Despite the potential paradigm breaking capability of microcavities to control chemical processes, the extent to which photonic devices change properties of molecular materials is still unclear, in part due to challenges in modeling hybrid light-matter excitations delocalized over many length scales. We overcome these challenges for a photonic wire under strong coupling with a molecular ensemble. Our simulations provide a detailed picture of the effect of photonic wires on spectral and transport properties of a disordered molecular material. We find stronger changes to the probed molecular observables when the cavity is redshifted relative to the molecules and energetic disorder is weak. These trends are expected to hold also in higher-dimensional cavities, but are not captured with theories that only include a single cavity-mode. Therefore, our results raise important issues for future experiments and model building focused on unraveling new ways to manipulate chemistry with optical cavities. Optical microcavities affect chemical processes via strong light-matter interactions, but the extent to which chemistry can be controlled with these materials remains unclear. Here, the author employs microscopic simulations of multimode photonic wire under strong coupling with a disordered molecular ensemble to investigate the influence of the cavity resonator on the local molecular density of states and excited-state diffusion. Much of the observed phenomenology stems from the hybridization of the collective material polarization and the resonances of an optical cavity, which leads to the formation of delocalized polariton modes when the energy exchange between the collective molecular excitations and the photonic structure is faster than the dissipative processes acting on each subsystem15. Polaritonic states are always accompanied by molecular states weakly coupled to light16. The latter are also sometimes called “dark states”, as the optical response of strongly coupled molecular ensembles is dominated by polaritonic excitations. The weakly coupled states form a reservoir containing the vast majority of the states of the system with significant molecular character18, and therefore, they play an essential role in equilibrium and non-equilibrium molecular phenomena in optical cavities23.Strong light–matter interactions hosted by nanostructures and optical microcavities can induce significant and qualitative changes to chemical processes18. Therefore, it is puzzling that significant changes in thermal reaction rates and branching ratios of some organic systems29 were observed under conditions of infrared collective strong coupling. The dominance of molecular reservoir modes over the polaritonic31 suggests there is no simple explanation of the cavity effect on thermal reactions based on transition-state theory34. This notion has motivated the hypothesis that cavity-induced changes to chemical reactions originate from dynamical effects of the electromagnetic environment on intramolecular dynamics37 .Polaritons dominate the optical response of a strongly coupled device, but the reservoir states are much more numerous43 where the cavity is modeled as a single bosonic mode and the molecular system has permutational symmetry, i.e., it consists of an ensemble of identical NM ≫ 1 two-level system with equal transition energy and dipole moment18. In these models, hybrid light–matter excitations  extending over the entire system emerge from the interaction of the homogeneous cavity field with the molecular bright mode corresponding to the totally symmetric combination of molecular states where a single molecule is excited. The other NM−1 molecular modes are degenerate and correspond to the dark reservoir discussed above. Permutationally invariant models deviating from this Tavis-Cummings (TC) picture via the introduction of exciton-phonon interactions46 have also been thoroughly investigated.Given the complexity of polaritonic systems with states delocalized over length scales of the order of the optical wavelength, much of what is known about collective strong coupling relies on quantum mechanical simulations of effective models17 who showed that the presence of energetic and structural disorder lead to weak photonic intensity borrowing by the reservoir states, but only minor changes to the TC picture when the collective light–matter interaction ΩR/2 (Rabi splitting) is much greater than the mean fluctuation σ of the material transition energies. Recently, Scholes revisited the polariton coherence protection in single-mode (0D) cavities47, whereas Botzung et al.48 provided quantitative localization properties of the reservoir states of an energetically disordered emitter ensemble under strong coupling to a single spatially homogeneous cavity mode. Both studies showed in agreement with ref. 17 that polariton coherence is largely unaffected by energetic disorder weaker than the collective light–matter coupling, but also noted that dark modes inherit spatial delocalization .The effects of material imperfections on polaritonic and dark states were probed in early work by Houdre et al.52. For example, in an FP cavity, each electromagnetic mode is characterized by its polarization (TE or TM), cavity order m = 1, 2, 3... , and the essentially continuous in-plane wave-vector q = . In systems such as FP cavities, the disorder is known to severely restrict polariton space-time coherence53 and may lead to weak or strong Anderson localization as well as diffusive and ballistic transport depending on the initial wave-packet, disorder strength, cavity geometric parameters, and magnitude of light–matter interactions54. Numerical simulations have shown that polariton wave functions can be localized over different length scales depending on their mean wave-vector57 .However, the majority of photonic materials employed in polariton chemistry research are multimode Fabry-Perot (FP) or plasmonic cavities with a continuous spectrumDespite their prominence, much less attention has been paid to multimode strong coupling effects on the reservoir states of structurally and energetically disordered molecular ensembles. These states form the majority of the excitations with significant molecular character, and therefore, they largely determine the equilibrium and transport properties of a molecular subsystem under collective strong coupling with an optical cavity.Here, we employ numerical simulations of the microscopic states of a multimode photonic wire under strong coupling with a disordered molecular ensemble to investigate the influence of the cavity on the local molecular density of states and the exciton return probability. These quantities essentially reflect the properties of the reservoir modes (since they are much more numerous than the polaritonic) and allow us to quantitatively probe the effects of multimode optical cavities on the molecular ensemble.59 and simple spectral overlap arguments explain these facts which provide another piece to the puzzle60 of the optical cavity effect on chemical reactions since some experimental observations suggest that the influence of photonic materials on chemical reactions is greater when the cavity-matter detuning vanishes61.Our results have significant implications for future model building in polariton chemistry since they provide detailed illustrations of qualitative shortcomings of single-mode representations of multimode devices. We also suggest practical principles to enhance cavity effects on molecular properties likely holding for generic systems, e.g., we find that for systems with equal collective light–matter interaction, polariton effects on the molecular ensemble are largest when the cavity is redshifted, the distance between molecules is largest and energetic disorder is weak. Wave function localization theoryO(μm) longitudinal and lateral confinement lengths Lz and Ly along the z and y axes, respectively62. The length of the long axis is L ≫ Lz, Ly. The molecules are distributed homogeneously along x and have equal y, z coordinates , where ny and nz are positive integers. The parameters L, Ly, Lz, and ϵ are chosen such that the vast majority of the molecules are only resonant with cavity modes in their lowest-energy band (nz = ny = 1). Therefore, we ignore all other bands, and only include photons with q ≡ kx, omit any reference to ky and kz and identify q as the light wave-vector degree of freedom. The empty cavity Hamiltonian is given by:q = 0 cavity mode has lowest energy We model an optical microcavity with see Fig. . Ideal rNM = L/a two-level systems with transition frequencies sampled from a Gaussian distribution (representing low-frequency fluctuations of the solvent environment around each molecule) with mean EM and variance σ2, so the transition energy for the ith molecule is Ei = EM + σi, where a, and by allowing the single-molecule transition dipole moment to deviate weakly from the mean value μ0 > 0. The ith molecule position is xi = (i − 1)a + Δxi (mod NMa), where Δxi = fia, and fi is sampled from a uniform distribution over f controls the maximal (1 + 2f)a and minimal (1 − 2f)a distances between neighboring molecules, respectively. The transition dipole moment of the ith molecule is given by the random variable μi sampled from a normal distribution with mean μ0 > 0 and variance R ≫ σ, polariton localization at low energies may be primarily driven by fluctuations in the molecular position and dipole moments (relative to a perfectly ordered system)54. Nevertheless, for the studied model, except for infinitesimal values of σ and simultaneous σμ/μ0 close to or greater than 1, energetic disorder plays a more important role, and therefore, we take σμ and f to be constant throughout this work. A detailed discussion of energetic and structural disorder effects on molecular observables is given in Supplementary Note χM and molecular local density of states entropy variation ΔS[ρM]  to HLM since its effects are negligible under the studied conditions65.Assuming R and a = 1/ρS, NM, and NC are free parameters. Any particular choice is equivalent to imposing low and high-energy cutoffs to the EM field. Specifically, L = NMa defines the resolution of the cavity in reciprocal space Δq = 2π/L, and NC defines the maximal cavity-mode energy L is larger than the longest coherence length of the system and NM, L→∞ with fixed ρ) independence of molecular observables to the number of included degrees of freedom can be imposed to obtain a minimal number of molecular and cavity modes. Such computationally optimal