The phase transition thermodynamics theory of nanoparticles

Powder Technology 308 (2017) 258–265

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Size dependence of phase transition thermodynamics of nanoparticles: A theoretical and experimental study

Wenjuan Zhang, Yongqiang Xue ⁎, Qingshan Fu, Zixiang Cui, Shuting Wang Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan 030024, China

⁎ Corresponding author. E-mail address: xyqlw@126.com (Y. Xue).

http://dx.doi.org/10.1016/j.powtec.2016.11.052 0032-5910/© 2016 Published by Elsevier B.V.

a b s t r a c t

a r t i c l e i n f oArticle history: Received 26 August 2016 Received in revised form 17 November 2016 Accepted 29 November 2016 Available online 9 December 2016

The phase transitions of nanoparticles are involved in almost every field, which present amazing difference com- paredwith the corresponding bulkmaterials. Indeed despite extensive studies into phase transition temperature, little is known about the relationships between the temperature at the maximum rate of phase transition, the phase transition enthalpy, the phase transition entropy and the particle size. Hence, it is urgent to complete the size dependence of phase transition thermodynamics of nanoparticles. In this paper, the general equation of thermodynamic properties of phase transitions for nanoparticleswas presented. Then the relations of the ther- modynamic properties of crystal transition and the particle sizewere derived based on a thermodynamicsmodel of crystal transition. The theoretical results indicate that the particle size of nanoparticles can remarkably influ- ence the phase transition thermodynamics: with the decreasing particle size, the phase transition temperature, the temperature at themaximum rate of phase transition, the phase transition enthalpy and the phase transition entropy decrease, which are linearly related to the reciprocal of particle size. In experiment, the phase transitions from tetragonal to cubic of nano-BaTiO3 with different sizes were determined by means of Differential Scanning Calorimetry (DSC); then the regularities of influence of particle size on the phase transition thermodynamics were obtained. The experimental results are consistentwith the above relations. The phase transition theory pro- vides a quantitative description of phase behavior of nanoparticles.

© 2016 Published by Elsevier B.V.

Keywords: Nano-BaTiO3 Crystal transition Size dependence Thermodynamics

1. Introduction

The phase transition plays a central role in awide variety of chemical processes. Consideration of phase transitions has typically focused on solid –liquid phase transition [1–3], whereas relatively little attention has been paid to the question of size dependence of crystal phase tran- sition. Although there exist experimental data on the phase transition behavior [4,5], little is known about the quantitative relationships be- tween the thermodynamic properties of crystal phase transition of nanoparticles and the particle size. Therefore, study on thermodynam- ics of crystal phase transition in nanoscale is vital from the theoretical as well as the practical point of view, which can provide theoretical and practical value for the control of the crystalline phase and the fur- ther development of new phase transition materials.

Presently, there are some studies devoted to investigating the parti- cle size effects on the crystal transition of nanoparticles. Zhong et al. [6– 9] discussed the size-driven phase transition of BaTiO3 and PbTiO3 by using a Landau-type phenomenological theory and the results show that phase transition temperature, heat and latent heat decrease with particle sizes decrease; the phase transition entropy was obtained by

ΔS=ΔQ/Tc. Köferstein et al. [10] studied the phase transition enthalpy from tetragonal to cubic for CuFe2O4 and the results suggest that the phase transition enthalpy decreases with the decrease of particle size: ranges from 1020 J·mol−1 of 36 nm to 1229 J·mol−1 of 96 nm. Prabhu et al. [11] studied the phase transition temperature from tetragonal to cubic for CuFe2O4 (15 nm, 50 nm and bulk) and the results indicate that the phase transition temperature decreases with the decrease of particle size. Jiang et al. [12] studied the phase transition entropy from tetragonal to cubic for nano-PbTiO3 and the results demonstrate that the phase transition entropy decreases with the decrease of particle size.

Nevertheless, the theory of phase transition thermodynamics of nanoparticles and the quantitative regularities of influence of particle size on crystal transition thermodynamic properties have not been re- ported yet.

In this account, our group presents a general theory of phase transi- tion, developed over the past decade. In this paper, the general equation of phase transition thermodynamics of nanoparticles was derived by defining the surface chemical potential and the relations between ther- modynamic properties of crystal transition and particle size were de- rived based on a thermodynamics model of crystal transition of nanoparticles. Furthermore, the theoretical relationship of temperature at the maximum rate of phase transition and particle size was derived

Fig. 1. Phase transition model.

259W. Zhang et al. / Powder Technology 308 (2017) 258–265

for the first time. In experiment, the crystal transition of nano-BaTiO3 was taken as a system, the regularities of particle size effect on the phase transition thermodynamic quantities were summarized, respectively.

2. The phase transition thermodynamics theory of nanoparticles

The chemical potential of a dispersed phase is composed of that of the bulk phase and the surface phase, it is shown as follows,

μ ¼ μb þ μs ð1Þ

And the surface chemical potential was defined as [13],

μs ≡ ∂Gs

∂n

� � T;p

¼ σ ∂A ∂n

� � T;P

ð2Þ

where σ, A and n are the surface tension, the surface area, and the amount of substance of the dispersed phase, respectively.

When dispersed phase α of pure substance turns into dispersed phase β, the change in molar Gibbs energy can be written as,

ΔβαGm ¼ μβ−μα ¼ ΔβαGbm þ σβ ∂Aβ ∂nβ

� � T;p

−σα ∂Aα ∂nα

� � T ;p

ð3Þ

where ΔαβGmb is the change in molar Gibbs energy of the phase transi- tions for the bulk substance from phase α to β, i.e. ΔαβGmb =μβb−μαb .

Applying the Gibbs-Helmholtz equation to phase transition, the Eq. (4) can be obtained,

∂ ∂T

ΔβαGm T

!” # p

¼ −Δ β αHm T2

ð4Þ

Substituting Eq. (3) into Eq. (4), the general equation of phase tran- sition enthalpy can be obtained,

ΔβαHm ¼ ΔβαHbm þ ∂Aβ ∂nβ

� � T ;p

σβ−T ∂σβ ∂T

� � p

” # −Tσβ

∂ ∂T

∂Aβ ∂nβ

� � T ;p

” # p

− ∂Aα ∂nα

� � T ;p

σα−T ∂σα ∂T

� � p

” # þ Tσα ∂∂T

∂Aα ∂nα

� � T ;p

” # p

ð5Þ

where ΔαβHmb is the molar enthalpy of the phase transitions of bulk substance.

Taking the partial derivative of thermodynamic basic formula against T, the phase transition entropy can be expressed as,

ΔβαS ¼ − ∂ΔβαG ∂T

! p

ð6Þ

Substituting Eq. (3) into Eq. (6), the general equation of phase tran- sition entropy can be derived as follow,

ΔβαSm ¼ ΔβαSbm− ∂Aβ ∂nβ

� � T;p

∂σβ ∂T

� � p −σ l

∂ ∂T

∂Aβ ∂nβ

� � T ;p

” # p

þ ∂Aα ∂nα

� � T;p

∂σα ∂T

� � p þ σα ∂∂T

∂Aα ∂nα

� � T ;p

” # p

ð7Þ

where ΔαβSmb is the molar entropy of the phase transitions of bulk substance.

When the two phases in the dispersed system are in equilibrium, ΔαβGm=0, Thus,

ΔβαG b m ¼ σα

∂Aα ∂nα

� � T;p

−σβ ∂Aβ ∂nβ

� � T;p

ð8Þ