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Thermodynamics of Mixing for Nano-Particle Binary Mixing
Abstract
The physical properties of nano-particles are obtained by theoretical studies of phase diagrams. The phase diagrams of binary mechanical mixing are influenced by particle size and shape and different from bulk. For instance, there are differences of phase diagrams in solid-liquid curve from macroscopic to nano-particles. The phase equilibrium is one of introduction the simplified mathematical equations in solid-liquid systems. Under the equilibrium conditions, the chemical potentials of solid and liquid phase are equal. In general the activities are introduced through chemical potentials and assumed differently as the various phase behaviors. Considering of melting temperature is to relate the liquid phase to crystalline phase. In nano-particles the phase transitions and melting temperatures vary with their size. For the ideal binary mixing, the energy difference between solid and liquid is the sum of he corresponding energy of the elements and the bulk melting temperature is directly proportional to the ratio of the surface to volume atoms. In order to express the contribution of surface tension to nano-particles, the energy of particles in term of radius relative to inter-atomic distance is introduced. For the ideal system, the earlier developments of equations neglect segregation of particles at the surface. However, the effects of segregation at the surface lead to further modifications of the phase diagrams in many works later. The differences between regular and ideal cases are also occurred. With the validations of thermodynamic limits, thermodynamics of binary mechanical mixing of nano-particles is studied from Gibb’s free energy to apply for the phase diagrams. The Gibb’s free energy in mechanical mixing of two elements is the sum of each individual free energy in ideal system.
Thermodynamics of Mixing for Nano-Particle Binary Mixing
Nano-particles are current subjects of much attention because of fundamental and applied interests seen in many experiments [1]. Mainly the studies are in the frameworks of solid state and technological works like vacuum evaporation, heterogeneous catalysis, synthesis of nanostructures and nano-electronics [2]. Recent advances in the synthesis and the characterization of size-selected nano-particles become possible to investigate their physical and chemical properties. To understand the physical and chemical properties of materials, it is important to study the phase diagram of the system. However, there are few works dealing with the phase diagrams of nano-structures. For instance, in inorganic materials, the decrease of Tm is linearly reciprocal with radius and depends on surface tension of the liquid and solid as the melting temperature varies with radius in the theoretical works of Pawlow from various models [3]. The phase diagrams are not only involved in physical properties of nano-particles via melting temperature depending on size and shape of particles and on chemical environment via surface tension, but also related those properties to the deposition of particles via interfacial force between layers of nano-particles, growth velocity and diffusions [2].
The physical properties of nano-particles are mainly influenced by phase diagrams. The understanding of phase diagram is obviously accomplished by studies of theoretical thermodynamics. The development of thermodynamics in nano-particles for binary mixing was studied since 1965 in term of size effect on the phase diagrams of binary system [4].
In order to theoretically understand the influence of particle size on phase diagrams of binary mixing systems, this paper starts step by step from thermodynamic fundamentals of the binary mixing of nano-particles in term of thermodynamic free energy of mixing. Then the energy of particles is the introduction of the effect of surface tension on nano-particles by expressing the radius of particles in term of inter-atomic distance. In the next step the relation between liquid and solid phase is considering the melting criteria. To simplify the mathematic equations, the phase equilibrium is taking an account to develop thermodynamics equations for applied into phase diagram [5].
1. Fundamental Principle
Since particles with diameter range from 1 nm to 100 nm behave more or less intermediate between solid and molecular states, however, when the number of atoms in the particles lies in the range of 1,000 or above, its properties gradually evolve from molecule to solid types [6,7,8]. In this case it is obvious that the effects of surface in form of cohesive interactions of such particles cannot be neglected [1,9,10]. For instance, the properties of nano-particles are intermediate between bulk and surface properties so that their phase diagrams probably vary different from surface to bulk with the radii [1]. The process of segregation at the surface of solids and liquids during phase separations is the main argument for different in phase behavior of surface from bulk [2].
