Neutron imaging of liquid-liquid systems containing paramagnetic salt solutions
Tim A. Butcher, G. J. M. Formon, P. Dunne, T. M. Hermans, F. Ott, L. Noirez, J. M. D. Coey
aa r X i v : . [ phy s i c s . a pp - ph ] J a n Neutron imaging of liquid-liquid systems containing paramagnetic salt solutions
T. A. Butcher, ∗ G. J. M. Formon, P. Dunne, T. M. Hermans, F. Ott, L. Noirez, and J. M. D. Coey School of Physics and CRANN, Trinity College, Dublin 2, Ireland Universit´e de Strasbourg, CNRS, ISIS UMR 7006, 67000 Strasbourg, France Laboratoire L´eon Brillouin (CEA-CNRS), Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France (Dated: December 19, 2019)The method of neutron imaging was adopted to map the concentration evolution of aqueous para-magnetic Gd(NO ) solutions. Magnetic manipulation of the paramagnetic liquid within a misciblenonmagnetic liquid is possible by countering density-difference driven convection. The formationof salt fingers caused by double-diffusive convection in a liquid-liquid system of Gd(NO ) andY(NO ) solutions can be prevented by the magnetic field gradient force. Paramagnetic liquids are created by dissolving saltscontaining transition-metal or rare-earth ions in asolvent . Magnetic levitation of objects immersed inparamagnetic liquids has been used for magnetohydro-static separation since the 1960s and nowadays findsapplication in biotechnology . Exposing a paramagneticsolution to an inhomogeneous magnetic field gives rise tothe magnetic field gradient force : F ∇ B = χ µ ∇ B . (1)This expression relates the force density to the mag-netic susceptibility of the solution χ , the magnetic fluxdensity B , and the permeability of free space µ .It is possible to trap aqueous paramagnetic salt solu-tions in the magnetic field gradient of a magnetized ironwire . Convection from these paramagnetic liquid tubesis inhibited by the magnetic field gradient force, althoughmixing by diffusion still prevails on a long time scale. Amagnetic field gradient can also initiate magnetothermalconvection in a paramagnetic fluid .At present, the possibility of extracting paramagneticions from a homogeneous aqueous solution with an inho-mogeneous magnetic field is garnering significant researchinterest . In response to the recent rare-earth crisis,this activity has been spurred by the idea of magneticallyseparating rare-earth ions, which was originally exploredby Noddack et al. in the 1950s . Recent studiesused Mach-Zehnder interferometers to relate changes inthe refractive index to an enrichment of magnetic ionsunderneath a permanent magnet . The two mostrecent of these showed that the observed magnetic en-richment is evaporation-assisted . According to Leiet al. , the heightened ion concentration ( ≤
2% bulkconcentration) in the evaporation layer is maintained bythe magnetic field gradient. This results in a modestlong-lived paramagnetic ion enrichment underneath themagnet. The magnetic field gradient force pales in com-parison with the force governing diffusion and is unableto appreciably influence the motion of individual ions onthese grounds ( RT ∇ c ≈ N / m ≫ F ∇ B ≈ N / m ; R : gas constant, T : room temperature, and ∇ c : concen-tration gradient) . In this study, we use neutron imaging to track the con-centration distribution of aqueous paramagnetic gadolin-ium(III) nitrate (Gd(NO ) ) solutions in a liquid-liquidsystem with a miscible nonmagnetic counterpart. Thisdirect method consists of measuring the attenuation ofa white neutron beam on passing through a sample andhas potential for applications in a variety of fields .Neutrons interact with the nuclei of the sample, whichmakes the measurement element-specific, allowing thedirect study of liquids that are both miscible and visu-ally indistinguishable under normal conditions. Gd isthe element with the highest neutron absorption crosssection ( σ a = 46 700 barn for thermal neutrons ). Inaddition, Gd possesses