number of modes can be strongly dependent on the molecular observable of interest and may also vary significantly with ΩR and σ.For a given ΩNC = NM in our simulations below, since in this case, the thermodynamic limit is reached with a small number of disorder realizations for the observables and range of parameters probed in our studies. A significant fraction of cavity modes is highly off-resonant with the molecular system in every case studied (NM ≥ 1001), and we checked that a smaller number of cavity modes 66 for the study of the effects of photonic devices on molecules.We employ NM = NC and ΩR. The study of lattice constant effects with fixed NM = NC is a distinctive feature of our work in relationship to refs. 56  and is motivated by the following question: are there significant differences in the polariton effects on molecules showing equal ΩR but different molecular densities (1/a in our photonic wire)? Note that a change in the lattice constant a to f × a (f > 0) leaves ΩR fixed under the mentioned conditions if an only if R ∝ μ0a−1/2), as expected since a reduced molecular density requires greater transition dipole moment per molecule to preserve a given Rabi splitting. Nonetheless, a potential unintended consequence of changing a with fixed NC in cavity lattice models is that the photon density of states ρC(ω) = ∑qδ(ω − ωq) is also modified. The thermodynamic limit, in fact, L, and only ρC(ω)/L is independent of L. In Fig. L as long as a sufficiently large number of modes is introduced in the theory. This suggests that the change in photon density of states that result from varying a (with fixed NC = NM and ΩR) is immaterial to our conclusions. In Supplementary Note In later sections, we characterize the dependence of local molecular observables on the energetic disorder, cavity detuning, and mean intermolecular distance (or photonic lattice constant) for fixed entclass1pt{minimaH be denoted by ψ and Eψ, respectively. The ensemble-averaged molecular local density of states (LDOS) gives the conditional probability that an excited molecule will be detected with energy E:EM and width σ. We quantify the photonic effect on ρM(E) by evaluating the cavity-induced change in its Shannon entropyS[ρM] allows us to quantify the cavity effect on the molecular ensemble energy fluctuations. Roughly speaking, ΔS[ρM] provides a measure of molecular excited-state delocalization in energy space. Therefore, we expect ΔS[ρM] to be a nondecreasing function of ΩR that is greater than or equal to zero, since polaritons have energies separated from the bare molecule excited-states by approximately ± ΩR/2  probability M is the t→∞ limit of PM(t):The mean probability that an initially excited molecule at time χM is simply related to ΠM via:The exciton escape probability M provides a measure of excited-state delocalization in real-space and coherent energy diffusion efficiency: in systems where all Hamiltonian eigenstates are delocalized Pnψ ∝ 1/NM for all ψ and n, and therefore, ΠM vanishes in the thermodynamic limit, whereas in noninteracting systems with maximally localized excited-states, each molecule corresponds to a Hamiltonian eigenstate, and therefore Pnψ(t) = δnψ and ΠM = 1. In our model, the case where ΠM = 1 corresponds to the molecules outside of the cavity (since we assume direct intermolecular interactions are insignificant), while we find ΠM→0 when energetic disorder vanishes. Dark states are expected to have a much large contribution to ΠM than polaritons since the latter inherit greater delocalization from their strong mixing with cavity modes.ΠM and χM into a sum of contributions from polariton and reservoir modes and gauge the sensitivity of polariton and dark-state delocalization to increasing disorder. To analyze our numerical simulations, we choose polaritons to consist of all states where the total photonic content is >15%, but <85%, whereas dark states are all eigenstates with total molecular content >85%. We briefly discuss this choice in the section “Density and energetic disorder dependence”.Note that for a disordered multimode system there is no unambiguous definition of polariton and dark states since there is no energy gap between the LP band and the reservoir modes which are weakly coupled to the cavity. However, given a definition of polariton and weakly coupled modes, it is possible to decompose ΠLy = 400 nm, Lz = 200 nm, and dielectric constant ϵ = 3. This gives the lowest-energy cavity mode with EC(0) ≡ ℏω0 = 2.0eV. The highest energy wave-vector is a = 10, 25, and 50 nm. To probe the thermodynamic limit of ρM(E) and ΠM, we take ΩR = 0.3 eV, EM = 2.0 eV, σ = 0.2 ΩR, f = 0.1, and σμ/μ0 = 0.05. To compute the molecular LDOS we employed 400 bins of width 5 meV spanning the interval .Before presenting a detailed quantitative study of the photonic effects on the mentioned molecular observables, we first show that with a relatively small number of molecules and modes we obtain robust predictions for cavity-induced changes in molecular properties. We set ρM(E) obtained with five realizations of a system with NM = NC = 1001 and 5001 and a = 10 nm. Despite the small number of realizations, Fig. NM = 1001 leads to ρM(E) nearly indistinguishable from that with NM = 5001.In Fig. NM = NC > 1001, both S[ρM] and ΠM depend only on the molecular density 1/a. This feature ensures that we have reached the thermodynamic limit for these observables, and justifies our utilization of NM = NC = 1001 in subsequent sections of this article.These observations are quantitatively confirmed in Fig. 57 . These qualitative features of polariton localization in photonic wires have been thoroughly investigated so we provide no detailed discussion, except to mention that there is a clear analogy between the polariton states obtained in photonic wires and the local exciton ground states and the quasi-extended exciton states of polymers68. This analogy is a consequence of Anderson localization universality according to which, in general, there are no extended states over a 1D system in the thermodynamic limit. Instead, any small amount of disorder leads to a breakdown of long-range order.Our computations reveal quasi-extended and localized low-energy polariton states in qualitative agreement with earlier work focused on disorder effects on 1D polaritonic statesIn photonic wires strongly coupled to a resonant material, wave function localization, and more generally, the expected distance-dependent decay of intermolecular correlations  emerge only when both disorder and multiple electromagnetic field modes are included in the light–matter Hamiltonian.χM . This behavior is not surprising: when the molecular system is perfectly ordered, all excitations are maximally delocalized across the entire system (including the weakly coupled molecular states with negligible photonic content) and both ΔS[ρM] and χM take their largest possible value. For even small values of σ/ΩR < 0.1, scattering induced by energetic disorder induces weak and strong localization of polaritonic and reservoir modes, respectively. This notion is corroborated by Fig. σ/ΩR ≤ 0.1. Conversely, the polaritonic contribution to the survival probability is minimal at small values of σ/ΩR, but becomes significant when the energetic disorder is sufficiently large, further confirming that both weakly and strongly coupled excitations become localized over distances smaller than the size of the system. These trends agree with general ideas of wave function localization theory, namely that exciton localization becomes more prominent, in general, as σ increases.In Fig. χM (Eq. ) as a fuS[ρM] and χM with the intermolecular distance is more intriguing, since in the thermodynamic limit polariton scattering induced by energetic fluctuations is expected to be independent of a when q is nearly conserved53. Moreover, while elastic scattering induced by structural fluctuations is expected to be stronger as a approaches 1/q, we show numerically in Supplementart Note σ vanishes, energetic disorder provides the dominant contribution to polariton and reservoir localization.The consistent increase of Δa for fixed ΩR may be understood by noting that if the cavity modes are integrated out in a path integral representation of the partition function of the system70, the effective action for the molecular system acquires retarded two-body intermolecular interactions mediated by the cavity with coupling constant proportional to the product of the magnitudes of the single-molecule transition dipole moments71. For systems with fixed NM and ΩR, the mean transition dipole moment is proportional to ρ ∝ 1/a). Therefore, when ΩR and NM are fixed, the cavity influence on each molecule becomes stronger with decreasing value of density. Evidence for this feature can be obtained from the dark-state contribution to ΠM in Fig. a is increased from 10 to 50 nm. Note that, while the polaritonic contribution to ΠM in Fig. a, the gain in escape probability is dominated by the reservoir modes since their contribution to ΠM is nearly one order of magnitude larger than the polaritonic.The origin of the enhancement of photonic wire effects on the molecular ensemble with q = 0 mode of the optical cavity29. This feature motivates our study of S[ρM] and χM as a function of cavity detuning. In Fig. In several chemical reactions where strong influence by optical cavities has been reported, the obtained data suggest that the photonic material maximizes its effect on the reactive system when a specific transition of the reactant is on-resonance with the χM and S[ρM] on the density follows the pattern discussed in the previous subsection.As shown by Fig. EC(0) < EM can be understood as follows. First, when the mean molecular excited-state energy is smaller than or equal to all allowed cavity-photon energies (EC(0) − EM ≥ 0), a significant fraction of the molecular excitons will be off-resonant with all cavity modes and the spectral overlap of photonic and molecular modes will be weaker  = EM and EC(0) < EM showing that a large number of molecules will be off-resonant with all cavity modes when EC(0) = EM). This notion is confirmed in Fig. NP becomes largest at negative detuning, whereas the number of reservoir modes NDS is minimized in redshifted cavities. Similarly, Fig. EC(0) − EM < 0). This feature can also be ascribed to the greater mixing of the reservoir modes with the cavity when the latter is redshifted.The maximization of the effects of multimode photonic devices on low-energy observable properties  of the molecular system