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Atomic mobility and thermodynamics of ultrafine clusters
COLLOIDS AND ELSEVIER
Colloids and Surfaces A: Physicochemicaland Engineering Aspects 108 (1996) 315 319
A
SURFACES
Atomic mobility and thermodynamics of ultrafine clusters I.P. Suzdalev a,, N.I. Shklovskaya b a Institute of Chemical Physics, Russian Academy of Science, Kosygina 4, 117977 Moscow, Russia b Institute of Surface Chemistry, National Academy of Science of Ukraine, Prospect of Science 31, 252028 Kiev, Ukraine Received 15 October 1994; accepted 1 October 1995
Abstract The atomic dynamics of ultrafine clusters of ferric hydroxide and 7Fe203 with average size 1-3 nm are reported. The clusters were synthesised by nucleation in solution in the micropores of sorbents and by thermal decomposition of ferric oxalate in air. Thermodynamic and phenomenological approaches to the dynamic state of the cluster are considered and the probability of the formation of specific solid-liquid states of the cluster is discussed. Keywords: Intracluster dynamics; M6ssbauer spectroscopy 57Fe; Nucleation of clusters; Thermodynamics
1. Introduction Ultrafine clusters of matter in the nanometer scale possess unique properties connected with the dynamic state, electron conductivity, magnetic properties, electron-phonon interaction etc. [1]. Intracluster dynamics and the effects of the decrease of melting point and the existence of separate melting and freezing points are of special interest. This behaviour is considered in terms of the increase of intracluster atomic mobility and with the appearance of a special solid-liquid state in the region between the melting and freezing points [2]. The present paper deals with the study of the atomic dynamics of ultrafine clusters of ferric hydroxide and y-ferric oxide of size 1-3 nm by MSssbauer spectroscopy and thermodynamic analysis. In order to increase the atomic mobility, the clusters were obtained in pores of sorbents or * Corresponding author. Paper presented at the XIII European Chemistryat Interfaces Conference held in Kiev, Ukraine, 11-16 September 1994. 0927-7757/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0927-7757(95)03429-3
were treated by surface active agents (surfactants) that decreased intercluster interaction and prevented cluster aggregation. Thermodynamic and phenomenological cluster models for intracluster atomic dynamics were developed and conditions for the existence of a specific solid-liquid state were discussed.
2. Experimental Ferric hydroxide clusters were synthesised by chemical reaction in pores of sorbents [3,4]. Polysorbs obtained by copolymerization of styrene and divinylbenzene were impregnated with 57FEC13 in water or ethanol solutions of concentrations 0.05, 0.025 or 0.0125 M. The nucleation of iron hydroxide clusters occurred after passing air mixed with N H 3 through a layer of Polysorb. Average sizes of pores were 7-75 nm and average sizes of clusters 1-4.5 nm. The average sizes of the clusters were determined both from M6ssbauer spectra and
316
I.P. Suzdalev, N.I. Shklovskaya/ColloidsSurfacesA: Physicochem.Eng. Aspects108 (1996) 315-319
by taking account of the concentration of iron salt inside the pores. Clusters of ferric oxide, 7Fe20 3, were synthesised by a solid-state thermal decomposition of iron oxalate Fe2(C204)a-5H20 in air [5]. This reaction involves initial loss of water molecules and CO2, then partial reduction of Fe 3÷ to Fe 2÷ in the temperature range td= 180-200°C. The nucleation of 7Fe20 3 clusters occurred when the temperature was increased further to 400°C. The final size of the cluster was fixed after the end of the solid-state chemical reaction at some final temperature ta by rapid quenching of the sample to room temperature. The estimation of cluster size was carried out by both M6ssbauer spectroscopy and by BET measurement of specific surface area with argon as an adsorbate. The separation of clusters was carried out by mechanical treatment and the application of surface active substances (SAS), or surfactants, followed by drying. As SAS we used solutions of ethanol, butanol, isopropyl alcohol, sodium dodecyl sulphate (SDS) and polyoxyethylenesorbitan3-stearate (Tween-65). The studies were carried out in a MOssbauer spectrometer of Wissell type (Germany) with a 5VCo(Cr) source.
