NETWORK DILATATION AND RELAXATION IN CHEMICALLY
STRENGTHENED ALKALI SILICATE GLASSES
BY
PATRICK K. KRESKI
A THESIS
SUBMITTED TO THE FACULTY OF
ALFRED UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN
GLASS SCIENCE
ALFRED, NEW YORK
SEPTEMBER, 2014 Alfred University theses are copyright protected and may be used for education or personal research only. Reproduction or distribution in part or whole is prohibited without written permission from the author.
Signature page may be viewed at Scholes Library, New York State College of Ceramics, Alfred University, Alfred, New York. NETWORK DILATATION AND RELAXATION IN CHEMICALLY STRENGTHENED ALKALI SILICATE GLASSES BY PATRICK K. KRESKI B.S. ALFRED UNIVERSITY (2007) M.S. ALFRED UNIVERSITY (2009)
SIGNATURE OF AUTHOR ______
APPROVED BY ______ARUN K. VARSHNEYA, ADVISOR
______MATTHEW M. HALL, ADVISOR
______ALASTAIR N. CORMACK, ADVISORY COMMITTEE
______ALEXIS G. CLARE, ADVISORY COMMITTEE
______NATHAN P. MELLOTT, ADVISORY COMMITTEE
______CHAIR, ORAL THESIS DEFENSE
ACCEPTED BY ______DOREEN D. EDWARDS, DEAN KAZUO INAMORI SCHOOL OF ENGINEERING
ACCEPTED BY ______NANCY J. EVANGELISTA, ASSOCIATE PROVOST FOR GRADUATE AND PROFESSIONAL PROGRAMS ALFRED UNIVERSITY
ACKNOWLEDGMENTS
Many thanks to Dr. Arun Varshneya for supplying the seeds for this thesis and advising throughout its development. Thanks to Saxon Glass for providing materials and equipment time. Thanks to Dr. Matthew Hall for administrative advising and support. Thanks to Dr. Alastair Cormack for enabling me to branch this project into molecular dynamics simulations. Also thanks to Dr. Alexis Clare and Dr. Nathan Mellott for participating on my thesis committee. Thanks to Dr. Tim Wong, visiting professor from 2008-2010, and Dr. Jinghong Fan for numerous discussions related to mechanics and finite element analysis. Thanks also to the university staff, including Gerry Wynick with microprobe analysis and Fran Williams with profilometer maintenance. Throughout this project a number of graduate students have generously provided their time to help with various aspects of this thesis. For proof-of-concept experiments, thanks to Matt Brophy with vapor deposition of metallic coatings and Brian Adams with sol-gel silica spin-coatings. For getting started with molecular dynamics simulations and related equipment up-keep, I am grateful to Laura Adkins and Bu Wang for their time and effort. Finally, thank you to my family and friends for their support over the years. Many thanks to Jessie for continual support and encouragement. A special thanks to my parents, Mary Ann and Ken.
iii TABLE OF CONTENTS
Page Acknowledgments ...... iii Table of Contents...... iv List of Tables ...... vi List of Figures...... viii Abstract ...... xii I. INTRODUCTION ...... 1 References ...... 5 II. MOLECULAR DYNAMICS SIMULATIONS OF ALKALI STUFFED SILICATE GLASS NETWORKS ...... 8 A. Introduction...... 8 1. Basic Principles ...... 8 2. Background...... 10 3. Foundation ...... 12 B. Simulation and Analysis Methods ...... 17 B. Results ...... 20 1. Molar Volume and LNDC ...... 20 2. Range I: Structural Unit ...... 21 3. Range II: Interconnection of Adjacent Structural Units ...... 28 4. Range III: Network Topology ...... 31 5. Elastic Properties ...... 32 6. NVT Stuffing ...... 36 C. Discussion ...... 37 1. A Note on Timescale, Temperature, and Boundary Conditions ...... 37 2. What Influences LNDC? ...... 38 3. Network Features of Stuffing Alkali Accommodation ...... 38 4. LNDC and Compression Maximum ...... 41 D. Conclusions...... 47 E. References...... 49 III. DIMENSIONAL SWELLING CHARACTERIZATION OF CHEMICALLY STRENGTHENED ALKALI SILICATE GLASSES ...... 52 A. Introduction...... 52
iv 1. Background...... 52 2. Dimensional Changes and Models ...... 55 B. Method ...... 59 C. Results ...... 67 1. Laboratory Dimensional Changes ...... 67 2. Laboratory Chemical Diffusion and Stress Profiles ...... 71 3. Laboratory Combined Results ...... 78 4. Finite Element Method Elastic Dimensional Changes ...... 82 5. Strain Model with Shear Flow (SPS model) ...... 86 6. Strain Model with Densification and Shear Flow (DSPS model) ...... 93 D. Discussion ...... 97 1. General Dimensional Swelling ...... 97 2. Elastic Finite Element Method ...... 97 3. Strain Models ...... 100 E. Conclusions...... 103 F. References...... 105 IV. SUMMARY AND CONCLUSIONS ...... 110 V. FUTURE WORK ...... 112 APPENDIX ...... 114 A. MD Simulation Cutoff Distances ...... 114 B. MD Simulation Qn Distributions ...... 115 C. Surface Profile Notes ...... 116 1. Substrate Deformation ...... 116 2. Edge Profile Averaging ...... 117 D. Edge Profiles After Chemical Strengthening ...... 119 E. Stress Profiles After Chemical Strengthening ...... 121 F. Edge Profile Comparisons with Elastic FEM ...... 123 G. SPS Model Inputs and Outputs Tables ...... 128 H. DSPS Model Inputs and Outputs Tables ...... 131
v LIST OF TABLES
Page Table I. Alkali-Oxygen Coordination Number and Si-O-Si Bond Angles ...... 15
Table II. Host Glass Compositions ...... 18
Table III. Pair Lengths with Network Formers ...... 23
Table IV. Pair Lengths with Network Modifiers ...... 23
Table V. Coordination Number for Network Formers ...... 26
Table VI. Coordination Number for Network Modifiers...... 26
Table VII. Intratetrahedral Bond Angles ...... 28
Table VIII. Intertetrahedral and Torsion Angles ...... 29
Table IX. Elastic Properties ...... 34
Table X. Maximum Compressive Stress Magnitude from Various Sources ...... 42
Table XI. Approximate Glass Compositions and Properties ...... 60
Table XII. Chemical Strengthening Temperature and Time Parameters ...... 61
Table XIII. Summary of Purpose for Coupons in Each Set ...... 64
Table XIV. Grinding and Polishing Schedule for EPMA Samples ...... 67
Table XV. Cutoff Distances for All Simulations ...... 114
Table XVI. Cutoff Distances Specific to SLS Simulations ...... 114
Table XVII. Cutoff Distances Specific to SMAS Simulations ...... 114
Table XVIII. Qn Distributions for All Simulations ...... 115
Table XIX. SPS Model Inputs for SLS Series ...... 128 vi Table XX. SPS Model Outputs for SLS Series, Part I ...... 128
Table XXI. SPS Model Outputs for SLS Series, Part II ...... 129
Table XXII. SPS Model Inputs for SAS Series ...... 129
Table XXIII. SPS Model Outputs for SAS Series, Part I ...... 130
Table XXIV. SPS Model Outputs for SAS Series, Part II ...... 130
Table XXV. DSPS Model Inputs for SLS Series ...... 131
Table XXVI. DSPS Model Outputs for SLS Series, Part I ...... 131
Table XXVII. DSPS Model Outputs for SLS Series, Part II ...... 132
Table XXVIII. DSPS Model Inputs for SAS Series ...... 132
Table XXIX. DSPS Model Outputs for SAS Series, Part I ...... 133
Table XXX. DSPS Model Outputs for SAS Series, Part II ...... 133
vii LIST OF FIGURES
Page Figure 1. Molar volume for host, stuffed, and CEAM glasses for various alkali silicates. Molar volume error bars are within the symbol height...... 13
Figure 2. Linear network dilatation coefficients for potassium-stuffed silicate glasses. .. 14
Figure 3. Alkali-oxygen coordination distribution for the 23NS series glasses...... 15
Figure 4. Ring size distributions for (A) 9NS, (B) 16NS, and (C) 23NS series glasses. .. 16
Figure 5. System molar volume for host, stuffed, and CEAM configurations of all series. Error bars are less than the symbol height. 9NS, 16NS, and 23NS from Kreski, et al.12 ...... 20
Figure 6. System LNDC versus mean coordination number of network former(s) “A” by bridging oxygen “BO”...... 21
Figure 7. System LNDC versus percent change in alkali-oxygen coordination number between stuffed and host glasses...... 27
Figure 8. SLS series: histograms of Si-O-Si angles versus all torsion angles for (A) host, (B) stuffed, (C) difference (stuffed minus host)...... 30
Figure 9. SLS series: probability distribution difference (stuffed minus host) for (A) Si- O-Si angles and (B) all torsion angles...... 31
Figure 10. Primitive ring size distributions for (A) SLS series and (B) SMAS series. .... 32
Figure 11. System LNDC versus (A) Poisson’s ratio of host glass and (B) change in Poisson’s ratio after SAA...... 35
Figure 12. System pressure computed from NPT expansion and elastic properties after SAA versus observed NVT pressure after SAA. The solid line represents a 1:1 relationship...... 37
Figure 13. “Bond angle LNDC” versus mean coordination number of network former(s) “A” by bridging oxygen “BO”. Error bars are less than the symbol height and width...... 41
viii Figure 14. Example of a two-mechanism stress relaxation system, where each mechanism is responsible for half of the total stress. Stretched exponential relaxation with a stretching exponent of 0.5 has been assumed. The relaxation times for each of the mechanism are given in the figure...... 46
Figure 15. Diagram of an interior slice with associated axes and labeled dimensions. .... 53
Figure 16. Coupon selective-surface chemical strengthening, swelling, and measurement depictions: (A) paste-coated regions indicated by shaded blocks, (B) dimensional swelling after chemical strengthening where solid blue coloration indicates chemically strengthened surface, (C) arrow and eye indicating the profile or measurement line. Note, diffusion is in the z-direction and swelling and chemical diffusion gradients are not drawn to scale...... 63
Figure 17. Edge profiles after edge chemical strengthening determined by white light profilometer for (A) SLS 450 °C series and (B) SAS 450 °C series...... 69
Figure 18. Step height versus square-root of time after selective-surface chemical strengthening determined by white light profilometer for (A) SLS series and (B) SAS series...... 71
Figure 19. Chemical diffusion profiles as integrated x-ray counts after chemical strengthening determined by electron microprobe analysis for (A) SLS exchanged at 450 °C for 16 hours and (B) SAS exchanged at 450 °C for 6 hours...... 73
Figure 20. Average, normalized concentration integrand for potassium versus square-root of time for (A) SLS series and (B) SAS series...... 75
Figure 21. Stress profiles σyy(z) versus position from edge (z-position) after chemical strengthening determined by polarized light microscopy for (A) SLS exchanged at 450 °C for various times and (B) SAS exchanged at 450 °C for various times...... 77
Figure 22. Step height versus normalized concentration integrand for (A) SLS series and (B) SAS series. The collective slope is (A) 0.030 μm/μm and (B) 0.019 μm/μm...... 79
Figure 23. Elastic strain integrand versus normalized concentration integrand for (A) SLS series and (B) SAS series. The collective slope is (A) -5.6x10-3 μm/μm and (B) -8.2x10-3 μm/μm...... 81
ix Figure 24. Stress profile comparison in the plane strain dimension between laboratory measurement σyy(z) and FEM σxx(z) for SAS-450-4...... 82
Figure 25. Edge profile comparison between laboratory measurements and elastic FEM simulations for SAS-450 series at (A) 1 hour, (B) 4 hours, and (C) 16 hours.84
Figure 26. FEM step height as percent difference from laboratory measurement versus natural logarithm of time for SAS series. Lines are drawn to guide the eye. 86
tot pla Figure 27. SPS model: (A) average total strain xx and (B) average plastic strain xx versus natural logarithm of time for SAS series. Lines are drawn to guide the eye...... 87
Figure 28. SPS model: Mean across chemical strengthening times for average total strain tot pla xx and average plastic strain xx versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature...... 89
Figure 29. SPS model: Mean across chemical strengthening times for plastic-to-total strain ratio Rpla versus chemical strengthening temperature. Error bars represent the range observed for each temperature...... 90
Figure 30. SPS model: Mean across chemical strengthening times for maximum initial max stress xx versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature...... 92
Figure 31. DSPS model: Plastic-to-total strain ratios for SAS for (A) deviatoric Rpla-D and (B) hydrostatic Rpla-H components versus natural logarithm of time. Lines are drawn to guide the eye...... 94
Figure 32. DSPS model: Deviatoric Rpla-D and hydrostatic Rpla-H plastic-to-total strain ratios versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature. .. 96
Figure 33. Elastic FEM stress profiles σyy(z) along the thickness centerline versus z- position from the edge for SAS-450 series. Note, only the tensile portion of the profile is shown...... 100
Figure 34. Substrate deformation at various stages of preparation for a preliminary trial coupon, (A) hill side and (B) valley side...... 117
x Figure 35. Example of the edge profile averaging method for SAS-450-1...... 118
Figure 36. Edge profiles after edge chemical strengthening determined by white light profilometer for SLS series...... 119
Figure 37. Edge profiles after edge chemical strengthening determined by white light profilometer for SAS series...... 120
Figure 38. Stress profiles σyy(z) after chemical strengthening determined by polarized light microscopy for SLS series...... 121
Figure 39. Stress profiles σyy(z) after chemical strengthening determined by polarized light microscopy for SAS series...... 122
Figure 40. Edge swelling comparison for SAS-250-239 and SAS-300 series between laboratory measurement (solid) and elastic FEM (dashed)...... 123
Figure 41. Edge swelling comparison for SAS-350 series between laboratory measurement (solid) and elastic FEM (dashed)...... 124
Figure 42. Edge swelling comparison for SAS-400 series between laboratory measurement (solid) and elastic FEM (dashed)...... 125
Figure 43. Edge swelling comparison for SAS-450 series between laboratory measurement (solid) and elastic FEM (dashed)...... 126
Figure 44. Edge swelling comparison for SAS-500 series between laboratory measurement (solid) and elastic FEM (dashed)...... 127
xi ABSTRACT
Glass chemical strengthening is an enabling technology for smart phones, tablets, and other personal electronic devices providing both enhanced strength and abrasion resistance. These enhanced properties rely upon the glass structure that results from “stuffing” a larger alkali ion into a smaller host alkali site within the glass. Incompatible expansion between the stuffed glass layers and the underlying substrate generates beneficial surface compression. The structural changes that take place during chemical strengthening, which influence compression development and retention, and thus mechanical performance, are not well understood. The present study utilized two approaches to improve understanding of alkali stuffed glass structures: molecular dynamics simulations were used to examine structural changes in the initial stages of stuffing alkali accommodation and laboratory measurements of dimensional swelling were used to observe elastic-plastic processes associated with chemical strengthening. Molecular dynamics simulations of potassium stuffing in soda-lime silicate and sodium aluminosilicate glasses produced network dilatation similar to that expected from laboratory stress measurements of glasses chemically strengthened by specialty techniques. Volume expansion per quantity of stuffing ion was found to increase with a decrease in network cross-polymerization, which also trended with Poisson’s ratio. Selective-surface chemical strengthening was used to form edge swelling and step swelling arrangements from which dimensional changes were examined with an optical profilometer. A strain model incorporating elastic, deviatoric plastic (shear flow), and hydrostatic plastic (densification) contributions was evaluated using inputs of step swelling, stress profiles, chemical diffusion profiles, and maximum strain from molecular dynamics simulations. Plastic strain averaged throughout the chemically strengthened layers accounted for about 70% and 40% of the total strain for soda-lime silicate and sodium aluminosilicate, respectively. The ratio of deviatoric plastic strain to total strain was observed to increase with increasing chemical strengthening temperature. Maximum compressive stress produced by traditional chemical strengthening was found to be much lower than the total strain indicated. Early relaxation modes were proposed to bridge the gap between maximum compressive stress produced by specialty chemical strengthening techniques versus that obtained traditionally. These results indicate notable improvements to maximum compressive stress can potentially be achieved through prevention of the various forms of relaxation.
xii
I. INTRODUCTION
Glass chemical strengthening is the process in which alkali-containing glasses undergo strengthening by ion exchange. Although glass is intrinsically mechanically strong, all glass surfaces contain flaws that are introduced by handling, or even simple exposure to a non-vacuum atmosphere.1-3 In the presence of applied tension, these flaws act as stress concentrators as proposed by Inglis4 and improved with incorporation of energy balance criterion by Griffith.5,6 These stress concentrators produce local tensile stress magnitudes that can be up to several orders of magnitude higher than the macroscopically applied stress.7 As such, glass articles fail at relatively low tensile stress magnitude compared to the intrinsic strength of the material. Introduction of compressive stress into a glass surface can act as a barrier that the tensile stress concentrations must overcome before flaw propagation occurs. The most common methods of introducing compressive stress into glass surfaces are thermal tempering and chemical strengthening.3,7,8 Of these methods, thermal tempering is by-far the most common and economical process because quenching of glass through its glass transition range can often be achieved simply with air jets. The technique can generate a maximum surface compression magnitude of 140 MPa in fully-tempered, typical soda- lime silicate products and has a protective “case depth,” i.e. the depth of the compressive stress layer, of about one-fifth of the thickness of the glass article.9 The maximum compressive stress magnitude is limited by the thickness of the article, thermal expansion coefficient, thermal diffusivity, and elastic constants of the glass. Thus, thin-wall products cannot be readily strengthened. Further, the process reliance on quenching through the glass transition range causes notable optical distortion, which is nearly unavoidable, and creates limitations for complex geometries.9 Chemical strengthening, or ion-exchange strengthening, relies on exchange of larger alkali ions from an external source for smaller alkali ions present within the host glass.7 As such, the method is limited to alkali-containing glasses. The temperature and time of the process is managed such that alkali interdiffusion takes place to the desired degree and that the relaxation of beneficial compressive stresses is limited. Glass
1 chemical strengthening often generates surface compression of 300-1,000 MPa and case depths of 10-1,000 μm.7,9 Thus, clearly the compressive stress magnitude is an advantage over thermal tempering, but the limited case depth is a disadvantage. As chemical strengthening relies upon the interdiffusion of alkali ions, the process is relatively slow for an effective compressive stress depth to develop, taking on the order of at least 30 minutes, but often several hours or more, which increases the cost of the process considerably.9 Finally, the high magnitude of surface compression paired with the largely unaltered glass structure of chemically strengthened glass tends to have better abrasion resistance than thermally tempered glass. In-depth comparisons of various glass strengthening methods can be found elsewhere.7-10 The earliest work in ion-exchange strengthening was performed by Kistler11, Acloque and Tochon12, and Sendt13 around the year 1962. Throughout the 1960s and 1970s glasses for chemical strengthening were developed, intended for markets such as windshields for automobiles,14 windscreens for aircraft,15-17 lenses for safety eyewear,14 and high strength bottles.18-20 Of these, the only commonplace application that took hold at the time was windscreens for aircraft laminates where strengthened glass reduces abrasion resistance and improves strength in strike scenarios, such as against birds.21 In the 1980s and 1990s, chemically strengthened glass substrates were developed as platters for hard magnetic disk applications in data storage22-25 and continue to be used through the present time. The high degree to which the glass surface flatness can be prepared and maintained after strengthening is advantageous over aluminum, in addition to the strength enhancement. In the mid-1990s chemical strengthening processes were developed for use with pharmaceutical glasses, particularly cartridges for autoinjectors applications where use of glass is critical for drug stability and shelf-life, and where a strengthened glass significantly lowers the probability of glass failure upon activation of a potentially life-saving device.26 As of approximately 2005 to 2007, chemically strengthened glasses started to become widely introduced as protective cover glasses for consumer electronic devices27,28 such as cell phones, smart phones, tablets, etc., where high abrasion resistance, such as from incidental contact with common pocket items such as keys, and high impact resistance to drop events are desirable. The successful match of benefits of chemical
2 strengthening with the requirements of these electronic device markets has translated into commonplace integration of protective cover glasses into these devices. As of mid-2013, the leading manufacturer of glass substrate for chemical strengthening claimed to have their protective cover glass in 1.5 billion devices worldwide.29 As electronic touchscreen devices continue to become thinner, chemically strengthened protective cover glasses will remain relevant and useful for some time due to their low cost, relative to alternatives such as sapphire, and capability for thin substrate dimensions. The future of glass chemical strengthening technology will likely include energy-driven applications such as protective cover glass for solar applications and light-weighting of curtain wall lites, automotive windows, and perhaps a return to light-weight glass containers.30,31 Currently, the science of chemical strengthening is well-understood in terms of the kinetics of the process, but poorly understood regarding the compressive stress development.30 To further advance the science of glass chemical strengthening, a number of issues for understanding have been outlined by Varshneya30, including the observation of an interior tension maximum just beyond the neutral stress point, the presence of a subsurface compression maximum for some glass compositions, and the disparity of compressive stress predicted from the molar volumes of as-melted glasses to that observed experimentally. Of these, the most critical item in terms of advancing the science and technology is improving understanding of the inherent capacity for compressive stress development. As such, the purpose of the present set of investigations is to understand what dictates the maximum compressive stress magnitude generated by glass chemical strengthening. Since the compressive stress is dependent upon the elastic expansion of the glass network during chemical strengthening, elastic and plastic components of network expansion are of interest. The investigations that follow are divided into two sections, one approaching the topic using molecular dynamics simulations and the other using traditional laboratory measurements of dimensional swelling. Molecular dynamics simulations provide an unprecedented level of network structural information, which is highly valuable for understanding this relatively unique phenomenon in glass science. Laboratory study pairing dimensional swelling with the underlying chemical diffusion profiles, stress profiles, and finite element simulations allows estimates of elastic and plastic fractions of network expansion to be made.
3 Jointly, these separate approaches complement one another in terms of covering widely different length and time scales, but also generate points for comparison in terms of anticipated initial stress. The long-term intention of these studies is to aid the understanding and advancement of the ion-exchange network expansion process, so that future generations of glass compositions can be developed that generate and retain high magnitudes of surface compression.
4 References 1. C. R. Kurkjian, P. K. Gupta, and R. K. Brow, "The Strength of Silicate Glasses: What Do We Know, What Do We Need to Know?," Int. J. Appl. Glass Sci., 1 [1] 27-37 (2010).
2. C. R. Kurkjian, Strength of Inorganic Glass. Plenum Press, New York, 1985.
3. W. C. LaCourse, "Strength of Glass"; pp. 451-512 in Introduction to Glass Science. Edited by L. D. Pye, H. J. Stevens, and W. C. LaCourse. Plenum Press, Alfred, New York, 1970.
4. C. E. Inglis, "Stresses in a Plate Due to the Presence of Cracks and Sharp Corners," Trans. Roy. Inst. Naval Arch., 55, 219–41 (1913).
5. A. A. Griffith, "The Phenomena of Rupture and Flow in Solids," Philos. Trans. R. Soc. London, A, 221, 163-98 (1921).
6. A. A. Griffith, "Theory of Rupture"; pp. 54-63 in Proceedings of the First International Congress for Applied Mechanics: Delft. Edited by C. B. Biezeno and J. M. Burgers. J. Waltman, Jr. Press, Delft, 1924.
7. A. K. Varshneya, Fundamentals of Inorganic Glasses, 2nd ed. The Society of Glass Technology, Sheffield, 2006.
8. F. M. Ernsberger, "Techniques of Strengthening Glasses"; pp. 133-44 in Glass Science and Technology, Vol. 5, Elasticity and Strength in Glasses. Edited by D. R. Uhlmann and N. J. Kreidl. Academic Press, New York, 1980.
9. T. P. Seward and A. K. Varshneya, "Inorganic Glasses: Commercial Glass Families, Applications, and Manufacturing Methods"; pp. 8.1-8.173 in Handbook of Materials for Product Design. Edited by C. Harper. McGraw-Hill, New York, 2001.
10. S. Karlsson, B. Jonson, and C. Stålhandske, "The Technology of Chemical Glass Strengthening: A Review," Glass Technol.: Eur. J. Glass Sci. Technol., Part A, 51 [2] 41-54 (2010).
11. S. S. Kistler, "Stresses in Glass Produced by Nonuniform Exchange of Monovalent Ions," J. Am. Ceram. Soc., 45 [2] 59-68 (1962).
12. P. Acloque and J. Tochon, "Measurement of Mechanical Resistance of Glass after Reinforcement"; pp. 687-704 in Colloquium on Mechanical Strength of Glass and Ways of Improving It. Union Scientifique Continentale du Verre, Florence, Italy, 1961.
5
13. A. Sendt, "Ion Exchange and Diffusion Processes in Glass"; pp. 307-32 in Advances in Glass Technology: Technical Papers. Edited by International Congress on Glass. Plenum Press, New York, 1962.
14. M. B. W. Graham and A. L. Shuldiner, Corning and the Craft of Innovation; pp. 260-8. Oxford University Press, Oxford, 2001.
15. D. W. Rinehart, Pittsburgh Plate Glass Company, "Method of Strengthening a Glass Article by Ion Exchange," U.S. Pat. 3357876, December 1967.
16. D. W. Rinehart, PPG Industries, Inc., "Ion Exchange Strengthened Glass Containing Phosphorus Pentoxide," U.S. Pat. 4055703, October 1977.
17. D. W. Rinehart, PPG Industries, Inc., "Lithium Containing Ion Exchange Strengthened Glass," U.S. Pat. 4156755, May 1979.
18. J. P. Poole, H. C. Snyder, and M. A. Boschini, Brockway Glass Company, "Tripotassium Phosphate Treatment for Strengthening Glass," U.S. Pat. 3607172, September 1971.
19. J. P. Poole, H. C. Snyder, and M. A. Boschini, Brockway Glass Company, "Method of Strengthening Glass and Increasing the Scratch Resistance of the Surface Thereof," U.S. Pat. 3743491, July 1973.
20. N. Weber, Brockway Glass Company, "Strengthened Glass Article and Method of Producing Same," U.S. Pat. 3218220, November 1965.
21. R. Gy, "Ion Exchange for Glass Strengthening," Mater. Sci. Eng., B, 149 [2] 159- 65 (2008).
22. A. Lenhart and K. R. Hub, Siemens Aktiengesellschaft, "Disc-Shaped Carrier Body for a Recording Medium and Method for Manufacturing Same," U.S. Pat. US4803106 A, February 1989.
23. F. Rifqi, S. Koch, and D. Jousse, Saint-Gobain Vitrage S.A., "Glass Substrate of Specified Composition Which Has Been Polished and Reinforced by Surface Ion Exchange; Hard Magnetic Disks," U.S. Pat. US 5780371 A, July 1998.
