Thermal and electrical properties of porphyrin derivatives
and their relevance for molecule interferometry
Sarayut Deachapunya, André Stefanov, Martin Berninger, and Hendrik Ulbricht Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
Elisabeth Reiger
Kavli Intstitute of Nanoscience, Delft University of Technology, Lorentzweg 1, NL-2628 CJ Delft, The Netherlands
Nikos L. Doltsinis
Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, D-44780 Bochum, Germany Markus Arndt
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
共Received 28 February 2007; accepted 9 March 2007; published online 24 April 2007兲
The authors present new measurements of thermal and electrical properties for two porphyrin derivatives. They determine their sublimation enthalpy from the temperature dependence of the effusive beam intensity. The authors study H2TPP and Fe共TPP兲Cl in matter-wave interferometry. Both molecules have nearly equal de Broglie wavelengths but different internal characteristics: only Fe共TPP兲Cl exhibits an electric dipole moment of about 2.7 D and the authors discuss its influence on the molecular interference pattern. The authors add an external electric force field to the interferometer and use it to measure the scalar polarizability. They compare their experimental values ␣共H2TPP兲=105±4±6 Å3 and ␣共Fe共TPP兲Cl兲=102±9±6 Å3 to ab initio calculations and they discuss the influence of thermal excitations on the polarizability. © 2007 American Institute of
Physics.关DOI:10.1063/1.2721563兴
I. INTRODUCTION
Porphyrins and their derivatives are widely used and well-investigated objects in physics and chemistry.1They are of interest because of their numerous potential applications from molecular electronics,2over sensors,3information stor-age elements,4to medical agents.5Our own research initially focused on the question of whether these biodyes could also reveal their quantum mechanical wave nature in spite of their structural complexity. This question was answered positively in a recent experiment.6The present study reverses the ques-tion and asks what we can learn about molecular properties by using a near-field matter-wave interferometer.
For this purpose, we first analyze in Sec. II the thermal characteristics of two porphyrin derivatives and show that they possess a sufficiently high vapor pressure for our sub-sequent interferometry experiments. We discuss the possible influence of internal molecular properties on the external de Broglie interference in Sec. III and then compare experimen-tal Talbot-Lau interference patterns for H2TPP and Fe 共TP-P兲Cl in Sec. IV. These two derivatives differ only slightly in their internal structure and de Broglie wavelength but are distinguished by the absence and presence of a permanent electric dipole moment. By exposing these molecules to an external field gradient—inside the same matter-wave interferometer—we can determine their scalar electric polar-izabilities with good precision and also discuss the influence of the electric dipole moment 共Sec. IV兲. We compare our
measurements and density functional theory共DFT兲 calcula-tions in Sec. V, taking into account the thermal evolution of the molecules at 650 K.
II. MOLECULAR SUBLIMATION ENTHALPIES
In order to determine the sublimation properties of the porphyrins, a resistively heated furnace generates an effusive molecular beam of temperature dependent flux. The mol-ecules are detected using electron impact ionization quadru-pole mass spectrometry共EI-QMS兲 in a differentially pumped vacuum chamber. The pressure in the source chamber varies with the furnace temperature, but it always remains in the low 10−6mbar region. In the detection chamber, the pressure is kept below 5⫻10−8mbar.
The sublimation enthalpy ⌬Hsub共T兲 can then be mea-sured by observing the molecular beam intensity I as a func-tion of the source temperature T, assuming that the enthalpy is constant over the interval of interest.
The QMS signal at the mass of the singly charged ion is then continuously recorded, while the source temperature is ramped at a rate of 1.5 K / min. In Fig.1we show the result-ing Arrhenius plots for meso-tetraphenylporphyrin 共H2TPP兲 for tetraphenylporphyrin-iron共III兲 chloride 共Fe共TPP兲Cl兲, as well as for a fragment that we discuss below. The Clausius-Clapeyron relation,
ln共IT兲 = const − ⌬Hsub/RT, 共1兲
then allows us to determine the sublimation enthalpy from the slope of a linear fit to the data共Fig.1兲.
