In a modified model for polymer melts [H.C. Öttinger, Phys.
Rev. E 50 (1994) 4891-4895], the effects of constraint release or
partially anisotropic Brownian forces have been incorporated in a
very natural way into the well-known reptation models of
Doi-Edwards and Curtiss-Bird. The stress tensor has been
formulated such that it satisfies all known consistency criteria
of thermodynamics and continuum mechanics. In the linear
viscoelastic regime, the modified model predicts a relaxation
modulus which is very similar to the one resulting from the idea
of "double reptation". Compared to the usual reptation
models, the closed-form expressions for the linear viscoelastic
material functions are in much better agreement with experimental
results and, in contrast to "double reptation", the
modified model is not restricted to the linear viscoelastic
regime. In the nonlinear regime, a more realistic shear-rate
dependence of the viscosity is found. Particularly promising
features of the new reptation model are the complex mixing rules
for polydisperse melts and the fact that the irreversible
contribution to deformations becomes more relevant. Recent
simulations show that inclusion of full Rouse mode effects,
together with the double Reptation principle can give a very good
prediction of the linear viscoelastic properties of polymer melts.
Another current project involves an extension of the usual Gaussian fluctuations in the stochastic models of polymer kinetic. Due to the self-similarity shown by the models of polymer physics, they are promising candidates for an approach via alpha-stable distributions and processes. The central-limit-theorem argument may be applied to support the choice of non-Gaussian noise when fluctuations with infinite moments are present. These distributions show long tails with power law decay, asymmetry, infinite moments and self-similarity. Furthermore many observed anomalies in polymer physics, like anomalous diffusion (in polymer melts and concentrated solutions) or non-exponential relaxation (in polymer melts and solutions, branched polymers, filled polymers and rubbers), have been modeled by ad hoc models with long tail probability distributions and self-similarity. Within this project we investigate a new approach to stochastic modeling and simulation in polymer kinetic theory via alpha-stable noise, which includes the Gaussian noise for alpha=2, and should give us a deeper physical and safer modeling of such anomalies.
We use fortran, c++ and c-compilers and ksh and perl-scripts to run our jobs. Results are animated using SciAn, inventor, AVS and showcase (Silicon Graphics). Sometimes we make use of the NAG-libraries, postprocessing is achieved with the help of the symbolic language such as Mathematica. Our parameter space for the problems at hand is large enough to allow for non-parallel computations with maximum efficiency. For the asgard architecture, we reformulate our problems to rely on low communication speed between processors.
The original FENE dumbbell kinetic theory is extended to describe concentration dependent shear-induced anisotropy in dilute polymer solutions. A mean field term is introduced into the model equations to take into account intermolecular forces. For the case of stationary shear flow the corresponding coupled non-linear relaxation equations for the components of the tensor of gyration are solved numerically. We present results for the shear and concentration dependence of different quantities related to the tensor, i.e.~the orientation angle, radius of gyration, the eigenvalues, and different pseudospherical components. They are in good qualitative agreement with data from light scattering experiments. Corresponding results for the rheological quantities are briefly discussed.
J. Non-Newtonian Fluid Mech. 62 (1996) 235-251
Tensile stress and birefringence in both real and model amorphous polymer melts have been measured during constant rate uniaxial elongational flow. We focus on investigations where deviations from the linear stress-optical behavior are pronounced. A rate-dependent contribution to the stress which is not directly related to the intramolecular conformations ('stress offset') is detected for both types of macromolecular fluids. Independent of the flow history during relaxation a linear stress-optical behavior is revealed. Nonequilibrium molecular dynamics (NEMD) computer simulations on the multibead anharmonic spring model are shown to provide insight into the molecular mechanisms underlying the viscoelastic behavior: during relaxation the intermolecular interactions become dominant in correlation with linear stress-optical behavior; the stress offset is shown to be very similar to the stress arising in the corresponding simple fluid; the total stress can well be approximated by a sum of three parts which are based on single-particle and single-link distribution functions only; the yield point behavior at high elongation rates reflects the transition from affine to non-affine motion of bonds and is understood without reference to strong inhomogeneities resulting from local plastic strain production, the chemical structure does not influence the qualitative behavior; distinct microscopic stress contributions under elongation and subsequent relaxation such as inter- and intramolecular, attractive and repulsive, kinetic and potential contributions are resolved.
