How to compare diffusion processes assessed by single-particle tracking and pulsed field gradient nuclear magnetic resonance Michael Bauer, Rustem Valiullin, Günter Radons, and Jörg Kärger Citation: J. Chem. Phys. 135, 144118 (2011); doi: 10.1063/1.3647875 View online: http://dx.doi.org/10.1063/1.3647875 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i14 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 15 Apr 2013 to 128.197.27.9. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
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How to compare diffusion processes assessed by single-particle trackingand pulsed field gradient nuclear magnetic resonanceMichael Bauer, Rustem Valiullin, Günter Radons, and Jörg Kärger Citation: J. Chem. Phys. 135, 144118 (2011); doi: 10.1063/1.3647875 View online: http://dx.doi.org/10.1063/1.3647875 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i14 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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THE JOURNAL OF CHEMICAL PHYSICS 135, 144118 (2011)
How to compare diffusion processes assessed by single-particle trackingand pulsed field gradient nuclear magnetic resonance
Michael Bauer,1 Rustem Valiullin,2,a) Günter Radons,1,b) and Jörg Kärger2
1Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany2Institute of Experimental Physics I, University of Leipzig, 04103 Leipzig, Germany
(Received 15 July 2011; accepted 19 September 2011; published online 13 October 2011)
Diffusion is one of the omnipresent phenomena in na-ture and involved in most physico-chemical and biologi-cal processes.1 Often media, where the molecules performtheir chaotic Brownian motion, do include different typesof compartments, regions of different densities, or domainssurrounded by semipermeable membranes. Diffusion proper-ties in these spatially separated regions may, in general, bedifferent. Altogether, this typically gives rise to very com-plex processes of diffusive mass transport including regimesof anomalous diffusion. To model such inhomogeneous sys-tems, they may be represented to consist of a number ofdomains with different local diffusivities subjected to ex-change processes between them. The most simple two-phaseexchange model with an exponential exchange kernel has of-ten been used to describe experimental results obtained usingpulsed field gradient nuclear magnetic resonance (PFG NMR)technique.2 Such examples include, e.g., diffusive exchangebetween two pools of guest molecules in zeolite crystals andsurrounding gas atmosphere3 and between extra- and intra-cellular water4 in biosystems.
Recently, a new type of experimental approach, namelysingle-particle tracking (SPT) has emerged.5 It provides analternative method for studying diffusion processes and formeasuring their properties as well as some properties of thesurrounding medium.6 In contrast to PFG NMR, where anensemble of diffusing particles is investigated, SPT only ob-serves individual tracer particles. In particular, fluorescent dye
molecules, such as rhodamine B, in a solvent, e.g., tetrakis(2-ethylhexoxy)-silane (TEHOS), which arranges in ultra-thinliquid layers,7 are excited by laser radiation. The emittedlight of the dyes is captured with a wide-field microscopeand recorded by a CCD camera. Hence, the obtained moviesshow diffusing spots representing a two-dimensional projec-tion of the three-dimensional motion of the dyes. From astatistical point of view, such processes are known as ob-served diffusion8–10 or hidden Markov models11, 12 leading,in general, to the loss of the Markov property. A trackingalgorithm detects the positions of the spots and connectsthem to trajectories.13 A basic quantity for the characteri-zation of diffusion processes is obtained by taking two po-sitions x(t) and x(t + τ ) from a trajectory separated by atime lag τ and by considering the rescaled squared displace-ment [x(t + τ ) − x(t)]2/τ . This quantity is a local or micro-scopic diffusivity which fluctuates along a given trajectoryor in an ensemble of diffusing particles. It is natural to ex-tract the corresponding distribution of diffusivities from ex-periments by forming histograms of the observed rescaledsquared displacements.14 Note that other definitions of dif-fusivity distributions may be found in the literature.15 For ho-mogeneous diffusion processes the distribution of diffusivitiesis independent of the time lag τ , whereas for heterogeneoussystems a non-trivial τ -dependence is observed. Therefore inanalyzing heterogeneous systems, the distribution of diffusiv-ities provides advantages over an analysis via mean squareddisplacements because in addition to its mean value it con-tains all information about the fluctuations.16 Furthermore,quantities such as the mean diffusion coefficient, obtained asthe first moment of the distribution of diffusivities, are well
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144118-2 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
defined, and thus time-dependent diffusion coefficients andtheir fluctuations can be calculated.
The objective of this work is to investigate the connec-tion between the two different techniques of measuring dif-fusion. SPT and PFG NMR are clearly related to each other,since both measure displacements of diffusing particles. Forinstance, the time lag between the observation of two posi-tions in SPT corresponds to the time interval between twogradient pulses in PFG NMR. In both SPT and PFG NMRthis time lag τ is a parameter, which can be tuned by vary-ing the time between snapshots and by altering the temporaldistance between gradient pulses, respectively. Furthermore,the signal attenuation in PFG NMR is related to the propa-gator in Fourier space, from which the distribution of diffu-sivities can be calculated. At first, it seems to be sufficientto compare the propagators obtained from both types of ex-periments directly. However, if diffusion processes with het-erogeneities or anomalous behavior are considered, access tothe propagator will be complicated or even hindered. In suchcases, the distribution of diffusivities offers a well-definedanalysis of the processes and a comparison of data from thetwo approaches is feasible. Moreover, it becomes possible tocontrast results from time-averaged and ensemble-averagedquantities and detect anomalous diffusion leading to ergod-icity breaking as reported recently.17 More generally, an im-provement in the analysis of heterogeneous diffusion couldbenefit from the link between single-particle analysis and en-semble methods. Hence, analytical expressions for one- upto three-dimensional processes are derived which transformPFG NMR signal attenuation into the distribution of single-particle diffusivities from SPT.
For simple systems with heterogeneous diffusion thetwo-region exchange model of PFG NMR offers an analyt-ical expression for the spin-echo diffusion attenuation.18 Inconjunction with our transformation, this model provides anexample, where the distribution of single-particle diffusivi-ties can be calculated exactly and also the non-trivial time-lagdependence can be investigated. In this context we considera two-layer liquid film on a homogeneous surface character-ized by two distinct diffusion coefficients.16 This two-layersystem corresponds exactly to the two-region exchange modelof PFG NMR. In particular, its condition of exponential wait-ing times is fulfilled since a change in the diffusion coeffi-cient is possible at any time and independent of the particles’current positions. For a system comprising an arbitrary num-ber of layers, exact asymptotic results for the dispersion ofparticles in the long-time limit have already been provided.19
We substantiate our findings by analyzing data from simu-lated single-particle trajectories of heterogeneous diffusion.To evaluate experimental limitations, we study the influenceof a signal attenuation bounded to a certain range of k-values.The impact on the distribution of single-particle diffusivitieswill also be pointed out.
The remainder of the paper is organized as follows. InSec. II, we recall the basic principles of PFG NMR and un-derline the differences to SPT experiments. In particular, wediscuss properties of both approaches and the connection be-tween them. In this context, we introduce the distribution ofsingle-particle diffusivities and provide expressions for the
well-known case of homogeneous diffusion. To apply our newconcepts to some more elaborated systems, we consider inSec. III heterogeneous diffusion in two-region systems, whereanalytical expressions of the PFG NMR signal attenuation ex-ist. We outline the principles of the simulation of such sys-tems in Sec. IV. In order to provide a simple relation betweensignal attenuation and distribution of diffusivities, we suggestan approximation in Sec. V to avoid the inconvenient Fouriertransformation. This approximation is then compared to theexact expressions of the relation in Sec. VI for simulated dataof the two-region system. Finally, in Sec. VII we address theissue of finite intensity of the magnetic field gradient pulses inthe PFG NMR experiment and illustrate its influence on ourexact transformation into the distribution of diffusivities.
