Physics Letters A 374 (2010) 2681–2687

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Physics Letters A

www.elsevier.com/locate/pla

Theoretical investigation of optical patterning of monolayers with subwavelengthresolution

Triet Nguyen, Michael Mansell, Alex Small ∗

Physics Department, California State Polytechnic University, Pomona, CA 91768, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 December 2009Received in revised form 31 March 2010Accepted 15 April 2010Available online 21 April 2010Communicated by R. Wu

Keywords:PhotolithographySuperresolutionMonolayers

We formulate a model of monolayer patterning via optically-controlled chemical reactions, with the goalof beating the diffraction limit in photolithography. We consider the use of the proven technique ofSTimulated Emission Depletion (STED) to selectively place a handful of molecules in a reactive excitedstate. We show that repeated optical excitation has a greater effect on pattern formation than increasingthe reaction rate, auguring well for experimental work. We also consider optically-controlled depositionof a soluble species via STED, and show that even for very large concentrations and excited state lifetimesthe full width at half maximum of the features formed is robust against the effects of diffusion andsaturation.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Due to diffraction, spatial resolution in photolithography haslong been limited to approximately half the wavelength of the lightused to initiate the chemical reaction in the photoresist. A varietyof techniques have been proposed to overcome this limit and op-tically write patterns onto surfaces with subwavelength resolution,including schemes based on multi-photon absorption [1], coher-ent control of transitions within the photoresist molecules (viacoherent population trapping) [2], and negative index materials [3].However, achieving very high resolution in a multi-photon absorp-tion scheme generally requires a reaction that only occurs afterthe simultaneous absorption of a large number of photons, mak-ing it difficult to realize resolutions significantly better than λ/4.Formation of patterns with subwavelength resolution via coher-ent control of transitions has only been realized in atomic vapors[4], which will not be suitable for most surface-patterning appli-cations. Negative index materials hold the promise for potentiallyunlimited resolution, but are still in their early stages of develop-ment.

Here we propose to take advantage of STimulated Emission De-pletion (STED), a technique that has been demonstrated to achieveλ/20 or better resolution in fluorescence microscopy [5,6]. Thebasic idea of STED is to first focus a pulse of light to a conven-tional diffraction-limited spot of size ≈ λ/2, and use it to raisethe molecules to an excited state. That pulse of light is then im-

* Corresponding author.E-mail address: arsmall@csupomona.edu (A. Small).URL: http://sites.google.com/site/physicistatlarge (A. Small).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.04.038

mediately followed by a second pulse with a “doughnut” profile(i.e. a superposition of TEM10 and TEM01 modes) that has a nodein the radial center of the intensity profile. The second pulse istuned to the frequency of the downward transition, causing all ofthe molecules except those at the very center (where the intensityis zero) to undergo stimulated emission and return to the groundstate. The result is that excitation is confined to a region with a ra-dius at least an order of magnitude smaller than λ. When used forimaging, STED enables sub-wavelength resolution due to the factthat any photons emitted by the molecules after the second pulsemust be from molecules near the node of the depletion pulse.

What we will study here is the feasibility of using the STEDconcept to control photochemistry, by taking advantage of the factthat excited states are often reactive. The concept was first pro-posed in 2004 by Hell [7], assuming a reactive ground state. Wewill assume a reactive excited state, since such a reaction is eas-ier to turn “off” and thereby control. We also consider the effectsof diffusion in this work. Recently, several groups have used theSTED concept to achieve subwavelength resolution in photolithog-raphy, either by controlling photoinitiators [8,9] or by renderinga nanoscale region of a photochromic material transparent, andusing it as a mask for lithography [10]. These promising resultsprompt us to ask if we can approach single-molecule resolutionin surface patterning via the STED concept, and produce patternedmonolayers. One promising finding from our work is that (unlikein the photoinitiator approach) even a very weakly reactive speciescan be used to achieve subwavelength patterning of surfaces, if theexposure conditions are properly controlled.

