# The Role of Filler-Matrix Interaction on Viscoelastic Response of Biomimetic Nanocomposite Hydrogels

Authors: Alireza S. Sarvestani; Xuezhong He; Esmaiel Jabbari
Journal: Journal of Nanomaterials
 (2008)

---

## Abstract

The effect of a glutamic acid (negatively charged) peptide (Glu6), which mimics the terminal region of the osteonectin glycoprotein of bone on the shear modulus of a synthetic hydorgel/apatite nanocomposite, was investigated. One end of the synthesized peptide was functionalized with an acrylate group (Ac-Glu6) to covalently attach the peptide to the hydrogel phase of the composite matrix. The addition of Ac-Glu6 to hydroxyapatite (HA) nanoparticles (50 nm in size) resulted in significant reinforcement of the shear modulus of the nanocomposite (∼100% increase in elastic shear modulus). The reinforcement effect of the Glu6 peptide, a sequence in the terminal region of osteonectin, was modulated by the size of the apatite crystals. A molecular model is also proposed to demonstrate the role of polymer-apatite interaction in improving the viscoelastic behavior of the bone mimetic composite. The predictions of the model were compared with the measured dynamic shear modulus of the PLEOF hydrogel reinforced with HA nanoparticles. This predictive model provides a quantitative framework to optimize the properties of reinforced polymer nanocomposites as scaffolds for applications in tissue regeneration.


---

## Body

## 1. Introduction

Synthetic
degradable and biomimetic polymer nanocomposites are an ideal replacement
material for orthopedics and dental applications because of minimum risk of
disease transfer, reduced stress shielding and particulate wear, and the
ability to couple polymer degradation with tissue regeneration. In particular,
injectable hydrogels seeded with cells and growth factors and coupled with
minimally invasive arthroscopic techniques are an attractive alternative for
treating irregularly shaped degenerated hard tissues. Marrow stromal cells,
isolated from the bone marrow, and growth factors can be placed in a supportive
hydrogel and injected into an osteochondral defect by a minimally invasive
arthroscopic procedure [1–5].After injection, the composite mixture hardens in situ, guiding the development of the
seeded cells into the desired tissue. Furthermore, the composite matrix provides
dimensional stability and mechanical strength, similar to that of the host
tissue, during regeneration. A variety of multifunctional composite materials have
been developed to mimic the organized nanostructure of the bone, which consists
of the collagenous matrix and mineralized apatite nanocrystals [6–8]. In
addition, the gelatinous bone matrix contains noncollagenous proteins (NCPs), which play a central
role in regulation of mineralization and the extent of mineral-collagen
interactions [9, 10]. One of the NCPs with bone specific functions is
osteonectin which has a strong affinity for both collagen and hydroxyapatite (HA),
and it is speculated to be a bone-specific nucleator of mineralization [11, 12]. It is believed that the first seventeen
NH2-terminal
amino acids of osteonectin are responsible for binding to the bone collagen
network [13], while a glutamic acid-rich sequence binds to the bone
HA nanoparticles, due to its high ionic affinity for calcium ions [12].In this work, we describe
the synthesis and rheological characterization of a multifunctional bone
mimetic nano-composite with a matrix-apatite adhesion mechanism similar to that
of the natural bone. We have synthesized a glutamic acid-rich peptide (a
sequence of 6 glutamic acids) derived from osteonectin, functionalized with an
acrylate group for covalent attachment to the matrix, using solid-phase Fmoc
chemistry [14]. The
biodegradable in situ cross-linkable
poly (lactide-co-ethylene oxide fumarate) (PLEOF) hydrogel and HA crystals were
used to mimic the gelatinous matrix and mineral phases of the bone, respectively. PLEOF is a degradable macromer consisting of ultra-low-molecular-weight poly (L-lactide) (ULMW PLA) and poly (ethylene
glycol) (PEG) blocks li nked by
fumaric acid. HA particles were treated with the synthesized glutamic acid
peptide with acrylate group at one chain-end (Figure 1), hereafter designated as
Ac-Glu6, and the nanoparticles were dispersed in the aqueous PLEOF mixture by
sonication. The polymerizing mixture was cross-linked with a neutral redox
initiation system and the gelation process was monitored by monitoring the
viscoelastic response in situ as
a function of gelation time with a rheometer. The measured rheological and
viscoelastic characteristics can be used to control the injectability in the in situ hardening phase and to predict mechanical properties in the
postgelation phase.Figure 1
Schematic structure of the Ac-Glu6 peptide used for
surface treating of HA nanoparticle. Terminal
acrylate group of the Ac-Glu6 provides an unsaturated group for covalent cross-linking
of the apatite particles to the PLEOF matrix.The rheometry results show
that the Ac-Glu6 peptide can significantly enhance the viscoelastic properties
of the hydrogel/HA nanocomposite. A molecular model is also proposed to demonstrate
the role of polymer-apatite interaction in improving the viscoelastic behavior
of the synthesized bone mimetic hydrogel. The scaling law of de Gennes for
equilibrium reversible polymer adsorption in good solvent conditions [15] is used to predict the equilibrium configuration of the adsorbed polymer layer on
the surface of HA particles. The relaxation and diffusion of the adsorbed
segments, and consequently their flow characteristics, are predicted using a
Maxwell type kinetic model. Predictions of the model are compared with the
measured dynamic shear modulus of the PLEOF hydrogel reinforced with HA
nanoparticles. This predictive model can provide a quantitative framework to
design and optimize the properties of reinforced polymer composites as
scaffolds for applications in tissue regeneration.


