measuring densities of solids and liquids using magnetic

S1

Measuring Densities of Solids and Liquids Using Magnetic Levitation

Fundamentals

SUPPORTING INFORMATION

Katherine A Mirica Sergey S Shevkoplyas Scott T Phillips Malancha Gupta

and George M Whitesides

Department of Chemistry amp Chemical Biology Harvard University Cambridge MA 02138

Corresponding author E-mail gwhitesidesgmwgroupharvardedu

S2

General Methods

The NdFeB magnets (5 cm times 5 cm times 25 cm) were purchased from KampJ Magnetics

(wwwkjmagneticscom) and aligned on top of one another 45 cm apart within aluminum

blocks Similar magnets can also be obtained from Applied Magnets wwwmagnet4lesscom at

a lower price The strength of the magnetic field within the device was measured using a hand-

held DC magnetometer (AlphaLab Inc wwwtrifieldcom) Calibrated density standards (plusmn

00002 gcm3 at 23degC) were purchased from American Density Materials (Stauton VA

wwwdensitymaterialscom) Spherical polymer samples were purchased from McMaster-Carr

(wwwmcmastercom) Polystyrene microspheres with precisely defined radii were supplied by

Duke Scientific Corporation (wwwdukescientificcom) Polysciences Inc

(wwwpolysciencescom) and Spherotech (wwwspherotechcom) All other samples and

reagents were purchased from Sigma Aldrich (Atlanta GA) and used without further

purification ldquoLevitation heightrdquo of samples was measured using a ruler with millimeter-scaled

marking Helium pycnometery measurements were performed by Quantachrome Instruments for

a fee on an Ultrapyc 1200e instrument

S3

Figure S0 A plot generated using numerical simulation using COMSOL Multiphysics showing

the dependence of the z-component of the magnetic field Bz on the separation between magnets

(d) for d = 25 35 45 55 65 75 mm along the centerline between the two magnets

S4

Figure S1 Photographs demonstrating levitation of a glass bead (density = 11500 plusmn 00002

gcm3) in different aqueous paramagnetic solutions (1M MnCl2 1M MnBr2 1M CuSO4 1M

GdCl3 1M FeCl2)

MnCl2 MnBr2 CuSO4 GdCl3 FeCl2

10 m

m

S5

Figure S2 Net Volumetric Magnetic Susceptibilities of Common Diamagnetic Substances1-7

-50x10-4

-40x10-4

-30x10-4

-20x10-4

-10x10-4

00

10x10-4

20x10-4

30x10-4

Net

Volu

me M

agne

tic S

usceptibili

ty (

SI

unitle

ss)

diamagnetic substance

hem

oglo

bin

DN

A

alb

um

in

chole

stero

l

aceto

ne

eth

anol

ben

zene

chlo

roben

zene

eth

ylene g

lycol

wate

r

dic

hlo

rom

eth

ane

chlo

rofo

rm

gly

cero

l

PM

MA

silic

on

pol

ysty

rene

pol

yeth

ylene

teflon

silic

a

sodiu

m c

hlo

ride

zinc

lead

dia

mond

silv

er

gol

d

indiu

m

pyr

oly

ticgra

phite

(|| axi

s)

gra

phite

rod

bis

muth

meta

l

pyrolytic graphite

( axis)

S6

Figure S3 Deliberate misalignment of the container with the centerline between the magnets

(red dotted line) Beads of different densities (from top to bottom 10500 10800 11000

11200 11500 gcm3) levitating in 1M MnCl2 align with the centerline between the magnets

regardless of the position of the container as long as the centerline is accessible within the

container (A and B) Inability of the beads to align with the centerline (C and D) does not result

in significant change in the levitation height of the beads Scale bar represents 10 mm

A

B

C

D

S7

Figure S4 Effect of tilting the experimental set-up on height at which the objects levitate A)

Photographs of beads of different density (from top to bottom 10500 10800 11000 11200

11500 gcm3) levitating in 1 M MnCl2 at different values of tilt angle θ The dimensions of the

container in which the beads levitate are 50 mm times 30 mm times 45 mm The container spans the

entire width and height of the magnets and is centered lengthwise between the magnets B)

