Langmuir 1999, 15, 2169-2175
2169
The Nature of the Melting Transition for a Mixed Monolayer System Physisorbed onto Graphite: Argon and Krypton Patch Impurities M. W. Roth Department of Natural Sciences, Texas A&M International University, 5201 University Boulevard, Laredo, Texas 78041 Received February 10, 1998. In Final Form: December 24, 1998 A constant density (F ) 1) constant temperature molecular dynamics method with periodic boundary conditions is utilized to examine the melting transition for argon impurity patches embedded in krypton monolayer matrixes (as well as for the complement system) deposited onto graphite for various values of argon impurity fraction X. The character and temperature Tm of melting are found to be dependent on the impurity fraction as well as adsorbate topology vis-a`-vis which species is the patch impurity and which is the matrix. No phase separation is observed, as the melting temperature of the matrix is coincident with that of the patch in all cases examined. Much of the behavior exhibited by the system in this study can be understood by vacancy formation arguments.
I. Introduction For many years the nature of quasi-two-dimensional (2D) melting has been of scientific interest. Although many significant milestones have been made with respect to the understanding of 2D melting1-16 and in particular that on a graphite substrate,17-19 there still remain some points of curiosity and debate. As outlined in many wellknown works, one prominent theory is the KTHNY theory of melting which ascertains that the transition takes place in two second-order steps, placing an orientationally ordered hexatic phase between the solid and isotropic fluid.3-6 Other theories predict that the dislocation/ disclination KTHNY transitions should be pre-empted by first-order processes such as in the Chui theory for grainboundary induced melting.2 In addition some relatively recent computational models of physisorbed atomic systems20 suggest that lattice defects and vacancies play a central role in determining the nature of melting. Although it is very difficult to determine the order of the melting transition in computer simulations and even in some (1) Brinkman, W. F.; Fisher, D. S.; Moncton, D. E. Science 1982, 217, 693. (2) Strandburg, K. J. Rev. Mod. Phys. 1988, 60, 161 and references therein. (3) Kosterlitz, J. M.; Thouless, D. J. J. Phys. C 1973, 6, 1181. (4) Kosterlitz, J. M. J. Phys. C. 1974, 7, 1096. (5) Halperin, B. I.; Nelson, D. R. Phys. Rev. Lett. 1978, 41, 121. (6) Nelson, D. R.; Halperin, B. I. Phys. Rev. B 1979, 19, 2456. (7) Abraham, F. F. Rev. Sect. Phys. Lett. 1981, 80, 340. (8) Marx, R. Rev. Sect. of Phys. Lett. 1985, 125, 2. (9) Morales, Juan J. Phys. Rev. E 1994, 49, 5127. (10) Morales, Juan J.; Velasco, Enrique; Toxvaerd, Soren. Phys. Rev. E, 1994, 50, 2844. (11) Kusner, R. E.; Mann, J. A.; Kerins, J.; Dahm, A. J. Phys. Rev Lett. 1994, 73, 3113. (12) Chen, Kun; Kaplan, Theodore; and Mostoller, M. Phys. Rev. Lett. 1995, 73, 4019. (13) Hu, Jun; MacDonald, A. H. Phys. Rev. Lett. 1993, 71, 432. (14) Naidoo, Kevin; Schnitker, Jurgen Mol. Phys. 1993, 80, 1. (15) Fujita, A.; Hikami, S. Phys. Rev. B 1995, 51, 259. (16) Yates, K.; Newman, D. J.; DeGroot, P. A. J. Physica C 1995, 241, 111. (17) McTague, J. P.; Frenkel, D.; Allen, M. P. Ordering in Two Dimensions; Elsevier North-Holland: Amsterdam, 1980; p 147. (18) Halperin, B. I. Ordering in Two Dimensions; Elsevier NorthHolland Publishing: Amsterdam, 1980; p 143. (19) Shrimpton, N. D.; Cole, M. W.; Steele, W. A. Manuscript in preparation. (20) Steele, W. A. Private communication. R. D. Etters and Kuchta, Bogdan. J. Low Temp. Phys. 1998, 111, 271.
