and Shape-Dependent Thermodynamic Properties of the Actual

Article Cite This: J. Phys. Chem. C 2018, 122, 15713−15722

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Research of Size- and Shape-Dependent Thermodynamic Properties of the Actual Melting Process of Nanoparticles Qingshan Fu, Zixiang Cui,* Yongqiang Xue,* and Huijuan Duan

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Department of Applied Chemistry, Taiyuan University of Technology, Taiyuan 030024, China ABSTRACT: Melting phase transitions of nanoparticles are often involved in the preparations, research studies, and applications of nanomaterials. However, because of the changing melting temperature of nanoparticles during the melting process, the current relations of melting thermodynamic properties fail to accurately describe their actual melting behaviors. In this study, accurate thermodynamic relations between integral melting enthalpy and entropy and the size of nanoparticles with different shapes (sphere, rod, wire, and regular polyhedrons) were derived for the first time through designing a thermochemical cycle. In the experiment, Ag nanospheres, nanowires, and nanocubes with different sizes were prepared by chemical reduction methods, and differential scanning calorimetry was employed to determine the melting temperature, the melting enthalpy and the melting entropy. The experimental results agree with the theoretical predictions, indicating that the melting thermodynamic properties decrease with the particle size decrease and present linear variations with the inverse particle size within the experimental size range. Moreover, the melting enthalpy and entropy of nanoparticles in identical equivalent diameters take the same sequence as that of melting temperature as To(wire) > To(sphere) > To(cube). The derived relations of melting thermodynamic properties can quantitatively describe the actual melting behaviors of nanoparticles, and the findings herein provide us a comprehensive understanding of the melting thermodynamic properties of nanomaterials in the whole melting process.

1. INTRODUCTION Nanomaterials present considerable unique electronic, magnetic, optical, catalytic, and thermodynamic properties compared with conventional micro- or milli-sized materials.1−5 Among them, the melting thermodynamics of nanoparticles has received considerable attention worldwide because of its scientific and industrial importance, such as the exploitation of the melting mechanism6,7 and the fabrication of nanostructured materials and devices with desirable properties.8,9 The phase field approach has been routinely used to study the melting of nanoparticles.10−14 Additionally, numerous theoretical models,15−17 molecular dynamics simulations,18,19 and experiments20,21 have revealed melting temperature depression for nanoparticles. To better explicate and predict the thermodynamic behaviors of nanoparticles in the melting process, it is of significant importance and urgency to have a comprehensive understanding of effects of size and shape on the melting enthalpy and entropy. Numerous theoretical efforts have been made to investigate the effects of size on the melting enthalpy and entropy of nanoparticles. The model for size-dependent depression of melting entropy of nanocrystals was first proposed by Jiang et al.22 based on the equation for the vibrational entropy of a bulk crystal at the melting temperature23,24 and the size-dependent melting temperature.25 Thereafter, the relation between melting entropy and melting temperature of nanoparticles was employed in many other theoretical models26−29 to describe the size-dependent melting enthalpy and entropy. © 2018 American Chemical Society

These models explicated various melting phenomena and behaviors of nanoparticles, but the approximations were introduced to the derived relations because of the assumption that the relation of vibrational entropy and melting temperature of the bulk is applicable to the nano and the treatment that the vibrational entropy of melting equals the total melting entropy. Guisbiers and Buchaillot30 and Lu et al.31 derived similar relations of the nano to bulk ratio of melting enthalpy and melting temperature of nanoparticles by using different approaches. However, both the approaches unreasonably predicted that the melting entropy of the nano equals that of the bulk. Obviously, the melting enthalpy and entropy in the current relations22,26−33 belong to the differential quantities at the initial melting temperature. These differential quantities refer to the changes when only quite a few nanoparticles melt and then were converted to the changes of 1 mol. However, on the one hand, unlike macroscopic systems, it has been demonstrated that the melting temperature of nanoparticles is not a constant but a temperature range;34,35 that is, the melting enthalpy and entropy of nanoparticles at different melting stages are variational. On the other hand, for the change processes of the bulks, the differential thermodynamic properties are equal to the integral thermodynamic properties, Received: April 1, 2018 Revised: May 17, 2018 Published: June 20, 2018 15713

DOI: 10.1021/acs.jpcc.8b03085 J. Phys. Chem. C 2018, 122, 15713−15722

Article

The Journal of Physical Chemistry C

Figure 1. Thermochemical cycle using Hess’ Law to predict the size-dependent integral melting enthalpy and entropy of nanoparticles.

