Thermal Stability of Nanocrystals Confined in Nanoporous Media

Aug 12, 2009 - based on the perspective of Lindemann's criterion for melting. According to this model, the depression or enhancement of thermal stabil...
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J. Phys. Chem. C 2009, 113, 16036–16041

Thermal Stability of Nanocrystals Confined in Nanoporous Media X. Y. Lang* and L. P. Han Institute for Materials Research, Tohoku UniVersity, Sendai 980-8577, Japan ReceiVed: May 25, 2009; ReVised Manuscript ReceiVed: July 19, 2009

Thermal stability of nanocrystals confined in nanopores is dramatically different from their bulk counterparts. To understand the underlying mechanism and factors that dominate the general trend of tunability, we have established a simple and unified model for size and interface effects on the thermal stability of nanocrystals based on the perspective of Lindemann’s criterion for melting. According to this model, the depression or enhancement of thermal stability of nanoconfined crystals has been illustrated to depend on the confluence effect of size and interfacial conditions. Model predictions are consistent with available experimental results for molecular gases (oxygen, neon, argon, and krypton), organic molecules (benzene, nitrobenzene, carbon tetrachloride, and cyclohexane), water, and metallic (gallium, indium, and mercury) nanocrystals confined in nanoporous media. [Tm(D) - Tm(∞)]/Tm(∞) ) -2δV/[DHm(∞)]

Introduction Fascinating physicochemical properties of nanocrystals have attracted great interest because of scientific and industrial importance during the past decades.1 Recently, thermal stability of nanocrystals nanoconfined in nanoporous (2-50 nm) media has been paid special attention because of its practical significance in many diverse scientific and engineering processes like catalysis,2 sensors,3 and fabrication of nanomaterials4 as well as characterization of porous media.5–7 Thermal stability of nanocrystals is characterized by melting temperature Tm(D), where D denotes the thickness of thin films, and diameter of nanoparticles or nanorods. It has been revealed that Tm(D) dramatically differs from bulk because of a high surface/volume ratio and interface effects.5–9 So far, there have been a number of experimental investigations carried out using different siliceous materials (mainly, MCM-41, SBA-15, and Vycor glass) and a wide variety of molecular gases such as hydrogen (H2),10,11 nitrogen (N2),12 oxygen (O2),11,13–15 neon (Ne),10,11 argon (Ar),11,16–20 krypton (Kr),12 carbon monoxide (CO),12,17 and carbon dioxide (CO2)21 as well as organic molecules such as carbon tetrachloride (CCl4),22–26 nitrobenzene (C6H5NO2),22,27 benzene (C6H6),28–30 cyclohexane (C6H12),28,29,31,32 water (H2O),33–37 and metallic indium (In),38 mercury (Hg),39–41 and gallium (Ga).42,43 Computer simulations and theoretical investigations have shed much light on the molecular details underlying the structural and dynamic behavior in this highly confined regime.6,9,25,44 The majority of experiments have demonstrated that Tm(D) is suppressed below Tm(∞), and that the lowering becomes greater as D is reduced, where ∞ denotes bulk size. For sufficiently large pores, the shift in freezing or melting temperature can be related to the size of pores on the basis of the Gibbs-Thomson thermodynamic equation that is obtained either by equating the free energies of the confined liquid and solids or by determining the temperature at which the chemical potential of the confined solid equals that of the bulk reservoir,45,46 * To whom correspondence should be addressed. E-mail: xylang@ wpi-aimr.tohoku.ac.jp.

(1)

