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The laplace law in finely divided powders
NanoStructured Materials. Vol. 6. pi. 791-194.1995 Copyright Q 1995 Elsevia Science Ltd Printed in Ihe USA. All [email protected] reserved 09659773195 $9.50 + .@I
Pergamon 09659773(95)00178-6
THE LAPLACE
LAW
IN FINELY
DIVIDED
POWDERS
P.Perriat
Laboratoire de Rktivite
des Solides,UA23 CNRS, BP138,21004 Dijon Cedex, France
Abstract -- Due to surface energy phenomena quantified by the Laplace law, finely divided materials should undergo a decrease of their cell parameters when the grain size decreases. However this theoretical prediction is in poor agreement with the experimental results obtained in several materials such as barium titanate. The reason for such a discrepancy is carried out, involving the distinction between surface stress and surface free energy. Finally, it is shown that the usual explanation for phase transitions in fine grains involving the additional internal pressure quantijied by the Laplace law has to be completely revised. EXPERIMENTAL
The properties of BaTiOg fine powders are dependingon their grain size distribution. For example, Fig. 1 exhibits the cell parametersas a function of the mean grain diameter d=2R. For grain size smaller than O.O8pm,the material is cubic and smaller the grain size, greater the cell parameter. Such a behaviour is in complete contradiction with the Laplace law which gives the internal pressure in the grain, Pr,t as a function of the external one, Pext and the 2Y As y is necessarily positive, such a law surface energy per unit area,y : P,,t = Pext + --R-.
would predict compressivestressesin the grain, leading to a decreaseof its cell parameter.
4.045 1
3.985
0.01
1 d(w)
0.1
10
Figure 1. The cell parametersof BaTi03, at room temperature,as a function of the mean grain diameterd in powders l (1) o (2) i- (3). 791
792
P kRRl4T
Another way for expressingthis contradiction is to notice that, due to the experimentalincrease of the cell parameter when the grain size decreases,the surface area of the grain, A, also increases,leading to a correlative increaseof the surfacefree energy of the grain F%A, which would have theoretically to bc m inimized. MODEL
To explain this contradiction, a model has been built in the framework of Physics of continuous media. The studied material, with one chemical componentand existing under one phase,has a spherical shapeand an elastic bchaviour. It is describedby the variables : E , the strain tensor defined from a standard configuration, T, the temperatureand c3, a parameter describing the perturbationinducedby the surface,precisely the distanceto the surface.The free energy density @can then be written : f” = f’(E,T&) and the free energy of the grain, F, is obtained by integrating f” : F = gf”dVo . V” is the grain volume in the standard VO configuration defined by the following conditions : (a) T=To, a standardtemperature,(b) the external forces appliedon the grain are equal to zero, (c) C,3=“0. The influence of the surface is expressedthrough two parameters: the strain induced by the surface and the surface excess chemical potential. The total strain can be written : g=&+ET +er3, where & is the elastic strain, &Tthe thermic strain and Ec3 the strain due to the surface. As the surface atoms have atomic bondings quite different from the atoms just 2s(b ) below the surface, Ec3 has to present a Dirac singularity on the surface : i?r3 = Et3 7. X0
In the latter relation, x0 is the volume atomic density. To take into account the energetic discontinuity of the surface atoms, the chemical potential (without its mechanical Ps* contribution) is written : l.t* = p,‘* +-.28([3) xO'/3
where l.tV*=l,t’*(T) is the usual atomic
chemical potential and its* =l.t’* (T) the excessenergy of the surface atoms. EXPRESSION
OF
SURFACE
STRESS
AND
FREE
ENERGY
From such hypotheses,the expressions for the surface stress tensor and the free energy are derived. The surface stress tensor, Ts, which consists in the Dirac singularity of the stress tensor can be written :
Dl
793
THE LAPIACELAW IN FINELYDIVIDEDPOWDERS
where Et= T.& is the singular part of the elastic strain and 3 the elasticity tensor. When , the free energy is given by the following formula :
F=-~F,,t.Vc
+TS.A,
+~fS:~~.Ao
+c~”*ooX V +c1’*ox,A 0
I21
where Ve=Tr( El)@ and Ae =2/3Tr( Ez)A” are the volume and surface elastic deformations. A0 is the surface of the grain in the standardconfiguration and xf the surface atomic density.