The earlier development of theoretical thermodynamics is to apply in phase diagrams of mixing system. Since understanding of melting temperatures from phase diagrams leads to further approach of such system properties in term of melting temperatures and phase transitions, subsequently the data prediction from the mathematical equations allows observers to obtain the estimated parameters for other properties of the system [11]. Apparently, for instance of applications in materials, the thermodynamics of mixing helps good preparation to use proper compositions and conditions such as amount of reagents and temperatures [2]. Also theoretical thermodynamics allows calculated predictions to estimate experimental data and to obtain understanding of phase diagrams such as melting temperatures. And mathematic equations allow estimated parameters to gain the material properties in term of transition states and mechanical properties of modulus [1].
2. Thermodynamics of Nano-Particle Mixing
The reasons based on the calculation of the isobaric Gibbs free energies of the phases means that consideration is relatively large particles with R > 3 nm, where:
a) the number of atoms, N, is such that the thermodynamical arguments remain valid;
b) the surface of the particle may be characterized by the surface tension. In the following, it is assumed that the surface tension corresponds to the solid–vacuum interface.
2.1. Thermodynamics of Nano-Particle Binary Mixing
When two elements are mixed, the Gibbs free energy of any binary mixture, Ax B, in bulk form is given by [11]:
gbulk = gbulk (x, hA, hB, sA, sB, T ) (1)
The Gibbs free energy of a binary mechanical mixture, mixing of two elements where Ax replaced by 1 and B replaced by 2, is given by [9]
gm = x1h1 + x2h2 − T(x1s1 + x2s2) (2)
where x1 and x2 are the atomic fractions of elements 1 and 2, respectively; hi and si are the corresponding enthalpy and entropy, respectively [10]. In the following, the energies are related to one atom or, in other words, they represent the total energy of the system divided by the number of atoms. The mixing causes an increase of the entropy via the configurational entropy of mixing:
Dsm = −k(x1 ln x1 + x2 ln x2). (3)
If the interactions between atoms 1 and 2 are essentially the same as in the pure components, the solution is called ideal, and the Gibbs energy is given by [9]
gid = gm − TDsm (4)
gid = x1m1 + x2 m 2 (5)
m I = hi − T si − kT ln xi : (6)
3. The Energy of a Particle
In order to introduce surface effects, it is easier to express the radius of the particle in units of inter-atomic distance. Then in the next step, consider the case of spherical particles further. Since the configuration entropy of mixing is dictated by the overall number of atoms 1 and 2 in the system, the surface adds no term to Dsm [1]. Hence, only hi are modified by the surface, via the surface tensions gi. If one assumes, in a first approximation, that there is no segregation, the energy of a particle for ideal solutions is given by;
gpart = gid + x1 gsurf,1 + x2 gsurf,2 (7)
gpart = x1 m part,1 + x2 m part,2 (8)
m part,i = m i + gsurf,I (9)
Then assume that the total number of atoms in the particle is equal to N. Let x and (1 − x) be the relative concentrations of atoms 1 and 2, respectively. Then
N gpart = x (N m 1 + fN2/3 g1) + (1−x) (N m 2 + fN2/3 g2) (10)
where f is a geometrical factor. And that leads to
N gpart = N gid + fN2/3 G(x) (11)
G(x) = x g1 + (1 − x) g2 (12)
These equations show that, as expected, the energy of the particle is always larger than that of the bulk material [7]. Then apply the equations to inorganic materials. In these cases, it is known that the surface tensions vary only slightly with temperature, T. When one assumes that i are independent of T, equations (11) and (12) show that, at fixed x, the energy of the particle is larger than that of the bulk by a quantity independent of T [6].
