Size-Dependent Thermodynamic Properties of Two Types of Phase

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C: Physical Processes in Nanomaterials and Nanostructures

Size-Dependent Thermodynamic Properties of Two Types of Phase Transitions of Nano-Bi2O3 and Their Differences Yuantao Wang, Zixiang Cui, Yongqiang Xue, Rong Zhang, and Aijie Yan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00008 • Publication Date (Web): 10 Jul 2019 Downloaded from pubs.acs.org on July 18, 2019

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The Journal of Physical Chemistry

Size-Dependent Thermodynamic Properties of Two Types of Phase Transitions of Nano-Bi2O3 and Their Differences Yuantao Wang, Zixiang Cui, Yongqiang Xue, Rong Zhang, Aijie Yan Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan, 030024, P. R. China

Abstract Thermostability of nanomaterials has crucial influences on their performances and applications, but the effect regularities of particle size on crystal transition and melting thermodynamics for the same kind of nanoparticle and the relationships of phase transition thermodynamics between the two types of phase transitions are unclear. In this paper, we proposed a core-shell model applied to crystal transition and melting of nanospheres, and on this basis the equations of size-dependent thermodynamic properties for crystal transition and melting were deduced. Experimentally, we prepared nano-Bi2O3 with different particle diameters by hydrothermal method and determined the crystal transition (from phase α to δ) and melting behaviors of nano-Bi2O3 by differential scanning calorimetry (DSC). The experimental results indicate that the effects of particle size on the two types of phase transitions thermodynamic properties of nano-Bi2O3 are dramatic. And we found that the effect regularities of particle size on thermodynamic properties of the two types phase transitions are uniform; with the particle size of nano-Bi2O3 decreasing, the temperatures, enthalpies and entropies of the phase transitions decrease, the thermodynamic properties are all linearly related with the reciprocal of particle diameter, and the experimental results of the two types of phase transitions are all consistent with the theoretical formulas. Surprisingly, we found that the two types of phase transitions are correlative. The differences in temperature and thermodynamic properties of phase transitions between the two types of phase  

Corresponding author. E-mail：[email protected] Corresponding author. E-mail：[email protected]

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The Journal of Physical Chemistry 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

transitions are also size-dependent and linearly related with the reciprocal of particle diameter. The results are consistent with the theoretical analysis. The theory and size-dependent regularities can provide important references for the preparations, researches and applications of nano-Bi2O3 and other nanoparticles.

Introduction As an important p-type semiconductor, nano-Bi2O3 has been extensively applied to various fields including gas sensor,1 supercapacitor,2-4 optoelectronics device,5 and photocatalysts.6-8 Nano-Bi2O3 has six main polymorphic forms, denoted by α-, β-, γ-, δ-, ε- and ω-Bi2O3.9 Among them, α-Bi2O3 is known as the most stable phase at room temperature and applied as the anode material for sodium-ion batteries.10 δ-Bi2O3 is stable at high temperature (between 729 °C and its melting point of 824 °C), and it exhibits the high oxygen ion conductivity due to the high number of oxygen vacancies.11-14 There are two types of phase transitions (crystal transition and melting) existed in Nano-Bi2O3. Guenther et al. investigated the melting of nano-Bi2O3 with the particle sizes between 6 and 50 nm and found a strong melting point reduction (-55% at 6 nm) of nano-Bi2O3.15 Qiu et al. reported the phase transitions of α- or β- to δ- and δ- to liquid phase for Bi2O3 nanowires with corresponding diameters about 100 and 10 nm at high temperature.16 People have discussed universal relationships of material properties at small scales,17-19 and many empirical relations of phase transition had been proposed.20-26 But the effect regularities of particle size on phase transition temperature and phase transition thermodynamic properties of nano-Bi2O3 and the relationship of phase transition thermodynamics between the two types of phase transitions are unclear. Herein, a core-shell model applied to crystal transition and melting of nanospheres was proposed, the equations for size-dependent crystal transition and melting thermodynamic properties (phase transition temperature, enthalpy and entropy) were deduced theoretically. Meanwhile, the influencing mechanism and regularities of particle size on phase transition thermodynamic properties were also discussed. Furthermore, we found that the differences in temperatures, enthalpies and entropies


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between the two types of phase transitions are all size-dependent and show linear relationships with the reciprocal of particle diameter, too. Experimentally, two types of phase transitions of nano-Bi2O3 were chosen as the research system. We prepared nano-Bi2O3 with different particle diameters by hydrothermal method at first, then determined two types of phase transitions behaviors of nano-Bi2O3 by differential scanning calorimetry (DSC) and obtained the phase transitions thermodynamic data of nano-Bi2O3 with different particle diameters. Discussed the effect of particle size on phase transition thermodynamic properties of two types of phase transitions and their differences, and compared with theories finally.

