# Spin-current manipulation of photoinduced magnetization dynamics in heavy
metal / ferromagnet double layer based nanostructures

Steffen Wittrock1, Dennis Meyer2, Markus Müller2, Henning Ulrichs2, Jakob
Walowski3, Maria Mansurova3,
Ulrike Martens3, and Markus Münzenberg3 Manuscript received April 21, 2017;
revised April 21, 2017. Corresponding author: S. Wittrock (email:
steffen.wittrock@u-psud.fr). 1Unité Mixte de Physique CNRS/Thales, Université
Paris Sud, 1 Avenue Augustin Fresnel, 91767 Palaiseau, France  2I.
Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-
Platz 1, 37077 Göttingen, Germany  3Institut für Physik, Ernst-Moritz-Arndt-
Universität Greifswald, Felix-Hausdorff-Str. 6, 17489 Greifswald, Germany

###### Abstract

Spin currents offer a way to control static and dynamic magnetic properties,
and therefore they are crucial for next-generation MRAM devices or spin-torque
oscillators. Manipulating the dynamics is especially interesting within the
context of photo-magnonics. In typical $3d$ transition metal ferromagnets like
CoFeB, the lifetime of light-induced magnetization dynamics is restricted to
about 1 ns, which e.g. strongly limits the opportunities to exploit the wave
nature in a magnonic crystal filtering device. Here, we investigate the
potential of spin-currents to increase the spin wave lifetime in a functional
bilayer system, consisting of a heavy metal (8 nm of $\beta$-Tantalum
(Platinum)) and 5 nm CoFeB. Due to the spin Hall effect, the heavy metal layer
generates a transverse spin current when a lateral charge current passes
through the strip. Using time-resolved all-optical pump-probe spectroscopy, we
investigate how this spin current affects the magnetization dynamics in the
adjacent CoFeB layer. We observed a linear spin current manipulation of the
effective Gilbert damping parameter for the Kittel mode from which we were
able to determine the system’s spin Hall angles. Furthermore, we measured a
strong influence of the spin current on a high-frequency mode. We interpret
this mode an an exchange dominated higher order spin-wave resonance. Thus we
infer a strong dependence of the exchange constant on the spin current.

###### Index Terms:

Spin Hall effect, spin current, magnetization dynamics, magnetooptical Kerr-
effect.

##  I Introduction

The spin-transfer effect describes the transfer of spin angular momentum to a
ferromagnet’s magnetization from an injected spin polarized current. Since its
prediction by L. Berger [1], this effect has experienced a high research
interest as it permits to manipulate and control the magnetization of a thin
ferromagnetic (FM) layer. Especially when the resulting spin transfer torque
is collinear with the damping torque, magnetic dissipation can be controlled.
Thereby the life time of spin wave dynamics can be drastically enhanced.

Among the possible methods of creating the necessary spin current, the
exploitation of the spin Hall effect has become a powerful mean since its
first observation only a decade ago [2, 3, 4, 5]. Governed by spin-orbit
coupling phenomena, it generates a spin current $j_{s}$ from a transverse
charge current $j_{e}$ without any need for neither a ferromagnet nor an
external magnetic field. The efficiency of the conversion process can be
described by the spin Hall angle (SHA) $\Theta_{SH}=j_{s}/j_{e}$.

Here, we investigate the photo-induced magnetization dynamics in a few
nanometer thin soft magnetic layer consisting of amorphous metallic cobalt
iron boron alloy (Co20Fe60B20) under influence of a strong spin current
generated by the SHE in an adjacent heavy metal film made of platinum (Pt) or
tantalum (Ta). The sputter conditions for Ta were chosen such that the film
has grown in the high resistive $\beta$-phase for which a high SHA has
recently been reported [6].

##  II Experimental

The samples consist of two functional thin layers (fig. 1(a)): $8\,$nm Pt or
$\beta$-Ta as a SHE-material generating the spin current and $5\,$nm of
ferromagnetic amorphous CoFeB, into which the spin current is being injected
in order to manipulate its magnetization dynamics. The layer stack is
complemented by $3\,$nm of Ru as a capping layer. CoFeB and Ta are deposited
by argon ion sputtering and Pt and Ru by e-beam evaporation. All preparation
steps are conducted in situ in ultra-high vacuum. Subsequent structuring of
the samples by e-beam lithography enables the electrical contacting and
generation of a high charge current density (fig. 1(b)).

(a) (b) (c)

Figure 1: Experimental characteristics. (a) Schematic processes in the
functional layer stack of a SHE material (blue) and the ferromagnetic CoFeB
(yellow). (b) Patterned sample structure, the width $x$ was $12\,\upmu$m for
the results shown here. The red marked area was excited by the laser spot. (c)
Pump-probe setup with ratio of powers $P_{pump}:P_{probe}=95:5$, time
resolution is realized by a delay stage, a double modulation technique of
photoelastic modulator (PEM) and chopper frequency was used [7].

We used time resolved pump-probe spectroscopy exploiting the magnetooptical
Kerr effect to excite and measure magnetization dynamics (schematical setup in
1(c)). The laser pulses with central wave length $\lambda=800\,$nm have an
autocorrelation length of $\Delta\tau\approx 80\,$fs and a repetition rate of
$250\,$kHz. The pump spot size was $\approx 60\,\upmu$m providing an optical
fluence of $F=15\,\nicefrac{\mathrm{mJ}}{\mathrm{cm^{2}}}$.

The incoming pump pulse induces the dynamics at $\tau=0$ by firstly generating
hot electrons, which thermalize on a timescale of $\tau\sim 100\,$fs due to
electron-electron-scattering. Further scattering events with phonons and spins
lead to energy transfer into the phonon and spin system giving rise to
ultrafast demagnetization [8]. Caused by the high change in temperature on the
time scale of $\sim 1\,$ps, the local anisotropy constant of the ferromagnetic
material and thus the effective magnetic field $\vec{H}_{eff}$ is changed.
With $\vec{H}_{eff}$ rereaching its equilibrium position after a few
picoseconds, the magnetization dynamics corresponding to the Landau-Lifshitz-
Gilbert-equation (LLG) is excited. In order to subsequently trigger a coherent
precession of the magnetization, an external field ($145\,$mT) was applied at
an out-of-plane angle of $\varphi=35\,^{\circ}$, transversal to $\vec{j}_{e}$.

##  III Results - nanosecond timescale

(a)

(b) (c)

Figure 2: (a) Typical measurement data and Kittel mode after substraction of
exponential background. The TRMOKE-signal is proportional to the
magnetization. Dependence of (b) Kittel frequency and (c) saturation
magnetization on the current density in the Ta-layer.

