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Langevin equation for the dipole vector of an accidental symmetric top molecule
Journal of Molecular Liquids 289 (2019) 111123
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Langevin equation for the dipole vector of an accidental symmetric top molecule Yu.P. Kalmykov a, S.V. Titov b,⁎, W.T. Coffey c, M. Zarifakis c,d a
Laboratoire de Mathématiques et Physique (EA 4217), Université de Perpignan Via Domitia, F-66860, Perpignan, France Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region 141190, Russia Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland d Electricity Supply Board, Generation, Asset Management, Dublin 3, Ireland b c
a r t i c l e
i n f o
Article history: Received 5 March 2019 Received in revised form 24 May 2019 Accepted 2 June 2019 Available online 6 June 2019 Keywords: Langevin equation Accidental symmetric top Autocorrelation function Rotational Brownian motion
a b s t r a c t The Langevin equation governing the time dependent dipole moment μ(t) of an accidental symmetric top (i.e., the direction of μ(t) does not coincide with the axis of symmetry) in an external electric field is directly derived from the Euler-Langevin equation. Results which have been previously obtained in J. Mol. Liquids. 1991, 49, 79 are regained as special cases. The scalar Langevin equation governing the longitudinal component of μ(t) along a space fixed axis is also derived. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The Langevin equation, i.e., the equation of motion of the relevant random variables, is a cornerstone of the theory of the Brownian motion [1] representing a powerful tool for the study of molecular dynamics in condensed media. For example, it is widely used to solve problems involving orientational relaxation of molecules in liquids [2] as well as been entirely complementary to the Fokker-Planck equation [3] describing the time evolution of the probability distributions underlying the random variables. However, if explicit knowledge of the distribution function is not required, as is true in most physical applications, then the dynamical approach via the Langevin equation alone is in general far simpler and more intuitive than that based solely on the Fokker– Planck equation [3]. In the present context the Langevin equation for the dipole moment of symmetric top molecules has already been derived in Ref. [4] representing molecules with two rotational axes of symmetry each having the same inertia and one unique axis of rotation with differing moment of inertia, viz. Ixx = Iyy ≠ Izz. The dipole moment μ of a symmetrical top molecule is usually directed along the axis of symmetry (axis z of the molecular frame). However, in certain cases, so called
accidental symmetry exists, meaning that a molecule has approximately equal moments of inertia about two axes [5,6]. Such slightly asymmetric top molecules are naturally formed by a mixture of isotopes as in, for example, asymmetric methyl chloride CH2DCl [6]. The other class of slightly asymmetric tops comprises rod-shaped molecules. Here, the dipole vector μ may be oriented at an angle β to the long axis of the molecule. An example is provided by mesogenic 10-TPEB molecules (C10H21–∅-∅–CH2-CH2–∅-N=C=S with ∅ = C6H4), where the angle β is markedly different from zero, viz., β = 42° ± 2° [7]. This case was not covered at all in Ref. [4]. Therefore we now show how to obtain the Langevin equation for the dipole moment μ(t) of an accidental symmetric top molecule subjected to an external electric field E(t) and also how to reduce the vector Langevin equation to scalar equations for the components of μ(t) in the laboratory coordinate system. The results of Ref. [4] for linear, spherical and symmetric top molecules also follow directly from the more general result given in this paper. An example of using the vector Langevin equation is given in the Appendix, where the dipole autocorrelation function (ACF) for the collisionless (free rotational) model of an accidental symmetric top molecule is derived. 2. The Langevin equation for the dipole moment vector
⁎ Corresponding author. E-mail address: [email protected] (S.V. Titov).
https://doi.org/10.1016/j.molliq.2019.111123 0167-7322/© 2019 Elsevier B.V. All rights reserved.
In deriving the Langevin equation for the rotational Brownian motion of μ(t) for an accidental symmetric top molecule we naturally
2
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
commence with the equation of motion of a rigid body [8], viz. the rate of change of angular momentum
into the sum of two vectors Ωz and ΩM (see Fig. 1), directed along the ez and M axes, respectively, [4].
d Mðt Þ ¼ Kðt Þ; dt
Ω ¼ Ωz þ ΩM ;
ð1Þ
where M(t) is the angular momentum of the rigid body representing the molecule and K(t) is the imposed torque. Now M(t) for a symmetric top molecule is related to the angular velocity vector Ω(t) via the inertia tensor I [8], i.e., M = I ⋅ Ω. However, in the rotational Brownian motion of the accidental symmetric top molecule the direction of the dipole moment is no longer coincident with the axis of symmetry [5,6]. We commence by writing the total imposed torque K(t) which is [4]. Kðt Þ ¼ −ξΩðt Þ þ ½μðt Þ Eðt Þ þ Λðt Þ:
ð2Þ
In Eq. (2) Λ(t) is the usual Gaussian white noise torque arising from the heat bath and is represented by a Wiener process, [μ(t) × E(t)] is the deterministic external torque arising from E(t), −ξΩ(t) is the frictional torque, and ξ is the friction coefficient. Thus with Eq. (2) for K(t) Eq. (1) now becomes the vector Langevin equation d Mðt Þ ¼ −ξΩðt Þ þ ½μðt Þ Eðt Þ þ Λðt Þ: dt
ð3Þ