II. SIGNAL ATTENUATION AND DISTRIBUTIONOF DIFFUSIVITIES
Diffusion measurement by PFG NMR is based on ob-serving the transverse magnetization of nuclear spins in a con-stant magnetic field. Offering the highest sensitivity and oc-curring in numerous chemical compounds, in most cases thenuclei under study are protons. By superimposing, over twoshort time intervals, an additional magnetic field with a largegradient, the displacement of the nuclei (and hence of themolecules in which they are contained) in the time span be-tween these two “gradient pulses” is recorded in a phase shiftof their orientation in the plane perpendicular to the magneticfield with respect to the mean orientation. Hence, the distri-bution of the diffusion path lengths appears in the distribu-tion of these phase shifts and, consequently, in the vector sumof the magnetic moments of the individual spins, i.e., in themagnetization.2, 20–22 Since it is this magnetization which isrecorded as the NMR signal, molecular diffusion leads to anattenuation of the signal intensity during the PFG NMR ex-periments which is larger then the displacements in the timeinterval between these two gradient pulses are.
The signal attenuation from PFG NMR may be shown toobey the relation2, 18, 20, 23
�(τ, k) =∫
dr p(r, τ ) exp(ikr) (1)
with the ensemble-averaged conditional probability density
p(r, τ ) =∫
dx p(x + r, τ |x) p0(x), (2)
where p(x + r, τ |x) is the stationary probability density of adisplacement r = (r1, . . . , rd )T in d dimensions in the time in-terval τ and p0(x) refers to the equilibrium distribution givenby the Boltzmann distribution. Further, τ is the time inter-val between the two gradient pulses and represents the diffu-sion time in the PFG NMR experiment. According to the PFGNMR experiment, signal attenuation is measured in the direc-tion of the applied field gradients. Thus, k in Eq. (1) is givenby k = k e, where e denotes the unit vector in that direction.The quantity k is a measure of the intensity of the field gra-dient pulses. Assuming an isotropic system, without loss ofgenerality, an arbitrary direction k = (k, 0, . . . , 0)T may beconsidered. Obviously, the scalar product in the exponential
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144118-3 Comparing diffusion data from SPT and PFG NMR J. Chem. Phys. 135, 144118 (2011)
of Eq. (1) picks out only the component r1 of the displace-ment r. Then, the signal attenuation
�1(τ, k) = �(τ, k = (k, 0, . . . , 0)T)
=∫ +∞
−∞dr1 p1(r1, τ ) exp(ikr1), (3)
depends only on scalar values corresponding to the chosen di-rection and p1(r1, τ ) is the projection of the probability den-sity, Eq. (2), on the considered direction, given by
p1(r1, τ ) =∫
· · ·∫
dr2 · · · drd p(r, τ ). (4)
In NMR, p1(r1, τ ) in Eq. (3) is known as the mean prop-agator, i.e., the probability density that, during τ , an arbi-trarily selected molecule has been shifted over a distance r1
in the direction of the applied field gradients. However, itshould be noted that for heterogeneous systems, such as sys-tems with regions of different mobility, p1(r1, τ ) may notbe called propagator since it cannot evolve the system dueto the loss of Markovianity. The reason is that, in general,p1(r1, τ ) of such systems does not satisfy the Chapman–Kolmogorov equation.24 Non-Markovian behavior, besidesothers, may also arise in systems which can be modeled bythe fractional Brownian motion25 or by certain fractional dif-fusion equations.26 Further, the mean propagator in Fourierspace as given by Eq. (1) corresponds to the incoherent inter-mediate scattering function. The details of this connection aregiven in the Appendix A for clarification.
In contrast to the PFG NMR technique, which isensemble-based as described above, SPT experiments allowto follow the trace of individual diffusing molecules. There-fore, by considering the displacement of a particular moleculein d dimensions it is natural to define a microscopic single-particle diffusivity Dt(τ ) by the relation
Dt (τ ) = [x(t + τ ) − x(t)]2/(2d τ ), (5)
where x(t) denotes the trajectory of an arbitrary stochasticprocess. Note that the term “microscopic” has been used be-fore by Kusumi and co-workers27 to characterize the short-time behavior of averaged squared displacements equiva-lent to the small τ limit of our mean diffusivity defined inEq. (10) below. In the context of Markovian diffusion pro-cesses this limit also corresponds via jump moments tothe diffusion terms appearing in Fokker-Planck equations.24
Here, we use the term microscopic in analogy to the statisticalphysics concept of microstates to distinguish it from ensem-ble based averages. For a given time lag τ , the microscopicsingle-particle diffusivity is a fluctuating quantity along a tra-jectory x(t) and we now ask for the probability p(D)dD that,under the so far considered conditions of normal diffusion,Dt(τ ) attains a value in the interval D . . . D + dD. Therefore,the distribution of single-particle diffusivities is defined as
p(D, τ ) = 〈δ (D − Dt (τ ))〉 , (6)
where 〈. . . 〉 denotes an average, which can be evaluated ei-ther as a time-average 〈. . .〉 = limT →∞ 1/T
∫ T
0 . . . dt , whichis accessible by SPT, or as an ensemble average, appropri-ate for NMR measurements. Note, however, that in SPT, T is
usually limited by the finiteness of the trajectory and compli-cations arise due to the blinking and bleaching of the fluores-cent dyes.28 However, advanced tracking algorithms in SPTreduce these effects13, 29 and arbitrary time lags τ betweensnapshots, which are only limited below by the inverse framerate of the video microscope, can be accomplished. For exper-imental SPT data, the distribution of diffusivities is obtainedby binning diffusivities into a normalized histogram accord-ing to Eq. (6).
For ergodic systems, as considered here, time averageand ensemble average coincide. By additionally assumingtime invariance, Eq. (6) can be rewritten as
p(D, τ ) =∫
dr δ
(D − r2
2d τ
)p(r, τ ) (7)
with the probability density, Eq. (2), given by p(r, τ )= 〈δ(r − r(τ ))〉. By performing the angular integration,Eq. (6) or Eq. (7) can also be expressed as
p(D, τ ) =∫ ∞
0dr δ
(D − r2
2d τ
)pr(r, τ ), (8)
in terms of the radial propagator pr(r, τ ), which is theprobability density p(r, τ ) integrated over the surface of ad-dimensional sphere with radius r.
The delta functions in Eqs. (7) and (8) simply describea transformation of the coordinates from displacementsto diffusivities. Hence, the distribution of diffusivities is arescaled version of the propagator. This becomes obviousby expanding for r > 0, the delta function in Eq. (8) asδ[D − r2/(2d τ )] = √
d τ/(2D) δ[r − √2d τ D], which
yields the relation
p(D, τ ) =√
d τ
2Dpr(
√2d τD, τ ). (9)
Furthermore, it should be noted that the distribution of single-particle diffusivities is closely related to the self part of thevan Hove function given in the Appendix A, which coincideswith p(r, τ ) given by Eq. (2) for identical particles. Hence,the distribution of diffusivities is also a rescaled version of thevan Hove self-correlation function and offers some beneficialproperties for our investigations.
