Nanocrystallization of CaCO3 at solid/liquid interfaces in magnetic field: A quantum approach

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FNanocrystallization of CaCO3 at solid/liquid interfaces

A.C. Cefalas a,*, S. Kobe b, G. Drazic b, E. Sarantopoulou a, Z. Kollia a, J. Strazisar c, A. Meden d

a National Hellenic Research Foundation, TPCI, Athens, Greeceb Department of Nanostructured Materials, Jozef Stefan Institute, Ljubljana, Sloveniac Department for Geotechnology and Mining, University of Ljubljana, Sloveniad Faculty of Chemistry and Chemical Engineering, University of Ljubljana, Slovenia

1. Introduction

The formation of precipitated deposits at the liquid/solidinterface of industrial flow systems is commonly known as‘‘scaling’’. It is one of the most common processes in nature andthe cause of water corrosion in industrial flow systems. Despite thesimplicity of chemical reaction of forming CaCO3 and otherprecipitants on the surfaces of the water flow elements, scaling byitself is a complex process as the deposition rate and the cohesionof the precipitated particles on the flow surfaces is proportional tothe strength and the type of the chemical bonding between theparticles and surfaces, which mainly depend on the chemicalcomposition of the particles, the electro-kinetic potentials of theagglomerations at the nucleation point, the type and roughness of

the flow surfaces, the presence of crystal growth modifierssurface functionalizers [1–5], etc. In addition, the strengthchemical bonding of crystal nanocomposites of similar chemstructure depends on their symmetry (crystallographic pgroup). It is now widely accepted that initial stage of nucleatiocrucial for driving the chemical reactions to a certain directwith sound environmental, technological and economical impas it forms the crystal seed, on which the scaling productsfurther continue to agglomerate in crystal form. As the strengtattachment of the precipitating nanocrystals on flow surfadepends on the symmetry of the initial crystal seed, the degrescaling depends as well on the initial stage of crystallization ofprecipitating products. In addition, the first stage of nucleationcrystallization depends besides the catalyzing atomic/molec

Applied Surface Science xxx (2008) xxx–xxx

A R T I C L E I N F O

Article history:

Received 30 July 2007

Accepted 15 April 2008

Available online xxx

PACS:

68.35.Rh

81.65.kn

47.27.wb

67.70.DV

61.14.�x

Keywords:

Calcium carbonate

Turbulent flow

Scaling

Solid–liquid interface

Water

Nanocrystals

Calcite

Aragonite

Vaterite

A B S T R A C T

With the application of 1.2 T external magnetic field, 90% of CaCO3 soluble molecules in water

precipitate on stainless steel 316 solid/liquid interface in the form of aragonite/vaterite. The magn

field increases locally the thermodynamic potentials at interface, favoring the formation of arago

than calcite, despite the fact that the field-free ground electronic state of aragonite is situated higher t

of calcite. A quantum mechanical model predicts that magnetic fluctuations inside the liquid can

amplified by exchanging energy with an external magnetic field through the angular momentum of

water molecular rotors and with the macroscopic angular momentum of the turbulent flow.

theoretical model predicts that the gain is higher when the magnetic field is in resonance with

rotational frequencies of the molecular rotors or/and the low frequencies of the turbulent flow and

aragonite concentration is increasing at 0.4 T in agreement with the experimental results. Contrar

calcite, aragonite binds weakly on flow surfaces; and hence the process has significant industrial

environmental impact.

� 2008 Published by Elsevier

Contents l is ts ava i lab le at ScienceDirec t

Applied Surface Science

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* Corresponding author. Tel.: +30 210 7273840; fax: +30 210 7273842.

E-mail address: [email protected] (A.C. Cefalas).

0169-4332/$ – see front matter � 2008 Published by Elsevier B.V.

doi:10.1016/j.apsusc.2008.04.056

Please cite this article in press as: A.C. Cefalas et al., Nanocrystalldoi:10.1016/j.apsusc.2008.04.056

blocks of flow surfaces, on additional factors, such as the preseof magnetic fields during the first stage of nucleation, andinitial state of atoms and molecules in the flow (enemomentum, state of flow, etc.) [6–18]. Furthermore, str

ization of CaCO3 at solid/liquid interfaces, Appl. Surf. Sci. (2008),

mailto:[email protected]

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http://dx.doi.org/10.1016/j.apsusc.2008.04.056


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netic fields can be generated within the turbulent flow of ionicids [19], which alter the local thermodynamical equilibrium ofliquid/solid interface. The most common example of industrialosive scaling agent is CaCO3, which naturally exists in threerent crystal forms. At normal conditions the most stable formlcite (rhombohedral–hexagonal unit cell), mostly found inre. Aragonite has lower symmetry with orthorhombic unitHardly ever CaCO3 is found in the less stable form of vaterite,

ch eventually undergoes phase transition to calcite oronite. Contrary to calcite, aragonite bounds weakly atid/solid interfaces and scaling can be minimized providedaragonite will be formed instead of calcite.raditional chemical methods of scale control involve either theprecipitation of the scale former with lime or soda ash, thetion of scale inhibiting reagents [21–23] or the replacement ofscale former with soluble ion exchange [24]. All of thesehods though effective in scale control, substantially change thesicochemical properties of solutions.s an alternative to chemical de-scaling, the magnetic anti-

ing has been reported as being effective by limiting the scalesition rate, removing existing scale, or producing a softer andtenacious surface effects [1–56]. Many reports claim large

ngs in energy, cleaning and process downtime costs from thellation of magnetic water conditioners in real systems [14].he investigations on the influence of magnetic field on scalingainly focused on the process of nucleation and/or crystal-ion of CaCO3 [10,32–35], the crystal structure of CaCO3

