Universal Size Dependence of Integral Enthalpy and Entropy for Solid

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Universal Size Dependence of Integral Enthalpy and Entropy for Solid-Solid Phase Transitions of Nanocrystals Zixiang Cui, Huijuan Duan, Qingshan Fu, Yongqiang Xue, and Shuting Wang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b07193 • Publication Date (Web): 17 Oct 2017 Downloaded from http://pubs.acs.org on October 18, 2017

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The Journal of Physical Chemistry

Universal Size Dependence of Integral Enthalpy and Entropy for Solid-Solid Phase Transitions of Nanocrystals Zixiang Cui, Huijuan Duan, Qingshan Fu, Yongqiang Xue*, Shuting Wang Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan 030024, China Email: [email protected]

Abstract: There is a significant effect of particle size on solid-solid phase transition of nanocrystals, but the precise relations of integral enthalpy and entropy of phase transition with particle size have not been reported. In this paper, we deduced the fundamental equations of the integral enthalpy and entropy of solid-solid phase transitions for nanocrystals with different particle sizes, respectively. These universal and precise thermodynamic relations, free of any adjustable parameter, can precisely describe the real process of phase transition. Experimentally, we researched the phase transition from tetragonal to cubic of nano-PbTiO3 with different diameters (24.8 nm, 35.6 nm, 41.9 nm, 54.2 nm, 63.7 nm and 70.1 nm). The regularities obtained from these experiments were consistent with the theoretical relations. There were linear relations between the integral enthalpy and entropy of solid-solid phase transition and the reciprocal of size of nanocrystals as the radius exceeds 10 nm. Our thermodynamic relations and the quantitative size-dependent regularities of solid-solid phase transition can provide references for explaining and confirming the solid-solid phase transitions involved in the real processes of preparation and application of nanocrystals.

1. Introduction Solid-solid phase transition of nanocrystals has captured the attention of a broad scientific community1-4, because crystal structure plays a much more important role in its functionalities in various fields, such as photocatalysis5, ferroelectricity6, adsorption7, and so on. However, there is a significant effect of particle size on solid-solid phase transition of nanocrystals. Thus a good understanding of the size effect on thermodynamics of solid-solid phase transition for nanocrystals is important for deepening one’s knowledge as well as identifying further applications. Quantities of researches focus on the size dependence on solid-solid phase transition temperature of nanocrystals8-10, it was found that the phase transition temperature increases with the 1

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particle size decreasing. A relatively few number of literatures reported the size effect of enthalpy and entropy of solid-solid phase transition for nanocrystals. Zhong et al.11 researched the phase transition of nano-PbTiO3 with different particle sizes, the enthalpy of phase transition declined sharply with the decrease of particle size. The decreasing tendency also was observed by Köferstein et al.12 and Asiaie et al.13. Aside from the size dependence on the enthalpy of phase transition, Jiang et al.14 also reported the size effect on the entropy of phase transition and found that there is an obvious decrease of the entropy of phase transition with the particle size decreasing. Although these qualitative trends of size effect on enthalpy and entropy of solid-solid phase transition for nanocrystals were presented, a theoretical description which can precisely describe the real process of phase transition is extremely necessary to understand the solid-solid transition of nanocrystals any further. Ma15 deduced the relation between the enthalpy of phase transition and particle size basing on the relations from Fatuzzo et al.16 by surface layer model. But the surface layer thickness is difficult to determine. Based on Zhong’s model17, Jiang et al.14 proposed an improved phenomenological theory by taking the size effects on phenomenological coefficients into account, but the new model cannot be used as the particle size was less than critical size. The theoretical relations of size-dependent enthalpy and entropy of solid-solid phase transition for nanocrystals were proposed in our previous work18, but these relations indicated the relations of differential enthalpy and entropy of phase transition for nanoparticles and particle size. These differential quantities represent that the change of much lower amounts was occurred and then was converted to the change of one mole. However, enthalpy and entropy of phase transition obtained from experiments are integral values, which represent the change of these quantities in whole phase transition. In the process of solid-solid phase transition, the change of enthalpy and entropy is inhomogeneous for the phase transition of nanomaterials of one mole in different stage of phase transition, which leads to the significant gap between the differential quantities and integral quantities. Therefore, these theories cannot describe the real process of solid-solid phase transition. In this paper, we deduce the relations of the effect of particle size on integral enthalpy and entropy of solid-solid phase transitions for nanocrystals. In experiments, we researched the phase transition from tetragonal to cubic of nano-PbTiO3 with different diameters, and the enthalpy and entropy of phase transition were determined. And the regularities of size effect on the enthalpy and entropy of phase transition were summarized. 2