3. Results and discussion
M6ssbauer spectra of iron hydroxide and 7FezO3 are standard doublets [ 3 - 5 ] . Neither broadening nor so called broad components of the spectra [6] were observed. Therefore diffusion of clusters with correlation times smaller than 10-Vs is absent. The variation of the line spectra intensity (e) of iron hydroxide clusters in 7-75 nm pores is shown on Fig. 1. It is easy to see a considerable decrease of e with decreasing pore diameter. This effect is connected with an increase in atomic mobility because e is determined by the M6ssbauer effect probability f , = exp( - ( x 2)/¢~2), where ( x 2) is the mean square displacement of an iron atom and ¢~2 the wavelength of 14.4 keV radiation divided by 2~. The sharp drop in e with decreasing cluster size is explained by the increase of ( x 2) and a phase transition connected with the melting of clusters in a pore containing solution. A similar effect of e could be caused also by the diffusion of
fo-
jr,
YO
o{a.,~
Fig. 1. Dependence of line intensity e for MOssbauer spectra on the pore sizes of Polysorbs at T=300 K: (1) concentration of NH3 in air 0.08%; (2) concentration of NH3 in air 0.008%. a cluster as a whole and by intracluster mobility. However the mobility of the whole cluster in a pore is connected unequivocally with the presence in the spectra of both narrow and broad components, i.e. diffusive, restricted motions [-6]. Therefore the drop in e and the increase of ( x 2) are connected with intracluster mobility. This effect can be understood in terms of a thermodynamic model of nucleation [3 ]. In Fig. 2, the Gibb's free energy is shown for cluster nucleation from solution in a pore:
AG=4/3~pr3(3a/pr- {A#+ln[no/(no- n)]})
(1)
where a is the surface free energy per unit surface area, A# = #1 - #s is the difference between chemical potentials of liquid and solid phases, no/N is initial concentration of solution, no and n = 4/3~pr 3 are the numbers of iron atoms in solution and in the cluster respectively, r and p are the radius and density of the cluster. The minimum critical volume of the cluster is n~n = 327zcr3K(~)/3p2A#3, the maxim u m volume is n~ax = noA#/(2 + A#), where K(c0 is the factor of heterogeneity and ct the contact angle of the cluster with the surface of the pore. The energy barrier of nucleation A GD depends on the heterogeneity as AG~ = 16rw3K(e)/3p2A# 2 and
LP. Suzdalev, N.1. Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
317
gf.
G
~o
3.o
2,o \\ r
g,o
2 3
5 Fig. 2. G as a function of cluster size and conditions of nucleation. (1) Cluster in solution of macro volume with concentration no/N; (2) clusters in micropore; (3) clusters in micropore with heterogeneity K(~),AG~ and AGR are direct and reverse activation barrier for the nucleation and decomposition of clusters; rmi., rma~ are minimum and maximum radii of the clusters respectively.
decreases with decreasing pore size (K(e)< 1). The energy barrier for decomposition of a cluster, AG R, diminishes too and AGO > AG R. The Gibbs free energy (Fig. 2) allows us to interpret the atomic mobility of clusters in terms of fluctuating reversible transitions from the solid phase (n. . . . rmax) through the barrier AG R to the liquid phase. The decrease of pore size corresponds to the reduction of cluster size from 3 to 1 nm (excepting larger pores of 75 nm) [3] and to the increase of the probability of fluctuations. It means that a unique solid-liquid state and higher atomic mobility appear for 1-2 nm clusters. This hypothesis allows us to explain the diminishing atomic mobility not only by the increase of cluster size but also by the loss of collective (solid state) properties after cluster conversion to trimer, dimer and single atom. So one can propose the existence of a minimum e at some cluster sizes. A similar minimum was observed for ~Fe203 clusters in the solid-state chemical thermal decomposition of iron oxalate. In Fig. 3 the areas of MSssbauer spectra (S) of 7Fe203 clusters at different temperatures of thermal decomposition are shown, A minimum S value is observed at td = 215°C, corresponding to minimal clusters with
Fig. 3. Dependence of spectral area S on temperature of thermal decomposition of ferric oxalate (full points). The decrease of S after treatment with various SAS is shown by open points; (1) ethanol; (2) isopropyl alcohol; (3) Tween-65; (4) butanol; (5) SDS.
