Chemical Reaction Dynamics within Anisotropic Solvents in Time-Dependent Fields

0.29 MB·English

by  Eli Hershkovits

Chemical Reaction Dynamics within Anisotropic Solvents in Time-Dependent Fields
Eli Hershkovits and Rigoberto Hernandez
∗
Center for Computational Molecular Science and Technology
School of Chemistry and Biochemistry
Georgia Institute of Technology
5
Atlanta, GA 30332-0400†
0
(Dated: February 2, 2008)
0
2 Thedynamicsoflow-dimensionalBrownianparticlescoupledtotime-dependentdrivenanisotropic

n heavy particles (mesogens) in a uniform bath (solvent) have been described through the use of a
a variant of the stochastic Langevin equation. The rotational motion of the mesogens is assumed
J to follow the motion of an external driving field in the linear response limit. Reaction dynamics

8 have also been probed using a two-state model for the Brownian particles. Analytical expressions
1 for diffusion and reaction rates have been developed and are found to be in good agreement with
numericalcalculations. When theexternalfield drivingthe mesogens is heldat constant rotational

] frequency,themodelforreaction dynamicspredictsthat theapplied fieldfrequencycanbeusedto
h
control theproduct composition.
c
e
m
I. INTRODUCTION presenceofatime-dependentdrivenmesogen.13 Another
- step toward this goal is the analytic and/or numerical
t
a solution of such. In the present work, the rigorous con-
The stochastic or Brownian motion of a particle in
t
s a uniform solvent is generally well-understood.1,2 The struction necessary for the first of these steps is not at-
. tempted. Instead, a naive phenomenological model de-
t dynamics is less clear when the solvents respond in a
a scribing the dynamics in lyotropic liquids has been con-
m non-uniform or time-dependent manner, although such
structed. Itservesasabenchmarkforthedevelopmentof
problems are not uncommon. For example, the dy-
- techniquesusefulinanalyzingthedynamicsofBrownian
d namical properties of a suspension in a liquid crystal
particles dissipated by an anisotropic solvent through a
n can be projected onto an anisotropic stochastic equa-
o tion of motion.3,4 Other examples may include diffu- time-dependent friction. In particular, the lyotropic liq-
c sion and reaction in supercritical liquids,5 liquids next uid is assumed to be nematic, i.e., the (calamitic) meso-
[ totheliquidvaporcriticalpoint6,7,8 andgrowthinliving gensareassumedtoberod-likeasisthecasewithmineral
1 polymerization.9 moieties in water14. The mesogens are further assumed
v The flow properties of liquid crystals have gener- tobeone-dimensionalandrigid,andaseriesofadditional
4 simplifying assumptions have been invoked. A physical
ally been analyzed from the perspective of macroscopic
3 system rigorously satisfying all these assumptions may
nematohydrodynamics.10 Therein, liquid crystals have
4 not exist, but the benchmark may still exhibit some of
1 been classified according to the presence or absence of
the importantdynamicsthathasbeenseeninrealliquid
0 solvent. Pure liquid crystals containing no solvent are
crystals in the presence of magnetic fields with time and
5 called thermotropic in part because they have exhibited
spaceinstabilities.15 Another steptowardunderstanding
0 strongtemperature-dependentbehavior. Asuspensionof
t/ nematogens (anisotropic molecules) within a simple sol- the dynamics in anisotropic liquids is the rigorous solu-
a tion of a thermotropic (nematic) model in which the di-
vent is known as a lyotropic liquid. The presence of ne-
m lute Brownianparticle diffuses or isomerizes in a solvent
matogens leads to different transport properties within
that consists exclusively of mesogens. It is based on the
- the solvent than would be seen in a pure simple liquid
d possible connection to a rotating nematic liquid system
alone. The additional complexity is a result of the cou-
n previouslyobserved,16,17andontheanalyticunderstand-
o plingbetweenthevelocityfieldandtheaveragedirection
ing of the dynamics in nematic liquids in a few special
c of the nematogens. As a result, the dynamics of a parti-
cases.11,18 For this thermotropic case, we don’t attempt
: cle in the liquid crystal is dissipated by a friction whose
v to develop a microscopic model of the friction and in-
i form is that of a tensor and not a scalar.11 The actual
X stead make assumptions based on the known properties
dragcanbe furthercomplicatedbythepresenceoftopo-
of isotropic liquids.
r logical discontinuities in the liquid.12 To our knowledge,
a
analytic solutions for the diffusion of Brownian particles
inthese generalenvironmentsarenotknown. The situa-
tion for a reactive solute is even less clear as no analytic
formalismhasbeenconstructed. Inthe presentwork,we In general,the complicated microscopic dynamics of a
constructaformalism—thatinsomelimits—fillsinthese subsystemcoupled to a many-dimensionalisotropicheat
gaps. bathcanbeprojectedontoasimplereduced-dimensional
One step toward understanding the dynamics in stochasticequationofmotionintermsofthe variablesof
anisotropic liquids would thus be the development of a the subsystem alone. In the limit when the fluctuations
lyotropic model consisting of a Brownian particle in the in the isotropic bath are uncorrelated, the equation of
2
motion is the Langevin Equation (LE),1 manifested in these models by way of a time-dependent
friction that is externally driven. The diffusion of free
q˙ = p (1a) Brownianparticles dissipated by a time-dependent envi-
p˙ = V′(q) γp+ξ(t). (1b) ronmentis describedinSec. IV. The numericalmethods
− − forcalculatingreactionsratesneededto extendthe solu-
where (q,p) are the position and momenta vectors in tions of these models to include nontrivial potentials of
mass-weighted coordinates (i.e. mass equals one), V(q) meanforcearepresentedinSec.VA. Analyticalapprox-
is the system potential, γ is the frictionand ξ is a Gaus- imationsforotherwise-rigorousrateformulasarederived
sian random force due to the thermal bath fluctuations. and compared to the the numerical results in Sec. VB.
