Molecular self-assembly: Quantifying the balance between intermolecular attraction and repulsion from distance and length distributions
Christoph Schiel, Maximilian Vogtland, Ralf Bechstein, Angelika Kühnle, Philipp Maass
MMolecular self-assembly:Quantifying the balance betweenintermolecular attraction and repulsion fromdistance and length distributions
Christoph Schiel, † Maximilian Vogtland, ‡ Ralf Bechstein, ‡ Angelika Kühnle, ∗ , ‡ andPhilipp Maass ∗ , † † Universität Osnabrück, Fachbereich Physik, Barbarastraße 7, D-49076 Osnabrück,Germany ‡ Fakultät für Chemie, Universität Bielefeld, Universitätsstraße 25, D-33615 Bielefeld,Germany
E-mail: [email protected]; [email protected]
Abstract
Molecular self-assembly on surfaces constitutesa powerful method for creating tailor-made sur-face structures with dedicated functionalities.Varying the intermolecular interactions allowsfor tuning the resulting molecular structures ina rational fashion. So far, however, the discus-sion of the involved intermolecular interactionsis often limited to attractive forces only. Inreal systems, the intermolecular interaction canbe composed of both, attractive and repulsiveforces. Adjusting the balance between these in-teractions provides a promising strategy for ex-tending the structural variety in molecular self-assembly on surfaces. This strategy, however,relies on a method to quantify the involved in-teractions.Here, we investigate a molecular model systemof 3-hydroxybenzoic acid molecules on calcite(10.4) in ultrahigh vacuum. This system of-fers both anisotropic short-range attraction andlong-range repulsion between the molecules, re-sulting in the self-assembly of molecular stripes.We analyze the stripe-to-stripe distance distri-bution and the stripe length distribution andcompare these distributions with analytical ex-pressions from an anisotropic Ising model with additional repulsive interaction. We show thatthis approach allows to extract quantitative in-formation about the strength of the attractiveand repulsive interactions.Our work demonstrates how the detailed analy-sis of the self-assembled structures can be usedto obtain quantitative insight into the molecule-molecule interactions.1 a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Introduction
Molecular self-assembly has attracted great at-tention due to the impressive structural andfunctional variability that can be achievedwith this versatile bottom-up method forsupramolecular material synthesis. A cleverdesign of the molecular building blocks allowscontrolling the resulting structures and tai-loring them to the specific needs of a givenapplication. The interaction of the moleculeswith the surface provides an additional way totune the molecular structure formation.
In the last decades, the subtle balance be-tween intermolecular and molecule-surface in-teractions has been explored to arrive at an im-pressive multitude of various structures, rang-ing from perfectly ordered two-dimensionalfilms over uni-directional rows to porousnetworks, and complex guest-host architec-tures. The vast majority of these studieshas focused on attractive molecule-molecule in-teractions such as hydrogen bonds, van-der-Waals forces, π − π interactions or electrostat-ics. In contrast, repulsive molecule-moleculeinteractions have only rarely been studiedfor steering the structure formation.
Inthe latter examples, the electrostatic repul-sion between permanent as well as adsorption-induced electrical dipoles has been discussed asa promising way to enhance the structural com-plexity in molecular self-assembly on surfaces.Intermolecular repulsion gives rise to the for-mation of homogeneously dispersed individualmolecules , extended rows with well-definedrow-to-row distances as well as islands andclusters with well-defined sizes.So far, however, the interplay between attrac-tive and repulsive interactions on the molecularstructure formation has barely been explored asa powerful strategy to control both the shapeand the size of self-assembled molecular struc-tures on surfaces. For systematic exploringthe balance between molecular attraction andrepulsion in molecular self-assembly, it is essen-tial to quantification the involved interactions.Here, we present a molecular model systemof adsorbed 3-hydroxybenzoic acid moleculeson a calcite (10.4) surface that provides both, anisotropic attraction and repulsion. For thissystem, the molecular self-assembly has beenshown to be governed by the balance betweenshort-range intermolecular attraction and long-range intermolecular repulsion.
