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Renormalization of critical exponents for surface diffusion
PHYSICS LETTERS A
Physics LettersA 161 (1992) 556—558 North-Holland
Renormalization of critical exponents for surface diffusion V.P. Zhdanov Institute ofCatalysis, Novosibirsk 630090, USSR Received 22 April 1991; accepted for publication 20 November 1991 Communicated by A.A. Maradudin
Surface diffusion is often studied at coverages and temperatures where continuous phase transitions occur in the adsorbate/ substrate systems. Employing the scaling hypothesis and Fisher renormalization, we show that the diffusion coefficient as a function of coverage does not have anyinfinite power-law or logarithmic anomalies at critical coverages.
Surface diffusion is of considerable intrinsic interest and is also important for understanding the mechanism of surface reactions, ranging from the simplest, like recombination of adsorbed particles on a surface, to the complex processes encountered in heterogeneous catalysis [1]. Its intrinsic interest arises from the dynamical and statistical features of particles in adsorbed overlayers. In particular, surface diffusion is often studied experimentally at coyerages and temperatures where continuous phase transitions take place in the adsorbate/substrate systems. Phenomenologically, diffusion is described by Fick’s laws. In particular, according to Fick’s first law the diffusion flux of particles driven by the concentration gradient of these particles is represented as J— DVc ‘1 ~ — —
/
‘
where D is the chemical diffusion coefficient. The latter coefficient is well known to be proportional to the first partial derivative of the chemical potential with respect to the coverage [1], (2)
D —~ ~,
where p~= t3F/80, and F is the free energy per site. For example, the diffusion coefficient for a square lattice can be written as [21 (we use kB = 1) 0 ~
D= ~
556
~ l?PA0,ekAo,Z,
(3)
where PAO,1 is the probability that a particle has near an empty site and that the arrangement of particles in othersites is marked by the index i, kAo,l is the rate constant for a jump between the corresponding sites, and l~is the jump length. As a rule, surface diffusion is strongly nonideal because of lateral interactions between adparticles and adsorbate-induced changes in the surface. In particular, several orders of magnitude in the coverage dependence of the diffusion coefficient are typical for real systems [1]. The dominant contribution to the latter dependence is from rate constants kAo,I. The effect of the factor ôji/80 is usually minor. However, the situation may be different at critical coverages corresponding to continuous phase transitions in the adsorbed overlayer. If, for example, one studies the coverage dependence of the diffusion coefficient at T
T
1 < T0 (fig. 1), the latter coefficient may have a singularity at 0=0k due to the factor O,u/ô0 (the other factors are continuous at 0=0k). To motivate our analysis we ask the following fundamental question: “What kind of singularities may be observed in real systems?” The mean-field approximation and phenomenological Landau theory are well known to predict a stepwise coverage dependence of the diffusion coefficient near the critical coverage (see, e.g., fig. 2). However, these approaches are too rough. More accurate results, as it has been recently pointed
out by Bolshov and Veshunov [4], may be obtained employing the scaling hypothesis. In particular, at 0= const, the singular contribution to the tempera-
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 161, number 6
PHYSICS LETTERS A
D(0)—.(I0—01I)~.
DISORDER
T
(7)
Thus, the critical exponent for the coverage depen-
/
ORDER
__________________
COVERAGE
i o2
a
0
04
Fig. 2. Diffusion coefficient as a function ofcoverage for a square lattice at T= 400 K. The results have been obtained [31 taking into account repulsive lateral interactions between adsorbed partides and surface reconstruction and employing the mean-field approximation and phenomenological Landau theory. The model considered predicts T0=435 K at 0~=0.5.At T=400 K, the continuous phase transition occurs at 0= 0.26 and 0.74.
ture dependence of the free energy can be represented near the phase boundary T~(O)as usual [5] F—~[IT— T~(0)I ]2~~a, (4) where a is the specific heat exponent. Taking into account that (see fig. 1)
and using eq. (4), one can easily derive a singular contribution to the coverage dependence of the free energy, F— (I0—0~I)2_a
.
