Beyond the Wilemski–Fixman theory - CORE

Time-dependent survival probability in diffusion-controlled reactions in a polymer chain: Beyond the Wilemski–Fixman theory

Goundla Srinivas

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India 560 012 K. L. Sebastian

Inorganic and Physical Chemistry Department, Indian Institute of Science, Bangalore, India 560 012 Biman Bagchi^a)

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India 560 012

Brownian dynamics simulation results of the time-dependent survival probability (S_p(t)) of a donor–acceptor pair embedded at the two ends of a Rouse chain are compared with two different theories, one of which is the well-known Wilemski–Fixman共WF兲theory. The reaction studied is fluorescence energy transfer via the Fo¨rster mechanism, which has a R^⫺6distance共R兲dependence of the reaction rate. It has been reported earlier关G. Srinivas, A. Yethiraj, and B. Bagchi, J. Chem.

Phys. 114, 9170 共2001兲兴 that while the WF theory is satisfactory for small reaction rates, the agreement was found to become progressively poorer as the rate is increased. In this work, we have generalized the WF theory. We suggest an approximate, reduced propagator technique for three-dimensional treatment共instead of 3N dimensions, where N is the number of monomers in the polymer chain兲. This equation is solved by combining a Green’s function solution with a discretized sink method. The results obtained by this new scheme are in better agreement with the simulation results.

I. INTRODUCTION

Reactions between any two sites of a polymer chain have been a subject of great interest.^{1– 8}Often these reactions occur via the distance dependent rate, such as fluorescence resonance energy transfer 共FRET兲 by the long distance Fo¨r- ster mechanism,^1,9–14electron transfer reactions,^15,16etc. The mechanism of long distance FRET is Coulombic and is usu- ally discussed in terms of Fo¨rster theory^1,2 which gives the following distance dependent rate of energy transfer,

k_f共R兲⫽k_rad

冉

^RRF

冊

^{6, 共1兲}

where R_Fis the Fo¨rster radius, defined as the D-A separation corresponding to 50% energy transfer. k_rad is the radiative rate, which is typically of the order of 10⁸ to 10⁹ cm^⫺1 for the commonly used chromophores in FRET experiments. Ac- cording to the above equation k_rad can be understood as the rate of energy transfer when the separation between the do- nor and the acceptor is equal to Fo¨rster radius 共i.e., R/R_F

⫽1兲. The Fo¨rster radius is usually obtained from the overlap of the donor fluorescence with the acceptor absorption and several other available parameters.^9,10

The dynamics of Fo¨rster energy migration has been in- vestigated traditionally via time domain measurements of the decay of the fluorescence 共due to excitation transfer兲 from the donor.^1,10As both k_radand R_Fare determined by the D-A pair, the rate of decay of the fluorescence intensity provides a direct probe of the conformational dynamics of the polymer.

Recently, this technique has been used in single molecule spectroscopy¹⁷of biopolymers^11–14and proteins,¹¹where the distance dependence of FRET provides relevant information about the conformation and dynamics of single biopolymers.

At any given time after the initial excitation, the fluorescence intensity is a measure of the ‘‘unreacted’’ donor concentra- tion, that is, of the survival probability, S_p(t).

The complexity of describing the dynamics of energy transfer of polymers in solution arises from the fact that, due to chain connectivity, the Brownian motion of the monomers on the polymer are strongly correlated. The many-body na- ture of polymer dynamics can be described by a joint, time- dependent probability distribution P(r^N,t) where r^Ndenotes the position of all the N polymer beads, at time t. The time dependence of the probability distribution P(r^N,t) can be described by the following reaction-diffusion equation,^4,5

⳵

⳵^{tP共rN,t兲⫽LB共rN兲P共rN,t兲⫺k共R兲P共rN,t兲, 共2兲} whereLB is the full 3N dimensional diffusion operator,

LB共r^N兲P共r^N,t兲⫽Dj

兺

⫽1

N ⳵

⳵^rj

P_eq共r^N兲 ⳵

⳵^rj

1

P_eq共r^N兲P共r^N,t兲 共3兲 the subscript ‘‘eq’’ denotes equilibrium, R is the scalar dis- tance between the two ends of the polymer chain, and D is the diffusion coefficient of a monomer The solution of Eq.