with δ ) γws - γwf, where γws and γwf are the corresponding wall-solid and wall-fluid surface tensions, V is the molar volume of the liquid phase, and Hm(∞) is the bulk latent heat of melting. However, an opposite trend in the shift of Tm(D) has been revealed in the rare cases that nanocrystals are confined in nanopores with strong interaction at a nanocrystal-wall interface, where Tm(D) increases with decreasing D.9,24,25,32,47 Such an increase is consistent with the Gibbs-Thomson equation, if one recognizes that the direction of the shift of Tm(D) depends on whether the pore wall favors the confined solid phase or confined fluid phase. If γws is greater than γwf, then the shift of ∆Tm(D) ) Tm(D) - Tm(∞) is predicted to be negative; otherwise, it is positive. The latter can be attributed to the strength of the nanocrystal-wall interaction; that is, the solid phase, rather than the liquid phase, wets the interior pore surface.24,25,32,47 In previous literature, although eq 1 provides reasonable descriptions of experimental results for size-induced undercooling in a melting transition with the aid of adjustable parameters γws and γwf,5,6 eq 1 is strictly valid only for a larger D. Furthermore, in the process of fitting with experimental results, the utilization of free parameters could mislead understandings of the related underlying mechanism, and of Tm(D). Insight into the nature of thermal stability of nanocrystals is important for understanding the new physics that occurs due to finite-size effects, surface forces, and reduced dimensionality.5,6 Thus, deeper and consistent insight into the mechanism behind the unusual observations and finding factors are highly desirable. In this contribution, a simple and unified model is established for size and interface effects on thermal stability of nanocrystals within nanopores. The model enables us to reproduce available experimental results of molecular gases, organic molecules, and metallic nanocrystals without any adjustable parameter. This unification certainly gives rise to the comprehension of thermal stability nanocrystals. Methodology It is well-known that the termination of lattice periodicity in the surface normal leads to two effects. One is the coordination

10.1021/jp904844s CCC: $40.75  2009 American Chemical Society Published on Web 08/12/2009

Thermal Stability of Nanocrystals

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number (CN) reduction of surface atoms, and the other is the creation of a surface potential barrier induced by this CN reduction.48 Both effects give rise to surface physical quantities Qs of crystals differing from internal counterparts Qv, where subscript s and v denote atoms or molecules located at the surface and within crystals, respectively. As a result of the surface effect and increasing number of surface atoms due to a decreasing D of nanocrystals, physical quantities of nanocrystals Q(D) deviate dramatically from their corresponding bulk counterparts Q(∞).48,49 On the basis of mean-field approximation, a Q(D) function can be given by48–50

Q(D) ) Qv(D) + [Qs(D) - Qv(D)]ns /n

(2)

where ns/n ∝ 1/D is the ratio of the number of atoms or molecules located at surface layer (ns) to the total number of atoms (n) of nanocrystals.49 If the cooperative coupling between the surface region and interior region is phenomenologically considered by taking the variation of Q(D) to be dependent on the value of Q(D) itself, a change in Q can give rise to Q(x + dx) - Q(x) ) (Rs 1)Q(x)dx,49,51 which is achieved by assuming that Qs/Qv are size independent, and Rs ) Qs(D)/Qv(D) ≈ Qs(∞)/Qv(∞), x ) ns/nv ) D0/(D - D0), with D0 denoting a critical size at which all the atoms or molecules of nanocrystals are located on their surfaces. In terms of the definition of D0 and the bulk boundary condition, two asymptotic limits should be satisfied, namely, Q(D)/Q(∞) f 0 and Q(D)/Q(∞) f 1, when D f D0 and D f ∞, respectively. Associated with these boundary conditions and integrating the above equation, one can obtain

Q(D)/Q(∞) ) exp[(Rs - 1)ns /nv] ) exp[(Rs - 1)/(D/D0 - 1)] (3) For low-dimensional nanocrystals, D0 is related to the dimensionality of nanocrystals (d), and the normalized surface (interface) area (c).52,53 In general, the former can be fractal.52 The latter results from the different potential energy in comparison with the interior, for example, c ) 1 for free nanocrystals, while c ) 1/2 for nanocrystals supported by substrates.53 Let h denote the atomic or molecular diameter in the usual crystalline lattice, the values of D0 for free nanoparticles (d ) 0), nanorods or nanowires (d ) 1), and thin films (d ) 2) are determined to be 6h, 4h, and 2h, respectively, according to the eqs 4πh(D0/2)2 ) 4π(D0/2)3/3, 2πh(D0/2) ) π(D0/2)2, and 2h ) 2(D0/2) because of all atoms or molecules localized on the surface of nanoparticles, nanorods, nanowires, and thin films. In short,

D0 ) 2c(3 - d)h

(4)

