Foreword

The Mechanics of Complex Materials is a ten-week undergraduate research program hosted by the Department of Theoretical and Applied Mechanics at the University of Illinois at Urbana-Champaign. The dual purposes of this program are (1) to introduce undergraduate students from a broad background to modern problems in the mechanics of materials and (2) to encourage the students to attend graduate school and pursue a career in research. Support for the program is provided by the National Science Foundation and the Department of Defense, through the Research Experience for Undergraduates.

The focus of the summer program was an individual research project, though weekly activities included both structured supplemental learning opportunities and informal social activities. A weekly seminar included topics on ethics in science and engineering and the development of scientific writing skills. Six faculty research mentors guided the students in their research projects, and several graduate students helped with the projects as well. The program culminated in a research program on August 6, 2004, at which the students first gave brief oral presentations of their work in a two-hour session in the morning, and then presented their results in a poster session in the afternoon. The papers within this volume contain details of the research presented.

Kimberly M. Hill and James W. Phillips, editors

December 2004

Table of Contents Nanoscale stamp deformation resulting from an adhered back layer ..............................................1

John M. Fulton Advisor: K. Jimmy Hsia

Ion-induced surface sputtering: One-dimensional numerical analysis of island formation ............7 Syed Hussain Advisors: Jonathan B. Freund and Harley T. Johnson

Effect of microcapsule size on tensile properties of self-healing composites ...............................15 Joseph H. Lai, Alyssa A. Rzeszutko, and Benjamin J. Blaisik Advisor: Nancy R. Sottos

Interstitial fluid effects on dense free-surface granular flow .........................................................27 Michael E. Ly and Jiafeng Zhang Advisor: Kimberly M. Hill

2D-to-3D transitions in dense free-surface granular flow .............................................................37 Sophie A. McGough and Jiafeng Zhang Advisor: Kimberly M. Hill

Microcapsule-induced toughening of bone cement .......................................................................45 Gina M. Miller and Jason M. Kamphaus Advisors: Scott R. White and Nancy R. Sottos

Punching elastic foams in the self-similar regime .........................................................................51 Renee Oats, Xiangyu Dai, and Igor A. Khoroshilov Advisor: Gustavo Gioia

Flow through a three-dimensional microvascular network due to disproportional heating ..........61 Lyle A. Shipton and Kathleen S. Toohey Advisors: Scott R. White and Nancy R. Sottos

Lennard–Jones fluid simulation: A study of fluid–solid interaction .............................................67 Paul K. Shreeman Advisor: Jonathan B. Freund

Time evolution of nickel aluminide bond coat surface rumpling ..................................................73 Daniel S. Widrevitz Advisor: K. Jimmy Hsia

Viscoelastic response of solid propellant matrix: A criterion for particle dewetting ....................79 Elizabeth A. Zimmermann Advisor: Petros Sofronis

1

Nanoscale Stamp Deformation Resulting from an Adhered Back Layer

John M. Fulton Junior in Materials Science and Engineering

University of Wisconsin–Madison

Advisor: TAM Prof. K. Jimmy Hsia

The primary objective of this project was to model nanoscale stamp deformation using the Westergaard stress function to determine the adhesion energy of the interface between the stamp material and a substrate. The stamp deformation was modeled by numerically solving an integral equation using Mathematica and Microsoft Excel. The program that was built using Mathematica gathered results based on a set of parameters associated with the stress function. The program then produced a plot of force vs. distance and the integrated value of the curve. Excel was used to analyze these results by creating several other plots. We found that as the distance between consecutive stamp punches increases, the punches cease to have a significant effect on each other’s energy. We were also able to formulate several plots relating distance from the center of stamp deformation to its edge vs. a nondimensional parameter, which can be used with experimental data to find the adhesion energy.

Introduction

Nanoscale stamps allow for a simple, inexpensive printing method known as micro-contact printing. These stamps can be made from different types of elastomer and consist of a pattern of very small punches and grooves, which are on the order of microns in size. During microcontact printing a special ink is applied to the stamps and then transferred to the substrate by placing the punches in contact with the surface. The advantages of microcontact printing include the ability to pattern curved surfaces and the ability to use the stamps with a large number of organic and biological materials.1 The major problem impeding the progress of microcontact printing is the deformation of the nanoscale stamps. It has been found that when a stamp is placed on a flat substrate, such as a silicon wafer or glass plate, the grooves within the stamp collapse upon themselves.2 As a result, the corners of the punches are no longer sharp. The accuracy associated with microcontact printing can be attained only with well defined stamp punches, and therefore it is important to find out not only why this deformation occurs, but also what steps can be taken to prevent it. The stamps we investigated are made of an elastomer known as poly(dimethylsiloxane), commonly called PDMS. This elastomer is a flexible material that has a typical shear modulus of less than 1 MPa. I studied an existing theory as to why these stamps deform in the article by Hui, Jagota, Lin, and Kramer.3 These authors suggested that the PDMS stamp punches collapse due to the force of their own weight. This theory, however, has been disproved by showing that the punches collapse whether the stamp is right side up or upside down. A second theory, which

JOHN M. FULTON

2

we decided to investigate, suggests that stamp deformation is a result of van der Waals forces between the stamp and adhered material. If this theory is correct, then a set of parameters consisting of size ratios affects the amount of deformation that will occur.

Methods of analysis

There are currently no existing profile equations that are specifically designed for nanoscale stamps, so Professor Hsia recommended that we model the stamps using the Westergaard crack-opening profile

( ) ( )

( ) ( )

2

2

1

1

2cos

2cos1

2cos

2cos1

cothtanh4)0,(2

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

⎟⎟⎠

⎞⎜⎜⎝

⎛++

−

⎟⎟⎠

⎞⎜⎜⎝

⎛′

=−

−

wax

waw

wab

waw

EPxv

ππ

ππ

π

⎟⎟⎠

⎞⎜⎜⎝

⎛

≤<

≤

axb

bx , (1)

where x denotes position along the stamp punch, a denotes distance between consecutive punches, b denotes distance from the center of stamp deformation to its edge, w denotes one half of the entire punch distance, and P denotes force on the stamp, as illustrated in Figs. 1 and 2. This function is known as the Westergaard stress function. We desired first to model the stamp deformation for the area bx ≤|| and therefore began with the 1tanh− dependence function in equation (1). We knew from experimental results that for bx ≤|| the stamp profile is always equal to the height, as shown in Fig. 2. We were therefore able to use this information to model the force on the stamp for bx ≤|| by integrating the function and applying linear algebra.

y

xa bw

h

Stamp

Adhered Material

y

xa bw

h

Stamp

Adhered Material

Fig. 1. A single stamp punch before

deformation. Fig. 2. The punch after deformation.

To solve the equation, we set boundary conditions and normalized the equation with respect to the width of the stamp punch, 2w. The kernel of the integral was singular, and in order to obtain a numerical value to the integral, the integral was separated into four new definite integrals, two of which were nonsingular and two of which were singular. The finite-difference method was then used to evaluate the integrals. The finite-difference method evaluates an integral by summing the area of many rectangles under the desired curve. The two nonsingular integrals were evaluated by summations and the two singular integrals were discretized. The crack profile equation was now in a form that could be used to compute numerical solutions.

Nanoscale Stamp Deformation

3

In order to perform the kinds of calculations that were needed, Professor Hsia recom-mended that a computer program called Mathematica should be used. A program was then built using Mathematica to gather results using the Westergaard stress function. The program operates by creating an nn × matrix, where n is specified by the user. Each entry in the matrix is an individual function that the program also evaluates. The matrix has a principal diagonal running from the top left to the bottom right. The diagonal entries consist of all four of the previously mentioned integrals. The top right and bottom left elements within the matrix consist of only the two nonsingular integrals. The matrix entries are evaluated by running a loop that substitutes each specific function into each individual entry. Once the matrix has been built, the program then populates two vectors of n elements—one consisting of length along the stamp and one consisting of stamp height. The program then sets the dot product of the matrix and the length vector equal to the height vector, computing a numerical value for each matrix entry. At this point the program has created a table consisting of nn × numerical values. A graph of force vs. stamp length for bx ≤|| is created from these values. The integral of the force on the stamp is needed to find the total system energy

wbdxxP

whU

wb

Γ−= ∫0

)(21 . (2)

Therefore one final addition was made to the program that produces the integral of the force, which is the strain energy. The four variables that can be altered in the program to obtain different results are a, b, w, and h. Only two variables, however, were changed throughout the computations, namely a and b. The values for h and w were kept constant at 1 and 0.05, respectively. The units of the variables were not needed because the function had been normalized by w, so that only the parameter ratios such as b/w were needed.

We also wished to simulate the stamp deformation for axb ≤< . A new program was

built by modifying the initial program. The 1coth− dependence function in equation (1) was entered, and we were quickly able to obtain a model for axb ≤< . This model was not used for any further analysis.

Results

The program was used to find the strain energy for values of a/w from 1 to 10 in steps of 0.5 and values of b/w from 0.05 to 0.95 in steps of 0.015. The results for each set of variables were recorded into a table in Microsoft Excel. When all the results had been obtained, a plot of strain energy vs. a/w was made, as shown in Fig. 3. We see from Fig. 3 that, as a/w increases, the resulting strain energy value changes less and less. I showed this to Professor Hsia and he suggested that we test a/w = 100. The a/w plot was increased to include 100 and it was found that the result differed from the previous point (a/w = 10) by less than 0.01%. This change is very small compared with the function itself, and therefore it was concluded that as the stamp punches become farther and farther away from each other, they cease to have a large impact on each other’s energy.

JOHN M. FULTON

4

0.165

0.175

0.185

0.195

0.205

0.215

0.225

0 2 4 6 8 10

a/w

Stra

in E

nerg

y

Fig. 3. Graph of strain energy vs. a/w for b/w = 0.05.

As mentioned earlier, the strain energy is only part of the total energy (equation (2)). The other major contribution is the secondary term, Γ , which is subtracted from the force integral:

2adh4

hEw

′=Γ

πγ

. (3)

This is a nondimensional constant that involves stamp height h, Young’s modulus E′ , adhesion energy adhγ , and stamp width w, as shown in equation (3). An individual plot of total energy vs. b/w was generated for each Γ from 0.545 to 3. A graph of five sample plots for Γ = 0.35, 0.545, 0.6, 0.725, and 0.95 from top to bottom is shown in Fig. 4.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

b/w

Ener

gy

Fig. 4. Several plots of total energy vs. b/w for a/w = 100.

The resulting minimum that appears in the center of the plot is due to the nondimensional parameter. As the value of Γ increases, the minimum energy is lowered. The next step was to create graph of b/w vs. Γ . The corresponding b/w value for each minimum energy in every b/w vs. energy plot was found and recorded in a new table. This procedure was done for five

Nanoscale Stamp Deformation

5

different a/w values: 0.1, 0.5, 1, 10, and 100. Plots of b/w vs. Γ were then made, as shown in Fig. 5.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3

Nondimensional parameter, Γ

b/w

Fig. 5. Plot of b/w vs. Γ . From left to right a/w = 0.1, 0.5, 1, 10 and 100.

(The plots for 10 and 100 appear to be the same line.)

Conclusions

We were able to model nanoscale stamp deformation successfully using the Westergaard stress function. Based on the plot of strain energy vs. a/w (Fig. 1) it is our conclusion that, as the stamp punches are placed farther away from each other, they cease have a large impact on each other’s total energy. We found that the change in the force integral was less than 0.01% when comparing the results for a/w = 10 and 100. We also found that there is a critical b/w value at which the total energy of the stamp punch experiences a minimum. The critical b/w value is influenced by the nondimensional parameter Γ . As the value of Γ increases, the critical b/w value also increases. From the results gathered, a graph of Γ vs. b/w was made (Fig. 3), which can be used to interpret experimental data. It is hoped that experimental data will be taken, with known a/w and measured b/w, and then compared with the graph. In this way the Γ value can be determined and the adhesion energy adhγ can be found. This method can be carried out for any set of materials and we believe it will be of aid to future nanoscale stamp research.

Acknowledgments

I would like to thank my advisor, Prof. K. Jimmy Hsia, for his guidance, aid in mathematical derivations, and overall support of my efforts. I would also like to thank Prof. Kimberly M. Hill for her aid in overseeing the program and TAM students, and Prof. James W. Phillips for his guidance in the written aspects of the project. I would also like to thank my

JOHN M. FULTON

6

fellow undergraduate researcher Daniel S. Widrevitz for his help in learning and writing the Mathematica program. I would also like to thank the National Science Foundation, the Department of Defense, the Department of Theoretical and Applied Mechanics and the College of Engineering, UIUC, for sponsoring the program. Finally I would like to thank the REU program for offering this experience.

References

1. Mirkin, C. A., and Rogers, J. A. 2001. Materials Research Society Bulletin 26(7), 506. 2. Menard, E., and Rogers, J. A. 2004. Collapse of PDMS stamps. Paper in preparation. 3. Hui, C. Y., Jagota, A., Lin, Y. Y., and Kramer, E. J. 2002. Constraints on microcontact

printing imposed by stamp deformation. Langmuir 18, 1395–1407.

7

Ion-Induced Surface Sputtering: One-Dimensional Numerical Analysis of Island Formation

Syed Hussain Sophomore in Mathematics and Physics

Loyola University

Advisors: TAM Prof. Jonathan B. Freund and MIE Prof. Harley T. Johnson

Inspired by epitaxial thin film systems, in which the film roughens via strain-induced surface diffusion, we have developed a mathematical model for surface erosion velocity similar to the Mullins equation, based on Zhang and Bower’s work. However, the primary objective of this project is to simulate surface diffusion along the surface of a thin film due to ion sputtering, using graphical software. Ion sputtering, the removal of atoms from the film surface through the impact of ions, tends to transfer energy and momentum to surface atoms from ion–atom collisions directly; this interaction increases the probability for surface diffusion. This project is inspired mostly by the works of Zhou, Cuenat, and Aziz, which are based on the self-organization of nanoscale structures during ion beam sputtering of germanium, and of Fascko et al., who recently discovered a new self-organization process for semiconductor nanostructures. The film of amorphous silicon is idealized as initially isotropic and elastic with shear modulus µ, Poisson’s ratio ν, and a thickness of the order of nanometers bonded to a crystalline silicon substrate. In this report we formulate a model for surface diffusion that is thermally activated, adopt a surface diffusion equation based on the Makeev–Barabasi equation of surface motion, and solve this equation numerically. An analytic solution for the modified surface diffusion equation (MSDE) is outlined, based on the method of separation of variables. The explicit Euler and implicit trapezoidal schemes for the MSDE are analyzed for stability and accuracy. Due to time limitations, the data record is incomplete and no graphical simulation based on data is available.

Introduction

Ion-induced sputtering, the removal of atoms from a film surface through the impact of ions, tends to transfer energy and momentum to surface atoms from ion–atom collisions directly, and thereby increases the probability for surface diffusion (Makeev and Barabasi, 1997). Because of recent technological advancements, scientists are able to study various means and applications of material sputtering. In order for a surface to roughen, thousands of atoms must congregate into quantum dots. Quantum dots are smaller than the micron-sized structures typical of current microelectronic circuits, and have electrical and optical properties that may be exploited to develop a wide range of devices, including light-emitting diodes, transistors, photovoltaic cells, quantum semiconductor lasers, and semiconductors (Zhang and Bower, 1999). Previous studies have been conducted to examine spontaneous, strain-induced roughening on a thin epitaxial film, which has been observed on several semiconductor heterostructures, such as germanium silicide.

SYED HUSSAIN

8

Most recently, German scientists, including Stefan Facsko, have successfully produced a self-organized, hexagonal array of quantum dots by emitting argon (I) ions onto a film of gallium antimonide (Fascko et al., 1999). Until recently, scientists have had limited abilities to refine the physical configurations of quantum dots to influence lasing frequency, wavelength, and intensity, and thus they seek to make silicon emit light most efficiently in the visible spectrum. Light-emitting diodes, such as those that emit the lowest wavelength light, have recently made a plethora of new technologies possible, such as full-color, flat-panel displays and ultrahigh-density optical memories. Quantum dots, which are desirable in physical configurations, are difficult to manufacture using standard lithographic techniques. It has been discovered that semiconductor quantum dots can produce laser light output at significantly low wavelengths comparable with those of other elements (Fascko et al., 1999). The results of this project can provide a mathematical basis for ion-induced surface sputtering, potentially a new venue for producing semiconductor dots that can be manipulated conveniently. The primary objective of this project is to simulate the surface diffusion of the atoms on a thin film of amorphous silicon due to a vertical flux of argon (I) ions, using graphical software. A variation of Mullins’ surface diffusion equation (SDE) will be derived to find a one-dimensional surface diffusion function of position and time describing the epitaxial thin film system of Zhang and Bower (1999). The SDE will be modified to incorporate ion-induced surface diffusion using considerations from Makeev and Barabasi (1997) and will be called the modified surface diffusion equation (MSDE). The MSDE will be solved using numerical analysis to obtain the surface diffusion function (SDF). Fortran 90 software will be used to calculate the surface height at discrete positions and times. The film will be idealized as initially isotropic and elastic with shear modulus µ, Poisson’s ratio ν, and thickness of the order of nanometers. The film will be considered bonded to a crystal silicon substrate, and displacements due to mechanical loading, such as tension and compression, will be considered small to avoid other influences on surface diffusion. It will be assumed that any roughness is sinusoidal in one dimension, and thus the node and trough regions will be eroded more than the crest regions since the crests are at the lowest chemical potential. Surface diffusion tends to create equilibrium positions for surface atoms at the crests and produce additional roughness at positions of initial imperfections. The surface is assumed to change shape by surface diffusion alone, and the resulting change per unit time (SDF) will be mapped to an asymptotic solution for small roughness amplitudes having a doubly sinusoidal profile in one dimension. Explicit and implicit schemes for time advancement will be considered to obtain the desired accuracy.

Problem formulation

Surface diffusion is partly driven by a variation in chemical potential sµ , where s is the one-dimensional position coordinate along the edge of the film. (Note: All quantities that are functions of s have the subscript s.) Surface diffusion, which is thermally activated, is modeled as follows. The total flux of ions at a point s is ( )s s sj µ= −Γ ∇ , where ∇ denotes the surface gradient operator, and sΓ is given by

exps s ss

D QkT kT

δ ⎡ ⎤Γ = −⎢ ⎥⎣ ⎦,

Ion-Induced Surface Sputtering

9

where sD represents surface diffusivity, sδ represents the thickness of the diffusion layer, k is the Boltzmann constant, and T denotes temperature. For a line element of length ds and a unit vector e lying in the film surface normal to ds, ( ) sj e d⋅ is the volume of material crossing the line element per unit time (Zhang and Bower, 1999). The component of velocity normal to the film is

4

24

( , )( ) s= ( )n s s sh s t hv j e d

t sφ γκ β∂ ∂

= ⋅ = Γ ∇ Ω − Ω =∂ ∂

,

a variation of the Mullins equation [ ]2

22

( , ) s sn s

D ch s tv ut kT s

γκ∂ ∂= = Ω −

∂ ∂, where sc is the

concentration of diffusing atoms, Ω is the atomic volume, u represents strain-energy density, γ

represents surface-energy density, and 2

2

( , )s

h s ts

κ ∂=

∂. According to Makeev and Barabasi

(1997), the average energy deposition at a point on the surface of the film follows Gaussian distribution and the velocity of erosion at that point depends on the total power contributed by all the ions deposited within a finite range of the distribution. The approximation Makeev and Barabasi derived for the surface height, modified for one-dimensional surface diffusion, satisfies

22 4

0 2 42Ix

x xh h h h hv v Dt x x x x

λγ∂ ∂ ∂ ∂ ∂⎛ ⎞= − + + + −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠,

where 0v is the erosion rate, hx

γ ∂∂

denotes isotropic motion of surface atoms along the x axis, xv

represents ion-induced surface tension terms, xλ represents the slope dependence of the erosion rate, and I

xD are the ion-induced surface diffusion coefficients. For simplicity, the term 2

2x h

xλ ∂⎛ ⎞

⎜ ⎟∂⎝ ⎠ will be eliminated, slightly reducing the accuracy of the equation, and s x= . Also,

having a coordinate reference frame, where the height of the film has the constant change

0h vt

∂= −

∂ changes the Makeev–Barabasi equation to

2 4

2 4I

x xh h h hv Dt x x x

γ∂ ∂ ∂ ∂= + −

∂ ∂ ∂ ∂,

which is the modified surface diffusion equation (MSDE) to be solved in this project.

