Experimental Physics: Using a Helium-Neon LASER and Mie scattering techniques to determine particle size distributions in homogeneous colloids

Transcription

1 Experimental Physics: Using a Helium-Neon LASER and Mie scattering techniques to determine particle size distributions in homogeneous colloids Aberystwyth University Abstract Diluted samples of polymer microspheres (3.9µm diameter), semi-skimmed milk (homogeneous, pasteurized) and emulsion paint (Dulux, pale-lilac) were illuminated by a Helium-Neon LASER of wavelength 633nm, and the subsequent Mie scattering behaviour of the light was recorded using LabVIEW to ascertain their particle size distributions. A Guinier and Porod analysis was conducted, the result of which showed that the particles in the samples had clumped together in larger conglomerates with an estimated average radius of 67.33µm. Limitations of the apparatus and samples were discussed, and possible improvements were suggested. Mr. Benjamen P. Reed

2 I. Introduction & Theory Light scattering is formally described as the redirection of electromagnetic radiation from an incident path due to an encounter with an obstacle or non-homogeneity [1]. The development of a theory of light scattering arose from attempts to explain the colour and polarization of incoming visible solar radiation. It was through the work of John Tyndall and later on, Lord Rayleigh, that a theory of light scattering for very small, dielectric, isotropic spheres was birthed [2]. This theory, known as Rayleigh scattering, adequately explained the scattering of visible EM radiation in the atmosphere and answered the age old question Why is the sky blue? However, Rayleigh scattering is by no means a general scattering theory, and it can only be used under strict boundary conditions, namely when the particle diameter is much smaller than the wavelength of the incident light (i.e. the size parameter x << 1) [3]. Mie theory is a far more comprehensive framework for light scattering and has no real limitations on the particle size; it can be used to model EM scattering from small particles (x << 1) but the mathematics is far more laborious and hence the Rayleigh model is more sensible in this case. The Mie theory is most practical in cases where the diameters of the particles in a system are comparable to the wavelength of the incident light (i.e. 2r particle ~ λ incident ) [4]. Figure 1 - As the particle size increases, the range of scattering angles decreases and forms an antenna-like lobe parallel to the incident light direction. The mathematics of Mie theory is advanced and is considered in most cases to be beyond the ability of all but the most precocious students. The Mie solution is obtained by expanding the incident wave into a Fourier series using functions that satisfy Maxwell s equations of electromagnetism in spherical coordinates. The derivation of the Fourier coefficients of this expansion (after applying the appropriate boundary conditions) is extensive; hence the author refers any curious reader to the appendix in I. Weiner s 2001 paper on Mie scattering in undergraduate labs [5]. Mie scattering can be used in many applications, and in this experiment, its capacity to determine particle radii in homogeneous colloids was the main focus (i.e. systems whose particles are distributed uniformly throughout a solution). Whilst applicable for this purpose, the more rigorous mathematical approach was substituted for a simpler method: Guinier analysis. This method involves calculating, for a given incident wavelength, the corresponding dependant scattering vectors for given angles of scattering. The scattering vector q is defined as the difference between the incident and scattered wavevectors, such that q = 4π λ sin ( θ ) (eqn 1) where λ is the wavelength of the incident light and θ is the angle of the scattered light [6]. The modulus of q is given in units of inverse length, usually nm -1. The intensity of scattered light is plotted against the values of the scattering vectors to produce a Guinier plot, from which information about the particle sizes and interfaces can be obtained. 2

