**5.4 Low temperature sintering**

140 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

confining the plume at the focal point of the ellipsoidal cell, further nanoparticle formation

Figure 12 is a schematic diagram of the apparatus with an ellipsoidal cell. The laser spot is intentionally shifted by a distance, *x*, from the central axis of the ellipsoidal cell, while the target surface is also intentionally inclined by an angle, *θ*, against a plane perpendicular to the central axis. Figure 13 shows some of the results for nanoparticles produced as a result of changing these parameters. The experimental results shown in Figure 13(a), which are obtained under the conditions *x* = 0.0 mm and *θ* = 0.0 °, represent monodispersed nanoparticles. When the target surface has no inclination but the laser spot is shifted *x* = 2

Fig. 12. Schematic of experiment demonstrating the importance of confinement

Fig. 13. Influence of shock wave confinement on deposited nanoparticles morphology in the

ellipsoidal cell (field of view:200×200nm)

experiments were carried out.

As mentioned above, nanoparticle size was found to be monodispersed in the ellipsoidal cell under appropriate conditions. We will now discuss a case in which the monodispersed nanoparticles were sintered under low-temperature conditions. This low-temperature sintering procedure could serve as a metal bonding technique.

Fig. 14. Two gold nanoparticles forming a neck and binding to each other.

The bonding of metal is an important process for the construction of fine mechanical parts and heat sinks. Conventional bonding methods such as diffusion bonding, melted alloy bonding, hot isostatic pressing and silver brazing cause thermal stress at the interface between two metals because of differences in thermal expansion between the bonded parts. This thermal stress in turn causes warping of the bonded material. Therefore, lowtemperature metal bonding is desired to overcome these problems. Since the melting point of metals decreases with decreasing particle size, metal nanoparticle paste has been used as

Thermodynamics of Nanoparticle Formation in Laser Ablation 143

where *t* is the time, *τ* is the characteristic time of particle growth by sintering, *a* is the surface area, and *af* the value of the surface area at a final size. The particle growth rate is dependent on *τ*, which is determined by two main types of the diffusion: lattice diffusion and the grain boundary diffusion. The characteristic time of the lattice diffusion, *τl*, is proportionate to the third power of the particle diameter, *d*, and temperature, *T*, and it is inversely proportional to the surface energy, *γ*, and the diffusion constant, *D*. Therefore, *τl* is expressed as (Greer, J.

3 3

 

where *k* is the Boltzman constant, *D*0 is the vibrational constant, and *ε* the activation energy for diffusion. If τ used in Eq.(18) is known, the final diameter, *df*, can be estimated from the

As shown in Eq. (19), the characteristic time *τl* seems to increase proportionally with temperature, but *τl* actually decreases with increasing temperature due to the large contribution of temperature in the exponential term of the equation. However, the characteristic time *τb* for grain boundary diffusion is always shorter than *τl* under lowtemperature conditions. As a result, if *τb* is used as the value of τ in Eq.(18), the final particle

Since a large τ value corresponds to an unfavorable degree of the sintering, it is necessary to reduce the value of *τ* in order to enhance the sintering process. It can be deduced from Eq. (19) that it is effective to not only increase temperature but also to decrease the diameter of the nanoparticles. From the viewpoint of low-temperature bonding, however, it is preferable to keep the temperature as low as possible and to decrease the size of the nanoparticles

*<sup>l</sup>* exp *kTd kTd*

size *d*f can be estimated by measuring the particle sizes at specified time intervals.

Fig. 15. Nanoparticle sintering at various temperatures (field of view:200×200nm).

In this chapter, several topics on the thermodynamics of nanoparticles formation under laser

Firstly, thermodynamics related to some general aspects of nanoparticle formation in the gas phase and the principles behind of pulsed laser ablation (PLA) was explained. We divided the problem into the following parts for simplicity: (i) nanoparticle nucleation and growth, (ii) melting and evaporation by laser irradiation, and (iii) Knudsen layer formation. All these considerations were then used to build a model of nanoparticle formation into fluid

correlation between the diameter and annealing time.