24. B. Speit, Schott Glaswerke, "Chemically Prestressable Aluminosilicate Glass and Products Made Therefrom," U.S. Pat. US5895768 A, April 1999.
25. S. F. Starcke, J. D. Amundson, and D. H. Piltingsrud, International Business Machines Corporation, "Edge Strengthened Substrate of a Data Storage Disk and Method for Fabricating Same," U.S. Pat. US5733622 A, March 1998.
6 26. "Saxon Glass - Home" (2008) Saxon Glass Technologies, Inc. Accessed on: July 2014. Available at
27. D. W. Pogue, "Gorilla Glass, the Smartphone’s Unsung Hero" (2010) The New York Times. Accessed on: July 2014. Available at
28. B. Gardner, "Glass Works: How Corning Created the Ultrathin, Ultrastrong Material of the Future" (2012) Wired.com. Accessed on: July 2014. Available at
29. "Corning® Gorilla® Glass Now Found on More Than 1.5 Billion Devices" (2013) Corning Incorporated. Accessed on: July 2014. Available at
30. A. K. Varshneya, "Chemical Strengthening of Glass: Lessons Learned and yet to Be Learned," Int. J. Appl. Glass Sci., 1 [2] 131-42 (2010).
31. W. C. LaCourse, A. K. Varshneya, and D. Alderson, "Containers for the 21st Century: Opportunities for Lightweighting," J. Non-Cryst. Solids, 73 [1–3] 389- 94 (1985).
7
II. MOLECULAR DYNAMICS SIMULATIONS OF ALKALI STUFFED SILICATE GLASS NETWORKS
A. Introduction
1. Basic Principles Molecular dynamics (MD) simulation is a computer modeling technique in which a collection of particles interact with one another via potentials and Newton’s equations of motion.1,2 These potentials dictate the forces between particles and are combined with current position and velocity to compute position and velocity a short time later. This calculation is performed for all of the particles in a system, the particles are moved to their new positions; and the process is then repeated to generate the next configuration. System trajectory provides information both on structural configurations and dynamics, which allows the calculation of an array of properties from thermodynamic, such as energy and heat capacity, to spatial, such as radial distribution functions. This technique is particularly useful for glasses because the medium- and long- range disorder can cause limitations for traditional materials characterization techniques in their applicability or interpretation of results.3 MD simulation allows detailed inspection of structure and dynamics on sub-microsecond timescales, although more advanced techniques can be used to attempt to lengthen the effective timescale.4 Common properties of glasses studied by MD simulation range across length scales from pair distances and coordination number to bond angle distributions to network topology. Brief details on the MD simulation method, as it pertains to the present study, are given here. Further details can be found in text books by Allen and Tildesley1 and by Leach.2 There are a number of common algorithms for propagating the system configuration. The leap-frog algorithm is a variant of the Verlet algorithm in which the positions and velocities are computed alternately at each half-step. The positions are given by:1,2
8 1 rt t rt tvt t (1) 2
where r is the position vector, v is the velocity vector, t is time, and t is time step. The velocities are given by:1,2
1 1 vt t vt t tat (2) 2 2
where a is the acceleration vector. System boundary conditions can greatly influence the resulting particle interactions and evolution of a simulation. Use of periodic boundary conditions1,2 can allow a small collection of particles to mimic a much larger system. This boundary method allows each particle to have a periodic image. When a particle exits one boundary, its periodic image seamlessly re-enters from the opposing boundary. In this manner the number of the particles in the system is preserved. Periodic boundary conditions in three dimensions (cubic) are often used to study bulk properties of materials. Long-range forces can be critical to the behavior and properties of a simulation. In brief, the Ewald summation1,2 is a method that addresses this problem by introducing an infinite array of particle images about a simulation cell. Each particle is allowed to interact with itself and all other particles within the cell and the surrounding images. In this way, the total energy is modified to include these extended range interactions. This method is commonly employed when charged species are present. The ensemble dictates the conserved quantities during a simulation. Within a NVT simulation, for example, number of particles N, volume V, and temperature T are held constant. There are various methods to maintain a constant temperature for this ensemble. One example is the Berendsen thermostat.2 This approach makes use of an external heat bath held at a target temperature. Particle velocities are scaled to maintain a constant relationship between the rate of change of temperature with time and the temperature difference between the system and the heat bath. Use of this type of thermostat helps prevent oscillation about the target temperature. Similarly, the
9 Berendsen approach can also be used to control system pressure.2 The temperature bath is replaced by a pressure bath, which is held at a target pressure. In this instance, rather than particle velocities, the system volume is scaled to maintain a constant relationship between the rate of change of pressure with time and the pressure difference between the system and the pressure bath.
2. Background During glass chemical strengthening, concurrent with the well-understood process of alkali interdiffusion,5 when a larger stuffing alkali ion occupies an alkali site formed by a smaller host alkali ion, the site expands. This “chemical expansion” is analogous to thermal expansion and is given by a linear network dilatation coefficient, abbreviated “LNDC,” represented by B(z):6
1 lnV z Bz 3 Cz (3) where V(z) is molar volume, or system volume, and C(z) is concentration of stuffing ion. Stress resulting from chemical stuffing with diffusion along the z-dimension is given as follows:7
zz z 0 (4)
EBzCz E H z z B z C z dz (5) xx yy 1 H1 0 where stress is a function of z-position, E is Young’s modulus, ν is Poisson’s ratio, and H is the substrate thickness in the diffusion direction. This assumes biaxial plane strain in the x-y plane, and compressive stresses are treated as negative (-) in sign and tensile stresses are treated as positive (+) in sign. A visual reference is given in Figure 15 within Section III. Traditionally, the LNDC has been studied in terms of molar volumes of as-melted glasses.8-11 For example, a pair of as-melted glasses could be a sodium silicate “host” and the potassium silicate with equal alkali concentration (compositionally-equivalent,
10 as-melted or CEAM). The LNDC predicted from as-melted molar volumes often over- predicts that observed by chemical strengthening by a factor of three to four,8-10 or in terms of stress a typical soda-lime silicate glass may have maximum surface compression of 500 to 700 MPa after potassium chemical strengthening, whereas that expected from as-melted molar volumes would be about 2,500 MPa. Various researchers8-10 have concluded that ion-exchange “stuffed” glasses are derivative of the host glass and have notably lower molar volume than their corresponding CEAM glasses. Altered elastic properties due to chemical strengthening are another method in which the observed stress can be modified and studies related to this are covered in Section III. Previous studies of stuffing alkali accommodation (SAA) in silicate networks by molecular dynamics (MD) simulations revealed lower molar volume for the stuffed glass than its CEAM glass for three binary sodium silicate glasses,12 various mixed sodium- potassium silicates,13 and various mixed sodium-potassium aluminosilicates,14,15 where potassium was the stuffing ion in all cases. Network structural features preventing the stuffed glass from achieving the volume of its CEAM-equivalent have been attributed in one case to topological confinement12 and in another case to under-coordination of stuffing alkali by oxygen.13 Note, care must be taken to distinguish between studies of potassium stuffing in sodium silicate versus potassium stuffing in mixed-alkali silicates, which may already contain some potassium. General features of SAA are observed to be similar between these two types of studies, but specific details of SAA may vary because of differences in the structural features of binary sodium silicate and mixed sodium- potassium silicate hosts. MD simulations have also been utilized to study stuffing, followed by reverse-exchange, i.e. removing the stuffing potassium ions and replacing them with the host sodium ions. Network changes induced by stuffing were found to be reversible after restoring the original host alkali ions, at least for the MD time scales of picoseconds to nanoseconds.12,14 The present study utilizes molecular dynamics simulations to examine stuffing alkali accommodation in glasses of commercial interest: one soda-lime silicate and one sodium aluminosilicate. Analysis of previously-studied binary sodium silicates is extended for a number of properties, particularly in discrimination of bridging and non- bridging oxygen. Five glass compositions with varied network structures are studied,
11 enabling structural features that highly influence the LNDC magnitude to be noted and discussed.
3. Foundation The molecular dynamics simulation study that follows expands and builds upon the directly-related initial study of potassium stuffing in sodium silicate glasses by the same technique published as P. K. Kreski, A. K. Varshneya, and A. N. Cormack, "Investigation of Ion-Exchange ‘Stuffed’ Glass Structures by Molecular Dynamics Simulation," J. Non-Cryst. Solids, 358 [24] 3539-45 (2012). Key findings of the referenced study were briefly covered in the background section and are summarized in greater detail within this section. Full details can be found in the referenced publication.
Three binary sodium silicate compositions of 9, 16, and 23 mol% Na2O, referred to here as 9NS, 16NS, and 23NS, respectively, were formed along with their CEAM compositions via MD simulation. The sodium-containing glasses, or “hosts”, were stuffed with potassium at 25, 50, 75, and 100% of the total alkali concentration by re- labeling random alkali ions from sodium to potassium. Following a 200 ps relaxation under NPT, P = 0, T = 300 K or under NVT, T = 300 K ensemble, the structural properties of the resulting system were compared to those of the host and CEAM configurations. For the 23NS system, 100% potassium-stuffing at 623 K was also performed and, after returning the stuffed system to 300 K, the resulting structural properties were compared with the host and CEAM configurations. For each of the three glass compositions, the molar volume of the NPT stuffed glass was found to increase approximately linearly with increasing concentration of stuffing potassium and the 100% stuffed glasses had molar volume that was intermediate of the host and CEAM glasses (Figure 1). Using Eq. (3), the LNDC for each of the stuffed glasses was computed (Figure 2). The LNDC was observed to increase with increasing alkali concentration in the host glass (9NS < 16NS < 23 NS). Further, the LNDC of each series displayed a weak to mild stuffing potassium concentration- dependence. These differences between the LNDC magnitude and the LNDC trends among the three binary silicate compositions were likely related to the varied network structure for these glass compositions, which is explored in greater detail in the following manuscript. The LNDC for 100% stuffed 23NS glass suggested a resultant biaxial stress 12 when constrained to an underlying glass substrate of approximately -2.2 GPa, which was substantially higher than that observed for laboratory potassium stuffing of 23NS (about -350 MPa). At the time, this discrepancy was considered attributable to the NPT boundary conditions utilized for the simulation, which differed from the laboratory glass, where the laboratory glass is effectively NVT in-plane and NPT out-of-plane.
Figure 1. Molar volume for host, stuffed, and CEAM glasses for various alkali silicates. Molar volume error bars are within the symbol height.
13
Figure 2. Linear network dilatation coefficients for potassium-stuffed silicate glasses.
Upon stuffing alkali accommodation (SAA), the three glass compositions had some common structural responses. First, the (K)O coordination number (CN), where K is the central atom, was similar to or slightly greater than that of the CEAM glass (examples are given for the 23NS series in Figure 3 and Table I). This suggested the local alkali environment within the stuffed glass was much closer to that of the CEAM glass than of the host, at least in terms of number of neighboring ions. The mean intertetrahedral (Si-O-Si) bond angle increased upon SAA for all compositions as well, often exceeding that of the respective CEAM glasses at 100% stuffing (an example is given for the 23NS series in Table I).
14
Figure 3. Alkali-oxygen coordination distribution for the 23NS series glasses.
Table I. Alkali-Oxygen Coordination Number and Si-O-Si Bond Angles Si-O-Si (Na)O (K)O Simulation Mean Bond Angle Mean CN Mean CN (degrees) 23NS Host 5.8 ± 1.2 - 151.3 23KS CEAM - 7.4 ± 1.7 152.2 23NS Stuffed 25% 5.5 ± 1.2 8.0 ± 1.5 151.7 23NS Stuffed 50% 5.7 ± 1.3 8.4 ± 1.6 152.2 23NS Stuffed 75% 5.5 ± 1.1 8.4 ± 1.7 152.7 23NS Stuffed 100% - 7.5 ± 1.6 153.1
In terms of network connectivity, the Qn distributions were not substantially altered by SAA for all of the 100% stuffed glasses, relative to their hosts. Examining network topology, the ring size distributions for the 100% stuffed glasses remained notably closer to that of the host glass after SAA (Figure 4). This is more evident for 16NS and 23NS, since the ring size distributions of the host and the CEAM glasses show larger differences for those glasses.
15 (A)
(B)
(C)
Figure 4. Ring size distributions for (A) 9NS, (B) 16NS, and (C) 23NS series glasses.
16
Collectively, the results suggested the potassium SAA process in silicate glass involves expansion of the host alkali site, during which time the mean (K)O CN increases from about 6 to about 8 through site expansion and perhaps oxygen atoms also being drawn to the site. Displacement of the oxygen atoms is enabled, in part, by Si-O-Si bond angle changes. The SAA process lacks the activation energy, both in thermal energy and in strain energy, required to break and reform Si-O bonds, at least for the temperatures and time scales within the study. This leaves the Qn distributions and ring size distributions very similar to the host glass. The overall stuffed glass structures have local features about the stuffing potassium ion similar to the CEAM glass, but network connectivity and topology similar to the host glass. It was proposed, as a consequence, that the achievable molar volume by alkali stuffing was limited, at least in part, by topological confinement. With increased temperature and/or time, it was suggested the stuffed glass structure would evolve toward that of the CEAM glass, such that the volume of the stuffed glass would eventually meet that of the CEAM glass. The initial molar volume resulting from SAA is highly important, as it provides a limit to the maximum achievable surface compression. Any Si-O bond breaking and reformation involved with the structural evolution might increase the molar volume, but in the presence of a constraining substrate would result in relaxation of the initial compressive stress state. The study in the present manuscript resumes where the prior investigation left off, expanding the gamut of examined properties and adding two additional glass compositions of industrial interest, which also complement the laboratory study in Section III.
B. Simulation and Analysis Methods A simple soda-lime silicate (SLS) glass and a multi-component sodium magnesium aluminosilicate glass (SMAS) served as the host glasses in this study. Compositions studied are given in Table II. In addition to the host configuration, one stuffed configuration and one CEAM configuration were studied for each system. Simulation details largely follow those used previously.12 Glasses were initially formed by random placement of 6,000 (SLS) or 6,001 (SMAS) atoms in a cubic simulation box
17 with length selected to target molar volume of laboratory glasses of the same composition. Laboratory molar volumes were estimated from the mean of seven to ten SciGlass database entries16 found to be obtained under suitable laboratory conditions. Simulations were conducted using the DL_POLY software package17,18 with cubic periodic boundary conditions and using potentials developed by Pedone, et al.19 Host configurations were formed by propagating the system under NVT (T = 6,000 K) ensemble for 40 ps, followed by cooling to 300 K at 5 K/ps in intervals of 0.01 K/time step. A relaxation was performed at under NPT (P = 0, T = 300 K) ensemble for 200 ps to generate the “host” configuration (containing sodium) or “CEAM” configuration (containing potassium). For all simulations, the leap-frog algorithm was utilized, the time step increment was 2 fs, electrostatic interactions were calculated by the Ewald sum technique, the van der Waals cutoff was 5.5 Å, and the cutoff for all other forces was 12 Å. For NPT simulations, the Berendsen NPT ensemble was used with relaxation times of 0.4 ps and 0.369 ps for the thermostat and the barostat, respectively. For NVT simulations, a Berendsen thermostat was used with relaxation constant of 0.4 ps.
Table II. Host Glass Compositions Concentration
Component (mol%) SLS SMAS
SiO2 75.0 68.4
Al2O3 - 10.5 MgO - 5.3 CaO 10.0 -
Na2O 15.0 13.4
K2O - 2.4
“Stuffed” configurations were formed beginning with the configuration at the end of the “host” relaxation step, relabeling all Na ions to K ions. The system was then propagated under NPT (P = 0, T = 300 K) ensemble for 200 ps. Chemical potential gradients or interdiffusion processes were not of interest and were largely avoided by this direct ion substitution method.
18 New analyses are included for the x Na2O · (100-x) SiO2 (mol%), x = 9 (9NS), 16 (16NS), 23 (23NS), simulation series studied previously.12 Ion-exchange stuffed configurations for the NS series will be limited to 100% exchanged simulations here. Simulation naming convention is the host glass label and the state (host, CEAM, or stuffed). For example, the stuffed configuration for 16 Na2O ∙ 84 SiO2 is simply referred to as “16NS stuffed.” Properties of ion-exchanged stuffed simulated glasses were compared to host and CEAM simulated glasses. All properties were determined from the final 20 ps of each simulation, sampled at 0.1 ps intervals. Analyses of pair lengths, coordination numbers, bridging-oxygen fraction, bond angle, torsion angle, Qn distributions, and ring size distributions were calculated using a cutoff length corresponding to the first minimum after the first coordination sphere in the pair distribution functions. For silicon-oxygen pairs and aluminum-oxygen pairs, this corresponded to a cutoff length of 2.20 Å and 2.40 Å, respectively, for all simulations. For alkali- and alkaline-earth-oxygen pairs, the cutoff length was dependent upon the particular species and simulation configuration. These cutoff lengths are listed in Appendix A. For all simulations, oxygen was examined as bridging oxygen (BO) and non-bridging oxygen (NBO) based upon the number of Si and/or Al atoms present within the first coordination sphere. Elastic properties were determined using GULP software package20 version 3.4, first performing an energy minimization under NPT, P = 0, T = 300 K conditions, followed by computation of the various elastic moduli from elastic constants obtained from the second derivatives of the energy density with respect to strain. Results are presented beginning with molar volume and LNDC, after which the order of presentation is by increasing length scale using the convention of Wright.21 Error bars and error estimates are representative of one standard deviation of the population for quantities directly from simulation, such as coordination numbers, or representative of the propagated error for derived values, such as LNDC, unless otherwise noted.
19 B. Results
1. Molar Volume and LNDC Molar volume of 100% potassium stuffed, NPT relaxed glasses for SLS and SMAS is intermediate of that of the host and CEAM end-members (Figure 5). In each case, stuffed molar volume is midway within the range formed between the end- members. This observation is consistent with results of other MD studies of 100% potassium stuffing in NS glass12 (also shown in Figure 5) and sodium aluminosilicate.15
Figure 5. System molar volume for host, stuffed, and CEAM configurations of all series. Error bars are less than the symbol height. 9NS, 16NS, and 23NS from Kreski, et al.12
The system LNDC, computed from Eq. (3) using host and stuffed system volumes and concentration of stuffing K2O, is displayed against mean (A)BO coordination number in Figure 6, where A represents Si for the NS series and the SLS series, and represents Si, Al, and/or Mg for the SMAS series. For networks based on Si alone, the system LNDC increases linearly with decreasing mean (A)BO CN. For SMAS, if Si and Al are considered in network-forming roles, it then falls into a similar trend to the other systems 20 examined. On the other hand, if Si, Al, and Mg are considered in network-forming roles, the SMAS system falls well outside the linear trend of the Si-only-based networks. Laboratory studies of chemical strengthening of generic SLS glass and glass with a similar composition to SMAS typically display maximum compressive stress values of about 550 MPa and 850 MPa, respectively (Appendix E). This equates to an elastic -4 -4 LNDC of approximately 4.4x10 and 6.9x10 per mol% K2O for SLS and SAS, respectively, of which the stuffed MD simulation LNDC values are between about 1.5 and 3 times greater. Thus, while the stuffed MD simulation produces a LNDC that is notably lower than that anticipated from difference of the host and CEAM end-members, the simulated-stuffed LNDC still exceeds the laboratory-determined elastic LNDC by a notable amount.
Figure 6. System LNDC versus mean coordination number of network former(s) “A” by bridging oxygen “BO”.
2. Range I: Structural Unit Mean pair lengths are compared between host, stuffed, and CEAM configurations, for structural pairs and modifying pairs in Table III and Table IV,
21 respectively. The mean was established by first taking the mean within each timestep examined, then taking the mean across the timestep means. The error represents one standard deviation of the mean across the timesteps. Pair lengths for network forming pairs are nearly identical among host, stuffed, and CEAM simulations (Table III). The similarly of pair lengths between host and CEAM configurations suggests there is little dependence of this feature on alkali identity. Further, observation of nearly identical values for the stuffed simulation implies the structural component pair lengths are largely unchanged upon stuffing alkali accommodation (SAA). Alkali-containing pair lengths show differences between host and CEAM configurations, where the CEAM simulation generally displays longer lengths (Table IV). Stuffed simulations for the NS series have similar pair lengths to the CEAM simulation for K-containing pairs, demonstrating the ability for these pair lengths to largely be satisfied after SAA. The SLS simulation, on the other hand, shows intermediate pair lengths for the same pairs, although the differences are largely within the standard deviations for the values. For the stuffed SMAS simulation, M-BO and M-M pairs have lengths that are near the CEAM simulation. For SLS and SMAS simulations, alkaline- earth containing pairs have few differences between host and CEAM simulations, and the stuffed pair lengths are nearly equivalent. Overall, the pair lengths showing the largest differences due to SAA are those of alkali-containing pairs and these pair lengths largely achieve a value quite near that of the CEAM glass, i.e. the local environments of the stuffed glasses have pair distances that are not unlike that of their CEAM counterparts.
22
. Pair Lengths with Modifiers Network Lengths Pair .
. Pair Lengths with Formers Network Lengths Pair .
III IV
Table Table
23
Mean coordination numbers (CN) are compared between host, stuffed, and CEAM configurations, for structural pairs and modifying pairs in Table V and Table VI, respectively. The central element is represented in parentheses. The (Si)BO and (Si)NBO CN are well defined, showing narrower distributions than other structural pairs (Table V). For the NS series, differences between host and CEAM values are quite small, with the stuffed system taking an intermediate value to the end members for 16NS. For 9NS and 23NS, the stuffed value is lower for (Si)BO and higher for (Si)NBO. The SLS and SMAS series show slightly larger CN differences between host and CEAM configurations, with the stuffed configuration retaining CN near the host value. The (Si)Si CN is nearly identical between host, CEAM, and stuffed configurations for all simulation series. The (BO)BO and (BO)NBO CN are similar for host and CEAM configurations for all series. The stuffed simulations generally show slightly lower (BO)BO CN and unchanged or very slight increases for (BO)NBO CN. These changes are small relative to their standard deviations and may not be noteworthy. For the SMAS series, the (Al)BO CN is lower for the CEAM glass than the host, and the stuffed value remains near the host. The (Al)NBO CN is quite low, indicating that nearly all of the NBO is preferentially associated with Si, as has been noted in literature.22,23 The (Al)Al CN is lower for the CEAM glass than the host, and the stuffed glass takes an intermediate value, showing greater flexibility than that observed for (Si)Si. The (M)O CN, where M represents Na or K, was examined previously for the NS series.12 Here, (M)O is deconvoluted into (M)BO and (M)NBO contributions (Table VI). For (M)BO, all compositions display a larger CN for the CEAM (M=K) simulation than the host (M=Na) simulation. Relative to their CEAM simulations, the stuffed simulations have K over-coordinated by approximately 0.6 BO for the NS series, under-coordinated by about 0.6 BO for SLS, and attain similar coordination for SMAS. (M)NBO is at least marginally different between the host and CEAM simulations for all glasses examined. The (M)NBO CN of the stuffed simulations is consistently very near that of their CEAM end-members, demonstrating the preference for proper coordination by alkali-bonded oxygen (NBO) over the network bridging oxygen (BO). Decreasing percent change in
24 (M)BO between host and stuffed configurations trends with increasing system LNDC, whereas the same is not evident for (M)NBO (Figure 7). The (M)M CN is consistently higher for the CEAM simulation than the host simulation for the NS series glasses and the stuffed simulations have values very near the CEAM simulation (Table VI). The SLS simulation series shows little difference between the host and CEAM (M)M CN, but the stuffed simulation has a value slightly under that of the host simulation. For the SMAS series, the host and CEAM (M)M CN differ by about one and the stuffed (M)M CN remains near the host. For SLS, the (Ca)BO CN is different between the host and CEAM configurations. The stuffed simulation has (Ca)BO CN equivalent to the CEAM simulation. (Ca)NBO for the same series is slightly lower for the CEAM simulation than the host, with the stuffed simulation showing slightly lower value than both end-members. For the SMAS series, (Mg)BO CN is slightly lower for the stuffed and CEAM configurations than the host. The same series has no discernible difference between (Mg)NBO CN between host, stuffed, and CEAM simulations. Network modifying pairs show larger variation between host and CEAM glasses than the network forming pairs. Network modifying pairs also have larger distribution widths as indicated by their larger standard deviations relative to the network forming pairs, indicating a greater variety of site configurations for the network modifying pairs. Note, while the majority of the CN alteration is most apparent about the alkali site, if weighted by the number of species in the system, then the minor CN differences between network forming pairs may bear additional significance, although the alkali- related changes remain dominant.
25
mers
r
. Coordination Number for Network Modifiers Network for Number Coordination .
. Coordination Number for Network Fo Network for Number Coordination .
V VI
Table Table
26
Figure 7. System LNDC versus percent change in alkali-oxygen coordination number between stuffed and host glasses.
The fraction of NBO relative to total oxygen has a maximum difference of 0.4% between the host, stuffed, and CEAM simulations for each series. Thus NBO populations are not observed to significantly change with SAA for the compositions, temperatures, and timescales studied here. Intratetrahedral bond angles with Si and Al as the central cations, and all BO and NBO considered collectively, show very minor differences between host, stuffed, and CEAM simulations for all series (Table VII). Intratetrahedral bond angles with Si have a slight broadening of the distribution width after SAA for most series. For SMAS, intratetrahedral bond angles with Al have a slight narrowing of the distribution width after SAA. On the whole, these changes are slight and do not allow a conclusion to be confidently drawn.
27 Table VII. Intratetrahedral Bond Angles
3. Range II: Interconnection of Adjacent Structural Units Consistent with previous observations,12 all simulation sets display higher intertetrahedral (A-O-A or βo) bond angle upon SAA (Table VIII). The values consistently exceed that of their CEAM simulations. Changes in the mean of a few degrees, despite the width of the A-O-A distributions, are significant. For example, MD simulations of silica in uniaxial tension showed24 a Si-O-Si bond angle change of about 0.4 °/GPa and separate MD simulations of silica under hydrostatic compression observed25 a Si-O-Si bond angle change of about -1.1 °/GPa. The A-O-A distribution width also generally narrows after SAA, with the exception of mixed Si-O-Al bridges. Intertetrahedral changes clearly have a prominent role in SAA.
The two smallest torsion angles (labeled α1 and α2) are examined across each BO for end-member and stuffed simulations (Table VIII). For Si-O-Si, Si-O-Al, and Al-O-Al bridges, a general trend of slight decrease in the smallest torsion angle, α1, relative to the host is observed after SAA. This value also moves away from the CEAM value. Slight narrowing of the α1 distribution is observed in most cases after SAA as well, but the 28 change is quite small, when it is present. The second smallest torsion angle, α2, does not show a consistent trend across the various simulation series after SAA, other than when
α2 increases, its width increases and when α2 decreases, its width decreases. This trend does not hold for the SMAS series.
