 119909- and 119910-directions is investigated The dimensionlessinteraction potential and interaction force of a lithium-ion from different initial sites are illustrated along 119909- and119910-directions in Figures 6 and 7 In two directions theperiods of interaction potential and interaction force havepath-independent period 119886 and radic3119886 respectively Howeverfor different diffusion paths along 119909- or 119910-direction theiramplitudes are varied that is the minimum diffusion energyand diffusion force are varied

4 Conclusions

By performing theoretical analyses of interaction potentialand interaction force for a single atomion on graphenesurface based on an analytical model the mechanisms ofadsorption and diffusion are revealedThe equilibriumheightand binding energy on three particular adsorption sitesare calculated and the adsorption stability is discussedThe order from low to high for both equilibrium heightand binding energy is H-site B-site and T-site and H-site is the most stable adsorption site The variations ofinteraction potential and interaction force of lithium-iondiffusing along some specific directions on graphene surfaceare investigated and it can be concluded that the diffu-sion of atomsions on graphene surface is path-dependentThe minimum diffusion energy and diffusion force alongdifferent diffusing path can be calculated by the analyticalmodel

The studies on the path-dependent diffusion ofatomsions on graphene surface are beneficial to improve


00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0

ylowast = 12radic3ylowast = 1radic3

ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)

Figure 6 The variations of interaction potential between a lithium-ion and graphene from different initial sites along (a) 119909-direction and(b) 119910-direction

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)

Figure 7 The variations of interaction force between a lithium-ion and graphene from different initial sites along (a) 119909-direction and(b) 119910-direction

the efficiency of drug delivery based on graphene high-performance rechargeable lithium-ion batteries and lithiumstorage in carbon materials which can be further appliedfor the interaction between nanoparticles or AFM tip andgraphene surface Of course there is a little inadequacyin present studies For example the graphene is assumed

to be infinite perfect and flat moreover the influence oftemperature is not taken into account In fact edge effecttemperature defect and corrugation of graphene and soforth can affect the adsorption and diffusion of atomion[30 36ndash40] Therefore these factors will be considered inour future works


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (Grants nos 11372216 11372217 and11502167) and the National Basic Research Program of China(973 Program Grant no 2012CB937500)

References

[1] L Huang Z P Wang J K Zhang et al ldquoFully printed rapid-response sensors based on chemically modified graphene fordetecting NO

2at room temperaturerdquo ACS Applied Materials

and Interfaces vol 6 no 10 pp 7426ndash7433 2014[2] J Ma W Jin H L Ho and J Y Dai ldquoHigh-sensitivity fiber-tip

pressure sensor with graphene diaphragmrdquo Optics Letters vol37 no 13 pp 2493ndash2495 2012

[3] J H Li J L Liu G R Tan et al ldquoHigh-sensitivity paracetamolsensor based on Pdgraphene oxide nanocomposite as anenhanced electrochemical sensing platformrdquo Biosensors andBioelectronics vol 54 pp 468ndash475 2014

[4] K Park ldquoControlled drug delivery systems past forward andfuture backrdquo Journal of Controlled Release vol 190 pp 3ndash82014

[5] Y ZhangH F Chan andKW Leong ldquoAdvancedmaterials andprocessing for drug delivery the past and the futurerdquo AdvancedDrug Delivery Reviews vol 65 no 1 pp 104ndash120 2013

[6] H C Zhang G Gruner and Y L Zhao ldquoRecent advancementsof graphene in biomedicinerdquo Journal of Materials Chemistry Bvol 1 no 20 pp 2542ndash2567 2013

[7] J Q Liu L Cui and D Losic ldquoGraphene and grapheneoxide as new nanocarriers for drug delivery applicationsrdquo ActaBiomaterialia vol 9 no 12 pp 9243ndash9257 2013

[8] J Hassoun F Bonaccorso M Agostini et al ldquoAn advancedlithium-ion battery based on a graphene anode and a lithiumiron phosphate cathoderdquo Nano Letters vol 14 no 8 pp 4901ndash4906 2014

[9] Z Ji F F Contreras-Torres A F Jalbout and A Ramırez-Trevino ldquoSurface diffusion and coverage effect of Li atomon graphene as studied by several density functional theorymethodsrdquoApplied Surface Science vol 285 part B pp 846ndash8522013

[10] A Buldum and G Tetiker ldquoFirst-principles study of graphene-lithium structures for battery applicationsrdquo Journal of AppliedPhysics vol 113 no 15 Article ID 154312 2013

