© 2015 Szu-Yin Chen
THE EFFECT OF PORE SIZE AND POROSITY ON CAPILLARY PRESSURE IN MICROPOROUS HYDROXYAPATITE SAMPLES
BY
SZU-YIN CHEN
THESIS
Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2015
Urbana, Illinois
Adviser:
Associate Professor Amy Wagoner Johnson
ABSTRACT
Microporosity (<50μm) in hydroxyapatite (HA) scaffolds is known to improve bone ingrowth. The mechanism for the improvement is thought to be in part due to capillary forces induced by the micropores. The micropore-induced capillary forces can self load cells into the scaffolds. In this study, the effect of micropore size and porosity on capillary forces was investigated. Rectangular shaped HA samples that were either 50% or 60% porous with
5μm, 12μm, 20μm or 50μm pores were fabricated. The samples were characterized by imaging the microstructure, analyzing the composition, measuring the pore size, pore fraction, sample thickness and quantifying the defects inside and outside of the samples.
Capillary rise tests were conducted on the samples and fluid height curves were obtained as a function of time. A model was implemented to determine the equilibrium heights and calculate the capillary forces. For 50% porous samples, 5μm samples had the highest equilibrium height and capillary pressure, followed by 12μm, 20μm or 50μm samples. The
60% porous 5μm samples had a faster initial rise, but a lower equilibrium height and capillary pressure than the 50% porous 5μm samples. The 60% porous 50μm samples also had a faster initial rise, a lower equilibrium height and capillary pressure than the 50% porous 50μm samples.
ii ACKNOWLEDGEMENTS
I would first like to thank my adviser, Prof. Amy Wagoner Johnson, for the excitement and encouragement throughout my graduate study. I learned a lot during the process because of your guidance and motivation. You inspired me in ways I never thought possible.
I would also like to thank my groupmates, especially Zach for analyzing most of the images and showing me statistical analysis, Jeff for making HA inks, Laurie for helping me with my way around the lab, and also Jie, Ashley, Naomi, Sam and Sheryl for your insightful advices.
Special thanks to Anthony for constantly clearing my confusions. In addition, I would like to thank the Kriven research group for lending their furnaces and the staff members at
Beckman Institute of Advanced Science and Technology and Materials Research Laboratory for assisting with XRD, SEM and micro-CT. Lastly, I would like to thank my parents, my brother and my friends for your love and support. I would never have done it without you all. This work was supported by National Science Foundation DMR award #11-06165.
iii TABLE OF CONTENTS
LIST OF FIGURES ...... vi LIST OF TABLES ...... ix LIST OF ABBREVIATIONS ...... xii
CHAPTER 1: INTRODUCTION AND BACKGROUND ...... 1 1.1 Motivation ...... 1 1.2 Materials for bone graft substitutes ...... 1 1.3 Porosity in bone graft substitutes ...... 2 1.4 Capillary action ...... 3 1.5 Capillary action in porous media ...... 4 1.6 Capillary action in HA scaffolds ...... 4 1.7 Mathematical models for capillary rise in porous media ...... 5 1.8 Thesis outline ...... 11
CHAPTER 2: MATERIALS AND METHODS FOR EXPERIMENTS ...... 12 2.1 Sample preparation ...... 12 2.2 Characterization of PMMA microspheres ...... 13 2.3 Sintering profile ...... 14 2.4 Characterization of microstructure and compositions ...... 15 2.5 Analysis of pore size and pore fraction ...... 16 2.6 Analysis of sample thickness and defects ...... 17 2.7 Capillary rise tests ...... 18
CHAPTER 3: EXPERIMENTAL RESULTS ...... 20 3.1 Sample size ...... 20 3.2 Characterization of PMMA microspheres ...... 20 3.3 Sintering profile ...... 21 3.4 Characterization of microstructure and compositions ...... 22 3.5 Pore size and pore fraction analysis ...... 26 3.6 Sample thickness and defect analysis ...... 28 3.7 Capillary rise tests ...... 32
iv CHAPTER 4: IMPLEMENTATION OF MODEL ...... 34
CHAPTER 5: DISCUSSION ...... 42
CHAPTER 6: CONCLUSIONS ...... 51
CHAPTER 7: FUTURE WORK ...... 53
REFERENCES ...... 54
APPENDIX A: PMMA MICROSPHERES SIZE ...... 62
APPENDIX B: HEAT TREATMENTS ...... 63
APPENDIX C: XRD ANALYSIS OBTAINED FROM RMS FOUNDATION ...... 74
APPENDIX D: THE RADIUS AND EQUILIBRIUM HEIGHT AT DIFFERENT CUT-OFF TIMES ...... 77
APPENDIX E: MATHEMATICAL MODELS ...... 80 E.1 The Washburn model ...... 80 E.2 The Hamraoui and Nylander model ...... 81 E.3 The Hamraoui and Nylander model with a constant ...... 82 E.4 Log fit ...... 83 E.5 The Fries and Dreyer model ...... 84
APPENDIX F: HA POWDER ...... 85
v LIST OF FIGURES
Fig. 1.1. a) Effect of void fraction on equilibrium height. b) Effect of void fraction on initial
rise...... 8
Fig. 1.2. Effect of perimeter on capillary rise...... 8
Fig. 1.3. Comparison of mathematical models for capillary rise in porous media...... 9
Fig. 2.1. Sintering profile. See Table 2.1 for details...... 14
Fig. 2.2. Process flow of imaging. Two regions of each sample were scanned (a), producing
two stacks of cross sectional images (b). 130 images from each stack were
analyzed (c)...... 18
Fig. 3.1. SEM images of M-100 (a), MBX-12 (b), MBX-20 (c) and MBX-50 (d) microspheres.
...... 21
Fig. 3.2. TGA analysis of sample...... 22
Fig. 3.3. Representative SEM images of NMP samples under heat treatment 1275°C 3hrs 1250°C 2hrs before thresholding (a), after thresholding (b), sample under heat treatment 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs before thresholding (c), after thresholding (d) and sample under heat treatment 1250°C 6hrs 1100°C 6hrs before thresholding (e) and after thresholding (f)...... 24
Fig. 3.4. Cross sectional SEM image of 50/5 (a), 50/12 (b), 50/20 (c) and 50/50 (d) samples
under heat treatment 1275°C 3hrs 1250°C 2hrs...... 25
Fig. 3.5. XRD diffraction pattern (black) of sample with heat treatment 1275°C 3hrs 1250°C
2hrs. HA peaks are indicated in red and β-TCP peaks are indicated in blue. The top
image shows 2θ from 20° to 45° and the bottom image shows 2θ from 45° to 70°.
...... 25
Fig. 3.6. Representative SEM images of 50/5 samples before thresholding (a) and after
vi thresholding (b), 50/50 samples before thresholding (c), after thresholding the
original SEM image (d) and thresholding after digitally removing the particles (d).
Particles are indicated by red arrows...... 27
Fig. 3.7. Representative micro-CT cross sectional image of five samples. Thickness
measurements are shown with purple lines. Inside defects are indicated in blue
circles. Outside defects are indicated in orange circles...... 28
Fig. 3.8. Average thickness and standard error for five samples of each pore size for the
50/5, 50/12, 50/20 and 50/50 samples. Significance differences shown with *
(p<0.05)...... 29
Fig. 3.9. Average number of defect for five samples of each pore size. Defects are sorted
between inside and outside defects. Error bars indicate standard errors.
Significance differences shown with * (p<0.05)...... 31
Fig. 3.10. Average defect area for five samples of each pore size. Defects are sorted between
inside and outside defects. Error bars indicate standard errors. Significance
differences shown with * (p<0.05)...... 31
Fig. 3.11. Average defect area fraction in each of the 260 images per sample shown for five
samples of each pore size. Defects are sorted between inside and outside. Error
bars indicate standard errors. Significance differences shown with * (p<0.05). .... 32
Fig. 3.12. Capillary rise heights as a function of time from experiments...... 33
Fig. 4.1. Capillary rise heights and curve fits. Experimental data are indicated as dots and
curve fits are indicated as lines...... 40
Fig. 5.1. Representative images of the interconnected pores in 5μm and 12μm samples (a)
and the interconnected pores in 20μm and 50μm samples (b)...... 44
vii Fig. D.1. The effective radius for 50/5, 60/5, 50/12 and 50/20 samples at different cut-off
times...... 77
Fig. D.2. The effective radius for 50/50 and 60/50 samples at different cut-off times...... 78
Fig. D.3. The equilibrium height for 50/5, 60/5, 50/12 and 50/20 samples at different cut-
off times...... 78
Fig. D.4. The equilibrium height for 50/50 and 60/50 samples at different cut-off times. ... 79
Fig. E.1. Capillary rise heights and curve fits from the Washburn equation. Experimental
data are indicated as dots and curve fits are indicated as lines...... 80
Fig. E.2. Capillary rise heights and curve fits from the Hamraouni and Nylander equation.
Experimental data are indicated as dots and curve fits are indicated as lines...... 81
Fig. E.3. Capillary rise heights and curve fits from the Hamraouni and Nylander equation
with a constant. Experimental data are indicated as dots and curve fits are
indicated as lines...... 82
Fig. E.4. Capillary rise heights and curve fits from log fit. Experimental data are indicated as
dots and curve fits are indicated as lines...... 83
Fig. E.5. Capillary rise heights and curve fits from the Fries and Dreyer equation.
Experimental data are indicated as dots and curve fits are indicated as lines...... 84
viii LIST OF TABLES
Table 2.1. Sintering Profile...... 15
Table 3.1. Average pore area and pore fraction of NMP samples under different heat
treatments. Standard deviation is shown...... 23
Table 3.2. Composition of samples for different heat treatments. The samples analyzed
in the first, second and RMS Foundation analysis are from three separate
batch of samples for the same heat treatment...... 23
Table 3.3. Average pore area fraction, radius, circularity and aspect ratio of pores in
50/5, 50/12, 50/20 and 50/50 samples. The average pore area fraction in
50/20 and 50/50 samples are obtained from digitally removing the particles.
Standard deviation is shown...... 26
Table 3.4. Average pore area fraction obtained from the original SEM image and area of
each particle in 50/20 and 50/50 samples. Standard deviation is shown...... 27
Table 4.1. The equilibrium height ratio, C/A ratio and A/C ratio of different cut-off times
for 50/5 samples to 60/5 samples. 50 indicate 50% porous samples and 60
indicate 60% porous samples. The final selection of the cut-off time is
highlighted in bold...... 37
Table 4.2. The equilibrium height ratio, C/A ratio and A/C ratio of different cut-off times
for 50% 50μm samples to 60% 50μm samples. 50 indicate 50% porous
samples and 60 indicate 60% porous samples. The final selection of the cut-
off time is highlighted in bold...... 38
Table 4.3. The equilibrium height of different cut-off times for 50/12 samples to 50/20
ix samples. The final selection of the cut-off time is highlighted in bold...... 39
Table 4.4. The cut-off time, effective radius, contact angle, equilibrium height and
capillary pressure for all samples...... 41
Table 5.1. ℎ , new equilibrium height, new capillary pressure, original equilibrium
height and original capillary pressure of samples...... 47
Table A.1. Diameter of PMMA microspheres...... 62
Table B.1. Microstructure and composition of samples under different heat treatments.
The final profile is shaded. Magnification differs between samples...... 63
Table B.2. Larger images of the microstructures. The final profile is shaded.
Magnification differs between samples...... 68
Table C.1. Phase composition. Standard deviations indicated in parentheses...... 74
Table C.2. Chemical compositions determined from XRD data...... 75
Table C.3. Chemical compositions. Hydroxyapatite determined from refinement of Ca
site occupancies for. Mg content in β-TCP determined from unit cell
distortion...... 75
Table C.4. Phase composition of samples under different heat treatments. Standard
deviations indicated in parentheses...... 76
Table C.5. Mg content in β-TCP of samples under different heat treatments. Results
determined from β-TCP peak shift...... 76
Table E.1. Cut-off time and equation fitted from the Washburn equation...... 80
Table E.2. Cut-off time and equation fitted from the Hamraouni and Nylander equation.
...... 81
Table E.3. Cut-off time and equation fitted from the Hamraouni and Nylander equation
x with a constant...... 82
Table E.4. Cut-off time and equation fitted from log fit...... 83
Table E.5. Cut-off time and equation fitted from the Fries and Dreyer equation...... 84
Table F.1. ICP results of HA powder...... 85
xi LIST OF ABBREVIATIONS
PLGA ...... Polylactide-co-glycolide
HA ...... Hydroxyapatite
SEM ...... Scanning Electron Microscopy
XRD ...... X-Ray Diffraction
Micro-CT ...... Micro-Computed Tomography
PMMA ...... Polymethyl-methacrylate
PEI ...... Poly-ethyleneimine
TGA ...... Thermogravimetric Analysis
β-TCP ...... β-Tricalcium Phosphate
MP ...... Microporous
NMP ...... Non-microporous
BSE ...... Back-Scattered Electrons
PBS ...... Phosphate Buffered Saline
MIP ...... Mercury Intrusion Porosimetry
xii CHAPTER 1: INTRODUCTION AND BACKGROUND
1.1 Motivation
The repair for bone defects is an increasingly important problem. Currently, the standards for defect repair are autografts and allografts [1]. An autograft, a graft from the patient’s own tissue, is considered to be the gold standard for bone defect repair due to its osteogenic, osteoinductive, osteoconductive and osteointegrative properties [2]. However, common drawbacks include surgery at the donor site, donor site morbidity and chronic pain
[3]. An allograft, a graft from another donor, reduces the surgery time and avoids morbidity, but complications including disease transmission and fracture occur [3]. Both standards have their limitations, thus the need of alternative solutions is apparent.