This bal-ance results in the formation of molecularstripes with a coverage-dependent stripe-to-stripe distance distribution. In order to determine the strength of theinvolved attractive and repulsive interactions,we consider an anisotropic Ising model withadditional long-range dipole-dipole interaction.This model is generally applicable to stripe for-mation induced by intermolecular interaction.Based on a mean-field treatment we derive an-alytical expressions for stripe-to-stripe distanceand stripe length distributions.The theory is compared with experimen-tal data obtained by atomic force microscopyimages. An analysis of these images yieldscoverage-dependent stripe-to-stripe distancedistributions as well as stripe length distri-butions. By fitting the theoretical predictionsto the distance and length distributions we ex-tract the strength of the attractive and repul-sive molecule-molecule interactions. Our workconstitutes an example of how the mesoscopicstructural information can be used for gainingquantitative molecular-level insights into thedriving forces at play.
All dynamic atomic force microscopy (AFM)images shown in this work were acquired witha variable-temperature atomic force microscope(VT-AFM XA from ScientaOmicron, Ger-many) operating under ultrahigh vacuum con-ditions ( p < − mbar). We used silicon can-tilevers purchased from NanoWorld (Neuchâtel,Switzerland) with an eigenfrequency of around300 kHz (type PPP-NCH). To remove contam-inations and a possible oxide layer, the can-tilevers were sputtered with Ar + at 2 keV for10 minutes prior to use.The calcite crystals (Korth Kristalle GmbH,Germany) were prepared ex situ by mild ul-trasonication in acetone and isopropanol for25 min each. Inside the chamber, the crystalswere degassed at about 580 K for 2 h. After thisdegassing step, the crystals were cleaved andannealed at about 540 K for 1 h. The quality ofthe crystal surface was then checked by collect-ing an image of typically 100 nm size.The 3-hydroxybenzoic acid (3-HBA) molecules(99 % purity) were purchased from Sigma-Aldrich and used after degassing for 10 minat a temperature higher than 320 K. A home-built Knudsen cell with a glass crucible wasused for sublimation. For the crucible usedhere, a temperature of 309 K resulted in a fluxof approximately 0.01 monolayers per minute(ML/min). During sublimation, the partialpressure in the chamber was in the range of × − mbar for 3-HBA ( m/z = 137 u/e) asmeasured with a mass spectrometer from MKS(e-Vision 2). For molecule deposition, the cal-cite sample was cooled to a temperature below220 K.The AFM measurements were performed ata sample temperature of 290 K. This temper-ature is chosen such that the dynamics are fast In the AFM, the temperature is read out at the sam-ple stage using a Pt100 sensor 3 cm apart from the sam-ple. According to the manufacturer, the temperaturedifference between the sample and the sample readoutposition in the AFM is smaller than 10 K. enough to ensure thermodynamic equilibriumbut slow enough to minimize effects on thestatistical analysis. The images were acquiredwith a pixel resolution of 4000 × × , yielding a resolutionof 0.375 × /Px.We present measurement series for three dif-ferent coverages, with multiple images mea-sured at the same location. The number of im-ages per coverage in each series differs since wehad to sort out some of the images due to exper-imental difficulties. The remaining 14 imagesresulted in total amounts of 1758254 stripe-to-stripe distances d and 17015 stripe lengths l .To obtain the stripe-to-stripe distance andthe length distributions from the AFM imageswe proceeded as follows. After a plane subtrac-tion and line-by-line correction, the imageswere calibrated and corrected for linear drift. Each image was segmented using a trainablemachine learning tool. Afterwards neighbor-ing pixels were connected and the connectedstructures fitted with a rectangle.
All rel-evant data of the fit rectangles (centroid posi-tion, length l and orientation) were collectedand reconstructed as line segments for furtheranalysis using the package SpatStat within theFigure 1: Representative atomic force microscopy (AFM) topography (z p ) images of 3-hydroxybenzoic acid (3-HBA) on calcite (10.4) from the three measured series I, II, and III attemperature 290 K and coverages (a) θ I = 0 . ML, (b) θ II = 0 . ML, and (c) θ III = 0 . ML.All images are cutouts with a size of 1150 × and a resolution of 3446 × We sort out stripes shorter than5 nm since these are difficult to distinguish fromwrongly fitted structures. For simplicity, wedo not exclude stripes limited by image edges.We define the stripe-to-stripe distance as thedistance between each 3-HBA dimer and itsnext-neighbor in [010] direction. Thus, we getone distance per molecular dimer but only onelength per stripe, which implies that the num-ber of measured stripe distances is much largerthan the number of stripe lengths.
