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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).

a b s t r a c t

a r t i c l e i n f o
Article 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.

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

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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

At the phase transition temperature, the relation of the thermody- namic properties of the phase transition for bulk phase is

ΔβαGm ¼ ΔβαHm−TΔβαSm ð9Þ

A general equation of phase transition temperature Tc can be obtain- ed by the simultaneous Eqs. (8) and (9),

Tc ¼ Δ β αH

b m

ΔβαS b m

þ 1 ΔβαS

b m

σβ ∂Aβ ∂nβ

� � T ;p

−σα ∂Aα ∂nα

� � T;p

” # ð10Þ

It can be seen from the Eq. (10) that the phase transition tempera- ture of a dispersed system depends on not only the properties of the bulk phase (ΔαβHmb and ΔαβSmb ) but also the properties of the surface phase (the interfacial tensions and the specific surface areas of the two phases).

Based on the general equation of phase transition of nanoparticles and in combination of a thermodynamics model of crystal transition, the relations of the thermodynamic properties of the crystal transition with the particle size were derived, respectively.

The crystal transition begins on the surface of nanoparticles [14]. As- sume that the phase transition shell (β phase) with width t surrounds uniformly the solid core. When the phase transition is in equilibrium, the radii of the solid core and the phase transition shell are rα and rβ, re- spectively. The schematic diagram of the crystal transition model is as follows (Fig. 1).

For spherical nanoparticles,

∂Aα ∂nα

� � T ;p

¼ 2Vα rα

ð11Þ

where 43πr 3 αρα þ 43πt3ρβ ¼ 43πr3βρα is the molar volume of α phase of

nanoparticles. The total mass in the phase transition process is constant, therefore

4 3 πr3αρα þ

4 3 π r3β−r

3 α

� � ρβ ¼

4 3 πr3ρα ð12Þ

where r is the radius of nanoparticle before phase transition, ρα and ρβ are the densities of the solid core and the phase transition shell, respectively.

Since

Aβ ¼ 4πrβ2 ð13Þ nβ ¼ 43π r

3 β−r

3 α

� � =Vβ ð14Þ

where Vβ is the molar volume of β phase of nanoparticles.

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

The partial derivative of Aβ against nβ can be obtained by simulta- neous Eqs. (12), (13) and (14)

∂Aβ ∂nβ

� � T;p

¼ 2Vβ rβ

1− ρβ ρα

� � ð15Þ

Taking the partial derivative of Eqs. (11) and (15) against T, respec- tively, the following equation can be derived,

∂ ∂T

∂Aα ∂nα

� � T ;p

” # p

¼ 4Vαα 3rα

ð16Þ

∂ ∂T

∂Aβ ∂nβ

� � T;p

” # p

¼ −2M 1 r2β

∂rβ ∂T

� � 1 ρβ

− 1 ρα

! þ 1 rβ

αα ρα

− αβ ρβ

!” # ð17Þ

For a general substance, ð∂Aβ∂nβÞT;p ¼ 0, Eqs. (15) and (17) can be ap- proximated as,

∂Aβ ∂nβ

� � T;p

¼ 0 ð18Þ

∂ ∂T

∂Aβ ∂nβ

� � T;p

” # p

¼ 0 ð19Þ

The phase transition enthalpy can be derived by substituting Eqs. (11), (16), (18), and (19) into the Eq. (5),

ΔβαHm ¼ ΔβαHbm− 2Vα 3rα

σα 1− 2 3 Tα

� � −T

∂σα ∂T

� � P

� � ð20Þ

For the general materials, (∂σ/∂T)pb0, and the order is 10−4 [15], the order of α is 10−5 [16,17], so the value in the square bracket in Eq. (20) is positive. Hence, the smaller the nanoparticle size, the smaller the phase transition enthalpy. And the phase transition enthalpy ex- hibits a good linear relationship with the reciprocal of the particle radius.