Table VIII. Intertetrahedral and Torsion Angles
Histograms of Si-O-Si angles versus all torsion angles do not show a readily identifiable difference between the SLS host (Figure 8A) and stuffed (Figure 8B) glasses, although their difference (stuffed minus host, Figure 8C) shows a general tendency for Si-O-Si population decreases for angles less than about 145° and population increases for angles greater than 145°. This is shown more clearly in one-dimensional histogram of the Si-O-Si distribution difference (Figure 9A). Torsion angle distribution difference, on the other hand, does not display a consistent trend (Figure 9B).
29 (A)
(B)
(C)
Figure 8. SLS series: histograms of Si-O-Si angles versus all torsion angles for (A) host, (B) stuffed, (C) difference (stuffed minus host).
30 (A)
(B)
Figure 9. SLS series: probability distribution difference (stuffed minus host) for (A) Si-O-Si angles and (B) all torsion angles.
The Qn distribution is nearly completely unchanged after SAA for all simulation series. Further, the Qn distribution differences between the host and CEAM configurations are generally less than 1%. These observations are consistent with previous results.12-14 Tabulated results are available in Appendix B.
4. Range III: Network Topology The primitive ring size distributions (RSD) of SLS and SMAS series glasses are shown in Figure 10. The host and CEAM configurations show distinct distributions, although this is considerably less evident for the SMAS series. Similar to previous observations for the NS-series glasses,12 the RSD of the stuffed SLS and stuffed SMAS glasses remains similar to the host after SAA.
31 (A)
(B)
Figure 10. Primitive ring size distributions for (A) SLS series and (B) SMAS series.
5. Elastic Properties Young’s modulus and Poisson’s ratio were taken as means over the three principal axes and are presented in Table IX. Across the various series, the CEAM
32 configurations have lower Young's modulus than the host configuration. There is no consistent change in Young's modulus after SAA among the series, although the stuffed Young's modulus tends to stay near the value of the host. Elastic moduli are primarily dictated by the interatomic bonding energy and atomic packing density.26 During SAA, Na has been replaced by K, which would presumably be less strongly bonded within the network, but the atomic packing density is also quite a bit higher than in the CEAM glass (Figure 5). These factors would tend to drive the Young’s modulus in opposite directions and it is not clear which, if either, factor would be dominant. Note, the lack of consistent change of Young’s modulus after SAA differs from that observed in literature. For sodium stuffing in lithium aluminosilicates, Young’s modulus increases of up to 15% have been observed by fiber-bending technique27 and by pulse-echo technique.28,29 MD simulation of potassium SAA under NPT conditions in mixed-alkali aluminosilicates, where Young’s modulus was determined from stress-strain curves in uniaxial tension, had observed Young’s modulus increases of up to about 6%.15 As the five glass compositions studied here differ from those found in literature, it is unclear whether increased Young’s modulus is a feature common to all ion-exchange stuffing processes, or whether there is dependence on the host composition. Poisson's ratio is generally similar between the host and CEAM simulations across the series, with the exception of the SMAS series. The system LNDC scales linearly with increasing Poisson’s ratio of the host glass (Figure 11A). All series show an increase in Poisson's ratio relative to the host, by 0.01-0.04 (~4-22%), after SAA. This is consistent with laboratory measurements of Poisson’s ratio by pulse-echo technique after sodium stuffing in lithium aluminosilicates.28,29 The system LNDC also scales linearly with decreasing change in Poisson’s ratio after SAA (Figure 11B), although the error associated with this trend is somewhat large. Both of the aforementioned trends apply to Si-based networks. Allowing Poisson’s ratio to be representative of atomic packing density,26,30,31 the following is suggested: (1) all glass systems examined have an increase in atomic packing density after SAA and (2) host networks with higher atomic packing density undergo less consumption of free volume upon SAA and, as a result, exhibit larger LNDC.
33 Table IX. Elastic Properties
34
(A)
(B)
Figure 11. System LNDC versus (A) Poisson’s ratio of host glass and (B) change in Poisson’s ratio after SAA.
35 6. NVT Stuffing An alternative set of boundary conditions to employ during MD simulation of SAA is that of fixed volume (NVT). In this case, the volume of the host is maintained and a pressure is exerted on the walls of the simulated box in response to SAA. As reported previously, after SAA this generates about 0.9, 2.0, and 3.3 GPa for 9NS, 16NS, and 23NS, respectively.12 For SLS and SMAS, these boundary conditions generate about 2.4 and 2.2 GPa, respectively. With known LNDC from the NPT SAA simulations (Figure 6), and with elastic properties of those systems (Table IX), the equivalent pressure P can be estimated:
E P 3BCK BC 1 2 (6) where B is the LNDC, C is the concentration of stuffing ion, E is Young’s modulus, and ν is Poisson’s ratio. Comparing the observed NVT pressure to computed NPT pressure, a nearly 1:1 relationship is observed (Figure 12), i.e. the stuffed glasses exhibit linear elastic behavior in compression. A full analysis of NVT SAA is not undertaken here, but it is noted that the intertetrahedral bond angle decreases under these NVT SAA conditions, opposite to that observed for NPT SAA conditions (Table VIII).
36
Figure 12. System pressure computed from NPT expansion and elastic properties after SAA versus observed NVT pressure after SAA. The solid line represents a 1:1 relationship.
C. Discussion
1. A Note on Timescale, Temperature, and Boundary Conditions MD simulations, at present, are inherently limited to very short time scales relative to those of typical laboratory measurements. This difference in time of observation is approximately 10 orders of magnitude. Therefore, the structural features noted here by SAA are limited to the earliest stages, if not the “initial” stage, of the SAA process. Taking advantage of the short-time limitation, a goal of this study is to examine the initial stages of SAA. To improve the probability of discriminating structural features inherent to initial SAA from thermal relaxation, a temperature of 300 K was chosen, rather than a typical chemical strengthening temperature of, say, 650-700 K. Earlier studies of SAA by MD simulation utilized,13 or at least examined,12 SAA within the slightly elevated temperature range with little difference in the resulting structural properties relative to those observed in this study. Note, neither NPT nor NVT simulation boundary conditions replicate those of the laboratory glasses, which are effectively NVT in-plane (within the diffusion plane) and 37 NPT out-of-plane, in the diffusion direction. Thus property changes observed with MD simulation of SAA are taken as approximations to that of the laboratory SAA process.
2. What Influences LNDC? Investigation of the system LNDC with varied host composition allows identification of network features that significantly influence the LNDC magnitude. For the present set of simulated glasses, the LNDC magnitude is strongly correlated with the mean number of BO per network forming cation (Figure 6) and with Poisson’s ratio of the host (Figure 11A). Lesser, but identifiable, correlations are also noted for increasing LNDC magnitude with decreasing changes of alkali-bridging oxygen coordination number after SAA (Figure 7) and with decreasing Poisson’s ratio change after SAA (Figure 11B). Based upon these observations, the LNDC magnitude is highly related to the degree of network cross-polymerization, as both the number of BO per network- forming cation and Poisson’s ratio are well-recognized as representative of degree of network polymerization or network dimensionality.26,31 The LNDC magnitude also has a relationship to the alkali site similarity between the host and the stuffed glass. Alkali sites that undergo less re-arrangement in terms of coordination by BO exhibit greater net expansion. This is presumably due to drawing BO to an alkali site as part of SAA is counter to site expansion, thus less net volume expansion is realized. The preceding discussion primarily applies to Si-based glasses. SMAS loosely falls within some of these observed trends (Figure 6), depending on whether Si and Al or Si, Al, and Mg are considered network formers, but has notable departures in other trends (Figure 7 and Figure 11). The source of these differences in SAA by Si-based glasses and SMAS glass is related to differences in the alkali site construction in these networks and is discussed in detail in the next sub-section.
3. Network Features of Stuffing Alkali Accommodation Common structural features of NPT simulation of SAA include nearest-neighbor pair lengths that are near that of the CEAM, (K)NBO CN similar to the CEAM and (K)BO near the CEAM (with slight over-coordination or under-coordination), no change in NBO concentration, and little change of intratetrahedral bond angle. At intermediate length scales common SAA features include consistent increases of intertetrahedral bond
38 angle and no change of Qn distribution. On topological scales, the ring size distribution after SAA remains similar to the host and in terms of macroscopic properties, Poisson’s ratio increases after SAA. Overall, the local response of the network is immediate and relatively complete, i.e. K-O pair lengths, (K)BO and (K)NBO CN, etc. are quite near that of the CEAM glass. Flexible second-neighbor network changes, namely to intertetrahedral angles, are also immediate, but arrive at transient positions that differ from the CEAM configuration. Finally, features tied to network bonding and topology, including number of NBO, Qn distributions, and ring size distributions, are relatively unchanged by the initial SAA process. Changes to these latter features require re- bonding of network bridges, a process for which there is apparently insufficient activation energy, both thermal and from network strain, to be triggered in the present simulations. Thus the immediate stage of SAA produces a stuffed glass with local features similar to the CEAM glass, but with network connectivity and topology of the host glass. These two scales are bridged by the flexibility of the interterahedral bond angles. Despite some common structural responses after SAA, the SMAS series does not closely follow many of the system LNDC trends of the Si-only-based glasses. In particular, based on the trend from Si-based networks, the high degree of network polymerization of the SMAS glass would be expected to have lower system LNDC than that observed (Figure 6). On the other hand, again from the trend of Si-based networks, the small percent change of (M)BO upon SAA of SMAS would be expected to generate a large system LNDC (Figure 7). Thus there is a difference in the response of SMAS to SAA, compared to Si-based networks. A notable difference between SMAS and the other glasses studied presently is the construction of the alkali sites. Within Si-based networks, the alkali sites are of the traditional NBO-type. Simply viewed, each alkali ion within the silicate network is bonded to one oxygen, breaking a network bridge. In practice, the actual charge compensation is more complicated and is described by the modified random network model.32 Alkali sites vary, with average CN of (M)NBO ranging from 2 to 3 and a number of BO are situated at a slightly greater distance for further charge compensation.33,34 In the case of SMAS, the presence of Al within the alkali silicate
39 - network forms the well-known [AlO4] tetrahedron/complex which is charge balanced by a local alkali ion,3 or by several shared alkali ions – here SMAS after SAA has (Al)K CN of about 3. For this particular glass, the composition is such that about 66% of the alkali ions are occupied in this manner. As such, the alkali ion is surrounded by a greater ratio of BO to NBO, about 2.5 times greater than in the 16NS system, allowing BO to play a larger role in the change of the alkali site during SAA. The LNDC magnitude is lower than anticipated based upon the small change of K(BO), but the LNDC magnitude is near that of 16NS, despite the high degree of network polymerization. Intertetrahedral bond angles (Si-O-Si, Si-O-Al, an Al-O-Al) for SMAS after SAA show larger changes than that of 16NS (Si-O-Si). Examination of “bond angle LNDC,” by treating host and stuffed intertetrahedral bond angles similar to system volumes in Eq. (3), shows a nearly constant value for the Si-based networks, but much higher bond angle change per mol% stuffing K2O for the SMAS glass (Figure 13). Note, the majority of the SMAS network bridges are of type Si-O-Si and Si-O-Al, and -4 their weighted mean bond angle LNDC is 3.2x10 per mol% K2O, which is 1.8 times greater than that of the NS and SLS glasses. The drastic increase in the intertetrahedral bond angles for SMAS with SAA demonstrates that these bond angles are likely critical in allowing the system expansion to be realized, but are also potentially responsible for limiting the maximum achievable system expansion. This mechanism of restraint by intertetrahedral bond angles may be related to the non-linear system LNDC observed versus concentration of stuffing K2O for other high alkali aluminosilicates by MD simulation,15 but this is left to future investigation.
40
Figure 13. “Bond angle LNDC” versus mean coordination number of network former(s) “A” by bridging oxygen “BO”. Error bars are less than the symbol height and width.
4. LNDC and Compression Maximum SLS and sodium aluminosilicate glasses, similar to SMAS, have been widely studied in literature, and are examined in detail for their compression development and relaxation characteristics in Section III. A comparison of maximum compressive stress magnitude (MCSM) from various sources is given in Table X. For SLS, from this MD study, the MCSM assumes 15 mol% stuffing K2O per the simulated composition, whereas most commercial SLS compositions have closer to 14 mol% stuffing K2O. If a 14 mol% is assumed, the SLS MCSM from this study is reduced to 1,560 MPa. With this allowance, the MD-determined MCSM compares favorably with the stress-optical measurements of Ohta and Hara35 and of Shaisha and Cooper,36 where each study utilized electric field-assist chemical strengthening to observe a sub-surface compression maximum. In light of the present study, the advantage of this chemical strengthening technique is the diffusion front is fully exchanged and is driven into the glass. The leading edge of this front represents a fully chemically strengthened layer which has had essentially no time to relax and likely represents the highest achievable MCSM less any 41 instantaneous relaxation mechanisms, which would presumably be observable within the time window of the present MD simulations. Relative to the SPS model and traditional chemical strengthening from Section III, the MD-predicted MSCM for SLS is considerably higher. In the former case, this is attributed to systematic under-estimation of MCSM by the SPS model, as the model did not consider densification as is detailed in Section III. In the latter case, traditional chemical strengthening profiles have a fully- exchanged surface layer that is maintained in a high compressive state for at least minutes, if not hours, during which time relaxation, thermally-activated or otherwise, takes place. For SMAS, the most comparable laboratory MCSM measurement is that of Svenson, et al.,37 involving a similar aluminosilicate composition to that simulated here, although the glass was hydrostatically compressed at 1 GPa at 600 °C, then cooled under compression to maintain the densified structure. The resulting specimen, when chemically strengthened, showed about 200 MPa greater surface compression than that of the non-compressed glass, as determined by optical ellipsometry. Again, the MD simulation, while showing better agreement for SMAS than for SLS, has higher MCSM than the SPS model and traditional chemical strengthening from Section III.
Table X. Maximum Compressive Stress Magnitude from Various Sources Maximum Compressive Stress Magnitude (MCSM)
Observed, or Computed from LNDC (MPa, biaxial) Traditional Section III Glass This MD Study Literature Chemical “SPS” Model Strengthening^ 1,125 with SLS 1,671±35 range about 1,400† about 550 900 to 1,300 1,100 with SMAS 1,413±92 range 1,240‡ about 850 800 to 1,400 †SLS-type, electric field-assist chemical strengthening35,36 ‡SMAS-type, compaction prior to chemical strengthening37 ^ from Section III, stress-birefringence measurements
42 One additional point for consideration regarding comparisons of chemical strengthening that has been performed at elevated temperature is that of thermal contraction. Tyagi and Varshneya28,29 noted that in-situ measurements of stress- birefringence exhibited about 20% lower compression magnitude at room temperature than at the chemical strengthening temperature. If the literature values in Table X are revised by a comparable percentage, the resulting agreement with MD-predicted MCSM is quite good for both SLS and SMAS. A major point of interest is why traditional chemical strengthening produces much lower surface compression than that predicted by MD simulation and observed by specialty chemical strengthening techniques. Often, even short traditional chemical strengthening times, for example 15 minutes, do not exhibit notable surface compression increases over that at one or two hours, i.e. traditional chemical strengthening roughly shows a plateau of surface compressive stress versus increasing time for low to moderate chemical strengthening temperatures. This is especially evident for the SAS glass studied in Section III, which maintains a surface compression magnitude near 800 MPa for chemical strengthening temperatures of 400 °C and 450 °C and times out to 16 hours. It seems one or more relaxation mechanisms exist and are active within the time range after initial SAA (the period observed by MD simulation) and prior to surface compression measurement of traditionally chemically strengthened glass. For convenience, these will be referred to as early relaxation modes (ERM). They are posited to occur, but are not directly witnessed. Key features of the ERM are: (1) they are limited in their relaxation capacity (i.e. they do not fully relax the compressive stress state) and (2) they are mostly complete or are considerably slower by the time surface compression measurements are made. A likely component to the ERM with the aforementioned features is volume viscosity,38 which has been suggested briefly by Varshneya.3 Glasses have a capacity for volume relaxation. Under an isothermal hold within the glass transition range, i.e. annealing, glass will undergo both instantaneous and delayed volume changes to approach its equilibrium volume at that temperature, commonly known as structural relaxation.38 Often, glasses that are formed by fusion will contract during annealing because the finite cooling rate after fusion limits the volume relaxation. This capacity for
43 volume contraction during annealing implies the initial network has greater free volume than its annealed counterpart. If these two hypothetical glasses were to each be traditionally chemically strengthened, it may be presumed that the annealed glass will display greater surface compression for two reasons. First, the network with less free volume will have a higher LNDC, as supported by the present MD simulations (Figure 11A, allowing Poisson’s ratio to be representative of the atomic packing density). Second, the network with less free volume will have less capacity to relax stresses generated by chemical strengthening via consumption of free volume (permanent densification). A direct example of the influence of heat treatment upon the compressive stress resulting from traditional chemical strengthening is given in the experiments of Allen, et 39 al. Sub-Tg heat treatments of sodium aluminosilicate glass, similar to SMAS, prior to chemical strengthening produced surface compression increases of about 10% relative to as-formed glasses after potassium chemical strengthening. Further, the compressive stress increase over as-formed glass was nearly constant with increasing chemical strengthening time from 0.5 to 32 hours, suggesting the stress relaxation time over this period was not substantially altered. That is, the compressive stress improvement involves alteration of the initial compressive stress magnitude and is apparently largely decoupled from the subsequent relaxation occurring with increasing chemical strengthening time (at 0.5 hours and beyond). The compressive stress resulting from sub-
Tg heat treatment remains about 300 MPa to 500 MPa less than that predicted by MD simulation and observed by specialty chemical strengthening techniques (Table X). A likely reason is, at least in part, suggested by Allen, et al.39 that the heat treatment, “allow[s] for only a subset of the relaxation modes to be activated [during chemical strengthening].” The relaxation modes eliminated by heat treatment would be assigned to volume relaxation in light of the present study and only a fraction of these relaxation modes can be consumed during the heat treatment because of its elevated temperature, relative to the chemical strengthening temperature, and limited time. Further examination of the ERM hypothesis relative to the literature references in Table X, allows the following observations to be drawn. Electric field-assist chemical strengthening35,36 produces a sub-surface compression maximum in which the leading
44 edge of the compressive stress profile is nearly in a “time-zero” state. Instantaneous relaxation mechanisms will have taken place, but stress relaxation by ERM, and by other later time-dependent processes such as traditional shear flow, have not yet occurred. Densification of glass prior to chemical strengthening37 activates similar network relaxations that would occur during relaxation of chemical strengthening stress by the ERM, resulting in reduction of free volume and decreased capacity to release stresses by compaction. Thus, the resulting structure when chemically strengthened shows higher compression magnitude. A consequence of the reduced network free volume is that the interdiffusion kinetics are severely hindered.37 This experimental observation indirectly supports the increased atomic packing density. Note, in addition to volume relaxation, the ERM could potentially contain stress- dependent or strain rate-dependent relaxation under biaxial stress conditions. A high magnitude of compressive stress or rapid straining may promote certain relaxation modes until a lower compression magnitude or strain rate is reached, at which time those relaxation modes would no longer be active. While the ERM hypothesis may bridge observations of “time-zero” stress between MD simulation and other sources, it is apparently active outside of the 200 ps time window examined presently and/or is dependent upon the system boundary conditions (e.g. ability for application of external stress). This leaves a time span of about 12 orders of magnitude across which one or more ERM processes may reside. As an example, consider the case where two independent relaxation mechanisms are present. The first relaxation mechanism, “a,” is associated with an ERM and exhibits volume viscosity characteristics, thus it is limited in its relaxation capacity, and the second relaxation mechanism, “b,” is associated with the relaxation generally observed during traditional chemical strengthening and exhibits shear viscosity characteristics. Assuming stretched exponential relaxation with a stretching parameter of 0.5 for both mechanisms 6 and relaxation times τa = 10 seconds and τb = 10 seconds, a representation of the stress relaxation versus time is given in Figure 14. As noted in the figure, the leading-edge compressive stress magnitude measurements obtained by electric field-assisted method may represent a “near-zero” time observation, say effectively from 0.01 to 10 seconds, which may avoid much of the stress relaxation by the ERM. Observations of stress after
45 traditional chemical strengthening represent a near-fully exchanged surface layer at a time of at least about 1,000 seconds or later. By this time, the ERM mechanism “a” has nearly completely been extinguished and only relaxation by shear flow “b” remains. This example is limited for simplicity. The ERM may contain multiple components, which themselves may be represented by volume viscous and/or shear viscous behavior. These ERM events can potentially be studied via elevated temperature MD simulations of SAA, with much longer observation times, and/or alternative boundary conditions during SAA.
Figure 14. Example of a two-mechanism stress relaxation system, where each mechanism is responsible for half of the total stress. Stretched exponential relaxation with a stretching exponent of 0.5 has been assumed. The relaxation times for each of the mechanism are given in the figure.
Note, Shaisha and Cooper36 suggested the presence of a “fast” relaxation process and/or a temperature-dependent LNDC, albeit in a somewhat different context than that discussed here. Their experiments of potassium stuffing in SLS by electric field-assist chemical strengthening displayed a strong MCSM temperature dependence, ranging from about 1,500 MPa at 200 °C to 700 MPa at 450 °C. Further, by varying the applied potential at 350 °C, the time required for equivalent exchange of ions could be varied, and the MCSM was nearly constant for exchange times of about two minutes to 46 120 minutes. Assuming their compressive stress measurements were in-fact from the leading edge of the diffusion front, rather than averages across the exchanged zone, one would have to conclude that SAA instantaneous relaxation is temperature-dependent which is in agreement with the free volume arguments presented in the preceding discussion. That is, as the chemical strengthening temperature is increased, the host glass structure will expand and have greater free volume. Silicate networks with greater free volume have less expansion upon chemical stuffing because that expansion is partly consumed by the additional free volume of the network at elevated temperature. The lack of relaxation at 350 °C for the time span of two to 120 minutes, again assuming the compressive stress measurements were made at the leading edge of the diffusion front, is consistent with the leading edge representing a “time-zero” state for the alkali stuffed material. For commercial glasses a balance of chemical strengthening temperature and time must be achieved. While SLS and 23NS show high LNDC, the high NBO concentrations cause these compositions to be somewhat soft in terms of their viscosity-temperature profiles, relative to SMAS for example, and are thus more susceptible to relaxation by viscous means. Observations within this study highlight the possibility for enhanced surface compression through understanding factors that dictate the initial LNDC and that potentially influence subsequent stress relaxation. This type of understanding will likely be a key to developing future generations of glasses optimized for chemical strengthening and to developing alternative techniques by which chemical strengthening can be imparted, each to optimize surface compression.
D. Conclusions MD simulations were used to study potassium stuffing in SLS and SMAS glasses, with expanded analysis of NS glasses studied previously using the same technique. Molar volume of potassium-stuffed SLS and SMAS glasses was observed to be intermediate of the host and CEAM glasses. Consistent with the NS glasses, SLS and SMAS glasses after potassium stuffing had nearest-neighbor features similar to their CEAM glasses, but network connectivity and topology similar to the host glasses. Intertetrahedral bond angle was found increase after NPT stuffing, where increases were
47 much greater for SMAS than for SLS and NS, and this was attributed to differences in the nature of alkali sites within those networks. The system LNDC was found to scale linearly with mean BO per network- forming cation and with increasing Poisson’s ratio of the host glass, where depolymerized networks showed greater LNDC. Lesser change of Poisson’s ratio and lesser change of the (M)BO coordination number after alkali stuffing were also observed to trend with increasing system LNDC. Maximum compressive stress magnitudes predicted from these MD simulations for SLS and SMAS were roughly consistent with those achievable by specialty chemical strengthening techniques, but remained much higher than that observed by traditional chemical strengthening. The presence of early relaxation modes was proposed to account for the maximum compressive stress magnitudes observed by traditional chemical strengthening.
48 E. References 1. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids. Clarendon Press, Oxford, 1987.
2. A. R. Leach, Molecular Modelling Principles and Applications, 2nd ed. Pearson Educated Limited, Harlow, England, 2001.
3. A. K. Varshneya, Fundamentals of Inorganic Glasses, 2nd ed. The Society of Glass Technology, Sheffield, 2006.
4. J. C. Mauro and J. Du, "Achieving Long Time Scale Simulations of Glass- Forming Systems," Comp. Theor. Chem., 987, 122-33 (2012).
5. A. K. Varshneya, "Chemical Strengthening of Glass: Lessons Learned and yet to Be Learned," Int. J. Appl. Glass Sci., 1 [2] 131-42 (2010).
6. O. Richmond, W. C. Leslie, and H. A. Wriedt, "Theory of Residual Stresses Due to Chemical Concentration Gradients," Transactions of the ASM, 57 [1] 294-300 (1964).
7. A. R. Cooper and D. A. Krohn, "Strengthening of Glass Fibers: II, Ion Exchange*," J. Am. Ceram. Soc., 52 [12] 665-9 (1969).
8. A. J. Burggraaf, "The Strengthening of Glass by Ion Exchange. Part 2. Stress Formation and Stress Relaxation after Ion Exchange in Alkali Aluminosilicate Glasses in Connection with Structural Changes in the Glass," Phys. Chem. Glasses, 7 [5] 169-72 (1966).
9. A. J. Burggraaf, "The Mechanical Strength of Alkali-Aluminosilicate Glasses after Ion Exchange," Philips Res. Rep. Suppl., 3, 1-106 (1966).
10. A. K. Varshneya, "Kinetics of Ion Exchange in Glass," J. Non-Cryst. Solids, 19, 355-65 (1975).
11. D. K. Hale, "Strengthening of Silicate Glasses by Ion Exchange," Nature, 217 [5134] 1115-8 (1968).
12. P. K. Kreski, A. K. Varshneya, and A. N. Cormack, "Investigation of Ion- Exchange ‘Stuffed’ Glass Structures by Molecular Dynamics Simulation," J. Non- Cryst. Solids, 358 [24] 3539-45 (2012).
13. A. Tandia, K. D. Vargheese, J. C. Mauro, and A. K. Varshneya, "Atomistic Understanding of the Network Dilation Anomaly in Ion-Exchanged Glass," J. Non-Cryst. Solids, 358 [2] 316-20 (2012).
49
14. A. Tandia, K. D. Vargheese, and J. C. Mauro, "Elasticity of Ion Stuffing in Chemically Strengthened Glass," J. Non-Cryst. Solids, 358 [12–13] 1569-74 (2012).
15. K. D. Vargheese, A. Tandia, and J. C. Mauro, "Molecular Dynamics Simulations of Ion-Exchanged Glass," J. Non-Cryst. Solids, 403, 107-12 (2014).
16. SciGlass-6.5, [Computer Program] ITC, Inc., Newton, MA, USA, 2005.
17. W. Smith, T. R. Forester, and I. T. Todorov, DL_POLY2, [Computer Program] STFC Daresbury Laboratory, Daresbury, Washington Cheshire, UK, 2009.
