5.5

ORDER PARAMETER APPROACH TOUNDERSTANDING AND QUANTIFYINGTHE PHYSICO-CHEMICAL BEHAVIOROF COMPLEX SYSTEMS

Ravi Radhakrishnan1 and Bernhardt L. Trout21Department of Chemistry and Courant Institute of Mathematical Sciences,New York University, New York, NY, USA2Department of Chemical Engineering, Massachusetts Institute of Technology,Cambridge, MA, USA

Many physico-chemical processes such as nucleation events in phasetransitions, chemical reactions, conformational changes of biomolecules, andprotein folding are activated processes that involve rare transitions betweenstable or metastable states in the free energy surface. Understanding theunderlying mechanism and computing the rates associated with such processesis a central to many applications.

For instance, the familiar process of nucleation of ice from supercooledwater is encountered in several scientific and technologically relevant pro-cesses. The formation of ice microcrystals in clouds via nucleation is a phe-nomenon that has a large impact in terms of governing global climatic changes.The key to the survival of Antarctic fish and certain species of beetles throughharsh winters is their ability to inhibit nucleation of intracellular ice with theaid of antifreeze proteins. At the other end of the spectrum, certain proteinassemblies called ice-nucleation agents are believed to be responsible for cat-alyzing ice nucleation, a phenomenon, which is exploited by certain bacteriato derive nutrients from their host plants. Controlling the formation and prop-agation of intracellular ice is finding importance in cryopreservation of naturaland biosynthetic tissues.

Similarly, one can cite many technologically relevant self-assembly andtransport processes in the context of fabrication of advanced materials forspecific applications in drug delivery, biosensing, and chemical catalysis.

A unifying feature among various activated events is that they can beunderstood in terms of transitions between a series of stable (global minimum)

1S. Yip (ed.),Handbook of Materials Modeling. Volume I: Methods and Models,1–14.c 2005 Springer. Printed in the Netherlands.

R. Radhakrishnan, B. L. Trout, 2005, “Order parameters in molecular simulations”, Handbook of Materials Modeling, Ed. S. Yip, Chapter 6, Kluwer Academic Publishers, Dordrecht, pp 1613-1626

2 R. Radhakrishnan and B.L. Trout

or metastable (local minima) basins in the free energy landscape separatedby free energy barriers or bottlenecks (also known as transition states). Themetastable states represent high-probability regions and the transition statesrepresent low-probability regions of phase space. For a system consisting ofN atoms, the free energy landscape F = −kBT ln Q is 3N-dimensional andrelated the configurational partition function Q given by,

Q =∫

dr1 dr2 · · · drN exp(−βH(r1, r2, . . ., rN )). (1)

Here, kB is the Boltzmann constant, T is the temperature, r1,r2, . . . , rN arecoordinates of the N atoms in the system, β = 1/kBT , and H is the classicalHamiltonian giving the energy of the system for a given configuration.

An activated process can be described by a set of pathways connecting therelevant metastable states in the free energy landscape (see Fig. 1).

A Monte Carlo path or a molecular dynamics trajectory that captures anactivated process is likely to be representative of a pathway for the transitionin the sense that the trajectory will show many characteristics that are uniqueto the pathway. However, an ensemble of molecular dynamics trajectories (orMonte Carlo paths) connecting the metastable states of the free energy land-scape, rather than a single trajectory or path best describes the mechanismof transition. Consequently, the intermediate states along the pathway can be

1.5

1

H(r

1,r 2

)

0.5

�0.5

�1.5

�280

6040

200 0 10

20 30 40 5060 70

�1

0

A B

r2r1

Figure 1. Model

Q5

Hamiltonian for a system with two degrees of freedom. The stable (A) andmetastable (B) states are shown along with the transition state. Three different paths connectingstates A and B are shown on the contour projection.

Order parameter approach of complex systems 3

characterized by unifying patterns (structural or energetic) that are common tothe ensemble of molecular dynamics trajectories.