Images in shown in panel A rotated by angle θ to emphasize the effect of tilting on the levitation

height of the beads

θθθθ = 0deg 5deg 9deg 15deg 20deg 40deg 60deg

A

B


S8

Figure S5 Photographs of polystyrene spheres of different radii levitating at t0 in aqueous 500

mM MnCl2 Spheres of R gt ~ 7 microm form well defined clusters at h spheres of ~ 2 microm lt R lt ~ 7

microm form diffuse clouds centered around h while spheres of R lt ~ 2 microm remain essentially

uniformly dispersed throughout the solution

R (micromicromicromicrom) = 76 51 38 15

5 1

0

15

2

0

25 3

0 m

m

z

S9

Sources and Magnitude of Errors for Calculating Densities of Objects Using Eqn (8)

Experimental parameters and constants B0 d g c and T

We measured the magnitude of the magnetic field at the center on the surface of the

bottom magnet with a magnetometer and found that in the configuration shown in Fig 1a 0B

was (0375 plusmn 0003) Tesla

The distance between the two magnets d remained constant in our experiments and was

equal to (45 plusmn 05) mm

The acceleration due to gravity g at the surface of the Earth ranges from 978 ndash 982

ms2 where the exact value depends on the latitude and other factors we used the average value

of 980 ms2 in our calculations and neglected this variation in our error analysis The typical

variation in g does not constitute a significant source of uncertainty in calculation of density

using eqn 8

We measured T using a thermometer with an accuracy of plusmn 1 degC

We did not measure c directlymdashinstead we calculated it (eqn S0A) based on the mass

of the salt m (g) measured using an analytical balance with accuracy of 31010 minustimes=mδ g (as

specified by the manufacturer of the analytical balance) the total volume of solution V (L)

measured using a volumetric flask with accuracy of 310080 minustimes=Vδ L (as specified by the

manufacturer of the volumetric glassware) and the molecular weight of the salt M (gmol) as

provided by the vendor We estimated the error associated with our calculation of concentration

cδ using eqn (S0B)8 and found it to be less than plusmn 0002 M for the range of concentrations we

used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V

MV MVδ δ δ = + minus

(S0B)

Magnetic Susceptibility of the Medium m

χ

The magnetic susceptibility of the suspending mediumm

χ depends on concentrations

and magnetic properties of all the species present in solution We calculated mχ using eqn (S1)

in which p

χ is the molar magnetic susceptibility of the paramagnetic salt (m3mol units of SI) c

is the concentration of the paramagnetic salt (molL) and 6109 minustimesminus is the bulk magnetic

susceptibility of water9 10

The dependence of the molar magnetic susceptibilities of the

paramagnetic salt p

χ on temperature follows the Curie-Weiss law given by eqn (S2) where T is

the temperature of the medium (expressed in ordmK) CW

C is the Curie constant (m3 K mol) and θ

is the Weiss constant (ordmK) of the paramagnetic salt10

The dependence of magnetic susceptibility

of the medium on temperature is then given by eqn (S3) in which we assumed that the magnetic

properties of water do not change significantly with temperature1 2

69 10m pcχ χ minus= minus times (S1)

CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ

θminus= minus times

minus (S3)

In eqns (S1 S3) we neglected the contribution of the magnetic properties of gasses

dissolved in the suspending medium For example air-saturated water at room temperature

contains up to 03 mM of dissolved O2 gas ( 61034492

minustimes=molar

Oχ )1 11

mdash the contribution of O2 to

S11

the magnetic susceptibility of the medium is 2

813 10Oχ minus= times which is at least three orders of

magnitude less than typical values of m

χ

Density of the Paramagnetic Solution ρm

The dependence of density of the medium mρ on the concentration of paramagnetic salt

in solution and on temperature of the solution is given by eqn (S4) where T is expressed in ordmC

( )W

Tρ is the density of pure water ( 0 99965W = 1

1 20438 10Wminus= times 2

2 61744 10Wminus= minus times ) and

AndashF are empirical dimensional parameters specific to the paramagnetic salt (for GdCl3

2105382 times=A 1101491 minustimesminus=B 3103861 minustimes=C 1103061 minustimesminus=D 0E = and 0F = for