18. W. Smith, C. W. Yong, and P. M. Rodger, "DL_POLY: Application to Molecular Simulation," Mol. Simul., 28 [5] 385-471 (2002).
19. A. Pedone, G. Malavasi, M. C. Menziani, A. N. Cormack, and U. Segre, "A New Self-Consistent Empirical Interatomic Potential Model for Oxides, Silicates, and Silica-Based Glasses," J. Phys. Chem. B, 110 [24] 11780-95 (2006).
20. J. D. Gale and R. L. Rohl, "The General Utility Lattice Program (GULP)," Mol. Simul., 29 [5] 291-341 (2003).
21. A. C. Wright, "Neutron and X-Ray Amorphography," J. Non-Cryst. Solids, 106 [1–3] 1-16 (1988).
22. Y. Xiang, J. Du, M. M. Smedskjaer, and J. C. Mauro, "Structure and Properties of Sodium Aluminosilicate Glasses from Molecular Dynamics Simulations," J. Chem. Phys., 139 [4] 044507/7 (2013).
23. M. Benoit, S. Ispas, and M. E. Tuckerman, "Structural Properties of Molten Silicates from Ab Initio Molecular-Dynamics Simulations: Comparison between CaO-Al2O3-SiO2 and SiO2," Phys. Rev. B, 64 [22] 224205/10 (2001).
24. A. Pedone, G. Malavasi, M. C. Menziani, U. Segre, and A. N. Cormack, "Molecular Dynamics Studies of Stress−Strain Behavior of Silica Glass under a Tensile Load," Chem. Mater., 20 [13] 4356-66 (2008).
25. B. Mantisi, A. Tanguy, G. Kermouche, and E. Barthel, "Atomistic Response of a Model Silica Glass under Shear and Pressure," Eur. Phys. J. B, 85 [9] 1-13 (2012).
26. T. Rouxel, "Elastic Properties and Short-to Medium-Range Order in Glasses," J. Am. Ceram. Soc., 90 [10] 3019-39 (2007).
50 27. J. D. Mackenzie and J. Wakaki, "Effects of Ion Exchange on the Young's Modulus of Glass," J. Non-Cryst. Solids, 38–39, Part 1, 385-90 (1980).
28. V. Tyagi, "Physical Properties of Ion-Exchange Strengthened Glasses"; M.S. Thesis. Alfred University, Alfred, NY, 1996.
29. A. K. Varshneya, "Physical Properties of Ion-Exchanged and Melt-Processed Glasses Differ," GlassResearcher, 10-11 [2-1] 21-6, 51 (2001).
30. G. N. Greaves, A. L. Greer, R. S. Lakes, and T. Rouxel, "Poisson's Ratio and Modern Materials," Nat. Mater., 10 [12] 823–37 (2011).
31. S. Yoshida, J. C. Sanglebœuf, and T. Rouxel, "Quantitative Evaluation of Indentation-Induced Densification in Glass," J. Mater. Res., 20 [12] 3404-12 (2005).
32. G. N. Greaves, "EXAFS and the Structure of Glass," J. Non-Cryst. Solids, 71 [1– 3] 203-17 (1985).
33. X. Yuan and A. N. Cormack, "Local Structures of MD-Modeled Vitreous Silica and Sodium Silicate Glasses," J. Non-Cryst. Solids, 283 [1–3] 69-87 (2001).
34. G. N. Greaves, A. Fontaine, P. Lagarde, D. Raoux, and S. J. Gurman, "Local Structure of Silicate Glasses," Nature, 293 [5834] 611-6 (1981).
35. H. Ohta and M. Hara, "Ion Exchnage in Sheet Glass by Electrolysis," Reports Res. Lab. Asahi Glass Co., Ltd., 20 [1] 15-31 (1970).
36. E. E. Shaisha and A. R. Cooper, "Residual Stress in Singly and Doubly Ion- Exchanged Glass," J. Am. Ceram. Soc., 64 [1] 34-6 (1981).
37. M. N. Svenson, L. M. Thirion, R. E. Youngman, J. C. Mauro, S. J. Rzoska, M. Bockowski, and M. M. Smedskjaer, "Pressure-Induced Changes in Interdiffusivity and Compressive Stress in Chemically Strengthened Glass," ACS Appl. Matter. Inter., 6 [13] 10436-44 (2014).
38. G. W. Scherer, Relaxation in Glass and Composites. Kreiger Publishing Company, Malabar, FL, USA, 1986.
39. D. C. Allen, A. J. Ellison, and J. C. Mauro, Corning Incorporated, "Heat Treatment for Strengthening Glasses," U.S. Pat. App. US 2013/0260154 A1, October 2013.
51
III. DIMENSIONAL SWELLING CHARACTERIZATION OF CHEMICALLY STRENGTHENED ALKALI SILICATE GLASSES
A. Introduction
1. Background Since nearly the advent of glass chemical strengthening, the factors responsible for compression magnitude have been studied. The chemical expansion analogy to thermal stress development proposed by Richmond, et al.1 and applied to glass chemical strengthening by Cooper and Krohn2 has been successful in modeling compressive stress profiles3-8 and has been useful to a lesser extent for compressive stress profiles with complex relaxation behavior.9-11 Within this framework, local compressive stress is a sum of the local incompatible elastic expansion and the interior tension of the glass article fulfilling the net zero stress state requirement of elasticity theory. For a plate subject to biaxial plane strain:12
zz z 0 (7)
EBzCz E H z z B z C z dz (8) xx yy 1 H1 0 where diffusion is in the z-dimension, B(z) is the linear network dilatation coefficient, C(z) is the concentration of the invading ion, E is Young’s modulus, is Poisson’s ratio, and H is the thickness of the plate. See Figure 15 for a visual reference of the axes and various dimensional quantities. Note, chemical expansion strain (B(z)C(z))xx is analogous to thermal expansion strain (α(z)T(z))xx, where linear network dilatation coefficient B(z) is analogous to linear thermal expansion coefficient α(z) and concentration C(z) is analogous to temperature T(z). Throughout this manuscript, compressive stresses are treated as negative (-) in sign and tensile stresses are treated as positive (+) in sign.
52
Figure 15. Diagram of an interior slice with associated axes and labeled dimensions.
The linear network dilatation coefficient, which will be abbreviated as LNDC and is represented by B(z) here, is defined as the strain per concentration of stuffing alkali ion:2
1 lnV z Bz 3 Cz (9) where V(z) is molar volume and C(z) is the concentration of the stuffing alkali ion. Historically, the LNDC has been examined with molar volumes from as-melted glasses,3,13-15 i.e. the sodium silicate “host” and the potassium silicate with equal concentration alkali (compositionally-equivalent, as-melted or CEAM). Researchers observed the LDNC computed from as-melted molar volumes far over-estimated the surface compression observed in practice, often by a factor of three to four.3,13,14 For example, in terms of surface compression, a typical soda-lime silicate glass may display a maximum surface compression of 500 to 700 MPa by potassium chemical strengthening, whereas the surface compression anticipated from molar volumes of as-melted glasses is on the order of 2,500 MPa. This lead various researchers3,13,14 to conclude that ion- exchange “stuffed” glasses are derivative of the host glass and have notably lower molar
53 volume than their corresponding CEAM glasses. This conclusion has been supported by a number of later experimental observations, both in the laboratory16-20 and via computer simulation.21-23 Beyond the linear network dilatation coefficient B(z), estimation of surface compression resulting from chemical strengthening (Eq. (8)) relies upon physical properties of Young’s modulus E and Poisson’s ratio ν. Mackenzie and Wakaki16 observed Young’s modulus to increase by about 10% for sodium-stuffed lithium aluminosilicate glass fibers by horizontal cantilever method. Later measurements by Tyagi and Varshneya18,19 noted Young’s modulus and Poisson’s ratio increases of up to approximately 14% and 25%, respectively, as determined by pulse-echo technique in chemically strengthened commercial lithium-sodium aluminosilicate glass where sodium was replacing lithium. A final factor that does not appear in Eq. (8), but can influence observed stress state, is the coefficient of thermal expansion (or contraction) of the material. Chemical strengthening is carried out at temperatures that are high enough for alkali interdiffusion to take place in a reasonable about of time, which tends to be in a temperature range of 375-500 °C for many commercial glasses. After chemical strengthening, the glass is cooled to room temperature. Thus, altering the glass by ion exchange at elevated temperature, but making use of the glass at room temperature allows thermal expansion factors to be relevant. Tyagi and Varshneya18,19 studied this phenomenon by in-situ measurements of stress-birefringence with a heated cell and a polarized light microscope during sodium chemical strengthening of a lithium-sodium aluminosilicate glass. They observed approximately 20% lower surface compression magnitude at room temperature than at the exchange temperature, which was verified by returning the specimen to the exchange temperature restoring the previously observed surface compression magnitude. Finite element analysis was used to back-estimate the effective thermal contraction coefficient, which was found to be approximately one and a half times that of the host glass. While these physical properties of Young’s modulus, Poisson’s ratio, and thermal contraction coefficient have been shown to vary with modification by chemical strengthening, the influence of their modification to the resultant compressive stress is small relative to the compressive stress discrepancy from difference of molar volume of alkali-stuffed glass and CEAM glass. As such, there is
54 much interest in the nature of alkali-stuffed silicate glasses, in particular surface compression magnitude, which is dictated by the molar volume resulting from alkali stuffing and the associated relaxation behavior at the time of stuffing and afterward.24,25
2. Dimensional Changes and Models Chemically strengthened glass layers are isolated to the surface regions of glass articles, and for potassium ion exchange in commercial soda-lime silicate, the depth of the invading ion often does not exceed 30 μm for sensible chemical strengthening times of 24 hours or less. This can pose some difficulty in characterization, and precision of characterization, of these layers by traditional means, such as layer density estimation by iterative removal of surface layers followed by Archimedes’ method of density measurement for the remaining portion of the specimen and pulse-echo technique for elastic properties, which when applied in the standard setup is not particularly surface sensitive. Therefore, other methods of studying these ion-exchanged layers are of interest. While the dimensional changes induced by chemical strengthening are quite small, these measurements have been made with varying degrees of success. A common measurement type is that of the step height. In this arrangement, adjacent chemically- strengthened and non-chemically-strengthened portions of the same glass surface are measured for their differential height, typically using a stylus profiler or optical interference-based methods. Examples involving ion exchange below the glass strain point follow. Valles-Villarreal, et. al.26 exchanged copper ions into soda-lime silicate glass at 350 C under an applied potential. Despite relatively short exchange times of less than two hours and case depths less than 10 μm, surface swelling of up to 60 nm was observed using an interferometric technique. Oven27 and Yin28 examined copper ion exchange replacing sodium ions in borosilicate glass, which produced swelling of about 40 nm as measured by both interferometric and stylus profilometer techniques. The authors noted that swelling height varied with the width of the glass strip that was exposed to exchange, where the waveguide width was varied between 10 μm and 90 μm. A second method by which dimensional changes induced by chemical strengthening can be monitored involves boundary effects. Typically, stress measurements of chemically strengthened glasses are made at the mid-point of a long edge in order to establish an approximate state of plane strain, from which reliable 55 calculations can be made. In this case, the free end parallel to the direction of invading ion diffusion is observed for perpendicular swelling. An early estimate of this deformation was demonstrated in the finite element analysis of Sane and Cooper29 and later by Jain and Varshneya.30 Laboratory measurements referencing this feature were demonstrated by researchers attempting to improve the flatness of substrates for hard magnetic disks for data storage.31 Of these dimensional features, several studies have utilized the step height approach to characterize chemically strengthened glasses. Early work by Burggraaf14 attempted to measure the step height induced by selective chemical strengthening of sodium aluminosilicate exchanged with potassium. Step height measurements made with a roughness meter were somewhat unreliable, but allowed for an estimated expansion in -4 the diffusion (free) direction of about 10 per mol% K2O in terms of the LNDC. This result had order of magnitude agreement with the LNDC estimated from stress- -4 birefringence measurements, which were about 6x10 per mol% K2O. Measurements of step height have enabled a materials behavior model to be assembled. Glebov, et al.32 used a step height measurement to study potassium replacing sodium exchange in K8 optical glass (alkali borosilicate33) as a function of chemical strengthening temperature and time. The researchers assembled a simple model allowing local strain to have elastic and plastic components, using step height observations and optical ellipsometry measurements of stress state as inputs. In the form of Eq. (8), the model simply splits the LNDC B(z) into elastic Bela(z) and plastic Bpla(z) components. Allowing the z-axis to be the diffusion direction, then within the x-plane the total (or
tot ela initial) strain xx z is equated to the sum of the elastic strain xx z and plastic strain
pla 32 xx z:
tot ela pla xx z 1 xx z xx z (10) where ν is Poisson’s ratio and the input elastic strain is of the biaxial plane strain condition, hence the (1- ν) term to convert to linear strain for the summation. Integrating over all of the chemically strengthened layers in the z-direction, the integral of Eq. (10) is:32
56 tot z dz 1 ela z dz pla z dz xx xx xx (11) which has limits of integration (not shown) from the case depth, i.e. depth of neutral stress point from surface, to the surface. Note, the strain state is assumed equivalent in
tot tot the x- and y-directions, i.e. xx z yy z. Thus similar equations to Eq. (10) and Eq. (11) can be expressed for strain within the y-plane and in the y-direction. A second equation is constructed by equating the dimensional swelling in the z-direction, called the “step height” here, to the integral of the free strain in the diffusion direction (z-axis):32
z tot z dz pla z dz pla z dz xx xx yy (12) tot z dz 2 pla z dz xx xx where Δz is the step height and, again, the limits of integration (not shown) are from the case depth to the surface. Glebov, et al. invoke volume conservation, so plastic strain is
pla limited to shear flow, thus the factor of two preceding the xx z integral due to equal plastic strain contributions from the x- and y-dimensions. With Eq. (11) and Eq. (12), the unknowns of integrated total strain and integrated plastic strain can be computed. The contributions to Δz are not known as a function of z-position and prevent initial strain and plastic strain from being determined versus z-position, but the solution to this system of
tot equations is sufficient to allow for estimates of average total strain xx , average plastic
pla pla strain xx , plastic strain-to-total strain ratio R , and, if the average concentration is
totmax max known, the total maximum strain xx or stress xx . Glebov, et al.32 applied this system of equations to their data set and observed the average plastic strain increased with increasing temperature between 400 and
520 °C, whereas the average total strain (referred to as initial strain by the authors) was nearly constant with increasing temperature, until about 520 °C. That is, the initial potassium accommodation by the glass network displayed a constant average expansion, of which the fraction that relaxed by shear flow increased with increasing temperature.
57 This lead to the conclusion that for low-temperature ion exchange, plastic strain is temperature-dependent and the initial volume change associated with alkali stuffing causes displacement of constituents about the alkali ion, but does not require thermal activation. Varshneya34 noted a potential modification to the model of Glebov, et al.32 in
pla plaD which the plastic strain xx z is split into plastic deviatoric xx z (volume
plaH conserving) and plastic hydrostatic xx z (shape conserving) contributions. In this approach, Eq. (11) is modified as follows:
tot z dz 1 ela z dz plaD z dz plaH z dz xx xx xx xx (13)
And Eq. (12) is modified as follows:
z tot z dz plaD z dz plaD z dz plaH z dz xx xx yy xx (14) tot z dz 2 plaD z dz plaH z dz xx xx xx
Note, the plastic hydrostatic strain is equivalent in-plane and out-of-plane, allowing densification to alleviate elastic strain, together with shear flow from the original model. Varshneya19,34 has also proposed local elastic-plastic yielding behavior during chemical strengthening. Upon the entry of a stuffing alkali ion to a host site, bond bending or stretching occurs consisting of independent hydrostatic and deviatoric elastic limits. Beyond the hydrostatic limit, dilatation no longer contributes to expansion, thus leads to densification. Beyond the deviatoric limit, shear no longer contributes to in- plane expansion, but volume is conserved. Alkali site expansion and stretching is proposed to leave the site in a higher energy configuration than the non-stuffed state, but this configuration resides in a local potential energy minimum from which escape is mediated by limited thermal energy of the ion-exchange process. A goal of the present work is to examine the applicability of these proposed models and mechanisms where possible.
58 B. Method Two glass compositions were utilized, one float-produced 2.2 mm soda-lime silicate (generic, unknown manufacturer), labeled SLS, and one fusion down-drawn 2.1 mm sodium magnesium aluminosilicate (Corning code 0317), labeled SAS. Approximate glass compositions are given in Table XI. Note, the differential scanning calorimetry-determined glass transition temperature (Tg) was 560 °C and 618 °C for SLS and SAS, respectively, for as-received glass heated at a rate of 10 °C per minute under flowing air. Throughout, the air surface of the SLS float glass was used as the primary surface, whereas this distinction was not necessary for the fusion down-drawn SAS. For each selected chemical strengthening temperature and time, four coupons were prepared to dimensions 16 mm x 8 mm x H, where H is the as-received thickness (Figure 15), using a slow-speed metallographic, low-concentration diamond-bonded blade lubricated with a kerosene reservoir. Sacrificial glass layers were affixed to the coupon glass at the entry and exit points of the cut, to limit edge degradation of the coupons. Two of every four coupons were prepared to a visual polish on the 16 mm x H edges by light grinding using 1000 grit silicon carbide metallographic abrasive and water against a float glass plate, then polishing with cerium oxide and water on a polishing cloth, affixed to a polishing wheel. This procedure netted coupons with a high-quality corner along the 16 mm x H edges. After all cutting, grinding, and polishing was completed, all coupons were thoroughly cleaned with methanol and Kimwipes™, then placed against setters of the same glass composition and annealed at 520 °C for 8 hours and 565 °C for 8 hours for SLS and SAS, respectively. Afterward, the annealing furnace was cooled at approximately 3 to 5 K per minute. Annealing for extended time near the strain point was chosen to help preserve the flatness of the 16 mm x 8 mm surfaces.
59
Table XI. Approximate Glass Compositions and Properties Component SLS‡ (mol%) SAS† (mol%)
SiO2 72 66.2
Al2O3 0 10.8 MgO 6 5.6 CaO 9 0.5
Na2O 13 13.5
K2O 0 2.3 Other 0 1.0 Property SLS* SAS* Elastic Modulus (GPa) 70 72 Poisson’s Ratio 0.22 0.22 Strain Point (°C) 490 576 Annealing Point (°C) 530 622 Softening Point (°C) 710 870 Stress-optical Coef. (TPa-1) 2.75^ 2.75^ ‡ SLS = generic float SLS, example composition after Seward and Varshneya35 † SAS = Corning 0317, composition from Dumbaugh36 * SAS and SLS properties from Seward and Varshneya35 ^ Stress-optical coefficient estimates after Varshneya12
Chemical strengthening was performed using a paste-based potassium nitrate source. Two compositions were utilized. For chemical strengthening performed at 350 °C and higher temperatures, a paste of weight percent composition 25% C&C ball clay (Spinks Clay Company), 25% potassium nitrate (technical grade), and 50% distilled water was used, prepared as a 1,300 g batch. Below 350 °C, a paste of weight percent composition 26% porcelain, 18% potassium nitrate (technical grade), 4% potassium hydroxide (technical grade), and 52% distilled water was used, prepared in a 120 g batch. A 70/30 mole percent eutectic mixture of potassium nitrate and potassium hydroxide was utilized to attempt to depress the melting point of the salt mixture within the paste. These pastes were thoroughly mixed using a hand-held drill with stir attachment prior to application to the coupons. For each set, two coupons were fully masked on their 16 mm x 8 mm faces using Scotch™ tape with approximately 2 mm overhang, leaving the 16 mm x H edges exposed to the paste. One coupon was half-masked on the 16 mm x
60 8 mm faces, leaving half of those faces exposed to the paste. The final coupon had all faces exposed to the paste. Paste was applied by dip-coating the coupons into the freshly- mixed paste. After paste coating, the coupons were placed on aluminum foil and dried at 85 °C for one hour. After low-temperature drying, the paste had become sufficiently rigid to remain adhered to the glass and to maintain its shape (leather-hard), at which time the tape masks were carefully removed using tweezers to avoid disturbing the adherent paste. This masking process was highly effective for applying the paste to the intended surfaces. Paste-coated coupon sets were transferred to separate fine-wire stainless steel mesh setters for chemical strengthening. A muffle furnace was first pre-heated to the target exchange temperature and then setters with coated coupons were placed in the furnace. The furnace temperature was independently monitored using a probe thermocouple placed within 25 mm of the coated coupons. The chemical strengthening temperatures and times used for each glass are given in Table XII. For each chemical strengthening temperature, multiple setters were often placed within the furnace, and then removed individually when the target time had elapsed for each setter. When the chemical strengthening time was one hour or less, setters were independently chemically strengthened to prevent adverse effects from temperature fluctuations. After removal from the furnace, setters with coated coupons were allowed to cool to room temperature, after which coupons were rinsed with tap water to remove the paste and dried with paper towels. Samples were immediately placed in polyethylene bags and labeled after washing and drying.
Table XII. Chemical Strengthening Temperature and Time Parameters Time (hours) Temperature (°C) SLS SAS 250 - 239 300 - 72, 108, 144, 216, 288 350 - 4, 8, 16, 32, 64, 143 400 9, 16, 25, 37, 46, 66 1, 2.25, 4, 6.25, 9, 16 450 4, 9, 16, 25, 37, 50 1, 2.25, 4, 6.25, 9, 16 500 1, 2.25, 4, 6.25, 9 0.25, 1, 2.25, 4, 6.25
61 Figure 16 schematically depicts (A) the coating locations, (B) resulting swelling after chemical strengthening, and (C) locations of sampled line profiles that are further detailed below. Table XIII also provides an overview of the coupon purpose for each set. From each set of coupons, coupons #1 and #2 were used for surface profile measurement of edge swelling resulting from 16 mm x H edge exchange. Surface profile measurements were made using a white light profilometer (Zygo Corporation model NV5000 5032). Prior to each measurement session, a system error profile of a certified silicon carbide reference flat (approximately 5.9 nm peak-to-valley error over a 25 mm aperture) was measured and used for all subsequent measurements to subtract any aberrations introduced by the profilometer optics. A step height standard (VLSI Standards Incorporated, 1.8 μm step) was also measured and the average of 15 measurements was used to adjust profilometer calibration constants that were susceptible to drift with varying ambient temperature and humidity. Note, drift from this type of error was often less than 3 nm. With use of the reference profile and calibration to the step height standard, the profiler generated height measurement accuracy of ±11 nm, precision of ±6 nm, and resolution of 1 nm or better. The lateral resolution was similar to that of a standard optical microscope, on the order of 5 μm for the 20x objective with 1x zoom (1.13 μm pixel resolution). Profiles of a 0.30 mm x 8.75 mm region (i.e. the 8 mm coupon dimension was exceeded to ensure the edge was captured) between the mid-points of the 16 mm edges of the 16 mm x 8 mm face of the coupon were stitched from multiple measurements using the profilometer software, where individual profiles had spatial dimensions of 360 μm x 270 μm with profile overlap of approximately 13% percent for stitch alignment. From the measured profile, a line profile of 25 μm width was extracted, drawn to be perpendicular to the 16 mm edge and drawn to avoid regions of obvious edge chip-out. Height values at each position along the profile were an average of the height values of the 25 pixels across the line width. Line profiles were leveled and translated to a common reference position by a procedure detailed in Appendix C.
62
Figure 16. Coupon selective-surface chemical strengthening, swelling, and measurement depictions: (A) paste-coated regions indicated by shaded blocks, (B) dimensional swelling after chemical strengthening where solid blue coloration indicates chemically strengthened surface, (C) arrow and eye indicating the profile or measurement line. Note, diffusion is in the z-direction and swelling and chemical diffusion gradients are not drawn to scale.
63 Table XIII. Summary of Purpose for Coupons in Each Set
Coupon ID Purpose Measurement Location
16 mm x 8 mm Edge swelling by white light Exterior surface, 1 & 2 profilometer Line connecting midpoints of 16 mm edges
16 mm x 8 mm 3 Step height by white light profilometer Exterior surface, Entire surface
Stress-birefringence by polarized light 8 mm x H microscopy with compensator & Interior surface, 4 Chemical diffusion by electron Line connecting the midpoints of microprobe analysis the 8 mm edges
Coupon #3 from each set was used for surface profile measurement of step height. The white light profilometer was again used, with initialization and calibration equivalent to that described above. The 5x objective with 0.4x zoom (11.3 μm pixel resolution) was utilized to assemble full-surface profiles of the 16 mm x 8 mm surfaces. Afterward, five lines of several millimeters in length by 0.28 mm (25 pixels) width were drawn across the step-containing mid-section of the full-surface profiles. Height at each position along the profile was the average height of the 25 pixels across the 0.28 mm width. Each line profile was leveled by linear line subtraction, and then the step height was taken as the difference of the height peak and valley points near the step region. The mean step height was established from the step heights of the five lines. From each set, coupon #4 was used for stress-birefringence thin slice fabrication and electron microprobe mount preparation. A slow-speed metallographic saw with low- concentration diamond-bonded blade lubricated with a kerosene reservoir was used to cut slices parallel with the 8 mm x H plane. The end of the coupon from the first cut was discarded. Several slices of 0.9 mm were extracted for stress-birefringence slice preparation. The remaining portion of the coupon was used for electron microprobe mount preparation. Slices for stress-birefringence measurement were mounted at the center of a 50 mm x 38 mm x 3 mm glass back-plane using thermoplastic cement with
64 10 mm x 10 mm x 1 mm microscope slide feet at each of the four corners of the back- plane. The back-plane arrangement allowed for improved thickness uniformity across the stress-birefringence slices. A final slice width (dimension W in Figure 15) of 150 μm to 350 μm with visually polished surfaces was obtained by grinding with 600 grit silicon carbide metallographic abrasive and water against a brass wheel, followed by grinding with 1000 grit silicon carbide metallographic abrasive and water against a float glass plate, and finally polishing with cerium oxide and water on a polishing cloth, affixed to a polishing wheel. Stress-birefringence slices were placed against 1 mm microscope slides with a 0.17 mm cover slip and mineral oil immersion fluid. Birefringence as a function of position from the edge was measured using a polarized light microscope (Olympus BX43) equipped with a Berek compensator. A 40x objective and 2.5 μm step size was used for stress profiles of less than 25 μm case depth, or 20x objective and 5 μm step size was used for stress profiles of 25 μm and greater case depth. Two compensation measurements were made at each location and their average was used in subsequent calculations. Care was taken to avoid regions of edge chip-out. The profile was measured near the mid-section of dimension L (Figure 15) to ensure the desired boundary conditions were maintained.37 Birefringence was converted to retardation using the manufacturer-provided lookup table for the Berek compensator. Retardation was then converted to biaxial stress assuming a biaxial plane strain condition, using the following equation:35
z z z xx yy 1 W (15) where δ is the retardation, ν is Poisson’s ratio, β is the stress-optical coefficient, and W is the optical viewing path length (equivalent to slice dimension W in Figure 15). Refer to Table XI for values for ν and β. Stress error was established by propagating error for the following values: birefringence compensation ±0.2°, Poisson’s ratio ±0.01, stress-optical coefficient ±0.03 TPa-1, and optical viewing path length ±2 μm. With the remaining portion of coupon #4, the interior cut surface (8 mm x H) was mounted facing outward in air-setting epoxy (Fulton Metallurgical Products Quickmount Resin and Hardener). Often several coupons were placed in one mount. Care was taken
65 to clearly label and track each coupon. After allowing the resin to setup for at least 10 hours, mounts were ground and polished using an autopolisher (Buehler model Automet 2 paired with Buehler model Ecomet 3). The preparation schedule is given in Table XIV. After grinding and polishing, mounts were dried at 80 °C for 8 hours, and then were carefully cleaned with methanol, followed by distilled water and dried with Kimwipes™. Mounts were either carbon coated under hard vacuum (approximately 30 nm carbon deposited) or sputter coated with gold-palladium under rough vacuum (approximately 10-15 nm gold-palladium deposited). The prepared mounts were used for the determination of potassium, sodium, and silicon chemical profiles by electron probe microanalysis (EPMA). An electron microprobe analyzer (JEOL model JXA-8200) was used to monitor counts from the following characteristic lines: potassium (Kα-1), sodium
(Kα-1), and silicon (Kα-1). Other pertinent measurement parameters were: beam spot size 2 μm, beam accelerating potential 15 keV, beam current 10 nA, dwell time 3 seconds, and step size 2 μm. Parameters were chosen, in part, to prevent electron beam-induced alkali migration.19 A mean counts versus position profile was established from the mean of 10 rows, where each row was positioned perpendicular to the 8 mm edge and extended a minimum of 10 μm into the mount and a minimum of 10 μm past the potassium diffusion depth. For the SAS series, stress profiles established from stress-birefringence measurements were input into elastic finite element simulations utilizing the thermoelastic framework of the finite element software package.38 Simulations utilized a plane strain boundary condition along the 16 mm coupon dimension. In-plane dimensions were one-quarter of the 8 mm x H cross-section. Materials properties of Young’s modulus and Poisson’s ratio were those given in Table XI. A suitable combination of coefficient of thermal expansion and temperature change were selected to insert the linear (not biaxial) elastic strain profile into the simulation.