It is conspicuous that these data are not described by a single line, as one might expect from Eq. 共1兲. As already discussed earlier,9 the use of the Clausius-Clapeyron equa-tion is restricted to either the molecular or the hydrodynamic flow regime at low and high source temperatures and densi-ties, respectively. In Fig.1we find that for a given molecule the slopes at both ends of the temperature scale are identical within the experimental uncertainty 共see also Table I兲.
Be-cause the ionization efficiency of porphyrins in the QMS is not known, we cannot extract the absolute vapor pressure from our measurement. The published values for the vapor pressure of H2TPP Refs.共10 and 11兲 are different but both of them lead to a transition to the hydrodynamic flow at around 625 K. There are no published values for the vapor pressure of Fe共TPP兲Cl; however, our measurements at 625 K using a quartz balance show similar signals for both molecules. If we assume an equal sticking factor on the quartz, this tends to show that the transition occurs in the same temperature range for both molecules. This indicates that the kink at around 600 K in Fig.1. corresponds to the transition region between the two flow modes.
One uncertainty in the determination of the sublimation enthalpies is associated with the sample temperature, which is known with a relative precision of better than 1 K and an absolute accuracy of better than ±10 K. Another contribution to the total uncertainty is related to the slight nonlinearity of the enthalpy curves, in particular, at the edges of the tem-perature intervals. Our fits are therefore limited to the inter-vals marked by the straight lines in Fig. 1. Our sublimation data for H2TPP are in the range of literature values.7–10 While we did not find any reference values for Fe共TPP兲Cl, it
is interesting to note that the literature enthalpy for Fe共TPP兲 alone11is by a factor of 2 smaller than our value for Fe共TP-P兲Cl.
This is particularly noteworthy since the enthalpies asso-ciated with Fe共TPP兲Cl at m=704 amu and Fe共TPP兲 at m = 668 amu in our own experiments are nearly identical within the error bars, as shown in Fig.1and TableI. This is in agreement with a recent finding of Feil et al.12that Fe共T-PP兲Cl evaporates as an intact molecule but undergoes a strong and well-defined fragmentation upon electron impact ionization when the electron energy rises above 30 eV. The dominance of Fe共TPP兲 over Fe共TPP兲Cl in our experiments, with EI⬃70 eV, is therefore in good agreement with the expectations. In the following we will therefore assume that the molecules at mass m = 668 amu also represent free-flying Fe共TPP兲Cl.
III. INTERNAL MOLECULAR PROPERTIES AND de BROGLIE WAVES
Already in our earlier work, we showed that complex molecules may reveal their quantum mechanical wave-particle duality in a near-field interferometer, such as that shown in Fig. 2. These experiments supported the view that porphyrins may be coherently delocalized over more than 2 m, i.e., thousand times their structural size.6,13
Our present study now uses a similar setup to compare H2TPP and Fe共TPP兲Cl, which differ only by two central at-oms as shown in Fig.3. Their polarizability is rather similar, but while H2TPP has neither an electron magnetic moment nor an electric dipole moment in its ground state, we calcu-late D = 2.7 D for Fe共TPP兲Cl 共see Sec. V兲.
We therefore first investigate the possible role of these internal properties for de Broglie 共center-of-mass兲 interfer-ence in the presinterfer-ence of material diffraction gratings, in which the molecules pass gold slits as narrow as 450 nm and as thin as 500 nm.
For pointlike objects without any internal properties, matter-wave interferometry is only determined by the mo-mentum p = mv, which is related to the de Broglie
wave-length through =h/mv. In addition to that, earlier experiments13,14 showed that, in particular, the scalar polar-izability can be rather relevant for quantum interferometry. A conservative interaction potential V共r兲 between the polariz-able molecule and the grating wall causes a wave dephasing. FIG. 1. The Arrhenius plot establishes a relation between ln共IT兲 and 1/T
and thus permits to extract the sublimation enthalpy from the slope of the linear fit to the data.
