DePaul University Via Sapientiae College of Liberal Arts & Social Sciences Theses and Dissertations College of Liberal Arts and Social Sciences 6-2011 The dependence of suspension viscosity on particle size, shear rate, and solvent viscosity Marc Pavlik DePaul University, MARCPAVLIK@COMCAST.NET Follow this and additional works at: https://via.library.depaul.edu/etd Recommended Citation Pavlik, Marc, "The dependence of suspension viscosity on particle size, shear rate, and solvent viscosity" (2011). College of Liberal Arts & Social Sciences Theses and Dissertations. 71. https://via.library.depaul.edu/etd/71 This Thesis is brought to you for free and open access by the College of Liberal Arts and Social Sciences at Via Sapientiae. It has been accepted for inclusion in College of Liberal Arts & Social Sciences Theses and Dissertations by an authorized administrator of Via Sapientiae. For more information, please contact digitalservices@depaul.edu. THE DEPENDENCE OF SUSPENSION VISCOSITY ONPARTICLE SIZE, SHEAR RATE, AND SOLVENT VISCOSITYA ThesisPresented inPartial Fulfillment of theRequirements for the Degree ofMASTER OF SCIENCEAugust 19, 2009BYMarc PavlikPHYSICS DEPARTMENTCollege of Liberal Arts and SciencesDePaul UniversityChicago, Illinois2TABLE OF CONTENTSLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14CHAPTER 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 15CHAPTER 2 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . 202.1 Newton’s Law of Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Particle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Stream Lines and Interacting Spheres . . . . . . . . . . . . . . . . . . 242.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Einstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.2 Mooney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.3 Krieger-Dougherty . . . . . . . . . . . . . . . . . . . . . . . . 302.4.4 Batchelor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.5 Brady . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32CHAPTER 3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 343.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1 Concentric Cylinder . . . . . . . . . . . . . . . . . . . . . . . 343.1.2 NESLAB RTE 7 Bath Circulator . . . . . . . . . . . . . . . . 353.1.3 Rheometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Glycerine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.2 Glass Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 Experiment Procedure . . . . . . . . . . . . . . . . . . . . . . 443.3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46CHAPTER 4 Data/Analysis . . . . . . . . . . . . . . . . . . . . . . . 474.1 Universal Trends in the Data . . . . . . . . . . . . . . . . . . . . . . 474.1.1 Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Solvent Viscosity Dependence . . . . . . . . . . . . . . . . . . 494.1.3 Angular Velocity Dependence . . . . . . . . . . . . . . . . . . 51TABLE OF CONTENTS – Continued34.1.4 Particle Size Dependence . . . . . . . . . . . . . . . . . . . . . 534.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2.1 Fitting Procedures . . . . . . . . . . . . . . . . . . . . . . . . 534.2.2 The Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.3.1 Fitting Parameter . . . . . . . . . . . . . . . . . . . . . . . . 764.3.2 Fitting Parameter 'm . . . . . . . . . . . . . . . . . . . . . . . 904.3.3 Fitting Parameter ˛ . . . . . . . . . . . . . . . . . . . . . . . 1114.3.4 Fitting Parameter ˇ . . . . . . . . . . . . . . . . . . . . . . . 116CHAPTER 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1215.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123APPENDIX A Code used to minimize 2 . . . . . . . . . . . . . . . 1244LIST OF FIGURES2.1 A car in a wind tunnel showing laminar flow. Notice the streamlinesdo not touch or cross while remaining parallel to each other. . . . . . 202.2 A cigarette showing both laminar and turbulent flow. The smokenearest the cigarette is laminar. The smoke furthest from thecigarette is turbulent as shown by the swirling smoke patterns. . . . . 212.3 Couette Flow. Simple experiment showing the effects of viscosity[1]. 232.4 Free-body Diagram for a sphere traveling in a fluid at terminal velocity. 242.5 Stream lines flowing around a sphere traveling in a fluid at terminalvelocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Three equal spheres flowing at a constant velocity in the directionof the arrows at various times showing how particle interaction canincrease and decrease the velocities of the spheres. . . . . . . . . . . . 272.7 Streamlines of Fluid relative to moving cloud. . . . . . . . . . . . . . 282.8 A singlet is where the two particles rotate independently and a dou-blet is where the two particles rotate together like a dumbbell. . . . . 313.1 Images of concentric cylinder (a) is the inside geometry, (b) is theoutside geometry, and (c) is the inside geometry inserted into theoutside geometry with the sample filling the gap. . . . . . . . . . . . 343.2 The NESLAB RTE 7 Bath Circulator. . . . . . . . . . . . . . . . . . 363.3 Cut away of the head of the AR 1500ex Rheometer showing the vari-ous components used to rotated the geometry used to take measure-ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 The AR 1500ex Rheometer from TA Instruments. . . . . . . . . . . . 383.5 A plot of viscosity as a function of temperature for the calibrationcurve of the custom concentric cylinder (black triangles with errorbars) and DOW Standard values for 98% (red squares) pure glycerin,99% (green stars) pure glycerin, and 100% (blue diamonds) pure glyc-erin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6 The digital microscope model number DC4-410. . . . . . . . . . . . . 413.7 Image of the particles (a D 51 m) under the microscope using MoticImages Plus 2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8 Histogram of particles showing the mean and standard deviation values. 43LIST OF FIGURES – Continued54.1 The Standard System. The standard system using the followingparameters of 0 D 3017 cP ˙ 4 cP, ! D 1:000 rad=s ˙ 0:001 rad=s,and a D 36 m ˙ 3 m with Mooney’s Equation 2.11 the greenline with values of D 2:8 ˙ 0:3 and 'm D 0:79 ˙ 0:11, Krieger-Doughtery’s Equation 2.12 the blue line with values of D 3:2˙ 0:4and 'm D 0:57˙0:10, Batchelor’s Equation 2.13 (Standard) the blackline with values of D 2:8˙0:3, 'm D 0:59˙0:07, and ˛ D 3˙1, andBrady’s Equation 2.18 the purple line with values of D 3:2 ˙ 0:4,'m D 0:61˙ 0:04, and ˇ D 5:4˙ 0:6. . . . . . . . . . . . . . . . . . . 484.2 The following system is for an angular velocity ! D 1:000 rad=s forthe particle with a radius a D 36 m and the three solvent viscosities.The black line and points are for the standard at S D 3017 cP ˙4 cP(Batchelor’s: D 2:8˙ 0:3, 'm D 0:59˙ 0:07, and ˛ D 3˙ 1), thegreen line and diamonds are for S D 1264 cP ˙ 3 cP (Batchelor’s: D 2:6 ˙ 0:8, 'm D 0:61 ˙ 0:10, and ˛ D 3 ˙ 1) and the red lineand stars are for S D 571 cP ˙ 1 cP (Batchelor’s: D 2:2 ˙ 0:3,'m D 0:59˙ 0:08, and ˛ D 3˙ 1) solvent viscosities. . . . . . . . . . 504.3 The following system varies the angular velocities for the particle withradius a D 36 m. The black line and points are for the standard for! D 1:000 rad=s and S D 3017 cP ˙4 cP (Batchelor’s: D 2:8˙0:3,'m D 0:59 ˙ 0:07, and ˛ D 3 ˙ 1), the green line and diamondsare for ! D 2:000 rad=s and S D 2988 cP ˙ 2 cP (Batchelor’s: D 2:7˙0:2, 'm D 0:60˙0:06, and ˛ D 3:1˙0:9), the blue line andstars are for ! D 3:000 rad=s and S D 2965 cP ˙ 4 cP (Batchelor’s: D 2:6˙ 0:1, 'm D 0:61˙ 0:07, and ˛ D 2:7˙ 0:8) and the yellowline and triangles are for ! D 4:000 rad=s and S D 2924 cP ˙ 5 cP(Batchelor’s: D 2:6˙ 0:3, 'm D 0:62˙ 0:10, and ˛ D 3˙ 1). . . . . 524.4 The following system shows a plot for all four radii for the angularvelocity of ! D 1:000 rad=s and S D 3017 cP ˙ 4 cP. The blackline and points are the standard with a D 36 m (Batchelor’s: D2:8˙0:3, 'm D 0:59˙0:07, and ˛ D 3˙1), the red line and squares arefor radius a D 26 m (Batchelor’s: D 3:0˙ 0:3, 'm D 1:05˙ 0:12,and ˛ D 4 ˙ 1), the blue line and stars are for radius a D 51 m(Batchelor’s: D 3:1˙ 0:3, 'm D 0:61˙ 0:07, and ˛ D 4˙ 2) andthe yellow line and triangles are for radius a D 71 m (Batchelor’s: D 2:6˙ 0:2, 'm D 0:57˙ 0:06, and ˛ D 3˙ 1). . . . . . . . . . . . 54LIST OF FIGURES – Continued64.5 Contour plot of 2, Equation 4.1, with initial parameters of a D36 m, S D 2913 cP and ! D 1 rad=s and coefficients values of D 2:8 ˙ 0:3, 'm D 0:79 ˙ 0:11, and 2 D 1:170 showing theprocess in which the coefficients are calculated. The red star, D4:62, 'm D 2:75, and 2 D 23:125, is the initial guess found froma least squares fit. The black box shows the steps the code took forminimizing 2, taking smaller steps as in nears the minimum valueof 2. The blue triangle represents the best fitting coefficient foundusing this programs with values at D 2:8 ˙ 0:3, 'm D 0:79 ˙ 0:11,and 2 D 1:170. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.6 Three dimensional plot of 2, Equation 4.1, with initial parametersof a D 36 m, S D 2913 cP and ! D 2 rad=s and coefficients valuesof D 2:8 ˙0:3, 'm0:79 ˙0:11, and 2 D 1:170 showing the bottomof the well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.7 The plot of relative viscosity as a function of concentration usinginitial parameters of a D 36 m, S D 2988 cP and ! D 1 rad=sshows the red line using coefficients D 2:8 and 'm D 0:79 fromall seven concentrations and the blue line uses coefficients D 2:5and k'm D 0:69 from just the four concentrations. The shaded areais bounded by the maximum and minimum values of the coefficients D 2:8˙ 0:3 and 'm D 0:79˙ 0:11, respectively, indicating the error. 594.8 The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the blackpoints are the standard and the green line is Mooney’s Equation 2.11using the calculate value of 'm D 0:79 ˙ 0:11 and D 2:8 ˙ 0:3. . . 614.9 The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the blackpoints are the standard and the blue line is Krieger-Dougherty’s Equa-tion 2.12 using the calculate value of Œ D 3:2 ˙0:4 and ' D 0:6 ˙0:1. 654.10 The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the blackpoints are the standard and the yellow line is Batchelor’s Equation2.13 using the calculate value of D 1:0 ˙ 0:8, ˛ D 7 ˙ 6 and'm D 0:6 ˙ 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.11 The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the blackpoints are the standard and the purple line is Brady’s Equation 2.18using the calculate value of D 1:2 ˙ 0:9, 'm D 0:6 ˙ 0:1, andˇ D 8 ˙ 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73LIST OF FIGURES – Continued74.12 Plot of vs a modeling the data to Mooney’s Equation 2.11 andBatchelor’s Equation 2.13. (a) Different solvent viscosities with anangular velocity ! D 1:000 rad=s, the red stars are for S D 3017 cP,the green triangles are for S D 1265 cP and the blue squares arefor S D 571 cP. (b) Different angular viscosities, the red stars arefor angular velocity ! D 1:000 rad=s and S D 3017 cP, the greentriangles are ! D 2:000 rad=s and S D 2988 cP, the blue squaresare ! D 3:000 rad=s and S D 2965 cP, and yellow diamonds are! D 4:000 rad=s and S D 2924 cP. . . . . . . . . . . . . . . . . . . . 794.13 Plot of vs amodeling the data to Krieger-Dougherty’s Equation 2.12and Brady’s Equation 2.18. (a) Different solvent viscosities with anangular velocity ! D 1:000 rad=s, the red stars are for S D 3017 cP,the green triangles are for S D 1265 cP and the blue squares arefor S D 571 cP. (b) Different angular viscosities, the red stars arefor angular velocity ! D 1:000 rad=s and S D 3017 cP, the greentriangles are ! D 2:000 rad=s and S D 2988 cP, the blue squaresare ! D 3:000 rad=s and S D 2965 cP, and yellow diamonds are! D 4:000 rad=s and S D 2924 cP. . . . . . . . . . . . . . . . . . . . 814.14 Plot of vs S modeling the data to Mooney’s Equation 2.11 andBatchelor’s Equation 2.13. (a) Different radii with an angular veloc-ity ! D 1:000 rad=s, the red stars are for a D 26 m, the green trian-gles are for a D 36 m, the blue squares are for a D 51 m, and theyellow triangles are for a D 71 m. (b) Different angular viscositiesfor the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the bluesquares are ! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. 834.15 Plot of vs S modeling the data to Krieger-Dougherty’s Equation2.12 and Brady’s Equation 2.18. (a) Different radii with an angularvelocity ! D 1:000 rad=s, the red stars are for a D 26 m, the greentriangles are for a D 36 m, the blue squares are for a D 51 m,and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angularvelocity ! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s,the blue squares are ! D 3:000 rad=s, and yellow diamonds are ! D4:000 rad=s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85LIST OF FIGURES – Continued84.16 Plot of vs ! modeling the data to Mooney’s Equation 2.11 andBatchelor’s Equation 2.13. (a) Different radii with a range of 2924 <S < 3017 cP, the red stars are for a D 26 m, the green triangles arefor a D 36 m, the blue squares are for a D 51 m, and the yellowtriangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP,the green triangles are for 1211 < S < 1265 cP and the blue squaresare for solvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . 874.17 Plot of vs ! modeling the data to Krieger-Dougherty’s Equation2.12 and Brady’s Equation 2.18. (a) Different radii with a range of2924 < S < 3017 cP, the red stars are for a D 26 m, the greentriangles are for a D 36 m, the blue squares are for a D 51 m,and the yellow triangles are for a D 71 m. (b) Different solventviscosities for the particle size a D 36 m, the red stars are for2924 < S < 3017 cP, the green triangles are for 1211 < S < 1265 cPand the blue squares are for solvent viscosity of 555 < S < 571 cP. . 894.18 Plot of 'm vs a modeling the data to Mooney’s Equation 2.11. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s,the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.19 Plot of 'm vs a modeling the data to Krieger-Dougherty’s Equation2.12. (a) Different solvent viscosities with an angular velocity ! D1:000 rad=s, the red stars are for S D 3017 cP, the green trianglesare for S D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94LIST OF FIGURES – Continued94.20 Plot of 'm vs a modeling the data to Batchelor’s Equation 2.13. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s,the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.21 Plot of 'm vs a modeling the data to Brady’s Equation 2.18. (a) Dif-ferent solvent viscosities with an angular velocity ! D 1:000 rad=s,the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.22 Plot of 'm vs S modeling the data to Mooney’s Equation 2.11. (a)Different radii with an angular velocity ! D 1:000 rad=s, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different angular viscosities for the particle sizea D 36 m, the red stars are for angular velocity ! D 1:000 rad=s,the green triangles are ! D 2:000 rad=s, the blue squares are ! D3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. . . . . . . . . 1004.23 Plot of 'm vs S modeling the data to Krieger-Dougherty’s model,Equation 2.12. (a) Different radii with an angular velocity ! D1:000 rad=s, the red stars are for a D 26 m, the green triangles arefor a D 36 m, the blue squares are for a D 51 m, and the yellowtriangles are for a D 71 m. (b) Different angular viscosities forthe particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the bluesquares are ! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.101LIST OF FIGURES – Continued104.24 Plot of 'm vs S modeling the data to Batchelor’s Equation 2.13.(a) Different radii with an angular velocity ! D 1:000 rad=s, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different angular viscosities for the particle sizea D 36 m, the red stars are for angular velocity ! D 1:000 rad=s,the green triangles are ! D 2:000 rad=s, the blue squares are ! D3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. . . . . . . . . 1024.25 Plot of 'm vs S modeling the data to Brady’s Equation 2.18. (a)Different radii with an angular velocity ! D 1:000 rad=s, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different angular viscosities for the particle sizea D 36 m, the red stars are for angular velocity ! D 1:000 rad=s,the green triangles are ! D 2:000 rad=s, the blue squares are ! D3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. . . . . . . . . 1044.26 Plot of 'm vs ! modeling the data to Mooney’s Equation 2.11. (a)Different radii with a range of 2924 < S < 3017 cP, the red starsare for a D 26 m, the green triangles are for a D 36 m, theblue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 1064.27 Plot of 'm vs ! modeling the data to Krieger-Dougherty’s Equation2.12. (a) Different radii with a range of 2924 < S < 3017 cP, thered stars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles arefor a D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 107LIST OF FIGURES – Continued114.28 Plot of 'm vs ! modeling the data to Batchelor’s Equation 2.13.(a) Different radii with a range of 2924 < S < 3017 cP, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles arefor a D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 1084.29 Plot of 'm vs ! modeling the data to Brady’s Equation 2.18. (a)Different radii with a range of 2924 < S < 3017 cP, the red starsare for a D 26 m, the green triangles are for a D 36 m, theblue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 1094.30 Plot of ˛ vs a modeling the data to Batchelor’s Equation 2.13. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s,the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.31 Plot of ˛ vs S modeling the data to Batchelor’s Equation 2.13. (a)Different radii with an angular velocity ! D 1:000 rad=s, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different angular viscosities for the particle sizea D 36 m, the red stars are for angular velocity ! D 1:000 rad=s,the green triangles are ! D 2:000 rad=s, the blue squares are ! D3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. . . . . . . . . 