when q ≈ 0 tend to localize over smaller distances than polaritons dominated by wave-vectors with larger magnitude, since in states with long-distance coherence the wave-vector uncertainty δq is much smaller than q54. Therefore, in devices where the molecules are only resonant with q ≈ 0 transitions (zero detuning), any small amount of disorder will lead to polariton modes with finite δq > q. The latter are localized over an interval with a length smaller than the wavelength, and any delocalization effects on the molecular system induced by light–matter hybridization will be weaker relative to the case where the molecules are resonant with cavity modes with larger q57. This effect is easier to notice for the systems with a = 25 and 50 nm in Fig. EC(0) − EM approaches zero from negative detuning, but the polaritonic contribution q = 0 in comparison to the redshifted case.In addition to the reduced light–matter spectral overlap, hybrid modes formed from strongly interacting cavity photons with Note that, although our numerical results are strictly valid for one-dimensional systems, both arguments employed to explain the observed trends are independent of the dimensionality of the photonic and molecular system. Therefore, the detuning behavior reported in Fig. Our observations do not contradict experimental results on polariton effects on chemical reactions, since we only investigated the influence of strong light–matter interaction effects on low-energy properties of the molecular system involving only the ground and first excited-states of the material, while the energy scales and relevant microscopic states governing thermal chemical reaction rates can be much higher .N > 1 resonances72. The zero-detuning condition is then satisfied only by one of the resonant cavity bands, whereas the remaining N − 1 have resonances at q ≠ 0. Further work is required to establish whether the inclusion of multiple polariton bands allows recovery by low-energy effective models of the approximate experimental trend that zero-detuning cavities exert a stronger influence on the local properties of a molecular system.Additionally, we assumed that only the lowest-energy cavity-photon band interacts with the molecular system whereas in infrared optical cavities it is typical for the molecular transitions to have Before concluding, we contrast the main results of this article with the behavior predicted by optical cavity models that include only a single mode.R, σ, and detuning, Fig. 31 upon analysis of the viability of a hypothetical mechanism for cavity effects on charge transfer rates73.In systems with equal ΩρM(E) (blue) and bare R and NM. The top panel  for multimode cavities where the intermolecular distances are in average equal to 10 nm and 25 nm, respectively.Additional examples of qualitative disagreements between single- and multimode-cavity theories are displayed in Fig. nel Fig. a, b contnel Fig. c, d and nel Fig. e, f paneρM(E) is indistinguishable from ρM(E). This feature leads to finite ΔS[ρM] for a multimode cavity . In addition, the single-mode cavity is completely insensitive to the intermolecular distance, whereas our multimode photonic wire captures the stronger effects of the electromagnetic field on molecular ensembles with greater dipole moment per molecule required to preserve the Rabi splitting when the molecular density is smaller (see discussion under “Density and energetic disorder dependence”).Figure 48, we also find that, except for a narrow interval of values of σ/ΩR where the escape probability does not change too much  relative to the molecular system. In negatively detuned devices, there exists enhanced spectral overlap between the cavity and the molecular system. In addition, polariton modes with significant molecular and photonic content have a mean wave-vector far from zero, and delocalization is protected from the small fluctuations induced by structural and energetic fluctuations.27. Several of the reported experiments imply that photonic materials are more effective at changing reaction rates when the molecular system is resonant with the lowest-energy mode of a cavity band (zero detuning). Our results suggest otherwise but do not contradict experiments, since we only assess low-energy properties of the molecular system, whereas the studied slow chemical reactions are complex events often involving higher excited-states than those included in our model.The detuning dependence of cavity effects on molecular properties is particularly relevant when viewed in the context of recent research on polariton effects on thermal chemical reactionsR), to be generic for low-energy polariton models. For example, while wave function delocalization is significantly more favored in 2D and 3D74 relative to the case studied in this paper, it remains true that the spectral overlap between excitons and the optical cavity will be weaker at zero or positive detuning when compared to negative. Dynamical disorder  induces exciton localization and ultrafast coherence decay75 further reducing the efficiency of intermolecular exciton transport reported here, but is unlikely to affect any of the qualitative trends reported.Increasing the number of molecules, cavity-photon modes, dimensionality, and introducing dynamical disorder and cavity leakage is unlikely to qualitatively change any of the trends reported in this work. We expect the behavior of molecular observables with cavity detuning, energetic disorder, and density , S[ρM], and ΠM as described in the main text. The criteria used to identify states as polaritonic or weakly coupled (dark) states is discussed under Results and Discussion and specified in the captions of Figs. Exact diagonalization was performed for multiple realizations of the random light–matter Hamiltonian in Eq. , and theSupplemental MaterialsPeer Review File


Disorder is an intrinsic attribute of any realistic molecularsystem.It is known to lead to localization, which hampers efficient transport.It was recently proposed that in molecular ensembles strongly coupledto photonic cavities, moderate disorder leads to delocalization andincreases of the transport and chemical reaction rates. Vibrationalpolaritons involve molecular vibrations hybridized with an infraredcavity. When the coupling strength largely exceeds the molecular inhomogeneity,polaritons are unaffected by disorder. However, in many experiments,such a homogeneous limit does not apply. We investigated vibrationalpolaritons involving molecular ensembles with systematically modifieddisorder. Counterintuitively, moderate disorder leads to an increasein Rabi splitting and the modification of the polariton bandwidths.Experimental spectroscopic data agree with a Tavis–Cummings-likemodel that suggests enhanced delocalization of the reservoir statesoccurs via the admixture of the cavity mode. Our results provide newinsights into the paradigm of disorder-induced cavity-assisted delocalizationin molecular polaritons. N molecular transitions strongly interacts with the “privileged”electromagnetic mode of an optical cavity.2 Polaritonsare envisioned to play revolutionary roles in emerging applicationsranging from quantum technologies to chemical catalysis;8 thus, they have been investigated extensively.11 Experimentalresults are frequently analyzed within the framework of cavity quantumelectrodynamics. Specifically, the Tavis–Cummings (TC) model,13 which describes the interaction between a cavity and an ensembleof degenerate molecular modes, predicts that a manifold of the reservoirstates, potentially with new and important properties, is formed inaddition to a pair of polariton states.17Molecular polaritons are hybridlight–matter states formed when an ensemble of 23 Disorder is an intrinsic property of any realistic molecular system,which one expects to affect the composition and dynamics of both polaritonsand reservoir states.25 When the molecule–cavity coupling strength largely exceedsthe molecular inhomogeneous bandwidth, the effect of the disordercan be neglected.26 However, when the ratiobetween the two is smaller than about an order of magnitude, whichis a frequent situation in polaritonic systems involving molecularensembles,29 the conclusions drawn, based on the application ofhomogeneous models, might not rigorously hold. Recent studies theorizethat reservoir states delocalize over multiple molecules in the presenceof disorder, thus facilitating efficient transport30 and vacuum-field catalysis,31 which is strikingly opposed to the typical effect that disorderhas in the absence of coupling to a photonic cavity.32 However, in most practical realizations of these systems,experimentally controlling the molecular disorder is formidable, andno direct experimental demonstration of the associated effects hasbeen reported.Recently, the TC modelwas generalized to account for moleculardisorder.25 is33 The values 19 where the molecular manifoldis transformed into a many-body superoscillator bright state j and Consider the first excitation tier of the systemschematicallyillustrated in es basis 1a,11,13,m,σm, and 26 In such a regime, 34 Since The eigenstates of the Hamiltonian 1 involve6 molecules in solution, which were coupled to an open cavity.Our experimental system allows for the independent systematic controlof both the molecular inhomogeneous bandwidth and its collective couplingstrength to the cavity. The former is achieved via controlling solute–solventinteractions by changing the solvent polarity,36 whereas the latter is achieved via controlling the concentrationof solute molecules.37 We demonstrate that,counterintuitively, in the inhomogeneous regime, an increase in moleculardisorder leads to an increase in polariton splitting and line widthmodification, as schematically illustrated in In the present work, we examined therole of disorder on polaritonsformed by the carbonyl stretching (CO) vibrational modes of W(CO)39 where individual dipolar antenna resonancescouple to the in-plane lattice diffraction order.41 Unlike the commonly used Fabry–Perot-style optical cavities,in our experiments, the optical extinction occurs only at the ALRfrequency while the nearby spectral region remains transparent, thusallowing direct spectroscopic access both to the polaritons and tothe reservoir transitions.38 A 3 ×3 mm nearly defect-free antenna array was fabricated by means of electron-beamlithography on a CaF2 window.44 The gold antennas(dimensions of 0.75 × 0.2 × 0.07 μm) were arrangedin a rectangular lattice with a longitudinal period  of DL = 1.35 μm and a