[11] G Kucinskis G Bajars and J Kleperis ldquoGraphene in lithiumion battery cathode materials a reviewrdquo Journal of PowerSources vol 240 pp 66ndash79 2013

[12] E J Yoo J Kim E Hosono H-S Zhou T Kudo and I HonmaldquoLarge reversible Li storage of graphene nanosheet families foruse in rechargeable lithium ion batteriesrdquo Nano Letters vol 8no 8 pp 2277ndash2282 2008

[13] H Yildirim A Kinaci Z-J Zhao M K Y Chan and J P Gree-ley ldquoFirst-principles analysis of defect-mediated Li adsorptionon graphenerdquo ACS Applied Materials and Interfaces vol 6 no23 pp 21141ndash21150 2014

[14] J J ZhouWW Zhou C M Guan et al ldquoFirst-principles studyof lithium intercalated bilayer graphenerdquo Science China PhysicsMechanics and Astronomy vol 55 no 8 pp 1376ndash1382 2012

[15] D Krepel and O Hod ldquoLithium adsorption on armchairgraphene nanoribbonsrdquo Surface Science vol 605 no 17-18 pp1633ndash1642 2011

[16] M Amft S Lebegue O Eriksson and N V SkorodumovaldquoAdsorption of Cu Ag and Au atoms on graphene includingvan der Waals interactionsrdquo Journal of Physics CondensedMatter vol 23 Article ID 395001 2011

[17] L Xian and M Y Chou ldquoDiffusion of Si and C atoms on andbetween graphene layersrdquo Journal of Physics D Applied Physicsvol 45 no 45 Article ID 455309 2012

[18] Y Xia Z Li and H J Kreuzer ldquoAdsorption diffusion anddesorption of hydrogen on graphenerdquo Surface Science vol 605no 21-22 pp L70ndashL73 2011

[19] H Tachikawa ldquoA direct molecular orbital-molecular dynamicsstudy on the diffusion of the Li ion on a fluorinated graphenesurfacerdquo Journal of Physical Chemistry C vol 112 no 27 pp10193ndash10199 2008

[20] H Tachikawa T Iyama and H Kawabata ldquoMD simulation ofthe interaction of magnesium with graphenerdquoThin Solid Filmsvol 518 no 2 pp 877ndash879 2009

[21] X B Zhao and P Liu ldquoBiocompatible graphene oxide as afolate receptor-targeting drug delivery system for the controlledrelease of anti-cancer drugsrdquo RSC Advances vol 4 no 46 pp24232ndash24239 2014

[22] J Rodrıguez-Lopez N L Ritzert J A Mann C Tan W RDichtel and H D Abruna ldquoQuantification of the surface dif-fusion of tripodal binding motifs on graphene using scanningelectrochemical microscopyrdquo Journal of the American ChemicalSociety vol 134 no 14 pp 6224ndash6236 2012

[23] W A Steele ldquoThe physical interaction of gases with crystallinesolids I Gas-solid energies and properties of isolated adsorbedatomsrdquo Surface Science vol 36 no 1 pp 317ndash352 1973

[24] W A Steele ldquoThe physical interaction of gases with crystallinesolids II Two-dimensional second and third virial coefficientsrdquoSurface Science vol 39 no 1 pp 149ndash175 1973

[25] J-A Ruan and B Bhushan ldquoAtomic-scale and microscalefriction studies of graphite and diamond using friction forcemicroscopyrdquo Journal of Applied Physics vol 76 no 9 pp 5022ndash5035 1994

[26] M Chi and Y-P Zhao ldquoAdsorption of formaldehyde moleculeon the intrinsic and Al-doped graphene a first principle studyrdquoComputational Materials Science vol 46 no 4 pp 1085ndash10902009

[27] M Chi and Y-P Zhao ldquoFirst principle study of the interactionand charge transfer between graphene and organic moleculesrdquoComputational Materials Science vol 56 pp 79ndash84 2012

[28] D Qian G J Wagner W K Liu M Yu and R S RuoffldquoMechanics of carbon nanotubesrdquo Applied Mechanics Reviewsvol 55 no 6 pp 495ndash533 2002

[29] Y Wang D Tomanek and G F Bertsch ldquoStiffness of a solidcomposed of C60 clustersrdquo Physical Review B vol 44 no 12pp 6562ndash6565 1991

[30] Y Chan and JMHill ldquoModelling interaction of atoms and ionswith graphenerdquo Micro and Nano Letters vol 5 no 5 pp 247ndash250 2010