experiments, the previous observations suggest that there is much that could be learned about melting by examining physisorbed mixtures whose components exhibit different types of melting signatures when pure. The purpose of this work is to better understand the difference in melting between two such systems, to better understand the dynamics of 2D melting in physisorbed systems, and to further delineate the role of adsorbate topology and boundary conditions in their melting transition. The two systems chosen are argon on graphite (Ar/ gr) and krypton on graphite (Kr/gr) not only because they exhibit markedly different melting properties but also because the potentials describing necessary interactions are of the same analytical form and are well-known and the systems are accompanied by a wealth of experimental data. The phase diagrams of Kr/gr and Ar/gr have been thoroughly studied and mapped out.19 Briefly put, Kr/gr is commensurate and exhibits a strongly first-order melting transition in the submonolayer regime (F < 1) which becomes more continuous21-24 as monolayer completion is approached at F ) 1. The Ar/gr system, on the other hand, is incommensurate and exhibits more continuous melting25-32 in the submonolayer coverage region up to completion at F ) 1.26.32 Some of the more recent work makes the interpretation that Ar/gr might melt in a twostage process.31 (21) Specht, E. D.; Mak, A.; Peters, C.; Sutton, M.; Birgenau, R. J.; D’Amico, K. L.; Moncton, D. E.; Nagler, S. E.; Horn, P. M. Z. Phys. B 1987, 69, 347. (22) Dupont-Pavlovsky, N.; Bockel, C.; Thomy, A. Surf. Sci. 1985, 160, 12. (23) Caflisch, R. G.; Berker, A. N.; Kardar, M. J. Vac. Sci. Technol. 1985, A3, 1592. (24) Caflisch, R. G.; Berker, A. N.; Kardar, M. J. Phys. Rev. B 1985, 31, 4527. (25) Chung, S.; Kara, A.; Larese, J. Z.; Leung, W. Y.; Frankl, D. R. Phys. Rev. B 1987, 35, 4870. (26) Chinn, M. D.; Fain, S. C. Jr. Phys. Rev. Lett. 1977, 39, 146. (27) Migone, A. D.; Li, Z. R.; Chan, M. H. W. Phys. Rev. Lett. 1984, 53, 810. (28) Larher, Y.; Terlain, A. J. Chem. Phys. 1980, 72, 1052. (29) Taub, H.; Carneiro, C.; Kjems, J. K.; Passell, L. Phys. Rev. B 1977, 16, 4551. (30) Abraham, F. F. Phys. Rev. B 1983, 28, 7338. (31) Zhang, Q. M.; Larese, J. Z. Phys. Rev. B 1991, 43, 938. (32) Chung, Thomas T. Surf. Sci. 1979, 87, 348.
10.1021/la9801716 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/26/1999
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Of more importance to the work at hand, there exist accounts of analytical, computer simulation, and experimental studies of melting in 2D systems with impurities. Nelson et al. examined the translational and rotational symmetries of randomly packed planar arrays of hard spheres (ball bearings) of two different sizes.33 They found that small impurity spheres were trapped in hexagonal formations of matrix constituents and that larger impurity spheres introduced disorder by trapping dislocations. A subsequent elastic continuum theory treatment34 of a solid crystalline matrix with random impurities by Nelson confirmed the ball bearing study and showed that large impurity concentrations obliterated the order of the solid phase but that a hexatic phase was seen, even persisting to very low temperatures, for lower impurity concentrations. Heat capacity measurements have been conducted on H2-D2 mixtures on D2 plated graphite,35 with emphasis on superfluid behavior. In this study, important thermodynamic parameters such as the temperature of the threephase melting region Tm and the high-temperature boundary Te of the two-phase liquid-vapor coexistence region were not linear functions of the impurity concentration X. The melting of 3He droplets in a phase-separated 3He-4He mixture was experimentally investigated,36 and the melting transition for the droplets took place at a higher temperature than the bulk. It was also found that the entire droplet did not melt because of the interaction with the surrounding solid 4He matrix at the boundary. Monte Carlo and molecular dynamics computer simulation techniques were used to