MnCl2 2100221 times=A 1109664 minustimes=B 2103071 minustimesminus=C 0106593 minustimesminus=D

1106311 minustimesminus=E and 3107744 minustimes=F )12

2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W

T c T Ac BcT CcT Dc Ec T Fc T

W W T W T Ac BcT CcT Dc Ec T Fc T

ρ ρ= + + + + + + =

= + + + + + + + + (S4)

Magnetic Susceptibility of the Sample χs

The typical values of sχ for diamagnetic substances are temperature-independent small

(about 6 61 10 10 10minus minusminus times minus times ) and vary little from substance to substance (see Fig S2 for typical

values of magnetic susceptibilities of common substances)1 2 9

We neglected these differences

in calculations of densities based on the empirical estimates of α and β (eqn 8) from the

calibration curves For the measurements of density via the direct application of eqn (8) we set

6105 minustimesminus=sχ (in the middle of the typical range of values of sχ for most diamagnetic

substances Fig S2) and estimated the accuracy of this assumption as 510s

δχ minus= (unitless)

S12

This type of oversimplification and the estimation of uncertainty may not be general for

all cases especially when strongly diamagnetic (| sχ | gtgt 10-5

) or slightly paramagnetic samples

are levitated In such cases either an accurate value of sχ should be used when calculating

densities of eqn (8) or a larger margin of error assumed for this type of density measurement 13

Levitation Height h

We measured the levitation heights of objects h using a rulermdashwe estimate the precision

of this measurement to be plusmn05 mm

S13

Error Analysis

To estimate the error of our measurements of density using the calibration curves we

assume that we know parameters α and β in eqn (8) exactly and consider sρ to be a function

of only one variable h we use eqn (S5) to propagate the uncertainty in h for these types of

measurements8 In eqn (S5) α is the calibration parameter from eqn (8) and hδ is plusmn 05 mm

ss

dh h

dh

ρδρ δ αδ= = (S5)

To estimate the error associated with the direct use of eqn (8) for measuring the density

we treat sρ as a function of several independent variables 0B d sχ T c and h each of

which is a source of independent random error (eqn S6)8 (Notice that

mρ and

mχ are not

independent parameters ndash they are calculated from T and c using eqns (S4) and (S3)

respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B

ρ ρ ρ ρ ρ ρδρ δ δ δχ δ δ δ

χ part part part part part part = + + + + + part part part part part part

(S6)

Eqn (S7) summarizes eqn (8) and eqns (S3) and (S4) in a form convenient for

calculating the partial derivatives in eqn (S6)

2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where

T c W W T W T Ac BcT CcT Dc Ec T Fc T

Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus

minus minus= + minus

= + + + + + + + +

= minus timesminus

(S7)

The partial derivates of m

ρ and m

χ with respect to T and c are needed for estimating

eqn (S6) and are given below (eqns S8A-B)

S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT

W W T Bc CcT Ec Fc TT

χθ

ρ

minuspart= minus minus

partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T

A BT CT Dc ETc FT cc

χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)

The partial derivatives of s

ρ with respect to 0B d sχ T c and h are given by eqns

(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus

part part part = minus minus + part part part

part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro

part = minus part (S9C)

2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro

minus minuspart minus= minus

part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro

part minus = minus part (S9F)

Finally to estimate the error s

δρ in the measurement of density of the sample s

ρ one

needs to substitute the expressions for the partial derivatives (eqns S9A-F) into eqn (S6) and

evaluate the resulting expression using the values of parameters 0B d sχ T c and h and

the tolerances of these parameters 0Bδ = plusmn 0003 (T) dδ = plusmn 05 times 10-3

(m) sδχ = plusmn 10-5

(unitless) Tδ = 1 (degK) cδ = plusmn 0002 (M) and hδ = plusmn 05 times 10-3

(m) respectively We provide

two examples of successful application of this procedure below

Dependence of sδρ on Tδ

We use eqn (S6) (and eqns (S9A-F)) to compare the effect of the uncertainty in

temperature (plusmn 1 degC vs plusmn 10 degC) on the accuracy of the measurement of density of polystyrene

spheres levitating in 350 mM solution of MnCl2 This type of estimate may be appropriate for a

user in a remote location with no immediate access to a thermometer for measuring temperature

