J. Phys. Chem. C 2010, 114, 2645–2654
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Mean-Field Theoretical Study of Bistability in Mixed Azobenzene-Alkylthiol Monolayers Chris R. L. Chapman and Irina Paci* Department of Chemistry, UniVersity of Victoria, Victoria, British Columbia, V8P 5C2 Canada ReceiVed: September 28, 2009; ReVised Manuscript ReceiVed: January 10, 2010
Azobenzene derivatives exhibit a well-known cis-trans photoinduced isomerization. At the gold surface, scanning tunneling microscopy studies of the azobenzene derivative N-(2-mercaptoethyl)-4-phenylazobenzamide have revealed similar isomerization behavior under the influence of an electric field. The reversible conversion of this adsorbate between a low-current and a high-current isomer shows promise for applications in molecular switching. This paper employs a mean-field theoretical approach to study bistability at medium to high surface coverages of chemisorbed mixed monolayers of azobenzene and dodecanethiol on gold surfaces. We find that the trans isomer dominates the monolayer under most sets of conditions, stabilized both by interactions between its total dipole moment and the field, and by lateral interactions with matrix molecules. Our results also highlight two important effects, which are often overlooked in discussions of bistability in self-assembled monolayers. First, polarizability effects can overcome destabilizing coupling between the permanent dipole moment and the applied field when fields are strong. We find that, at strong negative sample biases, the highly polarizable trans azobenzenes become most prevalent in the monolayer even though they are hindered by unfavorable permanent dipolefield coupling. Second, the more favorable lateral interactions between trans isomers and the dodecanethiol matrix lead to almost full conversion to the trans isomer in the monolayer, at densities well below those for which one would expect the footprint of the molecule to become important. I. Introduction Azobenzene and its derivatives exhibit a well-known cis-trans photoisomerization of the azo (sNdNs) group that leads to large geometrical changes in the otherwise rigid molecule. Coupled to its chromophore character due to an extended π system, this geometrical change has led to a broad range of practical applications for azobenzene derivatives. They include classical dye manufacturing and molecular sensors,1-3 photobiological switches,4,5 and the promotion of alignment changes in liquid crystals.6 Similar changes in azobenzene molecules tethered to metal surfaces may lead to applications such as switches in molecular electronics. In fact, adsorbed azobenzene derivatives have the ability to switch between two conductance states, that are usually attributed to cis and trans isomers. Scanning tunneling microscopy (STM) experiments suggest that self-assembled monolayers (SAMs) of adsorbed azobenzene undergo reversible cis-trans isomerization when exposed to UV light.7-13 Photoillumination studies by Levy et al.12 and Comstock et al.13 have shown that weakly bound, physisorbed tert-butyl azobenzene converts under UV illumination between a low conductance state and a high conductance state in closely packed SAM structures. The low conductance state is generally attributed to the trans isomer, oriented parallel to the substrate, whereas the high conductance state is attributed to the cis isomer, with one benzene ring parallel to the substrate and the other roughly perpendicular to it. Similar results regarding the isomerization of physisorbed azobenzene were obtained in STM studies without photoillumination14-16 and are attributed to the interaction between the tunneling electrons and the adsorbate. Strongly bound, chemisorbed azobenzene forms different closely packed structures, de* Electronic address:
[email protected].