18,31,34–36,25], the colloidal aqueous dispersions stability24,26,23,39], the change of the zeta potential (z) of colloidalicles [21,22,34,35,38,39] and the physical–chemical propertieshe water or its solutions, such as the surface tension, etc.44].he first systematic investigation on the effect of magnetics on zeta potential of suspensions of CaCO3, was preformed byov et al. [22]. It was shown that an externally applied magneticis lowering the zeta potential (z) of CaCO3 and the influence ofon zeta potential is increasing with time. The effect of

netic field in water flow systems under well-controlledratory conditions showed that there was a significant effecthe precipitation rate of CaSO4 crystals. Conductivity, calciumbility, and zeta potential were decreased, whereas the amounttal suspended solids was increased [22].urthermore Tombacz et al. studied the effect of weak magnetics on the aggregation state and electrophoretic mobility ofatite soils in flowing systems as a function of time andtrolyte concentration [23]. While enhanced aggregation ofatite was observed following application of magnetic fieldsnge in turbidity, scattered light intensity, and photonelation), little effect on aggregation state was observed forstatic systems or for the flowing systems in the absence ofnetic field. Mobility also increased during the initial phase ofc and dynamic application of magnetic fields. Changes in bothility and particle aggregation states indicated a significantndence on electrolyte concentration. These effects were

ussed in terms of a magnetohydrodynamic interactioneen the magnetic field and ions. Busch et al. [24] investigated

aggregation state of flowing colloidal dispersions of polystyr-latex microspheres. Comparable effects were not observedn cholesterol suspensions were recirculated in the absence of

The rates of precipitation and sedimentation of CaCO3 inmagnetic fields were significantly different from solution at zerofield intensities [25]. Also zeta potentials, nucleation rates andsurface tension of the entire in situ precipitated nucleus weresignificantly affected in comparison to the reference system.

Additional investigations by Chibowski et al. [26] confirmdifferent crystal forms and rates with the application of externalmagnetic fields and a systematic study on the nucleation of variouscrystals such as Na2CO3 in water and electrolyte solutions withmagnetic fields was done by Higashitani et al. [1]. Similarly, thesame authors investigated the effect of dynamic magnetic fields onNa2CO3, CaCl2 and CaCO3 precipitated from solutions, and theeffect of static magnetic field on the zeta potential and thediffusion of the colloid particles in electrolyte solutions [10]. All theabove studies clearly indicate the correlation between the crystalforms of the precipitating particles and the magnetic field and thestate of flow.

However, despite the large number of experimental results,which identified conditions of 90% reduction in CaCO3 scaleformation under magnetic fields [19–20,27–30], the interpretationof the process is still an open issue.

Indeed, up to now various mechanisms to interpret physicalanti-scaling have been proposed [23,31–35,25]. None of themhowever is fully comprehensive to account for all the experimentaleffects. Early work, which claimed changes in the structure ofwater resulting from magnetic exposure, has now been revised. Inaddition the influence of contaminants (Fe2+ or Zn2+, etc.)introduced by magnetically induced corrosion has been thor-oughly considering together with surface effects [50–56]. Otherworkers have more recently proposed models to support thetheory of enhanced nucleation in the bulk solution. However, onthe basis of current crystallization theories, any effects onheterogeneous nucleation would have nanosecond relaxationtime and homogeneous nucleation would be unlike, even afterexposure to a reasonably strong (0.5 T) magnetic fields.

Solely magnetically induced changes in crystal structures, i.e.calcite to aragonite, through hydrodynamic interaction betweenthe magnetic field and the flow [20,53] could interpret experi-mental results provided that strong magnetic fields can bedeveloped within the flow [57]. Magnetic induced changes oncrystal structure at interfaces, imply that strong magnetic fields aredeveloped locally, which alter the field-free thermodynamicpotentials of the system changing thus the crystal structure ofthe precipitating particles.

However, none of these theories are able to explain thereduction of scaling in a coherent way, because only classicaltheories have been applied up to now.

Along these lines and in order to explain the formation ofaragonite in the presence of magnetic fields at interfaces,calculations regarding the structure of the ground electronicstates for the two structural forms of CaCO3 were performed [19].Ab initio calculations for the ground electronic states of aragoniteand calcite suggest that the ground electronic state of aragonite isplaced as high as 28 eV above the ground electronic state of calcite.On the other hand, the slope of the ground electronic state ofaragonite is much stiffer than that of the calcite, and therefore theCa2+and CO3

2� ions should have higher kinetic energies toovercome the repulsive forces of the potential barrier. Therefore,the formation of calcite is energetically in favor of that of aragonite.

A.C. Cefalas et al. / Applied Surface Science xxx (2008) xxx–xxx

165166167168169170

netic field or when the suspensions were exposed to anvalent magnetic field in the absence of flow. For cholesterolensions, the increment of particle dimensions was mostounced between 0.15 and 1.0 T. Aggregation effects were alsorved for both types of suspensions, in the regions near tocal concentration of coagulation.

ase cite this article in press as: A.C. Cefalas et al., Nanocrystalli:10.1016/j.apsusc.2008.04.056

The theoretical results are in agreement with experiments thataragonite is formed at high temperature and pressure from melts.The required kinetic energy of the ions can be provided by anexternal magnetic field. In this case the localized free energy F perunit volume is larger than the free energy in the absence of externalmagnetic fields.



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Previous calculations suggest that the energy of 28 eV, which isrequired to bridge the gap between the ground electronic states ofcalcite–aragonite, can be provided by a magnetic field of 45 Twithin a typical internuclear distance of 0.5 nm between Ca2+ andCO3

2� ions. The formation of the aragonite was explained byassuming that energy transfer is taking place between themagnetic field and the turbulent flow and it was quantitativelydescribed with the Navier–Stokes and Maxwell’s equations [58].According to this model, the magnetic field can be amplified tohigh values as the kinetic energy of the turbulent flow istransferred to the magnetic field [19,20]. The transfer of energyis enhanced around conductive surfaces, and the theoreticalpredictions are in agreement with experimental results [41].

In this communication, we report on the influence of magneticfield on the initial state of nucleation of CaCO3 at the solid/liquidinterface of a stainless steel 316 tube under well-definedconditions by applying an external magnetic field of 1.2 T.Transmission and scanning electron microscopy and quantitativeX-ray diffraction analysis was used and it was found that under thepresence of magnetic fields 93.2% of CaCO3 mainly crystallizes inthe form of aragonite and vaterite rather than calcite, in agreementwith previous investigations [19,20].

