Generalized Gaussian Model for Uniaxial ... - ACS Publications

8746

J. Phys. Chem. B 2007, 111, 8746-8761

Generalized Gaussian Model for Uniaxial Rotational Motion: Application to the Calculation of Spectroscopic Responses Ferdinand Volino and Jean-Christophe Perrin* Structures et Proprie´ te´ s d’Architectures Mole´ culaires, UMR 5819 (CEA-CNRS-UJF), DRFMC/SPrAM, CEA-Grenoble, 38054 Grenoble Cedex 9, France ReceiVed: January 19, 2007; In Final Form: April 3, 2007

An exact model aimed at describing uniaxial rotational motions, based on a rotational adapted Gaussian statistics, is presented. In its simplest form, it depends on only two parameters, an order parameter which can vary from 1 (perfect order) to 0 (isotropic diffusion) and a time-dependent correlation parameter F which varies from 1 to 0 between initial and infinite times. This model yields closed form expressions for the correlation functions relevant to the main spectroscopic techniques (dielectric absorption, light and neutron scattering, NMR line shape, spin-lattice relaxation, etc.) for all values of the two parameters. According to the functional form postulated for F(t), in particular forms decaying as power laws at long times, one obtains shapes for the spectroscopic correlation functions and spectra that are similar to those experimentally observed in a large variety of complex systems (liquid crystals, polymers, gels, and amorphous and glassy materials), especially in confined geometries, which often resemble “stretched” exponentials. A simple way to introduce time coherent effects through a modification of F(t) is proposed. Examples of theoretical correlation functions and spectra are presented. Important remarks concerning the application of this model to the analysis of real data are made. This model is the rotational analogue of the Gaussian translational model developed recently (Volino et al. J. Phys. Chem B 2006, 110, 11217).

Introduction In the last two decades, there has been a growing interest in the study of the effect of confinement in many physical systems such as normal liquids, liquid crystals, colloids, and polymers1-7 or disordered complex systems such as amorphous and glassy materials.8 Among the various methods that can be used to characterize molecular arrangement and motions, spectroscopic techniques such as dielectric and infrared absorption, neutron and light scattering, NMR line shape, and relaxometry play an important role. Interpretation of the corresponding data requires the use of models to calculate theoretical responses to be compared with the experimental ones. In addition to the usual translational motion, rotational motion may also be affected by confinement or supercooling, for example, by rending it anisotropic or changing the degree of anisotropy compared to that of the bulk normal state. So far, fundamental theories such as the mode coupling theory (MCT) (often coupled with molecular dynamics (MD) simulations) and phenomenological models (PMs) have been employed to describe such anisotropic motions. However, MCT is essentially limited to the quantitative description of the high-temperature dynamics in glass-forming materials, and the formulation of the PMs is rather involved since it is based on the resolution of a rotational diffusion equation in the presence of an orienting potential in two9 or three10 dimensions. Such PMs have been widely used to interpret data, especially in the field of liquid crystals.11-18 We propose here a simple, exact, and in some aspects more general, model to describe anisotropic rotational motions of * Author to whom correspondence should be addressed. Present address: Energy Resources Engineering, Green Earth Science Building, 367 Panama St., Stanford University, Stanford, CA 94305. Phone: (650) 723-0611. Fax: (650) 725-2099. E-mail: [email protected].

uniaxial objects. In particular instances, it may advantageously replace the previous ones for practical applications due to the greater simplicity of its mathematics. The time correlation functions always appear as closed mathematical expressions, and the laws in the frequency domain obtained by LaplaceFourier transform can also be expressed formally at least in certain limits. This model is the rotational analogue of the Gaussian model developed recently to describe localized translational motions in one, two, and three dimensions1 and applied to analysis of quasi-elastic neutron scattering data from hydrated ionic membranes in terms of proton dynamics.2 Uniaxial Rotational Model General. Suppose a linear particle whose orientation can fluctuate about a fixed direction b n that we call the director, and let Ω be the two-dimensional variable, representing the polar and azimuthal angles (θ,φ), which defines its orientation in a orthonormal frame with z along b n. The model assumes that Ω is a two-dimensional random variable whose equilibrium distribution peq(Ω,S) has uniaxial symmetry about the z-axis. Here the letter S symbolizes the set of parameters that characterizes the existence of an orientational order. This distribution, which does not depend on φ due to the uniaxiality, is characterized by an infinite set of order parameters Sl, where l is the orbital quantum number that can vary from 0 to ∞. Introducing the Legendre polynomials Pl(cos θ),19,20 peq(Ω,S) can be written as ∞

peq(Ω,S) )

∑ l)0

2l + 1 4π

SlPl(cos θ)

(1)

The Pl(cos θ) values are related to the m ) 0 components of the spherical harmonics Yml (Ω), where m is the magnetic

10.1021/jp070474l CCC: $37.00 © 2007 American Chemical Society Published on Web 07/10/2007

Gaussian Model for Uniaxial Rotational Motion

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8747

quantum number that can take 2l + 1 integer values between -l and +l, via the formula

Pl(cos θ) )

x2l4π+ 1 Y (Ω) 0 l

(2)

The Yml (Ω) values have the important following properties

Y00(Ω) )

1 x4π

(3)

m -m Ym* l (Ω) ) ( -1) Yl (Ω)

∫ Yml (Ω)Yl′m′(Ω) dΩ ) δll′δmm′ Pl[cos(Ω B 1,Ω B 2)] )

4π

(4)

(orthonormality)

which depend on S and F, are (inter- and self-) correlation functions of Yml (Ω). mm′ The Cll′ (S,F) functions can be calculated using the equilibrium distribution peq(Ω) for F ) 0 (no correlation) and F ) 1 (complete correlation). For the former case, the two variables Ω1 and Ω2 are independent, and we have mm′ Cll′ (S,0) )

) SlSl′δm0δm′0

(5) mm′ Cll′ (S,1) )

∑

Yml (Ω1)Ym* l (Ω2) (addition theorem) (6)

In these expressions, (Ω B 1,Ω B 2) represents the angle between directions Ω1 and Ω2, and the asterisk symbolizes the complex conjugate. Through the use of eqs 2 and 5, it is easy to show that the distribution is normalized (∫ peq(Ω,S) dΩ ) 1) and that the Sl values represent the average values of the Legendre polynomials Pl(cos θ) using peq(Ω)

Sl ) 〈Pl(cos θ)〉 )

∫ Pl (cos θ)peq(Ω) dΩ

(7)

Since peq(Ω) must be a positive quantity, the functional form of Sl cannot be completely arbitrary. Although many functions can fulfill this requirement, it turns out that a Gaussian dependence is the only one that allows recovery of the standard isotropic limit for the correlation functions (see below). Since the Sl values are linked by such dependence, any Sl with a specific value of l * 0 can be chosen as parameter S. It seems natural to choose the popular “nematic” order parameter S2. In practice, Ω is a time-dependent variable. Let Ω1 and Ω2 be the possible values of Ω at two different times t1 and t2. The quantities Ω1 and Ω2 are equivalent random variables, and they are clearly not independent since for t1 ) t2 the two quantities are equal. Let F ) F1,2 ) F(t1 - t2) ) F(t) be the corresponding time-dependent correlation coefficient. The last equality holds because the random motion is assumed to be stationary. Note that F(t) is not a standard correlation function since it is not directly associated to a measurable quantity. It is only a timedependent coefficient that describes the memory of the initial orientation after a duration t. The problem is now to find the functional form for the probability density function p(Ω1,Ω2,S,F) for the variables Ω1 and Ω2. For this purpose, we develop this function in terms of products of spherical harmonics of Ω1 and Ω2. The spherical harmonics being a complete set of functions, this expansion is unique. We can write

∑

∫ Yml (Ω)Yl'm′*(Ω)peq(Ω) dΩ

(11)

mm′ The actual calculation of Cll′ (S,1) involves integrals of the m m′* form ∫ Yl (Ω)Yl′ (Ω)Yr0(Ω) dΩ. These integrals can be calculated using the composition relation of spherical harmonics and can be expressed in terms of the product of two particular Clebsch-Gordan or “3j” Wigner coefficients.21 From the properties of these coefficients, it appears that these integrals are nonzero only if m ) m′ and |l - l′| e r e l + l′, so for mm′ explicitly calculating Cll′ (S,1) using eq 11, it is sufficient to limit the infinite sum in eq 1 to this latter range. These quantities are tabulated, but their formal expressions are easily obtained for any values of the indices l and m with the help of software such as Mathematica.22 In Appendix 1, we give expressions of mm′ (S,1) until l ) 3. They are of the following form nonzero Cll′

l+l′

mm′ (S,1) ) δmm′ Cll′

mm σll′ ∑ r Sr r)|l-l′|

(12)

mm mm′ where the σll′ r values are numerical coefficients. The Cll′ (S,1) values have the following properties: (i) for all orders

mm′ -m-m′ (S,1) ) Cll′ (S,1) Cll′

(13a)

(ii) for perfect order, that is, Sl ) 1 mm′ (1,1) ) δmm′δm0 Cll′

(13b)

(iii) for zero order, that is, Sl ) δl0 mm′ (0,1) ) Cll′

1 δ δ 2l + 1 ll′ mm′

(13c)

They also satisfy the following sum rule over m for identical l indices

p(Ω1,Ω2,S,F) )

l,l′;m,m′

(10)