Besides the dominating coherent, in spatially homogeneous geometries called
Kittel mode oscillations, also incoherent phonons and magnons contribute to
the signal and give rise to a certain background [9, 8], which can be modelled
by exponential functions. Substracting it from the raw data (fig. 2)
highlights the magnetic oscillations. These are analysed with respect to
frequency and damping for different current densities passing through the SHE-
material.

###  III-A Kittel frequencies, magnetization, and Oersted field

The dependence of the Kittel frequency on $j$ is shown in fig. 2(b). The
mainly parabolic behaviour can be attributed to the reduction of the
saturation magnetization due to Joule-heating. The observable asymmetries for
opposite current directions can be related to the Oersted field produced by
the current, or to the presence of a field-like torque. Analysing the
asymmetries, an Oersted field of $H_{Oe}=aj$ with $a=(1.23\pm 0.04)\cdot
10^{-8}\,$m is determined which agrees well with theoretical estimations. Even
if present, the field-like torque must be much smaller than the effect arising
from the Oersted field. Taking also into account the in-plane applied magnetic
field component $H_{x}$, the saturation magnetization can be determined from
the Kittel formula $\omega=\gamma\mu_{0}\sqrt{H_{x}(H_{x}+M_{S})}$ (fig.
2(c)). Note that for the given field geometry, the magnetization’s out-of-
plane component is only around $2\,$%, as is estimated by micromagnetic
simulations; therefore this approximation is valid. Besides Joule-heating,
also the energy deposition of the pump pulse heats up the sample locally to
around $400$-$450\,$K at a time scale of up to $1\,$ns. Thus the saturation
magnetization at $j=0\,$A/m2 is slightly lower than the room temperature value
of $\mu_{0}M_{S}=1.63\,$T, determined by a vibrating sample magnetometer [10].
The Joule-heating effect leads to a temperature increase especially in Ta of
up to $200\,$K at a maximum current density of $j_{Ta}=9.3\cdot
10^{10}\,$A/m2.

###  III-B Effective Gilbert damping parameter of the Kittel mode

From the exponential decay time $\tau_{\alpha}$ of the Kittel mode (fig. 2(a))
and taking into account the full in-plane magnetic field
$H_{x}=H_{ext}\cos(\varphi)+H_{Oe}$, and the current dependence of the
magnetization, we calculate the effective Gilbert damping parameter using:

$\displaystyle\alpha_{Kittel}=\left(\tau_{\alpha}\gamma\mu_{0}\left(H_{x}+\frac{M_{S}}{2}\right)\right)^{-1}~{}~{}~{}.$
(1)

(a) (b)

Figure 3: Current dependence of the effective Gilbert damping parameter for
(a) the Ta based sample and (b) the Pt based sample. From the linear fit, the
SHE efficiency can be extracted.

Results for the two SHE-materials Ta and Pt are shown in fig. 3. We determined
the SHA $\Theta_{SH}$ by fitting the formula:

$\displaystyle\tilde{\alpha}=\alpha+j\cdot\Theta_{SH}\cdot\frac{\hbar}{2ed\mu_{0}M_{S}H_{x}}~{}~{}~{}.$
(2)

Note that this expression was derived from linearizing the Landau-Lifshitz-
Gilbert-Slonczewski-equation (LLGS).

Averaging a few datasets gives the SHAs of:

$\displaystyle\Theta_{SH}^{Ta}$ $\displaystyle=-0.043\pm 0.011~{}~{}~{},$ and
$\displaystyle\Theta_{SH}^{Pt}$ $\displaystyle=0.086\pm 0.012~{}~{}~{}.$

These results lie within the values obtained by other groups [11] and in
particular confirm the different sign to be expected for Pt [12, 13] and Ta
[14, 6].

###  III-C Spin-pumping effect

The spin mixing conductance $g^{\uparrow\downarrow}$ of a layer combination
can be estimated through comparison of the effective damping parameters
$\tilde{\alpha}$ of two layer stacks involving the same ferromagnetic
material. Using Ta and Pt as SHE-materials and CoFeB as the ferromagnet, the
spin mixing conductance of Ta can be calculated using:

$\displaystyle g_{Ta}^{\uparrow\downarrow}=-\frac{4\pi
M_{S}d\cdot\Delta\tilde{\alpha}}{\hbar\gamma}+g_{Pt}^{\uparrow\downarrow}$ (3)

$\Delta\tilde{\alpha}=\tilde{\alpha}_{Pt}-\tilde{\alpha}_{Ta}$ denotes the
difference of the effective magnetic damping constants of the $d=5\,$nm thick
CoFeB layer for the two different adjacent SHE-materials. Average values of a
few measurements give $\tilde{\alpha}_{Pt}=(1.20\pm 0.15)\cdot 10^{-2}$ and
$\tilde{\alpha}_{Ta}=(0.69\pm 0.05)\cdot 10^{-2}$ for $j=0$. The Pt/CoFeB spin
mixing conductance of $g_{Pt}^{\uparrow\downarrow}=(4\pm 1)\cdot 10^{19}\,$m-2
[18] was used as a reference value.

We evaluate the spin mixing conductance of Ta/CoFeB to
$g_{Ta}^{\uparrow\downarrow}=(1.9\pm 1.2)\cdot 10^{19}\,$m-2. The smaller
value for Ta/CoFeB compared to Pt/CoFeB is expected as similar values were
already obtained in Ta/NiFe [15] resp. Pt/NiFe [16] bilayers. In conclusion,
this analysis shows that TRMOKE is also a suitable method to determine the
spin mixing conductance which is an important parameter for heavy metal/FM
based bilayer systems.

##  IV Results - picosecond timescale

On the timescale of a few picoseconds, when the magnetization starts
relaxating again, we observed a strongly damped, ultrafast oscillation in the
terahertz regime (fig. 4(c)), usually existent for one or two periods (fig.
4). We identified this oscillation as the well-known perpendicular standing
spin-wave (PSSW) mode of first order $n=1$. Analysing its frequencies (fig.
4(c)) and taking into account the saturation magnetization gathered on the
nanosecond timescale, especially the exchange stiffness $A$ (fig. 4(d)) can be
obtained from:

$\displaystyle\omega=\gamma\mu_{0}\sqrt{\left(H_{x}+\frac{2A}{\mu_{0}M_{S}}k^{2}\right)\left(H_{x}+\frac{2A}{\mu_{0}M_{S}}k^{2}+M_{S}\right)}$
(4)

with $k^{2}=k_{z}^{2}=\left(\nicefrac{{n\pi}}{{d}}\right)^{2}$,
$n\in\mathbb{N}$ the quantized wavevector normal to the plane. The exchange
stiffness is shown in fig. 4(d) and found to depend on the spin current.

Note that pinning at interfaces can alter the effective wave length of the
exchange mode. Without further experimental evidence, assuming zero pinning is
the simplest model. Since the exchange constant fits to known values
determined from TRMOKE experiments on thicker films [17], we stick to this
model.