In a statistical sense, identifying the dynamical variables to quantify thepatterns and averaging over the different molecular configurations along thetransition pathway can yield insight into the relationship between the evolu-tion of the patterns and the free energy landscape. These dynamical variables,also referred to as order parameters, are quantities that can classify differ-ent metastable states according to their distinguishing characteristics (such assymmetries associated with different phases). In addition, the order parame-ters depend on the nature of intermolecular forces, solvent degrees of freedom,etc., and consequently, the chemical environment will impact their choice. Inthe simple example in Fig. 1, the intermediate states and the transition path-way are easily described in terms of the variables r1 and r2, which serve asgood order parameters.

As we shall see throughout this article, the information about the freeenergy landscape obtained from such an approach is useful in quantifyingthe rate of the activated process while the evolution of the order parame-ters along the pathway is useful in understanding the underlying mechanism.Furthermore, equilibrium properties of the system, and how they depend onthe control variables, can be inferred from the relationship between the orderparameters and the free energy landscape.

Conceptual understanding of the nature of the process, which arises fromthe relationship between the order parameters and the free energy landscape,generally goes by the name of “a phenomenological theory”. More specifi-cally, the phenomenology arises from identifying how the order parameters(and hence the rate of the process) are influenced by the chemical environ-ment, state variables, and control variables, so that the order parameters canthemselves be ascribed physical meaning and interpretation.

1. Relationship between the Order Parametersand the Free Energy Function

The use of order parameters to construct phenomenological theories(top-down approach) of phase transitions in condensed matter and solid-statesystems was pioneered by Landau, Ginzburg, De Gennes, and others. In thisapproach, the order parameter is chosen based on physical grounds andintuition, and the free energy functional (as a function of the order parame-ter) is constructed based on symmetry arguments. The literature in this classof problems is extensive, with applications ranging from superconductivityand superfluidity, magnetic and liquid-gas transitions, and theory of liquidcrystals. A comprehensive treatise on the study of phase transitions using

4 R. Radhakrishnan and B.L. Trout

phenomenological approaches, including universality associated with criticalbehavior and the re-normalization group method is provided in Refs. [1, 2].

Order parameters can also be used to construct the free energy functionalby the coarse graining (bottom-up approach) of the microscopic Hamilto-nian. For complex systems, for which the construction of an analytic freeenergy functional may be nontrivial, the free energy as a function of the orderparameters can be obtained via density functional theory [3], or via molecularsimulations. In this article, we discuss the latter approach.

Starting from a set of n order parameters (φ1, φ2, . . . , φn), the free energydensity, �[φ1, φ2, . . . , φn] (also called the potential of mean force or Landaufree energy), along the order parameters, are related to the microscopicHamiltonian by [1]

�[φ1, φ2, . . . , φn] =∫

dr1 dr2 · · · drN exp(−βH(r1, r2, . . . , rN ))

×δ(φ1 −φ′1)δ(φ2 − φ′

2) · · · (φn − φ′n). (2)

Here, δ is the Dirac delta function, and �[φ1, φ2, . . . , φn] is to be interpretedas �[φ1 = φ′

1,φ2 = φ′2, . . . , φn = φ′

n]. The free energy F is then given by

exp(−βF) =∫

dφ1 dφ2 · · · dφn exp(−β�[φ1, φ2, . . . , φn]). (3)

The domain of integration in the above equation covers the range of orderparameter values characterizing the particular state. In the example given inFig. 1, �[φ1, φ2] ≡�[r1, r2] = exp(−βH(r1, r2)), and the free energy of stateA is obtained by integrating over r1 in the range (0–30), and over r2 in therange (30–60). The value of the free energy is insensitive to the exact val-ues defining the domain of integration, as long as the region containing theminimum of the � function is included.

