Supporting Information for: Measuring the Relative Concentration of Particle Populations using Differential Centrifugal Sedimentation

Particle Populations using Differential Centrifugal Sedimentation Alexander G. Shard1, Katia Sparnacci,2 Aneta Sikora,1 Louise Wright,1 Dorota Bartczak,3 Heidi Goenaga-Infante3 and Caterina Minelli1* 1 National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK 2 Università del Piemonte Orientale ‘‘A. Avogadro’’, Viale T. Michel 11, I-15121 Alessandria, Italy 3LGC Limited, Queens Road, Teddington, Middlesex, TW11 0LY, UK

Where R is the distance from the centre of rotation, with fluid density and viscosity at that radius given by  R and  R respectively.  is the disk angular velocity, D is the spherical particle diameter and  P is the particle density.
Electronic Supplementary Material (ESI) for Analytical Methods. This journal is © The Royal Society of Chemistry 2018 Integration provides the normal sedimentation equation S.2, where the subscript 'a' represents an average value across the gradient, which is generally assumed to be the mean of the values at R 0 and R f . The detector covers an area approximated as a rectangle with width W and length R. The path length of light through the medium is L and the detection volume is the product of the three parameters. Measured light intensity, I, is recorded as a function of time and converted into absorbance, A = ln(I 0 /I) where I 0 is a reference intensity recorded when no particles are in the detection area.
Ignoring finer details such as: dispersity in the particle population, wall effects and particle diffusion, the time taken for particles in the population to cross the detection area, t, is given to a good approximation by equation S.3. Here, the subscript 'f' indicates the values at the average detector position.
With the same assumptions, the absorbance, A, will be constant during this time and is related to the individual particle extinction cross section, σ, the number concentration of particles in the analysis volume, N/(LWR), and the path length L by: A = σN/WR. Thus the integrated absorbance with time is given by equation S.4. The square brackets enclose terms that relate to the instrument, . For homogeneous, rigid, spherical particles the terms in curly brackets can be replaced with the reciprocal of the extinction efficiency, Q -1 and Q is obtained directly from Mie theory. Assuming, once again, that the physical diameter of the particle is the same as the sedimentation diameter. Note that error in R f and R 0 contribute significantly to uncertainty in the detected mass.
Here, we have neglected consideration of the finite area of the detector which will collect a fraction of light scattered at low angles. The manufacturer's software accounts for this and uses an adjusted value for Q, Q net , which will be smaller than Q calculated directly from Mie theory [CPS instruments application note 'conversion of CPS data into a weight distribution']. The value of Q net can be extracted from the instrument and compared to Q Mie calculated directly from Mie theory without accounting for forward scattered light hitting the detector. The ratio of these values for gold, silica and polystyrene are shown in Figure S.2. For particles less than 100 nm in diameter the two numbers are nearly identical and remain within 5% of each other until the particles are significantly larger than 200 nm diameter. Ratio of Q net calculated by the instrument software to Q Mie calculated using the same physical parameters. The difference between these values relates to the amount of forward scattered light hitting the detector, which is explicitly evaluated for Q net .
The conversion can be verified by calculating the ratio of absorbance to mass per diameter step directly from the data and showing that this is proportional to Q. See Figure S.3. The value of  calculated using typical values: R f = 45 mm, R 0 = 35 mm is approximately 4×10 9 m 2 . Silica has a density of approximately  P = 2×10 -6 g/m 3 . The slope on the graph should be approximately equal to 1/(  P ), which is ~1.2×10 -4 and within a factor of two of the slope found here. The reason for the difference between these values is not clear.  Figure S.3 Conversion factor for a DCS instrument. Absorbance data for silica particles in the range 40 nm to 200 nm were background corrected using a simple linear function and divided by the software output of mass per diameter step (g/m). These are plotted against Mie extinction efficiencies at selected diameters. The dashed line is a linear fit.
For a defined particle population with a narrow size distribution, M/D is integrated over D to obtain the total mass in that population injected into the centrifuge. Integrating S.7, assuming either that  is constant for that population or that a suitable average value can be found, provides the relationship shown in S.8. Whilst this result seems obvious, the analysis is necessary to identify the sources of error. The terms in S.8 are M, the total mass of particles in the population measured by integration of the converted absorbance versus time data. The diameter, D, is the physical diameter of the spherical particles and the density,  P , is the physical density. Note that D may not be identical to the measured diameter, D m , which is often in error due to shape effects (see next section) or incorrect densities. The input density of the particle,  X , is a source of uncertainty but, through the relationship to sedimentation time in S.2, is anticorrelated with D m .
The absorption cross section, σ, is the 'true' cross section of the particles and not necessarily the same as the cross section calculated by Mie theory, σ X . As well as errors that arise from possibly incorrect refractive indices, the cross section is calculated from the measured diameter, D m , which depends upon  X as noted above. For small (D < 50 nm) particles σ scales as D 6 for purely scattering materials and D 3 for absorbing particles. Thus, for small absorbing particles any errors in diameter cancel in S.8, leaving only the density term.
Thus, equation S.9 relates the mass of particles in the population to the measured mass, M m . All terms have the same meaning as S.1, except that D is now interpreted as the diameter of a sphere with an equal volume to that of the particle and a dynamic shape factor, , is introduced to express the ratio of the frictional force on the particle to that of a sphere of diameter D. This factor is propagated through equations S.2 to S.4 and in S.5 it (like the angular velocity term) cancels and the form of that equation is unchanged except for the definition of D. Because of the definition of D, S.6 is unchanged and so S.7 is also valid except for the conversion of the diameter and the cross section into Q because, except in limited circumstances, Mie theory is not valid for non-spherical particles. Therefore, the form of S.9 is unchanged and the D 3 terms are better replaced with particle volumes, V, to avoid confusion concerning the definition of D.