To describe the thermal stability of nanocrystals, in the light of the perspective of the Lindemann criterion, let Q(D) be the mean-square displacement of atoms or molecules σ2(D,T) at temperature T, which results in the extension of eq 3, namely, σ2(D,T)/σ2(∞,T) ) exp[(Rs - 1)/(D/D0 - 1)]. When T is much larger than the Debye temperature, the high temperature approximation gives rise to σ2(D,T) ) kBT/F(D), where F(D) is T-independent and a size-dependent vibrational force constant.49,54 As T is increased to the melting temperature of the bulk [Tm(∞)] or nanocrystals [Tm(D)], σ2[D,Tm(D)]/ σ2[∞,Tm(∞)] ) [F(∞)/F(D)][Tm(D)/Tm(∞)]. According to Lin-

demann’s criterion that a crystal melts when σ2/h2 ≈ c,49 where c is a constant dependent on the crystal structure, {σ2[D,Tm(D)]/ h2}/{σ2[∞,Tm(∞)]/h2} ) [F(∞)/F(D)][Tm(D)/Tm(∞)] ) 1.49 Note that h is assumed to be size independent because [h - h(D)]/h is negligible when D > 20 nm and [h - h(D)]/h ) 0.1-2.5%, when D < 20 nm.55 Therefore, Tm(D)/Tm(∞) ) σ2(∞)/σ2(D) ) exp[-(Rs - 1)/(D/D0 - 1)]

(5) with Rs ) σs2/σv2. For freestanding nanocrystals, Rs has been given by52 Rs ) 1 + 2Svib(∞)/(3R), with R being the ideal gas constant, and Svib(∞) the vibrational contribution of the melting entropy of crystals. While for nanocrystals nanoconfined in pores with interaction at the crystal-wall interfaces such as van der Waals force or hydrogen bonding, etc., subscript i denotes this interface, the corresponding Ri ) σi2(D)/σv2(D). Because Rs ) σs2(D)/σv2(D), Ri ) Rsσi2(D)/σs2(D). If bond strength E is assumed to be reversely proportional to σ2 because of F ∝ E,56,57 namely, σs2(D) ∝ 1/Es and σi2(D) ∝ 1/Ei,53 σi2(D)/σs2(D) ) Es/Ei or

Ri ) [1 + 2SVib(∞)/(3R)]Es /Ei

(6)

For this kind of case, only the interface effect needs to be considered because the side surfaces of nanocrystals have a small percentage of the total surface in comparison with that of interfaces. Substituting eq 6 into eq 5 induces the Tm(D) function of nanocrystals embedded in porous media

Tm(D)/Tm(∞) ) exp{-{[1 + 2Svib(∞)/ (3R)]Es /Ei - 1}/(D/D0 - 1)} (7) As a rule, melting entropy Sm(∞) consists at least of positional Spos(∞), vibrational Svib(∞), and electric Sel(∞),58 Sm(∞) ) Svib(∞) + Spos(∞) + Sel(∞). The idea of Spos(∞) of melting arises in connection with the positional disorder as a substance undergoes the melting transition. The number of particle species naturally plays a primary role involved in the disordering process. In the case of simple solids, only two particle species are present: atoms of the given substance and vacancies.58 In this case, Spos(∞) is given by,58 Spos(∞) ) -R(xAlnxA + xvlnxv) with xA ) 1/(1 + ∆Vm/Vm), where ∆ denotes the difference during the melting and xv ) 1 - xA are the molar fractions of the host material and vacancies, respectively. For metallic crystals, the type of chemical connection does not change during the melting transition. Thus, Sel(∞) ≈ 058 and Svib(∞) ) Sm(∞) - Spos(∞) or

Svib(∞) ) Sm(∞) + R(xAln xA + xvln xv)

(8)

For semimetals, Sel(∞) * 0, and Svib(∞) must be determined in a direct way, i.e., the Mott equation59

Svib(∞) ) 3Rln(υs /υ1) ) (3/2)R ln(µs /µ1)

(9)

where υ and µ denote characteristic vibration frequency and electrical conductivity, respectively. If the parameters in eq 9 are unavailable, the following equation can also be employed as a rough approximation,58

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Svib(∞) ) Sm(∞) - R

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(10)

TABLE 1: Parameters Utilized in Calculations of Eq 7 Tm(∞) (K)

For semiconductors, the melting is accompanied by a semiconductor-to-metallic transition, and the elements or compounds suffer contraction in volume rather than expansion for most metals. Thus, Sel(∞) strongly contributes to Sm(∞) and Spos(∞) , Sel(∞). Spos(∞) is thus negligible as a first order approximation.60 Namely,

Svib(∞) ) Sm(∞) - Sel(∞)

(11)

