Estimating diffusivity along a reaction coordinate in the high friction limit: Insights onpulse times in laser-induced nucleationBrandon C. Knott, Nathan Duff, Michael F. Doherty, and Baron Peters

Citation: The Journal of Chemical Physics 131, 224112 (2009); doi: 10.1063/1.3268704 View online: http://dx.doi.org/10.1063/1.3268704 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/131/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Correction factors for boundary diffusion in reaction-diffusion master equations J. Chem. Phys. 135, 134109 (2011); 10.1063/1.3634003 Collision limited reaction rates for arbitrarily shaped particles across the entire diffusive Knudsen number range J. Chem. Phys. 135, 054302 (2011); 10.1063/1.3617251 A simulation test of the optical Kerr mechanism for laser-induced nucleation J. Chem. Phys. 134, 154501 (2011); 10.1063/1.3574010 Kinetics of diffusion-limited catalytically activated reactions: An extension of the Wilemski–Fixman approach J. Chem. Phys. 123, 194506 (2005); 10.1063/1.2109967 Diffusion and reaction for a spherical source and sink J. Chem. Phys. 118, 4598 (2003); 10.1063/1.1543937

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Estimating diffusivity along a reaction coordinate in the high friction limit:Insights on pulse times in laser-induced nucleation

Brandon C. Knott,1 Nathan Duff,1 Michael F. Doherty,1 and Baron Peters1,2,a�

1Department of Chemical Engineering, University of California,Santa Barbara, California 93106-5080, USA2Department of Chemistry and Biochemistry, University of California,Santa Barbara, California 93106-5080, USA

�Received 25 August 2009; accepted 6 November 2009; published online 14 December 2009�

In the high friction limit of Kramers’ theory, the diffusion coefficient for motion along the reactioncoordinate is a crucial parameter in determining reaction rates from mean first passage times. TheEinstein relation between mean squared displacement, time, and diffusivity is inaccurate at shorttimes because of ballistic motion and inaccurate at long times because trajectories drift away frommaxima in the potential of mean force. Starting from the Smoluchowski equation for a downwardparabolic barrier, we show how drift induced by the potential of mean force can be included inestimating the diffusivity. A modified relation between mean squared displacement, time, anddiffusivity now also includes a dependence on the barrier curvature. The new relation provides thediffusivity at the top of the barrier from a linear regression that is analogous to the procedurecommonly used with Einstein’s relation. The new approach has particular advantages over previousapproaches when evaluations of the reaction coordinate are costly or when the reaction coordinatecannot be differentiated to compute restraining forces or velocities. We use the new method to studythe dynamics of barrier crossing in a Potts lattice gas model of nucleation from solution. Ouranalysis shows that some current hypotheses about laser-induced nucleation mechanisms lead to anonzero threshold laser pulse duration below which a laser pulse will not affect nucleation. Wetherefore propose experiments that might be used to test these hypotheses. © 2009 AmericanInstitute of Physics. �doi:10.1063/1.3268704�

I. INTRODUCTION

Highly damped barrier crossing dynamics arecharacteristic of nucleation,1 protein folding,2–10 andself-assembly.11–15 In Kramers’ theory16,17 the transmissioncoefficient in the limit of high friction becomes proportionalto a diffusion coefficient along the reaction coordinate.17

This reaction coordinate diffusivity can be an important fac-tor in discriminating between dominant kinetic pathways.18

For example, recent simulations by Sanz et al.19 suggest thatextremely slow dynamics over a low free energy barrier cancause a system to nucleate instead via a higher free energybarrier. These findings emphasize the importance ofdynamics,20,21 as well as the free energy landscape for un-derstanding polymorph selection and Ostwald’s rule ofstages.22,23

Kinetic selectivity is most reliably determined by com-paring the rates of nucleation from the metastable phase toeach more stable phase. In particular, the nucleation rate con-stant to form phase �i� is17,21

k�→i� = D�i����

exp�− �F�n�i���dn�i���

exp��F�n�i���dn�i��−1

,

�1�

where �F�n�i�� is the dimensionless free energy, D�i� is thediffusivity along the reaction coordinate leading to phase �i�,and the integrations are over the reactant well �� � and overthe region near top of the barrier �� �. To properly accountfor dynamics, the reaction coordinate n�i� should be uniquefor each pathway. This paper focuses on an accurate methodto determine the diffusivity near the free energy barrier top.We seek only a scalar diffusivity along a scalar reaction co-ordinate because of recently developed methods that can pro-vide dynamically accurate scalar reaction coordinates.24–28

Estimating the diffusivity from the mean squared displace-ment �MSD� is complicated because the MSD is not a linearfunction of time for motion over a free energy barrier.29 Thetrajectories gradually “accelerate” as they move away fromthe barrier top and this leads to upward curvature in a plot ofMSD versus time. Similarly, choosing a time that is too smallcan lead to nonlinear behavior in the MSD as a function oftime because of transient ballistic behavior.30 Thus both ex-tremes, short times and long times, lead to problems in usingthe Einstein relation to estimate diffusivity along a reaction

a�Author to whom correspondence should be addressed. Electronic mail:baronp@engineering.ucsb.edu.

