JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 3, pp. 637-648, Warsaw 2006 INNOVATIVE MODELING METHODS IN DAMAGE ASSESSMENT: APPLICATION OF DISSIPATIVE PARTICLE DYNAMICS TO SIMULATION OF DAMAGE AND SELF-HEALING OF POLYMER-COATED SURFACES Aleksandar S. Jovanovic University of Stuttgart/Steinbeis University Berlin, Germany e-mail: jovanovic@risk-technologies.com Nenad Filipovic University of Kragujevac, Serbia e-mail: filipovic@risk-technologies.com The paper presents an exemplary application of an innovativemodeling technique (Dissipative Particle Dynamics – DPD) as a possible tool for multi-scale modeling of the behavior of advanced engineering materials (e.g. coatedmaterials and/ormulti-materials). Themulti-scalemodeling of the behavior of advanced engineering materials is primarily related to development of new techniques and methods needed to bridge the gap between the atomistic scale and macro applications. The presented example of examination of the applicability of theDPDtodamageasses- sment dealingwith the problemof interaction of particles, polymers and surfaces, is an issue concerningmany applications in the colloid science. It has confirmed the applicability of the DPD as a method for linking themacro and atomistic scales.The imminent application of themethod and the example are in the area of ”self-healing” advanced engineering materials, thus in line with the objectives of EuMaT (cf. Appendix). Key words: modeling, engineeringmaterials, self-healing 1. Introduction Many practical issues in the colloid science concern interactions of particles, polymers, andsurfaces.The techniqueofDissipativeParticleDynamics (DPD) has been initially implemented to problems of controlling the effects of soot 638 A. Jovanovic, N. Filipovic and deposition (e.g. in diesel engine lubricants or surface fouling in chemical reactors), but now it is also a suitable alternative technique for simulating hy- drodynamic behavior of other colloid-polymer-solvent systems. Hoogerbrugge and Koelman (1992) suggested a new Dissipative Particle Dynamics (DPD) approach to simulate flow fields of incompressible complex fluids. A fluid is divided into mesoscale particles so that each particle contains a large number ofmolecules but is still much smaller thandimensions of the containing vessel. Based on concepts prevalent inMolecular Dynamics (MD) theory, a Langevin equation was postulated by which motion of the fluid particles can be calcu- lated. The forces exerted on every particle by its neighbors that are within a prescribed distance rc (radius of influence) from it, are normally divided into three categories; conservative repulsive forces that can be derived from a potential function, dissipative forces that stem fromviscous friction generated by relative translational motion of adjacent particles, and random forces that may be significant due to the mesoscale dimension of the particles. One of the controversial, unsettled aspects inDPDmodeling relates to the formulation of proper no-slip conditions near a rigid wall. The Lees-Edwards method (Lees and Edwards, 1972) in essence, circumvented the problem of how to avoid particles that can both penetrate and slide along rigid walls. Their ingenious suggestion that worked quite well for Couette flows, in which particles penetrating one wall should be reintroduced at the other wall, can hardly be qualified as a condition that should be applied locally at any rigid wall inmore complex flow systems. Amore advanced suggestion was to freeze the regions of fluidnear the rigidwall (Hoogerburgge andKoelman, 1992;Boek et al., 1996). This, however, resulted in possible particle penetration through thewalls due to a ’soft’ conservative potential (Revenga et al., 1998). To avoid such non-physical results, various methods were suggested that combine the freezing particle layer near the wall with specular, bounce-back orMaxwellian reflection of a particle reaching a rigid wall (e.g., Hoogerburgge andKoelman, 1992; Revenga et al., 1998). More recently, Pivkin and Karniadakis (to be published) suggested combining the above with an augmented conservative force at the rigid wall. Flekkoy and Coveney (1999), and in particular Flekkoy et al. (2000) sug- gested linking DPD and Molecular Dynamics (MD) equations that govern motion of mesoscopic and molecular size particles, respectively (a bottom-up strategy). However, reaching closure still required defining constitutive rela- tions prevailing at the continuum level (top-down strategy). Thus, the fric- tion coefficient was linked to the dynamic viscosity η of a Newtonian fluid perceived as a continuum and the ratio between the intersection length and Innovative modeling methods in damage assessment... 