The friction and the random force are connected via the Adiscussionofthevalidityofalloftheseapproachesand
fluctuation dissipation theorem, possible applications concludes the paper in Sec. VI.
2γ
ξ(t )ξ(t ) = δ(t t ), (2)
1 2 1 2
h i β − II. A NAIVE LYOTROPIC MODEL WITH
ROTATING EXTERNAL FIELDS
where the average is taken over all realizations of the
forces at the inverse temperature β[ (k T) 1]. The
≡ B − A naive model describing a particle propagated in an
LEcanrepresentthe genericproblemofthe escaperates
anisotropicsolventismotivatedinthissectioninthecon-
of a thermally activated particle from a metastable well
text of diffusive or reactive dynamics within a lyotropic
when the thermal energy is much lower then the bar-
solvent. The connection between the model and realiz-
rier height.2 The one-dimensional LE has been solved
able lyotropicsolventsis onlya loose one. No attempt is
in the asymptotic limits of weak and strong friction by
madeheretodoarigorousprojectionofthedetailedcom-
Kramers.2 A general solution for weak to intermediate
plex modes of the lyotropic solvent onto the subsystem
friction was found by Melnikov and Meshkov.19 This
dynamics. The lyotropic liquid is assumed to consist of
result was subsequently extended to the entire friction
rod-likemesogens anda uniform isotropicliquid solvent.
range in the turnover theory of Pollak, Grabert and
It is further assumed that there exists a single tagged
Ha¨nngi.20 The reactive rates for a multidimensional LE
motion characterized by an effective coordinate q that
have been obtained exactly in the strong21,22,23,24,25,26
describesthe subsystem—e.g., aprobeparticleorreact-
and weak friction24 limits and approximately in be-
ingpairofparticles—whosedynamicsisofinterest. This
tween these limits through a multidimensional turnover
taggedmotionistakentobeone-dimensionalforsimplic-
theory.27TheLEcanalsodescribethedynamicsofasub-
ity. The effective mass m associated with the tagged
system under an applied external force, and has led to q
subsystem is also assumed to be well separated from the
theobservationofsuchinterestingphenomenaasstochas-
smaller mass of the isotropic liquid, and the larger mass
tic resonance,28,29 resonant activation,30,31 and rectified
ofthe (anisotropic)rod-likemesogens. Consequentlythe
Brownian motion.32,33,34,35
tagged motion can be described as that of a Brownian
When the fluctuations inthe isotropicbathdo notde-
particle at position q experiencing a dissipative environ-
cay quickly in space or in time, the dynamics are known
mentduetotheinteractionswiththeisotropicliquidand
to be described by the the Generalized Langevin Equa-
the mesogens.
tion (GLE).36 The activated rate expression for a parti-
The model is further simplified by assuming that the
cle described by a GLE is also well-known.37,38 Less un-
mesogens of given concentration, c, do not interact with
derstood are the exact rates when the friction dissipates
eachother. Thisideal-soluteassumptioniscertainlyreal-
the subsystem differently at different times in a nonsta-
izedatlowenoughconcentrationsthatthemeanspacing
tionary GLE-like equation.5,9,13,39,40,41 Nonetheless, the
between mesogens is long compared to their effective in-
models developed in this work contain the flavor of this
teractiondistance. (Itwouldbeeasytoachievesuchcon-
nonstationarity in that the LE is driven by an external
centrations even at relatively high concentrations if the
periodic field through the friction rather than through a
interaction potentials are hard-core.) The ideal-solute
direct force on the system. Consequently the result of
mesogens will exhibit no orientational order in the ab-
this study also provide new insight into the dynamics of
sence of external fields.
systems driven out of equilibrium.
In real nematic liquids there are interactions between
The primary aim of the paper is the development of
the mesogens that result from cooperative forces. They,
analytical and numerical techniques to obtain the dif-
as well as boundary effects on the rods, are excluded
fusion and reaction rates of a subsystem dissipated by a
within the model of this work. The orientationof all the
time-dependentdrivenanisotropicsolventinvariouslim-
rods is firmly fixed by a magnetic field (homogeneous
its. A naive model for a nematic lyotropic liquid and its
director field) with inclination θ relative to the y axis:
various underlying assumptions is presentedin Sec. II as
one paradigmatic example for the accuracy of the meth-
H = H sinθ (3a)
ods described in this work. Another model based on x 0
H = H cosθ (3b)
an experimental system of the rotating nematic liquid y 0
is described briefly inSec. III. The anisotropicsolventis H = 0, (3c)
z
3
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friction,
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(cid:0)(cid:1) H
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)θ(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) γ(t)=γ0(πR2+2Rl cosωt), (5)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) | |
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) that is periodic in time. Including the dissipation of the
rotating mesogens will not change this friction, but will
FIG. 1: A Brownian particle with a diameter 2a moves with addafinitetemperaturetothebathduetorotationaldis-
velocity~vinsideamixtureofanisotropicliquidandcalamitic sipation. As long as this amount of heat is much smaller
mesogens. Themesogenshavealengthl ofthesameorderof
then the bath temperature, the friction in Eq. 5 is well-
a,anegligible width andconcentration, c. The mesogens are
definedandcanbeusedasthefrictionentirelydissipating
orientedbyanexternalmagneticfieldH~. Themagnetic field
the Brownian particle.
ischaracterizedbytheangleθrelativeto~v. Underthesecon-
ditions,theBrownianparticlecollideswith(πR2+2Rl|cosθ|)
mesogens per unit time.