We will consider two models of optically-controlled mono-layer formation. In the first (Fig. 1), the photoreactive species issurface-bound, and excitation either causes the excited surface-

2682 T. Nguyen et al. / Physics Letters A 374 (2010) 2681–2687

Fig. 1. Outline of the process. (a) A focused Gaussian beam of size ≈ λ raisesmolecules to the excited state. (b) A subsequent depletion beam with a node atthe center of its intensity profiles sends most of the molecules back to the groundstate, except those near the node. (c) Molecules remaining in the excited state eitherdetach from the surface or bind a species in solution, modifying the local surfaceproperties. This continues until eventual relaxation to the ground state.

bound molecules to dissociate (with a rate constant k) or attachto a species in solution. An example of the dissociation scenario isrhodopsin, which undergoes a dissociation reaction in its excitedstate [11,12]. An example of the attachment scenario is derivativesof azobenzene, which undergoes a photocontrollable cis–trans iso-merization and can be functionalized to attach to soluble speciesin one isomer or the other [13,14]. This model is also formallyequivalent to one in which surface-bound excited species bind toa species present in solution, if the soluble species are present inexcess (i.e. initial binding events do not reduce the local concentra-tion enough to significantly reduce the rate of subsequent bindingevents).

However, this model is only equivalent to a system of pho-toreactive molecules in solution that can attach to a surface ifduring the excited state lifetime τ they diffuse a distance muchless than λ. Since the diffusion coefficient for typical small organicsin liquids at room temperature is of order 10−9 m2/s [15], if STEDinitially confines the excited species to a region of size 10 nanome-ters (10−8 m) then the excited state lifetime would have to be nolarger than 10−7 seconds for this model to be applicable. The sec-ond model then explores the effect of a long-lived excited state ina diffusible and photoreactive species. In this model, we assumethat a population of excited molecules is created in solution witha Gaussian profile, and model the attachment of excited moleculesto the surface. We will show that getting a significant number ofmolecules to attach to a surface in a single activation cycle is achallenge for realistic reaction rates and concentrations, but thatthe width of the surface concentration profile is robust against theeffects diffusion and saturation.

2. Photoactivation of surface-bound species

2.1. Model

In modeling surface-bound photoactivated molecules, we willassume (for simplicity) that surface-bound molecules are initiallypresent on a surface with some concentration ci(r) (which willtypically start out uniform). For convenience, we will write theconcentration as c(r) ≡ csatφ(r) where csat is the density of sitesavailable on the surface and φ is the fraction of sites occupied. Wewill calculate the fraction of sites occupied by molecules in theground and excited states, denoted φg and φe . We assume that themolecules are first raised to the excited state by a pulsed excitationbeam tuned to the absorption maximum of the ground state, and

Fig. 2. Energy levels and transition rates for our photoactivated species. We as-sume that the rate of spontaneous radiative transitions from |3〉 to |1〉 is negligiblecompared with driven upward and downward transitions (with rates Ieσe ) and vi-brational transitions from |3〉 to |2〉 (with rate τ−1

v ).

we approximate its intensity profile at the surface as a Gaussian ofthe form:

Ie(r) = I1e−k21r2/2 (1)

where r is the radial distance from the origin, I1 is the intensity(units of photons per area per time) at the center of the beam,and k1 is the wavenumber (in the surrounding medium) of theexcitation beam.

We will assume that our molecules have 3 energy levels, asshown in Fig. 2. The reactive state is assumed to be |2〉. In reality,the two excited states shown will be part of a larger set of singletexcited states coupled by vibrational transitions, but the scheme inFig. 2 suffices for our purposes. We assume optical excitation fromthe ground state |1〉 to the highest state |3〉, and that the rate ofspontaneous radiative transitions from |3〉 to |1〉 is negligible com-pared to the rate τ−1

v for vibrational transitions from |3〉 to |2〉. Fororganic fluorophores, τv is often of order 10−11 to 10−13 seconds,while the radiative lifetime of the excited state is typically of ordernanoseconds or longer [16].

In the steady state, the kinetic equations describing the popu-lations of these states are:

d

dtN1 = σe Ie(N3 − N1) + N2/τ = 0 (2)

d

dtN2 = N3/τv − N2/τ = 0 (3)

d

dtN3 = σe Ie(N1 − N3) − N3/τv = 0 (4)

N1 + N2 + N3 = 1 (5)

where N1, N2, and N3 are the fraction of molecules in the 3 energylevels.

