## 1 Introduction
The size of the change depends, amongst other factors, on the particle size and concentration of the nanoparticles [9; 10]. This dependency is used in the UNPS measurement method [18]. Figure 1a shows the UNPS probe. The sensing part is located in the gap in the tip of this probe (see blue arrow). In Figure 1b this gap is filled with a dispersion. Figure 1b demonstrates the basics of the UNPS: the left transducer sends an ultrasonic wave through a glass wall, the dispersion, and another glass wall, to the right transducer that records the wave. It is beneficial to also use the right transducer as a source and the left transducer as a receiver. This results in four signals that are recorded: two transmission signals through the dispersion and two pulse-echo signals that have only travelled in the glass. [PERSON] et al. [19] demonstrate that these four signals make it possible to calculate the ultrasonic wave signal in the dispersion without any detailed knowledge of the wave behavior in the glass and transducers. The attenuation in the dispersion, \(\alpha\), as a function of frequency, \(f\), is obtained from the amplitude of the wave signal. The phase of the wave signal is used to obtain the speed of sound in the dispersion, \(c\). Besides the speed of sound and attenuation, we also measure the temperature in the dispersion. [PERSON] et al. [9] and [PERSON] et al. [10] both give detailed, complex equations that relate the speed of sound and attenuation to the particle size and concentration. We use these equations in our inversion. However, instead of these detailed complex equations we will discuss the first order approximation of these equations for low particle concentrations (<10 w %), as they are a lot more insightful. The speed of sound is given by:  
\[c(f)=c_{0}+\sum_{i}\ c_{0}\ L_{re}(f,d_{i})\ x(d_{i}), \tag{1}\]  
where \(c_{0}\) is the frequency independent speed of sound of the solvent, \(L_{re}(f\!d_{i})\) is a frequency, \(f\), and diameter, \(d_{i}\), dependent scalar, and \(x(d_{i})\) is the volume percentage of particles with a diameter that fits in the diameter bin \(i\) around \(d\). The diameter bins work such that the sum \(\sum_{i}\ x(d_{i})\) equals the total volume percentage of particles in the dispersion. Note that \(c_{0}\) describes the speed of sound in the absence of particles and that the second term on the right hand side adds a frequency and diameter dependent increment to the speed of sound. We will see later that this increment is a lot smaller than \(c_{0}\). The attenuation is given by:  
\[\alpha(f)=\beta f^{2}+\sum_{i}\frac{-2\pi f}{c_{0}}\ L_{im}(f,d_{i})\ x(d_{i}), \tag{2}\]  
Figure 1: (**a**) The ultrasound nanoparticle sizer (UNPS) probe with a diameter of 18 mm (image published with permission of Sonaxis); (**b**) sketch of the basics of the method to measure the speed of sound and attenuation of ultrasonic waves in a dispersion. where \(\beta\) is the attenuation coefficient of the acoustic waves in the solvent, and \(L_{im}(f,d_{i})\) is a frequency and diameter dependent scalar. Note that the term \(\beta f^{2}\) describes the attenuation in the solvent in the absence of particles, and that the second term on the right hand side adds a frequency and diameter dependent increment to the attenuation. Besides being dependent on frequency and diameter \(L_{re}\) and \(L_{im}\) depend on a few other parameters that depend on the material properties of the solvent and particles. For the silica water and Stober dispersions in this paper \(L_{re}\) and \(L_{im}\) are mainly depending on the particle density, the speed of sound in the particles, the solvent density, the speed of sound in the solvent and the viscosity of the solvent. The parameters mentioned are all dependent on temperature. Therefore, before the measurements begin \(L_{re}\) and \(L_{im}\) are calculated for a wide temperature range.