3 An experiment was conducted to determine the particle size distributions in various samples that were assumed to be homogeneous colloids. Samples of polymer microspheres, semi-skimmed milk and emulsion paint were illuminated with a Helium-Neon LASER (633nm) and the subsequent scattering angles of the incident light were measured. This undertaking was the collaborative effort of several authors, who have been acknowledged toward the end of this report. II. Experimental Procedure The apparatus used in this experiment was constructed by technicians at the Institute of Mathematics and Physics, at Aberystwyth University. The main unit consisted of: a Helium- Neon LASER; a beam splitter and reference detector; a stepper motor and stepper motor control box; a goniometer with scatter detector; and a LASER Power Supply Unit (PSU). Other equipment included a computer with LabVIEW installed, square acrylic glass cuvettes, isopropanol, and deionised water. Figure 2 illustrates the basic set up of the apparatus (an internal schematic of the main unit is available in appendix A). Detector attached to goniometer, angular range of 81 o To mains supply LASER PSU Activated with key Stepper Motor Control Box Δθ He-Ne LASER Beam Path To LabVIEW MiniLab 1008 Detector-MiniLab connection To Stepper Motor Control Box To LASER PSU Stepper motor responsible for moving the goniometer Cuvette containing sample Main Casing To LabVIEW MiniLab 1008 He-Ne LASER Figure 2 - Experimental Apparatus, rough schematic. Top diagram shows a top down view of the experiment, bottom diagram shows a side view. Both diagrams assume transparent casing. 3

4 The LASER was a class 3B LASER which was capable of damage to the human eye and skin. The LASER required two switches to be closed before emission took place. If the casing of the main unit was open in any place (e.g. sample loading hatch), then micro-switches ensured that the internal circuit was shorted and the LASER was turned off. Deionised water was used as the diluting substance for the samples as it was unlikely to react with the samples or contain minerals that may affect the scattering patterns. Deionised water is potentially harmful if ingested hence it was labelled clearly with appropriate warnings. Isopropanol was used for cleaning the cuvettes between data recordings. Isopropanol is an irritant and can cause damage to the skin, therefore it too was labelled clearly and kept in an appropriate container. The three materials/compounds used were: Polymer microspheres (3.90µm diameter), semiskimmed milk (1.7g/100ml fat) and emulsion paint (Dulux, pale lilac). All three samples were diluted using deionised water before being measured out into the cuvettes. Each sample had to be diluted to a different optimal concentration to allow for the most scattering. The microspheres were subjected to an ultrasonic bath to prevent clumping due to their age. Too dilute, and little or no scattering would take place. Too opaque, and multiple scattering would take place or the beam would not penetrate the sample and reach the detector. The chosen concentrations were 1:910 for microspheres, 1:90 for semi-skimmed milk, and 1:20,000 for emulsion paint (i.e. 1ml of test sample to 910ml of deionised water for example). These were determined by trial-andimprovement until the best possible data was collected. The main casing contained a MiniLab 1008 unit that communicated with a computer via a USB cable. From the computer, a LabVIEW virtual instrument called miescatter.vi dictated control of the main unit. This VI automatically recorded data points, so long as a file directory was chosen that the program could access. Running the VI prompted the goniometer to move to its starting position at the right-hand, most extreme angle relative to the incident beam path. The samples could then be loaded into the beam path and the sample hatch closed. With the main unit now sealed, the LASER could be turned on using the switch at the back of the unit, and arming the PSU unit. By selecting Scan on the VI, the goniometer moved through its angular range (450 steps), taking intensity readings and saving the data to a CSV file on the computer. This process was repeated three times for each sample, with the cuvette being cleaned with isopropanol and deionised water during each change. Before conducting tests with the samples, the apparatus had to be calibrated. The VI did not make the conversion from steps of the stepper motor, to degrees deflection of the goniometer. A constant was required to make this conversion, so it was necessary to find how many steps the motor moved through in a given change of angle. A diffraction grating of 13,500 lines per inch (1.85µm line spacing) was placed into the beam path and the program to record intensities was run, as shown in figure 3. This gave the classic diffraction grating pattern on the program, with zeroth, first and second peaks visible. The step difference between the two first order peaks was measured as 82 steps. Figure 3 - Diffraction grating pattern displayed on miescatter.vi 4