We assume that the pulse duration is shorter than the timescale for reactions, so the fraction of sites occupied by excitedmolecules (in state |2〉) immediately after the excitation pulse canbe found from solutions to the kinetic equations (3)–(5):

φe(r) = φg,iσeτ Ie(r)

1 + σe Ie(r)(τ + 2τv)

≡ φg,iI ′e(r)

1 + I ′e(r)(1 + 2τv/τ )(6)

where φg,i is the fraction of sites occupied by molecules in thegrounds state before excitation, σe is the absorption cross sectionof the ground state at the frequency of the excitation beam, andτ is the radiative lifetime of the excited state. We are assumingthat the time intervals between excitation events are long com-pared to the excited state lifetime. The last term in the denomina-tor will hereafter be neglected, because the lifetime of vibrationaltransitions is generally at least an order of magnitude smaller thanthe lifetime of radiative transitions.

T. Nguyen et al. / Physics Letters A 374 (2010) 2681–2687 2683

The subsequent depletion pulse that returns molecules to theground state via stimulated emission is approximated with a spa-tial profile of the form:

Id(r) = I2k22r2e−k2

2r2/2 (7)

where I2 also has units of photons per area per time and k2 isagain the wavenumber (in the surrounding medium) of the deple-tion pulse. If the pulse duration �t (as well as the time elapsedbetween the end of the excitation pulse and the beginning of thesecond pulse) is significantly shorter than the lifetime of the ex-cited state, then the concentration of molecules in the excited stateafter the STED beam is:

φe(r) = φg,iI ′e(r)

1 + I ′e(r)e−σs Id(r)�t

≡ φg,iI ′e(r)

1 + I ′e(r)e−I ′d (8)

where σs is the stimulated emission cross-section of the excitedstate, I ′e = σeτ Ie , and I ′d = σs Id�t .

For large I2, after the sequence of pulses the excitation is con-fined to a region close to the origin, where Id ≈ 0 because of itsr2 dependence. If I2 is small then φe(r) is a bell-shaped curvewith the usual width for STED microscopy, ≈ λ/

√1 + I2στ (as-

suming that the wavelengths of the two pulses are similar, as isusually the case). If, however, I ′e ∼ 0.1 or greater (i.e. the rate ofexcitation begins to approach the same order of magnitude as therate of spontaneous emission, a necessary condition for bringing alarge fraction of the surface-bound molecules into the reactive ex-cited state in each cycle) then I ′e/(1 + I ′e) begins to saturate andthe resolution changes. We can work out the relationship betweenresolution and excitation intensity by expanding Eq. (8) near theorigin to second order in r. The result is:

φe(r)/φg,i ≈ I ′11 + I ′1

(1 − r2

2

k21 + 2I ′2k2

2 + 2I ′1 I ′2k22

1 + I ′1

)(9)

where I ′1 = I1σeτ and I ′2 = I2σs�t . The first two terms in the nu-merator of the coefficient of r2 are familiar from traditional STEDmicroscopy. The I ′1 in the denominator and the other numeratorterm, however, occur because at high excitation intensity the up-ward transition saturates, and the concentration profile of excitedmolecules after the excitation pulse is wider than λ. When usingSTED for imaging, the excitation power may be low to avoid photo-bleaching, and these terms will not matter. However, if using STEDto pattern a surface, from a time/efficiency perspective it is usu-ally desirable to raise a very large fraction of the molecules (in thevicinity of a given site) to the excited state.

The result of all these effects gives the following scaling for thewidth �r of the concentration profile of excited molecules:

�r ∝ λ√2I ′2 + 1

1+I ′1

(10)

Note that we have assumed that the upward and downward tran-sitions are close in energy, so k1 ≈ k2 and hence the value of λ isapproximately the same for both beams. When this approximationis not valid, the equations can still be made to take this form bymaking minor rescaling adjustments to the intensity I ′2 in Eq. (9).The last term in the denominator is never larger than 1, whileI ′2 1 in most STED experiments, and so the dependence of reso-

lution on the intensity of the depletion beam is ∼ λ/

√I ′2. The last

term in the denominator would only matter in a situation whereI ′2 is small and I ′1 is large. In that case, due to the high excita-tion intensity even those molecules much more than λ away fromthe center would be raised to the excited state by the intense tail

of the excitation beam, and very few of them would return to theground state, reducing the resolution of the patterning approach.