III. A NEMATIC MODEL WITH EXTERNAL
ROTATING FIELDS
This strong field assumption —all the mesogens will ori-
ent uniformly in the direction of H~— also ensures that While the naive model described above does capture
there is no angular momentum transfer in collisions be- some of the features of liquid crystal diffusion, it is
tweenthe mesogensanddiffusingBrowniansolutes. The nonetheless too simplistic. Experiments of pure nematic
environmentisclearlyanisotropic,andaBrownianparti- liquids under a rotating magnetic field16,17 can serve to
clediffusingthroughitwouldexperiencedifferentdissipa- illustrate the possibility of solvent responses character-
tive forcesdepending onthe directionofits motion. The ized by time dependent viscosity. In these experiments,
suspended particle is assumed to have a spherical shape the homogeneous director field of a nematic liquid con-
with a radius R. The particle velocity v(t) is restricted finedbetweentwoparallelglassplateswasalignedinthe
to the x direction. The number of collisions per unit plane of the plates by strong magnetic field. The mag-
time between the Brownianparticle and the mesogens is netic field was also rotated at constant velocity within
simply (πR2+2Rl cosθ )vc. This result is illustrated in this plane. For manyofthe experimentalconditions,the
Fig.1. Furtherassu|ming|thateachofthemesogenshasa nematic liquidretaineduniformalignmentbut its homo-
thermaldistributionofvelocitiesandnotingthemassdif- geneous director field followed the magnetic field with a
ference between the mesogens and the environment, the constantphaselag. Finding anexpressionfor the viscos-
friction force onthe Brownianparticle is proportionalto ity in a nematic liquid is far more complicated then for
the number of collisions. This gives a friction coefficient: an isotropicliquid. It has five coefficients10 and depends
ontheorientationofthedirector,thevelocityandtheve-
locity gradient. This problem was only partially solved
γ =γ0(πR2+2Rl cosθ ), (4) for some special cases. One case obtains the effective
| |
viscosity in a suspension of small particles in a nematic
where γ0 is a proportionality constant characteristic of liquid.18 Thekeysimplificationsarethatthesmallparti-
the system. The viscosity of the isotropic liquid leads to clesareassumedtobenotmuchlargerthanthenemato-
additionaldissipationthatismanifestedasanadditional gens and with spherical shape. The friction coefficient
constant term to the overall friction. However, in those has the simple form,
cases when this isotropic friction is dominated by the
friction of Eq. 4, its effect is small and insufficient to f =a(Aδ +Bcos2θ), (6)
i ik
blurthe anisotropyofthe system. Forfurthersimplicity,
therefore,inwhatfollows,theisotropicfrictionduetothe where the expression for constant coefficients A and B
homogeneoussolventwillbe assumedto be zerowithout may be found in Ref. 18. A second case treats the limit
loss of generality as long as the actual isotropic friction in which the chosen particle in a nematic liquid has a
is weak in this sense. largesphericalshape.11 The resulting effective friction is
Theinstantaneousinclinationθ(t)hasalargeinfluence composed of an isotropic term and an anisotropic term
ontheshort-timedynamicsofaparticlewhosemotionis that depends on the angle between the director and the
measured only along an initially chosen x axis. Without particle velocity. The anisotropic expression is a little
4
morecomplicatedthanEq.6,butitsleadingorderterms
also involve sinθ and cosθ. In both of these cases, the 200.0
nematic liquid is assumed to be firmly oriented by a
strong external field and the friction force is taken to ω=0.1
be much larger than the elastic forces in the nematic 150.0
liquid. Thus the naive model described in the previous >
2
section does exhibit both a uniform constant term and 0)^
an anisotropic oscillatory term that are in qualitative— q( 100.0
−
though not quantitative—agreement with more detailed q(t)
models. <(
ω=1.0
50.0
ω=10.0
ω=0.0
IV. FREE BROWNIAN DIFFUSION IN AN
ANISOTROPIC SOLVENT 0.0
0.0 100.0 200.0 300.0
t
The motion of a free Brownian particle in the time-
dependent friction field of Eq. 5 can be described by the FIG. 2: The mean square displacement of a free Brownian
Langevin equation, particle in the naive lyotropic bath model has been obtained
by direct integration and through the use of the analytical
p˙ = γ φ(t)p+ψ(t)ξ(t), (7) expression in Eq. 13 at various frequencies ω of the driving
0
− rotating magnetic field. In the former integration method,
where the time-dependent coefficients, 100,000 trajectories were sufficient to obtain convergence. In
the latter, the average is taken over an ensemble of 100,000
particlesstartingattimet=0withinclinationperpendicular
φ(t) = ψ(t)2 (8a)
to the velocity, and overlays the results of the former within
ψ(t) = a+cos(ωt), (8b) theresolution of the figure.
havebeenchosentodescribetheperiodicbehaviorofthe
naive lyotropic model and the hydrodynamical friction The square mean displacement of the free particle after
terms in pure nematics as simply as possible. The noise time t is the double integral,
is related to the friction by the fluctuation dissipation
relation, (q(t) q(t ))2
0
h − i
t t
ξ(t)ξ(t′) = 2γ0δ(t t′). (9) = dt1dt2hp(t1)p(t2)i (13a)
h i β − Zt0 Zt0
1 t t1
= dt dt e[ γ0 G(t1) G(t2) ]
The strength a of the isotropic term has been chosen to 1 2 − { − }
β
be 1.05 throughout the illustrations in this work to em- (cid:20)Zt0 Zt0
t t
phasize the anisotropic effects, but different physically- + dt dt e[γ0 G(t1) G(t2) ] . (13b)
1 2 { − }
realizable strengths do not lead to different conclusions. Zt0 Zt1 (cid:21)
The solution to the equation of motion 7 is
A similar expression was developed by Drozdov and
p(t) = p exp[ γ G(t)] Tucker5 for the case of fluctuations in the local density
0 0
− of supercritical solvent. The result in Eq. 13 leads to
t
+ dt ψ(t )ξ(t )e γ0 G(t) G(t1) , (10) the diffusivity of the particle. As will be shown below,
1 1 1 − { − }
Zt0 the diffusivity in the time-dependent environment devi-
ates from the linear correlation known to result in the
where p0 and t0 satisfy the initial condition, p0 p(t0), constant friction environment.