114LIST OF FIGURES – Continued124.32 Plot of ˛ vs ! modeling the data to Batchelor’s Equation 2.13. (a)Different radii with a range of 2924 < S < 3017 cP, the red starsare for a D 26 m, the green triangles are for a D 36 m, theblue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 1154.33 Plot of ˇ vs a modeling the data to Brady’s Equation 2.18. (a) Dif-ferent solvent viscosities with an angular velocity ! D 1:000 rad=s,the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b)Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=sand S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.34 Plot of ˇ vs S modeling the data to Brady’s Equation 2.18. (a)Different radii with an angular velocity ! D 1:000 rad=s, the redstars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different angular viscosities for the particle sizea D 36 m, the red stars are for angular velocity ! D 1:000 rad=s,the green triangles are ! D 2:000 rad=s, the blue squares are ! D3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s. . . . . . . . . 1194.35 Plot of ˇ vs ! modeling the data to Brady’s Equation 2.18. (a)Different radii with a range of 2924 < S < 3017 cP, the red starsare for a D 26 m, the green triangles are for a D 36 m, theblue squares are for a D 51 m, and the yellow triangles are fora D 71 m. (b) Different solvent viscosities for the particle sizea D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are forsolvent viscosity of 555 < S < 571 cP. . . . . . . . . . . . . . . . . . 120A.1 Code used to minimize 2 . . . . . . . . . . . . . . . . . . . . . . . . 12413LIST OF TABLES3.1 Solvent Viscosity Table. A table showing all the solvent viscositiesused in this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Table of particle size, radius ratio and density with standard deviation. 423.3 Table of Actual % Volume using Equation 3.3 and Equation 3.4 forthe standard deviation using the appropriate values found in the tableand Table 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Table of the percent difference used to calculate the error bars for thecoefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2 Mooney Table. A table using the data to model Equation 2.11using a least squares-fit to calculate the values of k and with anaverage 2 D 2:492. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Krieger-Dougherty Table. A table using the data to model Equa-tion 2.12 using a least squares-fit to calculate the values of Œ and'm with an average 2 D 8:455. . . . . . . . . . . . . . . . . . . . . . 674.4 Batchelor Table. A table using the data to model Equation 2.13using a least squares-fit to calculate the values of , ˛, and 'm withan average 2 D 2:506. . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 Brady Table. A table using the data to model Equation 2.18 usinga least squares-fit to calculate the values of , ˇ, and 'm with anaverage 2 D 2:421. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7514ABSTRACTThe relative viscosity of a mono-disperse suspension of glass particles can vary dras-tically depending on a number of factors, including but not limited to the angularvelocity, the size of particles, the volume fraction of the suspension, and the solventviscosity of the host. Several models have been proposed in the literature describingthe relative viscosity as a function of concentration. This study will use volumeconcentrations of 20%, 25%, 30%, 35%, 40%, and 50% with solvent viscosities of3017 2924 cP ˙ 5 cP, 1265 1211 cP ˙ 3 cP and 571 555 cP ˙ 1 cP in aconcentric cylinder geometry. Suspensions of 26 m ˙ 3 m, 36 m ˙ 3 m,51 m ˙ 4 m, and 71 m ˙ 4 m radii will be sheared at angular velocities of1:000 rad=s ˙ 0:001 rad=s, 2:000 rad=s ˙ 0:001 rad=s, 3:000 rad=s ˙ 0:001 rad=s,and 4:000 rad=s ˙ 0:001 rad=s. The resulting relative viscosities will be used tostudy the functional dependence of the fitting parameters in the models proposed byMooney, Krieger-Dougherty, Batchelor, and Brady on particle size, solvent viscosity,and angular velocity.15CHAPTER 1IntroductionEvery morning many have dreams of racing to work at 55 miles per hour with no redlights to impede their voyage. Instead, they are stopped on the expressway staringat a sea of red brake lights wondering if they will ever make it to work on time.As many sit there motionless, some may begin to question why traffic comes to acrawl and can there be ways of improving traffic flow. One way of considering thistraffic problem is using fluid mechanics principles to make improvements to trafficpatterns that maximize traffic flow, thereby minimizing travel times. By consideringan expressway as a “pipe” and the cars as the “liquid,” we can use the principlesof fluid mechanics to increase the flow rate, or traffic flow. Ideally, we want tocreate a “pipe,” or expressway, that maximizes the flow rate under all conditionsallowing traffic to flow more openly and freely, creating less congestion and lowertravel times. Fluid mechanics can solve this problem making everyone’s voyage towork more enjoyable.Fluid mechanics is the branch of physics that deals with the principles that gov-ern the properties of a fluid and fluid-like materials. It is extremely mathematicallydense with many unsolved or partially solved problems. Consequently, something assimple as predicting today’s weather becomes extremely difficult. Research projects,such as the International Space Station Fluids Integrated Rack, seeks for a deeperunderstanding of the principles of fluid mechanics at macroscopic and microscopiclevels in macro-gravity. These research projects are looking for a better understand-ing into the properties of fluids, which will allow us to design better products; likemore fuel efficient cars, faster airplanes, and better roadways. These products canmake crucial improvements in our daily lives.A fluid can be any substance that will not sustain an applied shear stress. Ashear stress is a force applied parallel to the surface of an object. For example, place16your hand on the top of a table and push it parallel to the surface. The table doesnot deform. However, when applying the same stress to the surface of water in abath tub, we see the water easily deforms. A fluid is not limited to just liquids likewater; on the contrary, it can be a solid, like sand, or a gas, like air. Neither of thesesubstances can sustain a shear stress. Solids and liquids have a definite volume;whereas, a gas will expand to fill the volume defined by the size of the containerholding it.All fluids will conform to the shape of the container. If the volume of thecontainer is larger than the volume of the fluid placed in the container, the solidor liquid will occupy one distinct portion of the container. A single boundary willseparate the container’s occupied and unoccupied regions; however, a gas will expandfilling the entire volume of the container with boundaries defined only by the shapeof the container.Since a fluid can be a solid, a civil engineer designing an asphalt road needs totake into consideration the terrain. Over a long period of time, the asphalt will flowlike any other fluid; therefore, the engineer needs to design a proper roadbed andmixture of asphalt to ensure a long lifespan of the roadway.Generally, we think of fluids as a liquid. Maximizing the flow rate is of vitalimportance in many professions, like firefighting. A fire hose with insufficient waterpressure, or water flow, can put firefighters’ lives in jeopardy.A fluid can also be a gas, which becomes important when predicating the weather.For example, a simple cloud is understood to form when warm air rises and reaches alevel of moderately cool air, where the moisture in the air condenses forming the cu-mulus cloud. This is a simple model that is understood particularly well. However,little is understood about the more complex supercell that produces serve thun-derstorms and powerful tornadoes. This rotating cloud formation is a much morecomplex and unpredictable system; therefore, understanding it becomes extremelydifficult. By studying simple systems, like the cumulus cloud, we can get a deeperunderstanding of more complex systems, like the supercell.Changing a fluid’s temperature or pressure consequently changes the properties17of the fluid. The air mass changes temperature and pressure as the warm air risesand cools. Eventually, the air mass reaches the dew point which changes the vaporfrom a gas, water vapor, to a liquid, water.Changing a fluid’s density or velocity also changes the properties of a fluid. Thedensity is related to the temperature and pressure by the Ideal Gas Law. Whenthe pressure changes, a velocity is indirectly created. Conversely, when a velocityis induced, a pressure change is created. The velocity of a fluid also creates heatbecause of friction, whether it’s two fluids rubbing against each other or air moleculesflowing past each other. As we can see, the velocity and density are indirectly relatedto pressure and temperature, each of which can affect the properties of a fluid.A change in the temperature of a simple system, such as warm air, can cause thesystem to become complex. The cumulus cloud changes from a warm air mass, towater droplets, to rain because of a temperature change. The same is true in morecomplex systems. If you change one of the system’s parameters they become evenmore complex. For example, we don’t understand what happens when a warm airmass inside a supercell rises and meets a layer of colder air. It could weaken thesystem, strengthen the system, or be the onset of a tornado.The current hypothesis maintains the velocity of the wind creates a shear forcethat gets a column of air spinning. An upward draft bends the column of air towardsthe sky creating a funnel cloud which eventually creates a tornado. However, thereis no explanation of why or how the column of rotating air bends in the middle toform a funnel cloud. As the middle of rotating column rises, one half starts thetornado and the other half just seems to disappear.The reason we see the tornado is dust particles and other stuff rotating inside it.We know the velocity fields are changing as the tornado picks up stuff, but exactlyhow it is changing is a mystery to us. It could be weakening or strengthening thetornado, but again, we just don’t know.Simple fluids are a starting place when considering more complex fluids contain-ing suspended particles. A complex fluid is a mixture of two or more substances thatcan be separated. A mud puddle is an example of a complex fluid, where the dirt18particles are mixed with water creating a suspension. These systems are complexfor good reason. Their behavior is often unpredictable and mystifying. An exampleis spontaneous particle size separation, where there is a de-mixing of the solutionby particle size. Most of the information about complex systems is obtained, notby theoretical derivation, but by qualitative and empirical analysis. Next time youget an oil change, consider using high grade oil that contains silicate microspheresto reduce bearing friction. Not only does it reduce the bearing friction, but it helpsfight off the formation of sludge and deposits created by the damaging bi-products ofcombustion like fuel, water, acid, metals, and dirt. Exactly how the silicate spheresare reducing friction or cleaning the engine is a mystery, but we can see the effectsof the microspheres which demonstrates they are working. This is just one of themany mysteries of fluid mechanics.In this thesis, the effects on the viscosity of a suspension were studied usinga highly sensitive general-purpose rheometer. The rheometer applied a torque tothe suspension under investigation and calculated the viscosity. The suspensionswere composed of difference sizes and concentrations and measurements are takenat different angular velocities and solvent viscosities (temperatures). I used thedata obtained and explored the effects on the viscosity for different particle sizes,velocities, concentrations, and solvent viscosities comparing it to semi-theoreticalequations. Using the semi-theoretical equations, I was able to investigate the termsand definitions given by those equations determining if there were places for improve-ment or if new terms needed to be defined. Finally, as technology and equipmentimproves, testing and re-testing past theories not only will allow for a better under-standing of the current theories but will permit a better understanding of how theentire universe works.This study will explore the functional dependence of the Mooney, Krieger-Dougherty, Batchelor, and Brady models of relative viscosity versus concentrationon particle size, concentration, angular velocity, and solvent viscosity. The studywill use particle sizes of 26 m ˙ 3 m, 36 m ˙ 3 m, 51 m ˙ 4 m,and 71 m ˙ 4 m. The study will included concentrations of 20:0% ˙ 0:3%,1930:0% ˙0:4%, and 40:0% ˙0:6% by volume; however, the 36 m particles will alsouse 25:0% ˙0:3%, 35:0% ˙0:3%, and 50:0% ˙0:4%. Each of these concentrationsand particle sizes were sheared at angular velocities of 1:000 rad=s ˙ 0:001 rad=s,2:000 rad=s ˙0:001 rad=s, 3:000 rad=s ˙0:001 rad=s, and 4:000 rad=s ˙0:001 rad=s.Finally, each concentration, particle size, and angular velocity used a range of sol-vent viscosities from 3017 to 2924 centipoises ˙ 5 cP, 1265 to 1211 cP ˙ 3 cP and571 to 555 cP ˙ 1 cP.20CHAPTER 2Fluid DynamicsA fluid is a collection of molecules that are randomly arranged and held togetherby weak cohesive forces[2]. It is difficult to study the motion of each individualparticle of the fluid as a function of time. Instead, we describe the properties ofthe fluid at each point as a function of time. When fluids are in motion, their flowcan be characterized into two groups: laminar and turbulent flow. Laminar flowis defined by*r *vD 0, which results in the fluid following a smooth path, asshown in Figure 2.1. The curl is a vector operator that describes the rotation ofthe vector field[3]. When it is equal to zero, there is no rotation; hence, the flow isstreamline. Turbulent flow is defined by*r *v¤ 0, which results in the rotation ofthe streamlines and the production of vortices, as shown in Figure 2.2.The Reynolds number .Re/ is a dimensionless quantity that gives the ratio ofinertial forces to viscous forces. It is the product of velocity and some characteristiclength divided by viscosity.Re D U`where U is a velocityms, ` is a length .m/, is a densitykgm3, and is a viscositykgms. The value of the number determines whether a fluid is flowing in a laminar,transitional, or turbulent flow regime. For a low Reynolds number, the flow isFigure 2.1: A car in a wind tunnel showinglaminar flow. Notice the streamlines do nottouch or cross while remaining parallel to eachother.21Figure 2.2: A cigarette showing both laminar and turbu-lent flow. The smoke nearest the cigarette is laminar. Thesmoke furthest from the cigarette is turbulent as shownby the swirling smoke patterns.laminar and the viscous forces are dominant. For a high Reynolds number, the flowis turbulent and inertial forces are dominant[4].For concentric cylinders, the flow regime can be found using the followingReynolds number calculation:Re D !a.b a/(2.1)where ! is the inner cylinder’s angular rotation raterads, a is the diameter of theinner cylinder .m/, b is the diameter of the outer cylinder .m/, is the solventdensitykgm3, and is the fluid viscositykgms. For this project, there is a range forthe Reynolds number because the viscosity is affected by temperature and velocity.We used three different temperatures and four different angular velocities, yieldinga Reynolds number range of 0:21 < Re < 4:5. The Reynolds number for this projectis low, Re < 10, conferring the system is well within the laminar flow regime withthe streamlines flowing around an object, as shown in Figure 2.1.The equation of continuity is defined as the net mass flux through a closedsurface. For an incompressible fluid the net flux must vanish[2]. The net flux isgiven by —*v d *AD 0Using the divergence theorem, we get• *r *vd D 022therefore*r *vD 0 (2.2)Equation 2.2, the conservation of mass equation is valid for both laminar and tur-bulent flow. The continuity equation describes the velocity of cross-sectional area atany given point along the streamline. The streamlines will diverge before an objectand converge after the object changing velocity to keep the streamlines in laminarflow. The streamlines over the hood of the car converge and accelerate while thestreamlines over the trunk diverge and decelerate, as seen in Figure 2.1.The viscosity depends on several parameters, one being temperature. The varia-tion of viscosity as a function of temperature is given by the Arrhenius Equation[4] ŒT D A ExpBT(2.3)where is the viscosity .Pa s/, T is temperature .Kelvin/, and both A and Bare positive empirical constants unique to each substance. The Arrhenius equa-tion states the viscosity decreases exponentially with temperature. Molecules of aliquid at higher energies vibrate more causing the spacing between the moleculesto be increased, which, in turn, allows the molecules to flow more freely and lessobstructed.2.1 Newton’s Law of ViscosityConsider two parallel plates a distanceH apart. The bottom plate is held stationary,and the top plate is moving by some applied force,*F , with velocity,*U . A fluid fillsthe gap with no slipping at either surface, as shown in Figure 2.3. The force tomove the top plate is proportional to the velocity gradient, dudy. Imagine the fluidin-between the plates as individual infinitesimal thin layers, stacked on top of eachother. When the top plate moves at a velocity*U , it will force the first infinitesimallayer of fluid beneath the top plate to move. The first layer will move the secondlayer, and the second layer will move the third layer. Each infinitesimal layer of fluidhas a slightly smaller velocity as it approaches the bottom plate where the velocityis zero.23Figure 2.3: Couette Flow. Simple experiment showing the effects of viscosity[1].Experimentally a shear force, D FA, is applied to the top plate where A isthe surface area of the plate that is in contact with the fluid. The shear force isproportional to the velocity gradient dudy, where is the proportionality constantcalled viscosity with SI units Pa s D kgms .FA/ dudy yDH) FAD dudy yDH(2.4)Viscosity is defined as the resistance of the fluid to a shear force[4]. Newton’slaw of viscosity is defined asViscous Stress D Velocity Gradientor D @u@y(2.5)2.2 Particle FlowConsider a sphere with constant velocity moving in a viscous medium, as shown inFigure 2.4 . Terminal velocity occurs when the net acceleration equals zero, and thevelocity is constant. Stokes’ drag law defines the force on the sphere as it travelsthrough the medium asFd D 6aU (2.6)24where is the viscosity of the medium, a is the radius of the sphere, and U is thesphere’s velocity. The drag force is always opposing the velocity.Figure 2.4: Free-body Diagram for a sphere traveling in a fluid at terminal velocity.The buoyancy force is given byFb D 43a3g; (2.7)and the weight force is given byFw D 43a3sg; (2.8)where a is the radius of the sphere, is the density of the medium, s is the densityof the sphere, and g is the acceleration due to gravity. Setting Equation 2.6 and 2.7equal to 2.8 gives Stokes’ Velocity as the followingU D 29a2g .s / (2.9)2.3 Stream Lines and Interacting SpheresConsider the flow around a particle. For Re << 1, the Navier-Stokes equation@v@tC *v rvD g rp C r2 *U25can be reduced torp D r2 *UD *r *!where p is the fluid pressure, is the viscosity,*U is the flow velocity and*! is thevorticity defined as*!D*r *u .