In what follows, we use the spatial profile given in Eq. (8) asthe initial condition for φe(r, t) and φg,i − φe(r, t = 0) as the initialcondition for φg(r, t). These concentration profiles obey the follow-ing equations:

d

dtφe = −φe/τ − kdφe − kbφe (11)

d

dtφg = φe/τ (12)

where kd is the (first order) rate constant for dissociation fromthe surface, kb is the rate at which excited molecules photobleach,and φe/τ is the spontaneous emission rate. Solving these cou-pled linear equations gives the following result for the number ofmolecules in the ground state and attached to the surface at t τ(i.e. after all of the molecules still attached have returned to theground state):

φg(r)/φg,i = 1 − I ′e(r)1 + I ′e(r)

e−σs Id(r)�t (kd + kb)τ

1 + (kd + kb)τ(13)

Since bleaching and dissociation rates always enter this model ad-ditively, there is no need to vary these parameters separately tounderstand the behavior of the model. From a practical standpoint,however, it is desirable to have kd kb . For the remainder of thiswork, we will only treat the case kb = 0. If kb �= 0, the ratio ofdissociated to bleached molecules can be calculated from the ra-tio kd/kb .

2.2. Key results for photodissociation of surface-bound molecules

Eq. (13) can be used to calculate the number of molecules at-tached to the surface after illumination by excitation and depletionpulses of arbitrary beam profile, not just the profiles in Eq. (1) andEq. (7). For illumination by a series of alternating excitation anddepletion pulses, φg,i is whatever the local concentration was af-ter the previous exposure. We assumed simultaneous illuminationby several pulses of the forms in Eq. (1) and Eq. (7), centered atdifferent locations and (possibly) having different intensities. Wecalculated the local concentration after the exposures, and thenrepeated the calculations for a new set of beams. All calculationswere performed in Matlab.

For validation of the code, we also implemented a discretemodel, where arrays of molecules with λ/500 spacing were sim-ulated. A molecule’s probability of dissociating from the surface isequal to the right-hand side of Eq. (13) and is assumed to be unaf-fected by the status of its neighbors. At the end of each simulation,a site was either occupied or unoccupied. By running many simu-lations, counting the number of times that a molecule dissociatedfrom a site, and dividing by the number of simulations run, we ob-tained average concentration profiles. Many stochastic simulationswere performed and averaged, and the results were compared withcalculations from the concentration model of Eq. (13) to validatethe code. Because the stochastic simulations and the direct calcu-lations from Eq. (13) both use the same probability model, it isnot surprising that they gave the same results. We only used thestochastic simulations as a check on our code. All results reportedhere will be based on direct calculations from Eq. (13).

We will assume that the excitation beam has a wavelength of500 nanometers, and that the depletion beam has a wavelengthof 525 nanometers, representing common peaks in the absorptionand emission spectra of organic dyes in the visible region.

Our key finding comes when we compare the effects of in-creasing the number of exposures and increasing the rate constant.Eq. (13) is messy to manipulate analytically, but it is clear that

2684 T. Nguyen et al. / Physics Letters A 374 (2010) 2681–2687

Fig. 3. Results of increasing the reaction rate or the number of exposures. All threeconcentration profiles are normalized so that a concentration of unity correspondsto a saturated surface with no molecules removed.

raising it to a positive integer exponent (due to multiple exposureswith the same beam profile) can give an arbitrarily small valueof φg , while increasing kd arbitrarily cannot make φg any smaller

than 1 − I ′e1+I ′e

exp(−σs Id(r)�t). Moreover, the incremental reduc-

tion in concentration due to multiplying kd by a scaling factor s islinear in s, but the effect of going from 1 exposure to s exposuresis exponential in s. Note that we assume that all exposures aredone by a lens focused on the same site for the duration of severalconsecutive pulses, without moving. Jittering can, of course, be anissue in scanning microscopes, and thus limit the resolution, butthe situation we envision at least minimizes these issues some-what because there is no motion of the lens between exposures.

To get a sense of typical values, in Fig. 3 we consider the caseI ′1 = 1 (strong excitation) and I ′2 = 8, for different numbers of ex-posures and values of kdτ . In all three cases shown, the concentra-tion profile has a dip with a full width at half maximum (FWHM)of ≈ 50 nm (λ/10). If we first work with a weakly photoreactivespecies (kdτ = 0.1, i.e. reaction rate much slower than the decayof the excited state) only about 5% of the molecules are removedin a single cycle. Increasing kdτ to 1 (factor of 10 increase) re-sults in the removal of 25% of the surface-bound molecules in asingle step. On the other hand, if kdτ = 0.1 but the exposure pro-cess is repeated 10 times, 38% of the molecules are removed. Evenlarger disparities in favor of repeated exposure are found as thenumber of exposure cycles is increased. Because multiple expo-sures are likely to be almost trivial to implement, while the designof a more photoreactive species is likely to be a highly non-trivialtask in synthetic chemistry, this fractionation effect is encourag-ing.