THE JOURNAL OF CHEMICAL PHYSICS 131, 224112 �2009�

0021-9606/2009/131�22�/224112/9/$25.00 © 2009 American Institute of Physics131, 224112-1

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coordinate. It can therefore be difficult to choose an appro-priate trajectory duration for estimating a reaction coordinatediffusivity from the Einstein relation.29

This paper uses the Smoluchowski equation to accountfor the effects of curvature in the free energy surface on thediffusivity estimate. The final relationship between MSD andtime still requires a simple linear regression to obtain thediffusivity, but the relationship is slightly more complicatedthan the Einstein relation. Anticipating that the Einstein re-lation will continue to be widely used, we derive dimension-less bounds on the times for which Einstein estimates shouldbe accurate. Next, we use the Smoluchowski equation-basedapproach in a Potts lattice gas model of nucleation fromsolution. We then examine the dynamics of diffusion alongthe nucleus size coordinate to gain insight into pulse durationeffects in laser-induced nucleation. Based on our analysis,we propose simple experiments to test previously proposedtheories about the mechanism of laser-induced nucleation.

II. ESTIMATING THE DIFFUSIVITY ALONG AREACTION COORDINATE

Some investigators1 used the Einstein diffusionequation31 ��x�t�2=2Dt, where ��x�t�2 is the MSD alongthe reaction coordinate x�t� from the top of the free energybarrier, to estimate the diffusivity at the top of the reactionbarrier. Im and Roux29 extended this approach to account fordrift on a free energy surface by computing MSDs relative toan average drift displacement,

D =���x�t� − ��x�t��2

2t. �2�

The approach of Im and Roux accounts for the potential ofmean force only if the time is short enough that the force feltby the trajectories remains constant and long enough that thevelocity decorrelation time is exceeded.29 Pan, Sezer, andRoux recently applied a multidimensional generalization ofEq. �2�.32 Equation �2� has also been used in the coarse mo-lecular dynamics approach of Hummer and Kevrekidis.33

Following Berne et al.34 and Woolf and Roux,35

Hummer showed that the local diffusivity can be obtainedfrom an umbrella sampling simulation as36

D =���x�2bias

tx-corr, �3�

where ���x�2bias is the variance in x from the biased averageand tx-corr is the decorrelation time for x during an umbrellasampling simulation.37 Hummer’s approach circumventsproblems in choosing an appropriate trajectory length, butumbrella sampling with the true dynamics requires comput-ing the reaction coordinate and derivatives of the reactioncoordinate at every timestep. For nucleation, such frequentevaluations of the cluster size or its derivatives are not prac-tical. Hummer also showed that numerical discretization ofthe Smoluchowski equation could be used with likelihoodmaximization to estimate diffusivities from state-to-statehopping statistics.36

Ma, Nag, and Dinner38 further extended these ap-proaches starting with the solution for an inertial Langevin

equation for a harmonically restrained reaction coordinate.They showed how the choice of simulation time and dynami-cal memory effects influence the diffusivity estimate in theabove methods. Their analysis also provided a correlationfunction that approaches D in the high friction limit and atlong times.

Here we estimate the diffusivity at the top of an activa-tion barrier in the high friction limit by starting from theSmoluchowski equation for diffusion from the top of adownward parabola. Our approach does not require a re-straining potential, numerical discretization of the reactioncoordinate, or knowledge of the initial velocity along thereaction coordinate. For nucleation, the reaction coordinatemay be an integer size �n�, but the nuclei are often largeenough �compared with the particle size� that n can betreated as a continuous variable. Near the top of the freeenergy barrier, the free energy as a function of cluster size nresembles a downward parabola,

�F�n� = �F�n�� − 12�2�n − n��2, �4�

where n� is the critical nucleus size and −�2 is the curvatureof the free energy barrier at its maximum �i.e., −�2 is thesecond derivative of �F with respect to n, evaluated at n��.In what follows, we assume the dynamics along the reactionbarrier are governed by attachment and detachment ofmonomers.39–41 Treating n as a continuous variable, motionalong the reaction coordinate should follow a Smoluchowskiequation.17 Trajectories launched from the top of the barrierat time zero should then evolve according to

���n,t��t

=�

�n�D

��F�n��n

��n,t� + D���n,t�

�n� , �5�

with the initial condition ��n ,0�=��n−n��, where ��n , t� isthe probability density of the cluster distribution. Note alsothat D has units of inverse time.