[1]Using the Stokes stream function, , in spherical polar coordinates, the stream-lines around a particle are as followsur D 1r2 sin @ @u D 1r sin @ @rIn the case of axisymmetric flow, u' D 0, and the vorticity equation for the^'component is given by!' D 1r@@r.ru/ @ur@The vorticity equation for low Reynolds number flow is!' D 1r1sin @2 @r2C 1r2@@1sin @ @Using separation of variables and r2!' D 0, one finds the solution for as thefollowing [1] D Ur2 sin2 12 3a4rC a34r3where U is the flow velocity, r is the radial coordinate and a is the radius of thesphere. Figure 2.5 shows the streamlines flowing past a sphere. When the stream-lines are very close together they will be defined as compressed, and when thestreamlines are spread far apart they will be defined as relaxed. A suspension witha high concentration of particles will have tightly compressed streamlines; whereas,a suspension with low concentration will have relaxed streamlines.The introduction of more particles has some strange effects on particle interac-tions. First, two or more particles can flow faster than the Stokes velocity for asingle comparable particle.[5] The particles approach each other, aligning in sucha way that they appear to have the surface area of a single particle, but heavier26Figure 2.5: Stream lines flowing around asphere traveling in a fluid at terminal velocity.total mass. This is why they can flow faster than the Stokes’ velocity. Second, thestreamlines for a single particle can influence particles at great distances becausethe interactions decay like 1r. Finally, for a particle flowing at a constant velocity,the streamlines converge before the particle and diverge after the particle.Consider three equal particles flowing at a constant velocity in the downwarddirection with streamlines compressing and relaxing as they interact, as shown inFigure 2.6. At some time, t D 1, the three particles are all flowing at the samevelocity. At some later time, t D 2, the red and green particles have changedvelocities speeding up and catching the blue particle. At t D 4, the red and greenparticle have passed the blue one. The particles’ streamlines are interacting withone another such that they are changing velocities. At t D 6, the blue particlehas pushed the green particle out of the way by influencing the streamlines to thepoint that the red particle is flowing faster and the green particle is flowing slower.Finally at t D 8, the red and blue particles are flowing faster than the green particle,but there is no interaction here. This process keeps repeating itself with randomvelocities for each particle. Laminar flow does not allow the crossing of streamlinesbetween particles. Instead, the streamlines compress and diverge, changing thevelocity of a particle or particles. This explains why the viscosity is higher as theconcentration is increased. The stream lines are compacted closer together, makingthe flow more difficult, thus increasing the viscosity.[5]27Figure 2.6: Three equal spheres flowing at a constant velocity in the direction of thearrows at various times showing how particle interaction can increase and decreasethe velocities of the spheres.28Figure 2.7: Streamlines of Fluid relative to moving cloud.A finite number of particles with radius a can form a spherical cloud with alarger radius R that has a different density and viscosity, as shown in Figure 2.7.[5]The speed of the cloud can be much faster than the speed of a single particle and thelarger size of the particle creates a bigger wake for the hydrodynamic interactions.2.4 EquationsThe data collected will be modeled to Einstein’s, Mooney’s, Krieger-Dougherty’s,Batchelor’s, and Brady’s equations to obtain a better understanding of the relativeviscosity as a function of concentration. The models will look at different concentra-tions, solvent viscosities, particle sizes, and angular velocities to explore the affectsthose parameters have on the relative viscosity. Einstein’s equation starts out as asimple linear equation and Mooney’s, Krieger-Dougherty’s, Batchelor’s, and Brady’sequations all build upon each other adding more insight and detail between particleinteractions as the equations become more complex.292.4.1 EinsteinIn 1906, Albert Einstein wrote his Ph. D. dissertation on the molecular-kinetictheory of liquids, where the size of molecules of substances dissolved in an undisso-ciated dilute solution can be determined from the internal viscosity of the solutionand of the pure solvent[6]. By studying how a singular sphere and multiple spheressuspended in a fluid influenced the fluid’s motion, Einstein was able to write thefollowing equationr D 1C ' (2.10)where r D 0 is the relative viscosity, is the viscosity of the suspension, 0 is theviscosity of pure solvent, ' is the volume fraction of spheres, and is a coefficientto be determined. Einstein states for very small rigid spheres suspended in a liquid,the coefficient of viscosity increases by a fraction that is equal to 2:5 times the totalvolume of the spheres suspended in a unit volume, provided that this total volumeis very small. It has been well accepted that the value of D 2:5 is acceptable forlow concentrations, 0% < ' < 10%.2.4.2 MooneySince Einstein’s equation is only valid for low concentrations, Mooney consideredhigher concentrations. Mooney quickly realized that, at higher concentrations, theempirical data for relative viscosity appear to follow more of an exponential functionof concentration. Taking into account the variation in particle size and particleinteractions for higher concentrations, Mooney derived the following[7]r D Exp'1 k'(2.11)where r is the relative viscosity, ' is the volume fraction, k is a constant calledthe self-crowding factor, and is a fitting parameter chosen to agree with Einstein’svalue of 2:5.There is a limit to the maximum concentration, or self-crowding factor, becausethe curve is exponential; when the curve goes to infinity, we have reached maxi-mum concentration. This self-crowding factor at maximum concentration does not30allow the movement of neighboring particles because the suspension is acting likea solid. Consider a two-component system. Given spheres of size a1 in partial vol-ume concentration '1. Spheres of a2 crowd into the remaining free volume given by1 12'1.[7] To determine a value for the self-crowding factor, Mooney consideredthe packing densities.The minimum value of k comes from the least dense packing factor, which isthe sphere packing factor of 3p2, k D 3p2 1:35. The maximum value of kcomes from the densest cubic factor, which is 6, so k D 6 1:91. Mooney alsodetermined that the factor of k is not significantly affected by temperature, butrather more dependent on concentration because k D 1'm, where 'm is the maximumconcentration. Mooney determines the range of k to be1:35 < k < 1:91or0:52 < 'm < 0:742.4.3 Krieger-DoughertyKrieger and Dougherty began looking at the interactions between neighboring spher-ical particles the following two ways. First, how each particle can rotate separately,called a singlet. Second, how two particles can rotate (about one another) together,called a doublet. Figure 2.8 shows an example of singlets and doublets. Even thoughthe solution contains both singlets and doublets, most of the interactions are sin-glets because eventually the doublets will separate forming singlets[8]. Krieger andDougherty used Eiler’s equation,r D"1C 12Œ'1 ''m#2;and Mooney’s equation, Equation 2.11, to come up with the following equationr D1 ''mŒ'm(2.12)31Figure 2.8: A singlet is where the two particles rotate independently and a doubletis where the two particles rotate together like a dumbbell.where Œ is a fitting parameter based on the shape of the particles (called the“intrinsic viscosity” taken to equal 2:5 to agree with Einstein), ' is the volumefraction, and 'm is the maximum volume fraction. The parameter Œ D 2:5 becauseof the spherical shape of a single particle (singlet). However, the capsule shape oftwo particles rotating together as one (doublet) will increase the value of Œ becausethe particles are not rotating like a perfect sphere.2.4.4 BatchelorBatchelor expanded Equation 2.10 to include terms of order c2, where c is thevolume fraction. Again, he brought to light the fact that the particles interact witheach other, and those interactions will affect the viscosity of the entire system.[9]The value of c is taken from Mooney and written as the followingc D '1 ''m!Batchelor writes his equation as followsr D 1C c C ˛c2 (2.13)32The term ˛ is a fitting parameter based on the probability density, p*r ; t,which is determined by three-sphere interactions, Brownian motion, or the assump-tion of some particle’ s initial state[9]. The probability density represents thevector*r separating the centers of the two particles. The values can range from5:2 < ˛ < 7:6. The linear coefficient, , is taken to agree with Einstein.2.4.5 BradyBrady expanded upon the simple model for the rheological behavior of concentratedcolloidal dispersions consisting of the two phases for suspensions. Brady consideredthe suspensions of Brownian hard spheres and its two contributions to the macro-scopic stress, both hydrodynamic and Brownian stress, when deriving his equation.The hydrodynamic stress is the dynamic viscosity. The Brownian stress is respon-sible for the viscoelastic behavior of colloidal dispersions. The Brownian stress isproportional to the equilibrium radial-distribution function at contact, g.2/ dividedby the short-time self-diffusivity, DS0 .'/, both evaluated at the volume fraction'.[10]Physically, it is the short-time self-diffusivity that governs the viscoelastic re-sponse because, for small departures from equilibrium, particles only need to diffusea proportionally small distance to relax the structure back to equilibrium.As random close packing is approached, the number of particles “at contact” di-verges asg.2/ 1:21 ''m1as ' ! 'm (2.14)and the short-time self-diffusivity vanishes asDS0 .'/ 0:851 ''mas ' ! 'm (2.15)The particles are “located” into place by the strong hydrodynamic lubricationforces, thus the effective viscosity diverges as1 ''m233as random close packing is approached. Brady claims this is the essence behind thesingular behavior of the viscosity of Brownian dispersions.The hydrodynamic viscosity H scales in the dilute limit asH 52' as ' ! 'm (2.16)The zero shear-rate viscosity is given byr D 1C H C 125'2g.2/DS0 .'/(2.17)Substituting Equations 2.14, 2.15, and 2.16 into Equation 2.17 results in thefollowing equationr D 1C ' C ˇ'21 ''m2(2.18)where is a fitting parameter taken as 2:5 to agree with Einstein’ s Equation 2.10(section 2.4.1) and ˇ ' 28885is a combination of fitting parameters and 'm is themaximum concentration.34CHAPTER 3Experiment3.1 Apparatus3.1.1 Concentric CylinderThe concentric cylinder geometry used in this experiment is a system of two cylindersof common axis. The inner cylinder is defined as the inside geometry and is a solidpiece of Plexiglas, as shown in Figure 3.1(a). The outer cylinder is defined as theoutside geometry and forms a hollow cylindrical cup, as shown in Figure 3.1(b). Theoutside geometry holds the sample, at constant thermal equilibrium, and the insidegeometry, which applies the torque, is inserted into the sample, as shown in Figure3.1(c).The inside geometry, Figure 3.1(a), is made from a solid cylindrical piece of Plex-iglas with an outside diameter of 1 inch and a hollow cylindrical piece of Plexiglaswith an inside diameter of 1 inch and 1=4 inch thick walls, for a total outside diam-eter of 1:5 inches. The two pieces were glued together to form one solid cylinder.A lathe was used to turn the cylinder so that the outside diameter has a uniform(a) (b) (c)Figure 3.1: Images of concentric cylinder (a) is the inside geometry, (b) is the outsidegeometry, and (c) is the inside geometry inserted into the outside geometry with thesample filling the gap.35surface at 3:76 cm ˙ 0:01 cm. The roundness ratio is defined as the maximumradius divided by the minimum radius therefor 1 defines a perfect circle. The insidegeometry has a roundness ratio of 1:005. The geometry has an eccentricity causingthe outer surface to deviate ˙0:0001 cm from the mean for one full revolution. Themeasuring surface has a height of 6:02 cm ˙ 0:05 cm. The overall cylinder hasa height of 7:92 cm ˙ 0:01 cm. An M4 tap was used to thread the center of thiscylinder so that it could be attached to the AR 1500ex Rheometer’s spindle (Section3.1.3).The outside geometry is a cylindrical cup with an inside diameter of 2 inchesand a little over 2 inches deep. A 2 inch Plexiglas disk was attached to the bottomof the cylinder allowing one end of the cylinder to be closed and the other to beopen, as shown in Figure 3.1(b). The cylindrical cup is housed in a box, creatinga water jacket, allowing the sample to be held at a constant temperature using aNESLAB RTE 7 Bath Circulator (Section 3.1.2) which circulates water around thewater jacket. The hollowed cylinder has a diameter of 5:08 cm ˙ 0:01 cm and adepth of 5:59 cm ˙ 0:07 cm. The roundness ratio for the outside geometry is 1:002and has a total volume of 113:08 cm3 ˙ 0:01 cm3.The gap size is the distance between the outside and inside geometries. In thiscase the gap size is 0:66 cm˙0:01 cm. The custom concentric cylinder was calibratedand compared to known values for glycerin at specific concentrations (Section 3.2.1).The experimental values where consisted with the tabulated values.3.1.2 NESLAB RTE 7 Bath CirculatorThe NESLAB RTE 7 Bath Circulator, Figure 3.2, is a constant cool, pulsating heatunit that is used to keep the sample chamber at constant thermal equilibrium. TheNESLAB Bath has a temperature range 25:00ıC to C150:00ıC and temperaturestability ˙0:01ıC. The RTE 7 has a flow rate of 15 liters per minute which is fastenough to remove any heat generated. The unit has a 7 liter reservoir that holds thecooling agent, water. The water circulates around the outside geometry ensuringthe sample is held at constant thermal equilibrium. The NESLAB RTE 7 removes36Figure 3.2: The NESLAB RTE 7 Bath Circulator.the heat generated by the friction fromm the shearing of the suspension. It alsoremoves any heat introduced by the environment.The water jacket is a cubic box made out of 1=4 inch acrylic Plexiglas withoutside dimensions 12:7 cm 12:7 cm 11:4 cm used to stabilize the temperatureof the sample, as shown in Figure 3.1(c). The water jacket had a minimum of1 1=4 inch insulation of water surrounding the sample. The box is held together withPermabond Engineering Adhesive (910 Metal Bonding General Purpose DA2857).3.1.3 RheometerThe AR 1500ex, the heart of the experiment, is an advanced combined motor andtransducer (CMT) rheometer, shown in Figure 3.4. The head of the instrument isattached to a ball-slide that allows the vertical movement of the inside geometry.The head contains a drag cup motor, with an armature that forms the rotatingspindle of the rheometer, an air thrust bearing that supports the spindle, and anoptical encoder that determines angular position, as seen in Figure 3.3.[11]37Figure 3.3: Cut away of the head of the AR 1500ex Rheometer showing the variouscomponents used to rotated the geometry used to take measurements.The air bearing uses compressed air as the lubricating medium allowing a vir-tually friction-free application of torque for the attached geometry. A continuoussupply of air at 30 psi is needed to create this cushion of air allowing the geometryto rotate without friction. To ensure that the bearing rotation is steady throughouta full 360ı rotation, a rotational mapping is performed. The rotational mappingcombines the absolute angular position data from the optical encoder with micro-processor control of the motor calculating and storing the small mapping variations.The microprocessor then applies the stored mapped variations making the appro-priate baseline corrections to the applied torque as the spindle rotates. An initialrotational calibration is measured to calculate the moment of inertia of the spindleand geometry. This measurement is used to eliminate the torque caused by thespindle and geometry ensuring the torque measured by the rheometer is solely fromthe sample. The AR Instrument Software controls the rheometer and stores thisinformation implementing the corrections when applicable.The zero gap distance is defined when the bottom of the inside geometry touchesthe bottom of the outside geometry and is accurate to ˙0:01 m. This is done by38Figure 3.4: The AR 1500ex Rheometer from TA Instruments.lowering the inside geometry slowly until it touches the outside geometry. Thesoftware stores this zero gap location and all future measurements use this mark aszero.The rheometer’s hardware is capable of measuring torque to ˙0:1 Nm andangular displacement to 40 nRad. The microprocessor is capable of taking timemeasurements at 1 GHz or a time of 109 s per sample. The angular velocity forthe software has a resolution of 0:001 rad=s and an accuracy of 107 rad=s. Thesemeasurements are used to calculate the viscosity with a resolution of 104 Pa s andan accuracy of ˙106 Pa s.3.2 Sample3.2.1 GlycerineGlycerin, HOCH2CHOHCH2OH , is a colorless and odorless liquid, soluble inwater. The Material Safety Data Sheet indicates that the sample should be less39than 99:5% glycerin and a density of 1:2607 g=cm3. The density was measuredusing a graduated cylinder, ˙0:001 cm3, and a precision digital balance, ˙0:0001 g,made by Fulcrum Inc., model AGN200C. The measured mass density of glycerinwas g D 1:245 g=cm3 ˙ 0:001 g=cm3.The viscosity of glycerin as a function of temperature is shown in Figure 3:5. Therheometer calculated the viscosity for several different temperatures and is comparedto standard values from DOW Chemical standards for 100%, 99% and 98% pureglycerin.