Unfortunately, Eq. (3) is very difficult to directly apply to relaxation problems because of the complicated relations (involving the Eulerian angles) between Ω, E, μ, M, and the orientation of the body, therefore one must proceed as follows. First the geometry of the accidental symmetrical top molecule problem is shown in Fig. 1. The moving axes xyz are fixed in the molecule and z is the axis of symmetry. The dipole μ makes an angle β with the z axis and the axes x and y are chosen without loss of generality so that μ has the following components referred to the fixed axes μ ¼ ðμ sinβ; 0; μ cosβÞ:
ð4Þ
Now the angular velocity Ω and the angular momentum M in the molecular frame have components Ω ¼ ωx ; ωy ; ωz ;
ð5Þ
M ¼ Iωx ; Iωy ; Iz ωz ;
ð6Þ
ð7Þ
where ΩM ¼ M=I;
ð8Þ
and by geometry, Ωz and ΩM have the following components referred to the fixed axes xyz Ωz ¼ f0; 0; ð1−I z =I Þωz g;
ð9Þ
ΩM ¼ ωx ; ωy ; Iz ωz =I :
ð10Þ
ωx ¼ φ_ sinθ sinψ þ θ_ cosψ;
ð11Þ
ωy ¼ φ_ sinθ cosψ−θ_ sinψ;
ð12Þ
_ ωz ¼ φ_ cosθ þ ψ:
ð13Þ
Moreover, the components of the angular velocity ωx, ωy, ωz can be expressed in terms of the Eulerian angles θ, φ, ψ and their time derivatives. Recall that these angles connect molecular fixed axes xyz with laboratory fixed axes XYZ (see Fig. 2). Thus, we have [8]. The vector μ is constant relative to molecular fixed axes xyz, therefore the time dependence of μ(t) in the laboratory coordinate system XYZ is wholly due to rotation of the top and satisfies the kinematic equation [8]. μ_ ðt Þ ¼ ½Ωðt Þ μðt Þ:
ð14Þ
By differentiation of the above Eq. (14) and subsequently using Eq. (7), we then have the equation of motion of the random variable μ(t) h i h M i h z i _ μ þ ½Ω μ_ ¼ Ω _ μ þ Ω _ μ þ ½Ω ½Ω μ: €¼ Ω μ
where I = Ixx = Iyy and Iz = Izz are the principal moments of inertia of the body. Eqs. (5) and (6) show that both M, Ω, and the unit vector ez along the z axis lie in the same plane. Hence Ω may be decomposed
Fig. 1. Geometry of an accidental symmetric top molecule.
Fig. 2. Eulerian angles.
ð15Þ
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
Eq. (15) may be further simplified as follows. We note that the vec_ M ðtÞ satisfies the Langevin Eq. (3), which with Eq. (8) for ΩM(t) may tor Ω then be rewritten as M
_ þ ξΩ ¼ ½μ E þ Λ: IΩ
ð16Þ
_ z ðtÞ satisfies the kinematic relation (cf. Moreover, the vector Ω Eq. (14)) _ z ¼ Ω Ωz : Ω
ð17Þ
_ z from Eq. (17) into _ M from Eq. (16) and Ω Hence by substituting Ω Eq. (15), we have € þ ðξ=IÞ½Ω μ−½Ω ½Ω μ− Ω Ωz μ μ −1 ¼ I ð½½μ E μ þ ½Λ μÞ:
ð18Þ
Using Eqs. (7) and (14) and also the vector triple product [A × [B × C]] = B(A ⋅ C) − C(A ⋅ B), Eq. (18) then becomes € þ ðξ=IÞμ_ þ Ω2 μ− ΩM μ Ω−ðΩ μÞΩz μ ¼ I −1 μ 2 E−ðμ EÞμ þ ½Λ μ :
ð19Þ
Eq. (19) is the desired vector Langevin equation of motion for the random variable μ(t). Nevertheless, in practical applications, e.g., in interpreting experimental results on orientational relaxation of dipolar molecules in a liquid, it is not the dynamics of μ(t) itself that are of interest, rather the time variation of its projection onto some axis fixed in space say the Z axis. Hence, we need only an equation for the projection of μ(t) onto the Z axis of the laboratory coordinate system. Thus taking the general Eq. (19) and supposing for simplicity that E(t) is also applied along M z z that axis then the projections μZ, ΩZ , ΩM Z , ΩZ of μ, Ω, Ω , Ω onto axis Z have the following components μ Z ðt Þ ¼ μ ð cosβ cosθðt Þ þ sinβ sinθðt Þ sinψðt ÞÞ;
ð20Þ
ΩZ ðt Þ ¼ φ_ ðt Þ þ ψ_ ðt Þ cosθðt Þ;
ð21Þ
2 _ _ _ ΩM Z ðt Þ ¼ φðt Þ sin θðt Þ þ ðI z =I Þ φðt Þ cosθðt Þ þ ψðt Þ cosθðt Þ;
ð22Þ
ΩzZ ðt Þ ¼ ð1−Iz =IÞ φ_ ðt Þ cosθðt Þ þ ψ_ ðt Þ cosθðt Þ:
ð23Þ
Next rewriting the general Eq. (19) for the motion of μ(t) in terms of its projection onto axis Z and then using Eqs. (20)–(23) we finally have the following scalar equation
2 M € þ ðξ=IÞq_ þ ΩM q−ΩM cosβ q Z Ωz n o M M þ sinβ €r þ ðξ=I Þr_ þ ðΩÞ2 r− 2bΩM z q þ ΩZ Ωx ¼ E μ 2 −μ 2Z þ ½Λ μjZ =ðμIÞ;
3
3. Autocorrelation function of the dipole moment of a free accidental symmetrical top molecule The dipole autocorrelation functions (ACFs) of an ensemble of freely rotating molecules are often used in the analysis of orientational relaxation in polar liquids [9] and although the free rotation Ansatz does not strictly correspond to actual physical conditions, nevertheless it still is of use, e.g. in liquids consisting of polar molecules with large inertia and weak intermolecular interactions so that the rotational motion is dominated by inertia. Here the angular momentum of a typical molecule undergoes weak fluctuations and so differs but little from its initial value. Hence interpreting the rotation as free is a good approximation at least for short time intervals. Moreover, yet another class of models of molecular rotational motion (so called extended rotational diffusion) exists. In these, free rotation between strong collisions is assumed [9,10] so that their molecular orientational ACFs may be represented simply by the corresponding ACFs of a freely rotating molecule and the other parameters characterizing the collision process [9,10]. Taken in this sense, analysis of free rotation underpins the dynamics of molecules and thus the rotational absorption spectra of fluids. We now calculate from Eq. (19) the ACF of the longitudinal component of the dipole moment CFR(t) = 〈μZ(0)μZ(t)〉0 of freely rotating accidental symmetric top molecules, which is given by C FR ðt Þ ¼ cos2 βhqð0Þqðt Þi0 þ sin2 βhr ð0Þr ðt Þi0 ;
ð25Þ
where the angular braces 〈(⋅)〉0 denote the (equilibrium) average over all possible initial values (i.e., at time t = 0) of orientations {θ0, φ0, ψ0} and conjugate momenta {p0θ , p0φ, p0ψ}, namely Z hðÞi0 ¼
H0 dΓ: ðÞ e kB T −
ð26Þ
0 0 0 0 0 0 Here dΓ = Z−1 0 dθ dφ dψ dpθ dpφdpψ, Z0 is the partition function, kB is Boltzmann's constant, Т is the temperature, and
2 2 0 2 p0φ −p0ψ cosθ0 p0ψ pθ þ þ H0 ¼ 2 0 2I 2Iz 2I sin θ
ð27Þ
is the Hamiltonian of a freely rotating molecule. For the deterministic case of non-interacting molecules (free rotation) so that Λ(t), ξ and E(t) are all zero, Eq. (24) yields two simultaneous differential equations for q(t) = cos θ(t) and r(t) = sin θ(t) sin ψ(t), viz.
