Nonequilibrium autocorrelations in phase-ordering dynamics

Nonuniversal aging during phase separation with long-range interaction

Issues concerning the kinetics of phase transitions are not well established for the cases where the order parameter remains conserved with time, particularly when the interatomic interactions are long-range in nature. Here we present results on structure, growth, and aging from Monte Carlo simulations of the two-dimensional long-range Ising model. In our computer simulations, random initial configurations, for 50:50 compositions of up and down spins, mimicking high-temperature equilibrium states, have been quenched to temperatures inside the coexistence curve. Our analyses of the simulation data, for such a protocol, show interesting dependence of the aging exponent, λ, on σ, the parameter, within the Hamiltonian, that controls the range of interaction. These nonuniversal values of λ are compared with a theoretical result for lower bounds. For this purpose, we extracted information on relevant aspects of structural properties during the evolution. To estimate λ, as is necessary, we also calculated the average domain size and analyzed its time dependence to obtain the growth exponent α which also is nonuniversal. The trends in the values of λ and α, as well as an anomaly in structure, suggest that a crossover from the long-range to the short-range variety occurs at σ≃1. The location of this boundary and the nonuniversality provide a picture that is surprisingly different from that of the corresponding static critical phenomena. Furthermore, our results suggest an important scaling law combining α and λ.

Aging following a zero-temperature quench in the d=3 Ising model

Aging in phase-ordering kinetics of the d=3 Ising model following a quench from infinite to zero temperature is studied by means of Monte Carlo simulations. In this model the two-time spin-spin autocorrelator C_{ag} is expected to obey dynamical scaling and to follow asymptotically a power-law decay with the autocorrelation exponent λ. Previous work indicated that the lower Fisher-Huse bound of λ≥d/2=1.5 is violated in this model. Using much larger systems than previously studied, the instantaneous exponent for λ we obtain at late times does not disagree with this bound. By conducting systematic fits to the data of C_{ag} using different Ansätze for the leading correction term, we find λ=1.58(14), with most of the error attributed to the systematic uncertainty regarding the Ansätze. This result is in contrast to the recent report that below the roughening transition universality might be violated.

Automated characterization of spatial and dynamical heterogeneity in supercooled liquids via implementation of machine learning

A computational approach by an implementation of the principle component analysis (PCA) withK-means and Gaussian mixture (GM) clustering methods from machine learning algorithms to identify structural and dynamical heterogeneities of supercooled liquids is developed. In this method, a collection of the average weighted coordination numbers (WCNs‾) of particles calculated from particles' positions are used as an order parameter to build a low-dimensional representation of feature (structural) space forK-means clustering to sort the particles in the system into few meso-states using PCA. Nano-domains or aggregated clusters are also formed in configurational (real) space from a direct mapping using associated meso-states' particle identities with some misclassified interfacial particles. These classification uncertainties can be improved by a co-learning strategy which utilizes the probabilistic GM clustering and the information transfer between the structural space and configurational space iteratively until convergence. A final classification of meso-states in structural space and domains in configurational space are stable over long times and measured to have dynamical heterogeneities. Armed with such a classification protocol, various studies over the thermodynamic and dynamical properties of these domains indicate that the observed heterogeneity is the result of liquid-liquid phase separation after quenching to a supercooled state.

Growth and aging in a few phase-separating active matter systems

Via computer simulations we study evolution dynamics in systems of continuously moving active Brownian particles. The obtained results are discussed against those from the passive 2D Ising case. Following sudden quenches of random configurations to state points lying within the miscibility gaps and to the critical points, we investigate the far-from-steady-state dynamics by calculating quantities associated with structure and characteristic length scales. We also study aging for quenches into the miscibility gap and provide a quantitative picture for the scaling behavior of the two-time order-parameter correlation function. The overall structure and dynamics are consistent with expectations from the Ising model. This remains true for certain active lattice models as well, for which we present results for quenches to the critical points.

