Langmuir 1988,4, 305-320
305
Sequestering and the Influence of Domain Structure on Excimer Formation in Spread Monolayers Philip A. Politowiczt Physical Chemistry Laboratory, University of Oxford, Oxford OX1 3Q2,United Kingdom
John J. Kozak* Department of Chemistry and Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 Received April 3, 1987. In Final Form: August 26, 1987 We explore the interplay between the photophysical, reactive behavior of probe molecules distributed on a monolayer and the organizationalstructure of the monolayer itself. Focusing on experiments in which the microscopic kinetic behavior of a probe molecule in a monolayer was studied by monitoring the excimer-monomer steady-state photoexcitation of the probe, we examine theoretically the consequences of sequesteringand domain structure on excimer formation in the monolayer. By complementingthe results obtained in previous theoretical studies of diffusion-controlledrandom processes on finite, two-dimensional lattices of local coordination v = 3 and 4 with new results (presented here) on finite lattices of local coordination v = 6, we are able to quantify changes in the lifetime 7 of the diffusing excited-statemonomer induced by concentration effects, local packing of the monolayer, and the nature of the boundary properties of the domain which enforce sequestering in the spread monolayer system. For definiteness, we focus on a previously studied eltperimental system, 1 2 41-pyrene)dodecanoicacid (PDA) dispersed in oleic acid, and study how the ratio 12/11of intensities of excited-state dimer to excited-state monomer is influehced by the domain size in each type of domain (Le., for each type of local coordination). The extent to which competing (reactiveand nonreactive) de-excitationprocesses can influencethe efficiency of excimer formation is also studied quantitatively in this work. Finally, given that abrupt changes in the local coordination may occur when the monolayer, upon compression,passes from the liquid-extendedto the liquid-condensed phase, we estimate the magnitude of changes in the intensity ratio I z / I l that can be anticipated when the system crosses a phase boundary.
I. Introduction The reactivity of heterogeneous chemical systems may be governed not only by the molecular properties of its individual components but also by the spatial relationship and organization of these components. An experimental model which has received much attention recently in exploring the interplay between structure and function (or reactivity) is the spread monolayer at the air-water interface. Although the studies of Kuhn et al.14 and Leblanc et al.- are of great interest and importance, the principal motivation for the present study is the work of Sackmann et al.“17 and Patterson et al.1g23 Using the excimer-formation technique, these authors examined the microscopic kinetic behavior of a probe molecule P in bilayers24and in a monolayer25 by monitoring the excimer-monomer steady-state photoexcitation of the probe. The steady-state ratio of excimer to monomer intensities, 12/11,was determined as a function of the mole fraction of the probe molecule; these data were then used to extract information on the interaction of the probe species and to estimate the lateral diffusion constant of P in the assembly. To set the stage for the problem discussed in this paper, it is useful to summarize the approach taken by Galla et al.24and Loughran et al.25in interpreting their experimental data on excimer formation and quenching of probe molecules distributed in a bilayer or spread monolayer assembly. Suppose, for definiteness, we use the notation PIoto denote the ground-state monomer, P1* the excited-state monomer, and P2*the excited-state dimer. The general scheme is ‘Present address: .Research School of Chemistry, Australian National University, Canberra, Australia ACT 2601.
Pl0
-
(1)
k,’ + PI* 7 P2*
(2)
hv +
P10
P2*
PI*
kl’
ki’
Plo+ hv
Pl0 + PIo+ hv2
(3)
The design of the experiment is such that the concentration of Pl0is much greater than the concentration of P1*; (1) Bijcher, H.; Kuhn, H. Chem. Phys. Lett. 1970,6,183and references contained therein. The following three references are not exhaustive, but only representative of their recent work. (2)Kuhn, H.; Mbbius, D.; Bather, H. Physical Methods of Chemistry; Weissberger, A., Rossiter, A., Ed.; Wiley: New York, 1972;Vol. 1, p 577. (3)Mabius, D.;Kuhn, H. Zsr. J. Chem. 1979,18,375. (4)Penner, T. L.; Mijbius, D. J. Am. Chem. SOC.1982, 104, 7407. (5)Chapados, C.; Leblanc, R. M. Biophys. Chem. 1983,17,211,and references contained therein. The following three references are not exhaustive but only representative of their recent work. (6)Robert, S.; Tancrede, P.; Salesse, C.; Leblanc, R. M. Biochim. Biophys. Acta. 1983,730,217. (7)Tandrede, P.; Paquin, P.; Houle, A.; Leblanc, R. M. J. Biochem. Biophys. Methods 1983,7,299. (8)Frackowiak, D.;Hotchandani, S.; Leblanc, R. M. Photobiochem. Photobiophys. 1984,7,41. (9)Sackmann, E.;Trauble, H. J. Am. Chem. SOC.1972, 94,4482. (IO) Sackmann, E.; Triluble, H.; Galla, H.-J. Biochemistry 1973,12, 5360. (11)Galla, H.-J.; Sackmann, E. Ber. Busen-Ges. Phys. Chem. 1974,78, 949. (12)Galla, H.-J.; Sackmann, E. Biochim. Biophys. Acta 1974,339,103. (13)Sackmann, E. 2.Phys. Chem. (Munich) 1976,101,391. Gruler, H.; Sackmann, H. J. Phys. (Les Ulgs, Fr.) (14)Albrecht, 0.; 1978,39, 301. (15)Galla, H.-J.; Theilen, U.; H a r t ” , W. Chem. Phys. Lipids 1979, 23,239. (16)Kapitza, H.; Sackmann, E. Biochim.Biophys. Acta 1980,595,56. (17)Sackmann, E. In Biophysics; Hoppe, W., Lohmann, W., Markl, H., Zuegler, H., Eds.; Springer-Verlag: Berlin, 1983;p 425.
0143-1463/88/ 24Q4-Q3Q5$01.50/0 0 1988 American Chemical Society
306 Langmuir, Vol. 4, No. 2, 1988
Politowicz and Kozak
consequently, one identifies an effective first-order rate constant
k3'[P1O]
N
k3 E
1/13
where r3is the time required for the randomly distributed Pl0 and P1*to diffuse together and form the excimer. If it is assumed that reaction 2 occurs on the first encounter of Pl0 and P1*, with a reaction time T[ which is much shorter than the diffusion time i d , then 13
E
Td
Galla et identify the reciprocal of the time Td as a collision rate, vml. The other time scales that are important in the problem are the fluorescence decay time r1 of the monomer and the fluorescence decay time r2 of the excited-state dimer. In ref 24, r2 is denoted 7,' and the first-order rate constants k,' and k i are denoted kf and k(, respectively. Galla et al.24assume that "the dissociation of the complex is negligible". A rationale is given in Loughran et al.% Let the dissociation rate constant for the species P2* be expressed as
terface of a neutral aqueous solution, a = 0.75, whereas Galla and Sackmann12 report 0.80 for pyrene in model membranes. Once a (or K ) is determined, the implications of result (4) can be explored systematically. In particular, the strategy adopted in ref 24 and 25 is to use the theory of random walks in conjunction with result (4)to study lateral diffusion and reaction in bilayer and monolayer assemblies, respectively. One identifies 7d
I2 --=[l+---1 k2'
+ k4 7 2
I1 + I2
k3
-1
71
or, given the above arguments concerning the rate constant
k4 I2/I1
ll/ld
(4)
where, again, 12/11is the ratio of the intensity of the excited-state dimer to the intensity of the excited-state monomer. To relate the ratio I z / I 1to the experimentally determined ratio of excited-state quantum yields &/&, the proportionality constant a (or K in ref 24) in the relation
&/I1 = ff(42/41) must be determined; a is specific for each probe and depends on the wavelength dependence of the spectrometer sensitivity. For the systems studied in ref 25, 12-(l-pyrene)dodecanoic acid and oleic acid at the air-water in(18) Gonen, 0.; Levanon, H.; Patterson, L. K. Isr. J. Chem. 1981,21, 271. (19) Subramanian, R.; Patterson, L. K.; Levanon, H. Chem. Phys. Lett. 1982, 93, 578. (20) Chauvet, J.-P.;Agrawal, M. L.; Hug, G. L.; Patterson, L. K. Thin Solid Films 1985, 133, 227. (21) Agrawal, M. L.; Chauvet, J.-P.;Patterson, L. K. J. Phys. Chem. 1985,89, 2979. (22) Subramanian,R.; Patterson, L. K. J. Phys. Chem. 1985,89,1202. (23) Vaidyanathan, S.;Patterson, L. K.; Mobius, D.; Gruniger, H.-R. J . Phys. Chem. 1985,89,491. (24) Galla, H.-J.;Hartmann, W.; Theilen, U.;Sackmann, E. J. Membr. Biol. 1979, 48, 215. (25) Loughran, T.; Hatlee, M. D.; Patterson, L. K.; Kozak, J. J. J . Chem. Phys. 1980, 72, 5791.
(5)
where ? is the reciprocal of the jump frequency uj and ( n ) is the mean number of steps taken by the probe molecule before first encounter with the target. Combining the results, (4) and (5), yields an expression25linking the stochastic variable (n)to the intensity ratio (12/11)and the lifetime ratio (?/T,), viz.