1.2 Materials for bone graft substitutes
Many materials have been researched as potential bone graft substitutes, and the two major materials are polymers and ceramics. Natural polymers are biocompatible, biodegradable, and have low toxicity and low manufacture cost. However, they are also difficult to process, have weak mechanical properties and degrade rapidly [4], [5]. The chemical, mechanical and degradation properties of synthetic polymers are much easier to modify than natural polymers [4], [5]. Synthetic polymers can also be tailored into different shape and sizes with more flexibility [4], [5]. However, they have limited biological functions, can cause inflammatory responses and the degradation products can create an acidic environment [5], [6].
Ceramics in the form of calcium phosphate are favorable materials for bone graft
1 substitutes. They are biocompatible, osteoconductive and can form a strong interface with bone [7]. Although calcium phosphate ceramics are brittle, they can be improved by combining with other polymeric materials such as polylactide-co-glycolide (PLGA), chitosan and polyglactin fibers [8]–[10].
Of the calcium phosphates, hydroxyapatite (HA) has received at lot of attention as a sub-class of bone graft substitutes. It has a composition similar to the mineral phase in bone.
Osteoblasts can produce osteoid directly on the surface of HA and form an intimate bond of new bone [11]. It is biocompatible, osteoconductive and bioactive [11]–[13]. Despite its brittle behavior, HA inks can be fabricated into complex 3D HA scaffolds that maintain mechanical stability [10], [14].
1.3 Porosity in bone graft substitutes
Porosity and interconnected pores are crucial features for bone graft substitutes because they allow for a greater degree and faster rate of bone ingrowth [12], [15]–[20].
Bone ingrowth can not occur without porosity [21]. Pores in bone graft substitutes can be categorized as macropores, pores larger than 100μm, and micropores, pores smaller than
50μm. It is generally agreed that pores must be at least 100μm for bone ingrowth [17]. The macropores enable cells to penetrate and proliferate, they also allow nutrient and waste transport and vascularization [12], [16], [18]. Many studies have indicated that bone graft substitutes with both microporosity and macroporosity can further enhance bone ingrowth
[12], [15], [19], [22]–[25]. The combined porosity can improve bone formation, promote bioactivity, and increase the rate of bone regeneration [12], [15], [22]–[25].
There are various methods to produce micropores in HA bone graft substitutes. Some
2 common ones include incomplete sintering [26], slip foaming [22] and the incorporation of sacrificial porogens [17]. Incomplete sintering is controlled by varying the sintering time and temperature; slip foaming generates pores by using a foaming agent; the incorporation of sacrificial porogens burns out the sacrificial porogens, which are mixed into HA slurries, during sintering. The incorporation of sacrificial porogens provides exact control of the pore size and pore fraction, since it relates directly to the type and amount of sacrificial porogens added [17].
Mechanisms such as an increase in surface area, an increase in ionic solubility, surface energy and roughness have been proposed as the reasons for the enhancement of bone growth with microporosity [15], [25]. However, none of these studies have shown definitive proof of the proposed mechanisms. In a recent work by Polak et al. [27], [52], capillary force was reported as a mechanism for self-seeding in scaffolds with both macroporess and micropores. Results showed that the microstructure can generate capillary forces that draw in cells and that the pore size, pore fraction and interconnection size can affect the penetration distances [27], [52].
1.4 Capillary action
Capillary action is the result of intermolecular attraction within a liquid and a solid. The adhesive forces between a liquid and a solid combined with the surface tension of the fluid, a phenomenon produced by cohesive forces within the liquid, can drive a liquid upward in narrow spaces like capillary tubes [53]. The rise height of the liquid, or the capillary rise, depends on the surface tension of the liquid, the contact angle between the liquid and solid, the perimeter available for the liquid to wet, the density of the liquid, and gravity [28]. One
3 example of capillary action is the transport of fluid in plants, where water is drawn upward from the root to replace the water evaporated water from the leaves [53]. Another example is dry paper towels drawing in liquids into the narrow spacing between fibers [53].
1.5 Capillary action in porous media
Extensive research has been focused on the capillary rise in porous media. Fluid flow through porous media is often encountered in topics such as petroleum in reservoir rocks
[29], ground water hydrology [30] and soil science [31]. Works have been conducted on packed powder beds to develop a non-uniform capillary model [32], [33] or columns filled with beads to calculate the equilibrium height [33]. The fluid flows through the porous gaps between the powder beds or beads. Textile materials are another example of porous media that uses capillary forces. Studies have investigated the capillary flow in yarns to optimize the processes involving liquid and fiber interaction [34] and in filter papers to characterize porous media by studying the kinetics of liquid penetration [35]. Capillary action is also found in rocks, stones, mortars and concretes [36], [37][38], [39]. The penetration of water can gradually deteriorate the materials over a long period of time, thus understanding the mechanism of capillary rise can help in preserving the materials and selecting suitable materials for the construction of buildings and monuments.
1.6 Capillary action in HA scaffolds
As mentioned briefly in Chapter 1.3, works by Polak et al. [27], [52] established that capillary force is the mechanism for self-seeding in HA scaffolds that contain both macropores and micropores and that the penetration distances depend on the specific
4 characteristics of the microstructure. To investigate the capillary forces in different microstructures, HA scaffolds with either 5μm or 50μm diameter pores that had a pore fraction of 50%, 55% or 60% were used in capillary rise tests. The scaffolds were nominally
2.7mm x 48.9mm x 2.1mm and were HA blocks made up of orthogonal rods. The equilibrium heights were either measured or obtained by curved fitting in cases where the fluid heights exceeded the samples. Oh et al. [40] investigated the capillary action in HA scaffolds with macro-pores and micro-channels. The scaffolds were 1cm x 1cm x 4cm and had 50μm micro- channels with either 150μm, 350μm or 750μm macro-pores. They showed experimental and simulation results of the equilibrium heights. Bai et al. [41] showed that capillary forces facilitates the self-seeding of cells in a HA scaffold with gradient channel structures. The scaffold was 12mm x 5mm x 2mm and had increasing channel width from 4.54 ± 0.88μm to
8.14 ± 1.24μm to 11.8 ± 2.47μm.
1.7 Mathematical models for capillary rise in porous media
Mathematical models for predicting the time-dependent capillary rise in porous media were first developed by Lucas in 1918 [42] and Washburn in 1921 [43]. The Washburn equation shows that the penetration distance of fluid in a cylindrical tube is proportional to the square root of time,
ℎ = �, Eq. 1.1 where h is the penetration distance of fluid, γ is the surface tension, r is the tube radius, θ is the contact angle, η is the dynamic viscosity and t is the time. This equation can also be applied to porous media by idealizing the porous structure as a bundle of uniform cylindrical tubes. However, drawbacks of the model such as neglecting the effect of inertia and gravity
5 and assuming porous structures have a uniform pore size, have prompted research into modifications of the Washburn equation.
Marmur et al. [35] studied the capillary rise in a vertical cylindrical tube with gravity present. Dullien et al. [44] considered a porous structure as a tube with repeating step changes in diameter and developed an equation for the rate of capillary rise with nonuniform pores. Delker et al. [45] and Lago et al. [33] observed that the capillary rise in a column of glass beads followed the Washburn equation only for early times. Zhmud et al.
[46] developed different equations for different asymptotic regimes during the capillary rise.
Hamraoui et al. [47] reported a solution that converges to the equilibrium capillary rise for long times by considering the dynamic contact angle. Fries et al. [48] included the gravity term and used the Lambert W function1 to derive an analytical solution for capillary rise in porous media at longer times. Comparisons of the mathematical models are shown in
Fig.1.3.
Reed et al. [49] developed a macroscopic force balance for capillary flow in a circular tube that balanced the surface tension against viscous drag, inertia and gravity. The force balance was then simplified by assuming that the inertia is negligible, and solved to derive the expression for the time t required for the liquid to reach a height h above the reservoir:
� = ((ℎ + ℎ ) ln 1 − + ℎ), Eq. 1.2
Here, ℎ is the length of the tube that submerged below the liquid-air interface in the reservoir and ℎ is the equilibrium height. At infinite time, the surface tension forces are equal to gravity, and the force balance can be solved to find the equilibrium height,
1 The Lambert W function, W(z), is defined to be the function satisfying W z � ( ) = �. W(z) is real for −1/e ≤ z. [56]
6 ℎ = . Eq. 1.3
In Eq. 1.2 and 1.3, γ refers to the surface tension, r to the tube radius, θ to the contact angle, η to the dynamic viscosity, ρ to the liquid density and g to gravity.
Rajagopalan et al. [50] extended the model by Reed et al. [49] to capillaries of arbitrary cross section by taking into account C, the total perimeter available for liquid to wet, and A, the total cross-sectional area through which the liquid flows. The model thus becomes
� = ((ℎ + ℎ ) ln 1 − + ℎ). Eq. 1.4
The shape factor, �, comes from Reed’s [49] derivation to noncircular conduits. It is defined in a manner that its numerical value is equal to the product of Reynolds number and friction factor for a given conduit [51]. Generally, it can be determined by solving the equations of
Newtonian flow in the conduit geometry of interest [51]. It depends only on the cross- sectional shape of the conduit, which can be found in the work by Miller [51]. The equilibrium height and initial velocity is given by Eq. 1.5 and 1.6.
ℎ = Eq. 1.5
� � = 0 = Eq. 1.6
According to Rajagopalan’s work [50], the equilibrium height is proportional to C/A and the initial velocity is proportional to A/C. In their study, they looked at the capillary rise in fiber bundles of different void fraction and shape. The liquid was to wet the perimeter of the fibers and flow through the void between the fibers. The effect of A is demonstrated by holding C constant and increasing A by increasing the void space available for liquid flow. The result showed a higher equilibrium height, Fig. 1.1a, and a slower initial rise, Fig. 1.1b, for smaller void fraction. The effect of C was also studied by holding A constant and increasing C by
7 changing the shape of the fibers from circles to ellipses. The increased perimeter resulted in an increase in equilibrium height, as shown in Fig. 1.2. It can be seen from Fig. 1.1 and 1.2 that changing the void fraction resulted in a larger difference in equilibrium height than changing the perimeter and that the effect was also in a larger time scale, thus it can be concluded that the void fraction has a larger effect on capillary rise than the perimeter.
a) b)
Fig. 1.1. a) Effect of void fraction on equilibrium height. b) Effect of void fraction on initial rise. [50]
Fig. 1.2. Effect of perimeter on capillary rise. [50]
8 pyrex Validation/ of/ model shaped&sandstones. 4 igecplaymdlfte&with& Single&capillary&model&fitted& An&assembly&of&capillaries&model& Model&fitted&with&experimental&data& Model&fitted&with&experimental&data& Model&fitted&with&experimental&data& 4 data&on&capillary&rise&of&a& experimental& liquid& of&Brookfield& viscosity&standard& in&Whatman&filter& paper. 4 fitted&with& experimental&data&on& capillary&rise&of¶ffin&oil&in&sand. 4 on&capillary&rise&of&NaCl&solution& in& rod 4 on&capillary&rise&of&water&in&glass&tube& filled&with& glass&beads. 4 on&capillary&rise&of&water&in&glass& tube&filled&with& glass&beads&and&of& water&in&Berea&sandstones. Porous/media Assumed&as&a&single& Assumed&as&an&assembly& Assumed&as&repeating& Assumed&as&a&single& Assumed&as&a®ular& 4 cylindrical& capillary. 4 of&cylindrical&capillaries& of& various&radii&in¶llel. 4 capillary&element&with& step&changes&in&diameter,& similar&to&a&consolidated& porous&media. 4 capillary&or&an&assembly& of&vertical&capillaries. 4 packing&of&spheres. 4 Assumptions statically&in&a&piston 4 No&viscosity&effect. The&effect&of&gravity&was& The&effect&of&gravity&and& No&inertia& and&viscous& The&effect&of&gravity& The&meniscus&advances& 4 4 included. 4 viscous&drag&included. 4 effect. 4 included. 4 quasi like&motion& and&that&the& contact&line&is&maintained&in& plane. a&horizontal& water& 4 Washburn/equation oiiain/from/the/ Modifications/ Included&the&effect&of&gravity& Two&models&were&discussed.& Capillary&rise&in&porous&media& Capillary&rise&in&porous&media& At&late×&the&rise&is& Capillary&rise&in&porous&media& Model&was&based&on&capillary& 4 by&demonstrating& vertical& capillary&penetration. 4 into&a& One&was&the&penetration& single&vertical&capillary,&the& other& was&an&assembly&of& vertical&capillaries. 4 with&nonuniform& pores&was& investigated. 4 does¬&follow&the&Washburn& equation& at&late×. 4 governed&by&the&randomness&of& the&forces&on&the&air interface. 4 followed&the&Washburn& equation& for&early×,&but& not&for&later×.& 4 rise&in&a®ular&packing&of&spheres&and&&an&equation&for& the&equilibrium& height&was& developed. et# et# 1977& 1996& Authors Marmur& and&Cohen& 1997&[35] Dullien& al.# [44] Delker& al. [45] Lago&and& Araujo& 2001&[33] Fig. 1.3. Comparison of mathematical models for capillary rise in porous media.