## 2. Experimental Methods

### 2.1. Synthesis of PLEOF

PLEOF was synthesized by condensation polymerization of ULMW PLA [16] and PEG with
fumaryl chloride (FuCl), as shown in Figure 2. The procedure for synthesis is
described in a previous publication [16]. The weight ratio of PLA to PEG was 30/70 to produce a hydrophilic PLEOF
macromer. The structure of the macromer was characterized by 1H-NMR
and GPC. The synthesized PLEOF had Mn and PI values of 10.5 kDa and 1.7,
respectively, as determined by GPC.Figure 2
Schematic
structure of the PLEOF macromer. DEG in the structure of PLEOF is diethylene
glycol used as initiator in the synthesis of ULMW PLA by ring-opening
polymerization of L-lactide monomer.


### 2.2. Synthesis of Ac-Glu6 peptide

The functionalized Ac-Glu6
peptide, a negatively charged Glu-Glu-Glu-Glu-Glu-Glu peptide sequence with an
acrylate group at one chain-end (Figure1), was synthesized manually in the
solid phase on the Rink Amide NovaGel resin [14]. Briefly, the Fmoc-protected
amino acids were coupled to the resin in N,N-dimethylformamide
using N,N•-diisopropylcarbodiimide and N,N-dimethylaminopyridine as
the coupling agents. After coupling the last amino acid, the Fmoc-protecting
group of the last glutamine residue was selectively deprotected with
piperidine. One end was acrylated directly on the peptidyl resin by coupling
acrylic acid to the amine group of the last glutamine residue in the peptide
sequence. The resin was treated with 95% TFA/2.5% TIPS/2.5% water for 2 hours
to cleave the peptide from the resin. The solution was precipitated in ether,
the solid was purified by preparative HPLC (Waters, Milford,
Mass, USA), and the product was freeze-dried. The product was characterized by
mass spectrometry with a Finnigan 4500 spectrometer [14]. A similar procedure was used to synthesize the neutral Gly-Gly-Gly-Gly-Gly-Gly
(Gly6) peptide and positively-charged Lys-Lys-Lys-Lys-Lys-Lys (Lys6) peptide
sequences with an acrylate group at one chain-end (Ac-Gly6 and Ac-Lys6, resp.).


### 2.3.Preparation of the Hydrogel Nanocomposite

100 mg Ac-Glu6 peptide (Mn = 900 Da) was dissolved in 0.825 mL of
distilled deionized (DDI) water by vortexing and heating the mixture to 50°C. HA filler (Berkeley Advanced Biomaterials, Berkeley, Calif, USA) with
average size of 50 nm (measured by TEM), with volume
fraction ranging from 3 to 9% (ρHA = 3.16 g/cm3) was added to the PLEOF polymerizing mixture and the
resulting dispersion was sonicated for 5 minutes. Larger spherical particles,
with average diameter of 5 μm, were also used to investigate
the effect of particle size. The composite mixture was prepared by dispersing
the HA/Ac-Glu6 in PLEOF macromer (0.04 M) and methylene bisacrylamide cross-linker
(0.25 M). The neutral redox initiation system with equimolar concentrations
(0.03 M) of ammonium persulfate and tetramethylethylenediamine was used to
maintain the pH of the polymerizing mixture constant at 7.4.


### 2.4. Rheological Measurements

The composite
mixture was injected on the Peltier plate of the rheometer for rheological and
gelation measurements. The dynamic storage modulus (G′) was measured at 37°C by a TA instrument
AR2000 rheometer equipped with a parallel plate geometry (diameter = 20 mm). A
sinusoidal shear strain profile was exerted on the sample via the upper plate.
The time sweep oscillatory shear measurements were done at constant frequency
of 1 Hz and deformation amplitude equal to 1% for 3 hours.
Each measurement was immediately followed by an amplitude sweep in the range
increasing from 0.1% to 10% strain at frequency of 1 Hz. To reduce the effect
of particle aggregates disruption on the high strain nonlinear response of the
composite, each amplitude sweep measurement was repeated with 30 minutes relaxation
between the two runs and the results of the second run are reported here.
Measurement for the third time revealed no difference between the last two
measurements.


## 2.1. Synthesis of PLEOF

PLEOF was synthesized by condensation polymerization of ULMW PLA [16] and PEG with
fumaryl chloride (FuCl), as shown in Figure 2. The procedure for synthesis is
described in a previous publication [16]. The weight ratio of PLA to PEG was 30/70 to produce a hydrophilic PLEOF
macromer. The structure of the macromer was characterized by 1H-NMR
and GPC. The synthesized PLEOF had Mn and PI values of 10.5 kDa and 1.7,
respectively, as determined by GPC.Figure 2
Schematic
structure of the PLEOF macromer. DEG in the structure of PLEOF is diethylene
glycol used as initiator in the synthesis of ULMW PLA by ring-opening
polymerization of L-lactide monomer.


## 2.2. Synthesis of Ac-Glu6 peptide

The functionalized Ac-Glu6
peptide, a negatively charged Glu-Glu-Glu-Glu-Glu-Glu peptide sequence with an
acrylate group at one chain-end (Figure1), was synthesized manually in the
solid phase on the Rink Amide NovaGel resin [14]. Briefly, the Fmoc-protected
amino acids were coupled to the resin in N,N-dimethylformamide
using N,N•-diisopropylcarbodiimide and N,N-dimethylaminopyridine as
the coupling agents. After coupling the last amino acid, the Fmoc-protecting
group of the last glutamine residue was selectively deprotected with
piperidine. One end was acrylated directly on the peptidyl resin by coupling
acrylic acid to the amine group of the last glutamine residue in the peptide
sequence. The resin was treated with 95% TFA/2.5% TIPS/2.5% water for 2 hours
to cleave the peptide from the resin. The solution was precipitated in ether,
the solid was purified by preparative HPLC (Waters, Milford,
Mass, USA), and the product was freeze-dried. The product was characterized by
mass spectrometry with a Finnigan 4500 spectrometer [14]. A similar procedure was used to synthesize the neutral Gly-Gly-Gly-Gly-Gly-Gly
(Gly6) peptide and positively-charged Lys-Lys-Lys-Lys-Lys-Lys (Lys6) peptide
sequences with an acrylate group at one chain-end (Ac-Gly6 and Ac-Lys6, resp.).