Influence of roughening transition on magnetic ordering

In the literature of magnetic phase transitions, in addition to a critical point, the existence of another special point has been discussed. This is related to the broadening of the interface between two different ordering phases and is referred to as the point of roughening transition. While the equilibrium properties associated with this transition are well understood, the influence of this on nonequilibrium dynamics still needs to be investigated. In this paper we present comprehensive results, from Monte Carlo simulations, on coarsening dynamics in a system, over a wide range of temperature, in space dimension d=3, for which there exists a roughening transition at a nonzero temperature T_{R}. An advanced analysis of the simulation data, on structure, growth, and aging, shows that the onset of unexpected glasslike slow dynamics in this system, that has received attention in recent times, for quenches to zero temperature, actually occurs at this transition point. This implies that the structure and aging depend upon the final temperature, when the latter lies between 0 and T_{R}. This is a very interesting exception to universality in coarsening dynamics. The results also demonstrate an important structure-dynamics connection in the phase-ordering dynamics. We compare the key results with those from d=2, for which there exists no nonzero roughening transition temperature. The absence of the above-mentioned anomalous features in the latter dimension places our conjecture on the role of the roughening transition on a firmer footing.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Quasideterministic dynamics, memory effects, and lack of self-averaging in the relaxation of quenched ferromagnets

We discuss the interplay between the degree of dynamical stochasticity, memory persistence, and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasideterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including the mean-field, and the short-range random-field Ising model.

Aging exponents for nonequilibrium dynamics following quenches from critical points

Via Monte Carlo simulations we study nonequilibrium dynamics in the nearest-neighbor Ising model, following quenches to points inside the ordered region of the phase diagram. With the broad objective of quantifying the nonequilibrium universality classes corresponding to spatially correlated and uncorrelated initial configurations, in this paper we present results for the decay of the order-parameter autocorrelation function for quenches from the critical point. This autocorrelation is an important probe for the aging dynamics in far-from-equilibrium systems and typically exhibits power-law scaling. From the state-of-the-art analysis of the simulation results, we quantify the corresponding exponents (λ) for both conserved and nonconserved (order-parameter) dynamics of the model in space dimension d=3. Via structural analysis we demonstrate that the exponents satisfy a bound. We also revisit the d=2 case to obtain more accurate results. It appears that irrespective of the dimension, λ is approximately the same for both conserved and nonconserved dynamics.

Initial correlation dependence of aging in phase separating solid binary mixtures and ordering ferromagnets

Following quenches of initial configurations having long range spatial correlations, prepared at the demixing critical point, to points inside the miscibility gap, we study aging phenomena in solid binary mixtures. Results on the decay of the two-time order-parameter autocorrelation functions, obtained from Monte Carlo simulations of the two-dimensional Ising model, with Kawasaki exchange kinetics, are analyzed via state-of-the art methods. The outcome is compared with that obtained for the ordering in uniaxial ferromagnets. For the latter, we have performed Monte Carlo simulations of the same model using the Glauber mechanism. For both types of systems we provide comparative discussion of our results with reference to those concerning quenches with configurations having no spatial correlation. We also discuss the role of structure on the decay of these correlations.

Zero-temperature ordering dynamics in a two-dimensional biaxial next-nearest-neighbor Ising model

We investigate the dynamics of a two-dimensional biaxial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=-J_{0}∑_{i,j=1}^{L}[(S_{i,j}S_{i+1,j}+S_{i,j}S_{i,j+1})-κ(S_{i,j}S_{i+2,j}+S_{i,j}S_{i,j+2})]. For κ<1, the system does not reach the equilibrium ground state and keeps evolving in active states forever. For κ≥1, though, the system reaches a final state, but it does not reach the ground state always and freezes to a striped state with a finite probability like a two-dimensional ferromagnetic Ising model and an axial next-nearest-neighbor Ising (ANNNI) model. The overall dynamical behavior for κ>1 and κ=1 is quite different. The residual energy decays in a power law for both κ>1 and κ=1 from which the dynamical exponent z has been estimated. The persistence probability shows algebraic decay for κ>1 with an exponent θ=0.22±0.002 while the dynamical exponent for ordering z=2.33±0.01. For κ=1, the system belongs to a completely different dynamical class with θ=0.332±0.002 and z=2.47±0.04. We have computed the freezing probability for different values of κ. We have also studied the decay of autocorrelation function with time for a different regime of κ values. The results have been compared with those of the two-dimensional ANNNI model.

