J. Phys. Chem. B 2006, 110, 11217-11223
11217
Gaussian Model for Localized Translational Motion: Application to Incoherent Neutron Scattering Ferdinand Volino, Jean-Christophe Perrin,* and Sandrine Lyonnard Structures et Proprie´ te´ s d’Architectures Mole´ culaires, UMR 5819 (CEA-CNRS-UJF), DRFMC/SPrAM, CEA-Grenoble, 38054 Grenoble Cedex 9, France ReceiVed: February 21, 2006; In Final Form: March 28, 2006
A simple model based on Gaussian statistics, aimed at describing localized diffusive translational motion in one, two, and three dimensions is presented and used to calculate the corresponding incoherent neutron scattering laws. In the time domain, these laws are closed form mathematical functions. In the frequency domain, some of these laws can be expressed as an infinite series depending on one single index. Owing to this relative simplicity, such a model can advantageously replace previous models such as diffusion on a segment, inside a circle and inside a sphere with an impermeable surface, to analyze neutron quasielastic scattering data associated with molecular motions in confined media. It may also be more realistic when the confinement is defined by soft, ill-defined boundaries.
Gaussian Model
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 has supercooled liquids and polymers.1-5 Among the various experimental methods that can be used to investigate the molecular dynamical behavior in restricted geometries, spectroscopic techniques such as neutron scattering, light scattering, and NMR, play an important role. Interpretation of the corresponding data requires the use of models for molecular distribution and motions, to calculate theoretical spectra to be compared with the experimental ones. Confinement means that the molecular motions are restricted to finite volumes. As a consequence, usual models, such as random motion in infinite spacesleading to the usual diffusion equationsare no more sufficient to describe the physical situation. In other words, the spatial boundary conditions need to be explicitly taken into account. So far, several models have been proposed to describe diffusion on a segment,6 inside a circle,7 a sphere,8 and a cylinder.9 All of them have been widely used to interpret data.4,5,10-12 However, in these models, the boundary appears to be rather “hard” since it is impermeable, implying a discontinuity which may not correspond to the actual situations in which boundaries are softer. For neutron scattering, the theoretical expressions for the scattering laws in these cases often write as an infinite sum of terms often implying several indices. We propose here a simple model based on Gaussian statistics which may advantageously replace the impermeable boundary condition of the previous models. The neutron intermediate scattering law is a closed form function, and the scattering law in the frequency domain is an infinite sum which implies one single index, leading to (much) simpler mathematical formulas. In addition, complex dynamics can be easily introduced via slow decaying correlation coefficients such as power laws. * To whom correspondence should be addressed. Phone: +33 4 38 78 32 85. Fax: +33 4 38 78 56 91. E-mail:
[email protected].
Probability Density Functions. Contrary to the square well, the Gaussian well is represented by a continuous and infinitely derivable function, with no discontinuity in the first derivative, thus convenient to describe a localized motion in a region of space with a soft boundary. Suppose a particle that can move along the direction x about a fixed point taken as the origin, and let ux be the displacement from this origin. The model assumes that ux is a Gaussian random variable with variance 〈ux2〉. This variance quantifies the size of the region in which the particle is confined. The normalized equilibrium density probability function p(ux) writes:
p(ux) )
[ ]
ux2 1 exp [2π〈ux2〉]1/2 2〈ux2〉
(1)
In practice, ux is a time-dependent variable. Let u1 and u2 be the possible values of ux at two different times t1 and t2. The quantities u1 and u2 are equivalent random variables, and they are clearly not independent, since for t1 ) t2 the two values are equal. Let F1,2 be the correlation coefficient defined as
〈u1u2〉 ) 〈ux2〉F1,2
(2)
The joint probability density function for the two correlated Gaussian variables u1 and u2 is
p(u1,u2,F1,2) )
[
]
u12 + u22 - 2u1u2F1,2 1 exp (3) 2π〈ux2〉(1 - F1,22)1/2 2〈ux2〉(1 - F1,22) One can easily verify (for example, using Mathematica software) that p(u1,u2,F1,2), in addition to being a positive
10.1021/jp061103s CCC: $33.50 © 2006 American Chemical Society Published on Web 05/23/2006
11218 J. Phys. Chem. B, Vol. 110, No. 23, 2006
Volino et al.