Transposing these variables into the ones identified by Galla et al" (see the earlier discussion) allows one to make contact with their expression for the jump frequency u j , viz. uj =
where AE is the energy difference between the species Pl0, P1*, and the excited-state dimer P2* and r& the decay rate associated with the reverse reaction. Since PAE is large for this system, the reverse rate contributes negligibly to the dynamics, even if [Pl*] >> [P2*] (r;tv is very small compared to the transition rate of the excimer into 2Plo). Under steady-state illumination of the system (wherein the pumping rate exactly compensates the disappearance of P1*), it is shown in ref 25 that
= ?(n)
I' 1 kf (n)-Ky I r, k(
(where I ' = I , and I Il).The lateral diffusion coefficient may then be determined from the expression
D=
Y4ujX2
where X is the average jump length of the probe molecule on the two-dimensional array. Central to the studies described above is the evaluation of the stochastic variable ( n ) . We now review how ( n ) was evaluated in ref 24 and 25 and use this discussion as the springboard for the generalization developed in this paper. In ref 24 it is assumed that a given bilayer contains a mole fraction xh of label (probe) molecules. Galla et al.24 argue that if every Nth position of the lipid lattice is occupied by a label molecule, then a "sublattice" of 2N points contains on average two label molecules. They then adopt the following expression for the mean walk length ( n )of a species diffusing on a square-planar, infinite periodic lattice (see later text) ( n ) CIN In N (7)
-
where C1 = 1 / and ~ N = 2/xLa. In ref 25, a somewhat broader view is taken. I f f represents the fraction of probe molecules excited, xp is the mole fraction of (total) probe molecules in the monolayer, and no is the number of randomly moving probe molecules per unit cell, then
c4fxp
+
*..
1
Equation 8 is an expression which reduces for N large to the one (above) used by Galla et al.24in the special case where exactly half the probe molecules are excited (f = 1/2) and where there is but one diffusing probe molecule per unit cell (no= 1). Having outlined the main features of the analysis presented in ref 24 for bilayers and in ref 25 for monolayers, we now introduce the theoretical issues dealt with in this paper. First, in terms of their physical properties, although spread monolayers of fatty acids share many of
Excimer Formation i n Spread Monolayers the organizational characteristics of lipid bilayers, there are important difference^."^^^^^ Monolayers when compressed exhibit phase transitions that carry the system from a “gaseous” phase to a fluid (or liquid-expanded) phase to a “crystalline” (or liquid-condensed) phase.30 Although both of these transitions are similar to first-order phase transitions, the fluid-crystal transition in monolayers (in contrast to lipid bilayers) is characterized by a finite slope of the experimental isotherm in the coexistence region. Since a first-order transition should be characterized by flat isotherms in the coexistence region, several reasons have been put forward to account for the observed finite-slope isotherms in monolayer systems. Wiegel and KoxZ7cite (1)impurities in the monolayer, (2) the finite size of the Langmuir trough, (3) interaction of the head groups of the amphiphatic molecules with water molecules, (4) the possibility that experimentally pure monolayers may not exist (there is always a certain amount of the amphiphatic species in solution in the substrate), and (5) metastability. While all of the factors may be important, Albrecht, Gruler, and Sackmannl4 have argued that monolayer isotherms are not flat because there exist limited regions of cooperativity; they estimate that about 80-190 molecules comprise a cooperative unit. Nagle,2s in reviewing their evidence, although cautioning that the theory underlying the estimate of Albrecht et al.14 is “crude”, concludes that “the idea that some nonessential regularity associated with the surface phase disrupts the cooperativity of the transition is appealing”. The evident complexity of the above phase behavior of monolayer systems has provided a considerable challenge to theorists. While no attempt will be made here to review the status of these theories (since that is not the main point of this paper), the reader may wish to consult the reviews of G e r ~ h f e l dWiegel , ~ ~ and Kox:’ Nagle,28and the literature cited in Patterson et al.,31 in Tembe et a1.,32 and in Fisher.33 Rather, what we should like to explore quantitatively in this paper are some consequences of the original suggestion by Albrecht et al.14that there exist finite regions of cooperativity in certain regimes of the phase portrait. In particular, we shall examine how the possible existence of such regions may influence the efficiency of the excimer-monomer steady-state photoexcitation experiments described previously. To deal with the problem posed in the preceding paragraph, two issues will have to be addressed. Firstly, there is the question of the “packing” of the monolayer molecules on the surface. In the fluid-crystal coexistence regime or in the liquid-condensed region itself, the local ordering around a target molecule may assume several possible arrangements. If, as Albrecht et have suggested, in the condensed regime the head groups are packed (“probably”)in a hexagonal array, then the local coordi(26) Gershfeld, N. L. Annu. Rev. Phys. Chem. 1976, 27, 349. (27) Wiegel, F. W.; Kox, A. J. In Adv. Chem. Phys. 1980, 41, 195. (28) Nagle, J. F. Annu. Rev. Phys. Chem. 1980, 31, 157. (29) Ben-Shaul, A,; Gelbart, W. M. Annu. Reu. Phys. Chem. 1985,36, 179. (30) It should be noted in passing that the spread monolayer may also exhibit a further transition or structural change in the crystalline or liquid-compressed regime. In that region it is believed that the head groups are organized in a regular array (probably hexagonal according to Albrecht et al.“), while the C-C bonds of the lipid are in the all-trans state. The direction of the tails is either perpendicular to the surface or tilted, and in some substances there appears to be a further phase transition, probably of second order, between these two states. (31) Patterson, L. K.; MacCarthy, J. E.; Kozak, J. J. Chem. Phys. Lett. 1982, 89, 435. (32) Tembe, B. L.; MacCarthy, J. E.; Kozak, J. J. J. Phys. Chem. 1985, 87,‘4562. (33) Fisher, M. E. “Phase Transitions in Adsorbed Layers” In Proceedrngs of Faraday Symposium 20, in press.
Langmuir, Vol. 4, No. 2, 1988 307 nation number about a given (“target”) molecule will be three. If one imagines a two-dimensional lattice superimposed on this array of molecules, then the lattice parameter which describes this coordination is the valency u, which here would be u = 3. In a somewhat more compressed state, the coordination number could be four (u = 4) or even six (v = 6). The latter possibility (u = 6) would define a triangular lattice which, topologically, is the dual of the hexagonal lattice (u = 3). Physically, a triangular array of head groups (characterized by a coordination number u = 6) could form quite easily from an initial, hexagonal array upon compression. Instead of occupying the vertices of a hexagonal lattice (u = 3), the head groups could close up, with some shifting to the geometric center of the hexagons defining the array; this would generate a triangular array of valency u = 6. Hence it seems plausible that if different types of packing contribute to the “limited regions of cooperativity”, local orderings characterized by u = 3 through u = 6 should be considered. The second point that must be addressed in order to make contact with the experimental work is to account for the fact that the regions of cooperativity (as reflected, say, in the local organization of the monolayer molecules about a target molecule) are probably limited in spatial extent, Le., are of finite extent. Thus, on a length scale which is macroscopically small but microscopically large, there may exist “patches” of molecules characterized by coordination numbers spanning the range 3 5 u I6. Since the target molecule may be situated in different local environments, the lifetime measured experimentally of a probe molecule (coreactant) dfiusing in a condensed monolayer phase may be an average of diffusion-controlled reactive encounters between the probe and target molecules on juxtaposed arrays of finite extent, with each array characterized locally by a specific coordination (u = 3, 4, or 6). Thus, the overall theoretical problem to be dealt with here concerns the study of diffusion-controlled reactive processes on lattices of valency u = 3, and 6 of finite extent and the description of these processes when several types of ordering are present simultaneously. For infinite periodic lattices characterized by a given valency, the underlying lattice-statistical problem has had a long history and, in the Markovian approximation, was first treated systematically in a classic paper by Montr011.~~Specifically, Montroll’s work focused on the problem of random walks on infinite periodic lattices with traps, and analytic, asymptotic expressions were derived for a species diffusing on lattices of different types (i.e., valencies) in dimensions d = 2, 3 (he also presented a closed-form, analytic result in d = 1). A general procedure for dealing with finite lattice systems was described by Montroll and Shuler%in 1958, and subsequently an approach based on the theory of finite Markov processes (in conjunction with symmetry arguments) was developed to calculate the moments of the underlying probability distribution function governing reaction-diffusion processes in finite systems (see ref 36-41). In fact, as regards the present study, compre(34) Montroll, M. E. J.Math. Phys. (N.Y.)1969, 10, 753. (35) Montroll, E. W.; Shuler, K. E. Adu. Chem. Phys. 1958, I, 361. (36) Walsh, C. A.; Kozak, J. J. Phys. Rev. Lett. 1981, 47, 1500. (37) Walsh, C. A.; Kozak, J. J. Phys. Reu. B: Condens. Matter 1982, 26, 4166. (38) Politowicz, P. A.; Kozak, J. J. Phys. Rev. B: Condens. Mutter 1983,28,5549. (39) Musho, M. K.; Kozak, J. J. J. Chem. Phys. 1984, 81, 3229. (40) Politowicz, P. A.; Kozak, J. J.; Weiss, G. H. Chem. Phys. Lett. 1985,120, 388. (41) Politowicz, P. A.; Kozak, J. J. Chem. Phys. Lett. 1986, 27, 257. (42) den Hollander, W. Th. F.; Kasteleyn, P. W. Physica A Amsterdam 1982,112A, 523.
