Journal of Nanoparticle Research

, Volume 11, Issue 6, pp 1485–1499

Bistability, hysteresis and fluctuations in adiabatic ensembles of nanoparticles


    • NanoConsulting
  • F. D. Fischer
    • Montanuniversität Leoben, Institute of Mechanics
Research Paper

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


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.


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.

Concerning the discussion of adiabatic fluctuations, two idealized cases will be considered:
  • 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).

The thermodynamic assumption forming the basis for the considerations mentioned above is depicted in Fig. 1.
Fig. 1

Free enthalpy G as linearized function of the temperature T for the two phases with the bulk phase transformation temperature at Tcross

Figure 1 displays the free enthalpies G1 and G2 of the two phases in question as function of the temperature. Under isothermal conditions, at temperatures below Tcross, phase 1 is the stable one and above Tcross 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 Tcross. 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.”

Initially, all particles of an ensemble are forced to be in one phase at the temperature 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 ΔT, 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 fT),
$$ T^{*} = T + f\left( {\Updelta T} \right). $$

fT) is calculated for each specific case of enclosure. As the system is assumed to be equilibrated, fT) 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 pvi, with p the external pressure and vi the volume of one particle of the phase i, does not appear in the following derivations.

This is equivalent with a minimum of the free enthalpy. Therefore, in equilibrium, the free enthalpy of the ensemble Gen,
$$ G_{\text{en}} = \sum\limits_{i = 1}^{I} {\left( {n_{i} u_{i} + n_{i} c_{\text{p}} T} \right)} - \sum\limits_{i = 1}^{I} {\left[ {n_{i} s_{i} T_{i} + kT_{i} n_{i} \ln \left( {\frac{N}{{n_{i} }}} \right)} \right]} , $$
has to be minimized. The quantity cp stands for the heat capacity, and without loosing generality, to reduce the complexity of the equations, cp was assumed to be independent of the temperature and the phase. The enthalpy term ui contains the bulk enthalpy at Ti = 0 and the surface energy \( u_{\text{si}} = \gamma_{i} {{\uppi}}D^{2} \). The quantity si 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 Ti the temperature of the phase i. Lastly, this is the entropy production caused by the fluctuation process. The quantities ni are numbers of particles of the phase i. Equation 2 is valid with the side condition
$$ N = \sum\limits_{i = 1}^{I} {n_{i} } , $$
assuming a constant number N of particles per mol in the ensemble.
Following Eq. 1, the average, experimentally accessible temperature \( T^{*} \) in such an ensemble is
$$ T^{*} = \frac{1}{N}\sum\limits_{i = 1}^{I} {n_{i} } T_{i} \;\;{\text{or}}\;\;T^{*} = T_{\text{ref}} + \frac{1}{N}\sum\limits_{i = 1}^{I} {n_{i} } \Updelta T_{i} . $$

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

The minimum of the free enthalpy according to Eq. 2 is calculated by the use of the side condition (3) by applying the Lagrange multiplier λ as
$$ G_{\text{en}}^{*} = \sum\limits_{i = 1}^{I} {\left[ {n_{i} u_{i} + n_{i} c_{\text{p}} T_{i} - n_{i} s_{i} T_{i} - kT_{i} n_{i} \ln \left( {\frac{N}{{n_{i} }}} \right)} \right]} + \lambda \left( {N - \sum\limits_{i = 1}^{I} {n_{i} } } \right). $$
The minimum of the free enthalpy is found by solving the following system of equations:
$$ \left( {\frac{{\partial G_{\text{en}}^{*} }}{{\partial n_{i} }}} \right) = 0,\quad i \in \left\{ {1,2,3, \ldots I} \right\}. $$

