- Email: [email protected]
The transition from bulk to nano as a phase transition
ARTICLE IN PRESS
Physica E 33 (2006) 359–362 www.elsevier.com/locate/physe
The transition from bulk to nano as a phase transition M. Ghanashyam Krishnaa,, A.K. Kapoora, M. Durga Prasadb, V. Srinivasanc a
School of Physics, University of Hyderabad, Hyderabad 500 046, India School of Chemistry, University of Hyderabad, Hyderabad 500 046, India c Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600025, India b
Received 13 February 2006; accepted 6 April 2006 Available online 12 June 2006
Abstract It is demonstrated that the chemical potential of bosons trapped in a harmonic potential shows a discontinuity as a function of the number of particles in the system. In the model used, it is shown that if the number of particles is of the order of 106 or greater, bulk-like behaviour is exhibited by the system. This translates to a ratio of V/Vc4106 for bulk behaviour, where V is the crystallite volume of the experimental sample and Vc is the volume of the unit cell. Several experimental results covering a wide range of physical phenomena that corroborate the fact that such a number-induced phase transition indeed exists are presented. r 2006 Elsevier B.V. All rights reserved. PACS: 64.70.Nd; 81.07.b; 05.30.Jp Keywords: Phase transitions; Nanostructures
1. Introduction Phase transitions are characterized by discontinuities in the (higher order) derivatives of the Gibbs free energy. The Gibbs free energy G is a function of pressure (P), temperature (T) and the number of particles (N) in the system. In all treatments of phase transition, until to recently, G has been considered in terms of P and T alone. It is, however, possible that G has discontinuous derivatives with respect to N also. That this is indeed so appears to emerge from the work of Napolitano et al. [1] although their goal was different. They considered a collection of Bose particles trapped in a three-dimensional harmonic potential. Their motivation was to present a size-dependent definition for the critical temperature. Fixing the average number of particles, they calculate m (T), and then the specific heat CN(T). Using the criterion that dC N ¼ 0, dT
they determined the critical temperature Tc. A close examination of their data (Fig. 1 of Napolitano et al.) indicates that the derivatives of CN are discontinuous as N increases for fixed T. A study of this N dependent phase transition is the goal of the present work. We describe the model in Section 2 along with the results. Implications of these results are presented in Section 3. 2. Model system and numerical results The system under consideration is a three-dimensional isotropic harmonic oscillator. The energy levels are given by E n ¼ n_o; n ¼ 0; 1; 2 . . . . The degeneracy for the nth level is ðn þ 1Þðn þ 2Þ . (2) 2 The B.E. distribution for the average number of particles is gn ¼
ZðE n Þ ¼ Corresponding author. Tel.: +91 40 23134255; fax: +91 40 23010227.
E-mail address: [email protected] (M. Ghanashyam Krishna). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.04.003
(1)
1 , ðebðE n mÞ 1Þ
where b ¼ 1=kB T, kB is Boltzmann’s constant.
(3)
ARTICLE IN PRESS M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
360
Assuming a fixed number N of bosons we calculate m using the equation N¼
1 X
3.1. Emergence of nano-behaviour gn ZðE n Þ.
(4)
n¼0
The chemical potential m is calculated as a function of N for a fixed temperature. These results are presented in Fig. 1. As can be seen, m clearly undergoes a discontinuity between N ¼ 105 and N ¼ 107. A close look at the Fig. 1 of Napolitano et al [1]. shows that Cv/N shows a smooth change as T increases and a cusp-like behaviour that characterizes a phase transition emerges only after N4105, agreeing with our results. Since m ¼ ðqG=qNÞP;T , the discontinuity in m heralds a phase transition. Although it is well known that G is a function of N, phase transitions induced by change in N have not been considered in literature to date. 3000000
1.2Tc 2Tc
5000
2500000
0.999Tc
4000 µΝ
2000000
3000 2000 1000
µN
3. Discussion
0
1500000
2
3
4 5 log N
6
7
1000000 500000 0 2
3
4
5
6
7
8
log N Fig. 1. The variation in mN as a function of log N. It should be noted that m is small and negative and what is plotted is abs(m)N and N stands for mean N on the y-axis. The inset shows the variation for 0.999 Tc.