For the latter case, the two variables are the same (Ω1 ) Ω2 ) Ω), and we have

l

2l + 1m) -l

∫ Yml (Ω1)peq(Ω1) dΩ1 ∫ Yl′m′*(Ω2) peq(Ω2) dΩ2

l

x(2l + 1)(2l′ + 1) 4π

(iv) Cll'mm′(S,F)Yml (Ω1)

Yl'm′*(Ω2)

(8)

where the quantities

〈Yml (Ω1) Y l'm′*(Ω2)〉 )

x(2l + 1)(2l′ + 1) 4π

mm′ (S,F) (9) Cll′

Cmm ∑ ll (S,1) ) 1 m)-l

(14)

An Explicit Form for the Correlation Functions of the Spherical Harmonics. The problem is now to find an explicit mm′ (S,F) for any value of F between 0 and 1. There form of Cll′ are a priori many possible functional forms that have the correct limiting values. However, there is another limit that this form

8748 J. Phys. Chem. B, Vol. 111, No. 30, 2007

Volino and Perrin

must satisfy, namely, the zeroth-order limit (Sl ) δl0) for which the result is well-known. We have23 mm′ Cll′ (0,F(t)) )

δll′δmm′ exp[-l(l + 1)Drt] 2l + 1

(15)

where Dr is the isotropic rotational diffusion coefficient. We now show that the following expression

∑ σll′mmr(Sr)F r) |l-l′|

Dr ) a

(17)

Concerning the functional form of F(t), the only particular property that this function must possess is that the first term of its series expansion at small t/τ0 is linear, that is

[( ) ]

t t +o τ0 τ0

2

t/τ0 , 1

(18)

where a is a positive dimensionless constant that depends on the explicit form of F(t). Thus, the simple exponential form F(t) ) exp[-t/τ0], corresponding to a ) 1, is possible but not the “stretched” form exp[-(t/τ0)β] with β * 1, which does not vary linearly at short times. Below, we discuss more complex, slowly decaying forms of F(t) satisfying eq 18, which appear to be appropriate to describe collective long-range orientational fluctuations in anisotropic media. Concerning the functional form of Sl versus l, we first look at the high-order limit, when the orientation is always close to b n, that is, θ close to zero. In this case, we can expand the Legendre polynomials to first order in θ and write, after taking averages

Sl ) 〈Pl(cos θ)〉 ≈ 1 -

l(l + 1) 2 〈θ 〉 4

Pl[cos λ] )

This expression suggests that the disorder can be characterized by a single geometric parameter 〈θ2〉, which quantifies the magnitude of the polar disorder at equilibrium. We then look at zero order. In this limit, all of the order parameters should be zero (except S0, which is 1). It turns out that the simplest mathematical form that satisfies all of the requirements is the Gaussian function of l

Using eq 9, we obtain

[

]

(20)

This form implies that Sl can be expressed in terms of the second-order parameter S2 instead of 〈θ2〉 as

Sl ) S2 l( l+1)/6

(21)

Let us now take the limit of zero order (〈θ2〉 f ∞) in eq 16. mm′ According to eq 13c, in this limit, Cll′ (S,F) can be written as

〈θ2〉 2τ0

(24)

is identified as the usual isotropic rotational diffusion coefficient. With this definition, one recovers eq 15 for the isotropic limit. Thus the functional form of eq 20 is practically imposed by the zeroth-order limit. We have also verified numerically that with this form peq(Ω,S2) is definite and positive for all values of Ω and similarly for p(Ω1,Ω2,S,F) for all values of Ω1 and Ω2. (The mathematical proofs of these properties remain to be given.) It is instructive to remark that this limiting case of isotropic rotational diffusion obtained by taking the limit 〈θ2〉 f ∞ and τ0 f ∞ in such a way that the ratio tends to a constant is the rotational analogue of the infinite medium limit of the translational Gaussian model.1 An important question is to know if, given the forms of eq 1 for the equilibrium distribution function and eq 20 for Sl, the form of eq 16 for the pair distribution, on which all of the modeling is based, is one among possible other ones or if it is unique. By analogy with the one-dimensional problem (unipolar motion in a plane), we show in Appendix 3 that the form of eq 16 is probably unique. More Complex Correlation Functions. So far, we have expressions for the correlation functions of the spherical harmonics, which are given by eq 9. Of particular interest for practical purposes are the average values and correlation functions of the quantity Pl(cos λ), where λ is the angle between the particle direction Ω and a fixed direction Q B characterized by the orientation ΩQ (polar and azimuthal angles θQ and φQ) with respect to the director frame. This latter direction may be a momentum transfer, a polarization vector (case of scattering), or a static magnetic field (case of NMR). Using eq 6, we can write

θ , 1 (19)

l(l + 1) 2 〈θ 〉 Sl ) exp 4

(23)

where Dr, defined as

l

F(t) ) 1- a

(22)

This quantity is zero for all values of F except for F ≈ 1, that is, for t/τ0 , 1, where the expression appears as an indeterminate form 00. According to eqs 18 and 20, in this limit, we can write

(16)

is a possible explicit form, provided (i) that F depends on a single characteristic time τ0 and (ii) that the functional form of Sl versus l has Gaussian character. Incidentally, we note the following sum rule over the index m, which generalizes relation eq 14 2(1-F) ∑ Cmm ll (S,F) ) Sl m)-l

δll′δmm′ limSlfδl0 Sl2(1-F) 2l + 1

Sl2(1-F) ≈ exp[-l(l + 1)Drt]

l+l′

mm′ Cll′ (S,F) ) δmm′(Sl′Sl′)(1-F)

mm′ Cll′ (0,F) )

4π

l

∑ Yml (Ω)Ym* l (ΩQ) 2l + 1 m)-l

〈Pl[cos λ1]Pl′[cos λ2]〉 )

4π

x(2l + 1)(2l′ + 1)

(25)

×

mm Cll′ (S,F)Yml (ΩQ)Yl′m*(ΩQ) ∑ m

(26)

where we have used the property that the Cmm′ ll (S,F) values are zero for m * m′. If l gl′, then the values of m are limited to integer values in the range from -l′ to l′. Below, we apply this formalism to the calculation of correlation functions associated with several spectroscopic techniques. Incoherent Neutron Scattering Law for Anisotropic Uniaxial Rotational Motion With these latter formulas, one can easily express the incoherent neutron scattering law for a nucleus located at a



distance R from the center of mass of the fluctuating linear particle. One has

Is(Q B ,t) ) 〈exp[iQ B ‚R B 2] exp[-iQ B ‚R B 1]〉

(27)

B 2| ) R. Using the well-known expansion with |R B 1| ) |R ∞

exp[iQ B ‚R B] )

il(2l + 1)jl(QR)Pl(cos λ1) ∑ l)0

(28)

where the jl values are the spherical Bessel functions of order l, related to the Bessel functions of fractional order of the first kind Jν by jl(z) ) xπ/2z Jl+1/2(z), one can write, after some calculations ∞

Is(Q B ,t) )

2 l

(QR)

∑C

mm m m* ll (S,F)Yl (ΩQ)Yl (ΩQ)

[

l)0

∞

2

∑ 4π(2l + 1)j

]

m

+

π

∑ 4π x(2l + 1)(2l′ + 1) cos (l - l′) 2 j (QR)j (QR) × (29) ∑ C (S,F)Y (Ω )Y (Ω ) l>l′

mm ll′

m l

Q

m* l′

l

l′

Q

m

many systems have planar symmetry, which means that the north and south poles are equivalent equilibrium positions. In this case, the above formulas apply if the probability to jump from one pole to the other is negligibly small on the time scale of the experiment. If this is not the case, and this is particularly true when the order is small, then one must take into account the possibility for such a jump. Note that this situation is similar to the one that is met with the translational Gaussian model1 with the introduction of long-range diffusion. To calculate the correlation functions for this bipolar model using the above results, we make the following reasoning. Suppose, for example, that the orientation is close to the north pole at initial time. It will fluctuate about it during a certain time until it reaches by chance the angle θ ) π/2. At this point, the orientation has an equal probability to continue toward the south pole or to return toward the north pole. To calculate the mean time to exchange between the north and south pole regions, the value of the equilibrium probability density at θ ) π/2 is thus crucial. If the order is large, then this probability may be very weak but nonzero. It is reasonable to assume that this exchange time is proportional to the probability to reach the angle θ ) π/2. The ratio Rr defined as

It is observed that this function does not decay to 0 at infinite time, that is, for F ) 0. The elastic incoherent structure factor (EISF) Is(Q B ,∞) is obtained by substituting F ) 0 in this expression. Using eqs 3 and 7, one obtains

B ,∞) ) EISF ) Is(Q

(2l + 1)jl (QR)Sl Pl (cos θQ) + ∑ l)0

[

∞

2

2

2

]

π

x(2l + 1)(2l′ + 1) cos (l - l′) ∑ 2 l>l′

peq(S,π/2)

(34)

peq(S,0)

plays the role of a reduction factor to calculate the mean jump time τjump for exchanging north and south pole orientations. Since τ0 is the mean time to explore the angular range defined by 〈θ2〉 about one pole, we should clearly have

∞

2

Rr )

jl(QR)jl′(QR) ×

τjump ) τ0/Rr

(35)

SlSl′Pl(cos θQ)Pl′(cos θQ) (30) The limit of isotropic diffusion S ) 0 is obtained by assuming that Sl ) δl0 in these two equations. Using all of the above results, we obtain ∞

Is(Q B ,t)S)0 )

