Thermodynamics of Size Effect on Phase Transition Temperatures of

ARTICLE pubs.acs.org/JPCC

Thermodynamics of Size Effect on Phase Transition Temperatures of Dispersed Phases Zi-Xiang Cui,† Miao-Zhi Zhao,† Wei-Peng Lai,†,‡ and Yong-Qiang Xue*,† † ‡

Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, P. R. China Xi’an Modern Chemistry Research Insititute, Xi’an, 710065, Shanxi, P. R. China ABSTRACT: An equation for a phase transition in a dispersed system has been proposed, and the applications of the equation in various kinds of phase transitions have been discussed. The determinate relation between the interfacial tension and the radius of a droplet has been derived by the monolayer model. Applying the fusion transition equation and the interfacial tension relation, the melting temperatures of Au and Sn nanoparticles have been calculated, and the predicted melting temperatures are in good agreement with the available experimental data. The research results show that the phase transition equations can be applied to predict the temperatures of phase transitions of dispersed systems and to explain the phenomenon of metastable states; that the size of a dispersed phase has a remarkable effect on the phase transition temperatures, and the phase transition temperatures decrease with the radius of the dispersed phase decreasing; and that the depression of the melting temperature for a nanowire is half of that for a spherical nanoparticle with identical radius.

1. INTRODUCTION A wide range of fields, such as chemistry, chemical engineering, materials, environment, meteorology, and medicine, are involved with phase transitions of nanosystems. The theory of phase transition in classical physical chemistry does not take the surface effect into account; therefore it can not be applied to solve the problems of phase transitions of nanosystems. At present, the investigations on phase transition of nanosystems chiefly focus on the fusion, and many models, such as the empirical or semiempirical formulas,14 and approximate expressions57 have been put forward to predict melting temperatures or explain experimental observations. There are mainly three phenomenological models for the fusion of free-standing nanoparticles. Early in 1909, Pawlow8 developed a model in which the equilibration condition for a fusion system is formed by a solid particle, a liquid droplet having the same mass, and their saturating vapor phase; later the thermodynamic treatment of Pawlow’s model was improved by Hanszen.9 Rie10 proposed another approach in which the equilibration between an infinite liquid and the surrounding solid is considered as fusion equilibrium. Reiss et al.11 considered the fusion equilibrium between a thin liquid shell and a surrounding solid core and proposed the corresponding melting theory; subsequently, Curzon12 modified the theory. The three models have been applied successfully to predict and explain some melting temperatures of nanocrystals.1320 In addition, some melting models can also be obtained by other ways. On the basis of the empirical relation that the melting temperature of a solid is directly proportional to its cohesive energy,2123 the models for size-dependent melting and dimensiondependent melting were proposed, such as Safaei’s,24 Lu et al.’s,25,26 and Qi et al.’s,3,27 in which some differences exist in the expressions r 2011 American Chemical Society

of cohesive energy. Furthermore, Safaei et al.28,29 investigated the dependency of the surface-to- volume coordination number upon particle size, and obtained an analytical function of melting temperature. Vahdati-Khaki et al.30 developed a model to predict the size-dependent melting temperature of nanoparticles based on the cluster mean coordination number. Jiang et al.3133 developed a model of melting temperature according to the size-dependent amplitude of the atomic thermal vibrations of nanocrystals; since the model involves a microcosmic parameter (vibrational entropy), which is hard to determine, the application of the model is difficult. The melting temperatures predicted by these models are consistent with some experimental results.18,3436 However, these models are only applicable to metallic crystal. All of the above models can reveal that the melting temperature decreases with decreasing size of particles. However, some of the models are approximate; others are not convenient for application. In addition, all of the models can only be applicable to fusion transition of nanocrystals, and some fail to predict melting temperature well when particle size is smaller (e.g., the radius is smaller than 5 nm).3,14,19,20,3740 Moreover, the phase transitions of nanosystems can result in a lot of metastable states, such as supercooled liquid, superheated liquid, and supersaturated vapor. However, the phase transitions about the metastable states have not been expressed by corresponding equations yet. Furthermore, all of the thermodynamic relations between the melting temperature and the radius include the variable of surface tension. Many scientists13,16,38 treat surface tension as a constant in the calculation of the melting temperature. However, there are Received: July 15, 2011 Revised: October 12, 2011 Published: October 13, 2011 22796

dx.doi.org/10.1021/jp2067364 | J. Phys. Chem. C 2011, 115, 22796–22803

The Journal of Physical Chemistry C

ARTICLE

remarkable effects of size of small droplets or nanoparticles on the surface tension. Many approximate expressions including parameters have been obtained.4143 However, the parameters are difficult to determine, and most of the expressions can only apply well to bigger droplets or particles. The exact integral expression that is applicable to nanodroplets or nanoparticles has been derived.44 However, it is still hard to solve. In the present research, the equation for phase transitions is proposed, by which the specific thermodynamic equations for a variety of phase transitions of dispersed systems are derived. The determinate relation between interfacial tension and radius of small droplets and nanoparticles are also derived. The regularity and the extent of effect of size on phase transition temperatures are discussed, and the melting temperatures of Au and Sn nanoparticles are calculated.

