Molecular Plasmonics: Strong Coupling at the Low Molecular Density

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Molecular Plasmonics: Strong Coupling at the Low Molecular Density Limit Lihi Efremushkin, Maxim Sukharev, and Adi Salomon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b04096 • Publication Date (Web): 19 Jun 2017 Downloaded from http://pubs.acs.org on June 20, 2017

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The Journal of Physical Chemistry

Molecular Plasmonics: Strong Coupling at the Low Molecular Density Limit Lihi Efremushkin†, Maxim Sukharev‡ and Adi Salomon†* †Department of Chemistry, Institute of Nanotechnology and Advanced Materials (BINA), Bar-Ilan University, Ramat-Gan 5290002, Israel. ‡College of Integrative Sciences and Arts, Arizona State University, Mesa, AZ 85212, USA

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ABSTRACT The strong coupling between the molecular excited state and the plasmonic modes of silver hole arrays is studied both experimentally and theoretically for a resonance frequency very close to the asymptotic line of the plasmonic dispersion relation, in the nonlinear regime. It is shown that the strong coupling regime can be achieved between the two sub-systems at low molecular concentrations with negligible damping of the electromagnetic field. The results are supported by rigorous numerical simulations showing that the strong coupling is observed when the molecular transition lies within the nonlinear regime of the dispersion relation rather than the linear regime.

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INTRODUCTION Strong interaction between photonic modes and molecular states can lead to hybrid materials with new photophysical properties.1–5 In the strong coupling regime, damping rates are small compared to the light-matter interaction, and the exchange of energy between excited molecular states and photon states occurs on a very short time scale. Such interactions are expected to modify the dynamics of the molecular system and may lead to hybrid materials with unique optoelectronic properties.4–9 When optically coupled molecules and photonic modes form hybrid states, new pathways for energy redistribution may be opened, leading to new photochemistry occurring on metallic surfaces. These perspectives provide a new way to tailor the physical properties of a given molecular system, namely, to modify the molecular surface potential energy and control the energy-redistribution pathways, electron-transfer processes and radiative emission. For example, if coupling to the radiation field occurs, a change in the molecular emission can be observed; thus, when a coherent mixed exciton-plasmon (or exciton-cavity) mode is formed, a collective response of the whole system is observed. This is evident from the dependence of the Rabi-splitting values associated with the exciton-plasmon coupling on the molecular density.10 Traditionally, strong coupling has been studied using micro-cavity resonators and other solid-state systems,2 which demand cryogenic temperatures and state-of-the-art fabrication. Recently, the strong coupling regime has been realized with metallic nanostructures in which the plasmonic modes, coherent oscillations of the metal’s free electrons, replace the photonic modes of the microcavities.1,7,11,12 These metallic nanostructures have been found to reach the strong coupling regime at room temperature due to the confinement of the electromagnetic field to a deep subwavelength volume. The strong confinement of these photonic (plasmonic) modes, and the ability to control both their 3



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propagation and localization on the surface, at any given optical frequency, are the main reasons to study their interaction with molecules.13 So far, the strong coupling regime has been observed using subwavelength arrays and metallic nanoparticles.1,10,11,14–18 Most of the experimental reports are on an ensemble of excitons (molecules or quantum dots) and nanoparticles. Recently, the strong coupling regime has been realized using bowtie nanoantennas and semiconductor quantum dots.19 The enhanced electromagnetic field trapped in the very small volume between the particles is responsible for achieving the single quantum-emitter limit. The strong coupling regime was also realized using single molecules, where similar linear optical measurements were observed.20 Still, there are ambiguities in the interpretation of the optical linear measurements, which constitute the predominant method to observe the strong coupling regime in plasmonic systems. Specifically, the observed splitting between the two sub-systems, i.e. molecules and plasmons, may be due to enhanced absorption rather than strong interaction between excitons and plasmons.10 The strong coupling regime is typically observed using a relatively high concentration of Jaggregate molecules.1,9,11,14,17,21–24 The molecules are physically adsorbed onto a metallic structure (spin-coated or drop-cast), and the color of the thin molecular layer can be seen by the naked eye. In these cases, changes in the refractive index followed by strong molecular dipole transitions also play a role in the observed spectrum of the so-called hybrid system. Clearly, both oscillator strength and high concentration are critical for reaching the strong coupling regime at room temperature. We note that for very high molecular concentrations, strong Rabi-splitting values up to about 700 meV have been observed,25 which is about 30% of the molecular transition-state energy, yet a clear change in the nature of the molecular system has not been observed. Only a few studies 4