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2. Theoretical section The molar Gibbs energy (Gm) of nanoparticle can be defined as the sum of that of the bulk phase ( Gmb ) and the surface phase ( Gms ),

Gm = Gmb + Gms

(1)

where superscript b and s denote the bulk phase and the surface phase, respectively. The integral molar surface Gibbs energy can be expressed as follows,

Gms =

σA

(2)

n

where σ , A, n denote the surface tension, the surface area and the amount of substance, respectively. In the process of solid-solid phase transition of phase α to phase β, the change in molar Gibbs energy ( ∆αβ Gm ) can be obtained as follows:

∆αβ Gm =Gm ( β ) − Gm (α ) = ∆αβ Gmb +

σ β Aβ nβ

−

σ α Aα nα

(3)

β b b b where ∆α Gm = Gm ( β ) − Gm (α ) .

Applying the Gibbs-Helmholtz equation to phase transition,  ∂  ∆βα Gm    ∂T  T

 ∆βα H m  = − T2  p

(4)

where ∆αβ H m , T and p denote the enthalpy of phase transition, temperature and pressure. Combining Eq. (3) and Eq. (4), the accurate integral enthalpy of phase transition can be obtained,    ∂σ β    ∂Aβ   ∂σ α    ∂Aα  ∆βα H m = ∆βα H mb + Aβ σ β − T    − Tσ β   − Aα σ α − T    + Tσ α    ∂T  p   ∂T  p  ∂T  p   ∂T  p  

(5)

Basing on the essential thermodynamic relation,  ∂∆ βα Gm  ∆ Sm = −    ∂T  β α

where ∆αβ S m denotes the entropy of phase transition. The accurate integral entropy of phase transition can be obtained,

3


(6)


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 ∂σ β   ∂Aβ   ∂σ α   ∂Aα  ∆βα S m = ∆βα S mb − Aβ   −σβ   + Aα   +σα    ∂T  p  ∂T  p  ∂T  p  ∂T  p

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(7)

For the solid-solid phase transition of spherical nanocrystals, integral molar surface Gibbs energy can be written as follows: Gms =

3σ V r

(8)

where V and r denote molar volume and radius. In addition, the mass of both before and after phase transition remains unchanged, so for spherical nanoparticles, 4 3 4 π rβ ⋅ ρβ = π rα 3 ⋅ ρα 3 3

(9)

where ρ denotes densities. That is to say,

ρ = α rα  ρβ rβ

1

1

 3  Vβ  3  =     Vα 

(10)

Substituting Eq. (8) and Eq. (10) into Eq. (3), Eq. (5) and Eq. (7), respectively, the integral Gibbs energy, enthalpy and entropy of phase transition can be expressed as follows:

 3Vα   ρα β β b ∆ αGm =∆ α Gm + σβ  rα   ρβ  3V ∆βα H m = ∆βα H mb + α rα

  ρ α   ρβ 

2  3   − σ α   

2    3    ∂σ β  2  ∂σ α  2  σ β − T   − T α β  − σ α − T   − Tα α    ∂T  p 3  ∂T  p 3       

 3Vα  ρα β β b ∆ α Sm = ∆α Sm −  rα  ρβ 

2

3  

  ∂σ β  2   ∂σ  2  α   + αβ  −   − αα   ∂T  p 3   ∂T  p 3  

(11)

(12)

(13)

where α denotes the expansion factor. In general, the densities of the same materials with different crystal structures are nearly identical, namely, ρα / ρβ ≈ 1 . As a result, Eq. (12) and Eq. (13) can be written as follows: ∆βα H m = ∆βα H mb +

3Vα rα

   ∂σ β   ∂σ   2 α − T − α α  ( ) (σ β − σ α ) − T    − β α   ∂T  p  ∂T  p  3   4


(14)

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∆βα S m = ∆βα S mb −

3Vα rα

 ∂σ β   ∂σ  2  α   −  + (α β − α α )   ∂T  p  ∂T  p 3 

For general nanoparticles, the order of magnitude for σ, Vm, α, T,

( ∂σ

(15)

