#  Effective forces between active polymers

M. C. Gandikota and A. Cacciuto<EMAIL_ADDRESS>Department of Chemistry,
Columbia University
3000 Broadway, New York, NY 10027

###### Abstract

The characterization of the interactions between two fully flexible self-
avoiding polymers is one of the classic and most important problems in polymer
physics. In this paper we measure these interactions in the presence of active
fluctuations. We introduce activity into the problem using two of the most
popular models in this field. One where activity is effectively embedded into
the monomers’ dynamics, and the other where passive polymers fluctuate in an
explicit bath of active particles. We establish the conditions under which the
interaction between active polymers can be mapped into the classical passive
problem. We observe that the active bath can drive the development of strong
attractive interactions between the polymers and that, upon enforcing a
significant degree of overlap, they come together to form a single double-
stranded unit. A phase diagram tracing this change in conformational behavior
is also reported.

One of the defining features of active systems is their ability to transform
energy in their environment into translational or rotational motion Dey and
Sen (2017). This enhanced kinetics results in very exotic physical behavior
not achievable in systems in thermal equilibrium. A significant amount of work
has been done to understand the interplay between active and thermodynamic
forces (see for instance Zöttl and Stark (2016); Bialké _et al._ (2015);
Menzel (2015); Marchetti _et al._ (2013); Romanczuk _et al._ (2012); Bechinger
_et al._ (2016); Cates and Tailleur (2015); Marchetti _et al._ (2016);
Ramaswamy (2010); Klapp (2016); Mallory _et al._ (2018); Wang _et al._ (2013))
and to discover ways to exploit active forces to perform specific tasks at the
micro scale Frangipane _et al._ (2018); Di Leonardo (2016). Although most of
the literature has focused on the behavior of spherical colloidal active
particles, more recently, the way active particles interact with flexible and
deformable objects or the behavior of active filaments has been the subject of
intense scrutiny Loi _et al._ (2011); Kaiser and Löwen (2014); Harder _et al._
(2014); Ghosh and Gov (2014); Shin _et al._ (2015); Kaiser _et al._ (2015);
Samanta and Chakrabarti (2016); Eisenstecken _et al._ (2016); Chelakkot _et
al._ (2014); Isele-Holder _et al._ (2015, 2016); Kaiser _et al._ (2015);
Nogucci and Ishihara (2016); Bianco _et al._ (2018); Harder and Cacciuto
(2018) (see also Winkler _et al._ (2017); Winkler and Gompper (2020) for a
brief review on the subject and references therein). Fully flexible active
polymers are of particular interest. While there is model dependence for the
scaling of the radius of gyration as a function of Péclet number Bianco _et
al._ (2018), the scaling behavior for polymers made out of active Brownian
particles seems to remain unaffected by the action of active forces. Within
this framework, the radius of gyration of a fully flexible active polymer
follows Flory’s exponent (at least for weak to moderate activities), and the
active forces only affect the pre-factor of the scaling law Kaiser _et al._
(2015). Similarly, we have recently shown Das _et al._ (2021) how the coil-to-
globule transition of an active polymer with attractive interactions can also
be understood with a rescaling of the temperature. Yet, some of its scaling
behavior breaks down when polymers are placed under strong confinement Das and
Cacciuto (2019).

In this article, we study how adding activity changes the entropic forces two
polymers exert on each other within the framework of dry active matter. This
is a classic problem in polymer physics of passive polymers as it is of
crucial importance to understand the phase behavior of dense polymer solutions
Gennes (1979); Grosberg and Khokhlov (2002). What is somewhat surprising about
the interaction between two polymers in the infinite dilute limit is that the
free-energy cost of fully overlapping two self-avoiding flexible polymers is
finite and accounts for only few $k_{\rm B}T$, where $k_{\rm B}$ is the
Boltzmann’s constant and $T$ is the temperature of the system Grosberg _et
al._ (1982); Louis _et al._ (2000).

A simple way of rationalizing this result is to realize that overlapping two
chains of $N$ monomers and radius of gyration $R^{(N)}_{\rm g}$, is similar to
confining a single chain of double the original length ($2N$) into a spherical
cavity of radius equal to the radius of gyration of a single chain, i.e.
$R=R^{(N)}_{\rm g}$. The free energy cost associated with this operation in
units of $k_{\rm B}T$ is equal to $\beta F\sim(R_{\rm
g}^{(2N)}/R)^{d/(d\nu-1)}$ Grosberg and Khokhlov (2002); Cacciuto and Luijten
(2006) where $\beta=1/(k_{\rm B}T)$ is the inverse temperature, $\nu=3/(d+2)$
is the Flory scaling exponent and $d$ is the dimension of the embedding space.
By plugging $R_{\rm g}^{(2N)}\sim(2N)^{\nu}$ and $R\simeq R^{(N)}_{\rm g}\sim
N^{\nu}$, one obtains an estimate of the overlapping free energy $\beta F\sim
2^{d\nu/(d\nu-1)}$, which is clearly finite. Unfortunately, this quantity
cannot be directly measured in active systems as free energies cannot be
consistently defined; yet, their derivatives, i.e., pressures and forces the
polymers exert on each other can be readily measured numerically.

There are two distinct models that have been put forward to study flexible
active polymers in the context of dry active matter. One where a passive chain
is free to fluctuate in an explicit bath containing active particles Harder
_et al._ (2014), which we will refer to here as the ${\it explicit}$ model,
and the other where the action of the active bath is incorporated into the
chain by treating every single monomer as an effective independent Brownian
active particle Kaiser _et al._ (2015), which we will refer to as the ${\it
implicit}$ model.