TABLE I. Measured sublimation enthalpies⌬Hsubfor two porphyrin derivaties, as determined in our
experi-ment. The total count rate is given to provide a measure of the relative beam intensity. The mass peak associated with Fe共TPP兲wFe共TPP兲Cl-frgm. originates from the intact parent molecules Fe共TPP兲Cl undergoing fragmen-tation inside the ionization detector. Literature values for H2TPP range between 111 and 240 kJ/ mol共Refs. 7–11兲. For the other TPP derivative, ⌬Hsubis measured for the first time in this work.
Molecule Formula
Mass
共amu兲 共kJ/mol兲⌬Hsub
The molecule-wall interaction often turns out to be an undesired effect in quantum interferometry: The added phase ⌬depends on the transition time. For molecular beams of finite velocity spread, phase averaging may then strongly modify and also reduce the observed interference fringe visibility.13
In traversing a grating slit, the molecular center-of-mass wave function 共r兲 acquires a phase
⌬= − i ប
冕
0
V共r兲dt. 共2兲
Depending on the distance between the molecule and the grating wall, r, the attractive interaction V is best described
by the van der Waals potential at short distances,
VvdW= − C3/r3, 共3兲
the Casimir-Polder approximation at long distances,
VCP= − C4 r4 = − 3បc 3220r4 ␣, 共4兲
or a more elaborate transition formula valid for all distances.15
Typically the transition between VvdW and VCP occurs already at several tens of nanometers away from the surface, and it is therefore justified to use the Casimir-Polder approxi-mation for the description of our experiments. This molecule-wall interaction is associated with the particle’s po-larizability and quantum fluctuations of the molecular dipole moment.
For polar molecules with a permanent electric dipole moment D, we also have to consider the interaction with its own mirror image in the grating wall. If we neglect their rapid rotation at high temperature—which tends to reduce the mutual attraction—the maximum interaction potential reads16 VDD= − D2 40 1 共2r兲3. 共5兲
In order to assess the relative importance of these effects, we choose Systeme International units and assume a polarizabil-ity of ␣=共40兲共100 Å3兲, an electric dipole moment of D = 2.7 D = 8.9⫻10−30C m, and an average molecule-wall dis-tance of r = 200 nm. The Casimir-Polder energy then amounts to VCP= 1.5 neV, whereas the dipole-dipole poten-tial reaches only VDD= 0.07 neV. With regard to matter-wave interferometry and in the absence of any external electric fields, the influence of the molecular polarizability should therefore be clearly dominant.
IV. INTERFEROMETRIC MEASUREMENT OF THE MOLECULAR POLARIZABILITY
Because of its high sensitivity to external perturbations, a matter-wave interferometer may also serve for precisely determining ␣. It is important to explore this possibility, since the knowledge of biomolecular polarizabilities may also reveal information on molecular conformations, from small polyperptides17up to genuine macromolecules, such as DNA.18
For biomolecules, ␣ is often determined using optical light scattering and refractive index measurements19,20in sol-vents or on substrates. This always includes an interaction with a local environment, which can be avoided when work-ing with isolated particles in molecular beams. Classical Stark deflectometry of free beams in an inhomogeneous ex-ternal electric field has therefore become a very useful tool for the determination of molecular polarizabilities.21,22
Our Talbot-Lau deflectometer 共TLD兲, shown in Fig. 2, now combines a near-field matter-wave interferometer of the Talbot-Lau type13 with a pair of electrodes. The electrodes generate a constant electric force field in the transverse di-rection to the molecular beam to deflect the interference pat-FIG. 2. Setup of the Talbot-Lau deflectometer: The first grating selects a
spatial distribution of effective molecular sources, which then illuminate the second diffraction grating with sufficient transverse coherence to generate an interference pattern, i.e., a molecular density pattern, at the location of the third grating. The electrostatic deflector in front of the second grating shifts the molecular fringe patterns in proportion to the scalar molecular polarizability. The third grating acts as a spatially resolving mask in the detector. The inset illustrates how the electrical field leads to a lateral dis-placement of the interference pattern.