∂T ) p are 10-1-100

J/m2,19 10-5 m3/mol,20 10-5,21 102 K, 10-4 J·m-2·K-1,20 respectively. According to the orders of magnitude and Eq.(14), the size-dependent qualitative regularity of enthalpy of phase transition for nanocrystals mainly depends on the difference of surface tension for nanocrystals both before and after phase transition. If σ β > σ α , the enthalpy of phase transition for nanomaterials is higher than that for corresponding bulk materials, and there is a trend of increase for the enthalpy of phase transition with the particle size decreasing; otherwise, the opposite conclusion can be obtained. In addition, it can be calculated from the orders of magnitude and Eq.(14) that the orders of magnitude of the difference between enthalpy of phase transition of nanomaterials and that of corresponding bulk materials are 103, 102, 101, respectively, for r with the orders of 10-9, 10-8 and 10-7m, respectively. Hence, when the particle size reaches nanoscale, the effect of particle size on enthalpy of phase transition cannot be ignored. What’s more, as the radius exceeds 10 nm, the effect of particle size has little effect on the surface tension,22,23 so a linear relation can be obtained between enthalpy of phase transition of nanomaterials and particle size. By the same token, the orders of magnitude and Eq.(15) show that the size-dependent qualitative regularity of entropy of phase transition for nanocrystals mainly depends on the difference of the temperature coefficients of surface tension for nanomaterials both before and after phase transition. If ( ∂σ β / ∂T ) p > ( ∂σ α / ∂T ) p , the entropy of phase transition for nanomaterials is lower than that of corresponding bulk materials, and there is a trend of decrease for the entropy of phase transition with the particle size decreasing, otherwise, the opposite conclusion can be obtained. In addition, it can be calculated from the orders of magnitude and Eq.(15) that the orders of magnitude of the difference between entropy of phase transition of nanomaterials and that in corresponding bulk materials are 100, 10-1, 10-2, respectively, for r with the orders of 10-9, 10-8 and 10-7m, respectively. Hence, when the particle size reaches nanoscale, the effect of particle size on enthalpy of phase transition can not be ignored. Similarly, a linear relation between entropy of phase transition of nanomaterials and particle size can be obtained as the particle size exceeding 10 nm. 5



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Furthermore, σ, Vm, α and

( ∂σ

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∂T ) p are constants when the temperature is unchanged, so it

can be seen from Eq.(14) and Eq.(15) there are linear relations between the enthalpy of phase transition and entropy of phase transition, respectively, and particle size. In the same way, for the solid-solid phase transition of nanoparticles with other morphologies, corresponding equations and similar conclusions can be obtained. The general equations of enthalpy and entropy of solid-solid phase transition for nanomaterials are as follows: ∆βα H m = ∆βα H mb +

γ Vα  lα

  ∂σ β   ∂σ   2 α (σ β − σ α ) − T   −   − T (α β − α α )   ∂T  p  ∂T  p  3  

∆βα S m = ∆βα S mb −

(16)

γ Vα  ∂σ β  lα

  ∂σ α  2   −  + (α β − α α )   ∂T  p  ∂T  p 3 

(17)

where γ is shape parameter and l is characteristic length. The related parameters for nanoparticles with different morphologies were listed in table 1. Table 1 The l and γ for nanoparticles with different morphologies Morphology

Characteristic length (l)

Shape parameter (γ)

Sphere

Radius

3

Regular tetrahedron

Side length

6 6

Cube

Side length

6

Regular octahedron

Side length

3 6

Regular dodecahedron

Side length

12 25 + 10 5 / 15 + 7 5

Regular icosahedron

Side length

3 3 3− 5

(

(

)

)

Furthermore, for nanoparticles with different morphologies but in identical volume, the integral enthalpy and entropy of solid-solid phase transition for spherical nanoparticles are the minimum, and the more the morphology deviates from sphere, the larger the thermodynamic properties are.

3. Experimental section Preparation of nano-PbTiO3. In our experiments, the sol-gel method was employed for the synthesis of nano-PbTiO3 with varying particle sizes. A certain amount of lead acetate dissolved in acetic acid was added to equal 6


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molar butyl titanate in ethanol. The solution was heated in water bath to one temperature and was stirred strongly, until the oily sol was derived. Through standing and aging treatments for 18 h, the as-prepared sol was transferred into a clean culture dish and shake up. After drying and calcination, nano-PbTiO3 was obtained. Different reactant concentrations and calcination temperatures were used to adjust the particle size nano-PbTiO3.