equilibrium is the equality of chemical potentials of the two phases. Hence, by the employment of this strict criterion, we have derived another thermodynamic relation of the melting temperature of nanoparticles with different shapes43 ÄÅ ÉÑ ΔslHmb σsl ÑÑÑÑ γM ÅÅÅÅ σlv(ρs − ρl ) − T = l b + l b ÅÅ Ñ rlρρ rsρs ÑÑÑÑÖ ΔsSm ΔsSm ÅÅÅÇ (1) s l

while for nanoparticles, the two types of thermodynamic properties are not equal.36,37 Furthermore, the melting enthalpy and entropy obtained from experiments are integral values, which represent the quantities in the whole melting process. Therefore, the current theories cannot accurately describe the actual melting process of nanoparticles. Additionally, theoretical investigation of the shape effects on the melting enthalpy and entropy of nanoparticles is still lacking. In the experiment, differential scanning calorimetry (DSC) has been employed to determine the melting enthalpy of various nanoparticles. Liu and Wang38 determined the melting enthalpy and entropy of colloidal In, Sn, and Bi nanocrystals, and both the properties decrease as the nanocrystal size decreases. Jähnert et al.39 determined the melting enthalpies of water confined in cylindrical silica nanopores which decrease with the pore diameter decrease and present a linear dependence on the reciprocal of the pore size. Zhang et al.40 determined the melting behavior of 0.1−10 nm thick indium films, and the melting enthalpy was found to decrease with decreasing film thickness. Similar experiments have been conducted for the melting of nonpolar organic nanosolids41 and so forth, and the same qualitative influence regularities of the particle size on the melting enthalpy and entropy were observed. To our knowledge, few experimental investigations of the shape effects on the melting enthalpy and entropy of nanoparticles have been reported till now. Herein, accurate thermodynamic relations of integral melting enthalpy and entropy to the size of nanoparticles with different shapes were derived, and the influence of regularities and mechanisms of particle size and shape on the integral melting enthalpy and entropy were discussed. Taking the melting of Ag nanospheres, nanowires, and nanocubes with different sizes as examples, the melting temperature, the melting enthalpy, and entropy were determined by DSC, and the size and shape effects on the melting thermodynamic properties were summarized and compared with the theoretical analysis.

where ΔlsHbm and ΔlsSbm denote the molar enthalpy and molar entropy change of the bulk phase at the melting temperature, respectively; M is the molar mass; σlv and σsl are the tensions of the liquid−vapor and solid−liquid interfaces, respectively; rs and rl are the size of the solid core and the external size of the liquid shell, respectively; ρs and ρl are the densities of the phases solid and liquid, respectively; and γ is a shape parameter. In the melting process of nanoparticles into liquid completely, a thermochemical cycle can be designed as Figure 1, and the change in the integral Gibbs energy ΔlsGm at the initial melting temperature Ti is ΔslGm(Ti) = Gm(l) − Gm(s) = ΔslGmb(Ti) + ΔslGms(Ti) σA = ΔslGmb(Ti) − s s ns (2)

where Gm(l) and Gm(s) represent the molar Gibbs energies of phases solid and liquid, respectively; σs, As, and ns represent the surface tension, the molar surface area, and the amount of substance, respectively; the superscripts b and s represent the bulk and the surface phases, respectively. According to the Gibbs−Helmholtz equation for the melting of nanoparticles, [∂(ΔlsGm/T)/∂T]p = −ΔlsHm/T2, the melting enthalpy at Ti, ΔlsHm(Ti), is ÉÑ ÄÅ ÑÑ ij A s yz ÅÅÅÅ ∂ σ i y Ñ s l l b ΔsHm(Ti) = ΔsHm(Ti) − jjj zzz ÅÅÅσs − Ti jjj zzz ÑÑÑÑ j ns z ÅÅ ∂ T k { pÑÑÑÖ k {T , pÅÇ ÅÄÅ ÑÉ ÅÅ ∂ i A y ÑÑÑ ÅÅ jj s zz ÑÑ + TiσsÅÅ jj zz ÑÑ ÅÅ ∂T j ns z ÑÑ ÅÅÇ k {T , pÑÑÖ p (3)

