THE PHYSICAL PROPERTlES OF CYTOPLASM. A STUDY BY
MEANS OF THE MAGNETIC PARTICLE METHOI). PART II.
          THEORETlCAL TREATMENT

F. II. C. CRICK

Received February 9, 1950

A. INTRODUCTION

IN Part I (1) a method was described for measuring some of the physical properties
of the cytoplasm of chick cells in tissue culture by means of magnetic particles. The
cells were allowed to phagocytose these particles, which were then acted on by
magnetic fields, their movements being observed simultaneously under high magni-
fication.

In this paper the theoretical basis for the experimental methods used has been set
out. The results arc mainly standard pieces of magnetism and hydrodynamics, but
as they arc scattered about in the literature it was thought worth while to bring them
all together in one place.

The paper has been written for workers who may wish to USC the method thcm-
selves, or who wish to examine its foundations critically. I'or those only interested
in the results an extended summary of the theoretical conclusions has already been
given in Part I. An elementary knowledge of magnetism and hydrodynamics is
therefore assumed. There are occasional remarks from a more advanced standpoint,
but they arc not crucial to the main results.
The expcrimentai methods have been set out in part. I.  It will suffice hcrc to state
the gcncral theoretical probicms for which we require solutions. There arc three
main casts. They are

(1) Twisting: the permanent magnet case.
In this case the magnetic particle is turned into a little permanent magnet I)?
applying a large magnetic field momentarily. It is subsequently twisted by a much
smaller field applied in a direction roughly perpendicular lo its permanent magnrlic
moment.
(2) Twisting: the soft iron cast.
In this case the material is considcretl to have no hgstrrcsis. A magnetic ficltl is
applied at a small angle to the length of the particle, which is thus twisted.


(to I)ragging.

III Illis ww :I 1:1rgc fkltl, with a lnrgc field grattictit is apptirtt. `1%~ tn:IgwIic
tl:lrtirtc is I1i:lgneticatly satiiratctl, aiid after being twisted inlo linr, is tlraggml I)!
the field gra(licnt.

\Ve wish Lo calcrllatc the velocity (or the augular velocity) ol the partirlr iu lmm
of its six?, shalt, and magnclic I)ropcrtics, and thr physical prol)ertics and t~ountln-
rim of the iiirdiiini iu which the particle is cnrbrtitlcd.
\\`c tacklr the problcnis in the following order. M'e first show that we can ncgtrct
the> effects of incrlia, and by clcurcutary arguiuenls find how the forces vary with
scale. \Vc Illctl cousidrr the bchaviour of the ulcdiunl, discussing the cffccts of thr
shape of the particle, of boundaries, nut1 of non-newtonian and elastic behavioor.
Next WC give the formulae for the magnetic forces ou the particle, and theu in scc-
tion F we show how all the factors can bc combined to evaluate the velocity (or
angular velocity) of the particle for the three main cases.
l'inally we give some brief thcorctical notes on the production of large field gra-
dients and a unte on some comparative numerical values for the stress.

B. GENERAL CONSIDERATIONS

Incrlia \vilI tlcl:~y the approach to the steady state and will alter the linni
\clocily (lislribllliou. \\`c shall show that bOltI or thcsc rt~ccts C:III TIC ncg-
Iccl(4 `iii our c~sI)criniriils mainly l~c~aiisc the Ixirliclcs arc so Sninll.
`I'ticrc arc two I~roblcnis that can IK consitlercd srI~arntely. I:irslly, ttir
iu(srlia ol' thth lluitl, sccontlly the iiierlia of the particle.
l:or llio iii(~rlia 01' lhc Iliiitl lhr rclc*vaut characteristic of llic inotioit is llir
ralio or tlic incitia forks IO Ihr viscous I'OITCS (l~eynold's nuintwr). It i\
givc~ii 1,~              u p (I2

`j

(WI' I'or ~s::iliiI~l~ 111~ (`as0 01' :I11 osc~ill:~tiii~ sljlicw, (i), lJ;iragralJti X.54). `Illi~
fc~l~llllll:i :IljlJlics Strictly ollly to sllort lJ:irticlrs.
I I' Ihc aljovc l~ar;inicLcr is -: 1, the inertia of the liquitl is iicf.$igil~lr. .\II.

                                  1
other IISC of lhc lj:ir:tin~~tcr is that the \-aluch of - clrlincvl 1)~ pulling
          0          0 !`G
                                        u                   1,

rclu;ll to unity, gi\Ts lhe orrlrr of thr ti11Ir rcvliiiretl to aljlJIwach the Stcatlv st;111
\\`(b sh:ill 11ot t,c coiisi(lcrilig IJarliclrs I)i ggcr than 10 p in tliaiiirlcr, ?ro \\t

111:1\' put (1 = 5 x10-..' (`Ill. 11' UT lake as :I lower tmuntl l'or 7 the vaIu(b I'or
\v:llcr (0.01 lJoise), since lJiologica1 Iluicls can s~arwly Ijc less viscous, a11t1

put Q = 1. we obtain for the upper hound of  1
         0
                                      u the vallle $ IllilliSCc~JIltlS.
This is naturally wry 111uch sm:Itier than anything UT hnw mcasurctl. \\`(b
slioultl note in passing that if the margin were not so great 3 mow cxac~l
treatment woultl he advisable. Our formula gives the time for the particle
to reach a good fraction of its final vclocit~~, hut in certain cases the later
stages of the asymptotic aljproach to the final velocity may take much 1onp1
IhaIl the earlier stages.

These results only apply strictly to the case of an infinite fluid. \Ve can
give an argument which suggests that the efTcct of atltli11g fixed boundaries
will usually hc to decrease the time to approach the steady state. Consitlcr
Iw~ cases: firstly a particle in an infinite fluid, sccor~tlly the same particle
with fixed boundaries added to the Iluid. IAct the forces alJplietl to the part-
iclcs he such that the same steady velocity is attained in the two cases. \Vc
will assume that as a rough measure of the time to alJlJrwch ecluilihrium
w may take the ratio of the kinetic energy of the fluid to the rate of dissi-
pation of ctnergy, hoth {or fhe stetrtly stnfc.  `I`hc elfcct of fixed lJoundurir5
is lo incrrasc the rcsistauce anti thcrerorc the rate of dissipation of cricrgg.
`I'lic tJoi1ntlarics also reduce tile amourit of fluid in motion and over Inost ol
lllr volimic' tlccrcasc the Iluitl's velocity. The total kinetic cncrgy is thus
likely to tjr retliicetl. `I'hcrcfore the ratio rrfcrrecl to aho\-c \vill ho clccrc~:rsctl.
`I'0 (~Stilll:ltc the Clrcct Of the illCrti3 Of till! lKlrticl(! \v(f UJllSitlcr the' c:IS('
01 lhc dragging ol au iron sphe~.  `I'hc ratio of the inertia to the viwou\
forces is

II' \v(' tlcliuc ltic charactcristit: liiiic 1

                             u 1)~ tli(b c7lualiorl


     ?`he physienl propdies of fyfopltrsm. I'm? II.       509

\vherc I = magnetic induction of the Ijarticle
    (III
     -=
    d%  magnetic field gradient producing the force.
A force per unit volume has the dimensions nl I,-' T2. The only combina-

which nl)art I'ronl the numrrirai factor is the same form as Idore, csrrl~l
Ill:11 e' is now the density of the iron. It ran ensiiy be shown that exactly the
S:II~I(~ tyl~ 01' l~arametcr is involved in the case of rotation. Thus the clkcts
of lhc inertia of the l3articlc are negligible.

