Definition of Interface Parameter and Its Application on Estimating the

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C: Physical Processes in Nanomaterials and Nanostructures

Definition of Interface Parameter and Its Application on Estimating the Thermal Stability of Metallic Nanoparticles Xiaobao Jiang, Beibei Xiao, Rui Lan, Xiaoyan Gu, Hongchao Sheng, Hongyu Yang, and Xinghua Zhang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b07472 • Publication Date (Web): 25 Oct 2018 Downloaded from http://pubs.acs.org on October 29, 2018

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The Journal of Physical Chemistry

Definition of Interface Parameter and Its Application on Estimating the Thermal Stability of Metallic Nanoparticles Xiao Bao Jiang*a, Bei Bei Xiaob, Rui Lana, Xiao Yan Gua, Hong Chao Shenga Hong Yu Yang and Xing Hua Zhanga a

Department of Materials Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

b

School of Energy and Power Engineering, Jiangsu University of Science and Technology, 212003, China

Abstract: To describe the effect of interface structure on the thermal stability of nanoparticles (NPs) that embedded in a matrix, an interface parameter ψ is defined by considering both the interface energy γss and the interface stress fss. As ψ is introduced into a size-dependent melting model, the interface and size effects on melting temperature Tm(ψ,D) is found. Results show that fss predominates the elevation and depression of Tm(ψ,D) for coherent and incoherent interfaces, while γss is only a secondary effect. Furthermore, Tm(ψ,D) decreases following the increase of ψ. Namely, ψ < 0 for coherent interfaces, ψ = 0 for bulk, 0 < ψ < 1 for incoherent interfaces, and 1 < ψ for core-shell structure where the low melting point element is located at core. The success in the prediction of Tm(D) for metallic NPs confirm the validity of parameter ψ.

*

Corresponding author. Email: [email protected] 1

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1. Introduction: Melting in confined systems is an important subject in materials science. This is because the melting temperature Tm(D) (where D shows the diameter) can be related to many physical quantities such as Debye temperature, cohesive energy, diffusion activation energy, thermal expansion coefficient, vacancy formation energy, and specific heat 1-3 etc, which determine the physics, mechanics and chemical properties of materials while these are critical for nanodevice applications in the fields of microelectronics, nonlinear optics and solar energy. Investigations show that Tm(D) is size dependent. For freestanding nanoparticles (NPs), Tm(D) is suppressed as D decreases

4-8.

The large

surface-volume ratio and surface loss coordination atoms are believed to contribute to the variation of Tm(D). While for NPs embedded in a rigid matrix, the melting behavior is more complex. Experiments show that obvious superheating is observed at D < 20 nm when NPs and matrix form a coherent interface

9-15,

whereas the depression of Tm(D) is

observed for the incoherent interface structures 4,16-19. The remarkable contradiction in the variation tendency of Tm(D) observed above has attracted much attention and the corresponding theory works have been developed for the understanding of the intrinsic interface melting mechanism. Jiang proposed that the superheating and the enhanced melting entropy of embedded NPs occur when Tm of matrix is higher that of embedded NPs in the bulk, the matrix component has a smaller atomic diameter than the NPs, and the NPs and matrix form a coherent or semi-coherent interface

20-22.

The superheating is believed to be induced by the suppression of atomic

thermal vibrations at the interface. After that, Qi also clarified that the coherent or incoherent

interface

causes

the

surface-area-difference (SAD) model

variation 23.

of

Tm(D)

in

their

developed

In an extended bond-order-length-strength

(BOLS) model, Sun shows that the formation of compound or alloy at the junction interface can induce bond-strength gain and elevate Tm(D) 24,25. The larger electroaffinity at the interface can be verified by the variation of local valence density of state which is 2


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deeply trapped. By introducing the size dependence of melting enthalpy, Yang developed a thermodynamic model to describe the quasi-isochoric superheating of rigid matrix confined NPs by considering the pressure effect. They found that the pressure imposed by the rigid matrix predominate the extent of the superheating

26.

By considering the

contributions of cohesive energy, average coordination number as well as atomic bond strength, Omid has built Tm(D) function

27.

They found that the increasing of

superheating is attributed to the stronger bond of interface atoms. Furthermore, the classical nucleation theory has been introduced in Zhdanov’s work

28,

the roles of

liquid-solid interface on the matrix materials have also been considered to explain the superheating phenomenon

29-30.

Recently, the thermal stability of interface structure is

estimated by the interface energy. Results show that when the solid-solid interface energy γss is negative, superheating occurs, while the solid-liquid interface γsl is positive, the depression presents

31,32.

