Nanoparticles in the Mirror Are Smaller Than They Appear
Daniel Gopman, Dr. Eugene Belogay, and Dr. Korey Sorge*
of Physics, Florida Atlantic University
Background and Theory
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 (Fe3O4) 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.
1. Apply magnetic field and measure magnetization:
 Place nanoparticles in a Superconducting Quantum
Interference Device (SQUID),
Lognormal distribution
Distribution parameters:
• median µ0
• width 
Probability density function (pdf) of magnetic moment µ:
f (µ) = (1/√(2π)  µ) * exp(– ln (µ/µ0)2 / 22)
Facts:
 ln (µ/µ0) is 0-centered Normal (Gaussian) with
standard deviation  (hence the name lognormal).
 µ1/3 is also lognormal, with median µ01/3 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
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 µ0,  : min  (Mmeasured – Mmodel(Happlied) )2
assuming lognormal distribution of moments;
 approximating the integral for M with a discrete sum;
 changing variables: µ to z = ln (µ/µ0);
 using normal cdf, instead of pdf, in the weighted sum;
 (better precision) using µ f (µ) as weight, instead of f (µ).
0.75
0.5
 Interactively find a crude fit in Excel
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-10
-5
0
5
10
-0.25
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Goal
-0.75
Estimate the (distribution of) sizes in a given sample
of plastic-coated magnetic nanoparticles.
10 nm
Problem
Nanoparticles are so small — they are invisible!
nanoparticle
visible light
x
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.
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.
400
Magnetization ← Magn. moment ~ Volume ~ Diameter3
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.
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 <µ> (large-field saturation)
 Note: only the saturation magnetization M() = N <µ>
can be estimated in the lab, but not the number of
particles N and the mean moment <µ> separately.
Magnetization curves for six different
sizes of particles. The total ensemble
magnetization M is a pdf-weighted
sum (integral) of such curves.
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.
10%
5%
Single-size (bad)
Lognormal (good)
0%
-5% 0
10
20
-10%
Applied magetic field H (kOe)
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):
Dmagnet < Dmetal

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).
350
300
250
200
150
100
50
Measured data
Single-size fit (bad)
Lognormal fit (good)
0
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:
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.


 Compute best fit in Matlab, starting at the crude fit.
Magnetization M (μemu)
-1
Observe metal core diameter distribution via another
technique: Transmission Electron Microscope (TEM).

1
L(x)
Nanoparticles are smaller than visible light.
If we cannot see them,
how can we measure them?
Method
Residual (% M sat)
*Dept.
0
5
10
Applied magetic field H (kOe)
15
20
25
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.
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.
magnetic core (5.5 nm)
N
S
total metal core (10 nm)
4. Compute (lognormal) distribution of core diameters:
 Assume spherical magnetic core with diameter D.
 Using the spherical volume formula, V =  D3 /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
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.
Sixth Annual Harriet L. Wilkes Honors College Symposium for Research and Creative Projects Featuring the Chastain Honors Symposium Lecture Series
April 4, 2008