transverseperiod of DT = 3.8 μm, as shownin n-octane and (polar)1-chlorobutane solvents. In both solvents, the ALR appears at thesame frequency of –1 and has a qualityfactor of Q ∼ 115. The choice of the two fullymiscible solvents with similar refractive indices but with differentpolarities allows us to efficiently control the molecular disorderwithout modifying the frequency or bandwidth of the ALR.Experimentally,an open cavity was constructed with a high-qualityantenna-lattice resonance (ALR) of the half-wavelength infrared antennaarray,n-octane and 1-chlorobutane.As shown in –1 to –1 . The broadening reflects increasesin both the homogeneous and inhomogeneous line shape components withthe increasing solvent polarity. The increase in inhomogeneity indicatesthe presence of multiple solvation shell configurations created by1-chlorobutane around the CO group. The spectra fit well with theVoigt line shape, which was previously shown to describe the dynamicsof CO transition in W(CO)6 in aliphatic solvents.46 However, because fitting the linear spectrum to the Voigt profilemay produce homogeneous and inhomogeneous widths with a high covariance,we determined their values for each solvent mixture using 2D IR spectroscopy,as described in details in the Supporting Information. The resultsare shown in Figures S1 and S2, and allthe obtained parameters are summarized in Table S1. Changing the solvent composition from n-octane to 1-chlorobutane leads to an increase in the inhomogeneousbandwidth from –1 to –1, whereas thehomogeneous bandwidth changes from –1 to –1.The molecular inhomogeneous bandwidth was systematicallyvariedby preparing different mixtures of 6 and capped with an additional CaF2 window. The dispersion of the vibrational polaritons measured fordifferent incident angles, shown in n-octane and thatat 39 with a precision goniometer,as shown in Figure S3 of the Supporting Information.The antenna array was covered by a 3.6 μm- thick 20 mM solutionof W(CO)6 molecules48 and the detailed resolutionof the polariton line shapes . Polariton spectra were fit to two Lorentzian profiles, and thevacuum Rabi splitting 6 does not differ significantlyin these solvents, as demonstrated in Figure S5 of the Supporting Information.The background-subtractedspectra of the strongly coupled systemare shown in –1, whichwas followed by a slight decrease. In neat solvents, the Rabi splittingswere –1 and –1. These splittingvalues exceed the bandwidth of the uncoupled transitions of both theALR  andthe total bandwidth of the CO mode (–1 and –1) by lessthan 10-fold, indicating the inhomogeneous regime, as discussed above.The increase in the Rabi frequency is qualitatively reproduced bythe real parts of the eigenvalues of the Hamiltonian 6 solubility. These experimentalobservations were qualitatively reproduced by the imaginary part ofthe eigenvalues of the Hamiltonian in –1 and –1.Next,we analyze the polaritons’ bandwidths. For the 20mM solutions, bandwidths of Agreementbetween the experimental and theoretical results allowedus to use the latter to explore why there was an increase in Rabisplitting with an increase in the molecular disorder and the modificationof polariton line widths with an increase in the coupling strength.To this end, we inspected the compositions of the corresponding eigenstates j delocalized over the eigenstate 51While naturally 25 For the couplingstrength equivalent to that of the 20 nm W(CO)6 solutions,with an increase in disorder, –1 , then decreases toward the localized states when The average of PR over all the reservoirstates in the homogeneouscase is 52 and thenapproaches 53 however, the region of N = 40 000), which show thatfor –1 but only ∼3for –1. The maximallydelocalized states have PR values of 370 and 165.PR depends on the interplay between hes  see 5. As expOur calculationssuggest that delocalization of reservoir modesis closely correlated with the admixture of the cavity mode see 5. For boIn conclusion, we demonstrated that disorder within the molecularensemble, which strongly interacts with the optical cavity, affectsthe polariton spectrum, leading to a larger Rabi splitting and a modifiedpolariton bandwidth. We also showed that these phenomena are correlatedwith the delocalization of the reservoir states, which is enhancedby the admixture of the photonic mode into reservoir. Our experimentalresults, supported by theoretical models, indicate that in the regimeof a moderate disorder, coupling to the cavity can allow the well-knowneffect of localization to be overcome. Thus, our results provide newinsights into the strong interaction between matter and vacuum fields,which can be harnessed and utilized in future quantum technologies.


Investors nowadays post heterogeneous sentiments on social media about financial assets based on their trading preferences. However, existing works typically analyze the sentiment by its content only and do not account for investor profiles and trading preferences in different types of assets. This paper explicitly considers how investor sentiment about financial market events is shaped by the relative discussions of different types of investors. We leverage a large-scale financial social media dataset and employ a structural topic modeling approach to extract topical contents of investor sentiment across multiple finance-specific factors. The identified topics reveal important events related to the financial market and show strong heterogeneity in the social media content in terms of compositions of investor profiles, asset categories, and bullish/bearish sentiment. Results show that investors with different profiles and trading preferences tend to discuss financial markets with heterogeneous beliefs, leading to divergent opinions about those events regarding the topic prevalence and proportion. Moreover, our findings may shed light on the mechanism that underlies the efficient investor sentiment extraction and aggregation while considering the heterogeneity of investor sentiment across different dimensions. Investors' noise trading decisions are known to be subject to their sentiment or perception about the financial market, which in turn leads to more noise trading behavior, and excess financial market volatility  confirmed that companies could use social media outlets like Twitter to announce key information in compliance with Regulation Fair Disclosure  to identify thematic information of investor sentiment with variations in topic prevalence  is an unsupervised probabilistic model for extracting the latent thematic content within a collection of documents  combines the strengths of topic modeling with the expressive power by accounting for the heterogeneity of text corpora and the context related to topic expression . Therefore, we only retain tweets that contain at least one ticker symbol. The ticker symbol is also cross-referenced with the asset categories that StockTwits has officially maintained. In particular, for tweets with more than one ticker, we generate the category combination if these tickers are from different categories. For example, if one tweet mentions both AAPL and SPY, it will be associated with the category of . We only keep tweets that belong to the top 20 asset categories. Last, we further add the temporal dimension to the metadata of tweets using its timestamp.Standard text preprocessing techniques have been applied to all tweets, such as removing all numbers, non-ASCII notations, stop words and links, converting all characters to be lowercase, and eliminating tweets whose word length is less than two, and using the stemming and lemmatization method to obtain the root form of the words. Then, terms with a relatively high sparsity level of 0.995 are removed in order to keep only meaningful and frequent words in tweets.From the exploratory data analysis, we identify many news bots (hereby referred to as “robots”) that tweet about a broad coverage of assets using technical terminologies similarly. We choose to remove these robots from the data because they do not represent actual investor sentiment and may distort the results. More specifically, we first select the user IDs that have posted the top 1000 highest number of tweets. Then we compute the average similarity score of 100 pairs of randomly chosen tweets for each ID using fuzzy string matching. We consider users a robot if the average similarity is above a threshold (set as 0.6), thus removing their tweets from our corpus. As such, 4,590,854 tweets remain after we finish preprocessing steps.Topic modeling is known to produce less coherent topics when directly applying to short texts such as tweets due to the lack of word co-occurrence information in each short text . We denote the metadata as a matrix X, with each row (denoted xd) listing the values of metadata covariates for document d. The words of a corpus-level vocabulary list V are indexed by v ∈ {1, …, V}. The document d holds Nd empty positions, waiting to be filled by Nd words. To fulfill the word positions in a document, first, a specific topic-prevalence vector should be generated using the associated metadata :A graphical illustration of the data generative process of structural topic modeling is provided in zd,n, which indicates the associated topic for the word wd,n in the document, as shown in Equation (2):where Γ′ is a matrix of coefficients and Σ is a general variance-covariance matrix. Given the topic-prevalence vector, we can draw one specific k is chosen, the k-th entry of zd,n is unity and all other entries are zero.Note that, when a specific topic mv, and of deviations from different topics κk,v, covariates κyd,v and their interactions κyd,k,v in Equation (3):Topical content defines the distribution over terms associated with the topics as an exponential family model, added as a function of the marginal frequency of occurrence deviations for each term d,k,v is the probability of choosing vocabulary word v to fill a position in document d given topic k. Then a specific word wd,n is chosen from the vocabulary list in Equation (4):where βBzd,n =  is the overall probability of the words given topic k.where In short, the parameters mentioned above are estimated by solving the posterior likelihood maximization optimization problem via R's STM package  weights words by their overall frequency and how exclusive they are to the topic to determine the weighted harmonic mean as in Equation (6)w is the weight for exclusivity (which we set to 0.5 as a default) and ECDF denotes the corresponding empirical CDF function.where In conformance to the