(a)

(b) (c)

(d) (e)

Figure 4: (a) The first ten picoseconds of the excited dynamics including an
ultrafast, highly damped oscillation during remagnetization (b). (c-e) Current
dependence of certain derived parameters.

The Gilbert damping parameter of the PSSW mode (fig. 4(e)) can again be
determined from the oscillation’s exponential decay time $\tau_{\alpha}$:

$\displaystyle\alpha_{PSSW}=\left(\tau_{\alpha}\gamma\mu_{0}\left(H_{x}+\frac{2Ak^{2}}{\mu_{0}M_{S}}+\frac{M_{S}}{2}\right)\right)^{-1}~{}~{}~{}.$
(5)

Especially the exchange term $\sim Ak^{2}/M_{S}$ proves to be dominant due to
the small CoFeB thickness. We attribute the (at least for positive $j_{Ta}$)
linear behaviour of $\alpha_{PSSW}$ to the SHE. The curve’s slope (fig. 4(e),
$j_{Ta}\geq 0$) possesses the same sign as the Kittel mode damping
$\alpha_{Kittel}$. Furthermore, $\alpha_{PSSW}$ is found to be one order of
magnitude higher than $\alpha_{Kittel}$. The relative change from
$\alpha_{PSSW}\approx 0.04$ for a high positive current density up to
$\alpha_{PSSW}\approx 0.13$ for $j=0$ shows a strong dependence on the spin
current which is around three times higher than for the Kittel mode.

##  V Discussion

According to our experiments, the SHA for a Pt-based layer structure is almost
double the SHA of the Ta-system. A strong Joule-heating effect occurred
especially in the Ta based structure because of its high resistivity. The
latter also contributes to an eventual systematic reduction of the tantalum’s
SHA by up to $10\,$% assuming the capped Ru displays a small negative SHA of
$\Theta_{SH}\approx-0.001$ [19]. This would lead to a spin current with
opposing sign flowing into the CoFeB from above.

The strong spin pumping mechanism and thus increase of the magnetic damping
constant especially in the Pt sample is theoretically expected due to
platinum’s high spin flip probability. Adding an interlayer with high spin
diffusion length (such as Cu [20, 21, 22]) could solve this issue.

The interesting discovery of the fast, highly damped magnetic oscillation
within the first $10\,$ps of the remagnetization process is identified as the
PSSW mode of first order. It shows a strong dependence on the injected spin
current which influences its frequency, the exchange stiffness and the mode’s
damping. Note that usually, the PSSW mode is optically excitable and
observable in samples, whose layer thickness is higher than the optical
penetration depth which is $\sim 30\,$nm in our case. To obtain clear evidence
of this mode’s properties, further investigations with enhanced signal to
noise ratio are necessary.

##  VI Conclusion and Outlook

In this article we discussed the photoinduced magnetization dynamics under
influence of an injected spin current, generated by the SHE. The powerful tool
of time resolved magnetooptical Kerr effect, compared to more established
methods like BLS or ST-FMR, allowed to investigate nanosecond as well as
(sub-)picosecond dynamics and to get insights into high-frequency and non-
equilibrium dynamics so far not in the focus within this context.

We discussed the influence of Joule-heating on our samples and described the
linear manipulation of the Kittel mode’s Gilbert damping through a spin
current. The spin Hall angles which could be determined for the two different,
commonly used SHE materials Ta and Pt, lie within other reported values [11].

We reported a magnetic oscillation at the picosecond timescale which is found
to be highly dependent on the spin current. The exact behaviour of this mode
has to be further investigated in future experiments with enhanced signal to
noise ratio. Furthermore, the interesting timescales of the demagnetization
process will be adressed in future publications.

Standard methods such as ST-FMR do not allow to address such high-frequency
dynamics easily. Our approach enables us to enter this temporal regime, which
provides new insights on the action of spin torques on picosecond time scales.

## Acknowledgment

The authors acknowledge financial support by the DFG, within the CRC 1073
’Atomic scale control of energy conversion’.

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# Concentration Dependence of Elastic and Viscoelastic Properties of Aqueous
Solutions of Ficoll and Bovine Serum Albumin by Brillouin Light Scattering
Spectroscopy

Stephen J. Spencer, Venketesh Thrithamara Ranganathan, Anand Yethiraj, and G.
Todd Andrews  Department of Physics and Physical Oceanography, Memorial
University, St. John’s, NL, A1B 3X7, Canada

(August 31, 2024)

###### Abstract

The cellular environment is crowded with macromolecules of different shapes
and sizes. The effect of this macromolecular crowding has been studied in a
variety of synthetic crowding environments: two popular examples are the
compact colloid-like Ficoll macromolecule, and the globular protein bovine
serum albumin (BSA). Recent studies have indicated a significant component of
bound or surface-associated water in these crowders reduces the available free
volume. In this work, Brillouin light scattering experiments were performed on
aqueous solutions of Ficoll 70 and Ficoll 400 with concentrations ranging from
1 wt% to 35 wt% and BSA with concentrations of 1 wt% to 27 wt%. From the
dependence of spectral peak parameters on polymer concentration, we determined
fundamental solution properties: hypersound velocity, adiabatic bulk modulus
and compressibility, apparent viscosity, and hypersound attenuation. Existing
theory that ignores intermolecular interactions can only capture the observed
linear trends in the frequency shift up to a threshold concentration, beyond
which a quadratic term accounting for intermolecular interactions is
necessary. This likely indicates a transition from the dilute to semi-dilute
regime. In the Ficoll solutions (but not BSA) we see evidence for a central
mode, with a characteristic relaxation time of 20 ps, that we attribute to
exchange of the bound water.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Physical systems consisting of liquid water and macromolecules are ubiquitous.
The fluid medium inside a biological cell is an aqueous solution that consists
of macromolecules of different sizes and shapes which occupy a significant
volume (typically assumed to be 30 - 40$\%$ of the cell) [1]. It has been
understood for more than two decades that the crowded macromolecular
environment can affect biochemical reactions within the cell [2]. Any volume
other than the water volume is inaccessible to other molecules, and this
excluded volume affects molecular structure, motions and chemical kinetics. In
addition, for compact molecules with internal bound water, the accessible
water volume will be less than the total water volume. Thus, the simplest
picture of macromolecular crowding is entropic: the macromolecules and the
inaccessible water reduce the free volume and increase the excluded volume.

While there is an increasing realization of the role of other non-specific
interactions [3], experimental model systems have focused on aqueous solutions
with a simple crowder such as a polysaccharide (e.g., the compact Ficoll
macromolecule or the chain-like dextran) or a globular protein (e.g., bovine
serum albumin) [4, 5, 6]. Structure and dynamics in these crowders has been
reported extensively [7, 8, 9, 10, 11]. More realistic, heterogeneous crowding
media, such as bacterial cell lysate, have been employed as well [12, 13]. In
recent work, Ranganathan $et$ $al$. [14] have provided quantitative evidence,
via pulsed field gradient nuclear magnetic resonance spectroscopy, for water
that is bound and thus inaccessible to other molecules, implying that the true
excluded volume is larger than what is usually inferred.