≡
and the integrated friction G(t) is defined as In Fig. 2, the mean square displacement of a Brow-
nian particle whose motion is measured only along an
t
G(t)= dt ψ(t)2 . (11) arbitrary one-dimensional axis is plotted as a function
1 of time at various applied frequencies ω. The average
Zt0
behavior of the mean square displacement is linear with
The velocity correlation function is readily calculated to time, as in the constant friction regime, but it also con-
be tainsfluctuations(intime)aroundtheaveragewhosefre-
quency depends on the external field. It is important to
1
p(t )p(t ) = exp[ γ G(t )+G(t ) note that the overall slope of the mean square displace-
1 2 0 1 2
h i β − { ment depends on the frequency ω; that is, the diffusivity
2G(min(t ,t )) ] . (12) shows strong dependence on ω. Hence by changing the
1 2
− }
5
frequency of the external field, it becomes possible to
control the diffusivity of the Brownian particles. 800.0
The analytical result of Eq. 13 was used to check the
accuracy of the numerical integrator employed in prop-
agating particles in a time-periodic white noise bath. A 600.0
>
fourth-orderintegrator was developedbased on the Tay- 2
^
lormethodinRefs.13and42andisoutlinedinAppendix 0))
q(
Ano.nSstuacthioannaraylgporroibthlemmsisinexwthreicmhetlhyeuinsetfeuglraatsioancthimecekcfaonr <(q(t)− 400.0
bicealvreersyullotnugp. tTohteimneewstaelpgsoroifthsimzea,gδrte=es.w5,itihntthheeadnimaleynt-- γβ)1/( 200.0
sionless units of time defined in Eq. 7. In general, the
time step required to achieve a given accuracy decreases
as either the frequency or friction increases. 0.0
0.0 50.0 100.0 150.0 200.0 250.0
Theseresultsarelimitedtodiffusioninonedimension.
t
When studying the motion in the plane defined by the
rotatingmagneticfieldanaveragehastobetakenoverall
FIG.3: Thenormalizedaveragedisplacementofanensemble
the directions. The integrated friction function, Eq. 11,
offreeBrownianparticlesinthepresenceofaperiodicfriction
for a particle with the initial velocity inclinedwith angle
isdisplayedasafunctionoftime. Thedrivingfrequenciesare
ωτ relative to the magnetic field at the time, t=0, is
ω=10,1,0.1,0.2and0. Theresultforthefixedcase(ω=0)
hasbeencalculatedanalytically usingEq.15. Theremaining
2a
G(t+τ) = γ a2t+ sin(ω(t+τ)) results are obtained numerically by averaging over Brownian
0
ω particles with velocities in random inclination relative to the
(cid:20)
magnetic field at theinitial time, t=0. Notethat theslopes
t sin[2ω(t+τ)
+ + . (14) —viz.,thediffusion rate— increase with decreasing ω.
2 4ω]
(cid:21)
Aftersomeelementaryalgebra,theintegrationin13with
twoBrownianparticles. Neglectingthehydrodynamicin-
G as in Eq. 14 for the case of a constant magnetic field
teractionasbefore,thedynamicscanbedescribedbythe
leads to the average diffusion coefficient,
time-dependent Langevin equation,
1 a
D = , (15)
h i0 γ0β(a2−1)3/2 q¨=−∇V(q)−γ0φ(t)q˙+ψ(t)ξ(t), (16)
of a Brownian particle in a plane. The diffusivity of the where q is now a relative mass-weighted coordinate be-
Brownian particle in a rotating field at various frequen- tween the interacting particles, and V is the potential of
cieshasbeenobtainednumericallyandisshowninFigure meanforcebetweentheparticles. Theremainingsymbols
3. As canbe seen,thediffusivity isamonotonicdecreas- are the same as in the previous section. Phenomenologi-
ing function of the frequency. This result suggests the cal rate constants —e.g., transition from one metastable
use of the applied field frequency to controlthe diffusive state of the potential to another or to infinity— cannot
transport of the Brownian particle. becalculatedanalyticallywhenthepotentialisofamore
Inthea=1limitofthismodel,thereisadivergencein complex form than that of the harmonic oscillator. Di-
the averageddiffusion constant over all the directions at rect numerical evaluation of these rates is usually quite
constant magnetic field. This limiting behavior is a con- time consuming because the time scales involved in the
sequenceofatransitionfromdiffusivetoballisticmotion problem are widely varying. The reactive flux method
at the inclination in which θ = π. That is to say, that reduces much of the computation time by initiating the
it is an artifact of the model in so far as the physical trajectoriesatthebarrier.2Ithasbeenusedtoobtainre-
system it represents would never take on the value of active exact thermal escape rates in the stationary limit
a=1, and hence would not exhibit an infinite diffusion! bothnumericallyandexactly,andtoobtainapproximate
Nonetheless, the model above serves to demonstrate the rates under a variety of limiting approximations. In the
the accuracy of the numerical and analytical formalism presentcase,theproblemisnonstationaryatshorttimes
when a is far from 1. but retains an average stationarity at sufficiently long
times. The strategy is consequently to generalize the
rateformulaforstationarysystems. Itmustnowinclude
V. REACTION RATES IN A TIME-PERIODIC processesinwhichstationarityis requiredonly whenthe
FRICTION observables are integrated over a period equal to that of
the external periodic perturbation.
