Size and Shape Characterization of Prolate Particles

Transcription

1 Size and Shape haracterization of Prolate Particles W. PABST, E. GREGOROVÁ,. BERTHOLD, et al. Department of Glass and eramics, Institute of hemical Technology, Prague, zech Republic Institut für Geowissenschaften, Universität Tübingen, Germany PPS-Lecture ad Units 9 and Introduction Size and shape characterization of strongly anisometric particles and particle systems (platelet and fiber systems) poses severe theoretical and practical problems. Flaky particles (platelets): Shape information can be extracted from a comparison of sedimentation and laser diffraction data (PABST et al.,, ): π D L Ψ = DS D L = laser diffraction equivalent diameter (closely related to the true disc diameter), D S = sedimentation equivalent diameter (Stokes diameter), Ψ = LS shape factor (closely related to the aspect ratio). Introduction Elongated particles (fibers): Fiber orientation effects in the flow cell of laser diffractometers (alignment in the flow direction) can lead to strongly non-circular diffraction patterns and can have a strong impact on the results (BERTHOLD et al. ). Size and shape characterization of short-fiber systems: Only microscopic image analysis provides unbiased and direct quantitative information (PABST, BERTHOLD & GREGOROVÁ, submitted). Rheology of fiber suspensions: Fiber shape has a direct influence e.g. on effective viscosity. This influence can be quantified (PABST, GREGOROVÁ & BERTHOLD, submitted). Objectives Determination of the size and shape distribution of a polydisperse short-fiber system by image analysis. Quantification of the average shape. Measuring the effective viscosity of short-fiber suspensions in dependence of the fiber volume fraction via rotational viscometry. Fitting the concentration dependence of viscosity and extracting physically meaningful parameters. omparison with the predictions based on the average shape obtained from image analysis. Optical micrograph of wollastonite WM. Materials Two types of wollastonite, WM and HSV. SEM micrograph of wollastonite HSV. Microscopic image analysis Software: LUIA G version.8 (Laboratory Imaging, Prague, zech Republic). Short-fibers marked manually by rectangles (or fivepoint ellipses), - objects measured. Selected size measures: Projected area diameter, minimum and maximum Feret diameter. Selected shape measure: Aspect ratio (individual aspect ratios calculated from minimum and maximum Feret diameters). Transformation to volume-weighted distributions.

2 Microscopic image analysis Microscopic image analysis Absolute frequencies (q histogram) before first re-ordering Absolute frequencies (q histogram) after first re-ordering Number of objects [] Number of objects [] 8 8 Microscopic image analysis Microscopic image analysis Relative volumes (q histogram) before second re-ordering Relative volumes (q histogram) after second re-ordering 8 8 Microscopic image analysis Microscopic image analysis umulative (Q) curve without re-ordering umulative (Q) curve after first re-ordering only

3 Microscopic image analysis 8 Microscopic image analysis 9 umulative volumes (Q curve) after re-ordering (smoothing) Number of objects [] First re-ordering (q ) Number of objects [] Second re-ordering (q ) 8 Microscopic image analysis Microscopic image analysis umulative (Q) curves (log scale), comparison 9 8. WM HSV Wollastonite WM (cumulative curves in log scale) 9 8. Size [microns] Wollastonite HSV (cumulative curves in log scale) 9 8. Size [microns] Wollastonite WM (frequency curve in log scale) 9 8. LALLS equivalent diameter [microns] Wollastonite HSV (frequency curve in log scale) 9 8. LALLS equivalent diameter [microns] Red: projected area diam., green: Feret diam., blue: laser diffraction diam. Microscopic image analysis Microscopic image analysis Average aspect ratios: WM approx., HSV approx. (scatter %) Aspect ratio [] WM WM (nd) HSV HSV (nd) Aspect ratio []. Red circles: WM, green squares: HSV (full: st operator, empty: nd operator) Shape-size dependence curve of prolate (needle-like) particles measured via microscopic image analysis (size class average aspect ratios between 9 and 8, arithmetic average of individual particles approx.., grand arithmetic average of size class averages approx.. ±.8, median aspect ratio 8.9 ±.)

