Self-assembled monolayers of oligophenylenes stiffer than steel and silicon, possibly even stiffer than Si$_3$N$_4$

To quantify charge transport through molecular junctions fabricated using the conducting probe atomic force microscopy (CP-AFM) platform, information on the number of molecules $N$ per junction is absolutely necessary. $N$ can be currently obtained only via contact mechanics, and the Young's modulus $E$ of the self-assembled monolayer (SAM) utilized in the key quantity for this approach. The experimental determination of $E$ for SAMs of CP-AFM junctions fabricated using oligophenylene dithiols (OPDn, $1 \leq n \leq 4$) and gold electrodes turned out to be too challenging. Recent measurements (Z. Xie et al, J. Am. Chem. Soc. 139 (2017) 5696) merely succeeded to provide a low bound estimate ($E \approx 58\,$GPa). It is this state of affairs that motivated the present theoretical investigation. Our microscopic calculations yield values $E \approx 240 \pm 6\,$GPa for the OPDn SAMs of the aforementioned experimental study, which are larger than those of steel ($ E \approx 180 - 200\,$GPa) and silicon ($E \approx 130 - 185\,$GPa). The fact that the presently computed $E$ is much larger than the aforementioned experimental lower bound explain why experimentally measuring $E$ of OPDn SAM's is so challenging. Having $E \approx 337 \pm 8\,$GPa, OPDn SAMs with herringbone arrangement adsorbed on fcc (111)Au are even stiffer than Si$_3$N$_4$ ($E \approx 160 - 290\,$GPa).

While the aforementioned nuclear methods enable the direct determination of the surface coverage Σ, models developed in contact mechanics pose certain problems to reliably estimate the contact area A between the AFM tip and the self-assembled monolayer (SAMs) under investigation.To deduce A, various models were developed in contact mechanics [29,30,[37][38][39][40].To apply these models, SAM's Young modulus of elasticity E is a key quantity needed.Unfortunately, as noted earlier [24] and elaborated in Section 2, the experimental determination of E is problematic.
So, the best one can do at present is to theoretically compute E by investigating nanoelastic properties of the molecules of interest subject to compressive or tensile forces.It is this state of affairs that motivated the present theoretical investigation on molecules OPDn ≡ HS -(C 6 H 4 ) n -SH (1 ≤ n ≤ 4) of the benchmark oligophenylene dithiol family [1,24,31,[41][42][43][44].
While primarily aiming at providing values of E to be subsequently utilized in ongoing molecular elec-tronics studies, the results reported below also provide an explanation why a direct experimental determination of Young's modulus of SAMs based on aromatic oligophenylene molecules is currently too challenging.We found that OPDn SAMs are much stiffer than the lower bound estimate (E ≈ 58 GPa) deduced in recent experiments [24] may suggest.They are stiffer than steel and silicon, possibly also stiffer than Si 3 N 4 .

Method
Quantum chemical computations accomplished in conjunction with the present investigation used the GAUSSIAN 16 package [45] on the bwHPC platform.We performed geometry optimizations for molecules subject to axial mechanical forces (cf. Figure 1).These calculations were based on the density functional theory using the Becke three-parameter B3LYP hybrid exchange-correlation functional [46,47].For consistency with our recent [48][49][50][51] and ongoing works on related systems, we used 6-311++G(D, P) basis sets [52,53] although including diffuse basis functions is in fact necessarily required only at larger molecular elongations beyond the linear (elastic) deformation regime investigated in this paper.

