On Thermal Interface Materials With Polydisperse Fillers: Packing

Proceedings of the ASME 2018 International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems InterPACK2018 August 27-30, 2018, San Francisco, CA, USA

IPACK2018-8337

ON THERMAL INTERFACE MATERIALS WITH POLYDISPERSE FILLERS: PACKING ALGORITHM AND EFFECTIVE PROPERTIES *

Piyas Chowdhury IBM Research Albany, New York, USA Email: [email protected]

Kamal Sikka IBM Research Albany, New York, USA Email: [email protected]

Anuja De Silva IBM Research Albany, New York, USA Email: [email protected]

Indira Seshadri IBM Research Albany, New York, USA Email: [email protected]

ABSTRACT Thermal interface materials (TIMs), which transmit heat from semiconductor chips, are indispensable in today’s microelectronic devices. Designing superior TIMs for increasingly demanding integration requirements, especially for server-level hardware with high power density chips, remains a particularly coveted yet challenging objective. This is because achieving desired degrees of thermalmechanical attributes (e.g. high thermal conductivity, low elastic modulus, low viscosity) poses contradictory challenges. For instance, embedding thermally conductive fillers (e.g. metallic particles) into a compliant yet considerably less conductive matrix (e.g. polymer) enhances heat transmission, however at the expense of overall compliance. This leads to extensive trial-and-error based empirical approaches for optimal material design. Specifically, high volume fraction filler loading, role of filler size distribution, mixing of various filler types are some outstanding issues that need further clarification. To that end, we first forward a generic packing algorithm with ability to simulate a variety of filler types and distributions. Secondly, by modeling the physics of heat/force flux, we predict effective thermal conductivity, elastic modulus and viscosity for various packing cases. *

NOMENCLATURE TIM Thermal interface material BLT Bond line thickness, which is equal to L CTE Coefficient of thermal expansion r Radius T Temperature (oC or K) C, R Thermal conductance (W/K in SI unit) and thermal resistance (K/W) respectively L, A, V TIM Thickness and chip-side area and TIM volume (V = LA) respectively u Displacement F, q Total force (N) and heat flux (J) Effective viscosity of the particulate ηeff composite Viscosity of the matrix material ηmatrix N k p, k m E p, E m keff, Eeff α, ε

Total number of particles Thermal conductivity (W/mK) of particle (filler) and matrix respectively Elastic modulus (GPa) of particle (filer) and matrix respectively Effective thermal conductivity and elastic modulus of particulate composite Inter-particle heat/force transfer and network connectivity parameter (dimensionless) respectively

Corresponding author

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Copyright © 2018 ASME

φ , φi Δφi Φi P(r) σ, µ, µs

Total volume fraction of particles and volume fraction of i-th particle respectively Infinitesimal volume fraction increment

In a typical microprocessor, 90% of the heat emanates from computing cores with high power density [2, 4, 5] compared to the chip area populated with cache memories (Fig. 1). The principal concern in the die thermal management is to preclude these non-uniformly distributed hotspots surpassing a threshold, rather than maintain a prescribed average temperature throughout the die. From the materials design standpoint, tailoring the TIM microstructure to preferentially enhance thermal transport from such localized vulnerabilities poses a very promising yet inadequately addressed solution. The main emphasis in TIM research today is to reduce its overall bulk thermal resistance. The contact resistance induced by asperities is relatively small, which is typically mitigated by applying pressure. Improving TIM bulk properties, however, remains a highly empirical field, whereby new materials are invented through exhaustive trial-and-errors due to a lack of mechanistically based design principles [2]. For instance, one common strategy today is to fill the polymer matrix with conductive particles to the maximum fraction possible. This strategy generally results in increased thermal conductivity (keff), however, at the expense of elastic modulus (Eeff) and viscosity ( ηeff ). With a view to co-optimizing the

Maximum packing fraction of a polydisperse composite after addition of ith particle Probability density function of radius r Input standard deviation, mean, and smallest mean of particle size distribution