THE
TWO
WAYS
FOR
DERIVING
THE
LAPLACE
LAW
The Laplace law can be derived from mechanicalor thermodynamicconsiderations.On the one hand, the Laplace law is a direct consequencefrom the principle of virtual works which predicts the existence of an additional internal pressuredependingon the surface stress : 2 $. On the other hand, the Laplace law can be obtained from the relation : dTF=GW,,i which relates F and the work of the external forces, 6W,,t, during stretching at equilibrium. When stretching, the surface in the standardconfiguration A”, has to be preserved : dAO=O,whereas the elastic parametersare modified : dA, = f dV, # 0. One can then easily obtain the Laplace law which is found again to involve the surface stress : 2TS P tit= Pexi + R
SURFACE
STRESS
AND
SURFACE
EXCESS
131
FREE
ENERGY
Introducing Equation [3] in Equation [2], one obtains for no external pressure(Pext=O): F = p”* x”Vo + fAAo
with
f*=-x
l-v E
()y3 ,2 * 0 T +us xs
+$Tr(i:)TS
[41
where v is the Poisson coefficient, E the surface Young modulus and f* the surface excess free OV
P PERRIAT
794
l-v ($3 ,2 contains then a mechanical contribution, -x T , and a chemical one, ps *0xs . Only E the mechanical term has an influence upon the internal stresses given by the Laplace law. Consequently, in the case of pure chemical surface energy, there is no internal pressure, whereasfA#O. In the case of pure mechanicaleffets, f* is necessarily positive whereasthe sign of TS can be positive or negative. This is consistent with the fact that, contrary to TS, there are physical reasonsfor which fA>O. Thus, the cell parametersin the grain can decrease(when TS is positive) or increase (when TS is negative). In the same way, the surface area of the grain A=A,+AO is smaller than A0 when TS is positive or greater when TS is negative. In case of barium titanate, the experimental results are then explained by a negative surface stress TS. With v=O.33, E=lO1l Pa in the bulk and 1012Pa at the surface, x OV3=1010 m one finds from Fig. 1 : TS=-15 J/m2 and fA= 1.5 J/m2. The latter value is in good agreementwith the order of magnitudeof the surfaceenergieswhich arc generallyobserved. PHASE TRANSITIONS
INDUCED
BY SUPERFICIAL
PROPERTIES
To explain the phase transitions induced by the grain size, an interpretation involving the Laplace law is often given. It consists of two steps : (a) firstly, due to the surface free energy, there is a positive internal pressure in fine grains, (b) secondly, the phase transition is induced by this internal pressure. In the case of BaTi03, the tetragonal/cubic transition with grain size would then be explained by the value of Pint which would reach the critical value for the transition induced by the pressure in monocrystals (4). As a consequencefrom the present model, such an interpretation has to be completely revised : (a) firstly, the internal pressure in the grain is not positive but negative, (b) secondly, the contribution of the surface stress to the internal pressure has no influence at all upon the transition, this one depending only on the external pressureand on the surfacefree energies(5). CONCLUSION From a model built in the framework of Thermodynamics and Mechanics of continuous media, the distinction betweensurface stress and surface free energy has beencarried out. Whereasthe phase transitions with grain size are depending on the surface free energy, the Laplace law involves the surface stress tensor only. REFERENCES 1. G.Caboche,J.C.Niepce, Dielectric Ceramics Symposium. The Amer.Ceram.Soc.West Cost Annual Meeting, San Francisco (USA), November 1992 (Proceedings). 2. S.Malbe, J.C.Mutin, J.C.Niepce, J.Chim.Phys., &825(1992). 3. K.Uchino, E.Sadanaga,T.Hirose, J.Amer.Ceram.Soc.,22,1555(1989). 4. R.W&che, WDenner, H.Schulz, Mat.Res.Bull., Z,497(1981). 5. P.Perriat, J.C.Niepce, J.of High Temperatureand Chemical Processes(to be published)
Pergamon 09659773(95)00178-6
THE LAPLACE
LAW
IN FINELY
DIVIDED
POWDERS
P.Perriat
Laboratoire de Rktivite
des Solides,UA23 CNRS, BP138,21004 Dijon Cedex, France