€ðt Þ þ α 2 qðt Þ ¼ kl; q
ð28Þ
€r ðt Þ þ Ω2 r ðt Þ ¼ ½2bkqðt Þ þ lðωx ðt Þ;
ð29Þ
where ð24Þ
where q(t) = cos θ(t) and r(t) = sin θ(t) sin ψ(t) are the direction cosines, b = I/Iz − 1 and we have used ΩzZ = Ωzzq = bΩM z q. We first solve Eq. (24) for the particular case E(t) = 0 and vanishing stochastic terms ξΩ(t) and Λ(t) and then use the solution so obtained for the derivation of the autocorrelation function of the dipole moment of the free accidental symmetric top molecule.
2 α 2 ¼ ΩM ; 0 l ¼ ΩM Z ¼ pφ =I;
0 k ¼ ΩM z ¼ pψ =I; 2 2 2 Ω ¼ α þ ðI=I z Þ2 −1 k
ð30Þ
are time-independent parameters. To close the set of Eqs. (28) and (29), we also need an equation for the angular velocity component ωx(t). The
4
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
latter may be found from the Euler equations for a free rotating symmetrical top molecule [8], viz. _ x þ ðIz −I Þωy ωz ¼ 0; Iω _ y þ ðI−Iz Þωx ωz ¼ 0; Iω _z ¼0 Iz ω
ð31Þ
while the Hamiltonian, Eq. (27), and partition function Z become in the new variables 2 H 0 ¼ kB Tρ2 1 þ bx Z2π
Z2π
Eqs. (28), (29) and (31) have initial values (i.e., at time t = 0) of orientations {θ0, φ0, ψ0} and conjugate momenta {p0θ , p0φ, p0ψ} given by qð0Þ ¼ cosθ0 ; rð0Þ ¼ sinθ0 sinψ0 ; q_ ð0Þ ¼ −I −1 p0θ!sinθ0 ; p0φ −p0ψ cosθ0 0 cosθ0 sinθ0 cosψ0 ; r_ ð0Þ ¼ I −1 p0θ cosθ0 sinψ0 þ I −1 z pψ − 2 I sin θ0 !2 0 2 p0φ −p0ψ cosθ0 pθ þ : ω2x ð0Þ þ ω2y ð0Þ ¼ I I sinθ0
dφ0
Z¼ 0
¼
Z2π dϕ0
0
dϕ00
0
8π 7=2
ð38Þ Z1
Z1 dγ
−1
Z∞ dx
−1
2 2 ρ2 e−ρ ð1þbx Þ dρ
0
ð39Þ
1=2
ð1 þ bÞ
Noticing Eqs. (35) and (36), integrating Eq. (25) over the new variables yields the desired ACF 2 − C FR ðt Þ ¼ ½C 1 þ C 2 ðt Þ cos2 β þ C 3 ðt Þ þ C þ 4 ðt Þ þ C 4 ðt Þ sin β:
ð40Þ
ð32Þ Here Using the known integrals of the motion given by Eq. (30) and the constraint q2 þ r 2 þ s2 ¼ 1;
ð33Þ
where s = sin θ cos ψ denotes the third direction cosine, we have the solutions of Eqs. (28), (29) and (31), namely ωx ðt Þ ¼
qðt Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 α 2 −k sin bkt þ ϕ00 ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 α 2 −k α 2 −l α2
sinðαt þ ϕ0 Þ þ
ð34Þ
kl ; α2
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffii > −1 > þ 1 ln 1 þ b þ b −1; bN0; 1 < qbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffi C1 ¼ −1 3b > > jbj −1 arcsin jbj−1 ; −1=2bbb0; :
and the time-dependent functions C2(t), C3(t), and C± 4 (t) are defined as 2 − τ pffiffiffiffiffiffiffiffiffiffiffiffi Z1 2 2 2 2 1−x 1 þ bx −τ 2 1 þ bx 1þb C 2 ðt Þ ¼ dx; e 5=2 6 2 1 þ bx −1
ð35Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 r ðt Þ ¼ α 2 −l ðα−kÞ cos αt þ ϕ0 −bkt−ϕ00 2α 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þðα þ kÞ α 2 −l cos αt þ ϕ0 þ bkt þ ϕ00 þ 2l α 2 −k sin bkt þ ϕ00 :
ð41Þ
ð42Þ
2
ðbxÞ τ2 − 2 2 1 þ bx −τ 2 ðbxÞ 2 1 þ bx2 2 dx; 1−x e 5=2 2 1 þ bx −1
pffiffiffiffiffiffiffiffiffiffiffiffi Z1 1þb C 3 ðt Þ ¼ 12
ð43Þ
2
ð36Þ Here the integration constants ϕ0 and ϕ0′ are related to the initial values {θ0, φ0, ψ0, p0θ , p0φ, p0ψ} via Eq. (32). Eqs. (33), (35), and (36) for the direction cosines determine completely the orientation of the top with respect of the laboratory coordinate system i.e., the axis of the top rotates uniformly about the direction of the angular momentum M; simultaneously the top rotates uniformly about z axis. The angular velocities of these two rotations are, respectively, ΩM = M/I = α and ωz = Ik/Iz. Note also that b = -1/2, b = 0, and b → ∞ correspond to a disk, a spherical top, and a linear rotator, respectively. Next, we utilize the explicit solution of Eq. (19) given by Eqs. (35) and (36) and subsequently instead of averaging over the initial orientations θ0, φ0, ψ0 and momenta p0θ , p0φ, p0ψ in Eq. (25), we shall use the dimensionless variables
