{"id":32,"date":"2010-05-28T09:13:37","date_gmt":"2010-05-28T13:13:37","guid":{"rendered":"http:\/\/blogs.vassar.edu\/vaol\/?page_id=32"},"modified":"2013-07-15T10:04:01","modified_gmt":"2013-07-15T14:04:01","slug":"scattering","status":"publish","type":"page","link":"https:\/\/pages.vassar.edu\/vaol\/past-research-projects\/summer-2010\/scattering\/","title":{"rendered":"Scattering"},"content":{"rendered":"

This page written by Rebecca Eells ’12<\/em><\/p>\n

So far this project consists of observing the scattering of laser light by poly latex microspheres. \u00a0Eventually this work will build up to being able to predict the size and concentration of small particles. \u00a0This can be applied to dynamic systems such that the scattering from\u00a0C. elegan <\/em>poop and eggs can be measured and used to determine the size and concentration of the excrements.<\/p>\n

\"Picture<\/p>\n

5\/28\/10<\/strong><\/p>\n

Background research was conducted on the different types of scattering. \u00a0Rayleigh scattering is observed for tiny particles, less than one-tenth the wavelength of the light.<\/p>\n

\"Rayleigh
Rayleigh Scattering<\/figcaption><\/figure>\n

The microspheres are too large in comparison to the wavelength for Rayleigh scattering to be observed. Mie scattering is for larger particles. \u00a0A pattern should be produced in which there is more forward scattering than side scattering. \u00a0Mie scattering is not wavelength dependent.<\/p>\n

\"Mie
Mie Scattering<\/figcaption><\/figure>\n

My hypothesis is that Mie scattering will be observed for the microspheres. \u00a0The hypothesis can be tested by measuring the forward and 90 degree side scattering of the microspheres. \u00a0More forward scattering should be observed than side scattering. \u00a0Lasers of different wavelengths can be used to determine wavelength independence. \u00a0Statistically similar results should be obtained regardless of the wavelength of laser used.<\/p>\n

\"Set-Up
Set-Up Diagram<\/figcaption><\/figure>\n

The distance between the laser and the sample is 0.155m for the current set-up.<\/p>\n

Three samples were made along with a control cuvette. \u00a0The control cuvette consisted of distilled water. \u00a0Each sample contained the same volume of distilled water as the control cuvette. \u00a0The first sample contained 1 drop 0.11\u03bcm microspheres. \u00a0The second sample contained two drops and the third sample contained three drops.<\/p>\n

\"Preparing
Preparing the control cuvette of distilled water<\/figcaption><\/figure>\n

Intensity readings were taking measuring the forward scattering and the 90\u00b0 side scattering. \u00a0At this point there appears to be more forward scattering (as expected for Mie scattering) than side scattering.<\/p>\n

Another trial will be conducted using the o.11\u03bcm microspheres and the red laser to see if the results are the same as with the first trial. \u00a0If this is the case, laser of different wavelengths can be used to test for wavelength dependence.<\/p>\n

6\/1\/10<\/strong><\/p>\n

Mie scattering is very difficult to model computationally. \u00a0The Rayleigh-Debye theory, which is closely related to the Mie theory, provides an accurate model for light scattering of larger particles and is easier to compute.<\/p>\n

\"Rayleigh-Debye
Rayleigh-Debye Theory<\/figcaption><\/figure>\n

IRD=the intensity predicted by the Rayleigh-Debye theory<\/sub><\/p>\n

IR=the intensity predicted by the Rayleigh theory<\/sub><\/p>\n

P(\u03b8)=a dimensionless form factor<\/p>\n

if the spheres are very small then\u00a0P(\u03b8) goes to one and the equation becomes<\/p>\n

\"Rayleigh-Debye
Rayleigh-Debye theory for small spheres<\/figcaption><\/figure>\n

6\/2\/10<\/strong><\/p>\n

The Rayleigh theory of light intensity for scattering by a single particle is:<\/p>\n