Abstract -- Due to surface energy phenomena quantified by the Laplace law, finely divided materials should undergo a decrease of their cell parameters when the grain size decreases. However this theoretical prediction is in poor agreement with the experimental results obtained in several materials such as barium titanate. The reason for such a discrepancy is carried out, involving the distinction between surface stress and surface free energy. Finally, it is shown that the usual explanation for phase transitions in fine grains involving the additional internal pressure quantijied by the Laplace law has to be completely revised. EXPERIMENTAL
The properties of BaTiOg fine powders are dependingon their grain size distribution. For example, Fig. 1 exhibits the cell parametersas a function of the mean grain diameter d=2R. For grain size smaller than O.O8pm,the material is cubic and smaller the grain size, greater the cell parameter. Such a behaviour is in complete contradiction with the Laplace law which gives the internal pressure in the grain, Pr,t as a function of the external one, Pext and the 2Y As y is necessarily positive, such a law surface energy per unit area,y : P,,t = Pext + --R-.
would predict compressivestressesin the grain, leading to a decreaseof its cell parameter.
4.045 1
3.985
0.01
1 d(w)
0.1
10
Figure 1. The cell parametersof BaTi03, at room temperature,as a function of the mean grain diameterd in powders l (1) o (2) i- (3). 791
792
P kRRl4T
Another way for expressingthis contradiction is to notice that, due to the experimentalincrease of the cell parameter when the grain size decreases,the surface area of the grain, A, also increases,leading to a correlative increaseof the surfacefree energy of the grain F%A, which would have theoretically to bc m inimized. MODEL
To explain this contradiction, a model has been built in the framework of Physics of continuous media. The studied material, with one chemical componentand existing under one phase,has a spherical shapeand an elastic bchaviour. It is describedby the variables : E , the strain tensor defined from a standard configuration, T, the temperatureand c3, a parameter describing the perturbationinducedby the surface,precisely the distanceto the surface.The free energy density @can then be written : f” = f’(E,T&) and the free energy of the grain, F, is obtained by integrating f” : F = gf”dVo . V” is the grain volume in the standard VO configuration defined by the following conditions : (a) T=To, a standardtemperature,(b) the external forces appliedon the grain are equal to zero, (c) C,3=“0. The influence of the surface is expressedthrough two parameters: the strain induced by the surface and the surface excess chemical potential. The total strain can be written : g=&+ET +er3, where & is the elastic strain, &Tthe thermic strain and Ec3 the strain due to the surface. As the surface atoms have atomic bondings quite different from the atoms just 2s(b ) below the surface, Ec3 has to present a Dirac singularity on the surface : i?r3 = Et3 7. X0
In the latter relation, x0 is the volume atomic density. To take into account the energetic discontinuity of the surface atoms, the chemical potential (without its mechanical Ps* contribution) is written : l.t* = p,‘* +-.28([3) xO'/3
where l.tV*=l,t’*(T) is the usual atomic
chemical potential and its* =l.t’* (T) the excessenergy of the surface atoms. EXPRESSION
OF
SURFACE
STRESS
AND
FREE
ENERGY
From such hypotheses,the expressions for the surface stress tensor and the free energy are derived. The surface stress tensor, Ts, which consists in the Dirac singularity of the stress tensor can be written :
Dl
793
THE LAPIACELAW IN FINELYDIVIDEDPOWDERS
where Et= T.& is the singular part of the elastic strain and 3 the elasticity tensor. When , the free energy is given by the following formula :
F=-~F,,t.Vc
+TS.A,
+~fS:~~.Ao
+c~”*ooX V +c1’*ox,A 0
I21
where Ve=Tr( El)@ and Ae =2/3Tr( Ez)A” are the volume and surface elastic deformations. A0 is the surface of the grain in the standardconfiguration and xf the surface atomic density.