Numerical and analytic solution to the MSDE

The MSDE can be rewritten as

2 4

2 4

h h h ht x x x

α β∂ ∂ ∂ ∂= + + Γ

∂ ∂ ∂ ∂,

SYED HUSSAIN

10

where α γ= , xvβ = , and IxDΓ = − . The MSDE requires four constraints in x and one constraint

in t. We assume a solution to the partial differential equation of the form ( , ) ( ) ( )h x t X x T t= .

Substituting this assumed solution into the MSDE yields

2 4

2 4

1 1dT dX d X d X Kdt T dx dx dx X

α β⎛ ⎞

= + + Γ =⎜ ⎟⎝ ⎠

,

where K is a constant. The general solution to 1dT Kdt T

= is KtT De= , with D a constant. The

auxiliary equation for

2 4

2 4

1dX d X d X Kdx dx dx X

α β⎛ ⎞

+ + Γ =⎜ ⎟⎝ ⎠

is the depressed quartic equation 4 2 0r r r Kβ αΓ + + − = . With the substitution 4

r y β= −

Γ, the

equation

2

2 2 22 2 2K K Ky z z y y z zβ α⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − −+ + = − + − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟Γ Γ Γ Γ Γ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

is obtained, where z is a root to the associated resolvent cubic polynomial. Upon completing the square on the right side, a general solution for ( )X x can be obtained.

To solve the general cubic equation 3 2 0ax bx cx d+ + + = , we make the substitution

3bx ya

= − and find that

2 3

32

2 03 27 3b b bcay c y da a a

⎛ ⎞ ⎛ ⎞+ − + + − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠,

which is of the form 3y Ay B+ = , where A and B are constants. Upon making the substitutions 3A st= and 3 3B s t= − , a solution to 3y Ay B+ = is y s t= − , one root to the general cubic

equation. The quadratic formula can be used to solve the general cubic equation completely. To represent the MSDE numerically, we discretize the coordinate x into N uniformly spaced intervals with 1−= + ∆j jx x x for 1, 2,...,=j N and 00 Nx x x L= < < = (Moin, 2001). The finite difference schemes

4

2 1 1 14 4

4 6 4( )j j j j j

j

f f f f fd f O hdx h

− − + +− + − + −= + ,

2

1 12 2

2( )j j j

j

f f fd f O hdx h

− +− += + ,

and

Ion-Induced Surface Sputtering

11

1 1 2( )2

j jj

f fd f O hdx h

+ −−= +

will approximate the spatial derivatives of h in the MSDE, where ( )i if f x= . The MSDE can be represented in terms of the finite difference schemes as

2 1 1 24 2 4 2 4 2 4 44 2 6 4I I I I I

j x x x x x x x xj j j j j

dh D v D v D v D Dh h h h hdt

γ γ− − + +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − + − + − + + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

for 1, 2,..., 1= −j N . For convenience,

2 2 1 1 0 1 1 2 2j j j j jdh h A h A h A h A h Adt − − − − + += ⋅ + ⋅ + ⋅ + ⋅ + ⋅ .

Converting this ordinary differential equation into a system of 1−N equations yields dh Ahdt

= ,

where 1 2 1[ , ,..., ]nNh h h h −= is the surface height at time level n and A is the ( ) ( )1 1N N− × −

pentadiagonal matrix [ ]2 1 0 1 2, , , ,A B A A A A A− −= . The first-order Euler scheme for time advancement is

( )1 1n n n n nh h dh h t A h h t A+ = + ≈ + ∆ ⋅ ⋅ = + ∆ ⋅ .

Via a modified wavenumber analysis, the time step is bounded by

( )

4 6

2 4 2 4 2

42 2 32

x xtx x x

ββ α α β

∆ + ∆∆ ≤

Γ ∆ + ∆ + − ∆

for complex R iiλ λ λ= + corresponding to y yλ′ = , where Re( ) 0Rλ λ= < and Im( )iλ λ= is a real number. The second-order trapezoidal approximation is given by (Moin, 2001)

( )1 11 1 2( )2

n n

n n

t tn n n n n n

t t

dh th h dt h Ah dt h A h h O tdt

+ ++ +∆= + = + = + + + ∆∫ ∫ ,

which rewritten is

1

2 2n nt tI A h I A h+∆ ∆⎛ ⎞ ⎛ ⎞− = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠,

2 1 0 1 2[ , , , , ]

2tI A B φ φ φ φ φ± ± ± ± ±− −

∆± = ,

where 22 2tAφ ± −−

∆⎛ ⎞= ± ⎜ ⎟⎝ ⎠

, 11 2tAφ ± −−

∆⎛ ⎞= ± ⎜ ⎟⎝ ⎠

, 001

2tAφ ±

∆⎛ ⎞= ± ⎜ ⎟⎝ ⎠

, 11 2tAφ ±

∆⎛ ⎞= ± ⎜ ⎟⎝ ⎠

and

22 2tAφ ±

∆⎛ ⎞= ± ⎜ ⎟⎝ ⎠

. Finally,

1

1

2 2n n nt th I A I A h C h

−+

⎛ ⎞∆ ∆⎛ ⎞ ⎛ ⎞= − ⋅ + ⋅ = ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠,

SYED HUSSAIN

12

where C is a constant matrix. For the scalar equation y yλ′ = , the trapezoidal solution is given by

11

21

2

n n

t

y yt

λ

λ+

∆+

=∆

−.

For complex R iiλ λ λ= + , the amplification factor becomes

( )iA eB

θ ασ −= , 22

12 2

iR ttA λλ ∆∆ ⎛ ⎞⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

and 22

12 2

iR ttB λλ ∆∆ ⎛ ⎞⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

,

and because 0Rλ < , 1σ < for all 0t∆ > . Since the trapezoidal method is an implicit method, it is expected to be unconditionally stable for any time step size.

Conclusions

This research project required an undergraduate student to learn many new skills, including methods of numerical analysis, the finite-element method, and Fortran. In addition, this theoretical project required a basic knowledge of software programming. The sputtering of atoms, where attention is given to the sputtering yield and the velocity/angular distribution of the sputtered particles, is an important thin-film processing technique; yet ion-induced surface diffusion has not been sufficiently understood to be implemented for today’s technologies. Although the effects of ion-induced surface diffusion are documented, no theory has been devised to quantify these effects. Solving the modified SDE numerically potentially can introduce a first-order mechanism for this new phenomenon. Upon obtaining a solution of desired accuracy, the outlined analytic solution can be used to obtain a more accurate numerical solution. Finally, the approximated SDE reported by Makeev and

Barabasi (1997) can be expanded to include more complex terms, such as 2h

xφ ∂⎛ ⎞

⎜ ⎟∂⎝ ⎠, where φ is a

constant, to obtain a more accurate equation of motion for which numerical methods are best suited.

Acknowledgments

I am thankful to Prof. Jonathan B. Freund and Prof. Harley T. Johnson for their guidance and advice. Finally, I gratefully acknowledge the support of the National Science Foundation via the Summer 2004 Research Experiences for Undergraduates at the University of Illinois at Urbana-Champaign.

References

Makeev, M. A., and Barabasi, A.-L. 1997. Ion-induced effective surface diffusion in ion sputtering. Applied Physics Letters 71, 2800–2802.

Ion-Induced Surface Sputtering

13

Moin, P. 2001. Fundamentals of Engineering Numerical Analysis. Cambridge: Cambridge University Press.

Facsko, S., Dekorsy, T., Koerdt, C., Trappe, C., Kurz, H., Vogt, A., and Hartnagel, H. L. 1999. Formation of ordered nanoscale semiconductor dots by ion sputtering, Science 285, 1551.

Zhang, Y. W., and Bower, A. F. 1999. Numerical simulations of island formation in a coherent strained epitaxial thin film system. Journal of the Mechanics and Physics of Solids 47, 2273–2297.

15

Effect of Microcapsule Size on Tensile Properties of Self-Healing Composites

Joseph H. Lai Junior in Materials Science and Engineering

University of Michigan–Ann Arbor

Alyssa A. Rzeszutko Undergraduate Assistant in Aerospace Engineering, UIUC

Benjamin J. Blaiszik Graduate Assistant in Theoretical and Applied Mechanics, UIUC

Advisor: TAM Prof. Nancy R. Sottos

The effect of microcapsule size on the mechanical properties of a self-healing polymer composite is investigated. Three different sets of tensile bars were made from an epoxy with each set containing a different size microcapsule at a fixed concentration. Uniaxial tensile tests were performed to determine Young’s modulus and ultimate tensile strength. It was found that Young’s modulus decreased as microcapsule size increased. No correlation was found between ultimate tensile strength and the microcapsule sizes tested.

Introduction

In all composite materials, the properties of the filler material have a significant effect on the mechanical properties of the overall composite. White et al.1 have introduced a particle-dispersed polymer composite that has the capability of self-healing upon crack propagation. This composite consists of an epoxy matrix embedded with microcapsules containing a monomer healing agent. Grubbs catalyst is also dispersed throughout the epoxy matrix. Once a crack occurs, the crack propagates along the composite, rupturing the microcapsules and releasing the healing agent. As the healing agent flows along the crack plane, it comes into contact with the Grubbs catalyst and polymerization begins, sealing up the crack. This project analyzes the effect that microcapsule size has on the mechanical properties of the self-healing polymer composite. Composite tensile bar samples were produced containing microcapsules of various sizes at a fixed weight concentration and then mechanically tested for ultimate tensile strength and Young’s modulus.

Experimental procedure

Silicon-rubber molds for the tested tensile bars were cast in an aluminum mold using GE Silicones® RTV630 with 630B curing agent. Once six silicon-rubber molds were produced, composite tensile bars were then made using EPON® 828 epoxy resin (diglycidyl ether of

JOSEPH H. LAI

16

bisphenol A, DGEBA) with 12 pph Anacime DETA (diethylenetriamine) curing agent and microcapsules. A set of neat epoxy (no microcapsules) tensile bars were made first by stirring the EPON® 828 epoxy resin with the curing agent until homogeneous and then degassing the mixture for approximately 10 minutes. The longer the mixture is degassed, the more it is free of air bubbles. However, degassing longer than 10 minutes increases the likelihood of the mixture heating and curing too quickly. The epoxy quickly becomes more viscous or even cures instantaneously. If needed, microcapsules are added at this point and a second degassing is performed for approximately 5–10 minutes. Once degassed, the mixture is then poured at the top of a silicon-rubber mold, which is placed at an angle so as to allow the epoxy to fill the mold without creating any air bubbles. The sample is then placed at room temperature to cure for 24 hours and then in the oven set at 30°C to cure for another 24 hours. Composite tensile bars are produced with the addition of microcapsules stirred into the epoxy mixture after the initial degassing stage. Microcapsule production is a critical component in researching the effect of microcapsule size on a self-healing composite. Microcapsule production begins with an oil-in-water emulsion, by adding 25 mL of 5 wt% EMA copolymer into 200 mL of deionized water. The addition of EMA lowers the pH level to approximately 2.6. It is crucial that once the EMA is added into the deionized water that the pH falls to between 2.5 and 2.7; otherwise polymerization may not properly occur later on. Once the pH was in the appropriate range, the 3-bladed mixing propeller was lowered to approximately 2/3 from the level of the solution. Microcapsule size is regulated by mixer speed. The log-log graph2 in Figure 1 displays the relationship of microcapsule diameter size with mixer speed.

Fig. 1. Average microcapsule diameter as a function of mixing speed.2

For 180 µm, the mixing speed was set to 600 rpm; smaller capsules are produced by increasing the mixing speed. In this project, 1000 rpm and 2500 rpm were used to form 60 µm and 20 µm capsules, respectively. These speeds are slightly higher than those suggested by the graph in Fig. 1, but they resulted in a much larger yield of each expected capsule size. At 2500 rpm and at speeds higher than 1500 rpm in general, the mixer has a tendency to vibrate excessively, and to remedy this nuisance, the mixing blade should be raised so that it is just below the surface of the solution. Once the solution is mixing without any excessive vibration,

Tensile Properties of Self-Healing Composites

17

5 g of urea, 0.5 g ammonium chloride, and 0.5 g resorcinol are added to the solution. The addition of these compounds should not affect the pH level, except for the 5 g of urea, which should raise the pH by a value of 0.1. At this point, this pH should be between 2.6 and 2.8; otherwise an incorrect amount of urea was added. Once the pH level is within the necessary range, the pH is then increased to 3.5 with drops of sodium hydroxide (NaOH) in order for optimum polymerization of the microcapsule shell wall to occur. Incorrect pH levels throughout the process lead to large yields of urea-formaldehyde debris, unfilled capsules, or even no capsules. Surface bubbles or foam in the solution can result in poor quality microcapsules and so 1 drop of 1-octanol was added to eliminate them. Afterward, 60 mL of dicyclopentadiene (DCPD) healing agent is steadily streamed into the mixture to form an emulsion with EMA. After waiting 10 minutes for the DCPD to stabilize, 12.67 g of 37 wt% aqueous solution of formaldehyde is added into the mixing solution. Last, the mixture is covered and heated to 55°C at a rate of 1°C per minute. After 4 hours of continuous heating and agitation, the urea and formaldehyde have polymerized and the resultant slurry of microcapsules is poured into a coarse frit for drying. Once frit-dried, the clumps of microcapsules are then dumped in a weigh-boat lined with aluminum foil and dried at room temperature for approximately 24 hours or longer if needed. Clumps of microcapsules are completely dry when they are easily broken up by a slight gust of air. It is absolutely essential for microcapsules to be completely dry prior to sifting; otherwise sifting will result in microcapsule damage or rupture. Once completely dry, microcapsules are sifted for 10 minutes with appropriate sieves stacked up. Finally, microcapsules of the desired size are collected from the appropriate sieve. Each collection is examined under an optical microscope to check for purity of microcapsules. It was found that any more than two 10-minute sifts resulted in a progressively less pure yield of microcapsules, as an increasingly large amount of urea-formaldehyde debris became present with each collection. A size distribution is performed to verify that the capsule sizes are within the expected ranges. By using an optical microscope, an image of the produced capsules is taken, and the diameters are carefully measured out with ImageJ software.3 Once the diameters of at least 50 microcapsules have been measured out, the data are entered into Microsoft Excel and a histogram plot with a size distribution is generated. The methods for microcapsule collection, however, led to large quantities of urea-formaldehyde debris, especially for 20 and 60 µm capsules, and had to be modified. With 180 µm capsules, use of the original techniques did result in a pure yield of quality microcapsules (Fig. 2). During sifting, sieves of 350, 250, 180, and 125 µm were used and microcapsules in the 125 µm sieve were collected. From previous research, the amount of urea-formaldehyde debris present becomes progressively larger as the mixing speed used is increased. This debris is especially present in producing 60 and 20 µm capsules, as can be seen in Figs. 3 and 4. In an attempt to eliminate some of the debris, acetone was used to rinse the micro-capsules. However, rinsing microcapsules with acetone only allowed DCPD to leak out, leaving behind an empty, deflated urea-formaldehyde shell that can be seen as transparent under a microscope as in Fig. 5.

JOSEPH H. LAI

18

Fig. 2. Microphotograph of 180 µm capsules (5x objective).

Fig. 3. Microphotograph of 60 µm capsules with UF debris (5x objective).

Tensile Properties of Self-Healing Composites

19

Fig. 4. Microphotograph of 20 µm capsules with UF debris (10x objective).

Fig. 5. Microphotograph of 20 µm capsules after acetone rinse (10x objective).

JOSEPH H. LAI

20

Fig. 6. Microphotograph of purer 20 µm capsules (5x objective).

Black clusters are microcapsules lying above one another.

As a refinement of the previous microcapsule extraction techniques, a density separation was performed on the 20 µm capsules by pouring deionized water to the brim of the beaker. Since DCPD is less dense than water, good quality microcapsules filled with DCPD should float in water, while urea-formaldehyde, which is denser than water, should sink to the bottom of the beaker. With DCPD-filled microcapsules separated from unwanted remains of urea-formaldehyde, the top layer of capsules is carefully hand-spooned into another beaker of deionized water to remove further any unwanted urea-formaldehyde particles. The floating layer of quality microcapsules in the second beaker is scooped out and finally dumped into a frit for drying. Figure 6 shows a sample of purer microcapsules that were obtained from refining the original techniques of microcapsule extraction. Once frit dried and dried at room temperature, the clumped microcapsules are then sifted for 10 minutes, using a stack of 75 µm, 53 µm, 43 µm, and 38 µm sieves. Since 20 µm capsules are needed in this project, microcapsules in the container beneath the 38 µm sieve were collected. For extraction of 60 µm capsules, a slightly different procedure was used so that a larger and purer yield of microcapsules could be obtained. After 4 hours of continuous agitation and heating, the resulted slurry in the beaker was poured through a stack of 3 sieves: 75, 53, and 45 µm. Rather than immediately separating the different-sized capsules with Ro-Tap as originally done, a wet-sifting technique was performed by rinsing the set of sieves under a running faucet of deionized water, in order to remove better the urea-formaldehyde debris. After continuous washing of microcapsules, the capsules in the 53 µm

Tensile Properties of Self-Healing Composites

21

sieve were rinsed into a beaker and poured onto a filter paper that was dried by a vacuum. The collected microcapsules were then dried at room temperature for approximately 24 hours and then sifted with Ro-Tap for 10 minutes, using sieves of 1 mm, 500 µm, 75 µm, and 53 µm. Once sifted, the improved purity of the microcapsules could immediately be seen as they clearly were more free-flowing than the microcapsules produced with the original techniques. A sample of 60 µm capsules obtained with the refined techniques was examined, and clearly, the microcapsules appeared cleaner and more spherical than before, as can be seen in Fig. 7.

Fig. 7. Microphotograph of purer 60 µm capsules (5x objective). Image appears more magnified

than that in Fig. 3 because a new camera with a larger zoom was recently installed.

In making the tensile bar composites, approximately 2–2.5 g of microcapsules were needed for each pair of tensile bars, which would require 12–15 g of microcapsules to be produced for each set of 12 tensile bars. A single batch of 180 µm capsules was able to produce a sufficient yield. However, with 60 µm capsules, 5 batches of microcapsules were necessary just to produce an adequate amount. The 20 µm capsules required 6 batches. Once an adequate quantity of 180, 20, and 60 µm capsules were produced, tensile bar composites were then fabricated. Preparing the epoxy for making tensile bars should be done one cup at a time to prevent idle epoxy mixtures from curing instantaneously. Each cup represents one batch, which provides enough epoxy for a pair of tensile bars. Degassing more than one cup at a time led to the epoxy curing instantaneously. After the initial degassing stage, microcapsules were added into the epoxy and degassed for another 5 minutes. During this second degassing stage, the epoxy was monitored for signs of boiling, and if it did occur, the degassing process was terminated early. Boiling occurred when the surface of the epoxy started bubbling, rather than foam appearing at the surface of the epoxy; surface foam is indicative of normal degassing. Once degassed, the epoxy–microcapsule mixture was then poured at the top

JOSEPH H. LAI

22

of a silicon-rubber mold that was placed at an angle. The epoxy composite was then given 24 hours to cure at room temperature and another 24 hours to cure in the oven set at 30°C. The finished tensile bar composites were then individually measured for gage width and length and loaded in the Instron 8500 for mechanical testing. The data that resulted from the tests include strain (%), load (kN), stress (MPa), and time of load (sec).

Results and discussion

The tensile bars in each set were pulled until failure. While a few of the specimens broke in just the gage area as intended, some of the specimens only broke at the pinholes. Most of the specimens that broke in the gage area also broke in one or both of the pinholes. Some of the specimens that broke at the pinholes did not have any bubbles in that region. In the tensile bars that did have bubbles and broke at the pinholes, the crack was observed not to have started from those bubbles, as they were still present after the test. The results for a few of the tensile bars were discarded due to yield of improbable results or improper machine setup of the tensile bars. During the tests, the data for a few of the tested tensile bars were discarded due to improper loading procedure of the specimen or implausible results. Neat epoxy tensile bars represent microcapsules that are infinitely small. Figure 8 and the associated data in Table 1 show that Young’s modulus decreases slightly as microcapsule size increases. Therefore, the stiffness of the microcapsule composite increases with decreasing microcapsule size.

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

0 20 40 60 80 100 120 140 160 180 200

Microcapsule size (microns)

Youn

g’s

mod

ulus

(GPa

)

Fig. 8. Young’s modulus vs. microcapsule size.