(2l + 1)jl2(QR) exp[-l(l + 1)Drt] ∑ l)0

(31)

where Dr is given by eq 24. This is the well-known “Sears formula” for isotropic diffusion on the surface of a sphere.23 Let us return to the general case. If the sample is in a powder form, then one must perform an isotropic average over θQ in eq 30. Using the orthonormality property of the spherical harmonics relation eq 6 and the sum rule eq 17, we find the following result ∞

Is(Q,t)pow )

(2l + 1)jl2(QR)Sl2(1-F) ∑ l)0

(32)

The corresponding EISF Is(Q,∞)pow for a powder sample is obtained by substituting F ) 0 in this expression

∞

∞

EISFpow ) Is(Q,∞)pow )

This new characteristic time τjump is not a new independent parameter of the model since it can be deduced from τ0 using the equilibrium distribution. The problem is now to find the simplest way to introduce τjump in the unipolar model to obtain the correlation functions for the bipolar model. For this purpose, we use the following heuristic argument. If the particle is around the north pole at initial time, then the unipolar distribution given by eq 1 is approximately the correct one to describe its distribution until a time of the order of τjump. For much larger times this is no more true, and at infinite time, the distribution should be bipolar. Since the exchange (0,π) does not affect the values of the even (l ) 2p) order parameters but changes the signs of the odd (l ) 2p + 1) ones, this means that at infinite time the bipolar correlation functions do not contain odd order parameters. The simplest phenomenological way to introduce this property is to multiply all terms of the unipolar correlation functions involving odd values of l by the exponential function exp(-t/τjump). Applying this prescription, for example, to eq 32, we obtain the incoherent scattering law for the bipolar model (sample in powder form) as

(2l + 1)jl (QR)Sl ∑ l)0 2

2

(33)

Putting Sl ) δl0 in this formula, one recovers the EISF of the Sears model. Extension to the Bipolar Model. So far, the model is implicitly assumed to be unipolar in the sense that the preferred orientation is along the north pole (θ ) 0) only. In practice,

Is(Q,t)pow,bi )

+ ∑(4p + 1)j22p(QR)S2(1-F) 2p p)0 ∞

[

( )

2 2(1-F) (4p + 3)j2p+1 (QR)S2p+1 ] exp ∑ τ p)0

t

(36)

jump

It is interesting to consider the limit of large order (Sl ≈ 1) of this expression. We have


Volino and Perrin

∞

Is(Q,t)pow,bi,S≈1 )

∑(4 p + 1)j22p(QR) + p)0 ∞

[

∑(4p +

2 3)j2p+1 (QR))]

p)0

C(t) )

( )

exp -

t

τjump

(37)

Is,jump(Q,t)pow ) 1 1 t [1 + j0(2QR)] + [1 - j0(2QR)] exp (38) 2 2 τjump

)

Since eqs 37 and 38 correspond to exactly the same physical situation, they should be identical whatever the value of τjump. This leads to the following two mathematical identities, or sum rules, implying even and odd spherical Bessel functions ∞

1

(4p + 1)j22p(QR) ) [1 + j0(2QR)] ∑ 2 p)0

(39)

and ∞

1

2 (4p + 3)j2p+1 (QR) ) [1 - j0(2QR)] ∑ 2 p)0

(40)

Since these formulas are usually not given in usual mathematical textbooks, we have verified them numerically to a high accuracy. Similar but more complicated sum rules can be found by identifying the corresponding formulas in the general case, when the powder average is not performed. For nearly zero order, we have Rr ≈ 1, and thus τjump ≈ τ0. However, Sl2(1-F) reduces to the exponential functions as in eq 15. In this limit, we have

l(l + 1)Dr ≈ l(l + 1)〈θ2〉/τ0 . 1/τ0 ≈ 1/τjump

i

i

r p(0))]〉 (42) 〈exp[iQ B (r bp(t) - b

This limit corresponds to the simplistic model of a jump between two sites (here the north and south poles) distant by 2R. The scattering function for this two-site model is well-known and is given, for a powder sample, by23

(

[R 2cos2(i,f) + 〈βp ,f(t)βp′ ,f(0) 〉] × ∑ p,p′

(41)

because 〈θ2〉 (and τ0) are virtually infinite. Thus, exp(-t/τjump) can be replaced by 1 for all practical values of time t, and consequently the bipolar model reduces to the unipolar one, as it should. In Appendix 2, we give an approximate expression for the reduction factor Rr for large orders.

In this expression, Q B is the light momentum transfer, bı and Bf are the incident and scattered polarization unit vectors, and βpi,f is the component of the tensor of molecule p on band ı B. f Dynamic light scattering arises from the motion of the center of mass of the molecules that produces fluctuations of b rp and from the orientational motions that produce fluctuations of the tensor components. Thus, light scattering reflects complicated motions since it is always a combination of collective translational and rotational motions. In practice, since generally Qlmol , 1, where lmol is a molecular size, the broadening due to translation, which is Q-dependent, generally like Q2, is much smaller than that of rotation, which is independent of Q, and the separation between both contributions can be made by properly choosing the experimental energy window of the light-scattering instruments (correlators, interferometers, spectrometers, etc.). We shall be interested here in the rotational motions only. The collective aspect can be simply taken into account by defining a rotational coherence length ξR. Roughly speaking, we can say that the nξ molecules within the volume ξR3 fluctuate in a complete correlated way, so this volume can be considered as a single “large” molecule whose polarizability tensor is nξ times larger, and thus the scattered intensity is nξ2 times larger. This explains why light scattering by ordered systems such as nematics where the coherence length, associated with a finite value of S, is large, is much stronger than that of usual liquids where the spatial orientational correlation is very short range ( nξ ≈ 1). The only possible effect that would reduce the intensity is when the condition QξR , 1 is not fulfilled. In this case, the intensity reduction factor is of the order of 1/(1 + Q2ξR2)R, where R is a model-dependent exponent between 1 and 2. Here, we shall ignore this intensity aspect and focus on the correlation functions. To summarize, for the rotational part of the scattering, we only need to calculate the self-correlation functions 〈βi,f(t)βi,f(0)〉. Calculation of the Correlation Functions of the Polarizability Tensor Components. We first consider the diagonal, traceless, and uniaxial tensor β in the molecular frame whose matrix representation can be written as

[

Light Scattering by a Uniaxial Object General. Consider an ensemble of linear molecules characterized by a uniaxial polarizability tensor R. The average polarizability R ) Tr(R)/3, where Tr is the trace, is removed from this tensor to yield a traceless tensor β defined by β ) R - RI, where I is the unit tensor. If we further assume that β is uniaxial, then this tensor is diagonal in a molecular frame with z along the long axis. The specification of the x molecular axis is irrelevant due to uniaxiality. The components of this tensor in this frame are βxx ) βyy ) -r/3 and βzz ) 2r/3, where r ) ∆β ) βpara - βperp is the molecular polarizability anisotropy. The possibility to use the concept of molecular polarizability implies that the size lmol of the molecules is much smaller than the light wavelength. Assuming that translational and rotational motions are not coupled, the most general expression of the correlation function for light scattering by an ensemble of such molecules p is given by

]

- 1/ 3 0 0 0 -1/3 0 βmol ) ∆β 0 0 2/ 3

(43)

To obtain a representation in the laboratory frame, we first perform a rotation R1 to bring the molecular frame onto the “director frame” with z along the mean director b n and a second rotation R2 to bring the director frame onto the laboratory frame. To perform the rotations, we can use the rotation matrix R(R,β,γ) R(R,β,γ) ) cos γ cos β cos R - sin γ sin R -sin γ cos β cos R - cos γ sin R sin β cos γ cos γ cos β sin R + sin γ cos R -sin γ cos β sin R + cos γ cos R sin β sin R - cos γ sin β sin γ sin β cos β

|

|

where R, β, and γ are Euler angles relating two successive frames. For the first rotation R1, we can choose R1 ) 0 since β is uniaxial, and in these conditions, we have β1 ) θ and γ1 ) φ, where (θ,φ) ≡ Ω are the polar and azimuthal angles describing the orientation of the molecular axis in the director


J. Phys. Chem. B, Vol. 111, No. 30, 2007 8751 This quantity can be expressed as a function of the five geometrical parameters θn, φn, R, θi, and θf and the five timedependent quantities r, s, a, b, and c given by eqs 45-49 in terms of the second-order spherical harmonics. The correlation function C(βi,f) of βi,f is obtained by multiplying the value of βi,f for two values of Ω, say Ω1 and Ω2, and taking the statistical average. In this way, the correlation functions of all secondorder spherical harmonics will appear in the final result. In the most general case, the expression is rather involved although the calculation can be made without difficulty with software such as Mathematica,22 but it simplifies considerably for some particular geometries. We give below the results for the four geometries, called VV, VH, HV, and HH, at a 90° scattering angle. These geometries correspond to

Figure 1. Typical geometrical arrangement for a light-scattering experiment with a uniaxial material. The laboratory XY-plane is defined by the incident and scattered light momentum B ki and B kf, Z is perpendicular to it, θi and θf define the incident and scattered light polarizations with respect to Z, and b n (polar and azimuthal angle θn and φn) defines the uniaxial direction of the sample.

frame. In this director frame, β is represented by the following traceless and symmetric matrix

[

-r a b a -s c βbn ) R-1 β R ) ∆β 1 mol 1 b c r+s

]

(44)

s)