a large magnetic moment of7 µ B by virtue of unpaired 4f electrons. Consequently,solutions of Gd(NO ) are ideal candidates for neutronimaging of paramagnetic solutions, enabling the directobservation of their response to magnetic fields. Recentadvances in detector systems have provided the meansfor neutron imaging with both high spatial and temporalresolutions . Here, we monitor the interplay of con-vection, magnetic field gradient force, and diffusion byvariations in the neutron transmission profile.Neutron imaging experiments were carried out at theIMAGINE station located in the neutron guide hallof the Orph´ee reactor at the Laboratoire L´eon Bril-louin just before its final shutdown. The spectrumof the white neutron beam contained cold neutrons( λ = 2-20 ˚A), which emerged from a 10 mm pinhole andtravelled 2.5 m to the detector where the neutron fluxwas 2 × cm − s − . The detection system consisted ofa 50 µ m thick LiF/ZnS scintillator, with a resolution of18 µ m/pixel, coupled to an sCMOS camera. Recordedimages of 2560 × ×
39 mm . All images shown here were obtainedwith an acquisition time of 60 s. The spatial resolutionis on the order of 50 µ m.Quartz cuvettes with path lengths of 1 mm were filledwith the liquid solutions and placed 5 mm in front of thedetector (see sketch in Fig. 1(a)). The outside dimen-sions of the cuvettes were 40 mm × × × width × depth). Incoherent scattering bywater molecules ( σ inc = 160 . O ( σ inc = 4 . Detector
20 mmCuvette NdFeB magnet
Neutron beam ( (cid:1) =2-20 Å ) yx FIG. 1. (a) Sketch of the experimental setup (top view).(b) Calculated magnetic field gradient force distribution inthe cuvette for a 1 M aqueous Gd(NO ) solution in the fieldof a uniformly magnetized 20 mm Nd-Fe-B cube. imum contrast, the analyzed paramagnetic salt so-lutions were restricted to colorless and transparentGd(NO ) solutions. The neutron absorption cross sec-tion of Gd dwarfs the scattering cross section of D O( σ s = 19 . . For the study of magnetic effects, a cube-shaped Nd-Fe-B permanent magnet of side length 20 mmwas placed adjacent to the cuvettes (2 mm from the so-lution within) and shielded from the neutron beam witha boron carbide sheet. The horizontal magnetic field was B = 0 .
45 T at the surface of the magnet and B = 0 .
13 Tat a distance of 5 mm.An empty beam was recorded during each measure-ment session. This was necessary for normalization tothe intensity of the white beam I . Furthermore, theelectronic noise I df was subtracted from the image to ob-tain the transmittance: T = I − I df I − I df . (2)The final step of the image processing was the removalof noisy pixels by using an outlier filter.The Beer-Lambert law describes the attenuation of theneutron beam by the Gd ions in D O: I = I e − ǫcl , (3)with the molar neutron absorption coefficient ǫ , the Gd concentration c and the sample thickness l . Strictlyspeaking, ǫ depends on the neutron energy and the as-sumption of a single value for a polychromatic neutronbeam is a simplification. This approximation is not a con-cern, considering the fact that the neutron wavelengthsconstituting the beam (2-20 ˚A) lie within one order ofmagnitude of each other. A calibration of the transmit-ted intensity to the Gd concentration was performedby recording images of solutions in 1 mm path length c Gd (mol/L) T r a n s m i tt a n c e Ie −βc Ie −βc + b FIG. 2. Gd concentration calibration curve in a 1 mmquartz cuvette. Transmittance values were normalized to thatof a D O filled cuvette and follow the Beer-Lambert law (bro-ken line) up to 0.4 M. An offset b = 0 .
07 (solid line) is neededat higher concentration. cuvettes (see Fig. 2). The attenuation follows the Beer-Lambert law up to a concentration of about 0.4 M, whenthe beam is almost completely absorbed and the trans-mitted intensity originates predominately from incoher-ent scattering. An offset exponential fit with an extravariable ( b = 0 .