quantity, has all the required properties of a probability density function, namely,
(i)
∫-∞∞∫-∞∞p(u1,u2,F1,2) du1 du2 ) 1 (normalization)
(ii)
∫-∞∞p(u1,u2,F1,2) du1 ) p(u2)
(iii)
∫-∞∞p(u1,u2,F1,2) du2 ) p(u1)
(iv) if F1,2 ) 1 (complete correlation), p(u1,u2,1) ) δ(u1 - u2) p(u1) ) δ(u1 - u2) p(u2) (v) if F1,2 ) 0 (no correlation), p(u1,u2,0) ) p(u1) p(u2) (vi)
(vii)
∫-∞∫-∞u1 p(u1,u2,F1,2) du1 du2 ) ∫-∞∞∫-∞∞u22p(u1,u2,F1,2) du1 du2 ) 〈ux2〉 ∞
∞
2
∫-∞∞∫-∞∞u1u2p(u1,u1,F1,2) du1 du2 ) 〈ux2〉F1,2
∫-∞∞dq1 ∫-∞∞dq2C(q1,q2,F)
1 (2π)2
with
[
〈ux 〉 2 (q + q22 - 2q1q2F1,2) C(q1,q2,F1,2) ) exp 2 1
]
p(u1,u2,F1,2) ) p(r,s,F1,2) )
[
]
The distribution of the variable r ) u1 - u2 is obtained by integrating this equation on s:
∫-∞∞p(r,s,F1,2) J(r,s) ds
(7)
where J(r,s) is the Jacobian associated with the above variable change. It is easily shown that
J(r,s) ) 1 Performing the integration yields
(8)
]
[
exp 1 (2π)6
3
]
b u 12 + b u 22 - 2u b1b u 2F1,2 2〈ux2〉(1 - F1,22)
(10)
∫∫∫dqb1 ∫∫∫dqb2C(qb1,qb2,F1,2) exp[iq b 1b u 1] exp[-iq b2 b u 2] (11)
where
[
〈ux2〉 2 q 22 - 2q b 1b q 2F1,2) (q b1 + b 2
]
(12)
Finally,
[
]
(u b1 - b u 2)2 1 exp (13) [4π〈ux2〉(1 - F1,2)]3/2 4〈ux2〉(1 - F1,2)
(5)
(r + s)2 + s2 - 2F1,2(r + s)s 1 exp 2π〈ux2〉(1 - F1,22)1/2 4〈ux2〉(1 - F1,22) (6)
p(r,F1,2) ) p(u1 - u2,F1,2) )
[
1 b2,F1,2) ) p(u b1,u 2 2π〈ux 〉(1 - F1,22)1/2
p(u b1 - b u 2,F1,2) )
C(q1,q2,F1,2) represents the average value of product exp(iq1u1) exp(iq2u2). Application to Incoherent Neutron Scattering. As incoherent neutron scattering experiments probe the probability to find a particle at position u1 at time t if it was at position u2 at time 0, it is sufficient to know the distribution function of the single variable u1 - u2 to calculate the corresponding scattering function. To obtain this distribution, we perform the following change of variables: r ) u1 - u2; s ) u2, so that u1 ) r + s and u2 ) s. In terms of these new variables, eq 3 can be rewritten as
]
Note that this last expression is normalized only if it is considered as a function of the single variable u1 - u2. All of these formulas can easily be generalized to either isotropic or anisotropic motions occurring in two and three dimensions. For example, in the case of 3D isotropic motions, where 〈ux2〉 ) 〈uy2〉 ) 〈uz2〉 ) 〈u2〉/3, eqs 3, 4, 5, and 9 become
b2,F1,2) ) exp C(q b1,q
exp[iq1u1] exp[-iq2u2] (4)
2
[
(u1 - u2)2 1 exp (9) [4π〈ux2〉(1 - F1,2)]1/2 4〈ux2〉(1 - F1,2)
p(u b1,u b2,F1,2) )
Fourier Representation. The double Fourier transform of the distribution may be a useful way of rewriting the joint probability function. We have
p(u1,u2,F1,2) )
p(u1 - u2,F1,2) )
This last expression can be written as a simple Fourier integral:
u 2,F1,2) ) p(u b1 - b
1 (2π)3/2
∫∫∫C(qb,F1,2) u 2)] dq b (14) exp[-iq b(u b1 - b
where
b|2〈ux2〉(1 - F1,2)] C(q b,F1,2) ) exp[-|q
(15)
C(q b,F1,2) is the mean value of exp[iq b(u b1 - b u2)], that is, the selfcorrelation function of exp(iq bb u). Intermediate Incoherent Scattering Function The Gaussian model may be useful to describe in a phenomenological way a variety of physical situations in which the molecular motions are restricted in domains of finite size. Information about molecular dynamics under confinement can be obtained by neutron scattering experiments, the principle of which is to measure the projection of the motion along the wave vector transfer Q B direction. With the Gaussian model formalism, one can easily obtain the expression for the normalized intermediate incoherent neutron scattering function Is(Qx,t) ) 〈exp[iQx(ux(t) - ux(0))]〉 for a particle undergoing the random motion described by ux(t), with Qx being the component of Q B along x. The previous correlated variables u1 and u2 need here to be identified with the values of ux(t) at two successive times. Assuming that the random process is stationary, we can choose t1 ) t and t2 ) 0 (so that u1 ) ux(t) and u2 ) ux(0)). The