Although the diffusivity D will change along the reac-tion coordinate, as long as D remains roughly constant overthe range n=n���−1, approximating D as a constant shouldbe acceptable. Changing to the dimensionless variablesz=��n−n�� and �=�2Dt leads to

���z,����

=�2��z,��

�z2 − z���z,��

�z− ��z,�� , �6�

with the initial condition ��z ,0�=��z�. To our knowledge theGreen’s function solution is not available. However, to esti-mate the diffusivity, one only needs the moment-generatingfunction and, more specifically, the second moment of �.Analogous to the approach of Aris,42 Eq. �6� can be trans-formed into a series of ordinary differential equations for themoments

d

d��z2k��� = 2k�2k − 1��z2�k−1���� + 2k�z2k��� , �7�

where the time-dependent moments are defined as

�zk��� = �−�

�

zk�����z,��dz . �8�

224112-2 Knott et al. J. Chem. Phys. 131, 224112 �2009�

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The zeroth moment is unity for all time and all oddmoments are zero by symmetry. Equation �7� can be used asa recursion relation to obtain a simple ordinary differentialequation for the second moment. This has solution

�z2��� = e2� − 1, �9�

which, to leading order in time, looks exactly like theEinstein equation for diffusion in one dimension.31 However,the higher order terms in the exponential quickly becomeimportant and omission of these can lead to an overestima-tion of the diffusivity computed by a linear approximation to�z2��� versus 2�, as done with the Einstein relation. A betterprocedure for diffusive barrier crossings is to obtain the dif-fusivity from a linear fit of the form

ln���2�n�t� − n��2 + 1� = 2�2Dt . �10�

The error at long times in using the Einstein approxima-tion is �at least� the second term in the expansion of Eq. �10�.Also, Einstein’s result is accurate only when t is large com-pared to the time �−1 on which the velocity autocorrelationfunction for motion along the reaction coordinate decays.30

Note that in our simulations �and likely in any cluster size-based simulation of nucleation� that the ballistic time is veryshort. However, the nonzero ballistic time may be an impor-tant consideration for other systems exhibiting diffusive bar-rier crossings, such as biomolecular isomerizations, e.g., pro-tein folding, allosteric transitions, and reactions in solutionwhere the barrier crossing dynamics might be treated usingthe same analysis.43 This lower limit and the expansion ofEq. �10� give the following bounds for ensuring that theEinstein relation applied at the top of a free energy barriergives accurate results:

�2D�−1 �2Dt 1. �11�

Comparison of Eqs. �11�, �9�, and �4� reveals a general ruleof thumb: trajectories for estimating diffusivities via theEinstein formula should be short enough that they do not falleven a small fraction of 1kT from the barrier top! Finally, wenote that in some applications it may be useful to use theseries of moments from solving Eq. �7� to build aGram–Charlier44 approximation to Green’s function itself.

Appendix 1 shows how the analysis above can be ex-tended to estimate the diffusivity at points away from thebarrier top. However, many applications only require theMSD from the barrier top as a function of time, a relation-ship that is entirely contained in Eq. �10�. It is useful torearrange Eq. �10� to clarify the nonlinear relationship be-tween MSD and time.

2Dt

��n�t� − n��2=

ln���2�n�t� − n��2 + 1���2�n�t� − n��2

. �12�

In the Einstein relation, the right-hand side of this equation isunity. Deviations from this reveal the extent to which the freeenergy barrier’s curvature biases the random walk. This re-lationship is plotted in Fig. 1.

In the examples below we use Eqs. �10� and �12� for twoapplications to nucleation. First, we numerically estimate theattachment frequency �diffusivity along the nucleus size co-ordinate� for a Potts lattice gas model of nucleation from

solution. Then we use the curvature at the barrier top,Eq. �10�, and consequences of proposed mechanisms forlaser-induced nucleation to predict from these mechanismshow variations in laser pulse times should affect nucleationenhancement.