For simplicity we assume that all temperature-independent parameters and the value of 0B are

known exactly Table S1 summarizes the parameters used for and the result of this estimation

Dependence of sδρ on 0Bδ

Similarly we use eqn (S6) (and eqns (S9A-F)) to compare the effect of 0Bδ = plusmn 0003 T

(typical accuracy established by using a magnetometer) vs 0Bδ = 01 T (a safe assumption for

variation between NdFeB magnets supplied by various manufacturers) on the accuracy of the

density measurement of polystyrene spheres levitating in 350 mM MnCl2 For simplicity we

S16

assume that all parameters except 0Bδ are known exactly Table S2 summarizes the parameters

used for and the result of this estimation ndash while the uncertainty in the measurement of density

increases significantly from plusmn 00002 to 0008 gcm3 the measurement of density is still fairly

accurate and may be useful in less demanding situations

S17

Table S0 Densities common plastics and organic liquids that can be used as density standards

in magnetic levitation

sample density (gcm3)

organic polymers

high-density polyethylene 097

polydimethylsiloxane (PDMS) 104

polystyrene 105

poly(styrene-co-acrylonitrile) 108

poly(styrene-co-

methylmethacrylate)

114

nylon 66 114

polymethylmethacrylate 118

polycarbonate 122

neoprene rubber 123

polyethylene terephthalate

(mylar)

140

polyvinylchloride (PVC) 140

cellulose acetate 142

polyoxymethylene (delrin) 143

polyvinyledene chloride 170

polytetrafluoroethylene

(Teflon)

220

organic liquids

3-fluorotoluene 0997


fluorobenzene 1025

3-chlorotoluene 1072

chlorobenzene 1107

24-difluorotoluene 1120

2-nitrotoluene 1163

nitrobenzene 1196

1-chloro-2-fluorobenzene 1244

13-dichlorobenzene 1288


dichloromethane 1325

3-bromotoluene 1410

bromobenzene 1491

chloroform 1492

1-bromo-4-fluorobenzene 1593

hexafluorobenzene 1612

carbon tetrachloride 1630

tetradecafluorohexane 1669

25-dibromotoluene 1895

perfluoro(methyldecalin) 1950

iodomethane 2280

tribromomethane 2890

S18

Table S1 Dependence of uncertainty in measurement of density sδρ on uncertainty in Tδ

Parameter P Description Magnitude of P Pδ Pδ

experimental parameters

0B strength of magnetic

field at the surface of

the magnet

0375 T known exactly known exactly

d distance between

magnets

45 mm known exactly known exactly

T Temperature 23 degC plusmn 1 degC plusmn 10 degC

c Concentration of MnCl2 0350 M known exactly known exactly

Unknowns

sχ bulk magnetic

susceptibility of the

sample

-5 times 10-6

(SI unitless) known exactly known exactly

calculated parameters

)( Tcmρ density of

paramagnetic medium

10339 gcm3 plusmn 00003 gcm

3 plusmn 00026 gcm

3

)( Tcmχ bulk magnetic


medium

plusmn 56 times 10-6

known exactly known exactly

constants

g acceleration due to

gravity

980 ms2 na na

micro0 permeability of free

space

4π times 10-6

NA-2

na na

independent variable

h ldquolevitation heightrdquo of

the sample above the

bottom magnet


dependent variable

sρ density of sample 10483 gcm3 plusmn 00003 gcm

3 plusmn 0003 gcm

3

S19

Table S2 Dependence of uncertainty in measurement sδρ on ucertainty in 0Bδ





the magnet

0375 T plusmn 0003 T plusmn 01 T

d distance between

magnets


T Temperature 23 degC known exactly known exactly

c concentration of MnCl2 0350 M known exactly known exactly

Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium

10339 gcm3 known exactly

known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20

Table S3 Linear correlations for h and ρs at different concentrations of MnCl2 obtained from

the plot of these two parameters in Fig 5A

concentration of

MnCl2(M)

linear least squares fit

to Fig 5A

least squares linear fit

in the form of Eqn 8 α β

01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559

Figure S6 Dependence of α and β on [MnCl2] A) Plot correlating α with [MnCl2] least

squares linear fit y = 00041x - 00001 R2

= 0997 B) Plot correlating β with [MnCl2] least

squares linear fit y = 01889x + 10014 R2

= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21

Table S4 Details of experimental parameters used for obtaining densities of solids using

calibration curves in MnCl2 Table 2 compares the calculated densities from Tables S3-S5