308 Langmuir, Vol. 4, No. 2, 1988 hensive results have already been presented for random walks on finite two-dimensional lattices of valencies u = 3 and 4 (see ref 38 and 37, respectively). All that needs to be done here in order to treat the physical problem posed in this paper is to calculate the corresponding site-specific and overall walk length data for finite lattices of valency u = 6. Inasmuch as the formulation of the problem for finite triangular lattices requires a fairly careful discussion of primitive unit cells and attendant boundary conditions, we relegate this discussion to the Appendixes in order to not interrupt our concentration on the physical problem being studied here: monomer-excimer dynamics on spread monolayers. Accordingly, after presenting our formal results in the following section, where we document the variety of ways in which boundary conditions and concentration effects can influence reaction efficiency, we focus on the system 12-(l-pyrene)dodecanicacid dispersed in oleic acid at the air-water interface. By coupling the analysis reviewed earlier in this section with the results presented in section I1 [and in ref 37 and 38), we study in section 111how the ratio Iz/Il of intensities of excited-state dimer to excited-state monomer is influenced by concentration effects (domain size), local packing of the monolayer, and the nature of the boundary properties of the domain which enforce sequestering in the spread monolayer system. 11. Formulation and Results The lattice model introduced in this paper allows one to study two classes of monolayer diffusion-reaction processes. First of all, one can calculate the average number (n) of steps required for trapping (a measure of the mean lifetime) for a particle diffusing randomly on a finite lattice of N sites of coordination (valency) u in dimensionality d = 2, with N - 1 sites organized symmetrically about a centrally located, deep trap T (reaction center). Such a model describes the case where a reactant A diffuses through a reaction space until it encounters a fixed reactant B and undergoes there an irreversible reaction, A + B C. One can also calculate the average walk length ( n )for the case where the diffusing reactant A forms an activated complex, say [AB]*,at any of the N - 1sites of the system and with probability s forms the product (C) (with an attendant probability 1 - s that the activated complex falls apart and the species A resumes its migration in the reaction space). Note that in a finite system the size of the domain considered defines the local concentratione of reactants (and their ratio). Moreover, in such systems, one anticipates that the boundaries of the domain can exert a variety of influences on the motion of the particle. Both of these compartmentalization effects will be studied in this paper along with the consequences of introducing competing reaction centers. The efficiency of the underlying reaction-diffusion process in each situation will be monitored by calculating the characteristic, average walk length of the diffusing species. As mentioned in the Introduction, extensive results are already available for reaction-diffusion processes on finite, square-planar (u = 4) and hexagonal (u = 3) lattices. Accordingly, all that remains to be done to complete the body of information required here is to calculate the lattice-statistical averages ( ( n ) ) for diffusion on finite triangular lattices (d = 2, u = 6) subject to various boundary conditions. The study of boundary effects is also useful in gauging the importance of “boundary tension” (a lower dimensional analogue of the surface tension) between finite domains in a spread monolayer.
-
Politowicz and Kozak
Figure 1. Diagram of the site classification scheme for a section of the d = 2 triangular lattice units N = 7-397; the full lattice is generated by rotating this (l/& sector about the point T. This site classification scheme applies to confining, reflecting, and one class of periodic boundary conditions.
In this paper, we shall consider finite, planar triangular lattices subject to nontransmitting (confining), reflecting, and periodic boundary conditions. Confining boundary conditions refer to the case where a diffusing particle, upon encountering the boundary, either moves laterally, is rest at the same lattice site, or moves to the interior of the lattice unit in the next step. Reflecting boundary conditions require that the random walker, upon confronting the boundary, either moves laterally or is deflected to a nearest-neighbor, interior site of the lattice in the next step. In dealing with triangular lattices, there are actually two distinct ways of imposing periodic boundary conditions (see the discussion in Appendix A3). To distinguish these two ways (conventions), we shall refer to the first class of periodic boundary conditions as Yactivenand the second class as “passive”. For each of these cases (as well as for the boundary conditions noted above), in addition to reporting the overall average (n), it is instructive to report the site-specific walk lengths ( n)b. Here, ( n)i denotes the average walk length of a coreactant diffusing randomly from a particuEar site i to the reaction center. The symmetry distinct sites i surrounding the trap T for the case of active periodic boundary conditions are shown in Figure 1,and the data for ( n)i and (n) for this case are displayed in Table I. From these data, it cag be seen that the site-specific average walk lengths (n), usually increase as the radial distance of each site (class) from the’centraltrap increases. The magnitude of this effect is documented in Table I for each of the finite, triangular lattice units considered in the present study. In our symmetry classification scheme, each site class is in the same relative position with respect to the central site regardless of the size of the unit. Thus, the effect of increasing the domain size N on the efficiency of a reaction-diffusion process initiated at a specific site of the reaction space can be assessed immediately from data reported in the table. Also listed in Table I are the results of Monte Carlo simulations we have performed on the N = 19 and 37 lattice units, and these are found to be in very good agreement with results obtained by using the theory of finite Markov processes. It should be noted that the time required to obtain accurate Monte Carlo results is very great compared to the time required to obtain ( n ) t and ( n )by using the Markovian
Excimer Formation in Spread Monolayers
Langmuir, Vol. 4 , No. 2, 1988 309
Table I. Site-Specific and Overall Average Walk Length Results for d = 2 Regular Triangular Lattices with a Centrosymmetric Deep Trap (pT= I) Subject to the Active Periodic Boundary Condition Excluding Background Absorption (8
N (n)
7 2.00
19 12.13" 11.00 12.60 12.80
37 32.60* 26.00 32.29 33.41 34.47 34.94
61 64.81
91 109.74
47.00 59.78 62.45 66.88 68.38 68.64 69.15 69.77
74.00 95.10 99.81 108.48 111.72 114.46 115.30 116.95 116.61 117.56 118.25
= 0.00) 127 168.16 107.00 138.26 145.48 159.30 164.77 170.26 171.72 175.07 176.77 178.17 179.89 178.56 179.14 180.40 181.15
169 240.68 146.00 189.27 199.46 219.36 221.49 236.15 238.42 243.92 247.67 249.94 253.30 253.54 254.28 256.07 257.84 255.68 256.78 258.27 259.06
217 327.80 191.00 248.13 261.74 288.65 299.87 312.14 315.38 323.41 329.39 332.81 338.21 339.78 340.86 343.63 346.96 346.09 347.43 349.50 351.31 347.89 348.49 350.01 351.69 352.50
271 429.98 242.00 314.84 332.33 367.19 381.91 398.26 402.60 413.53 421.99 426.76 434.54 437.39 438.92 442.97 448.96 448.26 450.13 453.24 456.55 454.09 454.79 456.62 458.92 460.74 456.23 457.38 459.26 461.08 461.91
331 547.60 299.00 389.39 411.22 454.96 473.60 494.50 500.09 514.26 525.45 531.77 542.25 546.42 548.49 554.06 561.60 562.33 564.90 569.35 574.52 572.46 573.36 575.83 579.18 582.47 578.68 579.99 582.22 584.69 586.54 580.48 581.09 582.73 584.90 586.84 587.69
397 681.03 362.00 471.79 498.41 551.97 574.94 600.87 607.84 625.60 639.81 647.84 661.33 666.91 669.58 676.87 686.95 688.33 691.75 697.77 705.09 703.10 704.30 707.63 712.33 717.40 713.67 715.34 718.28 721.81 725.07 719.48 720.15 722.00 724.57 727.18 729.04 721.61 722.80 724.87 727.28 729.32 730.18
"Monte Carlo result is ( n ) = 12.13; 20000 walks initiated per lattice site. *Monte Carlo result is ( n ) = 32.63; 20000 walks initiated per lattice site.
Table 11. Overall Average Walk Length Results for d = 2 Regular Triangular Lattices with a Centrosymmetric Deep Trap (pT = 1) Subject to the Active Periodic Boundary Condition Including Background AbsorBtion (s 1 0.00) S
(n)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 0.90
2.00 1.90 1.82 1.74 1.67 1.60 1.54 1.48 1.43 1.38 1.33 1.25 1.18 1.11
1.05
(n) 12.13 7.86 5.81 4.61 3.82 3.25 2.84 2.51 2.25 2.04 1.87 1.60 1.39 1.23 1.10
(n) 32.60 12.73 7.90 5.72 4.49 3.69 3.13 2.72 2.40 2.15 1.95 1.64 1.41 1.24 1.11
(n) 64.81 15.52 8.81 6.15 4.72 3.83 3.22 2.78 2.44 2.18 1.97 1.65 1.42 1.24 1.11
(n) 109.74 17.06 9.24 6.34 4.82 3.89 3.26 2.81 2.46 2.20 1.98 1.66 1.42 1.25 1.11
approach described in the Appendix. While the calculation of all the ( n ) i and ( n ) data presented in Table I required 0.6 s of CPU time on an IBM-370/3033 computer, calculation of the Monte Carlo results for the N = 37 unit alone (see Table I) required about 30 min of CPU time on a VAX 11/780 computer.
168.16 17.95 9.48 6.44 4.87 3.92 3.28 2.82 2.47 2.20 1.99 1.66 1.42 1.25 1.11
240.68 18.49 9.61 6.50 4.91 3.94 3.30 2.83 2.48 2.21 1.99 1.66 1.43 1.25 1.11
327.80 18.85 9.70 6.54 4.93 3.96 3.30 2.84 2.48 2.21 1.99 1.66 1.43 1.25 1.11
429.98 19-09 9.76 6.56 4.94 3.96 3.31 2.84 2.49 2.21 1.99 1.66 1.43 1.25 1.11
547.60 19.26 , 9.81 6.58 4.95 3.97 3.31 2.84 2.49 2.22 1.99 1.66 1.43 1.25 1.11
681.03 19.39 9.84 6.60 4.96 3.98 3.32 2.85 2.49 2.22 2.00 1.66 1.43 1.25 1.11
We consider next the case where each of the N - 1 background sites of the system can react with the diffusing species with a probability s. Data for triangular lattice units N = 7-397 subject to the active periodic boundary condition with background absorption are reported in Table 11. In its effect on the overall reaction efficiency,
310 Langmuir, Vol. 4, No. 2, 1988
A
Figure 2. Diagram of the site classification scheme for a section of the d = 2 triangular lattice units N = 7-397; the full lattice is generated by rotating this (l/& sector about the point T. This site classification scheme applies to one class of periodic boundary
conditions (passive).
the consequence of introducing competing reaction centers is similar to that seen for the d = 2 ~quare-planar~? and hexagonal lattices38 and the d = 3 cubic lattice.37 In particular, the average walk length is drastically reduced for nonzero values of s and as shown previ~usly,~’ for all si 5 s > 0, ( n ) l / s as N m for any lattice of regular structure. We consider next the case of passive periodic boundary conditions (see Appendix A3). Displayed in Figure 2 are the symmetry-distinct sites i surrounding the trap T for this class of boundary conditions, and in Table I11 are reported the site-specific average walk length data for the N = 7-397 triangular lattice units. Examination of the ( n ) data shows that the trends described for the active periodic boundary condition case pertain here as well. Again, as a check on our method, we have performed Monte Carlo simulations, and the results for the N = 19 and 37 units are in very good agreement with the Markovian values of (n). Table IV presents the site-specific average walk length data for the case of confining (nontransmitting) boundary conditions for the N = 7-397 regular triangular lattice unita. Inasmuch as the site classification schemes for active periodic and confining boundary conditions are identical (see Appendix), comparison of the data in Tables I and IV provides a quantitative assessment of the effect of boundary conditions and system size on the efficiency of the underlying diffusion-reaction process. One finds that correspondingvalues of ( n)ifor a given lattice unit subject to active periodic versus confining boundary conditions tend to differ more and more with increase in distance from the central target molecule. This behavior can be understood by noting that the confining boundary condition ”influences” the particle’s motion by allowing it to remain near the lattice system boundaries, whereas the active periodic boundary focuses the diffusing particle inward (toward the central site). Our Monte Carlo result for the N = 19 triangular lattice unit is in very good agreement with the Markovian value of ( n ) ,while the value obtained for the N = 37 triangular lattice unit is certainly acceptable. (Larger u = 6 triangular lattices require more walkers per site in order to obtain good histogr ams.)