These results reveal that the smaller the capsule size, the more the composite’s elastic modulus will approach that of the neat epoxy, that is, an epoxy without any capsules. The ultimate tensile strength of each bar was also analyzed. Figure 9 and the associated data in Table 2 reveal that, if microcapsules are present, there is no relationship between ultimate

Tensile Properties of Self-Healing Composites

23

tensile strength of the composite and the tested sizes of its embedded microcapsules. However, as Fig. 9 reveals, a steep drop-off in ultimate tensile strength occurs once microcapsules 20 µm and larger are added. Current research is being done on the manufacturing of nanocapsules and whether the ultimate tensile strength will be more similar to that of the neat epoxy.

Young’s modulus (GPa) Bar no. Neat 20 µm 60 µm 180 µm

1 3.98 3.58 3.27 3.01 2 3.60 4.09 3.36 2.95 3 X* 3.05 3.60 3.43 4 3.58 3.93 4.04 3.00 5 3.68 3.27 3.52 X 6 3.78 3.57 3.12 3.28 7 3.31 3.22 3.28 3.92 8 3.52 3.33 3.25 3.69 9 2.95 3.85 3.42 3.24

10 3.91 3.20 X 3.59 11 3.42 3.47 3.69 3.45 12 3.39 3.50 3.42 3.83

Average 3.56 3.51 3.45 3.40 Std dev 0.29 0.32 0.26 0.34

*X denotes discarded results

Table 1. Young’s modulus of tested tensile composites.

0

10

20

30

40

50

60

Neat 20 60 180

Microcapsule size (microns)

Ulti

mat

e te

nsile

str

engt

h (M

Pa)

Fig. 9. Ultimate tensile strength vs. microcapsule size.

JOSEPH H. LAI

24

Ultimate Tensile Strength (MPA) Bar No. Neat 20 µm 60 µm 180 µm

1 34.0 20.8 26.2 28.8 2 47.3 32.0 21.3 26.6 3 X* 30.8 24.7 28.5 4 51.6 19.7 22.4 25.9 5 39.7 32.0 24.6 X 6 53.1 29.4 26.6 31.0 7 49.1 22.8 27.1 19.4 8 40.6 35.8 27.3 26.4 9 48.4 13.7 20.3 31.1

10 44.0 36.2 X 23.5 11 49.4 34.2 26.7 29.1 12 36.4 28.0 26.9 29.5

Average 44.9 27.9 24.9 27.2 Std dev 6.4 7.1 2.5 3.5

*X denotes discarded results

Table 2. Ultimate tensile strength of tested tensile composites.

Summary and conclusions

The effect on Young’s modulus and ultimate tensile strength from varying microcapsule size at a fixed concentration was investigated in this project. Microcapsules of sizes 180 µm, 60 µm, and 20 µm were studied. However, original extraction techniques of 60 µm and 20 µm led to an insufficiently low yield of microcapsules and a large yield of urea-formaldehyde debris. The original techniques were refined and a larger quantity of microcapsules with significantly less debris was obtained. Once a purer sample of microcapsules was obtained, a set of 12 tensile bars containing 5 wt% concentration of microcapsules was produced for each microcapsule size. Neat tensile bars (that is, tensile bars containing no capsules) were produced to represent tensile bars containing infinitely small capsules. Uniaxial-tensile tests were performed on the produced tensile bars to reveal information on Young’s modulus and ultimate tensile strength. Young’s modulus decreased slightly as the size of the microcapsule increased, indicating that the stiffness of the microcapsule composite will increase slightly as microcapsule size is decreased. No correlation was found between ultimate tensile strength and microcapsule sizes greater than 20 µm. A steep drop-off in ultimate tensile strength occurred between the neat epoxy and the rest of the tensile bars, suggesting that smaller-than-20 µm capsules may be required for the ultimate tensile strength to be more similar to that of the neat epoxy.

Acknowledgments

The author acknowledges the support of the National Science Foundation and the Department of Defense. The author would also like to thank Benjamin J. Blaiszik, Alyssa A. Rzeszutko, Michael W. Keller, and Kathleen S. Toohey for their help in microcapsule manufacturing and tensile bar fabrication. Mechanical testing was done in the Advanced Mechanical Testing and Evaluation Laboratory facilities at the University of Illinois at Urbana-Champaign with the assistance of Dr. Peter Kurath.

Tensile Properties of Self-Healing Composites

25

References

1. White, S. R., N. R. Sottos, P. H. Geubelle, J. S. Moore, M. R. Kessler, S. R. Sriram, E. N. Brown, and S. Viswanathan. 2001. Autonomic healing of polymer composites. Nature 409, 794–797.

2. Brown, E. N., S. R. White, and N. R. Sottos. 2004. Microcapsule induced toughening in a self-healing polymer composite. Journal of Materials Science 39, 1703–1710.

3. National Institutes of Health, U.S. Department of Health and Human Services. 2004. ImageJ—Image processing and analysis in Java. http://rsb.info.nih.gov/ij/

27

Interstitial Fluid Effect on Dense Free-Surface Granular Flow

Michael E. Ly Junior in Physics, Loyola University at Chicago

Jiafeng Zhang Graduate Assistant in Theoretical and Applied Mechanics, UIUC

Advisor: TAM Prof. Kimberly M. Hill

This study focused on the effect of the viscosity of an interstitial fluid on the kinematics of a dense, free-surface granular flow in a rotated drum. In many ways, the nature of the flow was similar to that of particles in air. The movement of the particles relative to one another was limited to a thin surface boundary layer, whose velocities and velocity fluctuations were greatest near the free surface and decreased to zero bead diameters from the surface. Additionally, for all viscosities, the structure of the flow for particles is dominated by a simple stratified laminar structure, again, similar to that for particles flowing in air. However, as the viscosity of the interstitial fluid was increased, certain changes were observed: the structure of the flow became more dominant, and its effect carried over into the solid-like substratum; as the viscosity increased, both the velocities and velocity fluctuations appeared to decrease near the free surface for low viscosities; finally, at the highest viscosities studied, the functional form of the velocity fluctuations appear to change as well.

Introduction

Granular materials appear in a wide range of industrial and natural settings. Half the products produced by the chemical industry and three-fourths of the raw materials used are at some point in granular form. Furthermore, 1.3% of U.S. electrical power production goes towards the grinding of ores and particles [1]. Additionally, landslides and ground failures cost more lives and money per year than all other natural disasters combined [2]. While there has been extensive research done in the field of granular flow, there is no agreement on the constitutive relations that govern dense flow. While individual particle–particle interactions are relatively simple to describe, a statistical description of the steady-state behavior of the material in bulk, when it is in dense flow, is elusive. Further, there is even less known about granular slurries, i.e. macroscopic particles in fluids such as water. To understand granular flow, researchers often use a rotating drum because it is a rela-tively simple system for controlled granular flow studies. It also resembles devices used in industry to mix and dry various products and materials. In a slowly rotated drum, at any given time, the particles flow only in a relatively thin boundary layer—see Fig. 1a. Most of the particles rotate in solid-like rotation with the outside of the drum (Fig. 1b).

MICHAEL E. LY

28

Recent studies of granular flow in a drum have illuminated certain details of dense free-surface granular flow. For example, Hill, Gioia, and Tota [3] found the structure in the boundary layer could be described by the most simple laminar flow; that is, it is dominated by layers of particles moving over one another, exchanging particles relatively infrequently. Signatures of this structure appeared clearly in other kinematic details of the flow including the volume fraction, the velocity, and the relationship between the local velocity and the diffusion perpendicular to the flow. Ott-Monsivais et al. [4] showed that this structure could, in fact, be used to model the velocity fluctuations, that the velocity fluctuations could be modeled as the jostling of the particles as they moved over one another in layers. When mixtures of granular materials are rotated in drums, the particles tend to segregate in a variety of patterns, from simple radially symmetric patterns perpendicular to the axis of the drum to banding along the length of the drum. Jain et al. [5] found that when water and soap were added to the granular system in a quasi-2D rotating cylinder, there was a greater rate of banding segregation when compared with that of the dry granular system, though the radial segregation was somewhat less pronounced. They pointed toward different rates of diffusion depending on the interstitial fluid, though it is believed that no studies have yet measured the dependence of the kinematics of the flow on the interstitial fluid; most studies have concentrated only on particles in air. The study reported here focuses on the relationship between the interstitial fluid and the kinematics of monodisperse granular systems in a rotated drum.

Experimental setup

For the experimental studies reported here, a thin transparent drum (D ≈ 300 mm, thickness ≈ 7 mm) was filled halfway with chrome steel spheres (d ≈ 2 mm, ρ ≈ 7.4 g/cc3), and the rest of the way with an interstitial fluid of water or glycerin–water mixture. A glycerin–water mixture was used because a slight change in concentration leads to a relatively large change in viscosity with a relatively low change in density. The concentrations of glycerin–water mixture used were 0%, 33%, and 66% glycerin; viscosities, respectively, were 1.005, 3.28, and 17.1 cP, while the densities varied only slightly. (See Table 1.) The viscosity of each solution was measured both before and after the experiment using a Brookfield viscometer. The viscosity of the fluid only changed by approximately 4% from before the experiment to after the experiment; the viscosity of the solution, reported in Table 1, is the average of these two values. The system was rotated at constant speeds using a Compumotor stepper motor. The speeds tested were 1, 2, and 3 rpm. These speeds allowed for a steady flow, slow enough that the surface wass flat and the beads were not ejected above the surface into the liquid, yet fast enough that there was a steady flow (and not merely intermittent avalanches).

ab

Fig. 1. As the drum rotates, there is (a) a thin flowing layer while (b) the rest of the particles

rotate as in solid body rotation.

Interstitial Fluid Effect on Dense Free-Surface Granular Flow

29

Table 1. Viscosities and densities of the interstitial fluids used % Glycerin Viscosity (cP) Density (g/mL)

0 33 66

1.01 ± 0.06 3.3 ± 0.3 17 ± 4

1.00 ± 0.02 1.18 ± 0.06 1.20 ± 0.05

Data were taken by focusing a high-speed digital camera in the middle of the boundary layer where the flow is most uniform (Fig. 2a) and taking pictures at the rate of 500 frames per second. A particle-tracking program [5] located the position of each particle in each image and tracked the particles from image to image (Fig. 2b). From these position measurements, instantaneous velocities (Fig. 2c), as well as such average fields as the density (Fig. 3a) and velocity (Fig. 3b) could be ascertained. (For more details, see Ref. [5].) Each concentration was tested at all three speeds and three experiments were performed under each set of experimental condition to determine the variability from run to run. As can be seen from Figs. 2d and 2e, there are only minor variations from run to run.

Fig. 2. (a) Diagram of drum with square where camera is focused. (b) Path of each particle

tracked superposed onto one photo. (c) Photo of beads that the camera sees. Instantaneous velocity vectors are placed on each bead.

66% Glycerin 1 rpm Volume Frction

-5

0

5

10

15

20

-0.05 0.05 0.15 0.25 0.35 0.45 0.55 0.65

Volume Fraction

dept

h (m

m)

66% Glycerin 1 rpm u average

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2 0.25u average (m/s)

dept

h (m

m)

Fig. 3. Results for 66% glycerin solution rotated at 1 rpm:

(a) volume fraction; (b) velocity in the u direction.

In Fig. 3a, the top of the flowing layer is set to be the point at which the volume fraction 1%f ≈ (as shown in Fig 2a). Any data above that point correspond to less than one particle,

and therefore are statistically unreliable. Three different experiments run under the same

~ 150d

~20d (a) (b) (c)

(a) (b)

MICHAEL E. LY

30

conditions illustrate the level of reproducibility of the results. In Fig. 3b, results from three identical experiments are shown to determine the level of reproducibility from run to run.

Experimental results

There are several similarities between the flow of particles in air, as reported by Hill, Gioia, and Tota [3], and the flow of particles in other (low viscosity) interstitial fluids. For example, for all cases, the volume fraction contains a characteristic oscillation indicative of the structure in the flow (Fig. 3a). The average velocity of the flowing layer throughout the boundary layer is parallel to the free surface; the velocity is a maximum near the free surface and decreases to zero approximately 9–10 bead diameters into the flowing layer. Also, as the rotation speed increases, the maximum velocity u parallel to the free surface increases, as does the depth of the flowing layer. While the layered structure of the flowing layer is not immediately apparent in the graph of the average velocity u , the derivative of the velocity with respect to depth, or shear rate, d / du y , reveals a clear signature of the layered structure, as was seen for particles in dry granular flow. The velocity fluctuations parallel and perpendicular to the free surface, defined by

( ) ( ) ( )u t u t u t′ = −

and

( ) ( ) ( ) ( )v t v t v t v t′ = − = ,

respectively, appear similar to those for particles in air as well. For example, the velocity fluctuation correlations u u′ ′ , v v′ ′ , and u v′ ′ are all maximum near the top, decreasing to zero within 9–10 bead diameters with the average velocity. The fluctuation in the direction of the average flow u u′ ′ is always greater than the fluctuations perpendicular to the average flow,

and u v′ ′ is always negative. In all cases, the velocity fluctuation correlations appear to increase with the frequency of rotation. While the general forms of these kinematic details are similar, regardless of the interstitial fluid, changing the viscosity of the interstitial fluid has some clear effect on the quantitative values of the Eulerian fields. For example, as the viscosity of the interstitial fluid increases, the maximum velocity near the top of the flow decreases, and the depth of the flowing layer increases (Fig. 5a). Correspondingly, the shear rate also decreases as the viscosity of the solution is increased, and the depth of the maximum shear rate drops (Fig. 5b). Similarly, the velocity fluctuations (Fig. 5c, 5d, 5e) also decrease in value as the viscosity of the interstitial fluid is increased. At the highest viscosities tested, the fluctuations are severely reduced. The shape of the curves is nearly flat at the top, and, in fact, the slope of the velocity fluctuations in the direction parallel to the flow u u′ ′ reverses near the surface, and a reduction in the fluctuations is apparent.

Interstitial Fluid Effect on Dense Free-Surface Granular Flow

31

1 part Glycerin 1 rpm Volume Fraction

-5

0

5

10

15

20

-0.05 0.05 0.15 0.25 0.35 0.45 0.55 0.65

dept

h (m

m)

33% Glycerin u average (m/s)

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

dept

h (m

m)

33% Glycerin du/dz (1/s)

0

0.005

0.01

0.015

0.02

0.025

0.03

-35 -25 -15 -5 5 15

dept

h (m

)

33% Glycerin <u'u'> (m2/s2)

0

5

10

15

20

25

30

0 0.001 0.002 0.003 0.004 0.005de

pth

(mm

)

33% Glycerin <v'v'> (m2/s2)

0

5

10

15

20

25

30

0 0.0005 0.001 0.0015 0.002 0.0025

dept

h (m

m)

33% Glycerin <u'v'>

0

5

10

15

20

25

30

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015<u'v'>

dept

h (m

m)

1rpm2rpm3rpm

Fig. 4. Results using 33% glycerin solution as the interstitial fluid. (a)Volume fraction at 1 rpm.

(b) u average velocity at 1, 2, and 3 rpm. (c) du/dz at 1, 2, and 3 rpm. (d) <u′u′> at 1, 2, and 3 rpm. (e) <v′v′> at 1, 2, and 3 rpm. (f) <u′v′> at 1, 2, and 3 rpm.

(a) (b)

(c) (d)

(e) (f)

1 rpm

2 rpm3 rpm

1 rpm

2 rpm 3 rpm

2 rpm

3 rpm 1 rpm

2 rpm

1 rpm

3 rpm

1 rpm

2 rpm 3 rpm

33% Glycerin <u′v′> (m2/s2)

MICHAEL E. LY

32

2 rpm u average (m/s)

0

5

10

15

20

25

30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

dept

h (m

m)

2 rpm du/dz (1/s)

0

0.005

0.01

0.015

0.02

0.025

0.03

-55 -45 -35 -25 -15 -5 5 15 25

dept

h (m

m)

2 rpm <u'u'> (m2/s2)

0

5

10

15

20

25

30

0 0.002 0.004 0.006 0.008 0.01

dept

h (m

m)

2 rpm <v'v'> (m2/s2)

0

5

10

15

20

25

30

0 0.0005 0.001 0.0015 0.002 0.0025 0.003de

pth

(mm

)

2 rpm <u'v'>

0

5

10

15

20

25

30

-0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015<u'v'>

dept

h (m

m)

0% glycerin33% glycerin66% glycerin

Fig. 5. Kinematics of the flowing layer as the viscosity of the interstitial fluid is varied:

(a) u average velocity, (b) du/dz, (c) <u′u′>, (d) <v′v′>, (e) <u′v′>. Rotation speed = 2 rpm.

(a) (b)

(c) (d)

(e)

0%

33%

66%

66%

33%0%

0%

66%

33% 0%

33%

66%

0% 33%

66%

2 rpm <u′v′> (m2/s2)

Interstitial Fluid Effect on Dense Free-Surface Granular Flow

33

Discussion

In many ways slow particulate flow in a viscous fluid resembles that in air, and the changes that were found by using a more viscous interstitial fluid might be expected. The flow of the particles appears limited primarily to a thin boundary layer, similar to the flow of particles in air, and the structure of the flow appears highly stratified. The speed and depth of the boundary layer both increase with rotation speed, as do the velocity fluctuations. Both the velocity and the velocity fluctuations of the particles decrease with an increase in the viscosity of the interstitial fluid. This result might be expected: as the viscous force increases, the maximum velocity should decrease. Further, considering the model described by Ott-Monsivais et al., where the velocity fluctuations are related to the shear rate, it is not surprising that the velocity fluctuations should also decrease with an increase in viscosity. On the other hand, the velocity does not decrease as much as might be expected from simple scaling arguments. If the maximum velocity simply scaled with the inverse of the viscosity of the interstitial fluid, the maximum velocity of the particles in the 66% glycerin mixture should be a factor of 17 less than that of the particles in water; instead the difference is not much more than a factor of two. This result might be understood qualitatively by noting that the interstitial fluid is most likely not stationary, but moving somewhat with the particles themselves. In fact, nuclear magnetic resonance measure-ments have shown that the motion of the interstitial fluid resembles that of different density (immiscible) fluids rotated without particles in a drum, as illustrated in Fig. 6. The details of how this similarity results in the profiles shown in Fig. 5 require a more complete study of how the particles and fluid move relative to one another. At low viscosities, the relationship between the velocity fluctuation correlations and the shear rate predicted by Ott-Monsivais et al., namely

2 2 2 2 2 2 2 21 1 12 2 2, , ,

i i i

i i u j i i v j i i u v jj j j

u u d v v d u v dξ γ ξ γ ξ ξ γ=∞ =∞ =∞

′ ′ ′ ′ ′ ′= = =∑ ∑ ∑

appears to hold. At high viscosities, however, the relationship appears to work less well. (See Fig. 7.) This apparent discrepancy at higher viscosities may be related to an increase in the ordered structure at the highest viscosities. This ordering is so striking that the particles in the drum were found to form a quasi-crystalline structure at the end of the flowing layer. After more than one rotation, this ordered structuring resulted in an intermittent avalanching for the slurry

ω oil

water

Fig. 6. Flow of two liquids in a drum, when the drum is rotated, as measured using nuclear magnetic resonance techniques, reported in Ref. [6]. This flow is similar to that of slurry flow in a drum, where instead of water in the picture above, the particles and fluid move together, and instead of oil in the top half, the fluid moves alone [7].

MICHAEL E. LY

34

systems as the different orientations of the “crystal” entered the flowing layer. In order to overcome the structured ordering, the drum was shaken such that the previous ordering was broken up, and the video of the flowing layer was then taken before half a rotation was complete—the time when the ordered structure would begin to enter the flowing layer. Nevertheless, the additional order in the flow might itself lead to changes in the nature of the flow, resulting in the severe drop in the fluctuations near the top surface.

Fig 7. Tests of the applicability of the model from Ref. [3] for slurries using,

as the interstitial fluid, (a) 0% glycerin and (b) 66% glycerin.

Summary and conclusions

We have studied the interstitial fluid effect on dense, free-surface granular flow in a pseudo-2D drum. The viscosity of the interstitial fluid was varied and the kinematics of the flowing layer was analyzed. It was found that the maximum velocity of the flowing layer decreased and its depth increased as the viscosity of the interstitial fluid increased. Distinct differences between slurry systems of high viscosities and dry systems were found in the velocity fluctuations and shear rate. It was found that there was a sharp decrease in the velocity fluctuations near the middle of the flowing layer when an interstitial fluid of higher viscosity was used. We also found a dominating ordered packing structure when an interstitial fluid was introduced. The ordered packing structure caused an intermittent avalanching, which can produce significant changes in the flowing layer. This study was preliminary research into the study of slurry flow. More research needs to be done to understand slurry flow.

