# Bistability, hysteresis and fluctuations in adiabatic ensembles of nanoparticles

## Authors

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DOI: 10.1007/s11051-009-9597-y

- Cite this article as:
- Vollath, D. & Fischer, F.D. J Nanopart Res (2009) 11: 1485. doi:10.1007/s11051-009-9597-y

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## Abstract

A presuppositionless thermodynamic analysis of the phase transformations of nanoparticles in an adiabatic enclosure leads to a series of predictions of the transformation behavior. These predictions are perfectly confirmed with experimental results, which have been difficult to be explained until now. The most important predicted and validated phenomena are: (i) a broad range of bistability or hysteresis in the vicinity of the transformation temperature, (ii) the width of this range increases with increasing particle size and with increasing temperature, and (iii) the transformation temperature may be higher than the one for bulk material. As in reality, an experiment can never be performed in an idealized isothermal or adiabatic environment; one always has a mixture of these conditions. This influences the results. The outcome of this analysis explains why different authors report, probably dependent on experimental conditions, widely scattering results.

### Keywords

Nanocrystalline materialTransformationThermodynamicsFluctuationBistabilityHysteresisTheoryModeling## Basic ideas

Structural and magnetic fluctuations are phenomena that are often observed in connection with nanoparticles. There is a series of experimental evidence related to melting, see e.g., Minima and Ichihashi (1986), Ohshima and Takayanagi (1993), Ajayan and Marks (1988), or habitus transformation, see e.g., Ajayan and Marks (1988) and Scherbarchov and Hendy (2005). Especially with respect to melting of individual nanoparticles, a series of molecular dynamic model calculations have shown that close to melting there is a temperature range, where both the solid and the liquid phase are stable, see e.g., Hendy (2005) and Scherbarchov and Hendy (2006). Additionally, there are reports of bistability and hysteresis in connection with melting and crystallizing of nanoparticles, see e.g., Stowell (1970), Cheyssac et al. (1988), Cheyssac et al. (1995), Tognini et al. (2000), Haro-Poniatowski et al. (2004), Xu et al. (2006), Xu et al. (2007), Tetu et al. (2008). Such phenomena have—until now—no conclusive explanation.

Fluctuations of isolated particles (Vollath and Fischer 2007) and isothermal aspects of the fluctuations of ensembles were discussed earlier (Vollath and Fischer 2008). The aim of this article is to elucidate the aspects in an adiabatic environment. Understanding of the adiabatic behavior is necessary, because most of the calorimetric methods to analyze phase transformations work in a more or less adiabatic way (Schlesinger and Jacob 2004). The same argument is valid for observations in the transmission electron microscope.

In order to study the adiabatic case, phase transformations of an ensemble of particles are considered. Kinetic aspects will not be included. In the adiabatic case, particles that change phases, change temperature. However, as it is necessary for the adiabatic case, the internal energy of the ensemble remains constant. One of the goals of this article is to estimate the number of particles found in both phases. Discussing isothermal or adiabatic processes is lastly a discussion using simplified models; in experimental reality, purely isothermal or adiabatic processes do not exist.

Each particle of the ensemble is in an adiabatic enclosure, the “Local Enclosure.” No flow of heat occurs within the ensemble or out of the system. Consequently, there is no temperature equalization between the particles. Hence, one finds different temperatures within the ensemble. Although this may sound to be strange, in a groundbreaking article Wales and Berry (1994) have proved that this situation of coexistence is possible in finite systems. Experimentally, this situation is realized in the case of phase transformation of particles that are embedded in a second phase with poor thermal conductivity (see e.g., Xu et al. 2006, 2007 or Tetu et al. 2008) or those coated either with a ceramic (Vollath and Szabó 1994) or with a polymer phase (Vollath and Szabó 1998).

The whole ensemble is in an adiabatic enclosure, the “Global Enclosure.” This model is connected to the assumption of an instantaneous temperature equilibration in the ensemble. This means at each moment, each particle of the ensemble has the same temperature. This is the standard assumption in adiabatic calorimetry, see e.g., Schlesinger and Jacob (2004).

