Francisco H. Sánchez G3M, Physics Department and La Plata Institute of Physics, FCE,UNLP, CONICET, Argentina Physics Building, 1905 Magnetic Nanomaterials. Some Biomedical Applications: Magnetofection, Magnetic Hyperthermia, and Ferrogels for Drug Delivery http://www.fisica.unlp.edu.ar/Members/sanchez/escola-de-magnetismo-vitoria-es- brasil-03-11-13-al-08-11-13
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Francisco H. Sánchez
G3M, Physics Department and La Plata Institute of Physics, FCE,UNLP,
CONICET, Argentina
Physics Building, 1905
Magnetic Nanomaterials.
Some Biomedical Applications: Magnetofection, Magnetic
Analysis of an interacting superparamagnet with theoretical expressions valid for non interacting systems (Langevin)
F.H. Sánchez
aparent
true
F.H. Sánchez
Analysis of an interacting superparamagnet with theoretical expressions valid for non interacting systems (Langevin)
Dipolar interactions
Hypothesis: dipolar interactions give rise to an aparent higher temperature
3
2
0d
D
d
*TTTa
con *kTD
1
kT
HL
M
THM
S
0),(
*
),( 0
TTk
HL
M
THM
S
N
M
kT S
2
0*
F.H. Sánchez
Dipolar interactions
*),( 00
TTk
HLN
kT
HLNTHM a
aa
TT
a/*1
1
NTTNNTT
NN aaaa /*1/*1
1
when
0* 0
0
3
0
3
0
3
T
SSD
aM
kT
dM
kTdkTdkT
T
T
*TT
*kTD 3
2
0d
D
F.H. Sánchez
Low field susceptibility
*3
2
0
TTk
N
N
M
kT S
2
0*
3
333*3312
0
2
0
2
0
2
2
0
2
2
0
2
0
S
S
SS M
TkN
N
M
kMN
kN
MN
TkN
N
kT
N
kT
331
2
0
SM
TkN
F.H. Sánchez
Allia et al. show that when a NP moments distribution exists former expression becomes:
3
32
0
SM
TkN
where
2
2
2
2
a
a
F.H. Sánchez
N
M SN
M
kT S
2
0*
*kTD
2
SM
T3
0/3 kN
slope
measurements
At different temperatures
HvsM
F.H. Sánchez
0 50 100 150 200 250 3000
2000
4000
6000
8000
10000
FT
GA
a
p(A
m2/k
g)
T (K)
Aparent moment variation with temperature
F.H. Sánchez
3
32
0
SM
TkN
FG
NPs
mass
mass
3/ mkg
2
2
nmd
nmD
J
FT
p
D
B
26
1.8
1037.1
9500
21
nmd
nmD
J
GA
p
D
B
32
1.9
1038.2
12600
21
F.H. Sánchez
65
Demagnetizing factor NDef in samples with disperse magnetic NPs
F.H. Sánchez, unpublished
ii
i
ii
i
ii
i
iM VHMHBE
222
1 00
Case 1: continuous material
Interacción among materials dipoles
mi
Hi iij
dip
ji rHH
i
dVHM
2
0
MNH D
Magnetostatic energy or magnetic self energy
66F.H. Sánchez
Demagnetizing factor NDef in samples with disperse magnetic NPs
D
i
j
ijd
D
ijij Ded
If NPs are in contact: 1(case of a continuos magnetic material)
Case 2: magnetic NPs disperse in non magnetic media
67
...2,3/22,3/2,1..