Brillouin light scattering spectroscopy has been recognized as a niche
technique for probing the mechanical and viscous properties in heterogeneous
biomaterials [15]. Brillouin scattering experiments on aqueous macromolecular
solutions and hydrogels typically report the dependence of spectral peak
parameters and derived elastic and viscoelastic properties on solute
concentration [16, 17, 18]. These parameters and properties are usually
measured over a large concentration range and include Brillouin peak frequency
shift and linewith, hypersound velocity and attenuation, apparent viscosity,
and various elastic and viscoelastic moduli. Attempts at using existing
theoretical models to describe the concentration dependence of these
quantities, however, have met with limited success. For example, Brillouin
studies of so-called “aqueous biorelevant solutions” [16] reveal a common
dependence of peak frequency shift on solute concentration for those solutions
containing macromolecules (lysozyme, bovine serum albumin, and gelatin) up to
40 wt%, implying that this behaviour is largely independent of the nature of
the solute. Application of the Reuss effective medium model for a two-
component system resulted in an equation for the shift as a function of
concentration that agreed well with most experimental data. A theoretical
expression relating peak linewidth to concentration that incorporated this
equation, however, was unable to fully reproduce the observed trends.
Moreover, the sparsity and relatively large concentration interval ($\sim 10$
wt%) between adjacent data points means that possible discontinuities in the
observed trends, such as might occur in proximity to the polymer overlap
concentration, were not accounted for in the model. In related Brillouin
scattering work on collagen hydrogels [19] the storage modulus of collagen was
extracted by fitting an expression for the concentration dependence of the
effective storage modulus from the Voigt model to a high-hydration subset of
the full experimental dataset for the concentration dependence of the gel
storage modulus. It was stated that the two-component Voigt model gave a
better fit than the Reuss model used in previous works [19]. No model was
advanced to describe the data trend over the entire measured concentration
range. Moreover, Brillouin scattering studies of cross-linked polyvinyl
alcohol hydrogels [20] found only crude qualitative agreement between the
observed dependence of Brillouin peak frequency shift and linewidth on gel
network volume fraction and that calculated from a theory incorporating
frictional damping and coupling between elastic waves in polymer network and
fluid [21].

In this paper we report on Brillouin light scattering studies of aqueous
solutions of polymers Ficoll 70, Ficoll 400, and Bovine Serum Albumin (BSA).
Solution elastic and viscoelastic properties were determined over the dilute
and semi-dilute ranges from the dependence of spectral peak parameters on
solute (polymer) concentration. The observed trends in these properties are
not consistent with existing theory but instead were found to be well-
described by expressions derived from a new model relating hypersound
frequency and solute concentration. The sensitivity of the Brillouin
scattering technique to changes in structure and water-macromolecule dynamics
also allowed the polymer overlap concentration and the relaxation time
associated with the hydration of Ficoll molecules to be determined. The extent
of the hydration, manifested in the unexpectedly low overlap concentration,
provides independent confirmation of recent results that suggest that the
effective volume fraction occupied by hydrated macromolecules in solution is
much larger than expected for the bare unhydrated variety. In characterizing
the viscoelastic properties of commonly-used experimental model systems over a
wide solute concentration range and advancing a model that incorporates
interparticle interaction, this study provides important new insight into the
physics of macromolecular crowding and biomacromolecular systems in general.

##  II Experimental Details

###  II.1 Sample Preparation

Solutions of Ficoll-70 ($m=70$ kDa) and Ficoll-400 ($m=400$ kDa) were prepared
under identical conditions by dissolving Ficoll powder in 99.9% pure D2O at
room temperature in small glass vials. A Scientific Industries Vortex Genie
was used to promote initial mixing of D2O and Ficoll, and subsequent
homogenization was performed using a Fisherbrand Homogenizer 850 with a rod
diameter of 7 mm and speed of 11,000 RPM. The homogenization sequence, which
was performed five times per sample, consisted of three minutes of mixing
followed by one minute of settling. The resulting solutions were clear and
colourless and had concentrations ranging from 1 wt% to 35 wt%, with a
noticeable increase in viscosity for those at the upper end of this range.

BSA solutions with concentrations of 1% w/w to 27% w/w were prepared by
dissolving BSA powder in 0.1 M phosphate buffer solution with pH of 7.0 at
room temperature and subjecting them to the same mixing and homogenization
procedure as used for the Ficoll solutions. Solutions with concentrations
$<20$ % w/w were clear and colourless while those with concentrations higher
than this value were somewhat cloudy.

###  II.2 Brillouin Light Scattering Spectroscopy

####  II.2.1 Apparatus

Brillouin light scattering experiments were performed under ambient conditions
using a 180∘ backscattering geometry. The incident light source was a Nd:YVO4
solid state laser with an emission wavelength of 532 nm and output power of
1.66 W. To minimize Fresnel reflection losses, a half-wave plate was used to
rotate the plane of polarization from vertical ($s$-polarized) to horizontal
($p$-polarized). Neutral density filters placed in the beam path were used to
reduce the power level at the sample to $\sim 100$ mW. Light was focused on
samples with a 5 cm lens of $f$-number 2.8. Scattered light was collected and
collimated by the same lens and subsequently focused by a 40 cm focal length
lens onto the 450 $\mu$m entrance pinhole of a six-pass tandem Fabry-Perot
interferometer for spectral analysis. The interferometer had a free spectral
range of 15 GHz and a finesse of $\sim 100$. A schematic of this experimental
setup can be found in Refs. [22, 23].

####  II.2.2 Quantities Derived from Spectra

Elastic and viscoelastic properties of the Ficoll solutions were deduced from
Brillouin spectra. Hypersound velocity was determined using the well-known
Brillouin equation applied to the case of a backscattering geometry,

$v=\frac{f\lambda}{2n},$ (1)

where $f$ is the measured Brillouin peak frequency shift, $\lambda$ is the
incident light wavelength, $n=n(x)$ is the concentration-dependent solution
refractive index, and $x$ is the concentration in weight percent. The latter
was obtained for each Ficoll 70 and Ficoll 400 solution using $\partial
n/\partial x$ relationships provided in the literature [24], however this was
found to be constant at $n=1.33$ for the full concentration range. This,
however, was not the case for solutions of BSA. At higher concentrations of
BSA, solutions became noticeably cloudy. As such, refractive indices for BSA
solutions were calculated using the relationship $\partial n/\partial C=0.190$
mL/g (with C expressed as g/mL), found by Tumolo $et$ $al$. [25].