639 distance between two interacting Voronoi cells. In this article, we adopt the strategy initiated by Hoogerbrugge and Koelman (1992) and further develo- pedbyEspanol andWarren (1995) andEspanol (1998), namely, that theDPD equations governing motion of identical mesoscopic particles, are postulated. 2. DPD model During more than a decade, a considerable number of articles focused on the proper formulation of constitutive equations governing different forces andhow their a priori unknown parameters are related to known phenomenological coefficients of a given fluid. A widely accepted formulation for equations of motion of aDPDparticle (e.g. Espanol andWarren, 1995; Groot andWarren, 1997; Novik and Coveney, 1997; Besold et al., 2000; Pivkin and Karniadakis, to be published) has the following form dri =vidt (2.1) dvi = ∑ j 6=i F ijdt= ∑ j 6=i (FCijdt+F D ijdt+F R ijd √ t) where dri and dvi are infinitesimal displacement and velocity changes me- asured relative to a Galilean coordinate system, that the particle i undergoes during the time increment dt. The forces FCij, F D ij and F R ij are conservative (repulsive), dissipative and randomforces (perunitmass of particle i) that the particle j exerts on the particle i, respectively, provided that the particle j is within the radius of influence rc of the particle i F C ij = aij ( 1− rij rc ) eij F D ij =−γ ( 1− rij rc )2 (vij ·eij)eij (2.2) F R ij = √ 2kBTγmi ( 1− rij rc ) ξijeij Here, aij is the maximum repulsion force per unit mass, rij is the distance between the particles i and j, eij is a unit vector pointing in the direction from j to i, vij = vi −vj is the velocity of the particle i relative to that of the particle j, mi is the mass of the particle i, γ stands for the friction coefficient, kB is the Boltzmann constant, T is the equilibrium temperature and ξij is a random number with zero mean and unit variance. In the case rij > rc, the particle j is assumed to exert no force on the particle i. Notice 640 A. Jovanovic, N. Filipovic that the conservative, dissipative and random forces that the particle j exerts on the particle i were assumed to depend upon the distance between these particles. The dissipative force also depends upon a single component of the relative velocity between the particles (vij ·eij), and that the direction of all the foregoing forces is along the line connecting the centers of two particles. If we define the following dimensionless variables r̂i = ri rc r̂ij = rij rc v̂ij = vij vT t̂= tvT rc (2.3) where vT = √ kBT/mi is the thermal velocity of the particle i, equation (2.1) can thus be expressed in the following dimensionless form dr̂i = v̂idt̂ (2.4) dv̂i = ∑ j 6=i aijrc v2T (1− r̂ij)eijdt̂− γrc vT (1− r̂ij) 2(v̂ij ·eij)eijdt̂+ + √ 2γrc vT (1− r̂ij)ξijeijd √ t̂ In the case when all particles have identical masses and all coefficients aij are equal, Eq. (2.4) depends on two dimensionless numbers only â= aijrc v2T γ̂= γrc vT (2.5) Thus, there is no need to assume that the trio; i.e. the mass of a particle m, kBT and rc are all unity, which has been an assumption made in many previous studies. It is sufficient to assume that vT and rc are equal to one and to vary γ and aij in order to obtain different DPD fluids. This would be equivalent to setting numerical values to the dimensionless variables â and γ̂. The boundary conditions would introduce at least two additional dimen- sionless parameters L̂= L rc V̂W = VW vT (2.6) where L and Vw scale the macroscopic size of the system and the velocity of its wall, respectively. More recently, Espanol (1998) suggested that additional components to dissipative and concomitant random forces be added. These included force components perpendicular to eij and the effect of particle rotation. The for- mer is a natural extension based on simple tensorial considerations and the Innovative modeling methods in damage assessment... 641 latter stems from the finiteness of the particles. Based onMD theory, Flekkoy andCoveney (1999) and Flekkoy et al. (2000) concluded that forces which are not collinear with eij exist at the DPD mesoscale. These modifications, ho- wever, have not thoroughly been tested, and their contribution to the solution accuracy is awaiting further exploration. 