Preliminary Remarks
As anticipated in Introduction, SAM's modulus of elasticity E ≡ E s represents an important challenge for experimentalists.Insufficient signal-to-noise ratio makes it impossible to reliable estimate an effective Young modulus for OPDn/Au from tiny differences in indentation depths measured with (OPDn/Au) and without (bare Au) SAM adsorbed on gold (C.D. Frisbie and Z. Xie, private communication).To this, from a more general perspective, one should still add the aggravating point that the exact determination of E merely from force-distance curves is impossible (see ref. [29], page 41 for details): the mutual dependence between the slope of the contact line and the jump-off-contact is expressed in terms of a parameter (λ in ref. [24]) for which (among other nontrivial things) information on SAM's modulus of elasticity E is required; see, e.g., ref. [29], page 41 for specific details.
To avoid misunderstandings, one should emphasize here that the aforementioned E = E s as used in contact mechanics calculations is a property of the SAM, which does not include the elastic interaction with the AFM tip.Contact mechanics does account for the tip-SAM interaction (as it should do), but it does it via the work of adhesion (γ in ref. [24]).The so-called "effective" Young's modulus entering formulas of the various contact mechanics models [29,30,[37][38][39][40] depends both on SAM's (E ≡ E s ) and tip's (E t ) moduli of elasticity; still these are "intrinsic" Young's moduli, which refer to tip and SAM isolated (i.e.elastically decoupled) of each other.
Having said this, and given the fact that elastic properties of metals (gold in ref. [24]) used for (coating) AFM tips are known, what we have to do is to focus on the elastic properties of monolayers of OPDn molecules which, albeit not (mechanically) coupled to the AFM tip, still have the same spatial structure as the real SAM.
In this context, one should note that OPDn molecules in SAMs utilized to fabricate the recently investigated CP-AFM junctions [24,31] do not resemble to those of ordinary (organic) solids or liquids.Both (ellipsometry and XPS [24,31,35]) experiments and theory [54][55][56] found that OPDn molecules stand nearly vertical on metal.The average intermolecular spacing deduced from the coverage (Σ ≃ 3.3 molec/nm 2 [24]) amounts to ∼ 5.5 Å.So, we are dealing with monolayers wherein parallel OPDn molecules are sufficiently apart of each other, and considering elastic properties of strands of OPDn molecules isolated of each other is a legitimate description.Parenthetically, this one dimensional picture in SAM's transverse direction is additionally supported by other pieces of recent experimental evidence [24,31] revealing a charge transport through individual OPDn moisolated isolated molecules weakly interacting among themselves in transverse direction.
The foregoing analysis made it clear that what we ultimately need to compute are elastic properties of isolated OPDn molecules.

Elastic Properties of OPDn and OPn Molecules
Constrained optimization imposing a fixed values of the distance L(X 1 , X 2 ) → L = L 0 + x between the two (non-hydrogen) X 1,2 atoms at the two molecular ends (X 1,2 --S for OPDn and X 1,2 --C for OPn) larger or smaller than the equilibrium value (L 0 ) straightforwardly allows the determination of the tensile or compressive forces F = F(x) (of opposite direction and equal magnitude) exerted on the X 1 and X 2 end atoms.
Results of these quantum chemical calculations at small deformations are depicted in Figures 2 and 3.The numerical values underlying these figures are collected in Table 1.They reveal a linear dependence (Hooke's law) on x of the force F producing molecule's mechan-ical deformation (cf. Figure 2) Its slope provides us with the molecule's elastic (spring) constant κ.As visible in Figure 2a, (straight) lines for longer OPDn molecules have larger slopes.This is in accord with the fact that, at a given deforming force F, homogeneous springs characterized by a lengthindependent specific stress K respond with elongations proportional to their length: x ∝ L 0 , κ ∝ 1/L 0 .In fact, OPDn molecules do not behave like homogeneous springs because the stiffness of their constituents is different: in the linear elastic regime presently considered (i.e. at small F), the interring C -C bonds are notably stiffer than the aromatic phenyl rings, which are in turns somewhat stiffer than the terminal C -S bonds.This results in values of the specific stress K slightly decreasing from the longer OPD4 to the shorter OPD1, as observable by inspecting the slopes of the lines depicted in Figure 3a.
For comparison purposes, in Figures 2b and 3b we also present results for the elastic properties of the parent (non-thiolated) molecules of the oligophenylene series OPn ≡ H -(C 6 H 4 ) n -H (1 ≤ n ≤ 4).The trend across the OPDn family noted above is also visible for OPn.Notice, in particular, that OP1 (≡ benzene) has the lowest specific stress; the stiffest interring C -C bond is missing there.
A variety of carbon-based molecular families M n [57][58][59][60][61] are known to exhibit even-odd alternation upon adding a repeat unit (phenyl ring in our case) M n → M n+1 .This is a particularly meaningful question because symmetry is found to alternate across the OPDn family; calculations indicate that the odd-number members OPD1 and OPD3 possess C i symmetry while evennumber members OPD2 and OPD4 possess C 2 symmetry.Nevertheless, the results depicted in Figure 4 reveal that this is not the case.The dependence on n is monotonic.Guided by the number of OPDn and OPn constituents, we were led to consider an analytical formula used for the fitting curves presented in Figure 4, which allows us to put the aforementioned monotonic dependence on n in more quantitative terms.
Like in cases of the rods or bars schematically depicted in Figure 1b, for the characterization in terms of Young's modulus of elasticity E via Hooke's law, a   cross section area A needs to be assigned In the specific case considered, the effective cross section area can be expressed in terms of the surface coverage Σ (cf. Figure 1b).Thanks to RBS and NRA, for OPDn SAMs used to fabricate the CP-AFM junctions of ref. [24], this quantity is available: This yields an average area per molecule The values of the Young's moduli deduced via eqs (3) and ( 4) are presented in Table 1.They show that the SAMs used to fabricate OPDn CP-AFM junctions of ref. [24] are very stiff.They are stiffer than steel (180 GPa for stainless AISI 302 and 200 GPa for structural ASTM-A36 [62]).
Let us also refer to two materials used for commercial AFM cantilevers: silicon and silicon nitride.These materials dominate the vast majority of applications [29].OPDn SAMs are stiffer than silicon: E Si ≈ 130 − 185 GPa [29].(Precise values of real materials depend on various factors, e.g., precise composition and crystallographic orientation).Values for silicon nitride are in the range E Si 3 N 4 ≈ 160 − 290 GPa [29].So, using the coverages measured in ref. [24] we can also conclude that the OPDn SAMs of ref. [24] can be even stiffer than silicon nitride.
Yet, this is not the whole issue.Notwithstanding that the OPDn SAMs fabricated on polycrystalline gold in ref. [24] were found to be characterized by exceptionally small statistical variations (hence implicitly good coverage), the surface coverage of OPDn SAMs with herringbone (hb) arrangement adsorbed on fcc Au (111) is higher (Σ hb = 4.63 molec/nm 2 [54]) than that of ref. [24] (Σ ≈ 3.3 molec/nm 2 ).This (hb) coverage corresponds to even higher values of E, which are also included in Table 1.Inspection of the last column of Table 1 reveals that OPDn SAMs with herringbone structure are definitely stiffer than Si 3 N 4 .