## 2.3.Preparation of the Hydrogel Nanocomposite

100 mg Ac-Glu6 peptide (Mn = 900 Da) was dissolved in 0.825 mL of
distilled deionized (DDI) water by vortexing and heating the mixture to 50°C. HA filler (Berkeley Advanced Biomaterials, Berkeley, Calif, USA) with
average size of 50 nm (measured by TEM), with volume
fraction ranging from 3 to 9% (ρHA = 3.16 g/cm3) was added to the PLEOF polymerizing mixture and the
resulting dispersion was sonicated for 5 minutes. Larger spherical particles,
with average diameter of 5 μm, were also used to investigate
the effect of particle size. The composite mixture was prepared by dispersing
the HA/Ac-Glu6 in PLEOF macromer (0.04 M) and methylene bisacrylamide cross-linker
(0.25 M). The neutral redox initiation system with equimolar concentrations
(0.03 M) of ammonium persulfate and tetramethylethylenediamine was used to
maintain the pH of the polymerizing mixture constant at 7.4.


## 2.4. Rheological Measurements

The composite
mixture was injected on the Peltier plate of the rheometer for rheological and
gelation measurements. The dynamic storage modulus (G′) was measured at 37°C by a TA instrument
AR2000 rheometer equipped with a parallel plate geometry (diameter = 20 mm). A
sinusoidal shear strain profile was exerted on the sample via the upper plate.
The time sweep oscillatory shear measurements were done at constant frequency
of 1 Hz and deformation amplitude equal to 1% for 3 hours.
Each measurement was immediately followed by an amplitude sweep in the range
increasing from 0.1% to 10% strain at frequency of 1 Hz. To reduce the effect
of particle aggregates disruption on the high strain nonlinear response of the
composite, each amplitude sweep measurement was repeated with 30 minutes relaxation
between the two runs and the results of the second run are reported here.
Measurement for the third time revealed no difference between the last two
measurements.


## 3. Experimental Results

Figure3 shows the variation
of the normalized storage modulus of the hydrogel composites (recorded at the
end of time sweep measurements) as a function of particle concentration for
different particle size and surface treatment. Nanoapatite composites (treated
and untreated) displayed far larger stiffness compared with microcomposites, at
the same volume fraction. Shear modulus of the nanocomposites with Ac-Glu6
linker was higher than those without the apatite linker. The modulus of the
composites with micron size particles did not appreciably change with the
addition of Ac-Glu6. The contribution of hydrodynamic effect to the modulus of
the composites can be predicted by Guth-Smallwood equation [17], G0′(Φ)=G0′(0)(1+2.5Φ),
where G0′(0) and G0′(Φ) are the storage modulus of the gel and  composite, respectively, and Φ is the filler volume fraction. The storage
modulus of the composites prepared with mi-cron size particles can be reasonably
predicted by Guth-Smallwood equation, as shown in Figure 3. However, the large
difference between the experimental results and prediction of Guth-Smallwood
equation for composites prepared with nanosize HA implies that the
reinforcement cannot be explained solely by hydrodynamic effects in
nanoparticulate systems.Figure 3
Dependence
of the shear modulus of PLEOF/HA hydrogel composites on the size of the
dispersed apatite particles. The normalized low-amplitude shear modulus predicted
by the presented theory (dashed line) and Guth-Smallwood equation (solid line) are
also compared with experimental data.To
provide further evidence for energetic affinity between the Ac-Glu6 peptide and
HA and its effect on the shear modulus of the nanocomposites, similar measurements
were performed on the PLEOF/HA composites treated with equal molar
concentrations of Ac-Gly6 and Ac-Lys6 in place of Ac-Glu6. Contrary to Glu6
peptide, Gly6 and Lys6 are neutral and positively charged, respectively.
HA/Ac-Gly6 and HA/Ac-Lys6 nanocomposites with 9 vol% apatite did not show a
significant change in storage shear modulus compared to that without HA surface
treatment, as shown in Figure4. The Ac-Lys6 is a positively charged sequence
and it is expected to interact with the phosphate groups on the apatite surface
in the same way that the negatively charged Ac-Glu6 sequence interacts with
calcium ions, but the results in Figure  4 do not support this expectation. The
charge ratio of ca2+ to PO43− groups in the HA
crystal, with atomic composition [Ca10(PO4)6(OH)2],
is greater than one and to meet the requirement for electroneutrality, the
negatively charged OH- groups compensate for the imbalance [18].
Electronic structure and interatomic potential-based calculations show that the
OH-
groups, in the bulk as well as the HA surface, are easily replaced by
negatively charged fluoride ions [18].
Furthermore, phosphate and fluoride ions have been demonstrated to alter the
mineral-organic interactions and influence the mechanical properties of the bone
[19]. It is well established that certain small anionic molecules and polymers
like poly (vinyl phosphonic acid) [20] and poly (acrylamide-co-acrylic acid) [21] displace
negatively-charged hydroxyl groups (and in some cases phosphate groups) to interact
and bond with calcium ions on the apatite surface. Based on these previous
results, we believe that the Ac-Glu6 sequence ionically interacts with the
apatite crystals by replacing the weakly bound hydroxyl groups (and perhaps the
surface bound phosphate groups) from the surface but the same mode of
interaction is not energetically favorable for Ac-Lys6, that is, Ac-Lys6 cannot
replace the positively-charged surface calcium ions to interact with the
apatite crystals. These results demonstrate
that the increase in adsorption energy and its effect on the overall
viscoelastic response of the nanocomposite are specific to the Ac-Glu6 peptide.
The reinforcement is amplified as the size of the nanoparticles is reduced from
5 μm to 50 nm, due to the higher surface area for ionic
interactions provided by nanoapatite fillers.
A molecular model is
developed for the viscoelastic behavior of filled hydrogels which accounts for
the effect of polymer/filler interaction energy. The model is used to predict
the viscoelastic response of PLEOF/HA hydrogel nanocomposite.Figure 4
Comparison of the shear modulus of PLEOF/HA
composites with 9 vol% untreated nanoparticles with nanoparticles treated with
Ac-Glu6, Ac-Gly6, and Ac-Lys6.


