Effect of size on dissolution thermodynamics of nanoparticles: A theoretical and experimental research

Materials Chemistry and Physics 214 (2018) 499e506

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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys

Effect of size on dissolution thermodynamics of nanoparticles: A theoretical and experimental research Zongru Li, Qingshan Fu, Yongqiang Xue*, Zixiang Cui Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan, 030024, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t


The dissolution model of nanoparticles was established.
The size-dependent dissolution thermodynamics were derived.
The influence regularities of nano-Cu dissolved in dilute acid were discussed.
The dissolution behavior of nanoparticles could be explained by the dissolution model.
It provided guidance for the preparations and applications of the nanomaterials.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 October 2017 Accepted 30 April 2018 Available online 2 May 2018

Dramatic differences in dissolution thermodynamics between nanoparticles and their bulk counterparts have been observed in which the particle size plays a crucial role. However, the quantitative influences of particle size on the dissolution thermodynamic properties of nanoparticles have not been systematically investigated. In this paper, the relations between dissolution equilibrium constant and dissolution thermodynamic functions with particle size were deduced by introducing the surface variables into the Gibbs function. Then the influence regularities of size on the dissolution equilibrium constant and the dissolution thermodynamic functions were obtained by measuring the solubilities of nano-Cu with different sizes in dilute acid at different temperatures. Theoretical analysis and experimental results reveal that with particle size decreasing, the dissolution equilibrium constant (K o ) increases, while the o ) and the standard molar dissolution Gibbs energy (Dr Gom ), the standard molar dissolution enthalpy (Dr Hm standard molar dissolution entropy (Dr Som ) decrease. Furthermore, when the diameter approaches or o , and D So present linear relations with the reciprocal of diameter, exceeds 20 nm, ln K o , Dr Gom , Dr Hm r m respectively. © 2018 Elsevier B.V. All rights reserved.

Keywords: Nanoparticle Size-dependent Dissolution Thermodynamics

1. Introduction

* Corresponding author. E-mail addresses: [email protected] (Z. Li), [email protected] (Q. Fu), [email protected] (Y. Xue), [email protected] (Z. Cui). https://doi.org/10.1016/j.matchemphys.2018.04.112 0254-0584/© 2018 Elsevier B.V. All rights reserved.

In the processes of preparation and application of nanomaterials, the dissolution of nanoparticles is often involved, such as the preparation of antibacterial agent composite materials [1], and heterogeneous nanocatalysts [2,3], as well as the improvement of

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Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

the solubility for the drug bioavailability [4e6]. Nevertheless, there are remarkable differences in the dissolution thermodynamic properties of nanomaterials compared with those of the corresponding bulk materials, which are caused by the nanometer size effect. So it is essential to investigate the size-dependent dissolution thermodynamics. Presently, with regard to effect of size on the dissolution of nanoparticles, it has been found that the solubility increases with the decrease of particle size [7e13]. Vogelsberger et al. found that the solubility product of nano-BaSO4 in the water is far greater than that of the bulk [14]. Lubej et al. observed that the solubility product of Cu2Cl(OH)3 increases with the particle size decreasing in the range below 100 nm, and then the molar dissolution Gibbs energy, enthalpy and entropy of Cu2Cl(OH)3 at 298 K were obtained [15]. Fan et al. harvested that the standard molar reaction enthalpy of peanut-like micro/nano CaMoO4 decrease with size decreasing [16]. As a matter of fact, far insufficient theoretical and experimental studies on the dissolution thermodynamics for nanoparticles have been reported, not to mention the quantitative influence regularities of particle size on the dissolution thermodynamic functions. Herein, the relations of dissolution equilibrium constant and dissolution thermodynamic functions with the particle size were derived. Then nano-Cu was taken as an example, and by measuring the solubilities of nano-Cu with different particle sizes at different temperatures, the effects of size on the equilibrium constant, the molar Gibbs energy, the molar enthalpy and the molar entropy for dissolution were investigated. Subsequently, the experimental results were compared with the theoretical analysis.