FIG. 3. Comparison of the structure of H2TPP共a兲 Fe共TPP兲Cl 共b兲. The
terns in proportion to ␣. The TLD is capable of operating with a rather wide molecular beam and has a high throughput and at the same time a high spatial resolution. Thanks to the nanostructured diffraction gratings, it may resolve molecular beam shifts even down to⌬s⬃10 nm.
The principles of Talbot-Lau interferometry13,23 and its application to metrology have already been explained before.24 In brief, the basic principles are as follows: The porphyrins are sublimated and emerge with a typical de Bro-glie wavelength of dB⬃5 pm. The longitudinal 共spectral兲 coherence length lc⬃2/⌬ is determined by the velocity selection,⌬/=⌬v/v⬃15%, and thus amounts to about 30 pm in the porphyrin experiments. The transverse coherence covers only a few tens of nanometers in width when the molecules encounter the first grating. But according to the van Cittert–Zernike theorem,25each of the many narrow and parallel slits of the first grating localizes the molecular posi-tion so well that the corresponding momentum uncertainty suffices to delocalize each single-molecule wave function co-herently over at least two neighboring slits of the second grating, 38.5 cm downstream of the first grating. Diffraction at this second nanostructure then creates an interference fringe pattern, again 38.5 cm further downstream, which is sampled by help of a third mask of identical period. This third grating is translated in the x direction by a low vibra-tion, UHV compatible piezo translation stage with solid state hinges. In order to minimize vibrations, the whole setup is sitting on an optical table and the turbo pumps are mechani-cally decoupled from the vacuum chamber by using damping bellows. Additionally, to minimize thermal drifts, the three gratings are mounted on a single steel bar connected to the chamber by only 12 contact points with less than 1 mm2 contact area for each point. Typical molecular interferograms are shown in Fig.4.
The Talbot-Lau interferometer is now complemented by two electrodes which create a constant force field Fx =␣共Eⵜ兲Ex, perpendicular to both the directions of the mo-lecular beam and the grating slits共Fig.2兲. The experimental
geometry is chosen such that an applied voltage of 5 kV will yield Fx/␣=共3.84±0.02兲⫻1013V2m−3, which varies by not more than 1% over the entire cross section of the porphyrin beam.
The interaction between the external electric field and the polarizability then leads to an interference fringe shift
⌬sx= ␣ m 共E ⵜ 兲Ex vy 2 d
冉
d 2+ K冊
, 共6兲where m is the molecular mass,vyis the longitudinal beam velocity, and d =共4.73±0.1兲 cm and K=共27±1兲 cm are the characteristic dimensions of our TLD, as indicated in Fig.2. All interference experiments are done using the same source and detector as in the sublimation studies. However, the final flux in the deflection experiments is smaller because of the longer flight path, the filtering by three nanogratings— required for coherence preparation, diffraction, and detection—and the velocity selection. The Fe共TPP兲Cl mea-surements were therefore recorded on the clearly identified mass peak at 668 amu, which showed a much stronger signal than the parent peak.
We first select a given velocity band共for details see Ref.
13兲, namely, v¯=222 m/s, ⌬v/v=22% for H2TPP and v¯ = 180 m / s, ⌬v/v=16% for Fe共TPP兲Cl. Each interference curve is then recorded by shifting the third grating in steps of several 10 nm and by measuring the number of passing mol-ecules at each of these steps. We vary the deflection voltage
U for each grating position, within the interval of 0 – 15 kV
for H2TPP and between U = 0 and 13 kV for Fe共TPP兲Cl. Two typical interference curves—with and without ap-plied deflection voltage—are shown in Fig.4for both H2TPP and Fe共TPP兲Cl. The data are fitted by the expected sinusoidal fringe shape, and they also include a linear term to take into account an overall decrease of the signal with time.
The curves show a slightly larger period than expected. The stretching factor is, however, consistent with the known thermal drift of about 2 – 3 nm/ min, which has already been observed in earlier experiments in this machine. The influ-ence of this drift on the polarizability measurements is strongly reduced by recording the curve shift for all deflec-tion voltages in a row before moving the grating to the next step of the interference curve.