## 4. Theoretical Model

The model is
based on the theory of reversible adsorption from a dilute polymer solution [15, 22]. Adsorption of the polymer chains from solution on the
solid surface takes place when the chains
energetically prefer the surface over the solvent. The average residence
time of each monomer on the solid surface is determined by the binding energy
between the monomer and particle surface. It has been shown that when the
contact energy per monomer is less than the thermal energy, kBT,
the adsorption process is reversible, that is, the adsorbed polymer chain
detaches from the surface after a finite residence time and the bonding site are
replaced with another polymer chain [22]. When
the binding energy is somewhat larger than kBT,
the adsorption becomes irreversible, and the adsorbed chains flatten and freeze
on the interactive surface [23].


### 4.1. Filler-Gel Interfacial Structure

The equilibrium
configuration of a chain segment (between two consecutive cross-link points),
near the filler surface with radius,Rf,
is schematically shown in Figure 5. The segment can reversibly adsorb on the
colloidal surface and form a polydisperse succession of loops, tails, and sequences of bound monomers (trains). Each segment with N monomers of size a occupies a spherical volume with a radius
comparable with the Flory radius, RF=aN3/5.Figure 5
Schematic diagram of the equilibrium configuration of
an adsorbed polymer segment (solid line) between two cross-link points on the
filler surface. The adsorbed chain consists of loops, tails, and sequences of
bonded monomers. The adsorbed
segment can detach from the surface at a number of points after the application
of deformation (dashed line).In order to describe
the structure of the adsorbed and fully-equilibrated polymer layer on the
filler surface, we used a modified version of de Gennes scaling theory [24] for reversible adsorption from dilute
solutions under good solvent conditions. The chain configuration in an adsorbed
layer is determined by the competition between excluded volume, surface energy,
and chain entropic effects. Assuming that the
loops are extended to an average thickness D from the surface, the fraction of monomers in direct
contact with the particle surface can be approximated by f≅a/D. As-suming that the conformational entropy and energetic affinity with the
surface are the only factors that determine the configuration of the adsorbed
layer, the free energy per segment, Ψ,
can be written as [25]
(1)Ψ≅kBT(RFD)5/3−fNΔEad.
Minimizing the free energy
with respect to D yields
(2)f≅(ΔEadkBT)3/2.


### 4.2. Dynamics of the Adsorbed Layer

The gel-particle energetic
attraction is modeled as a frictional interaction between the adsorbed monomers
and particle surface, in addition to the regular monomer-solvent and/or
monomer-monomer frictions. Therefore, the total friction
coefficient due to the hydrodynamic force acting on theith monomer is [26]
(3)(ξ)i=ξ1,ith monomer is adsorbed,(ξ)i=ξ0,ith monomer is not adsorbed,
where ξ1 is the friction coefficient due to
monomer-particle interaction and ξ0 is the friction coefficient corresponding to the self-diffusion of a
single monomer and it accounts for its friction with the solvent molecules and/or other nonadsorbed
monomers. Using an Arrhenius-type activation model for a monomer
of size a,
the friction coefficient is approximated by ξ0≅kBTτ0/a2,
with time constant τ0 defined by
(4)τ0=τ∗exp⁡(E0kBT),
where τ∗ is a constant. A similar activation model can
be used to estimate ξ1≅kBTτ1/a2,
where τ1 is defined as
(5)τ1=τ∗exp⁡(E1kBT).
Here, E1=ΔEad+E0 is the energy required to detach the adsorbed
monomer from the particle surface.Since a fractionf of the monomers in an adsorbed segment is in
contact with the particle surface, the total friction coefficient of the entire
adsorbed segment is given by
(6)ξa=N(fξ1+(1−f)ξ0),
where by using (4) and (5),
(7)ξ1=ξ0exp⁡ΔEadkBT.
For weakly attractive
surfaces, segments are partially adsorbed to the surface and exhibit their 3D
Rouse dynamics [27]. Hence, the relaxation
time of the adsorbed segment is
(8)τa≅RF2ξakBT=τf(fexp⁡ΔEadkBT+(1−f)),
where τf≅RF2(ξf/kBT) is the relaxation time of a free segment.The self-similar grid structure [22] describes the
ad-sorbed layer as a semi-dilute solution of the polymer with continuously
varying local concentration of the monomers, such that at any distance r from the surface, the local blob size is equal
to r.
Therefore, the equilibrium thickness of the layer is on the order of RF.
In a cross-linked system, due to the fixed-end constraint, segments cannot
diffuse independently like linear chains. Hence, the adsorption-desorption
process takes place between those segments which are located within the interphase
region with thickness RF around the fillers and with total population
density equal to Nfp+Na.
Here, Na is the number density of the adsorbed segments
and Nfp represents the number density of free segments
within the interphase zone, which are able to participate in the
adsorption-desorption process. The rate of attachment can be shown by the
following kinetic equation:
(9)dNadt=Nfp(τads)−1−Na(τdes)−1,
where τads and τdes are the characteristic times of adsorption and
desorption of the segments, respectively.The energy
required for detachment of an adsorbed segment is equal tofNΔEad.
In the presence of an applied macrodeformation, the tails of each segment move with
the bulk material (Figure 5). The detachment process is thus favored by the
resultant entropic tension exerted by the segment. Considering this effect, the
time constants associated with the attachment and detachment of the segments
follow the relation defined by
(10)τdes=τadsexp⁡[fNΔEad−δFakBT],
where Fa is the entropic force in the segment and δ is an activation length on the order of the
displacement required to detach the bound segment from the particle surface.
The desorption of a bound monomer with weak and short range interaction with
the adsorbing surface can be considered as a local process. It takes place when
the monomer diffuses a distance on the order of the equilibrium size of the
first blob in contact with the wall [28]. According
to the self-similar grid structure theory [22], the size
of the first blob in contact with the particle surface is on the order of the
size of a monomer. Therefore, δ,
the total displacement required to separate the entire segment with f fraction of adsorbed monomers, is a≤δ≤RF.