Characterization of nano-PbTiO3. Nano-PbTiO3 with varying particle sizes was examined using a Germany Bluker D8 Advance Powder diffractometer (Cu Kα, λ = 0.154178 nm). Particles size of samples was calculated by Scherrer formula based on the half peak width of characteristic diffraction peaks. Morphology of samples was observed using JSM-6701F scanning electron microscope (SEM).

Solid-solid phase transition of nano-PbTiO3 The differential scanning calorimetry (DSC) was used to follow the phase transitions. 40 mg (±10%) samples were weighed into aluminium crucibles and hermetically sealed in an N2 atmosphere, the crucibles filled with samples and a reference pan were placed into the furnace chamber. Nitrogen was used as the purge gas at a rate of 50 mL•min-1 in the furnace chamber to reduce any influence of water or air on the measurements during the experiment. The chamber was heated at a constant rate according to the programmed heating ramp (10 K•min-1) from room temperature to 723.15 K. The enthalpy and entropy of phase transition were obtained according to the literature 18. In brief, the endothermic peak was divided into several small trapezoids, the enthalpy and entropy of phase transition can be calculated as the following equations, ∆H = ∫ δ Q = ∑ δ Qi

∆S = ∫

δQ T

=∑

δ Qi Ti

(18) (19)

where Qi and Ti indicate the heat and the medium temperature of phase transition corresponding to each small trapezoid.

4. Results and discussion 4.1 Structure and morphology of nano-PbTiO3 Figure 1 shows the XRD patterns of nano-PbTiO3. The XRD patterns indicate that all samples 7



exhibit a tetragonal structure. The main characteristic peaks at 22.78 °, 31.45 °, 32.44 °, 39.19 °, 46.53 ° and 55.36 ° can be assigned to the (100), (101), (110), (111), (200) and (112) planes of crystal structure, respectively. These peaks are consistent with the database in JCPDS file (PDF No. 6-0452). In addition, no other peak was observed, which suggests that the samples were pure. The particles size is correlated with the half width at half maximum by the Debye-Scherrer equation24:

D=

kλ β cos θ

(20)

where D is the average diameter of nanoparticles, λ is the wavelength of X-ray, θ is the angle of diffraction, k = 0.9 (a shape factor) and β is the full width at half maximum of X-ray diffraction peaks. The average particle sizes of them were calculated by Debye-Scherrer formula via half-peak width of diffraction peaks derived from XRD. The average particle sizes of nano-PbTiO3 are 24.8 nm, 35.6 nm, 41.9 nm, 54.2 nm, 63.7 nm and 70.1 nm, respectively.

Intensity

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70.1 nm 63.7 nm 54.2 nm 41.9 nm 35.6 nm 24.8 nm

20

30

40

50

60

70

80

2θ/degree Figure 1 The XRD patterns of nano-PbTiO3 with different particle sizes SEM images of the as-prepared PbTiO3 are shown in figure 2. It was clear that the morphology of the as-obtained samples was nearly spherical and the size was almost identical with the calculation results by Debye-Scherrer’s formula.

8


Figure 2 SEM photos of the nano- PbTiO3

4.2 Size dependence the enthalpy of phase transition of nano-PbTiO3 70.1nm

63.7nm

Heat Flow(W/g)

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54.2nm 41.9nm 35.6nm

endo

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24.8nm

480 482 484 486 488 490 492 494 496 498

Temperature/°C

Figure 3 DSC curves of nano-PbTiO3 with varying particle sizes Figure 3 shows the DSC curves of nano-PbTiO3 with varying particle sizes, the enthalpy of phase transition can be obtained through the integration for the peak area of DSC from the initial temperature to the final temperature of phase transition. A plot of the enthalpy of phase transition versus particle size was shown in figure 4. It is obvious that the enthalpy of phase transition decreases with particle size decreasing, which is consistent with the results in literatures.11,14,15 It can be attributed to the participation of the larger surface energy in the solid-solid phase transition, and therefore fewer calories are absorbed. Due to the constant pressure environment, without non-volume work, the value of enthalpy of phase 9