66
Table XIV. Grinding and Polishing Schedule for EPMA Samples
Grinding / Force Per Minimum Platen Platen Speed Polishing Lubricant Sample Time Rotation (rpm) Surface (lbs) (minutes) Direction
240 grit SiC Counter- Water 3 320 12 paper clockwise
400 grit SiC Counter- Water 3 320 12 paper clockwise
600 grit SiC Counter- Water 4 360 12 paper clockwise
6 μm 6 μm polishing diamond 4 280 16 Clockwise cloth suspension
1 μm 1 μm polishing diamond 5 280 10 Clockwise cloth suspension
C. Results Throughout the figures and tables that follow, the series labeling convention is: (glass type)-(chemical strengthening temperature in degrees Celsius)-[chemical strengthening time in hours]-[edge number], where the segments in parentheses will always be given and the segments in square brackets will be given only when required. For example, SLS-450-16-1 refers to the SLS glass, chemically strengthened at 450 °C for 16 hours, edge #1. Error bars and error estimates are representative of one standard deviation for direct laboratory measurements or representative of the propagated error for derived values, unless otherwise noted.
1. Laboratory Dimensional Changes Edge dimensional swelling measured by white light profilometer displays increased swelling with increasing time of exchange for all temperatures and both glass
67 types (Figure 17 and additional figures in Appendix D). Note, the error for these profile measurements is approximately ±25 nm. Some inconsistencies in edge swelling are evident, for example: multiple profiles in the SLS-400 series, SLS-450-16, and SAS-400- 9. In most cases this is attributable to one of three factors: (1) migration of salt from the exchange medium of the coated edge surface onto the bare glass top surface, (2) inconsistent ion exchange within the coated edge surface due to poor contact with the exchange medium, and/or (3) deformation of the substrate (see Appendix C). Whereas the typical depth of exchange does not exceed 40 μm and 125 μm for SLS and SAS, respectively, the deformed edge portion of the glass is quite deep by comparison, often extending to roughly 500 μm. This provides a sense for the magnitude of incompatibility induced by sodium for potassium ion exchange in these glasses.
68
(A)
(B)
Figure 17. Edge profiles after edge chemical strengthening determined by white light profilometer for (A) SLS 450 °C series and (B) SAS 450 °C series.
Step height, also determined by white light profilometry, shows in an increasing trend with the square-root of time for each temperature series of both of the glass types (Figure 18), indicating a strong relationship with the underlying stuffing ion 69 concentration. Several measurements fall away from this trend. Applicable to all series is error from deformation of the substrate (again, see Appendix C). For low-temperature chemical strengthening of the SAS series, i.e. below 350 °C, a eutectic mixture of salts was used to depress the melting point of the mixture, but in the presence of the clay components the degree to which the salt mixing took place and its distribution within the applied coating may have been adversely effected, in turn hindering the alkali exchange process.
70
(A)
(B)
Figure 18. Step height versus square-root of time after selective-surface chemical strengthening determined by white light profilometer for (A) SLS series and (B) SAS series.
2. Laboratory Chemical Diffusion and Stress Profiles Potassium and sodium chemical diffusion profiles were determined via electron microprobe analysis from coupon #4 of each series, with the exception of SAS-300-288 71 which was not measured. Where possible, a chemical diffusion profile was determined from each of the opposing 8 mm edges of coupon #4, thus the majority of the chemical strengthening conditions had two chemical diffusion profiles determined, one from each edge. Example profiles are shown in Figure 19. The vertical dashed lines in the figure are representative of the positions of the “start” and “end” of the potassium diffusion profile and were used to establish the diffusion depth utilized later. Note, profiles were not corrected for the finite beam diameter effect,39,40 leaving some near-edge profiles with some visible convolution or rounding.
72
(A)
(B)
Figure 19. Chemical diffusion profiles as integrated x-ray counts after chemical strengthening determined by electron microprobe analysis for (A) SLS exchanged at 450 °C for 16 hours and (B) SAS exchanged at 450 °C for 6 hours.
For each potassium profile, the potassium peak counts were paired with the baseline potassium counts in the substrate interior to generate a profile normalized from
73 zero to one. These normalized chemical profiles were integrated from the surface to the diffusion depth, termed “normalized, integrated concentration” or “normalized concentration integrand,” for input into the strain models that follow. Note, the units for this quantity are micrometers. Where possible, the average of two integrated, normalized concentration profiles from separate edges was utilized. The normalized concentration integrand increases linearly with the square-root of time (Figure 20). This relationship is anticipated from the form of the solution to Fick’s second law in one-dimension with constant source, when integrated across position and where interdiffusion coefficient D is constant:41
Czdz Dt 2 (16) Co
where C(z) is the concentration, Co is the source concentration, and t is time.
74
(A)
(B)
Figure 20. Average, normalized concentration integrand for potassium versus square-root of time for (A) SLS series and (B) SAS series.
Two stress profiles were measured using optical birefringence, one from each of the 8 mm edges of coupon #4. After collection and compilation, profiles were inspected and one was selected for use in subsequent calculations. Examples of the 450 °C 75 temperature sets for SLS and SAS are shown in Figure 21. Additional figures for the other temperature series are given in Appendix E. Error for the stress measurements is typically 5%, with a maximum of 10% when compression magnitude less than about 200 MPa. Compressive stress magnitudes for SLS are generally in agreement with those cited in literature.42,43 Most of the SLS stress profiles display a sub-surface compression maximum, a feature often observed in chemical strengthening literature.9-11,42,44 Compressive stress magnitudes for SAS show reasonable agreement with literature values, allowing for the different chemical strengthening times and temperatures utilized.45,46 Each selected profile was converted from stress to elastic strain by dividing by the appropriate Young’s modulus given in Table XI, and was then integrated from the surface to the case depth for use as model input, termed “integrated elastic strain” or “elastic strain integrand.” Note, no correction was made for interior tension, as this was often negligible at less than 10 and 20 MPa for SLS and SAS series, respectively. The elastic strain integrands are given in the section that follows.
76
(A)
(B)
Figure 21. Stress profiles σyy(z) versus position from edge (z-position) after chemical strengthening determined by polarized light microscopy for (A) SLS exchanged at 450 °C for various times and (B) SAS exchanged at 450 °C for various times.
77 3. Laboratory Combined Results Prior to assessing the strain models, the model inputs of step height, normalized concentration integrand, and elastic strain integrand are briefly examined to show readily obvious relationships between these quantities. A linear relationship is observed between step height and normalized concentration integrand (Figure 22), with little temperature dependence, except for the SAS-300 series which may have been susceptible to poor exchange medium behavior as noted earlier in the results section. The slope for the SLS and SAS series is 0.030 μm/μm and 0.019 μm/μm, respectively, i.e. the SLS series exhibits greater step height per quantity of stuffing potassium than the SAS series by a factor of about 1.6.
78
(A)
(B)
Figure 22. Step height versus normalized concentration integrand for (A) SLS series and (B) SAS series. The collective slope is (A) 0.030 μm/μm and (B) 0.019 μm/μm.
Elastic strain integrand also displays an approximately linear relationship with increasing normalized concentration integrand (Figure 23). A temperature-dependent
79 relationship is anticipated as compressive stress relaxation is typically observed at longer chemical strengthening times, and is enhanced by increasing temperature.47 This is evident in comparison of the SLS-400 and SLS-450 series to the SLS-500 series, where the SLS-500 series shows notably lower elastic strain integrand per quantity of stuffing potassium. The same relationship is not readily observed for the SAS series. The collective slope for the SLS and SAS series is 5.6x10-3 μm/μm and 8.2x10-3 μm/μm, respectively, indicating the SAS series generates larger elastic strain integrand per quantity of stuffing potassium, which is expected.
80
(A)
(B)
Figure 23. Elastic strain integrand versus normalized concentration integrand for (A) SLS series and (B) SAS series. The collective slope is (A) -5.6x10-3 μm/μm and (B) -8.2x10-3 μm/μm.
81 4. Finite Element Method Elastic Dimensional Changes Stress profiles from photoelastic birefringence measurements (Section C.2.) for the SAS series were first converted from biaxial stress to linear elastic strain by multiplication by a factor of (1-ν)/E, where ν is Poisson’s ratio and E is Young’s modulus from Table XI. The finite element calculation modeled one-quarter of the 8 mm x H cross-section, assuming plane strain in the third dimension (16 mm dimension), and introducing the strain profile to the H face (z-plane) as depicted for coupons #1 & 2 in Figure 16 via appropriate selection of thermal expansion coefficient and temperature difference. Good agreement of the input stress profile to the modeled profile in the plane strain dimension was observed in all cases. An example is given in Figure 24. Stepping observed for the FEM stress profile is due to the finite domain width over which the elastic strain was specified and had no observable impact on the resulting deformation. Minor stress profile deviation of up to ±30 MPa was observed at longer chemical strengthening times, attributable to the interior tension that was not removed from the input elastic strain profile.
Figure 24. Stress profile comparison in the plane strain dimension between laboratory measurement σyy(z) and FEM σxx(z) for SAS-450-4.
82 With stress profile agreement established, the dimensional changes attributable to the stress state can be examined. The spatial deformation of the 8 mm x 16 mm (y-plane) surface shown in Figure 25 is observed to qualitatively agree with the general shape observed in prior studies.29,30 Comparison of the measured edge swelling (Section C.1.) to that predicted from elastic FEM shows similar shape and magnitude (Figure 25 and Appendix F). In general, elastic FEM edge profiles are taller in y-dimension magnitude for the shortest chemical strengthening times (at temperatures less than 500 °C) and somewhat shorter in y-dimension magnitude for the longer chemical strengthening times. The former observation is possibly attributable to over-estimation of the stress profile by the photoelastic birefringence technique, although other possibilities are introduced in the discussion section. The latter observation can readily be attributed to the relaxation of stress. Laboratory stress profiles used as model inputs only reproduce elastic deformation in the model, thus relaxed stress profiles input into the FEM model display less deformation than the laboratory edge profiles. At intermediate times, the edge deformation profiles are fairly similar in shape and magnitude. Laboratory edge profiles are consistently shorter near the edge than the elastic FEM profiles (Appendix F). Finally, the intermediate and longer time laboratory edge profiles generally show larger swelling magnitudes, which might be described as a bulge, from the edge to 200-400 μm from the edge (see, for example, Figure 25 B & C). This feature may be attributable to migration of salt from the exchange medium of the coated edge surface onto the bare top surface. Additional possibilities are introduced in the discussion section.
83 (A)
(B)
(C)
Figure 25. Edge profile comparison between laboratory measurements and elastic FEM simulations for SAS-450 series at (A) 1 hour, (B) 4 hours, and (C) 16 hours. Laboratory measurements have error of ±25 nm. 84
Dimensional swelling perpendicular to the edge surface (swelling along the z- dimension) of the FEM model was used for comparisons to step heights obtained by the arrangement depicted for coupon #3 in Figure 16C. The difference in boundary conditions was briefly examined with several FEM simulations mimicking the chemical strengthening arrangement used for coupon #3 in Figure 16C, and was found to be within about 20 nm of the dimensional swelling observed by the edge FEM arrangement. As such, the dimensional swelling perpendicular to the edge surface was used for comparison with the laboratory measured step height, allowing for an error of ±20 nm for the FEM “step height.” Simulated step height difference in percent relative to the laboratory measurement for the same stress profile is given in Figure 26. In general, the FEM step height ranges from marginally similar to about 25% less than that observed in the laboratory measurement. For some series, such as SAS-350 and SAS-450, the magnitude of the difference appears to increase linearly with the logarithm of time, which is likely attributable to stress relaxation with increasing time. Similar to the edge swelling comparison above, higher exchange temperatures and longer exchange times lead to stress relaxation. When this relaxed stress profile is input into the elastic FEM simulation, only the elastic deformation is reproduced, hence the increased departure between laboratory step height and FEM step height with increasing stress relaxation.
85
Figure 26. FEM step height as percent difference from laboratory measurement versus natural logarithm of time for SAS series. Lines are drawn to guide the eye.
5. Strain Model with Shear Flow (SPS model) Laboratory measurements, or quantities derived thereof, including step height and elastic strain integrand were input into the shear plastic strain model of Glebov, et al.32 The model is referred to as the SPS model hereinafter. Model details are covered in the Introduction. Tabulated inputs and outputs are given in Appendix G. The average total
tot pla strain xx and the average plastic strain xx were computed by dividing the strain integrands of these quantities by their associated potassium diffusion depths. The
pla and the xx for the SAS series are given in Figure 27 against the logarithm of time. The
displays a linear relationship for four of the five multi-time-point temperature series, whereas the does not display any readily identifiable trends among the temperature series.
86 (A)
(B)
tot Figure 27. SPS model: (A) average total strain xx and (B) average plastic strain pla xx versus natural logarithm of time for SAS series. Lines are drawn to guide the eye.
Averages over the chemical strengthening times for each temperature series for
tot pla the xx and the xx are given in Figure 28, where the error bars represent the range of
87 values observed across the various chemical strengthening times for that temperature. Similar to that observed by Glebov, et al.,32 the SLS series displays relatively constant
tot pla xx and increasing xx with increasing temperature. The SAS series, on the other hand, displays constant or mildly decreasing and near constant with increasing temperature. The large range covered by the , and in some instances by the , is attributable to the chemical strengthening time dependence of these quantities.
88
(A)
(B)
Figure 28. SPS model: Mean across chemical strengthening times for average tot pla total strain xx and average plastic strain xx versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature.
89 An alternative method of examining the plastic contribution is via the plastic-to-
pla pla total strain ratio R , i.e. the quotient of the average plastic strain xx to the average total
tot pla strain xx . Collapsing the R across the various times for each temperature series into mean values, with error bars representing the observed range, clearly displays increasing Rpla for both glass series (Figure 29), where the slope for SLS is about 4.5 times larger than that of SAS. Also notable is the span covered across the selected temperatures: SLS ranges from 0.5 to 0.7 and SAS ranges from about 0.2 to 0.4.
Figure 29. SPS model: Mean across chemical strengthening times for plastic-to- total strain ratio Rpla versus chemical strengthening temperature. Error bars represent the range observed for each temperature.
Finally, dividing the average total strain by the normalized concentration integrand produces a maximum strain value, which is converted to stress, assuming a biaxial plane strain condition, by multiplying by E/(1-ν) using values from Table XI. The
max maximum initial (or total) stress xx , as averages over all chemical strengthening times for each temperature series, is given versus temperature in Figure 30, where error bars represent the observed range. For SLS, this value ranges from about -900 to -1,300 MPa
90 with an average near -1,125 MPa and for SAS this value ranges from -800 to -1,400 MPa
max with an average near -1,100 MPa. While the upper end of xx values are high relative to that witnessed in traditional potassium chemical strengthening of commercial glasses, specialized techniques such as electric field-assist chemical strengthening and compaction prior to chemical strengthening have shown compressive stress magnitudes as high as about 1,400 MPa for SLS-type glasses5,48 and 1,240 MPa for SAS-type glasses.49
91
(A)
(B)
Figure 30. SPS model: Mean across chemical strengthening times for maximum max initial stress xx versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature.
92 6. Strain Model with Densification and Shear Flow (DSPS model) Laboratory measurements, or quantities derived therefrom, including step height and elastic strain integrand, were input into a strain model including shear flow and densification, as suggested by Varshneya.34 The model is referred to as the DSPS model hereinafter. Model details are covered in the Introduction. Tabulated inputs, intermediate values, and outputs are given in Appendix H. Within this model, the additional variable related to densification requires a maximum total (or initial) strain value to be assumed for a solution to the system of equations to be obtained. Maximum total strain values were selected from results of molecular dynamics simulations presented in Section II, which were assumed representative of the maximum total strain as the low temperature and short time scale of the MD simulations prevented viscous relaxation. In terms of stress (biaxial plane strain condition), these values were about -1,660 MPa and -1,360 MPa, for SLS and SAS, respectively, which were then converted to linear strain by multiplication by (1-ν)/E using elastic properties obtained from the stuffed molecular dynamics simulations (Table IX). Note, within the DSPS model calculations, Poisson’s ratio ν of 0.22 was utilized. Plastic-to-total strain ratios were computed as the quotient of the plastic strain integrand (deviatoric plastic or hydrostatic plastic) to the total strain integrand and are represented by Rpla-D and Rpla-H for the deviatoric plastic and hydrostatic plastic ratios, respectively. The Rpla-D and Rpla-H for the SAS series are shown versus logarithm of time in Figure 31. No consistent trend is observed for Rpla-D. The hydrostatic plastic-to-total strain ratio Rpla-H generally increases in a linear fashion, starting at shorter times with a value near zero.
93 (A)
(B)
Figure 31. DSPS model: Plastic-to-total strain ratios for SAS for (A) deviatoric Rpla-D and (B) hydrostatic Rpla-H components versus natural logarithm of time. Lines are drawn to guide the eye.
The deviatoric and hydrostatic plastic-to-total strain ratios averaged across chemical strengthening times for each temperature series are given in Figure 32, where the error bars represent the observed range. For the SLS series, the Rpla-D increases with
94 increasing temperature, which holds despite the large ranges in values observed. On the other hand, the Rpla-H appears to decrease between 400 and 450 °C, and then marginally increase at 500 °C. In light of the large ranges observed for this quantity, it is likely there is little significance in the movement of its average across temperature. For the SAS series, the Rpla-D shows a steady increase with increasing temperature. The Rpla-H has a wide range of values, but is generally flat near about 0.1 to 0.2. For both glass series, there is appreciable spread of the plastic-to-total strain components, particularly for the hydrostatic component, suggesting the model is highly sensitive to the inputs. The deviatoric component is generally greater than the hydrostatic component for both glass series. The deviatoric and hydrostatic plastic components for the SLS series are consistently larger than that for the SAS series (Rpla-D 0.3-0.45 for SLS versus 0.2-0.3 for SAS and Rpla-H 0.3-0.4 SLS versus 0.1-0.2 SAS).
95
(A)
(B)
Figure 32. DSPS model: Deviatoric Rpla-D and hydrostatic Rpla-H plastic-to-total strain ratios versus chemical strengthening temperature for (A) SLS series and (B) SAS series. Error bars represent the range observed for each temperature.
96 D. Discussion
1. General Dimensional Swelling Dimensional changes induced by chemical strengthening are complex phenomena that involve contributions from chemical concentration, stress, and associated relaxation processes. Acknowledging some difficulty introduced by substrate deformation (Appendix C), the step height generally develops in a predictable manner with the square- root of time (Figure 18) or normalized concentration integrand (Figure 22) indicating a close relationship to the total quantity of the stuffing ion. The elastic strain integrand also displays this relationship with increasing normalized concentration, although to a lesser extent than dimensional swelling on account of stress relaxation at higher chemical strengthening temperatures and at longer times (Figure 23). Pairing these observations, the SLS series shows a larger step height magnitude per quantity stuffing ion, but a lower elastic strain integrand per quantity stuffing ion than the SAS series. This highly suggests the SLS series exhibits greater stress relaxation by shear flow than the SAS series, even prior to analysis by strain models, for the chemical strengthening temperatures examined here. Note, this observation assumes that SLS and SAS have similar concentrations of Na2O available for exchange, which is reasonable per the compositions given in Table XI.
2. Elastic Finite Element Method Inserting stress profiles into finite element simulations allows for comparison of the predicted elastic edge swelling and step height to that observed in laboratory measurements. This comparative approach is most applicable to chemical strengthening at low temperatures and at short times due to the presence of stress relaxation at higher chemical strengthening temperatures and longer times. At higher temperatures and longer times, the relaxed stress profiles underestimate the observed edge swelling (Figure 25C and Appendix F) and underestimate step heights (Figure 26) because the FEM model is elastic-only and does not account for material history or relaxation (path dependence). Thus, swelling comparisons at low chemical strengthening temperature and short times are most appropriate.
97 Of the shortest chemical strengthening times for each temperature series, edge swelling magnitude is consistently over-predicted by the elastic FEM model (Figure 25A and Appendix F), with the exception of SAS-500-0.25. This suggests that either the stress profiles have been over-estimated at short times, as mentioned in the results, or that at shorter chemical strengthening times the elastic properties are different from those used in the model. In the case of notably different elastic properties, the lower edge swelling of the laboratory observation relative to the FEM model would imply the Young’s modulus of the chemically strengthened region is higher than that input into the model (i.e. less strain generated per equivalent stress). Higher Young’s modulus of ion exchange stuffed layers has been reported in literature, with estimated increases of up to 10-15%.16,18,19 A relaxation mechanism appears an unlikely explanation for the short- time edge swelling discrepancy for the following reasons. If a higher stress state had existed prior to the short time observation, relaxation of the stress state by shear flow would result in similar edge swelling to that predicted by FEM, but with greater step height than that predicted by FEM. Similarly, if relaxation had occurred by densification, the edge swelling would again be expected to result in a profile similar to that of FEM, and the step height would also be similar to that predicted by FEM (i.e. densification is shape conserving and equivalent in all dimensions, and therefore might not be readily observed by these dimensional changes). Thus, regardless of the stress relaxation mechanism, elastic incompatibility between the ion-exchange stuffed layers and the substrate dictates the edge deformation. FEM step heights range from similar to 40% of the laboratory step height for low-temperature and short time conditions (Figure 26). This range is too wide to provide further insight into the source of the edge swelling difference at short times. Overall, a difference in elastic properties within the stuffed layers is the most consistent explanation for the short-time edge swelling difference. At intermediate times, edge swelling generally agrees well in both shape and magnitude between laboratory and elastic FEM measurements (Figure 25B and Appendix F). This suggests the source of the edge swelling difference at short time, related to elastic properties or otherwise, is largely absent at intermediate times. Step height at intermediate times is generally similar to slightly under-predicted by elastic FEM (Figure 26), consistent with the edge swelling agreement. One interesting,
98 relatively persistent feature is the slight departure of the laboratory edge profile shape from the edge position to about 200 μm, although sometimes extending to about 400 μm, where the magnitude of the edge swelling is slightly higher than that of the elastic FEM models, appearing mildly “bulged” (Appendix F, intermediate chemical strengthening times). This could be due to migration of salt from the exchange medium on the paste- coated edge surface to the bare top surface. Alternatively, it may be related to shear flow of the underlying substrate upon which the chemically strengthened layers are attached. The in-plane incompatibility between the chemically strengthened layers and those non- exchanged layers below it may be sufficient to cause local shear flow of these adjacent non-exchanged layers. If present, this would represent a new stress relaxation component, which is absent from the models in this study. One supporting observation for this mechanism is present in the interior stress state where stress direction is parallel to the y-dimension and the profile is along the thickness centerline from elastic FEM.
Examination of the σyy(z) stress profile in this dimension shows peak interior tension from 25 MPa up to 60 MPa for the SAS-450 series (Figure 33). Such a tensile stress magnitude is sufficient to induce shear flow when held at temperatures of 450 to 500 °C for times of several hours and longer.50 Note, this interior stress profile with tensile peak is the result of the finite y-dimension of the substrate near the edge and is not present in instances of biaxial plane strain, and therefore would not be anticipated to have a significant contribution to the strain models discussed later. It is important to note that the preceding analysis pertains to the edge swelling shape and magnitude in terms of profile alignment to an assumed common minima location as detailed in Appendix C. To fully reconstruct the edge deformation, highly accurate and precise thickness measurements of the y-dimension of the coupon near the edge before and after chemical strengthening would be required.
99
Figure 33. Elastic FEM stress profiles σyy(z) along the thickness centerline versus z-position from the edge for SAS-450 series. Note, only the tensile portion of the profile is shown.