BACKGROUND Today there exists a mounting need for denser integration of semiconductor chips with the advent of smart electronics, high-performance supercomputers and mainframes for consumer, business and scientific computing. For these demanding applications, a TIM remains an essential part of the thermal solution [13], which facilitates heat transport from a chip to a heat sink (Fig. 1). A polymeric TIM microstructure consists of a soft matrix impregnated with conducting fillers. Its desired properties are threefold: (a) high thermal conductivity, (b) low elastic modulus and (c) low viscosity. While enhanced thermal conductivity enables higher power dissipation or cooler temperatures, a low elastic modulus is vitally needed alongside to reduce deformation and stress, which arises due to the CTE mismatch among attached components. Additionally, a reduced degree of viscosity is important for the ease of dispensing the TIM during assembly. Ensuring all these three attributes is essential for convenience of assembly and for long-term package reliability.

desired properties, it would be useful to assess a priori the effects of, say, the presence of predominantly large/small particles, homogenized versus preferential particle distribution, mixing

FIGURE 1: HEAT IS TRANSPORTED FROM A MULTI-CORE CHIP WITH NON-UNIFORM POWER DISTRIBUTION THOUGH THE TIM, WHOSE BULK THERMAL RESISTANCE ( BLT keff ) DICTATES THE HEAT TRANSPORT EFFICIENCY.

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different types of particles, etc. To that end, a unified modeling framework capable of generating TIM microstructure and predicting effective properties is needed. In this paper, we proffer such a model.

collective random walk [19, 20], combined static/dynamic approach [21, 22], and distinct element method [23, 24]. By contrast, the sequential addition of particles is computationally more feasible, however, generally with a limited capability for reaching high packing fractions. Random drop [25], drop-fall-shake [26], random walk/contact algorithms [27, 28] are some examples of this category. From the perspective of TIM application, it is the final structure generated that is of utmost practical importance, irrespective of the transient state or the packing methodology. Hence, we adopt a simple sequential algorithm for generating various TIM structures, to be utilized in predicting keff, Eeff and ηeff .

Recent literature notes significant advances in the theory of effective properties (keff, Eeff and ηeff ) of a particulate composite [6-9]. For the heat transfer problem, inter-particle thermal conductance formulations (originally derived by Batchelor and O’Brien [10]) are lately revisited, leading to more accurate predictions. The local conductance (or resistance) is used to create a global 3D network of interconnected nodes, from which nodal temperatures can be solved [11, 12]. Noting the fundamental similarities between steady-state heat and force exchange between particles, Subbarayan and coworkers proposed a similar spring network for the particulate structure for estimating the force flux through the bond line thickness (BLT) [8]. In this formalism, a displacement gradient from top to bottom boundaries is imposed, which in turn generates force flux transmitting across the BLT. With a view to modeling polydisperse viscosity, Dorr et al. have lately proposed a discrete model based on one-by-one addition of particles to the matrix [9]. They formulated a closed form analytical expression for ηeff by proposing a generalized framework to

Numerical Generation of Particle-laden TIM Structures The current packing algorithm involves two general steps: (i) first, selecting a particle type and then its radius from a certain distribution, and (ii) secondly, randomly generating its center coordinates within a pre-defined volume (V). The scheme, as illustrated in Fig. 2, is implemented in MATLAB. To generate a radius (r) from a Gaussian distribution with specified mean (µ) and standard deviation (σ), a random magnitude of the normal cumulative density function is generated such that the corresponding probability density is bounded by 3σ confidence level i.e. P(r) = P(µ-3σ ≤ r ≤ µ+3σ). Inversion of the cumulative density function provides P(r), which then can be inserted into Equation (1) to obtain the corresponding value of r.

adapt monodisperse viscosity correlation (e.g. Quemada equation [13]) for predicting polydisperse cases. Central to all these endeavors of predicting Keff, Eeff and ηeff is the numerical generation of a particle-laden microstructure, especially polydisperse one with different particle types.