Furthermore, the appearance of large magnetic fields, which arerequired to form aragonite, was interpreted by applying a quantummechanical microscopic approach for the first time to ourknowledge. The coupling of the magnetic field to the molecularsystem was described by quantizing the flow, the electromagneticfield, and the molecular system. A simplified two level atomic/molecular system, within the turbulent flow and in the presence ofa constant magnetic field, is described by the same set of equationsof motion, which describe the MASER amplifier. According to thismodel, one magnetic fluctuating mode at a given frequency v, canbe amplified to high values by absorbing energy from an existingconstant external magnetic field. The coupling and transfer ofenergy is taking place through the angular momentum of the threeentities: the water molecules, the magnetic field and the flow.Using the quantum mechanical approach, the amplification ofmagnetic field within the flow can be derived from first principlesbypassing thus the ad hoc assumptions of the macroscopichydrodynamic theory [19,20]. The theoretical model predicts thatthe gain is high when the magnetic field is in resonance with therotational frequencies of the molecular rotors or/and the lowfrequencies of the turbulent flow. The aragonite concentration isconstantly increasing with the intensity of magnetic field and thegain is high at 0.4 T, in agreement with the experimental resultswhere the aragonite concentration is higher than 80% at 0.6 T.Results are in agreement with resent experiments, which claimthat quantum effects within the flow should be taken intoconsideration to interpret CaCO3 nucleation [57].

2. Experimental

Using X-ray diffraction analysis on a Philips X-ray diffract-ometer it is possible to follow the influence of the appliedmagnetic field on the phase compositions. Sample solutions ofcalcium hydrogen carbonate (Ca (HCO3)2) were prepared bydissolving finely ground calcium carbonate powder of analyticalpurity in deionized water with the resistivity of R = 18 MV. Due toits very low solubility (3.8 � 10�9) CaCO was dissolved in the so

meter) for all the experimental trials, contrary to previous trwith changing magnetic field [19,20]. The fluid-flow rate0.87 m/s and Reynolds numbers in the turbulent region (�60The solution was re-circulated for 8 h. In addition paracomparative trials were performed without the field. Followthe experimental procedure with and without magnetic fiforced precipitation of the dissolved CaCO3 was accomplishedpassing nitrogen gas through the systems. The remaining sowere removed by filtering the suspension through 0.45 mm fimedium. Solid particles were separated from the solutioncentrifugation and were dried at 40 8C and 70% R.H.morphology of the precipitated particles was analyzed by usscanning electron microscopy (SEM - JEOL 5800) and X-diffraction analysis was performed to determine the crystal foFor the preliminary study of the nucleation and further cryslization of different forms of CaCO3 an analytical electmicroscopy was used. A sample for the TEM observationanalysis (JEOL 2000 FX, JEOL 2010 F (FEG)) was prepared by usiC/Cu grid foundered into the solution for different times (510 min after the beginning of the process). In a speciconstructed cell where various parameters could be controthe nano-sized particles were collected on the grid and examiunder the electron microscope. EDXS was used to characterizechemical composition. X-ray powder diffraction patterns ofsamples were recorded on a Siemens D-5000 diffractometer wthe reflection (Bragg-Brentano) geometry using graphite mochromatized Cu Ka radiation. Data were collected in the 2u rafrom 20 to 708 in steps of 0.048; the integration time was 30 sstep. The divergence and anti-scatter slits were fixed (18);receiving slits were 0.2 cm; a quantitative X-ray analysissupported by the software program. The mass fractions wdetermined by Rietveld refinement. The structural models othree phases were taken from the ICSD standards. To determthe accuracy of the experimental set-up five trials were performunder the same conditions for each experiment and the resultthe X-ray analyses were compared.

3. Results and discussion

The effect of static magnetic field on crystal morphologystructure of precipitating aggregates on the liquid/solid inface in water flow is shown in Figs. 1 and 2 for 0 and 1.2 T fiintensities, respectively. CaCO3 crystals under zero extemagnetic field precipitate at the liquid/solid interface


Fig. 1. SEM images of the precipitated CaCO3 particles at the solid/liquid interface in

flow systems. With zero external magnetic field, the CaCO3 nucleates mainly as

calcite. The bar size is 200 nm.

3

called ‘‘model water’’ by blowing CO2 through a porous frit. Theresulting solution is an equilibrium system CaCO3–Ca (HCO3)2–H2O. For the further precipitation of CaCO3, CO2 was removed byheating and the air was blowing through the solution (0.05–0.5 l/min). The solution was then exposed to a constant appliedmagnetic field (DC) of 1.2 T (field was measured using a Gauss-

Please cite this article in press as: A.C. Cefalas et al., Nanocrystallization of CaCO3 at solid/liquid interfaces, Appl. Surf. Sci. (2008),doi:10.1016/j.apsusc.2008.04.056


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ogeneous agglomerations of calcite (Fig. 1). The size of thetals at 1.2 T magnetic field was smaller, the agglomerationse well separated and exhibited needle-like and flowered aragonite structures (Fig. 2) which were not observed

ero magnetic field. Following TEM imaging (Figs. 3 and 4) ofprecipitating particles during the early stage of crystal-ion, calcite, aragonite and vaterite, were detected atificantly different concentration levels. The vaterite wasmposed to form 5–10 nm wide CaO particles under the

ct of the electron beam. The dark-field TEM image ofmposed vaterite is shown in Fig. 5. The upper inset of Fig. 5bit SAED-like patterns of decomposed vaterite crystales with spots (arcs) of CaO texturing in [1 1 0] zone axis.

les correspond to randomly oriented nanocrystals of CaO.lower inset indicates the results between the experimentalthe simulated SAED patterns for cubic (Fm3-m) CaO.

erent size agglomerations of aragonite were grown atand the images of Fig. 6 were taken at different times, a

which suggests that CaCO3 continues to grow in the crystalof the initial seed. The average crystal size of the needle-

shapes of aragonite crystals of left image of Fig. 6 isnm � 100 nm, while the average size after 30 min is

