## 1 Introduction
In the inversion we estimate \(x(d_{i})\), \(c_{0}\), and \(\beta\) by minimizing the difference between the measured speed of sound and attenuation and their estimates based on \(x(d_{i})\), \(c_{0}\), and \(\beta\). Note that by estimating \(c_{0}\) and \(\beta\), we have incorporated the changes in the speed of sound and attenuation of the solvent during the reaction. A change in temperature is incorporated by selecting a different set of \(L_{re}\) and \(L_{im}\). However, changes in the material properties of the solvent caused by something other than temperature, e.g., one of the substances in the solvent is reacting, are not taken into account in \(L_{re}\) and \(L_{im}\). Although we estimate the volume concentration of the particles per diameter bin, we deliberately do not output a particle size distribution. This is done because the accuracy and frequency bandwidth of the measured speed of sound and attenuation is more than enough to determine the mean diameter, but not enough to also estimate an accurate width of the particle size distribution. The blue circles in Figure 2 show the measured speed of sound and attenuation values for silica particles dispersed in water with a mean diameter around 474 nm and a mass concentration of 5 w %. The blue circles in Figure 3 also show the measured speed and attenuation, but now for silica particles in water with a mean diameter around 794 nm and a mass concentration of 5 w %. Note that the speed of sound and attenuation values in the two figures are frequency dependent, and that they are different in the two figures. E.g., the apex of the speed of sound (i.e., the frequency where the highest speed of sound is measured) is at a different frequency in the two figures. This clearly indicates that the speed of sound as function of frequency is depending on the mean diameter of the particles. Also note that the changes are subtle, for a 5 w % concentration the measured difference in Figure 2 between the highest and the lowest speed of sound values is only 0.7 m/s on a total speed of sound velocity of 1502 m/s. The inversion algorithm has simulated these changes in speed and attenuation (red line in Figure 2) by estimating the mean particle size and concentration of the particles that caused them. Figure 2: (**a**) Measured and estimated fit of the speed of sound and (**b**) attenuation for a dispersion containing standardized silica particles with a mean diameter around 484 nm and a weight concentration of 5 w %. By using the left and right transducers in Figure 1b both as sources and receivers [18], the UNPS can derive the speed of sound and attenuation with great accuracy, allowing it to relate the subtle changes in the speed of sound and attenuation to particle growth. In this paper we investigate whether this would qualify the UNPS as a real-time monitoring instrument for particle growth. The qualification of the UNPS has three requirements:  
1. The UNPS should be able to accurately measure particle diameter. 2. The UNPS should be able to accurately measure particle concentration. 3. The UNPS should be fast enough to follow the reaction. The qualifications 'accurately' and 'fast enough' are related to our ability to monitor the particle growth in the liquid phase bottom-up synthesis. An 'accurately measured diameter' and an 'accurately measured concentration' have values that are close enough to the true diameter for our purposes. Also, the variations in measuring the same sample multiple times should not vary too much to ensure that we see a smooth growth during monitoring. One measurement per minute is 'fast enough' to monitor the reaction. It gives us confidence that we can follow the growth, and are able to make a good decision when to act at the right moment. We test the three requirements with two experiments. Firstly, we perform static measurements on commercially available silica particles to check the accuracy of the UNPS. The particle size measurements are verified with TEM measurements.