studies of event discovery, one primary approach for understanding the heterogeneity of investor sentiment is to separate the breaking finance news from the ongoing investment-related conversations  annotating the content. More specifically, we focused on the documents that contribute the most to each topic, i.e., whose topic proportions (θ) obtained after model estimation are at least larger than 50%, and then explored the distributions of the metadata of these documents on topic composition. The results in The market sentiment shown in etc. For professional investors in the same topic, they mention more about “bank,” “finance,” “bac” (The Bank of America), “Genpact” , “American Express” , “KCG" , etc.Besides the divergence in investor sentiment, investors with various profiles tend to discuss financial markets with heterogeneous beliefs, resulting in different distributions regarding the topic prevalence and proportion. The investors with different trading experiences show considerable divergence on topics they are interested in, as shown in etc.). By contrast, day traders and swing traders are more likely to discuss topics using sentiment-related words such as “breakout,” “posit,” “resist,” “high,” “bottom,” “ready,” “upside,” “bull” in the topic of bullish terms and “put,” “bearish,” “fall,” “short,” “dump,” “weak,” “downside” in the topic of bearish terms.Moreover, investors with different trading strategies, analogous to those with different trading experiences, also have divergent focuses and word preferences across topics, as shown in In terms of asset categories extracted from document contents, we found strong heterogeneity likewise, as shown in Furthermore, we found that topics of important social events contain many company tickers that reveal the underlying associative asset categories under discussions of the event. For example, in the Ukrainian Crisis (2014) topic, investors mention assets of basic materials and customer goods heavily. One possible explanation is that the sanctions against Russia severely impacted these assets after the Ukrainian Crisis (2014) , Greek Debt Crisis and Tesla & Electric Vehicles), and show the estimated results in According to the regression results shown in The second topic of the Greek Debt Crisis is a hot topic for all types of investors with different profiles in terms of a large number of tweets . Based oFor the third topic on Tesla & Electric Vehicles, Although investor sentiment may have a significant heterogeneous effect on the financial market, the sentiment analysis in the finance industry typically produces aggregated investor sentiment index without accounting for the heterogeneity. For example, in the WallStreetBets vs. Wall Street saga, investor sentiment is highly driven by heterogeneous investor profiles, as the topic is disproportionately contributed by bullish sentiment from non-professional retail investors that mainly engage in day trading , POR Lisboa and POR Norte , and European Union's Horizon 2020 Research and Innovation Programme under the Marie Skłodowska Curie grant agreement No 812912 for the project BIGMATH.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Molecular weight distributions are widely used to evaluate the effects of aging or modifiers in bituminous binders. As with polymers, the most common techniques to obtain the distributions can be subdivided into two main groups, depending on whether or not they use a solvent. In the first group, the dimension of the molecules is evaluated in a diluted unperturbed state, while, in the second, the dimension derives from the bulk, where aggregated or interacting molecules may behave as single entities. However, the calibration curves used in the bulk are tuned in order to homogenize the results derived from the two approaches. This sort of contradiction, plus the high number of experimental uncertainties, suggest that the term “apparent” should be used for both distributions. These aspects are well known in the field of polymers but have received less attention in the case of bitumens, which are even more complex. This paper pinpoints the advantages and disadvantages of the two techniques, thus highlighting the most appropriate use. Bulk methods are preferred when evaluating properties that are strictly dependent on the microstructure, such as the level of aging and the effects of additives or modifiers. Diluted methods should be used when the molecular size matters, such as in quantifying the presence of polymers or rejuvenators. Both techniques should be used for comparative studies only. The molecular weight distribution (MWD) influences the rheological and mechanical properties of viscoelastic materials, such as bitumen. For this reason, there is an increasing use of the MWD in the scientific literature related to asphalt paving, where it is correlated with the composition, performance, aging susceptibility and compatibility with additives and polymers . The MWDIt would thus be helpful to obtain the distribution directly in bulk, as in the “inverse mechanical” approach, which uses rheological data, which are typically collected in the linear viscoelastic region. The idea is that the viscoelastic functions contain the complete spectrum of relaxation times, which can thus be deconvoluted into an MWD ,7,8. TheOne of the criteria adopted to derive such a correlation is to obtain a similar MWD from the GPC and inverse mechanical method. In other words, the calibration curves in the inverse mechanical approach are tuned in order to unify the results obtained with the two techniques ,10. ThisIn summary, the two main approaches are very different from both the theoretical and experimental perspectives, and, consequently, in this paper, we differentiate between “solution” (SMWD) and “bulk” (BMWD) molecular weight distributions. In fact, neither approach guarantees a reliable quantitative description of the molecular weight distribution, and both should be considered as “apparent” molecular weight distributions (AMWDs) ,12,13. BDespite the abovementioned uncertainties, the AMWD provides useful information and has been very effective in the study and characterization of bituminous binders. Based on this premise, this paper focuses on the following: (1) the adherence of the two distributions to the theoretical significance of the MWD; (2) the main differences between the two options; (3) the correlation between the two distributions; (4) guidance in selecting the most appropriate distribution.The idea of the MWD derives from polymeric materials that are composed (at least in the case of homopolymers) of molecules with the same chemical composition, but variable lengths and structures . Simple knowledge of the chemical formula and average molecular weight is therefore insufficient to fully characterize the material and its properties. For example, in the case of polyolefins, the average MW may not correspond to a probable chain length because it is common practice to develop bimodal distributions with a low-MW fraction that reduces the viscosity in the melt state, and a high one that improves the mechanical properties of the product. Processing can also be performed on several types of machinery, each with specific requirements. This is why, and especially for the so-called “commodities”, there are many commercial grades of the same polymer with almost identical chemical compositions, but different MWDs.number molecular weight distribution, which considers the molar fraction (n(M)) of molecules with molar mass (M), and the weight molecular weight distribution, which considers the weight fraction (w(M)). The two distributions clearly need to be normalized, and any average value can be easily derived from the moments of the distribution curves: There are several possible definitions for the MWD, and the interested reader can find an accurate description in any book on polymer science ,14. The i” repeating units, with I = 1, 2,…, ∞. If M0 is the molar mass of the repeating unit, then the two average molecular weights can be expressed as:iM = iM0 is the molar mass, and iN are the moles of molecules with i repeating units. Despite the continuous nature of Equations (1)–(3), the MWDs of a polymeric material are intrinsically discontinuous because the length of each molecule must be equal to an integer number of repeating units. It is thus common practice to use a discrete representation that takes into account fractions that correspond to molecules containing “i refers to a specific molar mass instead of the chain length. In addition, any experimental technique used to evaluate the distribution produces a finite set of data, which justifies the use of discrete expressions.In the case of bituminous materials, which are composed of molecules with different chemical compositions, the discontinuous nature of the distributions is different because there is no repeating unit that is common to all molecules. However, Equations (4) and (5) remain valid if x is the conversion of the functional groups. Equations (6) and (7) represent the “most probable distribution”, or “Flory distribution”, for which the following average values can be determined:Other definitions of the average molar mass and distributions specifically derive from the technique used for their determination . When referring to average values or distributions, it should therefore be specified which one is intended, as they can differ significantly from each other. For example, in the case of the step polymerization and stoichiometric feed of the reacting functional groups, the following theoretical expression can be easily derived from a statistical theory :(6)ni= of the distribution:The higher the difference between the number and weight average molecular weights, the more dispersed the distribution. Of course, the dispersion index does not contain information on the shape and modes of the curves. i-axis.The number molecular weight distribution is a monotonically decreasing function, which thus indicates that, at any value of the conversion, the most probable length is 1, corresponding to an unreacted monomer. In contrast, the weight average distribution shows a maximum in the The main idea of GPC is to separate the molecules based on their hydrodynamic volume in solution. The apparatus consists of a column filled with a stationary phase, with a distribution of pore sizes. When the polymer solution (mobile phase) flows through the column, the paths of the molecules inside the stationary phase depend on their dimension. Large molecules are excluded from