4 Microscopic image analysis umulative volume [%] 9 8. Size [microns] Volume-weighted size distributions (i.e. obtained from the number-weighted size distributions after transformation) of the prolate particle system; dotted curves: minimum (left) and maximum (right) Feret diameter (image analysis), thin full curve: projected area diameter, thick full curve with error bars: laser diffraction diameter onclusions Image analysis Although the scatter is large (approx. % within each size class and approx. % among the mean values), average aspect ratios are well reproducible and roughly size invariant, approx. for wollastonite WM and approx. for wollastonite HSV. The transformation procedure applied ( q -Q transformation from number-weighted to volumeweighted particle size distributions) is a generic technique, applicable to any short-fiber system. For short-fiber systems laser diffraction results are close to the minimum Feret diameter, i.e. significantly lower than the projected area diameter determined via image analysis. Suspension rheology Theory η Relative viscosity: η Intrinsic viscosity: [ η] lim Viscosity of dilute suspensions (Jeffery-Einstein): η r = + [ η] Viscosity of concentrated suspensions: = (Krieger) η = (Maron-Pierce) r (viscosity η, viscosity of the suspending medium η, volume fraction, critical volume fraction ) [ η] Suspension rheology Theory Dependence of the intrinsic viscosity on the aspect ratio [ η]( R) (Brenner formula, theoretically sound) [ η] = Q Q sin θ ( Q + Q ) sin θ cos + ( Q + Q ) sin θ sin B BP The five material constants B, Q, Q, Q, Q are all functions of the aspect ratio and P is the rotary Péclet number, i.e. the ratio between shear rate and rotary Brownian diffusion coefficient (a function of shape and size). For weak Brownian motion (P>> and P>>R ) the following approximate expression holds: [ η] =. +. ( R ). 9 Suspension rheology Theory Dependence of the critical volume fraction on the aspect ratio ( R) (Kitano relation, purely empirical ) =.. R Rotary Péclet numbers (shear rate s -, minimum Feret diameter µm, temperature ):. 9 and. 9 for wollastonites WM (average aspect ratio ) and HSV (aspect ratio ) in wt.% sugar solution (density.8 g/cm, viscosity mpas) weak Brownian motion. Suspension rheology Experimental Suspensions: Wollastonites (density.9 g/cm ) WM and HSV in wt.% sugar solution (density.8 g/cm, viscosity mpas), prepared by ultrasonication. Sedimentation times increased by a factor of approx. compared to pure water. Viscometric measurements: Rotational viscometer RV (Haake, Germany) with coaxial cylinder system Z. Schedule: s ramp up, s hold at a shear rate of s -, s ramp down. Viscosity measured at s -. The flow behavior is close to Newtonian.

5 Suspension rheology Results Suspension of wollastonite WM (R=) Suspension rheology Results Suspension of wollastonite HSV (R=) Relative viscosity [] Relative viscosity [] Solids volume fraction [%] Solids volume fraction [%] Measured values (empty squares, error bars), Krieger fit (dashed line), Maron-Pierce fit (full line) Measured values (empty squares, error bars), Krieger fit (dashed line), Maron-Pierce fit (full line) Suspension rheology Results Starch (R=) suspensions (for comparison) Suspension rheology Results ritical volume fraction and intrinsic viscosity; fit parameters obtained via the Krieger relation (left) or the Maron-Pierce relation (right) Relative viscosity [] Solids volume fraction [%] Measured values (circles, full: corn, empty: wheat), Krieger fit (dashed), Maron-Pierce fit (full line) Material Aspect Ratio Starch.. WM.. HSV.. = Krieger relation [ η ] [ η] Material Aspect Ratio [ η ] est. Starch.9.8 WM..99 HSV.. η = r Maron-Pierce relation Suspension rheology Results omparison of Kitano et al. data and fit with our data and fit ritical volume fraction [] =.. R (Kitano et al. 98) Kitano data Kitano fit Our data Our fit Aspect ratio [] =.. R (Pabst et al. ) onclusions - Suspension rheology Experimentally determined intrinsic viscosities (.,.8 and.9 for particle systems with average aspect ratios, and respectively) are significantly higher than theoretically predicted, which are.,.8 and.9, respectively (Brenner formula). This is in agreement with the empirical finding occurring in the literature that in practice the Batchinski formula =+. works better than the (theoretically well founded) Einstein formula =+. Probable reason: electrostatic (not steric) interactions

6 onclusions - Suspension rheology Experimentally determined critical volume fractions ( %, 8 % and % for particle systems with average aspect ratios, and respectively) are significantly lower than expected, which are %, 8 % and %, respectively (Kitano et al.). This shows that the frequently cited Kitano relation with the original values given by Kitano et al. =.. R need not be useful for real systems. Being purely empirical a more general validity cannot be expected. For our wollastonite suspensions we find =.. R Probable reason: polydispersity in shape (not in size) Acknowledgement Bilateral project Size and Shape haracterization of Particles in eramic Science and Technology, zech Ministry of Education, Youth and Sports (Grant ZE /) and German Bundesministerium für Forschung und Technik (BMFT). Project Tvorba předmětu harakterizace částic a částicových soustav, zech Ministry of Education, Youth and Sports (Grant FRVŠ / / Fb). Selected references: The support is gratefully acknowledged. [] Pabst W., Kuneš K., Havrda J., Gregorová E.: J. Eur. eram. Soc. (), 9. [] Berthold., Klein R., Lühmann J., Nickel K. G.: Part. Part. Syst. haract. (),. [] Pabst W., Kuneš K., Gregorová E., Havrda J.: Brit. eram. Trans. (),. [] Pabst W., Kuneš K., Gregorová E., Havrda J.: Key Eng. Mater. - (),. [] Pabst W., Berthold., Gregorová E.: J. Eur. eram. Soc. (),. [] Pabst W., Gregorová E., Berthold. J. Eur. eram. Soc. (), 9. [] Pabst W., Berthold., Gregorová E.: J. Eur. eram. Soc. (), 9.