transition is identical with that of the phase transition heat. Compared with the enthalpy of phase transition of bulk PbTiO3 (1910 J/mol),24 as the particle size of PbTiO3 is 70.1 nm, the depression value of enthalpy of phase transition is 247 J/mol, and as the particle size is reduced to 24.8 nm, the depression value reaches 976 J/mol. The size effect on the enthalpy of phase transition of nano-PbTiO3 cannot be ignored. In addition, according to Eq. (14), as the particle size reaches 10-8 nm, the order of magnitude of the difference between enthalpy of phase transition of nanomaterials and that of corresponding bulk materials can reach 102 J/mol. The experimental results are agrees with the above theoretical analysis. Moreover, there is a linear relationship between the enthalpy of phase transition and reciprocal of particle size, and the quantitative regularity is consistent with Eq. (14). Besides, the intercept of the line is the enthalpy of phase transition for bulk PbTiO3 with a value of 2089 J/mol which is close β to that determined by Rossetti24. That is to say, the linear relation between ∆α H m and 1/r is reliable.

In addition, based on this decreasing tendency and Eq. (14), we can confirm that the surface tension of nano-PbTiO3 with tetragonal phase is higher than that with cubic phase.

-1

1.8

Transition enthalpy / kJ•mol

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1.6

1.4

1.2

1.0

0.8

0.03

0.04

0.05

0.06 -1

r /nm

0.07

0.08

-1

Figure 4 The relation between the enthalpy of phase transition and the reciprocal of radius of nano-PbTiO3

4.3 Size dependence on the entropy of phase transition for nano-PbTiO3 The entropy of phase transition for nano-PbTiO3 with varying particle size can be obtained basing on Eq. (17). A plot of the entropy of phase transition versus particle size was shown in figure 5. 10


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It was found that the entropy of phase transition decreases with the particle size decreasing. The reason for the decreasing tendency can be summarized as the size effect. Because the fewer particle size is, the more serious the surface defects is, which leading to the increase of the randomness, the entropy of phase transition subsequently increases. And the entropy of phase transition decreased from 1.89 J•K-1•mol-1 to 0.851 J•K-1•mol-1 as the particle size decreased from 70.1 nm to 24.8 nm. The order of magnitude of the amplitude of descent is basically consistent with theoretical Eq. (15). The experimental results are consistent with our theoretical analysis. That is to say, our theory can give right guidance for experiments. In addition, there is a linear relationship between the entropy of phase transition and reciprocal of particle size. This quantitative regularity is consistent with theoretical Eq. (15). Besides, the linear relation of the logarithm of entropy of phase transition with the reciprocal of particle size has been reported14. But this relation, obtained by experimental results, is empirical and cannot be used to explain other solid-solid phase transition of nanosystems. And the data processing is not very precise. In literature 14, the entropy of phase transition is obtained by dividing enthalpy of phase transition by temperature of phase transition. In our paper, the entropy of phase transition of nano-PbTiO3 with varying particle size can be obtained basing on Eq. (17). Moreover, according to Eq. (15), it can be confirmed that the temperature coefficient of surface tension for nano-PbTiO3 with tetragonal phase is lower than that with cubic phase.

-1

2.0 1.8

-1

Transition entropy / J•mol •K

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


1.6 1.4 1.2 1.0 0.8 0.03

0.04

0.05 -1

r /nm

0.06

0.07

0.08

-1

Figure 5 The relations between the entropy of phase transition and the reciprocal of the sizes of PbTiO3 nanoparticles

5 Conclusion 11



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We deduced the universal and precise thermodynamic relations of the integral enthalpy and entropy of solid-solid phase transitions for nanocrystals with particle size, respectively. Both theoretical and experimental results show that there is a remarkable effect of particle size on the enthalpy and entropy of solid-solid phase transition for nanocrystals. The change trends of enthalpy and entropy in the process of solid-solid phase transition depend on the numerical magnitudes for the surface tension and the temperature coefficient of surface tension, respectively. In addition, there is a good linear relationship between the enthalpy and entropy, respectively, and reciprocal of particle size as the radius exceeds 10 nm. According to our theory, the change of the surface tension and the temperature coefficient of surface tension for nanocrystals with different crystal structure and the orders of magnitude of enthalpy and entropy of solid-solid phase transition can be confirmed. And our theory can provides a theoretical method to precisely describe the real process of solid-solid phase transition and give a general framework for understanding how the particle size quantitatively impacts integral thermodynamics on solid-solid phase transition.

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

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Universal Size Dependence of Integral Enthalpy and Entropy for Solid

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