A. Example I: Estimating the diffusivity„attachment frequency… in nucleation

A Potts lattice gas model45 was utilized to study nucle-ation from solution. This model simultaneously includes thesolubility and the first-order melting-freezing phase transi-tion of a solute. In this model, as with molecules in a crystal,the intermolecular interactions depend on the orientations ofthe nearest-neighbor solute particles. Each solute and solventcan take Q orientations, and solidlike pairs are defined asneighboring solutes with the same orientation. The particlesin this model interact according to the followingHamiltonian:

H = − �ij

��vi=vj=1��K −A

Q� + �si=sj

A�+ �vi=vj=0��K� −

A�

Q� + �si=sj

A��� . �13�

The summation labeled �ij extends over all nearest neigh-bors. The parameter vi is the occupation number of a latticesite; the model includes two species corresponding to vi=1�solute� and vi=0 �solvent�. The parameters K and K� controlthe solubility of the two species in each other, while theparameters A and A� control the melting-freezing transitionfor the pure solute and pure solvent. The variable si can takeon any discrete value from one to Q and specifies orientation.A representative free energy barrier for this system is shownin Fig. 2. The free energy is defined as46–48

�F�n� = − ln� �N�n��N�1�� , �14�

where �N�n� represents the average number of nuclei of sizen in the system. A cluster in this system is defined as acontinuous arrangement of nearest-neighbor lattice sites oc-cupied by solute molecules with identical orientation. Figure2 is computed by using the method of equilibrium pathsampling49,50 with the cluster size as the reaction coordinate.The simulations described in this paper examine the limitingcase of a structureless solvent, which is A�=0, along with the

FIG. 1. The relationship expressed by Eq. �12� showing the extent to whichthe barrier’s curvature biases the random walk. Both axes are dimensionless.

224112-3 Estimating diffusivity at transition state J. Chem. Phys. 131, 224112 �2009�

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following parameter values: Q=6, A=1.275kT, and K=K�=1.0kT.

Systems governed by the Hamiltonian in Eq. �13� arecapable of producing diverse and interesting phase behavior.Figure 3�a� shows the phase diagram for the parameter val-ues reported in this paper.51 Figure 3�b� shows that for ap-propriate choices of the Hamiltonian parameters, this systemcan exhibit a single eutectic, which is characteristic of crystalnucleation from solution of small molecules.52

An ideal off-lattice simulation of nucleation from solu-tion in a box of volume V would compute the probability togrow clusters of a given size with temperature and chemicalpotentials of solutes and solvents fixed. The number of sol-utes in the box and the number of solvents in the box wouldthen fluctuate during the sampling. Solutes would comprisemore of the total composition as the nucleus size approachedthe size of the simulation box. Such off-lattice simulationsare not currently possible because particle insertion is diffi-cult for real molecules in an explicit solvent. Another idealoption would be to fix the pressure, the temperature, and thetotal number of molecules, but to let the composition vary by

changing solvents into solutes and solutes into solvents. Thisprocedure gives the isobaric semigrand ensemble. Impressiveprogress has been made using this ensemble for simulationswith fluctuating composition in the condensed phase.53–56

Additional work is needed to move from spherical particlesand chain-of-bead molecules to real molecules, but semi-grand calculations provide a promising way to control super-saturation in simulation of nucleation from solution.

We use Monte Carlo simulations in the semigrand ca-nonical ensemble54,57,58 for the Potts lattice gas model. Thisensemble maintains fixed volume and temperature, as well asa fixed fugacity ratio between solutes and solvents, i.e., afixed chemical potential difference. Thus, the composition ofthe mixture may fluctuate while holding constant the totalnumber of particles. Three types of Monte Carlo moves areattempted in these simulations: particle translation �nearest-neighbor swaps�, particle reorientation �changing si for a par-ticle�, and particle identity exchange �solute to solvent andvice versa�. The acceptance probability for a particle transla-tion or reorientation is the Metropolis condition.59 The ac-ceptance probability for a solvent to solute particle identityexchange is given by

Pacc�vi = 0 → vi = 1� = min�1,f1

f0exp�− �Hvi=1 − Hvi=0�/kT�� ,

�15�

where f1 / f0 is the solute to solvent fugacity ratio and adjuststhe supersaturation.57

The accuracy of cluster size as reaction coordinate forthe Potts lattice gas system presented here is evidenced byFig. 4, a histogram of estimated committor probabilities�pB�x�, where x represents an atomic configuration�. Thepeak centered at pB=1 /2 indicates that the cluster size is agood reaction coordinate.25,26 It is possible that a more accu-rate reaction coordinate could be obtained, but for thepresent work this size coordinate is sufficiently accurate.