-
-
Politowicz
and Kozak
In Table V are presented data for ( n)iand for the overall walk length ( n )obtained when reflecting boundary conditions are imposed. Comparison of the data in Table V with those in Table I (for the active periodic boundary condition case) reveals a great similarity in the results obtained for ( n ) (with the values of ( n ) for reflecting boundaries consistently greater than correspondingvalues for active periodic boundaries). Moreover, the values of ( n ) lfor each boundary condition are in the same relationship to each other; viz., as N increases, the difference between ( n ) ,calculated by using reflecting versus active periodic boundary conditions decreases from exactly unity to near zero (0.09). Again, our Monte Carlo results for the N = 19 and 37 units are in excellent agreement with the numerical values generated by using the Markovian theory. Examination of the data in Tables 1and 111-V reveals that the four boundary conditions introduced fall into two categories, each characterized by similar (but not equal) values of ( n )for each value of N . One set consists of the passive boundary conditions, i.e., the passive periodic and confining cases. The other set consists of the active periodic and reflecting boundary conditions. Table VI shows explicitly (quantitatively)how changes in system size and degree of competitive trapping (background absorption) affect differences in (n)for the passive periodic versus reflecting boundary condition cases. For the case s = 0.0, the values of ( n ) pand ( n ) Rdiffer, for a given value of N , by decreasing amounts as N is increased, although the magnitude of the difference ( n ) p- ( n ) Rincreases with increasing N . The introduction of competing reaction centers increases dramatically the efficiency of the reaction-diffusion process, reducing (n)to values approaching 11s in the limit of large N. As s increases, for a fixed value of N , the magnitude of the difference ( n)p - ( n)Rdecreases for all N values in Table VI. The means by which increase in lattice size Nor introduction of competing reaction centers influences the efficiency of the diffusion-reaction process are different; the increase in system size N effectively drives the system size to a “thermodynamic”limit wherein the system boundaries play a vanishingly small role in influencing the diffusion behavior, whereas the introduction of competing reaction centers (background absorption) effectively disperses the influence of the central trap to each and every site in the system (overwhelming any residual effect the boundary may have on the overall diffusion-reaction process). Although background absorption is the more significant factor, often both factors work synergetically to reduce differences in reaction efficiency induced by different boundary conditions. For example, in the case s = 0.1, the differences between (n)pand ( n ) Rdecrease from 1.50 for N = 7 to practically zero for N > 331. In the following section we shall utilize thk numerical results reported and those obtained in our earlier work on u = 4 ~ q u a r e - p l a n a rand ~ ~ u = 3 hexagonal lattices38 to study the photophysical problem described in section I. Before proceeding, however, it is of interest to provide comparisons of the results obtained for these three types of d = 2 lattice structures for the (one) case for which analytic asymptotic results are available-the case of d = 2, infinite periodic lattices of unit-cell size N with a single deep trap. In a classic study, MontrolP4 proved that the quantity ( n ) can be expressed explicitly in terms of the variable N for each of the d = 2 lattices u = 3, 4, and 6. The expression derived takes the general form
( n )= ( N / ( N- l))(A,NIn N
+ AzN + A3 + A, (1/N) + ...I (9)
Langmuir, Vol. 4 , No. 2,1988 311
Excimer Formation in Spread Monolayers
Table 111. Site-Specific and Overall Average Walk Length Results for d = 2 Regular Triangular Lattices with a Centrosymmetric Deep Trap (pT = 1) Subject to the Passive Periodic Boundary Condition Excluding Background Absorption ( 8 = 0.00)
N
7
19
37
61
91
127
169
217
271
331
397
6.00
20.73"
6.00
18.00 21.82 22.36
46.54*
84.61
135.80
36.00 45.38 47.23 49.87 49.98 50.76
60.00 76.78 80.44 86.90 86.93 89.22 90.76 91.24 91.48 92.36
90.00 116.02 121.95 133.11 133.12 137.43 141.53 142.61 142.70 145.09 145.74 145.92 146.74 147.07 147.99
200.79 126.00 163.11 171.78 188.56 188.56 195.32 202.37 204.20 204.23 208.60 211.29 211.37 213.05 213.19 215.60 215.16 215.60 215.95 216.99 217.41 218.35
280.15 168.00 218.05 229.91 253.23 253.24 262.87 273.30 276.04 276.05 282.77 287.61 287.65 290.41 290.49 294.79 295.73 296.53 296.71 298.75 298.95 301.34 299.85 300.06 300.85 301.36 302.55 303.03 303.99
374.34 216.00 280.83 296.34 127.15 327.15 340.08 354.36 358.13 358.14 367.58 374.78 374.80 378.84 378.88 385.42 387.64 388.89 388.98 392.26 392.37 396.58 396.15 396.26 397.64 397.92 400.19 400.45 402.82 400.03 400.44 400.86 401.93 402.57 403.88 404.41 405.38
483.78 270.00 351.46 371.08 410.30 410.30 426.95 445.54 450.49 450.49 463.01 472.81 472.82 478.33 478.35 487.45 490.96 492.72 492.77 497.51 497.57 503.92 504.28 504.34 506.44 506.59 510.18 510.33 514.46 512.28 512.94 513.18 514.99 515.36 517.79 518.10 520.44 516.34 516.57 517.34 517.95 519.25 520.00 521.40 521.97 522.95
608.86 330.00 429.94 454.12 502.69 502.70 523.46 546.84 553.11 553.11 569.05 581.72 581.73 588.87 588.89 600.84 605.71 608.06 608.09 614.47 614.51 623.27 624.32 624.36 627.27 627.37 632.48 632.57 638.75 636.76 637.74 637.88 640.57 640.79 644.58 644.76 648.81 645.08 645.22 646.42 646.79 648.92 649.38 651.92 652.27 654.60 648.96 649.35 649.81 650.89 651.67 653.14 654.00 655.47 656.07 657.05
749.89 396.00 516.27 545.47 604.33 604.33 629.63 658.28 665.99 665.99 685.69 701.52 701.52 710.48 710.48 725.59 731.92 734.92 734.94 743.14 743.16 754.61 756.30 756.33 760.17 760.23 767.06 767.12 775.59 773.57 774.90 774.99 778.69 778.83 784.17 784.29 790.31 786.42 786.51 788.24 788.47 791.59 791.88 795.79 796.02 799.99 794.36 794.95 795.24 796.88 797.38 799.75 800.30 802.92 803.31 805.62 798.39 798.64 799.39 800.06 801.39 802.33 803.95 804.89 806.41 807.05 808.03
"Monte Carlo result is ( n ) = 20.73; 20000 walks initiated per lattice site. *Monte Carlo result is (n)= 46.59; 20000 walks initiated per lattice site.
reexamined the calculation of Montroll for square-planar Presented in Table VI1 are the coefficients Aifor the three lattices and found that although their results for AI lattice structures u = 3,4,and 6 aa computed by Montroll.94 In work presented recently, den Hollander and K a ~ t e l e y n ~ ~ through A3 were essentially the same as Montroll's results
312 Langmuir, Vol. 4, No. 2, 1988
Politowicz and Kozak
Table IV. Site-Specific and Overall Average Walk Length Results for d = 2 Regular Triangular Lattices with a Centrosymmetric Deep Trap ( p T = 1) Subject to the Confining Boundary Condition Excluding Background Absorption (8
N (n)
7 6.00
19 20.75“
37 46.56’
61 84.64
91 135.83
18.00 21.75 22.50
36.00 45.38 47.25 49.88 51.00
60.00 76.78 80.44 86.90 89.28 90.64 91.37 92.67
90.00 116.02 121.95 133.11 137.45 141.49 142.66 145.19 145.71 146.99 148.39
= 0.00) 127 200.82 126.00 163.11 171.78 188.56 195.33 202.35 204.22 208.64 211.28 213.15 215.74 214.99 215.69 217.35 218.82
169 280.17
217 374.37
168.00 218.05 229.91 253.23 262.88 273.30 276.05 282.79 287.61 290.47 294.85 295.65 296.58 398.92 301.54 299.77 301.07 303.01 304.52
216.00 280.83 296.34 327.15 340.08 354.35 358.14 367.59 374.77 378.87 385.45 387.60 388.91 392.35 396.68 396.11 397.76 400.43 403.07 399.80 400.49 402.28 404.44 405.98
271 483.82 270.00 351.46 371.08 410.30 426.95 445.53 450.49 463.01 472.81 478.35 487.46 490.94 492.74 497.56 503.97 504.26 506.51 510.32 514.60 512.15 512.98 515.19 518.10 520.75 516.22 517.53 519.72 522.05 523.62
331 608.89 330.00 429.94 454.12 502.70 523.46 546.84 553.11 569.05 581.72 588.88 600.85 605.70 608.07 614.50 623.30 624.31 627.32 632.57 638.83 636.69 637.76 640.69 644.77 648.99 645.01 646.54 649.21 652.31 654.97 648.68 649.36 651.21 653.73 656.20 657.79
397 749.93 396.00 516.27 545.47 604.33 629.63 658.28 665.99 685.70 701.52 710.48 725.60 731.91 734.93 743.16 754.63 756.29 760.20 767.11 775.64 773.53 774.91 778.77 784.29 790.42 786.38 788.31 791.77 796.03 800.22 794.18 794.95 797.09 800.13 803.38 806.04 798.22 799.53 801.85 804.65 807.23 808.83
“Monte Carlo result is ( n )= 20.77: 20000 walks initiated Der lattice site. bMonte Carlo result is ( n ) = 46.48; 20000 walks initiated per lattice site.