The thermodynamic equations applied to two types of phase transitions of nanospheres The chemical potential of nano-condensed phases is equal to the sum of the bulk chemical potential and the surface chemical potential,27, 28

   σA    G s   A  b b μ  μ b  μs  μ b     μ    μ σ   n T , p  n T , p  n  where  b and  s

(1)

are the bulk chemical potential and the surface chemical

potential, respectively, σ is the surface tension, A is the surface area and n is the amount of substance of a nano-condensed phase. The change in molar Gibbs energy during a phase transition from nano-condensed phase α into nano-condensed phase β can be written as  A   A   Gm  μ  μ   Gmb  σ     σ      n   n T , p   T , p

(2)

where α is the solid phase of a kind of crystal form, β is another crystal form or the melted liquid phase, σ  is the surface tension of phase α and σ is the surface tension of phase β. Herein, a model applied to two types of phase transitions (crystal transition and melting) is presented (see Fig. 1). This is a model for the phase transition from phase α to β of a nanosphere, the phase α is surrounded by generated phase β continuously.



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When the phase transition is in equilibrium, the radii of the phase α core and the phase β shell are r and r , respectively, and the thickness of the shell is t  t  r  r  .

phase β

rβ

rα

phase α

thickness of the shell t

Fig. 1 The core-shell model for two types of phase transitions of a nanosphere

The total mass in the phase transition process is consistent, 4 3 4 4  rα α    rβ3  rα3  β   r 3α 3 3 3

A  4r2

(3) (4)

4 nβ    rβ3  rα3  / Vβ 3

(5)

where r is the radius of the phase α before the phase transition ， V is the molar volume of phase β and ρ is the density. The partial molar surface area of the core (phase α) can be expressed as

 A  2V 2 M     r r ρ  n T , P

(6)

where V is the molar volume of phase α. For the shell (phase β),  Aβ   nβ

 2Vβ  β    1   r T , p β  α 

(7)

When the phase transition is in equilibrium, i.e., μ  μ , Eq. (2) can be written as   Aβ  A  βα Gmb σ α  α  σβ   n   nα T , p  β

  b  b    H m T   S m T , p

(8)

where the  H mb and  S mb are the changes in molar enthalpy and entropy of the


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corresponding bulk substance, respectively. By substituting Eqs. (6) and (7) into Eq. (8), the precise equation of phase transition temperature can be obtained,

 H mb 1  2σgV  ρ  2σ V  T    1     S mb  S mb  r  t  ρ  r 

(9)

where σ g and σ  are the interface tension between phase β and gas and the interface tension between phase α and β, respectively. In the process of the phase transition, the densities of different condensed-phases of the same substance are equal approximatively,29 i.e., ρ  ρ , so Eq. (9) can be written as

 H mb 2σ V T  b   b   S m   S m r

(10)

At the beginning of phase transition, the thickness t can be ignored, i.e., r  r , the relation between phase transition temperature and particle radius at the beginning of phase transition is obtained.

 H mb 2σ V T0   b   b   Sm   Sm r

(11)

Applying the Gibbs-Helmholtz equation to Eq. (2), the general equation of phase transition enthalpy can be obtained,

 H m   H mb 

2V r

  σ   2   σ  T  σ          T  P 3   

(12)

where  is the thermal expansion coefficient. Similarly, on the basis of thermodynamics basic formula,  β G  βα S m    α m   T 

(13)

the phase transition entropy can be expressed as

 S m   S mb 

2V r

 σ  2     σ γ   T  P 3 


(14)


For the majority of substances, the orders of 10-1~100 N·m-1 and 10-5 m3·mol-1,30

 σ



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 , σαβ and V are 10-5 K-1,28

T  p is a negative quantity of the order of

10-4 N·m-1·K-1.31 When the r  10 nm, the effect of size on σ is not significant and the σ can be regarded as a constant approximately.32, 33 It is obvious that the value in brace of Eq. (12) is positive, the phase transition enthalpy decreases with the decreases of particle size consequently. Similarly, the value in square bracket of Eq. (14) is negative, so the phase transition entropy decreases with the decreases of particle size. And the effect regularities of particle size on the two types of phase transitions are uniform. It also can be seen from Eq. (12) that the difference between  H mb and  H m ( H m ) is mainly depended on the item 2V σ  / r . The orders of