ð1 bxÞ τ2 − pffiffiffiffiffiffiffiffiffiffiffiffi Z1 2 2 2 2 2 1 þ bx 1 þ b 1 þ bx −τ ð1 bxÞ 2 C dx; ð44Þ ð t Þ ¼ ð 1 x Þ e 5=2 4 24 2 1 þ bx −1 pffiffiffiffiffiffiffiffiffiffiffiffi where τ ¼ t kB T=I . Eq. (40) is in full agreement with that of Ref. [11] obtained by another method. 4. Conclusion We have derived the Langevin Eq. (19) for μ(t) of an accidental symmetric top molecule. Now particular cases of this equation were already considered in [4], e.g. if μ(t) is parallel to the axis of symmetry (axis z). We have for an arbitrary vector A ðA μÞΩz ¼ A Ωz μ;
ð45Þ
sffiffiffiffiffiffiffiffiffiffiffi I ρ¼α ð0 ≤ρb∞Þ; x ¼ k=α ð−1≤xb1Þ; γ ¼ l=α ð−1 ≤γb1Þ; 2kB T ϕ0 ð0≤ϕ0 b2πÞ; ϕ00 0 ≤ϕ00 b2π ; φ0 ¼ φ0 ð0 ≤φ0 b2πÞ:
and from this equation we can then rewrite Eq. (19) exactly in the form used in Ref. [4]
The Jacobian of the transformation of variables for the integral in Eq. (26) is just
For a spherical top molecule (Iz = I) ΩM(t) = Ω(t), Ωz(t) = 0 and Eq. (22) becomes [4].
∂ θ0 ; φ0 ; ψ0 ; p0 ; p0 ; p0 φ θ ψ ¼ ρ2 ð2IkB T Þ3=2 ; ∂ φ0 ; ϕ0 ; ϕ00 ; ρ; x; γ
€ þ ðξ=I Þμ_ þ Ω2 μ−ðΩ μÞΩ μ ¼ I −1 μ 2 E−ðμ EÞμ þ ½Λ μ :
2 € þ ðξ=IÞμ_ þ ΩM μ− ΩM μ ΩM μ ¼ I−1 μ 2 E−ðμ EÞμ þ ½Λ μ :
ð37Þ
ð46Þ
ð47Þ
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
The projected Eqs. (24) and (19) for μ(t) may be solved as in Ref. [3] pertaining to the solutions of the Langevin Eqs. (46) and (47). The Langevin method just described permits analysis of the dynamics of μ(t) without using kinematic equations. Therefore, it markedly simplifies the calculation of experimentally measurable quantities (e.g., orientation correlation functions and their spectra), where knowledge of the dynamics of μ(t) is vital. Moreover, our results also hold for molecules (or single domain magnetic nanoparticles) having magnetic dipole moments on simply replacing μ(t) and E(t) by the corresponding magnetic quantities. References [1] M.C. Wang, G.E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Mod. Phys. 17 (1945) 323.
5
[2] W.T. Coffey, M.W. Evans, P. Grigolini, Molecular Diffusion and Spectra, Wiley, New York, 1984 (384 p). [3] W.T. Coffey, Yu.P. Kalmykov, The Langevin Equation, 4th ed World Scientific, Singapore, 2017 (902 p). [4] W.T. Coffey, Yu.P. Kalmykov, The Langevin equation for the dipole vector of a symmetric top molecule, J. Mol. Liquids. 49 (1991) 79. [5] T.G. Kyle, Atmospheric Transmission, Emission and Scattering, Pergamon press, New York, 1991 (287 p). [6] C.H. Townes, A.L. Schawlow, Microwave Spectroscopy, Dover, New York, 2003. [7] J. Jadżyn, P. Kędziora, L. Hellemans, et al., Relaxation of the Langevin saturation in dilute solution of mesogenic 10-TPEB molecules, Chem. Phys. Lett. 302 (1999) 337. [8] L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edition Pergamon Press, London, 1976. [9] R.E.D. McClung, On the extended rotational diffusion models for molecular orientation in fluids, Adv. Mol. Relaxation Processes 10 (1977) 83. [10] V.I. Gaiduk, Yu.P. Kalmykov, Dielectric relaxation and molecular motion in polar fluids: dynamic and kinetic approaches, J. Mol. Liquids. 34 (1987) 1. [11] A.G.St. Pierre, W.A. Steele, Time correlations and conditional distribution functions for classical ensembles of free rotors, Phys. Rev. 184 (1969) 172.