### 4.3. Macroscopic Properties

The classical Maxwell model [29]
is used to describe the viscoelasticity of the
matrix. It is assumed that the deformations are relatively small such that
geometric nonlinearities can be neglected and only the thixotropic
nonlinearities, due to polymer-filler interactions, are considered. At any
instant in time, a representative segment is either
adsorbed to the surface of the particle or it is free. Assuming that the
configurations of free and adsorbed segments evolve independently, the total
stress in the composite is therefore the sum of the stresses by the adsorbed (σa) and free (σf) segments, that is,
(11)σ=σa+σf.
The contribution of the
segments to the stress tensor is given by Kramers expression:
(12)σa=3Ga〈RaRa〉RF2,σf=3Gf〈RfRf〉RF2,
where Ga and Gf represent the stiffness of the adsorbed
segments and free segments (located out of the interphase zone), respectively. Ri⁢(i=a,f)   is the segment end-to-end vector and 〈⋯〉 shows the ensemble average. According to the
classical theory of rubber elasticity [30], a linear dependency is introduced
between the modulus and number density of the chains at constant temperature, that
is, Gi∝Ni⁢(i=a,f). Therefore, at steady-state conditions, we
have
(13)Ga=Gfpexp⁡[fNΔEad−δFakBT],
where Gfp shows the stiffness of the free segments
within the interphase zone.The mechanical
response of the network can be decoupled into two parts: a rate independent
response and a time dependent deviation from the equilibrium
[31], that is,
(14)σi=σie+σiv,i=a,f,
where σie and σiv stand for the rate independent and rate
dependent components of the stress, respectively. Using (12), the time
independent component of stress can be expressed by
(15)σie=GiF⋅FT,i=a,f,
where F is the deformation gradient tensor.In their
simplest form, the constitutive relations for the evolution of the rate
dependent stresses produced by the segments can be expressed by the Maxwell
(upper-convected) equations [29]:
(16)τaσ^av+σav−GaI=0,τfσ^fv+σfv−GfI=0,
where I is the identity tensor. Here, σ^ designates the upper-convected derivative of
the stress tensor given by σ^=∂σ/∂t−σ⋅Lef−LefT⋅σ,
where Lef=h(ϕ)∇v is the effective velocity gradient tensor and v is the velocity field.
Here, h(ϕ) accounts for the hydrodynamic interaction
between the particles with volume fraction ϕ.
This is based on a phenomenological
consideration that the effective velocity gradient experienced by the polymer
matrix is higher than the externally-applied velocity gradient, due to the
rigidity of the filler particles. The contribution of the
hydrodynamic effect is determined by the shape and volume fraction of the
particles [32]. At low filler concentrations, it is
represented by
(17)h(ϕ)=1+ζϕ,
where
the prefactor parameter ζ accounts for the particle geometry.