9 Validation/of/model Model&fitted&with&experimental&data&on& Showed&model&compared&to& Calculated&and&plotted&for&a&setup&using& - capillary&rise&of&dodecane&in&borosilicate& glass&capillary. - experimental&data&of&water&andðanol&rise&in&a&glass&capillary. - silicon&fluid&(0.93cSt)&in&a&0.1mm&radius&borosilicate&glass&capillary Porous/media Assumed& as&a&single& Assumed& as&a&single& N/A - capillary&or&an&assembly&of& vertical&capillaries. - capillary&or&an&assembly&of& vertical&capillaries. Assumptions No&inertia&effect. The& effect&of&gravity&and& No&inertia&effects. Dominant&forces&included& No&inertia&effects. Static&constant&angle&and& - - viscous&drag&included. - - surface&tension&and&gravity. - - tube&(or&pore)&radius. to&include&possible& ! Washburn/equation Modifications/from/the/ Developed&short&term&and&long& Introduced&a&retardation& The& influence&of&the&dynamic& The& frictional&effects&at&the& Included&the&effect&of&gravity&for& Used&the&Lambert&W&function&to& - term&asymptotic&solutions&for& different&time®imes. - coefficient& sources&of&energy&dissipation. - contact&angle&is&a&major&source&of& the&retardation. - moving&fluid&front&are& responsible&for&the&dynamic& contact&angle. - liquid&rise&at&longer×.& - derive&an&analytic&solution&in& terms&of&h(t). et# 2000& Authors Zhmud& al.# [46] Hamraoui& and& Nylander& 2002&[47] Fries&and& Dreyer& 2008&[48] Fig. 1.3. (cont.) Comparison of mathematical models for capillary rise in porous media.
10 1.8 Thesis outline
It has been established that the mechanism for improvement in bone ingrowth in micropores is capillary forces [27], [52], as mentioned in Chapter 1.6. In this study, we created samples with only micropores, so that we could investigate the capillary pressure and fluid rise without the influence of macropores. A sintering profile was also determined.
Samples of different pore size and pore fraction were characterized by imaging the microstructure with scanning electron microscopy (SEM), verifying the composition using X-
Ray Diffraction (XRD), measuring the pore size and pore fraction with ImageJ and measuring the thickness and quantifying the defects within the samples using micro-computed tomography (micro-CT). Capillary rise experiments were conducted on the samples. The fluid height to time curve was fitted with several models and the model developed by
Rajagopalan et al. was used to determine the equilibrium height, and thus the capillary pressure for each microstructure. By understanding the effect of pore size and porosity on capillary pressure, we can better tailor microstructures for different uses such as selecting specific cells to draw in or controlling drug release and potentially improve the repair for bone defects.
11 CHAPTER 2: MATERIALS AND METHODS FOR EXPERIMENTS
2.1 Sample preparation
All samples in this study were made from HA ink prepared from raw HA powder
(Riedel-de Haën) as described in detail by Hoelzle [54]. Details of the raw HA powder are in
Appendix F. Raw HA powder was first calcined at 1100°C for 10 hours. The calcined powder was then ball milled with grinding media in ethanol for 14 hours and dried at 125°C for 15 hours. Deionized water and Darvan 821A was mixed with the HA powder to form the ink.
5M NH4OH was also added to adjust the pH to 10. The solution was then centrifuged at 2000 rpm for 1 hour and the liquid from the solution was separated and removed. About 2 grams of the solution were dried at 35°C for at least 12 hours in order to calculate the solids loading of the slurry and the corresponding amount of additives needed. Deionized water,
5wt% methocel and 1-octanol were added according to the corresponding amount of additives needed [54]. Polymethyl-methacrylate (PMMA) microspheres were also added as
sacrificial porogens for the porosity in the samples. 1M HNO3 was then added to adjust the viscosity of the ink. Lastly, poly-ethyleneimine (PEI) was added when the desired viscosity was reached.
Samples were made by forming ink into wax molds. A glass slide (12.3 x 75.9 x 0.9mm) was placed on a piece of foil and melted wax was poured over the whole glass slide. After wax was cooled, the foil was ripped off and the glass slide was taken out, thus creating a cavity in the wax. To avoid having bubbles in the samples, ink was centrifuged before forming it in the cavity. 1ml syringes were used and the tip of the syringes were cut off to create an opening at the top end of the barrel. Syringes were rinsed with paraffin oil to
12 facilitate the loading of ink. Ink was loaded into the syringes and tapped vigorously to remove large bubbles that were visible. Then, the ink was centrifuged for 35 minutes at
3000 rpm to further remove bubbles. However, due to the long centrifuge time, it resulted in a gradient in the ink. Thus, the top and bottom portion of the ink was removed and only the middle portion was used. Each sample was made by applying two syringes of ink. Ink was squeezed out from the syringe and into the cavity, then filter paper was used to cover over the ink so that it could be pressed flat uniformly distributed within the cavity. In addition, a glass slide was pressed over the filter paper to further smooth out the ink. The ink was left to dry for a few minutes to make it easier to peel off the filter paper. Excess ink that overflowed out of the cavity was removed and paraffin oil was used to cover the samples to prevent them from cracking when drying. The samples were left to dry overnight, then any residues of filter paper were peeled off with tweezers before sintering.
2.2 Characterization of PMMA microspheres
Four different sizes of PMMA microspheres were used to make samples of four different pore sizes. Particle size analysis was done by independent companies to get an accurate size distribution. M-100 (Matsumoto Microsphere) and MBX-20 microspheres
(Sekisui Plastics Co.) were done by Aveka, MBX-12 microspheres (Sekisui Plastics Co.) were done by AgEng and MBX-50 microspheres (Sekisui Plastics Co.) were done by Aveka.
Samples made from M-100, MBX-12, MBX-20 and MBX-50 microspheres are hereon referred to as 5μm, 12μm, 20μm and 50μm samples, respectively.
13 2.3 Sintering profile
The sintering profile consists of increments of low temperature steps followed by high temperature heat treatments. The low temperature steps were used to ensure the burnout of organic additives and PMMA microspheres. A small sample was analyzed with thermogravimetric analysis (Q50 TGA) to determine the burnout temperatures. Burnout temperatures were selected based on the TGA analysis; temperatures were chosen at which the change in weight loss was greatest. High temperature heat treatments were determined from a series of trail-and-error based on literature in calcium phosphates. The goal was to achieve a fully dense microstructure with a composition of HA and β-tricalcium phosphate
(β-TCP). The final sintering profile that was used in capillary rise tests is shown in Fig. 2.1 and Table 2.1.
Sintering(Profile 1400
1200
1000 ) C 800
600 Temperature3(
400
200
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Time3(hrs)
Fig. 2.1. Sintering profile. See Table 2.1 for details.
14 Ramp (°C/hr) Temperature (°C) Hold (hr) 150 100 0.75 150 150 0.75 150 300 0.75 150 400 0.75 300 1275 3 300 1250 2 600 25 0 Table 2.1. Sintering Profile.
2.4 Characterization of microstructure and compositions
To qualitatively assess the microstructure, samples were coated with Au-Pd for 40 –
80s and imaged on the surface and in cross-section with scanning electron microscopy
(SEM) (JEOL JSM-6060LV) at a voltage of 5kV – 25kV, a magnification of 1000x – 2000x, a working distance of 9 – 22mm and a spot size of 40 – 45. The imaging conditions varied slightly between samples. To verify the composition of the samples, sintered samples were crushed into powder with a mortar and pestle, then analyzed with X-ray diffraction (XRD)
(Siemens/Bruker D-5000). The scans were done with CuKα radiation at 2θ from 20° to 70°, a scan rate of 0.5°/min and a step size of 0.05°. The results were analyzed with Jade 8.2 to quantify the percentage of the phases present. Three heat treatments that showed a dense microstructure and a composition of HA and β-TCP were chosen for further analysis of pore area and pore fraction. Microstructures from the three heat treatments were compared by imaging the surface of microporous (MP) samples and non-microporous (NMP) samples with SEM. Images of 10 random non-overlapping sections of each NMP sample were thresholded with ImageJ, then the thresholded areas were calculated to find the pore area and pore fraction. In addition, a new batch of samples that went through the same three
15 heat treatments were analyzed with XRD again to verify the composition. Another batch of samples that also went through the same three heat treatments were further analyzed at the RMS Foundation, Bettlach, Switzerland. They prepared the sample with manual grinding in aceton and heat treatment at 1000°C 15hrs 840°C 15hrs before analysis. The composition was determined from XRD and Rietveld refinement.
2.5 Analysis of pore size and pore fraction
For pore size analysis, 50% porous 5μm, 12μm, 20μm and 50μm samples were embedded in epoxy and held under vacuum for four hours at 30 inHg in a vacuum oven
(Napco 5851) to ensure that the epoxy fully penetrated the pores in the samples. The samples were then polished to a flat surface, and imaged using SEM (JEOL JSM-6060LV) with back-scattered electrons (BSE) at a voltage of 30kV, a working distance of 8 – 9mm and a spot size of 56. The magnifications used for pore size analysis were x1000 for 5μm samples, x750 for 12μm samples, x600 for 20μm samples and x300 for 50μm samples. 10 images of each sample were thresholded with ImageJ. Pores smaller than 0.5μm2, which is equivalent to a circle with diameter 0.8μm, were considered as sintering pores, and were not included in the pore size analysis. For 5μm and 12μm samples, the pore area fraction, circularity and aspect ratio were calculated from the thresholded areas. The circularity is defined as
4� with a value of 1 indicating a perfect circle and a value of 0 indicating an
infinitely elongated polygon [55]. The aspect ratio is defined as , which is the aspect ratio of the pore’s fitted ellipse [55]. The pore size was determined using the linear intercept method [52]. Five random lines were drawn on each image, and the average pore
16 size was obtained by dividing the line length by the number of points where the line intercepted the pores. For 20μm and 50μm samples, the area of each particle inside the pores and the pore area fraction obtained from the original SEM image were measured. The particles inside the pores were then digitally removed by filling the pores as whole pores.
The circularity, aspect ratio and the pore area fraction obtained from digitally removing the particles were measured. The pore size was also determined with the linear intercept method.
2.6 Analysis of sample thickness and defects
Five out of the twelve 50% porous samples of each pore size that were used for capillary rise tests were scanned with micro-CT (Zeiss MxCT-400) to determine the sample thickness. The scans were performed with an energy of 90kV, a pixel size of 11.5μm for the first two 50% 5μm samples and a pixel size of 23μm for all the other samples. A schematic showing the process is in Fig. 2.2. For each sample, a 3D scan was conducted on two regions,
Fig. 2.2(a), which produced two stacks of cross sectional images, Fig. 2.2(b). Each scanned region was 15.3mm in length and the center of the scanned region was roughly 20mm from each end of the sample. In all, about half of the sample was imaged. 130 images distributed in relatively uniform increments in each image stack were analyzed for each region, Fig.
2.2(c). Each image analyzed was roughly 1.5mm apart for the first two 50% 5μm samples and 3.06mm apart for all the other samples. A total of 260 images were analyzed for each sample. Four thickness measurements were taken for each cross sectional image, two towards the middle and two towards the sides. The same images were analyzed to characterize the defects. Defects were counted manually with ImageJ and sorted between
17 inside and outside defects. The area and area fraction of each individual defect were measured in each of the 260 images per sample.
Setof 5 samples.
Cross sectional image of 5 samples from (a). 20mm 3D#scanned#region:# 15.3mm
63mm 3D#scanned#region:# 15.3mm
20mm Cross#sectional# image#in#(c). Twostacks of cross sectional 130 images were analyzed images. from each stack. (a) (b) (c)
Fig. 2.2. Process flow of imaging. Two regions of each sample were scanned (a), producing two stacks of cross sectional images (b). 130 images from each stack were analyzed (c).
Statistical analysis was performed on OriginPro 2015 using one-way ANOVA with Tukey analysis to compare more than two groups. The thickness, total number of inside defects, total number of outside defects, average inside defect area, average outside defect area, average inside defect area fraction, and average outside defect area fraction were compared between 50% 5μm, 50% 12μm, 50% 20μm and 50% 50μm samples.
2.7 Capillary rise tests
50% porous 5μm samples (n=12), 50% porous 12μm samples (n=12), 50% porous 20μm samples (n=12), 50% porous 20μm samples (n=12), 60% porous 5μm samples (n=9) and 60%
18 porous 50μm samples (n=6) were used in the capillary rise experiments. These are hereon referred to as 50/5, 50/12, 50/20, 60/5 and 60/50 samples, respectively. Samples were heated at 600°C for 2 hours before the capillary rise experiment to burn off any potential contaminants and moisture and to ensure that samples all started at the same state. This was based on experience that sample wettability changes over time (data not shown). One end of the sample was sanded then glued on a glass slide and suspended above a container of 1x phosphate buffered saline (PBS) at a fixed height. Sanding was done to ensure that the long axis of the sample was perpendicular to the slide and that it was fixed well to the slide.
Images were taken every 2 seconds with a Nikon D5100 DSLR camera while a translational stage was used to raise the container of PBS until it just contacted the sample. A timer was also included in the image for reference.