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Langevin equation for the dipole vector of an accidental symmetric top molecule Yu.P. Kalmykov a, S.V. Titov b,⁎, W.T. Coffey c, M. Zarifakis c,d a
Laboratoire de Mathématiques et Physique (EA 4217), Université de Perpignan Via Domitia, F-66860, Perpignan, France Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region 141190, Russia Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland d Electricity Supply Board, Generation, Asset Management, Dublin 3, Ireland b c
a r t i c l e
i n f o
Article history: Received 5 March 2019 Received in revised form 24 May 2019 Accepted 2 June 2019 Available online 6 June 2019 Keywords: Langevin equation Accidental symmetric top Autocorrelation function Rotational Brownian motion
a b s t r a c t The Langevin equation governing the time dependent dipole moment μ(t) of an accidental symmetric top (i.e., the direction of μ(t) does not coincide with the axis of symmetry) in an external electric field is directly derived from the Euler-Langevin equation. Results which have been previously obtained in J. Mol. Liquids. 1991, 49, 79 are regained as special cases. The scalar Langevin equation governing the longitudinal component of μ(t) along a space fixed axis is also derived. © 2019 Elsevier B.V. All rights reserved.
1. Introduction The Langevin equation, i.e., the equation of motion of the relevant random variables, is a cornerstone of the theory of the Brownian motion [1] representing a powerful tool for the study of molecular dynamics in condensed media. For example, it is widely used to solve problems involving orientational relaxation of molecules in liquids [2] as well as been entirely complementary to the Fokker-Planck equation [3] describing the time evolution of the probability distributions underlying the random variables. However, if explicit knowledge of the distribution function is not required, as is true in most physical applications, then the dynamical approach via the Langevin equation alone is in general far simpler and more intuitive than that based solely on the Fokker– Planck equation [3]. In the present context the Langevin equation for the dipole moment of symmetric top molecules has already been derived in Ref. [4] representing molecules with two rotational axes of symmetry each having the same inertia and one unique axis of rotation with differing moment of inertia, viz. Ixx = Iyy ≠ Izz. The dipole moment μ of a symmetrical top molecule is usually directed along the axis of symmetry (axis z of the molecular frame). However, in certain cases, so called
accidental symmetry exists, meaning that a molecule has approximately equal moments of inertia about two axes [5,6]. Such slightly asymmetric top molecules are naturally formed by a mixture of isotopes as in, for example, asymmetric methyl chloride CH2DCl [6]. The other class of slightly asymmetric tops comprises rod-shaped molecules. Here, the dipole vector μ may be oriented at an angle β to the long axis of the molecule. An example is provided by mesogenic 10-TPEB molecules (C10H21–∅-∅–CH2-CH2–∅-N=C=S with ∅ = C6H4), where the angle β is markedly different from zero, viz., β = 42° ± 2° [7]. This case was not covered at all in Ref. [4]. Therefore we now show how to obtain the Langevin equation for the dipole moment μ(t) of an accidental symmetric top molecule subjected to an external electric field E(t) and also how to reduce the vector Langevin equation to scalar equations for the components of μ(t) in the laboratory coordinate system. The results of Ref. [4] for linear, spherical and symmetric top molecules also follow directly from the more general result given in this paper. An example of using the vector Langevin equation is given in the Appendix, where the dipole autocorrelation function (ACF) for the collisionless (free rotational) model of an accidental symmetric top molecule is derived. 2. The Langevin equation for the dipole moment vector
⁎ Corresponding author. E-mail address: [email protected] (S.V. Titov).
https://doi.org/10.1016/j.molliq.2019.111123 0167-7322/© 2019 Elsevier B.V. All rights reserved.
In deriving the Langevin equation for the rotational Brownian motion of μ(t) for an accidental symmetric top molecule we naturally
2
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
commence with the equation of motion of a rigid body [8], viz. the rate of change of angular momentum
into the sum of two vectors Ωz and ΩM (see Fig. 1), directed along the ez and M axes, respectively, [4].
d Mðt Þ ¼ Kðt Þ; dt
Ω ¼ Ωz þ ΩM ;
ð1Þ
where M(t) is the angular momentum of the rigid body representing the molecule and K(t) is the imposed torque. Now M(t) for a symmetric top molecule is related to the angular velocity vector Ω(t) via the inertia tensor I [8], i.e., M = I ⋅ Ω. However, in the rotational Brownian motion of the accidental symmetric top molecule the direction of the dipole moment is no longer coincident with the axis of symmetry [5,6]. We commence by writing the total imposed torque K(t) which is [4]. Kðt Þ ¼ −ξΩðt Þ þ ½μðt Þ Eðt Þ þ Λðt Þ:
ð2Þ
In Eq. (2) Λ(t) is the usual Gaussian white noise torque arising from the heat bath and is represented by a Wiener process, [μ(t) × E(t)] is the deterministic external torque arising from E(t), −ξΩ(t) is the frictional torque, and ξ is the friction coefficient. Thus with Eq. (2) for K(t) Eq. (1) now becomes the vector Langevin equation d Mðt Þ ¼ −ξΩðt Þ þ ½μðt Þ Eðt Þ þ Λðt Þ: dt
ð3Þ
Unfortunately, Eq. (3) is very difficult to directly apply to relaxation problems because of the complicated relations (involving the Eulerian angles) between Ω, E, μ, M, and the orientation of the body, therefore one must proceed as follows. First the geometry of the accidental symmetrical top molecule problem is shown in Fig. 1. The moving axes xyz are fixed in the molecule and z is the axis of symmetry. The dipole μ makes an angle β with the z axis and the axes x and y are chosen without loss of generality so that μ has the following components referred to the fixed axes μ ¼ ðμ sinβ; 0; μ cosβÞ:
ð4Þ
Now the angular velocity Ω and the angular momentum M in the molecular frame have components Ω ¼ ωx ; ωy ; ωz ;
ð5Þ
M ¼ Iωx ; Iωy ; Iz ωz ;
ð6Þ
ð7Þ
where ΩM ¼ M=I;
ð8Þ
and by geometry, Ωz and ΩM have the following components referred to the fixed axes xyz Ωz ¼ f0; 0; ð1−I z =I Þωz g;
ð9Þ
ΩM ¼ ωx ; ωy ; Iz ωz =I :
ð10Þ
ωx ¼ φ_ sinθ sinψ þ θ_ cosψ;
ð11Þ
ωy ¼ φ_ sinθ cosψ−θ_ sinψ;
ð12Þ
_ ωz ¼ φ_ cosθ þ ψ:
ð13Þ
Moreover, the components of the angular velocity ωx, ωy, ωz can be expressed in terms of the Eulerian angles θ, φ, ψ and their time derivatives. Recall that these angles connect molecular fixed axes xyz with laboratory fixed axes XYZ (see Fig. 2). Thus, we have [8]. The vector μ is constant relative to molecular fixed axes xyz, therefore the time dependence of μ(t) in the laboratory coordinate system XYZ is wholly due to rotation of the top and satisfies the kinematic equation [8]. μ_ ðt Þ ¼ ½Ωðt Þ μðt Þ:
ð14Þ
By differentiation of the above Eq. (14) and subsequently using Eq. (7), we then have the equation of motion of the random variable μ(t) h i h M i h z i _ μ þ ½Ω μ_ ¼ Ω _ μ þ Ω _ μ þ ½Ω ½Ω μ: €¼ Ω μ
where I = Ixx = Iyy and Iz = Izz are the principal moments of inertia of the body. Eqs. (5) and (6) show that both M, Ω, and the unit vector ez along the z axis lie in the same plane. Hence Ω may be decomposed
Fig. 1. Geometry of an accidental symmetric top molecule.
Fig. 2. Eulerian angles.
ð15Þ
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
Eq. (15) may be further simplified as follows. We note that the vec_ M ðtÞ satisfies the Langevin Eq. (3), which with Eq. (8) for ΩM(t) may tor Ω then be rewritten as M
_ þ ξΩ ¼ ½μ E þ Λ: IΩ
ð16Þ
_ z ðtÞ satisfies the kinematic relation (cf. Moreover, the vector Ω Eq. (14)) _ z ¼ Ω Ωz : Ω
ð17Þ
_ z from Eq. (17) into _ M from Eq. (16) and Ω Hence by substituting Ω Eq. (15), we have € þ ðξ=IÞ½Ω μ−½Ω ½Ω μ− Ω Ωz μ μ −1 ¼ I ð½½μ E μ þ ½Λ μÞ:
ð18Þ
Using Eqs. (7) and (14) and also the vector triple product [A × [B × C]] = B(A ⋅ C) − C(A ⋅ B), Eq. (18) then becomes € þ ðξ=IÞμ_ þ Ω2 μ− ΩM μ Ω−ðΩ μÞΩz μ ¼ I −1 μ 2 E−ðμ EÞμ þ ½Λ μ :
ð19Þ
Eq. (19) is the desired vector Langevin equation of motion for the random variable μ(t). Nevertheless, in practical applications, e.g., in interpreting experimental results on orientational relaxation of dipolar molecules in a liquid, it is not the dynamics of μ(t) itself that are of interest, rather the time variation of its projection onto some axis fixed in space say the Z axis. Hence, we need only an equation for the projection of μ(t) onto the Z axis of the laboratory coordinate system. Thus taking the general Eq. (19) and supposing for simplicity that E(t) is also applied along M z z that axis then the projections μZ, ΩZ , ΩM Z , ΩZ of μ, Ω, Ω , Ω onto axis Z have the following components μ Z ðt Þ ¼ μ ð cosβ cosθðt Þ þ sinβ sinθðt Þ sinψðt ÞÞ;
ð20Þ
ΩZ ðt Þ ¼ φ_ ðt Þ þ ψ_ ðt Þ cosθðt Þ;
ð21Þ
2 _ _ _ ΩM Z ðt Þ ¼ φðt Þ sin θðt Þ þ ðI z =I Þ φðt Þ cosθðt Þ þ ψðt Þ cosθðt Þ;
ð22Þ
ΩzZ ðt Þ ¼ ð1−Iz =IÞ φ_ ðt Þ cosθðt Þ þ ψ_ ðt Þ cosθðt Þ:
ð23Þ
Next rewriting the general Eq. (19) for the motion of μ(t) in terms of its projection onto axis Z and then using Eqs. (20)–(23) we finally have the following scalar equation
2 M € þ ðξ=IÞq_ þ ΩM q−ΩM cosβ q Z Ωz n o M M þ sinβ €r þ ðξ=I Þr_ þ ðΩÞ2 r− 2bΩM z q þ ΩZ Ωx ¼ E μ 2 −μ 2Z þ ½Λ μjZ =ðμIÞ;
3
3. Autocorrelation function of the dipole moment of a free accidental symmetrical top molecule The dipole autocorrelation functions (ACFs) of an ensemble of freely rotating molecules are often used in the analysis of orientational relaxation in polar liquids [9] and although the free rotation Ansatz does not strictly correspond to actual physical conditions, nevertheless it still is of use, e.g. in liquids consisting of polar molecules with large inertia and weak intermolecular interactions so that the rotational motion is dominated by inertia. Here the angular momentum of a typical molecule undergoes weak fluctuations and so differs but little from its initial value. Hence interpreting the rotation as free is a good approximation at least for short time intervals. Moreover, yet another class of models of molecular rotational motion (so called extended rotational diffusion) exists. In these, free rotation between strong collisions is assumed [9,10] so that their molecular orientational ACFs may be represented simply by the corresponding ACFs of a freely rotating molecule and the other parameters characterizing the collision process [9,10]. Taken in this sense, analysis of free rotation underpins the dynamics of molecules and thus the rotational absorption spectra of fluids. We now calculate from Eq. (19) the ACF of the longitudinal component of the dipole moment CFR(t) = 〈μZ(0)μZ(t)〉0 of freely rotating accidental symmetric top molecules, which is given by C FR ðt Þ ¼ cos2 βhqð0Þqðt Þi0 þ sin2 βhr ð0Þr ðt Þi0 ;
ð25Þ
where the angular braces 〈(⋅)〉0 denote the (equilibrium) average over all possible initial values (i.e., at time t = 0) of orientations {θ0, φ0, ψ0} and conjugate momenta {p0θ , p0φ, p0ψ}, namely Z hðÞi0 ¼
H0 dΓ: ðÞ e kB T −
ð26Þ
0 0 0 0 0 0 Here dΓ = Z−1 0 dθ dφ dψ dpθ dpφdpψ, Z0 is the partition function, kB is Boltzmann's constant, Т is the temperature, and
2 2 0 2 p0φ −p0ψ cosθ0 p0ψ pθ þ þ H0 ¼ 2 0 2I 2Iz 2I sin θ
ð27Þ
is the Hamiltonian of a freely rotating molecule. For the deterministic case of non-interacting molecules (free rotation) so that Λ(t), ξ and E(t) are all zero, Eq. (24) yields two simultaneous differential equations for q(t) = cos θ(t) and r(t) = sin θ(t) sin ψ(t), viz.