4. The Melting Criteria
Now looking at the melting of the particle, one has to consider the energy of the liquid phase, Gl(T), relative to that of the crystalline phase, Gc(T) [2]. Since, near Tm, we are well above the Debye temperature of the solid, the specific heat is approximately constant. Hence, for the elements, one has [7]:-
(Gl − Gc) a = C − B * T (13)
where C and B are constants for a given material. The subscript a states that we are dealing with very large materials, i.e. with R much larger than the inter-atomic distance. In equation (13), (C/B) is the bulk melting point, and C is the latent heat for melting [11]. By taking into account the roles of the solid and the liquid phases in equation (11), one obtains
N (gpart,s −gpart,l) = N(gid,s −gid,l) + fN2/3 (Gs(x)− Gl(x)) (14)
where the subscripts s and l refer to the solid and liquid phases, respectively. Substituting equation (13) into (14), algebra leads to
N (gpart,l − gpart,s) = N[x(C1 − B1T) + (1−x)(C2 − B2T)] + fN2/3 [x(g1,l − g1,s)
+ (1 − x) (g2,l −g 2,s)] (15)
N (gpart,l − gpart,s) = x [N (gpart,l − gpart,s)1] + (1 − x) [N(gpart,l − gpart,s)2]. (16)
This last equation is important, since it indicates that for ideal solutions, the energy difference between the liquid and solid phases of a binary-system particle is the sum of the corresponding energy of the elements. For elemental materials, the melting point is calculated from the previous equations, by taking N (gpart,l−gpart,s) = 0 [10]. One then obtains
Tm = Tm,1+f (gl−gc)/B* N1/3 = Tm,a [1−a/(2R)] (17)
where Tm,1 is the bulk melting temperature. The term (f/N1/3) is directly proportional to the ratio of surface to volume atoms. For inorganic materials a is positive, lying between 0.4 and 3.3 nm [10].
5. Phase Equilibrium
Then now look at binary systems. Under equilibrium conditions, the chemical potentials of element A in the solid (µA,s) and liquid (µA,L) phases are equal, as are the chemical potentials of element B [1]:
µA,s = µA,L (18)
µB,s = µB,L. (19)
One usually introduces the activities, a, through
µI = µ0i + kT ln(ai ). (20)
The energies of the particles may now be rewritten [2]
Ngpart = N[xµA + (1 − x)µB] = N[xµA,bulk + (1 − x)µB,bulk] + fN2/3G(x) (21)
Ngpart = Ngbulk + fN2/3G(x). (22)
By taking into account the roles of the solid and the liquid phases into the equations, one obtains
N(gpart,s − gpart,L) = N(gbulk,s − gbulk,L)+fN2/3(Gs(x) − GL(x)), (23)
where the subscripts s and L refer to the solid and liquid phases respectively.
6. Phase diagrams
6.1. Ideal solutions
For ideal solutions, the activities are the stoichiometric components:
aA,s = xs; aA,L = xL;
aB,s = 1 − xs; aB,L = 1 − xL. (24)
In this case, the liquid and solid curves are calculated from the two simultaneous equations obtained by expressing the equality of the chemical potentials in the two phases [10]:
kT ln (xsolid/xliquid) = CA [1 −(T/Tm,A)]
kT ln [(1 − xsolid)/(1 − xliquid)] = CB [1 −(T/Tm,B) (25)
where x solid and x liquid define the solid and liquid curves at a given T , respectively. Tm,A and Tm,B are the melting temperatures of elements A and B, respectively. There is as yet no evidence that the latent heat for melting changes with R [11]. However, then combining the previous equations to calculate the liquid–solid curves of small particles was discussed in many works [1,2,7,10].
6.2. Regular solutions
Similar discussions may be applied to so-called regular solutions, for which activities are written as follows [10]:
(26)
where i and j stand for A or B species and solid or liquid phases, respectively. Furthermore, αj represent the energy parameters in the solid (s) or liquid (L) phase. Expressing again the equality of the chemical potentials in the two phases, we can write the following equilibrium conditions:
kT ln ( x solid/ x liquid) = CA [1 − (T/Tm,A)] +αl(1 − x liquid)E2 − αs (1 − x solid)E2 kT ln (1 − x solid)/ (1 − x liquid) = CB [1 −(T/Tm,B)+αl(x liquid)E2 − αs(x solid)E2
(27)
Figure 6-1 Solid–liquid curves for the system Au–Cu in both a macro-scopic system (up)
and a nano-particular system including 106 atoms (down).
The equations (26, 27) have been applied to the binary Au–Cu system well known to present a congruence at 1164 K for an atomic composition x = 0.48 [1, 10]. Parameters of this system are reported in table 6-1. Figure 6-1 shows the theoretical liquid–solid curves calculated for the macroscopic material and the nano-particular system containing about 106 atoms. As can be seen from this graph, the effect of size reduction is mainly to shift the congruence point and the whole curve to a lower temperature. Furthermore, a small shift to the right in atomic composition at the congruence point can also be found, that reveals an enrichment in copper at the congruence point for the nano-particular system.