In experiments, the simplest realization of an active particle is obtained by
coating one hemisphere of a silica or polymer micro-particle with a thin layer
of platinum. Since the metal hemisphere can be rather heavy, most active
colloids readily deposit at the bottom of the solution, and perform what is
effectively a two dimensional Brownian active motion with the axis of
propulsion parallel to the surface that supports them. We therefore limit our
study to two dimensions, and when considering the explicit model, we envision
the passive polymer as a chain of colloidal particles having the same diameter
of the active colloids.

In this work, we will perform our measurements using both models. Our basic
model for a flexible, self-avoiding polymer consists of $N$ monomers of
diameter $\sigma$ linearly connected with stiff harmonic springs and subject
to thermal forces. Every monomer undergoes Brownian dynamics at a constant
temperature $T$. For the model with implicit active forces, a self-propelling
force is introduced via a directional velocity of constant magnitude $v_{p}$
directed along a predefined orientation unit vector $\boldsymbol{\hat{n}}$
centered at the origin of each monomer. For the model with explicit solvent,
the monomers are exclusively subject to thermal forces, however, $N_{a}$
spherical particles of diameter $\sigma$ are also added in a simulation box of
size $L$, and these follow the same active dynamics discussed above for the
monomers of the implicit model. The resulting equations of motion for both
species of particles are

$\begin{split}\frac{d\boldsymbol{r_{i}}(t)}{dt}&=\frac{1}{\gamma}\boldsymbol{f}(\\{r_{ij}\\})+v_{p}\,\boldsymbol{\hat{n}_{i}}(t)\,\delta_{t_{i},1}+\sqrt{2D}\,\boldsymbol{\xi}(t),\\\
\frac{d\boldsymbol{{\hat{n}_{i}}}(t)}{dt}&=\sqrt{2D_{r}}\,\boldsymbol{\xi}_{r}(t)\times\boldsymbol{\hat{n}_{i}}(t),\end{split}$
(1)

where $i$ is the particle index, $t_{i}$ is a binary index that can acquire
two values, 0 for passive particles and 1 for the active ones.
$\delta_{t_{i},1}$ is a Kronecker delta function which deactivates the self-
propulsion term for the passive particles. The translational diffusion
coefficient $D$ is related to the temperature and the translational friction
$\gamma$ via the Stokes-Einstein relation $D=k_{\rm B}T\gamma^{-1}$. Likewise,
the rotational diffusion coefficient, $D_{r}=k_{\rm B}T\gamma_{r}^{-1}$, with
$D_{r}=3D\sigma^{-2}$. The solvent induced Gaussian white-noise terms for both
the translational $\boldsymbol{\xi}$ and rotational $\boldsymbol{\xi}_{r}$
motion are characterized by $\langle\boldsymbol{\xi}(t)\rangle=0$ and
$\langle\xi_{m}(t)\xi_{n}(t^{\prime})\rangle=\delta_{mn}\delta(t-t^{\prime})$.
$\boldsymbol{f}(\\{r_{ij}\\})$ indicates the excluded volume forces for all
particles and the harmonic forces between the monomers of each polymers.
Excluded volume forces between any two particles are enforced via a Weeks-
Chandler-Andersen (WCA) potential
$U(r_{ij})=4\varepsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}+\frac{1}{4}\right].$
We use harmonic bonds between the monomers according to the potential
$U_{b}=k_{b}(r_{i,i+1}-\sigma)^{2}$. Here $r_{i,i+1}$ is the distance between
consecutive monomers along the chain. $k_{b}$ is set to
$3500k_{\rm{B}}T/\sigma^{2}$ to ensure polymer connectivity while
simultaneously minimizing bond stretching that could arise from the action of
the active forces. Finally, the hard repulsion between the monomers was
selected to be $\varepsilon=100k_{\rm{B}}T$, to prevent interpenetration of
the polymers. To confine the distance between the center of mass of the two
polymers, $\delta R_{\rm cm}$, to remain within a given distance $R$, we use a
boundary defined by the potential

$U_{c}(\delta R_{\rm cm}-R)=\left\\{\begin{array}[]{ll}0,&\delta R_{\rm
cm}\leqslant R\\\ k(\delta R_{\rm cm}-R)^{2},&\delta R_{\rm cm}>R\\\
\end{array}\right.$ (2)

with $k=2000k_{\rm{B}}T/\sigma^{2}$. Each monomer in a chain at position
$\boldsymbol{r}_{i}$ will experience a confining force

$\boldsymbol{f}_{i}=-\left(\frac{\partial U_{c}}{\partial\delta
R_{\text{cm}}}\right)\frac{\partial\delta
R_{\text{cm}}}{\partial\boldsymbol{r}_{i}}$ (3)

where $\boldsymbol{\delta
R}_{\text{cm}}=1/N\sum_{i=1}^{N}(\boldsymbol{r}^{(1)}_{i}-\boldsymbol{r}^{(2)}_{i})$,
and the upper indices refer to whether the monomer is part of the first or the
second chain.