TheassumptionsintroducedinSec.IIarealsoapplica- In all of the calculations performed here to illustrate
bletothedescriptionofthereactiveinteractionsbetween the approach, the potential has been chosen to have the
6
the form of a symmetric quartic potential,
1 t
V(q)=q4 2q2 , (17) λ¯(1) ≡ t t λ(t′)dt′ (19a)
− − 0 Zt0
1 n (t) N/2
a
in which the two minima represent two distinct = ln − . (19b)
−t t N/2
metastable states separated by a dimensionless barrier − 0 (cid:20) (cid:21)
of unit height. (Note that for simplicity, all observables The second equality was introduced by Pollak and
inthisworkarewrittenindimensionlessunitsrelativeto Frishman44 as a construction that can lead to long time
thechoiceofthiseffectivebarrierandtheparticlemass.) stability thereby ensuring a substantial plateau time.43
The reactive rate has been calculated for particles with The instantaneous flux can be found using the differen-
inversetemperatures,β =10,or20. Thesetemperatures tial expression44:
are low enough to give a well-defined phenomenological
1 d
rate when the reactive flux method is employed in the λ(t) = (na nc) (20a)
− n n dt −
constant friction case, but not so low that trajectories a− c
d
are needlessly slow even when one obtains the rate by = ln(n n ). (20b)
a c
direct methods. dt −
The numerical calculation of either of the direct rate
formulasrequiresthedirectintegrationofalargenumber
of trajectories all initiated in the reactant region. Con-
A. Rate Formula and Numerical Methods sequently, it will only be accurate when the numerical
integrator is accurate for times that are sufficiently long
to capture the rate process. This holds at the moderate
The standard approach for calculating reaction rates,
temperatures (near βV = 10) explored in this work for
“the reactive flux method,” assumes stationarity.2,43 In ‡
which Eqs. 19 and 20 lead to the same result. The first
establishing its validity, the rate formula needs to be
method was used in all of the direct calculations in this
checked by comparison with direct methods measuring
work because it tends to be more stable.
the phenomenologicalrates between reactantsand prod-
Thedirectmethodsaretimeconsuminganditisprac-
ucts. In this section, a direct approach for obtaining
ticallyimpossibletoapplythematlowtemperatures. As
rates in the nonstationary cases of interest to this work
was mentioned at the beginning of this section, the typ-
is reviewed and similarly validated. The results of this
ical solution of this problem is the use of the reactive
approachare subsequently used to motivate anaveraged
flux method. It samples only those states that traverse
reactive flux formula appropriate for the nonstationary
thedividingsurface. Forstationarysystems,thereactive
case. flux is2
In the direct approach, one simply calculates the rate
δ[q(0)]q˙(0)θ[q(t)]
of population transfer from the reactant population na k+ = h i , (21)
θ(q)
to the the product population nc. The initial popula- h i
tion is assumed to be thermally distributed entirely at where the characteristic equation θ[q(t)] for a trajectory
the reactantside. The latterassumptionis validbecause is 1 if the particle is in the right well at time t and zero
the Boltzmann distribution is the steady state solution otherwise, and the Dirac δ-function ensures that all the
of the system restricted to the reactantregion(App. B). particles are initially located at the barrier (at x = 0).
Assuming that a simple first-order master equation de- Theanglebracketsrepresenttheaveragingoverthether-
scribes the rate process (App. B), the population in the mal distribution of the initial conditions.
reactant well can be solved directly as, One might naively assume that the rate expression
in Eq. 21 might still hold in the nonstationary case of
n (t) n¯ t time periodic friction, Eq. 8. The direct and reactive-
a a
− =exp dt′λ′(t) , (18) flux rates at different frequencies and different friction
n (t ) n¯ −
a 0 − a (cid:18) Zt0 (cid:19) constants are compared in Table I. The two don’t al-
waysagreeandthedifferencecanbeasmuchasanorder
where the relaxation rate λ(t) = k+ +k− is the sum of of magnitude. This result should not be surprising be-
the forward(k+) and reverse(k−) rates, n¯a is the popu- cause of the nonstationarity of the problem. However,
lation in the left well at equilibrium. At equilibrium, for correlation functions for this system do become station-
a symmetric potential, na(t)= nc(t)=N/2, where N is ary when one averages over the period of the external
thetotalpopulationofBrownianparticles. Inanonequi- perturbation. This suggests that Eq. 21 should be fur-
librium bath, such as is seen in the model described in ther averagedoverthe initialtime during aperiodofthe
Sec.II,thathasoscillatorycomponentswithamaximum external field, yielding the average reactive flux rate,
recurrence time, then a phenomenological rate may still
2π
be obtained by averaging at sufficiently long times com- ω ω δ[x(τ)]x˙(τ)θ[x(t+τ)]
κ¯(t)= dτh i . (22)
pared to the maximum recurrence time. In particular, 2π θ
Z0 h i
7
ω the well. Its value is
Rates at γ =10 0 0.1 1 10
integral method 2 ×10−6 1.17×10−5 1.55×10−5 8.5 ×10−5 δ =γβs, (23)
reactive flux 2 ×10−6 2 ×10−6 2 ×10−6 2.3 ×10−6 where s is the action of a frictionless particle starting
with zero momentum at the barrier and returning back
ω
to the top of the barrier after traversinga periodic orbit
Rates at γ =1 0 0.1 1 10
(the instanton), i.e.,
integral method 1.7×10−5 2.2 ×10−5 2.9 ×10−5 3 ×10−5
reactive flux 1.6×10−5 1.6 ×10−5 1.6 ×10−5 2.5 ×10−5 s= ∞ p2(t)dt= q(∞) pdq . (24)
ω Z−∞ Zq(−∞)
Ratesat γ =0.05 0 0.1 1 10 The resulting rate is
integral method 3.3×10−5 1.55×10−5 1.47×10−5 2.05×10−5
k =k Υ, (25)
reactive flux 3 ×10−5 3.15×10−5 1.76×10−5 1.9 ×10−5 TST
wherek = ω e βV‡ isthetransitionstatetheoryrate
TST 2π −
TABLE I: The integral method of Eq. 19 is compared to the (ωisthefrequencyatthebottomofthereactantwelland
stationary reactive flux method of Eq. 21 in calculating the
V is the barrier height) and the depopulation factor Υ
‡
activated rate across the double-well potential in a rotating
is
fieldoffrequencyω. Allthecalculationsareperformedatthe
samebathtemperaturesuchthatβV‡ =10,andatthreedif- Υ(δ)=exp 1 ∞ ln 1 eδ(λ2+1/4) 1 dλ (.26)
ferentvaluesofγillustrativeofthelow,intermediateandhigh 2π − λ2+1/4
friction limits. Hereand elsewhere, all valuesare reported in (cid:26) Z−∞ h i (cid:27)
The nonstationary analytic rate expression can now
thedimensionless units of Eq. 16.