Interstitial Fluid Effect on Dense Free-Surface Granular Flow

35

Acknowledgments

We thank Varun Mittal for his computational assistance. Furthermore, we are thankful to the Department of Theoretical and Applied Mechanics, UIUC, for their facilities, as well as the Department of Defense and the National Science Foundation for their support.

References

1. Ennis, B. J., J. Green, and R. Davies. The legacy of neglect in the U.S. Chemical Engineering Progress 90, 32–43 (1994).

2. Bell, B. The liquid Earth. The Atlantic Monthly, January 1999. http://www.theatlantic.com/ issues/99jan/mudslide.htm.

3. Hill, K. M., G. Gioia, and V. V. Tota. Structure and kinematics in dense free-surface granular flow. Physical Review Letters 91(6) (2003).

4. Ott-Monsivais, S., K. M. Hill, and G. Gioia. Structure and velocity correlations in dense free surface granular flow. Fifth Annual Undergraduate Research Conference in Mechanics, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign (2004).

5. Crocker, J. C., and D. G. Grier. Methods of digital video microscopy for colloidal studies. Journal of Colloid and Interface Science 179, 298 (1996).

6. Nakagawa, M., S. A. Altobelli, A. Caprihan, E. Fukushima, and E-K. Jeong. Experiments in Fluids 16, 54 (1993).

7. Altobelli, S., New Mexico Resonance. Private communication.

37

2D-to-3D Transitions in Dense Free-Surface Granular Flow

Sophie Alice McGough Senior in Physics, University of California, Los Angeles

Jiafeng Zhang Graduate Assistant in Theoretical and Applied Mechanics, UIUC

Advisor: TAM Prof. Kimberly M. Hill

The 2D-to-3D transition of dense free surface granular flow in a slowly rotating drum is investigated. Velocity at the top of the flowing layer and velocity fluctuations at one bead diameter are plotted versus drum thickness and studied for trends. The velocity appears to vary somewhat periodically with the thickness of the drum for very narrow drums.

Introduction

Granular materials appear everywhere—from the salt on our dinner tables to the aspirin in our medicine chests. While granular flow is commonplace, the dynamics that gives rise to the details of the flow are poorly understood [1]. Newton’s laws adequately describe the interactions between granular particles, but in the case of granular flow, many particles interact at once and give rise to complex collective behavior. This behavior can be studied experimentally but it is currently unknown how the frictional and collision interactions give rise to such collective phenomena as boundary-layer granular flow and segregation. Many studies focus on highly energetic, low-density granular flows, similar to a dense gas [2]. Here we focus instead on dense free-surface boundary layer granular flow. (See Ref. [3–6].) This kind of flow is often studied in either small-diameter deep drums or larger-diameter shallow drums, though it is believed that no detailed comparison has been made between the two. Previous experiments conducted with nuclear magnetic resonance imaging (NMRI) show that particle velocity and other kinematic properties are qualitatively similar in the 2D and 3D regimes [7]. However, while the 2D and 3D regimes are qualitatively similar, one would expect the interactions between the beads and the walls to play an increasingly important roll in the particle dynamics as the drum becomes narrow. Here we study how particle kinematics vary in the transition from 2D to 3D flow in a rotating drum.

Experiment

A transparent drum of 11¼ in. (approximately 30 cm) diameter and thickness t is filled halfway full with either 2 mm or 3 mm plastic spherical beads and rotated about its axis at an angular velocity of two rotations per minute. See Fig. 1. The thickness of the drum is varied from experiment to experiment from about 1 to 7 bead diameters.

SOPHIE A. MCGOUGH

38

(c)

Fig. 1. Experimental setup. (a) Schematic of the front of the partially filled rotating drum. (b) Side-view schematic of drum with thickness t indicated.

(c) Photograph of the experimental setup.

When the drum is rotated, only beads in a thin boundary layer flow. Outside the flowing layer the beads move with the drum in a solid-like rotation. A Photron FastCam high-speed high-resolution digital camera is focused on the center of the boundary layer where the flow is most uniform and a total of 1024 images are captured at a rate of 500 images per second. A single light source illuminates the apparatus creating one primary reflection spot on each bead (Fig. 1c). These bright spots are located in each image (to within 1/100 of a bead diameter) and tracked from image to image using software developed from IDL [8]. The data are further processed to obtain such kinematic details as velocities and velocity fluctuations of the beads. Particle tracking is limited to beads near the transparent front plate of the drum. Studying the 2D-to-3D transition using two-dimensional analysis may not initially seem useful. However, results from NMRI experiments indicate that the particle dynamics are measurably different on a drum’s front plate when the thickness is varied [8]. Therefore, varying the drum thickness and tracking particles on the drum’s front plate should provide insight into the flow kinematics of the 2D-to-3D transition.

Results

Initial studies were performed to determine the effect of total drum thickness on flow kinematics, regardless of bead size. Table 1 gives the parameters for these initial studies. For all systems measured, certain general observations can be made, consistent with those from previous measurements in slowly rotated drums, such as those described in Refs. [4,6]. Over most of the boundary layer, the packing fraction of the beads is approximately 50%, slightly below what is considered to be random close packing. (See Figs. 2a, 2b.) Superposed on this is an oscillation of wavelength of about one bead diameter, evidence of the structure of the flow. (For details, see Ref. [3].) The average velocity is essentially parallel to the free surface; perpendicular to the free surface the average velocity is zero (Fig. 3a). We refer to these two directions as x and y, respectively, as noted in Fig. 1. Both velocity and velocity fluctuations tend to be highest at the

2D-to-3D Transitions in Dense Free-Surface Granular Flow

39

top of the flowing layer and decrease rapidly with increasing depth beneath the free surface. (See Fig. 3a, 3b.) There are also clear differences in certain kinematic details for the 2 mm and 3 mm bead systems. The graphs in Fig. 2 illustrate some of these differences. Near the free surface, the 2 mm beads move faster than the 3 mm beads for drum thicknesses of similar thicknesses in bead

Bead diameter, d (mm)

Drum thickness, t (in.)

Drum thickness, t (bead diameters)

2 1/4 3.13 2 5/16 3.97 2 9/16 7.14 3 5/16 2.65 3 9/16 4.76

Table 1 Summary of the initial experiments conducted. Each experiment was repeated three times. Drum thickness is expressed in both inches and bead diameters to give an intuitive sense of the thickness. Drum angular velocity is 2 rpm.

0

10

20

30

40

0 0.1 0.2 0.3 0.4Velocity in x-direction [m/s]

posi

tion

y-di

rect

ion

[mm

]

3 mm

2 mm

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

<f> Average Volume Fraction

shift

ed Z

[mm

]

0

5

10

15

20

25

30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

<f> Average Volume Fraction

shift

ed Z

[mm

]

dept

hbe

neat

hsu

rfac

e(m

m)

dept

hbe

neat

hsu

rfac

e(m

m)

0

5

10

0

5

10

15

0

10

20

30

40

0 0.002 0.004 0.006 0.008Velocity Fluctuation in x-direction [m2/s2]

posi

tion

in y

-dire

ctio

n [m

m] 2mm

3 mm

<f> (average volume fraction) <f> (average volume fraction)

Fig. 2. Details of the flowing layer for drums of thickness ~4d. (a and b): Volume fractions for (a) 2 mm beads and (b) 3mm beads, as a function of depth.

(c and d): Velocities and velocity fluctuation correlations parallel to the free surface for 2 and 3 mm beads.

(a) (b)

(c) (d)

SOPHIE A. MCGOUGH

40

diameters. This result is consistent with NMRI measurements in a longer drum [7]. For this reason, in the sections below, we separately present the velocity and velocity fluctuation profiles for the 2 mm and 3 mm beads.

Coarse variations in drum thickness

Figure 3 shows the velocities for all the experiments summarized in Table 1. Qualitatively, the shapes of these plots are similar. Velocity is large at the top of the flowing layer and declines rapidly to zero. There are subtle differences as the thickness of the drum is varied. However, a systematic trend is not apparent when the thickness of the drum is varied. In Fig. 4, velocity fluctuation correlations parallel to the free surface are presented for the experiments in Table 1. Again, there are some common trends for all five systems: The magnitudes of the velocity fluctuations are greatest near the top of the flowing layer and decline rapidly to zero with increasing distance from the free surface. However, there is no clear trend in the variation of the behavior with increasing thickness of the drum.

0

10

20

30

40

0 0.1 0.2 0.3 0.4 0.5Velocity of 2 mm beads in x-direction [m/s]

posi

tion

y-di

rect

ion

[mm

]

t = 1/4 inch t = 5/16 inch t = 9/16 inch

0

10

20

30

40

0 0.1 0.2 0.3 0.4 0.5Velocity of 3 mm beads in x-direction [m/s]

t = 5/16 inch t = 9/16 inch (a) (b)

Fig. 3. (a) The velocity of 2 mm beads along the x direction (parallel to the flowing layer) versus position along the y direction (perpendicular to the flowing layer) for drums of varying thickness. (b) The velocity of 3 mm beads along the x direction (parallel to the flowing layer) versus position

along the y direction (perpendicular to the flowing layer) for drums of varying thickness.

Figure 5 summarizes the variation of the kinematics with drum thickness with plots of the maximum velocity and velocity fluctuation for each system. In Figs. 5a and b, the velocity at the top of the flowing layer for 2 mm and 3 mm beads is summarized. In Figs. 5c and d, the velocity fluctuations at one bead diameter from the free surface are shown. Each data point corresponds to the average result for three separate trials with the same thickness. The error bars correspond to the standard deviation. This summary makes it even more clear that there is no apparent relationship between the kinematics of the granular flow and the thickness of the drum. However, for all data within a set, the data are taken for drum thicknesses near an integral number of bead diameters (for the 2 mm beads) or a half-integral number of bead diameters (for the 3 mm beads). Perhaps it is more important to investigate changes in the packing of the particles rather than changes in the ratio

/ ,t d i.e. the thickness normalized in bead diameters. This point is illustrated in Fig. 6. If /t d is expressed as an integer plus a fraction of a bead diameter, /t d n δ= + , perhaps δ rather than

2D-to-3D Transitions in Dense Free-Surface Granular Flow

41

n sensitively affects the flow dynamics. In the next section we delve into this question by investigating the change in kinematics with finer changes of drum thicknesses.

0

10

20

30

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Velocity Fluctuations of 2 mm Beads in x-direction

Posi

tion

in y

-dire

ctio

n [m

m]

1/4 inch drum 5/16 inch drum 9/16 inch drum

0

5

10

15

20

25

30

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

Velocity Fluctuations of 3 mm beads in x-direction

5/16 inch drum 9/16 inch drum (a) (b)

Fig. 4. Velocity fluctuation correlations parallel to the flowing layer for the (a) 2 mm and (b) 3 mm beads.

2 mm beads

0

0.1

0.2

0.3

0.4

0 2 4 6 8drum thickness (t) in bead diameters

Velo

city

in x

-dire

ctio

n [m

/s]

3 mm beads

0

0.1

0.2

0.3

0.4

0 2 4 6 8drum thickness (t) in bead diameters

y[

]

(a) (b)

Fig. 5. (a) and (b): The velocity of beads near the top of the flowing layer. Each point represents the average from three different experiments.

Error bars represent the standard deviation from the three experiments.

SOPHIE A. MCGOUGH

42

2 mm beads

0

0.002

0.004

0.006

0.008

0 2 4 6 8drum thickness (t) in bead diameters

Velo

city

Flu

ctua

tion

in x

-dire

ctio

n

3 mm beads

0

0.002

0.004

0.006

0.008

0 2 4 6 8 drum thickness (t) in bead diameters

(c) (d)

Fig. 5 (cont’d). (c) and (d): The velocity fluctuation correlations parallel to the free surface near the top of the flowing layer as a function of drum thickness.

Each point represents the average from three different experiments. Error bars represent the standard deviation of the mean from the three experiments.

Fine variations in drum thickness

To examine the effect of variation in the drum thickness of fractions of a bead diameter, we rotated 3 mm beads in a drum of thickness varying from just over 3 mm to approximately 14 mm (~1d–3.1d). In contrast to the experiments resulting from coarser variations in drum thickness, these results showed a strong dependence on the velocities with minor variations in the drum thickness. In Fig. 7, results are shown for thicknesses of the drum near 2 bead diameters In this narrow range of drum thicknesses, the maximum velocity of the boundary layer increases with the thickness of the drum. In Fig. 8, the velocities are shown for a much wider range of drum diameters,

(a) (b)

Fig. 6. Change in packing that would occur if the drum thickness is increased (a) from two to five bead diameters, (b) from two to two-and-a-half bead diameters.

0

3

6

9

12

15

18

21

24

27

30

0 0.2 0.4 0.6

1.9d2.0d2.1d2.3d

<u> (m/s)

dept

h(m

m)

Fig. 7 Average velocity <u> as a function of depth for four different drum thicknesses, shown in bead

diameters, where 1d = 3mm.

2D-to-3D Transitions in Dense Free-Surface Granular Flow

43

showing this variation to be cyclic. The results for velocities and velocity fluctuations can be summarized again by looking at the values near the free surface. The results, plotted in Fig. 9, again show the dependence to be cyclic, with a frequency a bit less than a bead diameter. The dependence of the maximum velocity on total drum thickness is striking, decreasing, as one might expect, as the drum thickness gets significantly larger than a bead diameter. The dependence of the velocity fluctuations on drum thickness looks somewhat more sporadic, but is still noticeable.

maximum velocity vs. drum thickness

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4

total drum thickness (in bead diameters)

max

imum

vel

ocity

(m/s

)

maximum <u'u'> vs. drum thickness

0

0.002

0.004

0.006

0.008

0.01

0.012

1 2 3 4

total drum thickness (in bead diameters)

max

imum

vel

ocity

(m/s

)

(a) (b)

Fig. 9. (a) Velocity of beads and (b) velocity fluctuation correlations parallel to the free surface near the top of the flowing layer as a function of drum thickness.

Each point represents the average from three different experiments. Error bars represent the standard deviation from the three experiments.

0369

12151821242730

<u> (each line represents an increment of 0.2 m/s)

dept

h (m

m)

1.1 1.4 1.7 1.9 2.1 2.6 2.8 3.1 3.7 4.2 1.2 1.5 1.8 2.0 2.3 2.7 3.0 3.5 4.0

Fig. 8. Average velocity <u> as a function of depth for several drum thicknesses. Thickness of drum (in bead diameters, d = 3 mm) as noted.

SOPHIE A. MCGOUGH

44

Summary and future plans

In the experiments described herein, the thickness of a rotating drum was varied in order to determine how this parameter is related to the kinematics of flow. Velocity at the top of the flowing layer and velocity fluctuations at one bead diameter were plotted versus drum thickness and subsequently studied for trends. The results showed a systematic relationship with drum thickness at very narrow drum thicknesses that decreases as the thickness becomes much larger than a bead diameter.

References

[1] Jaeger, H. M., and Nagal, S. R. March 1992. Physics of the Granular State. Science 255, 1523–1531.

[2] Savage, S. B. 1994. The mechanics of rapid granular flow. Advances in Applied Mechanics 24, 289–365.

[3] Hill, K. M., Gioia, G., and Tota, V. V. 2003. Structure and kinematics in dense free-surface granular flow. Physical Review Letters 91, 064302-1–064302-4.

[4] Choi, J., Kudrolli, A., Rosales, R. R., and Bazant, M. Z. 2003. Diffusion and mixing in gravity-driven dense granular flows. Physical Review Letters 92(17), 174301-1–174301-4.

[5] Jain, N., Ottino, J. M., and Lueptow, R. M. February 2002. Physics of fluids: An experi-mental study of the flowing granular layer in a rotating tumbler. American Institute of Physics 42(2), 572–582.

[6] Nakagawa, M., Altobelli, S. A., Caprihan, A., Fukushima, E., and Jeong, E.-K. 1993. Experiments in Fluids 16, 54.

[7] Results such as the flow profiles in smaller drums appear similar in long drums (Ref.[6]) and shorter drums of larger diameter (Ref. [3], [5]). However, no detailed comparison has been done.

[8] Crocker, J. C., and Grier, D. G. 1996. Methods of digital video microscopy for colloidal studies. Journal of Colloid and Interface Science 179, 298.

45

Microcapsule-Induced Toughening of Bone Cement

Gina M. Miller Senior in Aerospace Engineering

University of Illinois at Urbana-Champaign

Jason M. Kamphaus Graduate Assistant in Aerospace Engineering, UIUC

Advisors: AE Prof. Scott R. White and TAM Prof. Nancy R. Sottos

Acrylic bone cement is the primary material used in orthopedic implants to anchor metal prostheses to bone. By applying existing self-healing techniques to bone cement, it may be possible to extend the lifetime of the implant, thus reducing the occurrence of revision surgeries. This study includes a review of current literature on existing self-healing composites, duplication of previous researchers’ data, as well as a study of various geometries for the localized region of a tapered double-cantilevered beam bone cement specimen.

Introduction

Due to revolutionary work by Charnley,1 two-part self-polymerizing polymethyl methacrylate (PMMA) has become the primary material used to secure a metal prosthesis to bone in an orthopedic implant. Current commercially available bone cement has several weaknesses, including the high exothermic temperatures of the cement, the role of the cement in chemical necrosis of the bone, the stiffness mismatch between cement and bone, the weak zones in the implant-cement interface and cement-bone interface, shrinkage during curing, and inflammatory periprosthetic tissue responses with surrounding tissues.2 Creating a self-healing form of bone cement would possibly extend the lifetime of the implant. Extensive research of self-healing epoxy has been conducted at the University of Illinois. When a crack develops in a self-healing system, urea-formaldehyde microcapsules filled with the monomer dicyclopentadiene (DCPD) break and this healing agent is drawn into the crack plane by capillary action. When DCPD is exposed to embedded Grubbs’ catalyst, polymerization of the DCPD occurs, sealing the crack.3 The aim of this study is to apply these existing self-healing techniques to the commercially available bone cement, Surgical Simplex™ P manufactured by Stryker Orthopaedics, Chicago, Illinois. Preliminary studies show that the addition of microcapsules to this bone cement increases fracture toughness due to localized plastic deformation and crack pinning.4

GINA M. MILLER

46

Experimental design

Microcapsule preparation

To prepare urea-formaldehyde (UF) microcapsules filled with DCPD, 50 mL of 2.5 wt% poly(ethylene-co-malaic anhydride) is added to 200 mL of deionized water. The reaction mixture is stirred at 550 rpm, and urea, ammonium chloride, and resorcinol are added to the mixture. Using NaOH the pH of the reaction mixture is adjusted to be between 3.6 and 3.8. Using HCL, the reaction mixture pH is brought down to 3.5. Octanol is added to eliminate bubbles. Finally, DCPD and formaldehyde are added to the solution and the temperature is raised to 55°C. After four hours, microcapsules are vacuum filtered and washed with water. To complete the manufacturing process they are air dried and sifted.

Specimen geometry

Tapered double cantilevered beam (TDCB) specimens were utilized (Figure 1). The TDCB geometry developed by Mostovoy et al.5 allows for the determination of fracture toughness,

βmPK CC 2I = , (1)

independent of the crack length limited region, where PC is the critical fracture load and m and β are geometric terms. The value β is dependent on widths bn and b. The value m is theoretically determined by the relation

)(

1)(

33

2

ahaham += , (2)

where a is the crack length and )(ah is the specimen height profile. The value m can also be determined experimentally using the Irwin–Kies6 method,

dadCEbm

8= , (3)

where C is compliance and E is Young’s modulus.[3] Due to the expense and workability of bone cement, a localized TDCB geometry was used (Figure 2). In this configuration the localized region where the crack develops consists of commercial bone cement while the remainder of the sample consists of an epoxy resin matrix.

Specimen manufacture

Several sets of localized TDCB specimens consisting of EPON® 828 epoxy resin and Ancamine® DETA curing agent were manufactured and tested to failure. To ensure proper manufacturing technique, experimental fracture toughness was then compared with past researchers’ data to determine uniformity. A set consisting of three epoxy resin specimens with 20% by weight of 180 µm microcapsules in the localized section was manufactured. Two-piece silicon rubber molds were cleaned and prepared with a releasing agent. A silicon rubber spacer, rectangular in shape measuring approximately 9.5 cm x 1 cm x 7 mm was placed in the center section of the mold where the crack plane is located. To mix the epoxy, 12 g

Microcapsule-Induced Toughening of Bone Cement

47

of Ancamine® DETA curing agent was added to 100 g of EPON® 828 epoxy resin and hand mixed with a wooden stick until the mixture was homogeneous. The mixture was degassed for approximately 10 minutes. After degassing, the mixture was poured into three complete molds and placed in an oven at 30°C for 24 hours. The silicon rubber spacer was then removed from each specimen. The molds and specimens were prepared, and the empty center section was filled with an EPON® 828/DETA mixture in which 20% by weight of 180 µm microcapsules were added. The specimens were cured at room temperature for 24 hours followed by 24 hours in an oven at 30°C. Using the same manufacturing technique, two sets consisting of five specimens each with 5% and 10% by weight of 180 µm microcapsules in the localized section were prepared. Specimens were precracked and tested to failure.