Figure 1 displays the free enthalpies *G*_{1} and *G*_{2} of the two phases in question as function of the temperature. Under isothermal conditions, at temperatures below *T*_{cross}, phase 1 is the stable one and above *T*_{cross} phase 2 is the stable one. Therefore, one observes the phase transformation at this temperature. Provided the particles are sufficiently small, even under equilibrium conditions one observes both phases also at temperatures different from *T*_{cross}. Additionally, one has to look at cases, where the transformation uses one or more intermediate steps, see e.g., Wang et al. (2007) for an intermetallic system and Vollath and Wedemeyer (1990) for a ceramic one.

Adiabatic phase transformations were studied in the past. Most interesting in this context are articles by Umantsev and Olson (1993) and Umantsev (1997). These articles, based on one-dimensional simulations, conclude in the observation that the equilibrium phase diagram of adiabatically insulated systems differs significantly from the isothermal one. These differences should increase with decreasing grain size. In general, these studies lead to the conclusion that materials consisting of small grains may be unstable. Analyzing an ordering parameter has led to the conclusion that, in the case of nanocomposites, there may exist conditions for amorphization. Basso et al. (2008) analyzed magnetic systems in an adiabatic enclosure with respect to phase transformations. These authors use a Preisach model, see e.g., Mayergoyz (1991), with a superposition of a bistable feature corresponding to the two possible phases (magnetic states) in the system as basic assumption.

It is the special approach of this article to use elementary thermodynamics without any additional preconditions. Furthermore, it is important to mention that it is one of the basic suppositions of this article that the thermodynamic data of the materials in question are known. Furthermore, it must be mentioned that all transformations and equilibrations are assumed to occur instantaneously; kinetic phenomena are not taken into account. Additionally, due to lack of sufficient data, the dependency of the material data, such as the surface energy (Nanda et al. 2003), on the particle size is not taken into account.

## Thermodynamic treatment

### General considerations

The description of adiabatic phase transformations is quite difficult. In order to simplify this task, our considerations will be based on the following “gedanken experiment.”

*T*. In case that the ensemble is allowed to move along an adiabatic path into the thermodynamic equilibrium, the temperature of the ensemble will change. According to this equilibrium, a certain quantity of the particles will undergo the phase transformation. Therefore, the average temperature of the ensemble will change. In the global case, all particles in the ensemble are instantaneously transferred to the new temperature \( T^{*} \). In the local case, \( T^{*} \) is the new average temperature of the system. It is assumed that the change of temperature is instantaneous. We denote the temperature difference observed with Δ

*T*, if a single particle would transform adiabatically. The new temperature \( T^{*} \), which can be determined experimentally, is a function of the temperature

*T*and the temperature difference

*f*(Δ

*T*),

*f*(Δ*T*) is calculated for each specific case of enclosure. As the system is assumed to be equilibrated, *f*(Δ*T*) can be a fixed, but possibly also a temperature-dependent function.

At first, the most general case, where each phase is at a different temperature, will be discussed. Furthermore, to keep the thermodynamic considerations most general, it is assumed that there are several steps between the two stable phases. Therefore, it is assumed that *N* particles of the ensemble are found in *I* different phases. In an adiabatic system, equilibrium is defined by a maximum of the entropy; additionally, the internal energy of the system (not of the individual phases!) represented by the first sum in Eq. 2, is a constant. It is presupposed that there is no work done against an external pressure. Furthermore, for the following considerations the external pressure is set as zero. Therefore, the term *pv*_{i}, with *p* the external pressure and *v*_{i} the volume of one particle of the phase *i*, does not appear in the following derivations.

*G*

_{en,}

*c*

_{p}stands for the heat capacity, and without loosing generality, to reduce the complexity of the equations,

*c*

_{p}was assumed to be independent of the temperature and the phase. The enthalpy term

*u*

_{i}contains the bulk enthalpy at

*T*

_{i}

*=*0 and the surface energy \( u_{\text{si}} = \gamma_{i} {{\uppi}}D^{2} \). The quantity

*s*

_{i}is the entropy of the phase

*i*. The last term at the right side of the Eq. 2 represents the entropy of mixing with

*k*the Boltzmann constant and

*T*

_{i}the temperature of the phase

*i*. Lastly, this is the entropy production caused by the fluctuation process. The quantities

*n*

_{i}are numbers of particles of the phase

*i*. Equation 2 is valid with the side condition

*N*of particles per mol in the ensemble.