VH: θn ) 0; R ) π/2; θi ) 0, θf ) π/2 HV: θn ) 0; R ) π/2; θi ) π/2, θf ) 0 HH: θn ) 0; R ) π/2; θi ) π/2, θf ) π/2 The results are

C(βi,f)VV )

with

r)

VV: θn ) 0; R ) π/2; θi ) 0, θf ) 0

1 (-1 - 3 cos 2θ + 6 cos 2φ sin2 θ) ) 12 2π 2 0 2 Y (Ω) + Y2* 2 (Ω) + Y2(Ω) (45) 15 3 2

x [ x

]

1 (-1 - 3 cos 2θ - 6 cos 2φ sin2 θ) ) 12 2π 2 0 2 Y (Ω) - Y2* 2 (Ω) - Y2(Ω) (46) 15 3 2

x [ x

]

1 a ) - sin2 θ sin 2φ ) i 2

x2π15[-Y (Ω) + Y (Ω)]

(47)

b ) -cos θ sin θ cos φ )

x2π15[Y (Ω) + Y (Ω)]

(48)

x2π15[-Y (Ω) + Y (Ω)]

(49)

c ) -cos θ sin θ sin φ ) i

2* 2

1* 2

1* 2

2 2

1 2

1 2

For the second rotation R2, we can choose R2 ) 0 since the equilibrium orientational distribution is uniaxial, and in these conditions β2 ) θn and γ2 ) φn, where (θn,φn) ≡ Ωn are the polar and azimuthal angles describing the orientation of b n in the laboratory frame. One possible light-scattering arrangement is pictured in Figure 1 where we have chosen the laboratory frame such that X is along the incident wave vector B ki, Z is perpendicular to the scattering plane, and Y is perpendicular to both vectors. The scattering angle is R, and the angles of the incident and scattered polarization vectors bı and Bf with Z are θi and θf. With these notations, the component of β on the two polarization vectors can be written as -1 -1 βi,f ) (R-1 2 βb nR2)i,f ) (R2 R1 βmolR1R2)i,f

(50)

16π (∆β)2〈Y02(Ω1)Y0* 2 (Ω2)〉 ) 45 2 4(∆β) 2(1-F) 10 18 1 + (S2)F + (S4)F (51) S 45 2 7 7

[

C(βi,f)VH,HV )

C(βi,f)HH )

]

4π (∆β)2〈Y12(Ω1)Y1* 2 (Ω2)〉 ) 15 3(∆β)2 2(1-F) 5 12 1 + (S2)F - (S4)F (52) S2 45 7 7

[

]

4π (∆β)2〈Y22(Ω1)Y2* 2 (Ω2)〉 ) 15 2 3(∆β) 2(1-F) 10 3 1 - (S2)F + (S4)F S2 45 7 7

]

[

(53)

There is a fourth simple situation, namely, when the sample is in powder form. In this case, one must isotropically average over Ωn ) (θn,φn). The ultimate result is very simple, although the calculation is rather lengthy. We obtain

C(βi,f)pow ) )

2 2π[7 + cos 2(b, ı B)] f 〈Ym2 (Ω1)Ym* (∆β)2 2 (Ω2)〉 225 m)-2

∑

2 [7 + cos 2(b, ı B)](∆β) f

90

S2(1-F) 2

(54a)

Similar but generally more complicated expressions can be written for any geometrical arrangement. In the field of complex systems, it is usual to define the (classical) light-scattering susceptibility χ′′(ω) as ω times the real part of the LaplaceFourier transform of the normalized depolarized powder spectrum.8 According to this definition, we thus have

χ′′(ω) ) ω

cos(ωt) dt ∫0∞ S2(1-F(t)) 2

(54b)


Volino and Perrin

For high values of S, approximate expressions can be found by expanding the expressions eqs 51-54 around S2 ≈ 1. Since S4 ) S10/3 2 , we obtain

multiplied by exp(-t/τjump). In the limit of perfect order (π jump model), the result for a powder sample is simply

〈µE(t)µE(0)〉jump,pow )

C(βi,f,S ≈ 1)VV ≈ 4(∆β)2 (5[1 - 2(1 - S2) + (1 + F2)(1 - S2)2]) (55) 45 C(βi,f,S ≈ 1)VH,HV ≈

3(∆β)2 [5F(1 - S2)] 45

[

]

(57)

C(βi,f,S ≈ 1)pow ≈ 2 [7 + cos 2(b, ı B)](∆β) f 1 - (1 - S2)(1 - F) (58) 45 2

[

]

All of these expressions are also valid for the bipolar model since S2 is an even order parameter.

〈µE(t)µE(0)〉iso )

〈µE(t)µE(0)〉 ) µ2〈P1[cos λ1]P1[cos λ2]〉

(59)

where λ1 and λ2 are the angles between b µ and E B at time t and time 0. Using eq 26, we obtain

〈µE(t)µE(0)〉 )

4π

′′(ω) ) (0 - ∞)ω

′(ω) - ∞ )

3

µ

∑ m)-1

(63b)

〈µE(t)µE(0)〉 〈|µE(0)|2〉

cos(ωt) dt

(64a)

[

∫0∞

〈µE(t)µE(0)〉 〈|µE(0)|2〉

]

sin(ωt) dt (64b)

In these expressions, 0 - ∞ is the dielectric dispersion of the substance, n is the refractive index, and c is the velocity of light in a vacuum. (nc is the velocity of light in the substance.) For a powder sample and since S1 ) (S2)1/3, these expressions may be rewritten for the bipolar model as

′′(ω) )ω 0 - ∞

(

)

cos(ωt) dt

(

)

sin(ωt) dt (64d)

t - F(t)) exp ∫0∞ S(2/3)(1 2 τjump

′(ω) - ∞ ) 0 - ∞

[

(60)

where ΩE ) (θE,φE) represents the polar and azimuthal angles of the director b n in the laboratory frame. Using all of the above formulas, we finally obtain

∫0∞

(0 - ∞) 1 - ω

1-ω

m m* Cmm 11 (S,F)Y1 (ΩE)Y1 (ΩE)

µ2 exp(-2Drt) 3

The real part is given by

1

2

(63a)

In the simplest case, the relation between this correlation function and the imaginary part of the complex permittivity ′′(ω) of the studied substance, related to the absorption coefficient per unit length R(ω) by the relation R(ω) ) ω′′(ω)/nc, is

Dielectric Absorption For dielectric absorption, the calculation is similar but much simpler than in the case of light scattering. We ignore the rotational collective aspects, which can be taken into account exactly as for light scattering, as well as the translational effects. The relevant correlation function is the correlation function of the component µE of the molecular electrical dipole b µ on the electric field component E B of the electromagnetic wave, taken as the Z-axis of the laboratory frame. We have

)

The limit of zero order (isotropic diffusion) is obtained by taking the limit t , τ0 in the above expressions. Noting that in this limit Rr f 1 so that τjump ≈ τ0, one obtains

(56)

3(∆β)2 5 2 F (1 - S2)2 C(βi,f,S ≈ 1)HH ≈ 45 3

(

µ2 t exp 3 τjump

t exp ∫0∞ S(2/3)(1-F(t)) 2 τjump

(64c)

]

Equations 54b and 64c show that comparison between dielectric absorption and depolarized light-scattering data may be a powerful way to test at least the qualitative validity of the present model. Spin-Lattice Relaxation

〈µE(t)µE(0)〉 )

2

µ 2(1-F) S [(1 + 2(S1)F) cos2 θE + 3 1 (1 - (S1)F) sin2 θE] (61)

This expression depends only on the angle θE between the electric field and the director. For a powder sample, one must isotropically average over ΩE. The result is

Here we limit ourselves to the simple case of two 1/2 spins located on the long axis at a distance r or of a spin 1 such as the electric field gradient acting on it has uniaxial symmetry around the long axis. In this case, the expression of the spinlattice relaxation rate R1 of the Zeeman levels produced by the uniaxial orientational fluctuations is given by24

(62)

∫0∞ 〈Y1*2 (ΩQ(0))Y12(ΩQ(t))〉 cos(ωLt) dt + ∞ 2 4 ∫0 〈Y2* 2 (ΩQ(0))Y2(ΩQ(t))〉 cos(2ωLt) dt] (65)

Limiting expressions for S around 1 and 0 are easily obtained as above. These formulas hold for the unipolar model. For the bipolar model (fluctuations plus π jumps), since S1 is an odd order parameter, eqs 61 and 62 need only to be

where ωL is the Larmor pulsation and ωint is equal to 2pγ2/r3 or eQVzz/p for the mentioned dipolar and quadrupolar interactions, respectively. The angles ΩQ describe the orientation of the z molecular axis with respect to the laboratory frame, with the static magnetic field H chosen as the Z-axis. Introducing

〈µE(t)µE(0)〉pow )

µ2 2(1-F) S 3 1

R1 )

6π ω 2[ 20 int



the director frame with b n along the corresponding z-axis, we can write 2

Ym2 (ΩQ)

)

∑ (-1)

m-m′

2 D-m,-m′ (χn,θn,φn)Ym′ 2 (Ω)

(66)

m′)-2

where Ω ) (θ,φ) are the polar and azimuthal angles of the molecular axis with respect to the director frame, χn, θn, and φn are the Euler angles which bring the director frame onto the 2 are the 25 elements of the laboratory frame, and D-m,-m′ second-order Wigner matrix. Introducing eq 66 into eq 65, it appears that in the most general case each correlation function in eq 65 is the sum of 25 correlation functions of Ym2 (Ω). In fact, some of them are the same due to the symmetry of these functions. Further, due to the uniaxiality of the magnetic tensors and to the fact that only the direction of the static field is relevant for the laboratory frame, χn and φn are arbitrary, so they can be chosen equal to 0. Below, we give formal expressions of R1 for two ideal cases, namely, when the director is along the static field and when the sample is in powder form. Director along the Static Field. In this case, θn ) 0, and the expression reduces to eq 65 where we have put ΩQ ) Ω. Using eqs 52 and 53, we obtain