Sample [MnCl2]

(molL)

h

(mm)

calibration curve used Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 167 plusmn 05 37173664 +minus= sh ρ 10099 plusmn 00002



spherical polymers

polystyrene 0500 235 plusmn 05 493449 +minus= sh ρ 1047 plusmn 0001

nylon 66 1000 135 plusmn 05 294247 +minus= sh ρ 1137 plusmn 0002

polymethylmethacrylate 2000 233 plusmn 05 162117 +minus= sh ρ 1186 plusmn 0004

irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)

1000 281 plusmn 05 294247 +minus= sh ρ 1076 plusmn 0002

poly(styrene-co-

methylmethacrylate)


organic droplets

chlorobenzene 1000 185 plusmn 05 294247 +minus= sh ρ 1115 plusmn 0003

2-nitrotoluene 1000 52 plusmn 05 294247 +minus= sh ρ 1169 plusmn 0003

dichloromethane 3000 195 plusmn 05 126880 +minus= sh ρ 1324 plusmn 0006

3-bromotoluene 3000 125 plusmn 05 126880 +minus= sh ρ 1405 plusmn 0006

chloroform 3000 65 plusmn 05 126880 +minus= sh ρ 1479 plusmn 0007

S22

Table S5 Details of experimental parameters used for obtaining densities of solids using eqn (8)

in MnCl2 Table 2 compares the calculated densities from Tables S3-S5

Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers

polystyrene 0500 10492 84 times 10-6

235 plusmn 05 1047 plusmn 0001

nylon 66 1000 10994 177 times 10-6


polymethylmethacrylate 2000 11971 363 times 10-6


irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets

chlorobenzene 1000 10994 177 times 10-6


2-nitrotoluene 1000 10994 177 times 10-6


dichloromethane 3000 12923 548 times 10-6


3-bromotoluene 3000 12923 548 times 10-6


chloroform 3000 12923 548 times 10-6


S23


in GdCl3 Table 2 compares the calculated densities from Tables S3-S5

Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24

Figure S7 Plots showing the dependence of error in sρ on various experimental parameters at

constant temperature A) Dependence of sδρ on [MnCl2] in water at 23 degC Dependence of sδρ

on h over the entire vertical distance between the magnets (45mm) at B) 01 M MnCl2 and C)

50 M MnCl2 at 23 degC

0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ

)(combinedsδρ)( ms ρδρ)( ms χδρ)( ss χδρ)( 0Bsδρ

)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References

(1) Lide D R Ed CRC Handbook of Chemistry and Physics 89th

ed [Online] CRC Press

Boca Raton FL 2008

(2) Du Tremolet de Lacheisserie E Gignoux D Schlenker M Eds Magnetism Kluwer

Academic Publishers Norwell MA 2002

(3) Hirota N Kurashige M Iwasaka M Ikehata M Uetake H Takayama T

Nakamura H Ikezoe Y Ueno S Kitazawa K Physica B 2004 346 267-271

(4) Ikezoe Y Kaihatsu T Sakae S Uetake H Hirota N Kitazawa K Energy Conv

Manag 2002 43 417-425

(5) Kang J H Choi S Lee W Park J K J Am Chem Soc 2008 130 396-397

(6) Kuchel P W Chapman B E Bubb W A Hansen P E Durrant C J Hertzberg

M P Concepts in Magn Reson 2003 18A 56-71

(7) Simon M D Geim A K J Appl Phys 2000 87 6200-6204

(8) Taylor J R An Introduction to Error Analysis University Science Books Sausalito