a discrepancy was found between the values reported for the coefficient A4, a discrepancy they traced to a small error in Montroll’s calculation of A& We have performed additional calculations that lend support to the analysis of den Hollander and Kasteleyn. Since the data generated for ( n )using the method developed in ref 38 are guaranteed to be numerically precise, we have fitted our data on periodic square-planar lattices3’ with an expression of the form of (9). T h e coefficients Al ,throughA, were set at the values given by den Hollander and Kasteleyn (which are essentially the Montroll values), but the coefficient A4 was allowed to “float” so as to give the best representation of the data. The result of this calculation is given in Table VII, and it is seen that the coefficient A4 determined analytically by den Hollander and Kasteleyn is in excellent agreement with our “fitted” coefficient A4. Two further calculations were then performed. Inasmuch as Montroll did not report values of the coefficient A4 for triangular or hexagonal lattices, we repeated the above procedure for the case u = 3 and 6 (Le., we assigned A , through A3 to be the Montroll values and determined the A4 which gave the best fit of our data); reported in Table VI1 is the outcome of these calculations. The results in this table may be taken as the “state of the art“ representation of the mean walk length ( n ) as a function of the unit-cell size N for
infinite periodic lattices in d = 2. These analytic results will be mobilized later when we analyze the efficiency of photophysical processes in monolayer assemblies. 111. Discussion In this section, we couple the analysis reviewed in section I with the numerical results reported i s section I1 (and in ref 37 and 38) to analyze the encounter-controlled photophysical processes studied in ref 24 and 25. The average walk length calibrates the average time required for first encounter of the diffusing coreactant (the excited-state probe molecule Pl*)with the target molecule (the ground-state monomer Plo), the latter assumed to be symmetrically positioned with respect to the boundaries of the finite array. This lifetime is just the diffusion time Td introduced in section I. Consider the behavior displayed in Figure 3. Plotted there is the time Td E r as a function of domain size N for each of the arrays u = 3 , 4 , and 6 for the case where the diffusing coreactant is passively confined to the domain. In order to draw meaningful conclusions from the results presented in this figure, both the separation between nearest-neighbor locations (sites) and the transition rate (or jump frequency) between sites are assumed to be the same for each of the two-dimensional arrays considered. Given these constraints, it is evident
Excimer Formation in Spread Monolayers
Langmuir, Vol. 4, No. 2, 1988 313
Table V. Site-Specific and Overall Average Walk Length Results for d = 2 Regular Triangular Lattices with a Centrosymmetric Deep Trap (pT= 1) Subject to the Reflecting Boundary Condition Excluding Background Absorption (e = 0.00)
N
7 3.00 3.00
19 12.86" 11.57 13.29 13.71
37 33.11b 26.36 32.76 33.92 35.00 35.64
61 65.21 47.26 60.12 62.81 67.28 68.81 69.05 69.59 70.33
91 110.08
127 , 168.46
169 240.94
74.20 95.36 100.09 108.79 112.05 114.80 115.65 117.33 116.97 117.94 118.74
107.17 138.48 145.72 159.56 165.04 170.55 172.01 175.38 177.07 178.48 180.23 178.87 179.45 180.74 181.57
146.15 189.46 199.66 219.58 227.72 236.39 238.67 244.18 247.93 250.21 253.58 253.81 254.56 256.36 258.15 255.96 257.06 258.59 259.44
217 328.04 191.13 248.30 261.91 288.85 300.07 312.36 315.60 323.64 329.62 333.05 338.45 340.02 341.10 343.88 347.22 346.34 347.68 349.77 351.59 348.14 348.74 350.27 351.97 352.85
271 430.20 242.11 314.98 332.48 367.36 382.09 398.45 402.80 413.73 422.19 426.97 434.76 437.60 439.13 443.20 448.48 448.49 450.36 453.48 456.79 454.32 455.02 456.86 459.16 461.01 456.46 457.62 459.50 461.35 462.23
331 547.81 299.10 389.52 411.36 455.12 473.76 494.67 500.26 514.44 525.64 531.96 542.45 546.62 548.68 554.26 561.80 562.53 565.11 569.56 574.74 572.66 573.57 576.05 579.40 582.69 578.89 580.20 582.44 584.92 586.78 580.69 581.30 582.95 585.13 587.09 587.99
397 681.22 362.09 471.91 498.54 552.11 575.09 601.03 608.00 625.77 639.97 648.01 661.50 667.08 669.76 677.05 687.13 688.51 691.93 697.96 705,28 703.29 704.49 707.82 712.53 717.61 713.87 715.53 718.48 722.01 725.28 719.67 720.35 722.20 724.78 727.40 729.27 721.81 723.00 725.07 727.50 729.55 730.46
"Monte Carlo result is ( n ) = 12.84; 20000 walks initiated per lattice site. bMonte Carlo result is ( n ) = 33.11; 20000 walks initiated per lattice site.
Table VI. Comparison of Overall Average Walk Length Results for d = 2 Triangular Lattices Subject to the Passive Periodic (P) versus Reflecting (R) Boundary Condition with a Centrosymmetric Deep Trap ( p =~1) Including Background Absorption (s 10.00) s = 0.00 s = 0.05 s = 0.10 N (n)P (n)R difference,' % (n)P (n)R difference, % (n)p (n)R difference, % 7 19 37 61 91 127 169 217 271 331 397
6.00 20.73 46.54 84.61 135.80 200.79 280.15 374.34 483.78 608.86 749.89
3.00 12.86 33.11 65.21 110.08 168.46 240.94 328.04 430.20 547.81 681.22
50.00 38.0 28.8 22.9 18.9 16.1 14.0 12.4 11.1 10.0 9.16
4.80 10.39 14.09 16.19 17.39 18.11 18.58 18.89 19.11 19.27 19.40
2.73 8.14 12.79 15.54 17.06 17.95 18.49 18.85 19.09 19.26 19.39
43.2 21.7 9.23 4.02 1.86 0.913 0.468 0.249 0.136 0.0763 0.0428
4.00 6.94 8.31 8.96 9.30 9.50 9.63 9.71 9.77 9.81 9.84
2.50 5.95 7.92 8.81 9.24 9.48 9.61 9.70 9.79 9.81 9.84
37.5 14.3 4.70 1.68 0.648 0.267 0.116 0.0515 0.0235 0.0112 0.00508
"Percent difference is (((n)p- (n)~)/(n)p)100.
that the time 7 increases with decreasing coordination number u of the lattice. As noted previously,4l this behavior can be understood by recognizing that for lattices of a fixed N and a common metric (lattice spacing) increasing the valency of the lattice allows one to cluster more sites closer to the central, target molecule and hence to decrease the average number of steps required for reaction. For example, using the figures in ref 37 and 38 and
in the present paper one finds that for the common setting N = 169 (i.e., a centrally disposed target and 168 surrounding, nonreactive sites) one must be at an effective nearest-neighbor distance A (where A is the minimum number of lattice displacements required to pass from site i to the trap) of A = 7 for u = 3, of A = 6 for u = 4, and of A = 5 for u = 6 before (at least) 50% of the surrounding, nontrapping sites is encompassed. The ( n ) idata in these
314 Langmuir, Vol. 4 , No. 2, 1988
Politowicz and Kozak
Table VII. Coefficients in the Expression ( n ) = ( N / ( N - 1)) {AJVln N
+
A2N Lattices of Valency v = 3, 4, and 6
lattice structure
ref
square planar
a b C
hexagonal
0.
C
triangular
a C
A,
11"
0.318 309 886 0.318 309 886 0.318 309 886 34314" 0.413 496 672 0.413496 672 4312" 0.275 664 448 0.275 664 448
+ A 3 + A,(l/N)}for d = 2 Infinite Periodic
A2
'43
A4
0.195 056 166 0.195 062 532 0.195 062 532
-0.116 964 81 -0.116964779 -0.116 964 779
-0.051 456 50 0.484 065 704 0.481 952
0.066 206 98 0.066 206 98
-0.254 222 79 -0.254 222 79
6.063 299
0.235 214 021 0.235 214 021
-0.251 407 596 -0.251 407 596
-0.044 4857
"Montroll, E. W. J. Math. Phys. 1969, 10, 753. *den Hollander, W. Th. F.; Kasteleyn, P. W. Physica A (Amsterdam) 1982, 112A, 523. Present study.
IO00
800
1
P
t
4 00
5001
4001
L
/ d/
/ 60 120 180 240 300 360 420
'0
DOMAIN
SIZE ( N )
Figure 3. Plot of the characteristic lifetime T versus the domain size N for d = 2 dimensional finite arrays of local coordination u = 3, 4, and 6 and subject to passive (confining) boundary conditions. sources quantify the consequences of this spatial effect. From the results presented in Figure 4 one finds that the above trend persists when the boundaries of the domain actively confine the diffusing coreactant to the interior of the given domain. In this case, the coreactant, upon confronting the boundary, is displaced to an interior site once removed from the perimeter of the system, and the time T is much reduced, especially for small domains. This point can be brought out more clearly by plotting the ratio T(active)/r(passive)versus the domain size (N). As confirmed from the results displayed in Figure 5, the boundary conditions constraining the motion of a species diffusing in a finite domain influence dramatically the average time T in small systems but become of decreasing importance with increase in the size of the array. The latter behavior is certainly consistent with one's intuition, but the profiles displayed in Figure 5 quantify the size of the domain required before the two collision times are essentially the same (i.e., .r(active)/T(passive) > 0.90). Moreover, there is one feature of the results plotted in Figure 5 which is, at first sight, not obvious. Note that for each domain size considered the ratio ?(active)/ 7(passive) is largest for u = 3, followed by u = 6 and finally u = 4. This ordering is different from the systematic trends observed in Figure 3 for passive confinement and in Figure 4 for active confinement of the diffusing species and needs to be explained. Focusing again on the common setting
DOMAIN
SIZE ( N )
Figure 4. Plot of the characteristic lifetime T versus the domain size N for d = 2 dimensional finite arrays of local coordination u = 3, 4, and 6 and subject to active (reflecting) boundary conditions. loo
r
:
v:?,
30
! 100
200
DOMAIN
300
400
500
SIZE ( N )
Figure 5. Effect of interfacial boundary conditions on the characteristic time of excimer formation in a monolayer system. Plotted here is the percentage, % (r(active)/r(passive))lOO, versus the domain size N for arrays characterized locally by coordination numbers u = 3, 4, and 6.