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

As the adiabatic enclosure thwarts flow of heat between the individual particles, there is no temperature equalization between the particles, which means that the temperature of the particles depends on the phase. Consequently, one finds different temperatures in the ensemble. Hence, the concept outlined in Sect. 2.1 may be used directly. The additional side conditions, valid in this case, are caused by the adiabatic enclosure of each individual particle enforcing the constant internal energy ei as
$$ e_{i} = e_{j} \Rightarrow u_{i} + c_{\text{p}} T_{i} = u_{j} + c_{\text{p}} T_{j} ,\quad i \ne j,\;\;i,j \in \{ 1, 2, \ldots \ldots I\} , $$
is valid. This side condition allows calculating the temperature Ti of the different phases. With phase 1 as the reference phase, we have
$$ \Updelta T_{i} = T_{i} - T_{1} = - \frac{{u_{i} - u_{1} }}{{c_{\text{p}} }} \Rightarrow T_{i} = T_{1} - \frac{{u_{i} - u_{1} }}{{c_{\text{p}} }}. $$

Equation 8 is insofar of great relevance, as it leads to only one independent variable temperature, namely, the reference temperature T1. All the other temperatures are now expressed by the reference temperature T1 and a few material constants.

Inserting the temperature differences into Eq. 2 yields
$$ G_{\text{en}} = \sum\limits_{i = 1}^{I} {\left[ {n_{i} u_{i} + n_{i} c_{\text{p}} T_{1} - n_{i} s_{i} T_{1} } \right]} - \sum\limits_{i = 1}^{I} {\left[ {kT_{1} n_{i} \ln \left( {\frac{N}{{n_{i} }}} \right)} \right]} + \sum\limits_{i = 1}^{I} {n_{i} \Updelta T_{i} \left[ {c_{\text{p}} - s_{i} - k\ln \left( {\frac{N}{{n_{i} }}} \right)} \right]} . $$

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 T1. The second term describes the entropy production connected to the fluctuation process at T1. The sum of these two terms is identical to Gen obtained in the isothermal case. Since ΔTi 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.

The minimum of the free enthalpy is found by solving the following system of linear equations, which is derived from the condition (6) as
$$ \left( {\frac{{\partial G_{\text{en}}^{*} }}{{\partial n_{i} }}} \right) = u_{i} + c_{\text{p}} T_{i} - s_{i} T_{i} - kT_{i} \ln \left( {\frac{N}{{n_{i} }}} \right) + kT_{i} + \lambda = 0,\;\quad i \in \left\{ {1,2,3, \ldots I} \right\}. $$
Using the abbreviations \( X_{i} = \ln \left( {{N \mathord{\left/ {\vphantom {N {n_{i} }}} \right. \kern-\nulldelimiterspace} {n_{i} }}} \right) \) and \( \alpha_{i} = u_{i} + c_{\text{p}} T_{i} - s_{i} T_{i} + kT_{i} , \) Eq. 10 reduces to the system of linear equations
$$ kT_{i} X_{i} - \lambda - \alpha_{i} = 0,\;\quad i \in \left\{ {1,2,3, \ldots I} \right\}. $$
For our discussion, only the ratio of the number of particles in the reference phase with the index 1 and the final phase with the index I are of importance. This leads to
$$ kT_{1} X_{1} - kT_{I} X_{I} - {{\upalpha}}_{1} + {{\upalpha}}_{I} = 0. $$
After resubstituting of Xi one obtains
$$ kT_{1} \ln \left( {\frac{N}{{n_{1} }}} \right) - kT_{I} \ln \left( {\frac{N}{{n_{I} }}} \right) = \alpha_{1} - \alpha_{I} $$
and finally
$$ \left( {\frac{N}{{n_{1} }}} \right)\left( {\frac{{n_{I} }}{N}} \right)^{{\frac{{T_{I} }}{{T_{1} }}}} = \exp \left( {\frac{{\alpha_{1} - \alpha_{I} }}{{kT_{1} }}} \right). $$
Interestingly, similar as in the isothermal case (Vollath and Fischer 2008), the ratio of the particle numbers of any two phases is independent of the number of particles in the other phases. This result allows expanding a theorem derived for isothermal fluctuations in the adiabatic case.