The diffusivity 〈D〉 results as the mean of the microscopicsingle-particle diffusivities, Eq. (5). Therefore, for clarity, wedenote it in the following as mean diffusivity. According tothe definition of the distribution of diffusivities the mean dif-fusivity has to obey the relation
〈D(τ )〉 =∫ ∞
0dD D p(D, τ ). (10)
It is thus obtained as the first moment of the probability den-sity of diffusivities by a well-defined integration, avoiding anynumerical fit. Obviously it may also depend on the time lag τ .
In the special case of free diffusion of a particle, x(t +τ ) − x(t) = ∫ t+τ
tdt ′ ξ (t ′) is a fluctuating quantity taken from
one realization of the Gaussian white noise ξ (t) with vari-ance proportional to the diffusion coefficient. With Eq. (3),the mean propagator and the signal attenuation are seen to be
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144118-4 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
interrelated by the Fourier transformation.18, 23 In the case ofnormal diffusion in one dimension one has
p1(r1, τ ) = (4π〈D〉τ )−1/2 exp
(− r2
1
4〈D〉τ)
, (11)
where 〈D〉 stands for the diffusivity. To avoid confusion wedeviated from the usual way of denoting the diffusivity sim-ply by D. This is because we use this notation to refer tomicroscopic single-particle diffusivities Dt(τ ). By insertingEq. (11) into Eq. (3) the signal attenuation in PFG NMR ex-periments is seen to obey the well-known exponential relation
�1(τ, k) = exp(−k2〈D〉τ ). (12)
Let us now consider a molecular random walk in a two-dimensional plane. Equation (11) describes the probability ofa molecular displacement in any arbitrarily chosen direction.For the probability that radial molecular displacements arewithin the interval r . . . r + dr one obtains, therefore,
pr(r, τ ) dr = 1
4π〈D〉τ exp
(− r2
4〈D〉τ)
2πr dr . (13)
The mean squared displacement,
1
4τ〈r2(τ )〉 = 1
4τ
∫ ∞
0dr pr(r, τ )r2 = 〈D〉, (14)
obeys the well-known Einstein relation for normal diffusionin two dimensions. Inserting the corresponding propagatorof homogeneous diffusion in two dimensions Eq. (13) intoEq. (7) yields the distribution of single-particle diffusivities
p(D) = 〈D〉−1 exp(−D/〈D〉). (15)
In general, for homogeneous diffusion in d dimensions, thedistribution of diffusivities is found to be
pd (D) = Nd
1
D
(D
〈D〉)d/2
exp
(−d
2
D
〈D〉)
, (16)
where Nd can be obtained from the normalization conditionand is explicitly given by
Nd =
⎧⎪⎨⎪⎩
1/√
2π for d = 1
1 for d = 2
3√
3/√
2π for d = 3
. (17)
Since the system is governed by only one diffusion constant,the dependence on τ vanishes in Eq. (16). However for het-erogeneous diffusion, the distribution of single-particle diffu-sivities additionally depends on the time lag τ . Then, p(D, τ )cannot generally be expressed by a simple exponential func-tion as in Eq. (16).
By inserting Eq. (15), the first moment of the distributionof diffusivities, Eq. (10),∫ ∞
0dD D/〈D〉 exp(−D/〈D〉) = 〈D〉, (18)
is easily seen to be fulfilled for homogeneous systemsand equals the mean squared displacement obtained in Eq.(14). Hence, with p(D, τ ), which is a rescaled van Hoveself-correlation function, it becomes possible to determine
p(D
)
D
10-5
10-4
10-3
10-2
10-1
100
101
0 2 4 6 8 10
p(D
)
D
10-5
10-4
10-3
6 6.5 7 7.5 8 8.5 9
FIG. 1. Distribution of diffusivities from a simulated trajectory of a homoge-neous diffusion process in two dimensions. The distribution agrees well withthe exponential behavior expected from Eq. (15) and is independent of τ . Theinset depicts deviations between simulation and Eq. (15) for large D due toinsufficient statistics from finite simulation.
the mean diffusion coefficient of the system by ordinaryintegration.
With Eq. (15) for diffusion in two dimensions, the distri-bution of the single-particle diffusivities in homogeneous sys-tems is seen to result in an exponential. The semi-logarithmicplot of the number of trajectory segments governed by a par-ticular single-particle diffusivity versus these diffusivities iscorrespondingly expected to yield a straight line. Its negativeslope is defined as the reciprocal value of the mean diffusivity.Figure 1 depicts the distribution of diffusivities of a homoge-neous diffusion process in two dimensions. The data are ob-tained from simulations of a system with diffusion coefficient〈D〉 = 0.7 and gathered in a normalized histogram. For com-parison, the solid line represents the analytical expression,Eq. (15), and shows a good agreement with the histogram.The inset of Fig. 1 shows deviations between simulated dataand Eq. (15) for large D due to insufficient statistics originat-ing from the finite sample in simulation.
It is interesting to note that the shape of the distribution ofdiffusivities of homogeneous diffusion is similar to that of theattenuation function of PFG NMR diffusion measurements(Eq. (12)). One has to note, however, that now, in contrast toEq. (12), the mean diffusivity 〈D〉 appears in the denomina-tor of the exponent. From a semi-logarithmic plot of the PFGNMR signal attenuation versus k2, the mean diffusivity thusdirectly results as the slope rather than its reciprocal value.
In the simple cases of isotropic and homogeneous diffu-sion both the signal attenuation from PFG NMR and the dis-tribution of diffusivities from SPT resulted in well-known andeasily obtainable expressions. In the following we investigatea more elaborated two-region system exhibiting inhomoge-neous diffusion.
III. HETEROGENEOUS DIFFUSION INTWO-REGION SYSTEMS
Let us now consider molecular diffusion in an isotropictwo-region system. With the respective probabilities π i, the
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144118-5 Comparing diffusion data from SPT and PFG NMR J. Chem. Phys. 135, 144118 (2011)
molecules are assumed to propagate with either the diffusiv-ity D1 or D2 and to remain with the mean dwell times τm (m= 1, 2) in each of these states of mobility. Thus, the observeddiffusion process exhibits dynamic heterogeneities emergingas a time-dependent diffusion coefficient due to the exchangeof particles between two regions with different diffusion co-efficients. For such heterogeneous systems, the behavior ofthe distribution of single-particle diffusivities, in general, de-viates from the mono-exponential decay. This is attributed toa superposition of many different exponentials of the type ofEq. (15) originating from trajectory segments which includelayer transitions during the time lag τ . Thus, we denote thedistribution of single-particle diffusivities by p(D, τ ) empha-sizing its dependence on τ . Further, the superposition and ac-cordingly the characteristics of the distribution of diffusivi-ties strongly depend on the relation of dwell times and thetime lag τ between observed positions.14 For short time lagscompared to the dwell times, the exchange rates are very low.Then, the two diffusion processes can be separated into thetwo underlying processes. As a result, the probability densityis the weighted superposition of the mono-exponential decaysbelonging to homogeneous diffusion inside each region. Inthe opposite case, for time lags much larger than both dwelltimes, the observation only reveals a long-term diffusion pro-cess with the mean diffusion coefficient of the system. Hence,the probability density is given by a mono-exponential decayparameterized by this mean diffusivity.
