Synthesis and characterization of nanorods for magnetic rotational spectroscopyPavel Aprelev, Yu Gu, Ruslan Burtovyy, Igor Luzinov, and Konstantin G. Kornev Citation: Journal of Applied Physics 118, 074901 (2015); doi: 10.1063/1.4928401 View online: http://dx.doi.org/10.1063/1.4928401 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/118/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nanoscale rheometry of viscoelastic soft matter by oscillating field magneto-optical transmission usingferromagnetic nanorod colloidal probes J. Appl. Phys. 116, 184305 (2014); 10.1063/1.4901575 Structure and magnetic properties of Co-doped ZnO dilute magnetic semiconductors synthesized viahydrothermal method AIP Conf. Proc. 1461, 87 (2012); 10.1063/1.4736875 Magnetic field alignment of template released ferromagnetic nanowires J. Appl. Phys. 112, 013910 (2012); 10.1063/1.4730967 Template synthesis and characterizations of nickel nanorods AIP Conf. Proc. 1447, 405 (2012); 10.1063/1.4710051 Synthesis and magnetic characterization of Co-NiO-Ni core-shell nanotube arrays J. Appl. Phys. 110, 073912 (2011); 10.1063/1.3646491

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Synthesis and characterization of nanorods for magnetic rotationalspectroscopy

Pavel Aprelev, Yu Gu, Ruslan Burtovyy, Igor Luzinov, and Konstantin G. KornevDepartment of Materials Science and Engineering, Clemson University, Clemson, South Carolina 29634, USA

(Received 2 June 2015; accepted 30 July 2015; published online 19 August 2015)

Magnetic rotational spectroscopy (MRS) with magnetic nanoprobes is a powerful method for

in-situ characterization of minute amounts of complex fluids. In MRS, a uniformly rotating

magnetic field rotates magnetic micro- or nano-probes in the liquid and one analyzes the features

of the probe rotation to extract rheological parameters of liquids. Magnetic properties of

nanoprobes must be well characterized and understood to make results reliable and reproducible.

Ni and Co nanorods synthesized by electrochemical template synthesis in alumina membranes are

discussed in applications to MRS. We employ alternating gradient field magnetometry, X-ray

diffraction, and magnetic force microscopy to evaluate and compare properties of these nanorods

and study their performance as the MRS probes. It is shown that nickel nanorods do not seem to

violate any assumptions of the MRS rigid dipole theory, while cobalt nanorods do. VC 2015AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4928401]

I. INTRODUCTION

To address the challenges of in situ characterization of

rheological properties of materials, different experimental

methods have been proposed and developed.1 In many cases,

rheological characteristics of materials are inferred by

comparing the translational and/or rotational motions of dif-

ferent tracers against available models of particle-medium

interactions.2,3

Magnetic particles as active probes attract a lot of

attention because they can be put in motion by applying an

external magnetic field without mechanically disturbing the

medium under investigation. The idea of using magnetic

tracers to probe rheological properties of materials was origi-

nated from the pioneering work of Crick and Hughes.4,5

Crick and Hughes used magnetic particles to probe viscosity

and elastic reaction of cytoplasm. An applied rotating mag-

netic field exerts a torque on a magnetic particle, which is

balanced by the viscous and elastic torques acting from the

medium. Crick and Hughes studied the reaction of the

medium on a step-like pulse of the external magnetic field.

They monitored the particle relaxation to its equilibrium

position. Following their ideas, magnetic tracers, mostly

spherical micro- and nano-particles, have been used in differ-

ent applications.6–17

Anisotropic particles, such as wires, rods, and chains,

have attracted attention of microrheologists only recently.18–31

Magnetic nanorods have several advantages over spherical

nanoparticles: Due to their anisotropic shape, rotational

motions of nanorods can be easily tracked and analyzed from

the microscope images. Moreover, magnetization of a rod-

like particle is often codirected with the rod axis.32,33 This

fact significantly simplifies the models of nanorod rotation,

making rheological measurements reliable.

Magnetic rotational spectroscopy (MRS) takes advant-

age of a distinguishable behavior of rotating tracers as the

frequency of applied rotating field changes. Unlike many

methods based on the analysis of small oscillations, MRS

with magnetic nanorods enjoys analysis of full revolutions of

magnetic tracers, which is much easier to track using

inexpensive microscopes. Nanorods as thin as hundreds of

nanometers in diameter can be seen with dark field imaging.

Therefore, MRS with magnetic nanorods provides very accu-

rate data on submicron rheology of materials.19–22,34

MRS theory is well developed for a rigid magnetic

dipole aligned with the easy axis of the probe rotating in a

Newtonian viscous substance in plane with the rotating mag-

netic field. Therefore, the behavior of a magnetic rod with

magnetic moment m suspended in a fluid of viscosity g and

rotated with a uniformly rotating magnetic field B can be

accurately predicted. In experiment, the nanorod orientation

is easy to visually track. Therefore, assuming that magnetic

moment m is parallel to the nanorod axis, it is convenient to

introduce angle u that the magnetic moment makes with a

reference axis. The rotation frequency and phase of the mag-

netic field B are set by the user, so orientation of the field

is known at all times; at each time t, the field makes angle

a ¼ 2pft with the reference axis, where f is a constant fre-

quency of the field rotation. The angle between the magnetic

field and the rod, #, is expressed as a-u (see Fig. 1 for

illustration).