...5,2,2,1..
ij
ij
ebccei
esquareei
Is a normalized array describing the geometry of NPs arrangementije
F.H. Sánchez
d
d: distance between neighbors
“dilution “ factor
iii
i
ii
i
iM HHE
00
2
1
2
68
Identical NPs: .,6/,1;3 etcDV
F.H. Sánchez
4/13
1333333
j ij
ij
j
jijijj
ij
ie
s
Dvuuv
eDH
ii v
jijijjij vuuvs
3
Dipolar field on NP i:
33DH i
i
j ij
ij
ie
s3
defining
Just depends on the geometry of the NPs array, on the relative orientations between moments and relative orientations between them and the segment joining them. may take different orientations.
i
ijs
Sii MH adimensional
,,,, i
69F.H. Sánchez
.,6/,1;3 etcDV using
SSpp VMMV 3 “global” sample magnetization
Volume per particle
MH S
ii
Assumption for non saturated sample: same proportionality constant between H and M holds.
SDN MNMH DSD
Averaging i on sample
S
S
i
S
i MH
Saturated sample, i corresponds to the saturation (sij) configuraciónS
i
70
Aproximation done: MM S
iSi
F.H. Sánchez
Using magnetization of magnetic phase (NPs)pM
MM p
3
pDefp
S
ii MNMH
3
hence
3D
Def
NN
71F.H. Sánchez
Determined by shapeDetermined by shape and NP dilution
DDef NNif 1.016.3 D
Dd
DDef Nd
DN
3
or
1. NPs homogeneously distributed. ND is the demagnetizing factor which corresponde to sample shape.
2. NPs are aggregated in clusters, and clusters do not interact among them. ND is the demagnetizing factor which corresponds to cluster average shape.
3. NPs are aggregated in clusters, and clusters interact among them. ND is the demagnetizing factor which corresponds both to sample shape and to cluster average shape.
72F.H. Sánchez
Cases of interest
Case of interest 3
pccp nnn NP nuber
73F.H. Sánchez
3. NPs are aggregated in clusters, and clusters interact among them. ND is the demagnetizing factor which corresponds both to sample shape and to cluster average shape.
CECC Dd
CDCD
Dd IC
D
6cn
7pcn
DIC
CD
D
333 DnD ICpcC
s
D
pc
S
ECS
IC
sample
D Nn
N
c
D
S
IC
cluster
D NN
Leads to
333
ECIC
cluster
D
sample
D
IC
cluster
DDef
NNNN
74
This is the sought result, it gives the dependency of the effective demagnetizing factor in terms of cluster and sample ones, and the characteristic distances (dilution factors) of the problem.
F.H. Sánchez
Continuous NPs distribution
1;1 ECIC Continuous Material
muestra
75F.H. Sánchez
Particular cases
s
D
ECIC
c
D
s
D
IC
c
DDef N
NNNN
333
Cluster with almost no interior demagnetizing field
76F.H. Sánchez
H
33333
ECIC
s
D
ECIC
c
D
s
D
IC
c
DDef
NNNNN
Clusters of random shape with orientations randomly distributed.
3/1c
DN
If clusters are far away from each other
33
1
IC
DefN
If clusters are in contact
3
IC
s
DDef
NN
77F.H. Sánchez
If sample does not present demagnetizing effect
H
33333
131
3
1
EC
s
D
ICECIC
c
D
s
D
IC
c
DDef
NNNNN
33
11
3
1
ECIC
DefN
78
3/ ICDDef NN
Leads to
3
EC
c
D
s
Dc
DD
NNNN
F.H. Sánchez
Relationship between DDef NandN
79
Other considerations
pccp nnn
NPs Magnetization3D
Mp
p
cluster Magnetization3
IC
p
C
MM
333
ECIC
p
EC
CMM
M
CD
D
F.H. Sánchez
Sample Magnetization
c
D
s
D
EC
c
DD NNNN 3
1
c
D
s
D
c
DD
S
pS
IC
c
DD
c
D
s
D
EC
NN
NN
M
M
NN
NN
3
3
Lead to
33
1
ECICppS
S
M
M
M
M
80F.H. Sánchez
SMpSMy are measured. The dilution factors product can be estimated:
81
For an isotropic clusters orientation distribution
3
11
3
13
s
D
EC
D NN
3/1
3/1
3/1
3/1
3
3
s
D
D
S
pS
IC
D
s
DEC
N
N
M
M
N
N
Leads to
F.H. Sánchez
3/1c
DN
Analysis with Langevin model. Comparison with Allia’s proposition
dipap HHH
kT
HHLM
kT
HLMM
dipap
pSpSp
00
pDef
dip MNH
82
Assumptions dip
p HHM
kT
MNHL
VkT
MNHLMTHM
pDef
ap
p
pDef
ap
pSp
00,
F.H. Sánchez
83
p
pDef
ap
pkTV
MNHM
3
2
0
When Langevin function argument is << 1
Def
p
ap
p
NkTV
HM
2
0
3
F.H. Sánchez
Measured susceptibility of magnetic phase.