\"Picture<\/p>\n

R=the distance to the sample<\/p>\n

\u03b8=the scattering angle (either 90 or 180 degrees)<\/p>\n

n=refractive index<\/p>\n

d=diameter of the particle<\/p>\n

The cross section of one scattering particle is:<\/p>\n

\"Picture<\/p>\n

6\/3\/10<\/strong><\/p>\n

An equation is still needed for Rayleigh-Debye scattering for multiple particles.<\/p>\n

6\/4\/10<\/strong><\/p>\n

An equation was found that accurately predicted the scattering intensity of the 0.11\u03bcm spheres. \u00a0The equation is:<\/p>\n

\"Picture<\/p>\n

\"Picture<\/p>\n

The Rayleigh coefficient is equal to the cross section of the particle (nm2<\/sup>) multiplied by the particle density of the cuvette (particles\/nm3<\/sup>).\u00a0 The Rayleigh coefficient is in units of nm-1<\/sup> and accounts for the likelihood of scattering given the particle size and the particle density [2].\u00a0 The likelihood of scattering also depends on the distance the light must travel through the sample, or the length of the cuvette in nm.\u00a0 Thus, the Rayleigh coefficient must be multiplied by the length of the cuvette in nm.\u00a0 This results in a unitless value [1].\u00a0 But as the light passes through the sample, refraction may take place due to the particles (which is taken into account by the cross section formula) and the background medium (distilled water).\u00a0 So the refraction of water (unitless) must be included in the formula [1].\u00a0 The 1+cos2<\/sup>\u03b8<\/em> term accounts for the phase shift between the incident light and the scattered light based on the angle between the incident beam and the observed scattered light.\u00a0 Finally the incident intensity is the light that initially entered the cuvette, thus is the maximum intensity of light that could possibly be scattered.<\/p>\n

This equation does not take into account the form function, thus can only be used for particles that are small enough to be included in the Rayleigh parameter.<\/p>\n

References:<\/em><\/p>\n

[1] \u201cECE 532,3. Optical Properties.\u201d Laser Photomedicine and Biomedical Optics at the Oregon Medical Laser Center. 1998.<\/em><\/p>\n

[2] \u201cRayleigh Scattering-OPT Telescopes.\u201d Meade Telescopes, Celestron Telescopes and Telescope Accessories-OPT telescopes. 2008.<\/em><\/p>\n

However when data was taken using 0.5\u03bcm microspheres the equation did not yield a theoretical scattering intensity that accurately predicted the observed scattering intensity. \u00a0This may be due to an increased particle size such that another term needs to be included. \u00a0At first the form factor seemed to make sense, since it only goes to unity for small particles, however the addition of the form factor to the equation still did not yield an accurate prediction.<\/p>\n

6\/7\/10<\/strong><\/p>\n

For the 0.11\u03bcm microspheres and the equation above the predicted results are:<\/p>\n

Now using the 0.5\u03bcm and the keeping the equation the same the predicted results are:<\/p>\n

\"Table<\/p>\n

\"Table<\/p>\n

Now taking into account the form factor P(\u03b8) due to larger sphere size, the equation becomes<\/p>\n

\"intensity&formfactor\"<\/p>\n

\"\"<\/a><\/p>\n

Using the new equation including the form factor and the 0.5\u03bcm microspheres the predicted intensity becomes<\/p>\n

\"TAble<\/p>\n

While these values are a much closer prediction of the actual intensity, they are not close enough to be an accurate model. \u00a0It seems another variable must be included in the equation to account for the spheres becoming larger. \u00a0Another option is the equation is wrong entirely.<\/p>\n

6\/8\/10<\/strong><\/p>\n

Another set of data is going to be taken for both the 0.11\u03bcm and the 0.5\u03bcm. \u00a0The volume of each of the “drops” is going to be lowered because the detector appears to become saturated at the 96\u03bcl volume. \u00a0Also another data set will be taken using the green laser instead of the red laser to see how the wavelength affects the intensity values and the calculation of the predicted scattering intensity.<\/p>\n