Macromolecules 30 (1997) 526
In order to obtain numerical estimates for the properties of a general model for polymers in dilute theta solutions in its long-chain limit we follow a stochastic approach to polymer kinetic theory. The model takes into account configuration-dependent hydrodynamic interaction (HI) and simplifies to the Zimm bead-spring chain model in the limit of preaveraged HI, for which parameter-free `universal ratios' such as the ratio between radius of gyration and hydrodynamic radius are known. The Chebyshev polynomial method and a variance reduction simulation technique is used to implement an efficient Brownian dynamics simulation. We resolve the full dependence of several characteristic ratios vs. both chain length and hydrodynamic interaction parameter, we extrapolate their values to determine universal behaviors, and compare with analytical and experimental results.
J. Chem. Phys. 113 (2000) 4767.
The linear and nonlinear rheological behavior of dilute ferrofluids is determined from an underlying kinetic model and the dependence of the viscosity coefficient on the scalar orientational order parameters is obtained. In case of uniaxial symmetry, the antisymmetric contribution to the hydrodynamic stress tensor is of the same form as in the classical Ericksen-Leslie theory of uniaxial nematic liquid crystals and the linear magnetoviscosity is found to coincide with earlier results obtained by the so-called effective field method. While the assumption of uniaxial symmetry is fulfilled exactly in the limit of strong vorticity and weak magnetic field, the exact result for the linear magnetoviscosity shows corrections due to contributions from biaxial symmetry. Measures for the deviations from uniaxial symmetry are introduced and the generalization of the stress tensor in case of biaxial symmetry is obtained. The investigations are accompanied by numerical simulation of the kinetic equation and reveal that the assumption of uniaxial symmetry seems to be a good approximation for most values of the magnetic field and vorticity.
J. Chem. Phys. 116 (2002) 9078-9088.
The quasi-equilibrium approximation is employed as a systematic tool for solving the problem of deriving constitutive equations from kinetic models of liquid-crystalline polymers. It is demonstrated how kinetic models of liquid-crystalline polymers can be approximated in a systematic way, how canonical distribution functions can be derived from the maximum entropy principle and how constitutive equations are derived therefrom. The numerical implementation of the constitutive equations based on the intrinsic dual structure of the quasi-equilibrium manifold thus derived is developed and illustrated for particular examples. Finally, a measure of the accuracy of the quasi-equilibrium approximation is proposed that can be implemented into the numerical integration of the constitutive equation.
The rheological properties of a dilute suspension of ellipsoidal ferromagnetic particles in the presence of a magnetic field are studied on the basis of a kinetic model, where the flow and magnetic external fields couple in qualitatively different ways to the orientational behavior of the ferrofluid. In the uniaxial phase the stress tensor is found to be of the same form as in the Ericksen-Leslie theory for nematic liquid crystals in the steady state. Expressions for a complete set of viscosity coefficients in terms of orientational order parameters are worked out. In the low Peclet number regime, the viscosity coefficients are given as explicit functions of the magnetic field and a particle shape factor, where the shape factor may equally represent a non-spherical unit (agglomerate, chain) composed of spherical particles. Further, by considering the magnetization as the only relevant variable, a magnetization equation within an effective field approach is derived from the kinetic equation and compared to existing magnetization equations. The alignment angle of the magnetization and the first and second normal stress coefficient are studied for the special case of plane Couette flow. The assumptions employed are tested against a Brownian dynamics simulation of the full kinetic model, and a few comparisons with experimental data are made.
