On the origin of the great rigidity of self-assembled diphenylalanine nanotubes

The elastic properties of the nanotubes of self-assembled aromatic dipeptide diphenylalanine are investigated by means of Raman spectroscopy and a mass-in-mass 1D model. Analysis of nanotubes’ lattice vibrations reveals the essential contribution of the water in the nanochannel core of the tubes to the Young’s modulus and high water mobility along the channel. Direct measurements of the Young’s modulus performed by nanoindentation confirm the obtained results.

as piezoelectric sensors and actuators in micromechanical systems (MEMS). 10,15,16pecial attention was paid to the remarkably rigid structure of FF nanotubes.Its experimentally measured transversal Young's modulus varies from 19 to 27 GPa, 17,18 and is several times larger than that of other peptide fibrils (about 3.3 GPa 19 ).An attempt to shed light on the origin of such enormous stiffness was made using first-principles calculations. 20However, the obtained value of the Young's modulus was still less than a half of the experimentally observed one, even when using the Tkatchenko-Scheffler (TS) correction, which allowed describing accurately hydrogen and van der Waals bonding, essential for biomolecular crystals. 20n our opinion, the main reason for such divergence is neglecting the water in the nanochannel core of the nanotubes.Its presence there was confirmed by X-ray analysis, 21 photoluminescence, 22 and Raman measurements. 23,246][27] Each FF monomer creates hydrogen bonds with water molecules, 21 which may thus contribute to the mechanical properties of the nanotubes.Exceptionally high values of Young's moduli (up to 44 GPa) due to hydrogen bonds between molecules were observed in several amino acid crystals. 28 similar effect should be expected for the FF nanotubes.
In this work we developed a simple mechanical model of FF nanotubes taking into account the existence of water in the nanochannel core of the tubes.Analysis of nanotubes' lattice vibrations by Raman spectroscopy in the context of the proposed model allowed us to calculate effective elastic constants and to demonstrate the large contribution of water to the tubes' Young's moduli.
It is known that FF monomers in water-rich solution form helical nanotubes filled with water molecules via a self-assembly process. 21,23Water molecules are held in the nanochannel core by radially oriented hydrogen bonds with FF monomers, whereas the latter interact among themselves via both hydrogen bonds and van der Waals forces. 21,29he unit cell of the nanotube (one step of the helix) consists of two embedded interacting subsystems: an FF ring (6 monomers)  and water (24 molecules, 4 molecules per FF monomer 22 ) inside the ring (Fig. 1(a)).Such systems can be described by the equivalent mass-in-mass unit of the same size (Fig. 1(b)).The masses of the FF ring and of its water ''core'' are m 1 = 3.11 Â 10 À24 kg and m 2 = 0.72 Â 10 À24 kg, respectively.Interaction between the FF ring and the water subsystem for the X and Y directions is characterized by the same effective spring constant a.The sequence of aligned unit cells forms the nanotube (Fig. 1(c)).Longitudinal interaction between adjacent rings and water therein occurs with effective spring constants b and g, respectively.Individual nanotubes bond together by aromatic rings and form microtubes, which can be considered as a hexagonal crystal. 23,29he equations of motion for this system provide dispersive relations in the general form: where The sign in front of the square root corresponds to optical (+) or acoustical (À) branches.
It is worth noting that for k = 0 eqn (1) is reduced to a simple expression for vibrations of crystal with two atoms per primitive cell. 30Therefore, keeping in mind the application to Raman scattering, the frequency of the optical mode at the center of the Brillouin zone is: 30 This expression can be used for determination of the spring constant a, assuming o opt to be an effective frequency of lattice vibrations measured by Raman spectroscopy.The obtained result will depend on the polarization conditions of the measurements, and therefore different spring constants a ij relating to lattice vibrations in different directions in the nanotube can be obtained.
The effective elastic constants of FF nanotubes can be derived from the analysis of the Raman spectra.A comparison of the frequencies of acoustic vibrations obtained in the context of the used model with those calculated in continuous approximation gives a relation between microscopic spring constants a ij and macroscopic elastic constants C ij : 30 where a i is a corresponding lattice parameter (either a or c) and V is the cell volume (lattice parameters for the nanotube are: 21 a = 24.1 Å and c = 5.46 Å; therefore V = sin(p/3)a 2 c = 2718.47Å 3 ).
For hexagonal crystals, the elastic constants form a tensor with five independent components (Voigt notations are used): 31 C 11 , C 12 , C 13 , C 33 and C 44 .In the most common backscattering geometry of Raman measurements, a variation of the polarization direction allows determination of 4 components of the effective elastic tensor (Fig. 2).For % X(zz)X and % X( yy)X geometries, where polarization of a laser beam is parallel to the analyzer's orientation and is oriented either along or perpendicular to the tube axis, eqn ( 2) and ( 3) provide C 33 and C 11 constants, respectively.In the % X(zy)X geometry the polarizations of the laser beam and the analyzer are 901-crossed; therefore the constant C 13 can be found.For estimation of the C 12 constant the spectrum has to be measured on the end face of the tube in the % Z( yx)Z geometry.The C 44 constant cannot be determined by Raman spectroscopy.Raman spectra of all four geometries are presented in Fig. 3.
For estimation of effective elastic constants we used sufficiently long FF microtubes grown from water rich solution using a common procedure. 9,25Raman spectra were measured using the confocal Raman microscope Alpha300AR (WITec GmbH, Germany)  A typical Raman spectrum of FF microtubes consists of three separate regions: region 1 (from 10 to 375 cm À1 ) corresponds to lattice vibrations, whereas region 2 (from 375 to 1850 cm À1 ) and region 3 (from 2750 to 3250 cm À1 ) are due to vibrations of various functional groups (Fig. 3).Assignment of highfrequency spectral lines can be based on the analysis of published data, 32 whereas detailed analysis of low-frequency lines is hampered by a rather complicated tube structure and highly overlapping spectral lines.Deconvolution of this region into separate lines provides little information about the tube structure, regardless of the fact that the assignment of several low-frequency lines has been performed recently. 24he effective frequency of lattice vibrations can be calculated either by fitting the low-frequency spectral region by a single curve (Lorentzian or Gaussian) or by calculating the weighted average over this region.The second method is more precise since it takes into account low intensive lines, whereas fitting by a single curve mainly gives the maximum frequency in the region.