Our results indicate that the chemical potential shows a discontinuity as the number of particles in the system increases beyond 106. On carrying out a detailed survey of literature to see if such a phenomenon is observed experimentally, it was found that there are indeed several such examples, primarily in the domain of what might be termed as ‘nano-to-bulk phase transitions’. We shall analyse this, in the context of the model used in the current study, by examining a wide variety of physical phenomena where the volume is decreased slowly and nanostructures emerge at some point. A phenomenological approach considered by some of us recently [2] came to the conclusion that for V/Vco106, the transition to the nanophase is made where V is the volume of crystallites in the solid prepared experimentally and Vc the unit-cell volume. If the phenomenon is observed in solids for which the volume of the unit cell is known, it can be used as a natural unit of volume. In the case of liquids and gases, the coefficient ‘b’ in the Van der Waals equation is an experimentally verified quantity and is therefore the analogue of the volume of the unit cell. Indeed, the order of magnitude of this is the same as that of the unit cell. A representative set of examples of transitions to the nanophase in a wide variety of physical phenomena is shown in Table 1. The selection criterion is based on differences in physical phenomena. From these wide variety of examples it emerges that if the crystallite volume of the experimentally prepared sample is such that V/Vco106, the nano-effects are seen. This volume ratio can be converted into a ratio for the number of particles by substituting the number of atoms per unit cell. In Bravais lattices, it is well known that this number rarely exceeds four. Therefore it leads to an upper limit of 105 on N/Nc. A dimensional argument can also be given to this effect. The expectation value of x2 in the highest occupied level nF
Table 1 Examples of bulk-nanophase transitions in different physical phenomena Phenomenon (ref. no.)
Bulk state phase
Nano-state
V/Vc
Physical quantity that changes
Magnetism [3] Superfluidity [4] Bose–Einstein condensate [6] Superconductivity [7] Optical absorption [8]
Ferromagnetic (solid) Bulk 4He (liquid) 40 K Bose superfluid Bulk superconductor (solid) Solid with single crystal-like band gap Co (HCP) Fe-Ge (FCC) Cu (normal metal) La0.875Sr0.125MnO3 (multidomain solid) Ferroelectric Bulk
Superparamagnetic (solid) 4 He clusters
105 105 105 106 105
Coercivity Viscosity Superfluidity Transition Temperature Band gap
106 106 106 106
Elasticity Magnetization
106 106
Polarization Bond length
Structural phase transition [9] Structural phase transition [10] Superplasticity [11] Multi-domain to single domain [12] Ferroelectricity [13] Encapsulated water [14]
No superconductivity (solid) Solid with blue shifted band gap (quantum confinement) Co (FCC) Bc Cu (superplastic solid) La0.875Sr0.125MnO3 (single domain solid) Paraelectric Nanotube water
ARTICLE IN PRESS M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
3=2
V 0 nF
.
1200
300 TC (K)
as the calculations indicate the N for the phase transition is between 105 and 106. We now discuss some of the examples presented in Table 1. For instance, one of the oldest known examples of such a transition is the size-dependent ferromagnetic to superparamagnetic transition. Superparamagnetism is a well-documented phenomenon wherein a previously ferromagnetic material with a finite coercivity undergoes a transition to a state of zero coercivity below a certain critical diameter of particles. Superparamagnetism has been observed in compounds as well as alloys. A recent example is of a study on CoFe2O4 that is a partially inverse spinel type ferrite and a soft magnetic substance that has found application in a number of devices [3]. It has a lattice parameter of 8.39 A˚ and the volume of the unit cell is 590.59 A˚3. In this study, Rajendran et al. [3] have found that below a crystallite size of the order of 30 nm, the material shows a transition to superparamagnetic behaviour. The superparamagnetic behaviour is completely quantum mechanical in nature. From these values the ratio of the bulk volume to that of the unit cell can easily be calculated. Bulk CoFe2O4/Unit cell of CoFe2O4 ¼ 27 1018/ 590.59 1024105. Significantly, at this value of the ratio, all examples shown in the table, including these two, show a transition to nanostructured behaviour. In Bose–Einstein condensation, only when the number of particles in the cloud is 104 or more, the superfluid behaviour is observed [4]. The bulk to nanophase transition can occur in solid, liquid or gaseous state as permitted by the Van der Waals equation (for the bulk state). The saturated vapour pressure in equillibrium with droplets of liquids is higher than the saturated vapour pressure over the bulk liquid [5]. This relation, given by the Kelvin equation, predicts that the droplet size is 40 nm for a difference of 1% in the two vapour pressures for a surface tension of 20 mN m1. This size, significantly, represents about 105 molecules. Recently, 4He clusters in the liquid
1400
(a)
350
1000
(b)
250
800
200
NANO
BULK
600
150 400 100
200
50 0 100
0 101
102
103
104 105 V/VC
106
107
108
-200 109
Fig. 2. The variation in (a) critical temperature, Tc, of BaTiO3 showing transition from ferroelectric to paraelectric behaviour and (b) the coercivity of CoFe2O4 showing the ferromagnetic to superparamagnetic transition as a function of V/Vc The figure shows the phase transition at 105–106 in both cases. The figure has been replotted using data from the respective Refs. [3,13]. The lines are only a guide to the eye.