Knowledge of the hypersound velocity allowed the adiabatic bulk modulus to be
found from

$B=\rho v^{2},$ (2)

where $\rho$ is the mass density of the solution. The density of the solution
showed little variation over the full concentration range, and was
approximated to a constant 1110 kg/m3 for the purposes of fitting. The
adiabatic compressibility, $\kappa$, was also determined using the fact that
$\kappa=1/B$.

The apparent viscosity, $\eta$, and hypersound attenuation, $\alpha$, in the
solution were deduced from Brillouin spectral data via

$\eta=\frac{4}{3}\eta_{s}+\eta_{b}=\frac{\rho
v^{2}\Gamma_{B}}{4\pi^{2}f^{2}}=\frac{\rho\lambda^{2}\Gamma_{B}}{16\pi^{2}n^{2}}$
(3)

and

$\alpha=\frac{\pi\Gamma_{B}}{v},$ (4)

respectively [26], where $\Gamma_{B}$ is the Brillouin peak full width at
half-maximum (FWHM), $\eta_{s}$ and $\eta_{b}$ are the shear and bulk
viscosities, and the other quantities are as already defined.

##  III Results and Discussion

###  III.1 Spectra

####  III.1.1 General Features and Mode Assignment

Figure 1 shows a series of Brillouin spectra collected from the Ficoll 70
solutions and BSA solutions. Spectra of the Ficoll 400 solutions are similar
to Ficoll 70 spectra in all respects (see Figure S1 in the Supplementary
Material). A single set of Brillouin peaks was observed in all spectra, with a
frequency shift ranging from $\sim 6.8$ GHz to $\sim 8.4$ GHz. Although not
obvious from the spectra shown in Figure 1, there is also a broad, weak peak
at the center of the spectra obtained from Ficoll 70 and Ficoll 400 solutions
with solute concentrations $\geq 20$%. The presence of this peak was inferred
from the fact that the baseline intensity in the region between the central
elastic peak and the Brillouin doublet in spectra of high concentration
solutions was noticeably higher than that in spectra of the low concentration
solutions and also higher than that on the high frequency shift side of the
Brillouin doublet. An example of this for Ficoll 70 solutions with 3% and 30%
concentration is shown in Figure 2 A. In contrast, no central peak was
discernible in spectra of the BSA solutions.

The Brillouin doublet and the peak in the center of the spectra of high
concentration Ficoll solutions have different origins. The former is assigned
to the usual longitudinal acoustic mode propagating through the solution based
on the similarity of its frequency shift to that of the corresponding mode in
water [27]. The central peak was attributed to a diffusive relaxation mode
based on its zero frequency shift and the fact that the width of the peak
showed no significant change with changing concentration [28]. These
properties are consistent with other Brillouin and Rayleigh scattering studies
which have observed this relaxation mode in macromolecular solutions. [17, 18,
29, 30]

Figure 1: Normalized Brillouin spectra collected from solutions of (A) Ficoll
70 and (B) BSA of various concentrations (wt%). L represents a longitudinal
bulk mode.  Figure 2: (A) Brillouin spectrum of an aqueous solution of Ficoll
70 with a solute concentration of 30%. Solid lines - Best-fit Lorentzian
functions for central peak and Brillouin peaks. Dotted line - sum of central
peak and Brillouin peak best-fit Lorentzians. (B) Anti-Stokes Brillouin peaks
for aqueous Ficoll solutions with solute concentrations of 3% and 30%. 30%
concentration peak is shifted horizontally from 7.96 GHz to 6.86 GHz so that
the peaks overlap to highlight the slight asymmetry and significantly higher
baseline intensity of the peak for the 30% concentration solution compared to
that for the 3% solution on the low frequency shift side.

####  III.1.2 Extraction of Peak Parameters

Longitudinal acoustic mode peak frequency shift and linewidth were obtained by
fitting Lorentzian functions to the Brillouin peaks, with the latter being
processed prior to plotting by subtraction of the 0.3 GHz instrumental
contribution to the best-fit linewidth.

To obtain an estimate of the central mode linewidth, $\Gamma_{C}$, it was
first necessary to remove data from regions of the spectrum containing other
peaks so as to minimize its impact on any subsequent fit. This included the
region containing the central elastic peak and other data contained within the
central Fabry-Perot interferometer control window, $\pm 2$ GHz from the center
of the spectrum, and that containing the two longitudinal mode peaks, which
was typically a $\sim$ 6 GHz range around the longitudinal peak. The remaining
data was fitted to a Lorentzian function (see Fig. 2A), revealing a central
mode that is very weak and exceptionally wide, with a FWHM of $\sim 14$ GHz.
As can be seen by the high degree of overlap between the experimental data and
the dotted curve in Fig. 2A, the addition of this Lorentzian to the best-fit
Lorentzians for the longitudinal mode peaks results in a function that well
represents the original Brillouin spectrum (without the central elastic peak).

###  III.2 Longitudinal Acoustic Mode: Elastic and Viscoelastic Properties

####  III.2.1 Dependence of Brillouin Peak Parameters on Solute Concentration

Figure 3 shows longitudinal acoustic mode peak frequency shift and linewidth
versus concentration for the Ficoll 70, Ficoll 400, and BSA solutions. For all
solutions, both quantities increase monotonically with increasing
concentration, with the linewidth being much more sensitive than the shift to
changes in concentration. While the frequency shift for each solution
increases by $\sim 20$% over the range probed, the peak FWHM for the Ficoll
and BSA solutions increases by a factor of $\sim 4$ and $\sim 2$,
respectively.

For a given concentration, the frequency shift for Ficoll 400 solution is
slightly larger than that for Ficoll 70 solution. At low concentrations, the
peak linewidths obtained from the Ficoll 70 and Ficoll 400 solutions are
nearly equal, while for concentrations in excess of $\sim 15$ wt% the
linewidths begin to diverge, with that for Ficoll 400 being greater than that
for Ficoll 70 over this range. This subtle change in the relationship between
concentration and frequency or linewidth may represent the overlap
concentration at which the system physics changes as it transitions from the
dilute to the semi-dilute regime.