a

P(r) =

In the literature, the numerical schemes of modeling particulate composite structures generally fall into two broad categories [14-16]: (a) collective rearrangement, and (b) sequential addition. The collective rearrangement algorithm involves a predefined number of particles (i.e., with known size distribution and initial center locations) undergoing simultaneous positional updates. This method becomes computationally very expensive with increasing number of particles (N>104), as their positions need to be evaluated and updated at each infinitesimal time-step. Moreover, step-wise computation of particle-binding force field, oftentimes included in these models, further adds to computational burden. Nonetheless, collective rearrangement in general can yield structures with large packing fractions. Notable examples of this category include Lubachevski-Stillinger algorithm [17], Kansal-Torquato-Stillinger algorithm [18],

1 2πσ 2

−( r−µ )

e

2σ

2

2

(1)

A rejection criterion is enforced within an iteration loop, whereby any newly generated particle, if overlapped with pre-existing one(s), is discarded. The loop continues until a predefined volume fraction or number of particles is reached. Upon rejection, one can either generate a new center position (for an existing radius) or generate a new particle type, radius and then center location. This method, albeit apparently simplistic, proves quite efficient, in that various structures can be concocted within reasonable computational time.

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Fig. 3 presents CASE I, where three types of particles are packed (Type A, B and C). The particles are classified into different types so that we can subsequently ascribe different material properties (Al, ZnO, BeO, Al2O3, etc.) when predicting keff and Eeff. Each type has its own P(r) with certain µ and σ. Note that the dimensions are all expressed in units of the smallest mean particle radius (µs), which will be the convention henceforth for all other dimensional parameters. Note that the generated structures have multimodal particle distributions. The frequency of each type of particles is consistent with the corresponding input probability density distribution. Fig. 4 presents CASE II, where particles with larger mean radii are randomly packed. In Fig. 5, three more structures (CASE III, CASE IV and CASE V) are presented. For CASE III, IV and V, first the largest particles (orange colored) are randomly generated and placed inside the volume until a convergence condition is met (either in terms of a limiting volume fraction or number of rejection per accepted particle). Second and third types of particles are packed similarly. Simulation details are summarized in Table 1. Note that, for visualization convenience, only half of the total domain size (split at the mid-section) for CASE I and III is presented. The actual domain volume is indicated by the dark outline.

FIGURE 2: PACKING ALGORITHM FLOWCHART FOR GENERATING PARTICLE-LADEN TIM MICROSTRUCTURE.

FIGURE 3: THREE TYPES OF PARTICLES ARE PICKED FROM DISTINCT GAUSSIAN DISTRIBUTIONS. THE RESULTANT STRUCTURE IS A MULTIMODAL ONE.

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FIGURE 4: THREE TYPES OF PARTICLES ARE PACKED WITH SPECIFIED INPUT DISTRIBUTIONS LEADING TO A MULTIMODAL STRUCTURE.

FIGURE 5: THREE DIFFERENT CASES OF PARTICLE PACKING.

TABLE 1: SUMMARY OF PARTICLE PACKING SIMULATIONS. Structure

Volume fraction, ϕ (%)

Number of particles, N

Box size ( µ 3s )

Types of particle

CASE I CASE II CASE III CASE IV CASE V

62 65 60 62 66

450,431 1,000 166,791 94 373

285.71 x 285.71 x 142.86 714.28 x 714.28 x 357.14 142.86 x 142.86 x 142.86 142.86 x 142.86 x 142.86 142.86 x 142.86 x 142.86

3 3 3 2 3

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PREDICTING keff

thermal resistance (SI unit being K/W) between the particle i and the particle j can be expressed as:

The method of predicting keff consists of three steps [7, 11]: (i) modeling the local thermal resistances between two particles accounting for intra-particle and trans-matrix heat transport, (ii) constructing a global 3D resistance network connecting nodes (i.e. particle centers) and solving nodal temperatures with prescribed boundary conditions, and (iii) finally calculating the heat flux considering the boundary particles connected to the hot surface (or the cold surface) and then calculating keff from the Fourier’s conduction law.