CA 1997

(9) Andres U Magnetohydrodynamic amp Magnetohydrostatic Methods of Mineral

Separation John Wiley amp Sons New York NY 1976

(10) Hatscher S Schilder H Lueken H Urland W Pure Appl Chem 2005 77 497-511

(11) Weiss R F Deep-Sea Research 1970 17 721-735

(12) Sohnel O Novotny P Densities of Aqueous Solutions of Inorganic Substances

Elsevier New York NY 1985

(13) Paramagnetic samples will also levitate in this device provided that χm gt χs

S2

General Methods

The NdFeB magnets (5 cm times 5 cm times 25 cm) were purchased from KampJ Magnetics

(wwwkjmagneticscom) and aligned on top of one another 45 cm apart within aluminum

blocks Similar magnets can also be obtained from Applied Magnets wwwmagnet4lesscom at

a lower price The strength of the magnetic field within the device was measured using a hand-

held DC magnetometer (AlphaLab Inc wwwtrifieldcom) Calibrated density standards (plusmn

00002 gcm3 at 23degC) were purchased from American Density Materials (Stauton VA

wwwdensitymaterialscom) Spherical polymer samples were purchased from McMaster-Carr

(wwwmcmastercom) Polystyrene microspheres with precisely defined radii were supplied by

Duke Scientific Corporation (wwwdukescientificcom) Polysciences Inc

(wwwpolysciencescom) and Spherotech (wwwspherotechcom) All other samples and

reagents were purchased from Sigma Aldrich (Atlanta GA) and used without further

purification ldquoLevitation heightrdquo of samples was measured using a ruler with millimeter-scaled

marking Helium pycnometery measurements were performed by Quantachrome Instruments for

a fee on an Ultrapyc 1200e instrument

S3




S4



GdCl3 1M FeCl2)


10 m

m

S5


-50x10-4

-40x10-4

-30x10-4

-20x10-4

-10x10-4

00

10x10-4

20x10-4

30x10-4

Net

Volu

me M

agne

tic S

usceptibili

ty (

SI

unitle

ss)


hem

oglo

bin

DN

A

alb

um

in

chole

stero

l

aceto

ne

eth

anol

ben

zene

chlo

roben

zene

eth

ylene g

lycol

wate

r

dic

hlo

rom

eth

ane

chlo

rofo

rm

gly

cero

l

PM

MA

silic

on

pol

ysty

rene

pol

yeth

ylene

teflon

silic

a

sodiu

m c

hlo

ride

zinc

lead

dia

mond

silv

er

gol

d

indiu

m

pyr

oly

ticgra

phite

(|| axi

s)

gra

phite

rod

bis

muth

meta

l

pyrolytic graphite

( axis)

S6







A

B

C

D

S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S3




S4



GdCl3 1M FeCl2)


10 m

m

S5


-50x10-4

-40x10-4

-30x10-4

-20x10-4

-10x10-4

00

10x10-4

20x10-4

30x10-4

Net

Volu

me M

agne

tic S

usceptibili

ty (

SI

unitle

ss)


hem

oglo

bin

DN

A

alb

um

in

chole

stero

l

aceto

ne

eth

anol

ben

zene

chlo

roben

zene

eth

ylene g

lycol

wate

r

dic

hlo

rom

eth

ane

chlo

rofo

rm

gly

cero

l

PM

MA

silic

on

pol

ysty

rene

pol

yeth

ylene

teflon

silic

a

sodiu

m c

hlo

ride

zinc

lead

dia

mond

silv

er

gol

d

indiu

m

pyr

oly

ticgra

phite

(|| axi

s)

gra

phite

rod

bis

muth

meta

l

pyrolytic graphite

( axis)

S6







A

B

C

D

S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S4



GdCl3 1M FeCl2)


10 m

m

S5


-50x10-4

-40x10-4

-30x10-4

-20x10-4

-10x10-4

00

10x10-4

20x10-4

30x10-4

Net

Volu

me M

agne

tic S

usceptibili

ty (

SI

unitle

ss)


hem

oglo

bin

DN

A

alb

um

in

chole

stero

l

aceto

ne

eth

anol

ben

zene

chlo

roben

zene

eth

ylene g

lycol

wate

r

dic

hlo

rom

eth

ane

chlo

rofo

rm

gly

cero

l

PM

MA

silic

on

pol

ysty

rene

pol

yeth

ylene

teflon

silic

a

sodiu

m c

hlo

ride

zinc

lead

dia

mond

silv

er

gol

d

indiu

m

pyr

oly

ticgra

phite

(|| axi

s)

gra

phite

rod

bis

muth

meta

l

pyrolytic graphite

( axis)