07) captures the behavior, but quanti-tative statements cannot be readily made at concentra-tions higher than 0.5 M. The latest development of blackbody correction opens the possibility to quantify thecontribution of background and sample scattering to thetransmittance, but it would require a black body grid.The magnetic susceptibility of a 1 M Gd(NO ) heavywater solution ( χ = 322 × − ) is the sum of the dia-magnetic D O contribution ( χ D O = − × − ) and theparamagnetic Curie-law contribution of the Gd ions .This value and the magnetic field distribution of theNd-Fe-B magnet allow the computation of the magneticfield gradient force in the vicinity of the magnet (seeFig. 1(b)). The magnetic field was calculated by approx-imating the magnet as two uniform sheets of magneticcharge .In the case of an inhomogeneous solution comprising aparamagnetic and nonmagnetic component, a magneticfield gradient orthogonal to the concentration gradientalters the equilibrium state . The Gd(NO ) solutionclimbs up the side of the cuvette until the balance be-tween buoyancy ( F g = ∆ ρg ) and magnetic field gradientforces is re-established. This can be seen in Fig. 3. Here,100 µ L of 0.4 M Gd(NO ) solution ( ρ = 1180 kg m − )atthe bottom of a 1 mm path length cuvette was cov-ered with 400 µ L D O ( ρ = 1110 kg m − ). A magnet wasplaced at the side and the diffusion of the Gd(NO ) wasmonitored for 3 h. The magnetic field gradient drawsthe Gd(NO ) solution towards the magnet, althoughhomogenization by diffusion continues in its presence.An estimate for the diffusion coefficient D of 0.4 MGd(NO ) in D O can be obtained from the vertical con-centration profile by a fit with the solution of the one- (a) Δ t = 3 min (b) Δ t = 3 h Δ x = 18 mm Δ z = mm c G d ( m o l / L ) Δ x = 3 mm Δ x = 19 mm (c) Δ t = 3 min Δ t = 3 h FIG. 3. Neutron image of a 1 mm path length quartz cuvettewith 100 µ L 0.4 M Gd(NO ) solution overlain with 400 µ L ofD O. A 20 mm magnet cube to the right skews the Gd concentration profile. (a) After 3 min (b) after 3 h. (c) Fits ofEq. (4) to the vertical cross sections (broken lines in neutronimages) of the concentration profiles 3 mm ( ∇ B = 0) and19 mm ( ∇ B = 0) away from the magnet show good agreementand the diffusion coefficient D can be obtained. dimensional diffusion equation (see Fig. 3(c)): c ( z, t ) = c (cid:18) z √ Dt (cid:19) , (4)with Gd starting concentration c . The value of D = 1 . × − m s − obtained for the nonmagnetizedregion after 3 h is reasonable for rare-earth ions inwater . However, this value should be treated with cau-tion, as the initial interface was smeared by introducingthe liquids into the cuvette before the onset of diffusion.The diffusion coefficient from the fit for the magnetizedregion is higher at D = 1 . × − m s − , but the one-dimensional expression does not account for horizontaldiffusion from the warped concentration profile.The density difference between the Gd(NO ) solutionand the nonmagnetic liquid can be adjusted by additionof Yttrium(III) nitrate (Y(NO ) ), which is transparentto neutrons ( σ a = 7 . ), to the D O. Decreasingthe density difference leads to more vigorous magnet-ically induced migration and facilitates magnetic con-finement. If the density of the Gd solution is higherthan that of the Y solution, the removal of the mag-net before homogenization has taken place prompts a (cid:0)(cid:2) i g(cid:3) (cid:4) (b) : z = 12 mm (cid:5)(cid:6)(cid:7) (cid:8)(cid:9)(cid:10)6(cid:11)(cid:12)(cid:13)7 c G d ( m o l / L ) (cid:14) i (cid:15)(cid:16) (cid:17) (c) (cid:18) z = (cid:19) mm (cid:20) (cid:21) (cid:22) (cid:23)
12 1 (cid:24) (cid:25) x (m m ) (cid:26)(cid:27)(cid:28) (cid:29)(cid:30) (cid:31) i !" ( e ) $ z = % mm (&’ ) = (b) t = (c) t = (d) t = *+, - = /12 t =
11 min