The interface energy and pressure effects predominate the

variation of Tm(D) 32. This assumption is reasonable since the interface structure between NPs and matrix determines the nature of interatomic bonds. Due to the existence of matrix, there is an additive pressure on each bond at the interface, as a result, the bonds at the surface of NPs are stretching or contracting as reviewed above. In fact, this variation of bonds can be described by interface stress fss, where ss means the solid-solid interface. As referred to the definition of surface energy γsv and surface stress fsv (for unit area, γsv is the reversible work needed to form a new solid surface, and fsv is that induced by the elastic deformation

33,34.

Noted that fsv is the derivative of γsv regarding the strain

tangential to the surface, sv means the solid-vapor interface), the interface energy γss can be defined as the work needed to form a new solid-solid interface and the interface stress fss shows the reversible work induced by the elastic deformation which exerted by matrix. In previous works, γsv and fsv are found play important roles on the phase transition of free-standing semiconductor NPs 35-37. It is evident that for the embedded NPs, fss as well as γss plays important role on modulating Tm(D). 3



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As reviewed above, several questions arise: (a) What does the roles of fss and γss play on modulating the variation of Tm(D) for the embedded NPs? (b) How to estimate the surface and interface effects using an unified parameter? In this paper, an interface parameter ψ is defined and introduced into a size-dependent melting model to distinguish the surface and interface contributions to Tm(D). The competition relations between γss and fss on the evaluation or depression of Tm(D) are discussed. We find that Tm(D) decreases monotonously when ψ varies from -2 to 2. Either for the coherent or incoherent interface, our result shows that fss dominates the variation tendency of Tm(D), which differs from the conclusions reported in 31,32. 2. Theory: In previous works, the Tm(D) function has been built by introducing the size-dependent cohesive energy model as 38,39,   2 S b ( )  Tm ( D)  1 1  1   exp   Tm ()  12 D / D0  1 3R 12 D / D0  1 

(1)

Among which, D0 denotes the smallest size of nanocrystals, below which the crystal structure disappeared. At D0, all atoms are located on the surface. D0 can be calculated by the dimensionality of nanocrystals d and the diameter of atoms h as: D0 = 2(3-d)h. For NPs, d=0 and there is D0=6h. Sb(∞) is the solid-vapor transition entropy of bulk crystals, which can be determined by the latent heat of vaporization Hb(∞) and the evaporation temperature Tb(∞) as Sb(∞)=Hb(∞)/Tb(∞). R shows the ideal gas constant. This model has successfully described the melting behavior of free-standing NPs. For embedded NPs, Eq. (1) must be modified and the interface effect should be considered. Here, an interface parameter ψ is defined by considering the expression of Gibbs free energy of NPs and introduced into Eq. (1). For a solid NPs, assuming that the external pressure is constant, the Gibbs free energy G depends on temperature T and size D, which can be written as G(T, D). G(T, D) function can be expressed as 35-37, 4


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G (T , D)  Gv (T )  Ge ( D)  Gs ( D) .

(2)

Gv(T) shows bulk free energy only determined by temperature, which is related to the energetic state to the interior of NPs and thus is independent on size and interface condition. Gs(D) is the effect of size on surface energy while Ge(D) denotes the internal pressure Pin induced size-dependent surface stress. Gs(D) and Ge(D) can be determined by,

6 svVm , D 4f V Ge ( D)  PinVm  sv m , D Gs ( D )   s A 

(3) (4)

where A means the surface area per gram-atom, Vm denotes the volume per gram-atom. The surface stress fsv can be substituted by the solid-liquid interface stress fsl, which can be expressed as 33-34,

6 sl h . 8

f sl 

(5)

κ in Eq. (5) is the compressibility, which can be determined by κ=1/B and B is bulk modulus. γsl shows the interface energy of solid-liquid interface and is determined by 34,35,

 sl 

2hSvib H m , 3Vm R

(6)

where Svib, and Hm are the vibrational entropy and melting enthalpy. When the NPs are embedded in a matrix, the atomic thermal vibration or the average mean-square displacement of atoms at the surface of NPs will be suppressed due to the formation of interface, while that in the interior remains unchanged. The corresponding Gibbs free energy of the interface system should thus be modified by,

Gi (T , D)  Gv (T )  Gei ( D)  Gsi ( D) ,

(7)

where i denotes the interface case. Since solid-solid interface can be considered as the combination of two solid-liquid interface, A =3V/D and P=2fssA/(3V) = 2fss/D are 5