Def
ppSp
NVM
kT
2
0
31
Def
pS
p
p
NM
kTN
2
0
31
pp VN /1
84
Def
pS
p
p
NM
TkN
2
0
31
p
pS
p
p M
TkN
3
312
0
According to Allia
It’s concluded that pDefN 3
F.H. Sánchez
85
For a sample with a distribution of NP moment sizes, Allia shows
p
pS
p
p M
TkN
3
32
0
2
2
2
2
a
a
and are NP moments. The former are “aparent” values which are obteined when analizing without considering dipolar interactions. The latter are values “corrected” from
a THM ,
ppDef
pS
p
p
VNNM
TkN/1,
32
0
F.H. Sánchez
In the present case
p
pS
N
M
86
D
pS
D
pS
NVM
kTN
VM
kT32
0
3326
0
3
33
As a function of sample “global” magnetization for a homogeneous sample, SM
ppppp VVN 3/1/1 D
S
ppN
M
TkN
2
0
3
F.H. Sánchez
Measured sample susceptibility
3/1)3 AlliaND
87
Measuring M vs. H at several temperatures, and plotting / vs. T/MS2 ND and Npp can
be retrieved
2/ pSMT
ppN
DN
From straight line slope
Data from M vs H
pp
S
N
M
pp
D
N
MN
2
2
0
Max dipolar energy per NPpp
SD
N
MN
2
2
0
88
If there are clusters, SM
F.H. Sánchez
3266
0
33
3
IC
D
SpECICECIC
N
M
T
V
k
Def
pSp
NM
T
V
k
2
0
3
pp
ppm
p
m
pcc
pICEC
pICpcECcm NVV
n
V
nn
VVnnV
1133
33
using
3233
0
33
3
IC
D
SECIC
pp
ECIC
N
M
TkN
DEC
S
ppN
M
TkN3
2
0
3
89
DEC
S
ppN
M
TkN3
2
0
3
/
2/ SMT
ppN
DEC N3
F.H. Sánchez
H
H
parallel perpendicular
Shape effects
F.H. Sánchez
0 5000 10000 150000.0
0.5
1.0
1.5
parallel
M(A
m2/k
g)
H(A/m)
perpendicular
//
Ferrogel PVA/maghemite 15.7% mass concentration
91
Assuming a uniform distribution of NPs
//
//3
DefDef
DD
NN
NN
1.2
kgm /1013.1 33
//
kgm /1008.1 33
//
3
//
11
4
10
DefDef NN
0.80 =
0.06 = //
N
N
De la forma de la muestra
F.H. Sánchez
D
Dd
Estimation of from FG density and mass concentration of Fe oxide x, assuming a uniform
distribution of NPs
FG
oFe
m
mx
FG
FGFG
V
m
3
33
666
xxd
DD
xV
N
xV
mN
xV
m
FG
NP
FG
NPNP
FG
oFeFG
3/1
6
FGx
F.H. Sánchez
3/1
6
FGx
5.2
/1.1
157.0
/18.5
3
3
cmg
x
cmg
using
FG
For FG 10_1
F.H. Sánchez
D
Dd
Diet and magnetic materials …
97
Dipolar Energy
ppSDef VMN 20
2
Energy per particle for a homogeneous saturated sample