the small-size pores and follow a shorter path than small molecules. Therefore, the larger the molecule, the shorter its retention time, and vice versa. After the column, the solution passes through a detector, the signal of which gives the weight fraction of the solute as a function of the retention time. This is then converted into a weight molecular weight distribution by applying a calibration curve that correlates time and molar mass. “size exclusion”, which means that the stationary phase works as a molecular sieve, without chemical interactions with the mobile phase;The detector records all the molecules;A calibration curve is available.The three main assumptions of the method are:These points have been widely discussed in the literature; however, it is useful to underline some of the differences between polymers and bitumen.Unfortunately, a complete absence of interactions between molecules and the stationary substrate is almost impossible, which may strongly influence the retention time. In the case of homopolymers, all the repeating units have the same chemical composition, and thus the magnitude of the interactions is in proportion to the molecular size. In contrast, in a bituminous binder, the molecules differ in both dimension and chemical composition, with the latter usually subdivided into a small number of families based on their polarity, such as the saturates, aromatics, resins and asphaltenes (SARA) fractions proposed by Corbett . The difThere are many types of detectors, but the most common ones in the bituminous literature are based on the refraction index (RI), ultraviolet absorption (UV) and light scattering (LS). While an appropriate choice is straightforward when dealing with polymers, the same is not true for bitumen. The wide spectrum of chemical compositions can induce errors because a detector can have different sensitivities or can even be completely blind to specific molecules. For example, consider the case of a UV detector, which has to be set on a single wavelength, and for which the value should be tailored to the specific solvent and solute.An appropriate number of monodispersed samples with known MWs is needed to obtain a calibration curve, which, for a given solvent, correlates the retention time and MW. In order to prevent problems related to the abovementioned uncertainties, the ideal situation is to have samples of the same polymer that needs to be tested. Of course, this is rarely possible, and reliable standards are only available for polystyrene (commonly used in the case of bitumen) and poly(ethylene oxide). For a limited number of polymers, the problem of the standards can be overcome by using “universal calibration”, which is based on Einstein’s viscosity formula and is exploited in combination with the Mark–Houwink equation . The conA simple demonstration of the above considerations is given by the raw GPC spectra reported in The two distributions show evident differences due to the sensitivity of the detectors with respect to the chemistry of the bitumen molecules. The small peak at the elution time of around 6 min in the RI spectrum corresponds to buthylhydroxytoluene (BHT), which is added as stabilizer in the solvent (tetrahydrofuran). The position of the peak, based on the calibration curve determined using polystyrene standards, does not correspond to the real MW, which is equal to 220 Dalton. Moreover, the UV spectrum shows that a considerable number of bitumen molecules have a higher elution time. This would mean that they are smaller than 220 Dalton, which is not realistic for the heaviest fraction derived from oil distillation. These considerations confirm the apparent character of the distributions, due to both the uncertainties in detection and the stationary phase not working as a pure molecular sieve. These effects are very important when dealing with bituminous materials, which, compared with polymers, vary greatly in the chemical composition of their constituents. Moreover, the level of molecular association is linked to operating parameters such as the flow rate, concentration, temperature and age of the solution. Therefore, these parameters, as well as the composition and porosity of the stationary phase, need to be carefully tuned before recording the results  is the monodisperse phase angle, w(MW) is the weight-distribution function, ω is the reduced angular frequency, and H is the Heaviside function. The last step is a correlation between the MW and ω, which, in the case of polymers, is well described by a simple mass law:k and α are constants. Introducing the new variable x = log ω, and differentiating Equation (12), the weight distribution as a function of the phase angle assumes the form:The procedure suggested by Tuminello derives from the double-reptation mixing rule  and relaw(MW) and other viscoelastic functions can be obtained. The first paper to apply the inverse method to bituminous binders was by Zanzotto and coworkers [*G) [i is the imaginary unit, β is a constant λ, and μ are the relaxation times.Likewise, a correlation between the oworkers , who useers [G*) :(14)G*(ωk and α (Equation (12)) for bitumen by correlating the crossover frequency (cω) at 0 °C with The details are reported by Zanzotto et al. , who alsThe BMWD derived from this method clearly depends on the analytical form of the viscoelastic function. In the case of the fractional model, the δ−ω curve inevitably shows several inflection points that depend on the number of terms in Equation (14). Zanzotto et al. observed that, to obtain a representative description of the complex modulus and phase angle, the number of inflection points is quite limited, and some are directly observable in the experimental data. Nevertheless, this leads to a multimodal MWD that strongly depends on the number of terms used to fit the master curves.A more recent approach assumes that the cumulative molecular weight distribution (CMWD) is proportional to the δ(ω) . The hypThe authors adopted the frequency-molecular-weight correlation proposed by Zanzotto et al. (Equation (16)); thus, the master curve must be built with a reference temperature equal to 0 °C. Later, Cuciniello et al. proposed a different correlation, derived from GPC distributions instead of the osmometry average molecular weight .(18)logIn both Equations (13) and (17), the shape of the distribution is calculated from the first derivative of the phase-angle master curve. Therefore, the final MWD depends on the mathematical procedure adopted for the differentiation. Considering the abovementioned difficulties, when using the fractional model (Equation (14)), the use of “smoother” rheological models has been suggested, such as the 1S2P1D  or 2S2P1w(M) will be obtained, which has no physical meaning. This can occur in the case of polymer-modified binders, which will be discussed later. Second, converting a phase-angle master curve into a cumulative molecular weight distribution is formally correct when the curve covers the whole range from 0 to 90°. Additional considerations regarding the δ-method are necessary. The first derivative in Equations (13) and (17) implies that, if the phase-angle function does not monotonically decrease with the reduced angular frequency, then negative values of the While data in the terminal zone are easy to collect, the same is not true at high frequencies (low temperatures), and especially for bitumen, which becomes very brittle . The avaGPC has been used in several studies on bituminous binders, and the recent review by Ma et al. provides a thorough overview of the available literature . Here, wOxidative aging is one of the main causes of long-term pavement deterioration, and there is a vast amount of literature on this topic in the field of asphalt ,35,36. FIn Both distributions confirm that oxidation does not involve the less polar low-MW molecules, but mainly affects the aromatics, resins and asphaltenes. The asphaltenes undergo an increase in polarity and mutual interactions, which decrease their mobility and thus increase the apparent molecular weight. What is barely appreciable in solution becomes a well-visible effect in bulk. Consequently, several aging indexes have been proposed based on the evolution of the BMWD ,32.Many types of additives may be used to improve or restore the physical properties of bituminous binders, and, again, the SMWD and BMWD provide different information. Let us consider the case of rejuvenators, for example, which are vegetable oil added to an aged binder. The oil causes the appearance of a small shoulder (between approximately four and five minutes), as the interactions with the binder are hidden by the solvent. In this case, the GPC confirms, at least qualitatively, the presence of the additive, as the latter gives a signal at a specific elution time. In some cases , a quantitative determination of the additive can also be obtained from the spectrum.The BMWDs of the same two binders are given in As expected, the oil affects the whole bulk distribution because it replaces the lost light fractions in order to restore the original degree of aggregation in the binder. Therefore, the main advantage of a bulk distribution is that it provides useful information. In The use of polymers to modify bitumen is well known. The polymer-modified bitumen (PMB) obtained is a blend of two constituents with very different average molecular weights, and, thus, in theory, they are easy to identify in the MWD. This is why GPC can assess the presence of polymers when there is a rapid elution peak that corresponds to the polymer, which is clearly distinguishable from the signal associated with the bitumen molecules ,38. The weight fraction of the polymer corresponds to the relative area of the peak in the spectrum, but this calculation can be quite difficult due to the small amount of loaded polymer. Moreover, due to both the presence of short chains in the polymer and interactions with the stationary phase, the polymer and bitumen signals cannot be completely separated. This can happen, for example, in the case of HS-SEC, which confirms that the appropriate operating conditions are necessary, depending on the aims of the analysis. A moderate flow rate and a highly diluted solution favor the separation of the peaks and are preferable when polymer content is required. The role of GPC in PMB is thus similar to the above-described example of low-molecular-weight additives. In Another potential cause of uncertainty in SBS-based PMBs derives from the common practice of adding sulfur to improve their