Two different consistency criteria are considered for the stress tensor expressions of two different kinetic theory models for dilute polymer solutions. The criteria are material objectivity and thermodynamic consistency. Two separate approaches for checking thermodynamic consistency are considered, the first proposed by Grmela [Phys. Lett. 111A, 41-44 (1985)] and the second by Jongschaap [Lecture Notes in Physics (Springer, Berlin, 1991)]. The two models considered are a Hookean dumbbell model with internal viscosity (no linearization or rotation matrix approximations), and an inertial, elastic bead-spring chain. Both sets of criteria may be applied to these models without finding either a closed-form expression for the stress tensor, or solving any of the equations. We find that the Giesekus form for the stress tensor expression for a Hookean dumbbell with internal viscosity is both thermodynamically consistent and objective, whereas the Kramers form is not thermodynamically consistent. The Kramers form for the stress tensor is found to be thermodynamically consistent for an inertial, elastic bead-spring chain, but is not materially objective. We also find that the approaches of Grmela and Jongschaap are the same, although each requires a different ansatz for the relationship between thermodynamic quantities and polymer conformations.
J. Rheol. 38 (1994) 1909-1924.
Brownian dynamics simulations can be made more efficient by incorporating the idea of importance sampling. By introducing and compensating a bias in favor of those configurations which mainly contribute to the average of a given quantity of interest, the variance of the stochastic simulation results can be considerably reduced. This idea can be applied to general stochastic differential equations of motion and is hence not restricted to Brownian dynamics simulations. The construction of variance reduced simulations requires an approximate understanding of the dynamics described by the underlying stochastic differential equation. The basic procedure is first developed in general and then illustrated for the example of a Hookean dumbbell solution in start-up of steady shear flow. Particular emphasis is put on the development of a clearly structured formulation of the procedure which immediately allows for variance reduced simulations of nonlinear models. Possible applications in polymer kinetic theory and in the flow calculation of viscoelastic liquids are discussed.
Macromolecules 27 (1994) 3415-3423.
With the idea of a broad investigation of the flow behavior of dilute polymer solutions in mind, the dynamics of polymer chains with rigid constraints and hydrodynamic interaction is formulated in various equivalent ways. Starting from a very general diffusion equation of polymer kinetic theory, equivalent stochastic differential equations of motion both in terms of generalized coordinates and in terms of constraint conditions are derived. Then an efficient Brownian dynamics simulation algorithm is constructed rigorously, and a convenient expression for evaluating stresses in simulations is suggested. Furthermore, a modified simulation algorithm which is appropriate for infinitely stiff rather than rigid systems is discussed.
Phys. Rev. E 50 (1994) 2696-2701.
A Hookean dumbbell model for polymers in dilute solutions undergoing homogeneous flow is generalized to include arbitrary imposed temperature profiles. In order to obtain the ``nonisothermal diffusion equation'' for the probability density in polymer configuration space we generalize the approach of Schieber and Öttinger [J. Chem. Phys. 89, 6972-6981 (1988)] to Brownian motion out of equilibrium. In addition, we derive the polymer contributions to the mass-flux vector, stress tensor and heat-flux vector by means of the kinetic theory approach of Curtiss and Bird [Adv. Polym. Sci. (1996)] for the case of a slowly varying temperature gradient, and we find coupled constitutive equations for the mass, momentum and energy fluxes. For a simple steady shear flow it is then possible to calculate the heat-flux vector explicitly, at least for small temperature gradients and shear rates. We compare our approach and results with previous works on this subject, and we finally discuss some extensions.
J. Rheol. 40 (1996) 857-874.
Stochastic simulation techniques play an important role in investigating polymer dynamics. Such simulations can be made more efficient by applying variance reduction methods. We present and compare various variance reduction schemes, based either on importance sampling strategies or on the use of control variates. In order to demonstrate that variance reduced simulations are a powerful tool in polymer kinetic theory, we show that for Hookean dumbbells with hydrodynamic interaction up to two orders in computer time can be saved uniformly in time.
J. Chem. Phys. 105 (1996) 3316-3331.
Modeling and understanding the flow behavior of polymeric liquids on the kinetic theory level is not merely a great intellectual challenge but rather a matter of immense practical importance, for example, in connection with plastics manufacturing, processing of foods, and movement of biological fluids. The purpose of this brief review paper is to describe how the fundamental problems of kinetic theory can be "cracked" by means of a very powerful tool, namely by stochastic simulation techniques such as Brownian dynamics. Simulations give us access to a variety of rheological material functions in given homogeneous flows, and they hence provide important constitutive information for kinetic theory models. Simulations can even be combined with discretization techniques such as the finite element or finite difference method to obtain direct solutions to fluid dynamics problems, without any need for constitutive equations. Finally, simulations also provide the most direct access to ensembles of molecular configurations. When state-of-the-art simulation methods are used, we are now in a position to crack many of the most interesting and important problems in polymer kinetic theory.