[31] B J Cox N Thamwattana and J M Hill ldquoMechanics of atomsand fullerenes in single-walled carbon nanotubes I Acceptanceand suction energiesrdquo Proceedings of the Royal Society of LondonA vol 463 pp 461ndash476 2007


[32] J M Hill N Thamwattana and B J Cox ldquoMechanics ofatoms and fullerenes in single-walled carbon nanotubes IIOscillatory behaviorrdquo Proceedings of the Royal Society of LondonA vol 463 no 2078 pp 477ndash494 2007

[33] O Zworner H Holscher U D Schwarz and R WiesendangerldquoThe velocity dependence of frictional forces in point-contactfrictionrdquo Applied Physics A Materials Science and Processingvol 66 no 1 pp S263ndashS267 1998

[34] H Holscher U D Schwarz O Zworner and R WiesendangerldquoConsequences of the stick-slip movement for the scanningforce microscopy imaging of graphiterdquo Physical Review B vol57 no 4 pp 2477ndash2481 1998

[35] Y Zhang L Q Liu N Xi Y C Wang Z L Dong and U CWejinya ldquoFriction anisotropy dependence on lattice orientationof graphenerdquo Science China Physics Mechanics and Astronomyvol 57 no 4 pp 663ndash667 2014

[36] CUthaisar andV Barone ldquoEdge effects on the characteristics ofLi diffusion in graphenerdquo Nano Letters vol 10 no 8 pp 2838ndash2842 2010

[37] L Chen H Hu Y Ouyang H Z Pan Y Y Sun and F LiuldquoAtomic chemisorption on graphene with Stone-Thrower-Walesdefectsrdquo Carbon vol 49 no 10 pp 3356ndash3361 2011

[38] D Datta J W Li N Koratkar and V B Shenoy ldquoEnhancedlithiation in defective graphenerdquo Carbon vol 80 no 1 pp 305ndash310 2014

[39] Z H Aitken and R Huang ldquoEffects of mismatch strain andsubstrate surface corrugation on morphology of supportedmonolayer graphenerdquo Journal of Applied Physics vol 107 no 12Article ID 123531 2010

[40] W Gao and R Huang ldquoEffect of surface roughness on adhesionof graphene membranesrdquo Journal of Physics D Applied Physicsvol 44 no 45 Article ID 452001 2011

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


barrier-freeHowever the diffusion of aC atomonmonolayergraphene was far more difficult than that of the Si atom Bythe same methods the adsorption of Li atom on graphenewas investigated [9] Three different adsorption sites weretaken into consideration According to their researches thehollow site was found to be the most stable followed by thebridge and top sites Moreover for Li atom the minimumdiffusion energy path where Li atom migrates from a hollowsite to a neighboring hollow site should be a path throughthe bridge site between them What is more the adsorptiondiffusion and desorption of hydrogen on graphene werestudied by DFTmethod [18]They concluded that the atomichydrogen adsorbate produced by bombarding the surfacewith an atomic beam was not possible under equilibriumconditions and below room temperature diffusionwas frozenout and a beam-deposited atomic hydrogen adsorbate couldbe maintained though completely out of equilibrium

Apart from density function theory method MD simu-lation methods have also been widely used in the study ofgraphene interacting with atomion The diffusion processesof the lithium-ion on a fluorinated graphene (F-graphene)surface were investigated by means of a direct molecularorbital-molecular dynamics (MO-MD) method [19] It wasfound that the diffusion coefficient of lithium-ion on F-graphene surface was close to that of normal graphene (H-graphene) but the thermal behavior on the F-graphene sur-face wasmuch different from that on theH-graphene surfaceThe direction of lithium-ion diffusion could be controlledby fluorinated substitution (F-substitution) on the graphenesheet around room temperature But at higher temperaturesthe lithium-ion diffused freely on surface and edge region ofF-graphene The interaction of Mg atom with graphene wasalso researched by using MD simulations [20] It was foundthat the Mg atom vibrated strongly on the graphene surfaceeven at high temperature (1000K) the diffusion of Mg isdifficult