### 4.4. Model Predictions

The model is
used to predict the effect of HA surface adsorption energy on the overall
steady-state shear modulus of the PLEOF hydrogel composites. Assuming the
material is under oscillatory shear strain with frequencyω,
the dynamic strain can be stated as
(18)λ(t)=λ0sin⁡ωt.
For simplicity, only the affine (time independent) part of the
deformation is considered for evaluation of the entropic force in (13). For
oscillatory shear loading with small strain λ0,
the deformation gradient can be written as
(19)F=(1λ0sin⁡ωt0010001).
The average end-to-end vector of an adsorbed segment during a period of
oscillation can be obtained by
(20)R¯a=F¯⋅RF,
where the components of F¯ are the average of the absolute values of the
corresponding components in F over one period of oscillation. The mean
square end-to-end distance, given by (20), was used in Fa=(3kBT/RF2)R¯a(1−R¯a2/Ra,max⁡2)−1
the Warner approximation for the entropic force, to calculate the average
entropic tension in an adsorbed segment.Ra,max⁡/RF, c=ΔEad/kBT,
and δ/RF are the modelparameters which represent the
characteristic length of the polymer segments near the particle surface and the
interaction energy between the PLEOF segments and HA nanoparticles. These
parameters are independent of the filler concentration. Gfp/Gf is another fitting parameter which is
proportional to the volume fraction of interphase zone and number of those free
chains, located in the interphase zone, contributing to the
adsorption-desorption kinetics. Therefore, this parameter changes with the size
and concentration of nanoparticles. The magnitude of the shear modulus in the
low strain region is found to be sensitive to the values of c,
while the onset of nonlinearity in the viscoelastic response is controlled by δ and Ra,max⁡.The Flory radius of the segments between two
consecutive fumarate units in the PLEOF (i.e., potential cross-link points) is estimated
to be approximately 10 nm. The hydrodynamic factorζ is set equal to 2.5 considering the spherical shape
of HA nanoparticles. The best fit of the experimental results to the model was
obtained with Gf≅4 kPa and τf=0.001 second for the free segments.Other extracted
fitting parameters are li sted in
Table 1. The value of c for the surface-treated samples was found to
be higher than that of untreated samples, due to the stronger average
monomer-filler interaction in the presence of Ac-Glu6 peptide.Table 1
Fitted parameters of the proposed model for the hydrogel/nanoapatite composites.
Parameters independent from filler concentrationδ/RF=0.5, Ra,max/RF=1.5TreatedUntreatedc=0.02c=0.012Parameters variable with filler concentrationϕTreatedUntreatedGfp/GfGfp/Gf3 (vol%)0.250.336 (vol%)0.530.759 (vol%)0.930.76Figure3 also represents the predicted values of small
strain shear modulus of the Ac-Glu6 treated nanocomposites at different volume
fraction of nanoapatite particles. Fitting the model parameters with measured
storage modulus of microapatite composites results in a negligible value for Gfp/Gf,
and consequently the model prediction is fairly close to h(ϕ)Gf,
that is, the Guth-Smallwood equation. Hence, for large HA particles, relative
to the length of the interacting segment, the
reinforcement is dominated by hydrodynamic effects. It should be mentioned that
due to constant thickness of interphase zone (~RF), reduction of the filler size increases the
volume fraction of the interphase zone in the matrix. As a result, the values
of Ga and Gfp and consequently the overall shear modulus of the
composite increase.The model results and experimental values for the
shear storage modulus of surface treated and untreated composites are shown in
Figure6 as a function of strain amplitude. The model predictions qualitatively
follow the trends in the experimentally-measured values; however, there are
discrepancies, especially between the model results and experimental data of
untreated samples.Comparison of the experimental results with model
predictions (solid lines) for the storage
modulus of PLEOF/HA composites, prepared with 3–9 vol% of (a) treated
and (b) untreated nanoparticles as a function of strain amplitude (frequency =
1 Hz).
(a)(b)The higher adsorption
energy between PLEOF segments and HA nanoparticles in Glu6-treated samples (indicated
by parameterc=ΔEad/kBT in Table 1) can be attributed to the
strong ionic bond with high adsorption energy between the negatively-charged
glutamic acid sequences and the calcium ions on the surface of HA particles,
compared to the weaker polar interactions in the absence of Ac-Glu6. The
exponential dependence of Ga on adsorption energy, as shown by (13), implies
that increasing the polymer-surface interaction energy leads to a significant
enhancement in linear viscoelastic properties of the polymer composites. On the
other hand, lower modulus of the untreated samples indicates weaker
polymer-filler interfacial bonds in those samples which may give rise to
stronger tendency for HA particles to aggregate, as a result of
interparticle electrostatic or van der Waals’ interactions [33]. The network of locally aggregated particles follows a
different kinetic and relaxation pattern rooted in the stored elastic energy in
the strained clusters and the failure properties of filler-filler bonds [34].
This mechanism, which is not accounted in the proposed model, can be considered
as the major source of discrepancy between the experimental data of untreated
samples and the model predictions.


## 4.1. Filler-Gel Interfacial Structure

The equilibrium
configuration of a chain segment (between two consecutive cross-link points),
near the filler surface with radius,Rf,
is schematically shown in Figure 5. The segment can reversibly adsorb on the
colloidal surface and form a polydisperse succession of loops, tails, and sequences of bound monomers (trains). Each segment with N monomers of size a occupies a spherical volume with a radius
comparable with the Flory radius, RF=aN3/5.Figure 5
Schematic diagram of the equilibrium configuration of
an adsorbed polymer segment (solid line) between two cross-link points on the
filler surface. The adsorbed chain consists of loops, tails, and sequences of
bonded monomers. The adsorbed
segment can detach from the surface at a number of points after the application
of deformation (dashed line).In order to describe
the structure of the adsorbed and fully-equilibrated polymer layer on the
filler surface, we used a modified version of de Gennes scaling theory [24] for reversible adsorption from dilute
solutions under good solvent conditions. The chain configuration in an adsorbed
layer is determined by the competition between excluded volume, surface energy,
and chain entropic effects. Assuming that the
loops are extended to an average thickness D from the surface, the fraction of monomers in direct
contact with the particle surface can be approximated by f≅a/D. As-suming that the conformational entropy and energetic affinity with the
surface are the only factors that determine the configuration of the adsorbed
layer, the free energy per segment, Ψ,
can be written as [25]
(1)Ψ≅kBT(RFD)5/3−fNΔEad.
Minimizing the free energy
with respect to D yields
(2)f≅(ΔEadkBT)3/2.


