## 1 Introduction  
Synthetic membranes are heterogeneous solid phase barriers between two phases. The transport of molecular/ionic species of liquids or gases is carried out by contact with their surfaces under the influence of a driving force. Synthetic membranes can selectively transport one species of liquid or gas over another (selectivity or degree of separation) or regulate the transport of different species at different controlled rates (permeability or flux). The transport of molecular/ionic species across membranes is driven by differences in the size, shape, chemical properties, or electrical charge of the components of the mixture being separated. The separation properties of a membrane are determined by both the chemical and physical nature of the membrane materials, as well as the method of preparation [1].  
Among the various methods of ionic membrane production, grafting polymerization occupies a special place, which makes it possible to regulate their physicochemical and operational properties within a wide range [2; 3; 4]. The nature of the monomer, conditions of graft polymerization, and degree of grafting make it possible to modify the surface layer of the polymer material, its volume, and the uniform or gradient distribution of the -graft polymer over the thickness of the polymer matrix [4; 5]. There are various methods of graft polymerization [6]: \"direct\" graft; \"post effect\" graft under irradiation of polymer in a vacuum, inert atmosphere, or in air; and \"post effect\" radiation graft polymerization under irradiation of polymer in air. The latter method is simple and highly reproducible. In this case, the polymerization process is initiated by the alkoxy radicals generated during the decomposition of hydro- and diperoxides. As peroxide-reducing agents, to lower the polymerization temperature, salts of metals of variable valence are added to the monomer solution. As a rule, these are the iron(II) salts [5].  
In most cases, during the synthesis of such membranes, graft polymerization of St is carried out on films of polyolefins or polyfluoroolefins, followed by the chemical modification of graft polystyrene (PS) and the production of cation-exchange membranes. Such sulfocationite membranes obtained by the sulfonation of PE films with grafted PS have high selectivity when separating mixtures of ethylene with ethane [7; 8] and butenes with butane [9]. They are efficient as separation membranes of fuel cells [10].  
In the earlier works of [PERSON] et al. [11], we studied the effect of various parameters of post-radiation chemically grafted polymerization of St on a PE film on the kinetics of graft polymerization, the distribution of grafted PS along the cross-section of the PE film, thermochemical parameters and morphology of the phases of PS and PE, and energy characteristics of the surface of the grafted polymer. It was shown in [12] that the addition of small amounts (\(\sim\)1 vol. %) of DVB led to an increase in the degree of grafting of PS during radiation graft polymerization \"under the beam\" of St on a PE film.  
Of particular interest is the diffusion of water molecules, which determines the ionic conductivity [13]. The diffusion coefficients of water in ion-exchange membranes measured by pulsed field gradient \({}^{1}\)H NMR (PFG NMR) demonstrate two types of water molecules in the films with low DVB contents.  
The purpose of this work is to study the effect of DVB on the post-radiation chemical graft polymerization of St on the PE film, the distribution of grafted PS over the thickness of this film, the morphology of these films, and the properties of related sulfocationite membranes.