In the course of a microscopic simulation (such as molecular dynamics orMonte Carlo), the free energy density � is calculable by collecting histogramsof the distribution of the order parameters. If the sampling in the simulationsis ergodic (i.e., encompasses the relevant phase space) and sufficiently long,these histograms are proportional to the joint probability distribution of theorder parameters, P[φ1, φ2, . . . , φn]. The free energy density is related to theP[φ1, φ2, . . . , φn] by

β�[φ1, φ2, . . . , φn] = −ln(P[φ1, φ2, . . . , φn]) + Constant. (4)

In order to circumvent the problem associated with ergodicity, the his-tograms are evaluated in separate windows of the order parameter rangesusing the procedure of umbrella sampling [4]. The umbrella sampling can beunderstood as performing the simulations in an extended ensemble whose free

Order parameter approach of complex systems 5

energy � is related to the original thermodynamic ensemble (with a freeenergy F) by

� = F − ∑i

hiφi , (5)

where∑

i hiφi is chosen a priori as a weighting function W{φ1, φ2, . . . , φn}.The umbrella sampling scheme amounts to simulations being performedusing a modified Hamiltonian H ′ = H +βW{φ1, φ2, . . . , φn}). The probabil-ity distribution P ′[φ1, φ2, . . . , φn] in the modified ensemble defined by � isthe related to P[φ1, φ2, . . . , φn] by

−kBT ln(P ′[φ1, φ2, . . . , φn]) = −kBT ln(P[φ1, φ2, . . . , φn])

−W{φ1, φ2, . . . , φn}. (6)

In the simplest case, choosing W{φ1, φ2, . . . , φn} = 0 for φ1,min < φ1 <φ1,max,and ∞ otherwise, enables the calculation of P[φ1, φ2, . . . , φn] in the rangeφ1,min < φ1 <φ1,max. Performing such calculations over several windows cov-ering the entire range of φ1 (of relevance) enables an accurate calculation ofP[φ1, φ2, . . . , φn]. In addition, enhanced sampling methods such as configura-tional bias sampling [5], parallel tempering [6], density of states Monte Carlo[7], and methods based on Tsallis statistics [8] can be used to improve theaccuracy of the calculations.

2. Types of Order Parameters

Order parameters have been extensively used in conjunction with molec-ular simulations in applications involving solid and liquid-crystalline (LC)phases, in which one of the phases is characterized by long-range order. Insuch cases, the order parameter can be chosen on the basis of the symmetry ofthe ordered phase. A few examples are given in Fig. 2. For phases with long-range order (i.e., 〈φ(0)φ(r →∞)〉 = nonzero constant), the order parameterassumes a nonzero value. For disordered phases (i.e., 〈φ(0)φ(r)〉∼ exp(−r/λ),λ being the correlation length), the order parameter is zero for an infinite sys-tem. For phases with quasi-long-range order (i.e., 〈φ(0)φ(r)〉∼ r−η), the orderparameter in a finite system assumes a value intermediate between the disor-dered and ordered phases, with system size dependence characterized by theexponent η.

The Mermin order parameter [9] is introduced to quantify order in a two-dimensional crystal of circular disks, where the only close-packing possible ishexagonal, i.e., leading to a triangular lattice. More generally, for an “N-atic”order in two-dimensional systems, the pair correlation function g(r) ≡ g(r, θ)in a cylindrical coordinate system (r ≡ reiθ ) can be expressed in terms of aFourier series in angular (θ) space

6 R. Radhakrishnan and B.L. Trout

Figure 2. Order

Q5

parameters describing bond-orientational order in condensed phases.

g(r, θ) =∑

j

g j (r) exp(i N jθ), (7)

where the summation over j runs from 0 to ∞. The coefficients of expansion(i.e., the g j (r)s) are suitable order parameters. For a hexatic (6-fold) symme-try, the dominant order parameter is given by the first term in the expansion