0

*D D kT*

(19)

R., 2007)

before annealing.

**6. Summary** 

ablation were explored.

dynamics equations.

a low-temperature bonding material. However, the bonding strength of nanoparticle paste is relatively low. Since the sintering of monodispersed nanoparticles has been observed to effectively bond metals, it is important to elucidate this sintering phenomenon in order to optimize the strength of the metal bonding.

The TEM image in Figure 14 shows two gold nanoparticles bonding to each other. In crystallized metallic nanoparticles, bonding between the nanoparticles starts to form even at room temperature if the crystal orientations of the two particles are coincident at the interfaces as shown.

Even if the crystal orientations do not match, it is possible for nanoparticles to bond to each other by using a low-temperature sintering effect which lowers the melting point of the material making up the nanoparticles. In the sintering phenomena of two particles at a certain high temperature, melting, vaporization and diffusion locally occurring in the particle surface result in a fusion at the narrowest neck portion of the contact area between the two particles.

It is well known that the melting point of a substance decreases with decreasing the particle size of materials. The decrement of the melting point, Δ*T*, for a nanoparticle of diameter *d* is expressed as follows (Ragone, D. V, 1996):

$$
\Delta T = -\frac{4V\_s\mathcal{Y}\_{l-s}T\_m}{\Delta H\_m}\frac{1}{d}\tag{17}
$$

where, *V*s is the volume per mole, Δ*H*m is the melting enthalpy per mole, *γ*l-s is the interface tension between the liquid and solid phase, and Δ*T*m is the melting point for the bulk material. If we assume that the material is copper, Δ*T* is about 160 K for a copper nanoparticle having a diameter of 10 nm. We also assume that the interface tension, *γ*l-s, is half the value of bulk surface tension.

The decrease in the melting point results in a decrease in the sintering temperature and strengthens the diffusion bonding at relatively low temperatures. In general,diffusion bonding is enhanced by the sintering process, in which atomic transport occurs between the small bumps on the material surface. By irradiating nanoparticles onto the surface of the materials before bonding, the number of effective small bumps greatly increases.

In some experiments, the aggregation of the nanoparticles was found to be the smallest when the helium background gas pressure was suitable for the dispersion conditions. AFM images of nanoparticles formed under these conditions by the PLA method show that the size of the nanoparticles ranges from 10 nm to several tens of nm. Annealing at comparatively low temperature was performed on nanoparticles formed under these conditions. Figure 15(a) shows an AFM image of nanoparticles before annealing, and and Figures 15(b), 15(c), and 15(d) show them after annealing at 473 K, 573 K and 673 K, respectively. As can be seen from the images, nanoparticle size increased with annealing temperature.

According to sintering process theory, the final diameter of a nanoparticle, *df*, is dependent on the annealing temperature. Particle growth rate can be expressed using the surface area of a nanoparticle by (Koch, W. 1990):

$$\frac{da}{dt} \propto -\frac{1}{\tau}(a - a\_f) \tag{18}$$

a low-temperature bonding material. However, the bonding strength of nanoparticle paste is relatively low. Since the sintering of monodispersed nanoparticles has been observed to effectively bond metals, it is important to elucidate this sintering phenomenon in order to

The TEM image in Figure 14 shows two gold nanoparticles bonding to each other. In crystallized metallic nanoparticles, bonding between the nanoparticles starts to form even at room temperature if the crystal orientations of the two particles are coincident at the

Even if the crystal orientations do not match, it is possible for nanoparticles to bond to each other by using a low-temperature sintering effect which lowers the melting point of the material making up the nanoparticles. In the sintering phenomena of two particles at a certain high temperature, melting, vaporization and diffusion locally occurring in the particle surface result in a fusion at the narrowest neck portion of the contact area between