In the case of a two-region system, the PFG NMR spin-echo diffusion attenuation (and hence the Fourier transformof the mean propagator) has been shown to result as a super-position of two terms of the shape of Eq. (12) (Refs. 2 and 18)
�1(τ, k) = p′1(k) exp(−k2D′
1(k)τ )
+p′2(k) exp(−k2D′
2(k)τ ) (19)
with
D′1,2(k) = 1
2
(D1 + D2 + 1
k2
(1
τ1+ 1
τ2
)
∓{[
D2 − D1 + 1
k2
(1
τ2− 1
τ1
)]2
+ 4
k4τ1τ2
}1/2⎞⎠ ,
(20)
p′1(k) = 1 − p′
2(k),
p′2(k) = 1
D′2(k) − D′
1(k)(π1D1 + π2D2 − D′
1(k)). (21)
It should be noted that the primed quantities in Eqs. (20)and (21) depend on the intensity of the magnetic field gra-dient being related to k and, thus, on the Fourier coordinate.Therefore, Eq. (19) cannot be considered as a superposition ofseparated populations of the two regions. It is rather the totalinterference of spin-echo attenuations observed from both re-gions. Further, the initial condition of a process described byEqs. (19)–(21) has to be chosen in such a manner that for the
initial time t = 0, the particles are located at a given positionx and are already distributed stationarily between the regions.This is obvious since neither p′
1(k) nor p′2(k) depends on t
which would be necessary to converge to the stationary distri-bution. For any other initial distribution, Eq. (19) will only bevalid in the limit of t → ∞.
The signal attenuation can also be considered for the lim-iting cases. For τ → 0, i.e., τ τ 1, τ 2, the signal attenuation,
decomposes into the superposition of two signal attenuationscorresponding to each region. As discussed, two completelyseparated diffusion processes are observed. Hence, the inverseFourier transformation leads to a superposition of the distri-bution of diffusivities of each region. In contrast, for τ → ∞,i.e., τ τ 1, τ 2, the mixing of the two regions leads to the ob-servation of an effective mean diffusion process with a signalattenuation,
�1(τ, k) = exp(−k2(π1D1 + π2D2)τ ), (23)
containing the mean diffusion coefficient. Analogously, its in-verse Fourier transform, i.e., the distribution of diffusivities,is only characterized by the mean diffusion coefficient 〈D〉 =π1D1 + π2D2. A detailed deviation of the limiting cases isgiven in the Appendix C.
IV. SIMULATION OF TWO-REGION SYSTEMS
In order to simulate heterogeneous diffusion, we con-sider a system with two regions where particles propagatewith different diffusivities and can change their state of mobil-ity. Following the experiment with rhodamine in TEHOS,16, 30
this two-region system is modeled by a bi-layer system withlayer-dependent diffusion coefficients D1 and D2, respec-tively. Such processes can formally be described as compos-ite Markov processes31 or equivalently as multistate randomwalks,32, 33 which are known to be widely applicable. A recentbiophysical application consists of changes in the diffusivebehavior of molecules in membranes due to random changesof the molecules’ conformation.34 In the case of two states orregions the probability density of finding the particle at posi-tion x at time t is determined by the evolution equations
for each region with corresponding diffusion coefficients D1
and D2. Within each region the motion of the molecules is ac-complished by ordinary two-dimensional diffusion, i.e., ran-dom walkers experiencing shifts of the positions distributedaccording to a Gaussian with a variance defined by the dif-fusion coefficient in the region. The exchange between thesetwo diffusive regions is simulated by a jump process governedby a master equation with jump rates wnm, which describe atransition from region m to n (m, n = 1, 2). The inverse of the
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144118-6 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
jump rates wnm yields the mean dwell time τm,
τ1 = 1
w21and τ2 = 1
w12, (25)
for which particles remain in region m. Further, the stationarydistribution between the regions,
π1 = w12
w12 + w21and π2 = w21
w12 + w21, (26)
is also dictated by the jump rates. With the stationary distribu-tion the mean diffusion coefficient of the two-region systemis given by
〈D〉 = π1D1 + π2D2, (27)
which is the weighted average of the diffusion coefficientsbelonging to each region.19
To investigate the effects of heterogeneous diffusion,simulation of the two-region system is performed with the fol-lowing system parameters. The diffusion coefficients withineach of the two regions are given by D1 = 0.1 and D2 = 1.0.The jump rates w21 = 8 and w12 = 4 yield the dwell timesτ 1 = 0.125 and τ 2 = 0.25, respectively. Hence, the station-ary distribution between the regions results in π1 = 1/3 andπ2 = 2/3 and a mean diffusion coefficient 〈D〉 = 0.7 is ob-tained. The length of the time step in the simulation is chosento be �t = 0.01, which is much smaller than the dwell timesto ensure diffusive motion of the particles within the regions.
Simulation of Eq. (24) is depicted in Fig. 2(a). It showsthe trajectory of a particle in a bi-layer system, where the par-ticle jumps between the layers. In each layer, diffusion is gov-erned by a different diffusion coefficient denoted by the colorof the trajectory segments. Since in experiments with videomicroscopy only a two-dimensional projection of the processis observed, the trajectory is projected onto the x-y-plane inFig. 2(b). As a consequence, information about the layer isobscured and can only be identified due to the color coding inthe figure. Hence, in the projection it is unknown which diffu-sion coefficient currently governs the process. A descriptionof such observed diffusion processes by the Fokker-Planckequation with time-dependent diffusion coefficient would be-come possible if all trajectories jump synchronously. Sincein our bi-layer system the particles move independently, theprocess is more complicated. As a result of the projection, theobserved process does not possess the Markov property any-more since, in general, the Chapman–Kolmogorov equationcannot be satisfied. The simulation provides an approach tostudy properties of an N-layer system, which is closely relatedto a system where the diffusion coefficient varies continuouslywith the z-coordinate.
To avoid transient behavior in our simulation, the parti-cle positions are initialized with their corresponding station-ary distributions between the layers given by π i. It should benoted, however, that experimental results will be influencedby such transient effects if the tracer molecules require a suf-ficiently long time to distribute between the layers of the sol-vent. On the other hand, such slow relaxation is related to lowexchange rates leading to almost complete separation of thetwo diffusive regions.14 This would allow for an appropriatebi-exponential fit of our distribution of diffusivities although
-1
0
1
2
3
4
5
6
7
-2 -1 0 1 2 3 4 5 6 7
y
x
1
2
(a)
(b)
x
y
laye
r
FIG. 2. Single-particle trajectory from simulation of diffusion in a bi-layersystem. (a) The particle performs diffusion with corresponding diffusion co-efficients and jumps between the layers. (b) Projection of the trajectory shownin (a) onto the x-y-plane as usually observed by single-particle tracking. In-formation of the layer and the corresponding diffusion coefficient is lost inthe projection and can only be identified due to the color code.
the weights do not correspond to the stationary distributionsyet.