THE
TWO
WAYS
FOR
DERIVING
THE
LAPLACE
LAW
The Laplace law can be derived from mechanicalor thermodynamicconsiderations.On the one hand, the Laplace law is a direct consequencefrom the principle of virtual works which predicts the existence of an additional internal pressuredependingon the surface stress : 2 $. On the other hand, the Laplace law can be obtained from the relation : dTF=GW,,i which relates F and the work of the external forces, 6W,,t, during stretching at equilibrium. When stretching, the surface in the standardconfiguration A”, has to be preserved : dAO=O,whereas the elastic parametersare modified : dA, = f dV, # 0. One can then easily obtain the Laplace law which is found again to involve the surface stress : 2TS P tit= Pexi + R
SURFACE
STRESS
AND
SURFACE
EXCESS
131
FREE
ENERGY
Introducing Equation [3] in Equation [2], one obtains for no external pressure(Pext=O): F = p”* x”Vo + fAAo
with
f*=-x
l-v E
()y3 ,2 * 0 T +us xs
+$Tr(i:)TS
[41
where v is the Poisson coefficient, E the surface Young modulus and f* the surface excess free OV
P PERRIAT
794
l-v ($3 ,2 contains then a mechanical contribution, -x T , and a chemical one, ps *0xs . Only E the mechanical term has an influence upon the internal stresses given by the Laplace law. Consequently, in the case of pure chemical surface energy, there is no internal pressure, whereasfA#O. In the case of pure mechanicaleffets, f* is necessarily positive whereasthe sign of TS can be positive or negative. This is consistent with the fact that, contrary to TS, there are physical reasonsfor which fA>O. Thus, the cell parametersin the grain can decrease(when TS is positive) or increase (when TS is negative). In the same way, the surface area of the grain A=A,+AO is smaller than A0 when TS is positive or greater when TS is negative. In case of barium titanate, the experimental results are then explained by a negative surface stress TS. With v=O.33, E=lO1l Pa in the bulk and 1012Pa at the surface, x OV3=1010 m one finds from Fig. 1 : TS=-15 J/m2 and fA= 1.5 J/m2. The latter value is in good agreementwith the order of magnitudeof the surfaceenergieswhich arc generallyobserved. PHASE TRANSITIONS
INDUCED
BY SUPERFICIAL
PROPERTIES
To explain the phase transitions induced by the grain size, an interpretation involving the Laplace law is often given. It consists of two steps : (a) firstly, due to the surface free energy, there is a positive internal pressure in fine grains, (b) secondly, the phase transition is induced by this internal pressure. In the case of BaTi03, the tetragonal/cubic transition with grain size would then be explained by the value of Pint which would reach the critical value for the transition induced by the pressure in monocrystals (4). As a consequencefrom the present model, such an interpretation has to be completely revised : (a) firstly, the internal pressure in the grain is not positive but negative, (b) secondly, the contribution of the surface stress to the internal pressure has no influence at all upon the transition, this one depending only on the external pressureand on the surfacefree energies(5). CONCLUSION From a model built in the framework of Thermodynamics and Mechanics of continuous media, the distinction betweensurface stress and surface free energy has beencarried out. Whereasthe phase transitions with grain size are depending on the surface free energy, the Laplace law involves the surface stress tensor only. REFERENCES 1. G.Caboche,J.C.Niepce, Dielectric Ceramics Symposium. The Amer.Ceram.Soc.West Cost Annual Meeting, San Francisco (USA), November 1992 (Proceedings). 2. S.Malbe, J.C.Mutin, J.C.Niepce, J.Chim.Phys., &825(1992). 3. K.Uchino, E.Sadanaga,T.Hirose, J.Amer.Ceram.Soc.,22,1555(1989). 4. R.W&che, WDenner, H.Schulz, Mat.Res.Bull., Z,497(1981). 5. P.Perriat, J.C.Niepce, J.of High Temperatureand Chemical Processes(to be published)