Mesoscale Phenomena in Fluid Systems - American Chemical Society

Chapter 14

Mesoscale Simulations: Industrial Applications 1

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SimonMcGrother ,Lam Yeng Ming *, and Gerhard Goldbeck-Wood 1

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Downloaded by UNIV LAVAL on July 13, 2016 | http://pubs.acs.org Publication Date: August 17, 2003 | doi: 10.1021/bk-2003-0861.ch014

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Accelrys Inc., 9685 Scranton Road, San Diego,CA92121 Nanyang Technological University, School of Materials Engineering, Nanyang Avenue, Singapore 639798

The industrial importance of mesoscale phenomena is increasingly appreciated. Techniques for modeling such length and time scales are increasingly validated and available in standard software packages. Accelrys offers DPD (dissipative particle dynamics) and MesoDyn. Both tools coarse-grain the familiar atomistic representation of the molecule to gain orders of magnitude in both length and time scale relative to traditional atomistic scale simulation. The chemistry is captured through effective potentials based on the energy of mixing of the binary pairs in the system. The methods yield structural and dynamic information on phase morphology and can be used as input to finite element analysis. In this chapter the accuracy of such tools is assessed with particular emphasis on the communication between various length scales. Case studies from the fields of polymer science and surfactant science are discussed, recent advances are presented and ideas for future applications and directions are suggested.

© 2003 American Chemical Society

Case and Alexandridis; Mesoscale Phenomena in Fluid Systems ACS Symposium Series; American Chemical Society: Washington, DC, 2003.

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Introduction The need for mesoscale modeling is clear. Structures which form at length-, and over time-scales that are large compared to atomistic motion critically effect the properties and hence endOuses of a material. There are several schemes available to study fluids where the underlying materials are faithfully represented, but the dimensions of the system are coarse-grained in order that large-scale phenomena may be interpreted or predicted. This paper examines the current state of the art of mesoscale simulation and the applications that such modeling has found in industry. Materials' modeling is an increasingly integrated tool in the R & D portfolio. The unique insights available through simulation of materials at the quantum, molecular, mesoscale and finite element levels can provide discontinuous scientific advances. The most significant challenge presently facing the development of materials modeling is the communication between the length and time-scales spanned by the various methods. The envisaged analogy is with a microscope: ideally one would like to be able to zoom the modeling, seamlessly between the levels of detail. This paper will address recent efforts to integrate the scales. The principal focus will be on the derivation of accurate input parameters for mesoscale simulation, and the subsequent use of finite element modeling to provide quantitative information regarding the properties of the simulated mesoscale morphologies. In polymer modelling, traditional molecular dynamics may be used to obtain thermodynamic information about a pure or mixed system. Properties obtained using these microscopic simulations assume that the system is homogeneous in composition, structure and density: this is a limitation. The slow dynamics of polymers also dictates that smart methods are needed to probe phase space. A mesoscale modelling approach can overcome many of these limitations. These tools achieve longer length scales by uniting many atoms into a single bead, and longer time scales by integrating out the fast motions of the underlying particles leaving only soft, effective interactions. The familiar atomistic description of the molecules is coarse-grained leading to beads of fluid (representing the collective degrees of freedom of many atoms). These beads interact through pair-potentials which crucially, i f meaningful data are to be obtained, capture the underlying interactions of the constituent atoms. Complex self-assembling fluids that have long-range order can be studied. The use of atomistic modeling to parameterise mesoscale simulation by suggesting sensible coarse-graining and effective interactions between species will be discussed. The major hole in the technology remains the reverse mapping from the mesoscale to the atomistic, where no adequate method has been developed



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The primary output of mesoscale modeling is phase morphologies with sizes up to the micron level. These morphologies are of interest, but it is non-trivial to relate such morphologies to material properties. Finite element modeling can be used to predict physical and mechanical properties of arbitrary structures. Details of the link that has been established between Accelrys' MesoDyn and MatSim's Palmyra are given and highlighted with some recent validation work on polymer blends. These results suggest that the combination of simulations at multiple scales can yield important insights.Whilst much of a materials' properties can be determined or inferred from chemistry of the underlying molecules, a wide-range of polymeric systems have functionality that is dependent purely on the structures that the material adopts at the mesoscale. At this scale atomistic information is not necessary but bulk (e.g., fluid dynamics) simulation is too coarse. A perfect example is that variety of applications for HIPS, all of which are made from polystyrene, yet each has distinct application because of the structure that the material gains on processing. MesoDyn and D P D are coarse-grained dynamics methods that capture such structural information whilst retaining a link to the chemistry of the underlying materials. MesoDyn is a dynamic variant on mean-field Langevin density functional theory. The system evolves because of local gradients in the chemical potentials of the various species that make up the system. These gradients arise from a gaussian chain Hamiltonian decorated with an effective external potential to represent the non-ideal interchain interactions. D P D is a coarse-grained dynamics algorithm that accurately captures the hydrodynamic behavior of the fluid and the underlying interactions of the species. D P D is ideal for surfactant solutions and MesoDyn for polymer melts and blends. The question that needs to be answered is: can we pose a question purely in terms of chemical formulation and accurately predict "nanoseale" morphology and, consequently, realistic material properties? The strategy that we have employed to address this issue comprises three stages. Simulation at the fully atomistic level, employing force-fields and molecular dynamics; mesoscale simulation using parameters obtained at the previous level to define interactions and mesoscale-topology, and finally finite element study of the properties of a nano-scaled structure obtained from mesoscale modeling.