While rotating, the rod experiences a balance of two

torques that act upon it: the magnetic torque and the viscous

torque. The magnetic torque is the cross product of the

vectors of magnetization m and the magnetic field B, and

thus depends on the angle between them, #. The viscous

torque depends on the geometry and angular velocity of the

rod _u. The torque balance reads35–37

c _ue ¼ m� B; (1)

where e is the unit vector directed perpendicularly to the

plane of the particle rotation. Substituting angle uðtÞ through

angles aðtÞ and #ðtÞ, uðtÞ ¼ 2pft� #ðtÞ. The governing

equation thus takes on the following form:

0021-8979/2015/118(7)/074901/13/$30.00 VC 2015 AIP Publishing LLC118, 074901-1

JOURNAL OF APPLIED PHYSICS 118, 074901 (2015)

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c 2pf � d#

dt

� �¼ mB sin h; (2)

where the drag coefficient c is defined for an ellipsoid as

c¼ 2p3

gl3 1� 1

p4

� �2p2�1

2p p2�1ð Þ1=2ln

pþ p2�1� �1=2

p� p2�1ð Þ1=2�1

!�1

;

(3)

where g is the viscosity of the medium, l is the ellipsoid

length, and p is the aspect ratio, p ¼ l=d, where d is the

diameter of the nanorod. For ellipsoids with aspect ratios

larger than 2—and our probes have aspect ratios on the order

of 10—Eq. (3) can be simplified and to38

c ¼ 2pl3g3 2ln 2 l=dð Þ � 1ð Þ : (4)

Equation (2) has a steady state solution, # ¼ const if the

driving frequency is below the critical frequency defined as

2pfc ¼ xc ¼ mB=c. When the rotating frequency is below

fc, the rod rotates synchronously with the magnetic field

(Fig. 1(a)). When the rotating frequency becomes higher

than fc, the rod begins to rotate asynchronously with the

magnetic field (Fig. 1(b)).35–37

This theory assumes that the magnetic moment associ-

ated with the nanorod does not change its magnitude or

direction relative to the nanorod and is perfectly aligned with

the geometrical long axis of the nanorod. This puts

constraints on the materials that can be used as MRS probes.

Since the rod needs to behave as a magnetic dipole and has

its magnetic moment parallel to the long axis, its magnetic

structure should be specially designed to avoid creation of a

radial component of the magnetic moment.

In this paper, we discuss the applicability of two types

of magnetic nanorods as MRS probes. Nickel and cobalt

nanorods were synthetized and characterized for these pur-

poses. The template based electrochemical growth of nano-

rods was employed.19–21,23,34,39–42 In this method,

nanoporous alumina membrane was used as a template and

Ni and Co nanorods were electrochemically grown inside

pores of this membrane. The electrochemical growth of mag-

netic nanorods enables one to precisely control the size of

the nanorods.19,20,40,43 One can generate Ni, Co, permalloy,

and other metallic nanorods that can be used for MRS. We

discuss magnetic properties of nickel and cobalt nanorods

formed using template based electrochemical synthesis and

study specific features of MRS with these materials.

II. SYNTHESIS OF MAGNETIC NANORODS

The circular alumina membranes (25 mm diameter,

Whatman 6809–6022) were used as the templates for the

synthesis of metallic nanorods. The chosen membranes were

claimed to be 60 lm thick with pores of 200 nm in diameter

running perpendicularly to the membrane surfaces as illus-

trated in Fig. 2. The membrane porosity was approximately

e¼ 0.5.

For production of nanorods, procedure described in

details in Refs. 20, 41, 42 was followed. The membrane was

covered with a conductive gallium-indium eutectic liquid.

Covered membrane was placed on a copper plate,

76� 38� 4.2 mm3, and sealed with the water-proof tape. For

the synthesis of Ni nanorods, a mixture of NiSO4�6H2O

(100 g/l), NiCl2�6H2O (20 g/l), and H3BO3 (45 g/l) in water

was used. For the synthesis of Co nanorods, an aqueous mix-

tures of CoSO4�7H2O (100 g/l) and H3BO3 (45 g/l) were

used. A 1.5 V voltage was applied by the DC regulated

power supply (GW Instek pss-2005, Instek) to initiate the

electrochemical reaction. After the power was turned on, the

metallic ions started to come inside the pores and got depos-

ited on the cathode. The deposition process was conducted

for 12 min.

After the reaction was finished, the gallium-indium coat-

ing was removed using concentrated nitric acid (HNO3).

After the coating was removed, the membrane was rinsed

with water and placed into the 10 ml 6M NaOH aqueous

solution for at least 10 min until alumina was completely dis-

solved. The produced nanorods can be separated by decant-

ing the solution and then transferred into the desired solvents

(water, ethanol, etc.) by several centrifugation/decanting/

FIG. 1. (a) Synchronous and (b) asyn-

chronous regimes of rotation of a

nanorod.

074901-2 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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dispersion cycles. Ultrasound sonication was applied for

about 1 min to obtain a better dispersion of nanorods.

The synthesized nanorods have a narrow length distribu-

tion as shown in Fig. 3. Under the same conditions of

chemical deposition (1.5 V voltage for 12 min), nickel nano-

rods were almost twice as long as cobalt nanorods. The

length of nanorods was controlled by both the deposition

time T and current I(t). In our experiments, the circular

FIG. 2. SEM images (Hitachi S4800)

of the alumina membrane used for the

synthesis of magnetic nanorods: (a) top

view and (b) side view of the fractured

membrane.

FIG. 3. SEM images (Hitachi S4800)

and the length distribution of (a) nickel

and (b) cobalt nanorods synthesized

using electrochemical deposition

method. Frequency of the histogram is

defined as DN/N. DN is the number of

nanorods in a certain length interval

(e.g., 5 lm–6 lm), N is the total num-

ber of nanorods. The applied voltage

was 1.5 V and duration of reaction was

12 min for both cases. (c) Reaction

time was 25 min and (d) reaction time

was 60 min.