H m are 10-2 and 100 kJ·mol-1 for r with the orders of 10-7 and 10-9 m, respectively. Therefore, when the order of r is larger than 10-7, the effect of size on phase transition enthalpy can be neglected. Similarly, in Eq. (14), when the order of r is larger than 10-7 m, the order of the difference ( Sm ) between  Smb and  Sm is smaller than 10-2J·mol-1·K-1. In this case, it is reasonable to ignore the effect of particle size on the phase transition entropy. The difference between the crystal transition and melting temperatures can be expressed as Eq. (15), 1  2σ V 2σ V  T  Tm  Tc  (Tmb  Tcb )   l sl b   b   r   s Sm   Sm 

(15)

where Tm and Tc are the melting temperature and crystal transition temperature, respectively. As we can see from Eq. (15), the ΔT only depends on the particle size and it is linearly related to the reciprocal of particle size. Similarly, the differences of enthalpies and entropies between the two types of phase transitions (defined as (H ) and (S ) ) can be written as Eqs. (16) and (17), respectively.


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(H )   ls H m   H m   ls H mb   H mb 

2V r

  σ   2    σ sl  2  σ sl  T    σ     (16)   σ sl    σ   T   T  P 3    T  P 3  

(S )   ls S m   S m   ls S mb   S mb 

  σ  2 2V  σ sl  2    σ γ    σ sl γ   r  T  P 3  T  P 3 

(17)

It can be seen from Eqs. (16) and (17), the (H ) and (S ) only depend on the particle size and are linearly related to the reciprocal of particle size, too.

Experimental section Materials Bi(NO3)3·5H2O, urea and PVP were purchased from Tianjin fengchuan Chemical Reagent Co., Ltd., China, nitrate acid and glycol were purchased from Sinopharm Chemical Reagent Co., Ltd., China. All of the chemical reagents in our experiments were of analytical grade and used without further purification. Preparation of nano-Bi2O3 Nano-Bi2O3 was prepared by hydrothermal method. 0.182 g of Bi(NO3)3·5H2O was dissolved in 5 mL of nitrate acid (2.0 mol·L-1) (labeled A solution), 0.16 g of urea and 0.3g of PVP was dissolved in 25 mL of glycol (labeled B solution). Mixed solution A with B, and then transferred the mixed solution into a 50 mL autoclave and maintained at 150 °C for 3 h. After cooling to room temperature naturally, the products were collected by centrifuging and washed with distilled water and ethyl alcohol repeatedly, then dried in air at 60 °C for 2 h. Finally, the products were annealed at 320 °C for 2 h in air. Characterization method Phase identification of nano-Bi2O3 were performed by X-ray diffraction (XRD) analysis (Bruker D8 ADVANCE powder diffractometer, Germany, Cu Kα, λ=1.54178Å). The morphologies of the prepared nanoparticles were examined by field-emission scanning electron microscopy (FE-SEM, JSM-7001F). Crystal transition and melting behaviors of the samples with different particle sizes were



measured by differential scanning calorimetry (DSC, STA 449F3) at a heating rate of 10 °C/min in a nitrogen atmosphere. Results and discussion The XRD patterns (Fig. 2) of nano-Bi2O3 with different diameters can be well-indexed to Bi2O3 (JCPDS 78-1793) corresponding to a monoclinic crystal

(203) (421) (402)

Intensity (a.u.)

(002) (220)

(222) (400)

(201)

without any miscellaneous peaks and the crystallinity is high.

85.5 nm

75.9 nm 67.2 nm 58.4 nm 45.1 nm 20

30

40

50 2(deg.)

60

70

80

Fig. 2 XRD patterns of nano-Bi2O3 with different diameters 30

b

20 15

30

b

25

Percentage (%)

a

a

25

10 5

c

20 15

20

10

38

40

42

44

46

48

50

50

Particle diameter (nm)

52

54

56

58

60

62

60

64

64

66

68

70

72

74

30

e

20

15

10

e

25

Percentage (%)

d

25

Percentage (%)

62



30

d

10

0

0

52

15

5

5

0

c

25

Percentage (%)

30

Percentage (%)

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

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20 15 10 5

5

0

0 68

70

72

74

76

78

80

82


78

80

82

84

86

88

90

92


Fig. 3 SEM images and size distribution of nano-Bi2O3 with different diameters (a 45.1±3.5 nm, b 58.4±4.8 nm, c 67.2±4.9 nm, d 75.9±4.3nm, e 85.5±6.3 nm)