The solute concentration dependence of Brillouin peak frequency shift of some
so-called biorelevant aqueous solutions have been well-fit for concentrations
up to $x\sim 40$ wt% by the relation [16]

$f_{R}(x)=\frac{f_{w}}{\sqrt{1-x+xv^{2}_{w}/v^{2}_{s}}}=\frac{f_{w}}{\sqrt{1+\alpha
x}},$ (5)

where $\alpha\coloneqq v^{2}_{w}/v^{2}_{s}-1$. Also, the concentration
dependence of linewidth has been well fit up to $x\sim 20$ wt% by

$\Gamma^{B}_{R}(x)=Af_{R}(x)^{2}+Bx,$ (6)

respectively, where $A$ and $B$ are constants and $f_{w}$, $v_{w}$, and
$v_{s}$ are the Brillouin peak frequency shift for water, and the hypersound
velocities of water and the solute, respectively. The basis of Equation 5 is
the two-component Reuss model for which the effective elastic modulus $M$ of
the solution is given by

$\frac{1}{M}=\frac{\rho\mu_{s}}{\rho_{s}}\left[\frac{1}{M_{s}}-\frac{1}{M_{w}}\right]+\frac{1}{M_{w}},$
(7)

where $\mu_{s}=m_{s}/(m_{s}+m_{w})$ is the solute mass fraction and $M_{s}$
and $M_{w}$ are the elastic moduli of the solute and water, respectively. If
the density of the solution $\rho$ equals that of the solute $\rho_{s}$, then
Equation 7 simplifies to

$\frac{1}{M}=\frac{\mu_{s}}{M_{s}}+\frac{1-\mu_{s}}{M_{w}},$ (8)

Equation 5 can be obtained by from Equations 1 and 8 with the approximation of
fixed density and refractive index.

Fitting Equation 5 to the $\\{f,x\\}$ data for the Ficoll and BSA solutions
with $v_{w}^{2}/v_{s}^{2}$ as an adjustable parameter, however, yielded best-
fit relations $f(x)$ that show only mediocre agreement with experiment -
overestimating $f$ at low concentrations and underestimating it at higher
concentrations over the range probed. Substitution of these “best-fit”
expressions for $f(x)$ into Equation 6 give similar quality fits to the
$\\{\Gamma_{B},x\\}$ data shown in Figure 3. In addition, a fit of Equation 5
to only the low concentration data yields an excellent fit for $x\leq 10$% but
the resulting function does not describe the higher concentration data (see
dotted curve in Figure 3). This sub-optimal agreement between theory and
experiment is likely due to it not properly accounting for molecular crowding
arising from hydration of Ficoll or BSA as the solution concentration
increases, the onset of which occurs at the transition from the dilute to
semi-dilute regime. This crowding leads to increased polymer-polymer
interaction in the solution. As such, the hypersound frequency can no longer
be described by Equation 5; there are also contributions from volumetric and
entropic changes due to polymer-polymer interactions, specifically the onset
of contact and the possible formation of loosely packed regions of solute
molecules.

The failure of the above model to accurately reproduce the concentration
dependence of $f$ and $\Gamma_{B}$ for the Ficoll and BSA solutions, coupled
with the lack of other appropriate theoretical models, lead us to propose a
new model to describe the current experimental data. This model assumes that
the concentration dependence of $f$ and $\Gamma_{B}$ both increase smoothly
with concentration according to

$f(x)=f_{R}(x)+A_{1}x^{2},$ (9)

where $f_{R}$ is given by Equation 5, and

$\Gamma_{B}(x)=Af(x)^{2}+Bx+A_{2}x^{2},$ (10)

where the $A_{i}$ are fit parameters and $f(x)$ is given by Equation 9.
Furthermore, the second order term in Equation 10, $A_{2}x^{2}$ is a
phenomenological extension to van’t Hoff’s law, as explained below [31, 32].
From this, solute molecules in a dilute solution may be treated as an ideal
gas. As such, the linewidth is proportional to solution density, as expressed
in the equation

$\Gamma_{B}(x)\propto 1+c_{1}\rho+c_{2}\rho^{2}+c_{3}\rho^{3}+\cdots.$ (11)

At low concentrations, corresponding to lower density, the higher order terms
are insignificant. At higher concentrations, however, these higher order terms
begin to dominate [32]. The second order term of the virial equation is
attributed to interactions between solute molecules, the contribution of which
becomes more important at higher solute concentrations.

Figure 3 shows best-fits of Equations 9 and 10 to the $\\{f,x\\}$ and
$\\{\Gamma_{B},x\\}$ data for the Ficoll and BSA solutions. The fits reproduce
the trends of the experimental data very well. For reference, the best-fit
equations are given in Table 2.

Figure 3: Brillouin peak frequency shift (A) and linewidth (B) as a function
of concentration for aqueous solutions of Ficoll 70, Ficoll 400, and BSA.
Solid lines represent best fits of $f(x)=f_{R}(x)+A_{1}x^{2}$ and
$\Gamma_{B}(x)=Af(x)^{2}+Bx+A_{2}x^{2}$. Dashed lines represent frequency
relationship provided by Equation 5 for Ficoll 70 and BSA. Inset in (B) is
linewidth of Ficoll 70.

Although not obvious, there is also what could be a subtle change in behaviour
which is visible in both the frequency shift and linewidth, which is localized
around $\sim 15$ wt% concentration for both Ficoll solutions (denoted by the
arrow in the inset in Figure 3 for change in behaviour of Ficoll 70 peak
width). The subtle change in behaviour of frequency and linewidth around
$\sim$15% solute concentration is also observed in the BSA data. This change
in behaviour oversver may also be indicative of the overlap concentration, and
the transition from the dilute to semi-dilute regime.

####  III.2.2 Hypersound Velocity

Figure 4 (A) shows the evolution of hypersound velocity with concentration for
the Ficoll 70, Ficoll 400, and BSA solutions. There is a consistent increase
in velocity with increasing solute concentration for all solutions, as
expected from the increase in Brillouin peak frequency shift with increasing
concentration. As with the frequency data, the hypersound velocity in Ficoll
400 solutions is systematically higher than in Ficoll 70 solutions.
Furthermore, while the velocity of BSA solutions being larger than that of
Ficoll solutions can be attributed to the different solvent, the change in
velocity with respect to concentration was much lower for BSA than for both
Ficoll solutions. Figure 4 (A) shows curves for velocity as a function of
concentration which were derived by substituting the best-fit equation for
frequency as a function of concentration into Equation 1.

In the dilute regime, the hypersound velocity of macromolecular solutions can
be expressed in a manner similar to Equation 5, as hypersound velocity is
linearly proportional to phonon frequency [16]. However, as the concentration
increases beyond 10%, we begin to see a significant increase in packing of
solute molecules in solution due to crowding effects. This is demonstrated by
the deviation of Brillouin peak frequency and linewidth from previous theory.
It also is important to note that previous studies have shown that solute
hydration leads to volume fraction of solute being much larger than expected
than expected for bare (no bound water) solute molecules at similar weight
concentrations.[14]. This concept is further supported by Brillouin scattering
results in this work, as the onset of polymer-polymer interactions was seen at
concentrations as low as 10% by weight. This increase in macromolecular
packing leads to an increase in interactions between solute molecules [33].
This is the transition between the dilute and the semi-dilute limits.