When depositing 3-HBA molecules onto the(10.4) surface of calcite kept in ultrahigh vac-uum, the molecules self-assemble into double-rows as has been reported previously. Themolecular double-rows can be identified in AFMimages as stripes oriented along the [421] direc-tion of the calcite crystal, see Figure 1. Twomolecules, one out of each row, form the stripebasis with a periodicity of 0.8 nm. We call thisbasis a 3-HBA dimer. Each image in Figure 1 isa representative example from one of three se-ries I-III of measurements at a given coverage,where θ I = 0.08 ML [Figure 1(a)], θ II = 0.11 ML[Figure 1(b)] and θ III = 0.16 ML [Figure 1(c)].In Figure 2, the difference between the im-age shown in Figure 1 (b) and an image takensix hours before is shown. The areas markedin blue (red) are regions of disappearing (ap-pearing) molecules over time. From this dif-ference image, it becomes evident that themolecules are mobile at a sample temperatureof 290 K. More specifically, a total of ≈
30% ofthe molecules, including entire stripes, changeposition within the measurement time of sixhours, while about 70% of the structures do notchange. We can thus expect that the statisticsof a single image is not strongly affected by thelong measurement time of roughly 3 h for a sin-gle image.From the last and the first image of seriesIII with a time difference of 18 hours, we havegenerated the stripe-to-stripe distance distribu-tions shown in Figure 3(a), using a bin size of0.5 nm. Comparing these two distributions re- [421][010]
80 nm
Figure 2: Comparison of image shown in Fig-ure 1 (b) and an image taken six hours before,demonstrating the redistribution of molecules.Areas where molecules disappear (appear) aremarked in blue (red).veals no significant difference. Both distribu-tions exhibit a distinct maximum at a distanceof 10 to 12 nm, implying that the stripes arenot randomly placed on the surface. A ran-dom placement would result in a geometric dis-tribution. In addition have determined thestripe length distributions for the two images,which are shown in Figure 3 (b) for a bin sizeof 4 nm. Again the comparison of the respec-tive two length distributions yields no signifi-cant difference.To conclude, during 18 hours of measure-ment time appreciable rearrangements of themolecules occur, but the stripe-to-stripe andthe length distributions do not change. Hencethe stripe patterns can be regarded to reflectequilibrium structures. This justifies to ana-lyze all images of each measurement series toimprove the statistics.In Figure 4(a) we show the stripe-to-stripedistance obtained from all images in each series(four images for series I and II, and six imagesfor series III). These distributions are coverage-dependent, exhibiting a decrease of mean dis-tance ( ¯ d I = 24.1 nm, ¯ d II = 18.2 nm and ¯ d III =12.2 nm), standard deviation ( σ I = 12.2 nm, σ II = 8.5 nm and σ III = 4.0 nm) and position of themaximum with increasing coverage.4
100 200 300 400
Stripe length / nm C o un t s (b) Stripe-to-stripe distance / nm C o un t s (a) Figure 3: Comparison of histograms obtained from the first and last image of the series III. Theimages have a time separation of 18 hours. In (a) the counts of stripe-to-stripe distances in bins ofsize 0.5 nm are shown, and in (b) the counts of stripe lengths in bins of size 4 nm. The bins givethe histograms obtained from the first image, and the horizontal bars marked in blue indicate thecorresponding counts from the last image.The corresponding length distributions areshown in Figure 4(b). As explained above, thenumber of counts in each bin is much less thanfor the stripe distances. Overall, the lengthdistributions decrease monotonically for large l . For the two higher coverages (series II andIII), local maxima in the range l ≈ − nmappear. A corresponding maximum, however,is not clearly detectable at the lowest coverage (series I).The total numbers of stripes are N I =959 stripes/image, N II = 1370 stripes/image, N III = 1284 stripes/image in series I-III. Wedetermined mean lengths ¯ l ( ¯ l I = 70.3 nm, ¯ l II = 65.8 nm, ¯ l III = 104.4 nm) and respectivestandard deviations σ l ( σ l, I = 59.6 nm, σ l, II =49.0 nm, σ l, III = 92.2 nm) for the three series.While we can see and expect a global trend
Stripe-to-stripe distance / nm C o un t s / I m ag e (a) IIIIII
Stripe length / nm C o un t s / I m ag e (b) Figure 4: Histograms of (a) stripe-to-stripe distances and (b) stripe lengths obtained from allimages in the series I (0.08 ML, 4 images), II (0.11 ML, 4 images), and III (0.16 ML, 6 images)[color coding according to legend in (a)]. The bin sizes are as in Figure 3.5f increasing mean length and standard devi-ation with increasing coverage, both values aresmaller for series II compared to series I.