beobtainedbyanalogytotheformulationoftheaverage
reactive flux rate in which the rate is averaged over the
period of driving term. In particular, the energy loss,
(There is a formal proof in Appendix B.) A comparison
betweenthedirectratesandtheaveragereactivefluxrate
δ(τ)=γβ ∞ ψ(t+τ)2p2(t)dt, (27)
is presented in Table II. The numerics were performed
at a temperature (βV = 10) that is high enough to Z−∞
‡
is now obtained as a function of the possible initial con-
enable direct calculation of the rate within a few hours
figurations of the driving term which are, in turn, pa-
of CPU time on a current workstation. As can be seen
rameterized by the time lag τ relative to the start of an
fromthetable,thereisverygoodagreementbetweenthe
oscillation in the friction of Eq. 8. Trajectories in one
methods. Equation22isthecentralresultofthisarticle,
dimension can be calculated up to a quadrature directly
and represents the fact that the reactive flux method is
from energy conservation,
valid for the case of a time-dependent bath when a proper
averaging is taken over the period of the external field. t
q(t)=q(t )+2 dt E V(q). (28)
Thisresultiscriticalforthenumericalcalculationofrates 0
−
because the direct approaches are cost prohibitive when Zt0
p
the temperature is much smaller then barrier height. In Inthecaseofthedouble-wellpotentialdefinedbyEq.17,
thissectionthereactivefluxmethodhasbeengeneralized the instanton at E = V‡ —viz. the periodic orbit on
to include out-of-equilibrium systems in cases in which theupside-downpotential—canbeobtainedanalytically.
an external force perturbs a bath that is coupled to a The results for time,
reactive system. The resulting thermal flux is defined
1 √2+ 2 q2
only after averagingover the time period of the external t(q)= ln − , (29)
perturbation. Using the non averaged rate expression −2 pq !
wouldleadtoundefinedratesbecausethereactivesystem
and momentum,
is so far out of equilibrium. A detailed explanation can
be found in App. B. p(q) = √2q 2 q2 , (30)
−
as a function of q follow readpily. By substitution into
Eq. 27, the energy loss parameter is obtained directly
B. Analytical Approximations with respect to the time lag τ relative to the start of an
oscillation in the friction of Eq. 8, i.e.,
1. Weak Friction √2
δ(τ) = 2γβ dq a
For the stationary problem, Melnikov and Meshkov19 Z0 (
developed a perturbation technique to find the reactive ω √2+ 2 q2
flux at weak to moderate friction limit.20 The expansion +cos ln − +ωτ
"−2 p2 ! #)
parameter of the method is the energy loss δ that a par-
ticle starting at the barrier experiences while traversing √2q 2 q2 . (31)
× −
p
8
γ
ω .005 .05 .5 1 10
.1 3 ×10−6 1.55×10−5 2.3 ×10−5 2.1 ×10−5 1.15 ×10−5
(2.78×10−6) (1.51×10−5) (2.3 ×10−5) (2.1 ×10−5) (1.18 ×10−5)
.5 2.8 ×10−6 1.43×10−5 2.22×10−5 2.1 ×10−5 1.375×10−5
(2.62×10−6) (1.45×10−5) (2.23×10−5) (2.13×10−5) (1.41 ×10−5)
1.0 3 ×10−6 1.45×10−5 3 ×10−5 2.7 ×10−5 1.45 ×10−5
(2.7 ×10−6) (1.39×10−5) (3.01×10−5) (2.7 ×10−5) (1.47 ×10−5)
5.0 3 ×10−6 1.9 ×10−5 3.25×10−5 2.8 ×10−5 1.2 ×10−5
(2.7 ×10−6) (1.86×10−5) (3.26×10−5) (2.86×10−5) (1.14 ×10−5)
10.0 3 ×10−6 1.92×10−5 3.25×10−5 2.7 ×10−5 8.6 ×10−6
(2.7 ×10−6) (1.92×10−5) (3.35×10−5) (2.78×10−5) (8.87 ×10−6)
TABLE II: The integral method of Eq. 19 is compared to the average reactive flux method of Eq. 22 in calculating the
activatedrateacrossthedouble-wellpotentialinarotatingfieldoffrequencyω andvariousfriction constantsγ. Thepotential
andinversetemperature(βV‡ =10) arethesame asin TableI. Ateachentry,theintegralmethod result iswritten abovethe
more approximate average reactive fluxresult. To aid theeye, thelatter is also signaled by parentheses.
The nonstationary rate formula for a time-periodic driv- ω
ingfrictioncanthusbewrittenastheproductoftheTST Rates at γ =0.005 0.1 1 10
rateanda generalizeddepopulationfactoraveragedover MM (Eq.33) 8.54×10−2 8.54×10−2 8.54×10−2
τ, k¯(t) (Eq. 22) 7.5 ×10−2 7 ×10−2 7.5 ×10−2
1
Υ¯[δ] dτΥ δ(τ) , (32) TABLE III: The transmission coefficients for the escape rate
≡Z0 k across a quartic potential at βV‡ = 10 and γ = .005 ob-
in analogy with Eq. 25. (cid:0) (cid:1) tained using the average reactive flux method of Eq. 22 with
theanalytical Melnikov-Meshkovexpression 33 for δ¯. An en-
The validity of the analytical result of Eq. 32 for the
sembleof100,000 trajectories hasbeenpropagatedineachof
rate can be checked in the low friction regime in which
thereactive flux calculations.
