Chapter 3 Heat Capacity, Entropy, and
3.4 Melting Thermodynamics
3.4.3 Size-dependent Melting Thermodynamics of
The following consideration is at the heart of various models based on Linde-mann’s criterion for Tm(r) [30],
σ2(r) = σ2va(r) + [σ2sa(r)− σva2(r)]nsa/nva (3.77) where the subscripts “sa” and “va” denote atoms/molecules at the sur-face and located within the particle, respectively. nsa/nva= ζ = (4πr2h)/
[(4/3)πr3− 4πr2h] = r0/(r− r0) with r0 = 3h for a spherical or a quasi-spherical particle. Equation (3.77) states that the rms of the particle is the average of its “surface” and “bulk” values. For bulk crystals, atoms in the surface layers can oscillate with larger amplitude than atoms in the interior of the crystals, and the average amplitude of the whole crystal is independent of the size of the crystal. However, both σ2va(r) and σ2sa(r) are larger than the corresponding σva2 (∞) and σsa2(∞). It is assumed that σ2sa(r)/σ2va(r) = σ2sa(∞)/σ2va(∞) = αris size-independent although σsa2(r) and σ2va(r) are
size-3.4 Melting Thermodynamics 101
dependent. Moreover, since the cooperative coupling between the surface and the interior atoms/molecules of small particles may be important, it is phenomenologically considered that the variation of σ2(r) is dependent on the value of σ2(r) itself. Thus, a change in σ2 with ζ can be given by σ2(ζ + dζ)− σ2(ζ) = (αr− 1)σ2(ζ)dζ [30]. Integrating this equation yields
σ2(r)/σ2(∞) = exp[(αr− 1)ζ] = exp{(αr− 1)/[(r/r0)− 1]} (3.78) where r0, at which all atoms/molecules are located on the surface, can be extended for different dimensions d with d = 0 for nanospheres, d = 1 for nanowires and d = 2 for thin films. For a nanosphere and a nanowire, r has the usual meaning of radius. For a thin film, r denotes its half thickness. r0
is given by:(1) r0= 3h for d = 0 since 4πr20h = 4πr30/3; (2) r0= 2h for d = 1 since 2πr0h =πr20; and (3) r0 = h for d = 2 since 2h = 2r0. Note that for disk-like nanoparticle, its quasi-dimension has been defined as d = 1 since its ζ is between that of particles and that of thin films [31]. In short,
r0= c1(3− d)h. (3.79)
In Eq. (3.79), c1 is added as an additional condition for different surface states. c1= 1 for nanocrystals with free surface. When the interface interac-tion between the nanocrystals and the corresponding substrate is weak, such as thin films deposited on inert substrates, the film/substrate interaction is van der Waals forces while the inner interactions within the thin films are strong chemical bonds, c1 = 1 too. If this strength on the interface is com-parable with that within films, c1 varies somewhat. When these are similar, which is equal to the case that one of the two surfaces of the films disappears, c1= 1/2 is thus got (the side surfaces of the thin films are neglected due to the low thickness). For more complicated interfaces, c1 may be considered case by case between 1/2 and 1.
Since usually Tm(∞) > ΘD(∞), the high temperature approximation can be utilized, σ2(r, T ) = f (r)T , where f (r) is a T -independent but r-dependent function. Thus, at any T , σ2(r, T )/σ2(∞, T ) = f(r)/f(∞). Moreo-ver, when T = Tm, f (r)/f (∞) = {σ2[r, Tm(r)]/h2}/{σ2[∞, Tm(∞)]/h2}·
[Tm(∞)/Tm(r)] = Tm(∞)/Tm(r) in terms of Lindemann’s criterion. In the above equation, h is assumed to be a size-independent constant, namely, ΔVs = Vs(∞) − Vs(r) ≈ 0 or Δh = h(∞) − h(r) ≈ 0. It is known that Δh/h = ΔVs/(3Vs) = 0.1%− 2.5% when r < 10 nm and it is negligible when r > 10 nm. Thus, even r < 10 nm, [Vs(r)/Vs(∞)]2/3 ≈ 0.95 − 0.97. Note also that ΘD(r) function can be obtained as a generalization of relationship, ΘD2(∞) ∝ Tm(∞), i.e., ΘD2(r)/ΘD2(∞) = Tm(r)/Tm(∞). According to Eq.