3. Strain Models For the SLS series, the SPS model produced (Figure 28A) a similar trend in the
tot pla average total strain xx and the average plastic strain xx with increasing temperature to that observed by Glebov, et al.32 for K8 optical glass. The K8 composition is similar to 33 Schott BK7, and has approximate composition 75 SiO2 · 10 B2O3 · 10 Na2O · 6 K2O · 1 BaO (mol%).51,52 Based upon their compositions, both SLS and K8 have notable fractions of non-bridging oxygens, where K8 is expected to have a lower fraction because 12 of the formation of tetrahedral [BO4], and as such both are anticipated to exhibit some tendency for shear flow as well as densification, as suggested by microindentation experiments.53 The non-bridging oxygen fraction for SAS, on the other hand, is quite low, particularly if Mg takes the role of a pseudo-network former as has been suggested by MD simulations.54 Perhaps not unexpectedly, the SAS series average strain behavior
pla with increasing temperature is different from the SLS series. Fairly constant xx and
tot slightly decreasing xx with increasing temperature (Figure 28B) suggests that either the
100 average alkali site expansion is decreasing with increasing temperature, which is unlikely as similar behavior would then be anticipated for the SLS series, or there is a
tot temperature-dependent component influencing the xx that is not being adequately captured by the SPS model. A second indication of potential difficulties with the SPS model is the time dependence of the (Figure 27A). These problems are addressed with the introduction of densification in the DSPS model. Despite these shortcomings of the SPS model relative to the SAS series, the aggregate plastic-to-total strain ratios Rpla with increasing temperature (Figure 29) appear qualitatively reasonable, where the SLS series has a notably greater slope than the SAS series. The greater rate of increase of the Rpla with increasing temperature for SLS compared to SAS is expected in terms of the proximity of the chemical strengthening temperatures to the glass transition temperatures
(for 400-500 °C, 0.75-0.94Tg for SLS and 0.64-0.80Tg for SAS). Finally, the maximum
max initial stress xx (Figure 30) obtained from the maximum total strain produces values that, while high relative to traditional chemical strengthening, are attainable by low- temperature electric field-assist chemical strengthening5,48 and by compaction prior to chemical strengthening.49 Thus, overall, the SPS model appears to generate reasonable Rpla values and values for glasses with larger fractions of shear plastic flow during chemical strengthening, such as SLS, but produces questionable results for glasses with lower fractions of shear plastic flow, such as SAS. Incorporating densification into the SPS model generated the DSPS model. Within the DSPS model a maximum total strain must be assumed and, for the present analysis, the maximum total strain values were obtained from MD simulations as detailed in the results. The average total strain is constant within this model, thus outputs of the deviatoric Rpla-D (shear) and the hydrostatic Rpla-H (densification) plastic-to-total strain ratios are of primary interest. The time-dependence of the average total strain from the SPS model is now evident in the Rpla-H (Figure 31B) in the DSPS model. Interestingly, 55 near-Tg indentation creep experiments of Shang, et al. have also noted an increase of densification relative to shear viscous flow with increasing time under load as determined by residual indentation volume and pile-up volume measurements, although this observation was for a soda-lime silicate glass. These authors ascribe this as possibly due
101 to shear- or pressure-thinning that occurs under the high initial stress of indentation and decays as stress is relaxed, causing an increase in the relative densification to be observed at longer loading times. Based upon the fact that the initial compressive stress magnitudes by chemical strengthening exceed 1 GPa as predicted by the SPS model and MD simulations (Section II), a shear-thinning mechanism56 could potentially be present at early times during traditional chemical strengthening, limited to layers that have high concentrations of stuffing alkali ions that have been exchanged rapidly due to a high chemical potential gradient. If present, the mechanism would be localized to the first few microns of ion-exchanged layers and is not captured by the present Rpla-D trends which are representative averages throughout the ion-exchanged depth (Figure 31A). Collectively examining the deviatoric and hydrostatic plastic-to-total strain ratios for each temperature (Figure 32), both the SLS series and the SAS series are observed to display an increasing Rpla-D with increasing temperature. This is expected on the basis of 55 increasing shear flow with increasing temperature during sub-Tg indentation and sub-Tg three-point bend and uniaxial compression experiments.50 On the other hand, no trends of the Rpla-H with temperature were identified for either glass series. The wide range of observed values allows much room for interpretation of a trend for this component versus temperature. As noted earlier, the source of the wide range is, in part, attributable to the time dependence of this quantity. Again turning to sub-Tg microindentation experiments for soda-lime silicate,55 the ratio of densification to shear flow, from residual indentation volume and pile-up volume measurements, was reported to be consistently in favor of densification by at least a factor of about three or greater in the 400-500 °C temperature range. In separate, but similar, studies of room temperature microindentation with post- annealing recovered volume measurements, ratios for volume of densification to volume of shear flow were roughly 2:1 for SLS57 and 1:1 for SAS-like glasses.58 Here, the ratio of densification to shear flow is generally around 0.9:1 for SLS and 0.7:1 for SAS as averages across all temperatures (Figure 32). The lower tendency for densification by ion-exchange stuffing compared to microindentation is understandable as the introduction of the larger alkali ion into the silicate network lowers the capacity for network densification.
102 Comparison of the magnitudes of the Rpla-D and the Rpla-H between the SLS series and SAS series reveals the SLS series has higher Rpla-D and Rpla-H than SAS. Given that the glasses have similar Poisson’s ratios and thus presumably similar atomic packing density,59 the higher degree of network cross-polymerization in the SAS may be responsible for its lower plastic ratios than SLS. Overall, plastic contributions to total strain, averaged throughout the ion-exchanged layers, are about 70% for the SLS series and about 40% for SAS series within the DSPS model, and are slightly lower within the SPS model. In either instance, this represents about half of the total strain, which could potentially be retained as elastic strain, i.e. alkali oxide glass compositions could potentially regularly generate in excess of -1,200 MPa by chemical strengthening if the various relaxation mechanisms could be removed. Future developments in chemical strengthening technology, particularly in glass composition and in ion-exchange technique development, may preserve this elastic strain, providing some room for improvement of glass mechanical strength.24,60 The SPS and DSPS models are highly sensitive to the inputs. Refined techniques for determining step height and stress profiles would improve the quality of the outputs from the models. For step height measurement, potential improvement may be obtained from use of a thicker substrate that is less apt to deform during the chemical strengthening process, thus remedying that described in Appendix C. As for stress profiles, quantification of retardation with a manual birefringence compensator is somewhat subjective as it involves positioning the darkest portion of a fringe at the center of a crosshair. Automated compensation quantification via use of a liquid crystal compensator, for example, may yield improved accuracy of the measured stress profiles.
E. Conclusions Dimensional monitoring of edge swelling and step height during chemical strengthening provides some indication of the elastic-plastic processes occurring within ion-exchanged stuffed layers. Step height and integrated elastic strain were observed to scale with integrated concentration, where SLS displayed larger step height and lesser elastic strain per quantity of stuffing ion than SAS, indicating a greater tendency for stress relaxation by shear flow. For the SAS series, edge swelling compared to elastic
103 FEM agreed well at intermediate chemical strengthening times, where the discrepancy at short times was attributed, potentially, to modified elastic properties and the discrepancy at long times was related to stress relaxation. Subtle, but consistent, departure of the observed edge swelling from that predicted by elastic FEM was suggested as indicative of stress relaxation occurring within the non-ion-exchanged substrate. Step height and integrated elastic strain were used as inputs into simple strain models. A strain model allowing for plasticity by shear flow only, the SPS model, provided reasonable qualitative output, although a more appropriate model incorporating plasticity by densification and by shear flow, the DSPS model, is preferred as it generated more appropriate strain behavior with varied temperature and time for both glass compositions studied. When averaged throughout the ion-exchanged stuffed layers, the ratio of overall plastic strain to total strain was found to be about 70% for SLS and about 40% for SAS. The plastic strain fraction attributable to shear flow was generally higher than that due to densification for both glass series. The shear flow plastic strain fraction was observed to increase with increasing chemical strengthening temperature and displayed a greater rate of increase for the SLS series than the SAS series. Finally, it is suggested the large observed plastic strain fractions present opportunities for future development to mitigate relaxation and maximize elastic strain, resulting in higher surface compression by chemical strengthening, and improved glass strength.
104 F. References 1. O. Richmond, W. C. Leslie, and H. A. Wriedt, "Theory of Residual Stresses Due to Chemical Concentration Gradients," Transactions of the ASM, 57 [1] 294-300 (1964).
2. A. R. Cooper and D. A. Krohn, "Strengthening of Glass Fibers: II, Ion Exchange*," J. Am. Ceram. Soc., 52 [12] 665-9 (1969).
3. A. K. Varshneya, "Kinetics of Ion Exchange in Glass," J. Non-Cryst. Solids, 19, 355-65 (1975).
4. M. Abou-El-Leil and A. R. Cooper, "Fracture of Soda-Lime Glass Tubes by Field-Assisted Ion Exchange," J. Am. Ceram. Soc., 61 [3-4] 131-6 (1978).
5. E. E. Shaisha and A. R. Cooper, "Residual Stress in Singly and Doubly Ion- Exchanged Glass," J. Am. Ceram. Soc., 64 [1] 34-6 (1981).
6. E. E. Shaisha and A. R. Cooper, "Ion Exchange of Soda-Lime Glass with Univalent Cations," J. Am. Ceram. Soc., 64 [5] 278-83 (1981).
7. H. M. Garfinkel and C. B. King, "Ion Concentration and Stress in a Chemically Tempered Glass," J. Am. Ceram. Soc., 53 [12] 686-91 (1970).
8. H. A. Schaeffer and R. Heinze, "Stress Formation by Ion Exchange of Glass," Glastech. Ber., 47 [9] 199-208 (1974).
9. A. Y. Sane and A. R. Cooper, "Stress Buildup and Relaxation During Ion Exchange Strengthening of Glass," J. Am. Ceram. Soc., 70 [2] 86-9 (1987).
10. J. Shen and D. J. Green, "Prediction of Stress Profiles in Ion Exchanged Glasses," J. Non-Cryst. Solids, 344 [1–2] 79-87 (2004).
11. D. A. Krohn, "Stress Relaxation During Ion Exchange Strengthening of Glass Fibers," Glass Technol., 12 [2] 36-41 (1971).
12. A. K. Varshneya, Fundamentals of Inorganic Glasses, 2nd ed. The Society of Glass Technology, Sheffield, 2006.
13. A. J. Burggraaf, "The Strengthening of Glass by Ion Exchange. Part 2. Stress Formation and Stress Relaxation after Ion Exchange in Alkali Aluminosilicate Glasses in Connection with Structural Changes in the Glass," Phys. Chem. Glasses, 7 [5] 169-72 (1966).
105 14. A. J. Burggraaf, "The Mechanical Strength of Alkali-Aluminosilicate Glasses after Ion Exchange," Philips Res. Rep. Suppl., 3, 1-106 (1966).
15. D. K. Hale, "Strengthening of Silicate Glasses by Ion Exchange," Nature, 217 [5134] 1115-8 (1968).
16. J. D. Mackenzie and J. Wakaki, "Effects of Ion Exchange on the Young's Modulus of Glass," J. Non-Cryst. Solids, 38–39, Part 1, 385-90 (1980).
17. V. Tyagi and A. K. Varshneya, "Measurement of Progressive Stress Buildup During Ion Exchange in Alkali Aluminosilicate Glass," J. Non-Cryst. Solids, 238 [3] 186-92 (1998).
18. V. Tyagi, "Physical Properties of Ion-Exchange Strengthened Glasses"; M.S. Thesis. Alfred University, Alfred, NY, 1996.
19. A. K. Varshneya, "Physical Properties of Ion-Exchanged and Melt-Processed Glasses Differ," GlassResearcher, 10-11 [2-1] 21-6, 51 (2001).
20. F. Orgaz and J. M. Fernandez Navarro, "Prediction of Results from Strengthening Glass by Using the Misfitting Sphere Theory"; pp. 513-24 in Strength of Inorganic Glasses. Edited by C. R. Kurkjian. Plenum Press, New York, 1985.
21. P. K. Kreski, A. K. Varshneya, and A. N. Cormack, "Investigation of Ion- Exchange ‘Stuffed’ Glass Structures by Molecular Dynamics Simulation," J. Non- Cryst. Solids, 358 [24] 3539-45 (2012).
22. A. Tandia, K. D. Vargheese, and J. C. Mauro, "Elasticity of Ion Stuffing in Chemically Strengthened Glass," J. Non-Cryst. Solids, 358 [12–13] 1569-74 (2012).
23. A. Tandia, K. D. Vargheese, J. C. Mauro, and A. K. Varshneya, "Atomistic Understanding of the Network Dilation Anomaly in Ion-Exchanged Glass," J. Non-Cryst. Solids, 358 [2] 316-20 (2012).
24. A. K. Varshneya, "Chemical Strengthening of Glass: Lessons Learned and yet to Be Learned," Int. J. Appl. Glass Sci., 1 [2] 131-42 (2010).
25. J. C. Mauro, C. S. Philip, D. J. Vaughn, and M. S. Pambianchi, "Glass Science in the United States: Current Status and Future Directions," Int. J. Appl. Glass Sci., 5 [1] 2-15 (2014).
26. N. Valles-Villarreal, A. Villalobos, and H. Márquez, "Stress in Copper Ion- Exchanged Glass Waveguides," J. Lightwave Technol., 17 [4] 606-12 (1999).
106 27. R. Oven, "Surface Expansion of Channel Waveguides Formed by Ion Exchange in Glass," J. Appl. Phys., 100 [5] 053513/7 (2006).
28. R. Oven and M. Yin, "Assessment of Ion Exchanged Channel Waveguides in Glass Using Interference Microscopy," Opt. Commun., 260 [2] 506-10 (2006).
29. A. Y. Sane and A. R. Cooper, "Anomalous Stress Profiles in Ion-Exchanged Glass," J. Am. Ceram. Soc., 61 [7] 359-62 (1978).
30. V. Jain and A. K. Varshneya, "Finite-Element Analysis of Network Dilatation in Ion-Exchanged Glass Rods after Slicing," J. Am. Ceram. Soc., 70 [8] 595-98 (1987).
31. T. Miyamoto and H. Insono, Hoya Corporation, "Chemical Reinforced Glass Substrate Having Desirable Edge Profile and Method of Manufacturing the Same," U.S. Pat. 6595028 B1, July 2003.
32. L. B. Glebov, V. G. Dokuchaev, N. B. Nikonorov, and G. T. Petrovskii, "Change in the Volume of Glass after Low-Temperature Ion Exchange," Sov. J .Glass. Phys. Chem., 14 [2] 131-7 (1989).
33. L. B. Glebov and M. N. Tolstoi, "Design of Russian Optical Glasses"; pp. 823-6 in Vol. (Supplement) 2, Handbook of Laser Science and Technology. Edited by M. Weber. CRC Press, Ann Arbor, MI, 1995.
34. A. K. Varshneya, "The Physics of Chemical Strengthening of Glass: Room for a New View," J. Non-Cryst. Solids, 356 [44-49] 2289-94 (2010).
35. T. P. Seward and A. K. Varshneya, "Inorganic Glasses: Commercial Glass Families, Applications, and Manufacturing Methods"; pp. 8.1-8.173 in Handbook of Materials for Product Design. Edited by C. Harper. McGraw-Hill, New York, 2001.
36. W. H. Dumbaugh, Corning Glass Works, "Method for Making Glass Articles with Defect-Free Surfaces," U.S. Pat. 4102664, July 1978.
37. S. T. Gulati and H. E. Hagy, "Theory of the Narrow Sandwich Seal," J. Am. Ceram. Soc., 61 [5-6] 260-3 (1978).
38. ABAQUS 6.8, [Computer Program] Dassault Systèmes, Providence, RI, USA, 2008.
39. P. K. Gupta, "Finite Beam Diameter Effect in Electron Microprobe Analysis," J. Phys. D: Appl. Phys., 3 [12] 1919 (1970).
107 40. A. K. Varshneya and Z. Mencik, "Application of Iterative Deconvolution Technique to Ion-Exchange Study of Glass," J. Am. Ceram. Soc., 57 [4] 170-2 (1974).
41. A. K. Varshneya and M. E. Milberg, "Ion Exchange in Sodium Borosilicate Glasses," J. Am. Ceram. Soc., 57 [4] 165-9 (1974).
42. J. R. Varner and R. Lang-Egelkraut, "Influences of Ion-Exchange Temperature and Time on Stress Profiles and Strengths of Chemically Strengthened Glass"; pp. 465-70 in The Physics of Non-Crystalline Solids : Fourth International Conference. Edited by G. H. Frischat. Trans Tech Publications, Clausthal- Zellerfeld, 1976.
43. S. Gomez, M. J. Dejneka, A. J. Ellison, and K. R. Rossington, "A Look at the Chemical Strengthening Process: Alkali Aluminosilicate Glasses Vs. Soda-Lime Glass"; pp. 61-6 in 71st Glass Problems Conference: Ceramic Engineering and Science Proceedings, Vol. 32. Edited by C. H. Drummond John Wiley & Sons, Inc., New York, 2011.
44. M. J. C. Hill and I. W. Donald, "Stress Profile Characteristics and Mechanical Behavior of Chemically Strengthened Lithium Magnesium Aluminosilicate Glasses," Glass Technol., 30 [4] 123-7 (1989).
45. W. Bradshaw, "Stress Profile Determination in Chemically Strengthened Glass Using Scattered Light," J. Mater. Sci., 14 [12] 2981-8 (1979).
46. P. J. Dwivedi and D. J. Green, "Indentation Crack Shapes and Their Evolution During Subcritical Crack Growth in Ion-Exchanged Glasses"; pp. 473-90 in Fractography of Glasses and Ceramics III, Vol. 64. Edited by J. R. Varner, V. D. Frechette, and G. D. Quinn. The American Ceramic Society, Westerville, OH, USA, 1996.
47. A. K. Varshneya and W. C. LaCourse, "Technology of Ion Exchange Strengthening of Glass: A Review"; pp. 365-76 in Advances in Fusion and Processing of Glass Vol. 29. Edited by A. K. Varshneya, D. F. Bickford, and P. P. Bihuniak. American Ceramic Society, Westerville, Ohio, USA, 1993.
48. H. Ohta and M. Hara, "Ion Exchnage in Sheet Glass by Electrolysis," Reports Res. Lab. Asahi Glass Co., Ltd., 20 [1] 15-31 (1970).
49. M. N. Svenson, L. M. Thirion, R. E. Youngman, J. C. Mauro, S. J. Rzoska, M. Bockowski, and M. M. Smedskjaer, "Pressure-Induced Changes in Interdiffusivity and Compressive Stress in Chemically Strengthened Glass," ACS Appl. Matter. Inter., 6 [13] 10436-44 (2014).
108 50. J. Shen, D. J. Green, R. E. Tressler, and D. L. Shelleman, "Stress Relaxation of a Soda Lime Silicate Glass Below the Glass Transition Temperature," J. Non-Cryst. Solids, 324 [3] 277-88 (2003).
51. D. Ehrt and P. Ebeling, "Radiation Defects in Borosilicate Glass," Glass Technol., 44 [2] 46-9 (2003).
52. A. Matthai, D. Ehrt, and C. Russel, "Voltammetric Investigations of the Redox Behaviour in Fe, Ni, Co and Sn Doped Glass Melts of AR and BK7 Type," Glastech. Ber., 72 [2] 38 (2000).
53. S. Yoshida, J. C. Sanglebœuf, and T. Rouxel, "Quantitative Evaluation of Indentation-Induced Densification in Glass," J. Mater. Res., 20 [12] 3404-12 (2005).
54. A. Pedone, G. Malavasi, M. C. Menziani, U. Segre, and A. N. Cormack, "Role of Magnesium in Soda-Lime Glasses: Insight into Structural, Transport, and Mechanical Properties through Computer Simulations," J. Phys. Chem. C, 112 [29] 11034-41 (2008).
55. H. Shang, T. Rouxel, M. Buckley, and C. Bernard, "Viscoelastic Behavior of a Soda-Lime-Silica Glass in the 293–833 K Range by Micro-Indentation," J. Mater. Res., 21 [3] 632-8 (2006).
56. J. H. Simmons, "What Is So Exciting About Non-Linear Viscous Flow in Glass, Molecular Dynamics Simulations of Brittle Fracture and Semiconductor–Glass Quantum Composites," J. Non-Cryst. Solids, 239 [1–3] 1-15 (1998).
57. S. Yoshida, J. C. Sanglebœuf, and T. Rouxel, "Indentation-Induced Densification of Soda-Lime Silicate Glass," Int. J. Mater. Res., 98 [5] 360-4 (2007).
58. J. Kjeldsen, M. M. Smedskjaer, J. C. Mauro, and Y. Yue, "Hardness and Incipient Plasticity in Silicate Glasses: Origin of the Mixed Modifier Effect," Appl. Phys. Lett., 104 [5] 051913/4 (2014).
59. T. Rouxel, "Elastic Properties and Short-to Medium-Range Order in Glasses," J. Am. Ceram. Soc., 90 [10] 3019-39 (2007).
60. C. R. Kurkjian, P. K. Gupta, and R. K. Brow, "The Strength of Silicate Glasses: What Do We Know, What Do We Need to Know?," Int. J. Appl. Glass Sci., 1 [1] 27-37 (2010).
109 IV. SUMMARY AND CONCLUSIONS
A combination of molecular dynamics simulations and laboratory measurements of dimensional changes was used to study elastic-plastic processes during glass chemical strengthening. Molecular dynamics simulations with potassium replacing sodium in a soda-lime silicate glass and in a sodium aluminosilicate glass produced network expansion similar to that expected from stress measurements of glasses chemically strengthened by specialty techniques. Volume expansion per quantity of stuffing ion was found to increase with a decrease in network cross-polymerization and with increasing Poisson’s ratio. The immediate stages of stuffing alkali accommodation were found to involve potassium nearest-neighbor features similar to the as-melted potassium glass, but network connectivity and topology similar to the as-melted sodium host glass. Intertetrahedral bond angle adjustments were observed and were found to have a prominent role in the alkali accommodation process. Sodium aluminosilicates exhibited greater intertetrahedral bond angle changes attributable to the high degree of network polymerization and the alkali site similarity to the host. Dimensional changes after potassium chemical strengthening were studied with an optical profilometer. Comparison of vertical swelling of exchanged and adjacent non- exchanged regions on the same glass surface, i.e. step height, paired with the underlying stress profiles and chemical diffusion profiles, showed a soda-lime silicate glass to have greater swelling and less integrated stress per quantity of stuffing potassium ion than a sodium aluminosilicate glass. This indicated a greater tendency for relaxation by shear flow in soda-lime silicate. Measurement of edge deformation generated by selective- surface chemical strengthening produced deformation profiles largely representative of the underlying stress state as determined with the aid of elastic finite element analysis. At short chemical strengthening times, edge deformation was less than that anticipated from elastic finite element simulations, which was potentially due to altered elastic properties of chemically strengthened layers. At intermediate chemical strengthening times, slight, but persistent, deviation of the laboratory edge profile from that predicted by elastic finite element simulation was a potential sign of substrate shear flow. Using
110 maximum total strain inputs from the molecular dynamics simulations, paired with laboratory step height, stress profile, and chemical diffusion profile measurements, a strain model incorporating elastic, deviatoric plastic (shear flow), and hydrostatic plastic (densification) contributions was evaluated. Plastic strain as a percentage of total strain, averaged throughout the chemically strengthened layers, was found to be on the order of 70% for the soda-lime silicate glass and 40% for the sodium aluminosilicate glass. For both compositions, the fraction of deviatoric plastic strain to the total strain was observed to increase with increasing chemical strengthening temperature. These complementary techniques of simulation and laboratory measurements proved capable for studying the poorly understood elastic-plastic phenomena that occur during glass chemical strengthening. Results showed that traditional chemical strengthening produces much lower maximum compressive stress than that suggested by the total strain. Early relaxation modes were proposed to bridge the gap between maximum compressive stress produced by specialty chemical strengthening techniques versus that obtained traditionally. These results indicate notable improvements to maximum compressive stress can potentially be achieved through prevention of the various forms of relaxation.
111 V. FUTURE WORK
In terms of molecular dynamics simulations of stuffing alkali accommodation in silicate networks, three areas are of interest. The first is use of higher temperature during the simulation process, specifically greater than typical laboratory chemical strengthening temperatures, in order to accelerate relaxation processes and gather information on structural features of these processes. This approach may provide better opportunities for comparison with structural features of laboratory glasses, which have generally undergone notable relaxation. A second area of interest is alternative boundary conditions during simulation of stuffing alkali accommodation. Presently, NPT and NVT boundary conditions have been examined, but the typical laboratory glass has biaxial plane strain boundaries (NVT-like) with free expansion in the third dimension (NPT- like). Incorporation of these exact boundary conditions, while maintaining periodicity in all dimensions, would likely provide the best analog to laboratory glasses. The third area of interest is the study of system dynamics and/or kinetics. Gaining a quantitative sense for how alkali stuffing alters viscosity and alkali interdiffusion properties would surely be of value. From the laboratory perspective, opportunities exist to improve the measurements obtained by the current methods. For example, use of thicker substrates may help mitigate overall part deflection, improving the resulting step height measurements. Edge swelling was limited to surface profiles in the present study, although there would potentially be opportunities to estimate molar volume on a layer-by-layer basis if optical thickness measurement of sufficient spatial resolution and accuracy were used. The present study worked exclusively with traditional chemical strengthening and examined average properties across entire chemical strengthening profiles. Future work with fractional ion-exchange sources, that is, with a minor level of intentional impurity to influence the concentration of exchange into the glass, may allow for concentration- dependence of the plastic processes to be determined. Finally, performing dimensional studies with techniques that tend to bury stress profiles, such as chemical strengthening with electric-field assistance has advantages in terms of (1) a near “time-zero” stress
112 measurement at the leading edge of the chemical strengthening profile, (2) near-constant concentration throughout the chemically strengthened region, and (3) easily established history of each of the chemically strengthened layers. These serve as major simplifications by removing the concentration gradient and offer the opportunity for improved understanding of elastic-plastic processes. Finally, further study into surface compression loss upon cooling from the chemical strengthening temperature to room temperature would be of interest. Determining whether this is a universal feature of chemically strengthened glasses will help improve quantitative comparisons between laboratory and molecular dynamics studies, as well as potentially shedding light on the source of the phenomenon.
113 APPENDIX
A. MD Simulation Cutoff Distances
Table XV. Cutoff Distances for All Simulations
Table XVI. Cutoff Distances Specific to SLS Simulations
Table XVII. Cutoff Distances Specific to SMAS Simulations
114 B. MD Simulation Qn Distributions
Table XVIII. Qn Distributions for All Simulations
115 C. Surface Profile Notes
1. Substrate Deformation Substrate deformation occurred after cutting as-received glass, after annealing, and after chemical strengthening. The magnitude of the deformation was generally small after cutting as-received glass, but increased in subsequent annealing and chemical strengthening steps. As an example, line profiles of a preliminary trial coupon are provided in Figure 34. A single coupon had surface profiles of both 16 mm x 8 mm surfaces measured using an optical profilometer after each of the steps listed above. The as-received, cut to size coupon exhibits a “hill” side (Figure 34A), peaking at about 175 nm y-position near the 4,000 μm z-position midpoint of the coupon, and a “valley” side (Figure 34B), although the valley is much shallower than the hill. After annealing, the magnitude of the hill and valley features is enhanced by 75 to 100 nm (y-position). After chemical strengthening, the intended edge swelling is observed, although there is a marked difference in the overall profile shape for the hill and valley sides. The hill side shows a minimum near 1,000 μm z-position. The valley side shows a transition in slope near that same z-position, but does not display an extremum at this location. This hill- valley substrate deformation was common for both SLS and SAS glasses, with an occasional coupon taking a saddle or cylindrical shape. The hill-valley magnitude was usually somewhat small relative to the edge deformation induced by chemical strengthening, allowing an averaging procedure to be used between edge profiles from the hill surface and valley surface for later comparison against edge swelling predicted by elastic finite element simulations. Edge averaging is described in the next passage. Step height measurements were obtained from the 16 mm x 8 mm surfaces (Figure 16, coupon #3) and were also susceptible to errors introduced by substrate deformation.