By using the $f_{R}(x)$ expression of Equation 9, the hypersound velocity for
Ficoll 70, Ficoll 400, and BSA was calculated. Values for these velocities are
shown in Table 1. Velocities calculated for solutes used in this work ranged
between $\sim 2300$ m/s and $\sim 2900$ m/s, which is comparable to results
from previous studies on macromolecular solutions [16].

Table 1: Hypersound velocities for Ficoll 70, Ficoll 400, and Bovine Serum Albumin (BSA) solute, calculated using the equation $v{{}_{w}}^{2}/v{{}_{s}}^{2}-1=\alpha$, where $\alpha$ is the fit parameter in the denominator of $f_{R}(x)$ from equation 5.  Solute | Velocity (m/s)
---|---
Ficoll 70 | 2320
Ficoll 400 | 2850
BSA | 2580
Figure 4: Hypersound velocity (A), solution bulk modulus (B), apparent
viscosity (C), and hypersound attenuation (D) as a function of concentration
in aqueous solutions of Ficoll 70, Ficoll 400, and BSA. Solid lines are curves
are based on the frequency fits from Figure 3 and Equations 1, 2, 3, and 4,
respectively

####  III.2.3 Bulk Modulus & Adiabatic Compressibility

Figure 4 (B) shows that the bulk modulus for the Ficoll 70, Ficoll 400, and
BSA solutions increase with increasing solute concentration. This trend was
expected not only from the relationship between bulk modulus and hypersound
velocity given by Equation 2, but also from an intuitive perspective as with
an increase in concentration a greater proportion of the available volume is
occupied by the solute. This leads to the solution being less compressible and
therefore to a higher bulk modulus. For low concentrations $B$ increases
approximately linearly with $x$ as predicted by Equation 7 in the limit of
small $x$. While there is no obvious change in trend, at $x\sim 15$ wt% the
slope of $B(x)$ is noticeably larger and continues to increase with increasing
concentration. Following the reasoning in Sec. III.B.1, this is caused by the
solution transitioning from dilute to semi-dilute. The increased interaction
between solute molecules (e.g., entanglement) results in a decrease in
compressibility and, consequently, an increase in bulk modulus. This deviation
of the solution compressibility from a volumetric average of the
compressibilities of the constituents (i.e., the Reuss model) can be
attributed to entropic and volume contributions due to these interactions as
discussed in Sec. III.B.1 [33].

####  III.2.4 Apparent Viscosity

The apparent viscosity of the Ficoll and BSA solutions for a range of
concentrations in the dilute and semi-dilute regimes are shown in Figure 4
(C). The $\eta(x)$ curves in this figure were obtained from Equation 3 using
$\Gamma_{B}(x)$ given by Equation 6 with the best-fit $f(x)$ from Equation 3.
As can be seen, the apparent viscosity values for the two Ficoll solutions are
roughly equal at low concentrations but begin to diverge from eachother at
$x\sim 15$ wt%, behaviour similar to that observed for relative viscosity in
rheology studies of aqueous Ficoll 70 and Ficoll 400 solutions [14]. This
change in behaviour approximately at a concentration corresponding to the
dilute-to-semi-dilute transition. Within this region, the contribution to
viscosity from Ficoll becomes more dominant than that of D2O. It is important
to note that previous rheology studies have shown that the intrinsic
viscosities of Ficoll 70 and Ficoll 400 differ by $\sim 12\%$ [14].
Furthermore, the range of apparent viscoties presented in this work are
significantly different the ranges of shear viscosities shown in previous
rheology studies. While the very low concentration viscosities are comparable,
the shear viscosities presented in previous rheology work is larger than the
apparent viscosities drerived from Brillouin scattering by a factor of $\sim
100$ [14]. At low concentrations, apparent viscosity of BSA solutions is
slightly larger than that of either solution of Ficoll. The curve of Equation
3 for BSA, however, is considerably less steep than that of either solution of
Ficoll, demonstrating a lower intrinsic viscosity for BSA.

####  III.2.5 Hypersound Attenuation

Figure 4 (D) shows hypersound attenuation for all solutions at various solute
concentrations determined from Equation 4. The curves that appear in this
figure were obtained in a manner analogous to those for apparent viscosity.
The attenuation increases monotonically with increasing concentration for all
solutions. As was the case for apparent viscosity, hypersound attenuation in
the Ficoll 70 solution is approximately equal to that in the Ficoll 400
solution at low concentrations. As concentration increases, the solutions
transition from the dilute to the semi-dilute and the attenuation values for
the two solutions diverge. This is an indication that hypersound attenuation
is much more strongly correlated with D2O than with Ficoll in the dilute
limit. In the semi-dilute limit, the dependence on Ficoll concentration
becomes more important. Furthermore, similar to viscosity, at low
concentrations, where attenuation is more closely related to the solvent,
attenuation of BSA solutions is greater than that of either Ficoll solution.
Once again, however, the concentration dependence of BSA solutions with
concentration is less sharp than that of Ficoll.

Table 2: Empirical equations describing the concentration dependence of the elastic and viscoelastic properties of aqueous solutions of Ficoll 70, Ficoll 400, and Bovine Serum Albumin (BSA).  Quantity | Base Equation | Solution | Solution-Specific Equation
---|---|---|---
Hypersound Frequency (GHz) | $f(x)=f_{R}(x)+A_{1}x^{2}$ | Ficoll 70-D2O | $f(x)=(6.80/\sqrt{1-0.00653x})+0.000412x^{2}$
Ficoll 400-D2O | $f(x)=(6.82/\sqrt{1-0.00771x})+0.000314x^{2}$
BSA | $f(x)=(7.45/\sqrt{1-0.00665x})+0.000253x^{2}$
Brillouin Linewidth (GHz) | $\Gamma_{B}(x)=Af(x)^{2}+Bx+A_{2}x^{2}$ | Ficoll 70-D2O | $\Gamma_{B}(x)=0.0123x+0.000566x^{2}+0.0108f(x)^{2}$
Ficoll 400-D2O | $\Gamma_{B}(x)=0.00899x+0.000753x^{2}+0.0108f(x)^{2}$
BSA | $\Gamma_{B}(x)=0.00896x+0.000446x^{2}+0.0124f(x)^{2}$
Table 3: Estimated overlap concentration for aqueous solutions of Ficoll 70, Ficoll 400, and Bovine Serum Albumin (BSA) obtained in present work and previous studies. PW-C: Present Work - Overlap concentration estimated from visible change in behaviour observed in plot of Frequency shift vs. Solute Concentration and/or FWHM vs. Solute Concentration. PW-Q: Present Work - Overlap concentration estimated from the divergence of previous model resulting from removing quadratic term from Eq. 9.  Solution | Overlap Concentration (wt. %)
---|---
PW-C | PW-Q | Ref [14] | Ref [34] | Ref [35]
Ficoll 70 | $\sim 15$ | $\sim 10$ | 10-15 | - | 22.9
Ficoll 400 | $\sim 17$ | $\sim 20$ | 10-15 | 5.33 | 13.5
BSA | $\sim 13$ | $\sim 20$ | - | - | -