E G HI JKLM x = 18 mm z = 12 mm N z = mm z = O mm z = P mm c G d ( m o l / L ) z Q R STUVW XY Z[ \]^ _‘a bcdf FIG. 4. (a)-(f) Neutron images of 100 µ L 0.4 MGd(NO ) above 300 µ L 1.3 M Y(NO ) solution (∆ ρ =140 kg m − ) in a 1 mm path length quartz cuvette. The viewis restricted to the area below the surface in the vicinity of theliquid-liquid interface. (a) 2 min after the Gd(NO ) solutionis suspended above the Y(NO ) surface. (b)-(c) Double-diffusion imposes Gd salt fingers which protrude into theY(NO ) solution after 90 min and begin to sink due to theloss of buoyancy. The fingers have a width of 1.2 mm and per-sist for over 8 h (Multimedia view). (d) A cubic 20 mm mag-net at the side of the cuvette halts the instability growth anddestroys the stratification instantly ( t : time since magnetiza-tion; Multimedia view). (e)-(f) Once the magnet is removed,the control over the Gd(NO ) is relinquished and it fans out.The system snaps back into the stratified state in less than10 min and the cascading salt fingers homogenize the mixtureafter 2 h ( t : time since removal of magnet; Multimedia view).(g) Horizontal cross sections (broken lines in the neutron im-ages) of the salt fingers in (b), (c), and (e) show a periodicvariation of the Gd concentration by ≈ − . Theplotted data was smoothed with a Savitzky-Golay filter. buoyancy-driven Rayleigh-Taylor instability and theGd solution plunges to the bottom of the cuvette ina matter of seconds. A different situation arises whenthe density difference is inverted and the Gd solutionfloats above the Y solution. To investigate this, 100 µ L0.4 M Gd(NO ) solution was injected on top of 300 µ L1.3 M Y(NO ) solution ( ρ = 1320 kg m − ) in a cu-vette (see Fig. 4(a)). After 1 h the system was be-set by a salt-fingering instability due to double diffu-sive convection (see Fig. 4 (b)-(c) and animationsfor greater visibility). This phenomenon is encounteredat the interface of solutions that diffuse into each otherat unequal rates. The diffusivity of Y in the 1.3 Msolution exceeds that of the Gd in the 0.4 M solution.It follows that Y(NO ) will diffuse laterally into smallportions of Gd(NO ) solution that cross the interface.The increase in density due to the gained Y(NO ) makesthe Gd(NO ) solution plummet in form of 1.2 mm widefingers (see cross sections in Fig. 4(g)), which continueto leech Y(NO ) from their surroundings during theirdescent. These transport the Gd(NO ) advectively,two orders of magnitude faster than regular diffusionand trigger a stratification with neighboring fingers thatrise thanks to the buoyancy acquired by the loss ofY(NO ) . Hence, the usually stabilizing factor of dif-fusion can destabilize a system in which the density de-creases upwards. The stratification persists for over 8 h(see Fig. 4(c) and animations in supplementary mate-rial). Viscous friction between the liquid and the cuvettewalls plays a role in the horizontal scale of the individualfingers, which is inversely proportional to the distancebetween the cell walls . Thus, a horizontal expansionof the fingers beyond the gap width is achievable in thincuvettes. A magnet next to the cuvette erases the stratifi-cation and restabilizes the system by capturing the para-magnetic solution (see Fig. 4(d)). This does not reversethe mixing that has occurred and the Gd ions can beseen to continuously diffuse into the Y(NO ) solution.The magnetic field gradient merely prevents the collapseof the liquid-liquid interface. Nonetheless, the systemundergoes an immediate change upon its withdrawal (seeFig. 