Rij = Ri + Rmatrix + R j

(3)

Note that in Equation (3) there is no explicit consideration of particle-matrix interface resistance, the effects of which are assumed to be implicitly regulated by the parameter α . For instance, a high value of α means increased rij i.e. enhanced interparticle heat transport. The terms

Ri

and

Rj

represent the absolute thermal resistances of the frontal hemispheres of the individual particles, and can be expressed as below [12]:

(i) Inter-particle Thermal Resistance Model

Ri =

⎛ r + r2 − r2 ⎞ 1 i i ij ⎟ log ⎜ 2 2πk p ri ⎜⎝ ri − ri − rij2 ⎟⎠

(4)

Where, k p denotes the thermal conductivity of the particle i. The thermal resistance of the matrix Rmatrix as encompassed by the cylindrical conduit can be expressed as follows [12]:

FIGURE 6: MODEL FOR PARTICLE THERMAL EXCHANGE, WHEREBY CONFRONTING HEMISPHERICAL FACADES OF INDIVIDUAL PARTICLES ARE ASSUMED TO TRANSMIT HEAT VIA A CYLINDRICAL CONDUIT.

Rmatrix

We model the inter-particle thermal resistance after the analytical solutions by Batchelor and O’Brien [10], which has recently been further revisited by Subbarayan and co-workers [7, 8, 12]. In this model, the physical process of heat transfer between two particles (i and j) occurs via a cylindrical conduit (Fig. 6), whose radius rij is a

r ⎞⎤ α ⎡ ⎛ ⎢ log ⎜ 1+ α ij ⎟ ⎥ = πkmrij ⎢ ⎝ hij ⎠ ⎥ ⎣ ⎦

−1

(5)

Where, km is the thermal conductivity of the matrix. The inter-particle surface-to-surface distance hij is given by:

(

hij = Dij − ri + rj

function of the radii of individual particles ( ri and rj

)

(6)

respectively). Where,

rij = α

2ri rj ri + rj

Dij

is the inter-particle center-to-center

distance. Note that a particle i would also exchange heat with the flat boundaries (top and bottom). In that case, one of the radii in Equations (2) would be infinity, representing the radius of curvature for a flat surface. For example, if the index j denotes the surface, then rj → ∞ and hence rij = 2 α ri (which should be used in Equations (4) and (5) to obtain the corresponding resistance terms). The total thermal

(2)

Where, α is a tuning parameter, whose value (ranging from 0.1 to 0.5) is determined on an ad hoc basis for the current work. As in Fig. 6, the total absolute

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resistance between the particle i and the surface would be the sum of Ri and Rmatrix. (ii) Global Model

Thermal

Resistance

top boundary (Ttop) while the sidewalls are adiabatic (i.e. no thermal connection with any particle). Note that lines connect particles to top/bottom boundaries but not to sidewalls. By setting rj → ∞, the condition for a particle i exchanging heat with either top or bottom surface (denoted j) can be obtained from Equation (7) as follows:

Network

With the inter-particle thermal resistances determined, the collection of particles from the packing simulation is now ready for a conversion into a global resistance network. The network consists of nodes, each of which represents the particle center as connected to the neighboring particles. The particle i is thermally connected with the nearby particle j only if:

hij < ε

2ri rj ri + rj

hij < 2ε ri

(8)

To balance heat flux at any node i connected to its neighbors j (j = 1,2,3… N neighbor ; N neighbor being variable for each i as dictated by Equation(7)), the heat inflow must equal heat outflow. Thus we can write:

(7) N neighbor

∑

Where, ε is a tuning parameter (typically 1), and can be expressed (based on [9]) as follows:

system is expressed as a function of the total volume fraction ( φ ) and the maximum volume fraction achievable ( φmax ), which has a constant value of 0.64

i−1 ⎛ ⎞ Φ i = Φ i−1 ∑ Δφj ⎜ Φ i−1Δφi + ∑ Δφj ⎟ ⎝ ⎠ j=1 j=1 i

[13, 29-31]. However, these correlations can only capture either the breakdown (at large φ → φmax ) or

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−1

(19)