The case for non-newtonian fluids, whose "viscosity" varies with shear
is not quitr so clear VIII.  However the margin in our experiments is so big
that we can simIdy ronsitler the extreme case where the inner parts of IIIC
liquid move effectively as a solid, and the older parts as a newtonian liquid.
This is clearly similar to the movements of a hotly of increased radius in
water. In onr experiments the radius of the body is hounded by the size
of the cell, so that we again get a very small value for the time to reach cqui-
lihriuni.

The case for the visco-elastic medium is given on page 519.
The conciusion is the sa~iic. `I'lir~s for all possible cases in onr experiments
the slenciy state is reached in n time very much smaller than anything we
can measure.

2. Scale

\\`v sll:lIl IlCXut SIIOW, by sinildc tiinic~nsion:d arguments, how the forrcs
involvctl in clraggiiig anti tlvisling change with scale. \Ve only consitlrsr ii

range 0I' s~:ilc OV(`I` \vliic*lr the niagnctic Iuctors (r.g. y) C;III be consitlcrcd

c~c,llsl:lll1.

( :I) l)r:iggiiig

`I'lic~ ti(bilsily 01' lli(* Ib:irLic*le is c~lvarly not illvoivctl in lhc stc:ltly state contli-
lion. \\`c rc.slric.1 011rsclvc3 Lo a range of sralc over which the rlcrisity rbl llt(
Iliiitl can :ilw bv ip7orctl, for the rr:7sons giwn ahovc.
`I'hc 7mig7ictk ficxltl \vili l~ro~luw a l'orcc per unit volunlr givc17 by
                 ,tifJ

(f%

lion of a, u and 7 which will give this is

.

\\`hence we obtain

Thus, as the scale is reduced, the velocity decreases as the square of the
characteristic length. The time for the hotly to traverse its own length in-
creases linearly with the reciprocal of the characteristic length.

(h) Twisting

Here the magnetic conple per unit volume depends on

(JH) f(O)

where 0 is an :wglc.

This has the tlimrnsions AI I,-' T2 and from w (the typical angnlar ve-
locity), 7 and a WC cati only form the ronihinalion 7 0).

Sate that if the licluitl has I,ounciarics they, loo, lllrlst Iw sraioti 1'01. 1111~
:~IKIVV results to al)l)ty.

i~inniiy note tiint the mngnctir contlitions hn\rc 17ot I)ctl7 sc-alctl. Sv:lli7lc:
Ilir magnets l)roducillg the licit1 mnltrs no tlilfcWnW to Lhc ~~:lluc* of IIl(a licbl(i,
l~it clots alter the licid gradient.  `I'his rcsc~rvatiori is lhcrc~l'orc iii~t~ol~l:71iI
ill tlraggiiig but not in t\visting. it is easy lo see that il' \vr do sc*:ilc Ihr rllagtic'ls
I;)r lhc dragging C:ISC the tinic for ;I l):irlic*le to be tlr;igqtl its O\YII lviigll1 i\
i7iticpri7dri~t ol stair for both the nc\vtoiii:in :III~I the Iion-llc~\vtoiii:1n (*:isw.


;,I()                I'. 11. C. CSiCli

       t:.  `I'IIE FORCES ON A BODY IN A VISCOUS FLUID

1. l'trricrtiorf rcrilh slttrpc

`1'11ta v:lri:ilioll 11 ill1 S~I:I~W is r~:~lur:~lly more corii~~li~ated than the wrhlion
\\.illl W:IIC, ;III~I has oni!, IJWI~ ~v~rltctl out for slwcial cases, usually in an
itIlitiil(~ Ilriitl. It is l)ossitjlc to ol)t:rin tlir rrsult for the grncral ellipsoicl. hl
11 (* sl~:~ll 0111y rl110t~ tl10sc~ 1'01. 0va1.y cllipsoitls (IL` wvolution, 3s wr rrquirr
Ilrch111 jllcarcslv to gi\,cs s(,jiic itl(x:i of the g(-i1cr:il IJrhaviour. \\`r consitkr in
Illis sc~c~tiori tllc, ~`OI~~II~I~:IC l'or :I IJotly iniiilrrwd in an inlinitc ncwtoriian liq-
uitl, I(L:lving lo tlic t\vo l'oIlo\ving scclions tlic consideration of hormrlarirs
arid 01' non-iic,\vtoni:rrl hcliaviour.
\V(: shall 11s~ the L'ollo\viug notation for the ovary ellipsoid:

m:~,jor axis = 0
minor axes = h = c
cccentririty, E, given by 1 -c2 = bz
                     112

2(1 -3)
  e3 ~- (1 log :" - t.) by MO.

_ (1.2;c2) log 1+ e
(               l
                ~~ _-
       3     l-e I? 1 bY Al*

ant1

b2 - 106 ;s by x0.
e

(All logs aIT natural logs).

For an ovary cllipsoitl of revolution:
\\`c shall only wnsitlcr the case ol rotation about a minor axis. This 11:~
bceu solvctl by IMwartls (2), but owiug to au algcbraical slip towards th
cntl he oniils the factor 2/S. The correct result, in his notation, is, for thr
gcricral ellipsoitl,                     I      r) \

     The physicxl properlies

`I'liis bring :itl:iljtecl lo our notation,
rllipsoid, i~~comcs

\\`c cxprws this as couljle = 1;. 8 n q w u2 0 and w:ili~alc Ii numcri~ally 1'01
tlifkrcnt valrics of cc/h.
Sonic ralucs nre given in `I'able I.

                         T.4131.~ I
     ; 11  1.0 /  2.0 1  3.0 /  4.0 1  5.0 (  10.0 )  20.0

             k jl  1.0 1  0.84 I  0.91 1  1.00 j  1.10 (  1.60 I  2.5(1

In the limit (a/b) -+ 00, k --f i

(b) Dragging
For a sphere:

the force is Gnq a u with the usual notation
(7, para 337)

For au ovary ellipsoid of revolution
(i) in the direction of its major axis, a

  we have        force= Onl;lRv

  (7, paras 339 and 114)
This rctluccs to

R= ;.

ae

For the special case whew the cllipsoitl is wry loug, so that a >, 11
         R-3.       (I


512                17. II. C. Crick

NtItii(~ri(*:il valuc~s arc given in `I'al,le 11 helow.
(ii) iI1 Ihc tlirection of its minor axis, 0

(7, paras 339 and 114)

and whc11 (I + b

?J'Ilnleriral v;~lucs are given in Table 11.

( :I) l$ountl:iries :iw, in gcmeral, 11101`c iriiporl:int in dragging than in luisliy!.
This is riot surl)rising \vhcn we wmenilwr that the viscosity of a viscow
                                   1

lliiitl tllrollgll \vhic~ll :I sl)licrc is tlrrr~y!gd f:llls 0lT as a lollg \\'ay fro111 ll1r
                                            r
sl)licw (ill wiilrast to the vase \\.hCrC the inertia is important and visc.o+
rlcgligil,lp), \vljilr Ihc arigirl:~~~ \clocily iii lhr Iluitl round :I Iwtrrlillg sl)llc*~c~
f:1Ils 011' ;1s :-

The physical properties of c~ytopImm. Purl II.       513

`I'he folluwillg examples illustrate this point.  111 all casts n is the inner ra-
dius, 0 the outer radius.