116
(A)
(B)
Figure 34. Substrate deformation at various stages of preparation for a preliminary trial coupon, (A) hill side and (B) valley side.
2. Edge Profile Averaging For each coupon, line profiles were extracted from the measured surface profiles (Figure 16, coupons #1 & 2). Generally, one hill profile and one valley profile were 117 obtained. The line profiles were translated along the y-axis such that the profile minimum was relocated to the y-position of zero (see “Valley Side” and “Hill Side” in Figure 35). Then a point-wise average of the two profiles was taken at each z-position. Finally, the new, average profile was translated along the y-axis to re-align its minimum with the y-position of zero (“Average” in Figure 35). This procedure was used for all instances where reliable hill and valley line profiles were available.
Figure 35. Example of the edge profile averaging method for SAS-450-1.
118 D. Edge Profiles After Chemical Strengthening
Figure 36. Edge profiles after edge chemical strengthening determined by white light profilometer for SLS series.
119
Figure 37. Edge profiles after edge chemical strengthening determined by white light profilometer for SAS series.
120 E. Stress Profiles After Chemical Strengthening
Figure 38. Stress profiles σyy(z) after chemical strengthening determined by polarized light microscopy for SLS series.
121
Figure 39. Stress profiles σyy(z) after chemical strengthening determined by polarized light microscopy for SAS series.
122 F. Edge Profile Comparisons with Elastic FEM
Figure 40. Edge swelling comparison for SAS-250-239 and SAS-300 series between laboratory measurement (solid) and elastic FEM (dashed). 123
Figure 41. Edge swelling comparison for SAS-350 series between laboratory measurement (solid) and elastic FEM (dashed).
124
Figure 42. Edge swelling comparison for SAS-400 series between laboratory measurement (solid) and elastic FEM (dashed).
125
Figure 43. Edge swelling comparison for SAS-450 series between laboratory measurement (solid) and elastic FEM (dashed).
126
Figure 44. Edge swelling comparison for SAS-500 series between laboratory measurement (solid) and elastic FEM (dashed).
127
G. SPS Model Inputs and Outputs Tables
StDev 2.40E-04 2.20E-04 2.70E-04 1.80E-04 1.50E-04 1.00E-04 2.90E-04 2.30E-04 1.40E-04 1.30E-04 6.00E-05 1.20E-04 2.40E-04 1.90E-04 1.70E-04 1.30E-04 1.10E-04
-2.37E-03 -2.15E-03 -2.13E-03 -2.29E-03 -2.30E-03 -1.65E-03 -2.80E-03 -2.92E-03 -2.91E-03 -3.20E-03 -2.13E-03 -2.73E-03 -2.79E-03 -3.04E-03 -3.04E-03 -2.94E-03 -2.98E-03
Plastic Strain Average (unitless) Average Strain Plastic 0.40 0.30 0.40 0.13 0.15 0.11 0.18 0.08 0.29 0.14 0.16 0.18 0.30 0.06 0.11 0.15 0.15 StDev StDev 2.10E-04 2.90E-04 1.80E-04 1.80E-04 1.40E-04 1.00E-04 2.60E-04 1.20E-04 1.10E-04 7.00E-05 6.00E-05 6.00E-05 1.60E-04 1.10E-04 6.00E-05 7.00E-05 5.00E-05 3.70 4.63 5.10 4.61 5.90 7.22 4.57 6.02 8.21 9.13 4.52 5.69 8.69 9.87 11.67 13.61 12.93 -2.08E-03 -2.84E-03 -2.11E-03 -2.34E-03 -2.16E-03 -2.02E-03 -2.49E-03 -2.01E-03 -2.26E-03 -1.85E-03 -1.90E-03 -1.93E-03 -1.79E-03 -1.81E-03 -1.29E-03 -1.56E-03 -1.40E-03 Elastic Strain Average (unitless) Average Strain Elastic Normalized Concentration Integrand (micron) Integrand Concentration Normalized StDev 4.00E-04 5.00E-04 4.00E-04 3.40E-04 2.80E-04 1.90E-04 5.00E-04 3.30E-04 2.40E-04 2.00E-04 1.20E-04 1.60E-04 4.00E-04 3.00E-04 2.10E-04 1.80E-04 1.50E-04 StDev 5.00E-04 5.00E-04 6.00E-04 9.00E-04 1.00E-03 8.00E-04 9.00E-04 7.00E-04 9.00E-04 9.00E-04 1.40E-03 1.50E-03 6.00E-04 8.00E-04 8.00E-04 1.30E-03 1.30E-03 -4.50E-03 -5.00E-03 -4.20E-03 -4.63E-03 -4.46E-03 -3.66E-03 -5.30E-03 -4.93E-03 -5.17E-03 -5.05E-03 -4.02E-03 -4.66E-03 -4.60E-03 -4.86E-03 -4.32E-03 -4.50E-03 -4.37E-03 -2.66E-02 -3.64E-02 -3.25E-02 -4.20E-02 -4.43E-02 -5.43E-02 -3.19E-02 -4.39E-02 -6.37E-02 -6.40E-02 -8.76E-02 -9.65E-02 -2.76E-02 -3.95E-02 -4.12E-02 -5.40E-02 -6.09E-02 Initial Strain Average (unitless) Average Strain Initial
Ela Strain Integrand (micron) Integrand Strain Ela
StDev 5.00E-04 5.00E-04 2.40E-03 1.00E-03 4.00E-04 1.40E-03 9.00E-04 2.60E-03 1.00E-03 1.60E-03 8.00E-04 4.00E-03 6.00E-04 1.30E-03 3.10E-03 1.70E-03 2.20E-03 0.002 0.002 0.007 0.003 0.001 0.004 0.003 0.008 0.003 0.005 0.002 0.011 0.002 0.004 0.009 0.005 0.007
StDev
0.092 0.093 0.102 0.129 0.145 0.146 0.109 0.183 0.242 0.309 0.298 0.395 0.122 0.186 0.260 0.280 0.351 Step Height (micron) Height Step -2.37E-02 -2.15E-02 -2.55E-02 -3.21E-02 -3.68E-02 -3.46E-02 -2.80E-02 -4.96E-02 -6.41E-02 -8.64E-02 -7.66E-02 -1.07E-01 -3.35E-02 -5.17E-02 -7.59E-02 -7.93E-02 -1.01E-01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 StDev Integrated Plastic Strain (micron) Strain Plastic Integrated 10 10 12 14 16 21 10 17 22 27 36 39 12 17 25 27 34 StDev 6.00E-04 6.00E-04 2.40E-03 1.10E-03 6.00E-04 1.50E-03 1.00E-03 2.60E-03 1.00E-03 1.70E-03 1.00E-03 4.00E-03 6.00E-04 1.40E-03 3.10E-03 1.80E-03 2.30E-03 Depth (micron) Depth 9.05 4.02 9.00 1.00 2.25 4.00 6.25 9.00 16.00 25.00 36.90 49.00 65.75 16.00 25.02 37.05 50.00 Time (hours) Time C) ° -4.45E-02 -4.99E-02 -5.09E-02 -6.48E-02 -7.14E-02 -7.69E-02 -5.29E-02 -8.38E-02 -1.14E-01 -1.36E-01 -1.45E-01 -1.82E-01 -5.50E-02 -8.26E-02 -1.08E-01 -1.21E-01 -1.49E-01
400 400 400 400 400 400 450 450 450 450 450 450 500 500 500 500 500
. SPS Model Inputs for SLS Series Model SLS for Inputs SPS . . SPS Model Outputs for SLS Series, Part SLS for Model I Outputs SPS . Integrated Initial Strain (micron) Strain Initial Integrated
Temperature ( Temperature
XIX XX ID ID SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 Outputs
Inputs
Table Table
128
0.01 0.01 0.01 0.008 0.008 0.024 0.008 0.012 0.012 0.014 0.006 0.006 0.006 0.008 0.008 0.008 0.007 StDev 0.17 0.29 0.19 0.32 0.40 0.32 0.31 0.40 0.40 0.27 0.60 0.60 0.13 0.14 0.20 0.26 0.40 0.13 0.22 0.24 0.25 0.34 0.40 0.15 0.23 0.40 0.80 0.60 StDev 0.53 0.533 0.432 0.502 0.495 0.516 0.449 0.592 0.563 0.634 0.529 0.586 0.609 0.626 0.702 0.653 0.681 Plastic-to-Total Strain Ratio Strain Plastic-to-Total 7.28 8.48 5.09 6.81 9.80 5.60 6.66 7.49 10.11 10.66 14.00 13.40 19.66 27.90 10.26 11.90 16.50 21.20 10.73 14.20 18.84 23.07 28.06 36.10 17.11 22.80 29.20 39.20 0.01 0.01 0.01 0.008 0.008 0.024 0.008 0.012 0.012 0.014 0.006 0.006 0.006 0.008 0.008 0.008 0.007 StDev Normalized Concentration Integrand (micron) Integrand Concentration Normalized 0.47 0.467 0.568 0.498 0.505 0.484 0.551 0.408 0.437 0.366 0.471 0.414 0.391 0.374 0.298 0.347 0.319 StDev 1.60E-03 1.00E-03 1.30E-03 1.10E-03 1.80E-03 1.60E-03 1.80E-03 1.90E-03 2.20E-03 2.40E-03 4.00E-03 1.70E-03 1.60E-03 2.00E-03 3.30E-03 2.50E-03 3.50E-03 2.20E-03 2.50E-03 3.00E-03 3.20E-03 4.00E-03 4.00E-03 2.40E-03 2.50E-03 3.00E-03 3.50E-03 3.50E-03 Elastic-to-Total Strain Ratio Strain Elastic-to-Total StDev 7.00E-04 3.20E-04 6.00E-04 2.90E-04 1.70E-04 2.10E-04 3.10E-04 4.00E-04 3.00E-04 2.30E-04 1.20E-04 3.00E-04 5.00E-04 2.50E-04 4.00E-04 2.10E-04 1.90E-04
-8.77E-02 -6.56E-02 -1.08E-01 -8.60E-02 -1.08E-01 -8.79E-02 -1.07E-01 -1.13E-01 -1.46E-01 -1.80E-01 -2.37E-01 -9.27E-02 -7.90E-02 -1.22E-01 -1.59E-01 -1.69E-01 -1.74E-01 -1.15E-01 -1.53E-01 -1.92E-01 -2.18E-01 -2.82E-01 -3.34E-01 -9.82E-02 -1.70E-01 -2.63E-01 -3.18E-01 -3.25E-01
Ela Strain Integrand (micron) Integrand Strain Ela -6.40E-03 -4.66E-03 -5.00E-03 -6.96E-03 -6.24E-03 -4.79E-03 -6.13E-03 -8.20E-03 -7.80E-03 -9.45E-03 -6.56E-03 -7.83E-03 -7.40E-03 -9.09E-03 -8.70E-03 -8.03E-03 -7.82E-03 0.003 0.010 0.009 0.007 0.011 0.003 0.008 0.000 0.010 0.007 0.012 0.002 0.009 0.001 0.008 0.015 0.004 0.005 0.002 0.004 0.012 0.013 0.008 0.005 0.002 0.004 0.006 0.007 StDev
Plastic Strain Max (unitless) Max Strain Plastic
StDev 6.00E-04 4.00E-04 4.00E-04 2.60E-04 2.00E-04 1.30E-04 2.60E-04 1.20E-04 2.30E-04 1.10E-04 1.30E-04 1.10E-04 3.40E-04 1.20E-04 8.00E-05 1.20E-04 9.00E-05 0.127 0.102 0.210 0.153 0.183 0.098 0.136 0.182 0.253 0.364 0.536 0.113 0.155 0.213 0.249 0.316 0.366 0.179 0.268 0.349 0.443 0.541 0.684 0.183 0.372 0.449 0.557 0.771 Step Height (micron) Height Step 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 StDev -5.60E-03 -6.10E-03 -5.00E-03 -7.11E-03 -5.86E-03 -5.87E-03 -5.44E-03 -5.69E-03 -6.05E-03 -5.46E-03 -5.85E-03 -5.53E-03 -4.76E-03 -5.42E-03 -3.70E-03 -4.27E-03 -3.67E-03 15 19 26 26 37 12 20 22 34 50 72 14 20 28 41 49 69 27 38 58 72 91 25 53 76 120 108 125 Elastic Strain Max (unitless) Max Strain Elastic Depth (micron) Depth StDev 1.30E-03 7.00E-04 1.00E-03 5.00E-04 3.30E-04 2.70E-04 5.00E-04 5.00E-04 5.00E-04 2.90E-04 2.00E-04 3.40E-04 8.00E-04 2.90E-04 4.00E-04 2.60E-04 2.20E-04 4.02 8.00 1.00 2.27 4.02 6.27 9.05 1.00 2.25 4.02 6.25 9.00 0.25 1.00 2.25 4.00 6.25 72.00 16.13 32.02 64.05 16.00 16.00 239.30 108.02 144.40 216.37 142.82 Time (hours) Time C) ° -1.21E-02 -1.08E-02 -1.00E-02 -1.41E-02 -1.21E-02 -1.07E-02 -1.16E-02 -1.39E-02 -1.39E-02 -1.49E-02 -1.24E-02 -1.34E-02 -1.22E-02 -1.45E-02 -1.24E-02 -1.23E-02 -1.15E-02
250 300 300 300 300 350 350 350 350 350 350 400 400 400 400 400 400 450 450 450 450 450 450 500 500 500 500 500
. SPS Model Outputs for SLS Series, SLS for Model Outputs Part II SPS . . SPS Model Inputs for SAS Series Model SAS for Inputs SPS . Temperature ( Temperature
Initial Strain Max (unitless) Max Strain Initial
XXI XXII ID ID SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6 SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 Inputs
Outputs
Table Table
129
StDev
1.10E-04 1.90E-04 1.30E-04 1.00E-04 1.00E-04 1.10E-04 1.50E-04 7.00E-05 1.10E-04 6.00E-05 6.00E-05 9.00E-05 1.70E-04 6.00E-05 7.00E-05 1.10E-04 2.70E-05 7.00E-05 4.00E-05 3.50E-05 6.00E-05 5.00E-05 2.60E-05 9.00E-05 3.40E-05 2.60E-05 2.20E-05 2.40E-05
-1.30E-03 -8.90E-04 -1.61E-03 -1.10E-03 -8.90E-04 -8.20E-04 -8.80E-04 -1.42E-03 -1.36E-03 -1.49E-03 -1.63E-03 -9.70E-04 -1.56E-03 -1.40E-03 -1.01E-03 -1.25E-03 -1.11E-03 -1.11E-03 -1.30E-03 -1.14E-03 -1.27E-03 -1.18E-03 -1.18E-03 -1.42E-03 -1.51E-03 -1.07E-03 -9.54E-04 -1.38E-03 0.011 0.040 0.016 0.018 0.023 0.012 0.023 0.006 0.016 0.009 0.010 0.009 0.023 0.007 0.014 0.018 0.008 0.012 0.007 0.007 0.012 0.010 0.006 0.014 0.006 0.006 0.006 0.005 StDev Plastic Strain Average (unitless) Average Strain Plastic 0.222 0.250 0.333 0.299 0.282 0.125 0.173 0.262 0.289 0.347 0.388 0.158 0.336 0.291 0.250 0.319 0.362 0.250 0.292 0.307 0.349 0.327 0.351 0.316 0.376 0.284 0.293 0.405 StDev 3.20E-04 1.50E-04 1.30E-04 1.00E-04 7.00E-05 5.00E-04 2.20E-04 1.90E-04 1.10E-04 7.00E-05 6.00E-05 4.00E-04 1.70E-04 1.30E-04 1.00E-04 7.00E-05 5.00E-05 1.40E-04 1.00E-04 6.00E-05 5.00E-05 4.00E-05 3.20E-05 1.40E-04 6.00E-05 5.00E-05 3.30E-05 2.70E-05 Plastic-to-Total Strain Ratio Strain Plastic-to-Total 0.011 0.040 0.016 0.018 0.023 0.012 0.023 0.006 0.016 0.009 0.010 0.009 0.023 0.007 0.014 0.018 0.008 0.012 0.007 0.007 0.012 0.010 0.006 0.014 0.006 0.006 0.006 0.005 StDev -4.56E-03 -2.69E-03 -3.23E-03 -2.58E-03 -2.27E-03 -5.70E-03 -4.17E-03 -4.01E-03 -3.35E-03 -2.81E-03 -2.57E-03 -5.20E-03 -3.08E-03 -3.41E-03 -3.03E-03 -2.68E-03 -1.96E-03 -3.31E-03 -3.15E-03 -2.59E-03 -2.36E-03 -2.42E-03 -2.17E-03 -3.06E-03 -2.50E-03 -2.70E-03 -2.30E-03 -2.03E-03 Elastic Strain Average (unitless) Average Strain Elastic 0.778 0.750 0.667 0.701 0.718 0.875 0.827 0.738 0.711 0.653 0.612 0.842 0.664 0.709 0.750 0.681 0.638 0.750 0.708 0.693 0.651 0.673 0.649 0.684 0.624 0.716 0.707 0.595 StDev 4.00E-04 2.60E-04 2.20E-04 1.70E-04 1.30E-04 6.00E-04 2.90E-04 2.50E-04 1.70E-04 1.00E-04 9.00E-05 4.00E-04 2.80E-04 1.80E-04 1.30E-04 1.30E-04 5.00E-05 1.80E-04 1.20E-04 7.00E-05 8.00E-05 7.00E-05 4.00E-05 2.00E-04 8.00E-05 6.00E-05 4.00E-05 4.00E-05
Elastic-to-Total Strain Ratio Strain Elastic-to-Total
StDev 1.60E-04 4.00E-04 2.90E-04 2.30E-04 2.60E-04 2.30E-04 4.00E-04 1.50E-04 2.80E-04 1.40E-04 1.80E-04 2.80E-04 5.00E-04 8.00E-05 2.40E-04 3.10E-04 1.00E-04 1.60E-04 9.00E-05 1.00E-04 1.90E-04 1.60E-04 9.00E-05 2.60E-04 9.00E-05 9.00E-05 1.20E-04 9.00E-05 -5.90E-03 -3.59E-03 -4.85E-03 -3.68E-03 -3.16E-03 -6.50E-03 -5.05E-03 -5.43E-03 -4.72E-03 -4.30E-03 -4.19E-03 -6.10E-03 -4.64E-03 -4.81E-03 -4.05E-03 -3.94E-03 -3.08E-03 -4.42E-03 -4.45E-03 -3.73E-03 -3.62E-03 -3.59E-03 -3.35E-03 -4.48E-03 -4.01E-03 -3.77E-03 -3.25E-03 -3.41E-03 Initial Strain Average (unitless) Average Strain Initial -2.68E-03 -2.00E-03 -4.15E-03 -2.68E-03 -2.36E-03 -1.93E-03 -2.60E-03 -3.19E-03 -3.47E-03 -3.79E-03 -4.19E-03 -2.41E-03 -4.70E-03 -3.82E-03 -3.49E-03 -3.73E-03 -3.63E-03 -2.78E-03 -3.48E-03 -3.52E-03 -3.95E-03 -3.81E-03 -3.90E-03 -4.73E-03 -4.66E-03 -3.57E-03 -3.52E-03 -4.40E-03 StDev Plastic Strain Max (unitless) Max Strain Plastic 1.10E-03 3.40E-03 2.90E-03 2.40E-03 4.00E-03 1.00E-03 2.80E-03 5.00E-04 3.40E-03 2.50E-03 4.00E-03 8.00E-04 3.00E-03 7.00E-04 2.80E-03 5.00E-03 1.50E-03 1.70E-03 1.00E-03 1.70E-03 4.00E-03 4.00E-03 2.80E-03 1.80E-03 1.00E-03 1.60E-03 2.20E-03 2.70E-03 StDev 2.80E-04 2.30E-04 1.90E-04 2.10E-04 2.00E-04 9.00E-04 6.00E-04 4.00E-04 2.70E-04 1.40E-04 1.90E-04 1.30E-03 2.60E-04 2.00E-04 2.80E-04 1.70E-04 1.70E-04 1.90E-04 1.90E-04 1.60E-04 1.30E-04 1.40E-04 1.20E-04 3.20E-04 1.50E-04 1.80E-04 2.40E-04 1.20E-04 -1.95E-02 -1.69E-02 -4.20E-02 -2.86E-02 -3.30E-02 -9.80E-03 -1.75E-02 -3.13E-02 -4.63E-02 -7.45E-02 -1.17E-01 -1.36E-02 -3.11E-02 -3.92E-02 -4.16E-02 -6.10E-02 -7.68E-02 -2.99E-02 -4.94E-02 -6.63E-02 -9.10E-02 -1.07E-01 -1.41E-01 -3.55E-02 -7.98E-02 -8.13E-02 -1.03E-01 -1.73E-01 -9.40E-03 -6.04E-03 -8.31E-03 -6.29E-03 -6.01E-03 -1.35E-02 -1.22E-02 -9.00E-03 -8.53E-03 -7.14E-03 -6.63E-03 -1.28E-02 -9.25E-03 -9.30E-03 -1.05E-02 -7.97E-03 -6.40E-03 -8.33E-03 -8.43E-03 -7.96E-03 -7.36E-03 -7.85E-03 -7.21E-03 -1.02E-02 -7.75E-03 -8.99E-03 -8.48E-03 -6.46E-03 Integrated Plastic Strain (micron) Strain Plastic Integrated Elastic Strain Max (unitless) Max Strain Elastic StDev 1.30E-03 3.50E-03 2.90E-03 2.40E-03 4.00E-03 1.30E-03 2.90E-03 1.00E-03 4.00E-03 2.70E-03 5.00E-03 1.10E-03 3.10E-03 1.10E-03 3.20E-03 5.00E-03 2.20E-03 1.90E-03 1.50E-03 2.10E-03 4.00E-03 5.00E-03 3.40E-03 2.10E-03 1.50E-03 2.10E-03 2.70E-03 3.10E-03 StDev
3.30E-04 5.00E-04 4.00E-04 3.50E-04 4.00E-04 1.00E-03 8.00E-04 6.00E-04 4.00E-04 2.00E-04 3.00E-04 1.50E-03 5.00E-04 2.10E-04 4.00E-04 4.00E-04 2.10E-04 2.30E-04 2.10E-04 1.90E-04 2.30E-04 2.20E-04 1.70E-04 4.00E-04 1.90E-04 2.20E-04 3.30E-04 1.80E-04
. SPS Model Outputs for SAS Series, SAS for Model Part Outputs II SPS . . SPS Model Outputs for SAS Series, SAS for Model Part Outputs I SPS . -8.79E-02 -6.81E-02 -1.26E-01 -9.57E-02 -1.17E-01 -7.84E-02 -1.01E-01 -1.19E-01 -1.60E-01 -2.15E-01 -3.02E-01 -8.59E-02 -9.27E-02 -1.35E-01 -1.66E-01 -1.93E-01 -2.12E-01 -1.19E-01 -1.69E-01 -2.16E-01 -2.61E-01 -3.27E-01 -4.02E-01 -1.12E-01 -2.12E-01 -2.86E-01 -3.51E-01 -4.26E-01 -1.21E-02 -8.00E-03 -1.25E-02 -8.98E-03 -8.40E-03 -1.54E-02 -1.48E-02 -1.22E-02 -1.20E-02 -1.09E-02 -1.08E-02 -1.52E-02 -1.39E-02 -1.31E-02 -1.39E-02 -1.17E-02 -1.00E-02 -1.11E-02 -1.19E-02 -1.15E-02 -1.13E-02 -1.17E-02 -1.11E-02 -1.50E-02 -1.24E-02 -1.26E-02 -1.20E-02 -1.09E-02 Initial Strain Max (unitless) Max Strain Initial
Integrated Initial Strain (micron) Strain Initial Integrated
XXIII XXIV ID ID SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6 SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6 SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025 SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025 Outputs
Outputs
Table Table
130
H. DSPS Model Inputs and Outputs Tables
StDev 8.00E-04 7.00E-04 8.00E-04 2.30E-04 2.50E-04 1.90E-04 5.00E-04 2.10E-04 2.70E-04 1.30E-04 1.10E-04 1.40E-04 5.00E-04 1.40E-04 1.80E-04 1.60E-04 1.40E-04
-2.60E-03 -3.90E-03 -3.90E-03 -1.66E-03 -2.59E-03 -2.91E-03 -3.50E-03 -1.84E-03 -1.97E-03 -1.42E-03 -2.18E-03 -2.01E-03 -2.60E-03 -1.54E-03 -2.33E-03 -2.49E-03 -2.90E-03
StDev 7.00E-03 6.00E-03 8.00E-03 2.50E-03 2.90E-03 2.20E-03 3.40E-03 1.60E-03 5.00E-03 2.70E-03 3.10E-03 3.50E-03 6.00E-03 1.20E-03 2.10E-03 2.90E-03 2.80E-03 Hydrostatic Plastic Strain Average (micron) Average Strain Plastic Hydrostatic StDev 2.40E-04 2.20E-04 2.70E-04 1.80E-04 1.50E-04 1.00E-04 2.90E-04 2.30E-04 1.40E-04 1.30E-04 6.00E-05 1.20E-04 2.40E-04 1.90E-04 1.70E-04 1.30E-04 1.10E-04 -7.10E-02 -8.80E-02 -9.70E-02 -8.81E-02 -1.13E-01 -1.38E-01 -8.74E-02 -1.15E-01 -1.57E-01 -1.75E-01 -2.23E-01 -2.60E-01 -8.70E-02 -1.09E-01 -1.66E-01 -1.89E-01 -2.47E-01 Integrated Initial Strain (micron) Strain Initial Integrated -2.37E-03 -2.15E-03 -2.13E-03 -2.29E-03 -2.30E-03 -1.65E-03 -2.80E-03 -2.92E-03 -2.91E-03 -3.20E-03 -2.13E-03 -2.73E-03 -2.79E-03 -3.04E-03 -3.04E-03 -2.94E-03 -2.98E-03 StDev 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 Deviatoric Plastic Strain Average (micron) Average Strain Plastic Deviatoric -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 -1.91E-02 StDev 2.10E-04 2.90E-04 1.80E-04 1.80E-04 1.40E-04 1.00E-04 2.60E-04 1.20E-04 1.10E-04 7.00E-05 6.00E-05 6.00E-05 1.60E-04 1.10E-04 6.00E-05 7.00E-05 5.00E-05 Initial Strain Max (unitless) Max Strain Initial 0.40 0.30 0.40 0.13 0.15 0.11 0.18 0.08 0.29 0.14 0.16 0.18 0.30 0.06 0.11 0.15 0.15 StDev -2.08E-03 -2.84E-03 -2.11E-03 -2.34E-03 -2.16E-03 -2.02E-03 -2.49E-03 -2.01E-03 -2.26E-03 -1.85E-03 -1.90E-03 -1.93E-03 -1.79E-03 -1.81E-03 -1.29E-03 -1.56E-03 -1.40E-03 Elastic Strain Average (unitless) Average Strain Elastic 3.70 4.63 5.10 4.61 5.90 7.22 4.57 6.02 8.21 9.13 4.52 5.69 8.69 9.87 11.67 13.61 12.93 StDev
1.00E-03 1.10E-03 1.00E-03 5.00E-04 5.00E-04 3.30E-04 9.00E-04 4.00E-04 4.00E-04 2.60E-04 1.90E-04 1.90E-04 8.00E-04 4.00E-04 2.80E-04 2.80E-04 2.30E-04
Normalized Concentration Integrand (micron) Integrand Concentration Normalized -7.10E-03 -8.80E-03 -8.10E-03 -6.30E-03 -7.10E-03 -6.57E-03 -8.70E-03 -6.80E-03 -7.10E-03 -6.47E-03 -6.20E-03 -6.67E-03 -7.20E-03 -6.40E-03 -6.65E-03 -6.99E-03 -7.27E-03 StDev 5.00E-04 5.00E-04 6.00E-04 9.00E-04 1.00E-03 8.00E-04 9.00E-04 7.00E-04 9.00E-04 9.00E-04 1.40E-03 1.50E-03 6.00E-04 8.00E-04 8.00E-04 1.30E-03 1.30E-03
Initial Strain Average (unitless) Average Strain Initial
StDev 8.00E-03 6.00E-03 8.00E-03 2.70E-03 2.90E-03 2.70E-03 3.50E-03 3.10E-03 6.00E-03 3.20E-03 3.30E-03 5.00E-03 6.00E-03 1.80E-03 4.00E-03 3.40E-03 4.00E-03 -2.66E-02 -3.64E-02 -3.25E-02 -4.20E-02 -4.43E-02 -5.43E-02 -3.19E-02 -4.39E-02 -6.37E-02 -6.40E-02 -8.76E-02 -9.65E-02 -2.76E-02 -3.95E-02 -4.12E-02 -5.40E-02 -6.09E-02 Ela Strain Integrand (micron) Integrand Strain Ela 0.002 0.002 0.007 0.003 0.001 0.004 0.003 0.008 0.003 0.005 0.002 0.011 0.002 0.004 0.009 0.005 0.007 StDev -2.60E-02 -3.90E-02 -4.60E-02 -2.32E-02 -4.14E-02 -6.11E-02 -3.45E-02 -3.12E-02 -4.30E-02 -3.84E-02 -7.83E-02 -7.80E-02 -3.20E-02 -2.62E-02 -5.80E-02 -6.73E-02 -9.90E-02 0.092 0.093 0.102 0.129 0.145 0.146 0.109 0.183 0.242 0.309 0.298 0.395 0.122 0.186 0.260 0.280 0.351 Step Height (micron) Height Step Integrated Hydrostatic Plastic Strain (micron) Strain Plastic Hydrostatic Integrated 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 StDev StDev 5.00E-04 5.00E-04 2.40E-03 1.00E-03 4.00E-04 1.40E-03 9.00E-04 2.60E-03 1.00E-03 1.60E-03 8.00E-04 4.00E-03 6.00E-04 1.30E-03 3.10E-03 1.70E-03 2.20E-03 10 10 12 14 16 21 10 17 22 27 36 39 12 17 25 27 34 Depth (micron) Depth 9.05 4.02 9.00 1.00 2.25 4.00 6.25 9.00 16.00 25.00 36.90 49.00 65.75 16.00 25.02 37.05 50.00 Time (hours) Time
-2.37E-02 -2.15E-02 -2.55E-02 -3.21E-02 -3.68E-02 -3.46E-02 -2.80E-02 -4.96E-02 -6.41E-02 -8.64E-02 -7.66E-02 -1.07E-01 -3.35E-02 -5.17E-02 -7.59E-02 -7.93E-02 -1.01E-01 . DSPS Model Outputs for SLS for Model Outputs Series, Part I DSPS .