Table 6-1 Thermodynamical parameters of Au and Cu. The parameters αi
correspond to a mean value between the (111) and (110) faces.
7. Discussion
Nano-particles are obviously affected by cohesive properties of surface rather than bulk does. So understanding the properties in nano-particles, which have the intermediate state between solid and molecular state, is to study the phase diagram from thermodynamic approach.
Since the theoretical works of various models have been devised to describe the thermodynamic parameters and melting temperature with different radius of particles, then the studies of thermodynamics of binary mixing leading to phase behaviors of system are more or less the first principle to gain the fundamental knowledge evolving particles to molecules [10]. Under the ideal circumstance, the binary mixing is generally considered as an ideal solution, without segregation, and the figure 6.1 is the work shown the different of compositions in nano-particle comparable of bulk for binary fcc crystals [10].
Figure 7.1 : Surface compositions for ideal (b) and regular (c), (d) solution binary
nano-particles: bulk composition dependence fcc (111) surface (z1v = 3, z1l = 6, z1 = 12),
first nearest neighbor bonding: (a) without segregation, (b) ideal solution with segregation
DHsub = 7.5kT , (c) regular solution with segregation DHsub = 7.5kT and W = −0.1kT ,
(d) regular solution with segregation DHsub = 7.5kT and W = +0.1kT .
However, for the different shapes, thermodynamic approach of melting point depression might be smaller or larger than that of spherical shapes depending on dimensions [2]. Since nano-particles are intermediate between bulk and surface, it seems very obvious that the phase diagram will vary with the radii of those particles [1]. So the surface tension will play the important roles depending upon chemical environmental different of the particles.
Consequently, the interests in theoretical approach of the effect of particle size on phase diagram of binary mixing developed to explain the experimental data and to apply for phase diagrams of binary mixing of nano-particles. Also all of these works show thermodynamics of binary mixing not only depending on the particle size but also the dimension of particles [6]. That result implies to the phase behaviors and phase diagrams of binary mixing depend on both particle size and dimension. However, apparently the works show the particle shape-dependent phase behavior still limited [2].
8. Conclusion
For all of the works, which presented in this paper, the experimental data are well predicted by using the given theoretical parameters for ideal solutions. Also in general, the works obviously indicate that the thermodynamic approach strongly depends on size of particles [10]. And those will lead to changing in at lease here the phase diagram and other properties of such nano-particles from those of the bulk.
For instances, at the fixed temperatures between the highest melting point and the lowest one of such particles, the relative concentration in particle differs from that in bulk and that leads to changing in properties of phase of mixing [2]. At the given composition, the phase equilibrium and temperature will be different with the size difference. And the result will lead to the different in phase behavior of bulk and particles. Also for the ideal solution, the phase behavior will change in the same trend [1] in decrease of size, which is obviously due to mathematical approach in equation (17).
Figure 8-1 : The phase diagram of solid-liquid curve for ideal solution of Si.
An x is the concentration of Si, T is a temperature in K, and the upper curve
is a bulk materials, whereas the lower is a nano-particles (20 nm) [7].
However, some parameters of crystallographic faces and non ideal mixing didn’t show up in those works which might generally indicate that those theoretically mathematic models will serve well for some nano-mixing of binary system.
9. Critical Points for the Future Works
All of the above theoretical approach are about approximation and may be improved and extended by the following ideas.
a) The present works mainly considered the particles as spherical particles, which help calculate some mathematical equation more easily. However, that assumption might differ from the real situation of nano-particles which might have facet or rod-like shape instead.
b) In the first improvement in earlier works, using the assumption of ideal solution to approach and solve the equation or even simplify some equations. So if the future work will include the effect of non-ideal solution, the work might lead to new term in bulk which would modify the variation possibly predicted in interface such as in term of melting temperature, or shape of solid-liquid of phase diagram.
c) Since the size difference is much more effect on the surface. So the surface segregation or adhesion will be the way to look closely into the difference of the surface atoms from the core atoms which will lead to at least the huge different expectation in phase diagram of those atoms. For instance, the phase diagram in which the segregation is considered shown in the picture 9.1 is different from that of figure 7.1 in which the segregation was not concerned [1, 2, 10].