The temperature *T*_{ref} may be selected arbitrarily and the temperature difference is defined as \( \Updelta T_{i} = T_{\text{ref}} - T_{i} \).

*Lagrange*multiplier

*λ*as

The further mathematical treatment is different in the local and global enclosure cases.

### Case 1: each particle is in an individual adiabatic enclosure, the Local Enclosure

*e*

_{i}as

*T*

_{i}of the different phases. With phase 1 as the reference phase, we have

Equation 8 is insofar of great relevance, as it leads to only one independent variable temperature, namely, the reference temperature *T*_{1}. All the other temperatures are now expressed by the reference temperature *T*_{1} and a few material constants.

It is now interesting to analyze the individual terms in Eq. 9. The first term is identical to the one valid for non-fluctuating particles at the temperature *T*_{1}. The second term describes the entropy production connected to the fluctuation process at *T*_{1}. The sum of these two terms is identical to *G*_{en} obtained in the isothermal case. Since Δ*T*_{i} is a combination of material constants, the third term is independent of the temperature. Hence, it is, de facto, not an entropy term but describes the additional enthalpy due to the adiabatic enclosure, which can be considered as an adiabatic excess enthalpy.

*I*are of importance. This leads to

*X*

_{i}one obtains

**Theorem***In the isothermal and the local adiabatic case, in ensembles transforming between two stable phases, the population ratio of these two levels is independent of intermediate steps.*

For further discussions, this allows us, without loosing any generality, to reduce the considerations to two phases, denominated as phases 1 and 2.

*T*=

*T*

_{2}−

*T*

_{1}, one obtains from Eq. 13b

*T*= 0, this equation is identical to the one obtained for the isothermal case. In order to simplify further discussions, we replace the absolute particle numbers by the fractions \( c_{i} = \frac{{n_{i} }}{N},\; i \in \left\{ {1,2} \right\}, \) leading to

*T*

_{cross}, since

*α*

_{1}=

*α*

_{2}at

*T*

_{cross}. For \( T \ge T_{\text{cross}} \) the reference phase, being the more stable phase, changes from phase 1 to phase 2. Now, one may ask about the implications, if the right-hand side of Eq. 14b is set to one. This is only possible, if

*α*

_{1}−

*α*

_{2}= 0. In this case, we have to distinguish the following two cases:

- Phase 1 selected as the reference phase:$$ u_{1} + c_{\text{p}} T_{1} - s_{1} T_{1} + kT_{1} = u_{2} + c_{\text{p}} \left( {T_{1} + \Updelta T} \right) - s_{2} \left( {T_{1} + \Updelta T} \right) + k\left( {T_{1} + \Updelta T} \right). $$(15a)
- Phase 2 selected as the reference phase:$$ u_{2} + c_{\text{p}} T_{2} - s_{2} T_{2} + kT_{2} = u_{1} + c_{\text{p}} \left( {T_{2} + \Updelta T} \right) - s_{1} \left( {T_{2} + \Updelta T} \right) + k\left( {T_{2} + \Updelta T} \right). $$(15b)

*T*derived from Eq. 8 and \( T_{\text{cross}} = {(u_{2}} - {u_{1})} / {(s_{2}} - {s_{1})}, \) Eqs. 15a and b deliver two characteristic temperatures, namely

Figure 2 visualizes that at temperatures below *T*_{char2} an adiabatic transformation from the more stable phase 2 to phase 1 is impossible. This is because the shifted temperature *T* + Δ*T* of the transformed phase 1, located on the line representing *g*_{1}, would lead to a lower free enthalpy of the phase 1, which is in contradiction to the original assumption that phase 2 should be the more stable phase. Therefore, the range of phase 1 as reference phase is extended from *T*_{cross} to *T*_{char1}. The range of phase 2 as reference phase ends at *T*_{char2}. In Fig. 2, the reference phases are drawn as thicker lines. In the range between the characteristic temperatures, the reference phase depends on the side where these temperatures are approached. Obviously, the status of the system between the two characteristic temperatures is ambiguous; it depends on the status of the reference phase. In the range between the two characteristic temperatures, the system is bistable.