600 nm � 200 nm. In the case of the application of strongexternal magnetic fields, the amorphous agglomerating phasescontain higher amount of Si contaminants. Si was found in thecrystalline structure of vaterite, especially in samples investi-gated immediately after the application of the external magneticfield. This is in agreement with recent results, where the effectof magnetic fields in industrial flow systems was enhanced withthe addition of Si [21]. The presence of Si in our experiments wasdue to the use of silicon rubber tubes in certain parts of thecirculating line of water flow system. The SEM images of theinner pipe surfaces suggest that Si particles were peeled-off inthe flow at higher rates under the magnetic field. The diffractionpatterns of precipitated particles at zero and 1.2 T magneticfields, respectively, are indicated in Figs. 7 and 8. Thepredominating peaks of aragonite in Fig. 8 can be clearlyidentified, while in Fig. 7 mainly CaCO3 in the form of calcite ispresent. For all the experimental conditions with externalmagnetic field in water flow, the concentration of aragonite/vaterite crystals in the precipitating samples was higher than

. SEM images of the precipitated CaCO3 at the solid/liquid interface in flow

ms. With the application of 1.2 T external magnetic field, the CaCO3 nucleates

ly as aragonite with needle and flower-like shapes. The bar size is 200 nm.

Fig. 4. TEM image and diffraction pattern (SEAD) of aragonite crystals. The bar size is

2 mm.


. TEM image of calcite. The crystal form of calcite is different than of Fig. 1. The

ize is 100 nm.

Fig. 5. Dark-field TEM image of decomposed vaterite. Insets (upper): SAED patterns

of decomposed vaterite crystal where spots (arcs) indicate textured CaO in [1 1 0]

zone axis. Circles correspond to randomly oriented nanocrystals of CaO. Lower

inset: comparison of experimental and simulated SAED patterns for cubic (Fm3-m)

CaO. The bar size is 20 nm.

ase cite this article in press as: A.C. Cefalas et al., Nanocrystallization of CaCO3 at solid/liquid interfaces, Appl. Surf. Sci. (2008),i:10.1016/j.apsusc.2008.04.056


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90% and it was constantly increasing with the intensity ofmagnetic field (Table 1).

For describing the formation of aragonite (and/or vaterinstead of calcite in flow systems in the presence of magnetic ficalculations have to be carried out for the relative position ofground electronic states of different structural forms of CaCPreliminary calculations for the ground electronic statesaragonite suggest that it is placed 28 eV above the groelectronic state of the calcite [19,20]. However, the situatiocompletely different when CaCO3 is dissolved in water.required energy to reach the ground electronic state ofaragonite is provided by either steady state or time dependmagnetic fields.

In the absence of static magnetic fields electromagnfluctuations always accompany the flow of a conductive fland spontaneous magnetic fields can be amplified to lavalues. When a conductive fluid moves under external magnfield, electric currents accompany the flow, which is modifiOn the other hand, energy is feed back from the flow tomagnetic field and the complex situation is describedMaxwell’s, Navier Stocks and the equations of heat tran[19,20].

They are form a complete set of equations, which fdescribe the interaction of electromagnetic fields withturbulent flow and the transfer of energy. The main quesduring the exchange of energy is whether an electromagnfluctuation can be amplified or dumped by the turbulent flowthe lapse of time. When the magnetic field interacts withflow, its energy is dissipated within the flow, and the curretend to diminish the field. On the other hand, it can be sho[58] that when a fluid is in motion, the lines of the magn

Fig. 6. TEM images of different size agglomerations of aragonite at 1.5 T. The size of the left and right bars is 200 and 500 nm, respectively. The two images were tak

different times, a fact which suggests that CaCO3 continues to grow around the crystal form of the initial crystal seed. The average crystal size of the needle-like shap

aragonite crystals of left image is 200 nm � 100 nm, while the average size after 30 min is 600 nm � 200 nm.

Fig. 7. X-ray diffraction spectrum of precipitated CaCO3 with zero magnetic field.

Table 1The crystal form of precipitating CaCO3 at the solid/liquid interface of stainless


Fig. 8. X-ray diffraction spectrum of precipitated CaCO3 at 1.2 T.

304 pipes is strongly depended on the intensity of a static magnetic field and at

1.2 T, 93.5% of CaCO3 crystallizes in the form of aragonite and vaterite

Applied magnetic field (T) Calcite (%) Aragonite + vaterite (%)

0 90.2 9.8

0.6 17.50 82.5

1.2 6.5 93.5

Please cite this article in press as: A.C. Cefalas et al., Nanocrystallization of CaCO3 at solid/liquid interfaces, Appl. Surf. Sci. (2008),doi:10.1016/j.apsusc.2008.04.056


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es follow the lines of flow and the magnetic field isortional to the stretching of the lines. In a turbulence flowtwo points move apart in time, and therefore as the lines ofmagnetic forces are stretched the magnetic field is

ngthened in time as well. Under certain conditions theseopposite tendencies are balancing each other and a criterionbe provided distinguishing the two cases of damping orlification of spontaneous magnetic field. The thresholdition for amplification is [58]:

0 �1 (1)

re h is the coefficient of viscosity of the fluid, s is the fluiductivity, r is the fluid density and m0 the vacuum permittivity.exchange of energy between the magnetic field and the

ulent flow is given by

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1) states that in the case of amplification, the energy densityagnetic fields developed within a flow volume �l3

0, isortional to the kinetic energy of the flow within the

e volume. The dimension l0 is the length where thegy dissipation and the viscosity of the fluid becomeortant.he energy density of amplified spontaneous magnetic fieldsn in a turbulent flow near the liquid/solid interface withm surface roughness, is of the order of 105 J/m3, when the

city fluctuations within l0, are �10 m/s. Similarly, theolds number R is �10, and l0 � 0.1 mm. It is obvious

efore that for an energy density of the spontaneousnetic field of the order of �109 J/m3, which is required toge the energy gap between the two structures of CaCO3,tuations of velocity of the order of 103 m/s should be

rated within the turbulent flow. These values of velocitytuations in principle could be developed at the interface of

uctive surfaces from local ion acceleration and from localmal fluctuations. The key point in the above macroscopicallysical treatment is that the exchange of energy between thenetic field and the flow is taking place through the angularentum of the turbulent flow. A microscopic approach onmolecular level will require a quantum mechanical

tment. In this case, the exchange and transfer of energyeen the external magnetic field and a fluctuating magnetic

e is taking place through the angular momentum of theting water molecules.he process is described by the same set of equations as for themetric amplification and the theory of MASER by quantizingthe field and the molecular water rotors.

n the general case where a molecular system with momentuminteracts with an electromagnetic field, ~A : ð~E; HÞ, the totaliltonian H of the system can be written in a variety ofvalent forms,

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~rc j �~ETð0Þ

(4)

where the symbols have their usual meaning and the summation isover the external electrons. Since the ~rc j is of the order of Bohrradius, and the gradient operator is of the order of the wavenumberk of the field ð~ET; ~HÞ, the successive terms in the expansion of theexponential diminish rapidly and only the first non-vanishingterms of ~Hð0Þ and the first two non-vanishing terms of ~ET areretained.