S6







A

B

C

D

S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S5


-50x10-4

-40x10-4

-30x10-4

-20x10-4

-10x10-4

00

10x10-4

20x10-4

30x10-4

Net

Volu

me M

agne

tic S

usceptibili

ty (

SI

unitle

ss)


hem

oglo

bin

DN

A

alb

um

in

chole

stero

l

aceto

ne

eth

anol

ben

zene

chlo

roben

zene

eth

ylene g

lycol

wate

r

dic

hlo

rom

eth

ane

chlo

rofo

rm

gly

cero

l

PM

MA

silic

on

pol

ysty

rene

pol

yeth

ylene

teflon

silic

a

sodiu

m c

hlo

ride

zinc

lead

dia

mond

silv

er

gol

d

indiu

m

pyr

oly

ticgra

phite

(|| axi

s)

gra

phite

rod

bis

muth

meta

l

pyrolytic graphite

( axis)

S6







A

B

C

D

S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S6







A

B

C

D

S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S7







height of the beads


A

B


S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S8






5 1

0

15

2

0

25 3

0 m

m

z

S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S9












using eqn 8









used in this study

mc

MV= (S0A)

S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S10

2 2

2

1 mc m V


(S0B)


χ




in which p





paramagnetic salt p









CWp

C

Tχ

θ=

minus (S2)

69 10CWm

Cc

Tχ


minus (S3)




minustimes=molar

Oχ )1 11


S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S11




χ




( )W








2 3 2 3 2 3 2 2

3 2 2 3 2 3 2 3 2 2

0 1 2

( ) ( )m W



ρ ρ= + + + + + + =

= + + + + + + + + (S4)











S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S12






Levitation Height h



S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S13

Error Analysis





ss

dh h

dh





mρ and

mχ are not


respectively)

2 22 2 2 2

0

0

s s s s s ss s

s

T c h d BT c h d B



(S6)



2 2

0 0

2

3 2 2 3 2 3 2 3 2 2

0 1 2

6

4( ) 2( )

( )

9 10

s m s ms m

o o

m

CWm

B Bh

g d g d

where


Cc

T

χ χ χ χρ ρ

micro micro

ρ

χθ

minus


= + + + + + + + +

= minus timesminus

(S7)


ρ and m



S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S14

( ) 2

1 2 3 2 3 2

1 2

32 2

2

mCW

m

C c TT


χθ

ρ


partpart

= + + + + +part

(S8A)

2 1 2 1 2 2 1 23 3 3

2 2 2

m CW

m

C

c T


χθ

ρ

part=

part minuspart

= + + + + +part

(S8B)



(S9A-F) below

( )

2

0

2

2

1 2 3 2 3 2

1 2

4

2

32 2

2

s m m

o

mCW

m

B dh

T g d T T

where

C c TT


ρ χ ρmicro

χθ

ρ

minus


part= minus minus

partpart

= + + + + +part

(S9A)

2

0

2

2 1 2 1 2 2 1 2

4

2

3 3 3

2 2 2

s m m

o

m CW

m

B dh

c g d c c

where

C

c T


ρ χ ρmicro

χθ

ρ


part=

part minuspart

= + + + + +part

(S9B)

2

0

2

4

2

s

s o

B dh

g d

ρχ micro


2

0

2

4( )s s m

o

B

h g d

ρ χ χmicro

part minus=

part (S9D)

( )2

2 302( )4s s m

o

Bd hd

d g

ρ χ χmicro


part (S9E)

S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S15

02

0

8( )

2

s s m

o

dh B

B g d

ρ χ χmicro




ρ one




















S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S16





S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S17




organic polymers



polystyrene 105


poly(styrene-co-

methylmethacrylate)

114

nylon 66 114


polycarbonate 122

neoprene rubber 123


(mylar)

140






(Teflon)