Gaussian Model for Localized Translational Motion
J. Phys. Chem. B, Vol. 110, No. 23, 2006 11219
correlation coefficient is F1,2 ) F(t,0) ) F(t) with the conditions that F(0) ) 1 (complete correlation) and F(∞) ) 0 (no correlation). According to eq 15, we have
Is(Qx,t) ) exp[-Qx2〈ux2〉(1 - F(t))]
(16)
1D to 3D Motions. In practice, molecular motions can occur in one, two, or three dimensions, and generally, a powder average must be done. One must thus express Qx in terms of the actual geometry. If the motion is three-dimensional and isotropic, one can choose Q B along any direction, for example, x, so that Qx ) Q. If the motion occurs in a plane and is isotropic (in two dimensions), one can choose x along the projection of Q B in the plane. If θ2 is the angle between the normal to this plane and Q B , we have Qx ) Q sin θ2. If the motion occurs on a straight line (one dimension), x is along this line, and if θ1 is the angle between the line and Q B , we have Qx ) Q cos θ1. Isotropic powder averages of the scattering laws are obtained by averaging over θi (i ) 1,2) as follows:
Is(Q,t)pow )
∫0πIs(Qx,t) sin θi dθi
1 2
(17)
The results are the following: (i) three-dimensional case with 〈ux2〉 ) 〈uy2〉 ) 〈uz2〉:
Is(Q,t)pow ) exp[-Q2〈ux2〉(1 - F(t))]
(18)
(ii) two-dimensional case with 〈ux2〉 ) 〈uy2〉:
Is(Q,t)pow ) erfi[xQ2〈ux2〉(1 - F(t))] xπ exp[-Q2〈ux2〉(1 - F(t))] (19) 2 2 2 Q 〈u 〉(1 F(t)) x x (iii) one-dimensional case: 2 2 xπ erf[xQ 〈ux 〉(1 - F(t))] Is(Q,t)pow ) 2 xQ2〈ux2〉(1 - F(t))
(20)
In these expressions, erf is the error function defined as erf(z) ) 2/xπ∫z0 exp(-t2) dt and erfi is the so-called imaginary error function defined as erfi(z) ) erf(iz)/i, which is a real function if z is real. The numerical values of these functions are available from software such as Mathematica. The 2D and 3D formulas (eqs 18 and 19) can be easily generalized to the case of anisotropic (〈ux2〉 * 〈uy2〉 * 〈uz2〉) motions. Elastic Incoherent Structure Factor. When the motion is localized, the incoherent scattering functions do not tend to 0 at infinite time. The variation of the asymptotic value versus Q is the so-called elastic incoherent structure factor (EISF) whose theoretical expressions in the three cases can be obtained by putting F ) 0 in the above expressions of Is(Qx,t) or Is(Q,t)pow. Note that the EISF is independent of the particular functional form chosen for F(t), as it represents the behavior when the memory of its initial position is lost. Figure 1 represents the variation of the EISF versus Q〈ux2〉1/2 of a powder for the three-, two-, and one-dimensional cases. It is observed that the smaller the dimensionality, the slower is the decrease versus Q, as expected on general grounds. Correlation Coefficient. Details on how the particle loses memory of its initial position and thus information on the local interactions within complex confined media, for instance, are
Figure 1. Elastic incoherent structure factor (EISF) of a powder as a function of Q〈ux2〉1/2 in the case of 1D, 2D, and 3D models (eqs 27a, 28a, and 29a).