Fig. 3 shows the morphology and size distribution of as-prepared nano-Bi2O3 with different diameters. It is clear that the morphology of nano-Bi2O3 is spherical and the particle diameters are relatively uniform. As we can see from the charts, in Figs 3(a),


Page 9 of 19

3(b) and 3(c), the particle diameters distribute on both sides of X-axes; in Fig. 3(d), the distribution of particle diameter is in a nearly normal distribution and in Fig. 3(e), the particle diameter distributes mainly in the middle of X-axis.

a

b

Fig. 4 TEM image (a) and the corresponding SAED pattern (b) of nano-Bi2O3

As show in Fig. 4(a), the spherical nano-Bi2O3 is clearly visible and the particle size is 58.1±8.7 nm. Fig. 4(b) is the corresponding SAED pattern of nano-Bi2O3. The lattice spacings of 0.274, 0.160 and 0.141 nm correspond to the (220), (402) and (004) planes, respectively.

Heat Flow (a.u.)

85.5 nm

75.9 nm

67.2 nm

58.4 nm

Endo

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


45.1 nm

575 600 625 650 675 700 725 750 775 800 825 Temperature/C

Fig. 5 DSC curves of nano-Bi2O3 with different diameters

Fig. 5 shows the DSC curves of phase transitions (crystal transition and melting) of nano-Bi2O3 with different particle diameters. Nano-Bi2O3 exhibits a crystal transition (from phase α to δ) at about 725 °C,9 and the peaks near 810 °C are caused by the melting of nano-Bi2O3. As clearly shown in the curves, the peaks shift to lower temperature with the particle diameter decreasing. It is obvious that the crystal transition peaks of nano-Bi2O3 with small particle diameters have shoulder peaks, while the same situation does not appear in melting



Page 10 of 19

peaks in Fig. 5. The reason for this phenomenon is that the first three distribution charts of nano-Bi2O3 have two peaks and the beginning of crystal transition of nanoparticles with smaller diameters precedes the bigger ones. Hence two crystal transition peaks overlap partially result in the formation of shoulder peaks. As for melting, the effect of particle size on melting temperature is not as great as that on crystal transition temperature (see Table 1), so, the two melting peaks overlap almost completely and there are not shoulder peaks. The crystal transition and melting thermodynamic data of nano-Bi2O3 were calculated and summarized in Table 1, respectively. T0 is the temperature of the beginning of the phase transitions. Table 1 Thermodynamic data of crystal transition and melting of nano-Bi2O3 with different diameters No.

1

2

3

4

5

6

d/nm

45.1

58.4

67.2

75.9

85.5

∞

d-1/nm-1

0.0222

0.0171

0.0149

0.0132

0.0117

0

T0 (  ) /°C H  /kJ·mol-1

709.8

717.4

720.6

726.2

730.7

7309

13.82

19.03

21.41

21.87

22.91

-

13.37

16.91

19.89

20.58

22.24

-

T0 ( s  l ) /°C

806.1

807.9

809.8

810.6

812.2

8179

H sl /kJ·mol-1

6.260

8.778

9.956

11.77

13.42

2829

S sl /J·mol-1·K-1

8.596

9.134

9.879

10.77

12.49

25.629

 

S

/J·mol-1·K-1

As can be seen from Table 1, nano-Bi2O3 with different diameters exhibit different phase transition behaviors, the thermodynamic data of crystal transition and melting decrease with the decrease of particle diameter. We can also find that the effects of particle diameter on enthalpy and entropy of two types of phase transitions are similar, but the effect of particle diameter on crystal transition temperature is greater than that on melting temperature.


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830 820

Structure phase transition Melting The crystal transition temperature of bulk Bi2O3 The melting temperature of bulk Bi2O3

810

T0/C

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


730 720 710 700 0.0000

0.012 0.014 0.016 0.018 0.020 0.022 0.024

d-1/nm-1

Fig. 6 Relationships between T0 and the reciprocal of particle diameter

Fig. 6 exhibits the relationships between T0 and the reciprocal of particle diameter. The intersection points between the linear correlation equations and the Y-axis are close to the phase transitions temperatures of bulk Bi2O3, 9 and it also proves that size dependence of properties is continuous and there is no break point above 100 nm.34 We can see that T0 decreases with the decreases of particle diameter, which is consistent with the literature.35 The reason is that nano-Bi2O3 has larger external surface and higher surface energy as compare with the corresponding bulk materials, which reduced the energy required for the process of phase transition, so the phase transitions occur at relatively low temperature. In addition, the T0 exhibits a good linear relationship with the reciprocal of particle diameter, the experimental results are consistent with theoretical equation of Eq. (9) and the literature.36, 37 Fig. 7 shows the relationship between ΔT and the reciprocal of the particle diameter.