6\/14\/10<\/strong><\/p>\n

Data was taken for 0.11\u03bcm, 0.5\u03bcm, and 1.09\u03bcm. \u00a0Cuvettes with volumes starting at 20\u03bcl and going to 46\u03bcl were made in 2\u03bcl increments. \u00a0This data was graphed in Origin to see the relationship between concentration and the intensity of the 90 degree side scattering.<\/p>\n

\"0.11<\/p>\n

A linear fit was used to fit the data points.\u00a0 The R-squared value shows that the linear fit was very good, suggesting a linear relationship between intensity and concentration for the 0.11\u03bcm microspheres.<\/p>\n

\"0.5<\/p>\n

For the 0.5\u03bcm spheres a linear fit was first used.\u00a0 The data points didn’t appear to be linear, but there was a linear relationship for the 0.11\u03bcm spheres.\u00a0 The R-squared value suggested that the fit was indeed not linear.<\/p>\n

\"0.5<\/p>\n

Instead of using a linear fit, the same data points were used with a polynomial fit.\u00a0 The R-squared value is much closer to one, suggesting the polynomial was a good fit.\u00a0 The non-linear relationship suggests the increased particle size has led to a more complicated relationship between intensity and concentration.\u00a0 The larger the particle size, the further the scattering becomes from Rayleigh scattering (linear relationship) and the closer is gets to Mie scattering (more complicated relationship).\u00a0 Mie scattering takes into consideration interference in addition to the scattering from the particles.<\/p>\n

\"1.09<\/p>\n

When the 1.09\u03bcm sphere data was plotted I expected to see a non-linear relationship between the intensity and the concentration because the particles were even larger and interference should be present. \u00a0However, when I did a linear fit to the data in Origin, the R-squared value was very close to one, suggesting the data was indeed linear.<\/p>\n

The 1.09\u03bcm sphere data may not actually be reliable however. \u00a0When preparing my samples, I noticed large white flakes in the solution. \u00a0The solution may have precipitated out since the larger sized particles are probably harder to keep suspended.<\/p>\n

6\/15\/10<\/strong><\/p>\n

Using the same size spheres and the same concentrations as on 6\/14\/10, another set of intensity data was collected.\u00a0 The data was again graphed using Origin.\u00a0 This time both a linear and a polynomial fit was applied to all the data.<\/p>\n

\"Linear
Linear Fit of the 0.11 microsphere data<\/figcaption><\/figure>\n
\"Polynomial
Polynomial Fit for 0.11 microsphere data<\/figcaption><\/figure>\n
\"Linear
Linear Fit for 0.5 microsphere data<\/figcaption><\/figure>\n
\"Polynomial
Polynomial Fit for 0.5 microsphere data<\/figcaption><\/figure>\n
\"Linear
Linear Fit for 1.09 microsphere data<\/figcaption><\/figure>\n
\"Polynomial
Polynomial Fit for 1.09 microsphere data<\/figcaption><\/figure>\n

For both the 0.11\u03bcm and the 0.5\u03bcm spheres the polynomial fit resulted in a better R-squared value (it was closer to one).\u00a0 However for the 0.11\u03bcm spheres, the linear fit did result in an R-squared value close to one, while the 0.5\u03bcm spheres the linear fit did not.\u00a0 This suggests the 0.11\u03bcm spheres have an almost linear relationship between intensity and concentration.\u00a0 It also suggests that the 0.5\u03bcm spheres have a non-linear relationship between intensity and concentration.\u00a0 Only the 1.09\u03bcm sphere data resulted in a better linear fit than polynomial fit.\u00a0 This suggests that the 1.09\u03bcm spheres have a linear relationship between intensity and concentration.<\/p>\n

Previously the equation for predicted intensity (above) was used to accurately predict the intensities for the 0.11\u03bcm spheres.\u00a0 Using this equation again and the new data for the 0.11\u03bcm spheres, the intensities were predicted.\u00a0 The table below shows the volume of particles as well as the predicted and actual intensities.<\/p>\n