contained in the correlation coefficient, the choice of which should witness a relevant physical scenario about molecular motions. Any functional form can be assumed a priori for describing the correlation function F(t) provided that limit conditions are fulfilled. The simplest form is the simple exponential which introduces the characteristic time τ0:
( )
F(t) ) exp -
t τ0
(21a)
This time τ0 is the correlation time of the variable ux. It roughly represents the mean time for the particle to explore a segment of length 2〈ux2〉1/2. Such a simple exponential form is expected to hold in systems such as normal liquids where memory effects are presumably weak or negligible. However, in more complex systems such as supercooled liquids or polymers, where memory effects may be important due to enhanced interactions or confinement, correlation functions with slower decay rates should be introduced. A simple mathematical form which predicts a power law decrease at long times of F(t) is
(
F(t) ) F(t,β) ) 1 +
)
t βτ0
-β
(21b)
where β is an exponent such that 0 < β < ∞. For finite β, τ0 represents the time above which memory effects become important. For β f ∞, there are no more memory effects and the simple exponential form (eq 21a) is recovered. The power law (eq 21b) provides an empirical way to describe the possible slowing down of the relaxation at long times via the decrease of parameter β from infinity to a finite value. Figure 2a shows the variation of Is(Q,t)pow (eqs 18, 19, and 20) versus t/τ0 for 3D, 2D, and 1D motions assuming that F(t) is the simple exponential (eq 21a) and Q2〈ux2〉 ) 2. One observes that the lower the dimensionality of the motion, the higher is the limit at infinite time, all things being equal. This reflects the different variations of the EISF shown in Figure 1. Figure 2b shows Is(Q,t) as a function of t/τ0 in the 3D case assuming that the correlation coefficient is F(t,β) given by eq 21b with β ) 1/2,1,∞ and Q2〈ux2〉 ) 2. These functions are practically simple exponentials for t < τ0, but for t > τ0, they decay more slowly and resemble “stretched exponentials”, with the stretched character increasing as the exponent β decreases. In Figure 2b, it is seen that the Kohlrausch-Williams-Watts (KWW) function exp[-(t/τKWW)R] with τKWW ) 0.5τ0 and R ) 0.4 can represent reasonably well the long-time behavior of Is(Q,t) for
11220 J. Phys. Chem. B, Vol. 110, No. 23, 2006
Volino et al. (eq 21a), then [F(t)]n ) exp(-nt/τ0), and the corresponding normalized Laplace-Fourier transform, defined as the real part of 1/π∫∞0 [F(t)]n exp(-iωt) dt, is the Lorentzian:
Ln(ω) )
τ0 n π n2 + ω2τ
2
(25a)
0
In the case of eq 21b, the result is the real part of
βτ0 exp(iβωτ0) × EI(nβ,iβωτ0) π
(25b)
where EI is the exponential integral function defined by EI(s,z) ) ∫∞1 x-s exp(-zx) dx. The limit of eq 25b for β f ∞ is eq 25a. If the samples are in powder form, isotropic averages of the incoherent structure factors should be made as was previously done. We have
An(Qx)pow )
∫0πAn(Qx) sin θi dθi
1 2
(26)
The results for the three-, two-, and one-dimensional cases are the following: (i) three-dimensional case:
A0(Q)pow ) exp[-Q2〈ux2〉]
Figure 2. Intermediate incoherent scattering functions Is(Q,t) as a function of t/τ0 for Q2〈ux2〉 ) 2: (a) calculated with eq 21a for the 1D, 2D, and 3D models (eqs 18, 19, and 20); (b) calculated with eq 21b for different values of β for the 3D model (eq 20). The crosses represent the KWW function with τKWW ) 0.5τ0 and R ) 0.4. The dotted lines show the limits at t ) ∞.
β ) 1, which may explain why the KWW function is commonly used to analyze real spectroscopic data. Frequency Domain: Incoherent Scattering Law If one is interested in the frequency domain, as is often the case in practice, one should take the time Fourier transform of the above expressions. No closed form expression exists, so that one should either calculate them numerically or use series developments. In the latter case, the general expression (eq 16) can be written as follows:
Is(Qx,t) ) exp[-Qx2〈ux2〉] exp[Qx2〈ux2〉F(t)]
(22)
Expanding the second exponential, one can write
∑An(Qx) F(t)n
(23)
n)1
where the elastic (n ) 0) and quasielastic (n > 0) incoherent structure factors An(Qx) are given by
A0(Qx) ) exp[-Qx2〈ux2〉] (Qx2〈ux2〉)n n!