Proceedings of the XIIth International Congress on Rheology (Laval University, Quebec City, 1996) 293-296.
Greatly needed improvements to rheological models of polymer fluids can best be made by understanding and modifying a model on the molecular level, where the physics governing real fluids are created. For this purpose, stochastic simulations are an indispensable tool. The usefulness of stochastic simulations for kinetic theory models are well-known. In this paper, we show that the class of Rivlin-Sawyers, or K-BKZ, integral models is also subject to a molecular interpretation and can be simulated using stochastic techniques. We also illustrate how an existing molecular model can be modified, resulting in better predictions of viscoelastic behavior. For this we formulate a stochastic simulation of a reptation model modified to account for double reptation and tube-length fluctuations.
Macromol. Symp. 121 (1997) 187-203.
Polymer orientation in dilute solutions undergoing shear flow is investigated by means of flow birefringence using a quantity called "orientation resistance." In the framework of theoretical models it was found that the intrinsic orientation resistance is strongly influenced by polydispersity. Dilute solutions of two radical polymerized polystyrene samples with different molecular weights in both good and Theta solvents are investigated by flow birefringence. The experimentally obtained values for the intrinsic orientation resistance are in very good agreement with the values calculated in the framework of kinetic polymer models. These experiments demonstrate that the polydispersity effect is well understood and that flow birefringence constitutes an extremely precise method for determining molecular weights. From our experiments it can be inferred that the number-average molecular weight of a polymer sample can be measured to within a relative error of 3%. The use of the birefringence method for the determination of the molecular weight distribution of a polymer sample is suggested.
Macromolecules 30 (1997) 5527-5540.
For a number of well-known time-evolution equations for nonequilibrium systems we extract a common structure from these equations, referred to as a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). This fundamental structure is determined by four building blocks, two "potentials" (total energy and entropy) and two "matrices." We illustrate for various examples how three of the four building blocks can be determined in a rather straightforward manner so that, within our GENERIC approach to nonequilibrium dynamics, understanding of a given nonequilibrium system is reduced to determining a single "metric matrix," or friction matrix, either empirically or by more microscopic considerations. In formulating nonisothermal polymer kinetic theories, we show how the general structure provides a clear distinction between spring potentials of energetic and entropic origins in the various time-evolution equations.
Phys. Rev. E 56 (1997) 6633-6655.
We express Boltzmann's kinetic equation in the form of the recently proposed general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). This GENERIC formulation demonstrates that no dissipative potential is required for representing Boltzmann's kinetic equation in a general framework for nonequilibrium systems.
J. Non-Equilib. Thermodyn. 22 (1997) 386-391.
A detailed comparison is made between single and double generator formalisms for the thermodynamics and mechanics of complex fluids, expressed in either bracket or operator form. In the first part of this paper (Part I), we found a complete equivalence between single and double generator formalisms applied to macroscopic systems with respect to the physics described and the practical utility of the formalisms. For microscopic systems the conclusion is different: the double generator formalism is more natural to implement, and for some physical systems may be absolutely essential to the system description. As in Part I, the interrelationships between these two alternate approaches are examined and direct connections are determined for some specific examples of fluid systems: a nonisothermal dilute polymer solution, with and without a polymer contribution to the heat transport, and Boltzmann's kinetic equation. The splitting of the GENERIC dissipation matrix into mechanical and thermodynamic submatrices in the linear regime, as described in Part I, remains valid at the microscopic level. Furthermore, this splitting allows for the mathematical statement of a double generator dissipation bracket, which may subsequently be transformed to one involving only a single generator for many systems of interest. The Boltzmann equation appears to require the use of two generators since no thermodynamic potential can be defined for the system.