The phase transition entropy can be derived by substituting Eqs. (11), (16), (18), and (19) into the Eq. (7),

ΔβαSm ¼ ΔβαSbm þ 2Vα rα

∂σα ∂T

� � P þ 2 3 σαα

� � ð21Þ

For the common materials, the order of σα is 10−1– 100 [18], the order of σαα is 10−6– 10−5, which is far less than the order of (∂σα/ ∂T)P, so the value in the square bracket in Eq. (21) is negative. It is obvi- ous that the phase transition entropy decreases with the particle size decreases. The phase transition entropy and the reciprocal of the parti- cle radius have a line relationship.

Applying Eqs. (11) and (15) to Eq. (10), one gets,

Tc ¼ Δ β αH

b m

ΔβαS b m

− 2Vα ΔβαS

b m

σα rα

þ σβ rβ

1− ρβ ρα

� �� ð22Þ

When the crystal transformation is just beginning, rβ≈rα, and the initial phase transition temperature can be obtained,

Tc ¼ Δ β αH

b m

ΔβαS b m

− 2Vα

ΔβαS b mrα

σα þ σβ 1− ρβ ρα

� �� ð23Þ

As shown in Eq. (23) the phase transition temperature decreases with the decrease of particle size, and the phase transition temperature exhibits a good linear relationship with the reciprocal of the particle ra- dius. However, when the particle size is smaller (rb10 nm), the effect

of rα on σ becomes notable that the linear relationship may disappear [13].

Ignore the influence of temperature on ΔαβHmb and ΔαβSmb , then,

ΔβαH b m T;Kð Þ

ΔβαS b m T ;Kð Þ

≈ ΔβαH

b m T0;Kð Þ

ΔβαS b m T0;Kð Þ

¼ T0 ð24Þ

where T0 is the phase transition temperature of the corresponding bulk substance.

Eq. (23) can be simplified as,

Tc ¼ T0− 2Vα ΔβαS

b mrα

σα þ σβ 1− ρβ ρα

� �� ð25Þ

In general, Tc ¼ T0− 2σαVαΔβαSbmrα, Eq. (25) can be approximated to

Tc ¼ T0− 2σαVα ΔβαS

b mrα

ð26Þ

3. Experimental section

3.1. Preparation and characterization of nano-BaTiO3 of different sizes

The nano-BaTiO3 was synthesized by sol-gel hydrothermal method [19,20]. An amount of Ba(OH)2·8H2O was dissolved in the acetic acid (labeled A solution), and a stoichiometric Ti(OC4H9)4 was dissolved in the absolute alcohol (labeled B solution). The B solution was added slowly into A solution with a speed magnetic stirrer and water bath temperature (20 °C –60 °C), then ultrafine sol was obtained. The sol was aged 24 h to gel (precursor) at room temperature. The precursor was dried at 80 °C and sintered at 600 °C for 2 h at a heating rate of 5 °C/min in air atmosphere andwas taken out of the furnace for cooling naturally. After being grinded, the sample was separated into themixed solution of dimethylacetamide and ethanol for hydrothermal reaction. After washing and drying and changing the reaction conditions, the nano-BaTiO3 with different sizes was obtained. Barium hydroxide, acetic acid and absolute ethanol were purchased from Tianjin Wind Ship Chemical Reagent Technology Co., LTD (Tianjing, China). Tetrabutyl titanate was supplied by Tianjin Guangfu Fine Chemical Re- search Institute (Tianjin, China).

The nano-BaTiO3 was characterized using a Germany Bluker D8 Ad- vance Powder Diffractometer (XRD) (Cu Kα, k = 0.154178 nm), and theXRD spectrumwas shown in Fig. 2. Fig. 2 shows the synthesized sam- pleswere pure nano-BaTiO3 of tetragonal phase, which can be proved by the two peaks at (002) and (200) around 45°, while the cubic phase only had one peak (200). The average particle diameters of nano-BaTiO3were obtained from SEM and calculated by Nano Measure [21]. The average particle diameters of nano-BaTiO3 used in this phase transition experi- ment are 66.4, 76.6, 89.8, 96.2, 106.9 and 120.1 nm, respectively.

The SEM (JSM-6701F) pictures presented in Fig. 3 show the obtained nano-BaTiO3 with uniform morphology and particle size was nearly to spherical. The bar charts of size distribution obtained by nanomeasurer software are shown in Fig. 4, and it was observed that the size distribu- tion is homogeneous.