Figure 1. The fusion model of a nanoparticle.

Applying eq 4 to eq 5, one gets an expression about the phase transition temperature of the dispersed system: 2 3 ! ∂Aβ Δβα Hmb 1 4 ∂Aα 5 ð6Þ T ¼ β þ β σβ σα ∂nβ ∂nα T, p Δα Sbm Δα Sbm T, p

2. THE EQUATION FOR PHASE TRANSITIONS OF DISPERSED SYSTEMS Let a phase transition occur from phase α to β, where at least one of the two phases is a dispersed phase. When the two phases are in equilibrium, their chemical potentials must be equal, μα ¼ μβ

ð1Þ

The chemical potential of a dispersed phase is made of sections of both bulk phase and surface phase,45 ∂A μ ¼ μb þ μs ¼ μb þ σ ð2Þ ∂n T, p where μb and μs are chemical potentials of the bulk phase and the surface phase, respectively, and σ, A, and n are surface tension, surface area, and amount of substance of the dispersed phase, respectively. The superscripts b and s denote bulk quantities (which are equal to the corresponding quantities of the bulk substances) and surface quantities, respectively. When the phase transition of the dispersed phase occurs, the change in molar Gibbs energy can be written as ! ∂Aβ ∂Aα β β b Δα Gm ¼ μβ μα ¼ Δα Gm þ σ β σα ∂nβ ∂nα T, p T, p

ð3Þ

Equation 6 is applicable to various phase transitions of dispersed systems. It can be seen from the equation that the phase transition temperature of a dispersed system depends on not only the properties of the bulk phase (ΔβαHbm and ΔβαSbm) but also the properties of the surface phase (the interfacial tensions and specific surface areas of the two phases). It is obvious that when the particle size of the dispersed phase tends to infinite, both of the items (∂Aβ/∂nβ)T,p and (∂Aα/∂nα)T,p approach zero. Consequently, eq 6 becomes the classical thermodynamic relation of phase equilibrium, then T =ΔβαHbm/ΔβαSbm = T0, where T0 is the normal phase transition temperature.

3. APPLICATION OF THE PHASE TRANSITION EQUATION 3.1. Application to Fusion of Isolated Nanocrystals. Fusion of Nanoparticles. Here we assume that an isolated nanoparticle is

spherical and the melting begins on the surface. The liquid shell with a width of t surrounds the solid core. When the solidliquid phase is in equilibrium, the radius of the solid core is rs, and the outside radius of the liquid shell is rl (see Figure 1). For the solid core, ∂As 2Vs ¼ ð7Þ ∂ns T, p rs where V is the molar volume, and the subscripts s and l denote solid and liquid, respectively. The total mass in fusion process is constant, therefore 4 3 4 4 πrs Fs þ πðrl3 rs3 ÞFl ¼ πr 3 Fs 3 3 3

ΔβαGbm

where is the change in molar Gibbs energy of the phase transition of bulk substances from phase α to β. When the two phases in the dispersed system are in equilibrium, ΔβαGm = 0. Thus, ! ∂A ∂A β α σβ ð4Þ Δβα Gbm ¼ σα ∂nα T, p ∂nβ T, p

At the phase transition temperature of the dispersed system, the relation of the thermodynamic properties of the phase transition for bulk phase is Δβα Gbm

¼

Δβα Hmb

TΔβα Sbm

ð5Þ

whereΔβαHbm and ΔβαSbm are the changes in molar enthalpy and molar entropy of the phase transition of bulk substances.

ð8Þ

where r is the radius of nanoparticle before melting, and Fs and Fl are the densities of the solid core and the liquid shell, respectively. Since Al ¼ 4πrl2

ð9Þ

and nl ¼

4 πðrl3 rs3 Þ=Vl 3

ð10Þ

The partial derivative of Al against nl can be obtained by simultaneous eqs 8, 9, and 10: ! ∂Al 2Vl Fl ¼ 1 ð11Þ ∂nl T, p rl Fs 22797

dx.doi.org/10.1021/jp2067364 |J. Phys. Chem. C 2011, 115, 22796–22803

The Journal of Physical Chemistry C Applying eqs 7 and 11 to eq 6, one can obtain " ! # Δls Hmb 1 2σlv Vl Fl 2σ sl Vs 1 þ l T ¼ l Fs rs Δs Sbm Δs Sbm rs þ t