Page 6 of 25

reasonable 33. According to Eqs. (3) and (4), Gsi(D) and Gei(D) can be rewritten as,

3 ssVm , D 2f V Gei ( D)  ss m . D

Gsi ( D) 

(8) (9)

It is evident that for the embedded NPs, the interface effects are only determined by Gsi(D) and Gei(D). When we only consider the interface case, the interface parameter ψ can be defined as,



Gei ( D)  Gsi ( D) 3 ss  2 f ss  . Ge ( D)  Gs ( D) 6 sv  4 f sv

(10)

For coherent or semi-coherent interfaces, γss can be determined by 31,32,

 ss 

  Z ss Z Z ( b   ss )  Tm (1  ss ) Sb ()  ss S ss () , 2 Zb Zb  

(11)

where Z shows the coordination number, ε denotes the bond energy. For incoherent interface, since the liquid can be regarded as high concentration dislocation cores contact solid surface everywhere, f can be assumed as constant for both solid-liquid and solid-solid interface 40. According to Eq. (6), γss can be expressed by 31,32,

 ss (T ) 

4h S vib H m , 3V m R

(12)

where the averaged values of h, Svib, Hm, and Vm between the NPs and matrix are taken. fss for coherent or incoherent interface can be determined by introducing the averaged values of γsl, κ and h based on Eq. (5) as,

f ss 

6 sl h . 8

(13)

It is evident that ψ is a size independent parameter, it is only determined by the interface conditions γss, fss, and the inherent surface properties γsv and fsv of NPs in bulk. Combining Eqs (10) and (1), there is,

  2Sb  Tm ( D)  1 1  1   exp  . Tm ()  12 D / D0  1  3R 12 D / D0  1 6


(14)

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3. Discussions: Figure 1 shows the predictions of Eq. (14) for coherent matrix confined metallic NPs and the comparisons with experimental data for (a) Ag/Ni, (b) In/Al, (c) Pb/Al and (d) Pb/Cu. For coherent interfaces, due to the strong bonding nature of atoms between matrix and NPs, γss is negative 31,32, fss=∂Gss/∂A≈γss+A∂γss/∂A. As a results, fss Tmγss(D), which means that fss dominates the increase of Tm(D). For Au/SiO2 and Fe/SiO2, Tmfss(D) < Tmγss(D), which shows that the depression of Tm(D) is also mainly contributed by fss. In other words, the variation tendency of Tm(D) is determined by fss, which disagrees with the conclusions reported in 31,32 that Tm(D) is dominated by ss. This 7



Page 8 of 25

can be understood by the fact of |fss|>|γss|. When a solid-solid interface forms, more reversible work is used to participate in the deformation of NPs. In addition, the bond geometry distortion at the solid-solid interface also contributes to the storage of bond energy. Thus, during the melting, more energy is necessary to break these distorted bonds at the interface. Through comparing Figures 1 and 2, we found that the superheating in Figure 1 starts at about D fsi >0 and γsv > γsi >0 are understandable where ψ >1. During melting, the stability of atoms in the core is lower than that in the shell and pre-melting may occur in the interior of NPs. 4. Conclusions: In summary, an interface parameter ψ is proposed based on the physical quantities of interface energy γss and interface stress fss. By introducing ψ into a size-dependent melting model, superheating and depression of Tm(D) are predicted. We found that fss predominate the variation tendency of Tm(D) while γss secondary. Either superheating or depression of Tm(D) depends on the value of ψ: ψ < 0, superheating occurs; ψ = 0, Tm(D) equals to that of the bulk; 0 < ψ < 1, depression of Tm(D) induced by the incoherent interface is present; ψ = 1, this is the case of free standing NPs; ψ > 1 corresponds to a core-shell structure where low melting point atoms are in the core. The interface parameter ψ proposed in this work can not only describes the interface effect on thermal stability of NPs, but also reveals the potential physical origin of the novel properties of the confined systems, which provides guidelines for the design of quantum devices.

Acknowledgments We acknowledge supports by the China National Natural Science Foundation (Grant Nos. 21503097, 51401090 and 51705218) and the Jiangsu Provincial Natural Science Foundation (Grant No. BK20140518).

