IEEE TRANSACTIONS O N MAGNETICS, VOL. MAG-18, NO. 6, NOVEMBER 1982

STUDY OF PARTICLE B E H N I O U R I N HIGH FIELD MAGNETIC FLOCCULATION

M.R. Parker, R.P.A.R. van Kleef, H.W. Myron and P. Wyder

ABSTRACT

A recently developed dynamic theory of magnetic field- induced par t ic le -par t ic le aggrega t ion in suspensoids i s fur ther ex tended in t e rms of charac te r i s t ic ve loc i ty c o e f f i c i e n t s i n t h e manner of t he i ne r t i a l e s s t heo ry of H.G.M.S. This i s done by expressing the f locculation process in terms of familiar parameters such as magne- t i c v e l o c i t y and capture radius. Hybrid f loccula t ion of diamagnetic-paramagnetic pairs i s described. Also demon- s t ra ted here i s the importance of the suspensoid sett- l ing veloci ty fol lowing the occurrence of binary f loc- cu la t ion .

INTRODUCTION

By magnetic flocculation i s general ly meant t h e pheno- menon of magnetic f i e l d induced par t ic le agglomerat ion i n c o l l o i d s . J u s t how long this technique has been known to the minerals processing industry is not c lear but an ear ly re fe rence is made t o it by Taggart 111 i n connection with the de-watering of ferromagnetic slimes. The extension of the technique t o t h e c a s e of paramag- netic suspensions was f i rs t suggested by C.P. Bean on 21 January 1971 i n an informally published document en- t i t l e d ' S t a b i l i t y of Colloids in a Magnetic F i e l d ' [2] After a gap of about 10 years the basic ideas of this paFer were elaborated upon by Svoboda [31 to account f o r magnetic f locculation in particular mineral types such as haematite, siderite and goethi te . Basical ly t h e approach by Svoboda i s t ha t o f ex t end ing t he D.L .V.O. theory [4] of c o l l o i d a l s t a b i l i t y t o i n c l u d e , i n a d d i t i o n t o t h e d o u b l e l a y e r and London-van der Waals in te rac t ion te rms , a magnetic dipole-dipole in- t e r a c t i o n which above a c r i t i c a l 121 or th reshold [?] f i e l d promotes c o l l o i d a l i n s t a b i l i t y and consequent f loccula t ion .

In ear l ie r papers [51 , [61 we have already shown t h a t fo r quas i - co l lo ida l suspens ions o f l a rge r pa r t i c l e s (> l v m diameter) - ie . ' suspensoids ' - a dynamic theory of magnetic aggregation can be developed which contains cha rac t e r i s t i c ve loc i ty coe f f i c i en t s o f a form resem- b l ing c lose ly those found i n H.G.M.S.

In t h i s pape r t he ea r l i e r t heo ry i s deve loped t o show how a secondary potent ia l minimum of the type described by Svoboda d e s c r i b e s i t s e l f . Also the hybrid magnetic aggregation of paramagnetic-diamagnetic particle pairs i s examined f o r t h e f i r s t t i m e . F i n a l l y , it i s shown how, i n weakly paramagnetic suspensions, the process of magnetic aggregation into clusters of more than two p a r t i c l e s i s f r u s t r a t e d by t h e e f f e c t s o f p a r t i c l e s e t t l i n g .

THEORY

C o l l o i d a l s t a b i l i t y can be expressed in terms of a sys- tem of two s p h e r i c a l p a r t i c l e s where, i n t h e i n t e r e s t s

M.R. Parker i s with the Department of Physics, Universi- t y of Salford, Salford MS 4WT, England. R.P.A.R. van Kleef, H.W. Myron+ and P . Wyder are with the Research Ins t i tu te o f Mater ia l s and High F i e l d Mag- net Laboratory+, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands. ?ar t of t h i s work has been supported by the "Stichting voor Fundamenteel Onderzoek der I laterie" ( F O M ) with f inanc ia l suppor t of t h e "Nederlandse Organisatie voor Zuiver Wetenschappeli jk Onderzoek" ( Z W O ) .