## 4.2. Dynamics of the Adsorbed Layer

The gel-particle energetic
attraction is modeled as a frictional interaction between the adsorbed monomers
and particle surface, in addition to the regular monomer-solvent and/or
monomer-monomer frictions. Therefore, the total friction
coefficient due to the hydrodynamic force acting on theith monomer is [26]
(3)(ξ)i=ξ1,ith monomer is adsorbed,(ξ)i=ξ0,ith monomer is not adsorbed,
where ξ1 is the friction coefficient due to
monomer-particle interaction and ξ0 is the friction coefficient corresponding to the self-diffusion of a
single monomer and it accounts for its friction with the solvent molecules and/or other nonadsorbed
monomers. Using an Arrhenius-type activation model for a monomer
of size a,
the friction coefficient is approximated by ξ0≅kBTτ0/a2,
with time constant τ0 defined by
(4)τ0=τ∗exp⁡(E0kBT),
where τ∗ is a constant. A similar activation model can
be used to estimate ξ1≅kBTτ1/a2,
where τ1 is defined as
(5)τ1=τ∗exp⁡(E1kBT).
Here, E1=ΔEad+E0 is the energy required to detach the adsorbed
monomer from the particle surface.Since a fractionf of the monomers in an adsorbed segment is in
contact with the particle surface, the total friction coefficient of the entire
adsorbed segment is given by
(6)ξa=N(fξ1+(1−f)ξ0),
where by using (4) and (5),
(7)ξ1=ξ0exp⁡ΔEadkBT.
For weakly attractive
surfaces, segments are partially adsorbed to the surface and exhibit their 3D
Rouse dynamics [27]. Hence, the relaxation
time of the adsorbed segment is
(8)τa≅RF2ξakBT=τf(fexp⁡ΔEadkBT+(1−f)),
where τf≅RF2(ξf/kBT) is the relaxation time of a free segment.The self-similar grid structure [22] describes the
ad-sorbed layer as a semi-dilute solution of the polymer with continuously
varying local concentration of the monomers, such that at any distance r from the surface, the local blob size is equal
to r.
Therefore, the equilibrium thickness of the layer is on the order of RF.
In a cross-linked system, due to the fixed-end constraint, segments cannot
diffuse independently like linear chains. Hence, the adsorption-desorption
process takes place between those segments which are located within the interphase
region with thickness RF around the fillers and with total population
density equal to Nfp+Na.
Here, Na is the number density of the adsorbed segments
and Nfp represents the number density of free segments
within the interphase zone, which are able to participate in the
adsorption-desorption process. The rate of attachment can be shown by the
following kinetic equation:
(9)dNadt=Nfp(τads)−1−Na(τdes)−1,
where τads and τdes are the characteristic times of adsorption and
desorption of the segments, respectively.The energy
required for detachment of an adsorbed segment is equal tofNΔEad.
In the presence of an applied macrodeformation, the tails of each segment move with
the bulk material (Figure 5). The detachment process is thus favored by the
resultant entropic tension exerted by the segment. Considering this effect, the
time constants associated with the attachment and detachment of the segments
follow the relation defined by
(10)τdes=τadsexp⁡[fNΔEad−δFakBT],
where Fa is the entropic force in the segment and δ is an activation length on the order of the
displacement required to detach the bound segment from the particle surface.
The desorption of a bound monomer with weak and short range interaction with
the adsorbing surface can be considered as a local process. It takes place when
the monomer diffuses a distance on the order of the equilibrium size of the
first blob in contact with the wall [28]. According
to the self-similar grid structure theory [22], the size
of the first blob in contact with the particle surface is on the order of the
size of a monomer. Therefore, δ,
the total displacement required to separate the entire segment with f fraction of adsorbed monomers, is a≤δ≤RF.


## 4.3. Macroscopic Properties

The classical Maxwell model [29]
is used to describe the viscoelasticity of the
matrix. It is assumed that the deformations are relatively small such that
geometric nonlinearities can be neglected and only the thixotropic
nonlinearities, due to polymer-filler interactions, are considered. At any
instant in time, a representative segment is either
adsorbed to the surface of the particle or it is free. Assuming that the
configurations of free and adsorbed segments evolve independently, the total
stress in the composite is therefore the sum of the stresses by the adsorbed (σa) and free (σf) segments, that is,
(11)σ=σa+σf.
The contribution of the
segments to the stress tensor is given by Kramers expression:
(12)σa=3Ga〈RaRa〉RF2,σf=3Gf〈RfRf〉RF2,
where Ga and Gf represent the stiffness of the adsorbed
segments and free segments (located out of the interphase zone), respectively. Ri⁢(i=a,f)   is the segment end-to-end vector and 〈⋯〉 shows the ensemble average. According to the
classical theory of rubber elasticity [30], a linear dependency is introduced
between the modulus and number density of the chains at constant temperature, that
is, Gi∝Ni⁢(i=a,f). Therefore, at steady-state conditions, we
have
(13)Ga=Gfpexp⁡[fNΔEad−δFakBT],
where Gfp shows the stiffness of the free segments
within the interphase zone.The mechanical
response of the network can be decoupled into two parts: a rate independent
response and a time dependent deviation from the equilibrium
[31], that is,
(14)σi=σie+σiv,i=a,f,
where σie and σiv stand for the rate independent and rate
dependent components of the stress, respectively. Using (12), the time
independent component of stress can be expressed by
(15)σie=GiF⋅FT,i=a,f,
where F is the deformation gradient tensor.In their
simplest form, the constitutive relations for the evolution of the rate
dependent stresses produced by the segments can be expressed by the Maxwell
(upper-convected) equations [29]:
(16)τaσ^av+σav−GaI=0,τfσ^fv+σfv−GfI=0,
where I is the identity tensor. Here, σ^ designates the upper-convected derivative of
the stress tensor given by σ^=∂σ/∂t−σ⋅Lef−LefT⋅σ,
where Lef=h(ϕ)∇v is the effective velocity gradient tensor and v is the velocity field.
Here, h(ϕ) accounts for the hydrodynamic interaction
between the particles with volume fraction ϕ.
This is based on a phenomenological
consideration that the effective velocity gradient experienced by the polymer
matrix is higher than the externally-applied velocity gradient, due to the
rigidity of the filler particles. The contribution of the
hydrodynamic effect is determined by the shape and volume fraction of the
particles [32]. At low filler concentrations, it is
represented by
(17)h(ϕ)=1+ζϕ,
where
the prefactor parameter ζ accounts for the particle geometry.