study various aspects of binary Ar-Kr solutions in graphite pores.37 Diffusion coefficients for Ar were not surprisingly found to be higher than those for Kr. Addition of Kr into any pore also resulted in a lowering of the diffusion in that pore. Ma et al. recently investigated the melting of annealed Ar-CH4 mixtures on graphite in the submonolayer regime using ac calorimetric techniques.38 They observed miscibility for low impurity concentrations. When introducing CH4 impurities into Ar/gr, the sharp Ar melting anomaly decreases rather rapidly, virtually vanishing for CH4 concentrations greater that about 7%. Qualitatively similar results were seen for addition of Xe impurities. It is thought that the disappearance of the sharp peak in Ar/gr signals a higher order KTHNY-type transition and hence a hexatic phase as was observed in the earlier ball bearing and quenched random impurity studies.33,34 The dilution of CH4 films with Ar results in a monatonic broadening and shifting of the specific heat peaks, which is taken to be a confirmation that Ar/gr might exhibit first-order melting qualities as well as the widely accepted interpretation that its melting is in fact continuous. II. Computational Approach and Interaction Potentials An (N ) 256, F ) 1,T) molecular dynamics (MD) scheme with periodic boundary conditions (PBC’s) is utilized. The details of the MD scheme are provided in another of the author’s papers39 and is therefore omitted here for brevity. (33) Nelson, David R.; Rubinstein, Michael; Spaepen, Frans Philos. Mag. 1982, 46, 105. (34) Nelson, David R. Phys. Rev. B 1983, 27, 2902. (35) Bovie, Lawrence, J.; Vilches, Oscar E. J. Low Temp. Phys. 1998, 110, 621. (36) Haley, R. P.; Adams, E. D. Low Temp. Phys. 1997, 23, 461. (37) Klochko, A. V.; Piotrovskaya, E. M.; Brodskaya, E. N. Langmuir 1996, 12, 1578. (38) Ma, Jian; Carter, Elanor D.; Kleinberg, Hillary B. Phys Rev. B 1998, 57, 9270. (39) Roth, M. W. Phys. Rev. B 1998, 57, 12520.
Roth
In this mixture study, NAr is the number of argon atoms and N is the total number of atoms present in the computational cell. Results are presented for both Ar patches embedded in a Kr matrix and vice-versa. Initial conditions for a mixture fraction X ) NAr/N involves a patch of NAr Ar atoms in the center of the computational cell surrounded by NKr ) N - NAr Kr atoms, with N being the total number of atoms contained in the computational cell. Initial conditions for Kr impurity patches are analogous, and both cases are studied here for impurity fractions X ) 0, 0.25, 0.5, 0.75, and 1. The important interactions considered are adatomadatom (A-A) and adatom-graphite (A-gr) interactions. The A-A interaction is of the Lennard-Jones (LJ) form:
[( ) ( ) ] σij rij
Uij ) 4ij
12
-
σij rij
6
(1)
Here Uij is the potential energy of interaction between adatom i and adatom j, rij is the distance separating the two relevant atoms, and the potential parameters are (ij,σij) ) (Kr-Kr,σKr-Kr) ) (171.0 K, 3.60 Å) for Kr-Kr interactions, (ij,σij) ) (Ar-Ar,σAr-Ar) ) (120.0 K, 3.38 Å) for Ar-Ar interactions and (ij,σij) ) (Ar-Kr,σAr-Kr) ) (143.2 K, 3.49 Å) for Ar-Kr interactions as determined by combining rules which give the heteronuclear as a geometric mean and the heteronuclear σ as an algebraic mean:40
Ar-Kr ) (Ar-ArKr-Kr)1/2
(2a)
σAr-Ar + σKr-Kr 2
(2b)
σAr-Kr )
The A-gr interaction UijA-gr is derived from an adsorbate-carbon (A-C) atom-atom interaction between adsorbate atom i and carbon atom j which is of the LJ form as in eq 1 with (ij,σij) ) (Kr-C,σKr-C) ) (64.83 K, 3.22 Å) for Kr-C interactions and (ij,σij) ) (Ar-C,σAr-C) ) (54.46 K, 3.11 Å) for Ar-C interactions. To avoid a computationally intense lattice sum, Steele40 takes advantage of the graphite substrate symmetry in a well-known and widely used Fourier expansion: ∞
) E0i(zi) + UA-gr i
∑ Eni(zi)fn(xi, yi)
(3)
n)1
The analytical forms for its terms are
{
2σA-C6 2πqA-CσA-C6 1 - 4E0i(zi) ) 9 as 45d(zi + 0.72d) zi 6d(zi + d)5 Eni(zi) )
2πqA-CσA-C6 as
}
2zi2 + 7zid + 7d2
{( )( ) ( ) σA-C6 gn 30 2zi 2
(40) Steele, W. A. Surf Sci. 1973, 36, 317.