collective properties. A lower t, = 200 °C or 207 °C corresponds probably to the appearance only of separate Fe 3 + atoms, oxodimers, trimers and other primary fragments that could be atomic ironcontaining impurities in a solid-state matrix. The increase of td above 215°C leads to the growth of clusters and to a decrease in atomic mobility. Once more neither broadening nor diffusive motion were observed. The average cluster size was estimated from the specific surface but more precisely from the temperature dependence of spectra at 4.2-77 K by defining a so called "blocking" temperature Tbl. At Tb~ the M6ssbauer spectra consist of two parts: 50% paramagnetic doublet and 50% magnetic hyperfine structure (HFS) caused by superparamagnetic relaxation of ~Fe203 clusters with relaxation time ~. Thus the clusters at td=215°C and Tb1~25K correspond to an average size d E 1.5-2 rim, while td=225°C and Tbl ~ 4 0 K correspond to average size d ~ 2 - 3 nm. The calculations have been carried out by means of the equation of superparamagnetic relaxation ~= Co exp(Kv/kT), with an anisotropy constant K = ( 1 - 2 ) x 105 G m -3 [7] and % = 5 x 10 -9 S [8], where v is the average volume of the cluster.
318
I.P. Suzdalev, N.!( Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
The addition of SAS results in a decrease in S (Fig. 3) for clusters with dav= 1.5 nm. This phenomenon is connected with the separation of particles and with the weakening of their interaction. In term of thermodynamics the influence of SAS leads to a decrease in the free energy of the cluster in the solid state Gs = ~ss + #spy, where as is surface energy, s surface area, /~s chemical potential, p density and v volume of the cluster. Indeed after interaction with SAS the free energy of a cluster will be Gss = asss + l~spv, and because ass < as, Gss < Gs. This reduction in Gss for solid clusters with SAS, their stabilisation and separation cause the increase of the probability of solid-liquid transition (as zJGm, Fig. 2) and of the number of clusters in the liquid state in accordance with the reduced melting point for smaller clusters [1,2]. The increase of the probability of cluster transition to the liquid state is shown by the increase in atomic mobility for clusters with dav ~ 1.5 nm. The peculiarities of interaction of different SAS with clusters will be reported elsewhere. The increase of atomic mobility in nanometer clusters and the influence of SAS can be interpreted by a phenomenological model. A cluster is considered as a sphere consisting of two parts: a surface layer and an inner core. Every part of the cluster possesses characteristic properties, for example, an Einstein temperature O)E and mean square atomic displacements (XZ)s for surface and ( x 2)i for inner core. In this case ( x2)s = h/MC°Es cth(hc°Es/kT) ( X2)i = h/MC°E i cth(hc°E]kT)
(2)
where M is the atomic mass, k the Boltzmann constant and T temperature. If we use the Lindemann criterion [ 1,23 for the melting of a substance: ~X2)m ~ O.Ola 2 at the melting point (a is the mean atomic distance), the difference between the melting point of the surface layer and inner core can be understood. Calculations [9] and experiment [1,2] suggest that (X 2)s > ( X2)i in accordance with the decrease in the number of atomic bonds on the surface. These reasons permit us to propose a model of coexistent liquid and solid states in a cluster under some conditions. In Fig. 4 the dependencies
Y 3 11 7" Fig. 4. Dependence of ~X2) =f(T) for surface and inner core of cluster. (x 2) is the mean square displacementscorresponding to the Lindemann criterion, T,,s and Tm~are melting points of surface and inner core of the cluster. (1) Solid state; (2) solidliquid state; (3) liquid state. (x2)s = f ( T ) and (xZ)i = f ( T ) for surface and inner core of a cluster are shown. Because of the difference between ( x 2)s and ( x 2)i the value ( x 2 )m corresponds to different melting points: Tins for surface and Tmi for inner atoms, and the region of coexistence of solid and liquid state energies. This model predicts an increase of ( x 2) and of cluster atomic mobility with decreasing cluster size and that when T < Tmi the cluster is in the solid state, when T > Tins in the liquid state, and when Tmi < T < Tms in the solid-liquid state. The effect of SAS leads to a decrease in cluster interaction and the cluster surface becomes more free. This explains the increasing value of (X2)s shown by the shift of Tins and the extension of the region of the solid-liquid state. The decrease of surface energy after interaction with SAS is a well known effect that is connected with the effect of strength diminished by adsorption [ 10]. In our case we provide a microscopic view of this effect on the basis of the increase of the atomic mobility of a cluster and the appearance of solid-liquid state in it.