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show a modification in the nature of the molecular system and its dynamic behavior due to strong coupling to plasmonic modes.1,5,7,9 It is reasonable to assume that in such systems, only some of the molecules strongly interact with the plasmonic modes, whereas the other molecules are weakly coupled, and their interaction results in damping of the transmitted light. Here, we reach the strong coupling regime using a relatively low molecular concentration of 0.2 mM and an absorbance of about 0.015, an order of magnitude lower than those used in other studies. The reason we could achieve the strong coupling at such a low molecular concentration is that our molecular system has a transition state that lies whithin the nonlinear region of the plasmonic dispersion, very close to the asymptotic line. In that region, light–matter interactions are significantly enhanced, and a small modification in the refractive index of the dielectric medium in proximity with the plasmonic system should lead to strong modulations in the dispersion relation.26 EXPERIMENTAL Sample preparation: Sub-wavelength hexagonal hole arrays were milled by a focused ion beam (FIB, Helios 600, FEI) in sputtered silver films of 150 nm and 250 nm thickness on glass substrates. The diameter of the holes was kept constant at 166 nm as the array period varied from 350 to 400 nm (increments of 10 nm). 100-nm thick polyvinyl alcohol (PVA, 89,000–98,000 Mw, Aldrich) with and without the dye molecule H 2 TPPS 4 (4, 4’, 4’’, 4’’’-(Porphine-5, 10, 15, 20-tetrayl) tetrakis (benzenesulfonic acid) tetrasodium salt hydrate (Aldrich)) in basic form, pH = 6–7, was spincoated on the sample. Optical measurements:

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Spectral measurements and optical imaging of the hole arrays and molecular layer were conducted using an inverted light microscope in transmission mode (IX83, Olympus). The illumination source was a halogen lamp (100 W, Olympus). The spectroscopic data were obtained using a spectrograph (IsoPlane SCT320, Princeton Instruments), the LightField program and a CCD camera (PIXIS 1024, Princeton Instruments). When using the spectrograph, the measurements were performed with the X20 objective (N.A. 0.25) and 500-nm blaze gratings (grating density 1200 g/mm and 300 g/mm) in the spectral range of 370 to 480 nm and 380 to 520 nm respectively. Rabi-splitting values were extracted from the transmission spectra of the hybrid system for a set of molecular densities (see Table S1). The Rabi-splitting value is the difference in energy between the higher polariton and lower polariton, and was taken for the resonant case, i.e. when the plasmonic modes are resonant with the molecular transition. The UV-vis spectra, for the TPPS 4 -covered glass, were collected by a CARY Bio-100 spectrophotometer. PVA-coated glass was used as the reference. The molecular concentration (or density) was calculated (see Table S1) for an oscillator strength of 6.82×106 M-1cm-1, based on the Beer–Lambert law. The molecular layer thickness was ~100 nm, based on a cross-section measurement performed by FIB. Theory section: Optical properties of the studied silver hole arrays are simulated using the finite-difference timedomain approach.27 We numerically integrate Maxwell’s equations �⃗ 𝜕𝜕𝐵𝐵 = −∇ × 𝐸𝐸�⃗ , 𝜕𝜕𝜕𝜕 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 𝜀𝜀0

(1.1)

𝜕𝜕𝐸𝐸�⃗ 1 �⃗ − 𝐽𝐽⃗ = ∇ × 𝐵𝐵 𝜕𝜕𝜕𝜕 𝜇𝜇0 6


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where 𝐽𝐽⃗ corresponds to the current density in the metal regions or the polarization density in the spatial regions occupied by molecules. The dielectric function of silver is taken in the form of the Drude function 2 𝛺𝛺𝑝𝑝