2. THERMODYNAMIC RELATIONS OF THE MELTING THERMODYNAMIC PROPERTIES TO THE SIZE OF NANOPARTICLES WITH DIFFERENT SHAPES A general expression for the melting temperature of nanoparticles of arbitrary shape was obtained by Kaptay et al.42 based on the equality of Gibbs energies of phases solid and liquid, namely, ΔmG = 0. However, from the chemical thermodynamics point of view, the criterion for phase

Then, the general relation of the integral melting enthalpy ΔlsHm of nanoparticles in the whole melting process can be obtained in conjunction with Figure 1 15714


ÄÅ ÉÑ ÄÅ ÉÑ ÅÅ Ñ Å Ñ ÅÅ ∂ ji A s zy ÑÑÑ ÅÅ jj zz ÑÑ = 3ÅÅÅ ∂ ijjj Vs yzzzÑÑÑ = 2Vsαs ÅÅ Ñ ÅÅ jj zz ÑÑ r ÅÇ ∂T k r {ÑÑÖ p ÅÅ ∂T k ns {T , pÑÑ ÅÇ ÑÖ p

The Journal of Physical Chemistry C ÄÅ ÉÑ ÑÑ ij A s yz ÅÅÅÅ ∂ σ i y Ñ s l l b j z ΔsHm = ΔsHm(Ti) − jjj zzz ÅÅÅσs − Ti jj zz ÑÑÑÑ j ns z ÅÅ ∂ T k { pÑÑÑÖ k {T , pÅÇ ÅÄÅ ÑÉ ÅÅ ∂ i A y ÑÑÑ Tf ÅÅ jj s zz ÑÑ + TiσsÅÅ jj zz ÑÑ + Cp ,l(T )dT ÅÅ ∂T j ns z ÑÑ Ti ÅÅÇ k {T , pÑÑÖ p

(4)

where Tf represents the final melting temperature and Cp,l represents the isobaric molar heat capacity of the planar liquid. Similarly, according to the following equation

+

+

∫T

i

T

ΔslSm

dT

A s = 4πr 2

ns =

πr 2l Vs

(17)

(18)

ij A s yz jj zz = 2Vs(l + r ) jj n zz rl k s {T , p

(19)

Then, the partial derivative of (As/ns)T,p against T is obtained ÄÅ ÉÑ ÅÄÅ ÑÉ Ñ ÅÅ ∂ i A y ÑÑÑ 2Vs ÅÅÅÅ ij ∂r yz 4V α (r + l) ij ∂l yz ÑÑÑ ÅÅ jj s zz ÑÑ ÅÅ jj zz ÑÑ = 2 ÅÅÅljj zz − r jj zz ÑÑÑ + s s ÅÅ ∂T j ns z ÑÑ Å Ñ ∂ T ∂ T 3rl rl k { pÑÖÑ ÅÅÇ k { p ÅÅÇ k {T , pÑÑÖ p

(9)

where Vs and ρs represent the molar volume and the density of the solid phase, respectively. The molar volume of spherical nanoparticles is Vs = N4πr3/ 3, where N denotes the number of nanoparticles of 1 mol. Then, the partial derivative of Vs against T is obtained as

(20)

Therefore, the accurate expressions for the integral melting enthalpy and entropy of the nanorod are as follows ÅÄÅ ÑÉÑ 2Vs(l + r ) ÅÅÅ ij ∂σs yz ÑÑÑ l l b ΔsHm = ΔsHm(Ti) − ÅÅÅσs − Ti jj zz ÑÑÑ ÅÅ rl k ∂T { pÑÑÑÖ ÅÇ Ä É Å ÑÑ | l o o o 2Vs ÅÅÅÅ ij ∂r yz 4Vsαs(r + l) o ij ∂l yz ÑÑÑ o o + Tiσsm } o rl 2 ÅÅÅÅljj ∂T zz − r jj ∂T zz ÑÑÑÑ + o o o 3rl k { k { o o Å Ñ p p ÅÇ ÑÖ n ~

(10)

The partial derivative of r against T obtained from eq 10 is

(11)

+

Then, Vs/r and [∂(Vs/r)/∂T]p are obtained as

∫T

Tf

Cp ,l(T )dT

(21)

2Vs(l + r ) ij ∂σs yz jj zz rl k ∂T { p Ä ÉÑ Å | l Ñ o o o 2Vs ÅÅÅÅ ji ∂r zy 4Vsαs(r + l) o i ∂l zy ÑÑÑ o o j Å Ñ j z j z + σsm − + l r } Å Ñ j z j z 2 o o Å Ñ o o 3rl k ∂T { pÑÑÑÖ o rl ÅÅÅÇ k ∂T { p o n ~ Tf Cp ,l(T ) + dT Ti T (22) i