[12] The data agrees with < 99:5% pure glycerin from the MSDS. This datawas modeled to the Arrhenius Equation, Equation 2.3, using a least squares-fit todetermine the empirical coefficients resulting in the following: ŒT D 3:82 109 Exp7745T(3.1)where T is temperature in Kelvin. Calculating the standard deviation of the linegives line D 48 mPa s, where 1 centipoise.cP/ D 1 mPa s. Arrhenius Equation isflawed because it does not take into account the angular velocity dependence for thesolvent viscosity, which is seen in Table 3.1. Consequently, we will use the empiricaldata as the solvent viscosities.3.2.2 Glass ParticlesThe particles are precision grade soda-lime glass spheres (Class V) from MO-SCISpecialty Products, L.L.C. The manufacturer claims the particles are more than90% in the given range and more than 90% spherical. The refractive index for thesoda-lime glass particles is 1:5.The particles were measured using a digital microscope, shown in Figure 3:6, andImage-Pro c Plus image analysis software to find the maximum diameter, minimumdiameter, mean diameter, maximum radius, minimum radius, perimeter, radius ra-tio, and roundness of the particle.The digital microscope, model number DC4-410, uses a 1.2 Megapixel digitalcamera and Motic Images Plus 2.0 software to capture a numeral representation ofthe particles, shown in Figure 3.7. The images were loaded into Image-Pro c Plus,40Temperature.ıC/Angular Velocity.rad=s/SolventViscosity .cP/Standard Deviationsolvent Viscosity .cP/101 3017 ˙42 2988 ˙23 2965 ˙44 2924 ˙5201 1265 ˙32 1250 ˙23 1233 ˙24 1211 ˙2301 571 ˙12 566 ˙13 560 ˙14 555 ˙1Table 3.1: Solvent Viscosity Table. A table showing all the solvent viscositiesused in this experiment.Figure 3.5: A plot of viscosity as a function of temperature for the calibrationcurve of the custom concentric cylinder (black triangles with error bars) and DOWStandard values for 98% (red squares) pure glycerin, 99% (green stars) pure glycerin,and 100% (blue diamonds) pure glycerin.41Figure 3.6: The digital microscope model number DC4-410.which calculated the average size of each individual particle. The image of the parti-cles has been digitized creating very small regions called picture elements, or pixels.A 2000 m circle, made by Motic c was used to calibrate the software to measure0:180030 Pixels=m, which now allows the measurements of the individual parti-cles. Several pictures of each different size of particle were taken to obtain 50 to 200particles of each size for analysis. The software counts the particles and eliminatesanything touching the edge of the picture. It also eliminated anything too small ortoo large to be a measurable sphere. This would eliminate dust and glass spheresthat where touching each other. Table 3:2 lists all the information obtained fromthe computer software, including the mean radius, mean radius standard deviation,radius ratio, radius ratio standard deviation, and density.A histogram of the data is provided to show the mean radius distribution inFigure 3:8. The particles are physically sorted by using different screens calledmesh screens to allow particles through if the are too small, or hold them if theyare too big. This is the reason we have a mean radius. The particles with radius42Figure 3.7: Image of the particles (a D 51 m) under the microscope using MoticImages Plus 2.0.a D 71 m have a mesh size of 100= C 120 which selects particles with radius63 m < a < 75 m. The particles with radius a D 51 m have a mesh size of140=C 170 which selects particles with radius 45 m < a < 53 m. The particleswith radius a D 36 m have a mesh size of 200=C230 which selects particles withradius 32 m < a < 38 m. The particles with radius a D 26 m have a mesh sizeof 270=C 325 which selects particles with radius 23 m < a < 27 m.The density of the glass spheres, s, was calculated using displaced water in aMeanRadius.m/RadiusStandardDeviation.m/RadiusRatioRatioStandardDeviationDensity.g=cm3/ g=cm326 3 1.588 1.507 2.27 ˙0:0736 3 1.255 0.098 2.64 ˙0:0151 4 1.210 0.054 2.78 ˙0:0271 4 1.138 0.055 2.62 ˙0:04Table 3.2: Table of particle size, radius ratio and density with standard deviation.43Figure 3.8: Histogram of particles showing the mean and standard deviation values.graduated cylinder, ˙0:01 cm3, and the AGM200C precision digital balance. Thiswas done several times for all sets of particles and averaged together to determinea density for each size particle, shown in Table 3.2.3.3 ProcedurePreparing a sample with a desired percent volume of glass spheres mixed in glycerin,used the following equations%Vol D VspheresVTotalD VsVglycerin C VsVs D .%Vol/.Vg/.1 %Vol/ms D .s/.%Vol/.Vg/.1 %Vol/ (3.2)44%VolActual D ms.Vg/.s/Cms (3.3)Using the value of s from Table 3.2, measuring Vg from a graduated cylinder,and inserting the desired %Vol into Equation 3.2 yields the total mass of particles,ms, needed for the desired concentration. Using the actual ms along with the valuesof s and Vg in Equation 3.3 gives the actual percent volume for the sample. Table3.3 displays a list of all values obtained during the experiment.To find the error, we use propagation as follows[14]:VgshVgsi DsVghVgi2Cshsi2Vgs DsVghVgi2Cshsi2hVgsiVgsCms Dq2Vgs C 2ms%Volh%Voli DsVgsCmshVgs Cmsi2Cmshmsi2%Vol DsVgsCmshVgs Cmsi2Cmshmsi2h%Voli (3.4)The values from Equation 3.4 are displayed in Table 3.3 under the Actual %Volumecolumn as a standard deviation.Mixing the glycerin and particles was done by slowly rotating a graduated cylin-der, to avoid air bubbles in the suspension. Once the particles were evenly dis-tributed, the suspension was added to the outside geometry, and the inside geometrywas lowered to the proper gap size so that the experimental data could be taken.3.3.1 Experiment ProcedureThe first conditioning step used an angular velocity of 1:000 rad=s to stir the sus-pension for a total of one hour at a desired temperature using the RTE 7. After anhour, the first peak hold step held the rotation of the inner geometry constant at45MeanRadius.m/Desired%VolumeMass ofBeads .g/Volume ofGlycerin(cm3)Actual%VolumeTemp.oC/26 ˙3 20% 41.9307 74 ˙1 20.0% ˙0:3% 10,2026 ˙3 20% 46.4469 82 ˙1 20.0% ˙0:2% 3026 ˙3 30% 67.9696 70 ˙1 30.0% ˙0:4% 10,20,3026 ˙3 40% 77.0245 51 ˙1 40.0% ˙0:6% 10,20,3036 ˙3 20% 51.4479 78 ˙1 20.0% ˙0:2% 10,20,3036 ˙3 25% 66.8368 76 ˙1 25.0% ˙0:3% 10,20,3036 ˙3 30% 87.0697 77 ˙1 30.0% ˙0:3% 10,20,3036 ˙3 35% 105.1302 74 ˙1 35.0% ˙0:3% 10,20,3036 ˙3 40% 133.6593 76 ˙1 40.0% ˙0:3% 10,20,3036 ˙3 50% 117.4003 89 ˙1 50.0% ˙0:4% 10,20,3051 ˙4 20% 78.4964 113 ˙1 20.0% ˙0:1% 1051 ˙4 20% 86.8215 125 ˙1 20.0% ˙0:1% 20,3051 ˙4 30% 94.0888 34 ˙1 30.0% ˙0:4% 10,20,3051 ˙4 40% 120.8217 65 ˙1 40.1% ˙0:4% 1051 ˙4 40% 113.0428 61 ˙1 40.0% ˙0:4% 20,3071 ˙4 20% 47.8739 73 ˙1 20.0% ˙0:2% 10,20,3071 ˙4 30% 78.6526 70 ˙1 30.0% ˙0:3% 10,20,3071 ˙4 40% 113.6166 65 ˙1 40.0% ˙0:4% 10,20,30Table 3.3: Table of Actual % Volume using Equation 3.3 and Equation 3.4 for thestandard deviation using the appropriate values found in the table and Table 3.2.461:000 rad=s for five minutes taking a data point every five seconds. The second con-ditioning step increased the angular velocity to 2:000 rad=s for five minutes beforeproceeding to the next step. The second peak hold step holds the inner geometryconstant at 2:000 rad=s, again, for five minutes taking a data point every five sec-onds. The third conditioning step increases the angular velocity to 3:000 rad=s forfive minutes and proceeds to the third peak hold step keeping the angular veloc-ity constant at 3:000 rad=s for five minutes taking a data point every five seconds.Finally, the fourth conditioning step rotates the inner geometry at 4:000 rad=s forfive minutes proceeding to the final peak hold step using a constant angular velocityat 4:000 rad=s for five minutes taking a data point every five seconds. The entireprocess was repeated at different temperatures, concentrations, and particle sizes toobtain all reported data.3.3.2 DataData was collected at solvent viscosities of 3017 2924 cP, 1265 1211 cP and571 555 cP, which corresponds to temperatures 10ıC, 20ıC, and 30ıC, respec-tively. Data was collected at angular velocities of 1 rad=s, 2 rad=s, 3 rad=s, and4 rad=s. Data was collected at concentrations of pure glycerin, 20%, 30%, and 40%.Data was collected for particles with radii of 26m, 36m, 51m, and 71m. Forthe solution containing the particles with a radius of a D 36 m, additional concen-trations of 25%, 35%, and 50% were used. Table 3.3, gives a detailed breakdown ofconcentrations and particle sizes used in this experiment.47CHAPTER 4Data/Analysis4.1 Universal Trends in the Data4.1.1 StandardThe relative viscosity for a system with solvent viscosity 3017 cP ˙ 4 cP, particleradius a D 36 m ˙ 3 m and angular velocity ! D 1:000 rad=s ˙ 0:001 rad=sis shown in Figure 4.1. This system will be noted as the Standard System andwill be used to compare different solvent viscosities, angular velocities, and particlesizes to the different equations studied in Section 2.4. Looking at Figure 4.1, wesee Mooney’s Equation 2.11, as the green line with values for the coefficients of D 2:8˙0:3 and 'm D 0:79˙0:11, Krieger-Doughtery’s Equation 2.12, as the blueline with values for the coefficients of D 3:2˙0:4 and 'm D 0:57˙0:10, Batchelor’sEquation 2.13 (Standard), as the black line with values for the coefficients of D2:8˙0:3, 'm D 0:59˙0:07, and ˛ D 3˙1, and Brady’s Equation 2.18 as the purpleline with values of D 3:2˙ 0:4, 'm D 0:61˙ 0:04, and ˇ D 5:4˙ 0:6.All four of the models show a swift increase in relative viscosity values as theconcentration is slowly increased. The relative viscosity is defined by r D 0 where is the viscosity of the suspension and 0 is the viscosity of pure solvent. The datastarts with a small increasing linear slope between 0 < ' < 0:15. For ' > 0:15, therelative viscosity begins an ever more rapid increase as the concentration continuesto slowly increase. The vertical asymptote, defined as 'm, describes the limit ofa function approaching infinity and is the maximum packing concentration. Wefound the maximum concentration, 'm, to be different than the jamming point,'j . The jamming point is a point where the suspension is no longer acting like aliquid, but instead is acting like a solid at 'j 52%. For Batchelor’s Equation 2.13,48Figure 4.1: The Standard System. The standard system using the followingparameters of 0 D 3017 cP ˙ 4 cP, ! D 1:000 rad=s ˙ 0:001 rad=s, and a D36 m ˙ 3 m with Mooney’s Equation 2.11 the green line with values of D2:8 ˙ 0:3 and 'm D 0:79 ˙ 0:11, Krieger-Doughtery’s Equation 2.12 the blue linewith values of D 3:2 ˙ 0:4 and 'm D 0:57 ˙ 0:10, Batchelor’s Equation 2.13(Standard) the black line with values of D 2:8 ˙ 0:3, 'm D 0:59 ˙ 0:07, and˛ D 3˙ 1, and Brady’s Equation 2.18 the purple line with values of D 3:2˙ 0:4,'m D 0:61˙ 0:04, and ˇ D 5:4˙ 0:6.49'm D 59% ˙ 7%. We see the 'mLow D 52%, is a translation point and happens tobe within error of 'j 52%.The standard deviation for the relative viscosity for most data points are relativ-ity small. This obscures many of the error bars behind the size of the data symbols.The standard deviation for concentration has a maximum of ' D ˙0:4% for eachpoint and is also hindered by the size of the symbols.4.1.2 Solvent Viscosity DependenceSuspensions using an angular velocity ! D 1:000 rad=s ˙0:001 rad=s, particle radiusa D 36 m ˙3 m and solvent viscosities S D 3017 cP ˙4 cP, S D 1265 cP ˙3 cPand S D 571 cP ˙ 1 cP, are shown in Figure 4.2. Again, Batchelor’s Equation2.13 is used to aid in the visualization of the data points with the values of thecoefficients being found in Table 4.4.Looking at the changing solvent viscosities we see for increasing values thereis an increase in relative viscosities, which is shown in Figure 4.2. At the highersolvent viscosities, the movement of a single particle creates a larger wake causing amuch greater region of influence which in turn will affect particles at much greaterdistances. This makes it difficult for the particles to compact tightly together be-cause the closer you approach a particle, the wake caused by the larger approachingparticle pushes all nearby particles away. This is similar to trying to connect N-Npoles of a magnet. Just as the strength of the magnet affects the magnetic forcefield between the poles, so does a wake field affect the disturbance of the solventcausing it to repel or push a larger region of particles away with a larger wake field.This is in contrast to lower viscosities, where the movement of a single particle hasmuch less significant influence on a region or particles further away from it.[5] Forinstance, an individual particle at S D 3017 cP has a significant effect on a largerregion, for example ten times the radius of the particle. For the same individualparticle at S D 571 cP, the region of influence is much smaller, for example 5 timesthe radius. At lower solvent viscosities, the influence region between particles issmaller allowing the particles to compact more tightly together increasing 'm.50Figure 4.2: The following system is for an angular velocity ! D 1:000 rad=s forthe particle with a radius a D 36 m and the three solvent viscosities. The blackline and points are for the standard at S D 3017 cP ˙ 4 cP (Batchelor’s: D2:8˙ 0:3, 'm D 0:59˙ 0:07, and ˛ D 3˙ 1), the green line and diamonds are forS D 1264 cP ˙3 cP (Batchelor’s: D 2:6˙0:8, 'm D 0:61˙0:10, and ˛ D 3˙1)and the red line and stars are for S D 571 cP ˙ 1 cP (Batchelor’s: D 2:2˙ 0:3,'m D 0:59˙ 0:08, and ˛ D 3˙ 1) solvent viscosities.51Based on Hinch’s argument, we would expect at higher solvent viscosities a largervalue for the relative viscosities, as well as, higher values for 'm. The reason for thisis at higher solvent viscosities the wake field created by the particles is much largerthan the wake field created at lower solvent viscosities. We believe the wake fieldis a determining factor for particle packing. A larger wake field prevents particlesfrom closely interacting with each other because the large wake field is pushing anynearby particles away. Compared to a smaller wake field, the smaller field allowsfor nearby particles to closely interact with each other, thus permitting for moreparticles to compact more and lowering the values for 'm. This is seen in Figure 4.2and the data in Tables 4.2, 4.3, 4.4, and 4.5. The figure shows the relative viscosityversus concentration, for S D 3017 cP data, the black points, has the highestrelative viscosity, followed by S D 1265 cP data, the green diamonds, and lastthe S D 571 cP data, the red stars. We also see the values of 'm increasing from'm D 61% ˙ 4% at S D 3017 cP to 'm D 64% ˙ 9% at S D 1265 cP, thendecreasing to 'm D 61% ˙ 7% at S D 571 cP. The data in the tables shows anoverall decrease in relative viscosity and 'm as the solvent viscosity is decreased.4.1.3 Angular Velocity DependenceFigure 4.3 shows solvent viscosities for suspensions using angular velocities ! D1:000 rad=s ˙ 0:001 rad=s, ! D 2:000 rad=s ˙ 0:001 rad=s, ! D 3:000 rad=s ˙0:001 rad=s and ! D 4:000 rad=s ˙ 0:001 rad=s, particle radius a D 36 m, andcorresponding solvent viscosities 3017 cP ˙4 cP, 2988 cP ˙2 cP, 2965 cP ˙4 cP and2924 cP ˙5 cP, respectively, because the solvent viscosity has angular dependence.As shown in Table 3.1 on page 40, the solvent viscosity, S, depends on the angularvelocity, !. Again, Batchelor’s Equation 2.13 is used to aid in the visualization ofthe data points with the values of the coefficients being found in Table 4.4.While the angular velocity is increasing, the relative viscosity is decreasing and'm is increasing, as shown in Figure 4.3. We suspect, because we cannot see, theincreasing angular velocity is creating particle aggregation which forces the particlesto the outer edge of the cylindrical cup. This creates a gap that is filled with more52Figure 4.3: The following system varies the angular velocities for the particle with ra-dius a D 36 m. The black line and points are for the standard for ! D 1:000 rad=sand S D 3017 cP ˙ 4 cP (Batchelor’s: D 2:8 ˙ 0:3, 'm D 0:59 ˙ 0:07, and˛ D 3 ˙ 1), the green line and diamonds are for ! D 2:000 rad=s and S D2988 cP ˙ 2 cP (Batchelor’s: D 2:7˙ 0:2, 'm D 0:60˙ 0:06, and ˛ D 3:1˙ 0:9),the blue line and stars are for ! D 3:000 rad=s and S D 2965 cP ˙ 4 cP (Batch-elor’s: D 2:6 ˙ 0:1, 'm D 0:61 ˙ 0:07, and ˛ D 2:7 ˙ 0:8) and the yellow lineand triangles are for ! D 4:000 rad=s and S D 2924 cP ˙ 5 cP (Batchelor’s: D 2:6˙ 0:3, 'm D 0:62˙ 0:10, and ˛ D 3˙ 1).53of the pure glycerin and fewer particles thus lowering the relative viscosity. Thevalue of 'm increases from 61% ˙ 4%, 62% ˙ 4%, 63% ˙ 4%, to 65% ˙ 7% asthe angular velocity is increased. The particles are being compressed more tightlybecause of the centripetal force being applied, thus increasing 'm.4.1.4 Particle Size DependenceSuspensions using an angular velocity ! D 1:000 rad=s, particle radii a D 26 m ˙3 m, a D 36 m ˙3 m, a D 51 m ˙4 m and a D 71 m ˙4 m, and solventviscosity 3017 cP, are shown in Figure 4.4. Again, Batchelor’s Equation 2.13 is usedto aid in the visualization of the data points with the values of the coefficients beingfound in Table 4.4.The smallest size particle, a D 26 m, has the lowest viscosity and tries to fol-low a more linear function, as shown in Figure 4.4. It is suspected that the minutesize of the particle for a given concentration generates larger open space betweenneighboring particles producing a solution that mirrors suspensions of lower concen-trations. This results in the data resembling a more linear function as described byEinstein’ s model, Equation 2.10.The 'm values are appearing to be indiscriminate for the different particle sizes,such that, it varies from 'm D 61%˙4% for the 71 m and 36 m, to 'm D 116%˙8% for the 26 m, this can be seen in Figure 4.4. The particle size dependence hasthe most bizarre results and is the least understood at this time.4.2 Models4.2.1 Fitting ProceduresThe following section describes the procedure used to determine the goodness-of-fitof the data to the models and the values of the error for the coefficients. We alsoused a Taylor Series Expansion of Mooney’s Equation 2.11 and Krieger-Dougherty’sEquation 2.12 to keep the number of constraints consistent at two for both Batche-lor’s Equation 2.13 and Brady’s Equation 2.18.54Figure 4.4: The following system shows a plot for all four radii for the angularvelocity of ! D 1:000 rad=s and S D 3017 cP ˙ 4 cP. The black line and pointsare the standard with a D 36 m (Batchelor’s: D 2:8˙ 0:3, 'm D 0:59˙ 0:07,and ˛ D 3 ˙ 1), the red line and squares are for radius a D 26 m (Batchelor’s: D 3:0 ˙ 0:3, 'm D 1:05 ˙ 0:12, and ˛ D 4 ˙ 1), the blue line and stars are forradius a D 51 m (Batchelor’s: D 3:1˙0:3, 'm D 0:61˙0:07, and ˛ D 4˙2) andthe yellow line and triangles are for radius a D 71 m (Batchelor’s: D 2:6˙ 0:2,'m D 0:57˙ 0:06, and ˛ D 3˙ 1).The data was modeled to the equations in section 2.4 to find the coefficients,, k, 'm, ˛, and ˇ. In order to determine the goodness of fit the following threemethods were used: a least squares fit, a reduced chi-squared, 2, and a probabilitydistribution function (PDF) of 2. The initial guess for the coefficients were foundusing the “FindFit” function in Mathematica. This function uses a least squaresfit, a method that minimizes the discrepancy between the measured and calculatedvalues, to determine the initial values of the coefficients. The 2 and PDF of 2were used in conjunction to interpret the results and verify the goodness of fit.55The Reduced Chi-Squared equation is defined by the following:2 D2D 1NXiD1"ri r .xi ; yi/22xi C 2yi#(4.1)where D N nc is the degrees of freedom, N is the number of data points and ncis the number of constraints. The expected value of˝2˛ D 1. Values much largerthan 1 result from a large deviation from the assumed distribution indicating eitherpoor measurements, incorrect assignments of uncertainties, or an incorrect choice ofprobability function. Very small values of 2 are equally unacceptable and implysome misunderstanding of the experiment.However, if 2 >> 1 the integral test of the PDF of 2, given by:P2I D 12=2 .