TheoremIn 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.

After inserting ΔT = T2 − T1, one obtains from Eq. 13b
$$ \left( {\frac{{n_{2} }}{{n_{1} }}} \right)\left( {\frac{{n_{2} }}{N}} \right)^{{\frac{\Updelta T}{{T_{1} }}}} = \exp \left( {\frac{{\alpha_{1} - \alpha_{2} }}{{kT_{1} }}} \right). $$
For ΔT = 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
$$ \left( {\frac{{c_{2} }}{{c_{1} }}} \right)\left( {c_{2} } \right)^{{\frac{\Updelta T}{{T_{1} }}}} = \exp \left( {\frac{{\alpha_{1} - \alpha_{2} }}{{kT_{1} }}} \right). $$
In the isothermal case, the number of particles in both phases is equal at Tcross, since α1 = α2 at Tcross. 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 α1 − α2 = 0. In this case, we have to distinguish the following two cases:
  • Phase 1 selected as the reference phase:
    $$ u_{1} + c_{\text{p}} T_{1} - s_{1} T_{1} + kT_{1} = u_{2} + c_{\text{p}} \left( {T_{1} + \Updelta T} \right) - s_{2} \left( {T_{1} + \Updelta T} \right) + k\left( {T_{1} + \Updelta T} \right). $$
  • Phase 2 selected as the reference phase:
    $$ u_{2} + c_{\text{p}} T_{2} - s_{2} T_{2} + kT_{2} = u_{1} + c_{\text{p}} \left( {T_{2} + \Updelta T} \right) - s_{1} \left( {T_{2} + \Updelta T} \right) + k\left( {T_{2} + \Updelta T} \right). $$
Using the value for ΔT 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
$$ T_{{{\text{char}}1}} = \frac{{s_{2} - k}}{{c_{\text{p}} }}T_{\text{cross}} , $$
$$ T_{{{\text{char}}2}} = \frac{{s_{1} - k}}{{c_{\text{p}} }}T_{\text{cross}} . $$
The difference of these two characteristic temperatures is equal to the temperature difference as calculated by Eq. 8,
$$ \left| {T_{\text{char1}} - T_{\text{char2}} } \right| = \left| {\Updelta T} \right|. $$
Figure 2 displays this situation.
Fig. 2

Free enthalpy per particle as function of the temperature. Additionally, the two characteristic temperatures Tchar1 and Tchar2, separated by the temperature difference ΔT, are indicated

Figure 2 visualizes that at temperatures below Tchar2 an adiabatic transformation from the more stable phase 2 to phase 1 is impossible. This is because the shifted temperature T + ΔT of the transformed phase 1, located on the line representing g1, 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 Tcross to Tchar1. The range of phase 2 as reference phase ends at Tchar2. 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 si are larger than cp, generally, the characteristic temperatures are higher than Tcross. 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 c2 = 1 − c1 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\( T_{\text{char}}^{*} . \)

In an ensemble, using phase 1 as reference phase, the highest probability for fluctuations is given by
$$ \left( {\frac{{c_{{2{\text{char}}}} }}{{c_{{1{\text{char}}}} }}} \right)\left( {c_{{2{\text{char}}}} } \right)^{{\frac{\Updelta T}{{T_{\text{char1}} }}}} = \left( {\frac{{1 - c_{{1{\text{char}}}} }}{{c_{{1{\text{char}}}} }}} \right)\left( {1 - c_{{1{\text{char}}}} } \right)^{{\frac{\Updelta T}{{T_{\text{char1}} }}}} = 1. $$
If ΔT = 0, this point is found as \( c_{1} = c_{2} = 0.5. \) In the adiabatic case under discussion, ΔT 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 ΔT/Tchari being selected as variable, are plotted in Fig. 3.
Fig. 3

The characteristic concentrations c1 and c2 as a function of the ratios \( {{\Updelta T} \mathord{\left/ {\vphantom {{\Updelta T} {T_{\text{char1}} }}} \right. \kern-\nulldelimiterspace} {T_{\text{char1}} }} \) and \( {{\Updelta T} \mathord{\left/ {\vphantom {{\Updelta T} {T_{\text{char2}} }}} \right. \kern-\nulldelimiterspace} {T_{\text{char2}} }} \). These are solutions of Eq. 18

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.