2

Vs N 4πr = r 3 ÄÅ É ÅÅ ∂ i Vs yÑÑÑ ÅÅ jj zzÑÑ = N 8πr ijj ∂r yzz = 2 ijjj ∂Vs yzzz = 2Vsαs j z ÅÅ j zÑÑ 3 k ∂T { p 3 k ∂T { p 3r ÅÇ ∂T k r {ÑÖ p

(16)

Hence

hence

1 ij ∂Vs yz jij ∂r zyz = jj zz j z N 4πr 2 k ∂T { p k ∂T { p

Tf

A s = 2πrl + 2πr 2

(6)

(8)

ij ∂Vs yz i ∂r y jj zz = N 4πr 2jjj zzz k ∂T { p k ∂T { p

ÅÄÅ ÑÉÑ 3Vs ÅÅÅij ∂σs yz 2σsαs ÑÑÑ ÅÅjj zz + ÑÑ + r ÅÅÅk ∂T { p 3 ÑÑÑÑ ÅÇ Ö Cp ,l(T ) dT T

(15)

2.2. Rod and Wire. For the nanorod with section radius r and length l, the surface area and the amount of substance are

3

3V jij A s zyz jj zz = s j ns z r k {T , p

∫T

i

(7)

4πr 3Vs

=

Cp ,l(T )dT

ΔslSmb(Ti) +

2.1. Sphere. For spherical nanoparticles with radius r, the surface area and the amount of substance are

ns =

Tf

(5)

p

Cp , l(T )

∫T

i

one obtains the general relation of the integral melting entropy ΔlsSm of nanoparticles in the whole melting process as ÄÅ ÉÑ ÅÅÅ ∂ i A y ÑÑÑ i y ∂ σ A i y Å j z Ñ j z ΔslSm = ΔslSmb(Ti) + jjj s zzz jjj s zzz + σsÅÅÅ jjj s zzz ÑÑÑ j ns z k ∂T { ÅÅ ∂T j ns z ÑÑ p k {T , p ÅÅÇ k {T , pÑÑÖ Tf

(14)

Finally, the accurate expressions for the integral melting enthalpy and entropy of spherical nanoparticles are obtained as follows ÅÄÅ ÑÉÑ 3Vs ÅÅÅ 2Tiσsαs ÑÑÑ ij ∂σs yz l l b ΔsHm = ΔsHm(Ti) − ÅÅσs − Ti jj zz − ÑÑ r ÅÅÅÅ ∂T { p 3 ÑÑÑÑ k Ç Ö

∫

ij ∂ΔslGm yz jj zz = −Δl S jj zz s m k ∂T { p

Article

ΔslSm = ΔslSmb(Ti) +

(12)

(13)

where αs = (∂Vs/∂T)p/Vs represents the coefficient of volumetric expansion. Therefore

∫

15715


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The Journal of Physical Chemistry C Table 1. A, V, ε, and η of Nanoparticles with Different Shapes A

shape

πd3/6

3 l2

tetrahedron cube

6l2

octahedron

2 3 l2

ε

V

πd2

sphere

2 l 2/12 l3

2 l 2/3 2

3

dodecahedron

15 tan(0.3π)l

(15 + 7 5 )l /4

icosahedron

5 3 l2

5(3 +

5 )l 3/12

nanowire

∫T

Tf

Cp ,l(T )dT

ÄÅ ÉÑ ÅÅ ÑÑ 2 V ∂ σ 2 σ α i y Å Ñ ΔslSm = ΔslSmb(Ti) + s ÅÅÅÅjjj s zzz + s s ÑÑÑÑ r ÅÅÅk ∂T { p 3 ÑÑÑ Ç Ö Tf Cp ,l(T ) + dT Ti T i

∫

ij A s yz jj zz = εVs jj n zz l k s {T , p

ÄÅ ÉÑ ÅÅ Ñ ÅÅ ∂ ij A s yz ÑÑÑ ÅÅ jj zz ÑÑ = 2εVsαs ÅÅ jj zz ÑÑ 3l ÅÅ ∂T k ns {T , pÑÑ ÅÇ ÑÖ p

(23)