ARTICLE

ð12Þ

Equation 12 is a precise equation that describes the relationship between the melting temperature and the size of nanoparticles. ΔlsHbm,ΔlsSbm, V, and F are functions of T; σ is a function of T and r. Therefore, when the radius of the nanoparticles is a definite value, eq 12 can be solved by the iterative method. When the radius of the particles is bigger, the difference between T and T0 is little, then ΔlsHbm(T/K)/ΔlsSbm(T/K) ≈ ΔlsHbm(T0/K)/ΔlsSbm(T0/K) = T0; hence one can obtain the approximate equation 2Vs T0 σlv F σ sl 1 s þ ð13Þ T ¼ T0 l Fl rs Δs Hmb rs þ t Equation 13 can be rearranged into T 2Vs σsl σlv Fs þ ¼ 1 l 1 T0 rl Fl Δs Hmb rl t

liquid shell that surrounds the nanorod are rs and rl, respectively (see Figure 2). The thickness of the liquid shell is t, thus rl rs = t. For the solid nanorod, As ¼ 2πrs ðL 2tÞ þ 2πrs2

ð16Þ

and ð14Þ

where rl t = rs. Obviously, eq 14 is the same as the equations derived by Reiss et al.11 and Curzon.12 For a general substance (whose molar volumes of solid and liquid phases during phase transition are quite close to each other), Fs ≈ Fl, eq 14 can be approximated to T0 T 2σsl Vs ¼ l T0 Δs Hmb rs

Figure 2. The fusion model of a nanorod.

ð15Þ

ns ¼ πrs2 ðL 2tÞ=Vs

ð17Þ

Combining eqs 16 and 17, one can get " # ∂rl Vs L 2rl 2rs þ 6rs ∂rs T, p ∂As # ¼ " ∂ns T, p ∂rl rs L 2rl rs þ 3rs ∂rs T, p

ð18Þ

For the thin liquid shell that surrounds the solid nanorod, Al ¼ 2πrl L þ 2πrl2

ð19Þ

nl ¼ ½πrl2 L πrs2 ðL 2tÞ=Vl

ð20Þ

5

Equation 15 is the same as the equations of Couchman et al. and Skripov et al.6 It is obvious that the melting temperature decreases with the particle size decreasing. One can estimate the order of magnitude for ΔT/T0(ΔT = T0 T) by eq 15: when the particle size is bigger, the order of σsl for general metals is 100 J 3 m2; Vs = 105 m3 3 mol1; ΔlsHm = 104 J 3 mol1. Thus, for r = 106 m, 107 m, and 108 m, the orders of the ratios ΔT/T0 are 103, 102, and 101, respectively. When r > 107 m, the effect of size on T is so small that it can be ignored in general cases, but when r e 108 m, the effect becomes obvious. It can be seen from eq 15 that when the particle size is bigger, there is an approximately linear relationship between the melting temperature and the inverse of particle size, which is verified by the experimental results.4650 However, when the particle size is smaller (e.g., r < 5 nm), the effect of r on σ becomes notable,26,5153 the calculated results of the smaller particles by eq 15 deviate from the experimental observations. Thus, many researchers found that the linear relationship disappears when the particle size is smaller.20,27,3740,54 If one considers the outer thin liquid shell as the solid shell, the said relations can also be applied to solidsolid phase transition (i.e., crystal phase transformation). Applying the above phase transition equation (eq 6) to Pawlow’s and Rie’s phenomenological models,810 and with the approximate treatments, one can also obtain the equations derived by them. Fusion of Nanorods and Nanowires. Let the length of a nanorod be L and the radius before melting be r. When the melting is in equilibrium, the radii of the solid nanorod and the

Combining eqs 19 and 20, one gets

∂Al ∂nl

∂rl Vl ðL þ 2rl Þ ∂rs

¼ T, p

∂rl rl L ∂rs

rs L þ 2rs rl þ T, p

T, p

rs2

∂rl ∂rs

3rs2 T, p

ð21Þ The total mass in the fusion process is constant, therefore πr02 LFs ¼ πrs2 ðL 2tÞFs þ ½πrl2 L πrs2 ðL 2tÞFl The partial derivative of eq 22 with respect to rs is ! Fl 1 ðrs L 2rs rl þ 3rs2 Þ Fs ∂rl ! ¼ ∂rs T, p Fl 2 F 1 rs rl L l Fs Fs

ð22Þ

ð23Þ

Applying eqs 18, 21, and 23 to eq 6, one can obtain a precise relation between the melting temperature and the radius of the nanorod: " Δls Hbm 1 σlv BðL þ 2rl ÞVl T ¼ l þ l b b Δs Sm rl LB rs L þ 2rs rl þ rs2 B 3rs2 Δs Sm σ sl V ðL 2rl 2rs B þ 6rs Þs ð24Þ rs ðL 2rl rs B þ 3rÞ 22798