At first sight, the conclusions presented above look convincing. In fact, however, that is not the case. Indeed, critical exponents are customarily calculated (and measured) in extreme points (e.g., the Ising model with the zero external field corresponds to a lattice at 0=O~= these points,eqs. eq.(6)— (5) does notgas hold (see fig. ~).1) Near and consequently,
-
U
for the temperature dependence ofthe specific heat. Eq. of (7)the is the main result obtained by same Bolshov and dence diffusion coefficient is the as that Veshunov [4]. According to this equation, the coverage dependence of the diffusion coefficient is continuous at a <0. If however a>0, the diffusion coefficient has an infinite power-law anomaly. At a = 0, the anomaly is expected to be logarithmic, D(0)—~IlnIO—01 I (8)
/~
Fig. I. Schematic phase diagram of the adsorbate/substrate system.
L~
20 January 1992
.
Eqs. (2) and (6) then yield
(6)
(8) are not applicable. Critical exponents in other points can be calculated assuming that the nature of the transition is the same along lines with fixed external field (i.e., with a fixed value of the chemical potential for a lattice gas). This general hypothesis proposed by Fisher [6] is confirmed by the analytical results obtained for the “decorated” Ising models [6] and also for a dilute solution near the liquid—gas critical point [7]. For adsorption, one can control the conjugate variable, i.e., coverage. When the coyerage that would vary critically at a continuous phase transition is constrained, the calculated or measured exponents will change. In particular, employing the 0o the free isomorphism hypothesis formulated above, Fisher [6] hasalong shownlines that with beyond the coverage region near energy fixed can be represented as F—[IT-—T~(0)I]~2””~ ,
(9)
where a is the specific-heat exponent calculated or measured at O=0~.Accordingly, eqs. (6) and (7) can be rewritten as F— (I0—0~I)(2_a)/(I_a) and D(0)—.- (I0—0~I)a/(I_a)
(10) .
(11)
Eqs. (9)—(l 1) hold at a>0. On the other hand, if 557
Volume 161, number 6
PHYSICS LETTERS A
a <0, Fisher has shown that critical exponents should not be renormalized (i.e., we can use in eqs. (6) and (7) the same exponent both at 0=0~and at 00~). Employing the Fisher renormalization [61, one also can easily derive the following singular contribution to the diffusion coefficient at a = 0,
___________________
a 10
1 D(0)~ (12) Iln 0—Oil I Eqs. (7), (11) and (12) describe diffusion at 0~r 0~.Analyzing these equations, we can conclude that the diffusion coefficient is a continuous function of coverage at all values of a (see eq. (7) at a < 0, eq. (12) at a = 0, and eq. (11) at a> 0). Using the Fisher renormalization, one can easily prove that our latter statement is correct also at 0 0~.Thus, contrary to the prediction of Bolshov and Veshunov [4], the diffission coefficient as afunction ofcoverage cannot have any infinite power-law or logarithmic anomalies. In fact, we have no grounds to expect at 0~t0~even finite cusp-like anomalies caused by the factor ôjt/80. The latter conclusion is confirmed by the calculations of the diffusion coefficient for partides adsorbed on a square lattice (the Monte Carlo [8] and transfer-matrix [9,10] techniques) and on vertex sites of a honeycomb lattice (the renormalization group approach [11]). On the other hand, all the above mentioned studies [8—11] show that the factor öu/ô0 results in a finite cusp-like anomaly in the coverage dependence of the diffusion coefficient near O~at T< T0 (see, e.g., fig. 3). Finally, it is reasonable to note that from symmetry considerations [12] the continuous phase transitions in adsorbed overlayers on solid surfaces usually belong to one of four universality classes (Ising (a = 0), x—y with cubic anisotropy (nonuniversal), three-state Potts (a = ~), and four-state Potts (a = h)). For all these classes, a ~ 0. Thus, the Fisher renormalization discussed above is particularly important for real systems. This short paper has a long history which is connected with discussions and consultations with O.M. Braun, A.A. Chumak, A.V. Myshlyavtsev, A.Z. Patashinskii, A.A. Tarasenko and M.S. Veshunov. All these contacts are acknowledged.