Figure 9-1 Solid–liquid curves in a nano-particular Si–Ge system including 106 atoms.
In this strictly regular model (! = 0.09kT > 0) with segregation, the bulk (thick solid curve)
and surface (thin solid curve) compositions are considered in the nanoparticular case.
d) Also from those works one thing that can be concluded is the size influences the properties. And that will imply that the size effects depend directly upon the chemical environment of the nano-particles. That will lead to new approach if there would be added such effect in the future works.
10. References
1. Wautelet M, Dauchot J P and Hecq M 2003 Mater. Sci. Eng. 23 187
2. Vallee R, Wautelet M, Dauchot J P and Hecq M 2000 Nanotechnology 12 68
3. Pawlow P 1909 Phys. Chem. 65 1
4. Damodara Das V, Karunakaran D, 1990 J. Appl. Phys. 68 2105
5. Edelstein A S, Harris V G, Rolison D R, Kurihara L, Smith D J, Perepezko J and da Silava Bassani M H 1999 Appl. Phys. Lett. 74 3161
6. Garvie R C, 1965 J. Phys. Chem. 69 1238
7. Lai S L, Carlsson J R A and Allen L H 1998 Appl. Phys. Lett.72 1098
8. Link S, Burda C, Nikoobakht B, El-Sayed M A, 2000 J. Phys. Chem. B 104 6152
9. Steininger J 1970 J. Appl. Phys. 41 2713
10. Wautelet M, Dauchot J P and Hecq M 2000 Nanotechnology 11 6
11. Yeadon M, Ghaly M, Yang J C, Averback R S and Gibson J M 1998 Appl. Phys. Lett. 73 3208
11. Appendix
From the Free Energy of Mixing, one possible way to obtain the entropy is using the lattice model of binary system as figure a-1, which implies that if system is in the liquid state that means the system is not crystallized or frozen into glass, then the molecules are constantly switching positions.
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Figure a-1: The lattice Model of Binary Mixing.
So if the total number of arrangements (W), then the equation used to calculate the entropy (S) will be following:-
S = k ln W (a-1)
If considering a binary system, we imagine that we take all the molecules off the lattices, as following equation, we can calculate the number of the possible ways to successively place these molecules back to the lattice so as to fill up all the sites.
W = (n12!) / (n1!) (n2!) (a-2)
Then the entropy of mixing (DSm) is the (a-3) equation.
DSm = -R (na ln xa + nb ln xb) (a-3)
And R = k/n, then the (a-3) can be rewritten into (a-4)
DSm = -k (xa ln xa + xb ln xb) (a-4)
So the equation (3) is derived from the free energy of mixing as shown by (a-4).
The way how the equations (3) and (2) are derived in Appendix A and B is inspired from the fundamental concepts of Gibb’s Free Energy of Mixing, DGm of the whole system coming from the summations of all of the individual free energy in the system.
From the free energy of mixing, using the lattice binary model, the enthalpy of mixing (DHm) is following equation.
DHm = Z Dw12 n1 F2 (b-1)
when F1 = Volume fraction of solvent (b-2)
F1 = (n1V1) / (n1V1 + n2V2) (b-3)
F1 = n1 / (n1 + n2) (b-4)
and at V1 = V2, then F1 = x1.
If given Dw12 = interactions between pairs of solvent (1) and solute (2), then
Dw12 = w12 – ˝ w11 – ˝ w22 (b-5)
However, w12 = w21 = 2w11 = 2w22, then Dw12 = w11 = w22. (b-6)
And w11 = (x1h1 + x2h2)/x1 (b-7)
Given Z = number of coordinated lattices, then Z of n1 is equal to 1/n1.
So the enthalpy of mixing (DHm) is given as equation (b-8);
DHm = [1/n1] [(x1h1 + x2h2)/n1] [n1] [x1] (b-8)
DHm = x1h1 + x2h2 (b-9)
As the same as shown in equation (2).
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