ARTICLE

where B = (∂rl/∂rs)T,p. It can be seen from eq 24 that the main factors influencing the melting temperature of a nanorod are the interfacial tensions, the radii, and the length of the nanorod. If L .rl and L . rs, the nanorod becomes the nanowire, and eq 24 can be simplified to Δls Hbm 1 σsl Vs σlv ðVs Vl Þ þ ð25Þ T ¼ l rl Δs Sbm Δls Sbm rs Equation 25 is the relation between the melting temperature and the radius of the nanowire. When the radius of the nanowire is bigger, the difference between T and T0 is not obvious, then ΔlsHbm/ΔlsSbm ≈ T0. Combining Vl/Vs = Fs/Fl, one gets T Vs σ sl σ lv F ¼ 1 l þ 1 s ð26Þ T0 rl Fl Δs Hmb rs At the beginning of fusion, t , rs, so rs ≈ rl ≈ r, and eq 26 becomes T Vs Fs σ sl þ σ lv 1 ¼ 1 l ð27Þ T0 Fl Δs Hmb r

nanoparticles: T ¼

T 2σsl Vs ¼1 þ s T0 Δ l H m rs

and ∂Al ¼0 ∂nl T, p

ð32Þ

The heat of solidification Δsl Hm < 0, therefore, the freezing temperature decreases with the particle size decreasing, by which one can explain the phenomena of supercooled liquid. 3.3. Application to Condensation. When a great deal of vapor begins to condense, the radius of the liquid droplet formed is tiny; let the radius be rl, then ∂Al 2Vl ¼ ð33Þ ∂nl T, p rl Consequently, one can obtain the accurate relationship between the condensing temperature and the radius of the droplet: T ¼

Δlg Hmb Δlg Sbm

þ

2σlv Vl Δlg Sbm rl

ð34Þ

Similarly, eq 34 can be simplified to

ð28Þ

Equation 28 is similar to the equations for the fusion of nanowire derived by Gulseren et al.55 and Sankaranarayanan et al.56,57 with the assumption that Fs ≈ Fl. It can be seen from eq 28 that the melting temperature drops with the radius of nanowire decreasing, which has been confirmed by a lot of experiments and molecular dynamics simulations.5861 However, compared eq 28 with eq 15, the ratio of depression of melting temperature for a spherical nanoparticle and a nanowire in the same radius is 2:1, and such ratio was also obtained by other researchers’ equations.5557 The factor of 2 arises because of the change in the partial molar surface areas when the nanosized substance goes from a spherical cluster to a wire, which is consistent with the changes in curvature and Laplace pressure.56,57 The approximate ratio 2 is in good agreement with molecular dynamics simulations and/or experimental data for both pure metallic and bimetallic nanocrystals.5558 Moreover, the phase transition equation can also be applied to other tiny particles with regular shapes, such as the tetrahedral particles and the cubic particles. 3.2. Application to Solidification. Let a tiny solid particle crystallized from a large quantity of liquid be spherical. The solid particle is surrounded by the liquid, so the solid particle is a disperse phase and the liquid is a continue phase. Thus, ∂As 2Vs ¼ ð29Þ ∂ns T, p rs

ð31Þ

When the radius of the solid particle is bigger, the difference between T and T0 is little, and then ΔlsHbmΔlsSbm ≈ T0; hence eq 31 can be simplified to

At the melting process of a general substance, Fs ≈ Fl, then eq 27 can be changed into T0 T σsl Vs ¼ l T0 Δs Hm r

Δsl Hmb 2σsl Vs þ s b Δsl Sbm Δl Sm rs

T 2σ lv Vl ¼1 þ l b T0 Δg Hm rl

ð35Þ

The heat of condensation is negative (i.e., ΔlgHm < 0), therefore, the condensing temperature also decreases with the decrease in the particle size of droplet. The relation can be applied to explain the phenomena of the supersaturated vapor. In the same way, for the desublimation between a great deal of vapor and small solid particles, one can obtain equations and conclusions similar to those above. 3.4. Application to Vaporization. The initial stage of ebullience begins with the formation of tiny bubbles in the liquid. When the liquid and the tiny bubbles are in equilibrium, the corresponding temperature is the bubble point temperature. Let the radius of the bubbles be rg, then ! ∂Ag 2Vg ¼ ð36Þ ∂ng rg T, p

where the negative sign is because of the radius of the bubbles, rg < 0. In accordance with eq 6, the accurate relation between the bubble point temperature and the radius of the bubbles can be obtained: T ¼

g 2σlv Vg Δl Hmb g b g b Δl Sm Δl Sm rg

ð37Þ

Approximately, 2σlv Vg T ¼ 1 g b T0 Δl Hm rg

ð30Þ

Applying eqs 29 and 30 to eq 6, one can obtain an accurate equation between the freezing temperature and the size of

ð38Þ

For the vaporization, Δgl Hm > 0, but rg < 0, thus the bubble point temperature increases with the absolute value of the radius 22799