Note that eq 11 is invalid for some metallic mixing oxides; for instance, Fe3O4 consists of Fe2O3 + FeO, which undergoes semiconductor-to-metal transition due to the Fe2+ and Fe3+ ions order on the B sites at 122 K (so-called Verwey transition), while Tm is far above the Verwey temperature.61 Results and Discussion Panels a-d of Figure 1 show comparisons of Tm(D) functions between model predictions according to eq 7 and available experimental results of O2,11,13–15 Ne,10,11 A,r11,16–18,20 and Kr12 nanoconfined in porous media (MCM-41, SBA-15, and Vycor glass), respectively, where the utilized parameters are listed in Table 1. As shown in Figure 1, a general trend of Tm(D) functions of gas molecules confined in nanoporous matrix is size-independent evidently in the range of D > 20 nm. As D is further reduced, Tm(D) decreases. Finally, Tm(D) approaches zero for clusters small enough.11–18,20 This trend results from the combined effect of the molecule-wall interface interaction and the increasing number of interface molecules.6,44,49,53 On one hand, the former gives rise to not only the lattice termination at interface but also the disappearance of one of two free surfaces of films because of the interaction between the nanocrystal and

O2 Ne Ar Kr H2O CCl4 C6H5NO2 C6 H6 C6H12 Ga In Hg

11

54.4 24.611 84.011 115.9512 273.1566 250.024 278.822 278.528 279.528 303.042 429.765 234.439

Svib(∞) (J g atom-1 K-1) 60

R ln2 R ln260 R ln260 R ln260 7.37066 R ln260 3.36722 2.87228 R ln260 10.14b 7.59465 9.79965

h65 (nm) 0.3299 0.3130 0.3405 0.4040 0.3104a 0.5429a 0.5830a 0.5287a 0.5870a 0.2442 0.3251 0.3005

a For water and organic molecules, h ) [M/(FNa)]1/3 with F ) 1.0, and 1.594, 1.249, 0.8765, 0.6897 g cm-3 for H2O, CCl4, C6H5NO2, and C6H6, C6H12,67 respectively, where M is molecular weight, F is density, and Na is the Avogadro constant. b For Ga, Svib(∞) ) 10.14 J g atom-1 K-1 in terms of eq 10 with Sm(∞) ) 18.45 J g atom-1 K-1.65

pore wall. Thus, when the interactions between molecules at the surface and interface are assumed to be approximately equal to each other, Ri ) Rs ) 1 + 2Svib(∞)/(3R) > 1 in terms of eq 6 with Ei ≈ Es, and D0 ) (3 - d)h, according to eq 4 with c ) 1/2, which results in the intrinsic difference from the freestanding nanocrystals, i.e., the change of Tm(D) of supported films is weaker than that of free-standing ones. On the other hand, as D decreases, the increasing number of interface molecules brings forth the finite-size effect because the interface effect becomes more and more significant.45,48–51 According to this model, the reduced Tm(D) function of nanocrystals confined in nanopores should be attributed to the CN reduction induced by lattice termination at interfaces. This is also confirmed by the fact that the different degree of CN reduction of nanocrystals can lead to a different trend of the

Figure 1. Comparisons of Tm(D) functions of molecular gases between predictions of eq 7 and available experimental results for (a) O2, (b) Ne, (c) Ar, and (d) Kr. Necessary parameters are (a) for O2, D0 ) 0.6598 in terms of eq 4 with d ) 1 and c ) 1/2, and Ri ) Rs ) 1.462, according to eq 6 with the assumption that Es ≈ Ei. Symbols 9,11 0,13 2,14 and 315 denote the experimental results of O2 nanocrystals confined in cylindrical pores. Similarly, (b) for Ne, D0 ) 0.6260, and the symbols 0,10 and 911 denote the experimental results. (c) For Ar, D0 ) 2.043, 1.022, 0.6810, and 0.3405 in light of eq 4, with d ) 0 and c ) 1 for free nanoparticles ([18), and d ) 0, 1, 2 and c ) 1/2 for Ar nanocrystals confined in spherical (317), cylindrical (0,11 O20), and slit ()16) pores, respectively. (d) For Kr, D0 ) 0.808 for nanocrystals confined in cylindrical siliceous MCM-41 (912). Other parameters utilized in the calculations are listed in Table 1.