With conversion to quantum mechanics and termination of theexponential expansions, the Hamiltonian can be written:

H ¼ HE þ HR þ HI (5)

where HE is the Hamiltonian of the single molecule

HE ¼1

2m

Xj

ð~pc jÞ2� 1

2eX

j

’ð~rc jÞ �1

2Ze’ð0Þ (6)

~HR is the Hamiltonian of the field

HR ¼1

2

Zðe0

~E2

þm0~H

2

Þdt (7)

The vector potential operator ~A of the field in the secondquantization form is the sum of field harmonic oscillations andis given by the equation.

~Að~rc; tÞ ¼X

kl

�h

2e0tvk

� �1=2

ð~elÞðakl expðið~k�~rc �vktÞ

þ aþkl expð�ið~k�~rc �vktÞÞ (8)

where the summation of k is taking place over all the modes of thefield at frequencies vk and within a volume element t of theinteraction.~el are the polarization vector operators, and ða; aþÞklare the annihilation and creation field operators, respectively. TheHamiltonian of the field in this case is becoming

HR ¼X

k

�hvkaþk ak (9)

The interaction Hamiltonian HI of the molecule with the magneticfield contains four contributions:

H ¼ HED þ HEQ þ HMD þ HNL (10)

Where HED; ~HEQ ; HMD; HNL are the electric dipole, electric quan-drapole, magnetic dipole and non-linear interactions, respectively.

The magnetic dipole interaction is given by

HMD ¼em0

2mð~Hð0Þ �~Jc jÞ (11)

where ~J is the angular momentum operator of the molecularsystem.The non-linear term is proportional to the square of the


439440441442443

re the first term represents the kinetic energies of electronsthe energy of interaction with the field ~A, the second term isnergy of field, and the three last terms are the static potentialgy of the electrons and nucleus. The system of reference incase is the fixed coordinate system~rc : x; y; z rotating with theecule. In the coulomb gauge and after some elaborative


magnetic vector operator ~Hð0Þ

HNL ¼e2m2

0

8mð~Hð0Þ �~rc jÞ

2

(12)

The next step it will be to describe the energy transfer betweenthe magnetic field and the molecular rotor and to investigate the



j3iivesose

thatoysl tofor; si

and

18)

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20)

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21)

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oleing

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24)

omthe

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7

G Model

APSUSC 16849 1–10

UN

CO

RR

EC

TED

PR

OO

F

conditions under which the magnetic field is amplified. Theinteraction between the magnetic field and the rotor will bereferred to one coordinate system fixed in space, out of molecule(inertial frame of reference). The molecule is moving in the flowwith velocity ~V and angular velocity ~V (turbulent flow) and themolecular coordinate system ð~rc : x; y; zÞ rotates with angularvelocities (v1, v2, v3) along the molecular principal axis of inertia(I1, I2, I3). By referring to the moving frame of reference fixed on themolecule, the position vector is

~rI ¼~r0 þ~rc þ ðð~vþ ~VÞ �~rcÞ;

where ~v ¼ ðv1;v2;v3Þ: The transformation of coordinates in theinertial frame of reference includes three components; the first oneis of the fixed coordinate system, the~rc term (putting~r0 ¼ 0), thesecond term has two contributions from the fast rotation of themolecules around the molecular principle axis, the ~v�~rc term,and the slow rotation of the molecule in the turbulent flow, the~V�~rc term. The equation of motion of the position vector in theinertial frame of reference is given by the equationd~rI=dt ¼ ð1=i�hÞ½H;~rI�, where H is the total Hamiltonian of themolecular system in the fix frame of reference with momentum ~pc.

In the case where both the fast (molecular) ~v and slow (flow) ~Vrotations are taken into consideration, the momentum of themolecule in the inertial frame of reference is given by

~PI ¼ md~r

dt

� �I

¼ md~r0

dt

� �þm

d~r

dt

� �c

þmðð~Vþ ~vÞ �~rcÞ

¼ ~p0 þ ~pc þmð~V�~rcÞ þmð~v�~rcÞ (13)

where ~p0 is the momentum of the molecule in the flow, ~pc is themomentum of the molecule in the fixed on the body frame ofreference, mð~V�~rcÞ is the contribution due to the turbulence ofthe flow and mð~v�~rcÞ is the contribution due to the rotation of theatoms of the molecule.

Let us consider the situation where the molecular systeminteracts weakly with a magnetic field, such as in the case of adiamagnetic molecule. For the case of simplicity we consider thatboth ~V and ~V are zero. In this case the Hamiltonian of the systemwill be described only with two terms, the first one is thecontribution from the free molecular system and the second one isthe magnetic dipole interaction term.

H ¼ HE þ HMD

¼ 1

2m

Xj

ð~pc jÞ2� 1

2eX

j

’ð~rc jÞ �1

2Ze’ð0Þ þ em0

2mð~Hð0Þ

�~Jc jÞ (14)

In the case of non-polarized molecules and low concentration ofcharges Eq. (14) becomes.