(i) translation of a sphere in a fixed cylindrical tube

force = Gnqau 1 + 2.104% + ...
   (    1

(ii) rotation of a sphere in a fixed spherical shell

(iii) translation of a cylinder in a fixed cylinder

Force per unit length = 2 nq u (----.-- 1
               log a - log b

(iv) rotation of a cylinder in a fixed cylinder

couple per unit length = 4x7 a2 w (&p)*

For the translation of a sphere along the axis
the solution with the higher terms included is

\vhcrc (2 = ratliiis of sl)hcre
    b := radius of liilw.

of a fixed cylindrical tulw

This fornii~l:i is for llic c.nsc ~~~h~~n l~c~y1oltl's I~III~I~K~I~ is iiiliiiil(*silll:ll. (1:. )
`l'lilis \vlic11 b = 3 n 111~~ for11i11l:1 givw ;I rcbsislanc-c 01' 2 . 7 tiiilw Ill:11 1'01.
ali iiililiilc Iliiicl , ant1 for b = 4 n, ;I I'avtor of :ll,ollt 2 . II, so ll1:11 in(*rv:ls~*\
of lhis sort are wry ~~rol~;iblc ill ;I s111all cdl. Krigl1lmuring inc~liisions ~II:I\
well I1:ive cluite a large rlTcct.

For bodies of a shal)e not greatly clill'rrcnt from :I sphere, a gootl :1l)l)ro\-
ii11:ilion is to usr tlic al~ovc Torlliuln I:lliillg I'or "n" lllc \~:1111(~ for lll(b "c(liii\
alrut sl)licw"  in llic infiiiilr~ Iluitl c:isc. This :Il)l)i`osiIii;iliori C':III 0111~. I)(,
vvrv rougli for llic case of :I very c~lo~ig:~tc~tl I,otly, 0~` of :I \v:lll v(*ry (.Ios(* III
:I Imly.


51 1                 F. II. C. Crirlc

The physical properties of cyiopirtsm. Pclrt I I.       515

`l'hc rl1'rcl of lwuntlaric3 on the couJ)lc cxcrtctl on R compact twtly of rr-
vollllion rotating alwut ils axis of syniii~etry is fairly easily grasped, nntl ic
SIII:III iii~lvss Ihr l,ollnclaries are near the rcJu:ilorial bolt of' the hotly.
`I'lrc 1~11'cct ol the bountlarics on less restrictctl Jwdics (including our Jw-
tic'lrs) has not 1)wn worked out, but l~clow wc try to give a rough bound.
(i) I<otlics of re\olulion.
`I'hcsc have l~rcn treatctl hy .IcfTcry (0). He gives a gcncral trcatmcnt which
can IX tlesrribetl as follows: for any body of revolution rotating in an in-
finite fluid about its axis of symmetry WC can find a family of surfaces earl1
of \vhich rotates with constant angular velocity. \Ve can then always ol)-
tain the solution for the J~otly rotating within one of these surfaces, rcgartlctl
as :I fixed wall, by SUpriillpOSitlg a uniform counter-rotation on the wJ~oJr
system to twin6 the "wall" to rest.  This explains the form and intinMlc
relationship hct\vccn the fall-off of angular velocity and the increase of
coul~lc tluc to a hounclary in the two simple cases of rotation quoted almr.
.Jcll'ery gives the formulae for the case of an ellipsoid of revolution. .I
Iyl)ic.:il result, for a J)l:inctary ellipsoitl with an axial ratio of 2.24, sur-
roiintletl by :I ronfocnl plaiictary ellipsoidal shell such that the spciug at
1110 c~lii:ltor is 20 per cent ol' lhc body's ccJuatoria1 radius, shows thal lhr
shrill inrrc:lscs the viscous c011J~lc by a factor 0JT 1.9.
.J(~fJ'~ry II:IS also s~tl~c~l another illu~llinnting raw; that ol' a sphcrr rolatirlg
c*losca lo :I liscbtl J)l:lllc J,~rl)(~i)(lic~iil:lr to the axis 01' rotalion.  `l'hc rcslllts sl1on
ll1:11 111~ l)l:inv Ii:rs IO lw c~slrcwrly rlosc to lhv pole to Iin~e any ronsi~l~~r-
:ll,l(* vli'wl e.g. :I[ a tlisl:incr 01' 2 Jwr crilt 01' Ihr ratlills it incrcascs lhl~ w
sisl:iric*ch I,? olrl~ Ii Jwr (`(~111. `I'liis is Iwc:riisc~ Ihr Ill:l,ior lxirt ol' llic Yiwl,r-
(+oiil~l~ V(IIII(~ fr0111 Ilic cvlii:itclri:il 1wl1, \~Ilclc hth the :irin of the ~olll~l~~
:III(I Iliv vcLloc.ily :iw Ijig.

\\.I. lll:l~' IIIIIS (~(rll(~lll~l~' l1l:ll lllllwG IIIC I,cNllltl:ll~l' :l~'l"'o:ll'll~~s C~IOSC~ IO t1w
I,c,tly al l)oiills I`:ir l`i.olll lll(~ :isis of rolalion, 111~ iiicrc:isr in c011l)lc is iilililr(-l!
to 1,~ I jig.
(ii) OlhcI. Ibotlic3.
In gcii~~r;~l oiir lxirLicIc5 :irr nc)l lwtlics of rwolution. ant1 cvcii if thw wm
\\`(I (*oill~l iiot ce:isily tlislingirisli Ilic4r tlill`(~wllt :tllgulnr J)osilions lltltl(~r 1111~
mic~rosco~w. Al~l~r~~ntly no case has lwctn solrrtl which hrlps us hw.
\\Y(b l~rol~osc to oslim:itr Ihc ~~~uplc on an ovary elliJ~soitl ol' rcwluli~ln
rolaling insitlc :I lisctl sJ)hrric:ll shdl, radius rl (t! not too rlosr to n) a'i L'(d-
lo\\-S: \\'C' SllSl""'t Ill:lt SUC.11 a SllC'll \\ollltl IlO1 i~lCl'<'aSC the COU~'k 011 1llC' Cl-

lipid by more than it woulcl iucrcase that of a sJ)hcre OF radius n. This
iiiiplics that the couple would be increased hy no more than

8      f13
nrj"36.J -- ( 1
      d3

                    cl3
iieglecling terms higher than -~- ( 1
                    d3

Sow we have shown (page 511) that the couple on the ellipsoid in an infinite
fluid is alp-oximately given 1)~

              87cqa2bw

SO that the total couple could he written as

dierc p is probably l~etween (a/b) and 1.  However this estimate is litlle
iuorc fl1:1n a @CsS.

For a comparntivcly short particle the lwnntlarics are not likely to cause
a large variation in the couJde e.g. fur d one half greater than CC, and for ;I
lmlicle oE axial ratio 2 : 1, the collJ11c is probably not more than tlonJ~lrtl.
Fiiially we consitlcr Ilo\\ a Jixctl obslruclion at oiic entl 01' the l,:irlicl(~
:iITwls its hchaviour. X IY~II~~I wtilnat~ cati 1~ matIc by wnlJ)aring this (`a~(:
nil11 th:~t of a pirliclc of Iuirc the lc11gll1, :rncl ac*tctl "~""I by tlollhlc lll(~
nl:igiidic CWII~~IC~, will1 ol~slrii~lioris nwr ils mi(ltllc. It is rlc:tr Ihal in Illis
*w~Il~l case 111~ niotion woiiltl h:ircIIy ~JC altc~twl :it all I,v IIIc* ol~slr~lc~liorls,
35 llwy arc so near Ilic asis 0I' rot:llion. `t`hc viscous cc,iil,lc~ 011 111~ sc*vcbiitl
pirlicl~~ iii ;I ilc\\.loni:~rl Iluitl is \.(bry roi~gl~l! I lirilw llial 01' 111~ lirsl (\(YL
l,:lfic' ,512). 'l'lic~ix~fc~rc lliv :~r!g~~l:ir \~~~loc,ilv will lw Ii:il~c~l.
'I`liiis iii a11 inliiiilc I,c\\4oni:lI, liclilitl :III ol~slr~iic~lioli al 011v CIICI illc*rc*:,sc3
111~ rwisl:l~iw so Ill:11 al,0111 Iwiw IIIc~ co111,l~ II~WIS IO IN- :il)l)lic~tl lo ,givcb 111(~
\:iuw :iiig~~lar vclocily. It :1ls0, of vouw', ~~IYNI~I~VS :I sin:~ll Iwnslalioir (II' (IIV
p:irlic*lr.