Diffusion about the top of the free energy barrier hasbeen studied in this Potts lattice gas system. Figure 5 showsa comparison of the Einstein relation �with both axes multi-plied by the curvature� to the result derived above �Eq. �10��from the Smoluchowski equation �note that �n�t�=n�t�−n��.Figure 5 shows these results out to a time where the nuclei

FIG. 2. Free energy profile for the Potts lattice gas system as a function ofthe reaction coordinate, cluster size. Overlaid on this profile is a snapshot ofa critical nucleus in this system. The different colors of the spheres representdifferent orientations.

FIG. 3. Phase diagrams of the Potts lattice gas model �a� with the parametervalues used in the simulations reported in this paper: Q=6, A=1.275kT,K=K�=1.0kT, and A�=0 and �b� at conditions of Q=6, A=A�=1.275kT,and K=K�=A /Q, a system exhibiting a single eutectic point. X is the vol-ume fraction of solute. Also shown is a liquid-solid tie line.

FIG. 4. Histogram of pB estimates for the putative transition state withcluster size as reaction coordinate. The sharp peak at pB=1 /2 is evidence ofthe accuracy of this choice for reaction coordinate. The overlaid curve rep-resents the intrinsic pB distribution.

224112-4 Knott et al. J. Chem. Phys. 131, 224112 �2009�

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have fallen, on average, about 1kT down from the peak ofthe barrier. The plot using Eq. �10� remains linear for muchlonger times, thus enabling a linear regression for the diffu-sivity that is less sensitive to the trajectory duration. The nextsection considers one situation where detailed knowledge ofthe MSD along the reaction coordinate as a function of timemay be useful.

B. Example II: Pulse length effects in laser-inducednucleation

Recent experiments show that nanosecond laser pulsescan induce nucleation in supersaturated solutions ofglycine,60,61 urea,62,63 lysosyme,64 L-histidine,65 and potas-sium chloride.66 Some evidence suggest the phenomenondoes not involve a photochemical transition.62,67 For ex-ample, the effect seems to be intensity dependent but seemsto have little wavelength dependence.63 Furthermore, fordipolar molecules like urea and glycine, there is apparentpolarization dependence in certain supersaturationwindows.61,63 Several hypotheses have been proposed to ex-plain these results. The optical Kerr effect may cause mol-ecules to align with their most polarizable axis along theoscillating field direction.60,62 Another hypothesis66 is thatthe oscillating field stabilizes the growing nucleus �whichhas a different dielectric constant than the surrounding fluid�according to the model of Isard.68 For both scenarios thecritical nucleus size is reduced during the brief laser pulse, sothe pulse duration must be long enough for critical nucleiformed during the laser pulse to reach a postcritical size bythe time the pulse ends.

In some cases, free energy barriers to nucleation quali-tatively resemble the form proposed by the classical nucle-ation theory �CNT�.1,47,48,51,53,69 CNT proposes that the freeenergy change to form a cluster of size n is a competitionbetween a free energy increase for forming a surface and areduction in free energy for forming the more stable phase,41

F�n� = 32n2/3� + n� , �16�

where is the specific surface free energy of the nucleus, �is a shape factor, and � is the chemical potential difference

between the solid and the solution �� �0 for a supersatu-rated solution�. Note that the addition of the factor of 3/2 isbalanced by a factor of 2/3 that is absorbed by the shapefactor; this is done for notational convenience in the equa-tions that follow. Within the dielectric stabilization model,66

the free energy as a function of nucleus size n and laserintensity I has been modeled as a perturbation to CNT. Thisgives a modified free energy for a nucleus according to66

F�n,aI� = 32n2/3� + n�� − aI� , �17�

where the parameter a modulates the effect of the field in-tensity on the driving force for nucleation �aI�0�. Solvingfor the critical nucleus size as a function of aI gives

n��aI� = ���3/�aI − � �3, �18�

and expanding the barrier to second order around the criticalsize gives

F�n,aI� ���3

2�aI − � �2 −�aI − � �4

6���3 �n − nI��2, �19�

where nI� represents the critical nucleus size with the laser

field on. Figure 6 shows a parabolic approximation to thedimensionless free energy barrier at laser intensity I alongwith the free energy barrier without a laser.