Order parameter approach of complex systems 7

evaluated at the nearest neighbor distance, i.e., g1(r = rnn), which is the same asthe Mermin order parameter 〈6(r)〉 in Fig. 2. The Steinhardt order parameters[10] are a generalization of the above definition to three-dimensional systems.The pair correlation function g(r) ≡ g(r, θ, φ) in spherical polar coordinatesis expanded in terms of a Fourier series,

g(r) = go(r)

[1 + ∑

l

∑m

lm(r)lm(θ, φ)

], (8)

where the summation over l runs from 0 to ∞, and that over m runs from−1 to +1. The coefficients lm are related to the Steinhardt order parameters(Fig. 2), which are useful in differentiating between various crystal types inthree-dimensional systems.

For water-like molecules (which have a propensity for tetrahedral coordi-nation, owing to their hydrogen-bonding nature), the tetrahedral order parame-ter [11] in Fig. 2 measures the degree to which the nearest-neighbor moleculesare tetrahedrally coordinated with respect to a given molecule. The tetrahedralorder parameter is a three-body order parameter, which ensures local tetra-hedral symmetry around each (water-like) molecule. The tetrahedral orderparameter is sensitive to formation of structures of crystalline structures inwater.

Similarly, the nematic order parameter [1] in Fig. 2 quantifies the deg-ree of nematic order (i.e., parallel ordering of anisotropic molecules alongtheir longitudinal axis) in liquid-crystalline systems. In fact, the nematic orderparameter is characterized by a 2-fold symmetry (the longitudinal axis is aheadless director, i.e., with no up or down direction), and therefore is closelyrelated to Y2m term of the Steinhardt order parameter.

In the examples given in Fig. 2, the definitions of order parameters arebased on bond-orientational order, i.e., orientation of nearest neighbor (ormolecular) bonds. In each case, the order parameters quantify the degree ofcrystalline order in the system; therefore, the order parameters assume nonzero(distinct) values in the crystalline phase, which reduce (mostly to zero) in thedisordered phase.

3. Applications of Order Parameters

3.1. Quantification of Disorder

Torquato and coworkers [12], have used the three-dimensional Steinhardtorder parameters along with a translational order parameter t (based on theradial distribution function g(r)) to quantify the degree of disorder in densepacked materials. For a system of hard spheres in three dimensions, the authors

8 R. Radhakrishnan and B.L. Trout

computed order parameter maps (Q6 vs. t) for the liquid and crystal phases atequilibrium, and series of metastable states with jammed configurations (i.e.,those configurations in which a given particle cannot be displaced when therest of the particles in the system are fixed). The authors found that the trans-lational order parameter was always positively correlated with the bond orien-tational order parameter (i.e., an increase in one led to an increase in the other)for the hard sphere system. Additionally, the bond orientational order param-eter (being the more sensitive measure of the two) increased monotonicallywith increasing volume packing fraction for the jammed structures, for theentire range of packing fraction between the equilibrium liquid and crystal.The authors concluded that the concept of random close-packing is ill-definedbased on the observation that an infinitesimal increase in the bond-orientationalorder parameter can lead to an infinitesimal increase in the packing fraction.The results also supported the view that glassy structures were not merelyliquid-like structures with “frozen-in” disorder, because they were charac-terized by distinctly different values of Q6, intermediate between liquid andcrystal phases.