3.2. Phase transition experiment

The thermodynamic properties of phase transition from tetragonal to cubic of nano-BaTiO3 were measured by DSC (Q2000). Approximate- ly 5 mg nano-BaTiO3 was weighed out accurately and sealed in the alu- minum alloy crucible and placed on the concave of sensor disc. The measurement started from room temperature to 150 °C with a heating rate of 10 °C/min in nitrogen-flow (50 mL/min) atmosphere. Then the heat flow curve, namely DSC curve, was obtained. After the cell

20 30 40 50 60 70 80

66.4 nm

76.6 nm

89.8 nm

96.2 nm

106.9 nm

120.1 nm

ytis net

44.0 44.5 45.0 45.5 46.0 46.5

ytis n

et nI

b (200) (002)

66.4 nm

76.6 nm

89.8 nm

96.2 nm

106.9 nm

120.1 nm

Fig. 2. a The XRD patterns of nano-BaTiO3 with different sizes; b The XRD patterns of nano-BaTiO3 with different sizes from 44° to 46.5°.

261W. Zhang et al. / Powder Technology 308 (2017) 258–265

compartment cooled, the second measurement started. Repeat the DSC experiment with different sizes of nano-BaTiO3 at the same conditions.

3.3. Processing of experimental data

In the DSC curve, the intersection abscissa of the baseline and the tangent before heating effect is the initial phase transition temperature of nano-BaTiO3, namely Tc. The lowest point of absorption peak means the temperature at the maximum rate of phase transition, namely Tm. The intersection abscissa of the baseline and the tangent after heating effect is the end of the phase transition temperature.

A linear equation was obtained by two-point method (the start and end point of the endothermic peak). The area surrounded by the linear equation and the endothermic peak is divided into several parts, and each one is viewed as a small trapezoid. Taking the abscissa data measured in phase transition process into the linear equation, the corre- sponding ordinate value (denoted as y2) was obtained. The difference between the ordinate value of the DSC curve (labeled y1) and y2 is the bottom of the trapezoid, and the difference between the two adjacent abscissas is the height of the trapezoid. According to the trapezoidal

d e

a b

Fig. 3. The SEM images of the nano-BaTiO3 of different sizes (a 66

formula, each small area of a trapezoid can be got. The sum of all the area of the small trapezoid is the phase transition enthalpy. The phase transition enthalpy from tetragonal to cubic for the synthesized sample was obtained, unit for J/g.

ΔH ¼ Z

δQ ¼ X

δQi ð27Þ

Dividing the small area of a trapezoid by the corresponding average temperature and summing all these numbers together, the phase tran- sition entropy from tetragonal to cubic was obtained as follows.

ΔS ¼ Z

δQ T

¼ X δQi

Ti ð28Þ

4. Results and discussion

The thermodynamic properties of structural phase transition of nano-BaTiO3with different sizes aremeasured and calculated according to the DSC curves, shown in Table 1.

.4, b 76.6 nm, c 89.8 nm, d 96.2 nm, e 106.9 nm, f 120. 1 nm).

55 60 65 70 75 80 0

40 a

particle size / d

pe rc

en ta

ge /%

pe rc

en ta

ge /%

pe rc

en ta

ge /%

pe rc

en ta

ge /%

pe rc

en ta

ge /%

pe rc

en ta

ge /%

60 70 80 90 100 0

particle size / d

70 80 90 100 110 120 0

particle size / d

80 85 90 95 100 105 110 115 120 0

particle size / d

96 99 102 105 108 111 114 0

particle size / d

100 110 120 130 140 0

particle size / d

Fig. 4. The bar charts of particle size distribution of nano-BaTiO3 particles of different sizes: (a 66.4, b 76.6 nm, c 89.8 nm, d 96.2 nm, e 106.9 nm, f 120.1 nm).

)g/ W(/

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

4.1. Effect of particle size on phase transition temperature

The thermodynamic properties of phase transition of nano-BaTiO3 were analyzed by using differential scanning calorimetry, and the DSC curves were got by the recorded date. Fig. 5 shows the DSC curves of nano-BaTiO3 of different sizes recorded between 388 and 408 K.