Images were then analyzed with ImageJ. Measurements were taken at 0s, 2s, 6s, 10s, then followed by 10s increments until 120s, then 30s increments until the fluid line stopped changing. Contrast and brightness were adjusted for each image to see the fluid line more clearly. The height of the fluid in the samples was measured by averaging the maximum and minimum height in pixels for each time interval, then converted using the known sample width as the scale bar. The data from all of the samples within a group were plotted using
Excel and an average curve with standard deviation for each measured time was obtained for each group. A model was implemented to determine the equilibrium heights and will be described in more detail in Chapter 4.
19 CHAPTER 3: EXPERIMENTAL RESULTS
3.1 Sample size
The 50/5 samples were 63.03 ± 1.02mm x 9.79 ± 0.32mm, the 60/5 samples were 60.57
± 1.41mm x 9.38 ± 0.54mm, the 50/12 samples were 63.27 ± 1.13mm x 9.75 ± 0.29 mm, the
50/20 samples were 64.03 ± 0.65mm x 9.72 ± 0.32mm, the 50/50 samples were 64.24 ±
1.08mm x 9.86 ± 0.19mm, and the 60/50 samples were 61.03 ± 1.18mm x 9.69 ± 0.23mm after sintering. The 60/50 samples broke a lot during preparation and sintering. Most of the
60/50 samples had visible cracks or were slightly crooked after sintering.
3.2 Characterization of PMMA microspheres
The diameter of the microspheres was 5.96 ± 2.00μm for M-100 microspheres, 11.025
± 4.245μm for MBX-12 microspheres, 16.2 ± 6.77μm for MBX-20 microspheres and 47.277 ±
19.572μm for MBX-50 microspheres. Representative SEM images of the microspheres are shown in Fig. 3.1.
20 (a) (b)
(c) (d)
Fig. 3.1. SEM images of M-100 (a), MBX-12 (b), MBX-20 (c) and MBX-50 (d) microspheres.
3.3 Sintering profile
The TGA analysis is shown in Fig. 3.2. As can be seen from the figure, the derivative weight curve has maximum peaks at 148.57°C, 291.22°C and 350.50°C. Thus, the low temperature steps were chosen at 100°C for water burnout, then followed by 150°C, 300°C and 400°C for the burnout of organic additives and PMMA microspheres. The list of a series of high temperature heat treatments is shown in Appendix B.
21 Sample: HA File: C:...\20150202\HA_Analyzed.001 Size: 18.5910 mg TGA Operator: SpilaT33 Method: Ramp Run Date: 02Feb2015 13:35 Instrument: TGA Q50 V20.13 Build 39 110 0.4 291.22°C
148.57°C 100 0.3
11.87% 38.83%
90 0.2 350.50°C Weight (%) 80 0.1 Deriv. Weight (%/°C) 26.96%
70 0.0
60 0.1 0 200 400 600 800 1000 Temperature (°C) Universal V4.5A TA Instruments
Fig. 3.2. TGA analysis of sample.
3.4 Characterization of microstructure and compositions
The average pore area and pore fraction for the three heat treatments chosen for further analysis are shown in Table 3.1. The average pore area is equivalent to a circle with the diameter 0.800 ± 0.144μm, 0.870 ± 0.084μm and 0.882 ± 0.116μm for 1275°C 3hrs
1250°C 2hrs, 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs and 1250°C 6hrs 1100°C 6hrs heat treatments, respectively. The 1275°C 3hrs 1250°C 2hrs heat treatment has the smallest average area and fraction of pores, thus indicating a denser microstructure. The composition is shown in Table 3.2. The samples analyzed in the first, second and RMS
Foundation analysis are from three separate batch of samples for the same heat treatment.
Analyses conducted at U of I, the first and second analyses, show that all samples have a composition of HA and β-TCP, while the analysis conducted by RMS Foundation, Bettlach,
Switzerland show that the sample under the 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs heat treatment has a minor phase of α-TCP. Heat treatment 1275°C 3hrs 1250°C 2hrs was
22 selected as the final sintering profile for all the samples. Samples under such heat treatment show a dense microstructure, have the smallest pore fraction in NMP samples, and have a dominate phase of HA and a minor phase of β-TCP. Representative images of NMP samples under different heat treatments before and after thresholding are shown in Fig. 3.3. The cross sectional SEM image of the microstructure and XRD data are shown in Fig. 3.4 and Fig.
3.5, respectively.
Heat Treatment Average Pore Area (μm2) Pore Fraction (%) 1275°C 3hrs 1250°C 2hrs 0.778 ± 0.243 4.633 ± 1.012 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs 0.856 ± 0.171 5.331 ± 0.770 1250°C 6hrs 1100°C 6hrs 0.874 ± 0.201 6.836 ± 1.137 Table 3.1. Average pore area and pore fraction of NMP samples under different heat treatments. Standard deviation is shown.
Heat Treatment HA β-TCP α-TCP 1st analysis 96.20% 3.80% N/A 1275°C 3hrs 1250°C 2hrs 2nd analysis 94.80% 5.20% N/A RMS Foundation 94.60% 5.40% N/A 1st analysis 91.20% 8.80% N/A 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs 2nd analysis 94.30% 5.70% N/A RMS Foundation 89.51% 5.50% 4.99% 1st analysis 94.50% 5.50% N/A 1250°C 6hrs 1100°C 6hrs 2nd analysis 94.10% 5.90% N/A RMS Foundation 92.44% 7.56% N/A Table 3.2. Composition of samples for different heat treatments. The samples analyzed in the first, second and RMS Foundation analysis are from three separate batch of samples for the same heat treatment.
23 (a) (b)
(c) (d)
(e) (f)
Fig. 3.3. Representative SEM images of NMP samples under heat treatment 1275°C 3hrs 1250°C 2hrs before thresholding (a), after thresholding (b), sample under heat treatment 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs before thresholding (c), after thresholding (d) and sample under heat treatment 1250°C 6hrs 1100°C 6hrs before thresholding (e) and after thresholding (f).
24 (a) (b)
(c) (d)
Fig. 3.4. Cross sectional SEM image of 50/5 (a), 50/12 (b), 50/20 (c) and 50/50 (d) samples under heat treatment 1275°C 3hrs 1250°C 2hrs.
Fig. 3.5. XRD diffraction pattern (black) of sample with heat treatment 1275°C 3hrs 1250°C 2hrs. HA peaks are indicated in red and β-TCP peaks are indicated in blue. The top image shows 2θ from 20° to 45° and the bottom image shows 2θ from 45° to 70°.
25 3.5 Pore size and pore fraction analysis
Representative images of the embedded samples before and after thresholding are shown in Fig. 3.6. The average pore area fraction, pore radius, circularity and aspect ratio of
50/5, 50/12, 50/20 and 50/50 samples that had particles digitally removed are shown in
Table 3.3. The 50/5 samples have the smallest pore radius, followed by 50/12, 50/20 and
50/50 samples. The pore radius for 50/5 samples is similar to the M-100 PMMA microspheres, however, the pore radius for 50/12, 50/20 and 50/50 samples are smaller than the corresponding PMMA microspheres, which are mentioned in Chapter 3.1. The average area of each particle in 50/20 and 50/50 samples and the pore area fraction obtained from the SEM image are shown in Table 3.4. The pore fractions obtained from the
SEM image are smaller than the pore fractions obtained from digitally removing the particles for both 50/20 and 50/50 samples. The particles in 50/50 samples are larger than the particles in 50/20 samples.
Pore Fraction (%) Sample (obtained from digitally Pore Radius (μm) Circularity Aspect ratio removing the particles) 50/5 51.65 ± 1.87 3.01 ± 0.42 0.52 ± 0.22 1.99 ± 0.87 50/12 46.96 ± 1.72 4.60 ± 0.92 0.60 ± 0.23 1.91 ± 0.80 50/20 55.35 ± 2.71 6.34 ± 1.43 0.59 ± 0.23 1.99 ± 0.80 50/50 44.69 ± 2.40 15.34 ± 3.43 0.79 ± 0.22 1.70 ± 0.63 Table 3.3. Average pore area fraction, radius, circularity and aspect ratio of pores in 50/5, 50/12, 50/20 and 50/50 samples. The average pore area fraction in 50/20 and 50/50 samples are obtained from digitally removing the particles. Standard deviation is shown.
26 Pore Fraction (%) Sample Particle Area (μm2) (obtained from the original SEM image) 50/20 49.49 ± 1.99 14.73 ± 32.26 50/50 38.87 ± 1.65 60.42 ± 115.62 Table 3.4. Average pore area fraction obtained from the original SEM image and area of each particle in 50/20 and 50/50 samples. Standard deviation is shown.
(c)
(a)
30kV%%%%%%%%%%%%%%%%%x300%%%%%%%%%50μm%%%%%%%%%%%%%%%%%%%%09 56%%%%BEC
(d)
5kV$$$$$$$$$$$$x1,000$$$$$$$$$$$10μm$$$$$$$$$$$$$$$$$$$$09 56$$$$BEC
(b)
(e)
Fig. 3.6. Representative SEM images of 50/5 samples before thresholding (a) and after thresholding (b), 50/50 samples before thresholding (c), after thresholding the original SEM image (d) and thresholding after digitally removing the particles (d). Particles are indicated by red arrows.
27 3.6 Sample thickness and defect analysis
A representative micro-CT cross sectional image of the samples is shown in Fig. 3.7. The image shows examples of both inside and outside defects, and how the thickness of the sample was measured. The average thickness with standard error measured from 260 images for five samples of each pore size is shown in Fig. 3.8. The average thickness is 0.909
± 0.155mm for the 50/5 samples, 0.909 ± 0.122mm for the 50/12 samples, 0.915 ± 0.115mm for the 50/20 samples and 0.915 ± 0.115mm for the 50/50 samples. The thickness of the
50/12 samples is significantly different compared to the 50/20 (p=0.034) samples and the
50/50 samples (p=0.039).
Fig. 3.7. Representative micro-CT cross sectional image of five samples. Thickness measurements are shown with purple lines. Inside defects are indicated in blue circles. Outside defects are indicated in orange circles.
28 Average''Thickness
* * 1000
900
800
700
m) 600
μ 50%35μm 500 50%312μm
400 50%320μm Thickness3( 50%350μm 300
200
100
0
Sample
Fig. 3.8. Average thickness and standard error for five samples of each pore size for the 50/5, 50/12, 50/20 and 50/50 samples. Significance differences shown with * (p<0.05).
The average number of inside and outside defects is shown in Fig. 3.9. The 50/50 samples show a significantly larger number of inside defects than the 50/5 (p=0.0032),
50/12 (p=0.028) and 50/20 (p=0.009) samples. The difference is likely because the pore size generated by the PMMA beads is larger than the pixel size (23μm) for the 50/50, so some pores were counted as defects. The 50/5 samples also show a significantly larger number of outside defects than the 50/20 samples (p=0.038).
Fig. 3.10 shows the average area of the defects. The average inside defect area for 50/5,
50/12, 50/20 and 50/50 samples is equivalent to a circle with the diameter 124.19μm,
118.8μm, 110.65μm and 90.95μm, respectively. The inside defects have an average area on the order of 104μm2 for the 50/5, 50/12 and 50/20 samples, which is equivalent to a circle with diameter 112.84μm. The 50/50 samples show a smaller inside defect area compared to the 50/5 (p=0), 50/12 (p=0) and 50/20 (p=2.61 x 10-6) samples. This supports the reasoning
29 that some pores were mistaken as defects; here the pores that were counted as defects are the small defects in these samples. Therefore, these lower the overall average size of the defects relative to other samples. The 50/5 samples also show significant difference when compared to the 50/20 samples (p=0.0036) in terms of inside defect size. The outside defects are larger than the inside defects. The average outside defect area is equivalent to a circle with the diameter 252.31μm, 176.18μm, 239.67μm and 182.17μm for 50/5, 50/12,
50/20 and 50/50 samples, respectively. The 50/5 samples are significantly different compared to the 50/12 (p=3.31 x 10-8) and 50/50 (p=9.24 x 10-9) samples. The 50/12 samples are significantly different compared to the 50/20 samples (p=3.22 x 10-4). The 50/20 samples are significantly different compared to the 50/50 samples (p=4.08 x 10-4).
The average defect area fraction is shown in Fig. 3.11. Similarly, the outside defects have a larger area fraction than the inside defects. The inside defect area fractions show no significant difference between samples. For outside defect area fraction, the 50/5 samples are significantly different compared to the 50/12 (p=4.79 x 10-9) and 50/50 (p=9.82 x 10-10) samples, the 50/12 samples are significantly different compared to the 50/20 samples
(p=4.7 x 10-5) and the 50/20 samples are significantly different compared to the 50/50 samples (p=7.59 x 10-5).
30 Average'Number'of'Defect * * 120 * 110
100 * 90 80 70 50%15μm 60 50%112μm 50 50%120μm
Number1of1Defect 40 50%150μm 30 20 10 0 Inside1Defect Outside1Defect Sample
Fig. 3.9. Average number of defects for five samples of each pore size. Defects are sorted between inside and outside defects. Error bars indicate standard errors. Significance differences shown with * (p<0.05).
Average'Defect'Area * 60 * * 55 *
50
45 * 40
) * 2 35 50%.5μm μm
3 * 30 50%.12μm * 25 50%.20μm Area.(x10 20 50%.50μm 15
10
5
0 Inside.Defect Outside.Defect Sample
Fig. 3.10. Average defect area for five samples of each pore size. Defects are sorted between inside and outside defects. Error bars indicate standard errors. Significance differences shown with * (p<0.05).