3.2. Anomalies of Liquid Water

Errington and Debenedetti [13] have advanced a formalism to understandthe structure–property relationship in liquid water on the basis of order param-eter maps. Based on the values of the translational order parameter t and thetetrahedral order parameter ξ for liquid water at equilibrium, the authors tracedpaths of the system in ξ − t space. Each path was obtained at constant temper-ature as the density was gradually increased in their computer simulations.Unlike the hard sphere case (where Q6 and t are positively correlated andmonotonically increase with increasing volume fraction), the authors foundstructurally anomalous regions (of state space, e.g., in the temperature–densityplot) in water where “order” decreased with increasing density. The boundaryof the structurally anomalous region was identified by ξ and t extrema on theξ − t traces at different temperatures. The authors also found that the structuralanomaly in water was correlated with a transport anomaly (when the diffu-sion coefficient increases with increasing density), and with a thermodynamicanomaly (when the coefficient of thermal expansion is negative); in particu-lar, the anomalous regions occur as a cascade, i.e., the structurally anomalousregion encompasses the region characterized by the transport anomaly, whichencompasses the region showing the thermodynamic anomaly. In a followingwork, the authors also investigated the order parameter maps in a system ofLennard-Jones particles and found that the LJ system of particles displayedthe same qualitative behavior as the system of hard spheres, which further

Order parameter approach of complex systems 9

supports the view that the anomalies in water are directly related to thechemical structure of the water molecule.

3.3. Nucleation of Crystalline Forms from Liquid

In a pioneering study, Frenkel and coworkers [14] calculated the freeenergy barrier to crystal nucleation in a system of Lennard-Jones particlesin three dimensions, by employing the Steinhardt order parameters and usingthe formalism in Section 2. The authors showed that the path to nucleationof the stable face centered cubic phase (when the liquid is supercooled belowthe freezing temperature) can be described in terms of increasing values ofQ6, while simultaneously suppressing the increase of W4. The path in orderparameter space along which both Q6 and W4 simultaneously increase, leadsthe system into a metastable body centered cubic phase.

Radhakrishnan and Trout [15–17] extended the above approach to describeice-nucleation under a variety of homogeneous and inhomogeneous environ-ments including hexagonal ice in the bulk, cubic ice under an external elec-tric field and in a confined system, and clathrate hydrates in a super-saturatedaqueous solution containing the hydrophobic solute, CO2. In order to calcu-late the free energy barrier to nucleation, the authors employed the two-bodySteinhardt order parameters and the three-body tetrahedral order parametersfor the one-component systems, and additionally used a translational order pa-rameter based on g(r) for the two-component aqueous solution of CO2. Theauthors found that as the successive density modes in the liquid (quantifiedby peaks in the direct correlation function) became more correlated – owingto a decrease in temperature, influence of an external potential, or increasedinhomogeneity – the free energy barrier to nucleation decreased. Interpretingtheir results in light of density functional theory of freezing, the authors dis-covered an inverse correlation between the degree of coupling of the succes-sive density modes in the liquid phase and the free energy barrier to nucleation.

Rutledge and coworkers [18], using molecular dynamics simulations, havestudied crystal nucleation in a polymer melt. The time-scale to observenucleation of a crystal phase in a polymer melt is normally beyond the scopeof molecular dynamics simulations. The authors, however, found that under anapplied a uniaxial (extensional) stress, the barrier to nucleation reduces consid-erably, to the extent that crystal nucleation can be captured in the simulations.The crystalline domains in the simulations were identified based on the localvalue of the nematic order parameter. The authors concluded that the additionof a large deforming stress accelerates the crystallization process by drivingthe individual chains into a low energy torsional conformation and by aligningthem in a single direction, which leads to a lowering of the nucleation barrier.

10 R. Radhakrishnan and B.L. Trout

Shetty and coworkers [19] have proposed a new formalism to construct newtypes of order parameters to quantify local order in inhomogeneous systems(such as a crystal–melt interface) based on pattern recognition and geneticalgorithm.