𝜀𝜀(𝜔𝜔) = 𝜀𝜀𝑟𝑟 − 𝜔𝜔2 +𝑖𝑖𝑖𝑖𝑖𝑖 ,

(1.2)

where28 𝜀𝜀𝑟𝑟 = 8.926, Ω𝑝𝑝 = 11.585 𝑒𝑒𝑒𝑒, Γ = 0.203 𝑒𝑒𝑒𝑒, and 𝜀𝜀𝑒𝑒𝑒𝑒𝑒𝑒 = 𝜀𝜀𝑟𝑟 in (1.1). To account for the dispersion (1.2) in the time domain, we follow the conventional auxiliary differential-equation method,29 leading to the following time-dependent equation for the density current: 𝜕𝜕𝐽𝐽⃗ 𝜕𝜕𝜕𝜕

+ 𝛤𝛤𝐽𝐽⃗ = 𝜀𝜀0 𝛺𝛺𝑝𝑝2 𝐸𝐸�⃗ .

(1.3)

To account for the optical response of molecules doped in PVA, we propagate a set of rate

equations describing interacting two-level emitters:6 𝑑𝑑𝑛𝑛2 1 = �𝐸𝐸�⃗ ∙ 𝑃𝑃�⃗ � − 𝛾𝛾1 𝑛𝑛2 , 𝑑𝑑𝑑𝑑 ℏΩ0 2 �⃗

𝑑𝑑 𝑃𝑃 𝑑𝑑𝑃𝑃�⃗ + 𝛾𝛾 + Ω20 𝑃𝑃�⃗ = 𝜎𝜎(2𝑛𝑛2 − 𝑛𝑛0 )𝐸𝐸�⃗ , 2 𝑑𝑑𝑡𝑡 2 𝑑𝑑𝑑𝑑

(1.4)

where 𝑛𝑛2 is the number density of the excited molecules, 𝛾𝛾1 is the radiationless lifetime of the excited state, Ω0 is the molecular transition frequency, 𝛾𝛾2 = 𝛾𝛾1 + 2𝛾𝛾𝑑𝑑𝑑𝑑 , 𝛾𝛾𝑑𝑑𝑑𝑑 is the pure dephasing

rate, 𝑛𝑛0 is the total number density of the molecules, 𝑃𝑃�⃗ is the macroscopic polarization, and 𝜎𝜎 = 2Ω0 𝑑𝑑02

is the coupling parameter with the molecular transition dipole 𝑑𝑑0 .30 The coupling between

3ℏ

Maxwell’s equations (1.1) and the rate equations (1.4) is achieved via polarization current density �⃗

𝑑𝑑𝑃𝑃 𝐽𝐽⃗ = 𝑑𝑑𝑑𝑑 in the spatial regions occupied by molecules.

The system is excited by a plane wave at normal incidence generated using a total field/scattered

field approach.27 The periodic boundaries are imposed transversely to the incident field 7



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propagation. The absorbing boundaries in the form of convolutional perfectly matched layers (CPM) 31 are set on both sides of the array at a distance of 1 μm. The convergence of all spectra shown is achieved at the spatial resolution of 1.5 nm. The size of the simulation domain varies depending on the period. The typical size of the domain is 250×250×1440. We parallelize our codes using message-passing interface and propagate coupled Maxwell-rate equations on 72 processors at Thunder and Topaz SGI ICE X clusters. The typical execution time to obtain transmission/reflection spectra is about 15 hours. RESULTS AND DISCUSSION Figure 1 shows a modulation of surface plasmons (SPs) dispersion relation following interactions with molecular layer. The molecuar layer has a relatively sharp transition state. We examine three types of molecules with the same physical properties, such as oscillator strength, concentration and lifetime, but with different transition states as is shown in Figure 1a. A very strong modulation of the dispersion relation is observed when the molecular transition lies near 420 nm, a moderate one when the molecular transition is at 530 nm and a smaller one when the transition lies at 620 nm. In fact, for the case of Figure 1d, we note that the group velocity, 𝑣𝑣𝑔𝑔 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑

, is very small compared to the other cases (Figures 1e and 1f). When the group velocity

vanishes, strong light-matter interactions are dramatically enhanced.32 Therefore, the strong coupling regime should be more pronounced in this case.