SI:ilotl in this \v;iv it swnis J)rot)al,lr that 111~ wsult \3~~11ltl also :ilb1)ly IO
lllost non-nc\vlonian Jicluitls.

Sotc lhat it is not csscnlial for lhc ol~sli~u~lion to lorrc11 llic l):irlic.l(*, :is :I
WI`v close :il~pr~a~h mav ~HX~~~IKT sul'licitnt rcsislanw to rctluw llic :ingiil:it
docity apJ)rccial)ly.


        .I dS

                    VP -Lz sn
                     :*--
                          ' ds
        i-
                         . 17
                             sa
\~llere q! = experimental raluc ol the viscosity
     s = shearing stress
     %I = shearing stress at inner cylinder
  s,, = shcarmg stress at ouler cylinder

`I'll;11 is, if wc substitute the experimental values of angular velocity, etr in
the usual l'fJrlnUh for the viscosity, we obtain the above value of qe.  It ic
:lssiiiiicfl that the "viscosity" is a function of the instantaneous value of the
shear only.

`I'his solution is possible because the surfaces in the fluit which retain
their sh;lpc (i.c. inore as if solitl) arc surfaces of constant stress, ant1 our
ea11 thcrrl'orc give the stress distribution across the auuulus irrespective of
thr curve of ! ngainst s.
  0
         7     This condition also applies to the only other CM
which has horn worked out, namrlg that of the ilo\\, throqh a cnpillar!.
altholl~ll the :Ictual liJrniula given :rhJvc floes Ilot.
11'0 lhus sc(`, as is intilitivcly otJvifJus, that the drcct is to slur over lhr de-

tailed variations in the curve of  1 .
      0
                      r against s, ant1 to give an average ~11~.
The tlcl:iils arc usually got hy inaliing b nnly sliglltlg greater than (I, so tli:rl
tllC ;l\t!l':ljir is tLIliPl1 over only :I very sniall lmrlion uf the curve on r:if*l~ II,`.
cwion
` , anfl by r(~pc:llillg at tlilfcrcllt ralcs 01' she:lr to cowr a \viflr r;1np
\\`C arc Clc:lI~Iy lltl:l1Jlc to tlo anything of this sort iii f)ilr expcrinic~nls. I.I\c.rl
liar :I sl)hvrc~ ii1 a 1101i-ii~~\\.tolli:l1l liftuitl the prol~l~flli is ol' a flill'crcnt orfkr 14

flilli~rilly ;11if1 flr~vs not :il)lwar to Ilavc IJccn solYf*fl.  `l'hf~ slif~ar slrf59 1 :1rif5
I\-iIll I;ilit\lftr 1'roiii zc'ro to :I 1llaxi1llu1ll :inft it is not at all clear Ilo\\ it flislri-
hlcs itwIT iii llic iion-ti~\\.lfJllian case. `l'hr prf~l)lciii of the cllipsoifl is CT'C'II
11101'c llfqJdess.

flowwcr it alJ1Jcars cxtrcrnf*ly plausible that if wc sflbslitute the vxprri-
iiicntal ~alucs iii 1he lorniiila, the value or 11 flcrivctl \vill corrcslJfJiifl to sonIf
average raliic; that is, to sonic iJoint on lhe curve of q against s htworn tlIf>
m;isiniuni v:~luc or s ant1 some iiiiiiiniunl, prohal~ly zero. Rloreowr it s(`C111S
\-cry likely that the :ItJl)arrnt change of 7 with shear will bc less than lhc
iiiaxirlirilii ~liaiigc :kiig\vlicrc wilhiii lhis wgion.  Since we ~:iiii~ol rely on
our rxlwrimcnlal arrallgcnlcnts to mcasurc s111all tlilr'crcilws accuralclg,
w roncl~rflc that this mcthotl will only show up largr changes ol' "viscosity"
willi shear, and may conceal small changes.

\\`c note at this point a feature which may bc expected in the behaviour
of a non-spherical particle in a non-ncwtonian fluid of the type which be-
COIIICS very viscous at low shearing stresses.  Since the shear near the axis
is much less than that near the ends of the particle, the fluid may behave
almost as a solid at points near the axis, and also at points far from the par-
ticle, so that the llow may take place over a rather restricted region.

Il. Z'he forces on u body in a jelly

We can apply almost all the previous formulae to the case of a particle
in an isotropic elastic medium. Since we are only dealing with feeble jel-
lies we may take Poisson's ratio equal to 4. For small strains all the algebra-
ical results for rlelocity in a viscous newtonian fluid apply to the dejlecliou
in a hookian elastic medium, providing we substitute n, the rigidity modulus,
for r] the viscosity.

For example, the couple on a sphere rotating in a viscous liquid, which is

  8nrjU3W
7 =: viscosity

w : angular velocity
a = raflius of sl)hcrc

riiahles us to write down the couple on a sphere cndwtlflccl iii aI1 csl:lslif*
iwfliuni as (for small angles)
                    8 ?c Ii fl3 0

II = rigiflity 1ii0flulris
0 = ;III~LI~:II. flrllf~f~lif~i~


:`j 18

F. II. C. Cridi

H            4               F
             Time
           Fig. 1.

`I'0 clarify our terms consider next a simple v&o-elastic medium, \vhirlt
hehavcs as shown in figure 1 when a constant stress is suddenly applietl
and later sudclcnly removed.  In this figure the elastic properties arc re-
prescntcd by AU ant1 DE, the Gscous damping by the exponential rise AC
and fall DC, and the pseudo-viscous yield occurring along CD, l)y El;: `I'lle
relaxation time is HA.

If \v(: have :I l)article cmhctltletl in a simple visco-elastic medium \\hose
rc~lasalioii lime is wnst:int with stress, we can obtnin lhc relaxation time es-
]wrinicnl:illy mithout kno\viuc :llly of lhc dclaih of the particle, hountlarirs,
etc. by simply dividing the elastic tlcllcrtion (for small strains) hy the ror-
rcsponclitig pseudo-viscous yield ralc for auy given applictl couple. `I`llis
follo\vs from the siinilarity in form of the viscous :lu(l cxlastir wcflicic~nls
mculioncd al~ovc. If the rcl:~xntiorl time is a function of the stress, th(bre ii
no siinl)l(u scjlulion for Ihc fienrr:rl case.

\\`e have so f:lr neglcclctl inertia.  \\`c now lake it iuto a0wunt antI SIICN
Ili;it the lwriotl 01' I'lcc cjwillation is vrry shot?, anal thr d:inipiug high, ;ig:iill
Inaiuly I)wauw lhc lj:lrlic*lcs are so s111al1.

a2 e' cl2

                              n

\vhcrc: 1

     u is the orclcr ol' Ihc period of free oscillation, and Q' is the dc~~~il\
of 111~ l):irlic*lr~.