An(Qx) ) exp[-Qx2〈ux2〉]
(24a) n>0
(24b)
Calculation of the scattering law thus reduces in finding the time Fourier transform of [F(t)]n. If F(t) is the simple exponential
(27b)
(ii) two-dimensional case: 2 2 2 2 xπ exp(-Q 〈ux 〉) × erfi(xQ 〈ux 〉) A0(Q)pow ) 2 xQ2〈ux2〉
(28a)
An(Q)pow ) 2 3 2 2 n 2 xπ (Q 〈ux 〉) × 1F1(1 + n; /2 + n;-Q 〈ux 〉) (28b) 2 Γ(3/2 + n)
(iii) one-dimensional case:
A0(Q)pow ) 2 2 1 1 2 2 xπ erf(xQ 〈ux 〉) Γ( /2) - Γ( /2,Q 〈ux 〉) ≡ (29a) 2 2 2 2 2 Q 〈u 〉 2 Q 〈u 〉Γ(1) x x x x
An(Q)pow )
∞
Is(Qx,t) ) A0(Qx) +
(Q2〈ux2〉)n n!
An(Q)pow ) exp[-Q2〈ux2〉]
(27a)
Γ(1/2 + n) - Γ(1/2 + n,Q2〈ux2〉) 2xQ2〈ux2〉Γ(1 + n)
(29b)
In eq 28b, 1F1(a,b,z) is the Kummer confluent hypergeometric function (see a handbook of mathematical functions or the Mathematica software library for details on this latter function). Figure 3 shows the variation of the various powder structure factors for n ) 0 (EISF) and n ) 1, 2, 3, 4, and 5 for the three(a), two- (b), and one- (c) dimensional cases. For a practical calculation at a given Q value, one must limit the sums to an index such that the values of the incoherent structure factors An are very small. It is observed in the figures that the number of terms to be taken into account is larger for the smallest dimensionality, all things being equal. As a rule of thumb, the number of terms nmax to be considered in practical
Gaussian Model for Localized Translational Motion
J. Phys. Chem. B, Vol. 110, No. 23, 2006 11221 In this limit, with F(t) given by either eq 21a or eq 21b, we have
F(t) ≈ 1 -
t τ0
(30)
and eq 16 becomes
[
]
t Is(Q,t) ≈ exp -Qx2〈ux2〉 τ0
(31)
One recovers in this way the usual law for diffusion in an infinite medium, exp[-DtQx2t], where the translational diffusion coefficient Dt is defined as
〈ux2〉 Dt ) τ0
Figure 3. Incoherent structure factors An(Q)pow calculated for n ) 0 (EISF) and n ) 1, 2, 3, 4, and 5 in the case of the 3D (a), 2D (b), and 1D (c) models (eqs 27, 28, and 29).
calculations must be (much) larger than Q2〈ux2〉. In fact, the value of nmax also depends on the actual energy range of the experimental spectra that one wants to analyze with the corresponding model. The High-Q Limit of the Scattering Law The scattering law for diffusion in an infinite medium is recovered when the size of the explored domain is much larger than the largest available Q values, in practice when Q2〈ux2〉 . 1. In the frequency domain, this means that one must consider a very large number of terms in the series expansions. To overcome this difficulty, it is interesting to study the high-Q limit in the time domain. Consider eq 16. It is seen that, for very large Q2〈ux2〉, Is(Qx,t) will have significant values only when F(t) ≈ 1, that is, when t/τ0 , 1. One can thus develop F(t) in series around t/τ0 ) 0 and limit this expansion to the first term.