=2/Z 12x2.2/=2ex2=2dx2 (4.2)can be used to determine the acceptance of the predicted values, where .=2/ isthe Gamma Function. If the PDF of 2, P2I , is reasonably close to 1, thenthe assumed distribution describes the spread of the data points well. If the proba-bility is small, either the assumed distribution is not a good estimate of the parentdistribution or the data sample is not representative of the parent distribution.[15]In order to minimize 2, a program was used which sequentially varies , k, 'm,˛, and ˇ by a ı, ık, ı'm, ı˛, and ıˇ, respectively. This code can be found inAppendix A.1 and works the following way. First, the program starts with an initialguess for the coefficients using a least squares fit. This initial guess is shown in Figure4.5 by the red star, with values of D 4:62, 'm D 2:75, and 2 D 23:125. Next, thecode varied each of the coefficients individually by decreasing and increasing its valueuntil it minimized 2. A condition was written in the program stating if the valueof coefficient did not change the value of 2 or there was no change in coefficient,the next loop would add an additional significant figure to that coefficient, helpingto narrow in on the lowest 2. This can be seen in Figure 4.5 by the black squaresbecoming tighter as 2 is minimized. Finally, the program stopped looping wheneither the coefficient’s significant figures were greater than 109 or 2 < 109. Theblue triangle represents the best fitting coefficient found using this program with56values at D 2:76, 'm D 0:79, and 2 D 1:170, as shown in Figure 4.5. By lookingat the 2 and the PDF of 2, I was able to determine if the calculated values of thecoefficients were indeed a good fit to the equations studied.The contour plot of 2 shows that there are several values close to minimal 2,but only one minimum 2. Figure 4.5 shows the contour plot with initial parametersof a D 36 m, S D 2913 cP and ! D 1 rad=s and coefficients values of D 2:7˙0:2,k D 1:2 ˙ 0:2 or 'm D 0:81 ˙ 0:15, and 2 D 1:068. Another way of thinking ofthis is to find the bottom of a well, as shown by the three dimensional plot in Figure4.6. The figure shows there are several values that are all close to finding the localminimum of 2, but again only one minimum.The error for the values of the coefficients were calculated by considering theparticle a D 36 m and taking the percentage difference between all seven con-centrations (0%, 20%, 25%, 30%, 35%, 40%, and 50%) to just the four concentra-tions (0%, 20%, 30%, and 40%) that were used with the other three particle sizes(a D 26 m, a D 51 m, and a D 71 m). For example, the coefficients foundusing initial parameters a D 36 m, S D 2988 cP and ! D 2 rad=s at all sevenconcentrations are D 2:75919 and 'm D 0:787328 compared to D 2:50994 and'm D 0:687182 using just the four concentrations. The percent difference in is9.5% and 'm is 13.6% for that specific data set giving the values found in Table 4.2of D 2:8 ˙ 0:3 and 'm D 0:79 ˙ 0:11. Different solvent viscosities and angularvelocities had different percentages used to determine the error of the coefficient,see Table 4.1.Figure 4.7 shows a plot of relative viscosity as a function of concentration forthose previously mentioned initial parameters. The red line uses coefficients D 2:8and 'm D 0:79 from all seven concentrations and the blue line uses coefficients D 2:5 and 'm D 0:69 from just the four concentrations. The shaded area isbounded by the maximum and minimum values of the coefficients D 2:8˙0:3 and'm D 0:79˙ 0:11, respectively, indicating the error. This process was repeated foreach solvent viscosity to ensure that the standard error of the coefficients, , k,'m , ˛, and ˇ , are within reason given the data collected.57Figure 4.5: Contour plot of 2, Equation 4.1, with initial parameters of a D 36 m,S D 2913 cP and ! D 1 rad=s and coefficients values of D 2:8 ˙ 0:3, 'm D0:79 ˙ 0:11, and 2 D 1:170 showing the process in which the coefficients arecalculated. The red star, D 4:62, 'm D 2:75, and 2 D 23:125, is the initial guessfound from a least squares fit. The black box shows the steps the code took forminimizing 2, taking smaller steps as in nears the minimum value of 2. The bluetriangle represents the best fitting coefficient found using this programs with valuesat D 2:8 ˙ 0:3, 'm D 0:79 ˙ 0:11, and 2 D 1:170.58Figure 4.6: Three dimensional plot of 2, Equation 4.1, with initial parameters ofa D 36 m, S D 2913 cP and ! D 2 rad=s and coefficients values of D 2:8 ˙0:3,'m0:79 ˙ 0:11, and 2 D 1:170 showing the bottom of the well.The value of as stated by Einstein is for low concentrations. In this case takinga small ' and performing a Taylor Series Expansion of Mooney’s Equation 2.11 wecan obtain Batchelor’s Equation 2.13r D e'1 ''mr 1C '1 ''m!C 22 '1 ''m!2r D 1C '1 ''m!C ˛ '1 ''m!2This allows me to use the value of found from Mooney’s Equation in Batchelor’sEquation in order to keep the number of constraints constant at two. The values inTables 4.2 and 4.4 have identical values for and . The same is true for Krieger-Dougherty’s Equation 2.12 and Brady’s Equations 2.18, when performing a TaylorSeries Expansionr D1 ''m'm59Figure 4.7: The plot of relative viscosity as a function of concentration using initialparameters of a D 36 m, S D 2988 cP and ! D 1 rad=s shows the red line usingcoefficients D 2:8 and 'm D 0:79 from all seven concentrations and the blue lineuses coefficients D 2:5 and k'm D 0:69 from just the four concentrations. Theshaded area is bounded by the maximum and minimum values of the coefficients D 2:8˙ 0:3 and 'm D 0:79˙ 0:11, respectively, indicating the error.60S .cP/ ! .rad=s/ % Difference in % Difference in k3017 1 9.5% 13.6%2988 2 6.4% 12.0%2965 3 3.4% 11.0%2924 4 9.7% 18.3%1265 1 30.4% 27.4%1250 2 8.2% 14.7%1233 3 6.5% 13.1%1211 4 2.9% 9.5%571 1 14.2% 18.1%566 2 14.9% 19.5%560 3 16.0% 20.6%555 4 16.8% 20.9%Table 4.1: Table of the percent difference used to calculate the error bars for thecoefficients.r 1C ' C '22'm.'m C 1/r D 1C ' C ˇ '1 ''m!2Again, this allows me to use the same values of found from Krieger-Dougherty’sEquation in Brady’s Equation in order to keep the number of constraints at two.The values in Tables 4.3 and 4.5 have identical values for and .4.2.2 The EquationsMooneyThe values of and k, or 'm D 1k , is using the experimental data to model itto Equation 2.11 are shown in Table 4.2. The fitting parameter has a range of1:4 < < 3:2 and the fitting parameter k has a range of 0:44 < k < 1:97. The fittingparameter can be rewritten as 1kD 'm, which gives a range of 0:51 < 'm < 2:27(section 2.4.2). The Standard System’s parameters a D 36 m, ! D 1:000 rad=s,S D 3017 cP, resulted in coefficients of 'm D 0:79 ˙0:11 and D 2:8 ˙0:3. Usingthese values in Equation 2.11 results in the line shown in Figure 4.8.61Figure 4.8: The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the black points are thestandard and the green line is Mooney’s Equation 2.11 using the calculate value of'm D 0:79 ˙ 0:11 and D 2:8 ˙ 0:3.62a .m/ S .cP/ ! .rad=s/ m m 2 P2; 263017 1 3.0 0.3 2.04 0.28 0.280 0.8692988 2 3.1 0.2 2.27 0.27 0.615 0.7352965 3 3.0 0.1 1.93 0.21 0.726 0.6962924 4 2.9 0.3 1.86 0.34 0.766 0.6821265 1 2.3 0.7 1.01 0.28 0.002 0.9991250 2 2.4 0.2 1.04 0.15 0.025 0.9881233 3 2.4 0.2 1.05 0.14 0.081 0.9601211 4 2.5 0.1 1.11 0.11 0.198 0.906571 1 2.5 0.4 1.08 0.20 0.062 0.969566 2 2.5 0.4 1.07 0.21 0.083 0.960560 3 2.4 0.4 1.08 0.22 0.091 0.955555 4 2.4 0.4 1.08 0.23 0.126 0.939363017 1 2.8 0.3 0.79 0.11 1.170 0.9482988 2 2.7 0.2 0.81 0.10 1.068 0.9572965 3 2.6 0.1 0.81 0.09 1.603 0.9012924 4 2.6 0.3 0.83 0.15 2.192 0.8221265 1 2.6 0.8 0.81 0.22 2.168 0.8261250 2 2.6 0.2 0.82 0.12 1.808 0.8751233 3 2.5 0.2 0.80 0.11 1.743 0.8831211 4 2.5 0.1 0.82 0.08 1.400 0.924571 1 2.2 0.3 0.74 0.13 4.119 0.532566 2 2.2 0.3 0.76 0.15 4.359 0.499560 3 2.3 0.4 0.77 0.16 4.313 0.505555 4 2.4 0.4 0.78 0.16 4.188 0.523513017 1 3.1 0.3 0.84 0.11 4.063 0.1312988 2 3.1 0.2 0.86 0.10 3.188 0.2032965 3 3.0 0.1 0.86 0.09 2.685 0.2612924 4 3.0 0.3 0.86 0.16 2.528 0.2821265 1 2.4 0.7 0.67 0.18 5.007 0.0821250 2 2.4 0.2 0.69 0.10 5.218 0.0741233 3 2.4 0.2 0.69 0.09 5.594 0.0611211 4 2.4 0.1 0.70 0.07 5.002 0.082571 1 1.7 0.2 0.55 0.10 10.079 0.006566 2 1.8 0.3 0.57 0.11 9.846 0.007560 3 2.1 0.3 0.60 0.12 9.143 0.010555 4 2.5 0.4 0.65 0.14 8.889 0.01263a .m/ S .cP/ ! .rad=s/ m m 2 P2; 713017 1 2.6 0.2 0.74 0.10 0.522 0.7702988 2 2.7 0.2 0.81 0.10 0.573 0.7512965 3 2.8 0.1 0.88 0.10 0.354 0.8382924 4 2.9 0.3 0.93 0.17 0.344 0.8421265 1 2.0 0.6 0.63 0.17 3.530 0.1711250 2 2.2 0.2 0.68 0.10 2.868 0.2381233 3 2.7 0.2 0.81 0.11 2.147 0.3421211 4 3.2 0.1 1.06 0.10 2.257 0.323571 1 1.7 0.2 0.55 0.10 0.001 0.999566 2 1.5 0.2 0.52 0.10 0.298 0.862560 3 1.4 0.2 0.51 0.10 1.077 0.584555 4 1.5 0.3 0.52 0.11 1.198 0.549Table 4.2: Mooney Table. A table using the data to model Equation 2.11 using aleast squares-fit to calculate the values of k and with an average 2 D 2:492.64The data in Table 4.2 shows the values of slowly increasing as the solventviscosity and angular velocity are increased. The values of appear to have noobvious dependence on particle size. However, the average value of from Table 4.2is D 2:5 with D 0:4. This is the value that is written in Mooney’s Equation 2.11and seen in the other three equations. The values of 'm are linearly increasing asthe solvent viscosity is increased. The values of 'm are remaining relativity constantas the angular velocity is varied. Finally, the values of 'm are very random with noapparent dependence on particle size. The average value of 'm from Table 4.2 is'm D 0:90 with 'm D 0:38.In Section 4.2.1, I interrupted the results of 2 and P2; , which are shownin Table 4.2. This was done in order to determine the goodness of fit of data,ideally I wanted values for 2 1. The values where 2 1, such as, a D 36 m,S D 2988 cP, and ! D 2 rad=s are consider to be a great fits regardless of the valuesfound from P2; . For the values where 2 << 1, but with a P2; << 1,such as, a D 26 m, S D 1250 cP, and ! D 2 rad=s, are considered to be a goodfits. For values where 2 > 1 and P;2; < 0, such as, a D 71 m, S D 1250 cP,and ! D 2 rad=s are considered to be a poor fits to the data. Finally, for valueswhere 2 >> 1 and P2; << 1, such as, a D 51 m, S D 571 cP, and! D 1 rad=s are considered to be a bad fits to the data. The interpretation is alsotrue for the goodness of fit in Tables 4.3, 4.4 and 4.5.Krieger-DoughertyThe values of Œ, or , and 'm is using the experimental data to model it to Equation2.12 are shown in Table 4.3. The range of values for the fitting parameters are 1:8 <Œ < 3:3 and 0:41 < 'm < 1:27. The Standard System’s parameters a D 36 m,! D 1:000 rad=s, S D 3017 cP, resulted in coefficients of Œ D 3:2 ˙ 0:4 and' D 0:6 ˙ 0:1. Using these values in Equation 2.12 results in the line shown inFigure 4.9.65Figure 4.9: The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the black points are thestandard and the blue line is Krieger-Dougherty’s Equation 2.12 using the calculatevalue of Œ D 3:2 ˙ 0:4 and ' D 0:6 ˙ 0:1.66a .m/ S .cP/ ! .rad=s/ Œ Œ 'm 'm 2 P2; 263017 1 3.1 0.4 1.15 0.19 0.287 0.8662988 2 3.1 0.3 1.27 0.20 0.623 0.7322965 3 3.0 0.2 1.10 0.17 0.738 0.6912924 4 2.9 0.3 1.07 0.21 0.780 0.6771265 1 2.4 0.7 0.63 0.16 0.000 1.0001250 2 2.5 0.2 0.64 0.11 0.038 0.9811233 3 2.5 0.2 0.65 0.11 0.103 0.9501211 4 2.5 0.2 0.68 0.10 0.226 0.893571 1 2.6 0.4 0.66 0.13 0.080 0.961566 2 2.6 0.4 0.66 0.14 0.103 0.950560 3 2.5 0.4 0.67 0.14 0.112 0.946555 4 2.5 0.4 0.67 0.14 0.150 0.928363017 1 3.2 0.4 0.57 0.10 2.137 0.8302988 2 3.1 0.3 0.58 0.09 1.781 0.8792965 3 2.9 0.2 0.58 0.09 2.267 0.8112924 4 2.9 0.3 0.59 0.12 2.899 0.7161265 1 3.0 0.8 0.58 0.15 3.170 0.6741250 2 2.9 0.3 0.58 0.10 2.686 0.7481233 3 2.9 0.3 0.58 0.10 2.615 0.7591211 4 2.9 0.2 0.58 0.08 2.084 0.837571 1 2.6 0.4 0.55 0.11 5.924 0.314566 2 2.6 0.4 0.56 0.11 6.044 0.302560 3 2.7 0.4 0.57 0.12 5.904 0.316555 4 2.7 0.5 0.57 0.12 5.719 0.335513017 1 3.3 0.4 0.56 0.09 4.433 0.1092988 2 3.3 0.3 0.56 0.09 3.497 0.1742965 3 3.2 0.2 0.56 0.09 2.969 0.2272924 4 3.2 0.3 0.57 0.11 2.790 0.2481265 1 2.7 0.8 0.48 0.12 5.793 0.0551250 2 2.7 0.3 0.48 0.08 5.947 0.0511233 3 2.7 0.2 0.49 0.08 6.321 0.0421211 4 2.7 0.2 0.49 0.07 5.666 0.059571 1 2.1 0.3 0.43 0.08 11.967 0.003566 2 2.2 0.3 0.43 0.09 11.616 0.003560 3 2.5 0.4 0.45 0.09 10.591 0.005555 4 2.8 0.5 0.47 0.10 9.992 0.00767a .m/ S .cP/ ! .rad=s/ Œ Œ 'm 'm 2 P2; 713017 1 2.8 0.3 0.50 0.09 0.733 0.6932988 2 3.0 0.3 0.54 0.09 0.728 0.6952965 3 3.0 0.2 0.57 0.09 0.451 0.7982924 4 3.0 0.3 0.60 0.12 0.422 0.8101265 1 2.3 0.6 0.46 0.12 4.270 0.1181250 2 2.5 0.3 0.48 0.08 3.402 0.1821233 3 2.9 0.3 0.54 0.09 2.426 0.2971211 4 3.3 0.2 0.67 0.09 2.382 0.304571 1 2.1 0.3 0.43 0.08 0.083 0.960566 2 2.0 0.3 0.42 0.09 0.884 0.643560 3 1.8 0.3 0.41 0.09 2.141 0.343555 4 1.9 0.3 0.42 0.09 2.218 0.330Table 4.3: Krieger-Dougherty Table. A table using the data to model Equation2.12 using a least squares-fit to calculate the values of Œ and 'm with an average2 D 8:455.68The data in Table 4.3 shows the values of Œ slowly increasing as the solventviscosity is increased. The values of Œ are remaining relatively constant as theangular velocity is increased. The values of Œ appear to have no clear dependenceon particle size. The average value of Œ in Table 4.3 is Œ D 2:7 with Œ D 0:4.The values of 'm, again, are slowly increasing as the solvent viscosity is increased.However, the values of 'm are remaining relativity constant as the angular velocityis varied. Finally, the values of 'm are very random with no apparent dependenceon particle size. The average value of 'm in Table 4.3 is 'm D 0:60 with 'm D 0:18.The patterns are very similar to what we saw in Mooney’s Table 4.2, but the valuesfor the coefficients are somewhat different.BatchelorThe values of , 'm, and ˛ is using the experimental data to model it to Equation2.13 are shown in Table 4.4. The range of the fitting parameters values is 1:4 < < 3:2, 0:9 < ˛ < 4:4, and 0:46 < 'm < 1:14. The Standard System’s parametersa D 36 m, ! D 1:000 rad=s, S D 3017 cP, resulted in coefficients of D 2:8 ˙0:3,˛ D 3 ˙ 1 and 'm D 0:59 ˙ 0:07, using these values in Equation 2.13 results inthe line shown in Figure 4.10. The values in Table 4.4, are displaying the calculatedvalues for the coefficients , ˛ and 'm for each of the different parameters that wasvaried when modeled to Equation 2.13.69Figure 4.10: The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the black points are thestandard and the yellow line is Batchelor’s Equation 2.13 using the calculate valueof D 1:0 ˙ 0:8, ˛ D 7 ˙ 6 and 'm D 0:6 ˙ 0:1.70a .m/ S .cP/ !rads ˛ ˛ m m 2 P2; 263017 1 3.0 0.3 4.0 1.0 1.05 0.12 0.279 0.8702988 2 3.1 0.2 4.0 1.0 1.14 0.12 0.609 0.7372965 3 3.0 0.1 4.0 1.0 1.07 0.12 0.717 0.6992924 4 2.9 0.3 4.0 2.0 1.08 0.17 0.757 0.6851265 1 2.3 0.7 2.0 1.0 0.70 0.12 0.002 0.9991250 2 2.4 0.2 1.9 0.5 0.73 0.08 0.026 0.9871233 3 2.4 0.2 2.1 0.5 0.74 0.07 0.081 0.9601211 4 2.5 0.1 2.4 0.3 0.78 0.06 0.196 0.906571 1 2.5 0.4 2.0 1.0 0.74 0.10 0.063 0.969566 2 2.5 0.4 2.3 0.9 0.75 0.11 0.083 0.959560 3 2.4 0.4 2.2 0.9 0.76 0.11 0.091 0.955555 4 2.4 0.4 2.2 0.8 0.76 0.11 0.125 0.939363017 1 2.8 0.3 3.0 1.0 0.59 0.07 1.445 0.9192988 2 2.7 0.2 3.1 0.9 0.60 0.06 1.267 0.9382965 3 2.6 0.1 2.7 0.8 0.61 0.07 1.769 0.8802924 4 2.6 0.3 3.0 1.0 0.62 0.10 2.369 0.7961265 1 2.6 0.8 3.0 2.0 0.61 0.10 2.379 0.7951250 2 2.6 0.2 2.8 0.7 0.62 0.07 1.978 0.8521233 3 2.5 0.2 2.7 0.6 0.61 0.06 1.907 0.8621211 4 2.5 0.1 2.7 0.3 0.62 0.05 1.531 0.909571 1 2.2 0.3 2.2 0.9 0.59 0.08 4.316 0.505566 2 2.2 0.3 2.3 0.9 0.60 0.09 4.525 0.477560 3 2.3 0.4 2.4 0.9 0.61 0.09 4.472 0.484555 4 2.4 0.4 3 1 0.61 0.09 4.361 0.499513017 1 3.1 0.3 4.0 2.0 0.61 0.07 4.061 0.1312988 2 3.1 0.2 4.0 1.0 0.62 0.06 3.189 0.2032965 3 3.0 0.1 4.0 1.0 0.62 0.07 2.684 0.2612924 4 3.0 0.3 4.0 2.0 0.63 0.10 2.524 0.2831265 1 2.4 0.7 3.0 2.0 0.55 0.09 4.939 0.0851250 2 2.4 0.2 3.0 0.8 0.56 0.06 5.141 0.0761233 3 2.4 0.2 3.0 0.6 0.57 0.06 5.507 0.0641211 4 2.4 0.1 3.0 0.4 0.57 0.04 4.929 0.085571 1 1.7 0.2 1.9 0.8 0.50 0.07 9.737 0.008566 2 1.8 0.3 2.1 0.8 0.51 0.07 9.548 0.008560 3 2.1 0.3 3.0 1.0 0.52 0.07 8.947 0.011555 4 2.5 0.4 3.0 1.0 0.54 0.08 8.778 0.01271a .m/ S .cP/ !rads ˛ ˛ m m 2 P2; 713017 1 2.6 0.2 