Fig. 1. Tapered double cantilevered beam geometry3

(all measurements mm). Fig. 2. Tapered double cantilevered

beam geometry (sketch).

The bone cement Surgical Simplex™ P, manufactured by Stryker Orthopaedics, was used to create several localized TDCB specimens. The TDCB specimens were composed of EPON® 828 resin and DETA for the outer region and Surgical Simplex™ P in the localized region. To create each localized region of bone cement, 4 mL of liquid monomer at 0°C was added to 8 g of the polymer powder and hand mixed for 30 seconds. Due to the short working time of Surgical Simplex™ P, the mixture was poured and then manually administered to the localized region. After curing in an oven for 24 hours at 30°C, specimens were precracked and tested to failure.

Preliminary results

Epoxy specimens

Using pin loading, specimens were tested until failure. Critical load was measured and the fracture toughness was determined from Eqn. (1). These values were compared with those experimentally determined by previous researchers.As determined by Brown et al.3, fracture toughness is a function of microcapsule concentration, as can be seen in Fig. 3. Fracture toughness for samples with 5 wt%, 10 wt% and 20 wt% concentration of microcapsules is seen in Table 1. Experimental data are within reasonable agreement with previous researchers’ data, ensuring consistent manufacturing technique.

GINA M. MILLER

48

0

0.5

1

1.5

0 5 10 15 20 25

Frac

ture

Tou

ghne

ss

(KIc, M

Pa

m1/

2 )

Capsule Concentration (wt%)

Virgin

Healed

Fracture toughness (MPa·m1/2) 180 µm Microcapsules

Sample 5 wt% 10 wt% 20 wt%A 0.72 1.07 1.11 B * 1.05 1.09 C 0.84 ** 1.370 D 0.92 0.71 — E 1.07 1.37 —

Ave: 0.89 1.05 1.19 * Improper Loading **Sample broke during preparation

Fig. 3. Influence of microcapsule concentration on fracture toughness of 180 µm microcapsules.3

Table 1. Experimental fracture toughness.

Bone cement specimens

Several samples consisting of EPON® 828 resin and DETA for the outer region and Surgical Simplex™ P in the localized region were manufactured and tested to failure. The first sample was created with the standard localized tapered double-cantilevered beam geometry as seen in Figure 3. When tested, instead of fracture occurring throughout the center of the specimen, the localized section debonded from the outer region. Failure occurred at the epoxy/bone cement interface. Another specimen was created using the same geometry, but the interface was sanded using 400 grit sandpaper to attempt to create a better mechanical interface. When tested debonding at the epoxy/bone interface again occurred. Two additional localized geometries were created (Figs. 4 and 5). Instead of being rectangular in shape, one has a rectangular key pattern and the other a triangular key pattern. These were designed to create a better mechanical interface and to prevent delaminating at the interface. These samples exhibited the same behavior as debonding at the interface occurred.

Fig. 4. Alternate localized section geometry. Fig. 5. Alternate localized section geometry.

Rather then using EPON® 828 resin and DETA for the outer region of the TDCB sample, PMMA could be used. The debonding at the epoxy/bone cement interface shows that the epoxy/bone cement system is not ideal. Using PMMA, the base material for bone cement, for the outer region could prove to be a better match and prevent debonding.

Microcapsule-Induced Toughening of Bone Cement

49

Conclusions

Application of existing self-healing techniques to bone cement may prove to extend the life of orthopedic implants. Preliminary testing with bone cement samples resulted in debonding at the epoxy/bone cement interface. Future work includes the investigation of PMMA as an alternative to epoxy for the outer region of the TDCB specimen. Future work also includes manufacture of localized tapered double-cantilevered beam specimens with a variety of microcapsule size and microcapsule concentrations. The effects of diameter size and concentration of microcapsules should be analyzed individually and then in the self-contained, in situ system.

Acknowledgments

The author gratefully acknowledges the support of the National Science Foundation and the Department of Defense award EEC-0354102. The author also thanks Profs. Kimberly M. Hill and James W. Phillips for helpful discussions.

References

1. Charnley, J. Anchorage of the femoral head prosthesis to the shaft of the femur. Journal of Bone and Joint Surgery 43B, 28–30, 1960.

2. Lewis, G. Properties of acrylic bone cement: State of the art review. Journal of Biomedical Materials Research (Applied Biomaterials) 38, 155–182; 1997.

3. Brown, E. N., Sottos, N. R., and White, S. R. Fracture testing of a self-healing polymer composite. Experimental Mechanics 42(4), 372–379, 2002.

4. White, S. R, and Sottos, N. R. Unpublished data, 2003. 5. Mostovoy, S., Crosley P. B., and Ripling, E. J. Use of crack-line loaded specimens for

measuring plane-strain fracture toughness. Journal of Materials 2, 661–681, 1967. 6. Irwin, G. R., and Kies, J. A. Critical energy rate analysis of fracture strength. American

Welding Society Journal 33, 193-s–198-s, 1954.

51

Punching Elastic Foams in the Self-Similar Regime

Renee Oats Senior in Physics, Lincoln University

Xiangyu Dai and Igor A. Khoroshilov Research Assistants in Theoretical and Applied Mechanics, UIUC

Advisor: TAM Prof. Gustavo Gioia

We present experimental measurements on the punching of elastic polyether polyurethane foam specimens with a wedge-shaped punch. These measurements show that the mechanical response is linear up to a penetration of the punch of about 40% of the height of the specimen. To explain this surprising result, we study the cells of the foam and conclude that these cells are bistable elastic structures, and, therefore, that in our experiments the strain field in the foams must consist of a high-strain region close to the tip of the punch and a low-strain region far from the tip, the regions being separated by a sharp interface. By studying the self-similar growth of the interface during an experiment, we predict a linear response within the self-similar regime, in accord with the experimental measurements. We then apply the same theory to the case of a conical punch, predict a quadratic response within the self-similar regime, and verify our prediction by performing experiments with a conical punch. We conclude that in the self-similar regime the mechanical response depends only on the dimensionality of the punch: for a two-dimensional, wedge-shaped punch the response is linear whereas for a three-dimensional, conical punch the response is quadratic.

Introduction

Polymeric elastic foams are light cellular materials used in many applications.1 In this paper, we study a specific type of polymeric elastic foam, namely polyether polyurethane foams. These foams are used in packaging, for example, to protect merchandise in shipping, transport-ing, and handling. Polyether polyurethane foams are also used in car seats to provide comfort and safety to car occupants. In these applications and many similar ones, the purpose of the foam is to react to outside forces that impinge on the foam. For example, when a car occupant is seated in a car, his legs, back, and buttocks impinge on the foam of the seat. In this example, the occupant’s legs and back may be thought of as cylindrical punches, and the buttocks as spherical punches. Thus, it is of interest to determine what happens when a punch penetrates an elastic foam, and how the foam reacts mechanically. Here we investigate experimentally the mechanical behavior of polyether polyurethane foams being penetrated by a rigid punch (i.e., a punch made of a material that is much stiffer than the foams). We use two types of punch of simple geometry: a wedge-shaped punch and a conical punch. We also perform a theoretical analysis of the experiments and show the predictions to be in good agreement with the experimental measurements.

RENEE OATS, XIANGYU DAI, IGOR A. KHOROSHILOV, AND GUSTAVO GIOIA

52

Experimental setup

Figure 1 shows the experimental setup. The setup consists of an ATS testing machine that is used to drive the punch into the foam specimen at a fixed, constant velocity. During an experiment, the punch is attached to a load cell whose signal is fed to a computer running the LabVIEW software. This software computes the force (by multiplying the signal from the load cell by a calibration constant) and the penetration of the punch (by multiplying the fixed velocity of the punch by the elapsed time since the punch first came into contact with the foam). After performing these computations, the software displays the force versus the penetration in graphical form on the screen of the computer and in real time. To obtain photographs of the specimen being penetrated by the punch during the experiment, we use a high-resolution digital camera attached to a frame grabber mounted on the same computer.

Fig. 1. Photograph of the experimental setup showing the ATS testing machine,

the wedge-shaped punch, the load cell (the dark cylinder immediately above the punch), the foam specimen, and the computer running the LabVIEW software.

In our experiments we use cubic specimens of side 10 cm. The specimens are made of polyether polyurethane foams of three different apparent densities: 51.6, 57.7, and 80.2 kg/m3. (These foams are known to the manufacturer, General Plastics of Tacoma, Wash., by the codes EF-4003, EF-4004, EF-4005, respectively). In all cases we align the (vertical) axis of the punch with the rise direction of the foam. (Note on the rise direction: Polymetric foams are manufactured by promoting the growth of numerous gas bubbles within a solid or liquid layer of polymer. As a result of the growth of these bubbles, the layer expands anisotropically, mostly along the direction normal to the midplane of the layer. This direction is the rise direction. The rise direction of the foam is an axis of transverse anisotropy.)

Punching Elastic Foams in the Self-Similar Regime

53

Experiments with a wedge-shaped punch

Figure 2 shows a picture of the wedge-shaped punch penetrating the foam specimen in one of these experiments. In the picture, the tip of the punch has penetrated through about 25% of the height of the specimen (or 2.5 cm). From the picture, it is apparent that large strains and rotations must develop in the foam during these experiments, especially in a vicinity of the tip of the punch.

Fig. 2. Experiment with a wedge-shaped punch.

Figure 3 shows plots of the force versus the penetration for the three foams tested with the wedge-shaped punch. It is seen from these plots that the larger the apparent density of the foam, the larger the punching force for any given penetration. The plots of Fig. 3 reveal a striking feature of the mechanical response of the foams: the force varies linearly with the penetration up to a penetration of about 40% of the height of the specimen; then, there is a sudden change, and the mechanical response becomes nonlinear. Because of the constitutive nonlinearities and the large strains and rotations that develop during the experiment, the observed linear mechanical response over a broad range of penetrations would seem unimaginable. How to explain it? To answer this question, in the following section we turn to a theoretical analysis of the experiments.

Strain fields and the mechanical behavior of foam cells

Consider once more the picture of Fig. 2, which shows the wedge-shaped punch penetrating a foam specimen. Close to the tip of the punch, the strain in the foam is high; far from the tip, it is low. Now we might expect a smooth transition form the high strain that prevails close to the tip to the low strain that prevails far from the tip. Figure 4 illustrates this expectation in a graphical form.

RENEE OATS, XIANGYU DAI, IGOR A. KHOROSHILOV, AND GUSTAVO GIOIA

54

Fig. 3. Plots of the force vs. the penetration displacement

measured in three experiments with the wedge-shaped punch. The three experiments correspond to foams of different apparent densities (EF-4003, EF-4004, and EF-4005). For comparison, the height of the specimen is 10 cm.

High Strain

Low Strain

Punch

Smooth Transition

Fig. 4. Schematic of a strain field with a smooth transition

from a high strain close to the tip of the punch to a low strain far from the tip.

Implicit in the expectation illustrated in Fig. 4 is the assumption that the foam may be locally subjected to any value of strain. Yet this is not the case. In fact, polyether polyurethane foams (at least foams of low apparent density) have preferred values of strain.2 These foams may be locally subjected to a low value of strain or a high value of strain, but not to intermediate values of strain. The reason for this behavior must be found in the microstructure of the foams. A polyether polyurethane foam consists of a more or less regular array of cells, where each cell consists of a number of slender bars of similar length and cross section (Figs. 5a and b). Figure 5c shows an idealized version of this array of cells.2 Now the cells of a foam subjected to a compressive strain act structurally as slender, shallow arches. When a load is applied to a slender, shallow arch, the deformation of the arch increases smoothly as the load is increased, but only up to a certain limit load. Then, if the load is increased beyond the limit load, the

Punching Elastic Foams in the Self-Similar Regime

55

deformation of the arch jumps discontinuously to a much higher level. This mechanical phenomenon, called “snap-through buckling” by structural engineers, implies that foams have two preferred values of strain: a low value of strain associated with the configuration of the cells before snap-through buckling (Fig. 5c) and a large value of strain associated with the configuration of the cells after snap-through buckling (Fig. 5d). Because they undergo snap-through buckling when compressed, the cells of foams may be termed bi-stable elastic structures. Even though we may be unaware of it, bi-stable elastic structures are frequently encountered in everyday life. For example, the cap of a shampoo bottle is a bi-stable elastic structure: the cap is always either closed (as in Fig. 5e, which should be compared with Fig 5c) or open (as in Fig. 5f, which should be compared with Fig. 5d), but not partially open or partially closed.

Fig. 5. Microphotographs of an array of cells in a polyether polyurethane foam.

In (a) the rise direction is perpendicular to the plane of the photograph, and in (b) the rise direction is parallel to the vertical direction of the photograph.

(c) An idealized array of cells. (d) The idealized array of cells after snap-through buckling. The cap of a shampoo bottle and a foam cell are examples of bi-stable elastic structures:

(e) closed cap and (f) open cap.

In view of the considerations of the previous paragraph, we must revise the expected distribution of strains illustrated in Fig. 4, in which we had a smooth transition from the high strain that prevails close to the tip to the low strain that prevails far from the tip. Figure 6a illustrates the revised distribution of strains. Close to the tip of the punch, the strain in the foam is still high; far from the tip, it is still low. However, the transition from the high strain close to the tip to the low strain far from the tip is not a smooth transition. Instead, there is a high-strain region and there is a low-strain region, and the interface between these regions is sharp. Across the sharp interface that separates the high-strain region from the low-strain region, the strain jumps from one preferred value of strain to the other preferred value of strain.

RENEE OATS, XIANGYU DAI, IGOR A. KHOROSHILOV, AND GUSTAVO GIOIA

56

The self-similar regime

Let us now discuss how the sharp interface evolves during an experiment with a wedge-shaped punch. For simplicity, we assume that the sharp interface is a semi-cylinder of radius R, as shown in Fig. 6(a). As the penetration increases, we expect that R should increase too (Fig. 6(b)). Since there is no characteristic length in the wedge-shaped punch, the only prevailing lengthscale is the penetration d. Thus the radius R of the sharp interface must be proportional to the penetration d of the punch, as illustrated in Fig. 6. As the penetration increases during the test, the radius of the sharp interface increases in direct proportion to the penetration. This we call the self-similar regime, because in this regime the sharp interface remains similar to itself (it is always a semi-cylinder). The self-similar regime prevails until the sharp interface reaches the bottom of the specimen. After the sharp interface has reached the bottom of the specimen, a new length scale, provided by the height of the specimen, comes into play. Then, the sharp interface need not (and will not) continue to be a semi-cylinder, the self-similar regime comes to an end, and the mechanical response may undergo a sudden change. We recall that Fig. 3 did display a sudden change in mechanical response for a penetration of about 40% of the height of the specimen; we may now ascribe the observed change in mechanical response to the end of the self-similar regime. We may also estimate the higher preferred strain to be close to 0.4.

H

LSharp Interface

H

LSharpInterface

dR

d

R

Punch

Punch

(a)

(b) Figure 6. (a) Schematic of a strain field with a sharp transition

from a high strain close to the tip of the punch to a low strain far from the tip. The penetration displacement is d and the radius of the sharp interface is R.

(b) As the penetration increases, the radius of the sharp interface increases proportionally. This is the self-similar regime.

We now turn to a discussion of how the punching force varies within the self-similar regime. We seek to establish that within the self-similar regime the punching force is proportional to the surface area of the sharp interface. Even though it is possible to prove this rigorously, here we offer only a highly simplified and intuitive plausibility argument, as follows.

Punching Elastic Foams in the Self-Similar Regime

57

Consider a point on the sharp interface at any time within the self-similar regime. There is a foam cell on one side of the interface and a foam cell on the other side. One of these cells has already undergone snap-through buckling, whereas the other has not. Because the cells must be in equilibrium with one another, they must both be subjected to a same stress, which we may call the interface stress. Now if both cells are subjected to the same interface stress, but one cell has buckled and the other one has not, then it must be that the interface stress is the buckling stress, which corresponds to the limit load at which the cells undergo snap-through buckling. Given that the limit load is strictly a property of the cells, and therefore the buckling stress strictly a property of the foam, it follows that the interface stress cannot depend on the radius of the sharp interface. Thus we conclude that the interface stress at any point on the sharp interface is independent of the radius of the sharp interface within the self-similar regime. Consider now the punching force. The punching force can be obtained as an integral of the interface stress over the entire surface of the sharp interface. Because the interface stress does not depend on the radius of the sharp interface, this integral, and therefore the punching force, must be proportional to the surface area of the sharp interface. In conclusion, we have established that within the self-similar regime the punching force is proportional to the surface area of the sharp interface. In the next paragraph, we apply this result to the special case of a wedge-shaped punch. In the case of a wedge-shaped punch in the self-similar regime, the sharp interface is a semi-cylinder, whose surface area is proportional to the radius R. Since R is in turn proportional to the penetration, it follows that the punching force is proportional to the penetration (in the self-similar regime). This is precisely the mechanical response observed experimentally (Fig. 3).

Experiments with a conical punch

Consider now an experiment with a conical punch. The theory developed for the experiment with the wedge-shaped punch should also be applicable to this experiment. In the self-similar regime, the sharp interface is now a semi-sphere, whose surface area is proportional to 2R . Since R is in turn proportional to the penetration, it follows that the punching force should be proportional to 2d (in the self-similar regime). This is our prediction for an experi-ment with a conical punch. To verify our prediction, we perform experiments with a conical punch (Fig. 7). Figure 8 shows plots of the force versus the penetration for the three foams tested with the conical punch. Just as it was the case in the experiments with the wedge-shaped punch, there is a sudden change in the mechanical response for a penetration of about 50% of the height of the specimen. Figure 8(b) shows the same plots of Fig. 8(a), but in the log-log scale. The plots of Fig. 8(b) are linear of slope 2 up to a penetration of about 5 cm, indicating that the punching force is proportional to 2d in the self-similar regime, in accord with our prediction.

RENEE OATS, XIANGYU DAI, IGOR A. KHOROSHILOV, AND GUSTAVO GIOIA

58

Fig. 7. Experiment with a conical punch.

(a) Linear scale (b) Log–log scale

Fig. 8. Plots of the force vs. the penetration displacement measured in three experiments with the conical punch. The three experiments correspond to foams of different apparent densities (EF-4003, EF-4004, and EF-4005). For comparison, the height of the specimen is 10 cm.

Conclusions

The surprising mechanical response observed in experiments with wedge-shaped and conical punches may be traced to the behavior of the basic microstructural components of a polyether polyurethane foam: the foam cells. Because the foam cells are bistable elastic structures, the foam has two preferred values of strain, and the strain field consists of a high-strain region and a low-strain region separated by a sharp interface. The strain jumps from one preferred value of strain to the other preferred value of strain across the sharp interface. As the penetration of the punch increases during an experiment, the preferred values of strain do not change (since they are strictly properties of the foams); therefore, for the strain field to accommodate the increasing penetration, the interface must expand. For lack of a characteristic

Punching Elastic Foams in the Self-Similar Regime

59

length other that the penetration, the sharp interface must expand in a self-similar manner and in direct proportion to the penetration, in what we have termed the self-similar regime. The geometrical simplicity of the self-similar regime makes the punching experiments unexpectedly amenable to analytical treatment. The analysis indicates that the mechanical response depends only on the dimensionality of the punch. For a two-dimensional wedge-shaped punch, the response is linear, whereas for a three-dimensional conical punch, the response is quadratic. These straightforward predictions are in accord with the experimental results.

Acknowledgments

R. O. would like to thank the TAM Department at the University of Illinois at Urbana-Champaign for the opportunity to participate in the REU program and for their help and support during her stay in Urbana-Champaign. She would also like to thank all her advisors and mentors for having made this research possible.

References

[1] Gibson, L. J., and Ashby, M. F. Cellular Solids. Cambridge: Cambridge University Press, 1997.

[2] Gioia, G., Wang, Y., and Cuitiño, A. M. The energetics of heterogeneous deformation in open-cell solid foams. Proceedings of the Royal Society of London A 457, 1079–1096, 2001.