As shown in Fig. 5, the endothermic peak near 400 K is caused by the phase transition from tetragonal to cubic of nano-BaTiO3 of different sizes. There is an obvious endothermic peak for BaTiO3 of 120.1 nm, whereas is not obvious for BaTiO3 of 66.4 nm. The phenomena indicated that the phase transition shift to lower temperature and the phase tran- sition enthalpy decreases with the decreasing particle sizes.

The curves of the phase transition temperature versus the reciprocal of particle radius are shown in Fig. 6.

As shown in Fig. 6, the phase transition temperature decreases with the decrease of particle size, which is in accordancewith the conclusions in some literatures [22–26]. And the phase transition temperature ex- hibits a good linear relationshipwith the reciprocal of the particle radius, which is consistent with the conclusions in some literatures [11,27].

Table 1 The thermodynamic properties of structural phase transition of nano-BaTiO3 vary with size.

No. d/nm d−1/nm−1 Tc/K Tm/K ΔH/J·mol−1 ΔS/J·mol−1·K−1

1 66.4 0.0151 395.49 398.22 15.16 0.0503 2 76.6 0.0131 396.72 399.63 27.91 0.0739 3 89.8 0.0111 397.34 400.56 31.64 0.0997 4 96.2 0.0104 397.43 400.87 47.03 0.1412 5 106.9 0.0094 398.47 401.39 49.67 0.1520 6 120.1 0.0083 399.08 402.15 57.46 0.2113

Furthermore, the experimental regularity of the influence of particle size on the phase transition temperature agrees with the theory analysis of Eq. (23). The influence essence of particle size onphase transition tem- perature is that: the huge surface energy of nanoparticle that involved in the phase transition process reduces the phase transition energy and makes the phase transition happen at lower temperature. When the straight line of phase transition temperature versus the reciprocal of the particle diameters is extrapolated until the particle size tends to in- finity, the phase transition temperature of bulk -BaTiO3 is 403.08 K, which is in good agreement with the value of 403 K found by Zhong et al. [28].

387 390 393 396 399 402 405 408 411

Tc TmExo Up

Temperature/ K

ol F

tae H

66.4 nm

76.6 nm

89.8 nm

96.2 nm

106.9 nm

120.1 nm

Fig. 5. DSC curves of nano-BaTiO3 with different sizes.

0.004 0.005 0.006 0.007 0.008 395

396

397

398

399 T

c/ K 0.004 0.005 0.006 0.007 0.008

39.5

39.6

39.7

39.8

39.9

40.0

r-1/nm-1

10 -1

T c/

Fig. 6. The relation between the reciprocal of the nano-BaTiO3 sizes and the phase transition temperature.

263W. Zhang et al. / Powder Technology 308 (2017) 258–265

The curves of the Tm versus the reciprocal of particle radius are shown in Fig. 7.

It can be seen from Fig. 7 that the temperature at themaximum rate of phase transition decreases rapidly with the decreasing particle size, and there is a linear relationship between the Tm and the reciprocal of the particle size.

Regard the process of phase transition as a special case of the chem- ical reaction, the reaction kinetics theory of nanoparticles also can be applied to the phase transition process. For bulk materials [29],

Ab ¼ CT2 ð29Þ

where

C ¼ e2kR=hpo−eΔ‡S=R ð30Þ k is the rate constant, h is the Planck constant,Δ‡S is the activated entro- py. Ignored the influence of temperature on activated entropy, C is a constant, independent of temperature.

The logarithm form of Eq. (29) is,

lnAb ¼ lnC þ 2 lnT ð31Þ

0.004 0.005 0.006 0.007 0.008

398

399

400

401

402

403

Fig. 7. The relation between the reciprocal of thenano-BaTiO3 sizes and the temperature at the maximum rate of phase transition.