RESULTS AND DISCUSSION

Knowing the volume fraction of the i-th particle, 1 φi = 4 πri3 , the infinitesimal volume fraction 3 V increment Δφi can be obtained (after [9]) as below:

⎛ Δφi = φi ⎜ 1− ⎝

⎞

N

∑ φ ⎟⎠

Using the formulations in the preceding sections, we predict effective properties (keff, Eeff and ηeff ) of the simulated microstructures (denoted CASE I, II, III, IV and V). The results are summarized in Table 2. We use two sets of materials properties as input for matrix and fillers for comparison. These materials are chosen based on the widely used fillers and matrix materials in the microelectronics packaging industry. The predicted values are well within the order of magnitudes of the properties reported in the literature (both predictions and experimental measurements) [7, 8, 34]. This lends credence to the current approach. We note that there could be ± 10% variation in the tabulated keff and Eeff values due to parametric sensitivity. Nonetheless, if one set of parameters values are used consistently, the effective properties maintain the trend across various packing cases. Currently, the tuning parameters for the keff and Eeff models are set as: α = 0.5 (heat transport), 0.2 (force transfer), and ε = 0 .9 (both heat and force). We report average values from sufficient number of test runs (>10). From Table 2, the trend in the magnitudes of the effective properties ( keff , Eeff and

−1

(20)

j

j=i+1

Following the foregoing recipe, the viscosity for CASE V is predicted and presented in Fig. 10 as an example. The relative viscosity increases nonlinearly as particles are added one-by-one. Note that φ = ∑φi

at any instant of particle addition. A maximum of relative viscosity of 12.75 is reached at an aggregate volume fraction, φ ≈ 66%.

ηeff ) can be summarized as follows: CASE V> CASE I> CASE IV > CASE III > CASE II FIGURE 10: VISCOSITY AS A FUNCTION OF FILER VOLUME FRACTION FOR CASE V. TABLE 2: PREDICTIONS OF EFFECTIVE PROPERTIES FOR VARIOUS PACKING CASES. Composite structures

Volume fraction, ϕ (%)

CASE I CASE II CASE III CASE IV* CASE V

62 65 60 62 66

Same as above

Materials † (W/moC), (GPa) Matrix Silicone: km = 0.20 Em = 0.05

Epoxy: km = 0.3 Em = 4.0

Filler

Ag: Al: Al2O3:

kp = 420 Ep = 83 kp = 205 Ep = 69 kp = 25 Ep = 370

keff (W/moC)

Eeff (GPa)

1.95 1.47 1.51 1.58 2.10 2.98 2.20 2.31 2.37 3.15

0.20 0.09 0.13 0.17 0.22 12.90 6.71 8.01 10.79 14.23

ηeff /ηmatrix 9.53 8.23 8.40 8.90 12.75 Same as above

* For CASE IV, only two fillers, Ag and Al, are considered. † Filler properties are ascribed to three simulated particles types (A, B, C): Ag → A, Al → B, Al2O3 → C.

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From these trends, one can see that the predominance of the largest particle generally leads to increased keff , Eeff and ηeff . This observation is

properties. Additionally, the interface thermal resistance between a particle and the matrix is presently not considered, which leaves room for a potential future revisit. We note that various other combinations of parameters could also be input into the model. A detailed optimization study needs to be performed building on the current modeling foundation.

consistent with the earlier literature findings [6]. Note that a high keff is desired, unlike Eeff and ηeff . It can be inferred that, as a general strategy, the microstructures packed with large particles to their fullest extent along with smaller particles to increase the volume fraction would correspond to increased thermal transport. The superior thermal conductivity of CASE V (where the largest particles are placed in preferential locations) indicates the suitability of preferred filler placement strategy to address local hotspots in the microprocessor. Further research will be conducted mapping out many other packing cases to establish the trends in more detail and co-optimize the thermal conductivity (high), elastic modulus (low) and viscosity (within a certain range). Different combinations of materials will also be investigated. In the current paper, we have outlined the methods and demonstrated their capability for future endeavors.