\"Intensity<\/p>\n

The equation only accurately predicts the first two volumes.\u00a0 This is strange because before it was used to accurately predict 32\u03bcl, 64\u03bcl, and 96\u03bcl.\u00a0 Now it does not accurately predict 32\u03bcl.\u00a0 The intensities are much higher for this newer set of data.\u00a0 Before 32\u03bcl resulted in approximately 0.040V of intensity.\u00a0 Now 26\u03bcl results in approximately 0.040V of intensity.\u00a0 I’m still not sure what would cause this increase in intensity when the volumes, concentrations, and particle size are not changed.<\/p>\n

6\/17\/10<\/strong><\/p>\n

One reason the intensities may have been much higher than expected is that there is a slight amount of error associated with adding the microspheres to the cuvette.\u00a0 Either more\/less microspheres may have been added to the cuvette each time resulting in higher\/lower particle density, which affects the intensity of the side scattering.\u00a0 The data is consistent within itself, based on the R-squared values of both the linear and polynomial fit, meaning the error was either too many particles or too few particles every time.\u00a0 Since the intensity value was always higher than expected, more particles must have been added to the cuvette each time 2\u03bcl were added, resulting in a higher particle density and more side scattering.<\/p>\n

Another data set was taken using the 0.11\u03bcm spheres and the same volumes.\u00a0 However, this time a new cuvette was made for each volume instead of adding 2\u03bcl to an original cuvette.\u00a0 The Origin graphs for both a linear and polynomial fit are shown below.<\/p>\n

\"Linear
Linear fit for 0.11 microsphere data<\/figcaption><\/figure>\n
\"Polynomial
Polynomial fit for 0.11 microsphere data<\/figcaption><\/figure>\n

In this data, the uncertainty associated with the data points wasn’t always high or always low, there was a mix.\u00a0 This resulted in the data points being harder to fit, thus the R-squared values aren’t as close to one as for the previous data.\u00a0 However, the intensities are closer to what was “expected” given the equation and previous results.\u00a0 Below is a table showing the intensities of this data set as well as the predicted intensities based on the equation.<\/p>\n

\"Picture<\/p>\n

Based on this data, the equation seems to once again accurately predict the intensity for the 0.11\u03bcm spheres.\u00a0 From now on all samples will be made one cuvette at a time, instead of adding volume to an initial cuvette.\u00a0 Hopefully the high and low uncertainty will somewhat balance out this way.<\/p>\n

6\/25\/10<\/strong><\/p>\n

In Mathematica, a program was written to model the predicted scattering intensity from the 0.11\u03bcm spheres as concentration changes. \u00a0Below is a polar plot of the predicted intensities as the scattering angle changes from 0 to 360 degrees. \u00a0Each curve represents a different concentration of the particles for volumes 20-46\u03bcl.<\/p>\n

\"\"<\/a><\/p>\n

Also using Mathematica, a program was written to animate the graph of predicted scattering intensity. \u00a0The animation below shows how the predicted scattering intensity changes as concentration changes for the 0.11\u03bcm spheres.<\/p>\n

\"\"<\/a><\/p>\n

In Mathematica, a program was written to model the predicted scattering intensity from the 0.5\u03bcm spheres as concentration changes. Below is a polar plot of the predicted intensities as the scattering angle changes from 0 to 360 degrees. \u00a0The formula used to predict the intensities of the larger particles includes the equation for the form factor, which also varies as the scattering angle changes. \u00a0Each curve represents a different concentration of the particles for volumes 20-46\u03bcl.<\/p>\n

\"\"<\/a><\/p>\n

The animation below shows how the predicted scattering intensity changes as concentration changes for the 0.5\u03bcm spheres.<\/p>\n

\"\"<\/a><\/p>\n

In Mathematica, a program was written to model the predicted scattering intensity from the 1.09\u03bcm spheres as concentration changes. Below is a polar plot of the predicted intensities as the scattering angle changes from 0 to 360 degrees. \u00a0The formula used to predict the intensities of the larger particles includes the equation for the form factor, which also varies as the scattering angle changes. \u00a0Each curve represents a different concentration of the particles for volumes 20-46\u03bcl.<\/p>\n