In our simulations, $\sigma$ and $k_{\rm B}T$ are used as the units of length
and energy scales of the system respectively, while $\tau=\sigma^{2}D^{-1}$ is
the unit of time. All simulations were typically run for at least $10^{9}$
time steps with a time step ranging from $\Delta t=10^{-4}\tau$ to $\Delta
t=10^{-5}\tau$. We skip the first $2\times 10^{6}$ time steps to let the
system achieve a state of dynamic equilibrium. The strength of the active
forces is reported in terms of the Péclet number defined as
$Pe=v_{p}\sigma/D$. The positions of the random active particles in the
explicit model are randomly and uniformly distributed.

Figure 1: Reduced contact force $f_{w}^{R}$ as a function of $Pe$ for two
polymers of length $N=256$. The vertical dashed line indicates the crossover
point, $Pe^{*}$, between quadratic and linear behavior obtained as discussed
in the text. The solid line is a fit to the data with the function
$f(x)=ax/(1+b/x)$. The inset shows the linear relationship between
$\ell_{p}^{*}=(Pe^{*}/3)\sigma$ and $R_{\rm g}$ for $N=32,64,128,256$.

We begin our analysis with the ${\it implicit}$ model. We measure how the
force exerted on the boundary, $f_{w}=-\partial U_{c}/\partial\delta
R_{\text{cm}}$, in the fully overlapping regime, i.e. when the distance
between the centers of mass of two polymers is confined within $R=\sigma$ from
each other, depends on the strengths of the active forces, $Pe$. The results
are shown in Fig. 1, where we report the reduced force
$f_{w}^{R}=[f_{w}(Pe)-f_{w}(0)]/f_{w}(0)$ as a function of $Pe$ $-$ written
this way $f_{w}^{R}$ is effectively equivalent to the reduced pressure between
the polymers. The data displays a quadratic behavior for small activities and
a linear behavior for large $Pe$. The data for all polymer lengths considered
in this study, $N=32,64,128,256$, is accurately described when fitted to the
functional form $f_{w}^{R}=aPe/(1+b/Pe)$, previously proposed to describe the
pressure of active particles within a cavity as a function of $Pe$ Mallory _et
al._ (2014). The crossover Péclet number, $Pe^{*}$, is estimated using the
fitted parameter $b$. Crucially, we find that the crossover occurs when the
persistence length of the active force, defined as
$\ell_{p}=v_{p}/D_{r}=(Pe/3)\sigma$, becomes of the order of the radius of
gyration of the polymers, $R_{\rm g}$. This result is important because it
indicates that as long as $\ell_{p}$ is much smaller than $R_{g}$, activity
acts as an effective temperature $T\propto Pe^{2}$, and the phenomenology of
the parent passive system can be easily extended to incorporate the active
forces. In the opposite limit, such a mapping becomes inappropriate. The inset
in Fig. 1 shows that the crossover $\ell_{p}^{*}$ tracks with $R_{\rm g}$ as
$\ell_{p}^{*}\approx R_{\rm g}/2$. An analogous result was obtained for the
implicit model embedded in three dimensions (see Fig. S1 in the supplementary
material).

Next, we measure the full force curve between two polymers, each of $N=128$
monomers, as a function of $R$. The force, $f_{w}$, contains two
contributions, one due to the repulsion between the polymers, $f_{p}$, and the
other, the ideal term, $f_{id}$, due to the overall motion of each polymer’s
center of mass independently of the presence of the other polymer. Since we
are exclusively interested in extracting the effective polymer-polymer
interactions, we subtract the second contribution from the first and report
$f_{p}=f_{w}(R)-f_{id}(R)$. $f_{id}$ is computed in the same manner as
$f_{w}$, but now the excluded volume interaction between monomers from two
different polymers is turned off.

Figure 2: Force between two active polymers, $f_{p}$ as a function of their
separation $R$, for different Péclet numbers. The data is normalized with
respect to the radius of gyration of a passive polymer, $Pe=0$, of equal
length, $R^{\rm o}_{\rm g}$.

Figure 2 shows how $f_{p}$ depends on the polymer separation, $R$, for
different values of $Pe$. The overall shape of the force as a function of $R$
is reminiscent of that expected for two passive polymers Louis _et al._
(2000), where a steep repulsion observed as one moves from small to moderate
overlaps leaves space to a decay of the force for large overlaps. This is
consistent with a fully repulsive potential of mean force between the two
polymers that flattens as the two polymers develop a significant degree of
overlap.

The phenomenology is richer for the explicit model. Here, we considered two
$N=64$ passive polymers embedded in a square box of side length $L=100\sigma$
with periodic boundaries containing $N_{a}=600$ active spherical particles.
Each particle in the system interacts via the purely repulsive WCA potential
discussed above. This particular number density
$\rho=N_{a}/L^{2}=0.06/\sigma^{2}$ is sufficiently large to ensure the
induction of strong active fluctuations on the polymer, but not too large to
drive motility-induced phase separation in the fluid. The net force between
the polymers as a function of $R$, computed in a similar fashion as in the
previous case 111In this case $f_{\rm id}$ is computed by placing a single
polymer inside a cavity of size $R$, confined via the truncated harmonic
potential wall in Eq. 2 with $\delta R_{\rm cm}\rightarrow R_{\rm cm}$. is
shown in Fig. 3. The data indicates a strikingly different behavior than what
is observed for the implicit model.

Figure 3: Force between two passive polymers, $f_{p}$, in a fluid of active
particles, as a function of their separation $R$, for different Péclet
numbers. The data is normalized with respect to the radius of gyration of a
passive polymer, $Pe=0$, of equal length, $R^{\rm o}_{\rm g}$. The inset shows
$f_{p}$ for small values of $Pe$.