R1,aligned ) ∞ 3 10 3 ωint2 0 S2(1-F(t)) 1 - (S2)F(t) + (S4)F(t) cos(ωLt) dt + 2 40 7 7 ∞ 5 12 1 + (S2)F(t) - (S4)F(t) cos (2ωLt) dt (67) 4 0 S2(1-F(t)) 2 7 7

[∫

∫

[

[

]

]

]

Powder Sample. In this case, the result is very simple since the isotropic averages of the product of a Wigner matrix element with another conjugated one are zero for m * m′ and equal to 1/ for m ) m′. The final expression of R is 5 1

R1,powder )

3 ω 2[ 40 int

cos(ωLt) dt + ∫0∞ S2(1-F(t)) 2 ∞ cos(2ωLt) dt] 4 ∫0 S2(1-F(t)) 2

(68)

The usual limit of zero order (isotropic diffusion) is easily obtained by taking the limits S2 f 0 and t , τ0 in eqs 67 and 68, which become identical in this limit. We obtain

∫0∞ exp(-6Drt) cos(ωLt) dt + ∞ 4 ∫0 exp(-6Drt) cos(2ωLt) dt]

R1,iso )

3 ω 2[ 40 int

)

[

]

ωj ) ωj0P2[cos λ]

The fluctuation of the molecule about b n, now described by the time-dependent random variable Ω(t), will affect the line shape according to the time scale and amplitude of this motion. When fluctuations are fast compared to the spin-lattice relaxation time (adiabatic approximation), the line shape Ij(ω) of an ensemble of such magnetically uncoupled molecules is the time Laplace-Fourier transformation of correlation function of the transverse magnetization Gj(t) (free induction decay (FID)) given by24

Gj(t) ) exp[iωLt]〈exp[iYj(t)]〉

(71)

∫0t ωj(t′) dt′ ) ωj0 ∫0t P2[cos λ(t′)] dt′

(72)

where

Yj(t) )

Defining the real random function ξ(t) as

ξ(t) ) P2[cos λ(t)] - 〈P2[cos λ(t)]〉 ) P2[cos λ(t)] - S2P2[cos θQ]

(73)

ωj(t′) ) ωj0{S2P2[cos θQ] + ξ(t′)}

(74)

we can write

Equation 71 can then be expressed as

Gj(t) ) exp[it(ωL + ωj0S2P2[cos θQ])]〈exp[iXj(t)]〉

(75)

where

Xj(t) ) ωj0

∫0t ξ(t′) dt′

(76)

The quantity Xj(t) being the integral of a random function with zero average, it can be considered as a Gaussian random variable since it is the sum of random variables, the central limit theorem asserting its Gaussian character. This argument holds whenever t is large enough compared to the correlation time of ξ(t) to ensure that the random variables are nearly independent. Assuming that this is the case, we have

[

〈exp[iXj(t)]〉 ) exp -

]

〈Xj2(t)〉 2

(77)

A simple mathematical transformation then gives

2

24Dr ωint 6Dr 3 + ≈ ω 2 40 int (6D )2 + ω 2 (6D )2 + 4ω 2 16Dr r L r L if ωL , Dr (69)

(70)

2 〈Xj2(t)〉 ) 2ωj0

∫0t(t - τ)〈ξ(τ)ξ(0)〉 dτ

(78)

where the self-correlation function of ξ is given by

This is a well-known result.

〈ξ(t)ξ(0)〉 ) 〈P2[cos λ(t)]P2[cos λ(0)]〉 - S22P22(cos θQ) (79)

NMR Line Shape

Using eq 26, replacing Cmm 22 and the spherical harmonics by their explicit forms, we obtain

The model can be used to calculate NMR line shapes. We consider here the simplest case of an idealized “basic molecule” constituted by a segment, picturing its cylindrical symmetry axis oriented at an angle λ with respect to the static magnetic field H, itself having an orientation ΩQ with respect to the director b n. We assume that if λ is fixed, then the NMR spectrum is a single sharp line j at a pulsation ωj from the Larmor pulsation ωL given by

〈ξ(τ)ξ(0)〉 ) S2(1-F) 2 (7 + 10(S2)F + 18(S4)F)P22(cos θQ) + 3(7 + 5(S2)F 35 3 12(S4)F)cos2 θQ sin2 θQ + ‚‚‚ (7 - 10(S2)F + 3(S4)F) sin4 θQ 4 S22P22(cos θQ) (80)

[

]


Volino and Perrin

To summarize, the NMR line shape of a uniaxial object whose aligned basic spectrum is composed of a single line j characterized by pulsation ωj0 can be calculated in three or four steps with the above formalism by following the following procedure: (i) Select numerical values for the two independent parameters of the model, for example, the mean-square angular fluctuation amplitude 〈θ2〉 or the order parameter S2 and an expression for F(t), and choose the parameters of the experiment, namely, the pulsation ωj0 and the angle θQ between the static magnetic field and the director. (ii) Use eqs 78-80 to calculate 〈exp[iWj(t)]〉. (iii) Use eq 75 to calculate Gj(t). The spectrum is the real part of the Laplace-Fourier transform of Gj(t). If the aligned basic spectrum is composed of several lines, then the FID is simply the sum of the individual FIDs for each line, and the spectrum is the sum of the individual spectra. (iv) Finally, if the sample is a polydomain, with a distribution function F(θQ,φQ) for the orientation of the directors with respect to the static magnetic field, then the FID to be Laplace-Fourier transformed is Gj(t)poly defined as

Gj(t)poly )

∫ GθQ,j(t)F(θQ,φQ) dΩQ

standard theory of nematics predicts that that the normalized correlation function for the component x of the local nematic decays at long times as t-1/2 yielding spectral densities that diverge as ω1/2 for ω f 0. One simple function that possesses this property is25

F(t) )

xπ erf(xt/τ0) 2 xt/τ

where erf is the error function defined as erf(z) ) 2/xπ × ∫z0 exp(-x2) dx. In the standard theory of nematics, τ0 is the relaxation time of the shortest wavelength elastic modes. Here, it represents the mean time after which the relaxation behaves as t-1/2. In fact, many decaying power laws of the form t-β with β > 0 are a priori possible. A generalization of eq 82 is

[

()

(83)

where Gamma(β,z) is the incomplete gamma function defined by Gamma(β,z) ) ∫∞z xβ-1 exp(-x) dx. Note that Gamma(β,0) ) Γ(β) is the usual gamma function. At long times (t . τ0), F(t) decays as (t/τ0)-β, whereas at short times (t , τ0) it is a linear function of t. We have

F(t) ≈ 1 -

β t 1 + β τ0

t , τ0

(84)

It can be shown that eq 82 is identical to eq 83 with β ) 1/2. According to eq 24 the rotational diffusion coefficient Dr defined by this model is

Dr(β) )

2 β 〈θ 〉 β ln(1/S2) ) β + 1 2τ0 β + 1 3τ0

(85)

For β ) 1/2, which is the usual case for bulk nematics, we obtain

Possible Expressions for the Correlation Coefficient G(t) Purely Decaying Functions. So far, the above formulas have been expressed in terms of the coefficient F, which describes the correlation between the two random variables Ω1 and Ω2. Since these variables represent the orientations of one molecule at time t and time t ) 0, in the absence of coherent motion and at sufficiently large time F ) F(t) is a uniformly decaying function of time from 1 at t ) 0 to 0 at t ) ∞. The functional form is a priori arbitrary, the only condition being that it can be expanded in series around t ) 0 to recover the limiting case of the isotropic rotational diffusion (see above). The simplest idea is to assume that F(t) is a decaying exponential function with a characteristic time τ0. However, this form implies that the decrease of the experimental correlation functions is relatively fast at all times, and this may not be adequate to describe long time relaxations as those that seem to exist in complex systems such as liquid crystals or glass-forming liquids. In such systems, it is often observed that the experimental correlation functions decay as power laws or stretched exponentials. A close inspection of the above formulas as well as numerical calculations shows that this is possible only if F(t) decays at long times as power laws with possibly different exponents. A special form for such a law is suggested by the theory of long-range orientational fluctuations in nematics. The

( )]

t Gamma(β,0) - Gamma β, τ0 F(t) ) β t β τ0

(81)

The isotropic “powder” corresponds to F(θQ,φQ) ) 1/(4π). This formalism may be applied to simulate NMR spectra in oriented soft media due to chemical shift anisotropy or dipolar or quadrupolar interactions. For example, if the chemical shift tensor σ is uniaxial around the long axis, then one can use the above formulas taking ωj0 ) 2∆σ/3 where ∆σ ) σpara - σperp is the anisotropy. Similarly, for two 1/2 spins distant by b r along the long axis, we must consider the two pulsations ωj0 ) (3pγ2/2r3, γ being the gyromagnetic ratio of the spins. For a spin 1 submitted to a uniaxial electric field gradient Vzz along the long axis, the two pulsations are ωj0 ) (3eQVzz/2p, Q being the quadrupolar constant of the nucleus and e the elementary electric charge. According to the relative values of the parameters ωj0 and 1/τ0 and the value of S2, one can obtain a variety of FID curves and spectral shapes for different geometries, which resemble those found in real experiments.