To investigate the connection between spin-echo signaldiffusion attenuation, as measured by PFG NMR, and distri-bution of single-particle diffusivities, as assessed by SPT, wesimulated one particle. Next, we recorded squared displace-ments along the simulated trajectory of 107 time steps. Thesquared displacements are calculated from the changes of theparticle positions and are divided by the time lag τ elapsedbetween the observations of the two positions. Hence, we ob-tain scaled squared displacement with the dimension of a dif-fusion coefficient. The thus obtained diffusivity is a fluctu-ating quantity along a trajectory. Finally, we gather them in ahistogram counting their occurrences. The histogram containsdata from a moving-time average since the diffusivities orig-inate from single trajectories. Note that for ergodic systemsensemble averaging will yield identical results. After normal-izing the histogram, we obtain a probability density referredto as the distribution of diffusivities. The distribution of dif-fusivities contains all information about the diffusivities ofthe process and their fluctuations. Following the experiment,only a fraction of the time steps is available for the distribu-tion depending on the selected time lag. Thus, our resulting
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144118-7 Comparing diffusion data from SPT and PFG NMR J. Chem. Phys. 135, 144118 (2011)
distributions of diffusivities depicted in log-linear plots havetheir lower boundary at 10−3 since data below suffer from in-sufficient statistics.
V. APPROXIMATION OF DIFFUSIVITY DISTRIBUTIONS
Since an exact relation of the PFG NMR signal atten-uations to distributions of diffusivities requires the inverseFourier transformation, we are now going to use the set ofEqs. (19)–(21) for an approximation of the probability distri-bution of the single-particle diffusivities in a two-region sys-tem. We proceed in analogy with our treatment of the sim-ple system with only one (mean) diffusivity. In either casethe information about the probability distribution p(D, τ ) ofthe single-particle diffusivities D is clearly contained in thepropagator. For the system with one diffusivity this propa-gator is given by Eq. (11). Its Fourier transform (Eq. (12),which is nothing else than the PFG NMR spin-echo diffusionattenuation curve) was found to coincide with the shape ofthe probability distribution of the single-particle diffusivities(Eq. (15)) with the only difference that the mean diffusivity,which represents the slope in the semi-logarithmic attenua-tion plots, appears in the denominator of the exponent in thedistribution function p(D).
In the two-region system, the PFG NMR spin-echo dif-fusion attenuation (and hence the Fourier transform of thepropagator) is now found to be given by two exponentials(Eq. (19)) of the form of Eq. (12). Formally we may re-fer, therefore, to two populations with the relative weightsp′
i and the effective (mean) diffusivities D′i as quantified by
Eqs. (20) and (21). Following the analogy of our simple initialsystem, as a first attempt, the resulting probability functionof the single-particle diffusivities may be approximated by acorresponding superposition of two exponentials of the typeof Eq. (15),
p(D, τ ) � p(D, k(τ )) = p′1(k)
1
D′1(k)
exp(−D/D′1(k))
+p′2(k)
1
D′2(k)
exp(−D/D′2(k))
(28)
with the parameters p′i(k) and D′
i(k) as given by Eqs. (20) and(21). Since this approximation avoids Fourier transformation,a proper τ -dependence of k has to be chosen for the primedquantities. It should be noted that the transformation ofEq. (19) from the Fourier space will only result in a superposi-tion of two exponentials in real space if the primed quantitiesin Fourier space are independent of k. Hence, Eq. (28) couldonly serve as a rough approximation of the observed process.However, inserting Eq. (28) into Eq. (10), the mean diffusivityof the two-region system results in
〈D〉 = p′1(k)D′
1(k) + p′2(k)D′
2(k) = π1D1 + π2D2 (29)
with the second equality resulting from the application ofEqs. (20) and (21). This is exactly the result which is well-known19 and it should be noted that it does not depend on τ .
Further on, we may consider the limiting cases k → 0and k → ∞ which can be translated to r → ∞ and r → 0,
respectively. Intuitively, large displacements r → ∞ are re-lated to long observation times τ → ∞ and vice versa. Thisrelation is substantiated by keeping k2τ constant (see alsoEq. (32)) where k → 0 corresponds to τ → ∞ and viceversa. Due to this, the respective limits of p(D, k(τ )) andp(D, τ ) should coincide. As a result we obtain the expectedexpressions
limk→∞
p(D, k(τ )) = limτ→0
p(D, τ )
= π1D−11 exp(−D/D1) + π2D
−12 exp(−D/D2), (30)
and
limk→0
p(D, k(τ )) = limτ→∞ p(D, τ )
= 〈D〉−1 exp(−D/〈D〉). (31)
Since the diffusivities and probabilities D′i and p′
i occur-ring in Eqs. (19)–(21) depend on the Fourier coordinate k, wehave referred to the probability density in this context as anapproximated one, p(D, k(τ )). Hence, Eq. (28) in the givennotation is unable to provide an approximation of the proba-bility distribution function of the single-particle diffusivitiesover the whole diffusivity scale. This is in perfect agreementwith the previous results14 where it has been shown that thedistribution of diffusivities, in general, cannot be representedby a weighted superposition of the underlying homogeneousdiffusion processes. However, such an approximation of theprobability density might become possible by inserting an ap-propriately selected value for the Fourier coordinate. As a firsttrial, one may put
k−2 = 〈D〉τ , (32)
which ensures highest sensitivity with respect to the spacescale covered during the experiments. Note that in PFG NMRexperiments, the exponent in the signal attenuation, Eq. (12),is of the order of 1, which yields an easily observable PFGNMR spin-echo diffusion attenuation.
Figure 3 depicts the distribution of diffusivities from asimulated two-dimensional trajectory in a two-region systemwith mean dwell times τ 1 = 0.125 and τ 2 = 0.25 for threetime lags τ = 0.01, 0.2, and 1.0. Further, the approximationof the distribution of diffusivities from Eq. (28) is investigatedfor corresponding k. Thus, the limiting case of completelyseparated diffusion processes found for τ → 0 is simulatedwith τ = 0.01 τ 1, τ 2 and compared with Eq. (28) fork → ∞, i.e., Eq. (30). On the other hand, the second limit-ing case of mean diffusion emerging for τ → ∞ is obtainedfrom simulation with τ = 1.0 τ 1, τ 2 and comparison withEq. (28) for k → 0, i.e., Eq. (31). Fig. 3 clearly shows thatsimulated data from both limiting cases are recovered reason-ably by Eq. (28) for corresponding k. In contrast, the distri-bution of diffusivities reveals a more complicated behavior inthe intermediate exchange regime between the limiting cases.Since the time lag τ = 0.2 is in the order of the mean dwelltimes, neither a mean diffusion process nor a weighted su-perposition of completely separated processes is observed.In particular, the distribution cannot be approximated byEq. (28) with a given k(τ ). This is obvious, since with such an
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144118-8 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
τ=0.01k~→∞
τ=0.2k~≈2.67
p(D
,τ)
D
τ=1.0k~→0
10-3
10-2
10-1
100
101
0 1 2 3 4 5 6
FIG. 3. Comparison of distribution of diffusivities (colored histograms) froma simulated two-dimensional trajectory with numerical approximation viaEq. (28) (solid lines) of a two-region system for time lags τ = 0.01, 0.2, and1.0 and mean dwell times of τ 1 = 0.125 and τ 2 = 0.25. The limiting cases ofk → 0 and k → ∞ approximate the simulated data reasonably. However, forτ = 0.2 in the order of the dwell times an intermediate k ≈ 2.67, as suggestedin Eq. (32), does not approximate the density sufficiently.
estimate of k the dependence on k of the primed quantities ofEqs. (20) and (21) in Fourier space is neglected. Then, the in-verse Fourier transformation of Eq. (19) as well as the trans-formation to the distribution of diffusivities would yield asimple superposition of two exponentials again. In general,this does not provide appropriate results for arbitrary dwelltimes and time lags.14 As a consequence, a general expres-sion requires inverse Fourier transformation of the PFG NMRattenuation curve.