For small Péclet numbers the functional form of the polymer-polymer force is
similar for both models, however, beyond a certain Péclet number a net
negative force develops for the explicit model, the depth and range of which
increases with $Pe$. This is clear evidence of the development of an
attractive interaction between the polymers at short range following a
repulsion for intermediate separations.

This attraction develops as a result of condensation of active particles along
the perimeter of the polymers, which leads to an effective active depletion
force not unlike that previously observed between rigid bodies in an active
fluid (see for instance Harder _et al._ (2014); Ni _et al._ (2015); Leite _et
al._ (2016); Baek _et al._ (2018); Ray _et al._ (2014)). The net effect is
that at short separations the polymers pair up to form a single, double-
stranded fluctuating unit. This is in opposition to the conformations in the
implicit model, where such a pairing is not observed.

We should stress that the range, strength and sign of depletion interactions
between passive objects immersed in active baths are known to be strongly
shape dependent. In a two dimensional system of active depletants, passive
colloidal plates experience an effective long-range attractive force whereas
passive colloidal disks experience a short-ranged repulsive force Harder _et
al._ (2014). Unlike rigid objects, flexible polymers are free to fluctuate
between shapes ranging from an extended rod-like conformation to a coiled up
disk-like conformation. Thus, the result that emerges from our simulations is
rather non-trivial and neatly extends the results obtained for rigid objects
to include fully flexible ones.

The right side of Fig. 4 shows typical paired and unpaired conformations
associated with the two polymers at full overlap. Although once in a while we
observe some active particles in the interstitial space between the polymers,
they can easily wiggle out because of their active forces and because of
fluctuations between the polymer centers of masses that allow for some
breathing room along the chain. It should be noticed that we also performed
simulations with polymers of double the length ($N$=256) and even in this case
we observe pairing of the polymer chains. To characterize the transition
between the two conformations as a function of the Péclet number, we measured
the probability distribution of a contact parameter, $q$, defined as the
average number of monomers in one polymer that are within a distance of
$2\sigma$ from each monomer in the other polymer. With this definition
$q\in[0,4]$, where $q=0$ for fully unpaired polymers, and $q=4$ when they are
fully aligned.

Figure 4: Left side: Probability distribution of the contact parameter $q$ as
a function of Péclet number $Pe$. Right side: Snapshots from simulations
showing characteristic unpaired (for small $Pe$) and paired (for large $Pe$)
configurations of the two polymers in an active bath while their centers of
mass are confined to be within a distance of $R=\sigma$ from each other.

Figure 4 shows the shift of $P(q)$ as a function of $Pe$ when the center of
mass of the two polymers are confined to be within a distance of $R=\sigma$
from each other. A peak at small values of $q$ is visible for small $Pe$ and
the peak moves closer to $q$’s maximum value for large $Pe$. In between, for
$Pe\approx 40$, we see a broad distribution of $q$ indicating a region where
the polymer can move unobstructed from one conformation to the other. Although
the onset value of $Pe^{\dagger}\approx 40$ appears to be rather independent
of $N$, apart from possible finite size effects, we do expect $Pe^{\dagger}$
to be very sensitive to the overall number density of active particles. By
measuring $P(q)$ for different values of $Pe$, we can construct a phase
diagram tracing $Pe^{\dagger}$ for different values of active particles
density $\rho$. The results are shown in Fig. 5.

Figure 5: Phase diagram separating the paired and unpaired configuration of
two polymers in an active bath with the distance between their centers of mass
confined within a distance of $R=\sigma$ as a function of the density of
active particles $\rho$ and Péclet number $Pe$.

This diagram presents two interesting features. First, density and Péclet
number are inversely related to each other, and it appears that even for very
low densities it is possible to find a sufficiently large $Pe$ to drive the
pairing of the polymers. Second, our data shows a lower bound for $Pe\approx
35$ below which no pairing occurs, even for large particle densities. We
suspect this is because upon increasing the density, the number of active
particles condensing on the two polymers become quite significant. The
polymers act as crystallization seeds for the active particles that form
multiple crystalline layers around and between the contour length of the two
polymers, effectively hindering their mobility. We find that the shape of the
boundary is well described by the simple empirical equation
$\rho=0.27/(Pe-31.3)$, which predicts a lower bound for $Pe\approx 31$.

Although we haven’t performed systematic simulations of the full force-
separation curves as a function of $Pe$ for polymers embedded in a three
dimensional space, this is because most of the experiments with active
colloids are in two dimensions, it is nevertheless of interest to look at the
behavior in this case. Our sparse data in three dimensions for large Péclet
numbers suggest that both the implicit (see Fig. 6) and explicit models
generate a purely repulsive interaction between the polymers which increases
with the strength of the active forces. In this case the active depletion is
not present as it is harder for the polymers to capture active particles along
their contour length.

Figure 6: Reduced contact force $f_{w}^{R}=[f_{w}(Pe)-f_{w}(0)]/f_{w}(0)$ as a
function of Pe for two polymers of length $N=128$ in three dimensions. Here,
both polymers are in the fully overlapping regime i.e. the distance between
the centers of mass of the polymers are confined to be within a distance of
$R=\sigma$, the monomer diameter. The reduced contact force moves from a
quadratic regime at small Pe to a linear regime at large Pe. The solid line is
a fit to the data with the function $f(x)=ax/(1+b/x)$. The fitting parameter
$b=48.4$ is the $Pe^{*}$ at which the system moves across the two regimes. At
this crossover, $l_{p}^{*}=Pe^{*}/3\approx 2R_{g}$. Note that the pre-factor
is different from the case of two polymers in two dimensions.