For the equilibrated system of 3-HBA moleculeson calcite there it has been proposed that repul-sive interactions are caused by a charge transferbetween surface and molecules, leading to dipo-lar moments perpendicular to the surface. Asthe stripe formation is formed by dimers, it isconvenient to consider these as molecular unitsoccupying lattice sites. We refer to them as“particles”. The lattice sites correspond to theanchoring positions on the calcite surface.The analysis of AFM images shows that thestripes have a width of 2 nm and a periodic-ity of 0.8 nm. This can be represented by arectangular lattice with spacings a (cid:107) = 0 . a instripe direction and a ⊥ = 2 a perpendicular toit, where a = 1 nm sets our length unit.The interplay between attractive and dipo-lar interactions is described by the lattice gasHamiltonian H = − J (cid:88) i NN j n i n j + Γ2 (cid:88) k,l n k n l r kl . (1)Here n i are occupation numbers, i.e. , n i = 1 if the site i is occupied by a particle and zerootherwise. The sum over i and j is restricted tonearest-neighbor (NN) sites in stripe directioncorresponding to an anisotropic Ising model,and r kl is the (dimensionless) distance betweensites k and l . The interaction parameter J > quantifies the strength of the attractive nearest-neighbor interaction. The strength of the repul-sive dipole interaction is given by Γ = p π(cid:15) a , (2)where p is the dipole moment of one dimer and (cid:15) is the dielectric permeability of the vacuum.In the following two subsections, we discussanalytical approaches to get insight into equili-brated stripe patterns for Γ = 0 , and for Γ > based on approximate one-dimensional treat- ments. This allows one to determine the inter-action parameters J and Γ by fitting analyticalexpressions to match experimentally observedstripe distance and length distributions. Forconvenient notation in the following theoreticaltreatment, the stripe length l is given in unitsof a (cid:107) and the stripe distance d in units of a ⊥ . Γ = 0
In the absence of dipolar interactions, thestripe positions in perpendicular directionare uncorrelated. As a consequence, thestripe distance distribution Φ ( d ) is geomet-ric, Φ ( d ) = θ (1 − θ ) d − .For deriving the stripe length distribution, wecan focus on a one-dimensional row of stripes.A stripe of length l corresponds to an occupa-tion number sequence . . . , i.e. , a config-uration of two zeros separated by l ones. Wedenote the probability of such sequence by q l .Knowing q l , the stripe length distribution is ψ ( l ) = q l / (cid:80) ∞ l =1 q l .To determine q l , we introduce the condi-tional probabilities w ( n i +1 | n i , n i − , . . . , n ) offinding occupation number n i +1 if the occu-pation numbers n i , n i − . . . n are given. Inthe grand-canonical ensemble these satisfy theMarkov property w ( n i +1 | n i , n i − , . . . , n ) = w ( n i +1 | n i ) . Accordingly, q l = (1 − θ ) w (1 | w (1 | l − w (0 | , where the factor (1 − θ ) accounts for the first zero in the se-quence, and the product of w ( . | . ) is the Markovchain corresponding to the occupation numbersin the sequence. The conditional probability w (1 | is given by w (1 |
1) = χ (1 , /χ (1) with χ (1) = θ , and the joint probability χ (1 , is equal to the equilibrium nearest-neighborcorrelator C ( J ) = (cid:104) n i n i +1 (cid:105) eq = θ + 1 − (cid:112) θ (1 − θ )( e J − e J − . (3)Hence, the l dependence of q l is ∝ ( C/θ ) l , andfor the length distribution we obtain Ψ ( l ) = θ − C ( J ) C ( J ) (cid:18) C ( J ) θ (cid:19) l , (4)6n agreement with results earlier reported inRef. . For J → , C (0) = θ and we obtainthe geometric distribution Ψ ( l ) = (1 − θ ) θ l − . Γ > The dipolar interaction for Γ > leads to re-pulsion between pairs of particles belonging tothe same stripe as well as to different stripes. Ittends to shorten the stripes and to increase thestripe distances. Compared to the case Γ = 0 ,the stripe distance distribution is more stronglyaffected than the length distribution, becausethe latter is largely determined by the attrac-tive nearest-neighbor interaction J (if Γ < J ).In fact, one can expect that the length distri-bution for large l is still geometric