31 Average'Defect'Area'Fraction * 0.70% * * 0.65% * 0.60% 0.55% 0.50% 0.45% 0.40% 50%15μm 0.35% 50%112μm 0.30% 50%120μm Area1Fraction 0.25% 50%150μm 0.20% 0.15% 0.10% 0.05% 0.00% Inside1Defect Outside1Defect Sample
Fig. 3.11. Average defect area fraction in each of the 260 images per sample shown for five samples of each pore size. Defects are sorted between inside and outside. Error bars indicate standard errors. Significance differences shown with * (p<0.05).
3.7 Capillary rise tests
The capillary rise heights obtained from experiments are shown in Fig 3.12 as a function of time. All the capillary rise heights had a steep slope initially, then the slope gradually declined and leveled off in the end. The height curves all level off at around 55 –
60mm for all samples, which corresponds approximately to the height of the samples. The
50/5, 50/12, 50/20 and 50/50 samples were 63.03 ± 1.02mm, 63.27 ± 1.13mm, 64.03 ±
0.65mm and 64.24 ± 1.08mm in height after sintering. It can be observed that the height curves do not exceed 60mm. The 60/5 and 60/50 samples were slightly shorter, 60.57 ±
1.41mm and 61.03 ± 1.18mm, so the corresponding height curves bend over sooner and to slightly lower heights than the 50% porous samples. However, the time required for the fluid to reach the top end varies between samples. For the 5μm, 12μm and 20μm samples, the
32 times required are significantly less than the 50μm samples.
For 50% porous samples, the 5μm samples have the fastest capillary rise, followed closely by the 12μm samples, then a slightly slower rise for the 20μm samples. The capillary rise time for 50μm samples is much less than all the other samples. The 60/5 samples have a faster capillary rise than the 50/5 samples, the fastest among all the samples, and the 60/50 samples also have a faster capillary rise than the 50/50 samples.
Capillary(Rise(Heights 65 60 55 50 45 40 50%15μm1(n=12) 35 60%15μm1(n=9) 30 50%112μm1(n=12)
Height1(mm) 25 50%120μm1(n=12) 20 50%150μm1(n=12) 15 60%150μm1(n=6) 10 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. 3.12. Capillary rise heights as a function of time from experiments.
33 CHAPTER 4: IMPLEMENTATION OF MODEL
As described in Chapter 3.4, the height curves obtained from experiments were observed to plateau near the top end of the samples at between 55 – 60mm for all samples.
There were almost certainly some end effects at the top of the samples, which affected the capillary rise. The top end of the samples was sanded before being glued onto the glass slides, which could result in debris from sanding filling up the pores. The sanding was done in order to have a better surface to attach to the glass slide. It is also possible that the glue penetrated into the top end of the samples. The fluid line in the images analyzed was harder to see as it reached the top end of the samples. Because of these end effects, only part of the data could be used for the model, and a cut-off time for the data had to be determined.
It is likely that the fluid stopped rising because it reached the end of the sample and could possibly continue rising if samples had been long enough. The fluid was unable to rise to the equilibrium height for each pore size/pore fraction because the equilibrium height was higher than the sample height. Thus, a model was needed in order to obtain the equilibrium height.
In order to determine the equilibrium rise height, the model developed by Rajagopalan et al. described in Chapter 1.7 was implemented. Taking the equation for the model
� = ((ℎ + ℎ ) ln 1 − + ℎ), Eq. 4.1 which is the same equation as Eq. 1.4, rearranging �(ℎ) and solving for ℎ � , we obtain
ℎ = (ℎ + ℎ )(1 + � −� ). Eq. 4.2
In Eq. 4.2, W(x) refers to the Lambert W function. ℎ , the height submerged below the
34 liquid-air interface in the reservoir, is neglected since the samples were not initially submerged in the fluid reservoir. The curve fit was performed using OriginPro 2015 with Eq.
4.2. ℎ and were obtained from the curve fit. C/A was obtained from the
numerical value of , using a shape factor � of 16 for circular cylinder [51], and properties of water for which � = 0.072N/m, � = 8.9 x 10-4Pa s, ρ = 1000kg/m3, and g =
9.8m/s.
To find the effective radius and contact angle of the samples using the model, the pores were assumed to be a single circular tube, and the model developed by Reed et al. described in Chapter 1.7 was implemented. Taking the model
� = ((ℎ + ℎ ) ln 1 − + ℎ), Eq. 4.3 which is the same equation as Eq. 1.2, rearranging �(ℎ) and solving for ℎ � , we obtain
ℎ = (ℎ + ℎ )(1 + � −� ). Eq. 4.4
Using Eq. 4.4 for curve fit, and by neglecting ℎ , ℎ and can be obtained. The ℎ obtained from Rajagopalan’s model is the same as the ℎ obtained from Reed’s model.
From the equation of equilibrium height, ℎ = , and properties of water, the effective radius and contact angle can be found.
For the 50/5 and 60/5 samples, the curves start to plateau at approximately 300s to 360s, so data at times greater than 360s were not considered. Even though the 60/5 samples have both a larger void fraction and perimeter, the effect of void fraction on capillary rise reported by Rajagopalan still holds, since a larger effect was observed on capillary rise by changing the void fraction, as shown in Fig. 1.2 and 1.3 in Chapter 1.7. Therefore, the equilibrium height,
35 which is proportional to C/A, should be greater and the initial capillary rise, which is proportional to A/C, should be slower for 50/5 samples. Thus, the equilibrium height ratio and the C/A ratio should be greater than 1 when comparing 50/5 samples to 60/5 samples and the A/C ratio should be less than 1. Also, since the equilibrium height is proportional to C/A, the equilibrium height ratio and the C/A ratio should be the same. To summarize the criterion for acceptable cut-off times:
(1) The curves must not be fully plateaued.
(2) The equilibrium height for the samples with the lower pore fraction should be higher;
the equilibrium height ratio and the C/A ratio should be greater than 1 [50].
(3) The initial rate of height increase should be less for the samples with lower pore
fraction; the A/C ratio should be less than 1 [50].
(4) The equilibrium height ratio and C/A ratio should be the same [50].
Different cut-off times were selected between 150s and 360s and the model fit to the data for the two types of samples. The equilibrium height ratio, C/A ratio and A/C ratio between these cut-off times are compared in Table 4.1.
For cut-off times before or at 360s, the curves are not fully plateaued. For cut-off time at 150s, the equilibrium height ratio is less than 1, which does not meet the criteria listed above that were established based on Rajagopalan’s studies. For cut-off times from 180s to
360s, the C/A ratio are all greater than 1. All the A/C ratio are less than 1, meaning the 60/5 samples have a faster rise. This agrees with the criterion. For cut-off time at 240s, the equilibrium height ratio and the C/A ratio has the smallest difference, thus 240s is determined to be the cut off time for 50/5 and 60/5 samples.
36 50/5 vs. 60/5
( / ) ( / ) Equilibrium height ratio: C/A ratio: A/C ratio: Cut-off time (s) ( / ) ( / ) 150 0.91 1.10 0.91 180 1.08 1.22 0.82 210 1.28 1.35 0.74 240 1.46 1.45 0.69 270 1.58 1.52 0.66 300 1.70 1.59 0.63 330 1.86 1.69 0.59 360 1.90 1.73 0.58 Table 4.1. The equilibrium height ratio, C/A ratio and A/C ratio for different cut-off times for 50/5 samples to 60/5 samples. 50 indicates 50% porous samples and 60 indicates 60% porous samples. The final selection of the cut-off time is highlighted in bold.
For the 50/50 samples, the curve starts to plateau at around 1600s to 1700s, and 1300s to 1400s for the 60/50 samples. Thus, data at times later than 1700s were not considered for
50/50 samples and data at times later than 1400s were not considered for 60/50 samples.
Following the same approach for 5μm samples, the 50/50 samples were compared to the
60/50 samples in terms of the equilibrium height ratio, C/A ratio and A/C ratio. The same criteria 1-4 above for acceptable cut-off times was as followed. Different cut-off times were selected between 1050s and 1650s and the model fit to the data for the two types of samples.
The equilibrium height ratio, C/A ratio and A/C ratio between these cut-off times are compared in Table 4.2.
The curve for 50/50 samples is fully plateaued after 1700s and the curve for 60/50 samples is fully plateaued after 1400s. Thus, the first criterion is not satisfied for cut-off times after 1400s. For cut-off times from 1050s to 1350s, the equilibrium height ratio is greater than
1 only at 1350s. The C/A ratio is also greater than 1 at 1350s and the A/C ratio is less than 1.
37 Thus, 1350s is considered to be the cut-off time for 50/50 and 60/50 samples. Even though the equilibrium height ratio is not the same as the C/A ratio for cut-off time at 1350s, it is still considered to be the cut-off time since it is the only cut-off time that satisfies the first three criteria.
50/50 vs. 60/50
( / ) ( / ) Equilibrium height ratio: C/A ratio: A/C ratio: Cut-off time (s) ( / ) ( / ) 1050 0.86 1.07 0.75 1170 0.93 1.13 0.72 1290 0.98 1.19 0.69 1350 1.01 1.22 0.68 1410 1.03 1.24 0.67 1530 1.07 1.28 0.65 1650 1.10 1.31 0.64 Table 4.2. The equilibrium height ratio, C/A ratio and A/C ratio of different cut-off times for 50% 50μm samples to 60% 50μm samples. 50 indicate 50% porous samples and 60 indicate 60% porous samples. The final selection of the cut-off time is highlighted in bold.
For the 50/12 samples, the curve starts to plateau at around 480s to 510s. For the 50/20 samples, the curve starts to plateau at around 570s to 600s. Since the 12μm samples and the
20μm samples have the same pore fraction, it was not possible to use C/A to compare.
However, we assessed the cut-off time in part based on the fact that a smaller radius theoretically results in a higher equilibrium height. Thus, the criterion for acceptable cut-off times is as follows:
(1) The curves must not be fully plateaued.
(2) The equilibrium height for the samples with the smaller radius should be higher; the
38 equilibrium height for 50/12 samples should be higher than 50/20 samples.
Different cut-off times were assessed for 50/12 and 50/20 samples and the equilibrium height is compared in Table 4.3.
The curve for 50/12 samples is fully plateaued after 480s and the curve for 60/50 samples is fully plateaued after 570s. Thus, the first criterion is not satisfied for cut-off times after 480s. For cut-off times greater than 330s, the equilibrium height for 20μm samples exceeds 12μm samples, which does not satisfy the second criterion. For cut-off time at 330s, the equilibrium height is very close between the two samples. Thus, 300s was determined to be the cut-off time, since the equilibrium height is higher for 12μm samples and includes the most data points possible.
50/12 50/20
Cut-off time (s) ℎ (mm) ℎ (mm) 240 96.41263 86.06276 270 105.78414 95.39951 300 110.90427 104.02102 330 114.35598 112.1726 360 113.58904 119.11399 390 112.35273 125.21061 420 110.38827 129.72147 450 107.86106 132.29025 480 104.89019 133.85574 Table 4.3. The equilibrium height of different cut-off times for 50/12 samples to 50/20 samples. The final selection of the cut-off time is highlighted in bold.
A plot of the experimental data and curve fits up to 1980s is shown in Fig. 4.1. The cut- off time, effective radius, contact angle, equilibrium height and capillary pressure for all
39 sample types are summarized in Table 4.4. For 50% porous samples, the 5μm samples have the highest equilibrium height and capillary pressure, followed by 12μm, 20μm and 50μm samples. The 50/5 samples have a higher equilibrium height and capillary pressure than the
60/5 samples. The equilibrium height and capillary pressure is close between 60/5 samples and 50/12 samples. The 50/50 samples also have a higher equilibrium height and capillary pressure than the 60/50 samples, although the difference is less obvious.
The average contact angle is 87.67 ± 0.71°. The difference between samples is expected to be small, since the contact angle is a property of the material and fluid. The contact angle for water and a solid, non-porous HA substrate was measured to be 89.8 ± 5.0° from previous work [52].
Capillary(Rise(Heights(and(Curve(Fits 110
100 50%15μm1(n=12) 90 50%15μm1curve1fit1(240s)
80 60%15μm1(n=9) 60%15μm1curve1fit1(240s) 70 50%112μm1(n=12) 60 50%112um1curve1fit1(300s) 50 50%120μm1(n=12)
Height1(mm) 50%120μm1curve1fit1(300s) 40 50%150μm1(n=12) 30 50%150μm1curve1fit1(1350s) 20 60%150μm1(n=6) 10 60%150μm1curve1fit1(1350s)
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. 4.1. Capillary rise heights and curve fits. Experimental data are indicated as dots and curve fits are indicated as lines.
40 Sample Cut-off time (s) Effective radius (μm) � (deg) ��� (mm) P (Pa) 50/5 240 5.01 86.89 159.03 1558.49 60/5 240 7.29 86.91 108.88 1067.02 50/12 300 6.14 87.34 110.90 1086.82 50/20 300 5.58 87.74 104.02 1019.40 50/50 1350 4.49 88.72 73.31 718.44 60/50 1350 5.45 88.46 72.61 711.58 Table 4.4. The cut-off time, effective radius, contact angle, equilibrium height and capillary pressure for all samples.