4. Conclusion Ultrafine clusters of iron hydroxide and iron oxide, prepared by two different methods, have
I.P. Suzdalev, N.I. Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
similar peculiar properties. F o r some cluster size ( 1 - 1 . 5 n m ) , a m a x i m u m a t o m i c m o b i l i t y is o b s e r v e d that is c o n s i d e r e d in terms of two t h e r m o d y n a m i c a n d p h e n o m e n o l o g i c a l m o d e l s as the a p p e a r a n c e of a unique s o l i d - l i q u i d state of the cluster. The s e p a r a t i o n of clusters in pores of sorbents a n d by i n t e r a c t i o n with SAS, a n d the decrease of their i n t e r a c t i o n lead to the increase of the p r o b a b i l i t y of this unique state.
Acknowledgements This s t u d y was s u p p o r t e d by the Russian F o u n d a t i o n of F u n d a m e n t a l Science, G r a n t N 9 4 - 0 3 0 8 0 8 1 a n d by the I n t e r n a t i o n a l Science F o u n d a t i o n (Soros F o u n d a t i o n ) N M01000.
References [1] H. Haberland (Ed.), Clusters of Atoms and Molecules, Springer Series in Chemical Physics, vol. 52, Springer Verlag, Berlin, 1994.
319
[2] R.S. Berry, T.L. Beck and H.U Davis, in I. Prigogine and S. Rice (Eds.), Evolution of Size Effect in Chemical Physics, Part 2, Wiley, 1988, p. 75 138. I-3] I.P. Suzdalev, V.N. Buravtzev, V.K. Imshennik and S.V. Novichikhin, Khim. Fiz., 12 (1993) 555. [4] I.P. Suzdalev, V.K. Imshennik and S.V. Novichkhin, Nucl. Instrum. Methods, Phys. Res. B, 76 (1993) 421. 1-5] I.P. Suzdalev, Dynamic Effects in Gamma-Resonance Spectroscopy, Atomizdat, Moscow, 1979 (in Russian). I-6] A.S. Plachinda, V.E. Sedov, V.I. Khromov, I.P. Suzdalev, V.I. Goldanski, U. Nienchaus and F. Parak, Phys. Rev. B, 45 (1992) 7716. [7] B. Rodmaq, J. Phys. Chem. Solids, 45 (1984) 1119. 1-8] P. Gutlich, R. Link and A. Trautwein, M6ssbauer Spectroscopy and Transition Metal Chemistry, Inorganic Chemistry Concepts, Vol. 3, Springer Verlag, Berlin, 1978. 1-9] A.A. Maradudin and I. Meingailis, Phys. Rev. A, 133 (1969) 1188. 1-10] P.A. Rehbinder, Surface Phenomena in Dispersed Systems: Physical Chemical Mechanics, Nauka, Moscow, 1979 (in Russian).
Colloids and Surfaces A: Physicochemicaland Engineering Aspects 108 (1996) 315 319
A
SURFACES
Atomic mobility and thermodynamics of ultrafine clusters I.P. Suzdalev a,, N.I. Shklovskaya b a Institute of Chemical Physics, Russian Academy of Science, Kosygina 4, 117977 Moscow, Russia b Institute of Surface Chemistry, National Academy of Science of Ukraine, Prospect of Science 31, 252028 Kiev, Ukraine Received 15 October 1994; accepted 1 October 1995
Abstract The atomic dynamics of ultrafine clusters of ferric hydroxide and 7Fe203 with average size 1-3 nm are reported. The clusters were synthesised by nucleation in solution in the micropores of sorbents and by thermal decomposition of ferric oxalate in air. Thermodynamic and phenomenological approaches to the dynamic state of the cluster are considered and the probability of the formation of specific solid-liquid states of the cluster is discussed. Keywords: Intracluster dynamics; M6ssbauer spectroscopy 57Fe; Nucleation of clusters; Thermodynamics
1. Introduction Ultrafine clusters of matter in the nanometer scale possess unique properties connected with the dynamic state, electron conductivity, magnetic properties, electron-phonon interaction etc. [1]. Intracluster dynamics and the effects of the decrease of melting point and the existence of separate melting and freezing points are of special interest. This behaviour is considered in terms of the increase of intracluster atomic mobility and with the appearance of a special solid-liquid state in the region between the melting and freezing points [2]. The present paper deals with the study of the atomic dynamics of ultrafine clusters of ferric hydroxide and y-ferric oxide of size 1-3 nm by MSssbauer spectroscopy and thermodynamic analysis. In order to increase the atomic mobility, the clusters were obtained in pores of sorbents or * Corresponding author. Paper presented at the XIII European Chemistryat Interfaces Conference held in Kiev, Ukraine, 11-16 September 1994. 0927-7757/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0927-7757(95)03429-3
were treated by surface active agents (surfactants) that decreased intercluster interaction and prevented cluster aggregation. Thermodynamic and phenomenological cluster models for intracluster atomic dynamics were developed and conditions for the existence of a specific solid-liquid state were discussed.