(32)
The property that the first term of the development of F(t) given by eq 21b is linear in t and β independent emphasizes the suitability of such a function for the purpose of introducing memory effects in the model. On the contrary, a stretched exponential of the form exp[-(t/τ0)R] with R * 1 is not a possible form for F(t) at short times, since the first term of the small t expansion is not linear, and one cannot recover the infinite medium limit. Note that, according to this model, in an infinite medium, no information regarding the correlation coefficient F(t), and thus the possible collective motions, can be obtained in a scattering experiment, as the scattering law reduces to the DQ2 law. On the contrary, in a restricted geometry, the quantity Q2〈ux2〉 can be small enough so that Is(Q,t) has a measurable value until several τ0, yielding information on F(t) and thus on collective motions, if any. Since it is experimentally verified that the diffusion coefficient varies little between bulk and confined states, and since 〈ux2〉 quantifies the size of the explored domain, the correlation time τ0 must also vary with the size in such a way that the ratio 〈ux2〉/τ0 remains approximately constant. If the sample is infinite, both 〈ux2〉 and τ0 are infinite so that their ratio is constant and equal to the diffusion coefficient. This shows that the meaning of the correlation time τ0 in the Gaussian model is different from that of the usual molecular models where the description is microscopic. In the latter case, the characteristic time is implicitly assumed to be a property of the substance and translational diffusion is pictured as random jumps of the particles between neighboring sites. If the average distance between such sites is δ and the corresponding mean jump time is τmi, the diffusion coefficient Dt is defined as the ratio δ2/2τmi. To recover the “DQ2 law ” in this model, the distance and the time between elementary jumps are needed to be really zero. Despite this very different feature, the two descriptions are mathematically equivalent to describe usual continuous diffusion in an infinite medium and lead to the same diffusion coefficient:
Dt )
〈ux2〉 δ2 ) τ0 2τmi
(33)
The factor 2 appearing in the second denominator comes from the fact that 〈ux2〉1/2 should be considered as a “radius” while δ should rather be identified with a “diameter”. In real systems composed of molecules with finite size, and which are locally rather organized in order to preserve the average local symmetry, the motions rather occur by jumps of small but finite size and jump time is short but also finite. This
11222 J. Phys. Chem. B, Vol. 110, No. 23, 2006
Volino et al.
means that the DQ2 law cannot be valid when Q2δ2 ≈ 1. In the limit Q2δ2 . 1, the model reduces to the jump model between two neighboring sites, and in this limit, the scattering law is Is(Qx,t) ≈ exp[-t/τmi]. A simple but not unique (see ref 11) empirical way to take into account these finite jump distances and times in the above Gaussian model is to replace the diffusion coefficient Dt by an effective, Q-dependent diffusion coefficient Deff(Q):
Deff(Q) )
Dt 1 + Q2δ2
)
Dt 1 + 2DtQ2τmi
(34)
To summarize, the simple (no memory effects) Gaussian model contains five parameters, the “macroscopic” ones 〈ux2〉 and τ0, the “microscopic” ones δ and τmi, and the diffusion coefficient Dt. They are not independent, since they are related by eq 33, such that the model is in fact characterized by only three independent parameters. At least an additional parameter such as β needs to be introduced to take memory effects into account. The Low-Energy Limit: Introduction of Long-Range Translational Diffusion So far, the model describes the motion of a particle in a finite region of space, in that sense that the probability to find the particle at a distance of a few 〈ux2〉1/2 is completely negligible. However, suppose that several other nearly equivalent regions exist in the macroscopic medium. If the probability to find the particle at a distance equal to half the inter-region distance L is not completely negligible, the particle may continue and diffuse in the neighboring region, causing long-range diffusion. The simplest way to describe this phenomenon is to introduce a fourth parameter, namely, a long-range diffusion coefficient, Dlr. Assuming that this motion is independent of the local motion, the complete intermediate incoherent scattering law is the product of the scattering laws for the two motions. For the three-dimensional case, we have
Is,tot(Qx,t) ) exp[-Qx2〈ux2〉(1 - F(t))] exp(-DlrQx2t) (35) In the framework of this model, one may have an estimate of the distance L from the ratio of the long-range to local diffusion coefficients, by saying that this ratio is equal to the ratio of the equilibrium probability densities at ux ) L/2 and at the center (ux ) 0). According to eq 1, we have