It is well known that the melting point of a substance decreases with decreasing the particle size of materials. The decrement of the melting point, Δ*T*, for a nanoparticle of diameter *d* is

*V T <sup>T</sup>*

where, *V*s is the volume per mole, Δ*H*m is the melting enthalpy per mole, *γ*l-s is the interface tension between the liquid and solid phase, and Δ*T*m is the melting point for the bulk material. If we assume that the material is copper, Δ*T* is about 160 K for a copper nanoparticle having a diameter of 10 nm. We also assume that the interface tension, *γ*l-s, is

The decrease in the melting point results in a decrease in the sintering temperature and strengthens the diffusion bonding at relatively low temperatures. In general,diffusion bonding is enhanced by the sintering process, in which atomic transport occurs between the small bumps on the material surface. By irradiating nanoparticles onto the surface of the

In some experiments, the aggregation of the nanoparticles was found to be the smallest when the helium background gas pressure was suitable for the dispersion conditions. AFM images of nanoparticles formed under these conditions by the PLA method show that the size of the nanoparticles ranges from 10 nm to several tens of nm. Annealing at comparatively low temperature was performed on nanoparticles formed under these conditions. Figure 15(a) shows an AFM image of nanoparticles before annealing, and and Figures 15(b), 15(c), and 15(d) show them after annealing at 473 K, 573 K and 673 K, respectively. As can be seen from the images, nanoparticle size increased with annealing

According to sintering process theory, the final diameter of a nanoparticle, *df*, is dependent on the annealing temperature. Particle growth rate can be expressed using the surface area

*da*

*dt*

 <sup>1</sup> *f*

*a a*

(18)

materials before bonding, the number of effective small bumps greatly increases.

4 1 *slsm m*

*H d* 

(17)

optimize the strength of the metal bonding.

expressed as follows (Ragone, D. V, 1996):

half the value of bulk surface tension.

of a nanoparticle by (Koch, W. 1990):

interfaces as shown.

the two particles.

temperature.

where *t* is the time, *τ* is the characteristic time of particle growth by sintering, *a* is the surface area, and *af* the value of the surface area at a final size. The particle growth rate is dependent on *τ*, which is determined by two main types of the diffusion: lattice diffusion and the grain boundary diffusion. The characteristic time of the lattice diffusion, *τl*, is proportionate to the third power of the particle diameter, *d*, and temperature, *T*, and it is inversely proportional to the surface energy, *γ*, and the diffusion constant, *D*. Therefore, *τl* is expressed as (Greer, J. R., 2007)

$$
\tau\_l \propto \frac{kTd^3}{\gamma D} = \frac{kTd^3}{\gamma D\_0} \exp\left(\frac{\varepsilon}{kT}\right) \tag{19}
$$

where *k* is the Boltzman constant, *D*0 is the vibrational constant, and *ε* the activation energy for diffusion. If τ used in Eq.(18) is known, the final diameter, *df*, can be estimated from the correlation between the diameter and annealing time.

As shown in Eq. (19), the characteristic time *τl* seems to increase proportionally with temperature, but *τl* actually decreases with increasing temperature due to the large contribution of temperature in the exponential term of the equation. However, the characteristic time *τb* for grain boundary diffusion is always shorter than *τl* under lowtemperature conditions. As a result, if *τb* is used as the value of τ in Eq.(18), the final particle size *d*f can be estimated by measuring the particle sizes at specified time intervals.

Since a large τ value corresponds to an unfavorable degree of the sintering, it is necessary to reduce the value of *τ* in order to enhance the sintering process. It can be deduced from Eq. (19) that it is effective to not only increase temperature but also to decrease the diameter of the nanoparticles. From the viewpoint of low-temperature bonding, however, it is preferable to keep the temperature as low as possible and to decrease the size of the nanoparticles before annealing.

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Fig. 15. Nanoparticle sintering at various temperatures (field of view:200×200nm).
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