## 1 Introduction  
Nanotechnology is an emerging area of scientific research with a wide range of applications in microbiology and biotechnology [1]. Silver nanoparticles (AgNPs) are aggregates of a relatively small number of atoms, ranging from small sizes consisting of only a few, to larger nanoparticles containing more than 100,000 atoms [2]. Clusters are a type of nanoparticle on the scale range of nanometers containing a small number of atoms [3]; their properties differ from those of both molecules and atoms and are very dependent on size [4]. This characteristic is one of the objectives of the present study; thus, two sizes of AgNPs are examined. AgNPs display unique physical and chemical properties, such as adsorption capacity, fast diffusion rate, and changing surface characteristics [5]. Their application in nanobiotechnology, biosensors, cell structure imaging, and targeted drug delivery are well known [1]. Biological resources, including bacteria, plants, algae, fungi, and biomolecules, have been proved to be excellent candidates for AgNPs synthesis [6]. To determine numerous properties, namely size, geometry, shape, crystallinity, and surface area, different techniques, including transmission electron microscopy (TEM), scanning electron microscopy (SEM), and UV-vis spectroscopy, can be applied [4]. A property of interest of AgNPs reported in the literature is the biocide capacity against different types of microorganisms, which will be analyzed thoroughly in the state-of-the-art section. In this research, it is of particular interest the analysis of the growth inhibition capacity of a composite PUR and AgNPs material on the fungus _Trichophyton rubrum_, propriety that could present potential applications on products able to inhibit the proliferation of _Trichophyton_ fungi, responsible for diseases including _tinea pedis_ and _tinea curris_[7]. PUR is considered one of the most versatile polymeric materials due to the wide range of materials with different properties that can be obtained from it, such as rigid and flexible foams, coatings, and thermoplastic materials [8]. These materials are chemically complex and usually obtained via chemical reactions between diisocyanates and polyols, which can be polyester or polyether-type [9]. Derived from the versatility of the properties that can be obtained, there have been developed, for example, fire-resistant PUR nanocomposites [10]. In recent years, due to their excellent chemical stability, biocompatibility, and low cytotoxicity, some PURs have been widely used for biomedical applications [11]. However, the cytotoxicity of its raw materials can limit their use in certain applications, for example, the toxicity of 4,4-methylene diphenyl disocyanate (MDI) in the formulation [12]; few studies have evaluated the effect of the PUR chemical structure on the cytotoxicity. [PERSON] et al., indicate that the isocyanate index highly influences the chemical structure of polyurethane foams, also causing changes in cytotoxicity [13]. This parameter must be carefully considered to ensure biocompatibility [13; 14]. PUR nanocomposites have also been used to remove inks and dyes [15]. The current medical PUR research has been focused on a new approach: metals, such as silver, and copper oxide nanoparticles as a new type of biocide agent [16; 17]. Thus, the proposed methodology for this research focuses on the use of polyester PUR and silver nanoparticles of different sizes to obtain a composite material with antifungal properties. This paper is based on the analysis of three control variables of interest: the size and concentration of AgNPs, and the use of an ultrasound method during the mixing of AgNPs and PUR to evaluate the effect of these factors on the growth inhibition of the fungus _Trichophyton rubrum_ on culture medium in the presence of the composite material. Through a 2\({}^{3}\)-factorial design with five replicates, 40 experimental runs were evaluated in accordance with ASTM-G21-15-4 [18].