1. Sixth Annual Harriet L. Wilkes Honors College Symposium for Research and Creative Projects Featuring the Chastain Honors Symposium Lecture Series April 4, 2008 Nanoparticles in the Mirror Are Smaller Than They Appear Daniel Gopman, Dr. Eugene Belogay, and Dr. Korey Sorge* * Dept. of Physics, Florida Atlantic University Nanoparticles are smaller than visible light. If we cannot see them, how can we measure them? Introduction Magnetic nanoparticles have promising applications in medicine. Scientists hope to produce nanopraticles that detect diseases, deliver drugs to specific targets, and help us "look inside" cells. The small magnetic cores of such particles are usually coated with plastic shells, in order to prevent unwanted agglomeration and oxidation. Our nanoparticle is a small (10 nm; 1 nm = 10 -9 m) sphere of magnetite ( Fe 3 O 4 ) coated with a thick (100 nm) shell of polyacryclic acid. Lognormal distribution of sizes is typical. Size matters! Biomedical applications require magnetic nanoparticles of specific size, so it must be measured. Goal Problem Nanoparticles are so small — they are invisible ! Solution Idea Nanoparticles can be sized up indirectly, since the way they respond to an applied magnetic field depends on their magnetic moments, which in turn are related to their sizes: Magnetization ← Magn. moment ~ Volume ~ Diameter 3 Solution Method 1. Apply magnetic field and measure magnetization. 2. Fit magnetization data to magnetization model. 3. Estimate distribution of particle magnetic moments. 4. Deduce distribution of particle diameters. Nanoparticles are smaller than visible light. If we cannot see them, how can we measure them? Introduction Magnetic nanoparticles have promising applications in medicine. Scientists hope to produce nanopraticles that detect diseases, deliver drugs to specific targets, and help us "look inside" cells. The small magnetic cores of such particles are usually coated with plastic shells, in order to prevent unwanted agglomeration and oxidation. Our nanoparticle is a small (10 nm; 1 nm = 10 -9 m) sphere of magnetite ( Fe 3 O 4 ) coated with a thick (100 nm) shell of polyacryclic acid. Lognormal distribution of sizes is typical. Size matters! Biomedical applications require magnetic nanoparticles of specific size, so it must be measured. Goal Problem Nanoparticles are so small — they are invisible ! Solution Idea Nanoparticles can be sized up indirectly, since the way they respond to an applied magnetic field depends on their magnetic moments, which in turn are related to their sizes: Magnetization ← Magn. moment ~ Volume ~ Diameter 3 Solution Method 1. Apply magnetic field and measure magnetization. 2. Fit magnetization data to magnetization model. 3. Estimate distribution of particle magnetic moments. 4. Deduce distribution of particle diameters. Method 1. Apply magnetic field and measure magnetization:  Place nanoparticles in a Superconducting Quantum Interference Device (SQUID),  keeping constant temperature T = 250 K,  applying large magnetic field ( H = 25,000 Oersted),  varying field down to zero, stopping every 200 Oersted.  Measure magnetization ( M ) as a function of field ( H ). 2. Fit magnetization model to measured data: Use least-squares criterion for (non-linear) best-fit: Find µ 0,  : min  (M measured – M model (H applied ) ) 2  assuming lognormal distribution of moments;  approximating the integral for M with a discrete sum;  changing variables: µ to z = ln (µ/µ 0 );  using normal cdf, instead of pdf, in the weighted sum;  (better precision) using µ f (µ) as weight, instead of f (µ).  Interactively find a crude fit in Excel Import and normalize magnetization data into spreadsheet. Create interactive Excel sliders for all parameters to be fitted. Program formulas (Langevin, weighted sum, cdf, residuals). Vary parameters (by hand via sliders) to obtain a crude fit.  Compute best fit in Matlab, starting at the crude fit. 3. Repeat previous step for single-size particles:  Keeping  = 0 fixed, find a crude fit in Excel.  Keeping  = 0 fixed, compute best fit in Matlab. 