C) . DSPS Model Inputs for SLS Model SLS for Inputs Series DSPS . °
400 400 400 400 400 400 450 450 450 450 450 450 500 500 500 500 500
XXV XXVI Integrated Deviatoric Plastic Strain (micron) Strain Plastic Deviatoric Integrated Temperature ( Temperature ID ID SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 Outputs
Inputs
Table Table
131
0.070 0.040 0.050 0.024 0.017 0.014 0.026 0.025 0.026 0.015 0.010 0.018 0.040 0.015 0.020 0.014 0.012 StDev 0.370 0.440 0.480 0.264 0.367 0.443 0.395 0.271 0.276 0.220 0.351 0.301 0.360 0.241 0.350 0.357 0.399
Hydrostatic-Plastic-to-Total Strain Ratio Strain Hydrostatic-Plastic-to-Total
0.040 0.017 0.033 0.015 0.009 0.011 0.016 0.023 0.015 0.012 0.006 0.016 0.027 0.013 0.020 0.011 0.010 StDev StDev 2.40E-03 4.00E-03 2.80E-03 5.00E-03 6.00E-03 5.00E-03 4.00E-03 6.00E-03 5.00E-03 4.00E-03 9.00E-03 8.00E-03 1.80E-03 2.00E-03 2.90E-03 4.00E-03 5.00E-03 1.90E-03 3.20E-03 3.50E-03 4.00E-03 5.00E-03 6.00E-03 2.20E-03 3.30E-03 5.00E-03 1.10E-02 8.00E-03 0.340 0.244 0.262 0.364 0.326 0.250 0.321 0.431 0.408 0.494 0.343 0.409 0.387 0.476 0.457 0.420 0.409 -1.04E-01 -1.21E-01 -1.44E-01 -1.52E-01 -2.00E-01 -7.30E-02 -9.70E-02 -1.40E-01 -1.91E-01 -2.81E-01 -3.98E-01 -8.00E-02 -9.51E-02 -1.47E-01 -1.70E-01 -2.35E-01 -3.02E-01 -1.53E-01 -2.03E-01 -2.69E-01 -3.29E-01 -4.01E-01 -5.16E-01 -1.07E-01 -2.44E-01 -3.25E-01 -4.17E-01 -5.60E-01 Integrated Initial Strain (micron) Strain Initial Integrated Deviatoric-Plastic-to-Total Strain Ratio Strain Deviatoric-Plastic-to-Total StDev 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.032 0.021 0.022 0.013 0.010 0.007 0.013 0.006 0.012 0.006 0.007 0.006 0.018 0.006 0.004 0.006 0.005 StDev 0.706 0.680 0.739 0.628 0.694 0.693 0.715 0.702 0.684 0.714 0.694 0.711 0.751 0.716 0.807 0.777 0.808 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 -1.43E-02 Initial Strain Max (unitless) Max Strain Initial Plastic-to-Total Strain Ratio Strain Plastic-to-Total 0.17 0.29 0.19 0.32 0.40 0.32 0.31 0.40 0.40 0.27 0.60 0.60 0.13 0.14 0.20 0.26 0.40 0.13 0.22 0.24 0.25 0.34 0.40 0.15 0.23 0.40 0.80 0.60 StDev 0.032 0.021 0.022 0.013 0.010 0.007 0.013 0.006 0.012 0.006 0.007 0.006 0.018 0.006 0.004 0.006 0.005 StDev 0.294 0.320 0.261 0.372 0.306 0.307 0.285 0.298 0.316 0.286 0.306 0.289 0.249 0.284 0.193 0.223 0.192 7.28 8.48 5.09 6.81 9.80 5.60 6.66 7.49 10.11 10.66 14.00 13.40 19.66 27.90 10.26 11.90 16.50 21.20 10.73 14.20 18.84 23.07 28.06 36.10 17.11 22.80 29.20 39.20
Elastic-to-Total Strain Ratio Strain Elastic-to-Total
StDev 1.30E-03 7.00E-04 1.00E-03 5.00E-04 3.30E-04 2.70E-04 5.00E-04 5.00E-04 5.00E-04 2.90E-04 2.00E-04 3.40E-04 8.00E-04 2.90E-04 4.00E-04 2.60E-04 2.20E-04 Normalized Concentration Integrand (micron) Integrand Concentration Normalized StDev
1.60E-03 1.00E-03 1.30E-03 1.10E-03 1.80E-03 1.60E-03 1.80E-03 1.90E-03 2.20E-03 2.40E-03 4.00E-03 1.70E-03 1.60E-03 2.00E-03 3.30E-03 2.50E-03 3.50E-03 2.20E-03 2.50E-03 3.00E-03 3.20E-03 4.00E-03 4.00E-03 2.40E-03 2.50E-03 3.00E-03 3.50E-03 3.50E-03
-7.10E-03 -8.30E-03 -9.10E-03 -5.00E-03 -7.02E-03 -8.47E-03 -7.50E-03 -5.20E-03 -5.30E-03 -4.21E-03 -6.71E-03 -5.76E-03 -7.00E-03 -4.60E-03 -6.70E-03 -6.82E-03 -7.63E-03 -8.77E-02 -6.56E-02 -1.08E-01 -8.60E-02 -1.08E-01 -8.79E-02 -1.07E-01 -1.13E-01 -1.46E-01 -1.80E-01 -2.37E-01 -9.27E-02 -7.90E-02 -1.22E-01 -1.59E-01 -1.69E-01 -1.74E-01 -1.15E-01 -1.53E-01 -1.92E-01 -2.18E-01 -2.82E-01 -3.34E-01 -9.82E-02 -1.70E-01 -2.63E-01 -3.18E-01 -3.25E-01 Hydrostatic Plastic Strain Max (micron) Max Strain Plastic Hydrostatic Ela Strain Integrand (micron) Integrand Strain Ela StDev 7.00E-04 3.20E-04 6.00E-04 2.90E-04 1.70E-04 2.10E-04 3.10E-04 4.00E-04 3.00E-04 2.30E-04 1.20E-04 3.00E-04 5.00E-04 2.50E-04 4.00E-04 2.10E-04 1.90E-04 0.003 0.010 0.009 0.007 0.011 0.003 0.008 0.000 0.010 0.007 0.012 0.002 0.009 0.001 0.008 0.015 0.004 0.005 0.002 0.004 0.012 0.013 0.008 0.005 0.002 0.004 0.006 0.007 StDev 0.127 0.102 0.210 0.153 0.183 0.098 0.136 0.182 0.253 0.364 0.536 0.113 0.155 0.213 0.249 0.316 0.366 0.179 0.268 0.349 0.443 0.541 0.684 0.183 0.372 0.449 0.557 0.771 Step Height (micron) Height Step -6.40E-03 -4.66E-03 -5.00E-03 -6.96E-03 -6.24E-03 -4.79E-03 -6.13E-03 -8.20E-03 -7.80E-03 -9.45E-03 -6.56E-03 -7.83E-03 -7.40E-03 -9.09E-03 -8.70E-03 -8.03E-03 -7.82E-03 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 StDev Deviatoric Plastic Strain Max (micron) Max Strain Plastic Deviatoric 15 19 26 26 37 12 20 22 34 50 72 14 20 28 41 49 69 27 38 58 72 91 25 53 76 120 108 125 Depth (micron) Depth StDev 6.00E-04 4.00E-04 4.00E-04 2.60E-04 2.00E-04 1.30E-04 2.60E-04 1.20E-04 2.30E-04 1.10E-04 1.30E-04 1.10E-04 3.40E-04 1.20E-04 8.00E-05 1.20E-04 9.00E-05 4.02 8.00 1.00 2.27 4.02 6.27 9.05 1.00 2.25 4.02 6.25 9.00 0.25 1.00 2.25 4.00 6.25 72.00 16.13 32.02 64.05 16.00 16.00
239.30 108.02 144.40 216.37 142.82
. DSPS Model Series SAS for Inputs DSPS . . DSPS Model Outputs for SLS for Model Outputs Series, Part II DSPS . Time (hours) Time C) ° -5.60E-03 -6.10E-03 -5.00E-03 -7.11E-03 -5.86E-03 -5.87E-03 -5.44E-03 -5.69E-03 -6.05E-03 -5.46E-03 -5.85E-03 -5.53E-03 -4.76E-03 -5.42E-03 -3.70E-03 -4.27E-03 -3.67E-03
250 300 300 300 300 350 350 350 350 350 350 400 400 400 400 400 400 450 450 450 450 450 450 500 500 500 500 500
XXVII XXVIII Elastic Strain Max (unitless) Max Strain Elastic Temperature ( Temperature ID ID SLS-400-9 SLS-450-4 SLS-450-9 SLS-500-1 SLS-500-2 SLS-500-4 SLS-500-6 SLS-500-9 SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6 SLS-400-16 SLS-400-25 SLS-400-37 SLS-400-49 SLS-400-66 SLS-450-16 SLS-450-25 SLS-450-37 SLS-450-50 SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 Outputs SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025
Inputs
Table Table
132
0.023 0.035 0.026 0.025 0.025 0.070 0.060 0.040 0.030 0.014 0.021 0.110 0.040 0.014 0.025 0.026 0.014 0.016 0.015 0.013 0.016 0.015 0.012 0.029 0.013 0.016 0.023 0.013 StDev StDev 1.90E-04 3.20E-04 1.60E-04 2.10E-04 2.00E-04 4.00E-04 2.60E-04 2.90E-04 1.90E-04 1.00E-04 1.40E-04 6.00E-04 1.80E-04 8.00E-05 1.00E-04 1.30E-04 9.00E-05 1.10E-04 1.00E-04 7.00E-05 8.00E-05 7.00E-05 6.00E-05 1.20E-04 7.00E-05 7.00E-05 1.10E-04 7.00E-05 0.154 0.437 0.127 0.371 0.414 0.150 0.159 0.234 0.242 0.020 0.081 0.024 0.180 0.298 0.222 0.166 0.196 0.208 0.183 0.221 0.130 0.120 0.159 0.239 -0.080 -0.040 -0.070 -0.048 5.00E-04 1.90E-04 4.00E-04 2.10E-04 -1.06E-03 -2.79E-03 -7.10E-04 -2.17E-03 -2.23E-03 -9.20E-04 -8.90E-04 -1.31E-03 -1.34E-03 -1.20E-04 -4.30E-04 -1.00E-04 -8.70E-04 -1.30E-03 -1.26E-03 -8.80E-04 -9.10E-04 -9.50E-04 -8.10E-04 -9.50E-04 -6.00E-04 -5.10E-04 -6.10E-04 -1.07E-03 Hydrostatic-Plastic-to-Total Strain Ratio Strain Hydrostatic-Plastic-to-Total 0.011 0.029 0.021 0.016 0.018 0.016 0.030 0.011 0.019 0.010 0.013 0.019 0.032 0.006 0.017 0.022 0.007 0.011 0.006 0.007 0.013 0.012 0.006 0.018 0.006 0.006 0.008 0.007 StDev Hydrostatic Plastic Strain Average (micron) Average Strain Plastic Hydrostatic StDev 1.10E-04 1.90E-04 1.30E-04 1.00E-04 1.00E-04 1.10E-04 1.50E-04 7.00E-05 1.10E-04 6.00E-05 6.00E-05 9.00E-05 1.70E-04 6.00E-05 7.00E-05 1.10E-04 2.70E-05 7.00E-05 4.00E-05 3.50E-05 6.00E-05 5.00E-05 2.60E-05 9.00E-05 3.40E-05 2.60E-05 2.20E-05 2.40E-05 0.188 0.140 0.291 0.188 0.165 0.135 0.180 0.224 0.243 0.266 0.294 0.169 0.327 0.268 0.245 0.261 0.254 0.195 0.244 0.247 0.277 0.267 0.274 0.332 0.327 0.250 0.247 0.308 -1.30E-03 -8.90E-04 -1.61E-03 -1.10E-03 -8.90E-04 -8.20E-04 -8.80E-04 -1.42E-03 -1.36E-03 -1.49E-03 -1.63E-03 -9.70E-04 -1.56E-03 -1.40E-03 -1.01E-03 -1.25E-03 -1.11E-03 -1.11E-03 -1.30E-03 -1.14E-03 -1.27E-03 -1.18E-03 -1.18E-03 -1.42E-03 -1.51E-03 -1.07E-03 -9.54E-04 -1.38E-03 Deviatoric-Plastic-to-Total Strain Ratio Strain Deviatoric-Plastic-to-Total 0.019 0.016 0.013 0.014 0.014 0.060 0.040 0.030 0.019 0.010 0.013 0.090 0.018 0.014 0.020 0.012 0.012 0.013 0.013 0.011 0.009 0.010 0.009 0.023 0.011 0.012 0.017 0.008 StDev Deviatoric Plastic Strain Average (micron) Average Strain Plastic Deviatoric StDev 3.20E-04 1.50E-04 1.30E-04 1.00E-04 7.00E-05 5.00E-04 2.20E-04 1.90E-04 1.10E-04 7.00E-05 6.00E-05 4.00E-04 1.70E-04 1.30E-04 1.00E-04 7.00E-05 5.00E-05 1.40E-04 1.00E-04 6.00E-05 5.00E-05 4.00E-05 3.20E-05 1.40E-04 6.00E-05 5.00E-05 3.30E-05 2.70E-05 0.342 0.577 0.418 0.559 0.579 0.060 0.140 0.369 0.402 0.500 0.536 0.100 0.352 0.349 0.268 0.441 0.552 0.416 0.410 0.442 0.484 0.450 0.495 0.284 0.457 0.370 0.406 0.547 Plastic-to-Total Strain Ratio Strain Plastic-to-Total 0.019 0.016 0.013 0.014 0.014 0.060 0.040 0.030 0.019 0.010 0.013 0.090 0.018 0.014 0.020 0.012 0.012 0.013 0.013 0.011 0.009 0.010 0.009 0.023 0.011 0.012 0.017 0.008 StDev -4.56E-03 -2.69E-03 -3.23E-03 -2.58E-03 -2.27E-03 -5.70E-03 -4.17E-03 -4.01E-03 -3.35E-03 -2.81E-03 -2.57E-03 -5.20E-03 -3.08E-03 -3.41E-03 -3.03E-03 -2.68E-03 -1.96E-03 -3.31E-03 -3.15E-03 -2.59E-03 -2.36E-03 -2.42E-03 -2.17E-03 -3.06E-03 -2.50E-03 -2.70E-03 -2.30E-03 -2.03E-03 Elastic Strain Average (unitless) Average Strain Elastic 0.658 0.423 0.582 0.441 0.421 0.940 0.860 0.631 0.598 0.500 0.464 0.900 0.648 0.651 0.732 0.559 0.448 0.584 0.590 0.558 0.516 0.550 0.505 0.716 0.543 0.630 0.594 0.453 StDev 5.00E-04 4.00E-04 2.40E-04 2.80E-04 2.20E-04 6.00E-04 3.30E-04 4.00E-04 2.30E-04 1.40E-04 1.50E-04 7.00E-04 2.50E-04 2.00E-04 1.20E-04 1.20E-04 1.00E-04 2.20E-04 1.60E-04 1.00E-04 8.00E-05 7.00E-05 6.00E-05 1.90E-04 1.10E-04 9.00E-05 1.10E-04 8.00E-05
Elastic-to-Total Strain Ratio Strain Elastic-to-Total
I StDev 3.30E-04 5.00E-04 4.00E-04 3.50E-04 4.00E-04 1.00E-03 8.00E-04 6.00E-04 4.00E-04 2.00E-04 3.00E-04 1.50E-03 5.00E-04 2.10E-04 4.00E-04 4.00E-04 2.10E-04 2.30E-04 2.10E-04 1.90E-04 2.30E-04 2.20E-04 1.70E-04 4.00E-04 1.90E-04 2.20E-04 3.30E-04 1.80E-04 -6.90E-03 -6.40E-03 -5.55E-03 -5.85E-03 -5.40E-03 -6.10E-03 -4.86E-03 -6.40E-03 -5.61E-03 -5.61E-03 -5.53E-03 -5.70E-03 -4.75E-03 -5.23E-03 -4.14E-03 -4.81E-03 -4.38E-03 -5.67E-03 -5.33E-03 -4.64E-03 -4.57E-03 -4.40E-03 -4.30E-03 -4.28E-03 -4.61E-03 -4.28E-03 -3.86E-03 -4.48E-03 Initial Strain Average (unitless) Average Strain Initial StDev 1.10E-03 5.00E-04 1.00E-03 7.00E-04 -2.19E-03 -6.20E-03 -1.80E-03 -5.30E-03 -5.90E-03 -2.10E-03 -2.30E-03 -3.34E-03 -3.45E-03 -3.00E-04 -1.16E-03 -3.00E-04 -2.60E-03 -4.25E-03 -3.16E-03 -2.37E-03 -2.79E-03 -2.97E-03 -2.62E-03 -3.16E-03 -1.86E-03 -1.71E-03 -2.27E-03 -3.42E-03 2.70E-03 5.00E-03 4.00E-03 5.00E-03 7.00E-03 5.00E-03 5.00E-03 6.00E-03 6.00E-03 5.00E-03 1.00E-02 8.00E-03 4.00E-03 2.30E-03 4.00E-03 6.00E-03 6.00E-03 2.70E-03 3.50E-03 4.00E-03 6.00E-03 7.00E-03 7.00E-03 3.00E-03 4.00E-03 6.00E-03 1.10E-02 9.00E-03 Hydrostatic Plastic Strain Max (micron) Max Strain Plastic Hydrostatic StDev 1.60E-04 4.00E-04 2.90E-04 2.30E-04 2.60E-04 2.30E-04 4.00E-04 1.50E-04 2.80E-04 1.40E-04 1.80E-04 2.80E-04 5.00E-04 8.00E-05 2.40E-04 3.10E-04 1.00E-04 1.60E-04 9.00E-05 1.00E-04 1.90E-04 1.60E-04 9.00E-05 2.60E-04 9.00E-05 9.00E-05 1.20E-04 9.00E-05 6.00E-03 4.00E-03 5.00E-03 5.10E-03 -1.60E-02 -5.30E-02 -1.80E-02 -5.60E-02 -8.30E-02 -2.00E-02 -3.00E-02 -6.60E-02 -9.60E-02 -2.00E-03 -1.19E-02 -4.00E-03 -4.20E-02 -9.00E-02 -3.39E-02 -3.36E-02 -5.30E-02 -6.80E-02 -7.30E-02 -1.14E-01 -3.20E-02 -3.90E-02 -6.60E-02 -1.34E-01 Integrated Hydrostatic Plastic Strain (micron) Strain Plastic Hydrostatic Integrated -2.68E-03 -2.00E-03 -4.15E-03 -2.68E-03 -2.36E-03 -1.93E-03 -2.60E-03 -3.19E-03 -3.47E-03 -3.79E-03 -4.19E-03 -2.41E-03 -4.70E-03 -3.82E-03 -3.49E-03 -3.73E-03 -3.63E-03 -2.78E-03 -3.48E-03 -3.52E-03 -3.95E-03 -3.81E-03 -3.90E-03 -4.73E-03 -4.66E-03 -3.57E-03 -3.52E-03 -4.40E-03 StDev 1.10E-03 3.40E-03 2.90E-03 2.40E-03 4.00E-03 1.00E-03 2.80E-03 5.00E-04 3.40E-03 2.50E-03 4.00E-03 8.00E-04 3.00E-03 7.00E-04 2.80E-03 5.00E-03 1.50E-03 1.70E-03 1.00E-03 1.70E-03 4.00E-03 4.00E-03 2.80E-03 1.80E-03 1.00E-03 1.60E-03 2.20E-03 2.70E-03 Deviatoric Plastic Strain Max (micron) Max Strain Plastic Deviatoric StDev 2.80E-04 2.30E-04 1.90E-04 2.10E-04 2.00E-04 9.00E-04 6.00E-04 4.00E-04 2.70E-04 1.40E-04 1.90E-04 1.30E-03 2.60E-04 2.00E-04 2.80E-04 1.70E-04 1.70E-04 1.90E-04 1.90E-04 1.60E-04 1.30E-04 1.40E-04 1.20E-04 3.20E-04 1.50E-04 1.80E-04 2.40E-04 1.20E-04
-1.95E-02 -1.69E-02 -4.20E-02 -2.86E-02 -3.30E-02 -9.80E-03 -1.75E-02 -3.13E-02 -4.63E-02 -7.45E-02 -1.17E-01 -1.36E-02 -3.11E-02 -3.92E-02 -4.16E-02 -6.10E-02 -7.68E-02 -2.99E-02 -4.94E-02 -6.63E-02 -9.10E-02 -1.07E-01 -1.41E-01 -3.55E-02 -7.98E-02 -8.13E-02 -1.03E-01 -1.73E-01
. DSPS Model Outputs for SAS Series, SAS for Model Part Outputs DSPS . . DSPS Model Outputs for SAS Series, SAS for Model Part Outputs II DSPS .
-9.40E-03 -6.04E-03 -8.31E-03 -6.29E-03 -6.01E-03 -1.35E-02 -1.22E-02 -9.00E-03 -8.53E-03 -7.14E-03 -6.63E-03 -1.28E-02 -9.25E-03 -9.30E-03 -1.05E-02 -7.97E-03 -6.40E-03 -8.33E-03 -8.43E-03 -7.96E-03 -7.36E-03 -7.85E-03 -7.21E-03 -1.02E-02 -7.75E-03 -8.99E-03 -8.48E-03 -6.46E-03
XXIX XXX Integrated Deviatoric Plastic Strain (micron) Strain Plastic Deviatoric Integrated Elastic Strain Max (unitless) Max Strain Elastic ID ID SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6 SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025 Outputs
SAS-350-4 SAS-350-8 SAS-400-1 SAS-400-2 SAS-400-4 SAS-400-6 SAS-400-9 SAS-450-1 SAS-450-2 SAS-450-4 SAS-450-6 SAS-450-9 SAS-500-1 SAS-500-2 SAS-500-4 SAS-500-6
Table Table SAS-300-72 SAS-350-16 SAS-350-32 SAS-350-64 SAS-400-16 SAS-450-16 SAS-250-239 SAS-300-108 SAS-300-144 SAS-300-216 SAS-350-143 SAS-500-025 Outputs
133