共2兲, with the sink term 关last term on the right-hand side of Eq.共2兲兴given by the Fo¨rster expression^1,2for k(R) is highly nontrivial.

a兲Electronic mail: bbagchi@sscu.iisc.ernet.in

In two seminal papers, Wilemski and Fixman 共WF兲^4,5 presented a nearly analytic solution of the problem for any arbitrary sink. Pastor, Zwanzig, and Szabo^3,18tested the WF theory only for the average rate, by computer simulations when the sink is a Heaviside function. They found that the WF theory is efficient for the sinks with smaller radii. In spite of its importance, the WF theory has never been studied for a distance dependence rate, such as Fo¨rster energy trans- fer. Such a study is clearly important because the end-to-end probability distribution in polymer, peaks at a distance which scales as N^2␯. ␯⫽1/2 for the Rouse chain and 3/5 for the self-avoiding walk共SAW兲.¹⁹

Recently, we have carried out a Brownian dynamics simulation study of the dynamics of energy transfer in Rouse chain.⁸ The polymer molecule was modeled as an ideal Gaussian chain with N monomer units with segment 共or Kuhn兲length b. The donor and acceptor sites were assumed to be located at opposite ends of the polymer chain. The resonance energy transfer rate, k(R) was assumed to be given by,

k共R兲⫽ k_rad

1⫹共R/R_F兲⁶. 共4兲

Note that the above-mentioned form is different from the commonly used form of the Fo¨rster rate, k_f(R) given by Eq.

共1兲. The (R_F/R)⁶ distance dependence is not appropriate here, since it diverges at R→0, which is allowed in Rouse chain,^20,21but not in a real polymer, where the end-to-end distance共R兲never approaches zero, due to the excluded vol- ume forces. Thus, the modified form 关Eq. 共4兲兴 used is rea- sonable. The reason that we use Rouse chain is that this case can be treated easily in theory. For example, the theory of Wilemski and Fixman^4,5,22 can be readily applied to the Rouse chain, because the necessary Green’s function is avail- able in analytic form.

It was reported in Ref. 8 that the Wilemski–Fixman theory, unfortunately, does not provide a satisfactory descrip- tion for the following situations: 共a兲 When the rate k_rad is much larger than the rate of monomer diffusion, given by b²/D;共b兲when R_Fis close to the separation where the prob- ability of finding the chain ends is maximum. The above limitations of the WF theory were somewhat surprising and motivated the present work.

The main objectives of this paper are the following:共1兲 to present further Brownian dynamics 共BD兲 simulations of Eq.共2兲, with k(R) given by the Fo¨rster rate关Eq.共4兲兴;共2兲to present an alternative theoretical approach to treat the dy- namics of FRET in polymers.

The new theoretical approach employs a reduced, three- dimensional equation-of-motion. The solution of the three- dimensional equation-of-motion employs a novel reduction of the equation-of-motion to one-dimension and subse- quently uses a discretized sink model to obtain the survival probability. Henceforth, this method will be referred to as the three-dimensional reduced Green’s function method共RGF兲.

It is found that the agreement of the results obtained from the new scheme with the simulations is superior to that of Wilemski–Fixman. However, the agreement is still not

perfect in several limiting cases. The reason for this has been discussed.

Simulation details and the description of the Wilemski–

Fixman theory remain same as that in our previous study.⁸ The organization of the rest of the paper is as follows. In the next section, we introduce the RGF method. The implemen- tation of this method through the discretized sink method is also presented in the same section. In Sec. III, the simula- tions results are compared with the theoretical predictions.

Conclusions are presented in Sec. IV.

II. THEORY

A. An approximate equation for the reduced Green’s function

The formal solution of Eq.共2兲is

P共r^N,t兲⫽

冕

^{dr0NG共rN,t兩r0N,0兲P共r0N,0兲. 共5兲}

P(r₀^N,0) denotes the initial distribution and G(r^N,t兩r₀^N,0) is the Green’s function for the problem. This is exact, but not very usable as it involves all the r^N. One would like to have an equation involving only the relevant coordinate, R. We derive such an equation in the following. The derivation in- volves two steps. First, we derive an approximate equation involving only the end-to-end vector R. Then, using the fact that the sink function depends only on the magnitude of R, this is reduced exactly to a one-dimensional diffusion prob- lem.

As we are interested only in the dynamics of R, we refer to the remaining coordinates of the chain as the irrelevant coordinates and denote them by S. Instead of using the vari- ables r^N, one can use共R, S兲. We now write G(r^N,t兩r₀^N,0), in terms of 共R, S兲, as G(R,S,t兩R₀,S₀,0). The differential Eq.