There are relatively few experimental researches onadsorption and diffusion of atomsions on graphene surfaceThe viability of utilizing functionalized graphene oxide withgood solubility and stability biocompatibility and target-specificity was demonstrated as a drug nanocarrier for con-trolled loading and folate receptor-targeted drug delivery ofanticancer drugs [21] It revealed that mastering the stabilityof equilibrium for graphene was vital to investigate a newdrug delivery system for cancer therapy The possibilityof higher lithium storage capacity was explored by con-trolling layered structures of graphene nanosheet materials[12] It enlightened that the spacing equilibrium distanceof atomion on and between graphene sheets is critical forthe future large-scale high-capacity lithium-ion batteriesThe diffusion of a cobalt bis-terpyridine Co(tpy)

2-containing

tripodal compound (1∙2PF6) noncovalently adsorbed onthe surface of a single-layer graphene (SLG) was stud-ied by scanning electrochemical microscopy [22] It wasproved that surface diffusion is the main mechanism forthe observed decrease in the electrochemical response dueto radial diffusion of the adsorbed molecules outward fromthe microspots onto the unfunctionalized areas of the SLGsurface

In order to quantitatively understand atomic adsorptionand diffusion on graphene surface from the perspective ofatomic scale an analytical continuousmodel is put forward inthis work based on the Fourier expansion and Lennard-Jonespotential With the theoretical model equilibrium positionbinding energy and adsorption stability of the atomsions(Au Pt Mn2+ Na1+ and Li1+) on monolayer graphenesurface are calculated and analyzed The different metalatomsions selected in this work possess many applicationprospects For example Mn2+ is an ideal candidate for drugdelivery and Li1+ is a common element for high-energydensity rechargeable alkali batteries The dependence ofdiffusion energy and diffusion forces on the path is illustrated

2 Theoretical Model

A monolayer graphene is assumed to be perfect flat andinfinite compared to atomic scale (as shown in Figure 1(a))Due to its periodic honeycomb lattice structure a two-dimensional lattice vector parallel to graphene can be defined[23ndash25]

l = 1198971a1 + 1198972a2 (1)

where 1198971and 1198972are integers and a1 and a2 are the unit lattice

vectors in the plane (as shown in Figure 1(a)) It is evidentto know that the translation of a single adsorption atom byl will take it from one position above a surface lattice cellinto an equivalent position above another different latticecell (as shown in Figure 1(a)) Therefore the total interactionpotential 119906(r) takes the form of

119906 (r + l) = 119906 (r) (2)

where r is the position of the adsorption atom with respect tothe graphene surface

Thus the periodic nature of the total interaction potential119906(r) is a Fourier series

119906 (r) = sum119892

119908119892(119911) exp (119894g sdot 120591) (3)

where 119911 is perpendicular distance between the adsorptionatom and graphene surface 120591 is the two-dimensional trans-lation vector and g is the multiples of the reciprocal latticevectors b1 and b2

g = 2120587 (1198921b1 + 1198922b2) (4)

where 1198921and 119892

2are integers and b1 and b2 are defined by

a1 sdot b1 = a2 sdot b2 = 1

a1 sdot b2 = a2 sdot b1 = 0(5)

Equation (5) indicates that b1 and b2 are perpendicular to a2a1 and of length csc120593119886

1 csc120593119886

2 respectively where 120593 is


x

y

zAdsorptionatomion

a1

a2

(a)

1


3

2

(b)



119908119892(119911) =

1

119860

sum

11989711198972

sum

119896

int

119860


119896) 119889120591

=

1

119860

sum

119896


infin

0


(6)




119896

and t = 120591 + l +m119896


119906119904(r119894) = 4120576 [(

120590

119903119894

)

12

minus (

120590

119903119894

)

6

] (7)




1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(8)


2and119870



119906 (r) asymp 1199080(119911) + sum

119892gt0

119902

sum

119896=1

119908119892(119911) exp [119894g sdot (m

119896+ 120591)]

= 119862 (119911) + sum

119892gt0

119902

sum

119896=1

119863(119892 119911) exp [119894g sdot (m119896+ 120591)]

(9)


119892 =

1003816100381610038161003816g1003816100381610038161003816=

4120587

radic3119886

(119892

2

1+ 119892

2

2minus 11989211198922)

12

119862 (119911) = 1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119863 (119892 119911) = 119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(10)







119906 (119903)

= 119862 (119911)

+ 2sum

119892gt0

119863(119892 119911) exp (119894g sdot 120591) cos [21205873

(1198921+ 1198922)]

(11)


1+



x = 119909119886

(a1 minus a2)

y = 119910

radic3

(a1 + a2) (12)


119906 (119903) = 119862 (119911) minus 119863(

4120587

radic3119886

119911)

sdot [2 cos(2120587119910

radic3119886

) cos(2120587119909119886

) + cos(4120587119910

radic3119886

)]