074901-3 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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membrane has porosity 0.5 and the average length of the

synthesized nanorods L can be estimated through Faraday’s

law as

Q ¼ðT

0

I tð Þdt ¼ Lp4

D2eqF

MAz; (5)

where Q is the total charge, q is the density of metal, MA is

the molecular weight, F¼ 96485 C/mol is the Faraday con-

stant, z is the valence of metal ion, D¼ 25 mm is the mem-

brane diameter, and e¼ 0.5 is the porosity. The current I(t)was recorded during each experiment. For comparison, it is

convenient to introduce an average current as Ia¼Q/T. This

parameter characterizes the average growth rate of the

nanorods.

Table I shows that nickel and cobalt have almost same

molecular weight MA, density q, and valence z, hence the dif-

ference in the rod length is mainly attributed to the different

average current Ia. The average current changes between

materials, even though the synthesis was conducted at a con-

stant voltage. The theoretical value of the nanorod length

calculated through Faraday’s law matches well with the

value estimated from the SEM images.

The nanorod length significantly depends on the time of

electrochemical deposition. This dependence for nickel

nanorods has been studied by our group and documented in

Ref. 37. During the synthesis of Co nanorods, the applied

voltage was kept constant and only the deposition time was

varied. The deposition time was 12 min, 25 min, and 60 min,

respectively. As shown in Figs. 3(b)–3(d), the length of cobalt

nanorods increases steadily (3.14 lm, 8.07 lm, 19.2 lm) with

the reaction time. Thus, the length of nanorods can be con-

trolled by changing deposition time T. Due to a lower average

current Ia, cobalt nanorods appeared almost twice shorter than

nickel nanorods under the same experimental conditions (the

same voltage and reaction time).

The nanorod diameter only depends on the diameter of the

pores of the membrane. The average diameter of synthesized

nanorods was inferred from their characterization on atomic

force microscope (AFM). From the statistical analysis of the

AFM images from Section VI, a significant spread of the diam-

eters of the nanorods is observed: For Ni nanorods, the average

diameter was found to be d¼ 340 6 30 nm, and for Co nano-

rods, the diameter was estimated as d¼ 400 6 60 nm.

III. MAGNETIC ROTATIONAL SPECTROSCOPY

In this section, we demonstrate that the rotation of nickel

nanorods is well described by the rigid dipole theory, while

the rotation of cobalt nanorods does not follow theoretical

predictions, showing different frequencies and amplitudes.

Therefore, nickel nanorods are good candidates as probes for

MRS experiments, while cobalt nanorods should not be used

for these purposes.

By tracking the rotation of the nanorod in the asynchro-

nous regime and comparing its trajectory uðtÞ with that

given by the numerical solution of Eq. (2), one can judge

whether the nanorod behaves as a rigid magnetic dipole or

its magnetic structure is more complicated giving rise to

some deviations from the theory. This type of characteriza-

tion requires the knowledge of the magnitude and direction

of the magnetic field applied at all times as well as the

geometrical dimensions of the nanorods.

In order to create a well-controlled uniform magnetic

field with a minimal magnetic gradient, we built a special

five-coil rotating system. This system employed five air-core

electromagnets coupled with a triple axis magnetometer

(HMC5883L). A custom-created LabView program allows

one to control the field distribution by cancelling out any

ambient magnetic fields. The system is able to produce a

rotating magnetic field with the strength ranging from 50 lT

to 1 mT and noise level of 5 lT.

The nanorods were suspended in a Cannon S60 Certified

Viscosity Reference Standard with viscosity g¼ 0.11 Pa*s

measured at 24.4 �C. The rotation of nickel and cobalt nano-

rods was analyzed at different driving frequencies. The ampli-

tudes of the rotating magnetic field ranged from 100 to 800 lT.

The driving frequencies ranged from f¼ 0.5 Hz to f¼ 5 Hz.

An example of extracted and fitted nanorod trajectory

uðtÞ vs. time is presented in Fig. 4. The nanorod trajectory

was found by numerically solving Eqs. (3) and (4), consider-

ing the nanorod magnetic moment m as an adjustable

parameter. The details of fitting procedure can be found else-

where.20,23,40 It appears that the rotation of nickel nanorod

(left) follows the theoretical path almost perfectly, while the

rotation of cobalt nanorod (right) differs from the theoretical

predictions both in frequency and amplitude. Thus, the rota-

tion of the nickel nanorod can be fitted by the theory very

well, while cobalt nanorods cannot. This means that for the

rotation of nickel nanorods, the assumptions of the rigid

dipole model hold, while for cobalt nanorods, one or more of

the assumptions of MRS are violated.

This quantitative analysis of the rotation of the two

types of nanorods supports the hypothesis that nickel nano-

rods behave as a rigid magnetic dipole with magnetic

moment oriented along the long axis of the nanorod, while

cobalt nanorods do not follow this model.

In order to check the hypothesis that Ni nanorods behave

as the rigid magnetic dipoles, the magnetization of a single

nanorod was measured at different magnitudes of the magnetic

field (Fig. 5). The applied magnetic field was varied from

100 lT to 800 lT, with multiple measurements performed at

TABLE I. Material parameters for nickel and cobalt and the estimated length of nanorod from Faraday’s law.