Method description Many modeling methods are used individually and combined in industrial problem solving. These span the gamut of length and times from the sub-atomic to the macroscopic. With molecular modeling, the focus is on chemical defined species, with various levels of coarse-graining. A comparison of various levels of


230 "molecular" modeling: quantum, classical atomistic and mesoscale is shown in Table I

Table I. Comparison of modeling techniques


Length Fundamental Unit Time scale Dynamics

Quantum Angstroms

Atomistic nm

Mesoscale 100s of nm

Electrons/nuclei

atoms

fs Not appropriate

ns F=ma

Beads representing group of atoms ms Hydrodynamics

The primary techniques we used for mesoscale modeling are MesoDyn and D P D . In this study we also use finite element methods to obtain material properties of mesoscopically-structured materials

MesoDyn MesoDyn is a dynamic mean-field density functional theory for complex fluids (1). The free energy comprises an ideal term based on a Gaussian Chain Hamiltonian representation of the polymeric materials; a Gibbs entropy contribution favoring mixing and a non-ideal term accounted for using a meanfield approximation. The key approximation is that in the time regime under consideration the distribution functions are optimized (i.e., the free energy is minimal). Applying appropriate constraints the optimal distribution can be obtained and related back to the free energy. We are left with a simple expression for the non-ideal term (obtained by invoking the random phase approximation-RPA):

*

IJ

which assumes a local mean-field. However, because the mean-field must account for the inter-chain interactions a non-local mean-field is preferred. A suitable choice leads to: Fnid

lp] = l l l ^

le,Ar-A) {r)pÀr')drdf Pl

IJ

where E (jr — r'|) is a cohesive interaction defined by the same Gaussian kernel i}

as in the ideal chain Hamiltonian. This parameter is then directly related to a


231 calculable property, namely the Flory-Huggins interaction parameter χ. Further details are given in the chapter by Sevink, Zvelindovsky and Fraaije

DPD


Dissipative particle dynamics (DPD) uses soft-spheres to represent groups of atoms, and incorporates hydrodynamic behavior via a random noise which is coupled to a pair-wise dissipation. These terms are coupled so as to obey the fluctuation-dissipation theorem. The force acting on one of these soft-spheres comprises three pair-wise terms, the first is a soft conservative repulsion related to the energy of mixing of

Ω y

=

w

Mr^rij.

r

( ij )

3

J

v..

2kT

the bead with its neighbor, the second is a random noise term and the last is a dissipative or drag term. The noise and dissipation are coupled (by the fluctuation-dissipation theorem) and act as a thermostat. Further details are given in the chapter by Noro and Warren and the chapter by Broze and Case. The two methods overlap, but D P D is preferred where concentrations are low, and MesoDyn is ideal for systems that comprise polymer melts and blends.

Finite Element methods Standard solvers interpolate pure component data to yield very accurate results for complex, mixed systems. In this study we use GridMorph and Palmyra (2,3) from MatSim GmbH.

Parameterization.

Both D P D and MesoDyn rely on links with the underlying chemistry for them to be useful in industrial settings. With MesoDyn the use of a non-local mean field for the non-ideal contribution to the free energy leads to an expression involving a cohesive energy interaction between neighbor species. The choice of interaction is not unique, but cohesive energy is a readily calculable property (4,5) and therefore makes a sensible choice.