storage stability and morphology . Sulfur In contrast, the BMWD of a PMB provides useful information on the binder structure; however, the interpretation of the distribution may not be simple. If the polymer confers its elastomeric properties to the whole material, then it has a large effect on the relaxation time of the bitumen molecules. In the phase-angle master curves of PMBs, the elastic behavior induced by the polymer is manifested as a plateau in the intermediate viscoelastic zone. Before the plateau, at lower frequencies (higher temperatures), the material has a liquid-like behavior. At higher frequencies (lower temperatures), the rigid bituminous matrix tends to prevail in determining the overall mechanical properties, and the behavior shifts to solid-like. The extension of the time–temperature intervals in which these behaviors occur depends on the type of polymer, concentration and use of crosslinker. Because the plateau zone corresponds to low values of the first derivative of the phase angle in the frequency axis, it becomes the separation zone of a bimodal distribution in the molecular weight axis. The longer the plateau zone, the better the separation between the two modes of distribution. The effect of the bitumen–polymer interactions on the relaxation times is clearly shown when comparing the BMWD distribution of a base bitumen and those of the corresponding PMBs obtained with and without the use of sulfur .Without sulfur, the distribution of the PMB has the same range of molecular weights that correspond to the base bitumen, but the shape is very different. The presence of the polymer determines the abovementioned sharp separation between high- and low-molecular-weight zones, as the number of bituminous molecules involved in the network is proportional to the relative area of the second peak. When sulfur is added, the BMWD changes again: the second peak becomes larger and shifts significantly toward higher molecular weights, and the distributions show negative values between the low- and high-MW peaks. A larger area corresponds to a wider portion of the bitumen molecules involved in the polymeric network, which is the compatibilizing effect of sulfur. As a general observation, the more the peak is separated from the remaining part of the distribution, the higher the level of interactions and compatibility between the bitumen and polymer. The shift to the right is probably associated with the phase inversion, which is when the network assumes a sort of continuity and dominates the mechanical behavior. The whole material assumes an elastomeric character with higher relaxation times that reflect the behavior of the polymer. The negative peaks derive from the local minimum in the δ master curve. Although not acceptable from a mathematical perspective, the existence of negative weight fractions underlines a well-defined boundary between the prevalence of bitumen or polymer in determining the macroscopic behavior. Therefore, the negative values should be considered as important information on the material structure, instead of as a simple mathematical artefact.The various analogies between polymers and bitumen enable similar characterization techniques to be used for both materials. However, if polymers have a homogeneous composition, then bitumen molecules cover a sort of continuous spectrum, from small nonpolar paraffinic chains to large and rigid asphaltenes with polar functional groups. Based on their interactions and aggregation, these molecules are arranged in very complex internal structures, which are often identified as colloidal and are determined by the relative abundance of the different components. The internal structure is, therefore, highly sensitive to the material composition and can change quite easily. In both polymers and bitumen, the structure and cohesive energy can be altered by parameters such as temperature, mechanical stress, oxidative aging and additives. However, in bitumens, there is an unstable colloidal equilibrium. This is why their behavior changes from Newtonian liquid to brittle solid within a fairly limited temperature range. The concept of the MWD of bitumen is thus different from that of polymers. The dimension of a single molecule contains less information than that of a polymer because its mobility also depends on the composition. Therefore, in relation to processing and in-service life, the molecules cannot be considered as single entities, but must be evaluated along with all their surroundings. The consequence is that, for bitumen, the difference between the bulk and solution MWDs is somewhat larger than it is for polymers. The bulk distribution has a closer relationship with the internal structure of the materials and is thus preferable when evaluating those properties that strictly depend on it, such as the degree of aging and the effect of additives/modifiers. The solvent separates the molecules; thus, the SMWD should be used when the size of the molecules is important . Lastly, it is important to highlight that bulk and solution distributions clearly have many uncertainties, which means that, irrespective of the purpose, they should only be used to compare binders based on the same base bitumen.


The capability to share knowledge is considered one of the most relevant components of knowledge management. Moreover, there is little empirical evidence indicating how future human resources in the construction industry value the richness of knowledge sharing and the richness of their innovative behavior. The purposes of this study are (1) to determine which facilitators, from the point of view of master’s degree students related to engineering and construction management in Spain, most substantially influence knowledge sharing capability; (2) to test whether knowledge sharing capability (KS) positively influences innovative behavior (IB); and (3) demonstrating whether organizational innovation climate (OIC) is a factor that moderates the relationship between KS and IB. In this research, we have proposed a theoretical model and empirically tested the model in a sample of 253 master’s degree students in public universities in Spain. The findings support the proposed model, and the structural equation modeling (SEM) evaluation suggests that, among all the facilitators of KS, information and communication technologies (ICT) stand out among the other facilitators and have a more significant influence on KS. Furthermore, the research found a direct correlation between KS and IB and causal links between OIC and IB. Beyond the advances in science and technological development, the main driving force of companies is their human resources and how managers manage knowledge. The problem is not the generation of knowledge in the construction sector but the waste of valuable information for future projects . It is aPrevious studies, summarized below, have demonstrated the importance of KS facilitators. For example, Lin  found thKnowledge gaps indicate that only some components of the model have been analyzed by the literature separately. However, the literature has not analyzed organizational innovation climate (OIC) as a moderating variable between KS and IB. In addition, most studies need to address the views of graduate students.For further empirical evidence and a more profound understanding, this paper will examine the moderation role of the OIC in the KS-IB relationship and include reciprocity in KS facilitators. We conducted this study in the context of an educational setting linked to the construction industry with the objectives of (1) determining which facilitators, from the point of view of master’s degree students related to construction engineering and management in Spain, most substantially influence the ability to exchange knowledge; (2) testing whether KS positively influences innovative behavior; and (3) demonstrating whether an OIC is a factor moderating the KS-IB relationship. Thus, to fill the above research gaps, the present study was undertaken to elucidate some of the following research questions:RQ1. Which facilitator of KS is most important for graduate students?RQ2. How does KS influence IB?RQ3. What role does OIC play between KS and IB?In order to answer the previous research questions, this research implements structural equation modeling (PLS-SEM) to examine the correlation between the components of the research model from data collected from a survey administered to 253 students from public universities in Spain See . We chosFrom the organizational context, the literature review indicates a need to explore further the reciprocity derived from KS ,9,10,11.From the educational context, Su and Zhang  found thKankanhalli et al.  defined At the educational level, Su and Zhang  found thRecent studies have examined the impact of top management support on KS. For example, Lee et al.  further According to Eletter et al. , from anOrganizational culture and rewards are positively associated with knowledge sharing . Many stOn the other hand, some studies claim that reward systems have a negative effect. Bock and Kim  stated tDifferent experts recognize the importance between KS and ICT in their research. For example, Islam and Ashif  indicateIn the context of education, ICT has revolutionized the methods of teaching and learning. According to Jain and Gupta , the newThis study suggests the following hypotheses in light of a thorough evaluation of the literature on knowledge sharing enablers:H1a. Reciprocity is positively related to KS.H1b. Knowledge self-efficacy is positively related to KS.H1c. Top management support is positively associated with KS.H1d. There is a significant relationship between rewards and KS.H1e. Information and communication technologies have a positive impact on KS.Recently, there has been increasing interest in studying the relationship between KS and IB ,44. SeveKS enhances employees’ ability to innovate because information must go through an internalization process ,50 to maBased on the above review, we propose the following hypothesis:H2. Employee knowledge sharing positively influences their innovative behavior.The OIC is a precursor  and predSeveral studies have examined the correlation between OIC and IB ,53,57,58From an educational perspective, Wang et al.  demonstrH3. Innovative climate is positively associated with employees’ innovative behavior.Individuals may feel less psychologically confident in the face of new challenges or technologies and are more likely to embrace conservative strategies instead of innovative behaviors . An orgaOn the other hand, the study by Witherspoon et al.  took theTherefore, we hypothesize the following:H4. The OIC positively moderates the relationship between knowledge sharing and employees’ innovative behavior.The dataset consisted of 253 usable responses from graduate students in master’s degree programs related to engineering and construction management in Spain. It is a convenience sample, non-probabilistic, due to not having full access to a nationwide random sample . TherefoThe questionnaire consisted of four distinct parts: (1) respondent characterization, (2) KS enablers, (3) innovation climate factors, and (4) elements that shape innovative behav-ior. The first part included professional title, gender, and age questions. The items consid-ered for the second part focused on the five second-order latent variables . The third and fourth parts of the questionnaire corresponded to the OIC (3 items) and IB (3 items), respectively. We developed the 24 items considered in the questionnaire from a thorough literature review based on ,32,44,57Reflective measurement models have been the instrument of countless studies. However, applying these traditional models under some circumstances is often less fruitful . Our stuWe took the following steps to lessen the impact of variance bias/common method bias (CMV). First, we separated the endogenous variables from the exogenous variables. Second, we randomized the items, and finally, we performed a pretest. Like Hulland et al. , we tookStatistical Package for the Social Sciences (SPSS) version 25  and SmarThis study’s data analysis was divided into two sections. The measuring approach was first tested to ensure that the formative and reflective constructs were reliable and valid. Second, the structural model trajectories (significance and path coefficients) were evaluated with 5000 resampling iterations using the bootstrapping approach .Convergent validity represents the degree of agreement between different indicators of the same construct. When factor loadings, CR, and AVE coefficients surpass 0.5, this measure is feasible . Table 2p-values for the first-order (formative) dimensions are the primary recommendations for evaluating formatively modeled HOCs [p-values. In addition to the above, the same table should contain the variance inflation factor (VIF) to check for multicollinearity issues. According to Becker et al. [p < 0.01), followed by KS .The external weights and led HOCs ,77. Tablr et al. , VIF valThe statistical significance of path coefficients between exogenous and endogenous variables is investigated using structural models . A PLS-Sp < 0.01) among the proposed formative indicators. It implies that its items made an adequate contribution to the formative construct. Finally, The first factors to be evaluated in formative models are collinearity and the relevance of the formative indicators. The VIF is used to assess collinearity, as described in the previous section. Our study’s PLS algorithm results revealed no multicollinearity issues because all scores were less than three . We coulThe loading of each reflective indicator on its corresponding construct (>0.7), composite reliability value (CR > 0.7), Cronbach’s alpha (>0.7), average variance extracted value (AVE > 0.5), and Fornell–Larcker criteria were used to evaluate the reflective measurement model ,79. The 2 > 0 represents that it meets the parameter [2 = 0.300 represents its validity. To confirm the suitability of the PLS-SEM model for the standardized value, we employed the root-mean criterion (SRMR < 0.08) [2) of 41.9, which means that KS and OIC collectively explain 41.9% of the variance of IB  , which ff IB see .t = 3.217; p < 0.01). Therefore, H2 is accepted. In addition, the innovation climate in the organization  also shows a significant and positive effect on innovative behavior.t = 2.116; p < 0.01) establishes a significantly negative moderation on the relationship between KS and IB. When interaction effects are introduced into the model, the coefficient of determination changes (R2). 2 value from 0.419 to 0.429. This increase means that the variance explanation for IB is improved by introducing the interaction effect of OIC. According to Hair et al. [2 represents, as insignificant as it may seem for moderation, an essential role in the interaction effect.We examined in the paper the effect of OIC on the relationship between KS (second-order formative construct) and IB see . OIC  measures the accuracy with which the exogenous variables predict the endogenous variables. According to Aguinis et al. [2 = 0.02 denotes a minor effect, f2 = 0.15 is a medium effect, and f2 = 0.35 is a significant effect. 2 = 0.150), followed by the KS effect on IB (f2 = 0.055), which has the slightest effect. Similarly, the effect size of the interaction term, i.e., OIC (f2 =0.018) is limited. Aguinis et al. [The model’s effect size (fs et al. , the effs et al.  suggesteThis study joins recent efforts that call attention to the theoretical and methodological distinctions between formative and reflective measurement models to address the lack of formative measurement models in KS research .One of the goals of this study was to determine, from the viewpoint of graduate students enrolled in master’s degree programs in engineering and construction management in Spain, which facilitators had the most significant influence on the capabilities to exchange knowledge. A second objective was empirically determining whether KS positively impacted innovation behavior from the respondents’ perspective. Finally, it was necessary to demonstrate whether the relationship between KS and innovation behavior was supported by environments that promote innovation.Structural equation model analysis of the first set of hypotheses confirmed that, among the group of knowledge facilitators, only ICT use had a statistically significant effect on KS. The results revealed that reciprocity had no significant effect on KS. Therefore, they are not consistent with the arguments of Caimo and Lomi , RichardKnowledge self-efficacy was not found to be a factor that significantly impacts KS. This result is not consistent with the arguments of Bock et al. , CasimirOur research finds that top management support does not significantly impact KS. Therefore, it is not consistent with the arguments of Lo et al. , MeddourThe results revealed that rewards did not correlate significantly with KS. They support the arguments of Bock and Kim , Bock etThe results showed a positive and significant correlation between ICT use and KS. These findings are consistent with the claims made by Lin , Saenz eThe results showed that KS is a predictor of IB. They support the arguments of Akhavan et al. , Hansen Our results also showed that OIC did not have a positive or significant moderating effect. This conclusion reinforces the arguments of Witherspoon et al. , Chow etThe results of the measurement model test, including convergent validity, discriminant validity, variance inflation factor, and explanatory power, were satisfactory. The new generations recognize that among all the enablers studied, the use of ICT is the primary way to strengthen KS in the future of the construction sector. However, from the authors’ perspective, one of the main problems is idealization of ICT as an “end product” and not as a tool to expand into new horizons. The innovative behavior of individuals must be based on more than the exclusive use of ICTs, since ICTs alone are not synonymous with innovation. It is necessary to combine principles such as reciprocity, management support, and knowledge self-efficacy with new technologies so that they become the elements that transform both institutions and individuals.Moreover, the importance of the educational sector is emphasized since it is a fact that it is rapidly adapting to the use of new technologies in teaching. The educational sector should maintain sight of fundamental aspects such as the development of reciprocity, self-efficacy of knowledge, and developing social and creative skills in its educational programs. On the other hand, it must continue on the path of adaptation to new technologies. With these two aspects strengthened, education can transform the innovation culture of professionals to the extent of permeating it in the sector’s organizations when they enter the workplace.From the respondents’ perspective, KS and OIC are precursors to IB. It is clear that the behavior of individuals depends on human and environmental factors, regardless of their professional background. At the organizational level, future human resources in the sector must be immersed in an environment that allows for development, tolerates failure, and provides the means to open up the possibilities for IB. Organizations will likely transform openness to innovation if they work systematically on OIC.Finally, our study corroborates that OIC is not a factor that enhances the bidirectional relationship between KS and IB. We authors agree that deeper issues of individuals’ behavior need to be delved into to reduce self-interested knowledge hoarding.Although the work adds knowledge and value to the literature, it also has certain limitations. First, the cross-sectional design does not eliminate the possibility of a long-term causal correlation emerging due to changes in the psychology and confidence of individuals over time. Second, we derived facilitators from studies conducted in a professional setting before applying them in an educational setting. In addition, most respondents had less than three years of professional experience, which would condition the study’s results. It should be added that the topics addressed from the professional environment in students were with the firm intention of analyzing the students’ perception of these topics. A longitudinal study would overcome this limitation. Future research could replicate this study in other countries and combine quantitative and qualitative data, not only among students but also among teachers, to strengthen the view of the impact of KS on IB and add other facilitators of KS. In addition, future studies could examine how other variables may act as moderators or mediators of KS, e.g., organizational culture or the facilitating conditions between the dependent and independent constructs. Finally, the literature requires more research linking students’ vision to the work environment and understanding, from the beginning, what to do with human resources to improve the innovation aspect in the construction sector.