100 6 80 (a)
60 Tc (K)
V n3 N V0 F
1600
400
The Ptotal number of particles in the system, N ¼ nF ðn þ 1Þðn þ 2Þ=2 is of the order of n3F , which 0 leads us to the estimate
Coercivity (arb. units)
Taking the sample to spherical, the radius of the sample be 1=2 1=2 is of the order of x2 nF . The total volume of the 3=2 sample, V, is then of the order of nF . Next, the number of nodes in the highest level is nF1nF for large nF. Since the 1=2 wave function spans a length of the order of nF , the distance between two successive nodes is of the order of 1=2 nF . So the volume of the nodal cell, V0, is
state were obtained [6]. On laser cooling these, the nanostructure to bulk transition occurs when the clusters go to the superfluid state, whereas the clusters are gaseous at normal temperatures. These examples suggest that there is a phase transition when V/Vc is of the order of 106. This is even more evident from Figs. 2 and 3, where the data from some of the examples presented in Table 1 have been re-plotted as a function of V/Vc. These figures show that the ferroelectric to paraelectric transition in BaTiO3
4
Nano
40
Bulk 2 (b)
Volume ratio Vα/β
is given by 2 x a nF .
361
20
0 0 103
104
105
106
107
108
V/Vc Fig. 3. The variation in (a) superconducting critical temperature, Tc, of YBCO and (b) lattice parameters showing the structural phase transition in Co particles as a function of V/Vc. The figure has been replotted using data from the respective Refs. [7,9]. The lines are only a guide to the eye.
ARTICLE IN PRESS 362
M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
and appearance of superparamagnetism in cobalt ferrite (Fig. 2), collapse of superconducting transition in YBCO and structural phase transition in Co particles (Fig. 3) all occur when V/Vc is of the order of 106. At this point, some physical property of the system undergoes an abrupt change. Since these properties are derivatives of the Gibbs free energy, these are indeed phase transitions. For example, in the structural phase transition in Co particles presented in Fig. 3, the volume changes discontinuously indicating that it is a first-order phase transition. Similarly, the coercivity shows a discontinuous change in the collapse of ferromagnetism in cobalt ferrite depicted in Fig. 2 indicating again a first order phase transition. In all the examples quoted here, the nano to bulk transitions carry the signature of a phase transition. Since all transitions go to a different phase in their physical characteristic described by a different phenomenon from that determining bulk behaviour, we propose that the nano–bulk transition is a phase transition. Recent experiments suggest that the nanophase, as it should, is retained as temperature is raised, while the bulk undergoes a phase transition from solid to liquid [7,14]. Experiments performed on nano-water show that from 10 to 300 1K the large mean-square displacement of hydrogen, the physical parameter used by them, does not undergo any abrupt changes, while in the bulk, ice transforms into water and consequently this physical parameter changes abruptly. Recent experiments on carbon nanotubes show that the additional Raman lines, that are not present in the bulk phase, appear in the nanophase. To a first approximation it seems that the V/Vc even in this case is of the order of 106 [15]. This further confirms our conclusion that nano is indeed a new phase, distinct from the solid, liquid and gas phases that appear in the bulk. If V/Vc is greater than 105–106, depending on the temperature it will go to one of the bulk phases. Another example of this transition at 106 is the inverse Hall Petch effect predicted theoretically and observed experimentally in Cu [11,16]. 4. Conclusions In conclusion we have studied a model of Bose particles and showed that a phase transition can be induced in it by changing the number of particles in the system. This threshold number appears to be of the order of 106. We
have examined several experiments and presented a representative set of examples to demonstrate that such a number threshold is universally valid in almost all such systems. The examples selected had a bulk to nanophase transition. So we believe that a criterion for the definition of the nanophase has emerged from this study.