In this article, we measured the forces between two active polymers. We used
two different models to incorporate the role of active fluctuations into the
problem. In one model the fluctuations are implicitly incorporated into the
motion of the monomers, in the other, they are explicitly accounted for by
placing the polymers in an active bath of active particles. We find that in
the former case, the forces between two fully overlapping polymers can be
understood in terms of an effective temperature $T\propto Pe^{2}$ as long as
$\ell_{p}$ is smaller than $R_{\rm g}/2$. Deviations from this behavior occur
in the opposite limit where the forces grow linearly with $Pe$.

We observe a very different scenario when considering the explicit model. For
small Péclet numbers, similar to the previous case, the net effect of the
active bath is that of increasing the overall repulsion between the polymers.
However, as soon as $Pe$ becomes sufficiently large for active particles to
condense on the contour length of the polymers, a strong depletion attractive
force emerges and drives the polymers to fluctuate as a pair. Our results
extend our current understanding of how active forces affect polymer
fluctuations and set clear limits for their mapping into effective equilibrium
systems. It is also important to emphasize that the emergence of active
depletion force between fully flexible filaments is not a trivial result, and
will have important implications for the dynamics and morphology of polymers
solutions in an active bath.

An important limitation of our study is that it does not account for
hydrodynamic effects Zöttl and Stark (2016). At a sufficiently large
concentration of active particles, hydrodynamic interactions could, in
principle, drive orientational instabilities that could destabilize the paired
configurations. Whether this is the case requires more work in this direction,
using a more sophisticated description of the active colloids with explicit
squirmer models Blake (1971) and explicit hydrodynamic interactions. We expect
possible deviations to be dependent on the specific choice of swimming mode
(puller vs pusher), and on whether the monomers in the chains are free to
rotate or act as a fixed boundary against the torques applied by the squirmers
Das and Cacciuto (2019).

## Acknowledgements

A.C. acknowledges financial support from the National Science Foundation under
Grant No. DMR-2003444.

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# Random magnetic anisotropy in a new compound
$\mbox{Sm}_{2}\mbox{Ag}\mbox{Si}_{3}$

Baidyanath Sahu<EMAIL_ADDRESS>R. Djoumessi Fobasso  Highly
Correlated Matter Research Group, Department of Physics, University of
Johannesburg, PO Box 524, Auckland Park 2006, South Africa  André M. Strydom
Highly Correlated Matter Research Group, Department of Physics, University of
Johannesburg, PO Box 524, Auckland Park 2006, South Africa

###### Abstract

We report the experimental study on the structural and magnetic properties of
a new ternary intermetallic compound $\mathrm{Sm_{2}AgSi_{3}}$. The properties
of the sample were investigated in detail by X-ray diffraction,
dc$\textendash$magnetization, and heat capacity measurements. The
polycrystalline compound of $\mathrm{Sm_{2}AgSi_{3}}$ crystallizes in the
$\mathrm{ThSi_{2}}$-type tetragonal structure (space group ${I_{4_{1}}/amd}$).
The temperature$\textendash$dependent dc$\textendash$magnetization and heat
capacity results demonstrate that the compound undergoes ferromagnetic
behaviour with a Curie temperature ($T_{C}$) of 14 K. The large coercive field
in hysteresis loops and the thermomagnetic irreversibility in the
ferromagnetic region revealed that the compound exhibits a large magnetic
anisotropy. The magnitude of the applied field and the coercivity obtained
from the M-H loops corroborate with the thermomagnetic irreversibility in the
magnetization data. The magnetic contribution of the heat capacity reveals a
broad Schottky$\textendash$type anomaly above $T_{C}$, due to the presence of
the crystal electric field effect of $\mathrm{Sm^{3+}}$ in
$\mathrm{Sm_{2}AgSi_{3}}$.

Ferromagnet; Magnetic anisotropy; Thermomagnetic irreversibility; Schottky
peak.

††preprint: AIP/123-QED

The ternary intermetallic compounds $\mathrm{R_{2}TX_{3}}$ (R = rare earths, T
= transition metals, X = Si, Ge, Al, and In), are extensively studied for
their structural and magnetic properties. Most of the compounds with
$\mathrm{R_{2}TX_{3}}$ stoichiometry form in $\mathrm{ThSi_{2}}$ type both
hexagonal and tetragonal crystal structure. The magnetic R ions occupy the Th
position, while the T and X ions mutually occupy the Si positions in
$\mathrm{ThSi_{2}}$ R2TX3 ; Tran ; R2AgGe3 . Recently, the random magnetic
anisotropy in a polycrystalline $\mathrm{ThSi_{2}}$ type hexagonal compound
$\mathrm{Sm_{2}NiSi_{3}}$ was confirmed from the temperature dependence of
zero-field cooled (ZFC) and field-cooled (FC) magnetization and magnetic
entropy changes Sm2NiSi3-A ; Sm2NiSi3-F .

Samarium (Sm) based intermetallic compounds have gained considerable interest
because of their interesting properties such as superconductivity,
magneto$\textendash$resistance, magnetic and non magnetic behaviour SC1 ; SC2
; MR . Sm based ternary and binary compounds can show large magnetic
hysteresis due to their large magnetic anisotropy and may be used as a
permanent magnet RNi4B ; Anisotropy . A strong magneto-crystalline anisotropy
may be produced by $\mathrm{Sm^{3+}}$ ion as a consequence of the crystal
electric field (CEF) effect acting on the 4f$\textendash$electrons.