3 1 0.57 0.06 0.530 0.7672988 2 2.7 0.2 3.0 0.9 0.60 0.06 0.580 0.7482965 3 2.8 0.1 3.1 0.9 0.63 0.07 0.361 0.8352924 4 2.9 0.3 3 2 0.65 0.10 0.350 0.8401265 1 2.0 0.6 2 1 0.55 0.09 3.438 0.1791250 2 2.2 0.2 2.4 0.6 0.57 0.06 2.813 0.2451233 3 2.7 0.2 3.2 0.7 0.62 0.06 2.127 0.3451211 4 3.2 0.1 4.4 0.6 0.72 0.06 2.248 0.325571 1 1.7 0.2 1.0 0.4 0.47 0.07 0.000 1.000566 2 1.5 0.2 0.9 0.4 0.46 0.07 0.299 0.861560 3 1.4 0.2 0.9 0.3 0.46 0.07 1.047 0.592555 4 1.5 0.3 1.0 0.4 0.47 0.07 1.168 0.558Table 4.4: Batchelor Table. A table using the data to model Equation 2.13 usinga least squares-fit to calculate the values of , ˛, and 'm with an average 2 D 2:506.72The data in Table 4.4 shows the values of slowly increasing as the solventviscosity and angular velocity are increased. The values of appear to have nostrong dependence on particle size. The average value of from Batchelor’s Table4.4 is D 2:5 with D 0:4 (remember that a Taylor Series expansion of Mooney’sEquation 2.11 gave the values for in Batchelor’s Equation 2.13, Section 4.2.1).The values of 'm are slowly decreasing as the solvent viscosity is increased. Thevalues of 'm are remaining relativity constant as the angular velocity is varied. Thevalues of 'm are very random with no apparent dependence on particle size. Theaverage value of 'm from the Table 4.4 is 'm D 0:65 with 'm D 0:15. The valuesof ˛ are remaining relatively constant as the solvent viscosity and angular velocityare increased. The values of ˛ are very random with no apparent dependence onparticle size. The average value of ˛ from Table 4.4 is ˛ D 2:7 with ˛ D 0:9.BradyThe values of , 'm, and ˇ is using the experimental data to model it to Equation2.18 and are shown in Table 4.5. The values for the fitting parameters are 1:8 < < 3:3, 0:48 < 'm < 1:24, and 2:0 < ˇ < 7:4. The Standard System’s parametersa D 36 m, ! D 1:000 rad=s, S D 3017 cP, resulted in coefficients of D 3:2 ˙0:4,'m D 0:61 ˙ 0:04, and ˇ D 5:4 ˙ 0:6, which are used in Equation 2.18 to results inthe line shown in Figure 4.11. The values in Table 4.5, are displaying the calculatedvalues for the coefficients , ˇ and 'm for each of the different parameters that wasvaried when modeled to Equation 2.18.73Figure 4.11: The system using data points with a radius a D 36 m, an angularvelocity ! D 1:000 rad=s, and solvent viscosity of 3017 cP, the black points are thestandard and the purple line is Brady’s Equation 2.18 using the calculate value of D 1:2 ˙ 0:9, 'm D 0:6 ˙ 0:1, and ˇ D 8 ˙ 3.74a .m/ S .cP/ !rads ˇ ˇ m m 2 P2; 263017 1 3.1 0.4 6.1 0.6 1.16 0.08 0.275 0.8712988 2 3.1 0.3 6.3 0.3 1.24 0.08 0.606 0.7382965 3 3.0 0.2 6.1 0.1 1.17 0.08 0.715 1.1712924 4 2.9 0.3 5.9 0.7 1.18 0.14 0.755 0.6861265 1 2.4 0.7 4.0 2.0 0.80 0.11 0.003 0.9991250 2 2.5 0.2 4.3 0.2 0.82 0.06 0.023 0.9881233 3 2.5 0.2 4.5 0.1 0.83 0.06 0.078 0.9621211 4 2.5 0.2 4.7 0.2 0.86 0.04 0.193 0.908571 1 2.6 0.4 4.8 0.7 0.83 0.09 0.059 0.971566 2 2.6 0.4 4.7 0.7 0.83 0.10 0.079 0.961560 3 2.5 0.4 4.6 0.8 0.84 0.10 0.088 0.957555 4 2.5 0.4 4.6 0.8 0.85 0.10 0.122 0.941363017 1 3.2 0.4 5.4 0.6 0.61 0.04 1.145 0.9502988 2 3.1 0.3 5.2 0.2 0.62 0.04 1.032 0.9602965 3 2.9 0.2 4.8 0.1 0.63 0.04 1.541 0.9082924 4 2.9 0.3 4.9 0.6 0.65 0.07 2.107 0.8341265 1 3.0 0.8 5.0 2.0 0.64 0.09 2.100 0.8351250 2 2.9 0.3 4.8 0.3 0.64 0.05 1.748 0.8831233 3 2.9 0.3 4.7 0.1 0.64 0.04 1.684 0.8911211 4 2.9 0.2 4.7 0.2 0.64 0.03 1.353 0.929571 1 2.6 0.4 3.9 0.6 0.61 0.07 3.964 0.555566 2 2.6 0.4 3.9 0.6 0.62 0.07 4.206 0.520560 3 2.7 0.4 4.1 0.7 0.63 0.07 4.173 0.525555 4 2.7 0.5 4.4 0.8 0.63 0.07 4.065 0.540513017 1 3.3 0.4 7.3 0.8 0.65 0.05 3.979 0.1372988 2 3.3 0.3 7.3 0.3 0.65 0.04 3.117 0.2102965 3 3.2 0.2 7.1 0.1 0.66 0.04 2.621 0.2702924 4 3.2 0.3 6.9 0.9 0.67 0.08 2.469 0.2911265 1 2.7 0.8 5.0 2.0 0.58 0.08 4.869 0.0881250 2 2.7 0.3 5.2 0.3 0.59 0.04 5.086 0.0791233 3 2.7 0.2 5.2 0.1 0.59 0.04 5.460 0.0651211 4 2.7 0.2 5.3 0.2 0.60 0.03 4.880 0.087571 1 2.1 0.3 3.3 0.5 0.52 0.06 9.744 0.008566 2 2.2 0.3 3.6 0.6 0.52 0.06 9.528 0.009560 3 2.5 0.4 4.4 0.7 0.54 0.06 8.873 0.012555 4 2.8 0.5 5.0 1.0 0.56 0.07 8.664 0.01375a .m/ S .cP/ !rads ˇ ˇ m m 2 P2; 713017 1 2.8 0.3 5.3 0.5 0.61 0.04 0.494 0.7812988 2 3.0 0.3 5.9 0.3 0.65 0.04 0.547 0.7612965 3 3.0 0.2 6.0 0.1 0.68 0.04 0.337 0.8452924 4 3.0 0.3 6.2 0.8 0.71 0.08 0.329 0.8481265 1 2.3 0.6 4.0 2.0 0.57 0.08 3.439 0.1791250 2 2.5 0.3 4.5 0.2 0.60 0.05 2.792 0.2481233 3 2.9 0.3 5.8 0.1 0.66 0.04 2.094 0.3511211 4 3.3 0.2 7.4 0.3 0.77 0.03 2.219 0.330571 1 2.1 0.3 2.5 0.4 0.50 0.05 0.001 1.000566 2 2.0 0.3 2.2 0.3 0.48 0.06 0.301 0.860560 3 1.8 0.3 2.0 0.3 0.48 0.06 1.070 0.585555 4 1.9 0.3 2.3 0.4 0.49 0.06 1.176 0.555Table 4.5: Brady Table. A table using the data to model Equation 2.18 using aleast squares-fit to calculate the values of , ˇ, and 'm with an average 2 D 2:421.76The data in Table 4.5 shows the values of slowly increasing as the solventviscosity is increased. The values of are remaining relatively constant as theangular velocity is increased. The values of appear to have no obvious dependenceon particle size. The average value of from Table 4.5 is D 2:7 with D 0:4(again, this is exactly what we saw with the Krieger-Dougherty Section because ofthe Taylor Series expansion of Krieger-Dougherty’s Equation 2.12 gave the valuesfor in Brady’s Equation 2.18, Section 4.2.1). The values of 'm are again remainingrelatively constant as the solvent viscosity is increased. The values of 'm are slightlyincreasing as the angular velocity is increased. This is something we did not seebefore, even though the increase is quite small. The values of 'm are very randomwith no apparent dependence on particle size. The average value of 'm from Table4.5 is 'm D 0:69 with 'm D 0:18. The values of ˇ are remaining relatively constantas the solvent viscosity and angular velocity are increased. The values of ˇ are veryrandom with no apparent dependence on particle size. The average value of ˇ fromTable 4.5 is ˇ D 5 with ˇ D 1.4.3 Analysis4.3.1 Fitting Parameter In section 4.2.1 it was determined that a Taylor series expansion on Mooney’s Equa-tion 2.11 allowed us to use its value of in Batchelor’s Equation 2.13 and a Taylorseries expansion on Krieger-Dougherty’s Equation 2.12 allowed us to use its valueof in Brady’s Equation 2.18. Thus the following Figures 4.12, 4.14, 4.16 and areidentical for both Mooney and Batchelor and Figures 4.13, 4.15, and 4.17 will beidentical for Krieger-Dougherty and Brady.Radial Dependence on Mooney takes as a constant to agree with Einstein’s Equation 2.10 for very dilutesuspensions, (section 2.4.1). The parameter values in Table 4.2, is showing a rangeof 1:4 < < 3:2, which is not a constant as Mooney predicted. Mooney states that77the size of the particle can have an effect on the fitting parameters and k; however,he did not have the means to record the sensitive data needed to determine thoseexperimental results. James V. Robinson used mono-disperse spheres of a D 10 mto a D 30 m, but in order for the data to fit his equation he had to allow Einstein’svalue of 2:5 to vary from 3 to 5. This supports our hypothesis that is not a constantand depends on the some of the parameters we tested, as shown in Figures 4.12,4.14, and 4.16.The values from Mooney’s Table 4.2, is showing that appears to have randomvalues creating a scattered pattern of increasing and decreasing values at the differentradii for the different solvent viscosities, as seen in Figure 4.12(a). There does appearto be a trend with a value of 2:5 ˙ 0:5, but it is difficult to say that thereis a linearly dependence. However, we do see now and will see in future plotsthe values of slightly decreasing for decreasing solvent viscosity .S/. In Figure4.12(a), we see a pattern of decreasing values for decreasing S. This is seenby the red stars .S D 3017 cP/ having the highest values of , followed by thegreen triangles .S D 1265 cP/ having lower values, and finally followed by the bluesquares .S D 571 cP/ having the lowest values of .In Figure 4.12(b), we are looking at the values for different angular velocitiesat the different particle sizes with a similar solvent viscosity .2924 < S < 3017 cP/.We see the values of appearing to be constant. At particular particle size theredoes appear to be some consistency in the values of , but the order of the angularvelocities does change at the different sizes. This is indicating that keeping thesolvent viscosity constant and varying the angular velocity using various particlesizes has little, to no effect, on the values of . This leads me to believe that mightbe a constant is this case.The data from Table ?? and Figures 4.12 is showing that 2:5. Indicatingthat could be considered a constant. We especially believe this to be so at differentangular velocities because of the linear dependence at different solvent viscosities.We have determined that the solvent viscosity directly affects the hydrodynamicinteractions between particles, for example, higher solvent viscosities create a larger78wake or field of disturbance. A larger wake is more likely interfere with anotherparticle’s wake in a nearby region, which can change the velocity, as well as, thevalue . We have now seen that the angular velocity has little, to no affect on thehydrodynamic interactions, thus appearing to produce a more constant value for .79(a)(b)Figure 4.12: Plot of vs a modeling the data to Mooney’s Equation 2.11 andBatchelor’s Equation 2.13. (a) Different solvent viscosities with an angular velocity! D 1:000 rad=s, the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b) Different angularviscosities, the red stars are for angular velocity ! D 1:000 rad=s and S D 3017 cP,the green triangles are ! D 2:000 rad=s and S D 2988 cP, the blue squares are! D 3:000 rad=s and S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP.80Krieger and Dougherty derive their equation using parts from Mooney’s model,Equation 2.11 (Section 2.4.3). This explains why the data for the fitting parameterŒ in Table 4.3 is almost identical to the data for the fitting parameter in Table4.2. Since the data is almost identical, the figures and results are almost identical aswell. The data in Tables 4.3 and 4.5 along with Figure 4.13 shows that 2:5 andcould be a constant especially at different angular velocities and possibly, but lesslikely, at different solvent viscosities. The fitting parameter Œ is the exact samefitting parameter as and will be referred to as from here on out.Krieger and Dougherty considered the interactions of a two particle system andhow the flow of those particles rotating together or independently can affect theviscosity of the medium. Introducing us to the idea of a doublet they describehow two particles can rotate together as one, like a dumbbell (Section 2.4.3). Thedoublet is a similar concept to Hinch’s idea of a group of particles rotating togetheras a single particle of larger radius. Both ideas change the group velocity whichcauses the field of disturbance to change thus affecting neighboring particles andthe relative viscosity.Considering Figure 4.13(a), the values at the different radii for different S arescattered around a common 2:75˙0:5. Excluding the data for at a D 26 m,the ordering of the solvent viscosities behaves exacting like they did in Figure 4.12(a)for Mooney. As both figures show there is common pattern with the red stars thatrepresent S D 3017 cP having the largest values of , followed by the green trianglesthat represent S D 1265 cP having the lower values of , and finally followed bythe blue squares that represent S D 571 cP having the lowest values of . Thoughit is hard to say that here is a constant, it does appear to be a trend showing someconsistency.Figure 4.13(b), has an identical pattern we saw in Figure 4.12(b) with the valuesof appearing to be constant at the different radii for the varying angular velocitieswhen using a similar solvent viscosities .2924 < S < 3017 cP/. Again we see, theordering of the angular velocities to be scattered with no apparent pattern.81(a)(b)Figure 4.13: Plot of vs a modeling the data to Krieger-Dougherty’s Equation 2.12and Brady’s Equation 2.18. (a) Different solvent viscosities with an angular velocity! D 1:000 rad=s, the red stars are for S D 3017 cP, the green triangles are forS D 1265 cP and the blue squares are for S D 571 cP. (b) Different angularviscosities, the red stars are for angular velocity ! D 1:000 rad=s and S D 3017 cP,the green triangles are ! D 2:000 rad=s and S D 2988 cP, the blue squares are! D 3:000 rad=s and S D 2965 cP, and yellow diamonds are ! D 4:000 rad=s andS D 2924 cP.82Solvent Viscosity Dependence on The parameter values in Mooney’s Table 4.2, is showing that is increasing as thesolvent viscosity is increased for both the different radii and angular velocities, asshown in Figure 4.14. As we discussed in Section 4.3.1, the solvent viscosity directlyaffects the hydrodynamic interactions between particles, at higher solvent viscositiesthe field of disturbance is larger increasing the values of regardless of particle sizeor angular velocity.In Figure 4.14(a), we can see for a small range of solvent viscosities the valuesof have a scattered pattern at the various particle sizes, but overall the values of are increasing as the solvent viscosity is increasing. This is seen by the orderingthe symbols representing the different particle sizes at the small range of solventviscosities. When we take a larger step in solvent viscosity, we again see differentorder of symbols representing the particle sizes, but with higher values of .In Figure 4.14(b), if we look at one particular solvent viscosity we see thatvarying the angular velocities has almost no affect on the values of . This affectis so small that we can say that there is possibly no change. If we see no changethen we can say that appears to be constant at one particular solvent viscosity.However, we do see a pattern of increasing values for increasing solvent viscosities.This increase appears to have some linearly dependence on values. as the solventviscosities are increased at the various angular velocities.The solvent viscosity obviously affects the medium in which the particles interact.At higher solvent viscosities, a particle of radius a will create a large wake or field ofdisturbance. The same particle of radius a at lower solvent viscosities will create asmaller wake or field of disturbance. Again, larger wakes affect the nearby particlesmore and particles much farther away which is causing the increase in the values.83(a)(b)Figure 4.14: Plot of vs S modeling the data to Mooney’s Equation 2.11 and Batch-elor’s Equation 2.13. (a) Different radii with an angular velocity ! D 1:000 rad=s,the red stars are for a D 26 m, the green triangles are for a D 36 m, the bluesquares are for a D 51 m, and the yellow triangles are for a D 71 m. (b) Differ-ent angular viscosities for the particle size a D 36 m, the red stars are for angularvelocity ! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squaresare ! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.84The parameter values in Krieger-Dougherty’s Table 4.3 shows that is increasingas the solvent viscosity is increased for both the different radii and angular velocities,as shown in Figure 4.15. We see an identical pattern in Figure 4.14 for similarreasons, but with slightly different values for . Again, as shown in Figure 4.15(a),we see for a small range of solvent viscosities the values of are scattered at thedifferent particle sizes, but overall the values of are increasing for increasing solventviscosity. In Figure 4.15(b), we see for similar solvent viscosities varying the angularvelocities has little to no effect on the values of . Instead, appears constant forthe various angular velocities with a linear increase for increasing solvent viscosity.85(a)(b)Figure 4.15: Plot of vs S modeling the data to Krieger-Dougherty’s Equation2.12 and Brady’s Equation 2.18. (a) Different radii with an angular velocity ! D1:000 rad=s, the red stars are for a D 26 m, the green triangles are for a D 36 m,the blue squares are for a D 51 m, and the yellow triangles are for a D 71 m.(b) Different angular viscosities for the particle size a D 36 m, the red stars arefor angular velocity ! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, theblue squares are ! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.86Angular Velocity Dependence on The parameter values in Mooney’s Table 4.2, is showing having little dependenceon the angular velocity with possible constant values for the different radii andsolvent viscosities, as shown in Figure 4.16. As we can see, changing the angularvelocity has little to no effect on the values, compared to what we saw whenwe varied the different radii and solvent viscosities as seen in Figures 4.12(b) and4.14(a).In Figure 4.16(a), we see 2:75˙0:5 with the ordering of particle sizes varyingat the different angular velocities. Again, we see this pattern of varying values for in Figure 4.12(b).In Figure 4.16(b), we see 2:5˙ 0:25 with a pattern in the values of deter-mined by the values of solvent viscosities. Figure 4.16(b) shows a similar pattern wesee before in Figure 4.12(a), which showed an increased in values as we increasedthe solvent viscosity regardless of particle size or angular velocity. Again, we seea similar pattern, but this time by the red stars .S D 3017 2924 cP/ having thehighest values of , followed by the green triangles .S D 1265 1211 cP/ havinglower values, and finally followed by the blue squares .S D 571 555 cP/ havingthe lowest values of . This reiterates the fact that there is a linearly dependenceon solvent viscosities which we saw in Section 4.3.1.87(a)(b)Figure 4.16: Plot of vs ! modeling the data to Mooney’s Equation 2.11 andBatchelor’s Equation 2.13. (a) Different radii with a range of 2924 < S < 3017 cP,the red stars are for a D 26 m, the green triangles are for a D 36 m, the bluesquares are for a D 51 m, and the yellow triangles are for a D 71 m. (b)Different solvent viscosities for the particle size a D 36 m, the red stars are for2924 < S < 3017 cP, the green triangles are for 1211 < S < 1265 cP and the bluesquares are for solvent viscosity of 555 < S < 571 cP.88The parameter values in Krieger-Dougherty’s Table 4.3, show that havingno strong dependence on the angular velocity with possible constant values at thedifferent radii and solvent viscosities, as shown in Figure 4.17. The values andpatterns are identical to what we saw in Figure 4.16, with 2:75 ˙ 0:5 for thevarious particle sizes Figure 4.17(a) and 2:5 ˙ 0:25 for the different solventviscosities Figure 4.17(b). Again, the pattern of values are being determined bythe solvent viscosities with higher values at higher solvent viscosities, as seen inFigure 4.17(b).Summary on DependenceStudying the coefficient has given us some valuable information on how the differ-ent parameters can affect its value. At the different particles sizes, we see the leastunderstood pattern and least consistent values. This is something that needs to beexplored in future experiments to get a better understanding of exactly how particlesize affects the coefficient of . Considering the different solvent viscosities thereis definitely appears to be some linear dependence with higher solvent viscositieshaving higher values. Finally, there appears to be a linear dependence on theangular velocity, which can almost be called a constant.89(a)(b)Figure 4.17: Plot of vs ! modeling the data to Krieger-Dougherty’s Equation 2.12and Brady’s Equation 2.18. (a) Different radii with a range of 2924 < S < 3017 cP,the red stars are for a D 26 m, the green triangles are for a D 36 m, the bluesquares are for a D 51 m, and the yellow triangles are for a D 71 m. (b)Different solvent viscosities for the particle size a D 36 m, the red stars are for2924 < S < 3017 cP, the green triangles are for 1211 < S < 1265 cP and the bluesquares are for solvent viscosity of 555 < S < 571 cP.904.3.2 Fitting Parameter 'mRadial Dependence of 'mThe fitting parameter 'm is a geometric estimate based on packing density deter-mining the maximum concentration for a spherical objects, called the self-crowdingfactor and is related to 'm D 1k . The values in Mooney’s Table 4.2 are experimentalvalues with a range of 0:41 < k < 1:97 or 0:51 < 'm < 2:27 compared to the the-oretical values in Section 2.4.2 of 1:35 < k < 1:91 or 0:52 < 'm < 0:74. The valueof 'm is defined as the point where the viscosity goes to infinity. This suspensionwill reach a jamming point, 'j , before reaching 'm. The jamming point is wherethe fluid no longer flows, so long as the rheometer is applying a shear force. Thejamming point for a suspension is somewhere between 50% and 60%.[16]The parameter values in Mooney’s Table 4.2 shows 'm having little dependencefor particles with a radius > 36 m at different solvent viscosities and angularvelocities, seen in Figure 4.18. The values are somewhat constant indicating thatchanging the radii, possibly, has effect on the values of 'm. The values for 'm arewell outside of Mooney’s predicted range of 0:52 < 'm < 0:74, but he did a poor jobestimating the range because his approach was using geometric estimate for packingdensity to calculate the values of 'm.We suspect the particle a D 26 m has different results for 'm because its minutesize is closer to colloidal sized particle, a < 21 m. This particle size is affectingthe packing density somehow, but we do not truly understand why.In Figure 4.18(a), there seems to be a similar pattern we saw with the wherethe higher solvent viscosities produced higher values. Looking back at Figure 4.12(a)and 4.16(b) we see the red stars that represent S D 3017 cP have the largest valuesof 'm, followed by the green triangles that represent S D 1265 cP have the lowervalues of 'm, and finally followed by the blue squares that represent S D 571 cP havethe lowest values of 'm. We are again, excluding the a D 26 m from our analysisbecause it showing unexplainable results due to it being so close to a colloidal sizeparticle. Again, the higher the solvent viscosity the larger the field of disturbance is91created and the harder it becomes to tightly pack the particles, thus increasing 'm.In Figure 4.18(b), there appears to be a linear dependence on the radii whenvarying the angular velocity, if we excluded a D 26 m data. This linear dependencecan almost be considered to be a constant. If we compare this figure to Figure4.12(b), where we studied , we see some similarities in the way the data points areclumped tightly together and appearing to have relatively constant values.92(a)(b)Figure 4.18: Plot of 'm vs a modeling the data to Mooney’s Equation 2.11. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s, the red starsare for S D 3017 cP, the green triangles are for S D 1265 cP and the blue squaresare for S D 571 cP. (b) Different angular viscosities, the red stars are for angularvelocity ! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D 2:000 rad=sand S D 2988 cP, the blue squares are ! D 3:000 rad=s and S D 2965 cP, andyellow diamonds are ! D 4:000 rad=s and S D 2924 cP.93The parameter values in Krieger-Dougherty’s Table 4.3, is showing 'm havingno strong dependence on the radii for different solvent viscosities and angular ve-locities with radii > 36 m, as seen in Figure 4.19. The values also appear to beconstant telling us that the radii should have no effect on the values of 'm. Thisis surprising, again, because I expected the particle size would directly affect 'mvalues. Considered the following: smaller particles are more efficient in filling thevoided spaces with more smaller particles for a given finite volume, compared tolarger particles which are much less efficient in filling the voided spaces betweenneighboring particles because of their larger size in the same finite volume.Figure 4.19 has an identical pattern to Figure 4.18, but lower values for 'mbecause Kriger and Dougherty used a volume fraction to calculate its value instead ofgeometric packing densities, used by Mooney. Figure 4.19(a), shows a similar patternwe saw before with the values, where the higher solvent viscosities produced highervalues. Figure 4.19(b), there appears to be a linear dependence on the radii whenvarying the angular velocity, again excluding a D 26 m from the data. This lineardependence can almost be considered to be a constant.94(a)(b)Figure 4.19: Plot of 'm vs a modeling the data to Krieger-Dougherty’s Equation2.12. (a) Different solvent viscosities with an angular velocity ! D 1:000 rad=s, thered stars are for S D 3017 cP, the green triangles are for S D 1265 cP and theblue squares are for S D 571 cP. (b) Different angular viscosities, the red starsare for angular velocity ! D 1:000 rad=s and S D 3017 cP, the green trianglesare ! D 2:000 rad=s and S D 2988 cP, the blue squares are ! D 3:000 rad=s andS D 2965 cP, and yellow diamonds are ! D 4:000 rad=s and S D 2924 cP.95Studying Batchelor’s Equation 2.13, the fitting parameter 'm is the volume frac-tion where the viscosity becomes infinite. The range of values for the fitting param-eters is 0:46 < 'm < 1:14. We see that there are values greater than 100%, but 'mis a theoretical value and experimentally we reached the jamming point, 'j , beforewe will reach 'm. The patterns in the figures are similar to Krieger-Dougherty’s.The parameter values in Batchelor’s Table 4.4 shows 'm appearing to be linearlydependent on the radii for different solvent viscosities and angular velocities, asseen in Figure 4.20. I suspect the particle size will directly affect values of 'm, sincesmaller particles create smaller wakes thus allowing them to compact themselvesmore tightly compared to larger particles which create larger wakes preventing theparticles from compacting as tightly as the smaller ones.Figure 4.20 has an identical pattern to Figure 4.19, with similar values for 'mbecause both Krieger-Dougherty and Batchelor used a volume fraction to calculateits value. In Figure 4.20(a), we see a similar pattern, where the higher solventviscosities produced higher values. In Figure 4.20(b), there definitely is a lineardependence on the radii when varying the angular velocity, again almost appearingto be constant.For Brady’s Equation 2.18 the fitting parameter 'm is, also, a volume fractionwhere the viscosity becomes infinite. The range of values for the fitting parametersis 0:48 < 'm < 1:24. Again, we have values of 'm > 100%, but we know that wewill reaching the jamming point, 'j before we get to values that high.The parameter values in Brady’s Table 4.5 shows 'm appears to be linearly de-pendent on the radii at different solvent viscosities and angular velocities for particlesizes greater than 36 m, as seen in Figure 4.21. The values 'm are decreasing lin-early as the radii are increasing for both different solvent viscosities and angularvelocities. As I mentioned earlier, I suspect the particle size will directly affect val-ues of 'm, since smaller particles can compact themselves more tightly compared tolarger particles in the same finite volume.Figure 4.21 has an identical pattern to Figures 4.19 and 4.20, with similar valuesfor 'm because Brady used a volume fraction to calculate its value, just like Krieger-96(a)(b)Figure 4.20: Plot of 'm vs a modeling the data to Batchelor’s Equation 2.13. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s, the red starsare for S D 3017 cP, the green triangles are for S D 1265 cP and the blue squaresare for S D 571 cP. (b) Different angular viscosities, the red stars are for angularvelocity ! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D 2:000 rad=sand S D 2988 cP, the blue squares are ! D 3:000 rad=s and S D 2965 cP, andyellow diamonds are ! D 4:000 rad=s and S D 2924 cP.97Dougherty and Batchelor. In Figure 4.21(a), again, the higher solvent viscositiesproduced higher values. In Figure 4.21(b), there is little linear dependence on theradii when varying the angular velocity, again almost appearing to be constant.98(a)(b)Figure 4.21: Plot of 'm vs a modeling the data to Brady’s Equation 2.18. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s, the red starsare for S D 3017 cP, the green triangles are for S D 1265 cP and the blue squaresare for S D 571 cP. (b) Different angular viscosities, the red stars are for angularvelocity ! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D 2:000 rad=sand S D 2988 cP, the blue squares are ! D 3:000 rad=s and S D 2965 cP, andyellow diamonds are ! D 4:000 rad=s and S D 2924 cP.99Solvent Viscosity Dependence of 'mThe parameter values in Mooney’s Table 4.2, is showing 'm appearing to be linearlydependent on the solvent viscosity for the different radii and angular velocities forparticles with a radius greater than 36 m, as seen in Figure 4.22. The values of 'mappearing to be increasing slightly for increasing solvent viscosities for the variousradii as seen in Figure 4.22(a). This is expected because the higher solvent viscositiescreate a larger hydrodynamic interaction. These interactions will determine howwell the particles can compact, such as, at higher viscosities the interactions aregreater not allowing larger particles to compact as tightly together as at lower solventviscosity, thus varying 'm. In Figure 4.22(b), the fitting parameter 'm appears tobe constant for increasing solvent viscosities. As the angular velocity is varied thevalue of 'm is not changing.The parameter values in Krieger-Dougherty’s Table 4.3, shows 'm appearing tobe linearly dependent on the solvent viscosity for the different radii and angularvelocities, as seen in Figure 4.23. The values of 'm are slightly increasing for thevarious radii, especially for the larger particles, as seen in Figure 4.23(a). A patternemerges, which is seen in all four models, where the smallest particle size has thelargest value for 'm and the largest particle size has the lowest value for 'm. InFigure 4.23(b), the fitting parameter, 'm, appears to be constant for the increasingsolvent viscosities. This is expected since the solvent viscosity is not changing, thusthe field disturbance is remaining constant and keeping the values of 'm constant.The parameter values in Batchelor’s Table 4.4 shows 'm appearing to be linearlydependent on the solvent viscosity for the different radii and angular velocities, asseen in Figure 4.24. Again, the values of 'm are slightly increasing for the variousradii, as seen in Figure 4.24(a). This slight increase and ordering of the particle sizes,indicating that as the particles become larger the values of 'm become smaller, is seenin Figures 4.22(a) and 4.23(a), as well. In Figure 4.24(b), the fitting parameter, 'm,is appearing to be constant for the increasing solvent viscosities, as seen previously.The parameter values in Brady’s Table 4.5 shows 'm appearing to be linearly100(a)(b)Figure 4.22: Plot of 'm vs S modeling the data to Mooney’s Equation 2.11. (a)Different radii with an angular velocity ! D 1:000 rad=s, the red stars are fora D 26 m, the green triangles are for a D 36 m, the blue squares are fora D 51 m, and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squares are! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.101(a)(b)Figure 4.23: Plot of 'm vs S modeling the data to Krieger-Dougherty’s model,Equation 2.12. (a) Different radii with an angular velocity ! D 1:000 rad=s, thered stars are for a D 26 m, the green triangles are for a D 36 m, the blue squaresare for a D 51 m, and the yellow triangles are for a D 71 m. (b) Differentangular viscosities for the particle size a D 36 m, the red stars are for angularvelocity ! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squaresare ! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.102(a)(b)Figure 4.24: Plot of 'm vs S modeling the data to Batchelor’s Equation 2.13.(a) Different radii with an angular velocity ! D 1:000 rad=s, the red stars arefor a D 26 m, the green triangles are for a D 36 m, the blue squares are fora D 51 m, and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squares are! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.103dependent on the solvent viscosity for the different radii and angular velocities, asseen in Figure 4.25. The values of 'm are increasing slightly for the various radii,as seen in Figure 4.25(a). In Figure 4.25(b), the fitting parameter 'm appears to beconstant for increasing solvent viscosities.104(a)(b)Figure 4.25: Plot of 'm vs S modeling the data to Brady’s Equation 2.18. (a)Different radii with an angular velocity ! D 1:000 rad=s, the red stars are fora D 26 m, the green triangles are for a D 36 m, the blue squares are fora D 51 m, and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squares are! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.105Angular Velocity Dependence of 'mThe parameter values in Mooney’s Table 4.2, shows 'm having no little dependenceon the angular velocity for the different radii and solvent viscosities, as seen inFigure 4.26. Looking at Figure 4.26(a), the values of 'm appear to be constant,excluding a D 26 m because of its size being so close to a colloidal size and givingbizarre results. Again, we see, varying the angular velocity as little to no effecton the values of 'm. The reason for this is the solvent viscosity plays the morevital role in determining the values of 'm, which is seen in Section 4.3.2 and Figure4.22(b). Looking at Figure 4.26(b), the values of 'm are slightly decreasing as thesolvent viscosity decreases, where the red stars represent S D 3017 2924 cPhaving the highest values of 'm, followed by the green triangles that represent S D1265 1211 cP having the lower values, and finally followed by the blue squaresthat represent S D 571 555 cP having the lowest values of 'm. A similar patternis seen in Figure 4.12(a), but with values instead of 'm values.The parameter values in Krieger-Dougherty’s Table 4.3, is showing 'm appearingto be linearly dependent on the angular velocity for the different radii and solventviscosities, as seen in Figure 4.27. The values of 'm are remaining relativity constant,excluding a D 26 m, as seen in Figure 4.27(a). Looking at Figure 4.27(b), thevalues of 'm are slightly decreasing as the solvent viscosity decreases.The parameter values in Batchelor’s Table 4.4 shows 'm, again, to be appearinglinear dependent on the angular velocity for the different radii and solvent viscosities,as seen in Figure 4.28. The values of 'm are remaining relativity constant, excludinga D 26 m, as seen in Figure 4.28(a). Looking at Figure 4.28(b), the values of 'm,again, are slightly decreasing as the solvent viscosity decreases.The parameter values in Brady’s Table 4.5 shows 'm to be appearing linearlydependent on the angular velocity for the different radii and solvent viscosities, asseen in Figure 4.29. The values of 'm are remaining relativity constant, excludinga D 26 m, as seen in Figure 4.29(a). Looking at Figure 4.29(b), the values of 'mare slightly decreasing as the solvent viscosity decreases.106(a)(b)Figure 4.26: Plot of 'm vs ! modeling the data to Mooney’s Equation 2.11. (a)Different radii with a range of 2924 < S < 3017 cP, the red stars are for a D 26 m,the green triangles are for a D 36 m, the blue squares are for a D 51 m, andthe yellow triangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are for solvent viscosityof 555 < S < 571 cP.107(a)(b)Figure 4.27: Plot of 'm vs ! modeling the data to Krieger-Dougherty’s Equation2.12. (a) Different radii with a range of 2924 < S < 3017 cP, the red starsare for a D 26 m, the green triangles are for a D 36 m, the blue squares arefor a D 51 m, and the yellow triangles are for a D 71 m. (b) Different solventviscosities for the particle size a D 36 m, the red stars are for 2924 < S < 3017 cP,the green triangles are for 1211 < S < 1265 cP and the blue squares are for solventviscosity of 555 < S < 571 cP.108(a)(b)Figure 4.28: Plot of 'm vs ! modeling the data to Batchelor’s Equation 2.13. (a)Different radii with a range of 2924 < S < 3017 cP, the red stars are for a D 26 m,the green triangles are for a D 36 m, the blue squares are for a D 51 m, andthe yellow triangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are for solvent viscosityof 555 < S < 571 cP.109(a)(b)Figure 4.29: Plot of 'm vs ! modeling the data to Brady’s Equation 2.18. (a)Different radii with a range of 2924 < S < 3017 cP, the red stars are for a D 26 m,the green triangles are for a D 36 m, the blue squares are for a D 51 m, andthe yellow triangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are for solvent viscosityof 555 < S < 571 cP.110Summary on 'm DependenceThe literature of Krieger Dougherty, Batchelor, and Brady all refer to 'm as the max-imum allowable concentration for a suspension and is the volume fraction where theviscosity becomes infinite. This value is no longer a prediction based on the pack-ing densities of spherical objects as Mooney believed, but a theoretical calculation.However, the range of values for the fitting parameters is 0:41 < 'm < 1:27. Aswe can see it is a wide range with some values greater than 100%. This makes nophysical sense because a solid object is 100% of any given material. In order to makephysical sense we are going to define the jamming point, 'j , as the point where thefluid will no longer flows and somewhere around 'j 55%.