as in Eq. (4)for Γ = 0 . This is because for each Γ > thereis a characteristic length scale of induced corre-lations by the dipolar interaction. Consideringlong stripes to be composed of particle blockswith this length scale, the reasoning in the pre-vious subsection leading to Eq. (4) is applicablewith a renormalized C = C eff ( J, Γ) in Eq. (3).Hence, the length distribution in the presence ofdipolar interactions is expected to decay expo-nentially for large l and to show deviations fromthe geometric shape at small stripe lengths.Exact analytical solutions for the distanceand length distributions are not available inthe presence of competing attractive nearest-neighbor and dipolar interactions. We thereforerely on approximate treatments here.As for the stripe lengths, it is instructive tofirst analyze whether a single isolated stripe canhave an energetic minimum at a finite length.When increasing the length of this single stripefrom l to l + 1 , the energy changes by ∆ H ( l ) = − J + Γ l (cid:88) k =1 k . (5)For a minimum to occur, ∆ H ( l ) must be nega-tive for l = 1 and positive for l → ∞ . This im-plies < J/ Γ < ζ (3) ∼ = 1 . , where ζ (3) is theRiemann zeta function (Apéry’s constant). Ac-cordingly, a finite single stripe can form only ina narrow regime of the interaction parameters J and Γ . However, in a system of many interact- ing stripes at a given coverage, the stripes canmutually stabilize each other at finite lengthsfor a wide range of J and Γ .Due to the fast convergence of the sum inEq. (5), the energy change ∆ H ( l ) for attach-ing one further particle to a stripe becomes es-sentially constant for l (cid:38) . We thus can ex-pect Eq. (4) to hold for large l with C eff ( J, Γ) = C ( J eff ) , where J eff = J − ζ (3)Γ . (6)The corresponding approximate stripe lengthdistribution is referred to as ˜Ψ( l ) .We expect this distribution to have the sameasymptotic behavior as the true length distri-bution Ψ( l ) , i.e. Ψ( l ) ∼ ˜Ψ( l ) ∼ (cid:18) C ( J eff ) θ (cid:19) l , l → ∞ . (7)Deviations from ˜Ψ( l ) are expected to be signifi-cant for small l . If the effective nearest-neighborinteraction J eff is attractive, i.e. , J > ζ (3)Γ ,the energy change ∆ H ( l ) in Eq. (5) is neg-ative, implying that single particles or smallstripes are energetically unfavorable comparedto longer stripes. Accordingly, we expect Ψ( l ) to be smaller than ˜Ψ( l ) for small l .As for the stripe distance distribution Φ( d ) ,we can assume that it is governed by the dipolarinteraction between neighboring stripes in per-pendicular direction. Applying a mean-field ap-proach similar to that introduced in Ref. , wedivide the two-dimensional stripe pattern intomutually independent one-dimensional parallelbands in perpendicular direction. The bandsare considered to have the same width ¯ l , where ¯ l is the mean stripe lengths.For each stripe appearing in a band, we con-sider it to span the whole band, i.e. to havelength l = ¯ l . In one band, the interaction U ( d ) between two stripes at distance d with dipoledensity p/a (cid:107) is (integrating along both stripes7ith parametrization s and s ) U ( d ) = p π(cid:15) a (cid:107) ¯ l/ (cid:90) − ¯ l/ ds l/ (cid:90) − ¯ l/ ds | (cid:126)x ( s ) − (cid:126)x ( s ) | = p π(cid:15) da (cid:107) (cid:34)(cid:16) l d (cid:17) / − (cid:35) . (8)Hence, we have mapped each band onto a one-dimensional lattice occupied by particles withinteraction U ( d ) between neighboring stripes.The mean occupation of lattice sites is fixedby the coverage θ . In the presence of the purelyrepulsive U ( d ) , it can be viewed as resultingfrom a confinement pressure f which hindersthe particles to become infinitely separated andto give rise to a mean distance ¯ d . Our approxi-mation ˜Φ( d ) of the stripe distance distributionthus is given by ˜Φ( d ) = 1 Z exp ( − β [ f d + U ( d )]) , (9a)where Z = (cid:80) ∞ d =1 exp ( − β [ f d + U ( d )]) and f isfixed by the condition ¯ d = ∞ (cid:88) d =1 d ˜Φ( d ) . (9b) The parameters J and Γ are estimated by fit-ting ˜Φ( d ) from Eq. (9a) to the distribution Φ( d ) , and by fitting Eq. (7) to the tail of Ψ( l ) ,where Φ( d ) and Ψ( l ) are the distributions ob-tained in the experiments.We