As \( g_{2} - g_{1} \le kT \) is the condition for fluctuations of a single particle (Vollath and Fischer 2007), the existence of two possibilities, given by the Eqs. 15a and b, leads to the conclusion that in the local adiabatic case, there are two temperatures centering a region, where the system is bistable. Obviously, these two characteristic temperatures have a similar meaning as the crossing temperature in the isothermal case. As, in most cases, the entropies *s*_{i} are larger than *c*_{p}, generally, the characteristic temperatures are higher than *T*_{cross}. However, we must be aware of the fact that, in the case of an ensemble, these characteristic temperatures are just material constants. The related temperature \( T^{*} \) must be calculated from Eq. 4. Doing this, using phase 1 as reference phase, with \( T_{\text{char2}} = T_{\text{char1}} + \Updelta T \) and *c*_{2} = 1 − *c*_{1} the experimentally determined characteristic temperature is \( T_{\text{char1}}^{*} = T_{\text{char1}} + \Updelta T - c_{1} \Updelta T \). The same result is obtained, if phase 2 is selected as the reference phase. Finally, this means that only one single characteristic temperature \( T_{\text{char}}^{*} \) replaces the two characteristic temperatures for one isolated particle, each one centering a region, where the system is bistable. This allows formulating the following *theorem*:

*In the local adiabatic case, thermal fluctuations of isolated particles are centered around two characteristic temperatures, defined by*\( T_{\text{chari}} = \frac{{s_{i} - k}}{{c_{\text{p}} }}T_{\text{cross}} \)*with*\( i =\,\in \{ 1,2\} . \)*In between these temperatures, the system is bistable. In an ensemble, experimentally, in view of the average temperature, there exists only one characteristic temperature*\( T_{\text{char}}^{*} . \)

*T*= 0, this point is found as \( c_{1} = c_{2} = 0.5. \) In the adiabatic case under discussion, Δ

*T*is calculated from the material constants, see Eqs. 8 and 17. Therefore, it is a constant value for each ensemble (as long as we neglect the temperature dependency of the surface energy and the heat capacity). Solutions of Eq. 18, for the expression Δ

*T*/

*T*

_{chari}being selected as variable, are plotted in Fig. 3.

The range of the abscissa in Fig. 3 was selected in such a way that most of the experimentally possible values are covered. Generally, one can say that the concentration at the characteristic temperature may be in the range from 0.5, which is equivalent to the isothermal case, to 0.6.

*T*

_{1}and

*T*

_{2}, now denominated solely as

*T*, for particles with a diameter of 2 and 5 nm. As there is a severe lack of reliable material data for nanomaterials, a consistent set of data was designed. These data are presented in detail in Appendix 1.

*c*

_{1}depending on the sign of Δ

*T*. Furthermore, there is a range between these two branches, where both, the concentrations 0 and 1, are possible. Typically, ranges, where two different concentrations are possible, are an indication for a bistable behavior. However, as mentioned above, the temperature

*T*is just an aid to simplify the thermodynamic considerations. In order to come to a statement that can be compared with experimental results, it is necessary to use the average temperature \( T^{*} \) according to Eq. 4. Figure 5 shows the same data as depicted in Fig. 4, however, plotted versus \( T^{*} \).

Plotting the concentration *c*_{1} versus \( T^{*} \) leads to only one curve. In a limited temperature range, one realizes three different concentrations for one distinct temperature. A detailed analysis of Fig. 5 teaches that the temperature range with three phases, observed for one distinct temperature, increases with particle size. Additionally, the transition is smoother in the case of smaller particles.