5 The diffraction grating equation, which is a modification of Bragg s Law, was used to calculate the angular difference between the first order peaks [7]. nλ = d sin( θ) where n is the peak order, λ is the incident light wavelength, d is the gratings line spacing and θ is the angle of diffraction of the nth order peak. Using a line spacing of 1.85µm and an incident wavelength of 633nm, the angular difference between the 1 st order peaks was calculated as 40 o (2 s.f.). Dividing 40 o by 82 steps gave a conversion constant of 0.48 degrees per step. This however posed an issue. If the stepper motor moves through 450 steps at 0.48 degrees per step, then this gives a total angular range of 216 o. This was clearly incorrect as the casing restricted the goniometer to an angular range of about ±45 o (refer to figure 2). An internal schematic was obtained from the technician s office at IMAPS (Aberystwyth University) and upon inspection, a steps-to-degrees constant was found: 0.18 degrees per step. This gave an angular range value of ±40.5 o, which made much more sense. Further discussion with the technician who built the main unit, confirmed that the schematics conversion value was correct, hence it was used in the data analysis. III. Data Analysis As discussed, the three samples used in this experiment were: Polymer microspheres (3.9µm diameter), semi-skimmed milk (1.7g/100ml fat), and emulsion paint (Dulux, pale-lilac). To represent the data correctly, the datasets had to be centred so that the main peaks occurred at zero steps. The correction value for each sample was slightly different, possibly due to the cuvette not being completely incident to the beam path. The data sets were centred in Microsoft Excel, split in half (symmetrical about zero degrees), and then imported into SciLab as element vectors. From here, the steps were converted into degrees by multiplying them by the step-to-degrees constant (0.18 degrees per step). The modulus of the negative angles was calculated to flip them into the positive quadrant. The intensity vectors were then summed to smooth out any anomalies and accentuate any secondary peaks in the data that would represent distributions of particle sizes. The logarithm of the intensity values was calculated and plotted against scattering angle using SciLabs plot function. Figures 4 to 6 show the plots of log(intensity) against scattering angle for microspheres, milk and paint respectively. 5

6 Figure 4 - Microspheres (3.90um) summed data The microspheres produced an apparent maximum scattering angle of ±19.5 o (based on when the intensity falls to its lowest level) but there were no evident peaks to suggest a particle distribution. Figure 5 - Semi-skimmed milk (1.7g/100ml fat) summed data The milk data showed a maximum scattering angle of ±12.5 o and once again, there were no secondary peaks to show particle distributions. 6

7 Figure 6 - Emulsion paint (Dulux, pale-lilac) summed data The paint data suggested a maximum scattering angle of ±17.5 o and similar to the other two samples, no peaks or interesting features were observed. The data from each sample was then subjected to a Guinier analysis. The scattering angles were substituted into the Guinier equation (eqn 1) with a constant incident wavelength of 633nm to convert them into scattering wavevector values q (i.e. Q-space). The logarithmic (base 10) values of intensity and Q-space were then plotted in SciLab to produce a Guinier plot (figure 7), from which particles sizes can be obtained and a Porod analysis can be conducted. Porod region Guinier Regime (small angle scattering) Large angle scattering Figure 7 - Guinier Plot showing microspheres (red), milk (blue) and paint (green) 7

8 In figure 7, larger q-values correspond to smaller particles and smaller q-values correspond to larger particles. The Q-space is related to the particle size by the equation r = 2π q (eqn 2) where r is the radius of the particles at that value of q. There are three main areas on the Guinier plot that are of interest. The Guinier regime represents the region of data points that have plateaued at a set value of intensity. It is within this region that the average particle size exists and is found by extrapolating the linear Porod region to the point where it intersects with the Guinier regimes constant intensity. In this case, there is not enough data in the Guinier region to determine where the data plateaus to within a reasonable accuracy. However, using the left-most boundary of the Porod region as an indicator (where q ~ -4.03), a minimum possible average particle radius can be estimated for all three samples. Using the q-value of -4.03, a corresponding particle radius of 67.33µm was calculated. This is the approximate smallest possible average particle radius, and was much larger than expected, considering in the case of the microspheres, the particle radius was supposed to be 1.95µm. Usually with the Guinier plot, minima exist that are homologous to distributions of particles with same spherical radius. However figure 7 lacks these all important minima, and so a description of the particle distribution cannot be formulated. A possible range of particle radii can be stated, but this is can only be true assuming the particle radii range matches the range of data points. The possible range of particle radii in all the sample is ± µm ± 0.005µm, but with the data collected, this cannot be confirmed (full error analysis is detailed in appendix B). To understand this result better, a Porod analysis was conducted on all of the data sets. The Porod region corresponds to the linear area of a Guinier plot that obeys Porod s Law such that I(q) q α where I(q) is the intensity as a function of Q-space and α is the gradient of Porod region [8]. The gradient of the Porod region can be used to ascertain the nature of the interface between the particles and the solution they are suspended in. For α-values close to -4, this implies a smooth sharp interface. For values close to -2.3, the interface is rough or has a fractal nature. To find the Porod gradients for the three samples, a linear regression was performed on the Porod regions of all three Guinier plots. To reduce workload, the regression was performed in SciLab using the built in regress function. The α-values for all three samples is detailed in table 1. The error on the slope was calculated to be ± for all three samples. The Porod slope values for all three samples indicate a rough, fractal interface between the particles and the deionised water. Table 1: Porod slope gradients for microspheres, semi-skimmed milk and emulsion paint Sample α-values Error on the slope Microspheres ± Semi-skimmed milk ± Emulsion paint ± Average ±