3.4. Solvation in Biomolecules

The contribution of hydration in molecular assembly and enzyme cataly-sis has long been recognized. Many biomolecules are characterized by surfacescontaining extended nonpolar regions, and the aggregation and subsequentremoval of such surfaces from water is believed to play a critical role in thebiomolecular assembly in cells. Conventional views hold that the hydration shellof small hydrophobic solutes is clathrate-like, characterized by local cage-likehydrogen-bonding structures and a distinct loss in entropy. Using moleculardynamics simulations on the solvated polypeptide melittin, Cheng and Roskky[20] found that the hydration of extended nonpolar planar surfaces appears toinvolve structures that are orientationally inverted relative to clathrate-likehydration shells, with unsatisfied hydrogen bonds that are directed towards thehydrophobic surface. The authors employed bond-orientational order param-eters to classify the local structuring of the solvent. Based on the correlationbetween the observed values of the order parameters and the average bindingenergy (i.e., the interaction of a molecule with all other molecules in thesystem) of proximal water molecules in each surface set, they concluded thatthe clathrate-like and inverted clathrate-like structures are distinguished by asubstantial difference in the water–water interaction enthalpy, and that their rel-ative contributions depended strongly on the surface topography of the melit-tin molecule. Clathrate-like structures dominate near convex surface patches,whereas the hydration shell near flat surfaces fluctuates between clathrate-likeand less-ordered or inverted structures. The strong influence of surface topog-raphy on the structure and free energy of hydrophobic hydration is likely to bea generic feature, which may be important for many biomolecules.

3.5. Freezing of Inhomogeneous Fluids in Porous Media

Molecular simulations for simple fluids confined in model systems ofslit-shaped pores show a freezing behavior that is governed by the relativestrength of the fluid–wall interaction to the fluid–fluid interaction (quantifiedby a parameter α), and the pore width H . The shift in the freezing tempera-ture, Tf,pore − Tf,bulk, is found to be positive if the fluid–wall interaction is morestrongly attractive than the fluid–fluid interaction and negative if the fluid–wallinteraction is less attractive than the fluid–fluid interaction.

Order parameter approach of complex systems 11

Using the Mermin order parameters and the free energy formalism inSection 2, Radhakrishnan and coworkers [21] discovered the presence of sev-eral thermodynamically stable intermediate phases lying between the liquidphase and the solid phase in computer simulation studies of Lennard-Jonesmolecules in slit-pores. Their studies led to the conclusion that the contactlayers, i.e., the layers closest to the pore walls, freeze at a higher temperaturethan the inner layers for strongly attractive pores (large values of α), and thusthe intermediate phase has the structure termed “contact-crystalline”, i.e., thecontact layers are crystalline while the inner layers are liquid-like. For mod-erate values of α, the contact layers are liquid-like while the inner layers arecrystalline, and the intermediate phase exists as a “contact-liquid” phase. Theauthors also found that for repulsive and weakly attractive walls, the interme-diate (contact-layer) phase is at best metastable, and thus, only the liquid andcrystal phases are stable. Based on these observations, the authors constructed“global phase diagrams” which present a unifying picture of confined phasefreezing behavior in terms of the parameter α and the pore width H .

In a later study [22], the authors also found evidence for the existenceof a hexatic phase as an intermediary between the fluid and crystalline ones.The hexatic phase is a manifestation of the fact that, in a continuous symme-try breaking transition such as the freezing transition, the translational sym-metry and the rotational symmetry can break at two different temperatures.Thus, in the liquid to hexatic phase transition, the rotational symmetry is bro-ken and in the hexatic to crystalline transition the translational symmetry isbroken. Hexatic phases, which retain long-range orientational, but not posi-tional order, are known to occur in infinite quasi-two-dimensional systems; theauthors established their presence in the simulations using a system size scal-ing analysis. They also found that for pore sizes accommodating more thanthree adsorbed molecular layers, a “contact-hexatic” phase was stable phase(in the simulations), where the contact layers are hexatic, while the innerlayers are liquid-like.

4. Assumptions in the Order ParameterApproach and Outlook

As illustrated in this article, the order parameter approach can be gain-fully employed in studies of complex systems. In problems where an obvioussymmetry is involved, defining a suitable set of order parameters becomes aneasy task. The various examples described here demonstrate that these orderparameters can be used to associate a defining characteristic of the free energylandscape to the observed phenomenon. At the expense of increased physicalinsight and phenomenological understanding, several inherent assumptions gointo the order parameter approach described here, which we discuss below.