ARTICLE

of the bubbles decreasing, which can be applied to explain the phenomena of superheated liquid. When the absolute value of radius of bubbles is bigger, σlv for general liquid is on the order of 102 J 3 m2; Vg = 102 m3 3 mol1; Δgl Hm = 104 J 3 mol1. Thus, for |r| = 106 m, 107 m, and 108 m, the ratios ΔT/T0 are on the orders of 102, 101, and 100, respectively. It is obvious that the effect of size on bubble point temperature is considerable only when |r| is less than 107 m. Furthermore, for the vaporization of a droplet one can obtain T 2σlv Vl ¼ 1 g b ð39Þ T0 Δl Hm rl It can be seen from eq 39 that the temperature of vaporization for the tiny droplet decreases with the radius of the droplet decreasing. That is to say, the smaller the droplet is, the easier the vaporization occurs. In the same way, for the sublimation between a small solid particle and its vapor, one can obtain an equation and conclusion similar to those of the vaporization of the droplet. To sum up the above equations and conclusions, if the radius of the dispersed phase is more than zero (for solid particles and droplets), the phase transition temperature decreases as the radius decreases; however, if the radius of the dispersed phase is less than zero (for bubbles), the phase transition temperature increases as the absolute value of the radius decreases.

4. THE DETERMINATE RELATIONS BETWEEN INTERFACIAL TENSIONS AND RADIUS Applying the above equations to accurately calculate phase transitions temperatures, the relations between the interfacial tensions and the radius of a droplet or a particle must be obtained. The monomolecular layer model of the surface phase of a microdroplet is proposed, several basic assumptions for the monolayer model are as follows: (1) The microdroplet consisting of one kind of substance is spherical, and the radius of the microdroplet is rl. (2) The surface phase is monolayer where atoms (or molecules) arrange tightly and regularly; the space utilization ratio of the atoms (or the molecules) in the monolayer is the same as that in the droplet. (3) The outer surface of the monolayer is the liquidgas interface; the two phases consist of the same substance. For the droplet, the surface area A = 4πr2l , and 4 ð40Þ NA πra 3 ¼ uVl 3 where NA is Avogadro’s constant, ra is the atomic (or molecular) radius, u is the space utilization ratio of atoms (or molecules) in the droplet. The gross mass of surficial atoms (or molecules) on the droplet m0 is 4 3 4 πrl πðrl 2ra Þ3 3 3 m0 ¼ u 3 ð41Þ 3M 4 3 πra NA 3 where M is the molar mass of the substance. The surface density Γ of the substance on the boundary between the two phases is m0 r 3 ðrl 2ra Þ3 ¼ u3 l M Γ¼ A 4πrl2 ra 3 NA 3

ð42Þ

According to the common definition of the Tolman parameter δ, one can obtain δ¼

Γ r 3 ðrl 2ra Þ3 M ¼ u 3 l2 3 Fl Fg 4πrl ra NA ðFl Fg Þ 3

Applying eq 43 to the Gibbs equation,62 dσ lv 2δ 1 d ¼ 1 þ ð2δ=rl Þ rl σ lv

ð43Þ

ð44Þ

and integrating the equation from the bulk phase (rl f ∞) to nanophase (rl), one can obtain the relation between the surface tension and the radius of the droplet, σ lv ¼ σ l, ∞ expðσl1 þ σ l2 þ σl3 Þ

ð45Þ

where

1 1 1 þ σ l1 ¼ þ 3 6ra X 1=3 12ra 2 X 2=3

1 1 þ X 1=3 rl 2ra ln 1 þ X 1=3 2ra

ð46Þ ! 1 1 1 σ l2 ¼ 3 12ra X 1=3 24ra2 X 2=3 1 1 2 1 1=3 1 X þ X 2=3 rl 2ra rl 2ra ln 1 1 1=3 þ X þ X 2=3 4ra2 2ra

ð47Þ

1 1 1 0 2 1 1 1=3 1=3 2 1=3 X þ X C 2ra 4ra X B r Barctanrl prffiffiaffi σl3 ¼ pffiffiffi 1=3 þ arctan apffiffiffi 1=3 C @ A 1=3 3X 3X 3X

ð48Þ X ¼

πNA ðFα Fβ Þ 1 þ 3 8ra 4uM

ð49Þ

In eq 45, σl∞ is the surface tension of the planar liquid, which can be calculated by the following formula:63a T n1 σ l, ∞ ¼ A1 1 ð50Þ Tc where A1 and n1 are parameters, and Tc is the critical temperature of the substance. Equation 45 is the determinate equation that describes the relationship between the liquidvapor interfacial tension, the temperature, and the radius of a droplet. Applying the expression of liquidvapor interfacial tension (eq 45) to Tanaka et al.’s equation,64 the expression of solid vapor interfacial tension can be obtained: T n1 1 T T0 σ sv ¼ 1:25σlv A1 n1 1 Tc Tc expðσl1 þ σ l2 þ σl3 Þ

ð51Þ

Then, the expression of solidliquid interfacial tension can be obtained by Young’s equation, σsl = σsv σlv cos θ, where θ is 22800