2. Experimental Ferric hydroxide clusters were synthesised by chemical reaction in pores of sorbents [3,4]. Polysorbs obtained by copolymerization of styrene and divinylbenzene were impregnated with 57FEC13 in water or ethanol solutions of concentrations 0.05, 0.025 or 0.0125 M. The nucleation of iron hydroxide clusters occurred after passing air mixed with N H 3 through a layer of Polysorb. Average sizes of pores were 7-75 nm and average sizes of clusters 1-4.5 nm. The average sizes of the clusters were determined both from M6ssbauer spectra and
316
I.P. Suzdalev, N.I. Shklovskaya/ColloidsSurfacesA: Physicochem.Eng. Aspects108 (1996) 315-319
by taking account of the concentration of iron salt inside the pores. Clusters of ferric oxide, 7Fe20 3, were synthesised by a solid-state thermal decomposition of iron oxalate Fe2(C204)a-5H20 in air [5]. This reaction involves initial loss of water molecules and CO2, then partial reduction of Fe 3÷ to Fe 2÷ in the temperature range td= 180-200°C. The nucleation of 7Fe20 3 clusters occurred when the temperature was increased further to 400°C. The final size of the cluster was fixed after the end of the solid-state chemical reaction at some final temperature ta by rapid quenching of the sample to room temperature. The estimation of cluster size was carried out by both M6ssbauer spectroscopy and by BET measurement of specific surface area with argon as an adsorbate. The separation of clusters was carried out by mechanical treatment and the application of surface active substances (SAS), or surfactants, followed by drying. As SAS we used solutions of ethanol, butanol, isopropyl alcohol, sodium dodecyl sulphate (SDS) and polyoxyethylenesorbitan3-stearate (Tween-65). The studies were carried out in a MOssbauer spectrometer of Wissell type (Germany) with a 5VCo(Cr) source.
3. Results and discussion
M6ssbauer spectra of iron hydroxide and 7FezO3 are standard doublets [ 3 - 5 ] . Neither broadening nor so called broad components of the spectra [6] were observed. Therefore diffusion of clusters with correlation times smaller than 10-Vs is absent. The variation of the line spectra intensity (e) of iron hydroxide clusters in 7-75 nm pores is shown on Fig. 1. It is easy to see a considerable decrease of e with decreasing pore diameter. This effect is connected with an increase in atomic mobility because e is determined by the M6ssbauer effect probability f , = exp( - ( x 2)/¢~2), where ( x 2) is the mean square displacement of an iron atom and ¢~2 the wavelength of 14.4 keV radiation divided by 2~. The sharp drop in e with decreasing cluster size is explained by the increase of ( x 2) and a phase transition connected with the melting of clusters in a pore containing solution. A similar effect of e could be caused also by the diffusion of
fo-
jr,
YO
o{a.,~
Fig. 1. Dependence of line intensity e for MOssbauer spectra on the pore sizes of Polysorbs at T=300 K: (1) concentration of NH3 in air 0.08%; (2) concentration of NH3 in air 0.008%. a cluster as a whole and by intracluster mobility. However the mobility of the whole cluster in a pore is connected unequivocally with the presence in the spectra of both narrow and broad components, i.e. diffusive, restricted motions [-6]. Therefore the drop in e and the increase of ( x 2) are connected with intracluster mobility. This effect can be understood in terms of a thermodynamic model of nucleation [3 ]. In Fig. 2, the Gibb's free energy is shown for cluster nucleation from solution in a pore:
AG=4/3~pr3(3a/pr- {A#+ln[no/(no- n)]})
(1)
where a is the surface free energy per unit surface area, A# = #1 - #s is the difference between chemical potentials of liquid and solid phases, no/N is initial concentration of solution, no and n = 4/3~pr 3 are the numbers of iron atoms in solution and in the cluster respectively, r and p are the radius and density of the cluster. The minimum critical volume of the cluster is n~n = 327zcr3K(~)/3p2A#3, the maxim u m volume is n~ax = noA#/(2 + A#), where K(c0 is the factor of heterogeneity and ct the contact angle of the cluster with the surface of the pore. The energy barrier of nucleation A GD depends on the heterogeneity as AG~ = 16rw3K(e)/3p2A# 2 and
LP. Suzdalev, N.1. Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
317
gf.