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Figure 1: (a) Schematic illustration of the system used for the simulation. The system is composed of a flat semi-infinite silver film in one dimension placed between glass (fused silica) and a layer of H 2 TPPS 4 -doped PVA. (b) Simulated absorbance spectrum of dye molecules H 2 TPPS 4 with an excited state at 420 nm (blue line), at 530 nm (red line) and at 620 nm (brown line). (c-f) Dispersion curves of surface plasmons of a flat semi-infinite silver in one dimension: black line is for metal covered by a dispersionless polymer, blue line corresponds to metal covered by dye molecules resonant at 420 nm (blue line in (a)), red line is for molecules resonant at 530 nm (red line in (a)), and brown line is for molecules resonant at 620 nm (brown line in (a)). The green dashed line illustrates the boundary between the linear and non-linear regions. The dispersion curves are based on experimental data for silver (n and k) taken from Johnson and Christy.33

Following the explanation above, we chose to work with H 2 TPPS 4 , a porphyrin derivative monomer whose absorption band is at ~3 eV (420 nm) with an absorption coefficient of ε = 7×106 M-1cm-1. The absorption spectrum of a thin layer of 0.23 mM (for correlation to density see Table S1) H 2 TPPS 4 embedded in PVA is shown in Figure 2a. We deposited those molecular 9



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layers onto hexagonal silver hole arrays and studied their interaction with the molecular transition state. Figure 2b shows a schematic illustration of the device we used for this study: a layer of H 2 TPPS 4 -doped PVA is spin-coated on a hexagonal silver hole array.

Figure 2: (a) Experimental absorbance spectrum of H 2 TPPS 4 -doped PVA spin-coated onto glass. The peak position of the S 5 exciton band is at ~3 eV (420 nm). The absorbance is ca. 0.016. Inset: H 2 TPPS 4 molecule structure. (b) Schematic illustration of the system used. The system is composed of about 200 nm thick fabricated Ag hole array placed between SiO x and a layer of PVA or H 2 TPPS 4 -doped PVA of about 100 nm (green layer). Following previous studies,11,34 we chose to work with plasmonic modes of a hexagonal hole array. The coupling to the free-propagating light is represented by Bragg scattering, as expressed by the equation35–37 ,

(2)

where i and j are integers, k || is the incident light wave vector, k sp is the SPs wave vector, and G x and G y are the reciprocal lattice vectors for which |G x | = |G y | = 2π/p, where p is the lattice period. For the symmetry of our plasmonic system, the peak positions at a specific wavelength λ are described, to a first approximation, by the following expression35–37 𝜆𝜆𝑚𝑚𝑚𝑚𝑚𝑚 =

𝑝𝑝

4 � (𝑖𝑖 2 +𝑖𝑖∙𝑗𝑗+𝑗𝑗 2 ) 3

𝜀𝜀𝑚𝑚 𝜀𝜀𝑑𝑑

�𝜀𝜀

𝑚𝑚 +𝜀𝜀𝑑𝑑

,

(3)

where ε m and ε d are the permittivity of the metal and the dielectric material in contact, respectively, and the integers i and j are the scattering orders. According to Equation (3), tuning the plasmonic 10


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modes to certain frequencies can easily be attained by changing the array periodicity p and, therefore, can provide a systematic tool to study their interaction with molecular transition states.

Figure 3: Experimental transmission spectra of the hybrid system which consists of a hexagonal hole array covered with H 2 TPPS 4 -doped PVA with an Ag film thickness of (a) 250 nm and (b) 150 nm. The green line is a guiding line to indicate the molecular transition state (c) Rabi-splitting values as a function of the square root of the H 2 TPPS 4 density for two types of Ag film thickness: blue dots denote 250 nm and black dots denote 150 nm thick film (see also Fig. S3).

Figures 3a and 3b show the transmission spectrum of a hexagonal hole array, in which the plasmonic mode at normal incidence is located at ~3 eV. This mode is due to coupling of SPs propagating in the second Brillouin zone. When molecules are deposited, the transmission spectrum of the plasmonic device changes dramatically. The original peak (black line) splits into two new eigenmodes, the lower and upper polaritons, and a broadening of the peak is observed. We note that the splitting value is larger compared to the full width at half maximum of the original plasmonic mode, as expected for strong coupling systems.7 In Figure 3b, the splitting is nonsymmetric, whereas the lower polariton gives rise to a moderate enhancement in the transmission spectrum. Interestingly, almost no damping of the electromagnetic field is observed in this region, and even a small-to-moderate enhancement is indicated. Notably, damping does occur at lower