Equation 14b has, as it is a transcendent equation, no analytical solution. Therefore, the behavior of the solution is demonstrated using two numerical examples. Figure 4 displays the solution of Eq. 14b in relation to the temperatures T1 and T2, 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.
Fig. 4

Concentration c1 of phase 1 as function of the temperature T for the particle sizes 2 and 5 nm. These graphs are solutions of Eq. 14b

Interestingly, Fig. 4 shows two different branches for the concentration c1 depending on the sign of ΔT. 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 \( T^{*} \).
Fig. 5

Concentration c1 of phase 1 as function of the experimentally accessible temperature \( T^{*} \) for the particle sizes 2 and 5 nm

Plotting the concentration c1 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 general features of the graphs in Fig. 5 are replotted in Fig. 6, where the different ranges are indicated. Usually, one assumes that ranges like the one with positive slope values are considered as not accessible; hence, the temperature range of bistability is limited by the vertical tangents. (The basic ideas of bistability and hysteresis are explained in Appendix 2). Accordingly, the range of concentrations between the vertical tangents (range 2 in Fig. 6) is not accessible. The concentration ranges between 1 and 0, respectively, and the points of contact of the vertical tangents (ranges 1 and 3 in Fig. 6) are the ranges, where fluctuations occur. Looking at Fig. 5, we see the expected result that the range of fluctuations is reduced with increasing particle size.
Fig. 6

General features of the graphs displayed Fig. 5. In this graph, the different ranges are indicated. The range with positive slope values is usually considered as not accessible; this is the range of bistability. The range 2 of concentrations with positive slope is not accessible. The concentration ranges 1 and 2 are the ranges, where fluctuations may be observed

Furthermore, it is interesting to calculate the free enthalpy according to Eq. 9 as a function of the temperature \( T^{*} \) and the particle size. The results are plotted in Fig. 7 again for the particle sizes 2 and 5 nm.
Fig. 7

Free enthalpy as a function of the experimentally accessible temperature \( T^{*} \) for the particle sizes 2 and 5 nm. Similar as in the plot of the concentrations, one observes a range of bistability

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.

Figure 8 shows the range, where, in the adiabatic case, two phases are stable in comparison with the transformation temperature (Ttrans) 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 c1(\( T^{*} \)) 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.
Fig. 8

Comparison of the temperature as a function of the particle size in the case of isothermal and adiabatic conditions with local enclosure. Please realize the extended range of bistability in the adiabatic case, additionally; this range of temperatures is shifted significantly upwards as compared to the isothermal case

From Fig. 8 we learn a few important facts about adiabatic phase transformations of ensembles of nanoparticles in the case of the local enclosure:
  • 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

$$ \mathop {\lim }\limits_{\delta \to 0 - } \frac{{\partial^{n} G}}{{\partial T^{n} }} \ne \mathop {\lim }\limits_{\delta \to 0 + } \frac{{\partial G^{n} }}{{\partial T^{n} }},\quad n \ge 1. $$

In the isothermal case, this analysis is performed in the vicinity of Tcross 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 n1 particles in the thermodynamically stable phase and n2 in the less stable phase, the numbers n1 and n2 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.