Length (lm)

Material Molecular weight, MA (g/mol) Density, q (g/cm3) Valence, z Average current, Ia (A) Theory Experiment

Nickel 58.7 8.91 2 0.028 5.72 5.83

Cobalt 58.9 8.90 2 0.015 3.08 3.14

074901-4 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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400 lT for the uncertainty estimation. The experimental data

do not indicate any statistically significant dependency of mag-

netization of the applied magnetic field at magnetic fields

lower than 1 mT. This result is consistent with the hypothesis

that nickel nanorods behave as rigid dipoles.

Considering the absence of dependency of the magnet-

ization on the magnetic field, a bar plot of magnetization for

each nanorod was built (Fig. 6). The error bars in Fig. 6 are

associated with the different results of the MRS measure-

ments on each nanorod. The mean magnetization of the

material shown as the dashed line in Fig. 6 was calculated

using the average magnetizations of each nanorod. The devi-

ation in the magnetization values of each nanorod from the

FIG. 4. An example of extracted angle

of rotation of a nanorod versus time.

The extracted data points are marked

as the red squares, and the theoretical

fit is shown as a blue solid line. (a)

Nickel nanorod follows the theoretical

angular trajectory. (b) Cobalt nanorod

does not follow the theoretical trajec-

tory; it begins the motion at a low

slope but has a sharp transition to a

larger slope. (c) A close-up of the

nickel nanorod trajectory along with a

gallery of analyzed video frames.

FIG. 5. MRS magnetization data for a single nanorod at different magnetic

fields. Multiple measurements were performed at 400 lT to demonstrate the

uncertainty of the measurement. FIG. 6. A bar graph for average magnetization of each nanorod.

074901-5 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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mean magnetization is caused mostly by the scatter of diam-

eters across the nanorods. For instance, the rod with the di-

ameter lower than the mean diameter of all the rods would

artificially yield a lower magnetization measurement and

would lead to a data point that is below the mean magnetiza-

tion (i.e., nanorods #1 and #2); conversely, the rod with the

diameter greater than the mean diameter of all the rods

would artificially yield a higher magnetization measurement

and lead to a data point that is above the mean magnetization

(i.e., nanorods #3, #4, and #5).

Taking the average diameter of nickel nanorod as

d¼ 340 nm, we calculated the average nanorod magnetiza-

tion to be 160 6 44 kA/m under very weak applied fields. As

discussed in Appendix A, the uncertainty in determination of

the magnetization from the MRS experiments is mostly

caused by the uncertainty in determination of the nanorod di-

ameter. The estimates show that the 38 kA/m fluctuation of

the MRS magnetization out of the 44 kA/m total error can be

explained by the scatter of nanorod diameters.

Overall, this section demonstrates that nickel nanorods

do not seem to violate any assumptions of the MRS theory,

while cobalt nanorods do. In Section IV, we will use other

characterization methods to provide insight into the relation

of magnetic properties of nickel and cobalt nanorods with

their microstructure.

IV. CHARACTERIZATION OF NANORODS BYALTERNATING GRADIENT FIELD MAGNETOMETRY(AGM)

The alternating gradient field magnetometer (AGM

2900 Princeton Measurement, Inc.) was employed for char-

acterization of the synthesized nanorods. Nanorods were

synthesized under 1.5 V for 12 min. Dry powder of magnetic

nanorods of about 0.5 mg weight was placed on the probe for

each measurement. The hysteresis loops are shown in Fig. 7.

Figure 7 confirms that both types of nanorods are ferro-

magnetic with the well-defined remanence MR and coercivity

HC. The saturation magnetization appeared close to the val-

ues for the bulk materials (nickel: 4.5� 105 A/m, cobalt:

1.4� 106 A/m).

In order to interpret these hysteresis loops, we assumed

that the nanorods are formed by the single domain magnetic

crystals. Magnetocrystalline anisotropy of the hexagonal

cobalt crystals is uniaxial. On the other hand, nickel has a

cubic crystalline anisotropy.33 For nanorods with a high

aspect ratio, one has to take into account the demagnetization

field33 which can be written as en extra energy of an apparent

uniaxial anisotropy with the shape anisotropy coefficient

defined as K¼l0Ms2/4.44,45 For nickel, the coefficient of

crystalline anisotropy is at least one order of magnitude

smaller than K.45 Hence, the coefficient of crystalline anisot-

ropy of nickel nanorods can be safely neglected. Keeping

only the shape anisotropy for Ni nanorods, one arrives at the

energy contribution that has the same form as that of a hex-

agonal crystal. Even though a single domain cubic nickel

and hexagonal cobalt have different forms of the energy of

crystalline anisotropy, Ea, the high aspect ratio nanorods of

different materials are expected to exhibit mathematically

similar energy contribution, Ea=V ¼ K sin2h, where V is the

nanorod volume and h is the angle formed by the magnetiza-

tion vector with the field direction. For the two materials,

however, the anisotropy coefficients K will be different.