N = 169 for the three arrays (u = 3, 4, and 6), and using the lattice diagrams given in ref 37 for u = 4, in ref 38 for
Excimer Formation in Spread Monolayers P
lor
I,
Langmuir, Vol. 4, No. 2, 1988 315
6t 0
0.0
I 0.15
0.05 0.10 MOLE FRACTION PDA
Figure 6. Plot of the rates of excited-state dimer to excited-state monomer intensity, Zz/Zl as a function of total PDA concentration (mole fraction). The area per molecule in the spread monolayer is assumed fixed at 41 A2, with T~ set at 200 ns and the time i at 1ns. The boundary of the finite domain of local coordination v = 3,4, or 6 is assumed to influence passively the motion of the excited-state monomer, P1* (i.e., the array is constrained by confining boundary conditions). u = 3, and in Figure 2 of the present study for u = 6, one determines that the number of boundary sites for the u = 3 array is NB = 36, for the u = 6 array is NB = 42, and for the u = 4 array is NB = 48. From the nature of the active boundary conditions imposed, it is clear that differences between active versus passive boundary conditions will be controlled by the number of perimeter (or boundary) sites in each array. The percentage of the total number N of sites in this case that are boundary sites NB is 28.40% for u = 4, 25.85% for u = 6, and 21.30% for u = 3. Thus, arrays of local coordination u = 3 should be (relatively) less influenced by differences in domain boundaries than is the case for u = 6, and both of these coordinations should be (relatively) less influenced than the case for u = 4; this is indeed the behavior uncovered in our calculations and displayed in Figure 5. We now use (6) to calculate the ratio of excimer to monomer intensity & / I l ) as a function of the total concentration of 12-(l-pyrene)dodecanoic acid (PDA) (expressed as the mole fraction of PDA) in a spread monolayer of PDA and oleic acid at the air-water interface of a neutral aqueous solution.25 We adopt the previously reported estimates of T~ 200 ns and i 1 ns for a system characterized by an area per molecule of 47 A2and assume the passive confinement of the diffusing excitedstate monomer P1* to finite domains characterized by the coordination u = 3,4, or 6. The curves displayed in Figure 6 give the ratio 12/11 versus x for the three types of arrays considered in this paper; note that since the ratio 12/11 is inversely proportional to ( n )the ordering of the curves is just the reverse of the ordering displayed in Figure 3. Since the ratio 12/11 is bracketed by the values reported for u = 3 and 6, one can estimate from this figure for each mole fraction x experimental bounds on I z /Il. For example, if the total PDA concentration is x = 0.10, one estimates 7.1 ,< I z / I l 5 9.1, a result which is certainly consistent with the one found experimentally (see Figure 2 of ref 25). Whereas the above bound is obviously system depend-
-
-
.,p/
,
00
0 05 0.10 MOLE FRACTION PDA
0.15
Figure 7. Effect of interfacial boundary conditions on the intensity ratio, Zz/Zl, versus total PDA concentration. Plotted here is the percentage, % ((Z2/ZJA - (12/Z1!~)/(Z2/Z1)~)t00, versus mole fraction of PDA ( x ) for finite domains Characterized locally by coordination u = 3, 4, and 6.
ent, note that if one calculates 12/11for active versus passive boundary conditions and then constructs the percentage difference
+
the specific role of the system parameters T~ and is suppressed (assuming, of course, that in a given monolayer that T~ and i are sensibly independent of the local packing or coordination). In the resulting plot of this percentage versus x , given in Figure 7, it is seen that the ordering u = 3 versus u = 6 versus u = 4 is, as expected, the inverse of the ordering reported in Figure 5 for the time 7 versus the domain size N. If one replaces the host substrate (say, oleic acid) by another substrate which either enhances or compromises the sequestering of PDA, these results show that a systematic and measurable change in the ratio I z / I l versus concentration of PDA should result, with the maximum and minimum differences exhibited by the curves characterized by u = 4 and 6, respectively. The results displayed in Figures 3-5 document how the time T changes as a function of the domain size N , and those in Figures 6 and 7 show how the excimer-monomer intensity ratio, 12/11, changes with increase in mole fraction x of total PDA. It is evident that both T and 12/11are sensitive to the local coordination or packing of molecules in a given domain. Now, as reviewed in section I, there are two attitudes one can take concerning the organization of the monolayer in the liquid-expanded versus the liquid-condensed regime. One possibility is that the monolayer in the lower density regime is uniformly characterized by a given packing (coordination u); further compression of the monolayer may then result in a discontinuous transition to a different packing. Alternatively, one can regard the structure of the monolayer in the lower density phase to be a mixture of “patches” of individual coordination u = 3, 4, or 6 with interfaces between these boundaries, i.e., to be locally ordered with a uniform co-
Politowicz and Kozak
316 Langmuir, Vol. 4, No. 2, 1988 ordination but globally disorganized. Further compression of the monolayer in this case may simply result in a shifting of the number of molecules in each type of domain. To explore these two possibilities analytically, we take advantage of the analytic (asymptotic) results for ( n )reported in the previous section. Since for the particular case of a centrosymmetric trap (target molecule) the results for finite lattices subject to confining boundary conditions are safely approximated by the ones for infinite periodic lattices of unit-cell size N , we write r
- i((W
In N ) / ( N - 1))X {Cl(u=3)x(u=3) Cl(u=4)x(u=4) + C1(u=6)x(u=6)) (10)
+
where x(u=j) is the mole fraction of molecules whose local coordination is u = j in a given domain (with Cx(u=j)= 1) and where the constants Cl(u=j) are the coefficients Al(u=j) given in Table VII. Assuming that the quantities 7I and i are sensibly constant in a given system, the first possibility described above suggests that upon compression discontinuous changes in packing from u = 3 to 6 , say, should induce a discontinuous change in the ratio 12/11 when monitored as a function of increasing total pyrene concentration. For example, switching from xp = x(u=3) to x p = x(u=6) should be reflected by an abrupt change in the ratio 12/11by a factor Cl(u=6)/Cl(u=3) 0.48. Alternatively, if one assumes that there is a persistent patchwork of domains, each of definite coordination u, then compressing the monolayer may simply result in a synchronous modulation in size of each of these domains; compression should result in smooth changes in the quantity 7 (and hence I z / I 1 )with increase in the total pyrene concentration. In earlier it was pointed out that excimer formation is extremely sensitive to the number no of diffusing, excited-state monomers per domain. In particular, Monte Carlo simulations on systems of finite extent have shown that the mean walk length (and hence the mean lifetime of P1*) is, to a fair approximation, inversely proportional to the number n,. Conversely, it can be shown that if the survival of P1*is compromised by decay processes mediated by the substrate molecules M (here, oleic acid), the lifetime of P1* is sensitive to the number of substrate molecules comprising a given domain. Consider the process
U
Figure 8. Diagram of a portion of a d = 2 triangular lattice. Circles (vertices) indicate lattice sites, and connecting lines (bonds) indicate the paths accesible to the diffusing particle. The dashed hexagon shows the irreducible unit of the associated dual lattice. C2
-
PI* + M
k,
Plo+ M
+ hv
or for large excess of M
PI*
kl'
Plo+ hv
where k,' is a pseudo-first-order decay rate. We have studied the case where excimer formation occurs with unit efficiency at the central target molecule, but quenching occurs with measurable efficiency 0 < e, < 1 at the N - 1 satellite locations defining a finite domain of the monolayer assembly. For ei > 0 it is then possible that Pl* can be de-excited long before excimer formation can occur. In that case, the mean survival time of the species P1*will certainly decrease, and what we demonstrate in Table I1 (with e = s ) is how efficient this quenching can be for the most closed-packed domains (Le., for domains of coordination u = 6 ) . Notice that with increasing size N of the domain, even a 5% probability of quenching at each location (i.e., all ei = e = 0.05) will lead to an effective destruction of the species Pl* after only a few encounters. Notice further (compare Table VI) that this dramatic falloff in the concentration of excited-state monomer PI*
c2
\
L a t t i c e Unit Boundary
c2
I
/
1 '
'Irreducible Subunit Boundaries
Lattice Sites
Figure 9. Construction of the smallest d = 2 regular triangular lattice considered in this paper. Irreducible hexagonal lattice units are juxtaposed to form the N = 7 regular triangular lattice.
will occur regardless of the constraints imposed on the diffusing coreactant when it confronts the boundary of the domain or the packing assumed to characterize the monolayer (compare also Table V of ref 37 for u = 4 and Tables XI, IV, and VI of ref 38 for u = 3). A further insight can be extracted from the above calculation. One can imagine a limiting case where a monolayer is assembled from the species PIo alone. Upon photoexcitation of one PIomolecule in a given domain, one would then have the possibility of excimer formation at any of the locations accessible to the diffusing coreactant P1*. Consider the case where excimer formation occurs with unit efficiency (e = 1)at a central target molecule but with a reduced efficiency ei < 1 at the satellite sites. Physically, the factor e, might be interpreted as an orientational factor e, cos 8, where 0 is the angle between the transition dipoles of the diffusing and target species; values of ei < 1then correspond to less than optimal alignments of the transition dipoles. As is evident from the data reported in the tables cited above, one would expect (see eq 6 ) a marked change (enhancement) in the signal I z / I 1 even when the efficiency of excimer formation at the satellite location is only of the order of 5%. In conclusion, the objective of this study was to provide a quantitative assessment of sequestering and the influence of domain structure on excimer formation in spread monolayers. In order that our discussion be reasonably complete, it was necessary to solve first a stochastic problem (random walks on finite, d = 2, u = 6 lattices), the results of which could then be combined with those obtained in earlier studies (on d = 2, u = 4 and 3 finite lattices). In dealing with the statistical problem of finite systems, boundary conditions play a crucial role, and, for this reason, a detailed discussion of the influence of "perimeter" or "interfacial" effects was presented. Although the complexity of the spread monolayer system is
-
Langmuir, Vol. 4 , No. 2, 1988 317
Excimer Formation in Spread Monolayers
Figure 10. (a) Diagram of the d = 2, N = 19 regular triangular lattice unit with the primitive boundary. Dashed lines indicate the interior irreducible subunit boundaries. (b) Diagram of the regular triangular lattice unit in a with the common site boundary.