In recent years, the development of magnetic materials has posed a great
challenge to researchers. To the best of our knowledge, the existence of
$\mathrm{Sm_{2}AgSi_{3}}$ compound was not earlier reported. In this work, we
have reported the detailed synthesis process and magnetic properties of a new
compound $\mathrm{Sm_{2}AgSi_{3}}$. The large random magnetic anisotropy in
$\mathrm{Sm_{2}AgSi_{3}}$ is also demonstrated.

##  .1 Experimental Set-up

A polycrystalline sample $\mathrm{Sm_{2}AgSi_{3}}$ was prepared under ultra-
high purity argon atmosphere using the standard arc-melting technique. The
melted ingot was turned over and remelted five times to ensure a good
homogeneity. The melted sample was wrapped in a tantalum foil and sealed in a
vacuum quartz tube. The tube was kept in a furnace for annealing at 1373 K for
one week and then quenched in cold water. The sample was characterized by
powder X-ray diffraction (XRD) using $\mathrm{CuK_{\alpha}}$ radiation of a
Rigaku XRD machine. The Rietveld analysis of XRD patterns was carried out
using FULLPROF software rietveld ; fullprof . The temperature and field
dependence of dc$\textendash$magnetization measurement was performed using a
Dynacool physical properties measurement system (PPMS) made by Quantum Design,
USA. Heat capacity measurements were carried out using the same equipment.

##  .2 Results and Discussion

###  .2.1 X-ray Diffraction

Fig. 1 shows the room-temperature powder XRD pattern along with the Rietveld
refinement fitting profile of polycrystalline $\mathrm{Sm_{2}AgSi_{3}}$. The
Rietveld refinement result reveals that the compound form in a
$\beta$$\textendash$$\mathrm{ThSi_{2}}$$\textendash$type of tetragonal
structure with space group ${I_{{4}_{1}}/amd}$. The crystallographic
parameters from the refinement are listed in Table 1. The inset of Fig. 1
shows the schematic diagram of the crystal structure of
$\mathrm{Sm_{2}AgSi_{3}}$. The shortest distance between Sm atoms is 4.119 Å,
and is twice larger than the expected ionic radius of $\mathrm{Sm^{3+}}$
($\mathrm{r_{Sm}}$ = 0.958 Å). This suggests that there is a weak interaction
between the rare$\textendash$earth atoms in $\mathrm{Sm_{2}AgSi_{3}}$.

Figure 1: X-ray powder diffraction data for $\mathrm{Sm_{2}AgSi_{3}}$. Red symbols represent the experimental data and the black line represents the calculated data. The difference between experimental and calculated data is shown as a blue line. A set of vertical bars represents the Bragg peak positions of the tetragonal $\alpha$-$\mathrm{ThSi_{2}}$ type structure. Inset: The schematic representation of the tetragonal crystal structure of $\mathrm{Sm_{2}AgSi_{3}}$. The blue balls are for Sm and silver$\textendash$red color balls are for Ag$+$Si. Table 1:  The lattice parameters, unit cell volume and the atomic coordinate positions of $\mathrm{Sm_{2}AgSi_{3}}$ obtained from the Rietveld refinement of XRD patterns. The refinement quality parameter is $\chi^{2}$ = 5.7. a = b |  |  |  | 4.128(2) Å
---|---|---|---|---
c |  |  |  | 14.254(2) Å
V |  |  |  | 242.890(1) Å3
Atomic coordinates for $\mathrm{Sm_{2}AgSi_{3}}$ |  |  |  |
Atom | Wyckoff | $x$ | $y$ | $z$
Sm | $4a$ | 0 | 3/4 | 1/8
Ag | $8e$ | 0 | 1/4 | 0.2895(2)
Si | $8e$ | 0 | 1/4 | 0.2895(2)

###  .2.2 Magnetic Properties

Fig. 2 shows the temperature dependence of ZFC and FC dc$\textendash$magnetic
susceptibility ($\chi(T)$) with an applied field ($H_{app}$) of 1 T. The
dc$\textendash$$\chi(T)$ shows a typical paramagnetic to ferromagnetic
transition in this compound. The Curie temperature ($T_{C}$) was found to be
14 K from the peak of the d$\chi(T)$/d$T$ curve. The inverse
dc$\textendash$magnetic susceptibility (not shown in figure) does not follow
the Curie$\textendash$Weiss relation. Hence, the modified
Curie$\textendash$Weiss law, $\chi$ =
[$\mathrm{C}$/($T$-$\theta_{P}$)]+$\chi_{0}$, (where C is the Curie constant,
$\theta_{P}$ is the Weiss paramagnetic temperature and $\chi_{0}$ is the
temperature$\textendash$independent magnetic susceptibility which includes the
core-electron diamagnetism) was fitted on the $\chi(T)$ of Fig. 2 for $T$
$\textgreater$ 20 K. The least$\textendash$squares fit (LSQ) fit to the
experimental data yielded $\theta_{P}$ = 12 K and $\chi_{0}$ = 3.1 $\times$
$\mathrm{10^{-4}}$ emu/mole.Oe. The positive value of $\theta_{P}$ shows that
the dominant interaction is ferromagnetic in $\mathrm{Sm_{2}AgSi_{3}}$. The
effective magnetic moment ($\mu_{eff}$) value was calculated from the fitting
parameter value of C and found to be 0.57 $\mu_{B}$/$\mathrm{Sm^{3+}}$. A
small deviation of $\mu_{eff}$ from the theoretical value of free ion
$\mathrm{Sm^{3+}}$ [$g$$\sqrt{J(J+1)}$ = 0.85 $\mu_{B}$] may arise from
crystal field effects (CEF). The expanded region in low temperature of
$\chi(T)$ for different magnetic fields shows in the inset of Fig. 2. As seen
in the inset, there is a thermal magnetic irreversibility (TMI) shown between
the data collected in the ZFC and FC protocols. The TMI point is defined as
the temperature at which the $\chi_{ZFC}$ and $\chi_{FC}$ curves bifurcate
from each other. It is also observed that TMI depends on the $H_{app}$. It can
be believed that the large magnetic anisotropy may cause the TMI behavior in
$\chi(T)$ for high $H_{app}$ = 1.0 T.