H ¼ HE þ HMD ¼1

2m

Xj

ð~pc jÞ2þ em0

2mð~Hð0Þ �~Jc jÞ (15)

The eigenstates c(W, w, x) of the rotational part of the Hamiltonianof Eq. (15) satisfy the following equations,

Hangjcð#;’;xÞi ¼ �hvRjcð#;’;xÞi andXi

jcð#;’;xÞihcð#;’;xÞj ¼ 1(16)

j1ih2j. The effect of j1ih2j on another rotational eigenstatechanges the state to j1i if the j3i state is the original j1i, but gzero otherwise. In other words, j1ih2j is an operator whapplication to an atom in a rotational state h2j removes it fromstate and put it into state j1i. We can say that j1ih2j destrrotational state j2iand creates rotational state j1i. It is usuareplace the operator j1ih2j by the notation similar to that usedthe operators which create and destroy photons. We define sþito be the creation and annihilation operators, respectively,make the change of notation

sþi si ¼ jcð#i;’i;xiÞihcð#i;’i;xiÞj: (

The molecular Hamiltonian can be written

Hang ¼X

i

�hviRsþi si (

Let us find now the physical interpretation of the operators sThe matrix elements of the rotation operator D(W w x)eigenfunctions of Eqs. (18) and (19). The angular momentoperators are the generating functions of the finite rotations

Dð#’xÞ ¼ expix�h

Jz

� �exp

i’�h

Jy

� �exp

ixh

Jz

� �(

Therefore, the operators sþi si are the corresponding angmomentum creation and annihilation operators

sþ ¼ Jþ¼ Jx þ iJy

s ¼ J�¼ Jx � iJy

(

The magnetic interaction part HM of the total Hamiltonian H

H ¼ 1

2m~p

2

c þ1

2

Zm0

~H2

dt þ ie

m

� �Xk

�h

2e0tvk

� �1=2

ðð~el �~kÞ

� ~JcÞ � ðakl expðið�vktÞ � aþkl expðivktÞÞ (

is given by the equation

HM ¼iem0

2m

� �Xkl

�hc2

2m0tvk

� �1=2

½ð~elð~k �~JcÞ �~kð~el �~JcÞ�

� ðakl expðið�vktÞ � aþkl expðivktÞÞ (

Eqs. (22) and (23) have been derived provided that the dipapproximation is valid and only the terms ~k �~r�1 are survivthe exponential series expansion.

We change now the frame of reference from the fix coordinsystem to the inertial one and the angular momentum~Jc becom

~Jc ¼~JI þmð~r � ~vRÞ �~r ¼~JI þ~JvR (

The term ~JvR is the contribution to the angular momentum frthe internal rotations of the molecule. The matrix elements ofrotation operator D(W w x) are the eigenfunctions of the angmomentum operator~JI �~Jv. Eqs. (23) with the use of Eq. (24)after some algebra becomes

HM ¼iem0

2m

� �X �hc2

2m0tvk

� �1=2

½ð~elð~k � ð~Jc �~JvRÞ �~kð~el � ð~Jc


25)

527528

It can be proved easily with the use of the closure theorem that

Hang ¼X

i

�hviRjcð#;’;xÞihcð#;’;xÞj (17)

Let us consider now the effect of the general combinationjcð#1;’1;x1Þihcð#2;’2;x2Þj which we denote for simplicity


kl

�~JvRÞ� � ðakl expðið�vktÞ � aþkl expðivktÞÞ

þ iem0

2m

� �Xkl

�hc2

2m0tvk

� �1=2

½ð~elð~k � 2~JvRÞ �~kð~el � 2~JvR�

� ðakl expðið�vktÞ � aþkl expðivktÞÞ (



The

~JI ¼

¼

A si

~JvR ¼

¼

¼

wheopera diinfin

LZ-ax(X, Y

comsþa

I

HM ¼

ThestateGold

w ¼

whemolnumtheof alw j, w

throit is

Fsyst

H ¼

and

g ¼

whemolmod

EHamHam

527528

529530

531532533534535536537538539540541

542543544545

546547548549550551552553554555556

557558

559560561562563564565566

566567568569570571572573574575576577578579

580581582583

584585

586587

588589

590591

592593

594595596597598599600601602603604605606607608609610611

8

G Model

APSUSC 16849 1–10

Pledo

UN

CO

RR

EC

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F

operator ~JI ¼ ~Jc �~JvR is analyzed as follows

~Jc � ~JvR ¼ ~Jxiþ~Jy jþ~Jzk ¼ ð~Jþ � i~J�Þiþ ð~Jþ þ i~J�Þ jþ~Jzk

sþðiþ jÞ þ isð j� iÞ þ~Jzk (26)

milar expression is derived for the operator ~JvR

~JvxRiþ~JvyR jþ~JvzRk

ð~JvRþ � i~JvR�Þiþ ð~JvRþ þ i~JvR�Þ jþ~JvRzk

sþvRðiþ jÞ þ isvRð j� iÞ þ ~JvRzk (27)

re the operators sþvR; svR are the creation and annihilationators of the angular momentum operator~JvR having in general

fferent set of eigenvalues than ~JI. The generating function ofitesimal rotations in this case is the Dð~vÞ.et us suppose now that the magnetic field is applying along theis of the inertial system. In this case the vectors~el; ~k are in the) plane and in the ði; jÞ directions and only the ~Jx;

~Jy;~JvRx;

~JvRy

ponents will survive. In addition we will retain only the terms; sa

þ; sþvRa; svRa

þ.

n this case Eq. (25) becomes

iem0

2m

� �Xkl

�hc2

2m0tvk

� �1=2

fði� jÞsþakl expðið�vktÞ

þ ðiþ i jÞsaþkl expðivktÞ þ 2ði� jÞðsþvRakl expðið�vktÞ

þ 2ðiþ i jÞsvRaþklexpðivktÞg (28)

total transition probability w between molecular rotationals induced by the external magnetic field is given by the Fermien rule:

2phjh f RjHMjiRij2rðE f R

Þ (29)

re hfj and jii are the final and initial rotational states of theecule. It is plausible to consider transitions between a largeber of very closely spaced intermediate rotational states. In

last case, the transition probability is given by the summationl the transition probabilities between the intermediate states¼P

jw j. This is similar to multiphoton absorption of photonsugh intermediate electronic states in atoms and molecules andvalid for either time dependent or static magnetic fields.rom the previous equations, the total Hamiltonian of theem now becomes in the second quantization formX

k

�hvkaþk ak þ

Xi

�hviRsþi si

þ gXkl

ðði� jÞsþakl expðið�vktÞ þ ðiþ i jÞsaþkl expðivktÞÞ

( )