(
Dlr L2 ) exp Dt 8〈ux2〉
)
(36)
Conclusion The present Gaussian model for localized translational motion has allowed closed form expressions to be obtained for the intermediate incoherent scattering function in a variety of situations, in one, two, or three dimensions for virtually any functional form for the temporal correlation function of the displacement variable. The use of this model for analysis of incoherent neutron scattering data is thus relatively simple in the time domain, even in the most complicated cases. In the frequency domain, things are more involved, since one must use series developments which need necessarily to be truncated, leading to systematic errors, especially at high Q. Presumably, this model, which depends only on a few parameters, can replace advantageously the models cited in the Introduction, because it
is simpler, as it involves single-index series, it is possibly more physical due to the soft boundary, and also it allows one to easily introduce complex dynamics, for example, via a functional form for the correlation function F(t) such as eq 21b. The model can be easily generalized to anisotropic motions, including or not long-range diffusion occurring when the particle diffuses from one domain of restricted motion to another. Acknowledgment. We thank O. Diat, G. Gebel, and A. Guillermo for very helpful discussions. Appendix. Comparison with the Diffusion inside an Impermeable Sphere Model The Gaussian model scattering law calculated with a simple exponential correlation function (eq 21a) is a single-index series:
Ss(Q,ω) ) ∞
A0(Q) δ(ω) +
(nDt/〈ux2〉)
1
∑An(Q) × π
n)1
(37)
(nDt/〈ux2〉)2 + ω2
where the incoherent structure factors An(Q) are given by eqs 27a and b. It can be compared to the corresponding law for diffusion inside an impermeable sphere of radius a, a model8 which is a double-index series:
Ss(Q,ω) ) A0 (Q) δ(ω) + 0
1
∑
π [l,n]*[0,0]
(2l + 1)
Aln(Q)
(xln)2Dt/a2
[(xln)2Dt/a2]2 + ω2 (38)
where the incoherent structure factors Aln(Q) are given by
A00(Q) ) Aln(Q) )
[
[
]
3j1(Qa) Qa
2
(39a)
]
Qajl+1(Qa) - ljl(Qa)
6(xln)2 (xln)2 - l(l + 1)
(Qa)2 - (xln)2
2
(39b)
In these expressions, jk are the spherical Bessel functions of first kind. The xln are tabulated constants depending on the boundary conditions.8 The two models can be quantitatively compared using the EISF and the width of the quasielastic component of Ss(Q,ω), with this width being defined as the half width at half-maximum (hwhm) of the Lorentzian that best fits its shape. For the EISF, A00(Q) ≈ 1 - [(Qa)2/5] when Qa f 0 and A0(Q) ≈ 1 - Q2〈ux2〉 when Q〈ux2〉1/2 f 0, so that the EISFs of both models are almost superimposed at least in the low-Q limit if
a2 ) 5〈ux2〉
(40a)
For the hwhm, Γ f 4.332 96Dt/a2 in the case of the diffusion inside a sphere model and Γ f Dt/〈ux2〉 for the Gaussian model. Thus, the hwhm’s of both models are almost superimposed at least in the low-Q limit if
a2 ) 4.332 96〈ux2〉
(40b)
In Figure 4, we compare the two models assuming eq 40a. This figure represents (a) the EISF and (b) the hwhm expressed
Gaussian Model for Localized Translational Motion
J. Phys. Chem. B, Vol. 110, No. 23, 2006 11223
Figure 5. hwhm, expressed in Dt/〈ux2〉 units, of the Lorentzian curve that best fits the quasielastic component of the Gaussian model calculated with F(t) given by eq 21a, in the absence (δ ) 0, continuous line) and presence (δ ) 〈ux2〉1/2801/2, dashed line) of granularity as a function of Q2〈ux2〉. The gray line is the DtQ2 law. The low-Q and high-Q asymptotic limits are visualized with dotted lines.
The hwhm expressed in Dt/〈ux2〉 units is plotted as a function of Q2〈ux2〉. It is observed that in the low-Q limit Γ f Dt/〈ux2〉 ) 1/τ0 and in the high-Q limit Γ f (1/τ0) × (〈ux2〉/δ2) ) 1/2τmi. A very similar result would be obtained with the diffusion in a sphere model. References and Notes
Figure 4. (a) EISF and (b) hwhm expressed in Dt/a2 units, for the diffusion inside a sphere model (dashed line) and the Gaussian model (continuous line) with 〈ux2〉 ) a2/5. The DtQ2 law (gray line) is shown in part b. The correlation coefficient used in the Gaussian model is the simple exponential (eq 21a). The microscopic jump distance and time are assumed to be zero (no granularity). hwhm stands for the width of the Lorentzian line that best fits the quasielastic component in both models.
in Dt/a2 units as a function of Q2a2 for the two models, as well as the DQ2 law in Figure 4b. It is seen that the behaviors for the two models are very similar, except around (Qa)2 ≈ 10 where the transition is softer in the Gaussian model. Both models and the long-range diffusion law are superimposed in the limit (Qa)2 f ∞. For completeness, Figure 5 compares the results for the Gaussian model in the absence or presence of “granularity” characterized by a finite value of elementary jump distance δ.
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