5
K5(gnzi) -
gn 2zi
2
K2(gnzi)
}
(4)
(5)
Mixed Monolayer System
Langmuir, Vol. 15, No. 6, 1999 2171
{
f1(xi, yi) ) fn(xi, yi)|n)1 ) -2 cos
( ( )) ( )}
y 2π x+ + a x3 4π y cos a 3x3
(6)
Here as ) 5.24 Å2 is the area of the graphite unit cell, d ) 3.37 Å is the spacing between graphite planes, the Ki’s are modified Bessel functions of the second kind and are of order i, and the gn’s are the moduli of the nth reciprocal lattice vector of the substrate.40 For implementation of the MD method, the force on adsorbate atom i is calculated from the spatial gradient of the interaction energies:
{∑
[ ( )]
N
bi) ) - ∇ Bı F Bi(r
UA-A (r bi, b r j) 1 - θ ij
j)1
rij rc
(1 - δij) + UA-g (r b i) i
}
(7)
In eq 7, the Heaviside θ function indicates that lattice sums are taken out to fourth neighbors. III. Results
Figure 1. Order parameter O1 as functions of temperature T calculated for the pure Kr adlayer at X ) 0 (9), for Kr impurity patches (0), for Ar impurity patches (O) and for the pure Ar adlayer at X ) 1 (b). For the impurity patches the values of X are X ) 0.25, X ) 0,5 and X ) 0.75, moving from the X ) 0 curve to the X ) 1 curve. The legend is chosen this way in order to best illustrate differences between the Ar impurity patch systems and the Kr patch systems, as well as to emphasize trends in each system. Uncertainties are on the order of (0.01 away from melting and (0.015 in the transition region.
In this study translational and thermodynamic signatures of melting are monitored as functions of temperature. In all cases examined, the melting temperatures as determined from the individual parameters are in agreement, consistent with well-converged MD runs. Two translational order parameters are monitored as functions of temperature. The first is
O1 )
1
N
6
B‚rB eig 〉 ∑ ∑ 6Nj)1 s)1
〈
s
(8)
i
O1 is an indicator of the translational order of the adsorbate with respect to the substrate. O1 ) 1 for a static commensurate overlayer and drops sharply upon melting, approaching zero when the adatoms uniformly sample the (x,y) plane due to thermal motion. Clearly O1 alone cannot determine melting, so another order parameter which monitors the translational order of the adlayer itself is needed. The order parameter O2 is defined by
O2 )
1
N
N
6
B‚rB eik 〉 ∑ ∑ ∑ 3N(N-1)i)1 j)i+1 s)1
〈
s
ij
(9)
As with O1, it is well-known and widely used. O2 ) 1 for a static, commensurate overlayer and vanishes for either an infinite incommensurate lattice or for random sampling of positions in the (x,y) plane due to thermal atomic fluctuations. In conjunction, O1 and O2 are reliable indicators of melting. The details of the construction of O1 and O2 are readily available. O1 ) O1(T) for various X with both Ar and Kr impurity patches is presented in Figure 1. Since O1 and O2 give the same information as to Tm as well as relative sharpness and thermal extent of melting in this study, O2 ) O2(T) is not shown here. In addition it is important to mention here that since there is no phase separation observed in this study, O1 is presented for the entire adlayer. The order parameters are calculated for the separate adsorbate species but in all cases the melting temperatures coincided. Any differences in order parameter behavior between species will be mentioned where needed. Two thermodynamic quantities are monitored as functions of temperature. The configurational energy per
Figure 2. Configurational energy UC as functions of temperature T calculated for the same systems and using the same legend as in Figure 1. Uncertainties range from (2 K away from melting and (15 K near melting for X ) 0 to (3 K away from melting and (5 K near melting for X ) 1.
