## 1 Introduction  
Rare earth elements (REEs) are a group consisting of lanthanides as well as yttrium and scandium. Most part of these elements demonstrate unique physical properties that are used in a wide range of high-tech industries. However, in many cases (e.g., production of luminescent and magnet materials, additives in nuclear fuels, etc. ), implementation of an individual pure REE is required. From a chemical point of view, all of these elements possess very close chemical properties: the same oxidation state (+3), the same character of electronic shells ([Xe]4n5d16s2 for lanthanides) and close values of ionic radii. Separation of rare earth elements is a complex chemical and technological task. One of the preferable ways to separate REEs and to isolate individual elements is with liquid-liquid extraction. This method allows the organization of a continuous separation process and the possibility to deal with high concentration of metals (\(>\)100 g\(\cdot\)L\({}^{-1}\)) [1]. Separation of a REE is complicated by the fact that isolation of a certain individual rare earth element requires its separation from two neighboring elements. For this reason, the processing of a total REE concentrate is a multi-stage procedure where all REEs are divided into groups (e.g., \"light\" and \"heavy\" REEs). Then, components of these groups are separated with isolation of pairs of elements. The last step is a separation of an individual element. The whole separation scheme is determined by the content of certain groups of elements in the initial ore, the media used to dissolve the REE ore [2], the availability of suitable equipment and the technological and economic aspects of the REE production [3]. The highest content in the natural REE mixtures have elements of the \"light\" group La-Nd [2]. Among La-Nd, neodymium has the greatest value as a component of permanent magnets (Nd-Fe-B) [4; 5; 6] and lasers [7; 8]. Praseodymium based materials have a prospective use in catalysis [9; 10; 11]. A group of La-Nd can be separated from a whole mixture of REEs relatively easily due to the absence of promethium in natural REEs. Thus, a gap between Nd and Sm allowstheir easier separation than a pair of neighboring elements. Mutual separation of Ce and Nd can be simplified by oxidation of cerium from +3 to +4 oxidation state [12], so the separation of the neodymium-praseodymium pair is the most difficult part in isolation of pure Nd and pure Pr (for a long time, this pair was considered as one element called didymium [13]). The basis of any extraction system is an extractant, which defines the extraction efficiency (can be estimated as a distribution ratio, \(D\), see Section 2.2.1) and the selectivity (can be estimated as a separation factor, \(SF\), see Section 2.2.1). In the case of metal cations with close chemical properties (close values of \(D\) and small values of \(SF\)), technological realization of their separation requires a multi-stage countercurrent process. The distribution ratios and separation factors affect the main parameters of the separation process--number of stages, volume ratios of organic and aqueous phases, product yields, etc. [14]. One of the acute research topics is the search for new extractants with high selectivity towards a certain REE. The main drawback of this approach is a high cost of new extractants and their unavailability in amounts sufficient for industrial scale applications. On the other hand, there are many classes of commercially available extractants that are already used in industry. Organophosphorus extractants and quaternary ammonium bases (QABs) are some of the most studied extractants in hydrometallurgy [15]. In particular, in the treatment and separation of f-elements tri-n-butyl-phosphate (TBP) [16; 17; 18], soluble in organic phases phosphoric acids [19; 20; 21; 22], and mixtures of QABs [23; 24; 25] are widely used. Beside the extractant, other important factors are the medium REEs are extracted from and the presence of a salting-out agent.