3. DPD simulation model Thesimulations are conducted ina three-dimensional simulationbox, as shown in Fig.1. Awall of frozen particles is created at one end providing the surface. Fig. 1. DPDmodel for adsorption of colloidal particles Although the surface of the wall is square, the periodic boundary conditions make it appear infinite. Polymers are then attached, effectively ”chemically grafted,” to this wall. A rough spherical object is created at a given distance from the wall to simulate a colloidal particle. The box is filled with DPD particles which represent the solvent. The wall is divided into two ones along a plane parallel to the wall-fluid interface. The half to which the polymers are attached ismade up of particles identical to thosewhichmake up the colloidal particle, while the other half is made up of solvent particles. These solvent particles ensure that the colloidal particle can only reach the wall, and hence adsorb onto its surface by traveling through the layer of polymers. Another type of the structure is a polymer (Fig.2). Here, a bead-and-spring typemodel 642 A. Jovanovic, N. Filipovic is used to provide an additional force between adjacent particles in the chain. A Fraenkel spring is mainly used F p ij = k(rij −req)êij (3.1) where k is the spring constant and req is the equilibrium spring length. Fig. 2. Example of polymer chains 4. Results The adsorption has been defined with the spring constant k = 50N/m. The snapshot with the distribution of colloidal particles and solvent particles after 0.1s is presented in Fig.3. Grey dots represent colloidal Voronoi elements, darker dots represent solvent Voronoi elements. Our results closely matched those experimentally and numerically recorded byGibson et al. (1998) as it is shown in Fig.4. Fig. 3. Snapshot after 0.1s of colloidal particle adsorption Innovative modeling methods in damage assessment... 643 Fig. 4. Comparison of results for colloidal particle adsorbtion 5. Discussion and conclusions The simulation and modeling of material behavior and degradation mecha- nisms, dependence of material structure on manufacturing methods and ma- nufacturing processes, is a prerequisite for the design of materials and manu- facturing method as well as for the evaluation of life cycle performance. The simulation example presented here shows that the DPD model is potentially a very useful tool for simulation of the adsorption process on coated surfaces. The imminent application of the method and the example are in the area of ”self-healing” advanced engineering materials. The method presented in the paper allows doing it up to the point of calculating the probability of deposition as a function of a system of interaction of particles, polymers and surfaces. In that sense, the DPDmodeling can be used for non-invasive repair of damaged coatings (Fig.5). A significant step and the desirable direction of furtherwork in the area of DPD-based damage modeling and analysis would be the research on ”intelli- gent repairmaterials” (Jovanovic andJovanovic, 2006). Corresponding theore- tical considerations are already available (Zomaya, 2006), andwhen combined with the availablemodeling techniques (Fig.6) could significantly improve the current practice in terms of (a) costs, (b) reduced total effort of repair and (c) significantly less intrusive character of the repair, which can be decisive for certain types of applications. The ideal situation would be simulation of the total life cycle. This requ- ires a combination of several research activities which are separated today. For 644 A. Jovanovic, N. Filipovic Fig. 5. Deposition of the repair material at the damaged coating Fig. 6. Practical steps in the development of intelligent repair materials (Jovanovic and Jovanovic, 2006; Zomaya, 2006) many engineering materials, the correlation of a microstructure with techni- cal properties, e.g. creep, is known.However, themodeling of amanufacturing process to achieve the optimummicrostructure is in infancy.Engineersusema- terial data to predict lifetime of components.However, because of non-existing correlation between the microstructure and material data, safety factors are Innovative modeling methods in damage assessment... 645 included which will not exploit the potential of engineering materials. There- fore, the vision is to develop a virtual manufacturing chain with an optimized technology of materials and low scatter output. Appendix – Mesoscale damage analysis and modeling in the future European research (EuMaT) Themainpractical policyobjective ofEuMaT(EuropeanTechnologyPlatform of Advanced Engineering Materials and Technologies) is to assure optimum involvement of industry and other important stakeholders in the process of es- tablishingEuropeanR&Dpriorities in the area of advanced engineeringmate- rials and technologies (Jovanovic, 2005). EuMaT should improve coherence in existing and forthcoming EU projects, and introduce ”Radical Changes” and assure ”SustainableDevelopment” in the sector of advanced engineeringmate- rials and related technologies (Jovanovic, 2006). Damage analysis and damage modeling, including their application to phenomena of self-healing materials and structures are among the key factors for achieving someof themain overall performance targets of EuMaT envisaged for 2030, like for instance: • helping to reduce life-cycle costs of theprocess equipment and infrastruc- ture by 30% and energy consumption by 50% (more efficient