ACKNOWLEDGEMENT The authors gratefully acknowledge Dinesh Gupta, Sushumna Iruvanti, Xiao Hu Liu, Vijay Narayanan, Arvind Kumar and Rama Divakaruni at IBM Research for their support. RERERENCES [1] Bar-Cohen, A., Matin, K., and Narumanchi, S., 2015, "Nanothermal interface materials: technology review and recent results," Journal of Electronic Packaging, 137(4), p. 040803. [2] Prasher, R., 2006, "Thermal interface materials: historical perspective, status, and future directions," Proceedings of the IEEE, 94(8), pp. 1571-1586. [3] Narumanchi, S., Mihalic, M., Kelly, K., and Eesley, G., "Thermal interface materials for power electronics applications," Proc. Thermal and Thermomechanical Phenomena in Electronic Systems, 2008. ITHERM 2008. 11th Intersociety Conference on, IEEE, pp. 395-404. [4] Mahajan, R., Chiu, C.-p., and Chrysler, G., 2006, "Cooling a microprocessor chip," Proceedings of the IEEE, 94(8), pp. 1476-1486. [5] Hamann, H. F., Lacey, J., Weger, A., and Wakil, J., "Spatially-resolved imaging of microprocessor power (SIMP): hotspots in microprocessors," Proc. Thermal and Thermomechanical Phenomena in Electronics Systems, 2006. ITHERM'06. The Tenth Intersociety Conference on, IEEE, pp. 5 pp.-125. [6] Pietrak, K., and Wisniewski, T. S., 2015, "A review of models for effective thermal conductivity of composite materials," Journal of Power Technologies, 95(1), p. 14. [7] Kanuparthi, S., Subbarayan, G., Siegmund, T., and Sammakia, B., 2008, "An efficient network model for determining the effective thermal conductivity of particulate thermal interface materials," IEEE Transactions on Components and Packaging Technologies, 31(3), pp. 611-621. [8] Vaitheeswaran, P. K., and Subbarayan, G., "Estimation of Effective Thermal and Mechanical Properties of Particulate Thermal Interface Materials (TIMs) by a Random Network Model," Proc. ASME 2017 International Technical Conference and Exhibition on Packaging and Integration of

SUMMARY AND FUTURE The current work can be summarized as follows: 1) A simple algorithm for generating particulate TIM microstructures is proposed. We have generated five distinct cases of particle packing, which are then used for studying effective properties. 2) We have elaborated on models for computing keff, Eeff and ηeff, which are the three important attributes of a TIM that a design researcher should be able to assess. 3) The models are suitable for diverse particle types and size distribution. 4) We present predictions on TIM properties based on the simulated microstructures and the thermal, mechanical and rheological models. In future, further packing cases will be conducted to co-optimize the thermal conductivity (high), elastic modulus (low) and viscosity (within a certain range). To conduct simulations within reasonable computational time, the current packing cases were stopped at a lower volume fraction than the maximum possible. To generate TIM structure with higher volume fraction, one necessary undertaking would be to parallelize the particle packing, so that larger and more densely packed systems can be simulated using supercomputers. A parallel version would allow for greater flexibility in generating various structures, and studying their