61

Analysis of Fluid and Thermal Flow throughout a Three-Dimensional Microvascular Network

due to Disproportional Heating

Lyle A. Shipton Senior in Aerospace Engineering

University of Illinois at Urbana-Champaign

Kathleen S. Toohey Graduate Assistant in Theoretical and Applied Mechanics, UIUC

Advisors: AE Prof. Scott R. White and TAM Prof. Nancy R. Sottos

The implementation of microvascular systems for microscale cooling purposes has generated a significant amount of research in recent years. A method for fabricating microvascular systems using direct-write assembly has been developed. The method involves the creation of networks using a fugitive ink, infiltration of networks with an epoxy resin, and evacuation of the fugitive ink, leaving fully developed microvascular networks. The possible application of microvascular systems fabricated using the direct-write method is investigated for microscale cooling purposes. Once created, the networks are infiltrated with a fluid that exhibits suitable thermal conductivity and significant density change with change in temperature. The networks are sealed off using a light-sensitive polymer. It is hypothesized that when heated, temperature gradients across the network will cause the thermally conductive fluid to circulate. Fluid will transfer excess heat away from the heated surface, and disperse it into the environment. This process is beneficial because it does not require external pumping for heat dispersion.

Introduction

The application of microvascular systems to cooling on a microscopic scale has been a widely researched topic in recent years. The overwhelming majority of research associated with micro-cooling techniques seems to focus on the cooling of electronics. Increase in micro-processor speed leads to an increase in heat generation by the processor. Heat generation can result in a high temperature density across the chip surface that can severely inhibit processor performance and cause residual damage to the processor. Moreover, size reduction of electronic components places critical restrictions on conventional microprocessor cooling techniques. Microvascular systems offer a new compact method for cooling of such electronic components. Heat can be absorbed into a microvascular network by conduction. By forcing fluid circulation through microchannels within the network, heat can be transferred away from its source and dispersed into the environment. High surface-area-to-volume ratio, low thermal resistance, and small coolant volume make microvascular systems a viable option for convective cooling.1 Several current research projects have focused on the use of two-dimensional microchannels for heat dissipation. Wang and Peng experimented with single phase forced-

LYLE A. SHIPTON

62

convection heat transfer characteristics through rectangular cross-sectioned microchannels.2 They hypothesized that the heat transfer coefficient for a laminar flow through microchannels may be higher than that of turbulent flow through normal-sized channels. Their results indicated that heat flow through microchannels, whether turbulent or laminar, is highly irregular and complicated. Heat transition characteristics on the microscale are highly dependent on liquid temperature, velocity, and microchannel diameter. Peng and Peterson researched single-phase forced-convection heat transfer and flow characteristics of water in microchannels with rectangular cross sections.3 They attempted to determine the effects that geometric configuration has on flow and heat transfer in microchannel structures. The results of their experiment showed that channel shape plays a negligible role for laminar and turbulent conditions at microscopic hydraulic radii. Results also indicated that convective heat transfer of the network’s center channels is a function of the distance between channels. One drawback of microvascular networks has been the loss of pressure due to small hydraulic diameter of microchannels.1 Loss of pressure requires advanced pumping techniques in order to sustain fluid motion for convection. All previous experiments have relied on external pumping mechanisms to maintain flow through the microchannels. External devices conflict with small size limitations imposed on cooling systems. Sert and Beskok produced numerical analysis of an oscillating-flow-forced convection system used for heat spreading on a micro-scopic scale.1 The theoretical apparatus involved two reservoirs connected by microchannels running beneath a heat source. When electrostatically or piezoelectically activated, the membranes of each reservoir would expand and contract in a sinusoidal fashion, causing fluid to flow between the reservoirs. Heat was collected from the heat source via convection through the microchannels and dispersed into the environment through the walls of the reservoirs.

Fig. 1. Microvascular system design along with a specimen

created using direct-write assembly.

Recent research has focused on the direct-write assembly of three-dimensional microvascular systems.4 This method begins with the robotically controlled deposition of a fugitive ink on a two-dimensional plane. A three-dimensional network is created when several layers of fugitive ink are stacked on top of one another. After deposition, the network is infiltrated with an epoxy resin and allowed to cure for 24 hours. Once the resin is cured, the fugitive ink is removed from the system, leaving empty channels where the original network was located. The final product is a three-dimensional network with pervasive, interconnected cylindrical microchannels. The microvascular system design along with a microvascular system fabricated using the direct-write method are shown in Figure 1. Previously, networks created by

Flow throughout a Microvascular Network

63

direct-write assembly have been utilized in fluid mixing on a microscopic scale and autonomic self-healing materials. Another possible application for three-dimensional microvascular systems is microscale cooling. The proposed system would be created using direct-write assembly, and then filled with a fluid that exhibits density change with heat addition. Once filled, the network would be sealed off using photopolymerization. When applied to a heat source, the system will begin to conduct heat through its wall and into the fluid. When fluid is exposed to heating, the fluid will begin to expand away from the heat source, causing circulation within the system. As the heated fluid moves away from the heat source, it disperses heat into the environment. The system will continue to disperse heat using convection through the circulating fluid. The current research project involves the examination of three-dimensional microvascular systems for microscale cooling. The goal of the project is to test the thermal capabilities of a microvascular system created using direct-write assembly. In addition, the flow within the network will be examined.

Experimental setup

The proposed experiment involves two aspects: the testing of the thermal capabilities of a microvascular system and the analysis of flow within the system during constant disproportional heating. The microvascular system’s thermal capabilities will be examined and compared with several controlled scenarios. Prior to testing of the experimental network, three control experiments will be performed. The control experiments involve a solid cube of epoxy resin with dimensions equal to those of the experimental system, an empty microvascular system, and a microvascular system containing fluid characteristically unresponsive to heating. Each control system will be placed on top of a constant-temperature heat source where the temperature of the heated surfaces will be compared with the temperature of the surfaces exposed to the environment. Finally, the experimental network will be placed under similar conditions. The temperature across the network will be monitored. The temperature variation across the experimental network will be compared with the temperature variation across the controlled setups to determine cooling efficiency. Each network’s bottom surface will be exposed to a constant temperature heat source in order to monitor flow and temperature variation. Heat will be provided using a SpotIR model 4085 heater (Research Inc., Eden Prairie, Minn.). This heater uses infrared energy to heat areas with small diameters (6.4 mm). The temperature of heating should not exceed the boiling point of the fluid inside the network or the melting point of the epoxy used to construct the system. The system will be kept at a constant temperature using proportional-integral-derivative (PID) controlling. Thermocouples will be utilized for both controlling the temperature as well as monitoring temperature variations across the networks. One thermocouple will be responsible for PID control of the heat source. Additional thermocouples will be adhered to the surface of the networks in order to collect surface temperature data across the network. Both controlling and temperature analysis will be conducted using LabVIEW software (National Instruments Corp.). In a separate experiment, a camera will be utilized to observe flow within the system. The experimental network will be filled with the thermally conductive fluid containing flow

LYLE A. SHIPTON

64

visualization particles. These particles will be microcapsules filled with a thermochromatic dye for easy observation. Particles range in size from 2 to 10 µm. An Edmund Smith zoom microscope will be fitted with a charge-coupled device (CCD) camera in order to track particle movement in flow. A diagram of the setup is displayed in Figure 2.

CCD cameraZoom microscope

Parabolic heater

Specimen

Fig. 2. Experimental setup for testing of microscale cooling.

Microvascular system design

Networks are designed using IlliniCAD© software and created using a robotically controlled depositor. Networks are designed as rectangular parallelepipeds with dimensions 20 mm in length and in width and approximately 15 mm in height. Microchannels within the network are on the order of 200 µm in diameter. Previous experimentation indicated that actual microchannel diameter varies by about 8 µm.4 An example of a system’s microchannels is displayed in Figure 3. Each layer of the network is created in a horizontal plane and stacked vertically on top of the previous layer. Layers are stacked by approximately 85% of their height to insure contact between channels. Each layer consists of parallel rows of ink distributed with a 4:1 ratio with approximately 25 rows per layer. Consecutive layers are positioned perpendicular to one another. Every other layer is offset in the horizontal plane by 400 µm. A diagram of the network design is displayed in Figure 4. Due to the imposed size restrictions, networks have approximately 100 layers. Networks are infiltrated with an epoxy resin for curing. The microvascular systems used in the current experiment are infiltrated with EPON 828 epoxy resin (Shell Chemical) as well as EPI-CURE 3274, which serves as a curing agent. Previous experimenters found these polymers to be optically clear and of relatively low viscosity prior to curing. Once networks are cured, the systems will be cut to size and polished to exact dimensions. Consistent network design is imperative for reliable data acquisition. The same structural pattern is used to create all three networks used for testing. All networks undergo the same fabrication process and are subjected to identical experimental conditions. In all, three networks are created for use in the experiment. The systems are sealed off on all sides using a photo-polymerization technique. A small section is left for fluid to be injected into the system. The first system is left vacant of fluid. The second system is filled with a fluid that is resistant to density change while under heating. The third system is the experimental system consisting of a fluid that will change density with heat addition and that optically expresses temperature change.

Flow throughout a Microvascular Network

65

Fig. 3. Microchannels embedded within a microvascular network (scale bar: 200 µm).

800 µm

200 µm

Fig. 4. Diagram of network structural design.

Conclusion

Microvascular systems created using direct-write assembly are currently being considered for use in microscale cooling techniques. The direct-write method used for fabrication of three dimensional microvascular systems has previously been researched. An appropriate network design has been created for the purpose of micrscale cooling, and networks for the experiment are being fabricated. Flow visualization particles for use during flow tracking are being examined. Techniques for proper fluid infiltration as well as photopolymerization are also being considered. Theoretically, when networks are exposed to disproportional heating, a temperature gradient will be created across the network that will in turn cause the fluid inside to begin to circulate. Fluid circulation will transfer heat from the heated surface by convection and disperse heat into the environment. Actual system efficiency will be monitored using thermocouples. Microvascular flow within the system will also be examined using flow tracking particles.

LYLE A. SHIPTON

66

Acknowledgments

The authors gratefully acknowledge the National Science Foundation and the Department of Defense for their support in the current project. The author would also like to thank Prof. Kimberly M. Hill and Prof. James W. Philips for their guidance.

References

1. Sert, C., and Beskok, A., Oscillatory flow forced convection in micro heat spreaders. Numerical Heat Transfer Part A—Applications 42 (2002): 685.

2. Wang, B. X., and Peng, X. F., Experimental investigation on liquid forced-convection heat transfer through microchannels. International Journal of Heat and Mass Transfer 37, supp 1 (1994): 73.

3. Peng, X. F., and Peterson, G. P., Convective heat transfer and flow friction for water flow in microchannel structures. International Journal of Heat and Mass Transfer 39(12) (1996): 2599.

4. Therriault, D., Direct write assembly of three-dimensional microvascular networks, Ph.D. thesis, Department of Aerospace Engineering, University of Illinois at Urbana-Champaign (2003).

5. Gal-el-Hak, M., The MEMS Handbook (Boca Raton: CRC Press, 2002), Chap. 31. 6. Senton, D., Erickson, D., Li, D., Microbubble lensing-induced photobleaching (µ-BLIP) with

application to microflow visualization. Experiments in Fluids 35(9) (2003): 178. 7. Devasenathipathy, S., Santiago, J. G., and Takehara, K. Particle tracking techniques for

electrokinetic microchannel flows. Analytical Chemistry 74(15) (2002): 3704.

67

Lennard–Jones Fluid Flow Simulation: A Study of Fluid–Solid Interaction

Paul K. Shreeman Senior in Physics (Math minor)

Southern Illinois Univsity

Advisor: TAM Prof. Jonathan B. Freund

Solid–liquid interactions on the atomic level are not well-known phenomena. There are many factors involved in these processes. This study focuses on the attractive force between the solid and liquid atoms due to the Lennard–Jones (LJ) pair potential. The LJ atoms are easy to simulate, and the atoms’ properties are well known. We will be using LJ atoms for all types of atoms (solid, liquid, and vapor). The difference between the types of atoms will be due to the mass and the energy. We will apply uniform body force on liquid to create flow over the solid wall. The interaction of liquid and solid at the phase boundary are studied. The interest in this particular phenomenon was inspired by a study on evaporating menisci by Prof. Jonathan B. Freund. A layer of liquid atoms near the solid wall was frozen due to strong attraction force. This frozen behavior was an unknown phenomenon. The flow on the atomistic level between two solid walls has been studied, but results on a solid wall and vapor in equilibrium have not been reported.

Background

The LJ pair potential formula is

,4)(612

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

rrru σσε (1)

where r and ,, σε represent energy (minimum), length where the potential is zero, and distance, respectively. This formula will be used to find the force between atoms. The force is derived by using the relation,

,x

f∂∂

−=U (2)

from which we then obtain total force for ith atom,

∑≠= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

N

ijj

iji rrr1

713

,2148

xf σσ

σε (3)

where jiij XXX −= . The force will be truncated at 2.5 sigma to conserve the computation time, and shifted by the value of cutoff cr ,

PAUL K. SHREEMAN

68

otherwise

, if )()(0F

ffF=

<−= cc rr rr (4)

to remove the discontinuity at the cutoff. The force calculation is the most time-intensive algorithm in this simulation. The time integration of motion requires that we calculate the force twice each time step. The velocity–Verlet algorithm is

,

2

,

11

21

i

ni

nin

ini

ni

i

ni

ni

ni

mt

mtt

++

+

+∆+=

∆+∆+=

ffvv

fvxx (5)

where nii

ni

ni mt fvx and ,,,,1 ∆+ represent position of an atom, length of time elapsed, velocity of an

atom, mass of an atom, and the force on the atom, respectively. As we can see, equation (5) requires two force values. First, we will calculate the position of atoms due to initial position, velocity, and the calculated forces based on initial positions. The new position is then used to calculate 1+n

if . Finally, we calculate the new velocity. This is a single time-step cycle, and the calculations involved are very time-intensive due to calculating the force for every atom. We are using two efficiency algorithms called the cell-list and the neighbor-list to reduce the computation time even more. The cell-list divides the simulation into cells with dimensions of cr , as shown in Fig. 1.

rc

rc

Fig. 1. The cell-list.

The cell-list records the cell that atoms belong according to their positions. Once that has been accomplished, the force has to be calculated only within the cell and its neighboring cells. All other cells are out of the range (cutoff). The neighbor-list uses this list to generate the atom’s neighbors. For example, if we are calculating the first atom, we will be looking at all atoms that are within its range. We first look at what cell it belongs to and the neighboring cells. We then calculate the actual distance between the first atom and all other atoms that are found in that selected area. The first atom neighbor-list is generated by linking all selected atoms that are within the cutoff range. The force algorithm then calculates only the list that is linked to that particular atom. The burden of selection process is no longer on force but on the cell-list and neighbor-list algorithms. This process speeds up the computation considerably, especially if the number of atoms being simulated is in range of thousands or more.

Leonard–Jones Flow Simulation

69

An Andersen thermostat is used to control the simulation’s temperature. The algorithm randomly selects atoms based on a selected value of probability. It then assigns the atoms velocities based on the Maxwell–Boltzmann distribution. This approach is much less intrusive than other methods, such as re-scaling velocities. But since it alters the molecular dynamics directly, it is best to control the fluid temperature indirectly by using conduction from the Andersen-controlled wall. Unfortunately, due to lack of time, we were unable to continue past this stage in my program. The simulation has limited capabilities so we would need some kind of boundary conditions. Periodic boundaries are chosen for continuity. The “rebox” algorithm is

where i is the ith atom, and “:” represents dimensions (such as x, y, and z coordinates). Function FLOOR is a Fortran90 function that finds the greatest integer less or equal to its argument. The “Lb” is the length of “box” being simulated. The “X” denotes the position of the atom. The fact that the box is periodic means there is more than one vector possible between two atoms. Therefore, a code is necessary to ensure that the vectors between atoms are the shortest ones possible. The algorithm

in which “xij” represents the vector between two atoms and NINT is the Fortran90 function that finds the nearest integer to its argument. This algorithm will help ensure that the forces between two atoms are represented correctly.

Simulation

Because of the author’s lack of background in molecular dynamics, we decided to start with a simple program. The first program contained only the LJ potential, the velocity–Verlet equations, periodic boundaries, and nearest-vector algorithms with two atoms (Fig. 2). It was successfully simulated. I then was able to add several atoms. At this phase, all atoms are identical. I was able to implement successfully a solid–liquid relationship by assigning a solid’s value of ε that is ten times larger than the liquid/vapor value. With this option, I was able to simulate various few-atom interactions in different setups. It was observed that the liquid does get trapped by the solid’s force under certain conditions. It was similar to the concept of escape velocity, in which an atom has to have high enough kinetic energy to escape the attractive force. The solid wall was fixed by assigning the value zero to the force and velocity. In other words, regardless of calculations, the solid wall will not move under any force, but will still influence the liquid atoms. When the solid wall is absolutely fixed, it appeared that the liquid is less likely to get stuck. The next step was to implement the Andersen thermostat. I was able to implement the random algorithm that assigns the velocities. Unfortunately, I did not understand the Maxwell–Boltzmann distribution well enough to control the temperature of the simulation (Fig. 3).

do i = 1,Number_of_atoms

X(i,:) = X(i,:) - FLOOR(X(i,:)/Lb)*Lb

end do

xij(:) = X(i,:)-X(j,:) - Lb(:)*NINT((X(i,:)-X(j,:))/Lb(:))

PAUL K. SHREEMAN

70

Fig. 2. Simulation of 2 atoms.

Fig. 3. Simulation of 30 atoms.

I also attempted to write the cell-list and neighbor-list algorithms. It did not work as I expected. At that phase, I decided to study the phenomenon by modifying Dr. Freund’s program that was used in the study of menisci. I was able to simulate approximately a thousand atoms. This model has solid wall, liquid, and vapor atoms. However, I did not complete the necessary steps to create a realistic simulation for boundary solid–liquid interaction study that was derived from menisci study (Fig. 4).

Fig. 4. Simulation of 864 atoms.

Leonard–Jones Flow Simulation

71

Conclusion

Although I have been successful in generating some numerical solutions, I have not advanced far enough to safely reach any conclusions. I have observed that the atoms get trapped by an attractive force. However, under what exact conditions, I do not know. Further study is required to reach any kind of proper conclusion.

Acknowledgments

The author wishes to thank the Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, for this rare opportunity. Dr. Kimberly M. Hill has been wonderful coordinator who kept everything running smoothly as possible. She has shown her unwavering dedication to the program and its success. Dr. Jonathan B. Freund is acknowledged for his vast knowledge of molecular dynamics simulation, and his willingness to help a clumsy beginner. In addition, the author wishes to express his gratitude to the opportunity that Dr. Freund has given him to work with him and the project. The author also wishes to thank Dr. James W. Phillips for his wise consultation on many technical points of professional presentations and papers, and his enthusiastic interest in the REU program. Ms. Brett Long is gratefully acknowledged for her unparalleled performance with translating the spoken technical language into and from visual language.

References

Andersen, H. C., Molecular dynamics simulations at constant pressure and/or temperature. Journal of Chemical Physics 72, 2384–2393, 1980.

Frenkel, D., and S. Berend, Understanding Molecular Simulation: From Algorithms to Applications. San Diego: Academic Press, 1996.

Freund, J. B., The atomic detail of evaporating menisci, work in progress, 2004. Freund, J. B., TAM 470: Supplemental notes, unpublished, 2004. Haile, J. M., Molecular Dynamics Simulation: Elementary Methods. New York: Wiley, 1997. Rapaport, D. C., The Art of Molecular Dynamics Simulation. Cambridge: Cambridge University

Press, 1995. Redwine, C., Upgrading to Fortran90. New York: Springer, 1995. Thompson, P. A., and S. M. Troian, A general boundary condition for liquid flow at solid

surfaces. Nature 389, 360–362, 1997.

73

Time Evolution of Nickel Aluminide Bond Coat Surface Rumpling

Daniel S. Widrevitz Junior in Engineering Mechanics

University of Illinois at Urbana-Champaign

Advisor: TAM Prof. K. Jimmy Hsia

Seeking to understand the time-evolution of a nickel aluminide bonding coat, we subjected numerous samples to high temperature for varying lengths of time. The surface became somewhat regular, with grains growing out of an originally polished or amor-phous surface. This morphology appears both in SEM photographs and profilometer scans.