materials) • increase productivity of assets by reducing downtime by 25% (more re- liable materials) • protect the environment by containing processes (e.g. by recycling 95% of metallic and 70% on average of other advanced engineeringmaterials at the end of their useful life) • capture the existing knowledge and effectively train a future workfor- ce and develop capability and capacity to develop a new generation of materials. Innovativedamageassessment andmodelingof advancedengineeringmaterials are important also for EuMaT”horizontal” and ”life-cycle issues” (Fig.7) like simulation, testing, inspection, monitoring, characterization, standardization and qualification of materials and manufacturing processes, prediction of in- service behavior/characteristics and failure criteria, risk and impacts of new materials, training and education issues. 646 A. Jovanovic, N. Filipovic Fig. 7. Place of innovative damage assessment andmodeling for advanced engineeringmaterials in the EuMaT ”life cycle oriented” approach Themulti-scalemodeling of thebehavior of advanced engineeringmaterials is one of the five ”pillars” of EuMaT (Fig.8). InEuMaT it is primarily related to development of new techniques and methods needed to bridge the gap between the atomistic scale andmacro applications. Fig. 8. Five priorities (”pillars”) of EuMaT Acknowledgement This study was partly supported by the Humboldt Fellowship Program IV – SER/1114337 STP hosted by Steinbeis University Berlin/Stuttgart. Innovative modeling methods in damage assessment... 647 References 1. Besold G., Vattulainen I., Karttunen M., Polson J.M., 2000, To- ward better integrators for dissipative particle simulations, Phys. Rev. E, 62, 6, R7611-R7614 2. Boek B., Coveney P., Lekkererker H., van der Schoot P., 1997, Si- mulating the rheology of dense colloidal suspensions using dissipative particle dynamics,Phys. Rev. E, 55, 3124- 3133 3. Espanol E., 1998, Fluid particle model, Phys. Rev. E., 57, 2930-2948 4. Espanol E., Warren P., 1995, Statistical mechanics of dissipative particle dynamics,Europhys. Lett., 30, 4, 191-196 5. FlekkoyE.G.,CoveneyP.V., 1999,Frommoleculardynamics todissipative particle dynamics, Phys. Rev. Lett., 83, 1775-1778 6. 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Jovanovic A., 2006,EuMaT – Materials for Life Cycle, The Roadmap docu- ment of the European Technology Platform for Advanced EngineeringMaterials and Technologies, EuMaTConsortium,Published byMPAStuttgart,Germany 12. JovanovicA., Jovanovic S., 2006,Needs of industry in the area of technolo- gy transfer andpractical implementation,Plenary lecture at theNiSIS (Nature- Inspired Smart InformationSystems),Brainstormingmeeting June 2006,NiSIS Consortium, www.nisis.de 13. Lees A.W., Edwards S.F., 1972,The computer study of transport processes under extreme conditions, J. Phys. C, 5, 1921-1926 14. Novik K.E., Coveney P.V., 1997, Using dissipative particle dynamics to model binary immiscible fluids, Int. J. Mod. Phys. C, 8, 4, 909-918 648 A. Jovanovic, N. Filipovic 15. RevengaM., Zuniga I., EspanolP., Pagonabarraga I., 1998,Boundary models in DPD, Int. J. Mod. Phys. C, 9, 1319-1331 16. Pivkin I.V., Karniadakis G.E., 2005, A New Method to Impose No-Slip Boundary Conditions in Dissipative Particle Dynamics, www.dam.brown.edu/ scicomp/publications/Reports/Y2005/BrownSC-2005-02.pdf 17. Zomaya A.Y. (edit.), 2006, Handbook of Nature-Inspired and Innovative Computing Integrating Classical Models with Emerging Technologies, Springer Science, BusinessMedia, Inc. Innowacyjne metody modelowania w szacowaniu zniszczenia. Zastosowanie dynamiki dyssypatywnego układu punktów materialnych do symulacji zniszczenia samonaprawiających się powierzchni pokrywanych polimerami Streszczenie W artykule zaprezentowano przykładową aplikację innowacyjnej techniki mode- lowania DPD (Dissipative Particle Dynamics – dynamika dyssypatywnego układu punktów materialnych) jako możliwego narzędzia do wielo-skalowego modelowania zachowania się zaawansowanychmateriałów inżynierskich (z pokryciamiwierzchnimi, lub wieloskładnikowych). Modelowanie wielo-skalowe jest związane głównie z rozwo- jem nowychmetod badawczych potrzebnych w opisie zjawisk przy przechodzeniu od poziomu atomistycznego do makroskali. Przedstawiona analiza przydatności metody DPDwszacowaniuzniszczeniazawierającegoproblemoddziaływaniacząstekmateria- łu, polimerów i powierzchni elementów stanowi przykład aplikacji nauki o koloidach. Wpracypotwierdzono stosowalnośćDPDwefektywnym łączeniumakroskali z pozio- mematomistycznym.Spodziewane zastosowaniazaprezentowanejmetody i omówiony przykładwchodząw zakres zaawansowanych technologii samonaprawiających sięma- teriałów, których rozwój jest konsekwencją wytycznych EuMat (w Dodatku). Manuscript received May 10, 2006; accepted for print June 2, 2006