Aging phenomena during phase separation in fluids: decay of autocorrelation for vapor-liquid transitions

We performed molecular dynamics simulations to study relaxation phenomena during vapor-liquid transitions in a single component Lennard-Jones system. Results from two different overall densities are presented: one in the neighborhood of the vapor branch of the coexistence curve and the other being close to the critical density. The nonequilibrium morphologies, growth mechanisms and growth laws in the two cases are vastly different. In the low density case growth occurs via diffusive coalescence of droplets in a disconnected morphology. On the other hand, the elongated structure in the higher density case grows via advective transport of particles inside the tube-like liquid domains. The objective in this work has been to identify how the decay of the order-parameter autocorrelation, an important quantity to understand aging dynamics, differs in the two cases. In the case of the disconnected morphology, we observe a very robust power-law decay, as a function of the ratio of the characteristic lengths at the observation time and at the age of the system, whereas the results for the percolating structure appear rather complex. To quantify the decay in the latter case, unlike the standard method followed in a previous study, here we have performed a finite-size scaling analysis. The outcome of this analysis shows the presence of a strong preasymptotic correction, while revealing that in this case also, albeit in the asymptotic limit, the decay follows a power-law. Even though the corresponding exponents in the two cases differ drastically, this study, combined with a few recent ones, suggests that power-law behavior of this correlation function is rather universal in coarsening dynamics.

Direct test of the Gaussian auxiliary field ansatz in nonconserved order parameter phase ordering dynamics

The assumption that the local order parameter is related to an underlying spatially smooth auxiliary field, u(r[over ⃗],t), is a common feature in theoretical approaches to non-conserved order parameter phase separation dynamics. In particular, the ansatz that u(r[over ⃗],t) is a Gaussian random field leads to predictions for the decay of the autocorrelation function which are consistent with observations, but distinct from predictions using alternative theoretical approaches. In this paper, the auxiliary field is obtained directly from simulations of the time-dependent Ginzburg-Landau equation in two and three dimensions. The results show that u(r[over ⃗],t) is equivalent to the distance to the nearest interface. In two dimensions, the probability distribution, P(u), is well approximated as Gaussian except for small values of u/L(t), where L(t) is the characteristic length-scale of the patterns. The behavior of P(u) in three dimensions is more complicated; the non-Gaussian region for small u/L(t) is much larger than that in two dimensions but the tails of P(u) begin to approach a Gaussian form at intermediate times. However, at later times, the tails of the probability distribution appear to decay faster than a Gaussian distribution.

Ballistic aggregation in systems of inelastic particles: Cluster growth, structure, and aging

We study far-from-equilibrium dynamics in models of freely cooling granular gas and ballistically aggregating compact clusters. For both the cases, from event-driven molecular dynamics simulations, we have presented detailed results on structure and dynamics in space dimensions d=1 and 2. Via appropriate analyses it has been confirmed that the ballistic aggregation mechanism applies in d=1 granular gases as well. Aging phenomena for this mechanism, in both the dimensions, have been studied via the two-time density autocorrelation function. This quantity is demonstrated to exhibit scaling property similar to that in the standard phase transition kinetics. The corresponding functional forms have been quantified and the outcomes have been discussed in connection with the structural properties. Our results on aging establish a more complete equivalence between the granular gas and the ballistic aggregation models in d=1.