+

∫T

Tf

i

=

Cp ,l(T )dT

ÅÄÅ ÑÉÑ εVs ÅÅÅij ∂σs yz 2σsαs ÑÑÑ ÅÅjj zz + ÑÑ + l ÅÅÅk ∂T { p 3 ÑÑÑÑ ÅÇ Ö Cp ,l(T ) dT T

8.94

6.00

7.44

3 6

7.09

3(7 5 − 15) tan(0.3π )

6.58

3 3 (3 −

6.39

∫T

i

Tf

Tf

(24)

ΔslSm

5)

=

Cp ,l(T )dT ÅÄÅ ÑÉÑ ηVs ÅÅÅij ∂σs yz 2σsαs ÑÑÑ ÅÅjj zz + ÑÑ + de ÅÅÅÅk ∂T { p 3 ÑÑÑÑ Ç Ö Cp ,l(T ) dT T

(29)

ΔslSmb(Ti) +

∫T

i

Tf

(30)

For general metallic nanoparticles, the orders of σs, Vs, αs, (∂σs/∂T)p, and Cp,l are 10−1 to 100 N·m−1,44 10−5 m3·mol−1, 10−5 K−1,45 10−4 N·m−1·K−1,46 and 101 J·mol−1·K−1,47 respectively. Thus, the items containing Cp,l in eqs 29 and 30 can be roughly neglected according to the difference value (with order of 101 K)48−51 between Ti and Tf compared with the bulk melting enthalpy and entropy. According to the orders of the items in square brackets, the changes in integral melting enthalpy and entropy of nanoparticles are mainly determined by ηVsσs/de and ηVs(∂σs/∂T)p/de, respectively. In addition, for larger nanoparticles, the effect of the particle size on the surface tension can be reasonably neglected.52−54 Under these conditions, the effect of size on integral melting enthalpy and entropy depends on the specific surface area, and eqs 29 and 30 indicate that the integral melting enthalpy and entropy decrease with the particle size decrease and present linear variations with the inverse particle size. The effect of shape on the integral melting enthalpy and entropy is the combination effect of the specific surface area and the surface tension. For nanoparticles with an identical equivalent diameter, if the difference in surface tension is neglected, it is the specific surface area that plays a decisive role in the shape effect on the integral melting enthalpy and entropy. Together with Table 1, it is found that the more the shape of nanoparticles (all dimensions are nanoscale) deviates from the sphere, the larger the specific surface area is, and hence the smaller the integral melting enthalpy and entropy are.

(25)

(26)

(27)

ΔslSmb(Ti) +

∫T

i

where ε is a shape parameter. Therefore, the accurate expressions for the integral melting enthalpy and entropy of regular polyhedral nanoparticles are obtained as follows ÅÄÅ ÑÉÑ εVs ÅÅÅ 2Tiσsαs ÑÑÑ ij ∂σs yz l l b ΔsHm = ΔsHm(Ti) − ÅÅσs − Ti jj zz − ÑÑ l ÅÅÅÅ ∂T { p 3 ÑÑÑÑ k Ç Ö

ΔslSm

6 6

surface area of regular polyhedral nanoparticles to that of spherical ones with identical volume. The related parameters of nanoparticles with different shapes are listed in Table 1. Then the expressions for the integral melting enthalpy and entropy of nanoparticles with different shapes can be generalized as follows ÅÄÅ ÑÉÑ ηVs ÅÅÅ 2Tiσsαs ÑÑÑ ij ∂σs yz l l b ÅÅσs − Ti jj zz − ÑÑ ΔsHm = ΔsHm(Ti) − de ÅÅÅÅ ∂T { p 3 ÑÑÑÑ k Ç Ö

2.3. Regular Polyhedrons. For the melting of regular polyhedral nanoparticles (tetrahedron, cube, octahedron, dodecahedron, and icosahedron), the side length before melting is l. Similarly

+

6.00

4.00

If l ≫ r, the nanorod becomes the nanowire, and then the accurate expressions for the integral melting enthalpy and entropy of the nanowire are obtained based on eqs 21 and 22 ÄÅ ÉÑ Å Å ÑÑÑ 2 V ∂ σ 2 T σ α i y Å ΔslHm = ΔslHmb(Ti) − s ÅÅÅÅσs − Ti jjj s zzz − i s s ÑÑÑÑ r ÅÅÅ 3 ÑÑÑ k ∂T { p Ç Ö +

η

6.00

(28)