(3.78),
Tm(r)/Tm(∞) = σ2(∞)/σ2(r) =ΘD2(r)/ΘD2(∞)
= exp{−(αr− 1)/[(r/r0)− 1]}. (3.80) Based on Mott’s expression for ΔSvib(∞) of bulk crystals at Tm(∞) [5, 32] and the above model, ΔSvib(r) of metallic crystals and αr in Eq.
102 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
(3.80) could also be developed [33, 34]. ΔSvib(r) of nanocrystals can be obtained as a generalization of Eq. (3.42), i.e., ΔSvib(r) - ΔSvib(∞) = (3R/2) ln{[Tm(r)/Tm(∞)][Cs(r)/CL(r)]2/[Cs(∞)/CL(∞)]2}. Instead of treat-ing CL(r) and Cs(r) to be size-dependent, respectively, the ratio of CL(r)/
Cs(r)≈ CL(∞)/Cs(∞) is approximately taken as a size-independent value.
Hence,
ΔSvib(r)− ΔSvib(∞) = (3R/2) ln{[Tm(r)/Tm(∞)]. (3.81) Substituting Eq. (3.79) into Eq. (3.81), it reads
ΔSvib(r) = ΔSvib(∞) − (3R/2)(αr− 1)/[(r/r0)− 1]. (3.82) It is known that ΔSm(r) for metallic crystals is mainly vibrational in na-ture. Hence, although ΔSvib(r) represents only one of several contributions to ΔSm(r), one may suggest that ΔSm(r) follows the same size dependence as ΔSvib(r),
ΔSm(r) = ΔSm(∞) − (3R/2)(αr− 1)/[(r/r0)− 1]. (3.83) In Eq. (3.83), the smallest size of crystals is assumed to be 2r0 where a half of atoms of a crystal are located on the surface with ζ = 1. Tm(2r0) = Tm(∞) exp(1−αr) where both have almost the same short range order and the structure difference between crystal and liquid is little. As a result, melting disappears, ΔSm(2r0) = ΔSvib(2r0) = 0 is thus assumed for the smallest nanocrystal, which leads to
αr= 2ΔSm(∞)/(3R) + 1 = 2ΔSvib(∞)/(3R) + 1. (3.84) Eliminating the parameter αr from Eq. (3.83) by means of Eq. (3.84), one has
ΔSm(r)/ΔSm(∞) = 1 − 1/(r/r0− 1). (3.85) Equation (3.85) is remarkably simple and more importantly, free of any ad-justable parameter. Equation (3.85) has been supported by the experimental results of Sn [35] and Al [36]. It is also utilized for organic nanocrystals due to their similar melting nature, i.e., ΔSm(r) of organic crystals are essentially contributed by a vibrational part [23]. However, since organic crystals are molecular ones, h or r0 stated above must be newly defined. A simple gener-alization is that h is defined as the mean diameter of the organic molecule, which implies that a molecule in organic crystals takes a similar effect of an atom in metallic crystals. Hence, when a molecule is located on the sur-face of the organic crystal, the amplitude of the thermal vibration of the full molecule is larger than that of molecules within the crystal.
Since the shape of an organic molecule is usually not spherical, h as a mean diameter of a molecule is defined as h = 1
3
3 i=1
hiwhere hiis the length of the molecule along three axis directions. For the organic molecules not
3.4 Melting Thermodynamics 103
having any regular shapes, the direction of the longest size is defined as the x-axis and the shortest size is defined as another axial direction. Different choices of the axes only lead to little difference of h, which changes Tm(r) and Sm(r) functions little specially when r/h is large enough (for instance, usually r > 2 nm in experiments and h < 0.5 nm for most organic crystals).
After determination of h by consideration of the geometric shape based on the bond length and bond angle of organic molecules, ΔSm(r) of organic nanocrystals may be calculated in terms of Eqs. (3.80) and (3.85). Figure 3.9 presents a comparison between the prediction of Eq. (3.85) and experimental results of four organic nanocrystals [37].
Fig. 3.9 ΔSm(r) functions of benzene, chlorobenzene, heptane, and naphthalene.