Figure 2: Temperature dependence of the field-cooled (FC) and
zero$\textendash$field$\textendash$cooled (ZFC) dc$\textendash$magnetic
susceptibility ($\chi(T)$) at 1 T of $\mathrm{Sm_{2}AgSi_{3}}$ along with a
fit to the data (green line) of a modified Curie-Weiss law. Inset shows ZFC
and FC of $\chi(T)$ at low temperature region for various applied magnetic
fields.

Fig. 3 shows the field-dependent magnetization $M(H)$ loops at different
temperatures. The well$\textendash$defined hysteresis below $T_{C}$ is
observed in the ferromagnetic ordered region of $\mathrm{Sm_{2}AgSi_{3}}$. The
$M(H)$ loops do not saturate even at a high field value of 9 T at 2 K. The
spontaneous magnetization was calculated from $M(H)$ at 2 K, and the obtained
value is 0.25 $\mu_{B}$/Sm. This value is smaller by a considerable margin
than the expected saturation moment value for parallel alignment of
$\mathrm{Sm^{3+}}$ spin ($g$$J$ = 0.71 $\mu_{B}$, with $g$ = 2/7, and J =
5/2). The low magnetization value is attributed to the influence of the
magnetic anisotropy in saturation. As seen in the M-H loops, a large value of
coercivity ($H_{C}$) of 0.8 T at 2 K appears for a polycrystalline
rare$\textendash$earth based compound. This large $H_{C}$ may result from the
random magnetic anisotropy of the sample. For more details, the $M(H)$ loops
were measured at different temperatures. The $H_{C}(T)$ and remanence
magnetization was estimated from $M(H)$ loops for each temperature and found
to gradually decrease with increasing temperature. The temperature variation
$H_{C}(T)$ is plotted in inset (a) of Fig. 3. An exponential behavior
described by the relation $H_{C}(T)$ = ${Aexp(-BT)}$ is observed Pr3Ru4Al12 .
The fitting is shown as a blue line on the data. The best fitting parameters
obtained from the $H_{C}(T)$ $vs.$ $T$ curve are $A$ = 1.256 T and $B$ = 0.234
K-1 represent the $H_{C}$ at 0 K and the steep temperature respectively. The
steep temperature dependence of $H_{C}(T)$ in ferromagnet suggests that the
observed hysteresis of $\mathrm{Sm_{2}AgSi_{3}}$ is appropriate for large
magnetic anisotropy.

Figure 3: Hysteresis loops of $\mathrm{Sm_{2}AgSi_{3}}$ for different
temperatures between 2 and 12 K. Inset (a): the temperatures dependence of the
coercivity along with the fitting line explained in the text. Inset (b): the
high field magnetization data with the fitted line by using Eq. 1.

The magneto$\textendash$crystalline anisotropy constant of magnetic materials
are evaluated from the law of approach to saturation. The law of approach to
saturation is therefore applied on the $M(H)$ by fitting Eq. 1 on the high
magnetic field magnetization data as A1 ; A2 ; R2Ni7 :

$\displaystyle M=M_{S}\left[1-\frac{D}{H^{2}}\right]+\chi_{0}H^{1/2},$ (1)

where $M_{S}$ (in A/m) is the saturation magnetization. Here, $D$ is a
function of magnetic anisotropy energy and $\chi_{0}H^{1/2}$ denotes the term
related to para-process. The parameter $D$ is expressed as:

$\displaystyle
D=\frac{8}{105}\frac{K^{2}_{1}}{\mu^{2}_{0}M^{2}_{S}}\mathrm{(A/m)^{2}},$ (2)

where $K_{1}$ is the anisotropy constant in J.m-3 and $\mu_{0}$ is the free
space magnetic permeability. For polycrystalline material, it is assumed that
the overall possible orientations of the individual crystallites are averaged.
The value of $K_{1}$ was estimated by using the fitting parameters $B$ and
$M_{S}$ on Eq. 2. The obtained $K_{1}$ value is $|K_{1}|$ = 1.3$\times$105
J.m-3 at 2 K. The obtained value of $|K_{1}|$ is comparable with
9.2$\times$106 J.m-3 for $\mathrm{SmFe_{11}Ti}$ and 4.0$\times$106J.m-3 for
$\mathrm{Sm_{3}Fe_{5}O_{12}}$ SmFeTi ; garnet . The observed large hysteresis
loop in $M(H)$ and comparable value of $|K_{1}|$ revealed that a random
magnetic anisotropy exists in $\mathrm{Sm_{2}AgSi_{3}}$.