9 IV. Discussion At the conclusion of the experiment, a detailed description of the samples particle radii could not be ascertained. Using the Guinier regime in figure 7, an estimate of the average particle radii was calculated to be 67.33µm, however the error attached to this value is as large as the range of the Guinier regime as there was not enough data to perform an adequate extrapolation. Despite this error, the Guinier regime itself occurs over a range of q-values that correspond to particle radii much larger than expected for the samples being tested. In the case of the polymer microspheres, which essentially acted as a control sample, the average particle radii were much larger than the manufacturers value of 1.95µm (3.9µm diameter). It was suggested that the larger than expected radii could be the result of the particles clumping together to form larger conglomerates. To support this claim, it was decided that a Porod analysis should be conducted to ascertain the nature of the particle interfaces. If the particles had clumped together, then the result of the Porod analysis should indicate a rough fractal interface for all of the samples. Figure 8 illustrates this reasoning using microspheres as an example. Microspheres sample (homogeneous) with Porod gradient of µm Interface between particles and deionised water is smooth/sharp Microspheres sample (heterogeneous) with Porod gradient of >-4 n x 3.9µm Particles have clumped together to form a rough fractal interface Figure 8 - A graphical representation of the significance of the Porod slope gradient Table 1 from the data analysis section (III) states that the average Porod gradient for all three samples was ± This implies a rough fractal interface that is congruent to clumping of the particles in the three samples. This result fits well with the position of the Guinier regime in figure 7. Both pieces of evidence imply that the samples were heterogeneous and had clumped together to form larger particles. The microspheres sample was provided with the experimental apparatus and was already slightly dilute in some solution that was never identified. This diluting solution may have affected the composition or quality of the sample and caused clumping to occur, although there was no available way to test this. Another factor that may have contributed was the age of the sample. Both the microspheres and paint were over 5 years old, and this may have affected their ability to remain uniform throughout the sample. As for the semi-skimmed milk, it stated on the label that is was homogenized, but this homogeneity may have been disrupted by the addition of deionized water. The fat in the milk may have exhibited hydrophobic qualities that caused it to clump together, thus breaking the uniformity of the sample. 9