共2兲, is equivalent to the exact integral equation:

G共R,S,t兩R₀,S₀兲

⫽G₀共R,S,t兩R₀,S₀,0兲⫺

冕

0 t

dt₁

冕

^dR1

冕

^dS1

⫻G₀共R,S,t兩R₁,S₁,t₁兲k共R₁兲G共R₁,S₁,t₁兩R₀,S₀,0兲. 共6兲 G₀(R,S,t兩R₀,S₀,0) would be the Green’s function if the sink is not present. We now introduce the conditional prob- ability distribution P_eq(S兩R)⫽P_eq(R,S)/ P_eq(R), where we define the reduced equilibrium probability distribution for R by P_eq(R)⫽兰dSP_eq(R,S). Multiplying Eq. 共6兲 by P_eq(S₀兩R₀)dS₀dS, and integrating over all initial and final positions of the irrelevant coordinates S₀ and S, we get G共R,t兩R₀,0兲

⫽G₀共R,t兩R₀,0兲⫺

冕

^dS

冕

^0tdt1

冕

^dR1

冕

^dS1

⫻G₀共R,S,t兩R₁,S₁,t₁兲k共R₁兲

冕

^dS0

⫻G共R₁,S₁,t₁兩R₀,S₀,0兲P_eq共S₀兩R₀兲. 共7兲

In the Eq.共7兲, we have defined the reduced Green’s function for the R coordinate by

G共R,t兩R₀,0兲⫽

冕

^dS

冕

^{dS0G共R,S,t兩R0,S0,0兲}

⫻P_eq共S₀兩R₀兲,

with a similar definition for G₀(R,t兩R₀,0). Equation共7兲can- not be solved. So we introduce the approximation

冕

^{dS0G共R1,S1,t1兩R0,S0,0兲Peq共S0兩R0兲}

⯝P_eq共S₁兩R₁兲

冕

^dS1⬘

冕

^{dS0G共R1,S1⬘,t1兩R0,S0,0兲}

⫻P_eq共S₀兩R₀兲. 共8兲 With this, Eq.共7兲becomes

G共R,t兩R₀,0兲⫽G₀共R,t兩R₀,0兲⫺

冕

0 t

dt₁

冕

^dR1

⫻G₀共R,t兩R₁,t₁兲k共R₁兲G共R₁,t₁兩R₀,0兲. 共9兲 The approximation of Eq.共8兲has the property that it is exact in the limits t₁→0 or⬁. The physical meaning of the approximation is that if, from the equilibrium ensemble, one selects only those that have an end-to-end vector equal to R₀ and evolves them in time in presence of the sink, then the resultant probability for finding the system at R₁, S₁may be approximated by the product of two terms. They are 共i兲the exact probability that the end to vector has a new value R₁, and共ii兲 the conditional probability for finding the irrelevant coordinates at S₁ given that the end-to-end vector has the value R₁, calculated at equilibrium. This approximation would follow if one assumed that the irrelevant variables adjust to the instantaneous value of R. One expects that ap- proximation is reasonable for longer times, but would show deviations for shorter.

B. Reduction to a one-dimensional equation

The main idea used in the subsequent steps is that the Fo¨rster reaction rate depends only on the distance R between the donor and the acceptor. Thus, the above three- dimensional Eq. 共9兲 can be reduced to a one-dimensional equation. This further reduction, however, requires some al- gebraic manipulations described below.

In Laplace frequency plane Eq.共9兲takes the form, G共R,z兩R₀兲⫽G₀共R,z兩R₀兲⫺

冕

^{dR⬘G0共R,z兩R⬘兲}

⫻k共R⬘兲G共R⬘^,z兩R₀兲. 共10兲 By multiplying the above equation by d⍀R and integrating over all the orientations of R we get,

G共R,z兩R₀兲⫽G1,共R,z兩R₀兲⫺

冕

0

⬁共R⬘兲²dR⬘G共R,z兩R⬘兲

⫻k共R⬘兲G共R⬘^,z兩R₀兲, 共11兲 where we have defined two auxiliary functions:

G0共R,z兩R₀兲⫽

冕

^{d⍀RG0共R,z兩R0兲 共12兲}

and

G共R,z兩R₀兲⫽

冕

^{d⍀RG共R,z兩R0兲. 共13兲}

Our interest is in the survival probability,

S_p共t兲⫽

冕

^dR

冕

^{dR0G共R,t兩R0兲Peq共R0兲, 共14兲}

using the fact that P_eq(R₀) depends only on R₀, we can write this as,

S_p共t兲⫽4␲

冕

0

⬁

R²dR

冕

0

⬁

R₀²dR₀G共R,t兩R₀兲P_eq共R₀兲. 共15兲 Now it is convenient to define,

Pˆ共R,z兲⫽

冕

0

⬁

R₀²dR₀G共R,z兩R₀兲P_eq共R₀兲. 共16兲 Then, Pˆ (R,z) obeys the equation

Pˆ共R,z兲⫽1

zPeq共R兲⫺

冕

0

⬁共R⬘兲²dR⬘^G共R,z兩R⬘兲

⫻k共R⬘兲Pˆ共R⬘^,z兲. 共17兲 Equation共17兲is now solved by employing a discretized sink method, developed earlier.^23–26

C. Discretized sink method

In this method, the continuous sink curve is discretized into M number of intervals. Thus, in discretized sink notation the sink function 关defined by Eq.共4兲兴can be written as

k共R兲⫽k_rad

兺

_s ^{␦共R⫺Rs兲}

冉

^Rs6R⫹F6RF6

冊

^{. 共18兲}

Note that the summation is over all the sink points (s), where the populations are given by P(R_s,z). Equation 共17兲 can be discretized as

Pˆ共R,z兲⫽1

zPeq共R,z兲⫺

兺

_s ^{Rs2G0共R,z兩Rs兲}

⫻k共R_s兲Pˆ共R_s,z兲. 共19兲 By finding P(R_s,z) using this, a set of linear equations are generated. These can be written in matrix form as,

B•P⫽P₀, 共20兲

where the elements of matrices B, P, and P₀ are given by, B_mn⫽␦mn⫹k_nR_n²G0共R_m,z兩R_n兲, 共21兲

P_m⫽P共R_m,z兲, 共22兲

and

P_0m⫽P₀共R_m,z兲, 共23兲 and k_n⫽k(R_n).

Now note that G(R,t兩R₀), after averaging over the angles and using the Green’s function definition 共given in Appendix A兲, can be written in the following form

G0共R,t兩R₀兲⫽ 1

LRR₀␳共t兲

冑

_␲共1⫺⁶␳共t兲²兲

⫻sinh

冉

^{L23RR共1⫺0␳␳共共tt兲兲2兲}

冊

⫻e^{⫺3共R2⫹R02␳共t兲2兲/2L2共1⫺␳共t兲2兲}. 共24兲 By taking the Laplace transform of G0(R,t兩R₀) and substi- tuting in Eq. 共19兲, we obtainPˆ (R,z). The resultingPˆ (R,z) can be used in the following equation to obtain the survival probability in z-plane.

S_p共z兲⫽1

z

冋

^1⫺

兺

^{s k共Rs兲Rs2Pˆ共Rs,z兲}

册

^{. 共25兲}

Finally, the time-dependent survival probability can be ob- tained through the Laplace inversion,

S_p共t兲⫽L^⫺1共S_p共z兲兲. 共26兲 III. NUMERICAL RESULTS

The discretized sink method can be conveniently used to solve Eq. 共2兲 for a wide range of k_rad and R_F values. The method is fairly simple to implement and is not numerically intensive. In this section we present the results obtained by using RGF and compare with WF theory prediction, as well as with the BD simulation results.

In the reduced three-dimensional description the mutual diffusion becomes time-dependent. This can be expressed in terms of the end-to-end vector correlations function as fol- lows,

D共t兲⫽⫺L²

3

冉

^{⳵⳵tln␳共t兲}

冊

^{. 共27兲}

Although D(t) has not been directly used in the present study, it provides a useful measure of the end-to-end diffu- sion. In Fig. 1, we show the time dependence of the end-to- end diffusion as a function of reduced time for N⫽50. It decays monotonically with time from its initial value. The full line corresponds to the theoretical result 关obtained by using analytical ␳(t); see Appendix B兴. The dashed line is obtained by using simulated␳(t). As shown in the figure, the initial value, which correspond to the sum of the bare diffu- sion of the two ends (2D₀), decreases with time and satu- rates to a finite nonzero value in the long time limit 共see Appendix B兲. Asymptotically, D^sim(t) and D^theory(t) ap- proach somewhat different values in the t→⬁ limit. This may be due to the finite size effects.