Thus, the total energy of each nanorod making an angle

0 � u � p=2 with the direction of external field H takes the

form E ¼ V½K sin2h� l0MsH cosðu� hÞ�.33

Experimentally, we deal with a frozen assembly of

magnetic nanorods. Assuming their easy axes are randomly

oriented, and minimizing the total energy E with respect to

h, we obtained a series of magnetization curves M(H,u) for

each angle u. Thus, the average hysteresis loop for an assem-

bly of randomly distributed nanorods was generated as (the

averaging with the azimuth angle was not considered due to

the uniaxial symmetry of the problem)

M Hð Þ ¼

ðp=2

0

Ms cos hmin Hð Þ � u� �

sin uduðp=2

0

sin udu

: (6)

Taking the anisotropy coefficient K as an adjustable pa-

rameter, we performed the analysis of experimental hystere-

sis loops. In Fig. 8, the best fits (red) to the experimental

data were plotted on the same experimental graph to com-

pare the prediction of the single domain theory of magnetic

hysteresis.32,33 As evident from these graphs, it was not pos-

sible to fit these loops with the model of a single domain

nanoparticle. However, for the nickel nanorods, the theoreti-

cal hysteresis loop seems to lie much closer to the experi-

mental data points than that for the cobalt nanorods. The

obtained anisotropy constant K¼ 22 kJ/m3 is smaller than

the theoretical shape anisotropy constant for a long nickel

FIG. 7. Magnetic hysteresis loops of

nickel and cobalt nanorods measured

by the Alternating Gradient Field

Magnetometer (AGM 2900 Princeton

Measurements, Inc.). Both types of

nanorods were synthesized under 1.5 V

for 12 min. (a) The full hysteresis loop

and (b) close up for the low field

range.

074901-6 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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nanorod K¼l0Ms2/4¼ 75 kJ/m3, suggesting that the single

domain cylinder is a very approximate model.

On the other hand, the MRS data on Ni nanorods favor

the rigid dipole hypothesis. As follows from the MRS experi-

ments, the remanent magnetization of each nanorod is esti-

mated as 160 6 44 kA/m, which is in excellent agreement

with the remanent magnetization of the bulk sample of the

dried up nanorods measured with AGM—140 kA/m.

Thus, the rigid rod model does appropriately describe

the nanorod behavior in weak magnetic fields. This model,

however, cannot be associated with the model of coherent

rotation of magnetization vector as would be expected from

a single domain nanorod. Therefore, we hypothesized that

the nanorods are composed of multiple domains separated by

grain boundaries. In order to confirm this hypothesis, a series

of additional experiments were conducted.

V. X-RAY DIFFRACTION (XRD)

To identify the crystal structure as well as estimate the

crystallite size of the synthesized nanorods, XRD experi-

ments were conducted. Figure 9 shows the XRD data for the

nickel and cobalt nanorod powders obtained from the X-ray

diffractometer (Rigaku, Ultima IV). The main peaks for the

Co nanorods appear at 41.7�, 44.6�, 47.4�, 75.9�, 84.1�

which correspond to (100), (002), (101), (110), (103) planes

for the hexagonal close packed crystal lattice. There are also

some weak peaks in the spectrum marked by the dashed

lines. These minor peaks are probably due to the presence of

very small amount of cobalt hydroxide (Co(OH)2). The main

peaks for Ni nanorods appear at 44.6�, 52.0�, 76.5� corre-

sponding to (111), (200), (220) planes for the face centered

cubic crystal structure. The additional peak at 83.0� corre-

sponds to the aluminum stage. There is no such a peak for

cobalt, because a zero background stage was used for the

experiment with cobalt nanorods.

The size of the crystallites t can be estimated using the

Scherrer equation46

t ¼ KSkb cos#

; (7)

where KS is the shape factor and h (measured in radians) is

the full width at half maximum (FWHM) for the peak. For a

spherical crystallite with the cubic symmetry, KS� 0.94.47

The rigorous derivation of Eq. (7) can be found in Ref. 46. A

simple derivation is presented in Appendix B. The crystallite

size t calculated from Eq. (7) is summarized in Table II.

The estimated crystallite size is much smaller than the

particle size (200 nm in diameter, several microns in length).

Therefore, the synthesized cobalt and nickel nanorods are pol-

ycrystalline particles. Since each crystallite has at least one

magnetic domain, the magnetic nanorods cannot be consid-

ered single domain but can have complex multidomain struc-

tures. It should be noted that the Scherrer equation provides

only the lower limit of the crystallite size and should be con-

sidered as the orders of magnitude estimation because there

are other factors that will contribute to the peak broadening as

well. The profile of the instrumental peak, defects, and micro-

strains, all cause the peak broadening.

FIG. 8. Hysteresis loop (red solid

curve) for an assembly of the single

domain nanoparticles whose easy axes

are randomly oriented. The blue dots

are the experimental hysteresis loop.

The theoretical curves were calculated

using a uniaxial anisotropy with the

easy axis co-aligning with the rod’s

long axis. (a) Nickel and (b) cobalt.

FIG. 9. X-ray diffraction data for (a) cobalt and (b) nickel nanorod powders.

TABLE II. Summary of the XRD data interpreted with the Scherrer equation

with k¼ 0.159 nm and Ks¼ 0.94.

2# (deg)

Crystalline

plane (hkl)FWHM

b (deg)

Crystallite

size, t (nm)

Co 41.7 (100) 0.27 32

44.6 (002) 0.25 36

47.4 (101) 0.62 14

75.9 (110) 0.40 26

84.1 (103) … …

Ni 44.6 (111) 0.49 17

52.0 (200) 0.56 15

76.5 (220) … …

074901-7 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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In order to analyze the magnetic features of the synthe-

sized nanorods, Magnetic Force Microscopy (MFM) was

employed.

VI. MAGNETIC FORCE MICROSCOPY

MFM is a variation of Atomic Force Microscopy (AFM).

It is a powerful tool to characterize the magnetic nanostruc-

ture of the material. In MFM, the AFM tip is coated with a

thin (< 50 nm) magnetic film with very high coercivity, so

that the magnetization of the probe does not change during

imaging. Figure 10 schematically illustrates the AFM and

MFM action. Forced by a piezoelectric element, the cantilever

continuously oscillates about its equilibrium position. The

laser beam is used to track the motion of the probe.