Kinetics of polymer collapse: effect of temperature on cluster growth and aging

Using state of the art Monte Carlo simulations of a bead-spring model we investigate both the equilibrium and the nonequilibrium behavior of the homopolymer collapse. The equilibrium properties obtained via multicanonical sampling recover the well-known finite-size scaling behavior of collapse for our model polymer. For the nonequilibrium dynamics we study the collapse by quenching the homopolymer from an expanded coiled state into the globular phase. The sequence of events observed during the collapse is independent of the quench depth. In particular, we focus on finding out universal scaling behaviors related to the growth or coarsening of clusters of monomers, by drawing phenomenological analogies with ordering kinetics. We distinguish the cluster coarsening stage from the initial stage of primary cluster formation. By successful application of a nonequilibrium finite-size scaling analysis we show that at all quench temperatures, during the coarsening stage, the cluster growth is roughly linear and can be characterised by a universal finite-size scaling function. In addition, we provide evidence of aging by constructing a suitable autocorrelation function and its corresponding dynamical power-law scaling with respect to the growing cluster sizes. The predicted theoretical bound for the exponent governing such scaling is strictly obeyed by the numerical data irrespective of the quench temperature. The results and methods presented here in general should find application in similar phenomena such as the collapse of a protein molecule preceding its folding.

Role of initial state and final quench temperature on aging properties in phase-ordering kinetics

We study numerically the two-dimensional Ising model with nonconserved dynamics quenched from an initial equilibrium state at the temperature T_{i}≥T_{c} to a final temperature T_{f} below the critical one. By considering processes initiating both from a disordered state at infinite temperature T_{i}=∞ and from the critical configurations at T_{i}=T_{c} and spanning the range of final temperatures T_{f}∈[0,T_{c}[ we elucidate the role played by T_{i} and T_{f} on the aging properties and, in particular, on the behavior of the autocorrelation C and of the integrated response function χ. Our results show that for any choice of T_{f}, while the autocorrelation function exponent λ_{C} takes a markedly different value for T_{i}=∞ [λ_{C}(T_{i}=∞)≃5/4] or T_{i}=T_{c} [λ_{C}(T_{i}=T_{c})≃1/8] the response function exponents are unchanged. Supported by the outcome of the analytical solution of the solvable spherical model we interpret this fact as due to the different contributions provided to autocorrelation and response by the large-scale properties of the system. As changing T_{f} is considered, although this is expected to play no role in the large-scale and long-time properties of the system, we show important effects on the quantitative behavior of χ. In particular, data for quenches to T_{f}=0 are consistent with a value of the response function exponent λ_{χ}=1/2λ_{C}(T_{i}=∞)=5/8 different from the one [λ_{χ}∈(0.5-0.56)] found in a wealth of previous numerical determinations in quenches to finite final temperatures. This is interpreted as due to important preasymptotic corrections associated to T_{f}>0.

Evidence of aging and dynamic scaling in the collapse of a polymer

We investigate a newly framed two-time property for the nonequilibrium evolution dynamics during the collapse of a homopolymer via Monte Carlo simulations of a model polymer. Our results show evidence of aging effects, as observed in the slow dynamics of structural and spin glasses, along with the presence of a dynamic scaling of the autocorrelation function ∼x(-λ(c)) (x being the ratio of the cluster sizes at two different times). We estimate the value of λ(c) unambiguously by applying a finite-size scaling analysis to the numerical data. The value thus obtained obeys a bound which we predict via general theoretical arguments. The results presented should be of general validity and may trigger direct experimental verification in single-polymer dynamics.