G
~o
3.o
2,o \\ r
g,o
2 3
5 Fig. 2. G as a function of cluster size and conditions of nucleation. (1) Cluster in solution of macro volume with concentration no/N; (2) clusters in micropore; (3) clusters in micropore with heterogeneity K(~),AG~ and AGR are direct and reverse activation barrier for the nucleation and decomposition of clusters; rmi., rma~ are minimum and maximum radii of the clusters respectively.
decreases with decreasing pore size (K(e)< 1). The energy barrier for decomposition of a cluster, AG R, diminishes too and AGO > AG R. The Gibbs free energy (Fig. 2) allows us to interpret the atomic mobility of clusters in terms of fluctuating reversible transitions from the solid phase (n. . . . rmax) through the barrier AG R to the liquid phase. The decrease of pore size corresponds to the reduction of cluster size from 3 to 1 nm (excepting larger pores of 75 nm) [3] and to the increase of the probability of fluctuations. It means that a unique solid-liquid state and higher atomic mobility appear for 1-2 nm clusters. This hypothesis allows us to explain the diminishing atomic mobility not only by the increase of cluster size but also by the loss of collective (solid state) properties after cluster conversion to trimer, dimer and single atom. So one can propose the existence of a minimum e at some cluster sizes. A similar minimum was observed for ~Fe203 clusters in the solid-state chemical thermal decomposition of iron oxalate. In Fig. 3 the areas of MSssbauer spectra (S) of 7Fe203 clusters at different temperatures of thermal decomposition are shown, A minimum S value is observed at td = 215°C, corresponding to minimal clusters with
Fig. 3. Dependence of spectral area S on temperature of thermal decomposition of ferric oxalate (full points). The decrease of S after treatment with various SAS is shown by open points; (1) ethanol; (2) isopropyl alcohol; (3) Tween-65; (4) butanol; (5) SDS.
collective properties. A lower t, = 200 °C or 207 °C corresponds probably to the appearance only of separate Fe 3 + atoms, oxodimers, trimers and other primary fragments that could be atomic ironcontaining impurities in a solid-state matrix. The increase of td above 215°C leads to the growth of clusters and to a decrease in atomic mobility. Once more neither broadening nor diffusive motion were observed. The average cluster size was estimated from the specific surface but more precisely from the temperature dependence of spectra at 4.2-77 K by defining a so called "blocking" temperature Tbl. At Tb~ the M6ssbauer spectra consist of two parts: 50% paramagnetic doublet and 50% magnetic hyperfine structure (HFS) caused by superparamagnetic relaxation of ~Fe203 clusters with relaxation time ~. Thus the clusters at td=215°C and Tb1~25K correspond to an average size d E 1.5-2 rim, while td=225°C and Tbl ~ 4 0 K correspond to average size d ~ 2 - 3 nm. The calculations have been carried out by means of the equation of superparamagnetic relaxation ~= Co exp(Kv/kT), with an anisotropy constant K = ( 1 - 2 ) x 105 G m -3 [7] and % = 5 x 10 -9 S [8], where v is the average volume of the cluster.