(82)

0

Dr(β ) 1/2) )

〈θ2〉 ln(1/S2) ) 6τ0 9τ0

(86)

Note that for β f ∞, one recovers the simple exponential F(t) ) exp[-t/τ0]. Another possible form proposed in ref 1, very similar to eq 83, which also decays as (t/τ0)-β at large times and tends to the simple exponential function for β f ∞, is

(

F(t) ) 1 +

)

(87)

t , τ0

(88)

1 t β τ0

-β

In this case, we have

F(t) ≈ 1 -

t τ0

and

Dr(β) ) Dr )

〈θ2〉 ln(1/S2) ) 2τ0 3τ0

(89)

The practical interest of eq 87 compared to eq 8 is that a closed form expression for the Laplace-Fourier transform of any



power of (F(t))n exists, so theoretical complex spectra can be calculated rather easily when correlation functions can be expanded as integer series of F. We have

(

)

∫0∞ exp(-iωt) 1 + β1 τt0

-nβ

dt )

βτ0 exp(iβωτ0)EI(nβ,iβωτ0) (90) where EI is the exponential integral function defined by EI(s,z) ) ∫∞1 x-s exp(-zx) dx. Concerning the correlation functions of the spherical harmonics, which are nontrivial functions of F, it can easily be verified numerically that at long times (t . τ0) they decay as a power of F(t). Consider, for example, the correlation functions for light scattering eqs 51-54. Assuming β ) 1/2, using either eq 83 or 87, we find that C(βi,f)VV and C(βi,f)powder behave as (t/τ0)0 (in other words they tend to a constant value), C(βi,f)VH,HV decreases as (t/τ0)-1/2, and C(βi,f)HH as (t/τ0)-1. Concerning the corresponding spectral densities obtained by Laplace-Fourier transformations of the correlation functions, it is easy to show that for ωτ0 , 1 (low frequencies), these functions diverge as δ(ωτ0), (ωτ0)-1/2, and ln(1/ωτ0), respectively. More generally, if the long time decay of a correlation function behaves as (t/τ0)-β with 0 < β < 1, then the lowfrequency Laplace-Fourier transformations diverge as (ωτ0)β-1. For β > 1, the zero frequency divergence disappears, and the spectral densities tend toward a constant. The power law decay implying a divergence of the spectrum at low frequency is however not physical, so a cutoff at long times, characterized by a characteristic time τcutoff, must be introduced. The simplest way to introduce this feature in the model is to multiply the above expressions of F(t) by exp(-Rcutofft/τ0), where by analogy with eq 36, we have defined the reduction coefficient Rcutoff by

τcutoff ) τ0/Rcutoff

(91)

Formal expressions of the Laplace-Fourier transformation of (F(t))n also exist in this case. We have

(

)

∫0∞ exp(-iωt) 1 + β1 τt0

-nβ

(

exp -

)

nRcutofft dt ) τ0

βτ0 exp[β(nRcutoff + iωτ0)]EI[nβ,β(nRcutoff + iωτ0)] (92) Oscillating Functions. So far, we have assumed that the rotational motions are completely random, that is, no inelastic mode of pulsation ω0 (for example, a rotational wave such as a magnon in ferromagnets), is excited. If such excitation exists, then the correlation function F(t) necessarily acquires a coherent component at ω0. The simplest phenomenological way to introduce this effect in the formalism is to multiply F(t) by cos(ω0t), yielding an oscillatory behavior of F and consequently also of the experimental correlation functions. Numerical calculations show that the spectra are no longer single lines centered at ω ) 0 but are now composed of a number of lines with different intensities centered at (nω0, where n is an integer. The formalism predicts that for ω0τ0 . 1 the intensities of these lines are such that their envelope is a Gaussian function centered at ω ) 0. Note that this formalism ignores quantum effects, implying that ω0 , 2kBT/p. Some Examples of Time Correlation Functions Using the above formulas, it is possible to calculate the correlation functions for any spectroscopic technique treated in

Figure 2. Correlation factor F(t) given by eq 83 for different values of the parameter β versus t/τ0. The dotted curve corresponds to β ) 1/ , and the dashed curve to the simple exponential (β ) ∞). The value 2 β ) 1/2 corresponds to the standard elastic mode theory of nematics. For t/τ0 g 10, F(t) behaves as (t/τ0)-β.

this paper. To illustrate the kind of results that are predicted by this formalism, we show curves obtained for light scattering. The basic parameter of this model being the correlation coefficient F(t), we first present the corresponding curves. Figure 2 shows eq 83 versus t/τ0 for finite values of the parameter β as well as the simple exponential corresponding to β ) ∞. It is observed that for t/τ0 g 10 the pure exponential has almost completely vanished whereas the other functions behave as (t/ τ0)-β (power law decay). Figures 3a-c (full curves) show the light-scattering normalized correlation functions for a sample aligned perpendicularly to the scattering plane in the VV, VH (or HV), and HH geometries versus t/τ0 (eqs 51-53). Figure 3d is the corresponding curve for a powder sample (eq 54a). It is observed that a large variety of behaviors is predicted according to the geometry and the order parameter. Figure 4 shows the intensities of the corresponding scattered intensities in units of (∆β)2/45. For the aligned samples, it is observed that in the isotropic (S2 ) 0) limit the ratio of intensities for VV and VH (or HH) geometries is 4/3. As the order increases, the VV intensity increases fast, the HH intensity decreases fast, but the VH (or HV) intensity goes through a flat maximum around S2 ) 0.41 before tending to zero for perfect order. For the powder sample, the intensity is independent of the order but changes by a factor of 4/3, according to the angle between polarization vectors. The isotropic limits are well-known results. Some Examples of Spectra Figures 5a-d represent dielectric relaxation results in reduced units. Figures 5a and 5b show the variation of the imaginary part of the dielectric permittivity (Figure 5a) and the corresponding Cole-Cole plot (Figure 5b) for F(t) given by eq 87 with β ) 1/2 for various values of S2 (eq 64a). Similarly, Figures 5c and 5d show the same results for S2 ) 0.7 and different values of β. It is interesting to note that these curves, with two main maxima whose distances increase with S2, are very similar to those measured in real systems, in particular complex systems.8 The same kind of comments can be made with light-scattering susceptibility results (figures not shown). Figures 6a and 6b show the variation of the spin-lattice relaxation rate R1 versus the Larmor pulsation ωL in reduced units for an aligned and a powder sample, respectively, for different values of S2 (eqs 67 and 68). Memory effects similar


Volino and Perrin

Figure 3. Normalized rotational part of the light-scattering correlation functions for a sample aligned perpendicularly to the scattering plane vs t/τ0. The function F(t) is eq 82. The S2 values are, from top to bottom: 1, 0.8, 0.6, 0.4, 0.2, 0.1, 0.001. The dashed curve is the isotropic result calculated using a diffusion coefficient Dr given by eq 68 with S2 ) 0.001 and τ0 ) 1. (a) VV geometry; (b) VH or HV geometry; (c) HH geometry; (d) powder sample.

Figure 4. Light-scattering intensities in units of (∆β)2/45 in the geometries of Figure 3. Note the non-monotonous variation of the VH intensity.

to those existing in nematics are included via F(t) given by eq 82. For large order, the well-known ωL-1/2 dependence is observed as soon as ωLτ0 < 1, but the model predicts that this dependence still survives at low frequencies for S2 values as small as 0.001. Thus, this method may be able to detect orientational memory effects, even if the residual order is very weak, as may be the case in isotropic fluids in confined media, where the memory is induced by the interaction with the boundaries. For ωLτ0 > 1, R1 varies as ωL-2 (Lorenzian

dependence). If one introduces the long time cutoff characteristic time τcutoff defined by eq 91, then the low-frequency divergences in Figures 6a and 6b disappear and R1 becomes constant for ωLτ0 < Rcutoff. The introduction of this cutoff time in the line shape problem is presumably less important in practice due to resolution effects. Figures 7a-c show powder NMR spectra, calculated using eq 81 and the prescriptions given in the text, as a function the parameter |ωj0τ0| varying around the critical value 1 for three values of the order parameter S2 and assuming that F(t) is a pure exponential (β ) ∞). Figure 7d shows the variation of the spectra with the parameter β, assuming that F(t) is given by eq 83 for |ωj0τ0| ) 1 and S2 ) 0.6. The spectra are convoluted by a Lorentzian line with a half-width at halfmaximum of 0.01|ωj0| to mimic a resolution function. It is seen that the average width is roughly governed by S2, and the shape by |ωj0τ0|. A comparison between Figures 7a and 7d reveals that a decrease of β is roughly equivalent to an increase of τ0 of the simple exponential, so it appears difficult to detect possible memory effects from line-shape analysis. Concluding Remarks The present Gaussian-like model for uniaxial rotational motions, valid whatever the degree of orientational order, has allowed us to obtain exact closed form expressions for the correlation functions relevant to several spectroscopic techniques associated with rotational motions. The use of this model for analysis of data is thus relatively simple in the time domain,



Figure 5. (a) Variation of the imaginary part of the dielectric permittivity for F(t) given by eq 87 with β ) 1/2 for various values of S2 and (b) the associated Cole-Cole plot. (c) Same results for S2 ) 0.7 and different values of β and (d) the associated Cole-Cole plot (eq 64a).