232 For D P D Groot and Warren (6) established very clearly a link between the conservative repulsion and the Flory-Huggins parameter of the interacting species. In addition they make sensible arguments for the value of repulsion between beads of like kind (relating this value to the compressibility of the pure fluid). Finally they established bounds on the noise (and dissipation) at which the system behaves sensibly. The Flory-Huggins parameter χ, which captures the enthalpic contributions to the mixing free energy for two materials, can fitted from experimental data or calculated using atomistic scale simulation. Methods are described in the paper by Case and Honeycutt (7). One particularly promising approach is to fit χ to the enthalpy of mixing obtained from atomistic simulations of the bulk pure and mixed phases. The Compass forcefield has been shown to provide accurate predictions of cohesive properties such as energy of mixing (4,5). For both methods the molecules must be described, at a coarse-grained level, by connected beads. Ideally we would represent the real chain by a beadspring model with the same response functions. In practice, it is more convenient to use the ideas of Kuhn and represent a statistical unit of the chain by a single bead (8,9).

Validation Studies

1) Predicting phase diagrams for block copolymers in water Both mesoscale methods have been validated by comparison with experimental phase diagrams of block copolymers and their aqueous solutions (70-/2). Pluronic molecules (ethylene oxide, propylene oxide, ethylene oxide triblock copolymers) are used extensively in lubrication and drug delivery systems. They are benign, stable, cheap to manufacture and self assemble into complex phases depending upon the relative block lengths and the concentration of solution. Consequently they have enormous industrial value. When a typical pluronic is added to water the relative hydrophobicity of the core propylene oxide leads to meso-phase separation. Depending upon the concentration of pluronic in the solution various phases are observed. With Gaussian chain representations obtained by including several monomers into each bead, and interaction parameters obtained from experimental vaporpressure data MesoDyn predicts the correct phase at the correct point in phase space (77).


233 2. Calculation of critical micellar concentration (CMC)


The CMC is a characteristic concentration of a surface-active agent (surfactant) in solution defined by the appearance of aggregate structures such as micelles. Experimentally, CMC can be obtained from a discontinuity in the interfacial tension at an oil/water or water/air interface as a function of surfactant concentration. D P D can be used to measure the interfacial tension at a planar interface from the difference in transverse and planar pressures, and predict CMC (12). A t concentrations above CMC, micelles can be observed forming and detaching from the interface. To relate interfacial tension (DPD units) to real surface tension it is necessary to stipulate the size and mass of the bead, this fixes a time frame for the simulation and permits conversion to real units.

3. Effect of Temperature MesoDyn has been used to study the temperature dependence of a pluronic (P85) in water at a concentration of 27%. Experimentally such systems are seen to evolve from a spherical, micellar solution at 15°C to a rod-like dispersion at 70°C. The parameters for the pluronic plus water are known at 25°C, but little validation work exists to suggest whether such parameters can be used at other temperatures. From vapor pressure data (13) the temperature dependence of the Flory-Huggins parameter can be inferred. The usual expression is:

For ethylene oxide with water, 0^=2.85 and β= -439K, for propylene oxide in water (£=2.023 and β= -97.9K. These parameters are used to obtain χ at both temperatures. The phases observed using MesoDyn at these temperatures can be seen in color Plates l a and l b . These phases agree well with experiment (14), and the dependence on Τ of the χ parameter proves to be important.

Case Studies

1. Mesoscale modeling of drug delivery Pluronic solutions can be used to deliver hydrophobic drugs. The selfassembly of pluronics leads to hydrophobic regions where the drug resides. In a



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Plate la. Pluronic 85 in water at 15 from MesoDyn simulation. A predominantly spherical micellar emulsion is seen. (See page 4 of color insert.)

Plate lb. Pluronic 85 in water at 70 *€fromMesoDyn simulation. The system appears to be in a coexistence region between spherical and rod-like micelles. (See page 5 of color insert.)


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typical example haloperidol (an anti-psychotic) is dissolved in a solution of pluronic in water. From electron micrographs (Figure 1) the drug is seen to significantly distort the size and shape of the micelles formed, from a uniformly spherical distribution when no drug is present. MesoDyn simulations are show in Figure 2. When 1% drug is present significant distortion is seen. The behavior can be understood by comparing the contents of each micelle: some contain no drug, while others are quite drug rich. This distribution of concentrations contributes to the non-sphericity and sizedistribution obtained. The core size increases with concentration of drug, with good agreement between experimental and simulated values (See Figure 3). Mechanistically, it

Figure la. Experimental TEM image of a 10% solution of pluronic F127 in water.

Figure lb. Experimental TEM image of a 10% solution of pluronic F127 in water with a small amount of haloperidol present.



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Figure 2a. MesoDyn simulation snapshot of pluronic F127 in water at 24% concentration. Uniform, spherical micelles form with propylene oxide in the center

Figure 2b. MesoDyn simulation snapshot of pluronic F127 in water at 24% concentration with 1% haloperidol present. Significant size and shape distribution of the hydrophobic region is observed.