12 R. Radhakrishnan and B.L. Trout

4.1. Physical Significance of the Order Parameters

Based on the relationship between the order parameters and the free energylandscape, it is tempting to associate the order parameters with physical vari-ables. However, in most cases, the order parameters appear as coefficients ofexpansion of an extensive thermodynamic variable. Consequently, dependingon the choice of expansion, the definitions of the order parameters vary andare certainly not unique. Therefore, it is not guaranteed a priori that the ord-er parameters should represent physical variables. In the examples describedhere, the physicality of the order parameters can be understood on the basisof density functional theory. Since the free energy F is a unique functional ofthe spatially varying density ρ(r), and the order parameters in Fig. 2 are allbased on expansions of g(r) (where ρg(r) = ρ(r)), the order parameters canbe ascribed to physical variables, if the particular density mode they charac-terize can be identified. In general, for other choices of order parameters, thisconnection must be established.

Another underlying assumption is that the phenomena of interest (happen-ing in 3N-dimensional space) can be described in terms of a small number oforder parameters. Although a rigorous proof in support of this assumption maynot be possible, an argument based on a few general characteristics of physicalsystems may be put forward in its defense. If our objective is to correlate theequilibrium properties of the system using the order parameters, the order pa-rameters can once again be chosen as the coefficients of expansion of g(r). Onfirst glance, it appears that one needs to include an infinite set of order parame-ters to have a rigorous theory. However, most often, correlations in g(r) die outby molecular length scales (even in the ordered phase, at finite temperatures),and therefore only order parameters corresponding to density modes relevantto these length-scales need to be included, which reduces their number to afew. If the objective is to describe an activated process, only those dynamicalvariables corresponding to the slowest modes (which are identifiable by prin-cipal component analysis of a molecular dynamics trajectory) in the transitionpathway need to be included, therefore greatly reducing the number of relevantorder parameters.

4.2. Rate Processes

Within order parameter framework, we need to invoke additional assump-tions to compute the rates of activated processes. A straightforward and con-ceptually appealing description is using transition state theory, according towhich, the rate k is given by

k = A exp(−β F), (9)

Order parameter approach of complex systems 13

where F is the free energy barrier along the reaction pathway (i.e., the freeenergy of the transition state relative to the reactants), and A is a pre-factorrelated to the inherent frequency of barrier crossing. For an ideal gas, A =(βh)−1, h being the Planck’s constant. More generally, A is given by theinverse of the time-scale over which the order parameter correlation function〈φ(0)φ(t)〉 decays when calculated at the transition state. Both terms (i.e., Fand A) can be important in calculating the rate.

In closing, we note that methods for verifying the existence of a transitionstate in a multidimensional free energy landscape independent of the orderparameter formalism exist, which can validate the findings of the order para-meter approach [23]. Alternative approaches to treat activated processes, whichare independent of the existence of order parameters (and hence relatedapproximations) have also been developed [24]. Nevertheless, the orderparameter approach continues to be widely employed because of its simplicity,computational efficiency, and phenomenological appeal.

References

[1] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cam-bridge University Press, Cambridge, 1995.

Q1[2] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group,Addison-Wesley, New York, 1992.

Q2[3] D. Henderson (ed.), Fundamentals of Inhomogeneous Fluids, Marcel Dekker, NewYork, 1992.

[4] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press,Oxford, 1987.

Q3[5] D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms toApplications, 2nd edn., Academic Press, San Diego, 2001.

[6] R.H. Zhou, B.J. Berne, and R. Germain, “The free energy landscape for betahairpin folding in explicit water,” Proc. Natl. Acad. Sci. USA, 98, 14931–14936,2001.