pending on monolayer density, lateral interactions, and whether it is adsorbed on its own or from a mixtures containing alkythiols. At high densities, molecules in chemisorbed azobenzene monolayers are found “standing up”, nearly perpendicular to the surface. Experiments on such systems were performed by Kumar et al.7 and Weidner et al.9 Current understanding suggests that in these monolayers, the high conductance state is the trans isomer since the extended conjugated system is prevented by surrounding molecules from interacting with the substrate at these densities. Upon isomerization, the benzene ring located furthest from the surface moves downward, resulting in a state with lower conductance. Recent studies have suggested that adsorbed azobenzene can undergo isomerization without photoillumination or electron tunneling: the electric field created by the STM tip is sufficient to overcome the isomerization energy barrier.17-19 Other systems have also been shown to switch between conductance states when an electric field was applied.20,21 In these cases, isomerization is thought to occur as a result of differences between the field-dipole coupling energies of the two isomers. In cases where binding to the surface occurs through highly flexible alkyl spacers, dipole-field coupling may produce conformational changes in the spacer that lead to changes in the molecular height of the adsorbate.19 This adds complexity to an already challenging path toward developing an understanding of the switching process in these systems. Moreover, particular care must be taken to discern between different kinds of switching when performing STM studies of optical isomerism.17,22 In addition to the external field, azobenzene switching is strongly dependent on the density of the SAM, particularly in chemisorbed structures. The cis isomer has a significantly larger footprint than the trans, which leads to inhibition of the cis-trans
10.1021/jp909325d 2010 American Chemical Society Published on Web 01/27/2010
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isomerization process at high densities.9,10 On the other hand, Elbing et al.8 showed that in extended, fully conjugated azo systems such as azo-biphenyl moieties, the highly ordered highdensity monolayer can exhibit collective switching when illuminated. When these authors used spacers such as methyl substituents on the biphenyl rings, molecules in the lowerdensity monolayer switched independently, in a fashion similar to that observed for regular azobenzene. Furthermore, in mixed monolayers, lateral interactions between switching molecules and the matrix, which become stronger as the density increases, play a role in the stabilization of one or both of the conductance states.7,23 A mean-field approximation (MFA) was first developed and applied to materials that are capable of forming nematic phases by Cotter and Wacker24 and Gelbart.25 Their method was later extended and used in numerous applications, including polymeric thin films,26-28 surface-protein interaction studies,29 and mixed monolayers.30 In a MFA, pair interactions are replaced by their average value, thereby reducing an n-body problem to a single-body one.31 The resulting potential is then determined largely by the orientation and nature of the molecules, and the overall order of the SAM.24 Appropriately applied and for relatively small molecules like N-(2-mercaptoethyl)-4-phenylazobenzamide (AB in the following pages) and dodecanethiol (C12), a MFA can yield good qualitative agreement with experiment, and, more importantly, is capable of providing significant physical insight. In this paper, an investigation of the density and fielddependence of the ratio between cis and trans states in a mixed monolayer of AB and C12, on a Au(111) surface is described. In an experimental study by Yasuda et al.,17 this azobenzene derivative was observed to undergo conductance switching in the low-density defects of the alkylthiol monolayer. The compound exhibits an interesting response to applied electric fields, as its permanent dipole is located along the amide CdO axis, while polarizability response occurs along the extended conjugated azobenzene moiety. In the present work, the relative stability of the cis and trans states of AB embedded in the C12 monolayer was determined by minimizing the free energy of the system using a series of approximations. These allow for the derivation of analytic equations for the derivatives of the free energy as a function of the system parameters. Minimization is then performed numerically to extract the system composition under a set of physical conditions. One advantage over other statistical mechanical methods is the ease of interpretation of the results, which are based in a straightforward way on the mathematical formulation of the free energy. Another is that the method provides and minimizes the Helmholtz free energy directly. This removes the need for more complex methodologies such as thermodynamic integration in order to determine the free energy and analyze the system’s stability. The remainder of the paper is organized as follows. In section II, the derivation of the Helmholtz free energy from the configurational partition function is presented, and the main assumptions of the theory are described. Details of the specific formulation of the free energy for the system, and the means by which the attractive and repulsive parameters are calculated are also discussed. Numerical results, and an analysis of the impact of the electric field strength, density, and other parameters on the ratio between the two states in the monolayer are given in section III. A brief summary of the main conclusions is presented in section IV.