þ gX

kl

2ði� jÞðsþvRakl expðið�vktÞ þ 2ðiþ jÞsvRaþkl expðivktÞ

( )

(30)

�X

k

e2m0�hc2

8m2tvk

� �1=2

(31)

angular momentum, the third term is the interaction Hamiltonianbetween the magnetic field and the molecular system and the fifthterm is the interaction Hamiltonian between the magnetic fieldand the molecular system due to molecular rotations. The physicalinterpretation of Eq. (30) is that the magnetic field interacts withthe rotational levels of the molecular system through the angularmomentum of the molecules. The efficiency of the energy transferbetween magnetic field and the molecules depends depend on thecoupling with the angular momentum of the molecule. Eq. (30) issimilar to the equation, which describes the operation of a MASERamplifier where a large number of spin 1/2 particles interact withan external magnetic field B oscillating with frequency vjR alongthe z-axis and when the coupling coefficients Kaj, Kfj are small(k < vjR) the Hamiltonian of the system is given by the equation:

H ¼ �hvaþaþ 1

2

Xj

�hv jRsZj þ �h

Xj

Ka jðsþj aþ s�j a

þÞ (32)

The next step is to solve the Heisenberg equation of motion. Sincethe boson and angular momentum operators commute, we obtainin a usual way the Heisenberg equations of motion:

dadt¼ �iva� i

Xj

gs jf (33a)

ds jf

dt¼ �iv jRs jf þ igsZ

jfa (33b)

dszjf

dt¼ �2igðsþjfaþ s�jfa

þÞ (33c)

dsþjfdt¼ iv jRs

þjf � igsZ

jfaþ (33d)

da

dt¼ ivaþ � i

Xj

gsþjf (33e)

Where now the operators sþ;zjf are defined as

sþjf ¼ ði� jÞsþ þ 2ði� jÞsþvR

s jf ¼ ðiþ i jÞs þ 2ðiþ i jÞsvR

szjf ¼ sþjfs jf

(34)

Eqs. (33a–e) cannot be solved analytically because they containnon-linear terms such as sZ

jfaþ, etc. We must therefore make a

linear approximation if the model is going to describe linearcoupling and transfer of energy and magnetic field amplification.When the energy is transferred from molecules to the magneticfield, the two level rotor flips from its upper quantum state to thelower one followed with the emission of one rotational quantumand, which its energy, is transferred to the magnetic field. In thecase of a week coupling between the molecules and the magneticfield (non-magnetic liquid) the initial thermal equilibrium of sz

jf

entities does not change appreciably and it can be substituted byits constant equilibrium value sZ

jf0. For according to Eq. (33), thechange in sz

jf is proportional to g which makes the term gsZjfa of

the order of g2. We may therefore have a system of linear equationsby replacing sz

jf by its equilibrium value sZjf0. This approximation

decouples Eq. (33) and we are left with an infinite set of linearequations:


612613

614615

re the summation over i, k, l is over the number of theecules, which are within a small volume t, the number ofes of the field and of the two polarizations~el.q. (30) is our final result. The first term of Eq. (30) is theiltonian of the magnetic field, the second term is theiltonian of the two level quantized molecular system of ~J


dadt¼ �iva� i

Xj

gs jf (35)

ds jf

dt¼ �iv jRs jf þ igsZ

jf0a (36)



46)

ity.of ad toeldthe

y v,eldat-thethes of

47)

The

48)

ainis

torsAtus,in

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6�5

ciess ine ofs toeticthe

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fiedthetiveandtivethe

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9

G Model

APSUSC 16849 1–10

UN

CO

RR

EC

TED

PR

OO

F

Since the equations are linear the solution for a(t) is of the form:

aðtÞ ¼ uðtÞað0Þ þX

j

y jf ðtÞs�jf (37)

where a and s�jf are the operators in the Schrodinger picture att = 0. From the commutation relation of the operators we have:

½aðtÞ;aþðtÞ� ¼ 1 ¼ juðtÞj2 �X

j

jy jf j2sZjf (38)

We must solve Eqs. (37 and (38) by means of Wigner–Weisskopfapproximation [60]. By omitting the details, and for gðv;vi;v jÞthe result is:

uðtÞffi exp �ivt þ g2sZ

jf0t� �

(39)

y jf ðtÞffig e�ivi jRtf1� exp½iðvi jR �v1Þt þ ðg=2ÞsZ

jf0�ðvi jR �v1Þ � iðg=2ÞsZ

jf0

(40)

where

g ¼ 2pg2ðv1Þrðv1Þ (41)

Here r(v1) is the density of the angular momentum states ofmolecular rotor modulated by the magnetic field at frequency vand v1 ¼ vþDvsZ

jf0

The small frequency shift is given by:

Dv ¼ }Z 1�1

rðv jf Þg2 dv jfR

v jfR �v

( )(42)

} is the Cauchy’s principal value and

hsZjf0i ¼ tanh

�hv jf

2kTs(43)

where Ts is the temperature of the rotational states, which can beeither positive or negative. Eq. (39) describes the amplification ofan initially weak field a(0) in the presence of an external magneticfield B, through the coupling with a rotating molecular rotor sZ

jf0, inour case the water molecules. Eq. (39) has a general validity and itis not restricted to a specific system. The molecular parametersenter in Eq. (39) through the angular velocity terms vjfR, whichdepend on the specific molecular system.