558
20 January 1992
10
-
04
0.8 8
Fig. 3. Diffusion coefficient asa function ofcoverage for a square lattice at T= 400 K. The results have been obtained [9] taking into account repulsive lateral interactions between adsorbed partides and surface reconstruction and employing the transfer-matrix technique. The model considered predicts T
0 600 K at
00=0.5.
References [1] G. Ehrlich and K. Stolt, Annu. Rev. Phys. Chem. 31(1980) 603; A.G. Naumovets and Yu.S. Vedula, Surf. Sci. Rep. 4 (1985) 365; J.D. Doll and A.F. Voter, Annu. Rev. Phys. Chem. 38 (1987) 413; A. Kapoor, R.T. Yang and C. Wong, Catal. Rev. Sci. Eng. 31(1989) 129; G.E. Rhead, Int. Mater. Rev. 34 (1989) 261; R. Gomer, Rep. Prog. Phys. 53 (1990) 917; V.P. Zhdanov, Elementary Physicochemical processes on solid surfaces (Plenum, New York, 1991). [21G. Wahnstrdm and V.P. Zhdanov, Surf. Sci. 247 (1991) 74.
[31V.P. Zhdanov, Langmuir 5
(1989) 1044. [4] L.A. Bolshov and M.S. Veshunov, Zh. Eksp. Teor. Fiz. 95 (1989) 2039. [5] Shang-keng Ma, Modern theory of critical phenomena (Benjamin, New York, 1976); A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation theory ofphase transitions (Pergamon, Oxford, 1979). [6] M.F. Fisher, Phys. Rev. 176 (1968) 257. [7] A.Z. Patashinskii, V.L. Pokrovskii and SB. K1~okhlachev, Zh. Eksp. Teor. Fiz. 63 (1972) 1521. [8]D. Reed and G. Ehrlich, Surf. Sci. 102 (1981) 588; 105 (1981) 603. [9] A.V. Myshlyavtsev and V.P. Zhdanov, Poverkhnost 11 (1990)5. [10] A.V. Myshlyavtsev and G.S. Yablonskii, Poverkhnost 12 (1990) 36. [Ill A.A. Tarasenko and A.A. Chumak, Poverkhnost 11(1989) 98. [l2lM.Schick,Prog.Surf.Sci.ll (1981)245.
Physics LettersA 161 (1992) 556—558 North-Holland
Renormalization of critical exponents for surface diffusion V.P. Zhdanov Institute ofCatalysis, Novosibirsk 630090, USSR Received 22 April 1991; accepted for publication 20 November 1991 Communicated by A.A. Maradudin
Surface diffusion is often studied at coverages and temperatures where continuous phase transitions occur in the adsorbate/ substrate systems. Employing the scaling hypothesis and Fisher renormalization, we show that the diffusion coefficient as a function of coverage does not have anyinfinite power-law or logarithmic anomalies at critical coverages.
Surface diffusion is of considerable intrinsic interest and is also important for understanding the mechanism of surface reactions, ranging from the simplest, like recombination of adsorbed particles on a surface, to the complex processes encountered in heterogeneous catalysis [1]. Its intrinsic interest arises from the dynamical and statistical features of particles in adsorbed overlayers. In particular, surface diffusion is often studied experimentally at coyerages and temperatures where continuous phase transitions take place in the adsorbate/substrate systems. Phenomenologically, diffusion is described by Fick’s laws. In particular, according to Fick’s first law the diffusion flux of particles driven by the concentration gradient of these particles is represented as J— DVc ‘1 ~ — —
/
‘
where D is the chemical diffusion coefficient. The latter coefficient is well known to be proportional to the first partial derivative of the chemical potential with respect to the coverage [1], (2)
D —~ ~,
where p~= t3F/80, and F is the free energy per site. For example, the diffusion coefficient for a square lattice can be written as [21 (we use kB = 1) 0 ~
D= ~
556
~ l?PA0,ekAo,Z,
(3)
where PAO,1 is the probability that a particle has near an empty site and that the arrangement of particles in othersites is marked by the index i, kAo,l is the rate constant for a jump between the corresponding sites, and l~is the jump length. As a rule, surface diffusion is strongly nonideal because of lateral interactions between adparticles and adsorbate-induced changes in the surface. In particular, several orders of magnitude in the coverage dependence of the diffusion coefficient are typical for real systems [1]. The dominant contribution to the latter dependence is from rate constants kAo,I. The effect of the factor ôji/80 is usually minor. However, the situation may be different at critical coverages corresponding to continuous phase transitions in the adsorbed overlayer. If, for example, one studies the coverage dependence of the diffusion coefficient at T
T
1 < T0 (fig. 1), the latter coefficient may have a singularity at 0=0k due to the factor O,u/ô0 (the other factors are continuous at 0=0k). To motivate our analysis we ask the following fundamental question: “What kind of singularities may be observed in real systems?” The mean-field approximation and phenomenological Landau theory are well known to predict a stepwise coverage dependence of the diffusion coefficient near the critical coverage (see, e.g., fig. 2). However, these approaches are too rough. More accurate results, as it has been recently pointed
out by Bolshov and Veshunov [4], may be obtained employing the scaling hypothesis. In particular, at 0= const, the singular contribution to the tempera-
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 161, number 6
PHYSICS LETTERS A
D(0)—.(I0—01I)~.