In the MFM experiment, two consecutive scans were

employed. First, the probe was moving along the sample

surface and tapping the surface intermittently as shown in

Fig. 10(a). The height of the probe was adjusted to keep the

amplitude of the oscillation constant during the scan. This

way, the surface morphology of the sample surface was

obtained. Then, the probe was lifted 50 nm above the surface

level obtained in the first scan, Fig. 10(b). The probe would

not touch the surface during the characterization of magnetic

interactions.

The aim of this two-step scan for the MFM is to separate

the magnetic interactions from other interactions. In the

intermittent contact mode, when the probe is close to the

sample surface, the mechanical contact force dominates.48

Therefore, the surface morphology can be correctly obtained.

In the noncontact mode, the probe is suspended above the

surface, the Van der Waals interactions are much weaker

than the magnetostatic interactions, hence the magnetic

nanostructure can be probed.

The magnetic moment m of the MFM tip is always point-

ing in the z-direction, Fig. 11. The orientation of magnetization

in the sample can be parallel, antiparallel, or perpendicular to

the magnetic moment of the MFM tip. Therefore, the sample

magnetization will exert a force on the magnetic tip. The mag-

netic force F acting on the tip is written as F¼l0(m�r)H,

where H is the magnetic field generated by the sample. m�r is

replaced by m@/@z because m is directed in the z-direction.

Since the cantilever is oscillating in the z-direction, only z-

component of the magnetic force Fz¼ l0 m@Hz/@z will be

probed.

Close to the surface, the direction of magnetic field Hfollows the direction of the magnetization M of the sample.

The magnetic field is stronger when the tip is closer to the

surface, i.e., @jHzj/@z> 0. In case A, magnetization M as

well as the magnetic field H is parallel to the magnetic

moment m (Hz> 0). Therefore, the force between the probe

and the sample is attractive (@Hz/@z> 0, Fz> 0). In case B,

both magnetization M and magnetic field H are perpendicu-

lar to m (Hz¼ 0). The z-component of the magnetic force

will be zero (Fz¼ 0). Case C is exactly opposite to case A.

The magnetization M is antiparallel to m, leading to negative

field and field gradient (Hz< 0, @Hz/@z< 0). Force between

the probe and the sample is repulsive (Fz< 0).

On the other hand, the magnetic force Fz(z) is also a

function of the position of the cantilever. This force is

FIG. 10. (a) Schematic of action of the

atomic force microscope. (b) and (c)

Associated two-step scan used in the

magnetic force microscopy. (b)

Intermittent contact mode to obtain the

surface morphology of the sample. (c)

Noncontact mode to characterize mag-

netic interactions between the probe

and nanorods.

FIG. 11. The phase shift caused by the magnetic interactions between the

MFM tip and magnetic sample with different spin orientations.

074901-8 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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stronger when the cantilever moves closer to the sample sur-

face, i.e., djFz(z)j/dz> 0. For these three cases shown in Fig.

11 we have, case A: Fz(z)> 0, dFz(z)/dz> 0, case B:

Fz(z)¼ 0, dFz(z)/dz¼ 0, case C: Fz(z)< 0, dFz(z)/dz< 0. It is

the dFz(z)/dz term that determines the phase shift due to the

magnetic interaction. For case B, dFz(z)/dz¼ 0, u¼u0. For

case A, dFz(z)/dz> 0, u<u0, i.e., Du< 0. For case C,

dFz(z)/dz< 0, u>u0, i.e., Du> 0. By scanning over the

sample surface, we can identify the orientation of the mag-

netization in different regions. This phase shift is explained

in detail in Appendix C.

MFM images were obtained using Dimension 3100

(Bruker) atomic force microscope equipped with Nanoscope

IIIa controller utilizing MESP probes (Bruker). The scan rate

was set to 0.5 Hz and lift end height in “Lift” mode was

30 nm (Figs. 12 and 13).

From the phase images, one can notice a clear difference

between magnetic structure of Co and Ni nanorods. The

phase contrast originates from repulsive/attractive forces

acting on the magnetic tip moving above the sample at a lift

distance (30 nm in our case). Co nanorods reveal domain

structure inside the rod with the spin directions perpendicular

to the rod main axis. These images clearly show the contrast

between attractive (bright) and repulsive (dark) regions, indi-

cating different orientations of magnetization vector M in

the constituting crystallites. The structure is uniform in the

middle section of the rod but it gets perturbed at the ends. At

the end, the moment is still perpendicular to the main axis,

but South/North polarity tends to change direction.

On the contrary, just a faint barely visible structure is

noticed in the middle section of Ni nanorods contrasting

with the strong contrast of poles at the nanorod ends where

spins are oriented parallel to the main axis.

Based on these observations, we conclude that the cobalt

nanorods have multi-domain structure in agreement with the

results of the X-ray diffraction experiments. Furthermore,

the width of each magnetic domain is about 100 nm. The

crystallite size of cobalt (40 nm) extracted from the XRD

analysis is about twice the MFM estimates suggesting a

more complex magnetic structure of the nanorod material

where each magnetic domain most likely contains a few

crystallites. On the other hand, the structure of magnetic

features of nickel nanorods is seemingly kindred to that of a

single domain particle. The X-ray diffraction data for nickel

already proved that nickel nanorods are polycrystalline and

expected to have multi-domain structures. This contradiction

can be explained by a weak magnetocrystalline anisotropy of

nickel nanorods. The shape anisotropy is almost ten times

stronger than the magnetocrystalline anisotropy for nickel.33

As a result, the magnetic moment is weakly bonded to the

FIG. 12. Height (left) and phase (right)

images of Co nanorods.