In order to calculate the temperature difference ΔT we take again advantage of our “gedanken experiment.” As the internal energy of the ensemble is constant, the following relation is valid
$$ E = Nu_{1} + Nc_{\text{p}} T = \sum\limits_{i = 1}^{I} {n_{i} \left[ {u_{i} + c_{\text{p}} \left( {T + \Updelta T} \right)} \right] = \sum\limits_{i = 1}^{I} {n_{i} u_{i} + Nc_{\text{p}} \left( {T + \Updelta T} \right) = {\text{const}}} } . $$
Equation 20 gives us immediately for ΔT
$$ \Updelta T = \frac{{u_{1} }}{{c_{\text{p}} }} - \frac{{\sum\limits_{i = 1}^{I} {n_{i} u_{i} } }}{{Nc_{\text{p}} }}. $$
Equation 21 brings a new property into the system. In contrast to the local adiabatic case, now ΔT depends on the population of the different phases. Inserting expression (21) for ΔT into Eq. 5 delivers
$$ \begin{gathered} G_{en}^{*} = \sum\limits_{i = 1}^{I} {\left[ {n_{i} u_{i} + n_{i} c_{\text{p}} \left( {T + \Updelta T} \right) - n_{i} s_{i} \left( {T + \Updelta T} \right) - k\left( {T + \Updelta T} \right)_{i} n_{i} \ln \left( {\frac{N}{{n_{i} }}} \right)} \right]} + \lambda \left( {N - \sum\limits_{i = 1}^{I} {n_{i} } } \right) = \hfill \\ \sum\limits_{i = 1}^{I} {n_{i} \left[ {u_{i} + \left( {T - \frac{{\sum\limits_{i = 2}^{I} {n_{i} u_{i} } }}{{Nc_{\text{p}} }} + \frac{{u_{1} }}{{c_{\text{p}} }}} \right)\left( {c_{\text{p}} - s_{i} - k\ln \left( {\frac{N}{{n_{i} }}} \right)} \right)} \right]} + \lambda \left( {N - \sum\limits_{i = 1}^{I} {n_{i} } } \right). \hfill \\ \end{gathered} $$
Determining the minimum of the free enthalpy is, therefore, more difficult. In order to reduce the complexity of the evaluation, we reduce the number of phases to two. Using n2 = N − n1 the minimum is found as the solution of the equation
$$ \left( {\frac{{\partial G_{\text{en}}^{*} }}{{\partial n_{1} }}} \right) = 0, $$
yielding after some algebra
$$ 2\left( {s_{1} - s_{2} } \right)c_{1} - k\ln \left( {c_{1} } \right) - \frac{T}{{T_{\text{cross}} }}c_{\text{p}} - \left( {s_{1} - 2s_{2} } \right) = 0. $$
With respect to c1, 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 ΔT 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.
Fig. 9

Temperature difference ΔT as function of \( T^{*} \) according to Eq. 21. In this example, the low temperature phase was taken as the reference phase

As expected, at low temperatures, where the amount of transformation is insignificant, the temperature difference is zero. At higher temperatures, where the whole ensemble transforms, we find a constant value. Between these two extreme cases, we see a continuous increase of the absolute value of the temperature difference. Figure 10 displays the result for particles with a diameter of 2 and 5 nm. Necessarily, the concentration was plotted versus the experimentally measurable temperature \( T^{*} \) of the ensemble.
Fig. 10

Concentration c1 as a function of the experimentally accessible temperature \( T^{*} \) for two different particle sizes in the case of the adiabatic global enclosure. This graph displays the characteristic behavior of a hysteresis

At the first glance, it is striking that there are two curves in Fig. 10; one for ΔT > 0, observed in case of c1 as reference phase and a second one for ΔT < 0, observed in case of c2 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.

Figure 11 displays the values of the free enthalpy per particle as function of the temperature \( T^{*} \). In this figure, the transition range indicating the hysteresis in-between the two phases is very well visible. This hysteresis of the free enthalpy represents a clear different behavior compared to the local case.
Fig. 11

Free enthalpy as a function of the temperature \( T^{*} \) for the particle sizes 2 and 5 nm in the case of the global adiabatic enclosure. The hysteresis in the temperature range of the phase transformation is well visible

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

In the preceding chapters, a few novel phenomena during phase transformations under adiabatic conditions were predicted. The most important phenomena are:
  • 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.