(13)




written as

119906

lowast(119903)

= 120582119902(

2120590

lowast12

5119911lowast10

minus

120590

lowast6



119910


radic3

119910

lowast)]

(14)


120582 =

2120587

119860lowast

120572 =

120590

lowast12

30

(

119892

lowast

2119911lowast)

5

1198705(119892

lowast119911

lowast)

120573 = 2120590

lowast6

(

119892

lowast

2119911lowast)

2

1198702(119892

lowast119911

lowast)

(15)



119865

lowast

119909= minus2120587120582 (120572 minus 120573) [2 cos( 2120587

radic3

119910


119865

lowast

119910= minus

4120587120582

radic3

(120572 minus 120573)


119910


radic3

119910

lowast)]

119865119911

lowast= 4120582119902(

120590

lowast12

119911lowast11

minus

120590

lowast6

119911lowast5)

minus 120582 [(

11

2119911lowast+ 119892


5

2119911lowast+ 119892

lowast)120573]


119910


radic3

119910

lowast)]

(16)


119865

lowast

119905= minus2120587120582 (120572 minus 120573)


120588







minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 20


B-site B-site

ulowast

xlowast

(a)

minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 35302520


B-site


B-site

ulowast

ylowast

(b)


20

15

10

5

0

minus5

minus10

minus1510 15 20 25 30

minus7

minus8

minus9

minus10085 090 095 100 105 110 115

ulowast

zlowast

HBT




119911(119909

lowast 119910












minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)




1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502



1199060



0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




x

y

zAdsorptionatomion

a1

a2

(a)

1


3

2

(b)



119908119892(119911) =

1

119860

sum

11989711198972

sum

119896

int

119860


119896) 119889120591

=

1

119860

sum

119896


infin

0


(6)




119896

and t = 120591 + l +m119896


119906119904(r119894) = 4120576 [(

120590

119903119894

)

12

minus (

120590

119903119894

)

6

] (7)




1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(8)


2and119870



119906 (r) asymp 1199080(119911) + sum

119892gt0

119902

sum

119896=1

119908119892(119911) exp [119894g sdot (m

119896+ 120591)]

= 119862 (119911) + sum

119892gt0

119902

sum

119896=1

119863(119892 119911) exp [119894g sdot (m119896+ 120591)]

(9)


119892 =

1003816100381610038161003816g1003816100381610038161003816=

4120587

radic3119886

(119892

2

1+ 119892

2

2minus 11989211198922)

12

119862 (119911) = 1199080(119911) = 120576

2120587

119860

119902(

2120590

12

511991110minus

120590

6

1199114)

119863 (119892 119911) = 119908119892(119911)

= 120576

2120587

119860

[

120590

12

30

(

119892

2119911

)

5

1198705(119892119911) minus 2120590

6(

119892

2119911

)

2

1198702(119892119911)]

(10)







119906 (119903)

= 119862 (119911)

+ 2sum

119892gt0

119863(119892 119911) exp (119894g sdot 120591) cos [21205873

(1198921+ 1198922)]

(11)


1+



x = 119909119886

(a1 minus a2)

y = 119910

radic3

(a1 + a2) (12)


119906 (119903) = 119862 (119911) minus 119863(

4120587

radic3119886

119911)

sdot [2 cos(2120587119910

radic3119886

) cos(2120587119909119886

) + cos(4120587119910

radic3119886

)]

(13)




written as

119906

lowast(119903)

= 120582119902(

2120590

lowast12

5119911lowast10

minus

120590

lowast6



119910


radic3

119910

lowast)]

(14)


120582 =

2120587

119860lowast

120572 =

120590

lowast12

30

(

119892

lowast

2119911lowast)

5

1198705(119892

lowast119911

lowast)

120573 = 2120590

lowast6

(

119892

lowast

2119911lowast)

2

1198702(119892

lowast119911

lowast)

(15)



119865

lowast

119909= minus2120587120582 (120572 minus 120573) [2 cos( 2120587

radic3

119910


119865

lowast

119910= minus

4120587120582

radic3

(120572 minus 120573)


119910


radic3

119910

lowast)]

119865119911

lowast= 4120582119902(

120590

lowast12

119911lowast11

minus

120590

lowast6

119911lowast5)

minus 120582 [(

11

2119911lowast+ 119892


5

2119911lowast+ 119892

lowast)120573]