The lines are Eq. (3.85). , , , and denote the experimental results of ben-zene, chlorobenben-zene, heptane, and naphthalene, respectively. ΔSm(∞) values of benzene, chlorobenzene, heptane, and naphthalene in J·g-atom−1·K−1 are 2.842, 3.375, 3.042, and 2.920, respectively. r0 is taken as 2h (for the calculations of h values, see ref. [23]). The corresponding r0 values in nm are 0.7584, 0.8036, 0.9650, and 0.9024.
For nanocrystals embedded in a matrix with coherent or semi-coherent in-terfaces, it is expected that the msd value of interfacial atoms of nanocrystals (σ2ia(r)) falls between that of the interior atoms of nanocrystals (σ2va(r)) and that of the matrix (σM2(r)), under the assumption that σia2(r) of the interfacial atoms of the nanocrystals has an algebraic average value between σva2 (r) of interior atoms and σ2M(r) and combining the assumption of σva2 (r)≈ σ2(∞), αris determined as
αr={[σ2M(r) + σ2va(r)]/2}/σ2va(r) = [σM2(r)/σ2(∞) + 1]/2. (3.86) As stated above, the high-temperature approximation for σ2M(T ) at a T is utilized. Substituting T = TM(∞) and T = Tm(r) into Eq. (3.63), one gets
104 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
σ2M[Tm(r)]/σ2M[TM(∞)] = Tm(r)/TM(∞). According to the Lindemann cri-terion, or σ(∞)/h = c, σM(TM(∞))/hM= σm(∞)/h, where σM[TM(∞)] de-notes the rms displacements of atoms of the matrix at the corresponding Tm, and hMshows atomic diameters of the matrix, σM2[TM(∞)] = (hM/h)2σ2m(∞) or σ2M(Tm(r))/σ2m(∞) = (hM/h)2[Tm(r)/TM(∞)] and thus αr = {(hM/h)2 [Tm(r)/TM(∞)] + 1}/2 in terms of Eq. (3.86). Since the difference between Tm(r)/TM(r) and Tm(∞)/TM(∞) is little, αr is a weak function of r. As a first order approximation, Tm(∞) takes the place of Tm(r), or αr takes its smallest value. Finally, it reads
αr={[Tm(∞)/TM(∞)](hM/h)2+ 1}/2. (3.87) Substituting Eq. (3.87) into Eq. (3.80), the superheating of nanocrystals can be predicted by Eq. (3.80). As shown in Eq. (3.80), Tm(r) depends on r and αr. Only when αr < 1, which is related to the relative size of h and Tm(∞) between the nanocrystals and the matrix, does Tm(r)/Tm(∞) > 1 apply and Tm(r) increases as r decreases. Thus, necessary conditions for superheating of the nanocrystals embedded in the matrix are Tm(∞)/TM(∞) < 1 and there are the coherent or semi-coherent interfaces between them. Another sufficient, but not necessary condition is that the atomic diameter of the matrix is smaller than that of the nanocrystals (hM/h < 1). For the superheating case, when Tm(∞)/TM(∞) = (h/hM)2, αr = 1. However, this is physically impossible since Tm(∞) ∝ 1/h. If αr > 1, which is determined solely by ΔSvib(∞) for free nanocrystals, Tm(r) decreases with decreasing r. Tm(2r0) is the lowest melting temperature and vice versa.
The physical nature for depression and enhancement of Tm(r) may essen-tially be induced by different surface/interface conditions. For crystals with free-standing surface, the increase of the coherent energy of surface atoms of crystals is larger than that of the corresponding liquid, which renders that Tm(r) and ΔSm(r) drop. For crystals embedded in a more stable matrix with coherent or semi-coherent interfaces, the chemical bonds of the atoms on the coherent nanocrystals/matrix interface have more or less ionic characteristic, which leads to the enhancement of the bond strength on the interface. Since the surface melting of the nanocrystals is avoided, a superheating arises due to the suppression of thermal vibration of atoms on the coherent interface between the matrix and the nanocrystals. Thus, Tm(r) and ΔSm(r) increase.
