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migration of the ions in aqueous solution occurs predominantly through the diffusion Membranes 2021, 11, 940 process. The corresponding diffusion coefficients, D, of the ions were calculated as 1.696 × 10−9, 1.702 × 10−9, 1.690 × 10−9, and 1.692 × 10−9 m2/s for Li+, Na+, Mg2+, and Ca2+ ions, re- spectively. While the simulated D values did not yield exactly the same results as the ac- tual D values of the ions in aqueous media [37], the simulated values are of almost the 7 of 12 Figure 5. Representative simulation model of the counterions–aqueous system (a) Li+, (b+) Na+, (c+) Figure 5. Representative simulation model of the counterions–aqueous system (a) Li , (b) Na , Mg2+, an2+d (d) Ca2+ ion2s+at 298 K. The MSD–t curve of all the ions—blue (Li+), red (+Na+), black+(Mg2+), (c) Mg , and (d) Ca ions at 298 K. The MSD–t curve of all the ions—blue (Li ), red (Na ), black and green (Ca2+)—are shown in (e), whereas the log (MSD) vs. log (t) curve for Li+ ions is shown in (Mg2+), and green (Ca2+)—are shown in (e), whereas the log (MSD) vs. log (t) curve for Li+ ions is (f). All other atoms remain the same as depicted in Figure 1. shown in (f). All other atoms remain the same as depicted in Figure 1. same magnitude as the actual values, and are consistent with the size–charge correlation, in the order Mg2+ < Ca2+ < Li+ < Na+. 3.3. Interaction of Co-Existing Ions with the Polyelectrolyte Chain 3.3. Interaction of Co-Existing Ions with the Polyelectrolyte Chain ++ ++ The interaction of the polyelectrolyte chain with co-existing ionic systems Li /Na , The interaction of the polyelectrolyte chain with co-existing ionic systems Li /Na , ++2+2+ ++2+2+ Li /Mg , and Li /Ca , was further investigated. MD simulations were carried out in an Li /Mg , and Li /Ca , was further investigated. MD simulations were carried out in an NVTeennsseemblleeoonaassimimuulalattiionbbooxxccoonssiissttiingoffaa11:1:1rraattioiofoforrththeeioionnss,,ththeeppoolylyeeleleccttrroollyttee -− cchhaainin,,55000mollecullesofH2O,,andaballanceofCl ioinons.sT. Thehesismimuulaltaitoionncoconndditiitoionnssrreemaainineed 2 ththeessaameeaassprreeviiooussllyy..Fiigurre6presenttsthedistributiionofLi ioionnssaarroundttheeccharrged ggrroouuppss..TwTwoommajaojroprepaekaskwsewreroebosbersverevdeidnianllaplloptslo,tins,diincdatiicnagtintwgotwmoajmoraijnotreirnatcetiroanctrioen- grieognisoninsaicncoarcdcoarndcaenwceithwtihthe itohneiciohnyidcrhaytidonrastihoenllsh[3e8ll,s39[]3.8T,3h9e].shTahrpe psheakrparpoeuankda1r.o0u5nÅd c1o.r0r5esÅpoconrdrsestpooansdtrsotnogaesletrcotrnogstealetictraottsrtactitcioanttbraectwtioenenbtehtwe Oee-nattohme sOo-fathome cshoafrgtheedcghraorugpesd + agnrdouthpes Lani+diotnhse. LTiheiosnecso. nTdhepseeackonatd1p.6e5akÅaitn1d.i6c5atÅesinthdaitcatwteos tmhatjotrwcolomsea-jroarncgleosien-trearnagc-e + tionntesroactciuornbseotwcceuernbthetewOe-eantotmhesaOn-datohmesLianiodnsth.FeoLriothioenrsp.eaFkosraotthhigerhepresaekpsaraattihoingsh,err + aspepeaarraetdiownse,arkearpapnedarevdenwteuaklleyrdainmdineivsehnetdudalulye tdoimthiensishhieldidnugeoftothtehenesghaietilvdeincghaorfgtehs,e negative charges, possibly by molecules of water, thereby weakening the interactions possibly by molecules of water, thereby weakening the interactions with Li+ ions. The with Li+ ions. The snapshots of the attraction of the counterions towards the O-atoms are snapshots of the attraction of the counterions towards the O-atoms are presented in Figure presented in Figure 7, and indicate a moderat+e attraction for the Li+ ions, resulting in the 7,andindicateamoderateattractionfortheLi ions,resultingintheeffectivediffusionof effective diffusion of the ions within the membrane material. the ions within the membrane material. 3.4. Interaction of the Polyelectrolyte Chain with Mg2+/Li+ Ions The concentration of Mg2+ ions in most brine is often greater than those of Li+ ions [40–42]. This results in a competition in the diffusion of the ions during the recovery process. In order to suppress the interference of the Mg2+ ions, it is desired that the polyelectrolyte chain in the membrane material exhibits a high affinity and, consequently, a greater retar- dation for the competing ions. The interaction of the polyelectrolyte chain with Mg2+/Li+ ions was investigated in this study to explore the effect of a high concentration of Mg2+ ions on the diffusion of Li+ ions. Three PE–aqueous systems were constructed, comprising 5 polyelectrolyte chains, 2500 molecules of H2O, 20 Li+ ions, 50, 100, and 200 Mg2+ ions in the first, second, and third systems, respectively, and a balanced charge of Cl− ions. All other simulation details remained as previously described. The distribution of Li+ ions around the charged groups are reflected in Figure 8. Two distinct strong peaks are visible in the curve, similar to those presented in Figure 6. The intensity of the peak at 1.05 Å was strongest when the amount of Mg2+ ions was maintained at 50. This is due to the presence of sufficient charged groups, and the competitive adsorption between the Mg2+ and the +PDF Image | Diffusion of Monovalent Ions in Polyelectrolyte
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