Acknowledgements The authors acknowledge several discussions with our dear friend late Professor Bhaskar Maiya, who passed away unexpectedly in March, 2004. Numerous discussions and suggestions of Professor Subhash Chaturvedi are gratefully acknowledged. Financial support from CSIR, DST and UGC-UPE and SAP programs is also acknowledged.
References [1] R. Napolitano, J. De Luca, V.S. Bagnato, Phys. Rev. A 55 (1997) 3954. [2] M. Ghanashyam Krishna, V. Srinivasan, AIP Conf. Proc. 695 (2003) 108. [3] M. Rajendran, R.C. Pullar, A.K. Bhattacharya, D. Das, S.N. Chintalapudi, C.K. Majumdar, J. Magn. Magn. Mater. 232 (2001) 71. [4] R. Bruhl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T. Savas, J.P. Toennies, Phys. Rev. Lett. 92 (2004) 185301. [5] P.W. Atkins, Physical Chemistry, Oxford University Press, London, 1982. [6] C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92 (2004) 040403. [7] M.S. Multani, P. Guptasarma, Phys. Lett. 142 (1989) 293. [8] H. Arizpe-Chavez, R. Ramirez-Bon, F.J. Espinoza-Beltran, O. Zelaya-Angel, J.L. Marin, R. Riera, J. Phys. Chem. Solids. 61 (2000) 511. [9] O. Kitakami, H. Sato, Y. Shimada, F. Sato, M. Tanaka, Phys. Rev B 56 (1997) 13849. [10] S. Sarkar, C. Bansal, A. Chaterjee, Phys Rev. B 62 (2000) 3218. [11] Y. Wang, M. Chen, F. Zhou, E. Ma, Nature 419 (2002) 912. [12] A. Dutta, N. Gayathri, R. Ranganathan, Phys. Rev. B 68 (2003) 054432. [13] R. Bottcher, C. Klimm, D. Michel, H.C. Semmelbeck, G. Volkel, H.J. Glasel, E. Hartmann, Phys Rev B 62 (2000) 2085. [14] A.I. Koesnikov, J. Zanotti, C.-K. Loong, P. Thiyagarajan, A. Moravsky, R.O. Loutfy, C.J. Burnham, Phys. Rev. Lett. 93 (2004) 035503. [15] R. Saito, T. Takeya, T. Kimura, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B. 59 (1999) 2388. [16] K.A. Padmanabhan, H. Gleiter, Mater. Sci. Eng A 381 (2004) 28.