41 CHAPTER 5: DISCUSSION
Samples of different porosity and pore size were fabricated from HA ink and characterized in terms of microstructure, composition, pore size, pore fraction, thickness and defects. Differences in microstructure, pore size, thickness, defects and conditions of samples were described and will be discussed. 50/5, 60/5, 50/12, 50/20, 50/50, 60/50 samples were used in capillary rise tests and fluid height curves were obtained as a function of time. The conditions of the experiment setup were not perfectly controlled, which could also cause variability in the results. A model was implemented to determine the equilibrium height, capillary pressure, effective radius and contact angle for each sample type. The differences between measured pore size and calculated effective pore size and the assumptions that were made in implementing the model will also be discussed. Several other models were also explored to determine an ideal fit.
In Fig. 3.3, it can be seen that the 50% porous 5μm and 12μm samples have a similar microstructure, with more interconnections between pores, and the 50% porous 20μm samples and 50μm samples have a similar microstructure, with particles of HA inside the larger pores. The interconnected pores in the 5μm and 12μm samples could result in a larger pore size and perimeter. The larger pore size would increase the effective pore size. It would also increase C, the perimeter for liquid to wet, and A, the void area for liquid to wet. Since in Rajagopalan’s study a larger effect was observed on capillary rise by changing the void fraction, it is likely that the equilibrium height and capillary pressure would decrease and the initial rise would increase due to the increase in A. The particles inside the pores of 20μm and 50μm samples could decrease the effective pore size, but also add to the surface area,
42 which increases the perimeter. The particles would increase C and decrease A. The equilibrium height and capillary pressure would likely increase and the initial rise would decrease due to the decrease in A. All sample types also had smaller pores, sintering pores, in addition to the pores generated by the PMMA microspheres. These pores were on the order of 0.8μm in diameter. The sintering pores could decrease the effective pore size, but would increase C and A since there are more pores for liquid to wet. The increase in A would decrease the equilibrium height and capillary pressure, but increase the initial capillary rise.
However, the microstructure of 60/5 and 60/50 samples were not examined, so it is unclear whether it is possible to draw the same conclusions for the 60% porous samples.
The measured pore size obtained from the pore size analysis, Table 3.3, shows that for
12μm, 20μm and 50μm samples, the pore size is smaller than the size of the PMMA microspheres. This is expected, since the HA consolidates and shrinks, decreasing the final pore size after sintering. However, the 5μm samples show a larger pore size than the PMMA microspheres. This is likely due to the fact that the 5μm samples have more interconnected pores, and the interconnected pores would often be counted as one larger pore. Ideally, the aspect ratio of the pores should be close to 1, since the PMMA microspheres are perfect spheres, however, the results in Table 3.3 show otherwise. For the 5μm and 12μm samples, the large aspect ratio could be due to the interconnected pores for the same reason as the larger pore size. The aspect ratio is defined from the pore’s fitted ellipse, thus an interconnected pore could result in a fitted ellipse with large aspect ratio. The 20μm and
50μm samples also resulted in a large aspect ratio. It is likely that in the case where there are interconnected pores, although much less than the 5μm and 12μm samples, the pores are further apart and the interconnections are smaller, which could also result in a fitted
43 ellipse with large aspect ratio. These ellipses with large aspect ratios could increase the overall average aspect ratio of 20μm and 50μm samples. Representative images of the interconnections are shown in Fig. 5.1.
(a) (b)
Fig. 5.1. Representative images of the interconnected pores in 5μm and 12μm samples (a) and the interconnected pores in 20μm and 50μm samples (b).
As described in Chapter 3.5, the thickness varies between sample types; there was a significant difference between the 50/12 and the 50/20 and the 50/50 samples. However, it is likely that the thickness of samples does not affect the capillary rise results. The equation that was used for curve fits, Eq. 4.2, shows that the fluid height is dependent on the properties of water, the shape factor, the perimeter and void area for liquid to wet. Since the shape factor, the perimeter and void area for liquid to wet are related to the pores, and there is no other parameter that relates the fluid height to the geometry of the sample, it can be concluded that the thickness of samples should not affect the capillary rise results.
The inside and outside defects of samples also vary significantly between sample types.
44 The presence of the outside defects makes the thickness vary locally, thus the outside defects can be considered as an influence that contributes to the variation of thickness, and since the thickness does not affect the capillary rise results, the outside defects should not affect the capillary rise as well. However, the inside defects are one of the factors that could affect the capillary rise results.
The average size of the inside defects is larger than the pore size, as described in
Chapter 3.5, which could result in an increase in effective pore size and perimeter. The inside defects would increase C, the perimeter for liquid to wet, and A, the void area for liquid to wet. As a result of the increase in A, the equilibrium height and capillary pressure would decrease and the initial capillary rise would increase. However, the area fraction of the inside defects is much smaller than the area fraction of the sintering pores, it is possible that the sintering pores played a bigger role on effecting the equilibrium height and capillary pressure than the inside defects.
The ink used to make 60/50 samples were old inks (Shaq 8). The samples were more difficult to make, they tended to crack and break more frequently; even samples without any visible cracks often cracked or broke during or after sintering. Most of the 60/50 samples used in the capillary rise tests had large visible cracks and seem non-homogeneous because the samples had areas where the color was different. It is likely that during the process of making samples, two syringes of ink were formed into the wax molds, and the inks couldn’t merge together fully because the inks were drier and did not mix well, which resulted in the cracks after drying or sintering. The cracks were mostly observed to be in the middle of the samples, which is where the two syringes of ink merge together. An explanation for the non- homogeneous of the samples could also be that the ink was drier and was not able to mix
45 fully. The samples were also sanded more on the end where it’s glued. Some samples were not in a perfect rectangular shape, but a little crooked. In order to ensure that the end in contact with the fluid is horizontal to the fluid surface, the glued end had to be sanded more. Thus, the samples were shorter overall by about 2 – 3mm, as indicated in Chapter 3.3.
The 60/50 experiment also had the most uneven levels of capillary rise across the sample width.
The capillary rise test results could be influenced by issues with the experiment. As mentioned in Chapter 4, there could be end effects at the top end of the samples. The stability of the fluid reservoir and the effect of evaporation could be some influencing factors as well. The exact moment when the samples came into contact with the fluid could not be recorded precisely by taking images at 2s intervals, thus the measured height at t = 0 was not h = 0. This could be better improved by recording with a video.
The fluid height at t = 0 can be considered as ℎ , the height submerged below the liquid-air interface in the reservoir. By curve fitting with Eq. 4.2,
ℎ = (ℎ + ℎ )(1 + � −� ), Eq. 5.1
ℎ + ℎ can be obtained from the curve fit. By subtracting ℎ + ℎ with ℎ , a new equilibrium height can be found. The ℎ , new equilibrium height, new capillary pressure, original equilibrium height and original capillary pressure are listed in Table 5.1. The new equilibrium heights are less than the original equilibrium heights, so the new capillary pressure would be lower as well. However, the order of heights is still the same, where 50/5 samples have the highest equilibrium height and 60/50 samples have the lowest equilibrium height.
46 Sample �� New ��� New P Original ��� Original P (mm) (mm) (Pa) (mm) (Pa) 50/5 4.36 154.67 1515.766 159.03 1558.49 60/5 5.20 103.68 1016.064 108.88 1067.02 50/12 5.09 105.81 1036.938 110.90 1086.82 50/20 4.28 99.74 977.452 104.02 1019.40 50/50 3.67 69.64 682.472 73.31 718.44 60/50 3.19 69.42 680.316 72.61 711.58
Table 5.1. ℎ , new equilibrium height, new capillary pressure, original equilibrium height and original capillary pressure of samples.
The calculated effective radius of each sample type obtained from curve fit was also different from the results obtained from the pore size analysis. For 50/5 and 50/12 samples, the effective radius calculated from fitting the model, 5.01μm and 6.14μm, are larger than the radius measured from pore size analysis, which are 3.01μm and 4.60μm. However, for
50/20 and 50/50 samples, the calculated effective radius, 5.58μm and 4.49μm, are smaller than the measured radius, 6.34μm and 15.35μm.
One possible explanation for the differences in pore size is because in using this model, the micropores are idealized as cylindrical tubes. In works by Marmur et al. [35], Delker et al.
[45], Hamraoui et al. [47] and Fries et al. [48], the pores were also assumed as a single capillary or an assembly of capillary tubes in developing their models. The real porous medium is not capillary tubes of circular cross section, but rather a network of pores of varying diameter and interconnection size. The interconnected pores in the 50/5 and 50/12 samples could increase the calculated effective radius. The particles inside the pores of 50/20 and 50/50 samples could also be a reason for the differences in pore size. The particles could decrease the calculated effective radius. The sintering pores could decrease the calculated effective radius
47 as well, since the sintering pores are much smaller than the pores made by PMMA microspheres. The inside defects could also increase the calculated effective radius. A more accurate pore size analysis could also be obtained with mercury intrusion porosimetry (MIP).
Another explanation could be that the surface tension, dynamic viscosity and density of water were used in calculating the effective radius instead of the properties of PBS. The surface tension has been reported to be 0.067N/m [53], the dynamic viscosity to be 10.9 x 10-
4Pa s [53] and the density to be 998.45kg/m3 [54] for PBS. The surface tension, dynamic viscosity and density used for water are 0.072N/m, 8.9 x 10-4Pa s and 1000kg/m3. The surface tension and density are smaller for PBS, and the dynamic viscosity is larger, but the differences are small.
Several models including the Washburn equation [43], the equation developed by
Hamraouni et al. [47] with and without a constant, a log fit and the solution reported by
Fries et al. [48] were explored as possible curve fits and are discussed in Appendix C. The model developed by Rajagopalan et al. [50] was used as the final model because it fit best with our experimental data. The model by Rajagopalan et al. takes into account the effect of gravity and the effect of void area and perimeter for liquid to wet. The effect of gravity is essential to this study because the samples were in a vertical position. In considering the effect of void area and perimeter, the size distribution of the pores and the porosity of the samples were taken into account.
In the study by Polak [52], HA samples with various pore size and porosity were used in capillary rise tests to determine the equilibrium height and capillary pressure. For 50/5 and
60/5 samples, the equilibrium heights were determined from curve fit because the equilibrium heights were greater than the height of the samples. The equilibrium height of
48 50/50 samples was determined from the measured values. The 50/5, 60/5 and 50/50 samples had a lower equilibrium height, 56.9mm, 59.5mm and 22.3 ± 1.2mm, and capillary pressure than the 50/5, 60/5 and 50/50 samples in this study. The equilibrium heights and capillary pressures in this study are almost 2.8 times, 1.8 times and 3.3 times higher than that in Polak’s study. The 60/5 samples had a higher equilibrium height and capillary pressure than the 50/5 samples, which is different from the results in this study where the
60/5 samples had a lower equilibrium height and capillary pressure. One difference between this study and Polak’s study is the fabrication and size of the samples. The samples used in
Polak’s study were HA blocks made up of columns, rows and layers of orthogonal rods deposited from a machine whereas the samples in this study were made by forming ink into wax molds. The size of Polak’s samples were 2.7 x 48.9 x 2.1mm whereas the samples in this study were about 9 x 63 x 0.9mm. Another difference is the capillary rise experiment set up.
In Polak’s study, the samples were gripped from the sides instead of glued on the top, and water was used as the fluid instead of PBS.
In summary, C, the total perimeter for liquid to wet, is increased by the presence of interconnected pores, the particles in larger pores, the sintering pores and the defects in samples. A, the total void area for liquid to wet, is increased by the presence of interconnected pores, the sintering pores and the defects in samples, but decreased by the particles in larger pores. The equilibrium height would be affected by the changes in C and A, but the degree of the effects is unknown, so it is hard to tell how much the equilibrium height would be affected.
The order of the heights should, however, be the same, since the effect of C and A is present in every sample type. The equilibrium height would also be different when considering the height of the fluid at t = 0 as ℎ . The heights would decrease, but would still end up with the
49 same order of heights. The differences in the properties of water and PBS would not affect the equilibrium heights, but would change the calculated effective radius. Overall, the fluid height at t = 0 would be the main difference in equilibrium heights and capillary pressure, but the order of the heights and capillary pressure should still be valid.
50 CHAPTER 6: CONCLUSIONS
The effect of pore size and pore fraction of HA samples on equilibrium height and capillary pressure was investigated. PMMA microspheres were incorporated with HA inks to make rectangular-shaped samples with a range of pore sizes and fractions. A sintering profile was determined from trial and error in order to obtain samples with a dense microstructure and a composition of HA as the dominant phase and β-TCP as the minor phase. The pore size and pore fraction were measured and compared between samples. The thickness of the samples was also measured and compared. The defects in the samples were sorted between inside and outside defects and quantified in terms of number, size and area fraction, then compared between sample types. 50/5, 60/5, 50/12, 50/20, 50/50 and 60/50 samples were used in capillary rise tests. The fluid height curve was plotted as a function of time. The samples were not long enough for the fluid to reach an equilibrium height, thus the model developed by Rajagopalan et al. was implemented to find the equilibrium heights and capillary pressures. Criteria were set to determine a cut off point for the data points used in the implementation of model due to end effects of the samples. The cut off time was 240s for
50/5 and 60/5 samples, 300s for 50/12 and 50/20 samples and 1350s for 50/50 and 60/50 samples. For 50% porous samples, the 5μm samples had the highest equilibrium height,
159.03mm, followed by 12μm samples, 110.9mm, then 20μm samples, 104.02mm and 50μm samples, 73.31mm. The 60/5 samples had a faster initial capillary rise than 50/5 samples, but a lower equilibrium height of 108.88mm. The 60/50 samples also had a faster initial capillary rise than 50/50 samples and a slightly lower equilibrium height of 72.61mm. This is consistent with what is expected from the model by Rajagopalan et al. where it was stated that a smaller
51 void fraction results in a higher equilibrium height and a slower initial capillary rise.