In Fig.5we plot the observed interference fringe shift as a function of the applied voltage. A quadratic fit to these data then yields the experimental polarizability, according to Eq.
共6兲. The results are presented in Table II together with the theoretical values that we compute in Sec. V.
The error bar on␣includes the uncertainties of the elec-tric deflection field, the velocity distribution, the geomeelec-trical factors共L,d,K兲, as well as the isotopic mass distribution. It turns out that the electric field and the experimental geom-etry are still the main factors in the present setup and amount to a systematic error of ⌬␣/␣⬃6%. The other errors are FIG. 4. An external electric force field serves to deflect the Talbot-Lau interference fringes for polarization measurements. 共a兲 H2TPP, without
related to the accuracy of the high voltage measurement 共0.5%兲, the knowledge of the mean velocity 共1%兲, and the fit to the interference fringe visibility共3%兲.
The interference contrast also depends on the spread of the velocity distribution. Slow molecules will acquire a larger deflection than fast ones. Therefore, a broad velocity distribution causes a dephasing which reduces the contrast at high voltages. We therefore limit the maximum voltage to
U = 15 kV, where the fringe shift can still be read reliably.
In Fig. 6 we show the clear influence of this voltage dependent dephasing on the fringe visibility for both H2TPP and Fe共TPP兲Cl. The decay behavior is obviously different. The polarizabilities of both species are measured and com-puted to be nearly identical. However, the velocity distribu-tions in both experiments were different. The velocity spread of Fe共TPP兲Cl was somewhat smaller 共16%兲 than that of H2TPP共22%兲, and Fe共TPP兲Cl was also 25% slower than the lighter porphyrin derivative. The quadratic dependence of the fringe shift with vy explains already most of the observed difference.
At the prevailing temperature of 650 K, all molecules are in highly excited rotational states and enter the interfer-ometer with a randomly oriented principal rotation axis.
De-pending on the internal molecular temperature, the force of the electric field gradient on the dipole moments may there-fore broaden the molecular beam in both horizontal direc-tions and thus blur the interference pattern additionally. But better resolved velocities are required 共and possible兲 in the future to allow a distinction of the different effects related to the dipole moment and the polarizability from the reduction of the interference contrast.
V. FINITE TEMPERATURE AB INITIO POLARIZABILITY CALCULATIONS
In general the molecular polarizability may strongly vary with the molecular conformation, as has been shown, in par-ticular, for polypeptides.17 It is therefore important to com-pute the structural form and corresponding polarizability of the porphyrins for realistic temperatures in our experiments. We have carried out Car-Parrinello molecular dynamics simulations26–28 of H2TPP at 650 K in an orthorhombic pe-riodic unit cell of size 22⫻22⫻18 Å3 using a time step of 4 a.u. and a fictitious electron mass of 400 a.u. The elec-tronic structure was described by density functional theory using the PBE exchange-correlation functional29 with a plane-wave basis set truncated at 25 Ry in conjunction with Vanderbilt ultrasoft pseudopotentials.30The temperature was controlled using a Nosé-Hoover chain thermostat31–33 for each degree of freedom.
For ten structures picked at random from the trajectory 共one every 0.5 ps兲, the polarizability was calculated using two different protocols. Using the variational density func-tional perturbation theory approach34,35 implemented in the CPMDpackage,26the polarizability tensor was calculated in a large periodic box of size 24⫻24⫻20 Å3.
In order to be able to exploit this feature ofCPMD, we switched to the BLYP functional,36,37 norm-conserving pseudopotentials of the Troullier-Martins type,38and a plane-wave cutoff of 70 Ry. The resulting thermally averaged po-larizability is共117.5±2.5兲 Å3. In a second series of calcula-tions, the thermally averaged polarizability was computed to be 共112.4±2.4兲 Å3 with the GAUSSIAN package39 using the FIG. 5. Interference fringe shift as a function of voltage for共a兲 H2TPP and
共b兲 Fe共TPP兲Cl. The experimental data are shown as full circles. The error bars represent the statistical uncertainty of fits to interference curves, such as shown in Fig.4. The solid lines represent a quadratic fit to the data weighted by the uncertainty bars.