318
I.P. Suzdalev, N.!( Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
The addition of SAS results in a decrease in S (Fig. 3) for clusters with dav= 1.5 nm. This phenomenon is connected with the separation of particles and with the weakening of their interaction. In term of thermodynamics the influence of SAS leads to a decrease in the free energy of the cluster in the solid state Gs = ~ss + #spy, where as is surface energy, s surface area, /~s chemical potential, p density and v volume of the cluster. Indeed after interaction with SAS the free energy of a cluster will be Gss = asss + l~spv, and because ass < as, Gss < Gs. This reduction in Gss for solid clusters with SAS, their stabilisation and separation cause the increase of the probability of solid-liquid transition (as zJGm, Fig. 2) and of the number of clusters in the liquid state in accordance with the reduced melting point for smaller clusters [1,2]. The increase of the probability of cluster transition to the liquid state is shown by the increase in atomic mobility for clusters with dav ~ 1.5 nm. The peculiarities of interaction of different SAS with clusters will be reported elsewhere. The increase of atomic mobility in nanometer clusters and the influence of SAS can be interpreted by a phenomenological model. A cluster is considered as a sphere consisting of two parts: a surface layer and an inner core. Every part of the cluster possesses characteristic properties, for example, an Einstein temperature O)E and mean square atomic displacements (XZ)s for surface and ( x 2)i for inner core. In this case ( x2)s = h/MC°Es cth(hc°Es/kT) ( X2)i = h/MC°E i cth(hc°E]kT)
(2)
where M is the atomic mass, k the Boltzmann constant and T temperature. If we use the Lindemann criterion [ 1,23 for the melting of a substance: ~X2)m ~ O.Ola 2 at the melting point (a is the mean atomic distance), the difference between the melting point of the surface layer and inner core can be understood. Calculations [9] and experiment [1,2] suggest that (X 2)s > ( X2)i in accordance with the decrease in the number of atomic bonds on the surface. These reasons permit us to propose a model of coexistent liquid and solid states in a cluster under some conditions. In Fig. 4 the dependencies
Y 3 11 7" Fig. 4. Dependence of ~X2) =f(T) for surface and inner core of cluster. (x 2) is the mean square displacementscorresponding to the Lindemann criterion, T,,s and Tm~are melting points of surface and inner core of the cluster. (1) Solid state; (2) solidliquid state; (3) liquid state. (x2)s = f ( T ) and (xZ)i = f ( T ) for surface and inner core of a cluster are shown. Because of the difference between ( x 2)s and ( x 2)i the value ( x 2 )m corresponds to different melting points: Tins for surface and Tmi for inner atoms, and the region of coexistence of solid and liquid state energies. This model predicts an increase of ( x 2) and of cluster atomic mobility with decreasing cluster size and that when T < Tmi the cluster is in the solid state, when T > Tins in the liquid state, and when Tmi < T < Tms in the solid-liquid state. The effect of SAS leads to a decrease in cluster interaction and the cluster surface becomes more free. This explains the increasing value of (X2)s shown by the shift of Tins and the extension of the region of the solid-liquid state. The decrease of surface energy after interaction with SAS is a well known effect that is connected with the effect of strength diminished by adsorption [ 10]. In our case we provide a microscopic view of this effect on the basis of the increase of the atomic mobility of a cluster and the appearance of solid-liquid state in it.
4. Conclusion Ultrafine clusters of iron hydroxide and iron oxide, prepared by two different methods, have
I.P. Suzdalev, N.I. Shklovskaya/Colloids Surfaces A: Physicochem. Eng. Aspects 108 (1996) 315-319
similar peculiar properties. F o r some cluster size ( 1 - 1 . 5 n m ) , a m a x i m u m a t o m i c m o b i l i t y is o b s e r v e d that is c o n s i d e r e d in terms of two t h e r m o d y n a m i c a n d p h e n o m e n o l o g i c a l m o d e l s as the a p p e a r a n c e of a unique s o l i d - l i q u i d state of the cluster. The s e p a r a t i o n of clusters in pores of sorbents a n d by i n t e r a c t i o n with SAS, a n d the decrease of their i n t e r a c t i o n lead to the increase of the p r o b a b i l i t y of this unique state.
Acknowledgements This s t u d y was s u p p o r t e d by the Russian F o u n d a t i o n of F u n d a m e n t a l Science, G r a n t N 9 4 - 0 3 0 8 0 8 1 a n d by the I n t e r n a t i o n a l Science F o u n d a t i o n (Soros F o u n d a t i o n ) N M01000.
References [1] H. Haberland (Ed.), Clusters of Atoms and Molecules, Springer Series in Chemical Physics, vol. 52, Springer Verlag, Berlin, 1994.
319
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