Dimensionality dependence of aging in kinetics of diffusive phase separation: Behavior of order-parameter autocorrelation

Behavior of two-time autocorrelation during the phase separation in solid binary mixtures is studied via numerical solutions of the Cahn-Hilliard equation as well as Monte Carlo simulations of the Ising model. Results are analyzed via state-of-the-art methods, including the finite-size scaling technique. Full forms of the autocorrelation in space dimensions 2 and 3 are obtained empirically. The long-time behavior is found to be power law, with exponents unexpectedly higher than the ones for the ferromagnetic ordering. Both Cahn-Hilliard and Ising models provide consistent results.

Aging in ferromagnetic ordering: full decay and finite-size scaling of autocorrelation

Nonequilibrium dynamics in Ising and Ginzburg-Landau models were studied for a nonconserved order parameter that mimics ordering in ferromagnets. The focus was on the understanding of the decay of the two time (t, t(w); t > tw) order-parameter correlation function. For this quantity, a full form has been obtained empirically which, for t ≫ t(w), provides a power-law ∼ (ℓ/ℓ(w))(-λ), ℓ and ℓ(w) being the characteristic lengths at t and tw, respectively. This empirical form was used for a finite-size scaling analysis to obtain the exponent λ in space dimensions d = 2 and 3. Our estimates of λ and understanding of the finite-size effects, for the models considered, provide useful information on the relevance of thermal noise. The values of λ obtained are in good agreement with the predictions of a theory based on Gaussian auxiliary field ansatz.

Effects of density conservation and hydrodynamics on aging in nonequilibrium processes

Aging in kinetics of three different phase transitions, viz., magnetic, a binary solid, and a single component fluid, are studied via Monte Carlo and molecular dynamics simulations in three space dimensions with the objective of identifying the effects of order-parameter conservation and hydrodynamics. We observe that the relevant autocorrelations exhibit power-law decay in a ferromagnet and binary solid but with different exponents. At early time the fluid autocorrelation function nicely follows that of the binary solid, the order parameter being conserved for both of them, as opposed to a ferromagnet. At a late time the fluid data crosses over to an exponential decay which we identify as a hydrodynamic effect and we provide analytical justification for this behavior.

Aging and crossovers in phase-separating fluid mixtures

We use state-of-the-art molecular dynamics simulations to study hydrodynamic effects on aging during kinetics of phase separation in a fluid mixture. The domain growth law shows a crossover from a diffusive regime to a viscous hydrodynamic regime. There is a corresponding crossover in the autocorrelation function from a power-law behavior to an exponential decay. While the former is consistent with theories for diffusive domain growth, the latter results as a consequence of faster advective transport in fluids for which an analytical justification has been provided.

Influence of thermal fluctuations on the geometry of interfaces of the quenched Ising model

We study the role of the quench temperature Tf in the phase-ordering kinetics of the Ising model with single spin flip in d=2,3 . Equilibrium interfaces are flat at Tf=0 , whereas at Tf>0 they are curved and rough (above the roughening temperature in d=3 ). We show, by means of scaling arguments and numerical simulations, that this geometrical difference is important for the phase-ordering kinetics as well. In particular, while the growth exponent z=2 of the size of domains L(t) approximately t 1/z is unaffected by Tf, other exponents related to the interface geometry take different values at Tf=0 or Tf>0 . For Tf>0 a crossover phenomenon is observed from an early stage where interfaces are still flat and the system behaves as at Tf=0 , to the asymptotic regime with curved interfaces characteristic of Tf>0 . Furthermore, it is shown that the roughening length, although subdominant with respect to L(t) , produces appreciable correction to scaling up to very long times in d=2 .

Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas

We consider the dynamics of a gas of free bosons within a semiclassical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein "collapse" within this model. We find that at the Bose-Einstein condensation temperature Tc, the chemical potential mu(t) vanishes exponentially with a universal rate that we compute exactly. Below Tc, we show analytically that square root mu(t) vanishes linearly in a finite time t coll. After t coll, the mass of the condensate grows linearly with time and saturates exponentially to its equilibrium value for large time. We also give analytical results for the density scaling functions, for the corrections to scaling, and for the exponential relaxation time. Finally, we find that the equilibration time (above Tc) and the collapse time T coll(below Tc) both behave like -T -3 c ln|T-Tc|, near Tc.