The calculation of the weighted average over the low-frequency region of Raman spectra (Fig. 3) gives average values of the effective frequency 131.2 and 126.8 cm À1 for % X(zz)X and % X( yy)X geometries, respectively, and 134.8 and 141.9 cm À1 for % X(zy)X and % Z( yx)Z geometries, respectively.It is worth noting that edge filters, which are usually used in spectroscopic systems, cut part of the low-frequency region up to 50-100 cm À1 .However, this region is usually overlapped with a broad boson peak and shoulders of the Rayleigh line, and therefore information about the lattice vibrations here cannot be extracted anyway.We found that the spectral resolution of our instrument has little effect on the resulting elastic constants and even 30 cm À1 variation of the upper and lower limits of region 1 changes E T and E L by less than 0.3 GPa.
Thus, using the measured effective frequencies and estimated reduced mass of the ring, four independent components of effective elastic tensor C W ij of the water-filled nanotubes can be calculated.The obtained values are in line with those found in ref. 20 within a density functional theory calculation using the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation (Table 1).
The values of C 12 and C 13 are close to those calculated in ref. 20 for empty tubes (Table 1), whereas a higher value of C 11 demonstrates the pronounced effect of the water subsystem on the elasticity of the tube in a transversal direction.The low value of C 33 shows that water molecules are weakly bound to FF rings and can easily move along the nanochannel, thus making FF tubes interesting for various micro-and nanofluidic applications. 33At the same time, the longitudinal mobility of water has only a negligible effect on the longitudinal stiffness of the tube, which is therefore determined by the interaction between FF monomers in the adjacent rings only.This interaction cannot be taken into account in our model, since eqn (2) for the effective frequency of the lattice vibrations does not include the spring constant b.
To take into account both water and FF contributions, the total elastic constants of the FF nanotubes can be calculated: Here C W ij describes water contribution and is found in this work in the context of the proposed model.For C FF ij we used elastic constants calculated in ref. 20 in the ''PBE+TS'' approach, i.e. with the van der Waals interactions after the Tkatchenko-Scheffler scheme included on top of the PBE exchange-correlation.The inclusion of this interaction is essential for accurately describing the long-range interaction between FF rings.The obtained values of the total elastic constants are presented in Table 1.
The hardness of the FF microtubes can be described by transversal E T (across the tube) and longitudinal E L (along the tube) Young's moduli due to the anisotropy of the microtubes.E T and E L can be written in terms of elastic constants: 34 Total elastic constants C tot ij were used to calculate E T (Table 2), which is noticeably greater than that obtained in firstprinciples calculations 20 and is very close to the experimentally derived value, 19 GPa. 18This demonstrates that the water subsystem essentially contributes to the stiffness of the FF nanotubes in the transversal direction.This is, most probably, due to the radial network of hydrogen bonds between FF Table 1 Effective elastic constants of FF nanotubes (in GPa).Constants C FF ij describe interaction between FF monomers and are obtained in firstprinciples calculations in PBE and PBE+TS schemes. 20Constants C W ij are calculated in this work and show the contribution of water molecules present in the core of the nanotubes.The total effective elastic constants are monomers in the ring and water molecules.This effect is similar to that observed in several amino acid crystals. 28owever, the obtained value (16.77GPa) is far though from another experimental value (27 GPa). 17This experimental value was obtained by a bending beam model using an atomic force microscope (AFM) on nanotubes suspended across the microcavities.However, this method provides results strongly depending on the nanotube fixed at the edges of the cavity. 35,36o get more precise values of the transversal Young's modulus, we performed direct measurements of E T by the dynamical nanoindentation method.The nano-hardness tester NanoScan-4D (FSBI TISNCM, Russia) equipped with a Berkovich indenter (diamond pyramid) was used.The values of E T were determined from loading-unloading curves by the Oliver-Pharr method. 37Measurements were performed in more than 50 points of several FF nanotubes fixed at the solid substrate.Our tests showed an instrumental error less than 2%.
Two characteristic values of E T were revealed (see Table 2 and Fig. 4).The first value (10 GPa) is close to that obtained from first-principles modeling of empty nanotubes. 20The second value (25 GPa) is close to that measured in ref. 17.The dispersion of these characteristic values is larger than the instrumental error, reproducible and can be attributed to inhomogeneous filling of the nanotubes due to the high mobility of water molecules along the nanochannel. 38,39In this case the lowest values of the Young's modulus correspond to almost empty nanotubes, whereas completely filled tubes provide the largest E T (see insets in Fig. 4).It is worth noting that only a part of the water molecules in FF tubes is considered in our model.This corresponds to partially filled nanotubes, and therefore the value of E T calculated in the present work falls into the intermediate range (Fig. 4).Additional water molecules that can exist in the nanochannel core 40,41 introduce additional stiffness into the nanotube, and therefore higher values of Young's moduli can be observed.The highest observed value of E T is about 31 GPa (Fig. 4).
Finally, the value of E L calculated in this work is very close to that obtained in ref. 20 in PBE-TS approximation, since movable water molecules weakly contribute to longitudinal stiffness.Experimental measurements of this value are conjugated with high technical difficulties and are still to be done.
In summary, the essential contribution of water molecules inside the nanochannel of diphenylalanine peptide nanotubes to its transversal Young's modulus is demonstrated.Analysis of the low-frequency region of the Raman spectra of the nanotubes in the context of a mass-in-mass 1D model allowed refining four independent constants in the effective elastic tensor and to demonstrate the high mobility of water molecules along the nanochannels.Direct measurements of the transversal Young's modulus using a nanoindenter showed that its value strongly depends on the degree of nanotube filling by water, whereas the longitudinal Young's modulus is primarily determined by the interaction between peptide monomers.
The equipment of the Ural Center for Shared Use ''Modern nanotechnology'' UrFU was used