Comparison with experiments is not simple since, in reality, experiments are neither purely isothermal nor adiabatic. Additionally, in nature nothing happens immediately; therefore, all experiments have to some extent an adiabatic character. Furthermore, in cases, where the phase, undergoing a transformation, is embedded in a matrix, the behavior is primarily adiabatic. First indications of a hysteresis in the melting–solidifying process were found by Stowell (1970) using electron diffraction. A first detailed study of a hysteresis during phase transformations was reported by Cheyssac et al. (1988). These authors determined the optical reflectance of lead nanoparticles during melting. The size of the particles was in the range between 4 and 25 nm, however, with a relatively broad particle size distribution. Figure 12 displays the essentials of the experimental results. This figure shows that the reflectance in the liquid state is significantly higher than in the crystallized state.
Fig. 12

General behavior of the optical reflectance of lead in the temperature range, where melting or crystallization occurs. One clearly realizes the broad temperature hysteresis between melting and solidifying. This graph follows experimental data of Cheyssac et al. (1988)

The hysteresis visible in this figure is very broad. These results, more detailed and quantified for the three particle sizes, are displayed in Fig. 13.
Fig. 13

Optical reflectance of lead nanoparticles determined at a wavelength of 600 nm in the temperature range of melting and solidifying (Cheyssac et al. 1988). The data for the optical reflectance are stacked. This graph shows clearly an increasing width of the hysteresis with increasing particle size. The shift of the hysteresis loop to lower temperatures with decreasing particle sizes is clearly visible. This is exactly the behavior predicted in the preceding chapters

In Fig. 13, in addition to the particle sizes, the 1σ 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.

Most interesting for this discussion are more recent results of Xu et al. (2006, 2007) on melting and crystallization of germanium nanoparticles in silica matrix. These most interesting experimental results were obtained by the analysis of electron diffraction patterns. The use of the intensity of the diffraction patterns is insofar of great advantage, as this intensity is directly proportional to the amount of the phase. Additionally, in the liquid phase, there is no diffraction pattern observed. The germanium nanoparticles had an average size of 5.1 nm and a distribution full width at half maximum (FWHM) of 3.4 nm. Again, as in the previous example, the particle size distribution is relatively broad. Figure 14 displays these results. In this figure, the individual experimental data are not marked; instead, the scattering ranges were plotted as shaded areas. As abscissa, the reduced temperature T/Tmelt is plotted; where, Tmelt is the melting point of bulk germanium. The melting point of the bulk material is found at T/Tmelt = 1.
Fig. 14

Intensity of the electron diffraction patterns of germanium embedded in silica during melting and solidification (Xu et al. 2006, 2007). Besides the hysteresis, this graph clearly shows widely scattered intensities in the ranges, where fluctuations are expected (see Fig. 6)

Figure 14 shows a few striking features, which correspond perfectly with the results of the thermodynamic analysis within this article:
  • 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.

Besides the examples discussed above, hysteresis or bistability regions were often observed in connection with melting and crystallizing of nanoparticles. Further typical examples are the melting hysteresis of bismuth (Haro-Poniatowski et al. 2004) or gallium nanoparticles (Tognini et al. 2000). Similar results are also obtained with nanoparticles made of the intermetallic phase InSb (Tetu et al. 2008). Figure 15 displays these results. Similar as in Fig. 14, only the scattering ranges of the intensity of the electron diffraction rings and not the individual data are indicated in this figure.
Fig. 15

Intensity of the electron diffraction rings of InSb during melting and crystallization for nanoparticles with an average diameter of ca. 20 nm embedded in silica (Tetu et al. 2008)

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.


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 analysis of phase transformations of ensembles in an adiabatic enclosure led to a series of predictions concerning the experimentally observable behavior. The most important features, characteristic for adiabatic transformations of such ensembles, are:
  • 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.

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© Springer Science+Business Media B.V. 2009