Figure 13. Diagram of the periodic boundary for the N = 37 triangular lattice unit using the passive boundary condition; the revised site classification scheme is shown. Table VIII. Accessible Adjacent Sites for Each Site Class of the N = 37, d = 2 Regular Triangular Lattice Unit Subject to the Passive Periodic Boundary Condition" site class accessible adjacent sites T
106)
1 2 3
T,1(2), 2(2), 3
4b 56 6b
2, 3, 4, 5(2), 6 2 , 3 , 4 ( 2 ) ,6(2) 3 , 4, 5(2), 6(2)
3(2), 4, 5
1, 2(2), 4, 5, 6
a Accessible adjacent site occurrences greater than 1 are indicated in parentheses. Boundary sites.
has been laid for further studies of diffusion-controlled reactive processes in monolayers.
Figure 11. Diagram of the periodic boundary condition for the N = 37 triangular lattice unit using the common site boundary convention; the site classification is shown.
Acknowledgment. This paper is based in part on work supported by the North Atlantic Treaty Organization under a grant awarded in 1984 (to P.A.P); the kind hospitality of Professor J. S. Rowlinson is gratefully acknowledged. The research described herein was also supported in part by the Office of Basic Energy Sciences of the U.S. Department of Energy. This is document No. NDRL-2967 from the University of Notre Dame Radiation Laboratory. Suggestions by Professors M. E. Schwartz and W. C. Streider in the early stages of this work are also gratefully acknowledged. Appendix A
Figure 12. Diagram of the periodic boundary for the N = 37 triangular lattice unit using the passive boundary convention. The
original site classification scheme is shown, and the sites marked 4 lying along a boundary between two lattices units are those referred to in the text (compare Figure 13).
only approximately represented by the model elaborated here (e.g., we have not considered chain-chain interactions), it is our hope that at least some important features of the problem have been exposed and that a foundation
We summarize here our approach to the study of reaction-diffusion processes on two-dimensional finite lattices of valency u = 6. For the reader's convenience we have decoupled the discussion into several parts: the identification of the unit cell (simplex), classification of the point symmetry, boundary conditions on the finite array, and determination of the average walk length. 1. Identification of the Unit Cell (Simplex). Consider a plane covered with an array of contiguous equilateral triangles. The vertices of this array and the lines connecting these vertices define an extended triangular lattice in d = 2. A section of such a lattice is shown in Figure 8. If one examines any site of this lattice, the smallest irreducible subunit can be generated by constructing the perpendicular bisectors of the six walk paths leading to that lattice site; the smallest figure generated by the intersection of these bisector lines is a hexagon (see Figure 8). Juxtaposition of an infinite number of such irreducible subunits generates an infinite triangular lattice.
318 Langmuir, Val. 4, No. 2, 1988 The simplest two-dimensional regular polygon which can cover a plane is the equilateral triangle. The triangular lattice possesses the highest possible valency (u = 6) for a regular two-dimensional-coveringlattice. Triangular lattice units are constructed in a manner similar to that developed for the other two-dimensional lattice units.37938Starting from an irreducible subunit, the first triangular lattice unit is constructed by juxtaposing one “layer” of an irreducible lattice subunit about its periphery. This generates a lattice unit of seven total sites with a centrosymmetric site, as shown in Figure 9. Higher triangular lattice units are generated by adding more layers of irreducible subunits about the periphery of the original lattice unit. With this procedure, lattice units of 19, 37, 61,91,127,169,217,271,331, and 397 sites are generated; N = 397 is the largest lattice unit considered in this work. The N = 7 triangular lattice, the smallest lattice unit considered in this work, is of Da symmetry (see Figure 9). There are six C, axes of symmetry in the plane of the lattice unit, each with an attendant cr plane of symmetry. There is one c6 axis of symmetry normal to the plane of the lattice unit passing through its center. The six C2 axes of symmetry are subclassified into two groups as follows. Those C, axes of symmetry in the plane of the lattice that are collinear with the walk paths leading from the central lattice unit site are denoted primary C, axes; the remaining three C2 axes of symmetry, which bisect the angles between the primary C2 axes, are denoted secondary C, axes. These assignments are useful in specifying the point symmetry classifications and boundary conditions. Figure 9 also displays the lattice boundary as a composite of all of the irreducible subunit boundaries whose associated walk paths are not continued past the boundary; i.e., they are not connected to another interior walk path. Such a lattice boundary has the property that any walk path which intersects it is perpendicular to it. This lattice boundary is denoted as the primitive boundary (see Figure 9). Alternatively, the boundary of the lattice unit can be collapsed such that it is coincident with the walk paths connecting the outer layer of lattice sites. This lattice unit boundary convention is denoted the common site boundary; if two such lattice units are juxtaposed, the sites on the boundary between the lattice units will be common to both units. Figure 10 displays the N = 19 lattice unit using the common site boundary; no elements of lattice unit symmetry are lost using the latter boundary convention. 2. Classification of the Point Symmetry. For each of the triangular lattices studied, the centrosymmetric site will be considered as the target site (or trap) and denoted T. The symmetry of the lattice units is then used to classify the remaining N - 1sites of the unit with respect to the central site. Since the lattice unit has a C, symmetry axis normal to the center of the unit, the site classification is identical for each of six subsections of the lattice unit; only one sector is shown in Figure 1. In this figure is displayed the site classification scheme for the N = 7-397 triangular lattice units. From Figure 1, one can see that the site classification scheme generates classes of sites that are of multiplicity 6 and 12. Those sites which lie on either a primary or secondary C2 symmetry axis of the lattice unit have multiplicity 6, whereas those that do not lie on a C, symmetry axis have multiplicity 12. In the latter case, note that for any site not lying on a C, axis of symmetry there is a companion site and both are symmetrically disposed with respect to that Cz axis (either primary or secondary); this “pairing” arises for each of the six sectors which comprise the lattice unit. The classification scheme is
Politowicz and Kozak designed such that the lattice site classifications of the higher lattice units are supersets of those for the smaller units. The motivation for implementing this symmetry classification scheme is to reduce the number of unknowns (and hence equations) that must be considered in calculating the average walk lengths. From Figure 1it is seen that this classification scheme leads to a dramatic reduction in the number of unknowns; for example, for the N = 397 unit the number of explicit sites to be considered collapses from 397 to 42. 3. Boundary Conditions. One can envision two classes of boundary conditions. In the first, the boundary is passive, or has a minimal effect on the diffusing particle’s motion whenever it encounters the system boundary. On the other hand, there is a whole range of situations where the diffusing particle’s motion may be influenced significantly by the boundary; these latter boundary conditions will be referred to as active. With respect to the former class of boundary conditions, consider first the case where the diffusing species crosses the system boundary to a symmetrically equivalent site in an adjacent lattice unit. In an extended lattice system comprised of replications of the original lattice unit, this symmetry equivalence leads to periodic boundary conditions. As a second example of passive boundary conditions, consider the case where the diffusing species encounters the boundary but then is reset at the site from whence it came. This boundary condition is referred to as confining, or nontransmitting. For d = 2 triangular lattices, a periodic boundary can be generated by juxtaposing identical lattice units by using the “common site” convention, as shown in Figure 11 for the N = 37 lattice unit. Such a procedure reproduces an infinite regular triangular lattice. All symmetry elements of each unit are preserved, and the symmetry site classification scheme remains intact. The major disadvantage of such a scheme is that the boundary sites of any given lattice unit are shared by at least one, or as many as two, additional lattice units. This causes an ambiguity in the interpretation of the number N , which scales the system size for each lattice unit. A second problem arises from having the lattice unit boundary coincide with the outermost layer of lattice sites; in effect, the randomly diffusing particle is injected into the adjacent lattice unit by one layer of lattice sites, simply by crossing the lattice boundary. This is inconsistent with the interpretation of periodic boundary conditions as developed for the hexagonal, square-planar, and cubic lattices and in fact produces results which are rather more characteristic of an active, reflecting boundary than a strictly passive one. An alternative formulation of the periodic boundary condition is shown in Figure 12 for the N = 37 lattice unit. This formulation generates an extended periodic array but at the expense of having the secondary C, axes of symmetry of two, face-to-face, adjoining lattice units not collinear. This symmetry breaking causes all site classes of the lattice unit with multiplicity 1 2 to lose class equivalence and bifurcate into two new site classes of multiplicity 6. For example, consider the pair of sites labeled 4 in any of the lattice units of Figure 12; these are aligned at a lattice unit boundary adjacent to another lattice unit. This alignment causes the nearest-neighbor sites of sites 4 to become symmetry-distinct sites; hence, the nearest-neighbor sites must be classified separately. The consequence of reclassifying sites of multiplicity 12 is that one generates the configuration diagrammed in Figure 13 for the N = 37 lattice unit. The new site clas-
Langmuir, Vol. 4, No. 2, 1988 319