€ðt Þ þ α 2 qðt Þ ¼ kl; q
ð28Þ
€r ðt Þ þ Ω2 r ðt Þ ¼ ½2bkqðt Þ þ lðωx ðt Þ;
ð29Þ
where ð24Þ
where q(t) = cos θ(t) and r(t) = sin θ(t) sin ψ(t) are the direction cosines, b = I/Iz − 1 and we have used ΩzZ = Ωzzq = bΩM z q. We first solve Eq. (24) for the particular case E(t) = 0 and vanishing stochastic terms ξΩ(t) and Λ(t) and then use the solution so obtained for the derivation of the autocorrelation function of the dipole moment of the free accidental symmetric top molecule.
2 α 2 ¼ ΩM ; 0 l ¼ ΩM Z ¼ pφ =I;
0 k ¼ ΩM z ¼ pψ =I; 2 2 2 Ω ¼ α þ ðI=I z Þ2 −1 k
ð30Þ
are time-independent parameters. To close the set of Eqs. (28) and (29), we also need an equation for the angular velocity component ωx(t). The
4
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
latter may be found from the Euler equations for a free rotating symmetrical top molecule [8], viz. _ x þ ðIz −I Þωy ωz ¼ 0; Iω _ y þ ðI−Iz Þωx ωz ¼ 0; Iω _z ¼0 Iz ω
ð31Þ
while the Hamiltonian, Eq. (27), and partition function Z become in the new variables 2 H 0 ¼ kB Tρ2 1 þ bx Z2π
Z2π
Eqs. (28), (29) and (31) have initial values (i.e., at time t = 0) of orientations {θ0, φ0, ψ0} and conjugate momenta {p0θ , p0φ, p0ψ} given by qð0Þ ¼ cosθ0 ; rð0Þ ¼ sinθ0 sinψ0 ; q_ ð0Þ ¼ −I −1 p0θ!sinθ0 ; p0φ −p0ψ cosθ0 0 cosθ0 sinθ0 cosψ0 ; r_ ð0Þ ¼ I −1 p0θ cosθ0 sinψ0 þ I −1 z pψ − 2 I sin θ0 !2 0 2 p0φ −p0ψ cosθ0 pθ þ : ω2x ð0Þ þ ω2y ð0Þ ¼ I I sinθ0
dφ0
Z¼ 0
¼
Z2π dϕ0
0
dϕ00
0
8π 7=2
ð38Þ Z1
Z1 dγ
−1
Z∞ dx
−1
2 2 ρ2 e−ρ ð1þbx Þ dρ
0
ð39Þ
1=2
ð1 þ bÞ
Noticing Eqs. (35) and (36), integrating Eq. (25) over the new variables yields the desired ACF 2 − C FR ðt Þ ¼ ½C 1 þ C 2 ðt Þ cos2 β þ C 3 ðt Þ þ C þ 4 ðt Þ þ C 4 ðt Þ sin β:
ð40Þ
ð32Þ Here Using the known integrals of the motion given by Eq. (30) and the constraint q2 þ r 2 þ s2 ¼ 1;
ð33Þ
where s = sin θ cos ψ denotes the third direction cosine, we have the solutions of Eqs. (28), (29) and (31), namely ωx ðt Þ ¼
qðt Þ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 α 2 −k sin bkt þ ϕ00 ;
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 α 2 −k α 2 −l α2
sinðαt þ ϕ0 Þ þ
ð34Þ
kl ; α2
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffii > −1 > þ 1 ln 1 þ b þ b −1; bN0; 1 < qbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffi C1 ¼ −1 3b > > jbj −1 arcsin jbj−1 ; −1=2bbb0; :
and the time-dependent functions C2(t), C3(t), and C± 4 (t) are defined as 2 − τ pffiffiffiffiffiffiffiffiffiffiffiffi Z1 2 2 2 2 1−x 1 þ bx −τ 2 1 þ bx 1þb C 2 ðt Þ ¼ dx; e 5=2 6 2 1 þ bx −1
ð35Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 r ðt Þ ¼ α 2 −l ðα−kÞ cos αt þ ϕ0 −bkt−ϕ00 2α 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þðα þ kÞ α 2 −l cos αt þ ϕ0 þ bkt þ ϕ00 þ 2l α 2 −k sin bkt þ ϕ00 :
ð41Þ
ð42Þ
2
ðbxÞ τ2 − 2 2 1 þ bx −τ 2 ðbxÞ 2 1 þ bx2 2 dx; 1−x e 5=2 2 1 þ bx −1
pffiffiffiffiffiffiffiffiffiffiffiffi Z1 1þb C 3 ðt Þ ¼ 12
ð43Þ
2
ð36Þ Here the integration constants ϕ0 and ϕ0′ are related to the initial values {θ0, φ0, ψ0, p0θ , p0φ, p0ψ} via Eq. (32). Eqs. (33), (35), and (36) for the direction cosines determine completely the orientation of the top with respect of the laboratory coordinate system i.e., the axis of the top rotates uniformly about the direction of the angular momentum M; simultaneously the top rotates uniformly about z axis. The angular velocities of these two rotations are, respectively, ΩM = M/I = α and ωz = Ik/Iz. Note also that b = -1/2, b = 0, and b → ∞ correspond to a disk, a spherical top, and a linear rotator, respectively. Next, we utilize the explicit solution of Eq. (19) given by Eqs. (35) and (36) and subsequently instead of averaging over the initial orientations θ0, φ0, ψ0 and momenta p0θ , p0φ, p0ψ in Eq. (25), we shall use the dimensionless variables