## 4.4. Model Predictions

The model is
used to predict the effect of HA surface adsorption energy on the overall
steady-state shear modulus of the PLEOF hydrogel composites. Assuming the
material is under oscillatory shear strain with frequencyω,
the dynamic strain can be stated as
(18)λ(t)=λ0sin⁡ωt.
For simplicity, only the affine (time independent) part of the
deformation is considered for evaluation of the entropic force in (13). For
oscillatory shear loading with small strain λ0,
the deformation gradient can be written as
(19)F=(1λ0sin⁡ωt0010001).
The average end-to-end vector of an adsorbed segment during a period of
oscillation can be obtained by
(20)R¯a=F¯⋅RF,
where the components of F¯ are the average of the absolute values of the
corresponding components in F over one period of oscillation. The mean
square end-to-end distance, given by (20), was used in Fa=(3kBT/RF2)R¯a(1−R¯a2/Ra,max⁡2)−1
the Warner approximation for the entropic force, to calculate the average
entropic tension in an adsorbed segment.Ra,max⁡/RF, c=ΔEad/kBT,
and δ/RF are the modelparameters which represent the
characteristic length of the polymer segments near the particle surface and the
interaction energy between the PLEOF segments and HA nanoparticles. These
parameters are independent of the filler concentration. Gfp/Gf is another fitting parameter which is
proportional to the volume fraction of interphase zone and number of those free
chains, located in the interphase zone, contributing to the
adsorption-desorption kinetics. Therefore, this parameter changes with the size
and concentration of nanoparticles. The magnitude of the shear modulus in the
low strain region is found to be sensitive to the values of c,
while the onset of nonlinearity in the viscoelastic response is controlled by δ and Ra,max⁡.The Flory radius of the segments between two
consecutive fumarate units in the PLEOF (i.e., potential cross-link points) is estimated
to be approximately 10 nm. The hydrodynamic factorζ is set equal to 2.5 considering the spherical shape
of HA nanoparticles. The best fit of the experimental results to the model was
obtained with Gf≅4 kPa and τf=0.001 second for the free segments.Other extracted
fitting parameters are li sted in
Table 1. The value of c for the surface-treated samples was found to
be higher than that of untreated samples, due to the stronger average
monomer-filler interaction in the presence of Ac-Glu6 peptide.Table 1
Fitted parameters of the proposed model for the hydrogel/nanoapatite composites.
Parameters independent from filler concentrationδ/RF=0.5, Ra,max/RF=1.5TreatedUntreatedc=0.02c=0.012Parameters variable with filler concentrationϕTreatedUntreatedGfp/GfGfp/Gf3 (vol%)0.250.336 (vol%)0.530.759 (vol%)0.930.76Figure3 also represents the predicted values of small
strain shear modulus of the Ac-Glu6 treated nanocomposites at different volume
fraction of nanoapatite particles. Fitting the model parameters with measured
storage modulus of microapatite composites results in a negligible value for Gfp/Gf,
and consequently the model prediction is fairly close to h(ϕ)Gf,
that is, the Guth-Smallwood equation. Hence, for large HA particles, relative
to the length of the interacting segment, the
reinforcement is dominated by hydrodynamic effects. It should be mentioned that
due to constant thickness of interphase zone (~RF), reduction of the filler size increases the
volume fraction of the interphase zone in the matrix. As a result, the values
of Ga and Gfp and consequently the overall shear modulus of the
composite increase.The model results and experimental values for the
shear storage modulus of surface treated and untreated composites are shown in
Figure6 as a function of strain amplitude. The model predictions qualitatively
follow the trends in the experimentally-measured values; however, there are
discrepancies, especially between the model results and experimental data of
untreated samples.Comparison of the experimental results with model
predictions (solid lines) for the storage
modulus of PLEOF/HA composites, prepared with 3–9 vol% of (a) treated
and (b) untreated nanoparticles as a function of strain amplitude (frequency =
1 Hz).
(a)(b)The higher adsorption
energy between PLEOF segments and HA nanoparticles in Glu6-treated samples (indicated
by parameterc=ΔEad/kBT in Table 1) can be attributed to the
strong ionic bond with high adsorption energy between the negatively-charged
glutamic acid sequences and the calcium ions on the surface of HA particles,
compared to the weaker polar interactions in the absence of Ac-Glu6. The
exponential dependence of Ga on adsorption energy, as shown by (13), implies
that increasing the polymer-surface interaction energy leads to a significant
enhancement in linear viscoelastic properties of the polymer composites. On the
other hand, lower modulus of the untreated samples indicates weaker
polymer-filler interfacial bonds in those samples which may give rise to
stronger tendency for HA particles to aggregate, as a result of
interparticle electrostatic or van der Waals’ interactions [33]. The network of locally aggregated particles follows a
different kinetic and relaxation pattern rooted in the stored elastic energy in
the strained clusters and the failure properties of filler-filler bonds [34].
This mechanism, which is not accounted in the proposed model, can be considered
as the major source of discrepancy between the experimental data of untreated
samples and the model predictions.


## 5. Conclusions

The viscoelastic properties of a multifunctional bone
mimetic nanocomposite with a matrix-apatite adhesion mechanism similar to that
of the natural bone were investigated. A glutamic acid-rich peptide (a sequence
of 6 glutamic acids), derived from osteonectin, was functionalized with an
acrylate group (Ac-Glu) for covalent attachment to the matrix. The biodegradable in situ cross-linkable poly (lactide-co-ethylene
oxide fumarate) (PLEOF) hydrogel and HA crystals were used to mimic the
gelatinous matrix and mineral phases of the bone, respectively. HA
particles were treated with Ac-Glu6 and dispersed in the aqueous PLEOF mixture.
The polymerizing mixture was cross-linked with a neutral redox initiation
system and the gelation process was monitored in situ as a function of time with a rheometer. The rheometry
results showed that the Ac-Glu6 peptide significantly enhanced the viscoelastic
properties of the hydrogel/HA nanocomposite. A molecular model was developed to
predict the role of polymer-apatite interaction in improving the viscoelastic
behavior of the synthesized bone mimetic hydrogel nanocomposite. The scaling
law of de Gennes for equilibrium reversible polymer adsorption in good solvent
conditions was used to predict the equilibrium configuration of the adsorbed
polymer layer on the surface of HA particles. The relaxation and diffusion of
the adsorbed segments were predicted using a Maxwell type kinetic model.
Predictions of the model are compared with the measured dynamic shear modulus
of the PLEOF hydrogel reinforced with HA nanoparticles. The measured rheological and
viscoelastic characteristics and the predictions of the model can be used to
control injectability, in the in situ hardening phase, and to predict
mechanical properties in the postgelation phase for composite scaffolds used in
skeletal tissue regeneration.



---

*Source: 126803-2008-06-30.xml*