VI. EXACT RELATION BETWEEN SIGNALATTENUATION AND DISTRIBUTION OF DIFFUSIVITIES
In Sec. V, the approximation of the distribution of diffu-sivities by Eq. (28) was shown to reproduce the limiting casesof time lag τ as well as the correct mean value. Cases in be-tween the limits did not deliver appropriate results. In order toproduce proper results for arbitrary τ , we derive general for-mulae for the transformation of PFG NMR signal attenuationsto distributions of single-particle diffusivities.
Quite formally two steps have to be accomplished to de-rive a general expression of p(D, τ ) from �(τ, k). As a firststep, inverse Fourier transformation of Eq. (1) yields the prop-agator in real space. Further, the shift r between positions, asgiven by the propagator, can be translated into diffusivitiesvia scaled squared displacements leading to the distributionof diffusivities as defined in Eq. (7).
The two steps can be combined to directly obtain theprobability density from signal attenuation. Depending on di-mensionality d, the distribution of diffusivities is given by
p(D, τ ) =∫
dr δ
(D − r2
2d τ
)
× 1
(2π )d
∫dk �(τ, k) exp(−ikr). (33)
With the rescaled coordinates,
r′ = r√2d τ
and k′ = k√
2d τ , (34)
it is further simplified to
p(D, τ ) =∫
dk′ �
(τ,
k′√
2d τ
)1
(2π )dSd (k′,D), (35)
with Sd (k,D) being the Fourier transform of a uniform den-sity on the surface of a d-dimensional sphere of radius
√D,
Sd (k,D) =∫
dr δ(D − r2) exp(−ikr). (36)
Since Eq. (36) can be expressed analytically35 by
Sd (k,D) = πa+1Da2aJa(|k|√
D)(|k|√
D)−a (37)
with a = d/2 − 1 and Ja(x) denoting the Bessel functionof the first kind, the exact transformation of signal attenua-tions �(τ, k) to distributions of diffusivities p(D, τ ) is accom-plished without applying an inverse Fourier transformation.
For isotropic systems, the signal attenuation �(τ, k) de-pends only on the absolute value of k, i.e., the radial inten-sity of the field gradient k. Without loss of generality, an ar-bitrary direction k = (k, 0, . . . , 0)T may be considered andthe corresponding signal attenuation is denoted by �1(τ, k) =�(τ, k = (k, 0, . . . , 0)T). Then the following expressions areobtained for the distribution of diffusivities depending on thedimensionality of the system. For one-dimensional systems,Eq. (33) reduces to
p(D, τ ) = 1
π√
D
∫ ∞
0dk �1
(τ,
k√2τ
)cos(k
√D).
(38)The transformation for d = 2 can be written as
p(D, τ ) = 1
2
∫ ∞
0dk �1
(τ,
k√4τ
)kJ0(k
√D), (39)
and for d = 3 one obtains
p(D, τ ) = 1
π
∫ ∞
0dk �1
(τ,
k√6τ
)k sin(k
√D) (40)
using polar and spherical coordinates, respectively. The giventransformations move the whole dependence on time lag τ
to the signal attenuation. This is achieved by rescaling the kcoordinate by
√2d τ .
Hence, a signal attenuation of an ensemble diffusing ina two-dimensional plane measured by PFG NMR is trans-formed into a distribution of single-particles diffusivities viaEq. (39). For homogeneous diffusion, Eq. (39) yields the ex-pected probability of single-particle diffusivities, Eq. (15), byinserting the simple exponential relation, Eq. (12), as signalattenuation.
Furthermore, the limiting cases of time lag τ are repro-duced exactly by the presented transformations, Eqs. (38)–(40): For τ → 0 the distribution of single-particle diffusivi-ties for the given dimensionality results in the superpositionof two respective distributions of Eq. (16) denoting two sep-arated, homogeneous diffusion processes. On the other hand,
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144118-9 Comparing diffusion data from SPT and PFG NMR J. Chem. Phys. 135, 144118 (2011)
τ=1.0
τ=0.5
τ=0.2
p(D
,τ)
D
τ=0.0510-2
10-1
100
101
0 1 2 3 4
FIG. 4. Comparison of distributions of single-particle diffusivities from asimulated two-dimensional trajectory (colored histograms) with distributionsobtained by applying Eq. (39) for an exact transformation of the PFG NMRspin-echo signal diffusion attenuation, Eq. (19), of a two-region system fortime lags τ = 0.05, 0.2, 0.5, and 1.0 and mean dwell times τ 1 = 0.125 and τ 2= 0.25 (solid lines). The data agree well with each other for each τ . Further,the dependence on τ is apparent, which is typical for diffusion in heteroge-neous media.
for τ → ∞, the resulting distribution of single-particle dif-fusivities for the given dimensionality is also of the type ofEq. (16), respectively, and depends only on the mean diffusioncoefficient of the system. A detailed derivation of the limitingcases is given in the Appendix C.
To examine the transformations, the same parameters, forwhich the approximation via Eq. (28) failed, are used again,now applying Eq. (39) for an exact transformation of the PFGNMR signal attenuation relation, Eq. (19), into the distribu-tion of single-particle diffusivities. The results are depicted inFig. 4 and again the distribution of diffusivities from a simu-lated two-dimensional trajectory is given for comparison. Foreach of the chosen τ = 0.05, 0.2, 0.5, and 1.0, a perfect agree-ment is obvious, confirming the relation between the two ap-proaches. Moreover, Fig. 4 clearly illustrates how the distribu-tion of diffusivities depends on τ and reveals a transition froma non-exponential behavior to a mono-exponential decay. Forsmall τ corresponding to diffusion in separated regions, it de-viates considerably from a mono-exponential behavior. How-ever, for long-term observations (τ → ∞) only a mean dif-fusion process is observed due to averaging of the motion inboth regions. Consequently, this yields a mono-exponentialdecay of the distribution of diffusivities. This transition re-veals the heterogeneity of the diffusion process.14 Hence, inorder to characterize diffusive motion the distribution of dif-fusivities has to be investigated for its dependence on the timelag τ .
VII. INFLUENCE OF EXPERIMENTALLY BOUNDED k
PFG NMR spin-echo diffusion attenuation functions canonly be measured up to a finite intensity k of the magnetic fieldgradient pulses. However, to generate the distribution of diffu-sivities exactly, the signal attenuation has to be given over thewhole intensity scale. Hence, the effect of an experimentallybounded Fourier coordinate k has to be considered.
p(D
,0.2
)
D
kmax=2kmax=5kmax=8kmax=10kmax=12kmax=15
10-3
10-2
10-1
100
101
0 1 2 3 4 5 6 7 8
FIG. 5. Transformation of Eq. (39) of PFG NMR spin-echo signal diffu-sion attenuation by integration up to kmax (solid lines) due to experimentallybounded intensity k of the field gradient pulses. The distribution of single-particle diffusivities (colored histogram) from a simulated two-dimensionaltrajectory will only be obtained reasonably if k is given over the whole inten-sity scale. For smaller intervals of k deviations become clearly visible as wellas oscillations introduced by the inverse Fourier transformation.