even in the most complicated experimental conditions. In the frequency domain, calculations are more involved since no closed form exists, and as in the translational case,1 one must either perform numerical Laplace-Fourier transforms or use series developments, which need necessarily to be truncated, leading to systematic errors. Presumably, this model, which depends only on a few parameters, can replace advantageously the models cited in the Introduction, because it is simpler and also because it allows to easily introduce complex dynamics via functional forms such as eq 83 or 87, with or without cutoff at long time, for the correlation coefficient F(t). So far, we have assumed that the nematic director is fixed. It is possible that in some complicated cases slow translational motion induces a change of orientation of this director by transporting the molecules in a different environment. In this case, one must introduce a new time scale describing this feature. If the final orientation of the director becomes isotropic at infinite time, then this new time scale can be represented by a “long time isotropic rotational diffusion coefficient” Dr,long time, roughly equal to Dt/L2, where Dt is the translational diffusion coefficient and L a characteristic distance over which the director has rotated significantly. If this process can be described by the standard isotropic diffusion equation, then this feature can be taken into account in the model by multiplying the correlation functions of Yml by exp[-l(l + 1)Dr,long timet]. If not, then a more complex l-dependent decaying function probably needs to be introduced. Furthermore, it must be stressed that this model can only describe an ideal situation because real objects generally lack uniaxial symmetry. A third angle χ needs, in principle, to be introduced to specify completely their orientation with respect to the laboratory frame. This third angle can be easily introduced in the above description if one assumes that the motion described

by χ (in practice the rotation around the “long molecular axis”) is not coupled with the fluctuations of this axis. In this case, the complete correlation functions are combinations of the above correlation functions with the corresponding ones associated with the χ fluctuations, this combination depending on the particular spectroscopic technique considered. The correlation functions associated with χ may, for example, be calculated using the Gaussian one-dimensional rotational model developed in Appendix 3. A model where the rotation around the long axis is uniform and the fluctuations of the axis is isotropic and are described by two diffusion coefficients has been developed in refs 26-28 to calculate NMR line shapes. The uncoupled hypothesis is the main limitation of the present model. In the absence of decoupling, a formally more complex model must be developed, in which all of the distribution functions, in particular eqs 1 and 8, should be expanded in terms of Wigner matrix elements of the variables (χ,θ,φ) instead of the simpler spherical harmonics of the variables (θ,φ). It is possible that exact, but clearly more complex, expressions for correlation functions can also be found in this completely general case. Other complications occur when the molecules are nonrigid, as this is the case when they are composed of rigid chemical moieties linked by single covalent bonds. In this case, internal rotations around these bonds generally occur, in particular the large amplitude ones associated with symmetry operations, such as π flips of phenyl rings and/or exchange between levo and dextro conformations in racemic mixtures of identical molecules, as is often the case in relatively complex molecular materials such as liquid crystals. These motions produce significant averaging of the various molecular vectors or tensors and thus affect correlation functions and spectra. However, the time scale of these motions generally corresponds to sufficiently high frequencies, so the rigid object introduced above can, in this


Volino and Perrin Case l ) l′

C00 00(S,1) ) 1 1 C00 11(S,1) ) (1 + 2S2) 3 1 C11 11(S,1) ) (1 - S2) 3 1 10 18 1 + S 2 + S4 5 7 7

( ) 1 5 12 C (S,1) ) (1 + S - S ) 5 7 7 1 10 3 C (S,1) ) (1 - S + S ) 5 7 7 1 4 18 100 C (S,1) ) (1 + S + S + S 7 3 11 33 ) 1 3 25 C (S,1) ) (1 + S + S - S ) 7 11 11 1 21 10 C (S,1) ) (1 - S + S ) 7 11 11 1 5 9 5 C (S,1) ) (1 - S + S - S ) 7 3 11 33 C00 22(S,1) ) 11 22

2

22 22

4

2

00 33

11 33

4

2

4

2

4

22 33

6

4

33 33

2

6

6

4

6

Case l * l' Figure 6. Log-Log plot in reduced units of the spin-lattice relaxation rate vs Larmor pulsation (a) for an aligned sample and (b) for a powder sample, calculated according to eqs 67 and 68, respectively. The correlation coefficient F(t) is assumed to be given by eq 83 with β ) 1/ (or equivalently eq 82) for the case of nematics for various values 2 of the order parameter S2.

00 Cl0 (S,1) ) Sl

1 C00 21(S,1) ) (2S1 + 3S3) 5 C11 21(S,1) )

case, be identified with the average (over the internal motions) of the molecule. In fact, it is the existence of these internal motions that often allows us to consider the average molecule as a uniaxial object. Further, molecular vibrations also exist, but they are of a weak amplitude and appear at even higher frequencies (typically infrared frequencies), so they can be neglected in a first approximation or taken into account as a Debye-Waller factor. Finally, coupling between rotational and translational motions has been neglected. It is possible that the proposed model is not adequate to describe such a situation since the isotropic limit ignores this coupling. If the coupling is sufficiently strong, then the Gaussian character of the l dependence of Sl (eq 20) is no longer a necessary condition, and a more complex dependence can be introduced. These aspects will not be discussed further here. Despite this limitation, it is hoped that the results of the present work will be useful in spectroscopic studies of physical systems in which anisotropic rotational motions occur, in particular when the order is weak but non-negligible, a situation in which the formulas obtained with the purely Gaussian approximation are no longer valid.25 m′ Appendix 1. Expressions of C ll′ (S,1) Coefficients ll′ ll′ The expressions of all nonzero C mm′ (S,1) ) C -m-m′ (S,1) coefficients for 0 e (l,l′) e 3 in terms of Sl (calculated with Software Mathematica22) are:

x3 (S - S3) 5 1

1 C00 31(S,1) ) (3S2 + 4S4) 7 C11 31(S,1) ) C00 32(S,1) )

1 (27S1 + 28S3 + 50S5) 105

C11 32(S,1) ) C22 32(S,1) )

x6 (S - S4) 7 2

x2 (18S1 + 7S3 - 25S5) 105 1

(9S1 - 14S3 + 5S5)

21x5

Appendix 2. Calculation of the Reduction Factor Rr: Gaussian Approximation In the physics of liquid crystals, which is the archetypal bipolar uniaxial model, a purely Gaussian model, valid only when the order is high, is often used to calculate experimental responses.25,29 The advantage of this model is that calculations are easier since they do not involve infinite sums whose convergent character may pose some numerical problems when these sums become alternate infinite series, as is the case in the



Figure 7. Normalized powder NMR spectra calculated according to eqs 71-81 as described in the text for three values of S2, 0.6, 0.2 and 0.001, as a function of the parameter |ωj0τ0| varying between 0.1 and 10, as indicated in Figure 7a. The dashed curve is the static powder spectrum, corresponding to S2 ) 1. The pulsation ωj0 is taken to be equal to -104 rad/s for convenience. In parts a, b, and c, F(t) is assumed to be a simple exponential, corresponding to β ) ∞ in eq 83. In part d, F(t) is assumed to be given by eq 83, with β variable as indicated, |ωj0τ0| being fixed to 1 and S2 to 0.6. All spectra are convoluted by a Lorenzian line with a half-width at half-maximum of 0.01|ωj0| mimicking a resolution function.

present model for peq(S,Ω) when S is large and the polar angle θ close to π/2. In fact, using eq 1, it is impossible to calculate numerically this quantity for θ equal to π/2 when S2 is larger than about 0.9. It is thus interesting to find another method for this purpose. It turns out that a Gaussian function provides such a method. Numerical calculations show that peq(S,Ω) can fairly well be approximated in the range (0,π/2) by the following Gaussian function

( )

peq,G(S,Ω) ≈ peq(S,θ ) 0) exp -

θ2 ) 〈θ2〉

( ( ))

peq(S,θ ) 0) exp -

3θ2 1 2 ln S2

(A2-1)

The approximation is becoming better and better as the order increases, and both formulas appear to yield identical results for perfect order. Figure A2-1 shows peq(S,Ω)/peq(S,0) and peq,G(S,Ω)/peq,G(S,0) given by eqs 1 and A2-1 calculated for S2 equal to 0.7, 0.4, and 0.1. It is observed that the two shapes are almost identical for S2 > 0.7, so this Gaussian approximation for the equilibrium distribution is extremely good for S2 > 0.7 and reasonable until S2 ≈ 0.4, that is, in the range of usual liquid crystals. It is interesting to compare the merits of the exact and of the above Gaussian distributions for calculating the reduction factor Rr (eq 34). The first point to be noticed is that, as mentioned

Figure A2-1. Normalized equilibrium distributions peq(S,Ω)/peq(S,0) (full lines) and peq,G(S,Ω)/peq,G(S,0) (dashed lines) given by eqs 1 and A2-1, respectively, vs polar angle θ between 0 and π/2, calculated for S2 of 0.7, 0.4, and 0.1. It is observed that the two distributions tend to become identical as the order tends to be perfect.

above, it is not possible to calculate numerically Rr with the exact formula eq 1 for S2 > 0.9, which is a relatively large order. Figure A2-2 shows the ratio Rr/RrG of Rr values calculated with the exact distribution eq 1 and with the Gaussian one eq A2-1 versus S2 until S2 ) 0.9. It is observed that despite the fact that Rr varies by 15 orders of magnitude for S2 between 0.1 and 0.9, the ratio is almost constant and equal to about 1.25 within a few percent. This curve can thus be reasonably extrapolated until S2 ) 1. More precisely, it can be shown that