[7] F.G. Wang and D.P. Landau, “Efficient, multiple-range random walk algorithm tocalculate the density of states,” Phys. Rev. Lett., 86, 2050–2053, 2001.

[8] C. Tsallis, “Possible generalization of the Boltzmann–Gibbs Statistics,” J. Stat.Phys., 52, 479–487, 1988.

[9] N.D. Mermin, “Crystalline order in 2 dimensions,” Phys. Rev., 176, 250, 1968.[10] P.J. Steinhardt, D.R. Nelson, and M. Ronchetti, “Bond-orientational order in liquids

and glasses,” Phys. Rev. B, 28, 784–805, 1983.[11] P.L. Chau and A.J. Hardwick, “A new order parameter for tetrahedral configura-

tions,” Mol. Phys., 93, 511–518, 1998.[12] S. Torquato, T.M. Truskett, and P.G. Debenedetti, “Is random close packing of

spheres well defined?” Phys. Rev. Lett., 84, 2064–2067, 2000.[13] J.R. Errington and P.G. Debenedetti, “Relationship between structural order and the

anomalies of liquid water,” Nature, 409, 318–321, 2001.[14] R.M. Lyndenbell, J.S. Van Duijneveldt, and D. Frenkel, “Free-energy changes on

freezing and melting ductile metals,” Mol. Phys., 80, 801–814, 1993.

14 R. Radhakrishnan and B.L. Trout

[15] R. Radhakrishnan and B.L. Trout, “A new approach for studying nucleation phenom-ena using molecular simulations: application to CO2 hydrate clathrates,” J. Chem.Phys., 117(4), 1786, 2002.

[16] R. Radhakrishnan and B.L. Trout, “Nucleation of crystalline phases of water inhomogeneous and inhomogeneous environments,” Phys. Rev. Lett., 90, 2003,2003.

[17] R. Radhakrishnan and B.L. Trout, “Nucleation of hexagonal ice Ih in liquid water,”J. Am. Chem. Soc., 125, 7743, 2003.

[18] M.S. Lavine, N. Waheed, and G.C. Rutledge, “Molecular dynamics simulation oforientation and crystallization of polyethylene during uniaxial extension,” Polymer,44, 1771–1779, 2003.

Q4 [19] R. Shetty, F. Escobedo, D. Choudhary, and P. Clancy, “Characterization of order insimple materials. A pattern recognition approach,” J. Chem. Phys., 117, 4000–4009,2002.

[20] Y.K. Cheng and P.J. Rossky, “Surface topography dependence of biomolecularhydrophobic hydration,” Nature, 392, 696–699, 1998.

[21] R. Radhakrishnan, K.E. Gubbins, and M. Sliwinska-Bartkowiak, “Global phasediagrams for freezing in porous media,” J. Chem. Phys., 116, 1147–1155, 2002.

[22] R. Radhakrishnan, K.E. Gubbins, and M. Sliwinska-Bartkowiak, “Existence of ahexatic phase in porous media,” Phys. Rev. Lett., 89, 076101, 2002.

[23] P.G. Bolhuis, C. Dellago, and D. Chandler, “Reaction coordinates of biomolecularisomerization,” Proc. Natl. Acad. Sci. USA, 97, 5877–5882, 2000.

[24] P.G. Bolhuis, D. Chandler, C. Dellago, and P. Geissler “Transition path sampling:throwing ropes over rough mountain passes, in the dark,” Ann. Rev. Phys. Chem., 53,291–318, 2002.

Author Queries

1. In Ref. [2], the year cited in the text was 1997 but 1992 is given in thelist. Please check.

2. Please check the year in Ref. [3] (1991 was cited in the text but 1992 isfound in the list).

3. In Ref. [5], the year cited in the text was 1996 but 2001 is provided inthe list. Please check.

4. Please check the year in Ref. [19] (2003 was cited in the text but 2002is found in the list).

5. Figures are in color.