For regular polyhedral nanoparticles in identical volume, ε/l = η/de, where de denotes the equivalent diameter, and η denotes another shape parameter, that is, the ratio of the molar 15716


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3. EXPERIMENTAL SECTION 3.1. Materials. AgNO3 (≥99.8%), FeCl3·6H2O (≥99.0%), and ethylene glycol (≥99.0%) were purchased from Sinopharm Chemical Reagent Co., Ltd. Hydrazine hydrate (80 wt %, >98.0%), trisodium citrate (≥99.0%), Na2S·9H2O (≥98.0%), and polyvinylpyrrolidone (PVP) powder (Mw ≈ 10 000) were purchased from Aldrich. The water used was double-distilled water. All the chemicals were used as purchased without further purification. 3.2. Instruments. The X-ray diffraction (XRD) patterns of the as-prepared samples were determined by XRD-6000 (Cu Kα, λ = 0.154178 nm) at a scanning rate of 8°·min−1 in the 2θ range from 30° to 80°. The scanning electron microscopy (SEM) images were recorded on a JEOL JSM-6701F scanning electron microscope, using an accelerating voltage of 200 kV. Differential scanning calorimetry (DSC, NETZSCH 404F3) was employed to measure the melting thermodynamic properties. After the DSC instrument was calibrated with three high-purity standard samples (In 99.999%, Al 99.999%, and Ag 99.99%) at three heating rates (5, 10, and 15 K·min−1), nano-Ag were hermetically sealed into the aluminum oxide crucible and placed in the DSC cell and then heated at a rate of 10 K·min−1 from 20 to 1100 °C with the protection of highpurity argon at a rate of 50 mL·min−1. 3.3. Preparation of Nano-Ag with Different Shapes. 3.3.1. Preparation of Ag Nanospheres. In a typical synthesis, an aqueous solution of AgNO3 (0.03 M, 20 mL), containing 35 mg (6 mM) of trisodium citrate, was stirred in a boiling flask-3neck (250 mL) and heated up to 50 °C. To this solution, hydrazine hydrate (0.06 M, 130 mL) was rapidly injected, and then the mixture solution was stirred for 30 min more. After cooling to room temperature, the resulting suspension was allowed to cool in air and centrifuged, and the precipitation was washed with distilled water and cyclohexane two times in sequence and then stored in cyclohexane until the following characterizations. To prepare samples with different sizes, the concentrations of silver nitrate and trisodium citrate, the adding type, and the reaction temperature were changed (Table 2).

of AgNO3 to PVP, and the reaction time were changed (Table 3). Table 3. Different Reaction Conditions for Preparing Ag Nanowires sample

C(AgNO3)/M

C(PVP)/M

C(FeCl3)/μM

reaction time/min

G H I J

0.10 0.10 0.10 0.10

0.10 0.10 0.08 0.10

90 90 60 90

150 180 180 220

3.3.3. Preparation of Ag Nanocubes. In a typical synthesis, a 10 mL solution with a final concentration of 50 μM Na2S and 0.12 M (in terms of the repeating unit) PVP in ethylene glycol was prepared. While stirring vigorously, 10 mL of 0.12 M AgNO3 was rapidly added. After continuous stirring for 5 min, the mixture was transferred to a Teflon-sealed autoclave of 25 mL capacity and heated at 140 °C for 150 min. After the reaction, the product was obtained through the same follow-up process as that of Ag nanospheres and nanowires. To prepare samples with different diameters, the concentration of Na2S, the molar ratio of AgNO3 to PVP, and the reaction time were changed (Table 4). Table 4. Different Reaction Conditions for Preparing Ag Nanocubes sample

C(AgNO3)/M

C(PVP)/M

C(Na2S)/μM

reaction time/min

k l m

0.12 0.10 0.10

0.12 0.12 0.12

50 55 50

150 150 180

3.4. Data Processing. In the DSC curve (T as abscissa and heat flow as ordinate), as shown in Figure 2, the initial and

Table 2. Different Reaction Conditions for Preparing Ag Nanospheres sample C(AgNO3)/M A B C D E F

0.03 0.03 0.03 0.03 0.06 0.06

C(C6H5Na3O7)/M

adding types

reaction temperature/°C

0.006 0.003 0.006 0.002 0.012 0.012

injection injection injection injection injection slow drip

50 50 70 50 60 60

Figure 2. Diagram of the DSC curve of the melting of nanoparticles.

final melting temperatures are Ti and Tf, respectively, and the onset melting temperature is To. The melting enthalpy can be calculated by the sum of the area of each small trapezoid (ΔlsH = ∑(y1 + y2)δT/2), and the melting entropy by ΔlsS = ∑δHj/ Tj (where δHj and Tj indicate the melting enthalpy for each trapezoid and the average temperature corresponding to each trapezoid, respectively).