particle UC is given by
UC )
1 N
〈∑ ∑ N
N
i)1 j)1
[ ( )]
UA-A (r b i, b r j) 1 - θ ij
rij rc
(1 - δi,j) +
〉
(r bi) (10) UA-gr i where summation and terms are similar to those in eq 7. UC ) UC(T) is shown for various X with both Ar and Kr impurity patches in Figure 2. The points at which the internal energy per particle U(T) ) UC(T) + 1.5kBT is steepest for this system is the thermodynamic melting point, and in all cases they agree well with those melting points indicated by structural order parameters and the behavior of UC(T). The specific heat at constant area CA is calculated by evaluating smooth fits for the curves U(T) at the same discrete set of temperatures as UC(T) are calculated and subsequently finding the appropriate derivatives between points. CA is calculated in this manner because it is felt that since UC(T) is calculated at discrete points, a smooth representative of a quantity containing its derivative in a sum would be misleading. As a check
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Roth
Figure 3. Specific heat at constant area CA as functions of temperature T calculated for the same systems and using the same legend as in Figure 1. Uncertainties range from (10 NkB near melting to (1 NkB away from melting for X ) 0 to ( 1 NkB near melting and (0.75 NkB away from melting for X ) 1.
of goodness of fit, the change in internal energy ∆U along the entire curve, obtained by integrating the various CA(T) curves agree with that obtained directly from U(T) to within 3% in all cases. CA ) CA(T) for various X with both Ar and Kr impurity patches is presented in Figure 3. The maxima of CA(T), CAmax, provide a measure of the relative geometrical sharpness of the specific heat through melting (how sharp the transition appears to be based on the appearance of the specific heat curves), as the temperature extent of melting for all X are similar. The change in entropy upon melting, ∆S, is in this study a measure of the thermodynamic sharpness of the transition and is given by
∆S )
∫TT
2
1
CA(T) dT T
Figure 4. Illustration of the technique for calculating ∆Sf using the X ) 0.25 Kr impurity patch case as an example. The length of the vertical bar represents the difference in internal energies had the system made a first-order melting transition at T ) Tm. Table 1. Specific Heat Maxima CAmax, Melting Temperatures Tm, and Values of ∆Sf for Ar Impurity Patches (Ar), for Kr Impurity Patches (Kr) and for the Pure Adlayersa X
CAmax(NkB)
Tm (K)
∆Sf (NkB)
0 0.25
30.9 ( 10 17.3 ( 4 (Ar) 25.6 (Kr) 9.7 ( 2 (Ar) 12.1 (Kr) 8.3 ( 1 (Ar) 9.8 (Kr) 6.2 ( 1
105 ( 4 77 (Ar) 98 (Kr) 69 (Ar) 88 (Kr) 61 (Ar) 82 (Kr) 48 ( 7
1.36 ( 0.15 0.97 (Ar) 1.03 (Kr) 0.57 (Ar) 0.57 (Kr) 0.49 (Ar) 0.43 (Kr) 0.31 ( 0.15
0.5 0.75 1 a
Typical uncertainties are quoted.