DISORDER
T
(7)
Thus, the critical exponent for the coverage depen-
/
ORDER
__________________
COVERAGE
i o2
a
0
04
Fig. 2. Diffusion coefficient as a function ofcoverage for a square lattice at T= 400 K. The results have been obtained [31 taking into account repulsive lateral interactions between adsorbed partides and surface reconstruction and employing the mean-field approximation and phenomenological Landau theory. The model considered predicts T0=435 K at 0~=0.5.At T=400 K, the continuous phase transition occurs at 0= 0.26 and 0.74.
ture dependence of the free energy can be represented near the phase boundary T~(O)as usual [5] F—~[IT— T~(0)I ]2~~a, (4) where a is the specific heat exponent. Taking into account that (see fig. 1)
and using eq. (4), one can easily derive a singular contribution to the coverage dependence of the free energy, F— (I0—0~I)2_a
.
At first sight, the conclusions presented above look convincing. In fact, however, that is not the case. Indeed, critical exponents are customarily calculated (and measured) in extreme points (e.g., the Ising model with the zero external field corresponds to a lattice at 0=O~= these points,eqs. eq.(6)— (5) does notgas hold (see fig. ~).1) Near and consequently,
-
U
for the temperature dependence ofthe specific heat. Eq. of (7)the is the main result obtained by same Bolshov and dence diffusion coefficient is the as that Veshunov [4]. According to this equation, the coverage dependence of the diffusion coefficient is continuous at a <0. If however a>0, the diffusion coefficient has an infinite power-law anomaly. At a = 0, the anomaly is expected to be logarithmic, D(0)—~IlnIO—01 I (8)
/~
Fig. I. Schematic phase diagram of the adsorbate/substrate system.
L~
20 January 1992
.
Eqs. (2) and (6) then yield
(6)
(8) are not applicable. Critical exponents in other points can be calculated assuming that the nature of the transition is the same along lines with fixed external field (i.e., with a fixed value of the chemical potential for a lattice gas). This general hypothesis proposed by Fisher [6] is confirmed by the analytical results obtained for the “decorated” Ising models [6] and also for a dilute solution near the liquid—gas critical point [7]. For adsorption, one can control the conjugate variable, i.e., coverage. When the coyerage that would vary critically at a continuous phase transition is constrained, the calculated or measured exponents will change. In particular, employing the 0o the free isomorphism hypothesis formulated above, Fisher [6] hasalong shownlines that with beyond the coverage region near energy fixed can be represented as F—[IT-—T~(0)I]~2””~ ,
(9)
where a is the specific-heat exponent calculated or measured at O=0~.Accordingly, eqs. (6) and (7) can be rewritten as F— (I0—0~I)(2_a)/(I_a) and D(0)—.- (I0—0~I)a/(I_a)
(10) .