4. Compute (lognormal) distribution of core diameters:  Assume spherical magnetic core with diameter D.  Using the spherical volume formula, V =  D 3 /6, and the known magnetic density of magnetite, relate core diameter D to magnetic moment µ : Median: D = (6 µ / 250 π) 1/3 = 5.5 nm Std. deviation:  (ln D) =  (ln µ) / 3 = 0.44 Method 1. Apply magnetic field and measure magnetization:  Place nanoparticles in a Superconducting Quantum Interference Device (SQUID),  keeping constant temperature T = 250 K,  applying large magnetic field ( H = 25,000 Oersted),  varying field down to zero, stopping every 200 Oersted.  Measure magnetization ( M ) as a function of field ( H ). 2. Fit magnetization model to measured data: Use least-squares criterion for (non-linear) best-fit: Find µ 0,  : min  (M measured – M model (H applied ) ) 2  assuming lognormal distribution of moments;  approximating the integral for M with a discrete sum;  changing variables: µ to z = ln (µ/µ 0 );  using normal cdf, instead of pdf, in the weighted sum;  (better precision) using µ f (µ) as weight, instead of f (µ).  Interactively find a crude fit in Excel Import and normalize magnetization data into spreadsheet. Create interactive Excel sliders for all parameters to be fitted. Program formulas (Langevin, weighted sum, cdf, residuals). Vary parameters (by hand via sliders) to obtain a crude fit.  Compute best fit in Matlab, starting at the crude fit. 3. Repeat previous step for single-size particles:  Keeping  = 0 fixed, find a crude fit in Excel.  Keeping  = 0 fixed, compute best fit in Matlab. 4. Compute (lognormal) distribution of core diameters:  Assume spherical magnetic core with diameter D.  Using the spherical volume formula, V =  D 3 /6, and the known magnetic density of magnetite, relate core diameter D to magnetic moment µ : Median: D = (6 µ / 250 π) 1/3 = 5.5 nm Std. deviation:  (ln D) =  (ln µ) / 3 = 0.44 Observe metal core diameter distribution via another technique: Transmission Electron Microscope (TEM).  Capture TEM image of nanoparticle metal cores.  Estimate the distribution of metal core diameters (use image analysis software to produce a histogram).  Compare TEM metal core diameter distribution to SQUID magnetic core diameter distribution. Results 1. Not all nanoparticles are created equal Our nanoparticles do not have the same diameters:  The single-size fit is about 5 times worse than the distributed-size lognormal fit.  The single-size residual has a distinct curved shape, indicating that the single-size fit is not good.  The best-fit width (  = 0.44, see below) is far from 0, which means that core diameters are spread over one order of magnitude ( ~ 1 to 20 nm). 2. Part of the core is "dead" (not magnetic) The magnetic core (estimated by SQUID) appears smaller than the metal core (seen in TEM): D magnet < D metal  The SQUID lognormal distribution of diameters (smooth curve) is to the left of the TEM distribution of diameters (histogram).  The lognormal median SQUID diameter (5.5 nm) is about half the median TEM diameter (9.2 nm). Summary The combination of TEM with SQUID magnetometry and mathematical analysis provides valuable insight into the invisible magnetic structure of nanoparticles, which cannot be obtained by either technique alone. The fact that the effective magnetic size of a nanoparticle can be smaller than the size of its metal core is essential to applications that require particles in specific narrow magnetic ranges. Observe metal core diameter distribution via another technique: Transmission Electron Microscope (TEM).  Capture TEM image of nanoparticle metal cores.  Estimate the distribution of metal core diameters (use image analysis software to produce a histogram).  Compare TEM metal core diameter distribution to SQUID magnetic core diameter distribution. Results 1. Not all nanoparticles are created equal Our nanoparticles do not have the same diameters:  The single-size fit is about 5 times worse than the distributed-size lognormal fit.  The single-size residual has a distinct curved shape, indicating that the single-size fit is not good.  