ARTICLE

(σl/σ∞ = 1/(1 + 2δ/rl)), and the Gibbs equation65 (σl/σ∞ = exp(2δ/rl)) at T = 1000 K, respectively. The calculated results are shown in Figure 3. It is obvious in Figure 3 that when the radius of the nanodroplet decreases, the surface tension decreases. However, only when r < 10 nm does the effect of size on surface tension become notable; when rl f ∞, σlv/σ∞ f 1, namely, σlv tends to the surface tension of the planar interface of the liquid. The results calculated by eq 45 are contained between those calculated by Tolman’s approximate equation and Gibbs’ approximate equation, and the calculated results are quite close to each other.

the contact angle between the liquidvapor and the solidliquid interfaces. Reiss et al.11 have proved that when liquid and solid are the same material and are in equilibrium, the contact angle θ is zero and σsl = σsv σlv. Thus, one can obtain the expression of solidliquid interfacial tension as follows: T n1 1 T T0 σsl ¼ 0:25 σlv A1 n1 1 Tc Tc expðσl1 þ σ l2 þ σl3 Þ

ð52Þ

Equation 52 can be applied to calculate the solidliquid interfacial tension of a small particle surrounded by the liquid shell. In order to visually explore the effect of size on surface tension, we take nano-Au for instance and calculate the surface tensions of Au droplets with different sizes by eq 45, the Tolman equation42

5. THE CALCULATED RESULTS OF MELTING TEMPERATURE Applying the above equations, one can predict the phase transition temperature of nanosystems. In the following part, taking the fusion of Au and Sn nanoparticles as examples, we calculate the melting temperature of the nanoparticles with different sizes and make comparison with available experimental data. The Calculated Details. A lot of experimental5,7,20,66 and molecular dynamics simulation results67 have indicated that the melting of nanoparticles begins on the surface. Therefore, we take the thickness of the liquid shell t = 0 when we calculate the onset melting temperature, and such treatment agrees with that of Zhang et al.68 Considering the effects of temperature on melting enthalpy, melting entropy, molar volume and density, and the effects of temperature and size on interfacial tensions, the melting temperatures of Au and Sn nanoparticles with different sizes are calculated. The corresponding formulas and parameters are listed in Table 1. The calculated melting temperatures and the available experimental data of Au and Sn nanoparticles are shown in Figures 4 and 5.

Figure 3. The relation between surface tension and radius of Au at T = 1000 K.

Table 1. The Formulas and the Parameters Used in the Calculation of the Melting Temperatures calculated items melting temperature T

formulas for the calculation

parameters ΔβαHm(T0), ΔβαSm(T0), Fl, Fs, Vl, Vs, σlv, σsl

eq 12 at t = 0

melting enthalpy ΔβαHm Δβα Hm

¼ Δβα Hm ðT0 Þ þ

Z

T

ðCp, m ðβÞ Cp, m ðαÞÞdT

T0

ΔβαHm(T0);63b a, b, c, d63c

¼ a þ bT þ cT 2 þ dT 3 } Cp, m melting entropy ΔβαSm Δβα Sm

¼ Δβα Sm ðT0 Þ þ

Z

T

T0

Cp, m ðβÞ Cp, m ðαÞ dT T

¼ a þ bT þ cT 2 þ dT 3 }

ΔβαSm(T0);63b a, b, c, d63c

Cp, m density of liquid Fl

Fl ¼ A2 B2 ð1 T=Tc Þ

n2

A2, B2, n2, Tc63d

density of solid Fs

obtained by fitting the densities of the solid in different

F69

molar volume Vl, Vs

temperatures V = (M/F)

M63e

liquidvapor interfacial tension σlv eqs 4550

A1, n1, Tc63f

solidliquid interfacial tension σsl

A1, n1, Tc63f

eqs 4552 22801



ARTICLE

their vapor), sublimation (transition from solid particles to their vapor) and desublimation (transition from a great deal of vapor to solid particles) decrease with the size of dispersed phases decreasing. On the contrary, the bubble point temperature of a liquid increases with the absolute value of the radius of the bubbles in liquid decreasing. The depression of melting temperature for a nanowire is half of that for a spherical nanoparticle with identical radius. In summary, a phase transition temperature decreases with the radius of dispersed phase decreasing.

’ AUTHOR INFORMATION Corresponding Author

*Tel (Fax): +86-351- 6014476. E-mail: [email protected]. Figure 4. The melting temperatures of Au nanoparticles with different radii.

’ ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China (No. 20773092) and by the Program for the Top Science and Technology Innovation Teams of Higher Learning Institutions of Shanxi Province of China. ’ REFERENCES

Figure 5. The melting temperatures of Sn nanoparticles with different radii.

Figures 4 and 5 indicate that the melting temperatures of Au and Sn nanoparticles calculated by the precise equation are in good consistency with the corresponding experimental data.13,70,20,38 Even in the lower particle size (radius smaller than 5 nm) of Au, the calculated values agree well with the experimental data. In addition, applying these equations, the other phase transition temperatures of nanosystems are also able to be calculated.