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energy regime (Figure 3a at ~2.6 eV), probably due to light absorption by molecular transition states at lower energy levels. Those states interact only weakly with the plasmonic modes. We repeated these experiments for several molecular concentrations and hole arrays of different thickness and periodicity, and found that the Rabi-splitting values scale linearly with the square root of the molecular density, as expected (Figure 3c).2-3 The strong coupling in our studies is achieved at very low molecular concentrations and low optical densities (see also Fig. S3) compared to other studies (see Table 1). For example, Sugawara et al.14 observed a splitting of about 230 meV in a concentrated molecular film with relatively high optical density (absorbance of 0.9). This is not surprising, since the molecular transition state lies at 1.85 eV, in the linear regime of the dispersion relation, and therefore a large amount of molecules is needed. Indeed, when approaching the nonlinear regime, with the molecular transition state lying at ~2.5 eV (Salomon et al.11), the same Rabi-splitting values of 217 meV are observed with lower optical density (absorbance of 0.46).

Table 1: Comparison of Rabi-splitting values of different molecular plasmonic systems. For comparison with our system, the estimated Rabi-splitting for absorbance of 0.2 are in brackets. Plasmonic system

Molecular transition (eV)

Absorbance

12


Rabisplitting

Ref

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value Ω (meV) Hexagonal

2.9

0.2

Hexagonal

2.53

0.46

210 312

Current study [11]

(205)

Silver hole array

217 Square

2.53

0.46

[11]

(143) 250 Square

1.78

0.8

[22]

(125) Gold nanorods

1.99

0.2

Silver nanoprisms

2.11

2

190 295

[23] [24]

(93)

Nanoparticles Spherical nanovoid arrays of 1.85 nanostructured gold

230 0.9

[14]

(108)

The experimental results were simulated using homebuilt finite-difference time-domain codes that numerically integrate coupled Maxwell-rate equations. The simulations details are discussed in the theory section. The model is based on rate equations and can only be used to describe H 2 TPPS 4 qualitatively. First, we simulated the transmission spectra of a set of hexagonal hole arrays covered with a PVA layer. The transmission spectra are shown in Figure 4a. The transmission peak at 3.032 eV for a period of 330 nm corresponds to experimental results of a 360 nm period (see Fig. S1). The corresponding steady-state spatial distributions of the electromagnetic intensity are shown in Figures 4b and 4c. The high energy mode with a frequency near 3 eV is 13



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highly localized inside the PVA layer near the metallic surface, resulting in very efficient coupling. We shall note that the observed localized fields within the PVA layer are due to the fact that both PVA and substrate have the same refractive index; otherwise broadening of the mode would have been observed.

Figure 4: (a) Simulations of transmission spectra for the periodic array of holes for the periods of 330 nm (black line), 350 nm (red line), 340 nm (green line) and 360 nm (blue line). The hole diameter is 166 nm, the thickness of the silver is 150 nm, and the PVA layer on the input side is 100 nm thick. The output side (substrate) is simulated as a semi-infinite, nondispersive dielectric with a refractive index of 1.5, corresponding to glass. Panels (b) and (c) show the electromagnetic intensity distributions (normalized with respect to the incident intensity). We use a log-scale to plot

EM

intensity

distributions.

The

color

bar

is

log 10 (EM intensity / incident intensity). The plot is for the resonance of 3.032 eV and period of 330 nm. The incident field propagates along the z-axis from top to bottom in panel (c) and is horizontally polarized. The transverse distribution shown in panel (b) is evaluated inside the PVA layer at 50 nm above the metal.

Next, we simulate the interaction of the plasmonic modes near 3 eV with molecular transition energy at 3.05 eV. Figure 5a shows the transmission spectra of the hybrid system at various molecular densities. The strong coupling regime is clearly observed for all molecular densities. For the molecular parameters specified in the figure legend, the strong coupling is achieved at densities of 1025 m-3 and greater.7 Moreover, we observed the appearance of a third mode (orange 14


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and violet curves) for high molecular concentrations. The physics of this mode is related to the long-range dipole–dipole interactions enhanced by the plasmon field, and is discussed elsewhere.7 Upon extracting the values of the upper and lower polariton modes and plotting their differences as a function of the molecular density, we observe the conventional square-root dependence as is shown in Figure 5b.3