The ratio of rate constants when the laser is on and whenthe laser is off is

kI

k0=

DI��dn exp��F�n,0��D0��dn exp��F�n,aI��

exp�����3

2�� �2 −����3

2�aI − � �2� . �20�

The last approximation makes a quadratic approximation tothe free energies near the barrier top and assumes that diffu-sivity along the reaction coordinate �attachment frequency�scales with n2/3 �nucleus surface area�. If the diffusivity is aconstant instead, the ratio of curvatures will give a smallcorrection of the order of one. The correction will have noeffect on the following analysis.

In experiments, the laser pulse is only on for a timescaleof nanoseconds. During this time, nucleation would happenat an accelerated rate, but critical nuclei would form at thetop of the reduced barrier in Fig. 6, size nI

�, rather than at thefield-off critical size n0

�. For slow reaction coordinate diffu-

FIG. 5. A comparison of actual simulation data showing the Einstein rela-tion and the relationship given by Eq. �10� �with best fit line�. Note that�n�t�=n�t�−n�. For the Einstein relation, the vertical axis is related to thefree energy after a time t as measured from the top of the barrier �by Eq.�4��.

FIG. 6. A schematic illustrating the dimensionless free energy barrier reduc-tion due to the applied field. A free energy barrier of 21kT with n0

�=41 isreduced to a barrier of 16kT with critical size shifted to nI

�=29 due to theexternal field �both curves computed according to Eq. �17��. The red curve isa quadratic approximation to the top of the reduced free energy barrier.

224112-5 Estimating diffusivity at transition state J. Chem. Phys. 131, 224112 �2009�

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sivities and short pulse times, the laser pulse may turn offagain before a laser-induced nucleus has time to diffuse�i.e., grow� beyond the second barrier, the critical size n0

� inFig. 6. When the laser pulse turns off, the free energy land-scape immediately reverts to the higher barrier, thus restoringthe original free energy landscape. For this reason, interpre-tation of experiments with extremely short laser pulses mayrequire additional considerations involving the nucleus at-tachment frequency.

Consider a pulse of time T. Nuclei appear at the top ofthe reduced barrier at a rate 2kI during the pulse �kI is thelaser-on nucleation rate�. The nuclei then begin to diffusefrom the barrier top. The effective rate constant involves anintegral over the dynamics while the pulse is on, followed bya formal dynamical projection to the nucleus size after a longcommitment time �infinity for the parabolic barrier�. Onlythose nuclei that have time to cross the second barrier andthen remain on the product side of that barrier long after thelaser is switched off will contribute to the effective rate keff,

keff�I,T� =1

T�

0

T

dt2kI�−�

�

dnH�n − n0��P�n,��n�,T�

��−�

�

dn��I�n�,T − t� . �21�

Here H�x� is the Heaviside function and �I�n , t� is theGreen’s function for the distribution of nuclei with the laseron for initial condition �I�n ,0�=��n−nI

��. To a good ap-proximation, H�n−n0

��P�n ,� �n� ,T�=H�n�−n0��, i.e., nuclei

will tend to grow if they are on the product side of n0� at time

T, and they will shrink if they are on the reactant side of n0�

at time T. Thus, the integral can be approximately simplifiedto

keff�I,T�kI

2

T�

0

T

dt�−�

�

dnH�n − n0���I�n,T − t� . �22�

The effective rate constant in Eq. �21� has been normalizedhere by kI to obtain a correction factor that will range fromzero to unity depending on the pulse time T, the diffusivityD, and the curvature of the reduced free energy barrier �I

2 atthe transition state.

The integral in Eq. �22� can be computed from a simu-lation, but first we seek a simple estimate. Equation �10�predicts that in a time �T− t�, the nuclei have grown by atypical amount

��n�T − t� − nI��21/2 =

�exp�2�I2D�T − t�� − 1

�I. �23�

If half of the nuclei grow at this rate, and the other halfshrink at this rate, then the effective rate enhancement due tothe short laser pulse is

keff�aI,T�kI

1

DT�

0

DT

d�H��exp�2��I2� − 1

�I− �n0

� − nI��� ,

�24�

where we have converted to dimensionless time by scalingby the diffusivity D. Note that, in terms of CNT parameters,�I