Chapman and Paci II. Theory A. Derivation of Helmholtz Free Energy. Equilibrium statistics in condensed matter systems can be obtained by locating minima on the free energy surface. This surface is usually very complex. In statistical thermodynamics, the free energy of a system of N particles in a volume V can be calculated directly from the configurational partition function, QN, using the expression
βf ) -
F ln QN N
(1)
where f is the free energy per unit volume, F ) N/V is the number density, and β ) 1/kBT. The partition function is given by
QN )
∫ dΩN drN e-βU(r ,Ω ) N
N
(2)
where Ω and r denote the molecular orientations and positions, respectively, and U is the potential energy of the system. The Helmholtz free energy for a multicomponent system with restricted molecular conformations has previously been derived in the MFA for phase separations in SAMs,30 and solutions capable of forming nematic phases.24,32,33 Only the main derivation points relevant specifically to our system will be discussed here. Consider a k-component monolayer. Molecules are adsorbed on the substrate, with a distribution of molecular orientations. An external electric field is applied during the STM experiments. Here we approximate it to be uniform. The system is physically defined by its potential energy, as is the case in most classical statistical approaches. In our case, the potential energy is expressed as:
U ) Vattr + V* + Vads + Vfield
(3)
where Vattr is the sum of all the attractive intermolecular interactions and V* is the sum of intermolecular repulsions. Vads is the adsorption energy. Vfield is the coupling energy between molecular dipoles and the external field of the STM tip. The 6N-dimensional integral in eq 2 is often evaluated numerically, in approaches such as molecular dynamics or Monte Carlo simulations. It can also be evaluated analytically by making a series of approximations that allow for a separation of variables. We follow others24,30 in making the following approximations: (1) Orientation-dependent integrals were calculated by considering a number n of discrete orientations available to each molecule. This transformed the integrals over Ω into nested sums over discrete orientations. Molecules in the system had a specific distribution of orientations that formed the largest term in the partition function summation. In this largest term distribution, there is a number Nν, σ of molecules of type ν which have the orientation σ. This approximation to the largest term in the sum reduced the configurational partition function to a 3N-dimensional integral over positions. (2) Two-center integrals stemming from pair intermolecular attractions were evaluated by using the MFA. In it, the attractive potential Vattr(r, N1, ... Nn) was replaced by the average potential,
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V˜attr(N1, ... Nn), which is independent of molecular positions. This led to
k
βf ) F ln F + Fβ k
attr
V
F2β 2 ν)1 k
n
k
(
n
∑∑∑ ∑
1 ) 2V ν)1
σ,σ′ Nν,σNν′,σ′Aν,ν′
ν)1 σ)1 k n
n
(4)
σ)1 ν′)1 σ′)1
where V is the total area and the attraction parameter Aν,σ, ν′σ′ is the average potential of molecule ν with orientation σ felt by molecule ν′ with orientation σ′. By invoking the MFA, the attractive term is no longer a function of the intermolecular distance, and can thus be brought outside of the integral in eq 2. (3) Integrals involving the repulsive potential were also evaluated in a mean-field fashion, by replacing them with excluded area terms. These terms depend on molecule types and their orientation but not on the identity and position of the molecules. The excluded area is based on a hard core potential24,34
{
∞ if molecules overlap 0 otherwise
(6)
The repulsive interaction integral then becomes
∫ drNe-βV*(r) )
(
V - Vex V
)
N
(7)
k
n
k
∑ ∑ xν,σ ln xν,σ +
ν)1 σ)1
k
)
-
∑ ∑ xν,σ(µzν,σ + Rzzν,σEz)
(11)
Fβ(x1h1 + x2h2 + x3h3) + F2βA - F ln(1 - FB) FβEz(x1µz,1 + x2µz,2 + x3µz,3) -
(8)
Here, bν,σ, ν′σ′ is equal to the area excluded to molecules of type ν and orientation σ by the presence of molecules of type ν′ and orientation σ′. (4) The adsorption and field-dipole coupling energies are independent of position n
∑ ∑ Nν,σhν,σ
n
σ)1 ν′)1 σ′)1
where xν, σ is the mole fraction of molecules of type ν and orientation σ. B. Application to the cis-trans Isomerization of Azobenzene Embedded in Alkylthiol Monolayers. Our system was comprised of C12 and cis and trans azobenzene (denoted CAB and TAB, respectively) adsorbed on a Au(111) substrate. TAB and CAB interconvert, but the total proportion of azobenzene in the monolayer is constant. To simplify the free energy expression, each component was assumed to have a characteristic tilt angle. In other words, each molecule type adopts a uniform orientation. This assumption is consistent with the wellordered nature of alkanethiol35-37 and conjugated38,39 singlelayer SAMs, and leads mathematically to the removal of all sums over orientations in the equations of section II.A. Molecule orientation is assumed to be independent of density. This is not usually the case in experimental investigations, but has a minor impact on our qualitative conclusions. This is confirmed by our investigation of tilt angles, considered parametrically, presented in section III.C. This gives a Helmholtz free energy for the system of