Now, let us consider a two level angular momentum system inan external constant magnetic field in the Z-axis, and a weak RFfield B1 cos vt, B1 sin vt in the X-, Y-axis the mean value of the z

component of the spin operator hszi (in the direction of theconstant magnetic field) is given by,

hszi ¼ cos2 1

2Vt

� �þ sin2 1

2Vt

� �cos 2u (44)

where

V ¼ ðv�VBÞ2 þGRB1

m

� �2" #1=2

; VB ¼GRbB

�hm;

sin u ¼ GRbB1

�hmV;

(45)

B is the intensity of the static external magnetic field, m is the

For weak coupling Eq. (39) becomes:

aðtÞ ¼ exp �ivt þ g2

tanh�hGRbB

�hmkTs

� �tað0Þ

�

haðtÞþjaðtÞi ¼ v2 þ g2

tanh�hGRbB

�hmkTs

� �2

t2hað0Þþjað0Þi (

Where ha(t)+ja(t)i is the mean number of field quanta in the cavEq. (46) is the final result. It describes the time evolution

small electromagnetic field fluctuation a(0). It can be amplifielarge values by taken energy from the constant magnetic fithrough the coupling term g2. In the case of the absence ofexternal magnetic field, a(0) fluctuates at the random frequencas is expected. In the case of high external magnetic fið�hGRB>2bmkTsÞ, a(t)! exp(�ivt + (g/2)) a(0) and the fluctuing frequency v is modulating by the coupling betweenmagnetic field and the two level system through g. In this caseaverage number of quanta of the electromagnetic field, and thuthe magnetic field is given by the equation

haðtÞþjaðtÞ! hað0Þþjað0Þi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ g2

4

� �t2

s(

in agreement with the classical theory of quantum amplifiers.gain G in this case is,

G ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2 þ g2

4

� �t2

s(

For weak coupling and for a constant magnetic field, the gcoefficient G of a fluctuating magnetic mode at frequency vindependent from the rotational frequency of the molecular roand thus independent from the type of liquid medium.resonance v jfR ¼ v ¼ ðGRbB=�hmÞ, g!1, and hence G!1. Thwhen the intensity of a static external magnetic field isresonance with the rotational states of a molecular rotofluctuating magnetic mode at this frequency can be amplifiedhigh values. As an example, for the water molecule the 5�1!transitions involve levels with rotational frequenvjfR = 2.2 � 1011 MHz, for J = 5, 6, and then B � 0.4 T. This iagreement with our experimental results where for this valuthe external magnetic field the formation of aragonite startincrease (Table 1). The quantum model predicts that high magnfields can be amplified in the turbulent flow as well. In this caseenergy a magnetic fluctuation can be amplified from the extemagnetic field through the low frequency angular velocity Vthe turbulent flow. It is expected therefore a magnetic fieldturbulent flow to be amplified to its higher possible value whenexternal magnetic field is oscillating at the high frequency ofmolecular rotors and at the low frequencies of the turbulence

Finally, in order to have a more realistic quantitative modeloss mechanism should be introducing in addition to the amplimechanism and the feeding a small time fluctuating signal intocavity should be introduced. The cavity now acts as a negaresistance device by operating just below oscillating thresholdwe might get a high gain in line with the general theory of negaresistance amplifiers. The average number of quanta insidecavity at the time t is in this case,


49)

706707

permittivity of the medium, b is the Bohr’s magnetron equal to9.27 � 10�24 J T�1, GR is the gyromagnetic ratio depended on theangular momentum J of the molecular rotor [61].

In the case of resonance where v jfR ¼ v ¼VB�ðGRbB=�hmÞ andfor weak RF magnetic field, the angular momentum processesthrough many cycles before hszi changes its value [60].


haðtÞþjaðtÞi

¼ 1

2�hvh pi2 þv2hqi2 þ 2v�h

gL

tL � 1þ gG

tG þ 1

� �1� expð�htÞ

h

� �(



whe

hqi

hpi

whe

h ¼

gG

andmag

aðtÞ

Ts iintroexcitermduethetionfluctmolenou(losshigh

4. C

Bsystliquformthatenerstatefavoquanmodenermomthemaxbe eagrecrys

UncQ2

[

Ack

Aanal

Refe

[1]

3

4

5

706707

708709

710711712

713714715716717718719720721722723724725726

727728729730731732733734735736737738739740741742743744

745746

747748749

750751

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10

G Model

APSUSC 16849 1–10

Pledo

UN

CO

RR

EC

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PR

OO

F

re

¼ k

ffiffiffiffiffiffiffi�h

2v

r�

2ðv�v jfRÞ cos v jfRt � h sin v jfRt

ðh=2Þ2 þ ðv�v jfRÞ2

" #

¼ k

ffiffiffiffiffiffiffi�hv2

r�

h cos v jfRt þ 2ðv�v jfRÞsin v jfRt

ðh=2Þ2 þ ðv�v jfRÞ2

" # (50)

re k is a constant and

gL � gGhszi; gL ¼1

tL¼ exp

�hvkTL

� �;

¼ 1

tG¼ exp � �hv

kjTs j

� � (51)

the small input signal, which is provided by the constantnetic field fluctuation is

¼ A0expð�iv jfRtÞ (52)

s the angular momentum temperature, and TL temperatureduced by the various losses, which diphase the coherent

tation of the angular momentum of the rotors. The first twos of Eq. (49) are due to the signal while the last two ones are

to spontaneous emission and account for the quantum noise inmolecular amplifier. Eqs. (50) and (51) predict high amplifica-of a small fluctuation in a constant magnetic field when such auation is coupled to the rotational frequencies vjfR of the

ecular rotors. In the case of resonance (v = vjfR) and at longgh time intervals, ht� 1, ha(t)+ja(t)i � 16/h2 and for h 0es � gain), the number of zero field modes in the cavity arely amplified.

onclusions

y applying a moderate magnetic field of 1.2 T in water flowems, more than 90% of CaCO3 nano-crystallizes at the solid/id interface of non-magnetic stainless steel 316 pipes in the

of aragonite. Nucleation as aragonite is taking place providedmagnetic fields were developed within the flow. Transfer ofgy from the magnetic field to the flow drives the system to a

of higher energy during the initial stage of crystallization,ring crystallization of CaCO3 in the form of aragonite. Atum mechanical model predicts that a fluctuating magnetice inside the fluid can be amplified to high values by taken itsgy from an external magnetic field through the angularentum of the molecular rotors and/or the turbulent flow In

case of resonance, the amplification coefficient attains itsimum value and the concentration of aragonite is expected tonhanced in comparison to calcite. The theoretical results are inement with present and previous experimental data of nano-tallization of CaCO3 in water flow systems.

ited reference

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nowledgment

uthors would like to acknowledge Dr. Zoran Samardzija for theytical work on SEM/EDXS.

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Nanocrystallization of CaCO3 at solid/liquid interfaces in magnetic field: A quantum approach

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