first determine Γ , and hence p = (cid:112) Γ4 π(cid:15) a , by fitting ˜Φ( d ) to Φ( d ) with theexperimental ¯ l in Eq. (8). We then extract J eff by fitting the tail of Ψ( l ) which yields J viaEq. (6).Figure 5 shows fits of ˜Φ( d ) (circles, connectedby solid lines) to Φ( d ) (histogram) for each se-ries, using the method of least square. Theoptimal values of Γ (and corresponding p ) foreach coverage are listed in Table 1. In all three ! ! ! ) I ( d ) (a) ! ! ! ! ! ! ) II ( d ) (b) d / nm10 ! ! ! ) III ( d ) (c) Figure 5: Histograms of the measured distancedistributions Φ( d ) for the three different cover-ages (a) θ I = 0 . ML, (b) θ II = 0 . ML, and(c) θ III = 0 . ML in comparison with the fit-ted theoretical distributions ˜Φ( d ) (circles, con-nected by solid lines). Dashed black lines cor-respond to ˜Φ( d ) with the mean dipole momentof ¯ p = 6 . D. The inset in (a) shows the fit-ted ˜Φ( d ) for θ I (circles, connected by orangeline) compared to ˜Φ( d ) for p > = 8 . D and p < = 4 . D (black lines).8 able 1: Parameters obtained from fit-ting the theoretical model to the experi-mental stripe distance and length distri-butions for the three different coverages(series I-III).
Series θ / ML p / D Γ / meV J /eVI 0.08 7.0 31 0.32II 0.11 6.1 24 0.28III 0.16 5.8 21 0.28cases, the predicted curves match the experi-ment. The fitted dipole moment decreases from . D to . D with increasing θ . When fixingthe dipole moment to the mean ¯ p = 6 . D ofthese values, the corresponding ˜Φ( d ) are also ingood agreement with the experiment, as shownby the dashed black lines in Figure 5.The mean ¯ p differs by about 1 D from the opti-mal value for the smallest coverage. This raisesthe question on the sensitivity of the fitting withrespect to p . We thus analyze how ˜Φ( d ) devi-ates from Φ( d ) for even larger differences of p from its optimal value. For values p > = 8 . Dand p < = 4 . D larger and smaller by 2 D, ˜Φ( d ) is shown in the inset of Figure 5(a). As canbe seen from this inset, deviations to ˜Φ( d ) forthe optimal p value are now clearly visible. Wethus conclude that the error in our estimate isabout ± D.Taking ¯ p as the dipole moment of the 3-HBAdimer yields a dipole moment p/ = 3.2 D forthe single molecule, in fair agreement with ourformer estimate. Having determined Γ , we now analyze thestripe length distribution to determine J . Fig-ure 6 shows the measured length distributions Ψ( l ) for each series (circles). The distributionsare determined by using bins of varying sizewith approximately equal amount of events ineach bin. As expected, all distributions showan exponential decay for large l . The solidlines are fits to these exponential decays for l > nm. According to Eq. (7), the decaylength of Ψ( l ) ∼ e − l/l is l = a (cid:107) ln( C ( J eff ) /θ ) . (10) l / nm ! ! ! ! ! * ( l ) IIIIII
Figure 6: Stripe length distributions Ψ( l ) forthe three different coverages (I-III, circles) withfits to the exponential tails for l > nm (solidlines).The characteristic decay length l for eachexperimental distribution thus yields a value J eff via Eq. (10) in combination with Eq. (3).The interaction parameters J then follow fromEq. (6) and are listed in the fifths column ofTable 1. These values lie around . eV. Ourfinal estimate of the analysis is p = 6 . D ± Dand J = 0 . ± . eV.The interaction strength J is in the rangeof hydrogen-bonds and lower than ≈ . eVbetween two molecules in a carboxylic aciddimer. In summary, we have presented an approachto estimate the strengths of short-range attrac-tive and long-range repulsive interactions be-tween 3-HBA molecules on a calcite surface byan analysis of stripe-to-stripe distance distribu-tions Φ( d ) and stripe length distributions Ψ( l ) .Experimental distributions were determinedfrom an analysis of three AFM image serieswith different coverages 0.08 ML, 0.11 ML, and0.16 ML at a temperature 290 K. The measure-ments of theses series spanned time intervals ofup to 18 hours. A comparison between distri-butions of individual images in the same seriesstrongly suggests that the stripe patterns are in9hermodynamical equilibrium.The attractive interaction responsible for thestripe formation was considered to