FβEz2(x1Rzz,1 + x2Rzz,2 + x3Rzz,3) + Fβx3∆Hiso
σ)1 ν′)1 σ′)1
Vads )
n
σ,σ′ ∑ ∑ ∑ ∑ xν,σxν′,σ′bν,ν′
ν)1 σ)1
n
σ,σ′ ∑ ∑ ∑ ∑ Nν,σNν′,σ′bν,ν′
k
F 2 ν)1
n
βf ) F ln F + F(x1 ln x1 + x2 ln x2 + x3 ln x3) +
where the excluded area Vex is given by
1 2N ν)1
σ)1 ν′)1 σ′)1 k n k
F ln 1 FβEz
Vex )
∑∑
k
xν,σhν,σ + F
σ,σ′ ∑ ∑ ∑ ∑ xν,σxν′,σ′Aν,ν′
) V˜attr
V*(r) )
n
(9)
(12)
where xi, i ) 1-3 are the mole fractions of the components, the indices 1, 2, and 3 referring to C12, TAB and CAB, respectively. A and B are attraction and excluded volume terms, respectively, and are defined as
1 A ) (x21A11 + x22A22 + x23A33 + 2x1x2A12 + 2x1x3A13 + 2 2x2x3A23) (13)
ν)1 σ)1
and
(10)
1 B ) (x21b11 + x22b22 + x23b33 + 2x1x2b12 + 2x1x3b13 + 2 2x2x3b23) (14)
where the external field is considered aligned along the z axis, hν, σ is the molecular adsorption energy, Ez is the electric field strength along z, µzν, σ is the z-component permanent dipole ν, σ is the z, z-component of the polarizability moment, and Rzz tensor. With these approximations and some mathematical manipulation, the free energy becomes
In eqs 13 and 14, the cross terms are symmetric with respect to molecular exchange which leads to the factors of two inside the brackets. ∆Hiso is the free energy of the trans-to-cis isomerization reaction. The possibility of desorption is not included, which means that the mole fraction of C12 is a constant. Moreover, because the total fraction of azobenzene is unchanged, the mole fraction
k
Vfield ) -
n
∑ ∑ Nν,σEz(µzν,σ + Rzzν,σEz)
ν)1 σ)1
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Chapman and Paci
of CAB can be expressed as a function of the mole fraction of TAB. Mathematically, this means that
x1 + x2 + x3 ) 1
(15)
and
x2 + x3 ) xAB
(16)
where xAB is the total azobenzene mole fraction. Thus, the free energy can be expressed as a function of just one variable. Two effects are usually examined in discussions of azobenzene switching in STM fields: the coupling of the molecular dipole moment with the electric field, and the existence of the necessary free volume to accommodate the larger surface footprint of CAB. Where sufficient free volume exists, switching is thought to occur because of the destabilization of TAB. This destabilization is thought to be caused by coupling between the TAB permanent dipole moment and negative sample biases produced by the STM field.17 However, our investigations40 into the energetics of zero-density-limit switching showed that polarizability makes a very significant contribution to the molecular response to the field. For this reason, polarizability was included in eq 12 and, as discussed in section III.A, was found to have a profound impact on the overall monolayer response. Stationary states of eq 12 were found by minimizing the free energy with respect to x2, using the expression
( )
β
()
x2 ∂f ) 0 ) F ln + Fβ(h2 - h3) + F2βA' + ∂x2 x3 F2B' - FβEz(µz,2 - µz,3) - FβEz2(Rzz,2 - Rzz,3) 1 - FB Fβ∆Hiso (17)
where
A' ) 2x2A22 - x3A33 + x1A12 - x1A13 + (x3 - x2)A23 (18) and
B' ) -x2b22 - x3b33 + x1b12 - x1b13 + (x3 - x2)b23 (19) The minimization was performed numerically using the NAG subroutine E04BBF.41 It is based on cubic interpolation, and is appropriate for unconstrained searches for a minimum within the interval (0,1) of the minimization variable. Analytic first derivatives in addition to the free energy expression itself are used. The subroutine requires the function being minimized to be continuous. There are no discontinuities in the free energy expression in the full range of TAB mole fractions. Calculations were performed at the SHARCNET high performance computing consortium. C. Calculation of Parameters. The attractive interactions (A) terms and the excluded volume (b) terms were evaluated numerically. The attractive terms included contributions from both van der Waals and permanent dipole-permanent dipole potentials, whereas the excluded volume terms were determined
TABLE 1: Attractive Free Energy Contributions and Excluded Volume Terms for Pairs of C12, TAB and CAB Molecules, Calculated for AB Tilt Angles of 10°, 20°, and 30° a Aν, ν′ (kJ nm2 mol-1)
bν, ν′ (nm2)
ν, ν′
10°
20°
30°
10°
20°
30°
1, 1 1, 2 1, 3 2, 2 2, 3 3, 3
-26.82 -15.34 -14.17 -18.63 -15.32 -15.56
-26.82 -17.02 -15.80 -19.49 -16.24 -15.89
-26.82 -19.98 -17.73 -22.73 -18.27 -16.12
0.090 0.098 0.213 0.095 0.131 0.167
0.090 0.098 0.178 0.100 0.137 0.175
0.090 0.099 0.140 0.108 0.149 0.189
a In the first column, ν and ν′ refer to the molecule type, identified as 1 for C12, 2 for TAB, and 3 for CAB.