10


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Captions: Figure 1. The Tm(D) functions which describe the coherent matrix confined metallic NPs with: (a) , , ▲

47

for Ag/Ni; (b)◄11,► 13, ▼48, +49 for In/Al; (c)  50, 16, 

51

for

Pb/Al; (d)  52 for Pb/Cu. Figure 2. The Tm(D) functions which describe the incoherent matrix confined metallic NPs with (a) 

53

for Al/Al2O3, (b) 

15

for Au/SiO2, (c) 

14

for Fe/SiO2, (d) 

18

for

Sn/C. Figure 3. The Tm ss (D) , Tmf ss (D) and Tm(D) functions of embedded metallic NPs for (a) Ag/Ni; (b) Pb/Al; (c) Au/SiO2; (d) Fe/SiO2 Figure 4. The Tm(D) functions for (a) Au embedded in incoherent SiO2 matrix and free-standing NPs; (b) Fe embedded in incoherent SiO2 matrix and NPs with free surface. Figure 5. The Tm(ψ) function of metallic NPs, the dot-dash lines I and II show the bulk and free standing NPs respectively. Figure 6. The core-shell model for the case of ψ>1 in Figure 5.

11



Figure 1. 1600

900

800

1500

700

600

m

T (K)

Tm (K)

1400

1300 500 1200

400

(a) Ag/Ni 1100

0

10

20

1200

30

D (nm)

40

(b) In/Al 300

0 60 1000

50

20

40

D (nm)

60

80

100

900 1000

Tm (K)

800

Tm (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 25

800

700

600 600

(d) Pb/Cu

(c) Pb/Al 400

0

10

20

30

D (nm)

40

50

60

500

0

10

12


20

D (nm)

30

40

50

Page 13 of 25

Figure 2. 1400 1000 1200

1000

Tm (K)

Tm (K)

800

600

800

600 400 400

(b) Au/SiO2

(a) Al/Al2O3 200

0

20

40

60

80

100

120

D (nm)

2000

200

0

4

8

500

1500

450

20

400

1250

350

1000

300

750

(c) Fe/SiO2 500

16

Tm (K)

1750

12

D (nm)

550

Tm (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

10

20 D (nm)

30

(d) Sn/C 40

250

0

20

13


40

D (nm)

60

80

100


Figure 3.

1260

900

Tm(D)

Tm(D)

fss

fss

Tm (D)

1250

Tm (D)

800

ss

ss

Tm (D) Tm (D) (K)

Tm(D) (K)

Tm (D) 1240

700

1230

600

(b) Pb/Al

(a) Ag/Ni 1220

0

20

40

60

80

500

100

D (nm)

1400

0

5

10

15

20

D (nm)

2000

1800 1300

1200

Tm(D)

Tm (D) (K)

1600

Tm (D) (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 25

1400

fss

Tm (D) 1100

Tm(D) fss

Tm (D)

1200

ss

ss

Tm (D)

Tm (D)

1000

(c) Au/SiO2 1000

0

5

10

15

(d) Fe/SiO2 20

800

0

D (nm)

5

10

D (nm)

14


15

20

Page 15 of 25

Figure 4. 1600

1400

1200

Tm (K)

1000

800

600

Au/SiO2 Au nanoparticle

400

(a) 200 2000 1800 1600

Tm (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1400 1200 1000

Fe/SiO2

800

Fe nanoparticle

600 400

(b) 0

5

10

15

20

D (nm)

15


25

30


Figure 5.

1000

D = 5 nm

In/M Sn/M

800

I

Tm() (K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 25

600

II free surface

Sn

Tm () = 505 K In

Tm () = 429.75 K

400

200

0 -2.0

incoherent interface

coherent interface

-1.5

-1.0

-0.5

0.0



0.5

16


1.0

1.5

2.0

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Figure 6.

17



Page 18 of 25

Table 1 The necessary parameters used in Eq. (14). Ag/Ni

In/Al

Pb/Al

hn (nm)32

0.289

0.368

0.35

hM (nm)32

0.249

0.286

n(10-12 Pa-1)33

9.653

28.341

M

(10-12

Pa-1)33

γss (J m-2)32 fss (J

m-2)

γsv (J

m-2)45

fsv (J m-2)

Pb/Cu

Al/Al2O3

Au/SiO2

Fe/SiO2

Sn/C

0.286

0.288

0.248

0.372

0.192

0.184

13.298

5.848

7.257

642

8.86543

0.256 21.834

0.284 5.55632

16.1832 144

5.640

13.298

-0.704

-0.484

-0.677

-0.506

0.302

0.301

0.807

0.261

-2.866

-1.084

-1.169

-1.655

2.052

2.328

3.685

2.096

1.246

0.7046

0.593

1.143

1.506

2.41746

0.70946

2.592

0.826

0.940

1.920

3.505

3.640

1.748

18


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TOC Graphic core-shell structure

nanoparticle

Tm(D) ?

coherent interface incoherent interface

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Definition of Interface Parameter and Its Application on Estimating the

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