(11)
where T1 and T2 are the lower and upper limits of the melting transition, respectively. However, calculation of ∆S is rather inaccurate and inconsistent because it is difficult, even for sharp transitions, to choose T1 and T2. Therefore, a quantity defined as ∆Sf ) (Uh - Ul)/Tm is introduced which gives a more accurate and objective picture of the relative thermodynamic sharpness of the transitions. Here, ∆Sf is the change in entropy as if the system made a first-order transition to the fluid at T ) Tm. Uh is the internal energy of the fluid obtained by extrapolating the internal energy curve to T ) Tm, and Ul is the internal energy of the solid as obtained by a similar extrapolation. An illustration of this technique is provided in Figure 4 for the X ) 0.25 Kr impurity patch case. The specific heat maxima CAmax, the melting temperature Tm and ∆Sf are shown for various values of X for both Ar and Kr impurity patches in Table 1. Tm is determined in concert by the behavior of O1(T), O2(T), UC(T), and CA(T) as previously described. The melting temperature Tm vs X are shown for both Ar and Kr impurity patches in Figure 5. To help illustrate the distribution of vacancies as well as the extent of horizontal and vertical diffusion typically exhibited in the systems studied here, overhead snapshot configurations as well as sideview snapshot configurations looking along the x-axis are presented in Figure 6 at T ) 100 K with X ) 0.5 for both Ar and Kr impurity patches. IV. Discussion As is evidenced by all structural and thermodynamic data presented, the position and character of the melting
Figure 5. Melting temperature Tm as functions of impurity fraction X calculated for the same systems and using the same legend as in Figure 1. Uncertainties are on the order of (4 K at X ) 0 to (7 K for X ) 1. The legend is kept the same as in previous figures for consistency, but the points have been connected for viewing aid.
of Ar/gr (X ) 1) and Kr/gr (X ) 0) are consistent with previous theoretical and experimental studies.7,8,19-32 This work shows that Kr/gr gradually loses translational order in the registered solid phase up until melting, subsequently presenting a registered fluid, as O1 never drops to zero in the fluid phase in Figure 1. Ar/gr, however, exists as an incommensurate solid at low temperatures, as O1 clearly does not take on a zero-temperature value of unity. When Ar/gr is studied at monolayer completion (F ) 1.26), the structural order parameters vanish in the solid and therefore must contain reciprocal lattice vectors for the adlayer and not those used here in order to be useful. After melting, O1 indicates a registered fluid exists, as
Mixed Monolayer System
Langmuir, Vol. 15, No. 6, 1999 2173
Figure 6. Overhead snapshot configurations and sideview snapshot configurations looking along the x-axis for the X ) 0.5 Ar impurity patch and the X ) 0.5 Kr impurity patch (a-d, respectively) at T ) 100K. Black circles (b) represent Kr atoms and open circles (O) represent Ar atoms. The marker sizes do not reflect relative atomic sizes.
O1 never vanishes for any T. Inspection of UC(T) in Figure 2 confirms the presence of a fluid phase directly after melting, subsequently followed by a steep climb past roughly T ) 90 K not seen nearly as dramatically for any other X. Examination of atomic height distributions and pair distribution functions show that desorption is occurring in this high temperature regime, consistent with vaporization of the monolayer previously reported by experiment.38 It is striking to note that, for cases with small impurity patches (X e 0.25 for Ar patches and X g 0.75 for Kr patches), the position and character of melting for the system are much more sensitive to X than for larger impurity patches, as shown in Figure 5 and as seen by inspection of other graphical data presented. This suggests that localized sites of enhanced structural order (for Kr patches) or disorder (for Ar patches), even if internal to the monolayer system, can drastically influence melting. Such an observation is, in turn, suggestive that localized thermal fluctuations, nucleation sites, and lattice defects play exceedingly important roles in melting. In the small impurity patch regime, sensitivity of the melting temperature to patch size results in part because the patches have not developed an interior sufficiently large to support their own atom-atom interactions with correlation lengths nearly as large as those of the patch complement. In addition, inspection of Figure 6 shows that the Ar regions contain greater vacancies than the Kr regions. Just as it has been demonstrated that melting close to monolayer completion in other systems is very sensitive to density and to the presence of vacancies,19,20 it is easy to understand here why the melting temperature is sensitive to impurity fraction X for small Ar patch sizes. With the introduction of Ar into the Kr matrix comes vacancies just like are created when the density is decreased from monolayer completion. For the complement situation, however, inspection of Figure 6 shows that the Kr patch