Physica E 33 (2006) 359–362 www.elsevier.com/locate/physe
The transition from bulk to nano as a phase transition M. Ghanashyam Krishnaa,, A.K. Kapoora, M. Durga Prasadb, V. Srinivasanc a
School of Physics, University of Hyderabad, Hyderabad 500 046, India School of Chemistry, University of Hyderabad, Hyderabad 500 046, India c Department of Theoretical Physics, University of Madras, Guindy Campus, Chennai 600025, India b
Received 13 February 2006; accepted 6 April 2006 Available online 12 June 2006
Abstract It is demonstrated that the chemical potential of bosons trapped in a harmonic potential shows a discontinuity as a function of the number of particles in the system. In the model used, it is shown that if the number of particles is of the order of 106 or greater, bulk-like behaviour is exhibited by the system. This translates to a ratio of V/Vc4106 for bulk behaviour, where V is the crystallite volume of the experimental sample and Vc is the volume of the unit cell. Several experimental results covering a wide range of physical phenomena that corroborate the fact that such a number-induced phase transition indeed exists are presented. r 2006 Elsevier B.V. All rights reserved. PACS: 64.70.Nd; 81.07.b; 05.30.Jp Keywords: Phase transitions; Nanostructures
1. Introduction Phase transitions are characterized by discontinuities in the (higher order) derivatives of the Gibbs free energy. The Gibbs free energy G is a function of pressure (P), temperature (T) and the number of particles (N) in the system. In all treatments of phase transition, until to recently, G has been considered in terms of P and T alone. It is, however, possible that G has discontinuous derivatives with respect to N also. That this is indeed so appears to emerge from the work of Napolitano et al. [1] although their goal was different. They considered a collection of Bose particles trapped in a three-dimensional harmonic potential. Their motivation was to present a size-dependent definition for the critical temperature. Fixing the average number of particles, they calculate m (T), and then the specific heat CN(T). Using the criterion that dC N ¼ 0, dT
they determined the critical temperature Tc. A close examination of their data (Fig. 1 of Napolitano et al.) indicates that the derivatives of CN are discontinuous as N increases for fixed T. A study of this N dependent phase transition is the goal of the present work. We describe the model in Section 2 along with the results. Implications of these results are presented in Section 3. 2. Model system and numerical results The system under consideration is a three-dimensional isotropic harmonic oscillator. The energy levels are given by E n ¼ n_o; n ¼ 0; 1; 2 . . . . The degeneracy for the nth level is ðn þ 1Þðn þ 2Þ . (2) 2 The B.E. distribution for the average number of particles is gn ¼
ZðE n Þ ¼ Corresponding author. Tel.: +91 40 23134255; fax: +91 40 23010227.
E-mail address: [email protected] (M. Ghanashyam Krishna). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.04.003
(1)
1 , ðebðE n mÞ 1Þ
where b ¼ 1=kB T, kB is Boltzmann’s constant.
(3)
ARTICLE IN PRESS M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
360
Assuming a fixed number N of bosons we calculate m using the equation N¼
1 X
3.1. Emergence of nano-behaviour gn ZðE n Þ.
(4)
n¼0
The chemical potential m is calculated as a function of N for a fixed temperature. These results are presented in Fig. 1. As can be seen, m clearly undergoes a discontinuity between N ¼ 105 and N ¼ 107. A close look at the Fig. 1 of Napolitano et al [1]. shows that Cv/N shows a smooth change as T increases and a cusp-like behaviour that characterizes a phase transition emerges only after N4105, agreeing with our results. Since m ¼ ðqG=qNÞP;T , the discontinuity in m heralds a phase transition. Although it is well known that G is a function of N, phase transitions induced by change in N have not been considered in literature to date. 3000000
1.2Tc 2Tc
5000
2500000
0.999Tc
4000 µΝ
2000000
3000 2000 1000
µN
3. Discussion
0
1500000
2
3
4 5 log N
6
7
1000000 500000 0 2
3
4
5
6
7
8
log N Fig. 1. The variation in mN as a function of log N. It should be noted that m is small and negative and what is plotted is abs(m)N and N stands for mean N on the y-axis. The inset shows the variation for 0.999 Tc.
Our results indicate that the chemical potential shows a discontinuity as the number of particles in the system increases beyond 106. On carrying out a detailed survey of literature to see if such a phenomenon is observed experimentally, it was found that there are indeed several such examples, primarily in the domain of what might be termed as ‘nano-to-bulk phase transitions’. We shall analyse this, in the context of the model used in the current study, by examining a wide variety of physical phenomena where the volume is decreased slowly and nanostructures emerge at some point. A phenomenological approach considered by some of us recently [2] came to the conclusion that for V/Vco106, the transition to the nanophase is made where V is the volume of crystallites in the solid prepared experimentally and Vc the unit-cell volume. If the phenomenon is observed in solids for which the volume of the unit cell is known, it can be used as a natural unit of volume. In the case of liquids and gases, the coefficient ‘b’ in the Van der Waals equation is an experimentally verified quantity and is therefore the analogue of the volume of the unit cell. Indeed, the order of magnitude of this is the same as that of the unit cell. A representative set of examples of transitions to the nanophase in a wide variety of physical phenomena is shown in Table 1. The selection criterion is based on differences in physical phenomena. From these wide variety of examples it emerges that if the crystallite volume of the experimentally prepared sample is such that V/Vco106, the nano-effects are seen. This volume ratio can be converted into a ratio for the number of particles by substituting the number of atoms per unit cell. In Bravais lattices, it is well known that this number rarely exceeds four. Therefore it leads to an upper limit of 105 on N/Nc. A dimensional argument can also be given to this effect. The expectation value of x2 in the highest occupied level nF
Table 1 Examples of bulk-nanophase transitions in different physical phenomena Phenomenon (ref. no.)