3.3.2. Preparation of Ag Nanowires. In a typical synthesis, a 20 mL solution with a final concentration of 90 μM FeCl3 and 0.1 M (in terms of the repeating unit) PVP in ethylene glycol was prepared. While stirring vigorously, 20 mL of 0.1 M AgNO3 was rapidly added. After continuous stirring for 5 min, the mixture was transferred to a Teflon-sealed autoclave of 50 mL capacity and heated at 120 °C for 150 min. After the reaction, the product was obtained through the same follow-up process as that of Ag nanospheres. To prepare samples with different diameters, the concentration of FeCl3, the molar ratio

4. RESULTS AND DISCUSSION Figure 3 shows the XRD patterns of the as-prepared samples. It can be concluded that the nano-Ag are face-centered cubic structures, and the diffraction peaks can be indexed to (111), (200), (220), and (311) reflections. 15717


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Figure 3. XRD patterns of (a) Ag nanospheres, (b) Ag nanowires, and (c) Ag nanocubes with different sizes.

Figure 4. SEM images of Ag nanospheres: (a) 23.2, (b) 33.0, (c) 51.3, (d) 61.5, (e) 76.2, and (f) 120.3 nm; Ag nanowires: (g) 57.8, (h) 69.3, (i) 87.2, and (j) 95.2 nm; and Ag nanocubes: (k) 93.6, (l) 138.1, and (m) 165.3 nm.

Figure 5. DSC curves of the melting of nano-Ag with shapes of (a) sphere, (b) wire, and (c) cube.

Figure 5a−c shows the DSC curves (the downward peak indicates the endothermic peak) for the melting of Ag nanospheres, nanowires, and cubes, respectively, with different

Figure 4 shows the scanning electron microscopy images of nano-Ag which possess relatively uniform morphologies of sphere, wire, and cube and narrow particle size distributions. 15718


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Figure 6. Variation of To with the inverse particle size of nano-Ag with shapes of (a) sphere, (b) wire, and (c) cube.

Figure 7. Variation of ΔlsHm with the inverse particle size of nano-Ag with shapes of (a) sphere, (b) wire, and (c) cube.

Figure 8. Variation of ΔlsSm with the inverse particle size of nano-Ag with shapes of (a) sphere, (b) wire, and (c) cube.

surface energy just equals the heat needed for the melting of nanoparticles. In addition, when the particle size approaches to infinity, the melting enthalpy of bulk Ag (ΔlsHbm) is obtained with values of 10.98, 11.01, and 10.96 kJ·mol−1 based on the fitted lines of melting enthalpy of nano-Ag with different shapes. All the three values are very close to 11.30 kJ·mol−1,56 and because of the heat loss in the determination process of melting, the obtained ΔlsHbm is therefore lower but acceptable. Figure 8 reveals that the melting entropies of nano-Ag with different shapes also decrease with the reducing particle size and present good linear relations with the reciprocal of the particle size, which agree with eqs 16, 24, and 28, respectively. When ΔlsSm = 0, the critical values of the particle sizes are 1.24, 1.00, and 1.45 nm for Ag nanosphere, nanowire, and nanocube, respectively, which are in agreement with the theoretical prediction.32 The phenomenon indicates that because of the increasing entropy of solid particles with the particle size decrease, while the relatively small entropy of highly ordered liquid, the entropy of solid particles is equal to that of the liquid just at these critical particle sizes. In addition, on the basis of the fitted lines of melting entropy of nano-Ag with shapes of sphere, wire, and cube, the melting entropies of bulk Ag (ΔlsSbm) obtained herein are 9.01, 9.01, and 8.96 J·mol−1· K−1, respectively, which are also very close to 9.11 J·mol−1·K−1 (calculated by the equation ΔlsSbm = ΔlsHbm/T0, where T0 is the melting temperature of bulk Ag). Similarly, because of the loss