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of reduced a lgebraic complexi ty , their radi i are con- s idered to be equal and of value a . For spherical par- t ic les with very thin polar isat ion double layers the double layer potent ia l , OR, between the above-mentioned p a r t i c l e s may be expressed most simply by the exp-es- s ion [31, [ 4 1

where T = <a, K i s t h e Debye-Huckel parameter [3], [4], E y i s the r e l a t ive pe rmi t t i v i ty o f t he pa r t i c l e s , $o i s t h e i r s u r f a c e p o t e n t i a l , and ra = r /a i s t he no rma l i sed in t e r -pa r t i c l e s epa ra t ion . In t he absence of external f i e l d s and i n a s t ab le co l lo id , piR i s much la rger than t h e a t t r a c t i v e London-van der Waals interact ion poten- t i a l , '$L, for most in te r -par t ic le separa t ions . Neglec t - i ng r e t a rda t ion e f f ec t s [ S I , m d f o r an i d e n t i c a l p a i r o f p a r t i c l e s , gL may be expressed as

where A i s the so-called Hamaker constant [3], [4].

Clea r ly , i n any s t a b l e c o l l o i d a p o t e n t i a l b a r r i e r ex- ists which f rus t r a t e s t he t endency o f e i t he r pa r t i c l e t o come in to contac t wi th the o ther .

In the presence of an external magnetic induction f ield, Bo, t h e t o t a l i n t e r a c t i o n p o t e n t i a l o f t h e two p a r t i - c l e s , @T, can now be expressed as

where gM i s given by

- 4n x* Bf a6

911, @b* = 3 ( b )

This expression, which i s consis tent with the theory of H.G.M.S. and which i s expessed i n Kenne1 . l~ S I un i t s , i s at var iance with forms expressed b y both Svoboda [31 and Watson [ T I . I t follows from eqns. (3 ) and ( 1 4 ) t h a t an applied magnetic f ield can reduce the potential ener-

bar r ie r to zero thus a l lowing par t ic les to aggrega te .

A visual isat ion of the process of magnet ic f locculat ion can be accomplished by the cons t ruc t ion , in the iner - t i a l e s s limit, o f a force balance equation. Using the spa t i a l de r iva t ives 05 OR, 0~ and 0~ toge ther wi th the Stokes ' hydrodynamic drag force for spherical par t ic les ,

0018-9464/82/1100-1647$00.7501982 IEEE

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it i s possi'ole t o d e s c r i b e p a r t i c l e motion i n a l o c a l i - sed frame of reference in which t h e o r i g i n i s f i x e d t o one of t h e p a r t i c l e s and i? which the magnetic f ield d i r e c t i o n i s coincident wich the x-axis. The r a d i a l and azimuthal components of t h a t motion a re , r e spec t ive ly ,

and

Here

and

are respectively normalised characterist i .c magnetic, London-van der Waals and do.a-ale l aye r ve loc i ty coe f f i - c i e n t s .

RESULTS

A fami ly o f f loccula t ion t ra jec tor ies Tor pairs of para- magnetic p a r t i c l e s i s shown in F ig . 1 f o r one s e t of values of the constants of equat ions (7), (8) and (9). Here, a garamagnet ic par t ic le , in i t ia l ly pos i t ioned a t a point r i , 8 i i n t he x-y plar,e i n t h e f i r s t q u a d r a n t , precesses around the stationary particle clockwise, with an a lmost f ixed rad ius , un t i l it. approaches the x (magnet ic f ie ld) axis . A s it nears the x-axis it follows one of two a l t e r n a t i v e p a t h s . I f t h e i n t i a l i n t e r - p a r - z i c l e s e p a r a t i o n i s l e s s t h a n a f ixed (normalised) val- ue, r,,, the moving p a r t i c l e , upon approaching the x-axis moves rap id ly to the po in t (2a,0). I f rai > rea, t h e moving p a r t i c l e i s pushed fur ther a long the x-axis f ron t he s t a t iona ry pa r t i c l e . I n analogy with E.G.M.S. rca i s cal led the (normalised) capture radius . It should be noted that eqns. ( 5 ) and ( 6 ) and Fig. 1 a lso descr ibe f locc-da t ion t ra jec tor ies for pa i r s o f d iamagnet lc par -