Figure 5: Simulation of the experimental hybrid system for different molecular densities: Panel (a) shows theoretical transmission as a function of the incident photon energy calculated for the period of 330 nm. Transmission with no molecules in PVA is shown as a black line. Simulations with molecules doped in PVA for molecular concentrations are 1025 m-3 (red line), 3×1025 m-3 (green line), 5×1025 m-3 (blue line), 7×1025 m-3 (violet line) and 1026 m-3 (orange line). Panel (b) shows the values of the Rabi-splitting extracted from the transmission spectra as a function of the square root of the number density (m-3)1/2. The dashed line shows the linear fitting. Other parameters used in the simulations are the molecular transition energy (3.05 eV), the transitiondipole moment (10 Debye), the radiationless lifetime of the excited state (4.14×10-3 eV), the pure dephasing rate (1.25×10-2 eV), the PVA thickness (100 nm), the silver thickness (150 nm) and the hole diameter (166 nm). We numerically studied the same hybrid system but when the molecular transition state is at 2.33 eV (linear regime). The results are shown in Figure S2. These results show noticeably lower Rabi-splitting values for the same molecular densities and molecular dipole strength. Indeed it 15



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consisted with our study that coupling to plasmonic mode in the nonlinear regime leads to stronger interaction. Finally, we study the same hybrid system but when the plasmonic modes are modified to be on/off resonance with respect to the molecular transition states. The hexagonal hole-array periodicity is varied from 330 to 450 nm, while the hole diameter is kept constant, d=166 nm. The arrays are covered with 100 nm film of H 2 TPPS 4 doped in PVA, where the molecular layer adsorption is about 0.015, indicating a molecular density of 1.8×1020 (see Table S1). We note that the plasmonic modes were aligned to be either resonant or lower in energy with respect to the molecular transition state since the molecular transition state is very close to the plasmonic asymptotic line. The plasmonic system transmission spectra are shown in Figure 6a. The observed peaks are red-shifted with increasing of the periodicity, according to the geometrical parameters of the hole array (see Equations 2 and 3). Figure 6b shows a set of transmission spectra of the hybrid system with a clear splitting at the molecular level. When the SPs modes are resonant (blue curve), the splitting is symmetric, whereas when they are off-resonant, the splitting is asymmetric. At 400 nm periodicity, the higher polariton is barely observed, which is consistent with the fact that the molecular system is very close to the asymptotic line.

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Figure 6: Experimental transmission spectra of hexagonal hole arrays of Ag with periodicity varying between 350 nm (blue) and 400 nm (pink) (increments of 10 nm), coated with (a) PVA and (b) H 2 TPPS 4 -doped PVA. The thickness of the Ag film is 150 nm. The vertical line at 420 nm (~3 eV) is for the S 5 absorption band. By changing the periodicity, the transmission peak can be tuned to be on/off resonance with respect to the molecular exciton. When the transmission peak is resonant, splitting in the SP mode occurs when the surface is coated with the molecular system, indicating strong coupling between the molecular system and the SP modes. CONCLUSIONS In conclusion, we show that strong interaction between plasmonic modes and molecular transition states is enhanced when the molecular transition lies in the nonlinear regime of the dispersion relation. Thus, even a very low molecular density can reach the strong coupling regime. The coupling results in the formation of hybrid states that are different from the original uncoupled modes with respect to their energy levels, dispersion and dynamics. Therefore, it should be possible to use strong coupling to tune the properties of a material for a given application in areas such as photochemistry and optoelectronic devices.38

ASSOCIATED CONTENT The supporting information is available free of charge: 17



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The relation between the molecular absorbance and its concentration and density in the polymer, comparison between calculated and experimental transmission spectra, transmission spectra of molecules with transition states at the linear and non-linear regimes, and an example of a hybrid system with two molecular layers of 1 nm each. (PDF)

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] ACKNOWLEDGMENT The experimental part is supported by GIF grant No. 203785. The numerical modeling performed by M. S. is supported by the Air Force Office of Scientific Research under grant No. FA9550-15-1-0189 and the Binational Science Foundation under grant No. 2014113.

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Molecular Plasmonics: Strong Coupling at the Low Molecular Density

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