J. Non-Equilib. Thermodyn. 23 (1998) 334-350.
We formulate a set of Lorentz-covariant equations for an imperfect fluid with viscosity, dilatational viscosity, and anisotropic thermal conductivity that possess the full GENERIC structure of nonequilibrium thermodynamics. The GENERIC structure, which includes and goes beyond prior nonequilibrium generalizations of the second law of thermodynamics, is shown to provide a guideline for modifying previous phenomenological or kinetic-theory based equations of extended relativistic hydrodynamics. In the nonrelativistic limit, we discuss the form of the equations for viscous and viscoelastic fluids with anisotropic heat conduction.
Physica A 254 (1998) 433-450.
Diffusive systems respecting the fluctuation-dissipation theorem with multiplicative noise are studied on the level of stochastic differential equations. We propose an efficient simulation scheme motivated by the direct definition of the `Kinetic stochastic integral', which differs from the better known Itô and the Stratonovich integrals. This simulation scheme is based on introducing the identity matrix, expressed in terms of the diffusion tensor and its inverse, in front of the noise term, and evaluating these factors at different times.
J. Chem. Soc. Faraday Trans. 94 (1998) 1403-1406.
We present a self-consistent method for deriving moment equations for kinetic models of polymer dynamics. The Doi model of liquid-crystalline polymers with the Onsager excluded-volume potential is considered as an example. To lowest order, this method amounts to a simple effective potential different from the Maier-Saupe form. Analytical results are presented which indicate that this effective potential provides a better approximation to the Onsager potential than the Maier-Saupe potential. Corrections to the effective potential are obtained.
Phys. Rev. E 60 (1999) 5783-5787.
A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse-graining. Explicit entropy production formula is established. A new dissipative kinetic equation extending Vlasov's mean field approximation is derived.
Phys. Rev. E 63 (2001) 066124 (6 pages).
An express method to approximate trajectories of space-independent kinetic equations is developed. It involves a two-step treatment of relaxation through a quasi-equilibria located on a line emerging from the inital state in the direction prescribed by the kinetic equation. A test for the Boltzmann equation shows the validity of the method.
Transport Theory and Stat. Phys. 28 (1999) 271-296.
Models of complex reactions in thermodynamically isolated systems often demonstrate evolution towards low-dimensional manifolds in the phase space. For this class of models, we suggest a direct method to construct such manifolds, and thereby to reduce the effective dimension of the problem. The approach realizes the invariance principle of the reduced description, it is based on iterations rather than on a small parameter expansion, it leads to tractable linear problems, and is consistent with thermodynamic requirements. The approach is tested with a model of catalytic reaction.
Physica A 275 (2000) 361-379.
The potential role of resummation techniques in the kinetic-theory approach to subgrid turbulence modeling is discussed.
Physica A 280 (2000) 92-98.
The recently derived fluctuation-dissipation folmula [A.N. Gorban et al, Phys. Rev. E 63, 066124(1-6) (2001)] is illustrated by the explicit computation for McKean's kinetic model [H.P. McKean, J. Math. Phys. 8, 547 (1967)]. It is demonstrated that the result is identical, on the one hand, to the sum of the Chapman-Enskog expansion, and, on the other hand, to the exact solution of the invariance equation. The equality between all the three results holds up to the crossover from the hydrodynamic to the kinetic domain.
Phys. Rev. E 65 (2002) 026116(5).
We revisit recent derivations of kinetic equations based on Tsallis' entropy concept. The method of kinetic functions is introduced as a standard tool for extensions of classical kinetic equations in the framework of Tsallis' statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable relation between thermodynamics and kinetics caused by the deformation of macroscopic observables.
Phys. Rev. E 65 (2002) 036128 (4 pages).
A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse-graining. Explicit entropy production formula is established. A new dissipative kinetic equation extending Vlasov's mean field approximation is derived.
The power of the atomistically founded GENERIC approach to nonequilibrium systems is illustrated by coarse graining from the full N-particle phase space to the level of single-particle distribution functions. For a dilute gas, the elements of GENERIC can be evaluated without any fundamental assumptions. Boltzmann's famous kinetic equation is recovered.
The quasi-equilibrium or maximum entropy approximation is applied in order to derive constitutive equations from kinetic models of polymer dynamics. It is demonstrated in general and illustrated for an example how canonical distribution functions are obtained from the maximum entropy principle, how macroscopic and constitutive equations are derived therefrom and how these constitutive equations can be implemented numerically. In addition, a measure for the accuracy of the quasi-equilibrium approximation is proposed that can be evaluated while integrating the constitutive equations. In the example considered, it is confirmed that the accuracy of the approximation is increased by including more macroscopic variables. In steady elongational flow, it is found that more macroscopic variables need to be included above the coil-stretch transition to achieve the same accuracy as below.