y lun l ts of r/al I

0 1 2 3 L 5 6 7 8 xlunlts of r/a)

Fig . 1 . Clockwise precessional trajectories for bina- ry f loccula t ion of paramagnet ic par t ic les in a iocalised coordinate system.

15

10

5

0

-5

-1 0

-1 5

%*3

I

@ B r 0 1 7 T

@ B = 0 1 8 T

@ B = 0.19 T

@ B = 0 2 5 T

I I I l 1 1 1 1 1 1 1 1

0 5 10 'a Fig. 2 . Normalised on-axis (0 = 0) r ad ia l ve loc i ty of

haerna t i te par t ic le for var ious ex te rna l f ie ld values ( in the range 0.00 + 3.25 T ) .

t i c l e s . C l e a r l y , from eqn. (7) vpIa i s p o s i t i v e , even f o r p a i r s of pa r t i c l e s o f nega t ive (ne t ) volu;ne suscep- t i b i l i t y . A p r a c t i c a l example o f t h i s phenomenon i s described by us elsewhere. The p a r t i c l e t r a j e c t o r i e s o f Fig. 1 have mir ror equiva len ts in a l l 4 quadrants of the Cartesian coordinate system as we l l a s fu l l r o t a - t i o n a l symmetry around the magnetic f ield axis.

In h i s ca l cu la t ions on szrongly paramagnetic minerals, Svoboda [ 3 ! descr iSes the phenomenon of rnag3e;ic f loc- culation in secondary minima. Using Svoboda's d a t a f o r haematite the question of a secondary minimum has been inves t iga ted i n r e l a t i o n t o our dynamic model o f p a r t i - cle aggregation and i s shown in Fig. 2. A p l o t of r ad ia l ve loc i ty agains; i n t e r - p a r t i c l e s epa ra t ion ( i n t he 8 = 0 d i r e c t i o n ) i s shown l o r various values of t he app l i ed f i e ld i n t he r azge 0 T t o 0.25 T. Clea r ly , a t a c r i t i c a l f i e l d v a l u e o f around 0.17 T , the radial veloci5y acquires a negative value over a f in i t e r ange of i n t e r -pa r t i c l e s epa ra t ion . A t h igker f ie ld values , th is secondary minimurn i s replaced by primary minima.

MAGKJETIC AGGRCGATION OF PAR4RAMAGNETIC-DIANETIC PAIRS

The theory described here can al.so be applied, with OS- vious simplifying assumptions, t o t h e c a s e o f magnetic cross-f locculat ion between parmagnet ic a d diamagnetic pa i r s . In th i s c i rcumstance , wa i s negative - and it follows from eqns. ( 5 ) and ( 6 ) tha t the xoving par t i - c l e will precess now i n an ant ic lockwise direct ion. A t a c r i t i c a l v a l u e of e ( = e c )

€ c = a r c c o s ( l / J 3 ) ,

the leading term on the R . H . S . of eqn. ( 5 ) becomes

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negative which al lows, for values of r . < rac, capture t ra jec tor ies running c lose to the y -ax ls . F ig . 3 shows a family 0.f h y b r i d t r a j e c t o r i e s f o r t h e s i t u a t i o n i n which rac 5 . '

The formulae for the formation of paramagnetic-diamagne- t i c p a i r s i s

a1

when t h e London-van d e r Waals i n t e r a c t i o n i s neglected. By successive approximation eqn. ( 1 0 ) becomes proportio- n a l t o (VMa/VRa)2 [F ig . 4 ) . This is a f ac to r o f 17J2 smaller than the case of par t ic le cap ture for paramag- n e t i c or diamagnetic pairs.