Fig. 1
Fig. 1 The equivalent scheme of FF nanotubes.(a) Typical unit cell of an FF nanotube; (b) its mass-in-mass representation; and (c) nanotube formed by aligned interacting mass-in-mass units.Solid lines show interactions with spring constants a, b and g.The arrow shows the tube's axis direction.

Fig. 2
Fig. 2 Schemes of polarization directions for determination of the components of the effective elastic tensor in backscattering Raman measurements.Red arrow -direction of laser propagation, green arrow -laser polarization, and black arrow -polarization of analyzed light.

Fig. 3
Fig. 3 Polarized Raman spectra of FF nanotubes in various geometries.Three characteristic regions: (1) lattice vibrations, and (2) and (3) vibrations of various functional groups.Attribution of several spectral lines to functional group vibrations is presented.
. The research was made possible by the President of Russian Federation grant for young scientists (Contract 14.Y30.15.6554-MK), by the Government of the Russian Federation (Act 211, Agreement 02.A03.21.0006), and by the Ministry of Foreign Affairs of France within Metchnikov Scholarship (grant 851365A).Part of this work was developed in the scope of Project CICECO-Aveiro Institute of Materials (ref.FCT UID/CTM/50011/2013), financed by national funds through the FCT/MEC and, when applicable, cofinanced by FEDER under the PT2020 Partnership Agreement.

Fig. 4
Fig.4Values of E T measured directly by the nanoindentation method.The inset shows FF nanotubes with different degrees of filling by water: empty nanotubes, partially filled nanotubes, and completely filled nanotubes.Blue solid lines point out values obtained from the literature and calculated in this work. PBE+TS This journal is © the Owner Societies 2016