Υ(δ) δ. Taking the average over a period yields the
≈
result,
ω 2π/ω wherethefixedfrictionγ andstochasticforceξ(t)satisfy
Υ¯[δ] δ(τ)dτ the regular fluctuation dissipation relation (Eq. 2) and
≈ 2π
Z0 iωb is the imaginary frequency at the barrier. It was
= 8 a2+ 1 . (33) shown that the reaction rate for this case is36,45
3 2
(cid:18) (cid:19) λ ω
b 0
k = exp βV , (35)
This result is in good agreement with the averaged re- ‡
ω 2π −
b
active flux rate formula of Eq. 22, as shown in Table III
at a low friction value (γ = .005), βV = 10, and var- where ω is the frequency of the reactant well, and iλ
0 ‡ 0 b
ious frequencies. Even within this weak friction regime, is the imaginary eigenvalue of the homogeneous part of
as the friction increases, the approximation leading to Eq.34. Thelatterisrelatedtotheexponentialdivergence
Eq. 33 will break down. The direct evaluation of Eq. 32 in the trajectories near the barrier,
corrects this error, and also leads the rate to depend on
the frequency of the driven friction. q(t) eλbt. (36)
∝
At strong friction in the nonstationary problem, the
2. Strong Friction reaction rate expression is also dominated by the tra-
jectories in the barrier region. Equation 35 can still be
The reactionratein the overdampedregime ofthe LE usedfortherates,thoughnowλ(t)isthetime-dependent
is well known.2 The central idea is that the motion in eigenvalue of the homogeneous part,
phase space is stronglydiffusive in this regime. The rate
q¨+γφ(t)q˙ ω2q =0, (37)
is consequently dominated by the dynamics close to the − b
barrier. At the vicinity of the barrier top, the potential
of the nonstationary stochastic equation of motion. The
can be approximated by an inverted parabolic potential
solution of this equation is not trivial. A possible way
and the LE at the barrier can be written as
to solve the problem is found in Ref. 31. It is easier to
q¨= ω2q γq˙+ξ(t), (34) extract the eigenvalue numerically from the exponential
− b −
9
0.000040 1.0
γ=0.005
γ=0.005 γ=0.05
γ=0.05 γ=0.5
γ=0.5 0.8 γ=1
0.000030 γ=1.0 γ=5
γ=10.0 γ=10
0.6
k 0.000020 κ
0.4
0.000010
0.2
0.000000 0.0
−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0
ln(ω) ln(ω)
FIG.4: Theactivated escaperates k ofparticles in aquartic FIG. 5: The activated escape rates k of particles in a quar-
potential and solvated by an anisotropic time-dependent liq- tic potential and solvated by an anisotropic time-dependent
uidisobtainedasafunctionofthedrivingfrequencyω using liquid is obtained as a function of the driving frequency ω
two numerical methods described in this work. The numer- comparing the reactive flux approach to an analytical result.
ical direct rate of Eq. 19 is shown by dashed lines, and the (Thelatterisexpectedtobeaccuratehere—andnotinFig.4
averaged reactive flux rate of Eq. 22 is shown by solid lines. or Table II—because the inverse temperature has been in-
In the former, an ensemble of 250,000 initial conditions were creased to 20.) The solid line corresponds to the numerical
usedtoachieveconvergence,andthecorrespondingnumerical resultcalculatedusingtheaveragedreactivefluxratemethod
values are summarized in Table III. In the latter, the aver- ofEq.22andthedashedlinecorrespondstotheturnoverfor-
age was performed over an ensemble of 100,000 trajectories, mula in Eq. 39. The numerical calculations were performed
yielding the results in a wall-clock time that was an order of byaveraging over an ensemble of 250,000 trajectories.
magnitudefasterthanthatforthedirectratecalculations. In
all cases, theinverse temperature βV‡ is 10.
γ
ω .005 .05 .5 1 5 10
divergence of trajectories starting near the barrier top,
.1 (.14) (.514) (.64) (.61) (.424) (.325)
q(t) e tλb(t′)dt′ . (38) .13 .51 .62 .56 .381 .32
∝ R .5 (.14) (.52) (.62) (.61) (.432) (.373)
The periodicity of the time dependent coefficient in .13 .474 .625 .595 .44 .366
Eq. 37 leads also to a periodicity in λb(t). If λb is the 1.0 (.14) (.535) (.694) (.67) (.451) (.376)
time average of the time-dependent eigenvalue of Eq. 37
.132 .5 .72 .708 .457 .366
over a period, then for t much larger than the period,
Eq. 38 is analogousEq. 36 with λ¯ in the exponent. Re- 5.0 (.15) (.7) (.825) (.7) (.38) (.283)
b
placement of λ by λ¯ in the rate expression (Eq. 35) .147 .691 .825 .7 .372 .282
b b
provides goodagreementwith the averagedreactive flux 10.0 (.155) (.727) (.82) (.685) (.31) (.21)
rates as shown in the high friction columns of Table IV. .148 .720 .821 .683 .3 .219
TABLE IV: The average reactive flux method of Eq. 22 is
compared to the analytic approximation of Eq. 39 in calcu-
3. Weak to High Friction
lating the activated rate across the quartic potential in a ro-
tating field of frequency ω and various friction constants γ.