Bulk state phase
Nano-state
V/Vc
Physical quantity that changes
Magnetism [3] Superfluidity [4] Bose–Einstein condensate [6] Superconductivity [7] Optical absorption [8]
Ferromagnetic (solid) Bulk 4He (liquid) 40 K Bose superfluid Bulk superconductor (solid) Solid with single crystal-like band gap Co (HCP) Fe-Ge (FCC) Cu (normal metal) La0.875Sr0.125MnO3 (multidomain solid) Ferroelectric Bulk
Superparamagnetic (solid) 4 He clusters
105 105 105 106 105
Coercivity Viscosity Superfluidity Transition Temperature Band gap
106 106 106 106
Elasticity Magnetization
106 106
Polarization Bond length
Structural phase transition [9] Structural phase transition [10] Superplasticity [11] Multi-domain to single domain [12] Ferroelectricity [13] Encapsulated water [14]
No superconductivity (solid) Solid with blue shifted band gap (quantum confinement) Co (FCC) Bc Cu (superplastic solid) La0.875Sr0.125MnO3 (single domain solid) Paraelectric Nanotube water
ARTICLE IN PRESS M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
3=2
V 0 nF
.
1200
300 TC (K)
as the calculations indicate the N for the phase transition is between 105 and 106. We now discuss some of the examples presented in Table 1. For instance, one of the oldest known examples of such a transition is the size-dependent ferromagnetic to superparamagnetic transition. Superparamagnetism is a well-documented phenomenon wherein a previously ferromagnetic material with a finite coercivity undergoes a transition to a state of zero coercivity below a certain critical diameter of particles. Superparamagnetism has been observed in compounds as well as alloys. A recent example is of a study on CoFe2O4 that is a partially inverse spinel type ferrite and a soft magnetic substance that has found application in a number of devices [3]. It has a lattice parameter of 8.39 A˚ and the volume of the unit cell is 590.59 A˚3. In this study, Rajendran et al. [3] have found that below a crystallite size of the order of 30 nm, the material shows a transition to superparamagnetic behaviour. The superparamagnetic behaviour is completely quantum mechanical in nature. From these values the ratio of the bulk volume to that of the unit cell can easily be calculated. Bulk CoFe2O4/Unit cell of CoFe2O4 ¼ 27 1018/ 590.59 1024105. Significantly, at this value of the ratio, all examples shown in the table, including these two, show a transition to nanostructured behaviour. In Bose–Einstein condensation, only when the number of particles in the cloud is 104 or more, the superfluid behaviour is observed [4]. The bulk to nanophase transition can occur in solid, liquid or gaseous state as permitted by the Van der Waals equation (for the bulk state). The saturated vapour pressure in equillibrium with droplets of liquids is higher than the saturated vapour pressure over the bulk liquid [5]. This relation, given by the Kelvin equation, predicts that the droplet size is 40 nm for a difference of 1% in the two vapour pressures for a surface tension of 20 mN m1. This size, significantly, represents about 105 molecules. Recently, 4He clusters in the liquid
1400
(a)
350
1000
(b)
250
800
200
NANO
BULK
600
150 400 100
200
50 0 100
0 101
102
103
104 105 V/VC
106
107
108
-200 109
Fig. 2. The variation in (a) critical temperature, Tc, of BaTiO3 showing transition from ferroelectric to paraelectric behaviour and (b) the coercivity of CoFe2O4 showing the ferromagnetic to superparamagnetic transition as a function of V/Vc The figure shows the phase transition at 105–106 in both cases. The figure has been replotted using data from the respective Refs. [3,13]. The lines are only a guide to the eye.