phase. The molar dissolution Gibbs energy Dr Gm of nanoparticles at constant temperature and constant pressure can be derived as follows,

2. Theory analysis

where V and d denote the molar volume and the diameter of nanoparticle, respectively. Substituting equation (8) into equation (7), it becomes

2.1. The dissolution thermodynamics of nanoparticles The dissolution of nanoparticles includes two types, one is the physical dissolution, and the other is the chemical dissolution of nanoparticles. Both of them can be seen as a reaction that can be expressed as follows,

0¼

X

nB B

(1)

where nB is the stoichiometric number of component B in the dissolution system. Assuming that the nanoparticles are the only dispersed phase and introducing the interface variable to the Gibbs energy, the total Gibbs energy G can be seen a function of temperature T, pressure p, amount of substance n for component B and interfacial area for nanoparticles A.

G ¼ GðT; p; n; AÞ

(2)

When the state has an infinitesimal change, dG can be given as:

     X vG vG vG vG dG ¼ dT þ dp þ dn þ dA vT p;n;A vp T;n;A vn vA T;p T;p;A B

Dr Gm ¼



     X vn vG vA vn ¼ mbB þs vx T;p vx T;p vn T;p vx T;p B

where x is the extent of reaction expressed as equation (1), and mbB is the chemical potential of bulk phase of component B. The surface chemical potential ms is defined as follows,

ms ≡s

  vA vn T;p

(6)

where the superscript s denotes the surface phase. According to the definition of ms and the stoichiometric number n, equation (5) can be changed as follows,

Dr Gm ¼

X

nB mbB þ nms ¼ Dr Gbm þ Dr Gsm

According to the classical thermodynamic theory, equation (2) can be simplified to the following formula,

dG ¼ SdT þ VdP þ

X

mbB dn þ sdA

(4)

B

where mbB is the bulk chemical potential of component B and s is the specific surface Gibbs function. The superscript b denotes the bulk

(7)

B

where Dr Gbm and Dr Gsm are the molar dissolution Gibbs energy of the bulk and surface phases, respectively. For a spherical nanoparticle without endoporus, ms can be expressed as follows,

ms ¼

4sV d

(8)

Dr Gm ¼ Dr Gbm þ

4vsV d

(9)

As can be seen from equation (9), the molar dissolution Gibbs energy decreases with the decrease of the particle size. On the base of equation (9) and ½ vðG=TÞ=vTp ¼ H=T 2 , the molar dissolution enthalpy Dr Hm is gotten, b s b Dr Hm ¼ Dr Hm þ Dr Hm ¼ Dr Hm þ

" #   4vV vs 2T sa sT  d vT p 3 (10)

b and D H s are where a is the coefficient of thermal expansion, Dr Hm r m the molar dissolution enthalpy of the bulk and surface phases, respectively. Similarly, the molar dissolution entropy Dr Sm can be obtained by equation (8) and ðvG=vTÞp ¼ S,



(3)

(5)

Dr Sm ¼

Dr Sbm

þ

Dr Ssm

¼

Dr Sbm

4vV  d

"

vs vT



# 2 þ sa 3 p

(11)

As for metals, the order of magnitudes of s, T, ðvs=vTÞP , and a are 100 Jm2, 102 K, 104 Jm2 K1, 105 K1, respectively [17e20], so the value in the bracket of equations (10) and (11) are positive, and both the molar enthalpy and the molar entropy decrease with the decrease of the particle size. On account of the transformation of the surface energy into the heat energy in dissolution, more heat releases for an exothermic reaction, or less heat absorbs for an endothermic reaction, causing the decrease of molar dissolution enthalpy. As for the decrease of the molar dissolution entropy, it is

Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

due to the larger surface entropy of nanoparticles. Based on the formulaDr Gm ¼ RT ln K, the following equation can be obtained,

ln

K 1 1 4nsV ¼  Dr Gsm ¼  RT RT d Kb

(12)

where K is the dissolution equilibrium constant. It can be seen from equation (12) that the smaller of the particle size, the greater the standard equilibrium constant. And when the diameter of nanoparticle approaches or exceeds 20 nm, the effect of particle size on the surface tension can be neglected [21,22]. So it can be deduced from equation (12) that the logarithm of standard equilibrium constants is inversely proportional to the particle size. 2.2. The feasibility analysis of nano-Cu dissolving in dilute acid It is actually a redox reaction to dissolve in dilute acid of nanoCu, which has been widely recognized to be difficult in the traditional theory of physical chemistry. But the dissolution reaction may proceed when the particle size of Cu is reduced to nanometer scale due to its strong surface effect. With the particle size decreasing, the specific surface area of nano-Cu increases, which leads to an obvious increase of the surface energy. And then the external energy needed for the reaction is certainly reduced, and the reaction is more likely to occur.

501

accelerating voltage of 10.0 kV. Typical SEM images of the asprepared nano-Cu are shown in Fig. 2. It was clear that the morphology of the as-obtained samples was nearly to spherical, while the particle size of the nano-Cu observed from the SEM may be a little bigger than that of the calculated values based on the XRD patterns due to the slight agglomeration of the nanoparticles. 3.3. Dissolution experiment The reactions were performed in a beaker which was placed at a DF-101S thermostat bath with magnetic stirrer. For each reaction, 100 mg of nano-Cu with different sizes was added into 100 mL of dilute sulfuric acid (0.0500 mol/L) respectively. The stirring speed is 900r/min. It was heated up to a setting temperature and was kept constant for use. Then the pH values and the concentrations of copper ion were measured with the PXST-216 ion analysis apparatus. The values were recorded every 10 s until they are unchanged which meant dissolution equilibrium, and the equilibrium concentration of copper ion can be got. Three parallel reactions were conducted for minimizing experimental error. Similarly, the reactions were performed at different temperatures of 298.15 K, 303.15 K, 308.15 K and 313.15 K. The dissolution reaction ionic equation of nano-Cu was expressed as follow,

CuðsÞ þ 2Hþ ðaqÞ/Cu2þ ðaqÞ þ H2 ðaqÞ

(13)

3. Experimental 3.4. Data processing 3.1. Preparation of nano-Cu Nano-Cu was prepared by liquid phase reduction [23]. Firstly, the mixture of CuSO4$5H2O and PVP at a certain ratio was placed in the airtight reactor at a setting temperature, then the solution of H2O$N2H4 was dropped in the reactor slowly. Nano-Cu with different sizes was obtained by changing the concentrates of reactants and the temperature of the reaction. The obtained samples were centrifuged and washed with distilled water and ethanol for several times, and then dried at 303.15 K for 4 h under vacuum condition, and the corresponding average particle diameters are shown in Table 1.

No bubble escaped from the solution during the dissolution process, and the hydrogen exists in the solution. The standard dissolution equilibrium constant of nano-Cu in dilute acid can be obtained,

Ko ¼

The as-prepared samples were examined using a Germany Bluker D8 Advance Powder Diffractometer (Cu Ka, k ¼ 0.154178 nm). Fig. 1 shows the XRD patterns of nano-Cu. The average particles size of samples was calculated by Scherrer formula based on the half peak width of characteristic diffraction peaks. The main characteristic peaks at 43.3 , 50.6 and 74.1 can be assigned to the (111), (200) and (220) planes of crystal structure, respectively. In addition, no other peak was observed, which suggests that the samples were pure. Morphology of samples was observed using JSM-6701F scanning electron microscope (SEM), which was operated at an Table 1 The average particle diameters (d) of nano-Cu prepared under different conditions. No.