Putting ! = 1 niillisccoutl, fl = 3 1'1, ant1 e' = 4, bvc see that for the IX-
      u

rnmetcr to equal unity, 21 must he 1 clyne/cm2. `l'his is extremely fccl~lc (f(jr
a uornial gclntin gel n = 103 to 105 dynes/cm2).  For a stilTer rnctliurn the
linw is ol course shorter.

It is clear that for a substance as feeble as this the viscous damping would
in praclicc be important. \\`e will thcrcfore consider a particle in a medium
with tdh viscolls ant1 elastic properties, and lint1 the condition for crilical
damping. \\`c assume that the times involved arc so short that we can neg-
lect any pseudo-viscous yiclcl.
If WC work throug$ a particular case, such as a sphere untlcrgoing rotaq
oscillations about its axis, and neglect for simplicity the inertia of the me-
dium, we obtain the condition for critical damping as

  4v2
n--Z
- e' fl

e' = density of sphere.

\\`c could have ttcrivetl this, without the constant, in a rough and read)
niaunrr by cclualing the values of o2 which make lhc two previous climcn-
sionlcss parameters (page 506 uud page 518) equal to unity.

Pulling a = 3 IL, 7 = 0 . 01 poise, and e' = 4, wc obtain 11 = 1,100 dynes/
('1112. `I'liis is not high, IjIlt it is rather liighcr than our estimates. E\loreO\c~r 3%
nlwntly obsrrwtl it is highly unlikely that lhc "visrosily" corfficicnt iii hio-
I0gical malcri:lls is as low as thaf ol' witrr.
As rcg:~rtls llrr elTe& of lbountlarics,  a c~lowr rsnniinalio~i shows 111:il
\iiiw llic ralio 01 111~: viscous lo c~laslic forcc>s is intlclwu(l(~nt of Illc~nl, IIlk*
c~r)ntlilioli lor Il~c tl:iuil)irig rcsli1:iining crilical or grc:ttCl \vlieii lJcbtil~tl:iric~s
:iw :rtltlccl, rctlilcc3 IO Il~c collclilic~ll tlial Ill? r:ilio of lhc Oaiiil)iiig l'or(*(t% IO
1110 inci.lia I'orws shall not tlccw:wc~, xvliicli \vc li:i\.c :rlrc:itl~ slic~\\-ll (lb:lg(*
1,117) Lo Iw ljrolj:~lblc* iii iiiosl casts.

\\`c Ihits concliitle lIi:it llrc li~11e-lwriotl of frw osc*illatiou 01' 011r l):irlic,l(*\
is Iws thau a milli-scwntl (l~r01~~1l~Iy inucll less) :lu(l that Ilir tl;tiiilJirig is
rriliral or greater unless the rigirlity m0~l11lus is high antI lhc visrcjsil\. I()\\.
which is not the cast in our c~xpcrimcnts.
\\`e have only consiclrrccl the caw 01' sinall strains. I.:irgc slr:iins tll:l!'
wll iucreasc the :il)t)arent valt~c of the rigiclily ll~otlulrrs, c~al(*~llalC~tl Ilsillg
tlic simple theory. It is unlikely ho\vcvcr iu 0111 ~ascs lo iwwasc il sul`li-
cicnlly to alter our general conctusious.


\\`cs li:~vc tlvalt \vith this Imint about critical tlaniping hecaiw it is so~w-
tinlw srlt,yyst(vl that :I rcwnancc nwthod sl~oultl lw usctl. Apart front th
cslJcrimoJJtnl tliflicrJlties due to the very short time periods, it is clear lh;Jl
tllc cxtrcww1y low "Q" of the system wo~11d make this unprufitnhle.
It has also been suggested that the rcstorilig force 011 a particle coultl III-
i~Jcrc~asctl by magnetic* Jncans, though this would mask ally elastic cliwt
due to the Jiicdiunl. The siniplc theory shows that in the case of a qJhrre
this is equivalent tu the nJctliuJn IJaviJJg a rigidity modulus of

ZIH
24n tlyJlcs/cnl~.

A more exact treatment would be required if the method were seriously con-
templated. It is thus rJot iJnpossilJ1e that oscil1atiolJs could be produced. Thr
envelope of these oscillations is deternJi~Jet1 by the ratio ol the viscous 10
the inertia Curces, and tliis could be used to measure the former. As has
been sho\vn this involves Jnaking mcnsuremerJts in a time 11rohalJly of the
o~dtrr of ~JJicroscwJJtls. TlJis is not irilpussible. but it is certainly Jiot ens!.
The only ~lt~v~llltagc of suc:lJ a mcthotl is that it is not rlccessary to know the
magnetic forces accurately, although it is csserltial to know the exact formo-
I:Je for lhc viscous and the inertia forces. These coultl il ncrcssary bc chrrkctl
IJy calilJratiorJ experiments iii a kno\vn liquid. \\`e have IJot pursued Illi\
aiJproach l'urthcr.

E. THE FORCES ON A HODY IN A MAGNETIC FIELD

AssuJJJitJg that IlJcl lJ:JJlic.lc is JJJagneticallv l~onio~ctico~~s wc: 110lc II131
llic JiJ:Jgnc~lic c.c)JitlilioJJ is iJJtl(~lJ~~JJ~lcJJt 0T sc*alr,  :rntl 111~~lT1'01'1' 111~~ I`~Il~W.
                                                  (1 Ii

leer uJJit VOIUJIIC~ are also intlc~J~~n~lcnl ofw:~lc, over 3 J3np \vhrrc 1-J ;lll~l  dir
can Iw corisitl~~iwl wJJsl;JJJt.

The varialioil with slJalJe is ~iiorc conlplrx. Tlic cllipsoitl is the only IIII~I\
for \vhich llic J1Jag~Jdic colitlitions arc constant tlJrolJglJu~Jt the volutnc I'**r
:I uiiil'orni JilJlJlictl firbltl. As iii the ~Jydrodynamic cases, we \vill con%i~l~.r
0JJlv c~llitJsoitls ol' rcvolutioli.

\\`o lwow~~~l as l'ollo~vs: \vc first c*:Jlcul:ltr :I fador tl(~pcntling on Ihe >II:I~"
(II' tlJ(~ c~llilJs~~icl. LsilJg IlJis WC fintl I'rom Ihr lIjl1 CIII'VC of the m:llvri;il 1111
wlcvant V;IIIIC 01' II Ior :III elli~Jsoitl lwr~~~aii~~nll~ JJJagJJctisctl along il\ 111.~

jor axis. Frol~i tliis WC easily obtain the magnetic moment (Jf) of the cllip-
soid. `I'hc couple due to a small app1ied field, 11, perpendicular to the klJgth
of the ellipsoid, is then Mh.

`IIJe factors which are calculated for ellipsoicls are "(IeJnagnetisirJg fac-
lors." These express the amount by which the magnetisatioJJ of the clli1J-
soid produces a rcvcrscd magrietic field acting upon itself, and thus teIithlg
lo tlemagnetisc itself, and are defined by tbc equation

H's DZ

where II' = the rlemagnctising field protlucctl
   Z = the intensity of magnctisation
   D = demagrJetisatioJ1 co-efficient.

D will in general depend upon direction, ant1 will have three different values
corresponding to the three axes of the geJJera1 ellipsoid. Thr behaviour for
other directions can be found by compoulJtling I and H vectorially.

\\`e shall, as usual, only qnotc the formulae for an ovary ellipsoid of re-
volution.  They arc

& 106 i-T< - 1  for the major axis.