4(e)). Bereft of the confining magnetic field gradientforce, the boundary between the solutions is once again disrupted. The ensuing release of the paramagnetic liq-uid is accompanied by convective mixing of the solutionsamidst which the salt fingering instability can be wit-nessed anew. After two hours the system equilibrates ashomogenization sets in (see Fig. 4(f)).In conclusion, neutron imaging is a viable methodfor capturing quasi two-dimensional convective and dif-fusive processes in solutions containing Gd ions. Apre-existing concentration of paramagnetic fluid in someregion can be redistributed within a miscible liquidby the magnetic field gradient force, which counter-acts density-difference driven convection. Furthermore,double-diffusive convection in the system of magneticGd(NO ) and nonmagnetic Y(NO ) salt solutions issuppressed. This manifests itself in the stratificationby salt fingers when the magnet is absent. The im-plication of this is of great importance for the devel-opment of the magnetic separation of rare-earth ions,as even minor differences in diffusivity can precipitatesalt fingering instabilities. If left unchecked, these willmix the separated solutions. A prerequisite for the gen-eration of the liquid-liquid interface is a driving forcethat creates and preserves the concentration gradient ofthe paramagnetic ions. The magnetic field gradient isthen able to bestow stability upon the system. Drivingforces can range from the weak factor of evaporation tothe more substantial injection of electrochemical energy,which can drive convection . In view of improvementsin both imaging instrumentation and available neutronflux, higher resolution and frame rates are expected toimprove the neutron imaging of hydrodynamic processesin the future . This may prove valuable for the anal-ysis of ions in solutions.See supplementary material for time sequenced im-ages of liquid-liquid systems containing Gd(NO ) andY(NO ) solutions: undisturbed double-diffusive convec-tion and magnetic confinement of a Gd(NO ) drop.This work forms part of the MAMI project, which isan Innovative Training Network funded by the EuropeanUnion’s Horizon 2020 research and innovation programunder grant agreement No. 766007. ∗ [email protected] U. Andres, Mater. Sci. Eng. , 269 (1976). U. T. Andres, G. M. Bunin, and B. B. Gil,J. Appl. Mech. Tech. Phys. , 109 (1966). E. Turker and A. Arslan-Yildiz,ACS Biomater. Sci. Eng. , 787 (2018). G. Mutschke, K. Tschulik, T. Weier,M. Uhlemann, A. Bund, and J. Fr¨ohlich,Electrochim. Acta , 9060 (2010). P. Dunne, L. Mazza, and J. M. D. Coey,Phys. Rev. Lett. , 024501 (2011). P. Dunne and J. M. D. Coey, Phys. Rev. B , 224411 (2012). Note that this expression is only valid if B ≈ µ H . J. M. D. Coey, R. Aogaki, F. Byrne, and P. Stamenov,Proc. Natl. Acad. Sci. U.S.A. , 8811 (2009). D. Braithwaite, E. Beaugnon, and R. Tournier, Nature , 134 (1991). I. R. Rodrigues, L. Lukina, S. Dehaeck,P. Colinet, K. Binnemans, and J. Fransaer,J. Phys. Chem. C , 23131 (2019). X. Yang, K. Tschulik, M. Uhlemann, S. Odenbach, andK. Eckert, J. Phys. Chem. Lett. , 3559 (2012). B. Pulko, X. Yang, Z. Lei, S. Odenbach, and K. Eckert,
Appl. Phys. Lett. , 232407 (2014). I. R. Rodrigues, L. Lukina, S. Dehaeck,P. Colinet, K. Binnemans, and J. Fransaer,J. Phys. Chem. Lett. , 5301 (2017). Z. Lei, B. Fritzsche, and K. Eckert,J. Phys. Chem. C , 24576 (2017). A. Franczak, K. Binnemans, and J. Fransaer,Phys. Chem. Chem. Phys. , 27342 (2016). K. Ko lczyk, M. Wojnicki, D. Kuty la, R. Kowalik,P. ˙Zabi´nski, and A. Cristofolini, Arch. Metall. Mater. ,1919 (2016). K. Kolczyk-Siedlecka, M. Wojnicki, X. Yang, G. Mutschke,and P. Zabinski, J. Flow Chem. , 175 (2019). W. Noddack and E. Wicht, Ber. Bunsenges. Phys. Chem. , 893 (1952). I. Noddack and E. Wicht, Chem. Techn. , 3 (1955). W. Noddack, I. Noddack, and E. Wicht,Ber. Bunsenges. Phys. Chem. , 77 (1958). N. Kardjilov, I. Manke, A. Hilger, M. Strobl, and J. Ban-hart, Mater. Today , 248 (2011). E. Perfect, C.