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pade approximants, and invariance principle are compared both in linear and nonlinear situations.
In the last decade, minimal kinetic models, and primarily the Lattice Boltmann (LB) equation, have met with significant success for the simulation of complex hydrodynamic flows, ranging from slow flows in grossly irregular geometries to fully developed turbulence. In this Colloquium, we present a chronological survey of the main ideas behind the Lattice Boltzmann method, with special focus on the role played by the H theorem in shaping up compliance of the LB method with macroscopic evolutionary constraints (Second Law) as well as its numerical stability as a computational tool for fluid dynamics. Finally, we discuss the potential impact of lattice H theorems for generalized Lattice Boltzmann equations dealing with thermal and non-ideal fluid effects.
Derivation of the lattice Boltzmann method from the continuous kinetic theory [X. He and L. S. Luo, Phys. Rev. E 55, R6333 (1997); X. Shan and X. He, Phys. Rev. Lett. 80, 65 (1998)] is extended in order to obtain boundary conditions for the method. For the model of a diffusively reflecting moving solid wall, the boundary condition for the discrete set of velocities is derived, and the error of the discretization is estimated. Numerical results are presented which demonstrate convergence to the hydrodynamic limit. In particular, the Knudsen layer in the Kramers' problem is reproduced correctly for small Knudsen numbers.
A new method of reconstructing constitutive equations from microscopic simulations of kinetic models for polymeric solutions is presented. Microscopic simulations provide a set of data from which the constitutive equations are extracted ("learned"). In order to be consistent with non-equilibrium thermodynamics, we work within the Quasi-Equilibrium approximation.
Proc. XIII Int. Congress on Rheology (Cambridge, 2000) Vol. 2, 64-66.
Fluids with microstructure often show fascinating flow behavior.
In particular, the rheological behavior of ferrofluids
- stable suspensions of nano--sized ferromagnetic particles in
a carrier liquid -
can be manipulated by external magnetic fields.
Anisotropic viscosities are observed in ferrofluids
depending on the relative orientation
of the applied magnetic field to the flow field.
For a plane shear flow, these are the so called Miesovicz viscosities.
As has been noticed [1,2], the stress tensor
of ferrofluids can be described phenomenologically by the same set of
viscosity coefficients used to describe uniaxial
(or generally biaxial) nematic liquid crystals in the
presence of an orienting field [3-7].
The microscopic mechanisms leading to the anisotropic viscosities
are, however, rather different in these systems [2].
To understand the origin of various viscosity coefficients for
ferrofluids, a simple model system is studied here.
The model is formulated on the level of kinetic theory,
in accordance with kinetic models of nematic fluids.
General expressions for the viscosity coefficients are obtained from
the model.
For the special case of uniaxial symmetry, the expressions are worked
out in detail and compared with the result of phenomenological approaches
to uniaxial nematic liquid crystals as well as findings from
simulations, other theoretical approaches and experiments.
[1] S. Hess, J.F. Schwarzl, D. Baalss, J. Phys. Condens. Matter 2 (1990) SA279.
[2] S. Hess, T. Weider, M. Kröger, Magnetohydrodynamics 37 (2001) 297.
[3] M.I. Shliomis, Sov. Phys. 34 (1972) 1291.
[4] S. Hess, Z. Naturforsch. 31A (1976) 1034.
[5] N. Kuzuu, M.Doi, J. Phys. Soc. Jpn. 52 (1983) 3486.
[6] S. Hess, J. Non--Equilib. Thermodyn. 11 (1986) 175.
[7] M. Kröger, S. Sellers, J. Chem. Phys. 103 (1995) 807; Phys. Rev. E 56 (1997) 1804
[8] P. Ilg, M. Kröger, S. Hess, J. Chem. Phys. 116 (2002) 9078.
Proc. VI Europ. Conf. Rheol. (ESR, Erlangen, 2002)