119910


radic3

119910

lowast)]

(16)


119865

lowast

119905= minus2120587120582 (120572 minus 120573)


120588







minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 20


B-site B-site

ulowast

xlowast

(a)

minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 35302520


B-site


B-site

ulowast

ylowast

(b)


20

15

10

5

0

minus5

minus10

minus1510 15 20 25 30

minus7

minus8

minus9

minus10085 090 095 100 105 110 115

ulowast

zlowast

HBT




119911(119909

lowast 119910












minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)




1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502



1199060



0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119906 (119903)

= 119862 (119911)

+ 2sum

119892gt0

119863(119892 119911) exp (119894g sdot 120591) cos [21205873

(1198921+ 1198922)]

(11)


1+



x = 119909119886

(a1 minus a2)

y = 119910

radic3

(a1 + a2) (12)


119906 (119903) = 119862 (119911) minus 119863(

4120587

radic3119886

119911)

sdot [2 cos(2120587119910

radic3119886

) cos(2120587119909119886

) + cos(4120587119910

radic3119886

)]

(13)




written as

119906

lowast(119903)

= 120582119902(

2120590

lowast12

5119911lowast10

minus

120590

lowast6



119910


radic3

119910

lowast)]

(14)


120582 =

2120587

119860lowast

120572 =

120590

lowast12

30

(

119892

lowast

2119911lowast)

5

1198705(119892

lowast119911

lowast)

120573 = 2120590

lowast6

(

119892

lowast

2119911lowast)

2

1198702(119892

lowast119911

lowast)

(15)



119865

lowast

119909= minus2120587120582 (120572 minus 120573) [2 cos( 2120587

radic3

119910


119865

lowast

119910= minus

4120587120582

radic3

(120572 minus 120573)


119910


radic3

119910

lowast)]

119865119911

lowast= 4120582119902(

120590

lowast12

119911lowast11

minus

120590

lowast6

119911lowast5)

minus 120582 [(

11

2119911lowast+ 119892


5

2119911lowast+ 119892

lowast)120573]


119910


radic3

119910

lowast)]

(16)


119865

lowast

119905= minus2120587120582 (120572 minus 120573)


120588







minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 20


B-site B-site

ulowast

xlowast

(a)

minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 35302520


B-site


B-site

ulowast

ylowast

(b)


20

15

10

5

0

minus5

minus10

minus1510 15 20 25 30

minus7

minus8

minus9

minus10085 090 095 100 105 110 115

ulowast

zlowast

HBT




119911(119909

lowast 119910












minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)




1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502



1199060



0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 20


B-site B-site

ulowast

xlowast

(a)

minus82

minus84

minus86

minus88

minus90

minus92

minus94

00 05 10 15 35302520


B-site


B-site

ulowast

ylowast

(b)


20

15

10

5

0

minus5

minus10

minus1510 15 20 25 30

minus7

minus8

minus9

minus10085 090 095 100 105 110 115

ulowast

zlowast

HBT




119911(119909

lowast 119910












minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)




1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502



1199060



0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus6

minus7

minus8

minus9

minus13

minus14

minus15

00 05 10 15 20 25 30

ulowast

xlowast

Mn2+

AuPt

Na1+

Li1+

(a)

ylowast0 1 2 3 4 5

minus6

minus7

minus8

minus9

minus13

minus10

minus14

minus15

ulowast

Mn2+

AuPt

Na1+

Li1+

(b)

10 15 20 25 30

20

15

10

5

0

minus5

minus15

minus10

minus20

ulowast

Mn2+

AuPt

Na1+

Li1+

zlowast

(c)




1199110

H 0218047 0316354 0310880 0290234 0241717B 0226746 0319419 0314168 0294467 0248800T 0227549 0319769 0314540 0294932 0249502



1199060



0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

(a)Flowast t

0∘

15∘30∘

60∘

120588lowast

00 05 10 15 20 25 30

4

2

0

minus2

minus4

(b)






4 Conclusions




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




00 05 10 15 20 25 30

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast

ylowast = 0


ylowast = radic32

(a)

00 05 10 15 20 25 3530

minus82

minus84

minus86

minus88

minus90

minus92

minus94

ulowast

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)


Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast

ylowast = 0


ylowast = radic32

(a)

Flowast t

00 05 10 15 20 25 30

4

2

0

minus2

minus4

xlowast = 0

xlowast = 025

xlowast = 05

ylowast

(b)







Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






Acknowledgments


References



















































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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