Figures 2 and 3 show the comparison between the sur- vival probability obtained from the Brownian dynamics共BD兲 simulations and the theoretical prediction 关Eq. 共25兲兴, for k_rad⫽1. In both the figures, BD simulation results are shown by symbol, while the theoretical results are shown by line.

The RGF and WF theory predictions are shown by full and dashed line, respectively. In Fig. 2 the survival probability for R_F⫽1 is plotted against the reduced time. The represen- tation of Fig. 3 is the same as that of Fig. 2, except that this figure is plotted for a larger R_Fvalue, namely R_F⫽5.

FIG. 1. The time-dependent diffusion of end-to-end separation obtained by using the␳(t) from Brownian dynamics simulations共dashed line兲and the theory共full line; obtained by using the analytical␳(t)兲is plotted as a func- tion of reduced time for N⫽50.

FIG. 2. Results of S_p(t) are plotted against the reduced time for R_F⫽1 and krad⫽1 for N⫽50. The results obtained by simulation are shown with sym- bol, while the theoretical predictions are shown by line. The reduced Green’s function method共RGF兲 共full line兲results are in better agreement with simulation result over the prediction of the WF theory共dashed line兲.

Note that similar to the WF theory, the RGF results agree well with that of the simulation at smaller R_Fvalues. How- ever, the agreement with WF becomes progressively poorer with increasing R_F. On the other hand, the RGF results are in better agreement with that of the simulations, even at large R_F values 共shown in Fig. 3兲, where WF theory completely breaks down in the long times.

In Fig. 4 the survival probability for a larger radiative rate (k_rad⫽10) is plotted at R_F⫽1. The RGF results 共shown by full line兲are again in better agreement with the simulation results 共shown by symbols兲 than the WF theory 共dashed line兲. However, the agreement with the present RGF method is not perfect.

The precise reason for the failure of our RGF is not clear at present. However, it is important to note that the reduced three-dimensional equation used in this study is an approxi- mation. This is probably the main source of error in our treatment.

IV. CONCLUSIONS

In view of the limitation of WF theory at large rates⁸we have generalized the WF theory by reducing the 3N dimen- sional problem to three-dimensions. The resulting reduced Green’s function equation is solved by using the discretized sink method to obtain the S_p(t). Although the deficiency still

remains in few cases, the results obtained from the new scheme are in better agreement with that of the simulations when compared to WF theory results. Note that the dis- cretized sink method gives the same results as that of WF theory in the case of a delta function sink.

The techniques employed in this work could be em- ployed in other related fields. The distance dependent rate appears in several other chemical processes, where the rate of transfer is known to show an exponential distance depen- dence. One such example is the electron-transfer reactions. It is of interest to use the method employed here to that prob- lem as well. Finally, the simulations results obtained here should be analyzed by using the variational theory of Port- man and Wolynes.²² Work in these directions is under progress.

ACKNOWLEDGMENTS

This work is supported in part by the Council of Scien- tific and Industrial Research 共CSIR兲 and the Department of Science and Technology 共DST兲, India. One of the authors 共G.S.兲thanks CSIR, New Delhi, India for a research fellow- ship.

FIG. 3. Similar to Fig. 2 but for R_F⫽5. Though the agreement becomes poorer as RFis increased, RGF results共full line兲are in better agreement with the simulation result 共symbols兲 compared to WF theory prediction 共dashed line兲.

FIG. 4. Comparison of Sp(t) at a larger kradvalue namely krad⫽10 at RF

⫽1 for N⫽50. Symbols represent the simulation result while the full and dashed lines correspond to the prediction of RGF and WF methods, respec- tively.

APPENDIX A: THE REDUCED GREEN’S FUNCTION G₀„R,t円R₀,0…

In the Rouse model, one has the modes X_p␣, 共p

⫽1,2,... and ␣⫽x,y ,z兲 which are the normal modes of the chain. We do not include p⫽0, which is the center-of-mass motion of the chain. These modes have the propagator关see Doi and Edwards, Eqs. 共4.18兲to共4.22兲兴

G₀共X_p␣,t兩X_p⁰_␣,0兲⫽

冋

^{2␲kkpBT共1⫺exp共⫺2t␭p兲兲}

册

^⫺1/2

⫻exp

冋

^{⫺kp2k共XBpT␣共⫺1X⫺0pexp␣exp共⫺共⫺2tt␭␭pp兲兲兲兲2}

册

关see Eq. 共3.90兲 of Doi and Edwards 共We deviate slightly from their notation.兲兴.