Figure 5 illustrates the influence of finite k on the distri-bution of single-particle diffusivities obtained for τ = 0.2. Ifwith the maximal applied kmax the respective spin-echo signalis not sufficiently attenuated, the transformation of the signalattenuation from a finite interval will yield significant devia-tions from the expected probability distribution. As a conse-quence, the first moment, i.e., the mean diffusion coefficientof the system, is altered accordingly. Furthermore, due to thebounded signal attenuation, the inverse Fourier transforma-tion introduces oscillations since only a limited range of thespectrum contributes to the values in real space. The reasonis the integrand in Eqs. (38)–(40) which will only vanish forlarge k if �1 decays faster than the remainder.
This effect may clearly be identified in Fig. 5. In orderto obtain reasonable results, the signal must be attenuated to asufficient extent. Simulated data of two-dimensional diffusionprocesses have shown that the attenuation should fall below10−4 of its maximum at kmax to suppress oscillations. Thishas to be considered when dealing with experimental data.
The necessity of fast decaying �1 becomes especiallyimportant for large time lags τ . In the case of small time lags τ
→ 0 our rescaling of the k coordinate in Eqs. (38)–(40) leadsto k/
√τ → ∞ in the second argument of �1(τ, k/
√2d τ ).
Thus, for small τ , signal attenuation becomes more pro-nounced and reduces the influence of the bounded k. More-over, signal attenuation is closely related to the incoherentstructure factor,36 as demonstrated in the Appendix A, deal-ing with similar limitations. A possible solution is to split theintegral into two parts, integrating numerically up to the ex-perimental limit kmax and assuming an analytical expressionfor the remaining part.
Since the oscillations in the approximate densities ofFig. 5 seem to be induced by the hard cutoff at the wave-length k = kmax, a possible strategy in reducing these oscil-lations may lie in applying an appropriate window functionas in spectrum estimation procedures. We tested this option
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144118-10 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
p(D
,0.2
)
D
kmax=2kmax=5kmax=8kmax=10kmax=12kmax=15
10-3
10-2
10-1
100
101
0 1 2 3 4 5 6 7 8
FIG. 6. Same situation as in Fig. 5, but the densities are now obtained by re-placing the sharp cutoff at k = kmax by a smooth cutoff resulting from apply-ing a half Hann window. A considerable improvement is achieved, especiallyif the value kmax is not too small.
by applying a half Hann window to smoothen the cutoff. Thebest results were obtained for a window decaying from thevalue one at k = 0 to zero at k = kmax. The obtained resultsare very convincing if the cutoff value kmax is not too small ascan be seen in Fig. 6.
VIII. CONCLUSIONS
We investigated the connection between the signal atten-uation measured by pulsed field gradient nuclear magneticresonance and the distribution of single-particle diffusivitiesobtained from single-particle tracking. Due to their interrela-tions with the diffusion propagator of the system, the distri-bution of diffusivities is expressed by a general transforma-tion of the signal attenuation. In the special case of a systeminvolving two different states of diffusive mobility, the two-region exchange model of PFG NMR offers analytical expres-sions and allows for a comparison of analytical and simulateddata. An approximation of the distribution of single-particlediffusivities via two populations with relative weights avoidsthe inverse Fourier transformation. Even in this simple sys-tem, such an approximation will only yield appropriate resultsif the time lag is much larger or much smaller than the dwelltimes. These cases correspond to an observation of the meandiffusion of the system and a process of completely separateddiffusive motion without transition between the regions, re-spectively. Thus, in general, to obtain a proper distributionof single-particle diffusivities for diffusion in two-region sys-tems, the exact transformation of the respective NMR signalattenuations is necessary. Only in this way we found perfectagreement of the experimental and analytical data. However,since PFG NMR data in some systems cannot be measuredover a sufficiently large dynamic range, the inverse Fouriertransformation may introduce deviations and oscillations. Inthese cases, the data analysis has to be performed with careand may require the use of additional information.
In summary, the investigated connection between twopopular methods to experimentally observe and analyze dif-
fusive motion offers new approaches for the evaluation ofdata. Hence, the methods of analysis may benefit from eachother. This becomes especially relevant for systems with het-erogeneities, where the distribution of diffusivities exhibits adependence on the time lag. For more elaborated processes itmay even not become stationary and enables to assess non-trivial properties of such systems. Since the distribution ofdiffusivities can be measured easily and contains more infor-mation from the propagator than well-established methods, itshould be used for future analysis of experimental data.
ACKNOWLEDGMENTS
We gratefully acknowledge financial support from theDeutsche Forschungsgemeinschaft (DFG) for funding of theresearch unit FOR 877 “From Local Constraints to Macro-scopic Transport”. We also thank the anonymous referees fortheir valuable suggestions, which helped to improve the paperconsiderably.
APPENDIX A: CORRESPONDENCE BETWEENINCOHERENT INTERMEDIATE SCATTERINGFUNCTION AND SIGNAL ATTENUATION
The signal attenuation of PFG NMR and the incoher-ent intermediate scattering function as well as the dynamicstructure factor are closely related. In this appendix, their cor-respondence is illustrated briefly and further details can befound in Refs. 18, 37, and 38.
The observed motion of tracer particles can be analyzedby the self part of the van Hove time-dependent pair correla-tion function,
Gs(r, τ ) =⟨
1
N
N∑i=1
δ(r − (xi(τ ) − xi(0)))
⟩, (A1)
describing the correlation of N individual particles.39 Its spa-tial Fourier transformation,
S(k, τ ) =∫
dr Gs(r, τ ) exp(ikr), (A2)
leads to the incoherent intermediate scattering function,
S(k, τ ) = 1
N
N∑i=1
〈exp(ik(xi(τ ) − xi(0)))〉, (A3)
which is linked to the velocity autocorrelation function ofthe particles. Furthermore, the incoherent intermediate scat-tering function S(k, τ ) is related to the dynamic structure fac-tor S(k, ω) known from neutron scattering via Fourier trans-formation in τ , i.e., the power spectrum of S(k, τ ), where ω
denotes a frequency.For ergodic systems, S(k, τ ) can be obtained from an ar-
bitrary particle,
S(k, τ ) = 〈exp(ik(x(τ ) − x(0)))〉
= 1
V 2
∫ ∫dx dx′ exp(ik(x − x′) p(x, τ, x′, 0),
(A4)
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144118-11 Comparing diffusion data from SPT and PFG NMR J. Chem. Phys. 135, 144118 (2011)
where V is the normalization and p(x, τ, x′, 0) denotes thejoint probability of a particle to be located initially at x′ andat time τ at position x. The joint probability can be expressedby the conditional probability,
p(x, τ, x′, 0) = p(x, τ |x′, 0) p0(x′). (A5)
Since during time τ the particle accomplished a displacementr, its positions are interrelated by x = x′ + r. Due to transla-tion invariance, without loss of generality, x′ = 0 leads to thepropagator in Fourier space,
1
V
∫dr exp(ikr) p(r, τ ) = �(τ, k), (A6)
corresponding to the signal attenuation in PFG NMR as in-troduced in Eq. (1). Hence, signal attenuation and incoher-ent intermediate scattering function coincide. Furthermore,for identical particles without restrictions by the boundariesthe averaging over the particles in Eq. (A1) can be omittedand Gs(r, τ ) is equal to p(r, τ ) given by Eq. (2).
For isotropic systems the self part of the radial van Hovetime-dependent pair correlation function,
Gs(r, τ ) =⟨
1
N
N∑i=1
δ (r − |xi(τ ) − xi(0)|)⟩
, (A7)
considers only absolute values of the displacements. Again,for identical particles without restrictions by the boundariesan arbitrary particle can be considered and Gs(r, τ ) is equalto pr(r, τ ).