The trend of the free enthalpy as a function of the temperature, as depicted in Fig. 7, is similar to the one derived for a single particle as shown in Fig. 2. Again, one realizes a temperature range, where the association of free enthalpy to temperature is ambiguous. Within this range, three values exist for the free enthalpy for one distinct temperature \( T^{*} \). The relevant one depends on the phase of the ensemble.

Altogether, the considerations above leave us in a dilemma. As discussed above, ranges like the one with a positive slope values in Fig. 7 are considered as not accessible. On the other hand, the solutions of Eq. 18 are pointing exactly to concentrations in this part of the graph. However, because of the bistability of the system, this range of concentrations is not accessible. Hence, the range of concentrations, where fluctuations have the highest probability, is not accessible.

*T*

_{trans}) in the isothermal case. As expected from Eqs. 16a and b, the ranges, where fluctuations are expected, are shifted in the direction to higher temperatures as compared to the isothermal case. In this case, the range of bistability was defined as the temperature interval between the vertical tangents of the

*c*

_{1}(\( T^{*} \)) function shown in Figs. 5 and 6. The range of isothermal fluctuations is so narrow, that it is—at least in this graph—not much broader than the line thickness.

The temperature range of bistability is significantly higher than the temperature range of fluctuations in the isothermal case.

The width of the bistable range increases with increasing particle size.

For larger particles, the melting temperature of the solid phase may exceed the bulk melting temperature.

According to Ehrenfest, see e.g., Jaeger (1998), the order

*n*of phase transformation is calculated by the limits

In the isothermal case, this analysis is performed in the vicinity of *T*_{cross} as transformation temperature. It is important to note that in the case of isothermal phase transformations of ensembles of nanoparticles, the order *n* tends to infinity. Obviously, this definition cannot be applied to the adiabatic case since there exists no unique transformation temperature. As transformation temperature, one may be tempted to select \( T_{\text{char}}^{*} \). However, because of the bistability of the system, the concentration according to Eq. 18 is not accessible at \( T_{\text{char}}^{*} \). Therefore, the Ehrenfest definition of the order of transformation cannot be applied.

The numbers of particles in the different phases, calculated until now, are average values that are, as already mentioned, fluctuating. Therefore, the average temperature, determined with a thermometer integrating over the whole ensemble, is fluctuating, too.

### Case 2: the whole ensemble is in an adiabatic enclosure, the Global Enclosure

As kinetic processes are not taken into account, this case assumes an instantaneous temperature equilibration in the ensemble. Therefore, each particle of the ensemble has the same temperature. Implicitly, this assumption was made by Callen (1985) in his treatment on microcanonical systems

As mentioned in Sect. 2.1, our considerations are based on a “gedanken experiment.” However, in this case, the instantaneous equilibration leads to a change of the temperature of the whole ensemble. Lastly, this case conveys the impression to be identical with the isothermal one. However, as fluctuation processes are stochastic in nature, these considerations are valid for large ensembles only. The situation changes, if the number of particles is small. In an ensemble of *n*_{1} particles in the thermodynamically stable phase and *n*_{2} in the less stable phase, the numbers *n*_{1} and *n*_{2} are average values. The standard deviation *σ* of the number of particles in phase *i* is \( \sigma \left( {n_{i} } \right) = \sqrt {n_{i} } \). Since a transformation of a particle is connected to a temperature difference, the average temperature of the ensemble fluctuates. At a first glance, this temperature fluctuation of the ensemble is the only experimentally observable difference of an adiabatic enclosure compared to an ensemble in an isothermal temperature bath.

*T*we take again advantage of our “gedanken experiment.” As the internal energy of the ensemble is constant, the following relation is valid

*T*

*T*depends on the population of the different phases. Inserting expression (21) for Δ

*T*into Eq. 5 delivers

*n*

_{2}=

*N*−

*n*

_{1}the minimum is found as the solution of the equation

*c*

_{1}, Eq. 23a must be solved numerically. Therefore, the behavior of this system was studied using the same material data as in the local adiabatic case. At first, the temperature difference Δ

*T*according to Eq. 21 was calculated. Figure 9 displays the results for a 5 nm particle. For these calculations, the low temperature phase with the index 1 was selected as reference phase. As abscissa, the average temperature \( T^{*} = T + \Delta{T} \) was selected.