S.11
From S.10, the following relationships are also useful: D =  0.5 D m and therefore V =  1.5 V m .

S3: Repeatability and precision
To provide the context for the analysis of error in this paper, it is important to point out that the DCS technique provides excellent precision and repeatability. Figure S.7 overlays two repeat measurements of the same mixed particle system. There is some variability in the peak height and width, which are of the order of 5 to 10%, however the important parameters are the modal peak diameter and the area of each peak in mass. The modal diameter has a variability of <0.5% and, although the area varies between runs by ~6%, the ratio of peak areas, w m , varies by <0.5%. The latter is expected if the change in peak areas are correlated due to e.g. uncertainty in the injection volume. Therefore the measurement of particle size and relative concentration in DCS is not limited by the precision and repeatibility of the method. It is interesting to observe that when a technique with lower size resolution is used, for example DLS in Figure S4B, it is impossible to discriminate whether the distributions that the LSPR wavelengths increase going from Agg1 to Agg4, i.e. increasing sample agglomeration. The LSPR peak broadens and decreases in intensity accordingly, while the adsoption at 450 nm varies to a lower extent. The disparity between these methods may therefore be used to identify agglomeration or other non-sphericity in gold particle populations.
The DCS traces may be fitted with excellent precision using normal distribution functions to represent different particle populations, as shown in Figure S.6. Here, seven populations are used but only the primary, doublet, triplet and tetramer populations can be identified with any certainty. The fit enables us to confirm that, in the mixed samples Agg2 and Agg3, the data can be described as a linear combination of the Agg1 and Agg4 results with relative weightings very close to the volumetric mixing ratios. Therefore no further interactions occur between particle populations upon mixing. Furthermore, the relative mass concentrations of dimer to monomer can be extracted from the fits.  LAM=400e-9;% Wavelength of light in vacuum N(1)=1.6178; % real part of refractive index of first sphere K(1)=1.9493; % imaginary part of refractive index of first sphere N(2)=N(1); % real part of refractive index of second sphere K(2)=K(1); % imaginary part of refractive index of second sphere R(1)=5.0e-9; % Radius of first sphere R(2)=5.0e-9; % Radius of second sphere R12=20.0e-9; % Separation of spheres NODR1=2; % Convergence parameter NODRT1=2; % Convergence parameter NPNA=37; % Number of equispaced angles between 0 and 180 deg for which scattering matrix is calculated. % % Adjust such that LAM is wavelength in the medium % and N is ratio of refractive indices (so that n*XI will come out right)