Figure 1. Sketch of the mixed C12/AB monolayer.
solely from geometrical arguments. Their values are presented in Table 1 for three different values of AB tilt angles. In all calculations, it was assumed that the molecules are tilted in the same direction, as shown in Figure 1. The common tilt direction of SAMs while in well-ordered phases supports this assumption.36,37,42,43 The tilt angle of C12 was taken from an X-ray diffraction study of SAMs composed of dodecanethiols.44 There is no specific information in the literature about the tilt angle of AB molecules with short flexible headgroups when embedded in an alkylthiol monolayer. In simulations by Alkis et al.,45 azobenzene inclusions with a single methylene thiol linking group were found to follow roughly the molecular orientation of the host alkylthiol monolayer, with an average monolayer tilt, defined from the surface normal, of roughly 30°. Similar angles were reported for pure monolayers of azobenzene bound to the surface through long bulky flexible headgroups.9 Other more rigid, sp2-bound guest molecules bind perpendicular to the surface plane.46 For biphenyl-based pure monolayers with alkylthiol headgroups, the dependence of the tilt angle on the length of the alkyl spacer exhibited an odd-even effect:47 on Au, monolayers with an odd number of methylene groups had a roughly 20° tilt angle, while molecules with even length spacers were tilted at 30°. Most of the calculations presented in this paper were performed using a tilt angle of 30° for the AB moiety. We briefly investigate the effect of the AB tilt angle on the main conclusions of our study in section III.C. As averages of the attractive interaction potential, A terms were calculated by integrating the pair intermolecular potential over the entire surface where the potential is attractive, i.e.
Aνν′ )
∫-∞∞ ∫-∞∞ uνν′(x, y) dx dy, for all values uνν′(x, y) < 0 (20)
The interaction potential uνν′ is
Bistability in Mixed Azobenzene-Alkylthiol Monolayers
uνν′ ) uvdw + udd m
n
)
∑ ∑ 4εij i)1 j)i
1 4πε0
(
[( ) ( ) ]
µν · µν′ 3 rνν′
σij rij
-
12
-
σij rij
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+
3(µν · rνν′)(µν′ · rνν′) 5 rνν′
)
(21)
Rzz ν wν (Å) lν (Å) φν (°) µx (D) µy (D) µz (D) (10-21 C nm2 /V)
where ε0 is the vacuum permittivity constant, summation in the Lennard-Jones term is done over the atoms in the molecules ν and ν′, εij and σij are the regular Lennard-Jones parameters,48 andrij isthedistancebetweenatoms.WeusedtheLorentz-Berthelot combination rules49,50 to estimate the mixed Lennard-Jones parameters: εij ) (εiiεjj)1/2 and σij ) (σii + σjj)/2. In the dipole-dipole interaction terms of eq 21, the intermolecular separation vector rνν′ was estimated using the displacement between the amide nitrogen atoms on the azobenzene molecules, because the amide unit is primarily responsible for the magnitude of the permanent dipole moment. Polarizability contributions were neglected in the evaluation of the pair potential, for the sake of keeping the attractive interaction terms A independent of the applied electric field. This approximation is equivalent to the neglect of local field effects. There are two considerations that make this a valid approximation. First, the dipole-dipole potential contribution to the A terms was much less than (