Additionally, we considered the effects of exposing two nearbyregions simultaneously with beams centered at two differentpoints (Fig. 4). One might consider a scheme in which parallelismis achieved by sending light through a diffraction grating to pro-duce multiple spots, which are then focused simultaneously. Asin Fig. 3, we used the parameters I ′1 = 1, I ′2 = 8, and kdτ = 1.For beam separations of d = 500 nm (i.e. the excitation wave-length) the concentration profile has two distinct dips (top ofFig. 4), with 25% of the molecules removed at each spot. As thebeam separation d decreases, the concentration dip (i.e. amountof molecules removed) does not change significantly, until ap-proximately d = 0.6λ, where 21% of the molecules are removed(middle of Fig. 4). Interestingly, however, the region in which sig-nificant numbers of molecules are removed is slightly narrower.As the separation continues to decrease, when d = λ/2 (bottomof Fig. 4) the concentration dip is only half of its original value

Fig. 4. Results for performing simultaneous exposures at multiple sites with vari-able separations. Down to separations of ≈ 0.6λ the simultaneous exposures can beperformed without any significant degradation of quality.

(12% of the molecules removed instead of 25%). Taken together,these results show that while the use of STED can enable opti-cal monolayer patterning with feature sizes much smaller than thewavelength of light, the smallest possible distance between twofeatures being printed simultaneously is approximately 0.6λ, com-parable to the conventional diffraction limit. Thus, diffraction doesnot limit feature size, but it does limit the degree of parallelismpossible in this scheme, by limiting the achievable distance be-tween spots.

3. Diffusion of photoactivated species

3.1. Model and parameters

The model described above is applicable to photoactivatedmolecules attaching to a surface from solution as long as the life-time is sufficiently short that the molecules do not have enoughtime to diffuse outside of the initial activation region before re-turning to the ground state. Here we explore the effects of diffu-sion when the lifetime is long, to estimate how robust resolutionis against changes in wavelength, initial concentration in solution,or surface reactivity.

We will assume that excited (and hence reactive) moleculesare initially present in solution at a concentration Cs(r). The ef-fect of the excitation and depletion beams is to confine the excitedmolecules to a region with a Gaussian profile centered at the co-ordinates (r, z) = (0,0) (see Fig. 5). We approximate the initialconcentration profile Cs (the subscript s stands for “solution”) ofexcited molecules with:

Cs(r, t = 0) = C0e−(r2+z2/4)/R2(14)

where R is the (lateral) width (1/e2 measure) of the Gaussian andthe 1/4 coefficient on z2 reflects the fact that focused laser beamstend to be better localized along the lateral direction than the axialdirection.

We will denote the concentration of molecules attached to thesurface as c ≡ csatφ as above. We will assume that initially thesurface has no molecules attached, so φ(r) = 0 at t = 0. In solu-tion, the time dependence of the concentration profile of excited

T. Nguyen et al. / Physics Letters A 374 (2010) 2681–2687 2685

Fig. 5. Initial condition for binding of a diffusible, photoreactive molecule to a sur-face. The initial concentration profile is a Gaussian centered at (r = 0, z = 0). Thereactive surface is at z = 0 and the boundaries of the window are at 10× the Gaus-sian’s 1/e2 width R . The system is axisymmetric about the z axis.

molecules is governed by diffusion and decay from the groundstate, giving the following equation:

∂

∂tCs = D∇2Cs − Cs/τ (15)

At the surface, the concentration of attached molecules is governedby second order kinetics, with a maximum surface concentration,giving the following time dependence:

∂

∂tφ = konCs(1 − φ) − koff φ (16)

where 1 − φ is the fraction of sites unoccupied.These equations for the time evolution of φ and Cs must be

supplemented by boundary conditions. We assume that this reac-tion happens in a large cylindrical container, and we impose no-flux conditions at the radial edge and at the top surface (r = 10Rand z = 10R in our simulations). At the reaction surface (z = 0)we impose the condition that the flux of molecules from solutiononto the surface (given by Fick’s Law) be equal to the rate at whichmolecules accumulate on the surface:

D∂

∂zCs = ∂

∂tcsatφ = koncsatCs(1 − φ) − csatkoff φ (17)

We will assume that the molecules are tightly bound to the sur-face (i.e. binding energy kB T ) so that koff ≈ 0 and the off ratewill be ignored hereafter. To non-dimensionalize the equations,we will measure distances in units of the Gaussian concentrationprofile’s 1/e2 width R (50 nanometers or smaller is likely to betypical) and times in units of the diffusion time scale R2/D . Fortypical D values of 10−9 m2/s or smaller [15], our diffusion timescale will be of order 10−7 seconds or longer. Concentrations willbe measured in units of csat/R where csat (not to be confused withcapital C0) is the density of binding sites on the surface and willtypically be of order 1 per square nanometer, giving a value ofcsat/R ≈ 1026 m−3 or 3 × 10−2 moles per liter.

For convenience we define rescaled distances (r′, z′) ≡ (r/R,

z/R), rescaled time variables t′ ≡ t D/R2 and τ ′ = τ D/R2, and so-lution concentrations C ′

s ≡ Cs R/csat and C ′0 ≡ C0 R/csat . Since we

have defined surface concentrations as csatφ we already have adimensionless variable to work with for surface concentration. To-gether, we get the following equations for the time dependencesof φ and C ′

s:

Table 1Summary of dimensionless quantities.

Name Definition Meaning

(r′, z′) (r/R, z/R) Radial and axial distancest′ Dt/R2 Timeτ ′ Dτ/R2 Lifetime of excited stateφ Fraction of surface sites occupiedC ′

s Cs R/csat Conc. of excited molecules in solution

C ′0 C0 R/csat Initial maximum concentration of excited molecules

k′ (csat R/D)kon Rate constant for binding to surface

∂

∂t′ φ = R2

D

csat

RkonC ′(1 − φ) ≡ k′C ′

s(1 − φ) (18)

∂

∂t′ C ′s = (∇′)2

C ′s − C ′

s/τ′ (19)

where k′ ≡ RcsatD kon . The boundary condition on C ′

s at z′ = 0 isgiven by:

∂

∂z′ C ′s = k′C ′

s(1 − φ) (20)

Our dimensionless quantities are summarized in Table 1.

3.2. Results from model

As a first step, our model is easy to solve in the limit whereτ ′ 1, the excited state decays before molecules can diffuse a dis-tance comparable to the 1/e2 width R of the initial concentrationprofile, and the time dependence of the concentration in solutionis just an exponential decay. If the total amount of molecules de-posited is also small, then the (1−φ) term in Eq. (18) and Eq. (19)does not change significantly during the lifetime of the excitedstate, and so the final surface concentration is:

φ(r) =∞∫

0

C ′s(t = 0)e−t′ dt′ = C ′

s(t = 0)k′τ ′ (21)

In this case, the concentration profile on the surface is directlyproportional to the local concentration profile in solution. The 1/e2

width of the surface profile is hence equal to the width of the ini-tial concentration profile in solution, and so nanoscale resolutionon the surface is easy to obtain if the initial concentration pro-file in solution has a narrow width due to the use of stimulatedemission depletion. However, if either condition is violated, so thatτ ′ � 1 and/or φ 0.1–1 after the excited state has decayed, then thewidth of the surface concentration profile may be greater than thewidth in solution, reducing the resolution. In the case of long-livedspecies, the reduction in resolution comes from diffusion of thesoluble species, increasing the width of the concentration profilein solution over time. In the case of large surface concentrations,saturation effects will cause the deposition rate to drop near thecenter of the profile while remaining steady near the edges, broad-ening the profile.

To determine the magnitude of these effects, we conductedsimulations of a “worst case scenario”: We first determined thenecessary volume concentration (i.e. value of C ′

0) so that the peakconcentration on the surface is φ = 0.5, for a range of differentlifetimes τ ′ and reaction rates k′ . We then determine the surfaceconcentration profile’s full width at half maximum (FWHM) underthose conditions. Simulations were performed in COMSOL Multi-physics, using the Chemical Engineering module in axisymmetricmode. The boundaries of the computational window were given by0 � r′, z′ � 10 (see Fig. 5). The mesh comprised 18,021 elements,with a higher density near the z′ = 0 surface where the reactionoccurs.