*T*> 0, observed in case of

*c*

_{1}as reference phase and a second one for Δ

*T*< 0, observed in case of

*c*

_{2}as reference phase. The graphs in Fig. 10 have the general appearance of a hysteresis. In the literature, a behavior like that is often described as “bistable,” see e.g., Cheyssac et al. (1988). (For the definitions of bistability and hysteresis, please see Appendix 2.) In Fig. 10, two features are remarkable:

Hysteresis and fluctuation ranges are not clearly separated. With increasing particle size, this separation even is reduced.

The range of hysteresis and fluctuation is widening with increasing temperature.

In the graph of the free enthalpy, this hysteresis is very well visible, too.

As the course of the free enthalpy in the range of the phase transformation is—because of the hysteresis—ambiguous, the Ehrenfest definition of the order *n* of phase transformation is not applicable.

## Discussion and comparison with experimental results

Occurrence of bistable ranges or a hysteresis during phase transformations.

The possibility that the temperature of phase transformation of nanoparticles is higher than the one of bulk material.

To some extent, these phenomena are against general knowledge. Therefore, one may be doubtful about these predictions. However, extended search in the experimental literature has taught that these phenomena have already been observed. In nearly all the cases, the strange behavior was explained with changing material data.

*σ*scattering ranges of the particle sizes are indicated. Three features of these experimental results are remarkable:

The width of the hysteresis range increases with increasing temperature. This result is in accordance with the predictions made in the preceding chapters on adiabatic phase transformations.

The transformation temperature is reduced with decreasing particle diameter. This behavior is generally expected for phase transformations of small particles.

The hysteresis curve is non-symmetric. Such a behavior was not yet predicted in the previous considerations. Possibly, it is—at least to some extent—caused by the very broad particle size distribution.

The experiments described above were performed for lead particles produced by condensation on an alumina substrate. Essentially, the same results were obtained for lead and tin particles embedded in a silica matrix (Cheyssac et al. 1995). This teaches us that the experimental conditions are not critical for the results explained above.

*T*/

*T*

_{melt}is plotted; where,

*T*

_{melt}is the melting point of bulk germanium. The melting point of the bulk material is found at

*T*/

*T*

_{melt}= 1.

The diagram shows a broad hysteresis.

The melting range of the material is above the bulk melting point.

There is a significant scattering of the experimental results in the ranges, where one expects the pure solid or liquid phase. This corresponds perfectly with the fluctuation ranges indicated in Fig. 6.

The melting and crystallizing curves, displayed in Fig. 15, are very similar to the results displayed in Fig. 10 and differ completely from the ones displayed in Fig. 14. From this difference, we may conclude that the results on germanium particles (Xu et al. 2006, 2007) were performed predominantly under conditions of a local enclosure, whereas the similarity to Fig. 10 suggests a global enclosure for the experiments of Tetu et al. (2008) on InSb particles.

## Conclusions

A presuppositionless thermodynamic analysis of adiabatic phase transformations of ensembles of nanoparticles is presented. The only assumptions made are the validity of Gibbs’s thermodynamics and Boltzmann’s statistics. Kinetic phenomena are not discussed.

The transformations show a broad range of bistability or hysteresis.

The width of these ranges increase with increasing particle size.

In case of increasing temperatures, the transformation temperature may be higher than the one for bulk material.

The Ehrenfest definition of the order of phase transformation is neither in case of the local nor in the one of global enclosure applicable.

These predictions are—to some extent—unusual for nanoparticles. However, in literature one finds published experimental results supporting the outcome of the current thermodynamic analysis.

As shown in Fig. 8, the temperature ranges for isothermal and adiabatic transformations are wide apart. In general, for the evaluation of phase transformation experiments, be it X-ray diffraction or electron microscopy, one assumes isothermal conditions. However, we know that one never has purely isothermal conditions. Knowing the results of this analysis, it is not astonishing to find experimental results of different authors scattering in a broad range.