100 6 80 (a)
60 Tc (K)
V n3 N V0 F
1600
400
The Ptotal number of particles in the system, N ¼ nF ðn þ 1Þðn þ 2Þ=2 is of the order of n3F , which 0 leads us to the estimate
Coercivity (arb. units)
Taking the sample to spherical, the radius of the sample be 1=2 1=2 is of the order of x2 nF . The total volume of the 3=2 sample, V, is then of the order of nF . Next, the number of nodes in the highest level is nF1nF for large nF. Since the 1=2 wave function spans a length of the order of nF , the distance between two successive nodes is of the order of 1=2 nF . So the volume of the nodal cell, V0, is
state were obtained [6]. On laser cooling these, the nanostructure to bulk transition occurs when the clusters go to the superfluid state, whereas the clusters are gaseous at normal temperatures. These examples suggest that there is a phase transition when V/Vc is of the order of 106. This is even more evident from Figs. 2 and 3, where the data from some of the examples presented in Table 1 have been re-plotted as a function of V/Vc. These figures show that the ferroelectric to paraelectric transition in BaTiO3
4
Nano
40
Bulk 2 (b)
Volume ratio Vα/β
is given by 2 x a nF .
361
20
0 0 103
104
105
106
107
108
V/Vc Fig. 3. The variation in (a) superconducting critical temperature, Tc, of YBCO and (b) lattice parameters showing the structural phase transition in Co particles as a function of V/Vc. The figure has been replotted using data from the respective Refs. [7,9]. The lines are only a guide to the eye.
ARTICLE IN PRESS 362
M. Ghanashyam Krishna et al. / Physica E 33 (2006) 359–362
and appearance of superparamagnetism in cobalt ferrite (Fig. 2), collapse of superconducting transition in YBCO and structural phase transition in Co particles (Fig. 3) all occur when V/Vc is of the order of 106. At this point, some physical property of the system undergoes an abrupt change. Since these properties are derivatives of the Gibbs free energy, these are indeed phase transitions. For example, in the structural phase transition in Co particles presented in Fig. 3, the volume changes discontinuously indicating that it is a first-order phase transition. Similarly, the coercivity shows a discontinuous change in the collapse of ferromagnetism in cobalt ferrite depicted in Fig. 2 indicating again a first order phase transition. In all the examples quoted here, the nano to bulk transitions carry the signature of a phase transition. Since all transitions go to a different phase in their physical characteristic described by a different phenomenon from that determining bulk behaviour, we propose that the nano–bulk transition is a phase transition. Recent experiments suggest that the nanophase, as it should, is retained as temperature is raised, while the bulk undergoes a phase transition from solid to liquid [7,14]. Experiments performed on nano-water show that from 10 to 300 1K the large mean-square displacement of hydrogen, the physical parameter used by them, does not undergo any abrupt changes, while in the bulk, ice transforms into water and consequently this physical parameter changes abruptly. Recent experiments on carbon nanotubes show that the additional Raman lines, that are not present in the bulk phase, appear in the nanophase. To a first approximation it seems that the V/Vc even in this case is of the order of 106 [15]. This further confirms our conclusion that nano is indeed a new phase, distinct from the solid, liquid and gas phases that appear in the bulk. If V/Vc is greater than 105–106, depending on the temperature it will go to one of the bulk phases. Another example of this transition at 106 is the inverse Hall Petch effect predicted theoretically and observed experimentally in Cu [11,16]. 4. Conclusions In conclusion we have studied a model of Bose particles and showed that a phase transition can be induced in it by changing the number of particles in the system. This threshold number appears to be of the order of 106. We
have examined several experiments and presented a representative set of examples to demonstrate that such a number threshold is universally valid in almost all such systems. The examples selected had a bulk to nanophase transition. So we believe that a criterion for the definition of the nanophase has emerged from this study.
Acknowledgements The authors acknowledge several discussions with our dear friend late Professor Bhaskar Maiya, who passed away unexpectedly in March, 2004. Numerous discussions and suggestions of Professor Subhash Chaturvedi are gratefully acknowledged. Financial support from CSIR, DST and UGC-UPE and SAP programs is also acknowledged.
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