-L. Cheng, M. Kang, H. Z. Bilheux,J. M. Lamanna, M. J. Gragg, and D. M. Wright,Earth-Sci. Rev , 120 (2014). G. Burca, S. Nagella, T. Clark, D. Tasev, I. A. Rahman,R. J. Garwood, A. R. T. Spencer, M. J. Turner, and J. F.Kelleher, J. Microsc. , 242 (2018). n TOF Collaboration, M. Mastromarco, A. Manna, et al. ,Eur. Phys. J. A , 9 (2019). J. Brenizer, Phys. Procedia , 10 (2013). P. Trtik, M. Morgano, R. Bentz, and E. Lehmann,MethodsX , 535 (2016). R. Zboray and P. Trtik, Flow. Meas. Instrum. , 182 (2019). F. Ott, C. Loupiac, S. D´esert, A. H´elary, and P. Lavie,Phys. Procedia , 67 (2015). N. Kardjilov, F. de Beer, R. Has-sanein, E. Lehmann, and P. Vontobel,Nucl. Instrum. Methods Phys. Res , 336 (2005). P. Boillat, C. Carminati, F. Schmid, C. Gr¨unzweig,J. Hovind, A. Kaestner, D. Mannes, M. Morgano,M. Siegwart, P. Trtik, P. Vontobel, and E. Lehmann,Opt. Express , 15769 (2018). C. Carminati, P. Boillat, F. Schmid, P. Vontobel, J. Hov-ind, M. Morgano, M. Raventos, M. Siegwart, D. Mannes,C. Gr¨unzweig, P. Trtik, E. Lehmann, M. Strobl, andA. Kaestner, PLOS ONE , 1 (2019). E. P. Furlani,
Permanent magnet and electromechanicaldevices: materials, analysis, and applications (AcademicPress, San Diego, 2001) pp. 208–217. E. L. Cussler,
Diffusion: Mass Transfer in Fluid Systems (Cambridge University Press, 2009) p. 162. V. F. Sears, Neutron News , 26 (1992). Z. Huang, A. De Luca, T. J. Atherton, M. Bird, C. Rosen-blatt, and P. Carl`es, Phys. Rev. Lett. , 204502 (2007). V. Tsiklashvili, P. E. R. Colio, O. A. Likhachev, and J. W.Jacobs, Phys. Fluids , 052106 (2012). J. S. Turner and H. Stommel, Proc. Natl. Acad. Sci. U.S.A. , 49 (1964). M. E. Stern and J. S. Turner,Deep-Sea Res. Oceanogr. Abstr. , 497 (1969). J. Taylor and G. Veronis, Science , 39 (1986). M. Morgano, E. Lehmann, and M. Strobl,Phys. Procedia , 152 (2015). P. Trtik and E. H. Lehmann,J. Phys. Conf. Ser. , 012004 (2016). upplementary Material:Neutron imaging of liquid-liquid systems containing paramagnetic salt solutions
T. A. Butcher, , ∗ G. J. M. Formon, P. Dunne , T. M. Hermans , F. Ott , L. Noirez , and J. M. D. Coey School of Physics and CRANN, Trinity College, Dublin 2, Ireland Universit´e de Strasbourg, CNRS, ISIS UMR 7006, 67000 Strasbourg, France Laboratoire L´eon Brillouin (CEA-CNRS), Universit´e Paris-Saclay, CEA-Saclay, 91191 Gif-sur-Yvette, France ∗ [email protected](Dated: December 19, 2019) ADDITIONAL TIME SEQUENCED NEUTRON IMAGESS1. Unhindered mixing by salt fingering c G d ( m o l / L ) hi jkl mn op qrst FIG. S1. Two 1 mm path length quartz cuvettes with 75 µ L 0.4 M Gd(NO ) (left) and 100 µ L 0.5 M Gd(NO ) (right) above300 µ L of 1.3 M Y(NO ) . Gd(NO ) solutions were injected above the Y(NO ) solution surface in the left cuvette and belowit in the right cuvette. Salt fingers form in both systems and mix the solutions within 7 h. Left cuvette: The surface profile iscaused by capillary forces. The salt fingers begin to form at the side of the cuvette and propagate inwards. (Multimedia view) S2. Magnetic confinement of a drop of Gd(NO ) solution - l og ( T ) (a) (b) (c) M agne t M agne t M agne t (d) u x = 18 mm FIG. S2. 50 µ L 0.5 M Gd(NO ) solution ( ρ = 1200 kg m − ) above 800 µ L 1.3 M Y(NO ) solution ( ρ = 1320 kg m − ) in a 2 mmpath length quartz cuvette. The acquisition time was 80 s for this measurement. The calibration from Fig 2 is not valid forpath lengths above 1 mm. Therefore, the negative of the logarithm of the transmittance is shown. (a) The Gd(NO ) dropmigrates to the cubic 20 mm magnet at the side of the cuvette within 2 min. (b) The trapped drop in the magnetic field gradient(see Fig. 1(b)) gradually diffuses into the Y(NO ) solution over the course of 4.5 h. (c) Salt fingers appear immediately afterremoval of the magnet at ∆ t = 4 h 30 min. (d) At ∆ tt