52 CHAPTER 7: FUTURE WORK
Future work could include improvements on the sample preparation and experiments.
The samples used can be improved by fabricating samples with a more consistent thickness and less defects inside and outside. This might be improved by fabricating the samples with direct write deposition or with a machine rather than manually with hands. Longer samples can also be used in the capillary rise tests to obtain equilibrium heights by measurement.
The samples could also be gripped from the sides rather than the top in order to avoid end effects that were observed in the data. Samples should be characterized for specific surface area, and pore size and pore fraction using Brunauer-Emmett-Teller (BET) analysis and mercury intrusion porosimetry (MIP), respectively. Additionally, a more complete set of pore sizes and fractions could be evaluated by capillary rise such as the 12μm and 20 μm nominal pore sizes at volume fraction of 60% and an intermediate pore size between 20μm and
50μm.
Future studies could focus on seeding cells in the samples and investigating the penetration distances. Same type cells can be seeded into samples of different pore size and porosity or different cells can be seeded into samples of the same pore size and porosity.
Manipulating the interconnection size of pores can be another interesting direction.
Understanding the effect of interconnection size in capillary pressure can also improve the tailoring of microstructures. By studying how the difference in microstructures influences capillary pressure, we can contribute in optimizing the design of bone graft substitutes in future works.
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61 APPENDIX A: PMMA MICROSPHERES SIZE
PARTICLE SIZE DISTRIBUTION
Jackie Cordell Brad Herschler Sheryl Wang On bag (Aveka, 2008) (Aveka, 6/30/10) (AgEng, 5/28/13)
Mean Mean Mean Mean S.D. S.D. Range S.D. Median S.D. Median (µm) (µm) (µm) (µm)
M-100 5.96 2 2-14
MBX 5 5.468 1.865 6.22726 2.7729 5.68058
MBX 8C 8.092 2.797 9.59241 3.8396 9.06446
MBX 12 11.76 4.3 11.025 4.245 10.566
MBX 20 17.1 6.24 16.2 6.77 6-40
MBX 30 24.33905 10.3157 22.78149 23.294 12.986 24.177
MBX 40 39.59587 15.2345 37.35677 41.185 20.36 37.513
MBX 50 47.27663 19.5723 44.32228 Table A.1. Diameter of PMMA microspheres.
62 Appendix B: HEAT TREATMENTS
A series of heat treatments were conducted to optimize the sintered density and phases present in the material to be used in capillary rise tests. The heat treatments and XRD results are summarized in the table below and SEM images of the microstructures are included. Larger images of the microstructures follow the table.
Heat Treatment XRD result SEM image
HA β-TCP α-TCP Cross sectional image of 5μm samples
900°C 2hrs 100%
(P)
1200°C 2hrs 94.4% 5.6%
(Q)
1250°C 2hrs 94.3% 5.7%
(R)
1250°C 4hrs 94.9% 5.1%
(Y)
Table B.1. Microstructure and composition of samples under different heat treatments. The final profile is shaded. Magnification differs between samples.
63 Heat Treatment XRD result SEM image
HA β-TCP α-TCP Cross sectional image of 5μm samples
1275°C 2hrs 95.2% 4.8%
(T)
1275°C 4hrs 93.1% 6.9%
(X)
1300°C 2hrs 93.5% 6.5%
(S)
900°C 2hrs 1275°C 2hrs 94.8% 5.2%
(V)
900°C 3hrs 1275°C 4hrs 93.6% 6.4%
(W)
Table B.1. (cont.) Microstructure and composition of samples under different heat treatments. The final profile is shaded. Magnification differs between samples.
64 Heat Treatment XRD result SEM image
HA β-TCP α-TCP Cross sectional image of 5μm samples
900°C 2hrs 1300°C 2hrs 93.6% 6.4%
(O)
1275°C 2hrs 900°C 4hrs 96.1% 3.9%
(U)
1275°C 2hrs 1100°C 4hrs 94.0% 6.0%
(1)
1275°C 2hrs 1100°C 6hrs 94.4% 5.6%
(4)
1275°C 2hrs 1100°C 8hrs 94.4% 5.6%
(3)
Table B.1. (cont.) Microstructure and composition of samples under different heat treatments. The final profile is shaded. Magnification differs between samples.
65 Heat Treatment XRD result SEM image
HA β-TCP α-TCP Cross sectional image of 5μm samples
1275°C 2hrs 1250°C 6hrs 95.1% 4.9%
(5)
1275°C 2hrs 1250°C 10hrs 95.6% 4.4%
(6)
1250°C 4hrs 1100°C 4hrs 94.7% 5.3%
(2)
1250°C 6hrs 1100°C 6hrs 94.5% 5.5%
(7)
1275°C 2hrs 1250°C 4hrs 1100°C 6hrs 94.3% 5.7% (8)
Table B.1. (cont.) Microstructure and composition of samples under different heat treatments. The final profile is shaded. Magnification differs between samples.
66 Heat Treatment XRD result SEM image
HA β-TCP α-TCP Cross sectional image of 5μm samples
1275°C 2hrs 1250°C 3hrs 1100°C 6hrs 95.3% 4.7%
(9)
1275°C 2hrs 1250°C 2hrs 1100°C 6hrs 94.2% 5.8%
(10)
1275°C 3hrs 94.3% 5.7%
(13)
1275°C 3hrs 1250°C 2hrs 96.2% 3.8%
(16)
1100°C 10hrs 95.0% 5.0%
(12)
Table B.1. (cont.) Microstructure and composition of samples under different heat treatments. The final profile is shaded. Magnification differs between samples.
67 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
900°C
2hrs N/A
(P)
1200°C
2hrs N/A
(Q)
1250°C
2hrs N/A
(R)
1250°C
4hrs
(Y)
Table B.2. Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
68 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
1275°C
2hrs
(T)
1275°C
4hrs
(X)
1300°C
2hrs N/A
(S)
900°C
2hrs
1275°C N/A
2hrs
(V)
Table B.2. (cont.) Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
69 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
900°C
3hrs
1275°C
4hrs
(W)
900°C
2hrs
1300°C N/A
2hrs
(O)
1275°C
2hrs
900°C N/A
4hrs
(U)
1275°C
2hrs
1100°C
4hrs
(1)
Table B.2. (cont.) Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
70 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
1275°C
2hrs
1100°C
6hrs
(4)
1275°C
2hrs
1100°C
8hrs
(3)
1275°C
2hrs
1250°C
6hrs
(5)
1275°C
2hrs
1250°C N/A
10hrs
(6)
Table B.2. (cont.) Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
71 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
1250°C
4hrs
1100°C
4hes
(2)
1250°C
6hrs
1100°C N/A
6hrs
(7)
1275°C
2hrs
1250°C
4hrs
1100°C
6hrs
(8)
1275°C
2hrs
1250°C
3hrs
1100°C
6hrs
(9)
Table B.2. (cont.) Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
72 Sample SEM cross sectional image of NMP samples SEM cross sectional image of 5μm samples
1275°C
2hrs
1250°C
2hrs
1100°C
6hrs
(10)
1275°C
3hrs N/A
(13)
1275°C
3hrs
1250°C
2hrs
(16)
1100°C
10hrs
(12)
Table B.2. (cont.) Larger images of the microstructures. The final profile is shaded. Magnification differs between samples.
73 APPENDIX C: XRD ANALYSIS OBTAINED FROM RMS FOUNDATION
• Sample materials analyzed: (1) NMP HA ink (made on 12/4/14). (2) 50% M-100 HA ink (made on 1/30/15). (3) Crushed HA powder: from Laurie’s in vivo samples. (4) Sample 7: HA sample under heat treatment 1250°C 6hrs 1100°C 6hrs. (5) Sample 8: HA sample under heat treatment 1275°C 2hrs 1250°C 4hrs 1100°C 6hrs. (6) Sample 9: HA sample under heat treatment 1275°C 2hrs 1250°C 3hrs 1100°C 6hrs. (7) Sample 16: HA sample under heat treatment 1275°C 3hrs 1250°C 2hrs. (8) Sample Original: HA sample under heat treatment 900°C 2hrs 1300°C 2hrs. (9) Sample X: HA sample under heat treatment 1275°C 4hrs.
• Sample preparation: (1) Heat treatment at 1000°C 15hrs + 840°C 15hrs. (2) Manual grinding in acetone.
• Data processing: (1) XRD + Rietveld refinement. (2) Mg content in HA: Ca site occupancy refinement. (3) Mg content in β-TCP: Change of unit cell dimensions.
• Results:
«NMP HA Ink» «50% M-100 HA ink» Phase Crushed HA Powder 12/4/14 1/30/15 Hydroxyapatite 94.57 (0.51) 94.37 (0.54) 94.84 (0.60) β-TCP 5.30 (0.48) 5.46 (0.51) 4.86 (0.60) Periclase (MgO) 0.13 (0.06) 0.17 (0.06) 0.30 (0.09) Table C.1. Phase composition. Standard deviations indicated in parentheses.
74 «NMP HA Ink» 12/4/14 CaO (wt-%) P2O5 (wt-%) MgO (wt-%) Hydroxyapatite 56.41 43.22 0.37 β-TCP 50.69 46.30 3.00 Periclase (MgO) 0 0 100 Bulk 56.04 43.33 0.64
«50% M-100 HA ink» 1/30/15 CaO (wt-%) P2O5 (wt-%) MgO (wt-%) Hydroxyapatite 56.50 43.21 0.29
β-TCP 50.69 46.30 3.00
Periclase (MgO) 0 0 100
Bulk 56.09 43.31 0.60
Crushed HA powder CaO (wt-%) P2O5 (wt-%) MgO (wt-%) Hydroxyapatite 56.34 43.23 0.42
β-TCP 50.69 46.30 3.00
Periclase (MgO) 0 0 100
Bulk 55.90 43.25 0.85 Table C.2. Chemical compositions determined from XRD data.
Phase NMP HA Ink 12/4/14 50% M-100 HA Ink Crushed HA Powder 1/30/15
Hydroxyapatite (Ca0.991Mg0.009)5(PO4)3OH (Ca0.993Mg0.007)5(PO4)3OH (Ca0.990Mg0.010)5(PO4)3OH
β-TCP (Ca0.924Mg0.076)3(PO4)2 (Ca0.924Mg0.076)3(PO4)2 (Ca0.924Mg0.076)3(PO4)2
Bulk (Ca0.982Mg0.016)5(PO4)3OH (Ca0.983Mg0.015)5(PO4)5OH (Ca0.981Mg0.021)5(PO4)3OH Table C.3. Chemical compositions. Hydroxyapatite determined from refinement of Ca site occupancies for. Mg content in β-TCP determined from unit cell distortion.
75 Sample HA (wt-%) α-TCP (wt-%) β-TCP (wt-%) 7 92.44 (0.28) 0 7.56 (0.28) 8 89.51 (0.48) 5.50 (0.43) 4.99 (0.53) 9 92.18 (0.26) 7.57 (0.25) 0.25 (0.10) 16 94.60 (0.27) 0 5.40 (0.27) Original 93.19 (0.26) 6.61 (0.24) 0.20 (0.10) X 92.22 (0.31) 7.30 (0.30) 0.48 (0.21) Table C.4. Phase composition of samples under different heat treatments. Standard deviations indicated in parentheses.
Sample Mg content in β-TCP (mol-% relative to Ca content) 7 4.86 8 1.93 9 N/A 16 6.14 Original N/A X N/A Table C.5. Mg content in β-TCP of samples under different heat treatments. Results determined from β-TCP peak shift.
76 APPENDIX D: THE RADIUS AND EQUILIBRIUM HEIGHT AT DIFFERENT CUT-OFF TIMES
In determining the cut-off time for each sample type, it was also noted that different cut off times result in different effective radii and equilibrium heights. The 50/5, 60/5 and 50/12 samples have a minimum radius between 270s to 330s, 180s to 240s and 300s to 390s, respectively. The cut-off time that was used in the final analysis for each of these sample types falls in the corresponding time range, except for the 50/5 samples. The effective radius for 50/20 samples decreases with increasing cut off time, which is different from the other samples, as can be seen in Fig. D.1. For 50/50 samples, the effective radius seems to be similar for each cut off time, but for 50/60 samples, the effective radius increases with increasing cut off time, as shown in Fig. D.2. The equilibrium height for 50/20 samples increases with increasing cut-off time. As mentioned in Chapter 4, the equilibrium height for 50/5 samples exceeds 60/5 samples for cut off time at or after 180s. For 50μm samples, the equilibrium height for 50/50 samples exceeds 60/50 samples for cut-off times equal to or greater than 1350s. Comparisons are shown in Fig. D.3 and D.4.
Effective(Radius 9
8
7
6
) 5 50%,5μm μm
r,( 60%,5μm 4 50%,12μm 3 50%,20μm
2
1
0 150 180 210 240 270 300 330 360 390 420 450 480 Cut,off,Time,(s)
Fig. D.1. The effective radius for 50/5, 60/5, 50/12 and 50/20 samples at different cut-off times.