Correction to scaling in the response function of the two-dimensional kinetic Ising model

The aging part Rag(t,s) of the impulsive response function of the two-dimensional ferromagnetic Ising model, quenched below the critical point, is studied numerically employing an algorithm without the imposition of the external field. We find that the simple scaling form Rag(t,s)=s-(1+a)f(t/s), which is usually believed to hold in the aging regime, is not obeyed. We analyze the data assuming the existence of a correction to scaling. We find a=0.273+/-0.006, in agreement with previous numerical results obtained from the zero field cooled magnetization. We investigate in detail also the scaling function f(t/s) and we compare the results with the predictions of analytical theories. We make an ansatz for the correction to scaling, deriving an analytical expression for Rag(t,s). This gives a satisfactory qualitative agreement with the numerical data for Rag(t,s) and for the integrated response functions. With the analytical model we explore the overall behavior, extrapolating beyond the time regime accessible with the simulations. We explain why the data for the zero field cooled susceptibility are not too sensitive to the existence of the correction to scaling in Rag(t,s), making this quantity the most convenient for the study of the asymptotic scaling properties.

Response functions in phase-ordering kinetics

We discuss the behavior of response functions in phase-ordering kinetics within the perturbation theory approach developed earlier. At zeroth order the results agree with previous gaussian theory calculations. At second order the nonequilibrium exponents lambda and lambda(R) are changed but remain equal.

Analytical results for random walk persistence

In this paper, we present a detailed calculation of the persistence exponent straight theta for a nearly Markovian Gaussian process X(t), a problem initially introduced elsewhere in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. Resummed perturbative and nonperturbative expressions for straight theta are derived, which suggest a connection with the result of the alternative independent interval approximation. The perturbation theory is extended to the calculation of straight theta for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and nonperturbative expressions for the persistence exponent straight theta(X0), describing the probability that the process remains larger than X(0)sqrt[<X2(t)>].

Perturbation expansion in phase-ordering kinetics. II. N-vector model

The perturbation theory expansion presented earlier to describe the phase-ordering kinetics in the case of a nonconserved scalar order parameter is generalized to the case of the n-vector model. At lowest order in this expansion, as in the scalar case, one obtains the theory due to Ohta, Jasnow, and Kawasaki (OJK). The second-order corrections for the nonequilibrium exponents are worked out explicitly in d dimensions and as a function of the number of components n of the order parameter. In the formulation developed here the corrections to the OJK results are found to go to zero in the large n and d limits. Indeed, the large-d convergence is exponential.

Counting defects in an instantaneous quench

We consider the formation of defects in a nonequilibrium second-order phase transition induced by an instantaneous quench to zero temperature in a type II superconductor. We perform a full nonlinear simulation where we follow the evolution in time of the local order parameter field. We determine how far into the phase transition theoretical estimates of the defect density based on the Gaussian approximation yield a reliable prediction for the actual density. We also characterize quantitatively some aspects of the out of equilibrium phase transition.

Corrections to scaling in the phase-ordering dynamics of a vector order parameter

Corrections to scaling, associated with deviations of the order parameter from the scaling morphology in the initial state, are studied for systems with O(n) symmetry at zero temperature in phase-ordering kinetics. Including corrections to scaling, the equal time pair correlation function has the form C(r,t)=f0(r/L)+L(-omega)f1(r/L)+., where L is the coarsening length scale. The correction-to-scaling exponent omega and the correction-to-scaling function f1(x) are calculated for both nonconserved and conserved order parameter systems using the approximate Gaussian closure theory of Mazenko. In general omega is a nontrivial exponent which depends on both the dimensionality d of the system and the number of components n of the order parameter. Corrections to scaling are also calculated for the nonconserved one-dimensional XY model, where an exact solution is possible.