T/K

pH

c(CuSO4)/(mol/L)

t/h

d/nm

1 2 3 4 5 6

343.15 343.15 323.15 323.15 303.15 303.15

11 11 11 11 11 11

0.5 1.0 0.5 1.0 0.5 1.0

1 1 1 1 1 1

21.5 24.2 26.2 29.1 35.1 38.4

(14)

where the superscript o denotes the standard state. ai is the activity of ion i. The activity is expressed as follows,

ai ¼ 3.2. Characterization of nano-Cu

aCu2þ aH2 a2Hþ

ci g coi i

(15)

where ci is ion concentration, coi ¼ 1 mol L1, and gi is the ionic activity coefficient. Especially, aH2 is the activity of hydrogen molecule in the solution, and it can be seen that aH2 zcH2 for the trace amounts of generated hydrogen and gH2 z1. Because of the high ionic strength of the solution, the activity coefficients may be estimated from the extended DebyeeHückel equation [15]:

ln gi ¼ 

A

pffiffi zi 1 þ ao B I

2

pffiffi I

(16)

where A and B are the DebyeeHückel parameters [24], I is the ionic strength of the medium, zi is the ion charge, and ao is the ion-size parameter which the value used here is 3.5 Å. The standard molar dissolution Gibbs energy of nano-Cu in dilute acid can be derived as follows,

Dr Gom ¼ RT ln K o

(17)

where R is the gas constant, and T is the temperature of the standard state.

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Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

dissolution temperature increases, the time needed for CCu2þ to achieve stable becomes shorter, and the equilibrium CCu2þ becomes larger. 4.1. Effect of size on the standard dissolution equilibrium constant of nano-Cu The standard dissolution equilibrium constant K o of nano-Cu under different temperatures were calculated by equation (14), and the relation between the logarithm of standard dissolution equilibrium constant ln K o and the reciprocal of diameter 1=d are presented in Fig. 4. Notable linear trends between ln K o and 1=d are shown in Fig. 4a, and ln K o increases with the decreasing of the particle size, which is consistent with the theoretical analysis of equation (12). The intercept and the slope of the lines (Fig. 4a) at different temperatures are shown in Fig. 4b. Each y-intercept and slope corresponds to the values of ln K bo and 4ð4vN sN BN Þ=ðRTÞ in equation (12). Fig. 8a shows that the deviation of intercept (ln K b ) of the lines for experimental equilibrium constant from the calculated

Fig. 1. The XRD patterns of nano-Cu with different sizes.

o and entropy The standard molar dissolution enthalpy Dr Hm o can be considered as constants within the temperature range, Dr Hm

and they can be calculated by the slope and the intercept of the linear regression for ln K o on 1=Tas equation (18), respectively,

ln K o ¼ 

o Dr Hm

RT

þ

Dr Som

(18)

R

values (ln K b ) obtained from theoretical equation at different temperatures. The error bars show that errors are very small at different temperatures. 4.2. Effect of size on the standard molar dissolution Gibbs energy of nano-Cu We can obtain the standard molar dissolution Gibbs energy

4. Results and discussion

Dr Gom of nano-Cu at different temperatures through equation (17), and the relations between Dr Gom and 1=d are shown in Fig. 5. It can be seen from Fig. 5a that good linear trends between Dr Gom and 1=d. As the particle size decreases, Dr Gom decreases, which is

Fig. 3 illustrates the variation of concentration with time of nano-Cu with different sizes at different temperatures. As shown in Fig. 3, in the dissolution process of nano-Cu in sulfuric acid, the CCu2þ increases dramatically in the beginning and becomes slowly till unchanged as dissolution progresses. Moreover, the smaller the particle size is, the shorter the time needs for CCu2þ to reach stable, and the larger the equilibrium CCu2þ is. With the

consistent with the theoretical analysis of equation (9). As the size decreases the surface energy, on Weber's account, increases, which makes the dissolution Gibbs energy decreases. The reported that electrode potential of nanoparticles increase with the particle size decreasing [25], which provides theoretical basis for explaining the phenomena from the perspective of electrochemical. That is, it is insoluble in diluted acid for a negative electrode potential of bulkCu as a reactant, while it can be soluble when the electrode

a

b

c

d

e

f

Fig. 2. The SEM images of nano-Cu with different sizes, a: 21.5 nm, b: 24.2 nm, c: 26.2 nm, d: 29.1 nm, e: 35.1 nm, f: 38.4 nm.

Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

503

0.60 38.4 nm

35.1 nm

-1

cCu (mmol L )

-1 2+

2+

0.2 298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.1 0.0

0

100

200

300

-1

cCu (mmol L )

0.3

0.30

0.15

2+

0.4

-0.1

0.6

29.1 nm

0.45

cCu (mmol L )

0.5

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.00

400

0

t (s)

100

200

300

t (s)

0.4

0.2

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.0 0

400

100

200

300

t (s)

400

0.8

26.2 nm

24.2 nm

0.60

0.8

21.5 nm

0.6

0.6

0.15

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.00

0

100

200

t (s)

300

0.4

-1

0.2

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.0

0

400

100

200

t (s)

300

2+

-1

2+

2+

0.30

cCu (mmol L )

cCu (mmol L )

-1

cCu (mmol L )

0.45

0.4

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

0.2

0.0

0

400

100

200

300

t (s)

400

Fig. 3. The plots of CCu2þ at different temperatures vs dissolution time.

potential increases to become positive as the particle size decrease. The intercept and slope of the line (Fig. 5a) at different temperatures are shown in Fig. 5b. Each y-intercept and slope corresponds to the value of Dr Gom and 4vN sN VN in equation (6). Fig. 9b shows deviation of the intercept (Dr Gbm ) of the line for experimental Gibbs energy change from the calculated values (Dr Gbm;calc ) from theoretical equation different temperatures. The error bars demonstrate that there are small errors at different temperatures.

4.3. Effect of size on the standard molar dissolution enthalpy of nano-Cu o and entropy The standard molar dissolution enthalpy Dr Hm Dr Som can be considered as a constant which is independent on the

temperature since the temperature changes slightly (298.15e318.15K) [26]. The relations between ln K o and 1=T are shown in Fig. 6.. Fig. 6 displays that ln K o decreases as the temperature decreases, and the remarkable linear trends confirmed the assumpo and D So as constants in the ranges of tion that considers Dr Hm r m o experiments. Dr Hm can be calculated by equation (18), and the o and 1=d are shown in Fig. 7. relation between Dr Hm o, As illustrated in Fig. 6, the size has notable influence on Dr Hm with size decreases, the molar dissolution enthalpy decreases, and o and 1=d, which conexhibits a linear relationship between Dr Hm sists with the theoretical analysis of equation (10). With the size decreasing, the surface energy increases, which provides energy for the dissolution of nano-Cu, so the less energy is needed for the dissolution of nano-Cu. -11.5

-12.0

50

-12.5

I

-11.5

40

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

-12.0 -12.5

b

45

-11.0

lnK

o

-10.5

a

55

S (nm)

-10.0

0.025

0.030

0.035 -1

-1

d /nm Fig. 4. a: Plot of ln

Ko

0.040

0.045

-13.0

35

intercept slope 0.050

-13.5 295

300

305

T (K)

310

315

versus 1=d at different temperatures. b: Plot of intercept (I) and slope (S) versus temperature.

30 320

Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

298.15 K 303.15 K 308.15 K 313.15 K 318.15 K

r

26

-90 -1

-100

32.0

-110

31.5

-120

31.0

a 0.025

33.0

-1

28

27

-80

32.5

I (kJ·mol )

29

o

-1

Gm (kJ· mol )

30

33.5

0.030

0.035

0.040

-1

0.045

0.050

S (kJ·nm·mol )

504

30.5 295

b

intercept slope 300

305

-130

310

315

320

T (K)

-1

d (nm )

Fig. 5. a: Plot of Dr Gom versus 1=d at different temperatures. b: Plot of intercept (I) and slope (S) versus temperature.