(The logs arc JlZJtllJYll logs)

wlirrc (I = major axis,  b = c = Jiiiiior asw aiitl 1-c' =

IllM! I'ol~lrlrll:lc hcYllllc

that is  4    In
  z); = & = 3

C(X).  The Jiotation has bwn altrrctl.)
`I'hus if we consitler tlJc case ol 0 fixed and (I incrv:lsilig. \vc see llial


As is \\~(~I1 lill~l\Vll thcrc is a simpl(~ gl~:ll)hical construction to find the work-
ivg l)oiut OII the LIjrZ Curve for a hotly with a given D in a (parallel) estcrnni
licltl H,. The Curve of (11-H) against 11 iS pJotted, and a line is drawn from
the point H, on the H axis with SIOJVL- ";: (p-m)* The working point is the point

where this line cuts the curve. This Construction applies for any WIIJC of
~JJC :JppJioti Jiciti 11,. J:or a JMXmanent magnet we usuaiiy have H, = I);
iu this Case we are working in the top left-hand quadrant of the (B-H)
against H Curve.

Onc*e \ve have fo~~ntl the working value of (U--H) the magnetic momrnl
(Al) of the cllipsoitl is simply

whrrc V .= volunic of the cllipsoitl.

\\`c Consider the Case of an ClliJ)soitl of soft iron in a magnetic field indited
at a sri~:lll a~~gli: to ils m:ljor axis. 13~' soft iron \ve mean hrre a matrrid with
~10 hysteresis. \\`c clo not rcslrirt oursctvrs to the Cast of Constant peruu'-
ability, and will in fact consider a material wJiiCJi J~ccomes magnclicall!
saturatrtl.

I\`(: Calculate the raw of 311 ovary cllipsoicl of revolution (major axis =
(.I, minor nxcs = h = C) wJ~re the aJ$ietl field, H, makes and an,gh~ 4
\rith the m:ijor axis, and the intensity of magnctisation, I, makes an ;III$.
CL with the m:i,jor axis. In gcnernl tc =/= 0. I is Constant throughout the JHIII!

bolh in iiiagniluclc ant1 tlircClioil. \\`e SolvC 1)~ splitting ilrto co~rll)o~lf~~~ts
\Vc oM3in

I cos a = e : ! (H COS 0 - D, I cvs u)  along the major axis

I sill a: = lf4:A (H sin (I - I), Z sin a)  along

\\`c shall only ronsidcr the cases whcrc Ille :lnglCS
J)llt sin 0 -= 0 aJl(l ~`0s 0 = I, etC, and eliminating

a minor axis.

are small. \\`c thc~Wf0rC
I \vc obtain

Now to this apJ)roximation the couple (c) experienced JJ~ tJle particle iS
            C=IVH(O-~a)

where V = volume of particle.
That is

C=ZVHB 1-i
  ( 1

which we can write

`l'his is the rxJ)rcssion wc rCcJuirC.
II follows that

(9 if (y-1) 3 (i{) ( the suffix 2 rCfCrri$ to the l~roatl\vays-011 c:lSl~) lllr
t?rm in the brarket is clTectircJy constant as H varies, and

                 C yaries as Z H
                            as \\`C sJ1o11ltl CSl"Yl.


521               F. II. C. Crick
(ii) il \ve have a material of low permeability, or one which is saturnld
so that (p-1) hs lWconle low, \vc may have (p-1) < ( ) 1;" . This latter factor
                                              2

varicbs I'ro~n 3 I'or a sphere to 2 for a long ellipsoid (page 522). \\`riting lhc

expression [or the couple as

c = zVH(/L-l)B

we see that C varies as I H (p-1) approximately
that is C varies as I2 approximately.
Thus if the material is saturated the couple does not increase with the
field indefinitely, hut tends to a limit.
(iii) for short ellipsoids the couple is less than might he expected on simple
theory by the factor

3. Jhi!/!JiJl~J                                \\`c take the extreme case first. II' the cliipsoitl is long, so that Ihc slol~ 01

As we are ~01i~crned with ol~tainiii~ the niaximutn drag, wc will oilI!
give the c:isc \vli~rc the I):irtic*lo is in a magnetic ficici large enough to satnr:ll~~
it, The ni:~~nctic III~I~~I~~ (Jr) will norniallg 1X im the direction ol Ihr al'-
pIid licltl. The I'orcct oil the particle in the .2' direcfior~, I;;,, is given I)!

the (B-H) against H curve is 111url1 less Ihan  1
         ( 1 -E? 1 then (1~~11) will cll'wlivc~l)
                                      1

IIC c'onstant, ant1 JI will only increase tlue lo Ihe increase in volume, lh:1t is
~~r~~porlionni to ab 2. `l'he viscous couple, however, increases at ;I rate IN:-

The physical properties of cy1oplfr.m. Part II.

525

whcrc M,, Mll, If, arc the vectorial components of dl
ant1 Hz is the 5 component of the applied field H.
There are similar expression for Fu and F,.
                            3HZ

The force thus depends on terms of the form z rather than of the
          ( )
form (Hz%) which occur in certain other cases.
The force on the particle is not necessarily along its length. For example.
if the particle is in a magnetic field which lies in the y direction (Hz= Hz-O),
so that it, too, points in the y direction (M, = Mz = 0), there will never-
theless be a force on the particle in the x direction if

F. THE VARIATION IN RATE OF MOVEMENT WITH SHAPE

\\`e can now combine the results of the previous sections.

1. Tmisiing: the permnnent magnet cm-e

\\`e assume that an ovary ellipsoid is magnetisccl l~araiiel to its major axis
so that it 1)econies a pernianent magnet, of inagnctic moment $1, ant1 thal it
is then acted on 1)s a small magnetic field (II) pcrpendi~ular to its lcnglli.
This will produce a couple Jf 11 and if the cllipsoiti is immersed in a nc\v-
tonian liquid it \vill rotate \vith an :~npila~ velocity 0). `1'11~ prol)lcni we misli
IO salvo is, how Ijig is 0, anti ho\\. does it vary with shal)cn?

To obtain Ihc: v:iluc 01' w I'or any parlicul:ir case \vc nirrcly Il:iw to \VOI~IC
oiil the magnetic* aii(l the viscous r011~~lcs fro111 the l'orlnul:le gi\cii ii1 Ilrc~
I)rcvioris sections ant1 equate tlicm.  Ilr~\vcvrr it is usc1'111 lo get a c~u:llil:~-
live itica ol' ho\v w cl1;111gcs \\.itll sh:~l)e (\\c kno\v that it is intl~~l~~ndc~nl
14 wnlc) so KC silall SII~~~IM~ that h is kept ~onstanl anal cf :illo~vc~tl IO iri-
rr(qsr. `I'lle rl:ltllrc of tlie vari:ilioii tlr~wii~ls oil tlic n:ltllI'c 0l' Ille ni:igii(di(*
iii:il(dal.


lhc :rng~l:rr vcloc.ily \.:iricbs as

\\`I, IIIIIS c~\~:llll:ll~~ ( 1 -Ix I 1). "
       ;I I), k fI 1 1'01' Yill.iOllS \:llueS 01 cl/b.

         `I'AULE I\'
0 (1 i  1  1.0 1  2.0 /  3.1) /   4.0 1   5.0

      I'            I      I

It Can I)e SWn that the variation vilh shape is not very great, Ih~entuall~
Ihe :III~~U~LI~ \cioc*ity will fall olT, but 1)~ this time the approximatiorl IJSC(~
is uuliicdy to IJC still valid. Thus for a real (B-H) against fi curve the augu-
lar vclocily will eventually decrcasc with increasing (n/b). It may be roug]lly
constant over a range for (0/b) small, but this tlepentls 011 tile sliapc of lh
curve. The exact values can IJC calculatetl for ally given curve from the for-
mulae givcu.