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Electronic and Photonic Microsystems collocated with the ASME 2017 Conference on Information Storage and Processing Systems, American Society of Mechanical Engineers, pp. V001T002A020V001T002A020. [9] Dörr, A., Sadiki, A., and Mehdizadeh, A., 2013, "A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction," Journal of Rheology, 57(3), pp. 743-765. [10] Batchelor, G., and O'Brien, R., "Thermal or electrical conduction through a granular material," Proc. Proc. R. Soc. Lond. A, The Royal Society, pp. 313-333. [11] Yun, T. S., and Evans, T. M., 2010, "Threedimensional random network model for thermal conductivity in particulate materials," Computers and Geotechnics, 37(7-8), pp. 991-998. [12] Dan, B., Kanuparthi, S., Subbarayan, G., and Sammakia, B. G., "An Improved Network Model for Determining the Effective Thermal Conductivity of Particulate Thermal Interface Materials," Proc. ASME 2009 InterPACK Conference collocated with the ASME 2009 Summer Heat Transfer Conference and the ASME 2009 3rd International Conference on Energy Sustainability, American Society of Mechanical Engineers, pp. 69-81. [13] Quemada, D., 1977, "Rheology of concentrated disperse systems and minimum energy dissipation principle," Rheologica Acta, 16(1), pp. 82-94. [14] Torquato, S., Truskett, T. M., and Debenedetti, P. G., 2000, "Is random close packing of spheres well defined?," Physical review letters, 84(10), p. 2064. [15] Farr, R. S., and Groot, R. D., 2009, "Close packing density of polydisperse hard spheres," The Journal of chemical physics, 131(24), p. 244104. [16] Desmond, K. W., and Weeks, E. R., 2009, "Random close packing of disks and spheres in confined geometries," Physical Review E, 80(5), p. 051305. [17] Lubachevsky, B. D., and Stillinger, F. H., 1990, "Geometric properties of random disk packings," Journal of statistical Physics, 60(5-6), pp. 561-583. [18] Kansal, A. R., Torquato, S., and Stillinger, F. H., 2002, "Computer generation of dense polydisperse sphere packings," The Journal of chemical physics, 117(18), pp. 8212-8218. [19] Smith, L., and Midha, P., 1997, "Computer simulation of morphology and packing behaviour of irregular particles, for predicting apparent powder densities," Computational materials science, 7(4), pp. 377-383. [20] Smith, L., and Midha, P., 1997, "A computer model for relating powder density to composition, employing simulations of dense random packings of monosized and bimodal spherical particles," Journal

of materials processing technology, 72(2), pp. 277282. [21] Stroeven, P., and Stroeven, M., 2011, "Dynamic computer simulation of concrete on different levels of the microstructure," Image Analysis & Stereology, 22(1), pp. 1-10. [22] Stroeven, P., and Stroeven, M., 1999, "Assessment of packing characteristics by computer simulation," Cement and Concrete Research, 29(8), pp. 1201-1206. [23] Cundall, P. A., and Strack, O. D., 1979, "A discrete numerical model for granular assemblies," geotechnique, 29(1), pp. 47-65. [24] Yang, R., Zou, R., and Yu, A., 2000, "Computer simulation of the packing of fine particles," Physical review E, 62(3), p. 3900. [25] Visscher, W. M., and Bolsterli, M., 1972, "Random packing of equal and unequal spheres in two and three dimensions," Nature, 239(5374), p. 504. [26] Santiso, E., and MÜLLER, E. A., 2002, "Dense packing of binary and polydisperse hard spheres," Molecular Physics, 100(15), pp. 2461-2469. [27] Yang, A., Miller, C., and Turcoliver, L., 1996, "Simulation of correlated and uncorrelated packing of random size spheres," Physical review E, 53(2), p. 1516. [28] Al-Raoush, R., and Alsaleh, M., 2007, "Simulation of random packing of polydisperse particles," Powder technology, 176(1), pp. 47-55. [29] Eilers, v. H., 1941, "Die viskosität von emulsionen hochviskoser stoffe als funktion der konzentration," Kolloid-Zeitschrift, 97(3), pp. 313321. [30] Krieger, I. M., and Dougherty, T. J., 1959, "A mechanism for non-Newtonian flow in suspensions of rigid spheres," Transactions of the Society of Rheology, 3(1), pp. 137-152. [31] Mooney, M., 1951, "The viscosity of a concentrated suspension of spherical particles," Journal of colloid science, 6(2), pp. 162-170. [32] Einstein, A., 1905, "Eine neue Bestimmung der Molekulardimensionen." [33] Batchelor, G., and Green, J., 1972, "The determination of the bulk stress in a suspension of spherical particles to order c 2," Journal of Fluid Mechanics, 56(3), pp. 401-427. [34] Nie, S., and Basaran, C., 2005, "A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds," International Journal of Solids and Structures, 42(14), pp. 4179-4191.

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