Arrhenius Equation can be expressed as follow,

lnkb ¼ lnAb−Eba=RT ð32Þ

Eqs. (31) and (32) can be combined into,

lnkb ¼ lnC þ 2 lnT− E b a

RT ð33Þ

And [28]

lnk ¼ lnkb þ 2σVm RTr

ð34Þ

Applying Eqs. (33) to (34), one can obtain,

lnk ¼ lnC þ 2lnT− E b a

RT þ 2σVm

RTr ð35Þ

Ignored the influence of temperature on molar volume and surface tension, the derivative of both sides of the Eq. (35) against the temper- ature can be obtained,

d lnk dT

¼ 2 T þ E

b a

RT2 −

2σvm RT2r

ð36Þ

when d(lnk)/dT=0, the temperature at the maximum rate of phase transition can be obtained,

Tm ¼ σvmRr − Eba 2R

ð37Þ

As can be seen from Eq. (37) that the temperature at the maximum rate of phase transition exhibits a linear relationship with the reciprocal of the particle size, which is consist with above experiment result.

4.2. Effect of particle size on phase transition enthalpy

The curves of the phase transition enthalpy versus the reciprocal of particle radius are shown in Fig. 8.

The results presented in Fig. 8 show that the particle size of nanopar- ticle has a notable influence on the phase transition enthalpy. This is be- cause the huge surface energy of nanoparticle participates in the phase transition process which reduces the absorption or release of heat, that is, with the decrease of the particle radius, the surface energy increases and the phase transition enthalpy decreases, and the influence regula- tion is consist with that in melting [30,31], absorption [32,33] and reac- tion [34] process. What’s more, the phase transition enthalpy exhibits a good linear relationship with the reciprocal of nanoparticle radius. The experimental influence regularity of the particle size on the phase tran- sition enthalpy agrees with the theory analysis of Eq. (20).

4.3. Effect of particle size on phase transition entropy

The relation between the phase transition entropy and the reciprocal of particle size can be obtained in Fig. 9.

It can be concluded from Fig. 9 that the decrease of the particle size leads to the decrease in the phase transition entropy, which is consist with that in melting [35,36], reaction [37], decomposition [38] and dis- solution [39] process. With particle size decreasing, the surface defects of nanoparticles becomemore serious, and then the chaos of the system increases that resulting in the phase transition entropy decreases. And the phase transition entropy is linearly related to the reciprocal of par- ticle radius. The experimental influence regularity of the particle size on the phase transition entropy agrees with the theory analysis of Eq. (21).

0.004 0.005 0.006 0.007 0.008 10

Fig. 8. The relation between thephase transition enthalpy and the reciprocal of the particle sizes of BaTiO3 nanoparticles.

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

5. Conclusions

In conclusion, the general equation of phase transition thermody- namics of nanoparticles derived herein provides a foundation and guid- ance for the further research of the phase transition behavior. The Eqs. (20), (21) and (23) were derived based on the above general equation and a thermodynamic model of crystal transition. Furthermore, the re- lation between the temperature at the maximum rate of phase transi- tion and the particle size was interpreted by Eq. (37). The results show that the phase transition enthalpy, the phase transition entropy, the phase transition temperature and the temperature at themaximum rate of phase transition decrease with the decrease of particle size, and are linearly related to the reciprocal of particle size, respectively. This theory can quantitatively describe the size-dependent phase transition behavior of nanoparticles. The experimental results of phase transition for nano-BaTiO3 are consistent well with the above theory relations.

Acknowledgments

We are thankful to the National Natural Science Foundation of China (No. 21373147 and No. 21573157).

0.004 0.005 0.006 0.007 0.008

0.04

0.08

0.12

0.16

0.20

Fig. 9. The relation between the phase transition entropy and the reciprocal of the sizes of nano-BaTiO3.

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Size dependence of phase transition thermodynamics of nanoparticles: A theoretical and experimental study
1. Introduction
2. The phase transition thermodynamics theory of nanoparticles
3. Experimental section
3.1. Preparation and characterization of nano-BaTiO3 of different sizes
3.2. Phase transition experiment
3.3. Processing of experimental data
4. Results and discussion
4.1. Effect of particle size on phase transition temperature
4.2. Effect of particle size on phase transition enthalpy
4.3. Effect of particle size on phase transition entropy
5. Conclusions
Acknowledgments
References

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