The results of the two previous subsections have mo- The inverse temperature (βV‡ =20) is higher in contrast to
tivated the redefinition of the components of the rate theprevioustables. Ateachentry,themoreapproximateav-
formulain the lowandhighfriction limits of the nonsta- erage reactivefluxresult is written abovetheanalyticresult.
tionary time-periodic problem. Retaining these assign- To aid theeye, theformer is also signaled byparentheses.
ments in the stationaryturnoverrate formula20 suggests
the nonstationary turnover rate,
theinversetemperatureβ =20inTableIVandinFig.5.
λ¯ ω Υ¯[δ] 2 As can be seen, there is a very good agreement between
k¯ = b 0 exp βV . (39)
ω 2π Υ¯[2δ] − ‡ the numerical and analytic results at the very weak and
b (cid:0) (cid:1)
strong friction limits. Therein the results differ by no
Theprefactorsfromthenonstationaryturnoverrateare more than 5% throughout the frequency range; an er-
comparedtothosefromtheaveragedreactivefluxrateat ror margin smaller than the error bars in the numerical
10
calculations. At moderate friction and low frequencies, ear behavior could be used to enhance reaction diffusion
however,the differences —on the order of 10%— cannot processes,suchasclusternucleation,byuptoafeworder
beexplainedbyerrorinthenumericalcalculationsalone, of magnitude.
and may be significant. Corrections or improvements in In the extension of the naive model to more realistic
the approximations leading to the connection formula of nematic liquids,the cooperativeeffects ofthe liquidcan-
Eq.39arealsoofinterest,butnotpursuedfurtherinthis not be omitted. There are phenomenological difficulties
work. Recallthattheturnoverescaperateexpressionfor in defining friction and the fluctuation dissipation rela-
the LE with constant friction was obtained through the tioninliquidcrystals. Tothebestofourknowledgesuch
solution of the equivalent Hamiltonian formalism.20 A a theory is still not fully developed. The development
similar approach for the solution of a the Hamiltonian of such a theory based on microscopic assumptions is
equivalent46 ofthestochastictime-dependentbathprob- anextremelychallengingproblem. Boundaryeffects and
lemmayleadto afruitfulsolution. However,eveninthe elastic forces will create dynamical micro-domains char-
constant friction case,the turnover formula can give rise acterizedby differing uniform directorsina realnematic
to small systematic error. With these reservations, the under rotating magnetic field. The theory for dynamics
approximate rate formula can be used to obtain time- in nematics will have to deal also with the spatial inho-
dependent escape rates. It is clear from the results that mogeneities. These are among the challenges to future
there is a frequency effect on the reaction rates. For the workintryingtobetterunderstandthediffusivedynam-
specific example studied here, the effect can modify the ics in lyotropic liquids.
reaction rate
ACKNOWLEDGEMENTS
VI. DISCUSSION AND CONCLUSIONS
We are grateful to Prof. Rina Tannenbaum for stimu-
In this work, several techniques for obtaining the dy- latingdiscussionsandhelpfulsuggestions. Thisworkhas
namicsofinteractingBrownianparticlesthatarecoupled been partially supported by the National Science Foun-
to a time dependent thermal bath have been discussed. dation under Grant, Nos. 97-03372 and 02-13223. RH
Two models, one of dynamics in lyotropic liquids and is a Goizueta Foundation Junior Professor. The Center
onefordynamicsinpurenematicliquidunderaperiodic for Computational Science and Technology is supported
external field has been brought as examples of such sys- througha SharedUniversity Research(SUR) grantfrom
tems. The models include a new mechanism for stochas- IBM and Georgia Tech.
tic dynamics in which an external force is used to drive
the thermal bath. There is no net injection of energy
to the Brownian particles in the bath due to the driv- APPENDIX A: FOURTH-ORDER INTEGRATOR
FOR THE LE WITH PERIODIC FRICTION
ing force, hence they keep their equilibrium properties.
Yet observables such as reaction and diffusion rates are
modified. Theexistenceofasteadystatethatretainsthe A high-order integrator was developed for the regu-
equilibriumenablesonetoexpressout-of-equilibriumob- lar LE or GLE in Ref. 42. A modified algorithm for
servables with respectto averagingover the equilibrium. time- and space-dependent friction was developed for
This is the Onsagerregressionhypothesis (Appendix B). the explicit GLE with exponential memory kernel in
We used this to extend known methods for calculating the friction13. This appendix introduces the numerical
the reaction rates in the constant friction to nonstation- schemenecessaryforsolvingatimedependentstochastic
ary baths. Extensive computation effort was used to il- equation equation of motion of the form of Eq. 7.
lustrate the diffusive and reactive rates for an effective Afinitedifferenceschemeisusedtopropagatethesolu-
Brownian particle in the naive anisotropic liquid bath tion over a small time step. At each iteration, the prop-
model with rotating magnetic field. However, the nu- agator is expanded to fourth order with respect to the
mericalandanalyticaltoolsthathavebeenmodifiedand time step using a strong Taylor scheme.47 The resulting
developed in this work are appropriate for any model integratorcanbe decomposedintotwouncoupledterms:
with time-dependent friction.
The construction introduces new control parameters
into the problem; namely, the external force amplitude q(t+h) = qdet(p,q,t)+qran(p,q,t) (A1a)
andfrequency. We concentratedonthe latterandexhib- p(t+h) = p (p,q,t)+p (p,q,t). (A1b)
det ran
ited the frequency dependence of diffusion and reaction
rates in the naive model. This dependence is not lin- The deterministic terms that are collected within q
det
ear and changes dramatically with the friction strength. and p are those that remain in a fourth-order Taylor
det
The enhancements in the reaction rate and the diffusion expansion of the deterministic equation of motion after
coefficient are not the same, i.e., the maximum in the removing any term that includes a stochastic variable.
diffusion rate as a function of the external frequency is The deterministic propagator can be calculated numer-
not the same as the maximum in the rate. This nonlin- ically with any fourth-order deterministic scheme; the