be an effec-tive one with strength J between neighboring 3-HBA dimers, without resorting to details of themolecular structure. The long-range repulsiveinteraction is modeled as dipole-dipole interac-tion of characteristic strength Γ as previouslyproposed in Ref. . It is believed to be causedby a charge transfer between the surface and 3-HBA molecules. As these molecules have spe-cific anchoring sites on the calcite surface, thesystem could be described by a lattice gas cor-responding to an anisotropic Ising model withadditional dipolar interaction.Based on this model, we developed mean-fieldapproaches to derive approximate expressionsfor the stripe distance and length distributionswith J and Γ as parameters. Fitting these pa-rameters to the experimental distributions weobtained the estimates J = 0 . ± . eV and p = 6 . D ± D for the dipole moment p ∝ √ Γ of a 3-HBA dimer.The modeling approach presented here is ap-plicable also to other molecular systems self-assembling into stripe patterns, if the stripe for-mation is dominated by short-range attractivemolecule-molecule interactions. In general, onecan expect additional long-range electrostaticinteractions to be present. Their impact onthe structure formation depends on their type(e.g., dipolar, quadrupolar) and strength, butthe core of our methodology is independent ofthese features.The mean-field treatment, however, requiresthe formation of structures with long stripes ar-ranging into patterns with large overlaps be-tween neighboring parallel stripes. This re-quirement is fulfilled only if the repulsive inter-action is not too strong compared to the attrac-tive one, and if the coverage is not too small.The coverage must not be too high either be-cause otherwise the structure will no longer becomposed of individual stripes. For determin-ing the respective limits of our mean-field treat-ment, extensive simulations of the many-bodyproblem are needed, which is left for future re-search.As long as the aforementioned requirements are met, other types of interactions can be ac-counted for by minor adjustments of the mean-field approach. As for the stripe distance dis-tribution, only the effective interaction poten-tial U ( d ) between stripes in Eq. (8) needs to bemodified. As for the stripe length distribution,we expect a length scale to exist beyond whichcorrelations within a stripe can be renormalizedto an effective nearest-neighbor interaction be-tween segments. The interplay between attrac-tive and repulsive interaction in Φ( l ) can thenbe accounted for by one effective coupling pa-rameter analogous to J eff in Eq. (6).From a general point of view, it should bescrutinized whether a modeling with staticdipole moment is appropriate. Our use of astatic dipole moment here relies on the as-sumption of an approximately fixed amount ofcharged transferred between the surface andeach molecule. The results in Table 1 indicate adecreasing dipole moment with increasing cov-erage. This can be interpreted by a dynamicdipole moment which becomes smaller in orderto compensate for additional repulsive interac-tions with further molecules. A change of themolecule-surface interaction as a response to arepulsive interaction has been reported earlierin Ref. Dynamical dipole moments can be coped within a theoretical treatment by introducing amolecular polarizability for the molecules. Thisleads to varying dipole moments in dependenceof their local environment. How importantthese variations are, is presently unknown. Theuncertainties of the values in Table 1 and therather narrow coverage range 0.08-0.16 ML doesnot allow us to give a firm assessment on howstrong effects of a dynamical dipole momentare. Additional investigations with a widerrange of coverages are needed. Further experi-mental and theoretical research in this directionoffers promising perspectives to gain deeper in-sight into the impact of the interplay betweenrepulsive and attractive interactions on molec-ular self-assembly.
Acknowledgement
We thank the GermanResearch Foundation (DFG) for funding (KU1980/10-1) "Intermolecular Repulsion in Molec-10lar Self-Assembly on Bulk Insulator Surfaces".
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