Thermal Stability of Nanocrystals Tm(D) function.48 As shown in Figure 1c, where the comparisons of eq 7 and experimental results for Tm(D) functions of Ar nanocrystals with different d values are presented, the change of the Tm(D) function of nanocrystals confined in cylindrical porous media11,20 is weaker than that of one embedded in spherical pores17 but stronger than that of one confined between mica slits.16 These differences are induced by different interface/ volume ratios of nanocrystals, which are introduced via eq 4 with d ) 0, 1, and 2 for nanoparticles, nanorods, and thin films, respectively. It should be noted that because the molecular diameter of molecular gases is larger than the size of the constituents of the walls, the epitaxial mechanism should be ruled out.62 Similar results for organic molecules such as C6H6, C6H5NO2, CCl4, and C6H12 within highly ordered mesoporous materials MCM-41,27,29 SBA-15,29 and Vycor22,27 and controllable pore glasses22,23,27,28,30,31 can be observed in panels a-d of Figure 2, where the predictions of eq 7 are shown as solid lines. Whereas for CCl4 nanoconfined within carbon nanotubes24 and graphite slits25 as well as C6H12 between parallel mica plates,32 there exist interactions between molecules and pore walls at their interfaces such as CCl4-carbon,24–26 and hydrogen bonding between mica,32,53 stronger than the intermolecular interactions of CCl4 and C6H12, with Ei/Es ) 2.6726 and 3, respectively.53 These strong bonds significantly depress the thermal vibrations of molecular nanocrystals and gives rise to Ri ) [1 + 2Svib(∞)/ (3R)]Es/Ei < 1 in light of eq 6. Therefore, their melting points are well above their bulk counterparts, and Tm(D) functions increase with decreasing D in terms of eq 7. As shown in these plots, the dashed and dotted lines in Figure 2c and dashed line in Figure 2d, respectively, are in quantitative agreement with available experimental data for CCl4 within carbon nanotubes,24

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Figure 3. Comparison of the Tm(D) function of water between the model prediction according to eq 7 and experimental data (),33 2,34 9,35 O36), where Ri ) Rs ) 1.591, according to eq 6, assuming Es ≈ Ei, D0 ) 0.6208 in terms of eq 4 with d ) 1 and c ) 1/2 for nanocrystals confined in cylindrical pores.

CCl4 and C6H12 nanoconfined between graphite slits,25 and parallel mica plates,32 in the full size range. However, the Tm(D) function of H2O nanoconfined in MCM41 is always depressed in comparison with Tm(∞),33–36 although there also exists a strong interaction between H2O molecules and MCM-41 such as hydrogen bonding. This is understandable according to the present model because the H2O-MCM-41 interaction is assumed to be comparable to one of intermolecular H2O, namely, Es ≈ Ei, which results in Ri > 1 according to eq 6. The validity of eq 7 is also confirmed by the consistence between model predictions and experimental measurements shown in Figure 3. In addition to the aforementioned molecular gases, organic molecules, and H2O, metallic substances such as Ga,42,43 In,38

Figure 2. Comparisons of Tm(D) functions of organic molecules between model predictions according to eq 7 and available experimental results of (a) benzene, (b) nitrobenzene, (c) carbon tetrachloride, and (d) cyclohexane. Necessary parameters are (a) for benzene, Ri ) Rs ) 1.230, according to eq 6, assuming Es ≈ Ei, D0 ) 1.0574 in terms of eq 4 with d ) 1 and c ) 1/2 for nanocrystals confined in cylindrical pores (9,28 0, O,29 230). Similarly, (b) for nitrobenzene, Ri ) Rs ) 1.269 and D0 ) 1.166 with d ) 1 and c ) 1/2. Symbols denote the experimental results (9,22 sideways triangle,22 [27). (c) For carbon tetrachloride, Ri ) Rs ) 1.462, according to eq 6 with Es ≈ Ei and D0 ) 1.086 in light of eq 4 with d ) 1, 2, and c ) 1/2 for CCl4 nanoconfined in porous glass (9,22 b23), and Ri ) 0.5483, according to eq 6 with Es/Ei ) 0.015/0.0426 for CCl4 nanocrystals confined in carbon nanotubes ([24), graphite slits (025), and D0 ) 1.086 and 0.5429, respectively. (d) For cyclohexane confined in cylindrical porous media with the van der Waal force (9,28 0,29 g31), Ri ) Rs ) 1.462, assuming Es ≈ Ei and D0 ) 1.174 with d ) 1 and c ) 1/2, while C6H12 nanoconfined between parallel mica ([32), Ri ) Rs/3 ) 0.4874 with Es/Ei ≈ 1/3, assuming a hydrogen bonding interaction between C6H12/mica, and d ) 2 and c ) 1/2. Other parameters are listed in Table 1.