2=��aI−� �4 /3���3.Figure 7�a� shows Eq. �24� plotted for different values of

the laser field intensity, and Fig. 7�b� shows actual simulationresults for the effective rate enhancement as a function ofpulse length. One significant feature of these plots is that thepulse length required to produce significant rate enhance-ment increases with increasing intensity of the applied field.This is due to the fact that as the field intensity is increased,the distance between the two critical nucleus sizes is alsoincreased, thus increasing the distance the nucleus must dif-fuse on the free energy barrier before the laser pulse ends.Note that the trend seen in Fig. 7�a� with increasing fieldstrength holds for critical nucleus sizes of O�102� or larger,i.e., in the range of interest for most nucleation problems. Atvery small critical nucleus sizes, e.g., O�100�, this trend canreverse, requiring longer pulse times at weaker fieldstrengths. The simulation results presented in Fig. 7�b� shownumerical calculations of the keff /kI according to Eq. �22�.Within the proposed mechanisms for laser-induced nucle-ation examined here, the free energy barrier to nucleation isreduced while the laser pulse is on. This is reproduced in oursimulations by increasing the fugacity ratio, which is relatedto the chemical potential difference by a Boltzmann factor,thus increasing the driving force and reducing the free en-ergy barrier to nucleation. The field-off case representsf1 / f0=0.8187 �corresponding to n0

�=111�, and two field-oncases shown in Fig. 7�b� have f1 / f0=0.8437 �correspondingto n0

�=102 and �I2=0.0041� and f1 / f0=1.25 �corresponding

to n0�=42 and �I

2=0.017�. The values for these parameters�the difference between the field-on and field-off critical

FIG. 7. A comparison of �a� the relationship given by Eq. �24� and �b�simulation results for the Potts lattice gas model using Eq. �22�.

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sizes and the curvatures of the reduced barrier� are the sameas those used to produce the curves shown in Fig. 7�a�.

The minimum dimensionless pulse length to see an ef-fect of the applied laser is only a function of two parameters:the curvature at the peak of the reduced barrier and the dif-ference between the critical nucleus sizes of the field-on andthe field-off cases. The nucleation rate enhancement due tothe laser will become significant when the terms within theHeaviside function of Eq. �24� are approximately equal. Thisfinal approximation leads to a minimum pulse length �DT�min

required for rate enhancement, a result that could have beendirectly obtained from Eq. �12�,

2�DT�min

�n0� − nI

��2 =ln��I

2�n0� − nI

��2 + 1��I

2�n0� − nI

��2 . �25�

This relationship can be used to test a number of theories inwhich the laser perturbs the driving force for nucleation, e.g.,the optical Kerr hypothesis60,62,63 and the dielectric-stabilization hypothesis.66 In these theories the free energybarrier is reduced only while the laser pulse is on, thus in-creasing the rate of homogeneous nucleation. Because suchmechanisms are purported to increase the driving force fornucleation, the size of the critical nucleus should also bereduced while the laser pulse is on. The key observation inrelating these predictions to experiments where D and �I areunavailable is the expression for keff /kI as given in Eq. �24�.For fixed intensity, the curvature of the reduced barrier andthe difference in critical nucleus sizes are fixed, and thuskeff /kI should depend only on the duration of the laser pulses.Thus if the previously proposed theories are correct, then anexperiment in which the laser intensity is held constant, butthe duration of the pulses is varied over several orders ofmagnitude, should reveal a threshold pulse duration fornucleation rate enhancement as seen in Fig. 7. In such anexperiment, the intensity of each laser pulse should be con-stant, but as the laser pulse duration is reduced, the numberof pulses should be increased to maintain the total exposuretime. Figure 8 shows a schematic illustrating the ideal ex-periment.

Note that a threshold pulse duration emerges as a conse-quence of the optical Kerr mechanism60,62 and other mecha-nistic hypotheses66 that invoke a laser-induced reduction inthe free energy barrier. In fact, the primary conclusion from

our dynamical analysis is that the dynamical effects of shortlaser pulses should arise from any hypothesis that involvesthe combination of these three essential features:

�1� A lowered free energy barrier to nucleation while thelaser is on, with an immediately restored free energybarrier when the laser is turned off again �i.e., when thelaser pulse ends�.

�2� A laser-induced shift of the critical nucleus location inphase space to sizes or structures that are precritical inthe absence of the laser.

�3� A finite mobility at the bottleneck along the nucleationpathway.

Under some conditions, the rate-determining step ofnucleation is the crystallization of an amorphous “droplet,”as in two-step nucleation theory.51,70–72 The optical Kerr ef-fect explanation for laser-induced nucleation proposes thatenhanced alignment lowers the entropic contribution to thefree energy of activation, accelerating the rate-limiting orga-nizational step of nucleation.62,67 Our analysis above treatsspecific free energy barrier shapes motivated by classicalnucleation theory, but qualitatively similar conclusionswould be reached for an analysis of two-step nucleation.Namely, that the laser pulse duration must be long enough toreach a degree of structural order that will remain postcriticalonce the laser pulse ends and the unperturbed barrier is re-stored. Both scenarios involve a finite mobility at the nucle-ation pathway bottleneck, whether from a limited rate of ac-cretion of monomers as in classical nucleation theory, orfrom the rate of structural ordering of an amorphous nucleusas in two-step nucleation theory.