Excimer Formation in Spread Monolayers sification scheme removes the aforementioned problem at the expense of introducing a greater number of unknowns in the subsequent walk length calculations. The site classification scheme for all of the regular triangular lattice units considered in this paper, subject to this second periodic boundary condition, is shown in Figure 2. We shall denote this latter boundary condition as the passive periodic boundary condition, to differentiate it from the formulation of the active periodic boundary condition described previously. In comparisons of stochastic data among d = 2 lattices of different valency subject to periodic boundary conditions, the data pertaining to triangular lattices must be that corresponding to the passive periodic boundary condition. Consider next nontransmitting or “confining” boundary conditions for the regular triangular lattice. Since such a boundary condition imposed on a lattice unit is necessarily independent of the adjacent lattice structure, all lattice units subject to this condition can employ the site classification scheme depicted in Figure 1. For the d = 2 regular triangular lattice, the passive periodic and confining boundary conditions are not degenerate (in contrast to what one finds for square and hexagonal lattices with a centrosymmetric trap). Therefore, data for triangular lattices subject to confining boundaries are reported separately. Finally, consider the reflecting boundary condition for the d = 2 regular triangular lattice. The site classification scheme is displayed in Figure 1. In this case, whenever a particle moves from a lattice site adjacent to the boundary and encounters the boundary it is reflected to a lattice position one site further removed from the boundary from whence it started. Notice that the path taken is perpendicular to that particular segment of the lattice unit boundary. Thus, in most cases, the result of encounteringthe (reflecting) boundary is that the diffusing particle is injected to a lattice site once removed from the boundary. 4. Determination of the Average Walk Length. The site classification scheme for each of the triangular lattice units considered in this paper is the basis of our calculation of the average walk lengths using the theory of finite Markov processes. The calculational procedure followed here will be illustrated for the N = 37 regular triangular lattice unit in d = 2 subject to the passive periodic boundary condition. The site classification scheme for such a unit is shown in Figure 2 while three lattice units are shown juxtaposed in Figure 13. From Figure 13 it can be seen that this lattice unit consists of seven distinct site classes, denoted T and 1-6. Imposition of the passive periodic boundary condition generates a situation wherein each distinct site has a set of accessible adjacent sites unique to that site class. For unbiased, nearest-neighbor random walks, the information necessary to calculate the average walk length is just this set of accessible adjacent sites for each site class. These data can be obtained by inspection (from Figure 13) and are recorded in Table VIII. To study random walks on a d = 2 triangular lattice of N = 37 sites with a centrosymmetric deep trap and subject to passive periodic boundary conditions, one proceeds by formulating the transition probability matrix Q of the theory of finite Markov processes. From Markov theory,
it is known that if such a process continues long enough the process will terminate in an ergodic state. This means that Q k 0 as 12 m. Thus, the inverse of I - Q exists and is equal to N, the latter denoting the fundamental matrix of the random walk problem. In particular
- -
(I - Q)-’ = I
+ k2= l Q
~ = N
where I is the unit matrix. The individual matrix eleN = (I - Q)-1 = 6
6 63241803 66241803 78721803 71881803 72181803
6 ‘ 63181803 66361803 7188/803 78121803 73621803
1
6 62761803 67201803 7218/803 73621803 83701803
ments of N (Nij)define the average “visitation number”, Le., the number of times the diffusing particle lands on site j , having started at site i, before it is trapped. The row sums of N yield the average walk length before trapping for walks initiated at particular sites (classes) of the lattice. In particular
pi]
NS = 50.76
where 5 is a column vector of ones. The overall average walk length ( n ) for a diffusion process on a lattice of N sites is given by
( n )= Cai(n)i/Cai i
1
where the ( r ~ are ) ~ the site-specific average walk lengths defined for each site class of the lattice and ai is the multiplicity of the ith site class. Since for passive periodic boundary conditions all site classes are of multiplicity 6, the previous equation simplifies to
( n )= ( 6 / ( N- 1 ) ) Z ( n ) i i
Hence, for the N = 37 regular triangular lattice unit subject to this boundary condition, the overall average walk length is
( n )= f/s[36
+ 45.38 + 47.23 + 49.87 + 49.98 + 50.761 = 46.54
The above procedure can be adapted to deal with situations where there are competing reaction centers at the N - 1background sites. Let si represent the probability of the walker being absorbed at the ith background lattice site (class). In this study, the background absorption factor is assumed to be the same for all background sites. Thus, the probability that the walker continues its migration by stepping from any given background site to one of its u = 6 accessible adjacent sites is given by q = (1 - s)/6. For the case where the central trap is deep and the background site absorption probability is uniformly s, the transition probability matrix Q for the transient states of the N = 37, d = 2 regular triangular lattice subject to the passive periodic boundary condition can be formulated. Again, it is related to the (inverse of the) fundamental matrix N of the theory of finite Markov processes. One proceeds as before by inverting the above matrix to obtain an ex-
1
Langmuir 1988, 4 . 320-326
320 1 - 2q Q =
kiq
-2q 1 -2q -9
-4
0 L
-4 !q
-24 1 - :’q
IJ
pression for N and subsequently the ( n ) ,and overall average walk length ( n ) .
Appendix B From the data listed in Table I one finds that the value of (n),(the average walk length required for trapping for walks starting at the site adjacent to the central deep trap) is equal to an integer for all lattice units considered: ( n )1 = N - N B / -~ 2 (B1) where NB is the number of sites in each lattice unit adjacent to its boundary. We note that the value of ( n ) lfor a regular, infinite periodic lattice in d dimensions is given by ( n ) ,= N - 1 (B2) a formula derived originally by M ~ n t r o l land, , ~ ~ at first glance, it would appear that there is a formal discrepancy between the two results, (Bl) and (B2). However, in formulating the first (active) periodic boundary condition one juxtaposes lattice units to form an extended structure in which the boundary sites of each lattice unit are shared by other units. In particular, for any triangular lattice unit of NB boundary sites and N t,otal sites, N B - 6 of the sites
of the unit are shared between the unit and its adjacent units and six boundary sites are shared among three lattice units; these sites correspond to those along a face of the lattice boundary or at a vertex, respectively. Thus, the common site boundary convention causes a reduction in the total number of lattice sites in any given unit to a value N’ < N.Since NB - 6 lattice sites along the boundary faces are shared between two units, (NB - 6)/2 sites should not be included in the total number of sites for a lattice unit. Similarly, of the six sites at the vertices of the lattice unit boundary, (2 X 6)/3 sites should not be included in the total lattice site count. Combining these terms, one obtains
N‘ = N
-
NB/ 2 - 1
033)
If one now reexpresses the Montroll result in terms of N’, 1.e. ( n ) , = N’-- 1
(B4)
where N’is the total number of distinct lattice sites in each unit, then substitution yields the relation B1. Finally, for a triangular lattice unit composed of a central site surrounded by 1 concentric single layers of lattice sites, one finds (see Figure 9 and 10) that
NB = 61
(B5)
Substitution of this result into (Bl) yields an interesting relation for ( n ) l :
( n ) l = N - 31 - 2 (B6) Registry No. PDA, 69168-45-2; oleic acid, 112-80-1.
Preparation and Electrical Properties of Lightly Substituted Phthalocyanine Langmuir-Blodgett Films Michiya Fujiki” and Hisao Tabei NTT Basic Research Laboratories, Tokai, Ibaraki, 329-11, Japan Received July 21, 1987. In Final Form: September 2, 1987 The electrical properties and characteristics of Langmuir-Blodgett (LB) films of several phthalocyanines (Pc’s) containing tert-butyl, isopropyl,and cyano groups are examined. These films are prepared by lifting the horizontal substrate through the water-air interface. The forcearea data, &-band spectra of the films, and d-spacing in X-ray diffraction patterns of the powders suggest that these Pc’s take one-dimensionally assembled structures and edge-on configurations to the water-air interface. The in-plane conductivities of undoped LB films of the Pc’s containing cyano groups are highly sensitive and reversible. The film conductivity increases steeply by 5 orders of magnitude when the film is exposed to active gases such as iodine, triethylamine, and n-butanethiol vapors. Conductivity responses of the LB films to the gases relate to the estimated ionization potentials and electron affinities of the films.
Introduction Phthalocyanine (Pc) thin films have been of particular interest because of their photo and electrical (1)Lever, A. B. P. Adu. Inorg. Chem. Radiochem. 1965, 7, 27-114. (2) Moser, F. H.; Thomas, A. L. The Phthalocyanines; CRC: Boca Raton, FL, 1983. (3) Loutfy, R. 0.;Sharp, J. H. J . Chem. Phys. 1979, 71, 1211-1216. (4) Tam. C. W. ADDZ.Phvs. Lett. 1982. 40. 183-185. ( 5 ) ArisLima, K.; Hfratauk‘a,H.; Tate, A.; Okada, T. Appl. Phys. Lett 1985.46. 279-281. Honeybourne, C. L.; Ewen, R. J.; Hill, C. A. S. J . Chem. Soc.. Faraday Trans. 1. 1984,80, 851-863. (7) Sadaoka, Y.; Yamazoe, N.; Seiyama, T. Denki Kagaku 1978, 46, 597-602 (in Japanese). (8) Moskalev, P. N.; Kirin, I. S. Russ. J . Phys. Chem. (Engl. Transl.) 1972, 46, 1019-1022.
(4
~~
0743-7463/88/2404-0320$01.50/0
However, unsubstituted Pc’s generally have some limitations in arranging and organizing Pc moieties into a desired crystal structure and thickness onto solid substrates, because Pc’s usually exhibit polymorphisms in the solid state and poor solubility in common organic solvents, despite the fact that Pc’s have high chemical and thermal stability. Thin Pc films can be made by vacuum evaporation, spin casting, and dispersion in a polymer binder. Recently, the Langmuir-Blodgett (LB) technique has been recognized as a useful way of tailoring thin films of molecular thickness, since various surface-active molecules can form high-quality LB films.1° Several groups have (9) Gutierrez, A. R.; Friedrich, J.; Haarer, D.; Wolfman, H. IBM J . Res.
D ~ u 1982, . 26, 198--208.
0 1988 American Chemical Society