The anisotropy field plays a crucial role in determining the FC magnetic
susceptibilities ($\chi_{FC}$) and the ZFC magnetic susceptibilities
($\chi_{ZFC}$) at a given field value. The temperature variation of anisotropy
field and $H_{app}$ for measuring $\chi(T)$ plays a major role in determining
the degree of TMI joy1 . Joy $et~{}al.,$ joy1 ; joy2  have proposed an
empirical relation between $H_{app}$, $H_{C}$, $\chi_{FC}$ and $\chi_{ZFC}$
for a ferromagnetic system:

$\displaystyle\chi_{ZFC}=\chi_{FC}\frac{H_{app}}{H_{app}+H_{C}},$ (3)

This Eq. 3 indicates that the $\chi_{ZFC}$ and $\chi_{FC}$ should overlap in
that temperature regime for $H_{app}$ $\gg$ $H_{C}$. However, the difference
between the $\chi_{ZFC}$ and $\chi_{FC}$ values is large for $H_{app}$ $\ll$
$H_{C}$. Hence, it can be assume that the hysteresis between $\chi_{ZFC}$ and
$\chi_{FC}$ arises even at high field $H_{app}$ of 1.0 T due to the magnetic
anisotropy in the sample. However, the $\chi_{ZFC}$ for H = 0.5 T starting
from negative values of at low temperatures is due to this anisotropic sample
being cooled in a net negative trapping field. This trap magnetic field
commonly exists in the PPMS superconducting solenoid magnet PPMS .

###  .2.3 Heat Capacity

The temperature variation of heat capacity ($C_{p}(T)$) of
$\mathrm{Sm_{2}AgSi_{3}}$ and of the isostructural non$\textendash$magnetic
compound $\mathrm{La_{2}AgSi_{3}}$ are depicted in Fig. 4. A value of 147
J/(mol.$\textendash$K) is attained at 300 K, very closed to the
Dulong$\textendash$Petit limit 3nR = 149.65 J/(mol.$\textendash$K) (n = 6 is
the number of atoms per formula unit, R stands for the gas constant). At low
temperatures, $C_{p}(T)$ exhibits an anomaly in $\mathrm{Sm_{2}AgSi_{3}}$ for
the characteristic of a magnetic phase transition. The transition temperature
of $\sim$13 K is defined from the peak of $C_{p}(T)$, and is consistent with
the $M(T)$ data.

Figure 4: (Left scale) Temperature dependence of zero field heat capacity
($C_{P}$) of $\mathrm{La_{2}AgSi_{3}}$, and $\mathrm{Sm_{2}AgSi_{3}}$. (Right
scale) Temperature dependence of magnetic entropy of
$\mathrm{Sm_{2}AgSi_{3}}$. Inset: (left with bottom scale) temperature
dependence for magnetic contribution of ($C_{P}$) for
$\mathrm{Sm_{2}AgSi_{3}}$ at low temperature region. (left with top scale)
$T^{3/2}$ dependence of magnetic contribution of heat capacity for
$\mathrm{Sm_{2}AgSi_{3}}$.

The magnetic contribution of heat capacity ($C_{4f}$) for
$\mathrm{Sm_{2}AgSi_{3}}$ was estimated by subtracting the $C_{P}(T)$ data of
$\mathrm{La_{2}AgSi_{3}}$ from that of $\mathrm{Sm_{2}AgSi_{3}}$ data and is
shown in the inset of Fig. 4. As seen in the inset of Fig. 4, $C_{4f}(T)$
exhibits a broad hump above the transition temperature, indicating a Schottky
type anomaly caused by crystalline electric field (CEF) present in this
compound. The magnetic entropy $\rm{S_{\mathrm{4f}}}$ was estimated using the
formula ${S_{\mathrm{4f}}}$ = $\int(C_{\mathrm{4f}}/T)dT$. The temperature
variation of ${S_{\mathrm{4f}}}$ is shown in the right-hand axis of Fig. 4.
The magnetic entropy is very close to Rln2 at $\rm{T_{C}}$, indicating the
presence of a doublet ground state in the magnetic system. ${S_{\mathrm{4f}}}$
increases above $T_{C}$ and displays a tendency to saturate above 50 K.
${S_{\mathrm{4f}}}$ saturates around a value of Rln6, indicating that the
highest excited crystal field level contributes to magnetic entropy.

###  .2.4 Summary

In this work the $\mathrm{Sm_{2}AgSi_{3}}$ compound was synthesized and found
to be stoichiometric and crystallizing in the $\beta$-$\mathrm{ThSi_{2}}$-type
tetragonal structure. The experimental results indicate that the compound
undergoes a paramagnetic to ferromagnetic phase transition at $T_{C}$ = 14 K.
The large $H_{C}$ confirms that the compound possesses a large large
anisotropy. The presence of magnetic anisotropy plays a role in the TMI
between $\chi_{ZFC}$ and $\chi_{FC}$, and are related through $H_{C}$. The
$C_{4f}$ results indicate that the Schottky type anomaly is present due to the
crystalline electric field (CEF), which may result in a large magnetic
anisotropy. The present results pave the way for future promising research of
magnetic properties in new $\mathrm{R_{2}TX_{3}}$ series of compounds.

## Acknowledgements

This work is supported by a Global Excellence and Stature (UJ-GES) fellowship,
University of Johannesburg, South Africa. DFR thanks OWSD and SIDA for the
fellowship towards PhD studies. AMS thanks the URC/FRC and the SA-NRF (93549)
for financial support of UJ for assistance of financial support.

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