Introduction

Thermal barrier coatings (TBCs) are widely used in industry to protect engine components from high temperature. These coatings do so by producing a heat gradient between the combustion reactions and the engine components. Coupling the effect of a TBC with active cooling systems provides multiple benefits. As Bose et al.1 note, coated engines can be run at higher temperatures (improving performance and efficiency), and components experience less creep and crack damage from thermal effects. It is of great industrial value to extend the working lifetime of TBCs. Downtime for repair and maintenance on systems using TBCs is costly both in lost revenue and in the need for skilled labor. Specific parts of engines experience spallation first, and so can require the disman-tling of the entire assembly as one set of parts loses protection out of sequence with the others. While the introduction of TBCs has greatly reduced the frequency of repairs, there is still great room for improvement. A TBC consists of four principal layers. The innermost layer is the structural metal. A bonding coat (BC) is laid on top of the base metal, ameliorating thermally induced mismatch stresses. As described by Padture et al.,2 a third layer of thermally grown oxide (TGO) grows in between the BC and the outermost layer, which is a ceramic. The ceramic layer provides the actual insulating properties of a TBC. Failure of a TBC occurs due to cracks propagating between the TGO and the ceramic layer, causing spallation. In an earlier investigation by Panat3 it was noted that an exposed BC rumples when held at high temperature. It is likely that this phenomenon plays a significant role in TBC failure by creating stress in the BC–TGO interface. The motivation for this project was to investigate the time evolution of the rumpling features by exposing samples of BC to high temperature for a variety of times. The principal methods of observation consisted of SEM photography and profilometry. Due to irregular oxidation on platinum aluminide BCs, only nickel aluminide BCs were examined.

DANIEL S. WIDREVITZ

74

Experimental procedure

All sample material originated from a 2 cm × 5 cm brick coated with nickel aluminide on the top and sides. The bottom side was coated with platinum aluminide. For testing purposes this brick was cut down by the use of a diamond-bit slow saw into 5 mm × 5 mm cubes. Some of these cubes were polished down to 1 µm via diamond paste abrasive. Markings were scratched into non-critical surfaces to orient the sample. The intent was to heat the sample bricks up to 1200°C and hold them at that temperature in a high vacuum for a specified period of time. To do so, the samples were first cleaned with acetone and ethyl alcohol to remove any accidental biological impurities and then placed into a quartz tube. This tube was then hooked up to a two-stage vacuum device. The primary stage consisted of a rotary-pump that could achieve vacuum levels around 310− torr. A diffusion pump would then continue down to the 610− torr range. In earlier experiments, vacuum levels of

810− torr were reported using the same apparatus, but this level was later found to be difficult to achieve and unnecessary. With a proper vacuum established, the sample would be heated via radiant heat within the quartz tube. The radiation emanated from a symmetrical series of heating elements circling around the quartz tube, providing a (presumably) even level of radiation on every face of the sample. An electronic timer would regulate the temperature, and shut off the heating elements once a preset time was reached. Once the heating cycle was completed, the quartz tube would be removed from the vacuum apparatus, and the sample would be carefully removed from the tube. Initial observations would be made via a stereoscopic light microscope, followed by SEM observations, and several profilometer scans.

Results and discussion

Stereoscopic observations

Stereoscopic observation revealed striking differences in the surface morphology of samples. A clear progression of surface rumpling appeared when samples were lined up in order of ascending time spent in the chamber. Between one and three hours of exposure at 1200°C, rumpling would become fully exhibited. (The one-hour samples exhibited a proto-rumpling surface.) A 25-hour sample showed large clear rumpling features, while those of a ten-hour sample were smaller. The 100-hour samples did not resemble the others in the least, possibly due to oxidation. The preliminary alignment of sample faces was found to have an effect on the final surface. Faces normal to the vertical plane exhibited far clearer rumpling than those otherwise aligned.

Scanning electron microscope observations

The difference between a rumpled and an unrumpled surface are particularly obvious when observed under an SEM. Figure 1 illustrates the microscopic surface differences.

Time Evolution of Surface Rumpling

75

Figure 1. (a) 25-hour sample at 300× magnification;

(b) 25-hour sample at 3000× magnification; (c) as-deposited sample at 3000× magnification.

Figure 1a exhibits the classic appearance of fully developed rumpling. The surface features have soft contours and large grain sizes (relative to samples heated for shorter periods of time). Figure 1b and 1c exhibit the difference between an as-deposited surface and one that has rumpled. SEM observation also reveals that even after ten hours of exposure, the surface is still transitioning towards what is observed on the 25-hour sample. Figure 2 clearly shows the grains found in Figure 1a and 1b forming amid amorphous structures.

Figure 2. Ten-hour sample at 300× and 3000× magnification.

The progression from an entirely amorphous as-deposited surface to the orderly grained surface of the 25-hour sample is clear in Figure 2. While both surface types are visible, the original surface shows signs of melting. At three hours the transition is also visible. The initial surface has softened in Figure 3, and proto-grains are visible as semi-circular protrusions. Some crystallization is also present on this surface. From a macroscopic position the surface already resembles a rumpled state.

DANIEL S. WIDREVITZ

76

Figure 3. Three-hour sample at 1000× magnification.

Beginning with the one-hour samples to the 25-hour sample, the evolution of the surface is that of one going from amorphous structures to an organized grained surface. The full transition period as illuminated by SEM photography is between 10 and 25 hours. The large-scale rumpling visible by light microscope is therefore a larger process than is visible from direct top-down SEM photography. The surface of the 100-hour sample (Figure 4) exhibited a highly crystallized surface; as a result, the grains in the 25-hour sample must also be in a transition state.

Figure 4. One-hundred-hour sample at 1000× magnification.

The rumpling typical of the post-one-hour samples fails to appear in the 100-hour sample. It seems that the mechanisms at work produce surface morphology beyond those so far explored, or that the 100-hour sample became oxidized. The experimental setup was compromised with the partial collapse of the vacuum tube while heating the 100-hour sample, and could have become porous to oxygen.

Time Evolution of Surface Rumpling

77

Profilometer observations

Numerical analysis of the surface evolution was the main focus of this investigation. Numerous profiles were made of the various sample surfaces to facilitate this analysis. Doing so revealed indications of surface change visually, and somewhat mathematically. However, Figure 5 illustrates the difficulty in working with this type of data.

(a) 1-hour sample (b) 10-hour sample (c) 100-hour sample

Figure 5. Profiles of samples.

The three graphs are very similar when viewed along the entire length of the sample. Some slight variation in periodicity or amplitude was noted, but was not found to be strongly consistent. Calculating the average absolute value of the first and second derivatives (by simple point-to-point approximation of slope) did exhibit an upward trend relative to increasing exposure time. A closer examination of the profiles as in Figure 6 reveals more.

1 Hour Sample

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

0.0015 0.00155 0.0016 0.00165 0.0017 0.00175 0.0018 0.00185 0.0019 0.00195 0.002

Sample position (m)

Surf

ace

Hei

ght (

kA)

10 hour Sample

-30000

-20000

-10000

0

10000

20000

30000

40000

50000

60000

0.0015 0.00155 0.0016 0.00165 0.0017 0.00175 0.0018 0.00185 0.0019 0.00195 0.002

Sample position (m)

Sam

ple

Hei

ght (

kA)

100 Hour Sample

-160000

-140000

-120000

-100000

-80000

-60000

-40000

-20000

0

20000

40000

60000

0.0015 0.00155 0.0016 0.00165 0.0017 0.00175 0.0018 0.00185 0.0019 0.00195 0.002

Sample position (m)

Sam

ple

Hei

ght (

kA)

(a) 1-hour sample (b) 10-hour sample (c) 100-hour sample

Figure 6. Profiles of samples (expanded scale).

Figure 6a presents a surface with many features on two scales (about 50 µm and about 10 µm). Figure 6b has a more regular surface curve without most of the smaller set of features, while Figure 6c is both somewhat regular, and has two scales of features (again about 50 µm and about 10 µm). Despite the somewhat suspect nature of the 100-hour sample, the trend up to 25 hours is consistent. (The ten-hour sample is smoother than the one-hour sample, but rougher than the 25-hour sample.)

100 H S l

160000

140000

120000

100000

80000

60000

40000

20000

20000

40000

60000

0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 0 0016 0 0018 0 002

S l iti

S l H i ht

10 h S l

30000

20000

10000

10000

20000

30000

40000

50000

60000

0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 0 0016 0 0018 0 002

S l iti

S l H i ht

1 H S l

40000

30000

20000

10000

10000

20000

30000

40000

50000

0 0002 0 0004 0 0006 0 0008 0 001 0 0012 0 0014 0 0016 0 0018 0 002

S l iti

S f H i ht

DANIEL S. WIDREVITZ

78

Conclusions

Nickel aluminide BC surfaces evolve when exposed to high temperature for long lengths of time. In the period between one and three hours, they develop a wavy surface characteristic. On a smaller scale, grains grow between one and 25 hours of exposure. Profilometry scans reveal the initial softening of the surface and the more distinct surface aberrations found later. Additionally, the orientation of the sample in the heating chamber affected the progress of the surface evolution. This result implies either that a difference in exposure to the heater existed, or more likely that the orientation of internal stress is a factor. It would be valuable to examine whether there is a difference in temperature between the faces. Performing Fourier transforma-tions on the profilometer data would indicate any periodicity or regularity in the surface morphology. Investigation of the diffusion of elements within the BC during the surface evolution might help explain what is feeding the surface evolution. Finally, a method of determining the surface stress and strain would go far towards correlating the current data and its relevance to the TBC delamination problem.

Acknowledgments

The author would like to acknowledge Prof. K. Jimmy Hsia and Dr. Rahul P. Panat for the opportunity and knowledge necessary to work on this project. The author would also like to thank Prof. S. Balachandar, Steve Burdin, Prof. Kimberly M. Hill, Jim Mabon, and Prof. James W. Phillips for help along the way. The author is extremely grateful for the support of the UROP program, the REU program, the Aerospace and TAM departments at the University of Illinois at Urbana-Champaign, and the facilities of the Center for Microanalysis of Materials, University of Illinois, which is partially supported by the U.S. Department of Energy under grant DEFG02-91-ER45439.

References

Bose, S., and J. DeMasi-Marcin, Thermal barrier coating experience in gas turbine engines at Pratt & Whitney. Journal of Thermal Spray Technology 6(1), 1997, 99–104.

Padture, N. P., M. Gell, and E. H. Jordan, Thermal barrier coatings for gas-turbine engine applications. Science 296, 2002.

Panat, R. P., On the rumpling instability in thermal barrier systems. Ph.D. dissertation, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, 2004.

79

Viscoelastic Response of Solid Propellant Materials: A Criterion for Particle Dewetting

Elizabeth A. Zimmermann Senior in Civil and Environmental Engineering

University of Illinois at Urbana-Champaign

Advisor: TAM Prof. Petros Sofronis

The micromechanics of dewetting in solid propellant materials is analyzed at a unit cell. The energetics of particle decohesion is established by accounting for the viscoelastic response of the polymeric binder. With the use of finite-element calcula-tions, the critical hydrostatic stress for dewetting is calculated as a function of the loading rate, particle size, and particle–binder interfacial energy.

1. Introduction

A solid propellant is a three-phase composite material consisting of metal fuel particles (e.g. aluminum), oxidizer particles (e.g. ammonium perchlorate), and a polymeric binder (Fig. 1). The overall constitutive response of a solid propellant is extremely complex due to the differing constitutive responses of the particle and binder phases. The behavior of the binder is linearly viscoelastic whereas the behavior of the particles is linearly elastic. In addition, straining of a solid propellant may bring about decohesion along the particle–binder interface, a phenomenon known as dewetting. Dewetting complicates the overall constitutive response of the material even further as it may result in the formation of voids from which shearing instabilities may initiate. Development of a dewetting criterion is vital to the overall description of the constitutive response of solid propellants. Such a criterion can be used in conjunction with rigorous homogenization theory to establish the effective response of the propellant by

Oxidizer particle

Void

Fuel particle

Binder

Fig. 1. Typical solid propellant microstruture.

Voids can form by dewetting at oxidizer particles.

ELIZABETH A. ZIMMERMANN

80

accounting for the evolution of its microstructure. Previous attempts (Anderson and Farris, 1988) to establish dewetting criteria are based on a linearly elastic analysis of the binder, these criteria are unrealistic because they do not account for the viscous component of the binder’s response. The purpose of the present work is to analyze the phenomenon of dewetting by properly treating the binder as linearly viscoelastic. The energetics of dewetting is analyzed first for the case of linearly elastic constituents, and then the analysis is extended to the case of a linearly viscoelastic binder.

2. The unit cell: Energetics of dewetting

The particles in the solid propellant are assumed to be arranged periodically; the analysis is then carried out at a unit cell (Fig. 2). The unit cell is comprised of a particle embedded in a spherical shell that represents the binder. The binder is strained hydrostatically by radial displacement increments applied at r b= ; that is, dewetting is studied under displacement-controlled conditions. In the present work dewetting is identified with complete decohesion of the particle–binder interface when the applied displacement reaches a critical value, cru . It is emphasized that the analysis could have been carried out under stress-controlled conditions; that is, traction increments could have been applied at r b= . However, as is well known from fracture mechanics, the energy released upon dewetting is independent of the boundary conditions used. Therefore, the displacement-controlled test has been chosen in view of the fact that the calculation of the energy release upon dewetting in this case does not involve calculation of the work done by the external tractions upon dewetting as is the case in the traction-controlled test.

a

bParticle

Binder

Fig. 2. Unit cell model of solid propellant used in establishing a dewetting criterion.

The cell was loaded by a constant radial velocity at r b= .

2.1. Elastic particle in an elastic binder

Both particle and binder are assumed linearly elastic characterized by corresponding bulk and shear moduli. Upon straining and prior to dewetting, the elastic strain energy stored in the cell is

before before beforetotal binder particleU U U= + , (1)

Viscoelastic Response of Solid Propellant Materials

81

where binder

beforebinder dij ij

V

U Vσ ε= ∫ is the energy stored in the binder, particle

beforeparticle dij ij

V

U Vσ ε= ∫ is the

energy stored in the particle, ijσ and ijε are respectively the components of the stress and strain tensors, binderV is the volume of the binder, and particleV is the volume of the particle. Instantaneously, at the time the particle decoheres completely, the outer boundary of the binder at r b= is subjected to the same displacement as that just prior to dewetting (displacement-controlled dewetting) and the inner boundary of the binder at r a= is traction free. Similarly the outer boundary of the particle is traction free. Therefore, part of the elastic energy stored in the binder prior to dewetting is released. The strain energy that remains stored in the binder is

binder

afterbinder dij ij

V

U Vσ ε= ∫ , where now the stress and strain tensors are calculated by solving the

elastic boundary-value problem for the binder under cru u= at r b= and zero traction at r a= . The particle, by unloading completely upon dewetting, releases all of its energy, that is,

afterparticle 0U = . Hence the total strain energy of the cell after dewetting is

after aftertotal binderU U= . (2)

Using Eqs. (1) and (2), one calculates the energy released by the system upon dewetting as

( )before after before after beforetotal total binder binder particleU U U U U= − = − +G . (3)

Using standard elastic calculations, one can show that the energy released by the particle upon dewetting, before

particleU , is equal to the work done to unload the particle once dewetting has occurred.

Similarly, the change in strain energy of the binder upon dewetting, before afterbinder binderU U− , is equal to

the work done to unload the inner boundary of the binder during dewetting. In other words, an alternative way to calculate the energy released upon dewetting is by determining the work needed to annul the tractions exerted at the particle–binder interface when the condition cru u= is reached. The necessary condition for dewetting to take place is that the energy released, G , be greater than or equal to the energy needed to create the two new surfaces:

( )22 4 aπ γ≥G , (4)

where γ is the energy required for complete interfacial debonding. Equation (4) can be used to calculate the critical displacement cru for dewetting or the related radial tractions at r a= and r b= . The energy release rate is calculated from Eqn. (3) by using the solutions for the appropriate elastic boundary-value problems for the binder and the particle.

2.2. Elastic particle in a viscoelastic binder

The viscoelastic nature of the binder means that its behavior is time-dependent; that is, the binder’s response depends on its deformation history. Unlike the case of elastic behavior studied in the previous subsection, the amount of strain energy stored in the viscous binder is

ELIZABETH A. ZIMMERMANN

82

strain-rate dependent, and therefore the related energy released upon dewetting depends on how fast or slow the system is being brought to the dewetting conditions, cru u= .

The constitutive response of the binder was modeled after the experiments by Ozupek and Becker (1992) in which the relaxation of a high-elongation solid propellant was described by the Prony series

( ) ( )eq1

expN

i ii

G t G G t τ=

= + −∑ , (5)

where eqG gives the long-time elastic equilibrium modulus, t is the time, iG is the shear

modulus corresponding to the relaxation time constant iτ , and 8=N . Of course, this Prony series equation involves the effect of the particles in the composite propellant material, and is not just an equation for the pure binder material. However, in view of the absence of experimental data for the pure binder material, it can be assumed that Eqn. (5) is a good descriptor of the viscoelasticity of the binder. Indeed by direct scaling of the constants, the pure binder response may be captured. The bulk modulus, K, of the binder is on the order of gigapascals. Such a high value of the bulk modulus in comparison with the values for the iG indicates binder incom-pressibility. The constitutive response of the particle was again assumed linearly elastic and isotropic. For the study of the energetics of dewetting, the particle–binder system is loaded by a constant velocity on its outer boundary at r b= . As has already been discussed, the energetics of the elastic–binder dewetting cannot be directly applied to the present case of the linearly viscoelastic binder due to the dependence of the binder’s response on loading history. In other words, it cannot be set that the energy released by the binder upon dewetting is before after

binder binderU U−

even if beforebinderU and after

binderU are calculated properly by accounting for the viscoelastic nature of the binder. On the other hand, one can assume that unloading of the binder upon dewetting happens at such a fast rate that the viscous component of the material’s response has no time to contribute to the deformation. Under this assumption, the instantaneous response of the binder upon dewetting can be considered as purely elastic. Thus, the energy released upon dewetting is

released beforebinder particleU U= +G , (6)

where releasedbinderU is the elastic energy released by the binder upon unloading and before

particleU is the corresponding elastic strain energy released by the particle and is calculated as discussed in the previous subsection. Since the unloading process of the binder is considered elastic, released

binderU is equal to the work done on the binder’s inner boundary in eliminating the corresponding boundary tractions:

( )( )released int 2 after before1binder cr2 4U a u uσ π= − , (7)

where intcrσ is the interfacial traction at the onset of dewetting, beforeu is the radial displacement

of the interface just prior to dewetting, and afteru is the radial displacement of the interface

Viscoelastic Response of Solid Propellant Materials

83

furnished by the solution of the elastic boundary value problem for the binder under cru u= at

r b= and traction equal to intcrσ− at r a= .

3. Dewetting stress for an elastic particle embedded in a viscoelastic binder

The spherical particle–binder model was used for the calculation of all parameters needed for the determination of the energy release stated in Eqn. (6). The cell was loaded by a constant velocity on its outer boundary, at r b= , and the problem was solved by the general-purpose finite-element code ABAQUS. Due to spherical symmetry, only a quarter had to be meshed in the finite-element procedure. See Fig. 3. The oxidizer and fuel particles of the solid propellant microstructure differ in size; therefore, numerical simulations were performed for a particle of size 200 ma µ= and 20 mµ . The standard volume fraction of particles, 0.71f = , in the solid propellant was then used to obtain the outer radius of the binder, b . For each particle size, finite-element calculations were performed at five different applied velocities at r b= (macroscopic strain rates). In view of the absence of experimental data on the particle–binder interfacial energy, the standard value of 21.0 J mγ = was assumed.

ParticleBinder

Fig. 3. Mesh of the particle–binder system.

After post-processing the data, the energy that would be released, G , from the particle–binder system upon dewetting was calculated and monitored at the end of each displacement increment using Eqn. (6). Once the energy released met the condition set by Eqn. (4), the system

ELIZABETH A. ZIMMERMANN

84

was said to be at a critical state at which dewetting could be assumed to initiate. The energy released, G , by the particle–binder system is plotted versus time in Figs. 4 and 5 and normalized by the energy needed to create the new particle–binder surfaces that are formed upon dewetting; therefore, when the energy released divided by the surface energy is equal to unity, the system has reached a critical time at which dewetting may occur. The critical time for dewetting was found for all the cases analyzed. These critical times were used along with Figs. 6 and 7 to determine the critical value crσ of the applied traction at r b= for all macroscopically applied strain rates and particle sizes. This critical traction is defined to be the critical stress for dewetting. The critical stress, that is, the positive hydrostatic traction on the outside boundary of the particle–binder system at the time of dewetting, increases slightly as the strain rate increases, as shown in Fig. 8. Also, as the particle size decreases, the critical stress is found to increase.