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can be envisaged that the haloperidol aggregates quickly into small units and the slower moving pluronic self-assembles around these units. This leads to a range of drug concentrations in the micelles. Control of size distribution might be achieved by addition of a dispersant (some component which prevents or delays the aggregation of drug). A more homogeneous solution of drug would be better distributed between the various micelles. Since the drug must cross the bloodbrain boundary, it is imperative that aggregate size distribution is adequately controlled.

Figure 3. Core radii of self-assembled micelles in a solution of pluronic in water as a function of concentration of pluronic. Experimental values are shown as diamonds, simulated values as squares.

2. Mesoscale and Finite Element Simulation of Polymer Blends The structures predicted by mesoscale modeling give rise to diverse and interesting material properties. Such properties can be calculated with Finite Element Methods such as Palmyra from MatSim GmbH. These methods require as input the structure of the material, and the property of the pure components that comprise the mixture. Using standard solvers the finite element code can then predict the property for the structured material. As a test case we studied the oxygen diffusion through a material designed to act as a gas separation membrane. A binary blend of polystyrene and polybutadiene was simulated with MesoDyn using parameters obtained from atomistic level modeling. These


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polymers tend to phase segregate and large domains form with little interface. Upon addition of a diblock copolymer of both species (styrene and butadiene), the blend is compatibilized and the interfacial tension is lowered. The resulting morphology is far more complex with much smaller domains, more interfacial zones and frustrated regions. Snapshots of the uncompatibilized and compatibilized blends are show in color Plate 2. Both of these structures were analyzed for oxygen diffusion using MatSim's Palmyra. The pure component oxygen diffusions for polystyrene and polybutadiene were obtained using classical dynamics with fully atomistic interactions (and again the Compass force-field (4)). It is also possible to extract such information from structure property relationships such as those proposed by Bicerano (75). The results of the F E M calculation are given in Table II.

Table II - predicted oxygen diffusion rates System Without Compatibilizer With Compatibilizer

Oxygen diffusion (Dow Units) 970 1040

The compatibilized blend shows increased diffusion of oxygen, which can be attributed to an increase in the number of channels that the oxygen can choose to diffuse through. The finite element method also indicates the diffusion rates for various spatial regions. Color Plate 3 shows flux rates predicted by Palmyra for the blend. A l l the flux is seen to be in the polybutadiene rich regions, with maximum rate in the "necking" regions. The finite element calculation also indicates that there is no preferred direction for flux (diffusion in each Cartesian coordinate is equal), indicating that our system size is probably sufficiently large to represent the bulk system. This study therefore has used interaction energies and diffusivities obtained from atomistic scale simulations to parameterize mesoscale methods and inform finite element tools, in order that mesoscopically calculated structures can be analyzed for diffusion rates of the true material. This is an exciting development that we intend to pursue.

Conclusion Mesoscale modeling methods are increasingly being applied to predict materials properties and behavior, as shown by the validation and case studies in this chapter and in other chapters in this book.



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Plate 2a. Snapshot from MesoDyn simulation of polystyrene and polybutadiene equi-molar mixture. The surfaces indicate where each component falls to half its maximum density (unsurprisingly these surfaces coincide). (See page 5 of color insert)

Plate 2b. Snapshot from a MesoDyn simulation of an equi-molar mixture of polystyrene and polybutadiene with 10% of the system comprising a symmetric diblock copolymer of styrene and butadiene. This copolymer acts as a compatibilizer, lowering the interfacial tension and leading to a more complex morphology (See page 6 of color insert.)



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Plate 3. Flux rates for a 2d slice through the simulation cell, obtained using Palmyra. Blue implies zero flux, white corresponds to maximum flux. Oxygen does not permeate the styrene rich regions (See page 6 of color insert.)


241 The use of accurate classical force-fields to generate material parameters, coupled with the novel ability to simulate system on a longer length and timescale using mesoscale techniques has been demonstrated. The novel approach herein is to take the simulated mesoscale morphology and to use finite element methods to predict realistic material properties based on the morphology obtained. This now gives the modeler a direct route from the atomistic description of the system to a trust-worthy estimate of the properties of a material. The material properties of course depending on processing and formulations, but these effects are not beyond the scope of multilevel simulation.


Acknowledgements We are most grateful to Albert Widmann-Shupack and Andrei Gusev of MatSim GmbH for their collaboration and support.

References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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Mesoscale Phenomena in Fluid Systems - American Chemical Society

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