The best-fit width (  = 0.44, see below) is far from 0, which means that core diameters are spread over one order of magnitude ( ~ 1 to 20 nm). 2. Part of the core is "dead" (not magnetic) The magnetic core (estimated by SQUID) appears smaller than the metal core (seen in TEM): D magnet < D metal  The SQUID lognormal distribution of diameters (smooth curve) is to the left of the TEM distribution of diameters (histogram).  The lognormal median SQUID diameter (5.5 nm) is about half the median TEM diameter (9.2 nm). Summary The combination of TEM with SQUID magnetometry and mathematical analysis provides valuable insight into the invisible magnetic structure of nanoparticles, which cannot be obtained by either technique alone. The fact that the effective magnetic size of a nanoparticle can be smaller than the size of its metal core is essential to applications that require particles in specific narrow magnetic ranges. Background and Theory Lognormal distribution Distribution parameters: median µ 0 width  Probability density function (pdf) of magnetic moment µ : f (µ) = (1/√(2π)  µ) * exp(– ln (µ/µ 0 ) 2 / 2  2 ) Facts:  ln (µ/µ 0 ) is 0-centered Normal (Gaussian) with standard deviation  (hence the name lognormal ).  µ 1/3 is also lognormal, with median µ 0 1/3 and width  /3.  µ f (µ) is also a lognormal pdf (shifted and scaled). Langevin function (sigmoid shape) L(x) = 1/tanh(x) – 1/x Facts:  L(–x) = – L(x)  L(0) = 0, L' (0) = 1/3  L(  ) = 1 Magnetization model (equal moments) The magnetization M (emu) of N nanoparticles, each of magnetic moment µ (emu), subjected to magnetic field H (Oersted) at constant temperature T (Kelvin) is: M(H) = N µ L(µH / kT), where L is the Langevin function; M(  ) = N µ (large-field saturation)  Note: only the saturation magnetization M(  ) = N µ can be estimated in the lab, but not N and µ separately. Magnetization model (distributed moments) If the magnetic moment of N nanoparticles is distributed with pdf f (µ), then the total ensemble magnetization is: M(H) = N  µ L (µH / kT) f (µ) dµ, M(  ) = N  µ f (µ) dµ = N ( l arge-field saturation)  Note: only the saturation magnetization M(  ) = N can be estimated in the lab, but not the number of particles N and the mean moment separately. Background and Theory Lognormal distribution Distribution parameters: median µ 0 width  Probability density function (pdf) of magnetic moment µ : f (µ) = (1/√(2π)  µ) * exp(– ln (µ/µ 0 ) 2 / 2  2 ) Facts:  ln (µ/µ 0 ) is 0-centered Normal (Gaussian) with standard deviation  (hence the name lognormal ).  µ 1/3 is also lognormal, with median µ 0 1/3 and width  /3.  µ f (µ) is also a lognormal pdf (shifted and scaled). Langevin function (sigmoid shape) L(x) = 1/tanh(x) – 1/x Facts:  L(–x) = – L(x)  L(0) = 0, L' (0) = 1/3  L(  ) = 1 Magnetization model (equal moments) The magnetization M (emu) of N nanoparticles, each of magnetic moment µ (emu), subjected to magnetic field H (Oersted) at constant temperature T (Kelvin) is: M(H) = N µ L(µH / kT), where L is the Langevin function; M(  ) = N µ (large-field saturation)  Note: only the saturation magnetization M(  ) = N µ can be estimated in the lab, but not N and µ separately. Magnetization model (distributed moments) If the magnetic moment of N nanoparticles is distributed with pdf f (µ), then the total ensemble magnetization is: M(H) = N  µ L (µH / kT) f (µ) dµ, M(  ) = N  µ f (µ) dµ = N ( l arge-field saturation)  Note: only the saturation magnetization M(  ) = N can be estimated in the lab, but not the number of particles N and the mean moment separately. 10 nm nanoparticlevisible light We think that there is a "magnetically dead" layer of atoms at the surface of the metal core, possibly due to some surface interaction between the nanoparticles or due to the nature of the chemical synthesis. N S magnetic core (5.5 nm) total metal core (10 nm) Magnetization curves for six different sizes of particles. The total ensemble magnetization M is a pdf-weighted sum (integral) of such curves. Estimate the (distribution of) sizes in a given sample of plastic-coated magnetic nanoparticles.