Conclusion
To the best of author's knowledge, this is the first work reporting results for Young's moduli of elasticity of SAMs based on benchmark species (as the case of the presently considered family of oligophenylene dithiols OPDn) routinely used for fabricating CP-AFM molecular junctions.The fact that the present values E OPDn ≈ 240 ± 6 GPa are much larger than the lower bound E exp OPDn ≈ 58 GPa extracted from recent experiments [24] is noteworthy.
On one side, it provides a rationale for the difficulty encountered by experimentalists to measure a reliable value of E; the OPDn SAM high stiffness makes the deformation measurable for OPDn/Au in excess to that for bare gold too small to be accurately estimated.
On the other side and more importantly from a practical standpoint, the presently reported values of E make it possible to update/refine the numbers N of OPDn molecules per junction.At a given AFM tip and load (typically compressive F = 1 nN) which is necessarily applied to render conduction through CP-AFM junctions possible, the contact area A for stiffer SAMs (read E OPDn ≈ 240 ± 6 GPa, the presently reported values, cf.Table 1) is significantly smaller than for softer SAMs (read E exp OPDn ≈ 58 GPa, as used in ref. [24]).Work is underway, but what one can state for sure is that the ensuing values of N will be even smaller than those (N ∼ 80) estimated in ref. [24], which, surprisingly, already appeared much smaller and reproducible than those (up to N ∼ 10 3 [20]) claimed in earlier literature [20] on CP-AFM junctions.Further, since what is directly accessible in CP-AFM experiments is a junction's transport property (e.g., conductance G junc , current I junc ), updated/refined Ns' also translate into updated/refined values of transport property per molecule (G 1 = G junc /N, I 1 = I junc /N).This, in turn, makes a quantitative comparison with transport properties measured using single-molecule (e.g., STM) testbeds more adequate.) and Young's moduli E (panels a and b, respectively) of oligophenylene dithiol molecules (OPDn) and their parent (OPn) non-thiolated species calculated microscopically as described in the main text and fitted as indicated in the inset.

Figure 1 :
Figure 1: Schematic representation of the method to compute the elastic constant of a molecule.

Figure 2 :
Figure 2: Results for the mechanical (here, F > 0, tensile or compressing F < 0) force as a function of molecular length L: (a) OPDn and (b) OPn.

Figure 3 :
Figure 3: Results for the mechanical (here, F > 0, tensile or compressing F < 0) force as a function of molecular length L: (a) OPDn and (b) OPn.

Figure 4 :
Figure4: Results for the specific strengths K (cf.eq(2)) and Young's moduli E (panels a and b, respectively) of oligophenylene dithiol molecules (OPDn) and their parent (OPn) non-thiolated species calculated microscopically as described in the main text and fitted as indicated in the inset.

Table 1 :
[54]tic properties of OPDn and OPn molecules calculated microscopically as indicated in the main text: spring constants (κ), specific stresses K, and Young's moduli E. The values of the Young's modulus E correspond to two values of the surface coverages Σ: (a) value measured via Rutherford backscattering (RBS) and nuclear reaction analysis (NRA) for OPDn SAMs used to fabricate molecular CP-AFM junctions in ref.[24](Σ = 3.3 molec/nm 2 ) and (b) value for OPDn SAMs with herringbone structure (Σ = 4.63 molec/nm 2 , cf. ref[54]and citations therein).