The :lhvr results apply strictly to the special ellipsoitls rhoscn.  It RC(`III~
rcason:~blc to aSsumC 1h:lt in the region \\here the shape of the ellipsoitl i,
making ;I large tlilfercnce the approxinlution for a ~J(J(I~ of arbitrary sh:lp
will not Iw as good as for ranges Ivhcrc the ellipsnitl's shape is having Mlk
olTect on the ungulur velocity. lIowc\er it is not easy to put a figure lo 11~
usefulness 01 the approximation.

\f'C have not pursued this furlhrr, as the prol)lcm is complicated ant1 I{(
have in any C;IW in our actual cxpcrimcnts taken an average ralur. Ifgwntrr
accuracy is wquirod the solutions for tbc viscous forces ant1 the tlcm:ipid.
ising ro-cflicicnls for the general ellipsoid arc availahlc, ant1 might givr ;I
hcllcr itlc:i of llie cll'wts of irrrgul;ir stlapr.

?. l'wistin~g : the soft iron uisc

`l'he qualitative results for the mrresponding prol~lrul in the soft iron C:ISO
c'nu easily i~c seen. For very long ovary ellipsoitls the angular velocity will

fall 0fT rather faster than

as in the previous case. For almost spherical

ones it will again lye small. Somewhere in l~etwecn there \vill he a nixxiliiirm,
tlcpcnding OII the propertics of Ilie uiatcrial and the size of the applktl lic,ltl.
The exact wlucs can hr calculated for particular cases from the fornlul;1(
given.

II sccnls pr0b:iblc that for actual I)articles of irregular shape wc shall gc4
si!nilar clfects to those calculated for the clli~~soitl. That is, for very short
parliclcs w';' shall get sniallc*r couplw than might I)c cxp~~tcttl 011 tile simple
theory, and for larger applied fields the couple tending to a maximum
iusteatl of increasing indefinitely. In the case 0r any particular material thr
evaluation of a few cases for the ellipsoid should give a good idea of the
general behaviour, though the reduction in couple due to shortness is likclj
lo, IJC less important for irregular bodies.
`l'hc beatment will not apply to materials which show hysteresis.

\Ve will only consider the cases of an ovary ellipsoid of revolution being
moved either parallel or perpendicular to its length. Other directions L';III
IN solved by compounding vcctorially. \Ve consider the rclcvant field gr:r-
dk?Il~ SS fiXCd, alld hVCS@+C hO\V th! VdOCity Of IIIOVeIllCIlt dCllCIldS 011 thC'
dimensions. For our case the magnetic force, for a given value of (U-II)
at saturation. depends only 011 the volume, not 011 the shape. \Ve have ul-
ready shown (page 509) that the clfect of size is to make the velocity vary us
the sqnare of the characteristic length, so that it only remains to invcsligntc*
shal~c variations. As before the formulae will give an exact solution for any
rhosen case.
(a) dragging parallel to the major axis.
The formula for the viscous rcsislance ant1 a selection of values arc giwrl
ou page 512. l'hcse show that for a fixed I>, the drag increases with (1, initiull!
                  .I

rather slowly, say as  i-
        1 0, and gradually incrcascs to rather slower Ihan ft.
`faking ii as a typical value, the velocity 0r the parliclc \vill roughly IK lm-
Iwrtional to                h2 (1
                            :, .
                I'a bi


528

F. H. C. Crick

     The physical properties of cytoplasm. Part II.       529

the shape of a truncated cone, semi-angle a, with the particle at the apex of
thecone, which we will take as the Origin. \\`e will assume that the dircclion
of magnetisation is everywhere parallel to the axis of the cone. The solution
of this problem, which is quite straightforward, gives the field gradient :II
the origin as

(h) draggiilg parallel to a minor axis.
`l'hc formula ant1 a selection of the values are given on page 512. Thm
show that if b is fixed and a increased, the drag increases initiali! a Iit&
fastrr than the previous case, so that we may take v/a as typical, giving

velocity - ----.
  SF6

\Vhat is wanted in fact in both cases is a good rstimate of the rohnnt d
the I)article, plus an approximate estimate of a and b. This couclusion iw
likely to stand for particles having irregular shapes.

G. PRODUCING FIELD GRADIENTS

\\`c first note that since

0 H,

we cnilnot get a 1argC value of __
                     8X  without either one or both of the dbr

two being large, and of opposite sign. This implies that the lines of fom
rannol be parallel in such a region. They must either diverge or he 1wnL
It (*an 1~ shown that a magnetically saturated particle can ncvcr lw im

lru(b stable quilibriuni under the influences of magnetic forces rrlonr. Tti&
I'ollo~~ simply 1))~ rcgarclin g the particle as having a (fixctl) surfarc +21i-
1)ution ol' magnrtic poles, and applying the appropriate analognc of I-j!,.
shaw's `l'hcor~~ni (5, 374 and 167). `l'hc l)articlc will in fact bc Glhrr in now
sl:t~)lc cquilibriunl or bc moving to\vards one of the lIlagllC!tS prottriciag 11
licbltl.

\\`c nest \vish to show, quile generally, that a very large firltl gra~lic9rlrrln
111i1y be l~rodu~ccl (leaving asiclc clcc*tric: currents for the mntnrnl) hy lla\iW
fcrronl:tgnclic* malcrial mnr the particle.  This is perhaps obviouc fin &-
mcnsional grouncls. A magnet of a given shrrpe and of a given w:dvri:d mdl
produrc the same fioltl at corresponding points, irrespective of sralr. `Ihw
clrarly, thr srnallcr lhe magnet, the grcatcr lhc field gradient. As thcrc it J~I
upprr limit 10 Ihc size of (B-H) for magnetic materials, thcrc must rWW P
time whrn Lhc gradirnt can only be inrrrnsed by making everything sn~,?ll~fl

\\`c c-an illuslratc this by calculating the result for an ideal polc+:cc ill

67cI

z=. = z sin2 a CO3 a

where Z = intensity of magnetisation of the pole-piece, which is as-
       sumed to be uniform.

  q, = distance of pole-piece from the particle at the origin.
There are three points to notice about this answer. Firstly that the cxpres-

rrile this maximum (putting 4ni= B-H) as

(B-H) 181/j
.-- ----~~
  X0   50 1/i

(B, H refer to the polepiece)

Secondly that we can

JO that the field gradient at the origin is of the furm

p.- H)
    __-
   3'0

Whcrc p is a constant a bit less than 1. This form of result is very grncral.
`fhirtlly we note that if WC hatI not continurtl the pole to infinily, l)ut stol)-
p+tl it at the point q, \vc shoultl have obtained

mhich shows ihat as long as x1 is several times .T~, the result is no1 scnsili\.cb
Is its csart value.  This ol~viously I'ollo~~s from tlic fact Ihat \YC! arc inI{,-
galin:: an expression of the form  B - H
      ( 1
            -7r;i- t111'011~11 a wlumr. l'l~llS tlislanl
rMriI~utions have hardly any clfrct, bcrausc of the upper limit to (11 I!
                   L .
It is not necessary, liowcvcr, to produce the magndc gr:lclic~nt rli~~ ! `!;
mitll 111~ primary magnet. \\`c can product a large uniform licbhl, nntl (`(III-
sillrr thr gratlicnt ncai a small body of soft iron plarctl in Ihis licItI. l:or
*inlljlic4y WC will consider the c'asc whcrc this body is a sl)hcre. This ih