␭p⫽3␲^2kBT

␨^{N2b2 p2⫽3D}

冉

^Nb␲

冊

^2p2,

and

k_p⫽6␲^2kBT Nb² p².

The full propagator is a product over all the p modes. Now R may be expressed in terms of the vectors X_p as

R⫽⫺4

兺

^p⬘Xp.

The prime in兺p⬘ indicates that we need to sum over only the odd values of p. We now evaluate the function G₀(R,t兩R₀,0). By definition,

G₀共R,t兩R₀,0兲

⫽

冕

^dS

冕

^{dS0G0共R,S,t兩R0,S0,0兲Peq共S0兩R0兲}

⫽

冕

^dS

冕

^{dS0G0共R,S,t兩R0,S0,0兲Peq共R0,S0兲/ Peq共R0兲.}

共A1兲 Now consider the numerator of the above equation. It is:

I⫽

冕

^dS

冕

^{dS0G0共R,S,t兩R0,S0,0兲Peq共R0,S0兲.}

This may be written in terms of the normal modes as I共R,R₀兲⫽

兿

^a

兿

^⬘p

冕

^dXp␣␦

冉

^R␣⫹4

兺

^p⬘Xp␣

冊

⫻

冕

^{dXp␣0 ␦}

冉

^R0␣⫹4

兺

^{⬘p Xp␣0}

冊

⫻G₀共X_p␣,t兩X_p⁰_␣,0兲P_eq共X_p⁰_␣兲.

It is easy to evaluate using the integral representation of the Dirac delta function and the G₀(X_p␣,t兩X_p⁰_␣,0) given pre- viously. Then, one gets

I共R,R₀兲⫽

兿

_␣ ^{共1/2␲兲2}

冕

^du␣

冕

^{dv␣exp关iu␣R␣}

⫹iv_␣R_0␣兴

兿

^⬘p

冕

^dXp␣

冕

^{dXp0␣G0共Xp␣,t兩Xp0␣,0兲}

⫻G₀共X_p␣,⬁兩0,0兲exp关iu_␣X_p␣⫹iv_␣X_p⁰_␣兴.

On performing the integration over X_p␣ and X_p⁰_␣, and then over u_␣ andv_␣, we get

I共R,R₀兲⫽e^{⫺共R2⫹R02⫺2R•R0␳共t兲兲/4k共1⫺␳共t兲2兲}

共4␲^k

冑

¹⫺␳共t兲²兲³ , 共A2兲 where

k⫽8k_BT

兺

^{p⬘1/kp⫽Nb2/6⫽L2/6,}

␳共t兲⫽兺⬘p^exp共⫺␭pt兲/k_p 兺p⬘^1/kp

⫽ 8

␲²

兺

^{p⬘exp共⫺␭pt兲/ p2.}

Now the equilibrium distribution P_eq(R₀) may be obtained from the above-mentioned, by integrating over R. Thus,

P_eq共R₀兲⫽

冕

^{dRI共R,R0兲⫽}_共^e2^⫺共R

冑

␲^02兲/4kk兲³. 共A3兲 Using Eqs.共A2兲and共A3兲in Eq.共A1兲gives

G₀共R,t兩R₀,0兲⫽

冉

^{2␲L2共13⫺␳共t兲2兲}

冊

^3/2

⫻exp⫺3共R⫺R₀␳共t兲兲²

2L²共1⫺␳共t兲²兲, 共A4兲 which is the expression used in the paper.

APPENDIX B: LIMITS FOR THE TIME DEPENDENT DIFFUSION,D„t…

By substituting␳^{(t) in Eq.}共27兲we get, D共t兲⫽4D₀

N _l;odd

兺

^{exp共⫺␭lt兲. 共B1兲}

From the above-mentioned expression we can obtain both short-time and long-time limits for D(t) as follows. At t

⫽0, since the summation in the previous equation can be replaced by N/2, we get

D共t⫽0兲⫽2D₀. 共B2兲

On the other hand, in t→⬁ limit, since only the first term in the summation survives, we can write,

lim

t→⬁

D共t兲⫽⫺L²

3 共⫺␭1t兲 共B3兲

by substituting␭1 value, we get, lim

t→⬁

D共t兲⫽␲^2D0

2N . 共B4兲

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