III. CONCLUSIONS

Starting from the Smoluchowski equation, we derive arelationship between elapsed time and the mean squared dis-placement along the reaction coordinate for trajectories thatoriginate at the barrier top. The relation remains linear farlonger than the Einstein relationship because it includes theeffects of downward curvature of the free energy barrier. Thenew relation provides a more accurate method for calculatingdiffusivity along the reaction coordinate, an important com-ponent of the rate constant in the high friction limit of Kram-ers’ theory. Because the transmission coefficient can be com-puted from the diffusivity along the reaction coordinate, thisapproach also provides a simple alternative for computingtransmission coefficients in diffusive barrier crossings.73,74

We demonstrated the usefulness of this new relationship insimulations of nucleation from solution with a Potts latticegas model.

The relationship between mean squared displacementand time was also applied to gain insight into the previouslylittle-explored effects of pulse duration in laser-inducednucleation. Within the context of mechanisms that have beenproposed in the literature, we derive an expression for thedependence of the effective nucleation rate enhancement onthe laser pulse duration. Our analysis shows that the currenttheories predict rate enhancement will be sensitive to pulseduration for very short laser pulses even when the total laser

FIG. 8. A schematic illustrating a proposed experiment in which the laserpulse duration is varied while keeping laser intensity and total exposure timeconstant. Other variables including wavelength and laser beam cross-sectional area should also be held constant. The time period between laserpulses is orders of magnitude longer than a laser pulse; this is not drawn toscale.

224112-7 Estimating diffusivity at transition state J. Chem. Phys. 131, 224112 �2009�

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exposure time is held constant by changing the duration andnumber of pulses. The dynamical effect emerges because ofa finite mobility along the nucleus size coordinate �or simi-larly along a structure coordinate in the case of two-stepnucleation�. This finite mobility implies that the laser pulsemust persist sufficiently long for a laser-induced criticalnucleus to grow to a nucleus size �or degree of crystallinestructure� that will remain postcritical when the laser pulseends and the original barrier is restored. Our predictions areconsequences of the optical Kerr hypothesis and of generaldynamical considerations about finite barrier crossing timesthat should apply whether nucleation occurs in one or twosteps.

If current theories like the optical Kerr hypothesis arecorrect, the effect of laser pulse duration with constant totalexposure time, intensity, and wavelength should show athreshold laser pulse duration. Based on our findings, weproposed experiments to test hypothesized mechanisms inthe literature for laser-induced nucleation.

ACKNOWLEDGMENTS

The authors thank Gregg T. Beckham, Todd M. Squires,and Horia Metiu for many discussions and helpful sugges-tions. The authors are especially grateful to M. Scott Shelland David A. Kofke for their assistance with orientationalstates in semigrand Monte Carlo. The authors also thankAaron Dinner for a critical reading of the manuscript. Thiswork was partially supported by the National Science Foun-dation under Grant No. CBET-0651711.

APPENDIX: MEAN SQUARED DISPLACEMENT ON ASLOPED OR CURVED POTENTIAL OF MEANFORCE

For points initially displaced from the top of the barrier,but where the local curvature in a parabolic approximation tothe free energy landscape is still downward, the previousanalysis is trivially extended to yield

ln� �z2��� + 1

�z2�0� + 1� = 2� , �A1�

where, as before, z=��n−n�� and �=�2Dt. Here the vari-ance at time zero is not necessarily zero. Also note that themean displacement from the barrier top grows with time as

ln� �z����z�0�� = � . �A2�

For the parabolic free energy surface, Expression �A2� canbe rearranged to show that it is equivalent to the familiarrelation between drift velocity of the mean and the meanforce,

��n�t

= − D� ��F

�n�

�n. �A3�

Expressions �A1� and �A2� both reduce the diffusivity calcu-lation to linear regression for the evolution of displacementsfrom the top of a barrier.

Finally, we note that our analysis can also be used tostudy the evolution of small displacements from a minimum.One only needs to change the sign of the last two terms inEq. �6� and repeat the analysis from there, because near aminimum the free energy locally looks like �F�n�=�F�nmin�+ �1 /2��2�n−nmin�2.

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