530

I<`. Ii. C. Crick

      The physical properties of cytoplasm. Pnrt II.       531

distant coils. \\lt will not give a general treatment, but will give one sitnple
example. Consider the conical polepiece of page 529. Instead of a cone of
magnetic material, imagine that this cone is a former up11 which a coil is
wound. \Vhat is the current through such a coil which w~ould produce tllc
same lirltl-gratlicnt as the magnet tlitl? By considering the equivalent mag-
netic shell we arrive at the simple answer that

cxtrrmcly easy, :ls for points outsiclc it hcharcs exactly as a doublet of
slrcngth                                              .
                    11 -Ii -13-L  3
                     4n  3 Cl

(M rrl'crs to the l)articIe, 11 and Ii to lhe soft iron sphere.)
For the case ~vhrbrc the field is pcq)cndicular to Ihc joining line, wc have
a rf~plsior~ of

It is thus 1)ossihle to control the direction ol' the Corcc to some extent by al-
triing the direction of the applied field.  It shoultl bc noted that the force

falls oil' as  1
       1.4 7 so Ihal it \vill vary ral)idly with the position ol' the particle.
`I'0 siim III), Ihrb nl;isiIlluil1 glntlic~nl \vill usu:illy hr of the form

nhcrc (U&If) is 01th value 1'01. the iron in the immrcliatc Ginit\-, .`I'~ is the
distoncr 01' the nearest iron rrom the I)artic.lc, and p is a constant depending
in :I ~orilpli~atctl wny oil tllc coIlIiguration, but approaching a value ol` tllcb
ortlrr ol' 1 in well-tlcsignctl cases. The more distant parts of the ningnc~fic
circuit tlo not all'cct the gratlicnt tliwctly, but only in so far as they clctcar-
Illiiir (11CZI) in thr iron nc:ir the ~~:~~Ii~lr.

\\`I' nirisl consitlcr I)rically the l)ossil)ilily ol' l)ro(luc*ing high firld gradicnls
1)y clccli% currents iii air-corNI coils. \\`c first obscrvc that we rcquircl :I
srr.s/rrir~cd forrr for our purl~osrs; n short pulse is in gcnrral not sul'licicnl.
The liniilalion is thc~i~~foi~r the steady Iicalitig cll'd: cithc*r the small risr in
temperature \vhich the culture \vill tolerate, \\hich would be importanl fol
coils close to, 01` lhc rise in temperature ol' the coil itself for larger, IlloI'C

11 i = I where II = number of turns per cm
       i -: curwilt in coil (c.ni.11.)
       I = magnetic intensity of the iron.

Notv WC can easily make f = 103, so that if we had 1 turn per mm (II =
10) i must be 10 2, or lo? amps. This will clearly give an enormous amount
of heat. The margin in the calculation is so big that more precise considera-
tions would not be appropriate. IS&fly we note that the heating effect al-
lows (II i) to increase as 1312, where 1 is a characteristic length, so that air-
core coils can only compete with magnets if they are both very large, which,
as WC have shown is the case which produces low field gradients. Thus,
in general, magnets arc much better than air-cored coils for our purpose.

H. SOME NUMERICAL VALUES FOR THE STRESS

Although WC have argued that the method is a very poor one for finding ho\\
the "viscosily" of a noll-lie\vtoni:In licluitl varies with stress, it is clear that
if the r:lngr of strrsscs is very wide indeed we may expect quite consitlcrablc
tlilTcrrnces in hchnvic~ur.  It is thcrel'orr usct'ul to compare the maximum
values of the strcsscs due to twisting and dragging niagnc~tirally, and due to
gravity. To simplify matters, since 11-e are only roncrrncd \vith orders of
magnitude, we will consider the case of a sphere, taking its radius (r) as I ~1.

1. tlvisling of ;I sphere magnetically.
  `l'hc ni:Ixinium stress in this case is

For B = 225 and H = 45 oerstctts \vc get

- 400 dy11es/ct11~.


I'or II - 1500 and ::if 7 104 oersteds/cm (say)

               antI r : 10-J Cl11
\vc grt
           - 40 rlyles/clIl~.

(e - eo) `1;. r

            \vhcrc p = density of particle
             e. = density of liquid.

There arc t\vo cases of intcrcst.
(a) for magnetic particles under gravity.

4. Exact formulae are quoted, for the three cases most often encountered,
for an ovary ellipsoid of revolution in an infinite newtonian liquid. Refc-
rences to the general ellipsoid are given.
5. The efTects of the irregular shape of a particle, of boundaries, and of
non-newtonian and elastic behaviour of the fluid are discussed qualita-
tively.

6. Some theoretical notes are given on producing large gradients of mag-
netic field.

    Taking e = 4 e,, = 1 n = 1 r = 10e4 cm            7. Some comparative numerical values of the stresses in certain biological
W'C get                                  applications are evaluated.

  1
=-
10 tlyt1es/ct112.

.

(I)) for natural iwlusions of the cell, in a centrifuge.
        Take, arbitrarily, (e - eO) = 0. 1.

   \\`c obtain for the maximum stress

II
300 dynes/cm2

`I'lic plillt mx: \\isli lo britlg out is 1101 IIICIT~~ that tllc strrsscs prO(ltlc(~l
(luring magii(!tic twisting aI'<' ratlicr bigger than in magiictic (lraggillg, lllii
lhat boll1 arc rnormously biggrr lhan Ihc cll'cct 0T gravity. hIoreo\-cr, thcsr

The physical properties of cytoplasm. Part II.

533

strcsscs arc only equaled when centrifuging natural inclusions of the same
size by centrifugal fields of the order of lo5 times gravity.
Finally we must emphasize that these results only apply for particles of
the chosen size, as can bc seen from the factor r in the later expressions.

SUMMARY'

1. The paper gives the theory of the magnetic particle method, in which
SOI~C of the mechanical properties of a fluid are estimated by observing the
movements of magnetic particles in it due to applied fields.
2. For the very small particles likely to be used in biological systems the
inertia can be neglected.

3. The elrect of scale is derived for particles of irregular shape in a new-
tonian liquid.

ACKNOWLEDGEMENTS

The author wishes to thank Dr. EIonor B. Fell for the hospitality of the Strange-
ways Research Laboratory, the Medical Research Council for a Studentship, and Mr.
G. Kreisel for many helpful and characteristic suggestions on presentation.

1.
2.
3.
i:
Ii.
7.
a.

REFERENCES

Crucx, I:. Il. C., and Ht~arm~, A. F. W., Exp. Cell Res. 1, 37 (1850).
Emvanas, Quarf. J. Molhs. XXVI, 70 (1893).
FAX&J, O., quoted in C. W. OSEEN'S Hydrodynamik, p. 198, Leipzig, 1927.
CANS, Silzungsb. K. U. Acad., IWnchen, 191 (1911).
JEANS, J., Electricity and hlagnetism.  5th Ed. Camb. 1Jniv. l'ress, 1925.
JI~ITEIIY, G. lj., l'roc. London i~fnlh. Sot. (2) XIV, 327 (1915).
LAMB, ft., Hydrodynamics, 6th Ed. Camb. Univ. Press, 1932.
I\laxwe~~, J. C., A treatise on Electricity and Magnetism, 3rd Ed. Vol II, Oxford Univ. Press,

1892.

1 A more extended summary has been given in Part I (1, p. 79).
37 -503704