6. CONCLUSIONS The phase transition equation can be applied to various kinds of phase transitions of dispersed systems, and the precise equations for the phase transitions can be obtained by the phase transition equation. The determinate relations between the interfacial tensions and the radius of a droplet and a particle are also derived by the monolayer model. The phase transition equations are able to be applied to predict the phase transition temperatures of dispersed systems and to explain the phenomenon of metastable states. The size of a dispersed phase in a dispersed system has noticeable effect on phase transition temperature. All temperatures of fusion, solidification (transition from a large quantity of liquid to solid particles), condensation (transition from a great deal of vapor to droplets), vaporization (transition from tiny droplets to

(1) Luo, E. Z.; Cai, Q.; Chung, W. F.; Altman, M. S. Appl. Surf. Sci. 1996, 92, 331–334. (2) Zhao, M.; Zhou, X. H.; Jiang, Q. J. Mater. Res. 2001, 16, 3304–3308. (3) Qi, W. H.; Wang, M. P. Chem. Phys. 2004, 88, 280–284. (4) Liu, X.; Yang, P.; Jiang, Q. Mater. Chem. Phys. 2007, 103, 1–4. (5) Couchman, P. R.; Jesser, W. A. Nature. 1977, 269, 481–483. (6) Skripov, V. P.; Koverda, V. P.; Skokov, V. N. Phys. Status Solidi A 1981, 66, 109–118. (7) Xue, Y. Q.; Zhao, Q. S.; Luan, C. H. J. Colloid Interface Sci. 2001, 243, 388–390. (8) Pawlow, P. Z. Phys. Chem. 1909, 65, 545–548. (9) Hanszen, K. J. Z. Phys. 1960, 157, 523–553. (10) Rie, E. Z. Phys. Chem. 1923, 104, 354–362. (11) Reiss, H.; Wilson, I. B. J. Colloid Sci. 1948, 3, 551–561. (12) Curzon, A. Ph.D. Thesis, University of London, London,1959. (13) Buffat, P.; Borel, J. P. Phys. Rev. A 1976, 13, 2287–2297. (14) Borel, J. P. Surf. Sci. 1981, 106, 1–9. (15) David, T. B.; Lereah, Y.; Deutscher, G.; Kofman, R.; Cheyssac, P. Philos. Mag. A 1995, 71, 1135–1143. (16) Castro, T.; Reifenberger, R. Phys. Rev. B 1990, 42, 8548–8556. (17) Mitome, M. Surf. Sci. 1999, 442, L953–L958. (18) Kimberly, D.; Dhanasekaran, T.; Zhang, Z. Y.; Meisel, D. J. Am. Chem. Soc. 2002, 124, 2312–2317. (19) Zhao, S. J.; Wang, S. Q.; Cheng, D. Y.; Ye, H. Q. J. Phys. Chem. B 2001, 105, 12857–12860. (20) Lai, S. L.; Guo, J. Y.; Petrova, V.; Ramanath, G.; Allen, L. H. Phys. Rev. Lett. 1996, 77, 99–102. (21) Rose, J. H.; Ferrante, J.; Smith, J. R. Phys. Rev. Lett. 1981, 47, 675–678. (22) Guinea, F.; Rose, J. H.; Smith, J. R.; Ferrante, J. Appl. Phys. Lett. 1984, 44, 53–55. (23) Safaei, A. J. Nanopart. Res. 2010, 12, 759–776. (24) Safaei, A. J. Phys. Chem. C 2010, 114, 13482–13496. (25) Lu, H. M.; Li, P. Y.; Cao, Z. H.; Meng, X. K. J. Phys. Chem. C 2009, 113, 7598–7602. (26) Lu, H. M.; Han, F. Q.; Meng, X. K. J. Phys. Chem. B 2008, 112, 9444–9448. (27) Qi, W. H. Phys. B 2005, 368, 46–50. (28) Safaei, A.; Shandiz, M. A.; Sanjabi, S.; Barber, Z. H. J. Phys. Chem. C 2008, 112, 99–105. (29) Shandiz, M. A.; Safaei, A. Mater. Lett. 2008, 62, 3954–3956. 22802