77 Effective(Radius 7
6
5
4
50%250μm 3 60%250μm
Equilibrium2Height2(mm) 2
1
0 1050 1110 1170 1230 1290 1350 1410 1470 1530 1590 1650 Cut2off2Time2(s)
Fig. D.2. The effective radius for 50/50 and 60/50 samples at different cut-off times.
Equilibrium)Height 180
160
140
120
100 50%35μm 60%35μm 80 50%312μm 60
Equilibrium3Height3(mm) 50%320μm
40
20
0 150 180 210 240 270 300 330 360 390 420 450 480 Cut3off3Time3(s)
Fig. D.3. The equilibrium height for 50/5, 60/5, 50/12 and 50/20 samples at different cut-off times.
78 Equilibrium)Height 90
80
70
60
50
50%350μm 40 60%350μm 30 Equilibrium3Height3(mm)
20
10
0 1050 1110 1170 1230 1290 1350 1410 1470 1530 1590 1650 Cut3off3Time3(s)
Fig. D.4. The equilibrium height for 50/50 and 60/50 samples at different cut-off times.
79 APPENDIX E: MATHEMATICAL MODELS
Several mathematical models related to or derived from the Washburn equation were used to try to model the data from the capillary rise tests. Appendix E summarizes results for these models. The final model used was the model developed by Rajagopalan et al. E.1 The Washburn model [43]
Data points were fit with the Washburn equation, ℎ = � , and was obtained from the curve fits. The cut-off times and equations are shown in Table E.1. The curve fits are shown in Fig. E.1.
Sample Cut-off time (s) Equation (h: mm)
50/5 240 ℎ = 9.349� 60/5 240 ℎ = 11.913� 50/12 300 ℎ = 8.785� 50/20 300 ℎ = 6.905� 50/50 1350 ℎ = 2.390� 60/50 1350 ℎ = 3.091� Table E.1. Cut-off time and equation fitted from the Washburn equation.
Capillary(Rise(Heights(and(Curve(Fits(5 Washburn(Equation 160 150 140 50%15μm1(n=12) 130 50%15μm1curve1fit1(240s) 120 60%15μm1(n=9) 110 60%15μm1curve1fit1(240s) 100 50%112μm1(n=12) 90 50%112um1curve1fit1(300s) 80 50%120μm1(n=12) 70 Height1(mm) 60 50%120μm1curve1fit1(300s) 50 50%150μm1(n=12) 40 50%150μm1curve1fit1(1350s) 30 60%150μm1(n=6) 20 60%150μm1curve1fit1(1350s) 10 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. E.1. Capillary rise heights and curve fits from the Washburn equation. Experimental data are indicated as dots and curve fits are indicated as lines.
80 E.2 The Hamraouni and Nylander model [47] Data points were fit with the Hamraouni and Nylander equation,
ℎ = ℎ (1 − � ),
where � is a nondimensional retardation coefficient. ℎ and were obtained from the curve fits. The cut-off times and equations are shown in Table E.2. The curve fits are shown in Fig. E.2.
Sample Cut-off time (s) Equation (h: mm) 50/5 240 ℎ = 44.8(1 − exp −0.013� ) 60/5 240 ℎ = 48.44(1 − exp −0.015� ) 50/12 300 ℎ = 46.91(1 − exp −0.012� ) 50/20 300 ℎ = 41.89(1 − exp −0.011� ) 50/50 1350 ℎ = 46.88(1 − exp −0.004� ) 60/50 1350 ℎ = 52.05(1 − exp −0.004� ) Table E.2. Cut-off time and equation fitted from the Hamraouni and Nylander equation.
Capillary(Rise(Heights(and(Curve(Fits(5 Hamraouni(and(Nylander(Equation 65 60 50%15μm1(n=12) 55 50%15μm1curve1fit1(240s) 50 60%15μm1(n=9) 45 60%15μm1curve1fit1(240s) 40 50%112μm1(n=12) 35 50%112um1curve1fit1(300s) 30 50%120μm1(n=12)
Height1(mm) 25 50%120μm1curve1fit1(300s) 20 50%150μm1(n=12) 15 50%150μm1curve1fit1(1350s) 10 60%150μm1(n=6) 5 60%150μm1curve1fit1(1350s) 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. E.2. Capillary rise heights and curve fits from the Hamraouni and Nylander equation. Experimental data are indicated as dots and curve fits are indicated as lines.
81 E.3 The Hamraouni and Nylander model with a constant [47] Data points were fit with the Hamraouni and Nylander equation with a constant,
ℎ = ℎ 1 − � + �,
where � is a nondimensional retardation coefficient and c is the constant. ℎ , and c were obtained from the curve fits. The cut-off times and equations are shown in Table E.3. The curve fits are shown in Fig. E.3.
Sample Cut-off time (s) Equation (h: mm)
50/5 240 ℎ = 45.91 1 − exp −0.0075� + 7 60/5 240 ℎ = 47.24 1 − exp −0.0090� + 7.669 50/12 300 ℎ = 46.3 1 − exp −0.0067� + 7.92 50/20 300 ℎ = 41.93 1 − exp −0.0063� + 7.252 50/50 1350 ℎ = 44.25 1 − exp −0.0020� + 8.747 60/50 1350 ℎ = 47.91 1 − exp −0.0029� + 7.235 Table E.3. Cut-off time and equation fitted from the Hamraouni and Nylander equation with a constant.
Capillary(Rise(Heights(and(Curve(Fits(5 Hamraouni(and(Nylander(Equation(with(a(Constant 65 60 50%15μm1(n=12) 55 50%15μm1curve1fit1(240s) 50 60%15μm1(n=9) 45 60%15μm1curve1fit1(240s) 40 50%112μm1(n=12) 35 50%112um1curve1fit1(300s) 30 50%120μm1(n=12)
Height1(mm) 25 50%120μm1curve1fit1(300s) 20 50%150μm1(n=12) 15 50%150μm1curve1fit1(1350s) 10 60%150μm1(n=6) 5 60%150μm1curve1fit1(1350s) 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. E.3. Capillary rise heights and curve fits from the Hamraouni and Nylander equation with a constant. Experimental data are indicated as dots and curve fits are indicated as lines.
82 E.4 Log fit Data points were fit with log fit in Excel, ℎ = ��� � + �, where a and c are constants and were obtained from the curve fits. The cut-off times and equations are shown in Table E.4. The curve fits are shown in Fig. E.4.
Sample Cut-off time (s) Equation (h: mm) 50/5 240 ℎ = 8.4871�� � − 7.311 60/5 240 ℎ = 9.4338�� � − 7.5524 50/12 300 ℎ = 8.8266�� � − 8.2208 50/20 300 ℎ = 7.7701�� � − 7.0471 50/50 1350 ℎ = 8.4598�� � − 15.627 60/50 1350 ℎ = 9.9176�� � − 19.436 Table E.4. Cut-off time and equation fitted from log fit.
Capillary(Rise(Heights(and(Curve(Fits(5 Log(Fit 70 65 50%15μm1(n=12) 60 50%15μm1curve1fit1(240s) 55 60%15μm1(n=9) 50 60%15μm1curve1fit1(240s) 45 50%112μm1(n=12) 40 50%112um1curve1fit1(300s) 35 50%120μm1(n=12) 30 Height1(mm) 50%120μm1curve1fit1(300s) 25 50%150μm1(n=12) 20 50%150μm1curve1fit1(1350s) 15 60%150μm1(n=6) 10 60%150μm1curve1fit1(1350s) 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. E.4. Capillary rise heights and curve fits from log fit. Experimental data are indicated as dots and curve fits are indicated as lines.
83 E.5 The Fries and Dreyer model [48] Data points were fit with the Fries and Dreyer equation,
ℎ = (1 + � −� ),
where � = , � = and � is the angle between an inclined tube and the
liquid surface. and were obtained from the curve fits. The cut-off times and equations are shown in Table E.5. The curve fits are shown in Fig. E.5.
Sample Cut-off time (s) Equation (h: mm)
50/5 240 ℎ = 159.03(1 + � −exp (−1 − 0.00022� ) 60/5 240 ℎ = 108.88(1 + � −exp (−1 − 0.00067� ) 50/12 300 ℎ = 110.90(1 + � −exp (−1 − 0.00047� ) 50/20 300 ℎ = 104.02(1 + � −exp (−1 − 0.00041� ) 50/50 1350 ℎ = 73.31(1 + � −exp (−1 − 0.00038� ) 60/50 1350 ℎ = 72.62(1 + � −exp (−1 − 0.00056� ) Table E.5. Cut-off time and equation fitted from the Fries and Dreyer equation.
Capillary(Rise(Heights(and(Curve(Fits(5 Fries(and(Dreyer 110
100 50%15μm1(n=12) 90 50%15μm1curve1fit1(240s) 60%15μm1(n=9) 80 60%15μm1curve1fit1(240s) 70 50%112μm1(n=12) 60 50%112um1curve1fit1(300s) 50 50%120μm1(n=12)
Height1(mm) 50%120μm1curve1fit1(300s) 40 50%150μm1(n=12) 30 50%150μm1curve1fit1(1350s) 20 60%150μm1(n=6) 10 60%150μm1curve1fit1(1350s)
0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Time1(x102s)
Fig. E.5. Capillary rise heights and curve fits from the Fries and Dreyer equation. Experimental data are indicated as dots and curve fits are indicated as lines.
84 APPENDIX F：HA POWDER
• Label on bottle: Sigma-Aldrich Quality by: Riedel-de Haën 04238-1KG Lot# SZBC3390V Hydroxyapatite puriss., meets analytical specification of Ph. Eur., BP, FCC, E341, ≥90% (calculated on glowed substance)
CAS: 1306-06-5, HCa5O13P3, MW: 502.31 assay: ≥90% (calculated on glowed substance) assay: 35-40% Ca basis ≤8% loss on ignition, 800°C - - 2- chloride (Cl ): ≤1500mg/kg, fluoride (F ): ≤50mg/kg, sulfate (SO4 ): ≤5000mg/kg As: ≤2mg/kg, Cd: ≤1mg/kg, Cu: ≤20mg/kg, Fe: ≤400mg/kg, Hg: ≤1mg/kg, Pb: ≤2mg/kg, Zn: ≤20mg/kg
• ICP analysis conducted at U of I: Atomic Mass Element Raw HA 13 M-100 15 M-100 # ppm mg/L ppm mg/L ppm mg/L 20 40 Ca 351370 379771 362840 15 31 P 143950 148685 148460 12 24 Mg 2880 2980 2928 14 28 Si 470 508 2240 26 56 Fe 2675 1102 1125 13 27 Al 230 848 956 3 7 Li 232 38 88 Sr 153.8 166.4 161 11 23 Na 856 165 136.4 25 55 Mn 126 97.4 96.4 22 48 Ti 17.2 89.6 60.3 42 98 Mo 5.54 50.2 39.8 Table F.1. ICP results of HA powder.
85 Atomic Mass Element Raw HA 13 M-100 15 M-100 # ppm mg/L ppm mg/L ppm mg/L 24 52 Cr 678 28.8 29.6 23 51 V 16.7 16.4 18.9 30 64 Zn 26.4 21.3 14.5 29 63 Cu 67.3 12.7 12 27 59 Co 3710 46.6 11 92 238 U 1.172 11.2 10.55 47 106 Ag 40 14.9 6.54 56 138 Ba 12.4 6.84 4.68 74 184 W 0.043 2.69 3.96 40 90 Zr 3.74 4.22 3.43 50 120 Sn 2.01 1.56 2.15 39 89 Y 1.85 2.12 1.76 33 75 As 3.7 2.34 1.59 53 127 I 1.2 0.907 1.56 58 140 Ce 0.905 1.5 1.35 57 139 La 0.865 1.34 1.01 83 209 Bi 0.013 0.382 0.897 82 208 Pb 0.562 0.888 60 142 Nd 0.347 0.824 0.724 90 232 Th 0.279 0.564 0.648 51 121 Sb 0.318 0.625 0.405 41 93 Nb 0.164 0.422 0.322 66 164 Dy 0.128 0.095 0.315 31 69 Ga 0.198 0.107 0.246 21 45 Sc 0.617 0.183 62 152 Sm 0.122 0.173 59 141 Pr 0.167 0.203 0.168 64 158 Gd 0.152 0.169 0.15 70 174 Yb 0.062 0.096 37 85 Rb 0.316 0.089 81 205 Tl 0.12 0.036 0.07 Table F.1. (cont.) ICP results of HA powder.
86 Atomic Mass Element Raw HA 13 M-100 15 M-100 # ppm mg/L ppm mg/L ppm mg/L 55 133 Cs 0.014 0.014 0.061 78 195 Pt 0.044 0.089 0.043 32 74 Ge 0.105 0.057 0.034 68 166 Er 0.06 0.06 0.033 65 159 Tb 0.04 0.024 0.031 63 153 Eu 0.045 0.091 0.03 45 102 Rh 0.023 0.034 0.022 71 175 Lu 0.016 0.008 0.016 67 165 Ho 0.065 0.016 0.008 49 115 In 0.002 0.007 4 9 Be 3605 5 11 B 9500 28 58 Ni 92.5 1.27 34 80 Se 17.2 35 79 Br 3.5 69 169 Tm 0.024 72 180 Hf 0.045 0.118 75 187 Re 0.017 0.017 Table F.1. (cont.) ICP results of HA powder.
13 M-100: HA sample under heat treatment 1275°C 3hrs (has β-TCP) 15 M-100: HA sample under heat treatment 1275°C 3hrs 1250°C 3hrs (has α -TCP)
87