APPENDIX B: RELATION BETWEEN EVOLUTIONEQUATIONS AND PFG NMR SIGNAL ATTENUATION
For Eq. (24), i.e., the evolution equations of the proba-bility density to find a particle at position x at time t, the mo-ments of the random variable x can be obtained via the charac-teristic functions. By introducing the vector p(k, t) compris-ing the characteristic functions of each region and the matrixW(k) consisting of the elements
W(k)nm = wnm +(
−Dnk2 −∑
l
wln
)δnm, (B1)
the Fourier transform of Eq. (24) can be written elegantly as
d
dtp(k, t) = W(k) p(k, t), (B2)
where
p(k, t) = exp(t W(k)) p(k, 0) (B3)
is easily seen to be the solution. For the two-region system theinitial distribution p(k, 0) = (π1, π2)T is given by the equilib-rium distribution between the regions.
Applying the spectral decomposition the matrix exponen-tial in Eq. (B3) for the two-region system can be written as
exp(t W(k)) =2∑
α=1
exp(tμα(k)) Aα(k), (B4)
where
μ1,2(k) = 1
2(−D1k2 − D2k2 − λ ± D(k)) (B5)
denotes the eigenvalues and,
A1,2(k) = 1
2D(k)
(D(k) ± η(k) ±2w12
±2w21 D(k) ∓ η(k)
),
(B6)represent the corresponding matrices from the dyadic productof the right- and left-eigenvectors with
λ = w21 + w12, (B7)
η(k) = −D1k2 + D2k2 − w21 + w12, (B8)
D(k) = {(D1k2 + D2k2 + λ)2 − 4D1D2k4
−4D1k2w12 − 4D2k2w21}1/2. (B9)
Finally, the signal attenuation obtained from PFG NMRcorresponds to the projection of the characteristic function
�(τ, k) = (1 1
)exp(τ W(k))
(π1
π2
), (B10)
where k = k e is measured in the direction of the applied fieldgradient denoted by the unit vector e. Since for isotropic sys-tems an arbitrary direction can be considered, Eq. (B10) re-sults in the expressions given in Eqs. (19)–(21) for the two-region system.
APPENDIX C: EXACT TRANSFORMATION OFLIMITING CASES
By choosing k = k e, the isotropic signal attenuations fordimensionality d in Eqs. (38)–(40)
�1
(τ,
k
u√
τ
)with u =
⎧⎪⎨⎪⎩
√2 for d = 1
2 for d = 2√6 for d = 3
,
are considered in an arbitrary direction of the applied fieldgradient with intensity k. The exponent of Eq. (B10) is givenby
τ W(
k eu√
τ
)= τ
(−w21 w12
w21 −w12
)− k2
u2
(D1 0
0 D2
).
(C1)Based on these expressions the limiting cases are discussedseparately.
1. Limiting case τ → 0
In the limiting case of τ → 0, only the diagonal matrixon the right-hand side of Eq. (C1) remains. Hence, the ma-trix exponential can be expressed by the exponentiation of the
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144118-12 Bauer et al. J. Chem. Phys. 135, 144118 (2011)
diagonal elements and Eq. (B10) reduces to
�1
(τ,
k
u√
τ
)
= (1 1
)⎛⎜⎝ exp(− k2
u2 D1
)0
0 exp(− k2
u2 D2
)⎞⎟⎠
(π1
π2
), (C2)
yielding a superposition of two exponentials correspondingto separated regions. This is in agreement with the previousfindings since for short times τ no exchange between the re-gions occurs. Obviously, this result is not restricted to the two-region system but holds for an arbitrary number of diffusionstates.
Applying the presented transformations Eqs. (38)–(40)for dimensionality d to the obtained signal attenuation resultsin a distribution of diffusivities which is the superposition oftwo distributions of diffusivities for homogeneous diffusionin each region as given by Eq. (16), respectively.
2. Limiting case τ → ∞In the limiting case of τ → ∞, the situation is more com-
plicated. Arguing analogously to the case of τ → 0 does notresult in an appropriate expression. If the diagonal matrix onthe right-hand side of Eq. (C1) is neglected, the signal attenu-ation will reduce to 1 yielding only its normalization. Hence,this limiting case is addressed by involving the spectral de-composition. The matrices Eq. (B6) are given by
A1,2
(k e
u√
τ
)τ→∞−−→A1,2(0),
A1(0) = 1
λ
(w12 w12
w21 w21
), (C3)
A2(0) = 1
λ
(w21 −w12
−w21 w12
). (C4)
Due to the projection in the signal attenuation Eq. (B10)(1 1
)A2(0) = (
0 0), (C5)
the contribution from A2(0) vanishes. Thus, for τ → ∞only eigenvalue μ1 contributes to the spectral decomposition.Moreover, μ1 = 0, which explains that the contribution fromthe diagonal matrix in Eq. (C1) cannot be neglected.
Then, according to Eq. (B4), the exponential ofτ μ1(k e/(u
√τ )) is required, which is given by
τ μ1
(k e
u√
τ
)= 1
2
(−a − λτ +
√(a + λτ )2 − b − cτ
)(C6)
with
a = D1k2
u2+ D2
k2
u2, (C7a)
b = 4D1D2k4
u4, (C7b)
c = (4D1w12 + 4D2w21)k2
u2. (C7c)
The square root in Eq. (C6) can be rewritten as√(a + λτ )2 − b − cτ
= λτ
√1 +
(2a
λ− c
λ2
)1
τ+ a2 − b
λ2
1
τ 2
= λτ
(1 + 1
2
(2a
λ− c
λ2
)1
τ+ O
(1
τ 2
)). (C8)
After further simplification, Eq. (C6) reduces to
τ μ1
(k e
u√
τ
)� 1
2
(−a − λτ + λτ + a − c
2λ
)
= − c
4λ, (C9)
which results in
τ μ1
(k e
u√
τ
)� −(π1D1 + π2D2)
k2
u2
= −〈D〉 k2
u2, (C10)
by applying Eq. (C7c) and Eqs. (26) and (29). Hence in thelimiting case of τ → ∞, the signal attenuation,
�1
(τ,
k
u√
τ
)= exp
(−〈D〉 k2
u2
), (C11)
depends only on the mean diffusion coefficient of the two-region system.
By integrating the signal attenuation Eq. (C11) for thelimiting case τ → ∞ with the presented transformationsEqs. (38)–(40) for dimensionality d, as expected, the respec-tive distributions of diffusivities, Eq. (16) are obtained, whichcorrespond to homogeneous diffusion with the mean diffusioncoefficient 〈D〉.
To conclude, the derivation of the two limiting cases re-veals the properties of the distribution of single-particle dif-fusivities and its dependence on τ . Starting from the limit-ing case τ → ∞, where only eigenvalue μ1 contributes, theweight of μ2 increases for decreasing τ . This is reflected inthe distribution of diffusivities by the dependence on τ as pre-sented in Fig. 4. It describes the transition from a mean diffu-sion process to two completely separated diffusion processesfor τ → ∞ and τ → 0, respectively. It should be noted that forthe self part of the van Hove function the limiting cases cannotbe determined. However, for the distribution of diffusivities,which is a rescaled van Hove self-correlation function, bothlimits are well-defined.
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