exhibits more diffusion for edge atoms which seems to contradict the graphite pore study where addition of Kr decreased the overall diffusion and mobility.37 Inspection of Figure 6, however, shows clearly that the impurity patches are vertically separated from the matrix and, in conjunction with the higher mobility of Ar, possess dynamic and therefore weaker boundary conditions and therefore exhibit the diffusion observed. Since boundaryto-interior ratios are large for small patches the sensitivity of the melting transition to Kr impurity patch size is also larger for smaller patches, which is consistent with the aspect of the pore study37 where the structure of the interior region (droplets) was greatly affected by the surrounding matrix. Although the patches are clearly vertically separated when Ar is on the inside as well, the lower diffusion of the Ar impurity patches suggests that the dynamic nature of the boundary conditions are more important than the vertical separation. Such sensitivity has been seen in many 2D studies, but no partial melting is seen for the 2D case here, unlike in the pore study.37 Once the patches have a sufficiently large patch interior (X g 0.25 for Ar patches and X e 0.75 for Kr patches), the melting temperature and melting character changes much more gradually with X until the pure adlayer is reached. The patches interact with their compliments most strongly at the boundary, which in this regime becomes less important to the interior with increasing patch size. Since the amount of heteronuclear atom-atom interactions on the boundary increases with increasing patch size, the actual boundary interactions per se are not as influential on the system as are the topology of the patch and patch complement regions vis-a`-vis which is interior and which is exterior. Also as the pure monolayer is reached, the opportunity of the matrix to form vacancies is inhibited because the matrix becomes restricted in area and
2174 Langmuir, Vol. 15, No. 6, 1999
connectivity, actually lining the perimeter of the patch for X ) 0.75 (Ar patches) and X ) 0.25 (Kr patches). Therefore the vacancy formation within the area of the patch is far more influential that that taking place around the perimeter, which, whether of like or different atoms, simply serves as a boundary for the patch. As far as purely topological effects are concerned, we may compare and contrast what happens to melting for a given X with Ar patches and with Kr patches. Inspection of all data presented shows that, for a given X with 0.25 e X e 0.75, the system with Ar patches melts about 20K lower than with Kr patches. Also, inspection of Figures 1-3 show that the transitions are somewhat more Arlike for Ar impurity patches and more Kr-like for Kr impurity patches at the same X. These results support the previous notion that the system melts closer to Tm for the pure adlayer of the species contained in the patch at F ) 1. In addition it is again supported that the ability for the patch to establish an interior and the nature of lattice defects in that interior have a strong effect on the melting transition for the entire system and in fact is more influential on melting than either the adatom-adatom boundary interactions or the behavior of the patch complement, although the importance of the latter is emphasized as well. Furthermore, examination of Table 1 (in conjunction with the graphical data) shows that the melting transition is much sharper geometrically by inspecting the shape of the curves in Figures 1-3 for a given patch type closer to X ) 0 and more continuous as X ) 1 is approached, as evidenced by the trends in CAmax with X for both patch types. The values for ∆Sf for a given patch type decreases with increasing Tm also supporting the notion that the melting transition is becoming more continuous thermodynamically as the transition involving pure Ar/gr is approached, since there is less entropy difference between the solid and the fluid phases. Comparing CAmax for Ar patches and Kr patches for a given X, it is noticeable that the geometric sharpness for the latter is much greater, as is consistent with previous discussion of the importance of the patch interior. It is interesting to note, however, that for a given X, ∆Sf for Ar patches are similar to those for Kr patches. Although the transitions are geometrically sharper for Kr patches, the slightly elevated values of ∆Sf for Ar patches are consistent with the observation that Ar patches embedded in Kr are very much out of character in that they are forced to be much closer to commensurate than for the pure Ar solid, which is confirmed by the inhibition of edge diffusion for Ar impurity patches shown in Figure 6 and the behavior of the order parameters in Figure 1. For Ar patches around X ) 0.05, O1 is actually higher for the patch than for the Kr patch complement. This effect occurs most close to the boundary of the patch, and although the Ar order