(11)
Eqs. (9)—(l 1) hold at a>0. On the other hand, if 557
Volume 161, number 6
PHYSICS LETTERS A
a <0, Fisher has shown that critical exponents should not be renormalized (i.e., we can use in eqs. (6) and (7) the same exponent both at 0=0~and at 00~). Employing the Fisher renormalization [61, one also can easily derive the following singular contribution to the diffusion coefficient at a = 0,
___________________
a 10
1 D(0)~ (12) Iln 0—Oil I Eqs. (7), (11) and (12) describe diffusion at 0~r 0~.Analyzing these equations, we can conclude that the diffusion coefficient is a continuous function of coverage at all values of a (see eq. (7) at a < 0, eq. (12) at a = 0, and eq. (11) at a> 0). Using the Fisher renormalization, one can easily prove that our latter statement is correct also at 0 0~.Thus, contrary to the prediction of Bolshov and Veshunov [4], the diffission coefficient as afunction ofcoverage cannot have any infinite power-law or logarithmic anomalies. In fact, we have no grounds to expect at 0~t0~even finite cusp-like anomalies caused by the factor ôjt/80. The latter conclusion is confirmed by the calculations of the diffusion coefficient for partides adsorbed on a square lattice (the Monte Carlo [8] and transfer-matrix [9,10] techniques) and on vertex sites of a honeycomb lattice (the renormalization group approach [11]). On the other hand, all the above mentioned studies [8—11] show that the factor öu/ô0 results in a finite cusp-like anomaly in the coverage dependence of the diffusion coefficient near O~at T< T0 (see, e.g., fig. 3). Finally, it is reasonable to note that from symmetry considerations [12] the continuous phase transitions in adsorbed overlayers on solid surfaces usually belong to one of four universality classes (Ising (a = 0), x—y with cubic anisotropy (nonuniversal), three-state Potts (a = ~), and four-state Potts (a = h)). For all these classes, a ~ 0. Thus, the Fisher renormalization discussed above is particularly important for real systems. This short paper has a long history which is connected with discussions and consultations with O.M. Braun, A.A. Chumak, A.V. Myshlyavtsev, A.Z. Patashinskii, A.A. Tarasenko and M.S. Veshunov. All these contacts are acknowledged.
558
20 January 1992
10
-
04
0.8 8
Fig. 3. Diffusion coefficient asa function ofcoverage for a square lattice at T= 400 K. The results have been obtained [9] taking into account repulsive lateral interactions between adsorbed partides and surface reconstruction and employing the transfer-matrix technique. The model considered predicts T
0 600 K at
00=0.5.
References [1] G. Ehrlich and K. Stolt, Annu. Rev. Phys. Chem. 31(1980) 603; A.G. Naumovets and Yu.S. Vedula, Surf. Sci. Rep. 4 (1985) 365; J.D. Doll and A.F. Voter, Annu. Rev. Phys. Chem. 38 (1987) 413; A. Kapoor, R.T. Yang and C. Wong, Catal. Rev. Sci. Eng. 31(1989) 129; G.E. Rhead, Int. Mater. Rev. 34 (1989) 261; R. Gomer, Rep. Prog. Phys. 53 (1990) 917; V.P. Zhdanov, Elementary Physicochemical processes on solid surfaces (Plenum, New York, 1991). [21G. Wahnstrdm and V.P. Zhdanov, Surf. Sci. 247 (1991) 74.
[31V.P. Zhdanov, Langmuir 5
(1989) 1044. [4] L.A. Bolshov and M.S. Veshunov, Zh. Eksp. Teor. Fiz. 95 (1989) 2039. [5] Shang-keng Ma, Modern theory of critical phenomena (Benjamin, New York, 1976); A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation theory ofphase transitions (Pergamon, Oxford, 1979). [6] M.F. Fisher, Phys. Rev. 176 (1968) 257. [7] A.Z. Patashinskii, V.L. Pokrovskii and SB. K1~okhlachev, Zh. Eksp. Teor. Fiz. 63 (1972) 1521. [8]D. Reed and G. Ehrlich, Surf. Sci. 102 (1981) 588; 105 (1981) 603. [9] A.V. Myshlyavtsev and V.P. Zhdanov, Poverkhnost 11 (1990)5. [10] A.V. Myshlyavtsev and G.S. Yablonskii, Poverkhnost 12 (1990) 36. [Ill A.A. Tarasenko and A.A. Chumak, Poverkhnost 11(1989) 98. [l2lM.Schick,Prog.Surf.Sci.ll (1981)245.