074901-9 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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crystal axis and would like to follow the long axis of the

rod due to the shape anisotropy. The shape anisotropy of

cobalt is comparable with its crystalline anisotropy, hence

the magnetic crystallites form a more complex magnetic

nanostructure.

VII. CONCLUSION

Nickel and cobalt nanorods were synthesized by tem-

plate electrochemical deposition in porous alumina mem-

brane and studied using X-ray diffraction, magnetic force

microscopy, alternating gradient field magnetometry, and

magnetic rotational spectroscopy.

Magnetic rotational spectroscopy was performed on the

nickel and cobalt nanorods. The rotation of nickel nanorods

agreed with the theory of rotation of a rigid dipole and the

magnetization was not dependent on the applied magnetic

field. This suggests that the nickel nanorods behave as rigid

magnetic dipoles. The rotation of cobalt nanorods, however,

did not agree with the theory of a rigid dipole. This suggests

that cobalt nanorods have multidomain complex magnetic

structure and these nanorods have a richer scenario of

rotation in a viscous fluid.

The hysteresis loops of the synthesized nanorods were

also measured using alternating gradient field magnetometer.

Both nickel and cobalt nanorods appeared ferromagnetic.

However, the hysteresis loops cannot be explained by the

model of a single domain magnet. The nickel nanorods how-

ever, showed better agreement with the single domain model

than cobalt nanorods.

The X-ray diffraction experiment identified the crystal

structure of these materials: fcc for nickel and hcp for cobalt.

The crystallite size was also estimated using the Scherrer

equation. The crystallite size for nickel is approximately

20 nm and 40 nm for cobalt, indicating that both nickel and

cobalt nanorods should be considered polycrystalline and

multi-domain materials.

The magnetic force microscopy confirmed the multi-

domain structure for the cobalt nanorods. The domain width

was found to be of the same order of magnitude as the crys-

tallite size obtained by XRD. According to the MFM images,

nickel nanorods appeared as the single domain rods. This

behavior was attributed to a weak crystalline anisotropy of

Ni relative to the rod shape anisotropy.

Overall, nickel nanorods do not seem to violate any

assumptions of the MRS rigid dipole theory, while cobalt

FIG. 13. Height (left) and phase (right)

images of Ni nanorods.

074901-10 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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nanorods do. Even though nickel nanorods are multidomain,

they behave like rigid dipole particles for reasons outlined

above. Cobalt nanorods, however, are multidomain magnetic

particles that have multiple poles and do not behave like

rigid dipoles. Thus, nickel nanorods are good candidates for

the MRS probes, while cobalt nanorods are not.

ACKNOWLEDGMENTS

We greatly acknowledge support of the U.S. Air Force

Office of Scientific Research, Grant No. FA9550-12-1-0459

managed by Dr. Ali Sayir. We appreciate the help of Dr. Chen-

Chih Tsai and James Townsend with nanorod imaging.

APPENDIX A: UNCERTAINTY CALCULATION FORNANOROD MAGNETIZATION MEASUREMENT WITHMRS

The critical condition of Eq. (2) when expressed through

magnetization reads

xc¼MB

cg; where M¼m

4

pd2land c¼

4

3

l

d

� �2

ln2l

d�1

� � : (A1)

Solving for magnetization,

M ¼ xccgB

: (A2)

The uncertainty of square of magnetization is thus calcu-

lated by taking the square of the total derivative under the

assumption of linearly independent variables (all covariances

equal zero)

DM ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidM

dc

� �2

Dc2 þ dM

dxc

B

0@

1A

2

Dxc

B

2

vuuut ; (A3)

DM ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixc

Bg

� �2

Dc2 þ cgð Þ2Dxc

B

2

s; (A4)

where D xc

B is a random error associated with the experiment

and is calculated to be D xc

B ¼ 70 s�1T�1 from the measure-

ments, provided that magnetic field measurements have

much smaller error than that of frequency, Dxc; dc is a

combination of a random error in diameter and a systematic

error in length. The random error in diameter arises due to

the fact that the actual diameter of a given rod is unknown.

The systematic error in length arises from the fact that as the

rod rotates, its rotation comes slightly out of plane. As the

rod comes out of plane of rotation, less viscous drag acts on

the rod, effectively shortening the nanorod. Using AFM, we

measured the surface profiles of the rods and found a varying

diameters, thus estimating dr to be dr¼ 30 nm. Using experi-

mental data, we extracted the apparent length of the rod at

different times, thus estimating dl to be dl¼ 0.6 lm. dc is

calculated from the uncertainty in length, dl, and uncertainty

in diameter, dr, in the following way:

Dc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidcdl

� �2

Dl2 þ dcdd

� �2

Dd2

s; (A5)

dcdd¼ �

16l2 ln 2ð Þ þ lnl

d

� �� 1

� �

d3 2ln 2ð Þ þ 2lnl

d

� �� 1

� �2; (A6)

dcdd¼

16l ln 2ð Þ þ lnl

d

� �� 1

� �

d2 2ln 2ð Þ þ 2lnl

d

� �� 1

� �2: (A7)

For a rod of length 10 6 0.5 lm, with diameter

340 6 30 nm, Eqs. (A3)–(A5) yield Dc ¼ 44. Thus, from

Eq. (A4), for g¼ 0.12 Pa*s, average critical frequency

xc¼ 4 Hz, average driving magnetic field B¼ 400 lT, we

get DM¼ 38 kA/m.