The Journal of Physical Chemistry C (30) Mirjalili, M.; Vahdati-Khaki, J. J. Phys. Chem. Solids 2008, 69, 2116–2123. (31) Jiang, Q.; Zhang, S.; Zhao, M. Mater. Chem. Phys. 2003, 82, 225–227. (32) Jiang, Q.; Zhang, Z.; Wang, Y. W. Mater. Sci. Eng., A 2000, 286, 139–143. (33) Zhu, Y. F.; Lian, J. S.; Jiang, Q. J. Phys. Chem. C 2009, 113, 16896–16900. (34) Allen, G. L.; Bayles, R. A.; Gile, W. W.; Jesser, W. A. Thin Solid Films 1986, 144, 297–308. (35) Kim, H. K.; Huh, S. H.; Park, J. W.; Jeong, J. W.; Lee, G. H. Chem. Phys. Lett. 2002, 354, 165–172. (36) Li, Y. J.; Qi, W. H.; Huang, B. Y.; Wang, M. P.; Xiong, S. Y. J. Phys. Chem. Solids. 2010, 71, 810–817. (37) Chernyshev, A. P. Mater. Lett. 2009, 63, 1525–1527. (38) Wronski, C. R. J. Appl. Phys. 1967, 18, 1731–1737. (39) Bachels, T.; Guntherodt, H. J.; Schafer, R. Phys. Rev. Lett. 2000, 85, 1250–1253. (40) Xiao, S. F.; Hu, W. Y.; Yang, J. Y. J. Chem. Phys. 2006, 125, 184504/1184504/4. (41) Bogdan, A.; Kulmala, M. J. Aerosol Sci. 1995, 26, 335–336. (42) Samsonov, V. M.; Shcherbakov, L. M.; Novoselov, A. R.; Lebedev, A. V. Colloids Surf. 1999, 160, 117–121. (43) Samsonov, V. M.; Sdobnyakov, N. Y.; Bazulev, A. N. Surf. Sci. 2003, 532535, 526–530. (44) Xue, Y. Q.; Yang, X. C.; Cui, Z. X.; Lai, W. P. J. Phys. Chem. B 2011, 115, 109–112. (45) Xue, Y. Q.; Gao, B. J.; Gao, J. F. J. Colloid Interface Sci. 1997, 191, 81–85. (46) Shibuta, Y.; Watanabe, Y.; Suzuki, T. Chem. Phys. Lett. 2009, 475, 264–268. (47) Lubashenko, V. V. J. Nanopart. Res. 2010, 12, 1837–1844. (48) Mottet, C.; Goniakowski, J.; Rossi, G.; Ferrando, R.; Baletto, F. Ann. Chim. 2005, 30, 303–313. (49) Shi, F. G. J. Mater. Res. 1994, 9, 1307–1313. (50) Mastny, E. A; de Pablo, J. J. J. Chem. Phys. 2007, 127, 104504/ 1104504/8. (51) Guisbiers, G.; Buchaillot, L. J. Phys. Chem. C 2009, 113, 3566–3568. (52) Vladimir, M. F.; Edgar, D. Z. J. Non-Cryst. Solids 2000, 265, 105–112. (53) Samsonov, V. M.; Sdobnyakov, N. Y.; Bazulev, A. N. Colloids Surf., A 2004, 239, 113–117. (54) Nanda, K. K.; Sahu, S. N.; Behera, S. N. Phys. Rev. A 2002, 66, 013208/1013208/8. (55) Gulseren, O.; Ercolessi, F.; Tosatti, E. Phys. Rev. B 1995, 51, 7377–7380. (56) Sankaranar, S. K. R. S.; Bhethanabotla, V. R.; Joseph, B. Phys. Rev. 2006, 74, 155441/1- 155441/12. (57) Sankaranar, S. K. R. S.; Bhethanabotla, V. R.; Joseph, B. J. Phys. Chem. C 2007, 111, 2430–2439. (58) Kim, E. H.; Lee, B. J. Met. Mater. Int. 2009, 15, 531–537. (59) Zhang, W. X.; He, C. J. Phys. Chem. C 2010, 114, 8717–8720. (60) Miao, L.; Bhethanabotla, V. R.; Joseph, B. Phys. Rev. B 2005, 72, 134109/112. (61) Hui, L.; Wang, B. L.; Wang, J. L.; Wang, G. H. J. Chem. Phys. 2004, 121, 8990–8996. (62) Gibbs, J. W.. Collected Works; Longmans Green and Company: New York, 1928; Vol. I, pp 219ff. (63) Yaws, C. L. Chemical Properties Handbook; McGraw-Hill Book Co.: Beijing, 1999; pp (a) 212; (b) 154, 157; (c) 56, 83, 78, 104, 81, 107; (d) 185, 207, 210; (e) 154, 157; (f) 212, 234, 237. (64) Tanaka, T.; Har, S. Z. Metallkd. 2001, 92, 1236. (65) Schmelzer, J. W. P.; Gutzow, I.; Schmelzer, J., Jr. J. Colloid Interface Sci. 1996, 178, 657–665. (66) Sheng, H. W.; Lu, K.; Ma, E. Acta Mater. 1998, 46, 5195–5205. (67) Zhao, S. J.; Wang, S. Q.; Ye, H. Q. J. Phys. Soc. Jpn. 2001, 70, 2953–2957.

ARTICLE

(68) Zhang, Y. N.; Wang, L.; Bian, X. F. Acta Phys.-Chim. Sin. 2003, 19, 35. (69) Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’Handbook, 6th ed.; McGraw- Hill: New York, 1984; pp 3128 (Chinese version). (70) Sambles, J. R. Proc. R. Soc. London, A 1971, 324, 339–351.

22803


Thermodynamics of Size Effect on Phase Transition Temperatures of

Recommend Documents