The coherent interface can exist between the same or different atomic structures related to definite epitaxial relations between the nanocrystals and matrix where the atomic distances, on the interface are similar. For instance, for Pb/Al system, they have the same structure, this relation is given by (111)Al//(111)Pb, and [110]Al//[110]Pb. When structures are differ-ent, such as Pb/Zn and In/Al systems, their epitaxial relations are given by (0001)Zn//(111)Pb, [1120]Zn//[110]Pb, and{111}Al//{111}In, [110]Al//[110]In
respectively.
Figure 3.10 presents Tm(r) of semimetal In nanocrystals in different di-mensions and different surroundings. Tm(r) is indeed a function of d especially
3.4 Melting Thermodynamics 105
when r is small. When r/r0 > 5− 10, exp(−x) ≈ 1 − x. Equation (3.80) is simplified as Eq. (3.74) with d = 1. Since Tm(r, d = 0) < Tm(r, d = 1) <
Tm(r, d = 2), Eq. (3.74) is a good approximation of Eq. (3.80) when the di-mension effect on Tm(r) is neglected. This is true when r is large enough.
In addition, according to Eq. (3.87), αr decreases as Tm(∞)/TM(∞) and hM/h decrease, both are essential and determine the superheating tendency of nanocrystals. The nonlinear relationship between Tm(r) and 1/r in Eq.
(3.80) implies that besides ζ, the interior atoms of nanocrystals have ad-ditional contribution to Tm(r). This result shows good evidence that the macroscopic rules cannot simply be extended to the microscopic size range with a linear relationship of 1/r when ζ is large.
Fig. 3.10 Tm(r) of semi-metal In nanocrystals in terms of Eqs. (3.80), (3.81), (3.85), and (3.89) shown as solid lines. The dash line presents Tm(∞) = 429.75 K. For In, ΔSvib(∞) = 6.58 J·mol−1·K−1 in terms of Eqs. (3.31) and (3.32) where ΔVm/Vs = 2.7% and ΔSm(∞) = 7.59 J·mol−1·K−1 with ΔHm(∞) = 3.26 KJ·mol−1, h = 0.3684 nm. For Al, TM(∞)= 933.47 K and hM = 0.3164 nm. The symbols , , and denote experimental results with d = 2, d = 1, and d = 0 for In nanocrystals with free surface or deposited on inert substrates. The symbol shows experimental results for the In/Al system. (Reproduced from Ref. [12] with permission of Bentham Science Publisers Ltd.)
An emphasis should be again laid on that although the surface melt-ing phenomena have not been directly considered in Eq. (3.80), they have been included in ΔSvib(∞) value since ΔSvib(∞) is experimentally deter-mined in the existence of surface melting. Thus, Eq. (3.80) correlates to Eqs.
(3.69)–(3.71) well but without fitting parameters. Hence, Eq. (3.80) is more convenient to predict Tm(r) function.
Another thermodynamic function of melting besides Tm(r) and ΔSm(r) is ΔHm(r). From the general thermodynamics,
ΔHm(r) = Tm(r)ΔSm(r). (3.88)
106 Chapter 3 Heat Capacity, Entropy, and Nanothermodynamics
Based on Eqs. (3.80), (3.83) and (3.88), ΔHm(r) function is determined, ΔHm(r)
ΔHm(∞)=
1− 1
(r/r0)− 1
exp
−2Svib(∞) 3R
1 (r/r0)− 1
. (3.89)
ΔHm(r) function as a general one is suitable for all kinds of the first order transition and has been extended to the second order transition, such as glass transition. For the contents of that see Chapter 5.
A comparison between Eq. (3.89) and experimental results for ΔHm(r) function of In nanocrystals is shown in Fig. 3.11. A pretty agreement between them is got. Since both of Tm(r) and ΔSm(r) functions are linearly propor-tional to 1/r, ΔHm(r) should drop more considerably than this linearity as r decreases especially when r is small.
Fig. 3.11 ΔHm(r) functions of disk-like In nanoparticles in terms of Eqs. (3.89) shown as the solid line and the symbol denotes the corresponding experimental results where ΔHm(∞) = 3.36 kJ·mol−1, d = 1 and other parameters are the same as the caption of Fig. 3.10. (Reproduced from Ref. [12] with permission of Bentham Science Publishers Ltd.)