2686 T. Nguyen et al. / Physics Letters A 374 (2010) 2681–2687

Fig. 6. Concentration of reagent in solution required to achieve a peak surface cov-erage of φ = 50% for different excited state lifetimes and reaction rates.

As shown in Fig. 6, the concentration required to achieve amaximum of 50% surface coverage decreases (not surprisingly) asthe reaction rate increases. For long lifetimes (relative to the char-acteristic diffusion timescale R2/D) the required concentration islargely independent of the excited state lifetime, as after a timeof order R2/D most of the excited molecules have diffused awayfrom the surface. The most important point is that if the reactionrate constant k′ is small (< 1), so that the characteristic reactiontime is longer than the characteristic diffusion time, highly con-centrated solutions C ′

0 > 102 are required. Since a value of C ′0 = 1

corresponds to approximately 3 × 10−2 M, values of C ′0 larger than

100 are unrealistic in most cases. This does not rule out the fea-sibility of working with weakly reactive species, as the logic ofusing multiple cycles (as above) still applies. Interestingly, due tothe effect of surface binding on the concentration profile in solu-tion and the non-linearity of Eq. (18) and Eq. (19), the requiredconcentration is not directly proportional to the inverse of the rateconstant k′ , although it does decrease as k′ increases.

With those required concentrations established for different ex-cited state lifetimes and reaction rate constants, we next examinethe FWHM of the surface concentration profiles. (We used FWHMrather than 1/e2 with simulation data because it is easy to in-fer from a visual examination of a graph.) As shown in Fig. 7,the FWHM of the surface concentration profile (normalized to R ,the 1/e2 width of the initial Gaussian profile of molecules in solu-tion) is quite robust against large changes in excited state lifetimeand reactivity. In the ideal case, where the profile on the surfacematches the initial Gaussian profile in solution, the FWHM wouldbe

√log 2R = 0.833R (from the condition e−r2/R2 = 0.5). For short

excited state lifetimes (less than the characteristic diffusion time)the FWHM is close to this value (within 25%), because most of theattachment to the surface happens before the profile of the ex-cited molecules in solution can be broadened by diffusion. Evenfor excited state lifetimes two orders of magnitude longer than thediffusion time scale, however, the width FWHM is only about twicethe ideal value. The key reason for this is that at long times mostof the excited molecules have diffused away from the surface. Wetherefore conclude that while diffusion effects may pose problemsfor attaching large numbers of molecules to the surface, they willnot significantly broaden the surface concentration profile.

Fig. 7. FWHM of surface concentration profile for different excited state lifetimesand reactivities.

4. Conclusions

In conclusion, we have formulated theoretical models of optical-ly-controlled patterning of monolayers via STimulated EmissionDepletion in two different scenarios: Excitation of a surface-boundspecies for either selective removal or attachment to a solublespecies, and excitation of a soluble species that attaches to a sur-face while in its excited state. Achieving significant attachment ofa soluble species during its excited state lifetime is likely to re-quire either very fast reaction rates or very high concentrations,due to the diffusion timescale being so short (≈ 10−7 seconds).However, diffusion effects do not significantly impair resolution,and repeated cycles of exposure and reaction can be used toaccumulate molecules on a surface with very high spatial reso-lution.

Moreover, reactions involving an excited species that is alreadybound to the surface can also lead to surface patterning with veryhigh resolution, but this concept faces no constraints due to diffu-sion. One promising way of realizing this concept is with a reactionin which an excited surface-bound species (e.g. a metastable struc-tural isomer) binds a diffusible molecule while in its excited state;due to the potentially long lifetimes of such molecules, significantattachment could be realized in a single cycle of activation andreaction. An additional benefit of working with surface-bound ex-cited species is that repeated exposure has an exponential effecton the number of unreacted molecules remaining on the surface,while increasing the rate constant has only a linear effect. Becauseincreasing the number of exposures is almost trivial (requiringmultiple laser pulses, provided that the point of focus is stablethroughout the process) and does not require the trial and error ofsynthesizing a photoreactive molecule with a larger rate constant,there is great promise for patterning surface-bound molecules viaSTED.

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