Time / AVG

Ene

rgy

rele

ased

/Sur

face

ener

gy

0 0.0001 0.00020

0.5

1

1.5

2

5

τ

0.1

1

10

Fig. 4. Plot of the normalized energy 28 aπ γG released on dewetting as a function of normalized

time AVGt τ at various applied macroscopic strain rates and a particle size a = 200 µm. The critical time for dewetting is found when the normalized released energy is equal to unity.

The parameter AVGτ is the average relaxation time of the binder, equal to 270 s.

4. Conclusions and future work

The critical stress at which an elastic particle dewets from a linear viscoelastic binder has been calculated by using a unit cell model. This dewetting criterion is more realistic than previous attempts because the true viscous response of the solid propellent’s binder has been accounted for. It has been found that, as the size of the particle decreases, the critical stress needed to dewet the particle increases. Also, the critical dewetting stress increases as the macroscopic strain rate, to which the particle–binder system is subjected, increases.

Viscoelastic Response of Solid Propellant Materials

85

Time / AVG

Ene

rgy

rele

ased

/Sur

face

ener

gy

0 0.0002 0.0004 0.0006 0.00080

0.5

1

1.5

2

τ

0.11

5

10

Fig. 5. Plot of the normalized energy 28 aπ γG released on dewetting as a function of normalized

time AVGt τ at various applied macroscopic strain rates and a particle size a = 20 µm. The critical time for dewetting is found when the normalized released energy is equal to unity.

Time / AVG

0 0.0001 0.00015 0.00020

1

2

3

4

5

6

7

8

9

10

τ0.00005

0.1

1

5

10

Fig. 6. Plot of the normalized applied traction 0Gσ on the cell at r b= as a function of

normalized time AVGt τ at various applied macroscopic strain rates and a particle size a = 200 µm. A critical stress for dewetting crσ can be found for each macrosocpic strain rate from the critical

time calculated from Fig. 4. The parameter 0G denotes the unrelaxed elastic shear modulus of the binder, which is equal to 2.924 GPa.

ELIZABETH A. ZIMMERMANN

86

Time / AVG

0 0.0001 0.00015 0.00020

10

20

30

τ0.00005

0.1

1

5

10

Fig. 7. Plot of the normalized applied traction 0Gσ on the cell at r b= as a function of

normalized time AVGt τ at various applied macroscopic strain rates and a particle size a = 20 µm. A critical stress for dewetting crσ can be found for each macrosocpic strain rate from the critical

time calculated from Fig. 5.

Strain rate ( )0 2 4 6 8 10

6

8

10

12

14

16

18

20

22

24

20200

Particle s ize ( )µm

0 1 2 3

23.851

23.852

0 1 2 3

7.5425

7.5426

Fig. 8. The normalized critical stress for dewetting cr 0Gσ

plotted versus the applied macroscopic strain rate.

The critical stress was seen to increase only slightly with strain rate (see Fig. 8); the dependence is hypothesized to be stronger as the value of γ increases. In the future the calculation will be done for different values of γ to determine its influence on the critical stress.

Viscoelastic Response of Solid Propellant Materials

87

Also the calculation will be performed for different-shaped particles since the true shape of the oxidizer and fuel particles is not perfectly spherical. Another possible source of strain-rate dependence that needs investigation is related to the unloading of the binder upon dewetting. Possible viscous response by the binder during dewetting brings about an additional strain-rate contribution to the critical stress. Lastly, the dependence of the critical dewetting stress will be calculated under a superposed macroscopic shear stress.

Acknowledgments

This work was supported by NASA under the NASA Space Grant Program Work Force Development Grant and the Center for Simulation of Advanced Rockets, funded by the U.S. Department of Energy through the University of California under subcontract number DOE/LLNL/B523819. The author would like to acknowledge the support and guidance received from Fengbin Xu, graduate student in Theoretical and Applied Mechanics, and the author’s advisor, Prof. Petros Sofronis.

References

Anderson, L. L., and Farris, R. J., 1988. A predictive model for the mechanical behavior of particulate composites. Polymer Engineering and Science 28, 522–528.

Nicholson, D. W., 1979. On the detachment of a rigid inclusion from an elastic matrix. Journal of Adhesion 10, 255–260.

Ozupek, S., and Becker, E. B., 1992. Constitutive modeling of high-elongation solid propellants. Journal of Engineering Materials and Technology 114, 111–115.

List of Recent TAM Reports

No. Authors Title Date

972 Sofronis, P., and I. M. Robertson

Atomistic scale experimental observations and micromechanical/ continuum models for the effect of hydrogen on the mechanical behavior of metals—Philosophical Magazine (submitted)

June 2001

973 Pushkin, D. O., and H. Aref

Self-similarity theory of stationary coagulation—Physics of Fluids 14, 694–703 (2002)

July 2001

974 Lian, L., and N. R. Sottos

Stress effects in ferroelectric thin films—Journal of the Mechanics and Physics of Solids (submitted)

Aug. 2001

975 Fried, E., and R. E. Todres

Prediction of disclinations in nematic elastomers—Proceedings of the National Academy of Sciences 98, 14773–14777 (2001)

Aug. 2001

976 Fried, E., and V. A. Korchagin

Striping of nematic elastomers—International Journal of Solids and Structures 39, 3451–3467 (2002)

Aug. 2001

977 Riahi, D. N. On nonlinear convection in mushy layers: Part I. Oscillatory modes of convection—Journal of Fluid Mechanics 467, 331–359 (2002)

Sept. 2001

978 Sofronis, P., I. M. Robertson, Y. Liang, D. F. Teter, and N. Aravas

Recent advances in the study of hydrogen embrittlement at the University of Illinois—Invited paper, Hydrogen–Corrosion Deformation Interactions (Sept. 16–21, 2001, Jackson Lake Lodge, Wyo.)

Sept. 2001

979 Fried, E., M. E. Gurtin, and K. Hutter

A void-based description of compaction and segregation in flowing granular materials—Continuum Mechanics and Thermodynamics, in press (2003)

Sept. 2001

980 Adrian, R. J., S. Balachandar, and Z.-C. Liu

Spanwise growth of vortex structure in wall turbulence—Korean Society of Mechanical Engineers International Journal 15, 1741–1749 (2001)

Sept. 2001

981 Adrian, R. J. Information and the study of turbulence and complex flow—Japanese Society of Mechanical Engineers Journal B, in press (2002)

Oct. 2001

982 Adrian, R. J., and Z.-C. Liu

Observation of vortex packets in direct numerical simulation of fully turbulent channel flow—Journal of Visualization, in press (2002)

Oct. 2001

983 Fried, E., and R. E. Todres

Disclinated states in nematic elastomers—Journal of the Mechanics and Physics of Solids 50, 2691–2716 (2002)

Oct. 2001

984 Stewart, D. S. Towards the miniaturization of explosive technology—Proceedings of the 23rd International Conference on Shock Waves (2001)

Oct. 2001

985 Kasimov, A. R., and Stewart, D. S.

Spinning instability of gaseous detonations—Journal of Fluid Mechanics (submitted)

Oct. 2001

986 Brown, E. N., N. R. Sottos, and S. R. White

Fracture testing of a self-healing polymer composite—Experimental Mechanics (submitted)

Nov. 2001

987 Phillips, W. R. C. Langmuir circulations—Surface Waves (J. C. R. Hunt and S. Sajjadi, eds.), in press (2002)

Nov. 2001

988 Gioia, G., and F. A. Bombardelli

Scaling and similarity in rough channel flows—Physical Review Letters 88, 014501 (2002)

Nov. 2001

989 Riahi, D. N. On stationary and oscillatory modes of flow instabilities in a rotating porous layer during alloy solidification—Journal of Porous Media 6, 1–11 (2003)

Nov. 2001

990 Okhuysen, B. S., and D. N. Riahi

Effect of Coriolis force on instabilities of liquid and mushy regions during alloy solidification—Physics of Fluids (submitted)

Dec. 2001

991 Christensen, K. T., and R. J. Adrian

Measurement of instantaneous Eulerian acceleration fields by particle-image accelerometry: Method and accuracy—Experimental Fluids (submitted)

Dec. 2001

992 Liu, M., and K. J. Hsia Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading—Journal of the Mechanics and Physics of Solids 51, 921–944 (2003)

Dec. 2001

993 Panat, R. P., S. Zhang, and K. J. Hsia

Bond coat surface rumpling in thermal barrier coatings—Acta Materialia 51, 239–249 (2003)

Jan. 2002

994 Aref, H. A transformation of the point vortex equations—Physics of Fluids 14, 2395–2401 (2002)

Jan. 2002

List of Recent TAM Reports (cont’d)

No. Authors Title Date 995 Saif, M. T. A, S. Zhang,

A. Haque, and K. J. Hsia

Effect of native Al2O3 on the elastic response of nanoscale aluminum films—Acta Materialia 50, 2779–2786 (2002)

Jan. 2002

996 Fried, E., and M. E. Gurtin

A nonequilibrium theory of epitaxial growth that accounts for surface stress and surface diffusion—Journal of the Mechanics and Physics of Solids 51, 487–517 (2003)

Jan. 2002

997 Aref, H. The development of chaotic advection—Physics of Fluids 14, 1315–1325 (2002); see also Virtual Journal of Nanoscale Science and Technology, 11 March 2002

Jan. 2002

998 Christensen, K. T., and R. J. Adrian

The velocity and acceleration signatures of small-scale vortices in turbulent channel flow—Journal of Turbulence, in press (2002)

Jan. 2002

999 Riahi, D. N. Flow instabilities in a horizontal dendrite layer rotating about an inclined axis—Journal of Porous Media, in press (2003)

Feb. 2002

1000 Kessler, M. R., and S. R. White

Cure kinetics of ring-opening metathesis polymerization of dicyclopentadiene—Journal of Polymer Science A 40, 2373–2383 (2002)

Feb. 2002

1001 Dolbow, J. E., E. Fried, and A. Q. Shen

Point defects in nematic gels: The case for hedgehogs—Archive for Rational Mechanics and Analysis, in press (2004)

Feb. 2002

1002 Riahi, D. N. Nonlinear steady convection in rotating mushy layers—Journal of Fluid Mechanics 485, 279–306 (2003)

Mar. 2002

1003 Carlson, D. E., E. Fried, and S. Sellers

The totality of soft-states in a neo-classical nematic elastomer—Journal of Elasticity 69, 169–180 (2003) with revised title

Mar. 2002

1004 Fried, E., and R. E. Todres

Normal-stress differences and the detection of disclinations in nematic elastomers—Journal of Polymer Science B: Polymer Physics 40, 2098–2106 (2002)

June 2002

1005 Fried, E., and B. C. Roy Gravity-induced segregation of cohesionless granular mixtures—Lecture Notes in Mechanics, in press (2002)

July 2002

1006 Tomkins, C. D., and R. J. Adrian

Spanwise structure and scale growth in turbulent boundary layers—Journal of Fluid Mechanics (submitted)

Aug. 2002

1007 Riahi, D. N. On nonlinear convection in mushy layers: Part 2. Mixed oscillatory and stationary modes of convection—Journal of Fluid Mechanics 517, 71–102 (2004)

Sept. 2002

1008 Aref, H., P. K. Newton, M. A. Stremler, T. Tokieda, and D. L. Vainchtein

Vortex crystals—Advances in Applied Mathematics 39, in press (2002) Oct. 2002

1009 Bagchi, P., and S. Balachandar

Effect of turbulence on the drag and lift of a particle—Physics of Fluids, in press (2003)

Oct. 2002

1010 Zhang, S., R. Panat, and K. J. Hsia

Influence of surface morphology on the adhesive strength of aluminum/epoxy interfaces—Journal of Adhesion Science and Technology 17, 1685–1711 (2003)

Oct. 2002

1011 Carlson, D. E., E. Fried, and D. A. Tortorelli

On internal constraints in continuum mechanics—Journal of Elasticity 70, 101–109 (2003)

Oct. 2002

1012 Boyland, P. L., M. A. Stremler, and H. Aref

Topological fluid mechanics of point vortex motions—Physica D 175, 69–95 (2002)

Oct. 2002

1013 Bhattacharjee, P., and D. N. Riahi

Computational studies of the effect of rotation on convection during protein crystallization—International Journal of Mathematical Sciences, in press (2004)

Feb. 2003

1014 Brown, E. N., M. R. Kessler, N. R. Sottos, and S. R. White

In situ poly(urea-formaldehyde) microencapsulation of dicyclopentadiene—Journal of Microencapsulation (submitted)

Feb. 2003

1015 Brown, E. N., S. R. White, and N. R. Sottos

Microcapsule induced toughening in a self-healing polymer composite—Journal of Materials Science (submitted)

Feb. 2003

List of Recent TAM Reports (cont’d)

No. Authors Title Date 1016 Kuznetsov, I. R., and

D. S. Stewart Burning rate of energetic materials with thermal expansion—Combustion and Flame (submitted)

Mar. 2003

1017 Dolbow, J., E. Fried, and H. Ji

Chemically induced swelling of hydrogels—Journal of the Mechanics and Physics of Solids, in press (2003)

Mar. 2003

1018 Costello, G. A. Mechanics of wire rope—Mordica Lecture, Interwire 2003, Wire Association International, Atlanta, Georgia, May 12, 2003

Mar. 2003

1019 Wang, J., N. R. Sottos, and R. L. Weaver

Thin film adhesion measurement by laser induced stress waves—Journal of the Mechanics and Physics of Solids (submitted)

Apr. 2003

1020 Bhattacharjee, P., and D. N. Riahi

Effect of rotation on surface tension driven flow during protein crystallization—Microgravity Science and Technology 14, 36–44 (2003)

Apr. 2003

1021 Fried, E. The configurational and standard force balances are not always statements of a single law—Proceedings of the Royal Society (submitted)

Apr. 2003

1022 Panat, R. P., and K. J. Hsia

Experimental investigation of the bond coat rumpling instability under isothermal and cyclic thermal histories in thermal barrier systems—Proceedings of the Royal Society of London A 460, 1957–1979 (2003)

May 2003

1023 Fried, E., and M. E. Gurtin

A unified treatment of evolving interfaces accounting for small deformations and atomic transport: grain-boundaries, phase transitions, epitaxy—Advances in Applied Mechanics 40, 1–177 (2004)

May 2003

1024 Dong, F., D. N. Riahi, and A. T. Hsui

On similarity waves in compacting media—Horizons in Physics, in press (2003)

May 2003

1025 Liu, M., and K. J. Hsia Locking of electric field induced non-180° domain switching and phase transition in ferroelectric materials upon cyclic electric fatigue—Applied Physics Letters 83, 3978–3980 (2003)

May 2003

1026 Liu, M., K. J. Hsia, and M. Sardela Jr.

In situ X-ray diffraction study of electric field induced domain switching and phase transition in PZT-5H—Journal of the American Ceramics Society (submitted)

May 2003

1027 Riahi, D. N. On flow of binary alloys during crystal growth—Recent Research Development in Crystal Growth, in press (2003)

May 2003

1028 Riahi, D. N. On fluid dynamics during crystallization—Recent Research Development in Fluid Dynamics, in press (2003)

July 2003

1029 Fried, E., V. Korchagin, and R. E. Todres

Biaxial disclinated states in nematic elastomers—Journal of Chemical Physics 119, 13170–13179 (2003)

July 2003

1030 Sharp, K. V., and R. J. Adrian

Transition from laminar to turbulent flow in liquid filled microtubes—Physics of Fluids (submitted)

July 2003

1031 Yoon, H. S., D. F. Hill, S. Balachandar, R. J. Adrian, and M. Y. Ha

Reynolds number scaling of flow in a Rushton turbine stirred tank: Part I—Mean flow, circular jet and tip vortex scaling—Chemical Engineering Science (submitted)

Aug. 2003

1032 Raju, R., S. Balachandar, D. F. Hill, and R. J. Adrian

Reynolds number scaling of flow in a Rushton turbine stirred tank: Part II—Eigen-decomposition of fluctuation—Chemical Engineering Science (submitted)

Aug. 2003

1033 Hill, K. M., G. Gioia, and V. V. Tota

Structure and kinematics in dense free-surface granular flow—Physical Review Letters, in press (2003)

Aug. 2003

1034 Fried, E., and S. Sellers Free-energy density functions for nematic elastomers—Journal of the Mechanics and Physics of Solids 52, 1671–1689 (2004)

Sept. 2003

1035 Kasimov, A. R., and D. S. Stewart

On the dynamics of self-sustained one-dimensional detonations: A numerical study in the shock-attached frame—Physics of Fluids (submitted)

Nov. 2003

1036 Fried, E., and B. C. Roy Disclinations in a homogeneously deformed nematic elastomer—Nature Materials (submitted)

Nov. 2003

1037 Fried, E., and M. E. Gurtin

The unifying nature of the configurational force balance—Mechanics of Material Forces (P. Steinmann and G. A. Maugin, eds.), in press (2003)

Dec. 2003

List of Recent TAM Reports (cont’d)

No. Authors Title Date 1038 Panat, R., K. J. Hsia,

and J. W. Oldham Rumpling instability in thermal barrier systems under isothermal conditions in vacuum—Philosophical Magazine, in press (2004)

Dec. 2003

1039 Cermelli, P., E. Fried, and M. E. Gurtin

Sharp-interface nematic–isotropic phase transitions without flow—Archive for Rational Mechanics and Analysis 174, 151–178 (2004)

Dec. 2003

1040 Yoo, S., and D. S. Stewart

A hybrid level-set method in two and three dimensions for modeling detonation and combustion problems in complex geometries—Combustion Theory and Modeling (submitted)

Feb. 2004

1041 Dienberg, C. E., S. E. Ott-Monsivais, J. L. Ranchero, A. A. Rzeszutko, and C. L. Winter

Proceedings of the Fifth Annual Research Conference in Mechanics (April 2003), TAM Department, UIUC (E. N. Brown, ed.)

Feb. 2004

1042 Kasimov, A. R., and D. S. Stewart

Asymptotic theory of ignition and failure of self-sustained detonations—Journal of Fluid Mechanics (submitted)

Feb. 2004

1043 Kasimov, A. R., and D. S. Stewart

Theory of direct initiation of gaseous detonations and comparison with experiment—Proceedings of the Combustion Institute (submitted)

Mar. 2004

1044 Panat, R., K. J. Hsia, and D. G. Cahill

Evolution of surface waviness in thin films via volume and surface diffusion—Journal of Applied Physics (submitted)

Mar. 2004

1045 Riahi, D. N. Steady and oscillatory flow in a mushy layer—Current Topics in Crystal Growth Research, in press (2004)

Mar. 2004

1046 Riahi, D. N. Modeling flows in protein crystal growth—Current Topics in Crystal Growth Research, in press (2004)

Mar. 2004

1047 Bagchi, P., and S. Balachandar

Response of the wake of an isolated particle to isotropic turbulent cross-flow—Journal of Fluid Mechanics (submitted)

Mar. 2004

1048 Brown, E. N., S. R. White, and N. R. Sottos

Fatigue crack propagation in microcapsule toughened epoxy—Journal of Materials Science (submitted)

Apr. 2004

1049 Zeng, L., S. Balachandar, and P. Fischer

Wall-induced forces on a rigid sphere at finite Reynolds number—Journal of Fluid Mechanics (submitted)

May 2004

1050 Dolbow, J., E. Fried, and H. Ji

A numerical strategy for investigating the kinetic response of stimulus-responsive hydrogels—Journal of the Mechanics and Physics of Solids (submitted)

June 2004

1051 Riahi, D. N. Effect of permeability on steady flow in a dendrite layer—Journal of Porous Media, in press (2004)

July 2004

1052 Cermelli, P., E. Fried, and M. E. Gurtin

Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces—Journal of Fluid Mechanics (submitted)

Sept. 2004

1053 Stewart, D. S., and A. R. Kasimov

Theory of detonation with an embedded sonic locus—SIAM Journal on Applied Mathematics (submitted)

Oct. 2004

1054 Stewart, D. S., K. C. Tang, S. Yoo, M. Q. Brewster, and I. R. Kuznetsov

Multi-scale modeling of solid rocket motors: Time integration methods from computational aerodynamics applied to stable quasi-steady motor burning—Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit (January 2005), Paper AIAA-2005-0357 (2005)

Oct. 2004

1055 Ji, H., H. Mourad, E. Fried, and J. Dolbow

Kinetics of thermally induced swelling of hydrogels—International Journal of Solids and Structures (submitted)

Dec. 2004

1056 Fulton, J. M., S. Hussain, J. H. Lai, M. E. Ly, S. A. McGough, G. M. Miller, R. Oats, L. A. Shipton, P. K. Shreeman, D. S. Widrevitz, and E. A. Zimmermann

Final reports: Mechanics of complex materials, Summer 2004 (K. M. Hill and J. W. Phillips, eds.)

Dec. 2004