\"\"<\/a><\/p>\n

The animation below shows how the predicted scattering intensity changes as concentration changes for the 1.09\u03bcm spheres.<\/p>\n

\"\"<\/a><\/p>\n

7\/7\/10<\/strong><\/p>\n

An equation was found that accurately predicts the scattering from the 1.09 microspheres. \u00a0The equation is:<\/p>\n

\"\"<\/a><\/p>\n

This equation is the same as the previous Rayleigh equation used for the 0.11 microspheres, except the form factor has been added as well as a constant of one-eighth. \u00a0All variables are as previously defined. \u00a0Below is \u00a0chart showing the volumes, as well as the actual and predicted intensities for the 1.09 microspheres.<\/p>\n

\"\"<\/a><\/p>\n

7\/13\/10<\/strong><\/p>\n

As soon as we return from Delaware State University I’m going to take data using the green HeNe laser (wavelength is approximately 543 nm). \u00a0Maybe the 0.5 microsphere intensities will be better predicted at this wavelength. \u00a0However, the green laser wavelength is even closer to the size of the 0.5 microspheres than the red laser. \u00a0Perhaps we will just notice more of the abnormalities in the predictions associated with the 0.5 sized microspheres. \u00a0This would still be good, because it’d help add evidence that the abnormalities are being caused by the particle size being approximately the same size as the wavelength.<\/p>\n

7\/21\/10<\/strong><\/p>\n

Instead of using a green laser to take 0.5 sphere data, I used a yellow laser of wavelength equal to 594 nm. \u00a0The initial intensity data set was predicted by the Rayleigh-Debye equation I have been using. \u00a0I’m going to take another data set to make sure that the data is consistent.<\/p>\n

7\/22\/10<\/strong><\/p>\n

I took another set of data and it appears consistent with the data from the previous day. \u00a0Below is a chart of actual intensity and predicted intensity for the 0.5 spheres using the yellow laser.<\/p>\n

\"\"<\/a><\/p>\n

Using this data a graph was made showing the actual and predicted intensities as well as the error bars and a linear fit. \u00a0Both the actual and the predicted intensities have a linear fit, however the fit is better for the actual intensities. \u00a0The data set error bars overlap which seems to indicate that the scattering intensity can be predicted using my equation.<\/p>\n

\"\"<\/a><\/p>\n

I’m also going to redo the graphs of the 0.11 and 1.09 sphere data to include both actual and predicted intensities as well as error bars. \u00a0I will also show the charts used to create these graphs.<\/p>\n

\"\"<\/a>
0.11 microsphere data chart<\/figcaption><\/figure>\n

\"\"<\/a><\/p>\n

\"\"<\/a><\/p>\n

\"\"<\/a><\/p>\n

For both the 0.11 and 1.09 spheres, the uncertainty associated with the actual intensities was a best guess. \u00a0Since this data was previously taken there was no way to know the actual uncertainty associated with each data point. \u00a0However, the detector very rarely gave readings that fluctuated more than 0.002 Volts in either direction.<\/p>\n

Yet again the error bars appear to overlap suggesting that for the 0.11 and 1.09 sized spheres the scattering intensity can also be predicted.<\/p>\n

\"Picture<\/strong><\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

This page written by Rebecca Eells ’12 So far this project consists of observing the scattering of laser light by poly latex microspheres. \u00a0Eventually this work will build up to being able to predict the size and concentration of small particles. \u00a0This can be applied to dynamic systems such that the scattering from\u00a0C. elegan poop … Continue reading “Scattering”<\/span><\/a><\/p>\n","protected":false},"author":264,"featured_media":0,"parent":805,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-32","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/pages\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/users\/264"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/comments?post=32"}],"version-history":[{"count":130,"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/pages\/32\/revisions"}],"predecessor-version":[{"id":40,"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/pages\/32\/revisions\/40"}],"up":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/pages\/805"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/vaol\/wp-json\/wp\/v2\/media?parent=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}