[16]Overall the fitting parameter behaves similar in all four models, the only majordifference seen is the actual values of 'm. This pattern of similarity is indicatingthat all four models are modeling the same information for 'm giving us a goodunderstanding what is happening. There does appear to be a strange result fora D 26 m in all four models, again, probability because of its small particle size.A major effect that is seen is the values of 'm is the slight decrease in values as thesolvent viscosity decreases, indicating that there is some sort of linearly dependence.When studying the effects of 'm on particle size and angular velocity, we again seesomewhat of a pattern having little dependence that appears to have a constantvalue.1114.3.3 Fitting Parameter ˛The fitting parameter ˛ is an estimate from numerical calculations for a suspen-sion of identical rigid spherical particles using the probability density, p*r ; t,determined by three-sphere interactions, Brownian motion, and assumption of someparticular initial state.The parameter values in Batchelor’s Table 4.4 shows ˛ varying slightly as theparticle size is increased for both the different solvent viscosities and angular veloc-ities. However, in Figure 4.30(a), the pattern of higher solvent viscosities havinghigher ˛ values has been seen in previous figures (Figures 4.12(a) and 4.13(a)). Theorder has the highest solvent viscosities having the largest ˛ values and the lowestsolvent viscosities having the smallest ˛ values. In Figure 4.30(b), we see varyingthe angular velocity has little effect on the values of ˛ at the individual particlesizes, only the particle sizes affecting the values of ˛. Comparing Figures 4.12 to4.30, we see the patterns are almost identical, but with different values for and ˛.The parameter ˛ is an estimate based on numerical calculations as stated byBatchelor[9], for a particular particle size. The size of the particle directly affectsthe hydrodynamic interactions between particles, for example larger particles createa larger wake or field of disturbance. A larger wake is more likely to overlap anotherparticle’s wake in a nearby region, which can change the velocity, as well as ˛.We see ˛ appearing to be linearly dependent on the solvent viscosity for differentradii, as shown in Figure 4.31(a). The different radii have increasing ˛ values forincreasing solvent viscosities. However, ˛ is also appearing to be independent of thesolvent viscosity for increasing angular velocities, as shown in Figure 4.31(b). This isindicating that ˛ might be a constant, with the size of the particle determining thevalue of ˛. Again, comparing Figures 4.14 to 4.31, we see the patterns are almostidentical, but with different values for and ˛.The reason for the similar patterns is the solvent viscosity affects the mediumin which the particles interact. At higher solvent viscosities, a particle of radiusa will create a large wake or field; whereas, the same particle of radius a at lower112solvent viscosities will create a smaller wake. Again, the large wake affects thenearby particles which is increasing the value of ˛.Finally we see ˛ appearing to be linearly dependent on the angular velocity withconstant values for the different radii and solvent viscosities, as shown in Figure 4.32.The angular velocities are not changing the values of ˛. Instead, the values of ˛are being determined by the particle’s size, as seen in Figure 4.32(a). The differentsolvent viscosities are not affected by the different angular velocities, but the differentsolvent viscosities are changing the values of ˛, as seen in Figure 4.32(b). Again,comparing Figures 4.16 to 4.32, we see the patterns are almost identical, but withdifferent values for and ˛.Summary on ˛ DependenceOverall, ˛ appears to looking very similar to with a relationship / ˛. The exactrelationship is not understood, but there appears to be some sort of affiliation.As we saw with , the pattern see for the different radii is very inconsistent anddifficult to explain. However, at different solvent viscosities there is definitely alinear dependence with higher solvent viscosities having higher ˛ values. We see alinear dependence on the angular velocity, again, we can almost call it a constant.113(a)(b)Figure 4.30: Plot of ˛ vs a modeling the data to Batchelor’s Equation 2.13. (a)Different solvent viscosities with an angular velocity ! D 1:000 rad=s, the red starsare for S D 3017 cP, the green triangles are for S D 1265 cP and the blue squaresare for S D 571 cP. (b) Different angular viscosities, the red stars are for angularvelocity ! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D 2:000 rad=sand S D 2988 cP, the blue squares are ! D 3:000 rad=s and S D 2965 cP, andyellow diamonds are ! D 4:000 rad=s and S D 2924 cP.114(a)(b)Figure 4.31: Plot of ˛ vs S modeling the data to Batchelor’s Equation 2.13. (a)Different radii with an angular velocity ! D 1:000 rad=s, the red stars are fora D 26 m, the green triangles are for a D 36 m, the blue squares are fora D 51 m, and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squares are! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.115(a)(b)Figure 4.32: Plot of ˛ vs ! modeling the data to Batchelor’s Equation 2.13. (a)Different radii with a range of 2924 < S < 3017 cP, the red stars are for a D 26 m,the green triangles are for a D 36 m, the blue squares are for a D 51 m, andthe yellow triangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are for solvent viscosityof 555 < S < 571 cP.1164.3.4 Fitting Parameter ˇThe fitting parameter ˇ is a combination of the short-time self-diffusivity equationand the equilibrium radial-distribution function for the Brownian stresses. The pa-rameter values in Brady’s Table 4.5 is showing ˇ appearing to be linearly dependenton the radius. As the solvent viscosity and angular velocity are varied, we see a ran-dom increase and decrease in ˇ for increasing radii, as shown in Figure 4.33. We seeˇ reaching a maximum value at a D 51 m, it also changes directions to a lineardecrease in ˇ for increasing radii. Comparing Figures 4.12 to 4.33, the figures arealmost identical with similar patterns and with similar values.The parameter ˇ has a direct correlation to the particle size because it is thecoefficient determined by the equilibrium radial-distribution function at contact,g.2/, divided by the short-time self-diffusivity, DS0 .'/. The size of the particledirectly affects the hydrodynamic interactions between particles, for example largerparticles create a larger wake or field of disturbance. A larger wake is more likely tooverlap another particle’s wake in a nearby region, which can change the velocity,as well as ˇ.We see that ˇ has various values at the different radii for both different solventviscosities and angular velocities. Again in Figure 4.33(a), we see at higher solventviscosities the values of ˇ are at their maximum and at lower solvent viscosities thevalues of ˇ are at their minimum, which we saw with , 'm, and ˛. We previoussaw with the other fitting parameters, the angular velocity having little effect on ˇ,which is seen in Figure 4.33(b) as well.In Figure 4.34(a), we see for increasing ˇ values a increase in solvent viscosities,indicating the ˇ might be linearly dependent on solvent viscosity. As well as, ˇappearing to be linearly dependent of the solvent viscosity for increasing angularvelocities, as shown in Figure 4.34(b). Not only is ˇ appearing to be linearly de-pendent of the solvent viscosity for increasing angular velocities, it appears to be aconstant, with the solvent viscosity determining the value of ˇ. Again, comparingFigures 4.14 to 4.34, the figures are almost identical with similar patterns and with117similar values.The solvent viscosity affects the medium in which the particles interact. Athigher solvent viscosities a particle of radius a will create a large wake or field.Compared to the same particle of radius a at lower solvent viscosities will create asmaller wake. Again, the large wake affects the nearby particles which is increasingthe value of ˇ.Finally, we see that ˇ appearing to be linear dependent on the angular velocitywith constant values for the different radii and solvent viscosities, as shown in Fig-ure 4.35. For increasing angular velocities, the values of ˇ are remaining constantfor the different particle sizes, as seen in Figure 4.35(a). The values of ˇ are beingdetermined by the particles size as previously seen. For increasing angular velocitiesthe values of ˇ are remaining constant for different solvent viscosities, as seen inFigure 4.35(b). The values of ˇ are being determined by the solvent viscosities, aspreviously seen. Comparing Figures 4.16 to 4.35, the figures are almost identicalwith similar patterns and with similar values.Summary on ˇ DependenceOverall, ˇ appears to have a similar pattern previous seen with , but with largervalues with possible relationship of pˇ. The exact relationship is not under-stood, but there appears to be some sort of affiliation. As we saw with and ˛, thepattern seen with the ˇ values for the different radii is very inconsistent and diffi-cult to explain. However, at different solvent viscosities there is definitely a lineardependence with higher solvent viscosities having higher ˛ values. Finally, we see alinear dependence on the angular velocity, again, we can almost call it a constant.118(a)(b)Figure 4.33: Plot of ˇ vs amodeling the data to Brady’s Equation 2.18. (a)Differentsolvent viscosities with an angular velocity ! D 1:000 rad=s, the red stars are forS D 3017 cP, the green triangles are for S D 1265 cP and the blue squares are forS D 571 cP. (b) Different angular viscosities, the red stars are for angular velocity! D 1:000 rad=s and S D 3017 cP, the green triangles are ! D 2:000 rad=s andS D 2988 cP, the blue squares are ! D 3:000 rad=s and S D 2965 cP, and yellowdiamonds are ! D 4:000 rad=s and S D 2924 cP.119(a)(b)Figure 4.34: Plot of ˇ vs S modeling the data to Brady’s Equation 2.18. (a)Different radii with an angular velocity ! D 1:000 rad=s, the red stars are fora D 26 m, the green triangles are for a D 36 m, the blue squares are fora D 51 m, and the yellow triangles are for a D 71 m. (b) Different angularviscosities for the particle size a D 36 m, the red stars are for angular velocity! D 1:000 rad=s, the green triangles are ! D 2:000 rad=s, the blue squares are! D 3:000 rad=s, and yellow diamonds are ! D 4:000 rad=s.120(a)(b)Figure 4.35: Plot of ˇ vs ! modeling the data to Brady’s Equation 2.18. (a)Different radii with a range of 2924 < S < 3017 cP, the red stars are for a D 26 m,the green triangles are for a D 36 m, the blue squares are for a D 51 m, andthe yellow triangles are for a D 71 m. (b) Different solvent viscosities for theparticle size a D 36 m, the red stars are for 2924 < S < 3017 cP, the greentriangles are for 1211 < S < 1265 cP and the blue squares are for solvent viscosityof 555 < S < 571 cP.121CHAPTER 5Conclusions5.1 ResultsIn this experiment, we explored the dependence of , Œ, ˛, ˇ, 'm on solvent vis-cosity, angular velocity, particles size, and concentration of particles. We exploredseveral theoretical and semi-theoretical equations showing the relative viscosity asa function of concentration. We examine the various fitting parameters from thoseequations to see how well to model to the experimental data, showing some surpris-ing results.In order to obtain are results we used an advanced combined motor and trans-ducer AR 1500ex Rheometer to create constant shearing force applying it to a sus-pension using different solvent viscosities, angular velocities, particles sizes to seewhat effects it has on the relative viscosity. We used solvent viscosities ranging from555 3017 cp. We used angular velocities of 1 rad=s, 2 rad=s, 3 rad=s, and 4 rad=s.We used particles with radii of 26 m, 36 m, 51 m, and 71 m. Finally, we usedparticle concentrations of 20 50%. Using all of these parameters we compiled ourdata to studied the results.We saw that there was an angular velocity dependence showing that ArrheniusEquation 2.3 is flawed. It only takes into account temperature and not angularvelocity, where we clearly saw the solvent viscosity is heavy depended on angularvelocity. Also, we saw that as the solvent viscosity was increased the relative vis-cosity was increased, as well. The angular velocity was appear to be independenton the relative viscosity, whereas increasing the angular velocity caused little to noeffect on the relative viscosity. Finally, we saw a strong dependence on the relativeviscosity as the particle size was varied.The value of 2:5 taken from Einstein’s model, Equation 2.10, based on the spher-122ical shape of the particle, has always been taken as a constant. The values found inthis study range from 1:4 < < 3:3 . We found to be dependent on particle sizeand independent of angular velocity and solvent viscosity. We did see smaller rangesfor both Batchelor’s model, Equation 2.13 and Brady’s model, Equation 2.18. Apossible relationship is found between D ˛ Dpˇ.I suspected the particle size, angular velocity, and solvent viscosity would effect'm and was surprised when the data showed differently. Especially particle size,base on the fact that gap distance puts a limitation and a finite number of particlesthat can be placed in-between the gap. For example, a gap size to radius ratioequaling a whole number, 6100 m50 mD 122 would pack better than a ratio equaling afraction, 6100 m51 mD 119:6. The difference between the ratio of those two particlesis small, 0:4, but adding multiple layers will quickly add up resulting in more fluidand less particles for the suspension changing the relative viscosity.We also define the jamming point, which is where the system would cease to actlike a fluid and is much less than 'm. The values of 'm are theoretical values locatingthe vertical asymptotes, several occurred at concentration greater than 100%. Therange of 0:41 < 'm < 2:27 became smaller as more free fitting parameters wereadded to the models. The values of 'm are linearly independent of particle size andindependent of solvent viscosity and angular velocity.Determining which equation modeled best to the data is not a particular easytask; as a matter of fact, it is quite difficult and almost subjective. By calculat-ing and comparing the average 2 for the different models in Mooney’s Table 4.22 D 2:492, Krieger-Dougherty’s Table 4.32 D 3:087, Batchelor’s Table 4.42 D 2:506, and Brady’s Table 4.52 D 2:421we see that the values are prac-tically identical. If I had to pick an equation which models the data best, I wouldstart by eliminating Krieger-Dougherty’s Equation 2.12 based on solely the 2 value.Next, I would eliminate Mooney’s Equation 2.11 based on the average 'm D 0:90being to high, which reiterates the importance of the new term I define 'j 0:52(Section 4.1), because its value is much higher than the jamming point. For thesame reason, I eliminate Brady’s Equation 2.18 based on the 'm D 0:69 and it123being much more complicated then Batchelor’s Equation 2.13.5.2 Future WorkStudying how the relative viscosity depends on particle sizes is a future step thatneeds to take place in order to full understand what is going on with the relativeviscosity. Using more particle sizes would better define the pattern seen in theFigures 4.12-4.21, and whether we are truly seeing a trigonometric or Gaussiancurve. We also need to look to see if there is any the relationship between thecoefficients and gap size. The ability to control the relative viscosity for a particularsize particle by varying the gap size has an enormous impact on the designing ofsystems where shape and size become a major issue.124APPENDIX ACode used to minimize 2Figure A.1: Code used to minimize 2125REFERENCES[1] Andreas Alexandrou. Principles of Fluid Mechanics. Prentice-Hall, Inc., UpperSaddle River, New Jersey 07458, 2001.[2] Raymond A. Serway. Physics For Scientists & Engineers with Modern Physics,Fourth Edition. Saunders College Publishing, Orlando, Florida 32887-6777,1996.[3] James Stewart. Calculus Fourth Edition. Brooks/Cole Publishing Company,Pacific Grove, CA 93950, 1999.[4] Edward G. Hauptmann Rolf H. Sabersky, Allan J. Acosta and E. M. Gates.Fluid Flow a First Course in Fluid Mechanics. Forth edition edition.[5] E. J. Hinch. Fluid Flow. Prentice Hall, 1999.[6] Albert Einstein. A new determination of molecular dimensions. Annalen derPhysik (ser. 4), 19:289–306, 1906.[7] M. Mooney. The viscosity of a concentrated suspension of spherical particles.Journal of Colloid Science, 6:162–170, 1951.[8] Irvin M. Krieger and Thomas J. Dougherty. A mechanism for non-newtonianflow in suspensions of rigid spheres. Transactions of The Society of Rheology,3:137–152, 1959.[9] G. K. Batchelor and J. T. Green. The determination of the bulk stress in a sus-pension of spherical particles to order c2. Journal Fluid Mechanics, 56(3):401–427, 1972.[10] John F. Brady. The rheological behavior of concentrated colloidal dispersions.The Journal of Chemical Physics, 99:567–581, 1993.126[11] TA Instruments, New Castle, DE 19720. AR 1500ex Rheometer Operator’sManual, 2007.[12] Viscosity of Aqueous Glycerine Solutions. Midland, Michigan, 2009.http://www.dow.com/glycerine/resources/table18.htm.[13] Mario F. Triola. Elementary statistics. Addison Wesley, San Francisco, CA94111, seventh edition edition, 1998.[14] John Mandel. The Statistical Analysis of Experimental Data. Free Dover Math-ematics and Science Catalog, Mineola, NY 11501, 1964.[15] Philip R. Bevington and D. Keith Robinson. Data Reduction and Error Analysisfor the Physical Sciences. McGraw-Hill, New York, NY 10020, third editionedition, 2002.[16] Brent Allen May Jarvis. Particle concentration dependence of viscosity. DePaulJournal, 2009.

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