Volino and Perrin

Cnn′(S,F) ) 〈exp[inφ1] exp[-in′φ2]〉 ) (Sn-n′)F(SnSn′)1-F (A3-5) We now show that this form is unique if one assumes that as in the three-dimensional case the n dependence of Sn is Gaussian, this form being practically imposed by the zeroth-order limit. We have

[

Sn ) exp -

]

n2〈φ2〉 2

(A3-6)

The relation between S2 ) S and the mean-square fluctuation 〈φ2〉 is S ) exp[-2 〈φ2〉]. Using eq A3-6, Cnn′(S,F) can be rewritten as Figure A2-2. Ratio Rr/RrG of the ratios of probability densities at θ ) π/2 and θ ) 0, calculated with the exact equilibrium distribution eq 1 and with the Gaussian distribution eq A2-1, for the values of the secondorder parameter S2 between 0.1 and 0.9 (full line). The dashed line represents eq A2-2.

for 0.4 < S2 < 1, Rr can very well (precision of 0.01%) be represented by the following empirical formula

[

(

Rr ≈ 1.252 + 0.0117 exp -

S2 0.476

)]

( ( ))

exp -

3π2 1 8 ln S2

0.4 < S2 < 1 (A2-2) This expression can be extended until S2 ≈ 0.1 if a precision of only 2% is accepted. For S2 smaller than 0.1, the exact calculation can be easily made since the infinite sum in eq 1 converges rapidly. In fact, as S2 decreases below 0.1, Rr/RrG goes through a maximum equal to 1.31 for S2 ≈ 1.1 × 10-4 and then decreases monotonically to the value 1 for S2 ) 0. Appendix 3. Anisotropic Rotational Motion in a Plane The aim of this appendix is to give the main formulas for a much simpler case than the one treated in the paper, namely, for anisotropic rotational motion in a plane, and to show that the proposed solution for the correlation function is unique. Here we assume a linear particle whose orientation can fluctuate in a plane about a fixed direction called the director and let φ be the instantaneous orientation angle with respect to the director. It is evident that the equivalents of eqs 1, 8, 9, and 11 are

peq(φ,S) )

peq(φ1,φ2,S,F) )

∞

Sn

n)-∞

2π

∑

∞

Cnn′(S,F)

n,n′)-∞

(2π)2

∑

exp[inφ]

(A3-1)

exp[inφ1] exp[-in′φ2] (A3-2)

Cnn′(S,0) ) Sn′n′

(A3-3)

Cnn′(S,1) ) Sn-n′

(A3-4)

By analogy with eq 16, the explicit form of the correlation function Cnn′(S,F) is thus

[

]

〈φ2〉 2

Cnn′(S,F) ) exp -(n2 + n′2 - 2nn′ F)

(A3-7)

Let us now consider the new pair distribution function peq,G(φ1,φ2,S,F) obtained from eq A3-2 by replacing the double sum over n and n′ by integrals. This replacement implies that we are now considering that the angular variable φ is no longer restricted to an interval 2π but can vary between -∞ and ∞. These two descriptions are clearly equivalent from a physical point of view since φ represents the orientation of the linear object. We have

peq,G(φ1,φ2,S,F) )

∫-∞∞ dn ∫-∞∞ dn′

Cnn′(S,F) (2π)2

exp[inφ1] exp[-in′φ2] (A3-8)

Using eq A3-7 in this expression and performing the integration yields

peq,G(φ1,φ2,S,F) )

[

]

φ12 + φ22 - 2φ1φ2F 1 exp (A3-9) 2π〈φ2〉(1 - F2)1/2 2〈φ2〉(1 - F2) This is the pair distribution of two equivalent correlated Gaussian random variables with variance 〈φ2〉, whose equilibrium distribution is1

peq,G(φ,S) )

1

x2π〈φ2〉

[ ]

exp -

φ2 2〈φ2〉

(A3-10)

This pair distribution is known to be unique.30 Now, if we calculate the correlation functions 〈exp[inφ1] exp[-in′φ2]〉 using eq A3-9, then we obtain exactly eq A3-7. This means that although the two distributions eqs A3-2 and A3-9 are different since the variables are not allowed to explore to the same range of values, they are equivalent from an operational point of view since they predict exactly the same correlation functions for the exponential functions exp[inφ] that form a complete basis for developing any function of the angular variable φ. We can thus conclude that if eq A3-7 holds, and this is imposed by both the high-order and the zeroth-order limits, then it is unique. The extension to the bipolar model can easily be made by analogy with that made for the three-dimensional case treated above. Given the close similarities between this formalism and the more complex one developed in the text and the fact that the equilibrium distribution in eq 1 is numerically very close to a Gaussian one (cf. Appendix 2), we can infer that our expression

Gaussian Model for Uniaxial Rotational Motion in eq 16 for the three-dimensional case is probably also unique. It may be possible that an exact demonstration exists since the spherical harmonics in eq 8 can be expressed in terms of Legendre functions Pml (cos θ). Since these functions can be defined for any real or complex function of l and m, it is a priori possible to replace the sums by integrals in eq 8 and consequently define a similar peq,G(Ω1,Ω2,S,F). The mathematical problem of calculating this integral appears too complex at the present stage and has not been explored here. References and Notes (1) Volino, F.; Perrin, J.-C.; Lyonnard, S. J. Phys. Chem. B 2006, 110, 11217. (2) Perrin, J.-C.; Lyonnard, S.; Volino, F. J. Phys. Chem C 2007, 111, 3393. (3) Jorgensen, J. D.; Shapiro, S. M.; Majkrzak, C. F. Proceedings of the International Conference on Neutron Scattering. Physica B 1998, 241243, 1-360. (4) Dianoux, A. J.; Petry, W.; Richter, D. Dynamics of Disordered Materials II. Physica A 1993, 201, 1-452. (5) Zorn, R.; Frick, B.; Bu¨ttner, H. Proceedings of the International Workshop on Dynamics in Confinement. J. Phys. IV 2000, 10, 3-346. (6) Frick, B.; Koza, M.; Zorn, R. Proceedings of the 2nd International Workshop on Dynamics in Confinement. Eur. Phys. J. E 2003, 12, 1-194. (7) Nagele, G.; D’Aguanno, B.; Akcasu, A. Z. Proceedings of the Workshop on Colloid Physics. Physica A 1997, 235, 1-305. (8) Fractals, Diffusion and Relaxation in Disordered Complex Systems; Coffey, W. T., Kalmykov, Y. P., Eds.; Advances in Chemical Physics 133; Wiley: Hoboken, NJ, 2006. (9) Dianoux, A. J.; Volino, F. Mol. Phys. 1977, 34, 1263. (10) Nordio, P. L.; Busolin, P. J. Chem. Phys. 1971, 55, 5485.

J. Phys. Chem. B, Vol. 111, No. 30, 2007 8761 (11) Dianoux, A. J.; Hervet, H.; Volino, F. J. Phys. (Paris) 1977, 38, 809. (12) Volino, F.; Dianoux, A. J. Phys. ReV. Lett. 1977, 39, 763. (13) Nordio, P. L.; Rigatti, G.; Segre, U. J. Chem. Phys. 1972, 56, 2117. (14) Humphries, R. L.; James, P. G.; Luckhurst, G. R. J. Chem. Soc., Faraday Trans. 1972, 68, 1032. (15) Moro, G.; Nordio, P. L. Mol. Cryst. Liq. Cryst. 1984, 104, 361. (16) Vold, R. R.; Vold, R. L. J. Chem. Phys. 1988, 88, 1443. (17) Berggren, E.; Tarroni, R.; Zannoni, C. J. Chem. Phys. 1993, 99, 6180. (18) Dong, R. Y. Nuclear Magnetic Resonance of Liquid Crystals; Springer-Verlag: New York, 1994. (19) Abramovitz, M.; Stegum, I. Handbook of Mathematical Functions; Dover Publication: New York, 1968. (20) Ayant, Y.; Borg, M. Fonctions Spe´ ciales a l’Usage des EÄ tudiants en Physique; Dunod Universite´: Paris, 1971. (21) Edmonds, A. E. Angular Momentum in Quantum Mechanics; Princeton University Press: Princeton, NJ, 1957. (22) Wolfram Research, Inc. http://www.wolfram.com. (23) Be´e, M. Quasielastic Neutron Scattering: Principles and Applications in Solid State Chemistry, Biology, and Materials Science; Adam Hilger: Bristol, U. K., 1988. (24) Abragam, A. Les Principes du Magne´ tisme Nucle´ aire.; Presses Universitaires de France: Paris, 1961. (25) Ferreira, J. B.; Ge´rard, H.; Galland, D.; Volino, F. Liq. Cryst. 1993, 13, 645. (26) Campbell, R. F.; Meirovitch, E.; Freed, J. H. J. Chem. Phys. 1979, 83, 525. (27) Baram, A.; Luz, Z.; Alexander, S. J. Chem. Phys. 1976, 64, 4321. (28) Vega, A. J.; English, A. D. Macromolecules 1980, 13, 1635. (29) Vold, R. L.; Vold, R. R.; Warner, M. J. Chem. Soc., Faraday Trans. 1988, 84, 997. (30) Kittel, C. Elementary Statistical Physics; Wiley: New York, 1958.

Generalized Gaussian Model for Uniaxial ... - ACS Publications

Recommend Documents