99 96

T/C

93 90 87 84 81 78 0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

d-1/nm-1

Fig. 7 Relationship between ΔT and the reciprocal of particle diameter

As we can see that the two types of phase transitions temperatures of nano-Bi2O3 are related. The difference between the crystal transition temperature and melting temperature increases with the decreases of particle diameter and the ΔT shows a good linear relationship with the reciprocal of the particle diameter. The results are identical with Eq. (15). That is, the smaller the diameter is, the bigger the difference in crystal transition temperature and melting temperature is bigger and the temperature range of application will be wider. Utilizing the relationship of the two types of phase transitions, the regularities and phenomena of crystal transition can be predicted via melting, and vice versa. The discovery has important potential application value. 24

Crystal transition Melting

22 20

H/kJmol-1

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

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18 16 14 12 10 8 6 0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

d-1/nm-1

Fig. 8(a) Relationships between crystal transition enthalpy, melting enthalpy and the reciprocal of particle diameter


Page 13 of 19

Fig. 8(a) shows the relationships between the phase transitions (crystal transition and melting) enthalpy of nano-Bi2O3 and the reciprocal of particle diameter. The crystal transition enthalpy is higher than melting enthalpy of nano-Bi2O3 with the same particle diameter, this situation is identical with the literature. It is obvious that the enthalpies of crystal transition and melting decrease with the decreases of particle diameter and are linearly related to the reciprocal of particle diameter. The results are identical with the theory analysis of Eq. (12) and literature.38, 39 24 Crystal transition Melting

22 20

S/(Jmol-1K-1)

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


18 16 14 12 10 8 0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

d-1/nm-1

Fig. 8(b) Relationships between crystal transition entropy, melting entropy and the reciprocal of particle diameter

As illustrated in Fig. 8(b), when the particle diameter decreases, the crystal transition entropy and melting entropy decrease, respectively. The reason is that surface entropy of nanoparticles become bigger with the decreases of the particle diameter,40, 41 The entropy of initial state increases while the entropy of final state is constant, hence the entropy change decreases. And there are linear relationships between the phase transitions entropies and particle diameter, too, which is in agreement with Eq. (14).



14

a kJmol-1

12

10

8

6 0.010

0.012

0.014

0.016

0.018

-1

0.020

0.022

0.024

-1

d /nm 12

b

10

S)/Jmol-1K-1

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

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8

6

4 0.010

0.012

0.014

0.016 -1

0.018

0.020

0.022

0.024

-1

d /nm

Fig. 9 Relationships between (a) Δ(ΔH), (b) Δ(ΔS) and the reciprocal of particle diameter

Fig. 9 shows the relationships between Δ(ΔH), Δ(ΔS) and the reciprocal of particle diameter, respectively. We can see that the enthalpies and entropies of two types of transitions are correlative, too. And they are all linearly related to the reciprocal of particle diameter. The results are identical with Eqs. (16) and (17).

Conclusions The phase transition model can be applied to two types of phase transitions and explain the influence regularities and mechanism of particle size on phase transition temperatures and thermodynamic properties of two types of phase transitions satisfactorily. By analyzing the experimental results, we found that the effects of particle size on the phase transition temperatures and thermodynamic properties of two types of phase transitions are consistent with the theory. The influence


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regularities of particle size on phase transition thermodynamics of two types of phase transitions are uniform. The phase transitions (crystal transition and melting) temperatures, enthalpies and entropies of nano-Bi2O3 decrease with the particle diameter decreasing and there are linear relationships between these thermodynamic properties and the reciprocal of particle diameter. In addition, we found that the two types of phase transitions of nano-Bi2O3 are correlative. With the particle diameter of nano-Bi2O3 decreasing, the difference in two types of phase transitions temperatures increases and the differences in enthalpies and entropies of two types of phase transitions decrease. And the differences in temperatures, enthalpies and entropies between two types of phase transitions show linear relationships with the reciprocal of particle diameter, too. The theory can describe the behaviors of phase transitions of nanoparticles quantitatively, explain and predict size-dependent temperature and thermodynamic properties of phase transitions of nanoparticles. More importantly, it plays a significant role in solving the problems of phase transitions in the field of nanomaterials.

Acknowledgements The authors are very grateful for the financial support from the National Natural Science Foundation of China (No. 21573157 and No. 21373147).

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Size-Dependent Thermodynamic Properties of Two Types of Phase

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