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Lang and Han Considering the mathematical relationship of exp(-x) ≈ 1 -x, when x is small enough as a first approximation, under the condition that D . D0, eq 7 can be simplified as

Tm(D)/Tm(∞) ) 1 - {[1 + 2Svib(∞)/(3R)]Es /Ei - 1}D0 /D (12) Equation 12 indicates that the most important size effect for nanocrystals in nanopores is still related to the interface/volume ratio or 1/D and suggests a progressively increasing role of the interface layer with decreasing D. However, as the size of the nanocrystals further decreases to the size being comparable with a critical size about several nanometers, the difference between eqs 7 and 12 is evident because of energetic changes of internal molecules or atoms. Combining eqs 1 and 7 leads to

δ(D) ) [DHm(∞)/(2V)]{1 exp{-[1 + 2Svib(∞)/(3R)]Es /Ei - 1}/(D/D0 - 1)} (13)

Figure 4. Comparisons of Tm(D) functions of metallic nanocrystals between model predictions of eq 7 and experimental evidence for (a) Ga (9,42 043), (b) In (9,238), and (c) Hg (9,39 O40) confined in cylindrical porous glass, where Ri ) Rs ) 1.813, 1.609, and 1.786, according to eq 6, assuming Es ≈ Ei and D0 ) 0.4884, 0.6502, and 0.0601 in terms of eq 4 with d ) 1 and c ) 1/2, respectively.

It is evident that δ(D) depends on not only the thermodynamic nature of substances but also the size of nanocrystals.63,64 For a given substance, the shifts of the δ(D) function reveal similar trends of the Tm(D) function, i.e., the δ(D) function may decrease or increase with increasing size as shown in Figure 5, with predictions in eq 13 for CCl4 nanoconfined in MCM-41, carbon nanotubes, and graphite slits, and C6H12 within MCM-41 and mica slits, respectively. However, size dependence of δ(D) function had been neglected in the Gibbs-Thomson equation. This is the reason why the Gibbs-Thomson equation can only fit well with experimental data for a larger D. Obviously, as D > 10D0, in terms of eq 12, δ(D) approaches a limited value, namely,

δ ) {[1 + 2Svib(∞)/(3R)]Es /Ei - 1}D0Hm(∞)/(2V)

(14)

Figure 5. δ(D) functions according to eq 13 for C6H12 (solid lines) and CCl4 (dashed lines) in porous glasses, mica slits, carbon nanotubes, and graphite slits. The parameters are Hm(∞) ) 26.73, 2.635 KJ mol-1 for CCl4 and C6H12, respectively. Other parameters are listed in Table 1.

and Hg39,40 are shown in panels a-c of Figure 4, respectively, where the depression of Tm(D) functions with decreasing D can be found. For In and Hg in cylindrical porous glasses, the difference between the model predictions of eq 7 and corresponding experimental results are smaller than 5%, while for Ga the difference is very large. The reason may be related to the size of the crystallites being larger than the pore diameter42,43 the structural transformations of confined Ga being different from the bulk structure.42,43 As shown in these figures, this model illustrates the depressed or enhanced thermal stability of nanocrystals with decreasing D, which is dependent on molecule or atom-wall interactions. When the molecule (atoms)-wall interaction is stronger than intermolecular or interatomic one, the thermal stability of nanocrystals is enhanced; otherwise, it is depressed in comparison the bulk counterparts.

The predictions of eqs 13 and 14 are consistent with the qualitative explanation according to the Gibbs-Thomson equation, i.e., if δ > 0, then the shift of ∆Tm(D) ) Tm(D) - Tm(∞) is predicted to be positive, while the opposite occurs when δ < 0.24,25,32,47 Compared with the Gibbs-Thomson equation, this unified model without any adjustable parameter can be utilized to predict the depression or enhancement of thermal stability of nanocrystals confined within nanoporous materials with different dimensions. Furthermore, the adjustable parameter that appeared in eq 1 could be quantitatively determined and a more exact physical meaning of these parameters may be found. Conclusions In summary, thermal stability of nanocrystals confined in nanoporous media has been modeled on the basis of Lindemann’s criterion for melting. According to this model, dramatical changes of thermal stability from the bulk stem from the lattice termination of nanocrystals, and their Tm(D) functions may decrease or increase with decreasing D, which is dependent on the molecules (atoms)-walls interaction at interfaces. When the interaction at the interface is weak, the Tm(D) function decreases, and thermal stability is depressed as D is reduced. However, when the interface interaction is strong, it is enhanced. Model predictions are in agreement with available experimental

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