parameters’ exceeding those for Kr is generally not the case, values of O1 for Ar patches are in all cases higher than for the pure Ar monolayer (X ) 1). The interior of the patch still supports vacancies, however, so the fluid is much less affected by the Kr matrix than is the solid. Hence there is a greater difference in entropy between the solid and fluid phases for the Ar patches. It is interesting to note that, for all X, melting occurs over very similar temperature ranges, indicating that the temperature extent of melting might be mainly influenced by steric factors, although further investigation is warranted to support or detract from a definite conclusion. Since this study examines impurity patches which exhibit little diffusion it seems difficult to draw direct parallels to the experimental Ar/CH4 work38 of Ma, which
Roth
involved annealed samples at submonolayer completion, and the work of Nelson et al.,33,34 which involves impurities distributed in a random fashion within the matrix. In those works it is central that many impurities are allowed to be surrounded by matrix constituents, which is not the situation in this study. Consistent with the work of Ma, however, a continuous evolution of the melting transition from one atomic type to the other is seen here, but since this work deals with patches the continuous evolution may in part involve an averaging between two different types of transitions and not two similar ones. Clearly further work on rare gas impurity/matrix mixtures with different topological distributions of impurity is certainly desirable, and much longer runs as well as much larger computational cell sizes would be useful in better characterizing the system’s diffusion and phase separation observed. V. Conclusions The major conclusions of this work are as follows. i. In the regime of small patches (X g 0.25 for Ar patches and X e 0.75 for Kr patches), Tm is very sensitive to X and hence to patch size, not only because exterior-to-boundary ratio considerations must be made but also because the patches are not large enough to establish an interior large enough to support large correlation lengths. In addition the character of the boundary conditions and vertical impurity-matrix separation are very influential on melting. The higher mobility of Ar results in larger edge diffusion for Kr patches. Also, vacancy formation in the patch interior is a crucial factor in determining the nature of melting. ii. For the regime involving larger patches, the transition evolves continuously and much more slowly with X toward the pure adlayer transition. Boundary interactions are not as influential on melting as is the system topology vis a vis the patch and patch complement regions. In this regime, vacancy formation in the patch interior is crucial in determining the melting character for the entire system. In addition, close to the pure monolayer cases, the host atoms are restricted to the perimeter of the patch and vacancy formation is suppressed. iii. The impurity patches play an important role in melting for the entire system, as Tm for Ar patches is roughly 20 K lower than that for Kr patches for a given X with 0.25 e X e 0.75, and the transitions are more Ar-like for Ar patches and more Kr-like for Kr patches. Enhanced sensitivity of Tm to X for small patches indicate that spots of localized enhanced structural order or disorder is important in the melting mechanism. iv. Ar patches embedded in Kr matrixes are very much out of character as they are forced closer to registry. Coupled with the Ar patch strongly influencing melting for the entire system, the values of ∆Sf are somewhat higher for the system with Ar patches than would be expected from geometric sharpness alone. v. The melting transition becomes more continuous from X ) 0 to X ) 1, and pure adlayer transitions agree well with experiment. This evolution with composition is monatonic. vi. The thermal extent of melting is similar for all X, indicating that the temperature range over which melting takes place may depend mainly on steric factors and not so strongly on adsorbate species. vii. After about T ) 90 K in the pure Ar adlayer (X ) 1), desorption is observed and is consistent with vaporization observed in previous experimental work.
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viii. In addition to allowing for a better understanding of long-range fluctuations and lattice defects in melting, further computer studies involving very long runs and very large computational cell sizes would be useful in further delineating the system’s diffusion and phase separation behavior, respectively.
computer-time allocation grants from Texas A&M at College Station, and by generous allocations of computational time from Eastern New Mexico University. Also the author wishes to thank William Steele and Jian Ma for helpful discussions as well as the referees for useful contributions.
Acknowledgment. This work has been supported by a grant from Texas A&M International University, by
LA9801716