220

organic liquids



fluorobenzene 1025


chlorobenzene 1107


2-nitrotoluene 1163

nitrobenzene 1196





3-bromotoluene 1410

bromobenzene 1491

chloroform 1492







iodomethane 2280


S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S18






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


3 plusmn 00026 gcm

3



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0003 gcm

3

S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S19






the magnet


d distance between

magnets




Unknowns

sχ bulk magnetic


sample

-5 times 10-6



)( Tcmρ density of

paramagnetic medium


known exactly



medium



constants


gravity

980 ms2 na na


space

4π times 10-6

NA-2

na na




bottom magnet


dependent variable


3 plusmn 0008 gcm

3

S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S20



concentration of

MnCl2(M)


to Fig 5A



01 h = -3664ρs + 3717 ρs = -00003h + 10145 -00003 10145

05 h = -449ρs + 493 ρs = -0002h + 1098 -0002 1098

10 h = -247ρs + 294 ρs = -0004h + 1190 -0004 1190

15 h = -160ρs + 206 ρs = -0006h + 1287 -0006 1287

20 h = -117ρs+ 162 ρs = -0008h + 1385 -0008 1385

25 h = -934ρs + 138 ρs = -0011h + 1478 -0011 1478

30 h = -808ρs + 126 ρs = -0012h + 1559 -0012 1559





= 0999

00000

00020

00040

00060

00080

00100

00120

00140

000 100 200 300 400

[MnCl2]

αα αα

080

090

100

110

120

130

140

150

160

170

000 100 200 300 400

[MnCl2]

ββ ββ

A

B

S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S21



Sample [MnCl2]

(molL)

h

(mm)


(gcm3)

glass beads




spherical polymers




irregularly-shaped

polymers


poly(styrene-co-

acrylonitrile)


poly(styrene-co-

methylmethacrylate)


organic droplets






S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S22



Sample [MnCl2]

(molL)

Density of

MnCl2

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0100 10081 10 times 10

-6 167 plusmn 05 1010 plusmn 0001

ρ = 11000 gcm3 1000 10994 177 times 10

-6 220 plusmn 05 1101 plusmn 0002

ρ = 11500 gcm3 1500 11486 270 times 10

-6 219 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 1000 10994 177 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

1000 10994 177 times 10-6

281 plusmn 05

1078 plusmn 0003

poly(styrene-co-

methylmethacrylate)

1000 10994 177 times 10-6

140 plusmn 05

1132 plusmn 0003

organic droplets











S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S23



Sample [GdCl3]

(molL)

Density of

GdCl3

solution

(gcm3)

mχ

(unitless)

h

(mm) Calculated sρ

(gcm3)

glass beads

ρ = 10100 gcm3 0050 10100 8times 10

-6 260 plusmn 05 1009 plusmn 0001

ρ = 11000 gcm3 0400 10950 125times 10

-6 202 plusmn 05 1102 plusmn 0002

ρ = 11500 gcm3 0600 11426 192times 10

-6 205 plusmn 05 1152 plusmn 0003

spherical polymers



nylon 66 0400 10950 125 times 10-6




irregularly-shaped

polymers



poly(styrene-co-

acrylonitrile)

0250 10589 75 times 10-6

127 plusmn 05

1076 plusmn 0003

poly(styrene-co-

methylmethacrylate)

0530 11260 168 times 10-6

209 plusmn 05

1132 plusmn 0002

organic droplets











S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S24





0000

0005

0010

0015

0020

0025

0 5 10 15 20 25 30 35 40 45

h (mm)

δρ

δρ

δρδρ

s

0

0001

0002

0003

0004

0005

0006

0 5 10 15 20 25 30 35 40 45

h (mm)

δρδρ δρδρ

s

0000

0005

0010

0015

0020

0 1 2 3 4 5 6 7

[MnCl2] (molL)

δρδρ δρδρ

s

combined

δρ

δρ

δρ

δρ

δρ

δρ


)(dsδρ)(hsδρ

A

B

C

01 M MnCl2

50 M MnCl2

S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








S25

References



Boca Raton FL 2008






Manag 2002 43 417-425






CA 1997








measuring densities of solids and liquids using magnetic

Documents