4.4. Effect of size on the standard molar dissolution entropy of nano-Cu

-1

Hm (kJ·mol )

60

50

r

o

The standard molar dissolution entropy Dr Som can be calculated by equation (18), and the relations between DSom and 1=d are shown in Fig. 8.. As is shown in Fig. 8, size has significant effect on the molar dissolution entropy of nano-Cu. With size decreasing, molar dissolution entropy decreases. The linear relations between DSom and 1=d coincides with the theoretical analysis of equation (11). As size decreases the surface deficiency becomes serious and the degree of chaos increases, which leads the dissolution entropy, by its very nature, decreases.

40

5. Error analysis

30

According to equations (12) and (9) and Figs. 4b and 5b, some parameters are calculated and listed in Table 2. Fig. 9 shows the deviation of intercepts (thermodynamic quantities of bulk Cu) of the fitted lines for experimental values from the theoretically calculated values, and small errors at different temperatures are observed.

0.025

0.030

0.035 -1

0.040

0.045

0.050

0.045

0.050

-1

d (nm ) o and 1=d. Fig. 7. The relation between Dr Hm

6. Conclusions

90

The influence regularities of particle size of nano-Cu on the

Sm(J·mol ·K )

80

-10.5

60

-12.0 -12.5

21.5 nm 24.2 nm 26.2 nm 29.1 nm 35.1 nm 38.4 nm 0.00315

0.00320

50

r

lnK

o

o

-11.0 -11.5

70

-1

-1

-10.0

40 30 0.00325 -1

-1

0.00330

0.00335

T (K ) Fig. 6. The relation between ln

Ko

and 1=T.

0.025

0.030

0.035

-1

0.040

-1

d (nm ) Fig. 8. The relations between Dr Som and 1=d.

Z. Li et al. / Materials Chemistry and Physics 214 (2018) 499e506

6.0

0.012 0.006 0.000

b

m,cal

5.2

b

10 ( G m- G

cal

)/lnK

b

cal

)/ G

b

m,cal

a

5.6

-0.006

b

4.8

b

(lnK -lnK

505

295

3

4.4 300

305

T (K)

310

315

320

b

-0.012 -0.018 295

300

305

310

315

320

T (K)

Fig. 9. Deviation of the intercept a:ln K b , b:Dr Gbm the line for experimental values from the corresponding calculated values from theoretical equation.

Table 2 Some parameters calculated from slope of the fitted lines compared with the corresponding one of theoretical formula at different temperatures. T (K)

½ð4vN sN VN Þ=ðRTÞ=ðnmÞa ð4vN sN VN Þ=ðkJ:nm:mol1 Þb Note

298.15

303.15

308.15

313.15

318.15

48.65 120.59

51.69 130.27

43.85 112.34

36.10 94.00

33.23 87.88

a, b

: Calculated from equations (12) and (9), respectively based on the slopes of the line in Figs. 3b and 4b.

dissolution equilibrium constant and the dissolution thermodynamic functions obtained in experiment agree with the thermodynamic relations deduced theoretically. The results show that the particle size has significant effect on the dissolution equilibrium constant and the dissolution thermodynamic functions; with the decrease of particle size, the dissolution equilibrium constant increases, while the standard molar dissolution Gibbs energy, the standard molar dissolution enthalpy and the standard molar dissolution entropy decrease; and that when the diameter approaches or exceeds 20 nm, the logarithm of the equilibrium constant, the molar Gibbs energy, the molar enthalpy, and the molar entropy, respectively, are linearly related with the reciprocal of diameter. The influence regularities of particle size on the dissolution thermodynamics are able to quantitatively predict the dissolution behaviors of nanoparticles and to provide valuable guidance for the preparations, researches and applications involving the dissolution of nanoparticles. The dissolving in dilute acid of nano-Cu indicates that the acid environment should be avoided when use the nanoCu as the high-performance catalyst, and the particle size could be well-controlled when wish to improve the drug bioavailability of nano-Cu. Reasonable use of this type of reaction can bring us much more surprise. Acknowledgements We are so grateful for the support from the National Natural Science Foundation of China (No. 21373147 and No. 21573157), which provide financial help in testing and characterization of the materials prepared. References [1] J.R. Morones, J.L. Elechiguerra, A. Camacho, The bactericidal effect of silver

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