sizes. Obviously, the initial melting temperature, the melting temperature at the maximum rate, and the final melting temperature decrease with the particle size decrease. Figure 6 illustrates that the To decreases with the particle size decrease. In addition, it is evident that there is a good linear relation between To and the reciprocal of the particle size. That is because when the solid and liquid phases reach equilibrium, ρs ≈ ρl, and for larger nanoparticles, ΔlsHbm(T)/ ΔlsSbm(T) ≈ ΔlsHbm(T0)/ΔlsSbm(T0) = T0. Then, eq 1 can be approximated as T = T0 − γVsσsl/(ΔlsSbmr). Together with the fact that the effect of size on the surface tension can be reasonably neglected for larger nanoparticles,52−54 the melting temperature presents linear variations with the reciprocal of the particle size. In addition, the melting temperature of bulk Ag can be obtained with values of 962.6, 963.2, and 962.9 °C based on the fitted lines of melting temperature of nano-Ag with shapes of wire, sphere, and cube. All the three values are very close to 961.8 °C.55 Figure 7 shows that the melting enthalpies of nano-Ag with different shapes decrease with the reducing particle size and present good linear relations with the inverse particle size, which agree with eqs 15, 23, and 27. When the melting enthalpy ΔlsHm = 0, the critical values of particle sizes are 1.38, 1.23, and 1.97 nm for Ag nanosphere, nanowire, and nanocube, respectively, which coincide with the theoretical prediction.32 The phenomenon means that when the particle size decreases, the surface energy increases, and at these critical values, the 15719


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Figure 9. Comparison of the (a) melting temperature, (b) melting enthalpy, and (c) melting entropy of nano-Ag with different shapes.

with identical volume, it is found that the more the shape of nanoparticles deviates from sphere, the smaller the melting thermodynamic properties are. The smaller the particle size, the greater is the effect of shape on the melting thermodynamic properties. The findings herein are capable of completing our understanding of the size- and-shape dependent melting thermodynamic behaviors of nanoparticles in combination with the corresponding differential relations.

of heat in the determination process of melting, the obtained ΔlsSbm is therefore lower. To ascertain the effects of shape on the melting thermodynamic properties of nanoparticles, Figure 9a−c compares the melting temperature, the melting enthalpy, and entropy of nano-Ag with different shapes. Figure 9a shows that the melting temperature of nano-Ag with shapes of sphere, wire, and cube decrease with the decreasing particle size, and the smaller the particle size, the greater the effect of shape on the melting temperature. At the same equivalent diameter, there is To(wire) > To(sphere) > To(cube). That is because at the same equivalent diameter, the specific surface area and the surface energy of Ag nanocubes are the largest, and the heat needed for the melting is the least, that is, the melting is easiest, and therefore the melting temperature of Ag nanocubes is lowest. In addition, only the dimension to crossrange is nanometer scale, the melting temperature of Ag nanowires is lower than that of bulk Ag but larger than that of Ag nanospheres. In Figure 9b, it is obvious that with the particle size decrease, the melting enthalpies of Ag nanospheres, nanowires, and nanocubes decrease, and the shape effect on the melting enthalpy becomes greater. For nano-Ag with different shapes but the same equivalent diameter, the melting enthalpy follows the sequence: ΔlsHm(wire) > ΔlsHm(sphere) > ΔlsHm(cube). That is because at the same equivalent diameter, the surface energy of Ag nanocubes is the largest, and the heat needed for the melting of Ag nanocubes is the least, and therefore, the melting enthalpy of Ag nanocubes is the smallest which is in accordance with eq 29 and literature results.32 Similarly, the melting enthalpy of Ag nanospheres is smaller than that of Ag nanowires. Figure 9c shows that the melting entropy of nano-Ag with different shapes decrease with the decreasing particle size, and the smaller the particle size, the greater the effect of shape on the melting entropy. At the same equivalent diameter, there is ΔlsSm(wire) > ΔlsSm(sphere) > ΔlsSm(cube). That is because the surface defects are more serious (more confusion), and the surface entropy is larger, while the entropy of the final melting state (flat liquid) is a constant, and therefore, the melting entropy of Ag nanocubes is the smallest which coincides with eq 30 and literature results.32 Similarly, the melting entropy of Ag nanowires is larger than that of Ag nanospheres.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Z.C.). *E-mail: [email protected] (Y.X.). ORCID

Qingshan Fu: 0000-0002-9459-7682 Zixiang Cui: 0000-0001-8323-9612 Yongqiang Xue: 0000-0002-5707-2596 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are very grateful for the financial support from the National Natural Science Foundation of China (nos. 21573157 and 21373147).

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REFERENCES

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