ð1 bxÞ τ2 − pffiffiffiffiffiffiffiffiffiffiffiffi Z1 2 2 2 2 2 1 þ bx 1 þ b 1 þ bx −τ ð1 bxÞ 2 C dx; ð44Þ ð t Þ ¼ ð 1 x Þ e 5=2 4 24 2 1 þ bx −1 pffiffiffiffiffiffiffiffiffiffiffiffi where τ ¼ t kB T=I . Eq. (40) is in full agreement with that of Ref. [11] obtained by another method. 4. Conclusion We have derived the Langevin Eq. (19) for μ(t) of an accidental symmetric top molecule. Now particular cases of this equation were already considered in [4], e.g. if μ(t) is parallel to the axis of symmetry (axis z). We have for an arbitrary vector A ðA μÞΩz ¼ A Ωz μ;
ð45Þ
sffiffiffiffiffiffiffiffiffiffiffi I ρ¼α ð0 ≤ρb∞Þ; x ¼ k=α ð−1≤xb1Þ; γ ¼ l=α ð−1 ≤γb1Þ; 2kB T ϕ0 ð0≤ϕ0 b2πÞ; ϕ00 0 ≤ϕ00 b2π ; φ0 ¼ φ0 ð0 ≤φ0 b2πÞ:
and from this equation we can then rewrite Eq. (19) exactly in the form used in Ref. [4]
The Jacobian of the transformation of variables for the integral in Eq. (26) is just
For a spherical top molecule (Iz = I) ΩM(t) = Ω(t), Ωz(t) = 0 and Eq. (22) becomes [4].
∂ θ0 ; φ0 ; ψ0 ; p0 ; p0 ; p0 φ θ ψ ¼ ρ2 ð2IkB T Þ3=2 ; ∂ φ0 ; ϕ0 ; ϕ00 ; ρ; x; γ
€ þ ðξ=I Þμ_ þ Ω2 μ−ðΩ μÞΩ μ ¼ I −1 μ 2 E−ðμ EÞμ þ ½Λ μ :
2 € þ ðξ=IÞμ_ þ ΩM μ− ΩM μ ΩM μ ¼ I−1 μ 2 E−ðμ EÞμ þ ½Λ μ :
ð37Þ
ð46Þ
ð47Þ
Y.P. Kalmykov et al. / Journal of Molecular Liquids 289 (2019) 111123
The projected Eqs. (24) and (19) for μ(t) may be solved as in Ref. [3] pertaining to the solutions of the Langevin Eqs. (46) and (47). The Langevin method just described permits analysis of the dynamics of μ(t) without using kinematic equations. Therefore, it markedly simplifies the calculation of experimentally measurable quantities (e.g., orientation correlation functions and their spectra), where knowledge of the dynamics of μ(t) is vital. Moreover, our results also hold for molecules (or single domain magnetic nanoparticles) having magnetic dipole moments on simply replacing μ(t) and E(t) by the corresponding magnetic quantities. References [1] M.C. Wang, G.E. Uhlenbeck, On the theory of the Brownian motion II, Rev. Mod. Phys. 17 (1945) 323.
5
[2] W.T. Coffey, M.W. Evans, P. Grigolini, Molecular Diffusion and Spectra, Wiley, New York, 1984 (384 p). [3] W.T. Coffey, Yu.P. Kalmykov, The Langevin Equation, 4th ed World Scientific, Singapore, 2017 (902 p). [4] W.T. Coffey, Yu.P. Kalmykov, The Langevin equation for the dipole vector of a symmetric top molecule, J. Mol. Liquids. 49 (1991) 79. [5] T.G. Kyle, Atmospheric Transmission, Emission and Scattering, Pergamon press, New York, 1991 (287 p). [6] C.H. Townes, A.L. Schawlow, Microwave Spectroscopy, Dover, New York, 2003. [7] J. Jadżyn, P. Kędziora, L. Hellemans, et al., Relaxation of the Langevin saturation in dilute solution of mesogenic 10-TPEB molecules, Chem. Phys. Lett. 302 (1999) 337. [8] L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edition Pergamon Press, London, 1976. [9] R.E.D. McClung, On the extended rotational diffusion models for molecular orientation in fluids, Adv. Mol. Relaxation Processes 10 (1977) 83. [10] V.I. Gaiduk, Yu.P. Kalmykov, Dielectric relaxation and molecular motion in polar fluids: dynamic and kinetic approaches, J. Mol. Liquids. 34 (1987) 1. [11] A.G.St. Pierre, W.A. Steele, Time correlations and conditional distribution functions for classical ensembles of free rotors, Phys. Rev. 184 (1969) 172.