Y ["ruts 01 rfal

6 0

5 .O

LO

30

2.0

1 0

L.0 5 0 6 0 7.0 YYn/VH

,

Fig. h. Dependence o f normalised f locculation capture radius as a function of (vpda/vfia)2.

INFLUENCE OF SETTLING VELOCITY

In previous publ icat ions / 4 ] , [5 ] we have indicated t h a t , i n weakly paramagnetic suspensions, the formation of l a rger par t ic le aggrega t ions i s absent and magnetic f loccula t ion i s r e s t r i c t e d t o t h e f o r m a t i o n of binary p a i r s . We a re , a t p re sen t , i nves t iga t ing t he l i ke l ihood of f loccula t ion of paramagnetic monomers with dimers, and t h i s w i l l be the ' subjec t o f a fu ture publ ica t ion . For the moment it i s of i n t e r e s t t o n o t e t h e compara- t i v e s i z e dependence of v ~ ~ , V R ~ , v~~ and v s a , where

i s the (normal i sed) se t t l ing ve loc i ty o f the suspensoid , n is i t s v i s c o s i t y and p e i s t h e e f f e c t i v e p a r t i c l e den- s i ty . This s i z e dependence i s i l l u s t r a t e d i n F i g . 5 f o r the minerals haemati te , s ider i te and goethi te . The ve- l o c i t i e s V L ~ . V R ~ , vIqa and "sa from eqns. ( 7 ) t o ( 9 ) and ( 1 1 ) can be seen t o be proport ional t0 .a-3, a-2, a0 and a , respec t ive ly . It i s ev ident tha t , when vSa v~~ t h e probability of observing higher forms than binary floc- cu la t ion i s negl ig ib le , s ince monomers w i l l acquire , following the occurrence of binary aggregation, a d r i f t veloci ty of -+ v s a p a r a l l e l t o t h e x - a x i s of the coordi- nate system of 'Fig. 1, and r e l a t i v e t o t h e b i n a r y p a i r f ixed at the o r ig in of that axis system. It i s ' c l e a r . from Fib. 5 t h a t v s a becomes s i g n i f i c a n t i n t h e p a r t i c l e s ize range of in te res t ( > lum) in p rac t ica l sed imenta t ion problems.

vo (PI

105

roo

165

1 6

a (m)

Fig. 5. Size dependences of V L ~ , V R ~ , v ~ ? and vsa for par t ic les o f haemat i te ( - ) s i d e r l t e (----) and goeth i te (-.-).

CONCLUSIONS

An outline has been given here of a simple dynamical theory o f magnetic flocculation in paxamagnetic/diamag- netic suspensoids. The concepts o f c h a r a c t e r i s t i c ve- l o c i t y and of capture radius, are of conceptual value here as they a re in H.G.M.S.

REFERENCES

A.F. Taggart, 'Elements of Ore Dressing' John Wiley, New York, 1951. D.R. Kelland, (private communication) J. Svoboda, I n t . J. of Mineral Processing, 8, 377, 1981. E.J.W. Verwey & J.Th.G. Overbeek, 'Theory o f t h e S t a b i l i t y o f Lyophobic Colloids ' , Else- v i e r , Amsterdam, 1948. M.R. Parker, R.P.A.R. van Kleef, H.W. Myron and P. Wyder, J. Magnetism and Magnetic Mater ia ls ( t o be publ ished) . R.P.A.R. van Kledf, H.W. Myron, M.R. Parker and P. Wyder. Proc. World F i l t r a t i o n Congress 111 ( t o be publ ished) . J.H.P. Watson, Proc. 6th I n t . Conf. on Cryogenic Engineering, 223, Grenoble 1976.