###  III.3 Central Mode: Relaxation

####  III.3.1 Origin

Figure 2 shows the central peak present in spectra of high concentration
Ficoll solutions. Such peaks were not observed in BSA spectra. This peak was
observed in the spectra of solutions with $x\geq 20$%, from which it was noted
that its intensity increases with increasing concentration for both Ficoll 70
and Ficoll 400 solutions. The width of this central peak, however, shows no
systematic change with change in concentration. This central peak was
attributed to a relaxation mode due to D2O molecules within the hydration
shells of Ficoll, a phenomenon which has been observed in Brillouin and
Rayleigh scattering experiments on other high concentration macromolecular
solutions [18, 17, 29, 36]. This phenomenon occurs when D2O molecules briefly
bind to the hydration shell of the Ficoll molecules before returning to the
bulk solvent. This relaxation time is, therefore, also considered the
residence time for solvent molecules within the hydration shell [37].

Table 4: Hydration relaxation time for aqueous Ficoll solutions (present work) and other aqueous macromolecular solutions. Solution | Macromolecule | Relaxation
---|---|---
Structure | Time (ps)
35 wt% Ficoll 70 - Pres Work | Globular | 22 $\pm$ 2
35 wt% Ficoll 400 - Pres Work | Globular | 18 $\pm$ 2
DNA [18] | Chain | 40
Hyaluronic Acid [29] | Chain | 50
Polyethylene glycol 600 [30] | Chain | 60

####  III.3.2 Relaxation Time

Relaxation times associated with the central peaks were determined from the
linewidths using the relationship $\tau=1/\pi\Gamma_{C}$ [38]. These times
were found to be $22\pm 2$ ps and $18\pm 2$ ps for the Ficoll 70 and Ficoll
400 solutions, respectively, and are representative of the duration that D2O
molecules spend within the hydration shells of Ficoll before returning to the
bulk solution. Table 4 compares these times to hydration relaxation times for
other macromolecular solutions. As shown, the relaxation time for both
varieties of Ficoll solution are lower than those for aqueous solutions of DNA
or hyaluronic acid [18, 29]. This is a reasonable result because relaxation
time is directly proportional to polymer chain length and Ficoll is spherical
in shape [14] while DNA and hyaluronic acid are long chain polymers [28]. It
should also be noted that Ranganathan et al [14] found that the NMR relaxation
rates had reached their saturation value by concentration of 35%, with
Ficoll-400 having higher rates than Ficoll-70. This is consistent with the
smaller hydration timescale observed here.

###  III.4 System Properties and Dynamics

Table 2 contains a summary of the dependence of hypersound frequency and
Brillouin peak linewidth on concentration given by Equations 3 and 10. All
elastic and viscoelastic properties explored in the present study increase
monotonically with increasing solute concentration. These increases can be
attributed to an increase in packing of solute molecules within the solution.
For aqueous solutions of Ficoll or BSA, there are two components which impact
the volume fraction. The first is the natural addition of solute molecules to
solution as concentration increases. The second component to consider is the
hydration of solute molecules. This process is seen to primarily occur in
higher concentration solutions, but makes a large contribution to the volume
fractions of Ficoll 70 and Ficoll 400. A change in volume fraction due to
hydration also occurs in BSA, but to a lesser degree than in Ficoll. As such,
there is a much more dramatic increase in volume fraction of these
macromolecules compared to solutions where there is no hydration of solute
molecules. Because of this, Ficoll solutions used in this study range from
dilute solutions to nearly maximum packing of randomly distributed spheres,
and BSA solutions also experience a very high degree of packing at the highest
concentrations. From the Brillouin scattering results, it is apparent that BSA
experiences less packing compared to Ficoll. Table 2 shows fit equations for
frequency versus solute concentration. The second order term for BSA,
corresponding to polymer-polymer interactions, is $\sim$20% lower than that of
either type of Ficoll. This increase in volume occupied by globular molecules
results in a reduction in solution compressibility and a corresponding
increase in bulk modulus. Furthermore, the increase in solute volume fraction
makes the aqueous solution more viscous and the packing of spheres increases
the attenuation of sound within the solution.

To further discuss the effects of hydration, above 15% concentration a
relaxation mode was observed in Brillouin spectra for both Ficoll 70 and
Ficoll 400 solutions. This relaxation was caused by hydration of Ficoll,
specifically due to an exchange of D2O between the bulk solvent and the
hydration shell of Ficoll. This addition of relaxation due to hydration
further validates the larger solute volume fraction due to hydration, compared
to bare solute at similar mass concentrations, by demonstrating a fundamental
change in the mechanics of the solution as concentration surpasses the overlap
concentration.

##  IV Conclusion

In the present study Brillouin light scattering experiments were performed on
solutions of Ficoll 70 and Ficoll 400 dissolved in D2O with concentrations
ranging from 1% to 35 wt%, and BSA dissolved in phosphate buffer with
concentrations ranging from 0% to 27%. Brillouin spectra for all such
solutions exhibited a single Brillouin peak which was attributed to a
longitudinal bulk mode. The frequency shifts and linewidths of these Brillouin
peaks were used to calculate hypersound velocity, attenuation, and bulk
modulus, all of which exhibited an increase with increasing concentration. For
the solutions studied in this work, the relationship between hypersound
frequency and solute concentration cannot accurately be described by models
which have been previously presented for non-interacting macromolecular
solutions. A model was derived to describe the change in hypersound frequency
in macromolecular solutions which incorporates solute-solute interactions.

Finally, a central peak was observed in high concentration spectra for both
Ficoll 70 and 400 but was not observed in BSA spectra. This peak was
attributed to relaxation associated with hydration of Ficoll by D2O, and the
occupation time of D2O within the hydration shell. The widths of this central
peaks were used to calculate relaxation times of $\sim 22$ ps and $\sim 18$ ps
for Ficoll 70 and Ficoll 400 solutions, respectively.

This work provides physical insight into the interaction of macromolecules and
water in crowded macromolecular environments, which are of immense importance
in biological systems. Further, it provides quantitative spectroscopic
signatures for bound or surface-associated water. Concentration dependence,
with complementary work on temperature dependence, of aqueous
biomacromolecular solutions is important because of the wide range of solution
concentrations found in naturally occurring biological systems. This work also
further establishes Brillouin spectroscopy as a valuable probe of the
elasticity and viscoelasticity of aqueous biomacromolecular systems.

###### Acknowledgements.

AY and GTA acknowledge the support of the Natural Sciences and Engineering
Research Council of Canada (RGPIN-2019-04970 and RGPIN-2015-04306,
respectively).

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