APPENDIX B: X-RAY DIFFRACTION

XRD is widely used to characterize the crystal structure as

well as the crystallite size of the material. Figure 14(a) shows

schematically the working principle of an X-ray diffractometer.

h is the incident angle of the X-ray beam defined with respect

to the sample surface. For the reflected beam, the detector is

positioned at the same angle h. During the experiment, the

angle h is varied step by step in a certain range and the inten-

sity I(h) of the reflected beam is measured by the detector.

FIG. 14. (a) Schematic of an X-ray dif-

fractometer and (b) schematic of the

Bragg diffraction.

074901-11 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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Bragg’s law is a simplified model of diffraction, describing the

diffraction of X-ray beam by crystals. As shown in Fig. 14(b),

two crystal planes are separated by distance d, and the path dif-

ference between the two beams reflected by the two parallel

planes is 2dsinh. The Bragg angle h is the same angle defined

in Fig. 14(a). Bragg proposed that when the phase difference is

a multiple of the X-ray wavelength (constructive interference),

a peak will appear in the intensity spectrum I(h). This condition

is expressed by Bragg’s law48

nk ¼ 2d sin h; (B1)

where n is a integer and k is the wavelength of the X-ray.

For a certain crystal structure, the spacing d between crystal

planes is determined by the orientation of the plane defined

by the Miller indices hkl. Therefore, different peaks in the

spectrum I(h) correspond to the different crystal planes.

The idea for the derivation of Scherrer equation is as fol-

lows. Assume that the crystal has Nþ 1 crystalline planes,

the size of crystallite t will be Nd. Bragg’s equation (B1) can

be written in the form nk¼Ndsinh¼ tsinh for the two

boundary planes of the crystallite. Taking derivative on both

sides, one obtains

Dnk ¼ t cos hDh: (B2)

If one takes Dh¼ b, Dn¼KS, Eq. (B2), one arrives at

the Scherrer equation.

The Scherrer equation also indicates that the greater the

FWHM b, the smaller the crystallite size. Thus, small crys-

tallites broaden the peak. One way to understand this de-

pendence is to consider the crystal as a diffraction grating.

The size of the crystallite t is proportional to the number of

parallel planes Nþ 1 that interact with the X-ray. The total

reflection from the crystallite will be the superposition of the

beam reflected by each individual planes. The phase differ-

ence between the two beams reflected by the plane #1 and

plane #N will be 2p(N�1)dsinh/k. Summing up the reflected

beams by all the planes, we can write the intensity I(h) as

I hð Þ /XN

n¼1

einc

2

/ sin Ncð Þsin c

2

c ¼ 2pk

d sin h: (B3)

Fig. 15 shows how the function I(h) varies for different

N. In calculations, we used c¼ sinh for simplicity. The

graphs in Fig. 15 clearly show that with the increasing

number of crystalline planes N, the peak becomes shaper and

shaper. The full width at half maximum b is smaller for

larger N, i.e., for the larger crystallite.

APPENDIX C: MAGNETIC FORCE MICROSCOPY

The phase dependence can be explained by modeling

small oscillations of the cantilever as a forced oscillation of

harmonic oscillator

€z þ 2d _z þ x20z ¼ F0 cos xt=mf þ FzðzÞ=mf ; (C1)

where d> 0 is the damping coefficient, x0¼ (k/mf)1/2 is the

natural frequency of the oscillator, k is the effective stiffness

of the cantilever, and mf is the effective mass. F0 is the driv-

ing amplitude and x is the driving frequency of the piezoele-

ment. Fz(z) is the magnetic force acting on the cantilever as

discussed above. The magnetic force here is written as a

function of the position of the cantilever. This force is stron-

ger when the cantilever moves closer to the sample surface,

i.e., djFz(z)j/dz> 0. For these three cases shown in Fig. 9 we

have, case A: Fz(z)> 0, dFz(z)/dz> 0, case B: Fz(z)¼ 0,

dFz(z)/dz¼ 0, case C: Fz(z)< 0, dFz(z)/dz< 0. For small

oscillations, we can Taylor expand magnetic force near the

equilibrium position z0 of the oscillator keeping only the first

order term

Fz zð Þ � Fz z0ð Þ þ Fz0 z� z0ð Þ þ :::; Fz

0 ¼ dFz zð Þdz

z¼z0

: (C2)

The equilibrium position z0 satisfies the relation:

kz0¼Fz(z0). The general solution for equation (C2) is written as

zðtÞ ¼ z0 þ e�dtz1ðtÞ þ Am cosðxtþ uÞ; (C3)

where Am is the amplitude of the oscillation, u is the phase

and they satisfy the following relation:

Am ¼F0=mfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k � Fz0ð Þ=mf � x2

� �2 þ 4d2x2

q ; (C4)

tan u ¼ 2dxx2 � k � Fz

0ð Þ=mf

: (C5)

As t!1, the second term on the right hand side of equa-

tion (C3) disappears and only a harmonic oscillation is observed.

Before the measurement, the piezoelement was tuned to operate

at the natural frequency of the oscillator, i.e., x¼x0. As a

result, the amplitude and phase can be rewritten as

Am ¼F0=mfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Fz0=mfð Þ2 þ 4d2x2

0

qtan u ¼ 2dx0

Fz0=mf

: (C6)

For case B, dFz(z)/dz¼ 0, u¼p/2. For case A, dFz(z)/

dz> 0, u< p/2, i.e., Du< 0. For case C, dFz(z)/dz< 0,

u> p/2, i.e., Du> 0. By scanning over the sample surface,

we can identify the orientation of the magnetization in differ-

ent regions.

FIG. 15. Dependence of the width of the peak on the number of crystalline

planes based on the diffraction grating model.

074901-12 Aprelev et al. J. Appl. Phys. 118, 074901 (2015)

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