TABLE II. Comparison of experimental and computed polarizability values. The experimental uncertainties include statistical and systematic uncertain-ties, respectively. The theoretical polarizabilities were calculated by ab initio methods as described in the text. For H2TPP we give a thermal average of
the polarizability, corresponding to a temperature of 650 K, and the standard deviation between various thermally accessible molecular configurations. Molecule Formula Mass共amu兲 ␣expt共Å3兲 ␣mod共Å3兲
H2TPP C44H30N4 614 105± 4 ± 6 112.4± 2.4
Fe共TPP兲Cl C44H28ClFeN4 704 102± 9 ± 6 113.4
FIG. 6. Reduction of the interference fringe visibility as a function of the applied deflection voltage. At finite spread of the longitudinal velocity ⌬vy= 10% – 20%, the interaction between the polarizability and the external field gradient leads to a spreading of the interferograms, according to Eq. 共6兲, and therefore to a reduced fringe contrast. H2TPP共full circles兲 shows a
same ten structures, the same functional 共BLYP兲, and the 6-31+ G*basis set. It is interesting to note that for the opti-mized structure we obtain a value of 116.5 Å3 with CPMD and 111.4 Å3 withGAUSSIAN.
We conclude that the effect of thermal fluctuations on the polarizability is comparatively minor, even at such el-evated temperatures. We therefore calculated only the polar-izability of an optimized structure for the case of Fe共TPP兲Cl. For the geometry optimization, we used spin-unrestricted DFT, assuming a multiplicity of 6, with the BLYP functional and the 6-31G* basis set. The polarizability was computed using the 6-31+ G* basis set yielding a value of 113.4 Å3. This calculation also predicts the dipole moment of Fe共TP-P兲Cl to be 2.7 D, while it vanishes for H2TPP.
VI. CONCLUSION
We have successfully used near-field Talbot-Lau interfer-ometry for determining the scalar polarizability of two por-phyrin derivatives for the first time. Our experimental results are in good agreement with our ab initio calculations. The experimental accuracy is presently limited to a few percent, given by the finite knowledge of the electric field distribution and the molecular velocity distribution.
Earlier experiments using far-field Mach-Zehnder interferometry40reached even an accuracy of better than one permille for the polarizability of neutral sodium atoms. But such an instrument is not scalable to large masses and small de Broglie wavelengths.
In contrast to that, the Talbot-Lau deflectometer pre-sented here has the advantage of being scalable to higher masses while preserving a high transmission and high spatial resolution. The accuracy can be improved in future experi-ments: Calibrating the field and all geometric factors with atoms of known polarizability, such as cesium,41 sodium,40 or lithium,42will improve the field values to better than 1%. The velocity resolution can also be improved by more than a factor of 10 in the future, for instance, by changing from effusive beams for small biomolecules to pulsed laser desorption and pulsed photoionization detection as typically used for larger polypeptides. Since the molecular velocity enters quadratically into the molecular beam shift, the in-creased resolution of ⌬v/v⬍1% will be very important in future experiments, in particular, for discriminating the ef-fects of polarizability from that of the static permanent elec-tric dipole moment. At such a high v resolution, nonpolar
molecules are expected to show a flat curve in a visibility-versus-voltage diagram, whereas polar molecules would still show a loss of interference contrast. Improving⌬v also has the additional advantage that higher deflection voltages can still be used and larger shifts may be reached. Future im-proved experiments with an absolute accuracy of 1% will thus be important for validating competing numerical models for complex molecular systems.
ACKNOWLEDGMENTS
This work is supported by the Austrian FWF through SFBF1505, STARTY177-2. One of the authors共S.D.兲 is sup-ported through a Royal Thai Government scholarship. Two
of the authors 共A.S. and E.R.兲 were supported by the Euro-pean Commission in the RTN network QUACS 共HPRN-CT-2002-00309兲. Another author 共N.L.D.兲 gratefully acknowl-edges the Centre for High Performance Computing of the RWTH Aachen.
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