10 Due to the lack of data in the Guinier regime (figure 7), the average particle radius could not be accurately determined. A smaller angular increment may have given more data points on which to perform a Guinier analysis on. Furthermore, a better quality detector would have allowed for a larger sample time per step without saturating the detector. This may have allowed the formation of minima on the Guinier plot, allowing distributions of particle sizes to be ascertained. During the research stage of the investigation, it was common theme among other Mie scattering experiments to polarize the incident LASER beam by using a Brewster window hence reducing the amount of light that is absorbed by the sample. Had a similar method been employed, the intensities measured by the detector may have more prominent. The cuvettes that were used for the experiment may also have had an effect. They were 10mm in width, which is extremely large in comparison to the particles. In that distance, a multitude of secondary scatterings or reflection could have taken place, meaning that only a small percentage of light actually reaches the detector. Once again, other experiments that were found during research often use disk shaped cuvettes with thicknesses of only a couple of millimetres. This reduces the amount of multiple scattering considerably, and the results of these experiments often heralded far more conclusive results. V. Conclusions An experiment to determine particle size distributions in homogeneous colloids, using Mie scattering techniques has been conducted. Three samples were tested: polymer microspheres (3.9µm diameter), semi-skimmed milk (homogeneous, pasteurized) and emulsion paint (Dulux, pale-lilac). Whilst unsuccessful to provide accurate particle radii values, a Guinier and Porod analysis was able to explain why. Clumping of the particles took place in all three samples, which ultimately returned larger particle radii than expected during the data analysis. This was supported by an average Porod gradient of ± for all samples. It was suggested that the age of the microspheres and paint, and the hydrophobic nature of milks fat, caused this clumping to occur and result in more heterogeneous samples. Possible improvements to the experiment apparatus were discussed including: polarizing the incident light; using a disk shaped cuvette with a smaller thickness; and upgrading the detector to allow for longer sample times per step. Acknowledgements The author would first like to extend their gratitude and acknowledge the efforts of their collaborators: Miss. R. E. Cooper, Miss. J. K. Maddocks, and Miss. C. J. Barratt. In addition, the author would like to thank Dr. R. Winter (for advice on the data analysis), Dr. D. Langstaff (for performing alterations on the miescatter.vi ), Mr. D. Lewis (technician; for constructing the apparatus and providing an internal schematic), Mr. M. Evans (postgraduate; for providing deionised water from the material lab), and finally, Mr. S. Fearn (for general lab assistance). 10

11 References 1. Formal definition of light scattering - D. W. Hahn (2009), Light Scattering Theory, Department of Mechanical and Aerospace Engineering; University of Florida (dwhahn@ufl.edu) 2. Development of the theory of light scattering 3. Rayleigh Scattering M. Kerker (1969), The Scattering of Light and other Electromagnetic Radiation, Academic Press, Inc. (London), pp ISBN: L. P. Bayvel and A. R. Jones (1981), Electromagnetic Scattering and its Applications, Applied Science Publishers Ltd., pp ISBN: X (March 3, 2013) 4. Mie theory boundary conditions A. J. Cox, A. J. DeWeerd & J. Linden (2002), An experiment to measure Mie and Rayleigh total scattering cross sections, Department of Physics, University of Redlands, Redlands, California 92373, (March 4, 2013) 5. Mie solution of a uniform, uncharged sphere of arbitrary radius and refractive index I. Weiner, M. Rust & T. D. Donnelly (2000), Particle size determination: An undergraduate lab in Mie scattering, Harvey Mudd College, Department of Physics, Claremont, California 91711, (March 4, 2013) 6. Guinier analysis, scattering vector equation (March 4, 2013) 7. Diffraction Grating Equation (March 9, 2013) 8. Porod analysis using linear regression E. Gilbert (2005), Introduction to Small-angle Scattering, SANS Instrument Replacement Research Reactor, Ansto, (March 9, 2013) Figure

12 A Internal Schematic of Mie Scattering Apparatus 12

13 B Error Analysis The possible error in the stepper motor is its resolution. Since it moved in discrete steps, the error was chosen to be ±1 step. The conversion from steps to degrees is calculated by multiplying the step number by Since 0.18 is a scalar, the error in the stepper motor can also be multiplied by This gives an angular error of ±0.18 o. The trigonometric functions in SciLab give answers in the form of radians. To convert degrees to radians, the degrees are multiplied by the constant (360/2π). Again, this is a scalar so the error in degrees can be multiplied by this scalar to find the error in radians σ rad, which is ±1.75x10-3 radians. The error in q, σ q is found using σ q = 16π 2 2 σ rad cos 2 (θ) λ 2 where λ is the incident wavelength of the Helium-Neon LASER, and θ is the scattering angle in radians. The error in the particle radius was dependent on the value of q as per equation 2. The error in the particle radius σ r is found using σ r = 4π 2 σ q q 4 Error analysis was performed using methods from Dr. Balázs Pinter s lecture slides on Error Analysis (2012). The author would like to extend thanks to Dr. Pinter for making his lecture slides available. 13